Enzyme Mathemat ics
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 10
Editors: J. L. LIONS, Paris G. PAPANICOLAO...
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Enzyme Mathemat ics
STUDIES IN MATHEMATICS AND ITS APPLICATIONS VOLUME 10
Editors: J. L. LIONS, Paris G. PAPANICOLAOU, New York R. T. ROCKAFELLAR, Seattle
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
ENZYME MATHEMATICS
JEAN-PIERRE KERNEVEZ Universitede Technologie de Compiegne
1980
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
ONorth-Holland Publishing Company, 1980 All rights resewed. No part of this pubtication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN 0 444 86122 x
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD
Sole distributors for the U.S.A. and Canada: ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE NEW YORK, N.Y. 10017
Library of Congress Cataloging in Publication Data
Kernevez, J e a n P i e r r e . Enzyme mathematics. ( S t u d i e s i n mathematics and i t s a p p l i c a t i o n s ;
v. 10) Includes bibliographies. 1. Immobilized enzymes--Mathematical models. I. Title. 11. S e r i e s . QP601. K36 574.19'25'0724 80-22449 ISBN 0-444-86122-X
PRINTED IN THE NETHERLANDS
PREFACE
In this book, the main goal is to present mathematical models for enzymes immobilized within membranes, to show how these models represent the biochemical phenomena, and to derive some interesting conclusions from these studies. In this preface we first want to give very general indications on what enzymes are and how they behave. Then we shall give some details on the contents of the book. Enzymes are molecules which catalyze the biochemical reactions in the metabolic pathways of living organisms, whence their importance. Most of the biochemical reactions are catalyzed by enzymes, each enzyme being highly specific for one type of reaction. But in living cells reaction interacts with diffusion because enzymes act within structured systems: either bound to cell organelles such as mitochondria, or linked to membrane structures where they catalyze metabolic reactions taking a part in the transfer of metabolites, or embedded within the cytoplasm, where the viscosity is high and the convection effects negligible. On the other hand, the complexity of living cells is such that it is difficult to model the whole system, where s o many phenomena play a role. There are many enzyme reactions, many kinds of transport, by diffusion, electrical migration, convection or other active processes. Whence the idea of embedding enzyme molecules within artificial membranes in order to study the interaction of enzyme reaction and diffusion in a well defined context. Such membranes, where enzymes are physically confined and localized in a certain region of space with retention of their catalytic activities, and where enzymes can be used repeatedly and continuously, are called immobilized enzyme systems.
V
vi
PREFACE
In these enzymatically active artificial membranes the interaction of diffusion and reaction produces macroscopic spatial, temporal, or functional structures in a self-organized way. The subject of this book is to study these phenomena. More precisely this interaction can give rise to concentration profiles (Chapter l), active transport (Chapter Z ) , multiple steady states and hysteresis (Chapters 3 and 4 ) , pattern formation (Chapters 5 and 6), oscillations and wave front propagation (Chapter 7). Biological phenomena such as active transport across membranes, short term memory, pattern formation in embryos, biological clocks, and transmission of information in living systems may have a similar origin. Each chapter is devoted to the description of a precise system, together with the numerical and mathematical methods employed for its analysis. The mathematical technicalities are postponed until the last sections of each chapter, so that non-mathematicians may read the first sections, dealing with the description of the phenomena, the numerical methods employed for their simulation, and the results of these numerical simulations. The first chapter is an introduction to enzyme kinetics and enzyme membranes. As a first example we present a simple case, represented by the equation: St
- sxx
+
us/(l
+
I s l ) = 0,
t > 0, 0 < x < 1 ,
with boundary conditions s(0,t)
= a
and s(1,t)
= B
and initial condition \s(x,O) = 0 Here u , a , and B are positive parameters and s = s(x,t) denotes the concentration of substrate at (x,t) in a one-dimensional membrane. It is the occasion to point out the important notion of concentration profile, and to show how simple the numerical algorithms may be, whose trivial generalizations enable one to solve more complex systems numerically. We also give a method for proving existence and uniqueness of a positive solution for the evolution equations which describe
vi i
PREFACE such diffusion-reaction systems.
The second chapter describes a “glucose pump” which has the property of transporting glucose from one side of a membrane to the other against the concentration gradient, i.e. from a region of low concentration to a region of high concentration. Chapter 3 deals with the possibility of multiple steady states in a system with hysteresis, governed by the equations:
- -d2s dx2
+
a
2 - 1 + s + ks2
-
0,
O < X < l ,
(0.2)
s(0)
= s(1)
=
A.
We study the S-shaped curve of solutions as the parameter A varies, both numerically (by the Kubicek method of continuation) and mathematically (by applying the implicit function theorem and results of Crandall and Rabinowitz). Lastly, we justify the consideration of sequential steady states as the parameter varies by proving the s o called quasi-steady state hypothesis, Assemblages of cells are presented in Chapter 4 and are the occasion to discuss bifurcation phenomena and numerical methods for exploring a whole connected set of solutions including bifurcating branches. Here we mainly employ H . B . Keller’s methods. Finally we analyse the effect of imperfections.
Chapters
5 and 6 are devoted to pattern formation in enzyme systems. More precisely, chapter 5 deals with an assemblage of N cells, each cell containing an enzyme which catalyzes a reaction involving two reactants, S and A. These reactants are supplied by an external reservoir to each cell by diffusion across a membrane. In the absence of diffusion coupling between cells, there exists a stable steady state, with uniform concentrations of S and A in all cells. But if we allow two neighboring cells to communicate by diffusion, this steady state becomes unstable and new stable steady states arise, which are no longer uniform, but with different concentrations of S and A in the cells. Chapter 5 thoroughly studies this bifurcation of structured steady states from the branch of uniform steady states.
viii
PREFACE
Chapter 6 studies the same kind of behavior, that is diffusion-driven instability, in one or two dimensional spatial domains. Here the governing equations are: st
- As
at
- BAa
- (so-s)]
X[R(s,a)
+
=
- a(ao-a)]
+ A[R(s,a)
in Q
0 =
0
(0.3)
R(s,a)
\ with
= pas/(l
+
Is1 + ks2)
zero-flux boundary conditions,
where Q is a bounded domain in Rn (n = 1 , 2 , or 3 ) , more precisely an enzyme membrane, and the conditions on the parameters s o , ao, a, p , and B are such that, in particular, the following holds: (0.4)
there is exactly one solution, R(f,Z)
-
(so-S)
R(f,Z)
-
a(ao-Z) = 0
@,a),
to the algebraic system,
= 0
-
(0 5)
and (0.6)
is a stable node for the dynamical system:
(2,Z)
ds
+
R(s,a)
-
(so-s)
= 0
(0.7)
da dt + R(s,a) - a(ao-a)
=
0
It is plain that (f,Z) is a trivial steady-state solution of (0.3). By steady state solution of (0.3) we mean a solution to the stationary problem: -As
A[R(s,a)
+
-BAa
+
X[R(s,a)
- (so-s)l - a(ao-a)]
=
0 =
in il 0
with zero-flux boundary conditions.
PREFACE
ix
The important fact is that, as in the preceding chapter, this equilibrium solution ( S , i i ) may become unstable as A crosses a critical value, and new spatially non-uniform stable steady state solutions arise. We point out the similarity of the observed phenomena with morphogenesis in embryos. However by no means do we claim that our S-A system is responsible for pattern formation in embryos. However, the analogy is so striking and the possible consequences so important that this single phenomenon should justify our study of immobilized enzyme systems. In chapter 7 we present experimental and numerical observations of oscillating enzyme systems, and numerical results about wave front propagation in the so-called S-A system. At the end of each chapter we give references, admittedly incomplete, to literature on the topics discussed, our aim being to give an idea of the biochemical and biological background without being exhaustive, and to direct the reader to additional information on numerical and mathematical methods. The intention of the book is two-fold: first to show some striking phenomena such as pattern formation, and secondly to present some numerical and mathematical methods in a simple form. The numerical methods described include finite difference and finite element analysis of reaction-diffusion problems, and procedures for solving bifurcation and nonlinear eigenvalue problems. These numerical continuation procedures are two-step methods, as described by Kubicek and H.B. Keller: a predictor step employing Euler or Adams-'Bashforthmethod, and a corrector step using Newton-Raphson method. The numerical methods for bifurcation of steady-states are those of H.B. Keller. The mathematical methods deal with existence and uniqueness of a positive solution for the evolution equations governing our systems, and existence, multiplicity and stability of their steady state solutions. We show how it is practical to use upper and lower solutions, as employed by Amann or Sattinger.
X
PREFACE
Special emphasis is put on analysis of a continuum of steady state solutions as some parameter varies, and bifurcation of steady state solutions, for which we mainly refer to Crandall and Rabinowitz, and H.B. Keller. It is a pleasure to acknowledge the contribution of Daniel Thomas. I am much indebted to him for the model systems hereafter described, He provided not only their description, but also the equations, the exact form of the rate expressions, a n d , which is very important, the numerical values of the parameters. Moreover his enthousiasm was the motivation of this work.
I would like to thank Jacques-Louis Lions whose teaching and encouragements have had such a major influence on my work. Thanks go to all those who contributed or are contributing to the numerical and mathematical work on these systems: H.T. Banks, B. Bunow, C.M. Brauner, M.C. Duban, D. Dubus, J . Henry, G. Joly, C. Milan, P. Penel, J.P. Puel, J.P. Quadrat, L. Tartar, B. Viot, and J.P. Yvon. Finally, a special thanks goes to Cindy Moreau for her excellent typing of the manuscript and her fine draftsmanship on the figures.
CompiBgne 1980
Jean-Pierre Kernevez
TABLE OF CONTENTS
CHAPTER 1
- DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
1
1.1 - Motivation for Studying Artificial Enzyme Membranes 1.2 - Enzyme Kinetics 1.3 - Enzyme Membranes 1.4 - Simple Irreversible Monoenzymatic Reactions 1.5 - Numerical Solution 1.5.1 Explicit scheme 1.5.2 Implicit schemes 1.6 - Mathematical Methods References CHAPTER 2
- GLUCOSE
PUMP
1 2 8 9 16 16 17
21 32 35
2.1 - Description 2.2 - Modeling 2.3 - Numerical Simulations 2.4 - Mathematical Analysis Re ferenc e s CHAPTER 3 3.1 3.2
35 38 43 44 55
- MULTIPLE STEADY STATES AND HYSTERESIS
- The Zero-dimensional System - The One-dimensional System
3.2.1 Experimental set up 3.2.2 Marhematical modeling 3.2.3 Numerical simulations 3.2.4 Theoretical justification 3.3 - Kubicek Method of Continuation 3.4 - Mathematical Analysis of the S-shaped Curve 3.5 - Stability 3.6 - Justification of the Quasi-steady-state Hypothesis References
xi
57 59 64 65 65 67 70 74 79 98
1 a2 109
TABLE OF CONTENTS
xii CHAPTER 4
-
4.2 4.4 4.5
111
- Multiple Cell Assemblages
4.1 4.3
- ASSEMBLAGES OF CELLS AND BIFURCATIONS Two Cell
114 115
Assemblage
- Nonisolated Solutions and Bifurcation - Numerical Analysis of Bifurcations - Imperfections
118 131 136
References CHAPTER 5 5.1
- THE URICASE SYSTEM AND PATTERN FORMATION
143 145
- Presentation of the Uricase System
149
Case of a single cell 5 . 1 . 2 The trivial steady state (s",ii) 5.1.3 N cells 5 . 2 - Stability of the Trivial Steady State 5.3 - Bifurcation from the Trivial Steady State 5.4 - Numerical Exploration of the Steady States References
149
5.1.1
CHAPTER 6 6.1
- PATTERN FORMATION IN A MONOENZYME MEMBRANE
152 153
155 160 163 168 169
- Pattern Formation in Fruitfly Drosophila Imaginal Disks
172
Garcia-Bellido observations 173 6 . 1 . 2 The model o f Kauffman, Shymko, and Trabert for 173 morphogenesis in Drosophila 6 . 1 . 3 Sprey's observations 176 Description of the S-A System 179 6 . 2 . 1 Modeling o f the S-A system 179 6 . 2 . 2 The spatially uniform steady-state uo = ( ? , Z ) 1a 2 Linear Stability Analysis of the Spatially Uniform Steady 6.1.1
6.2
-
6.3
-
6.4
6.5
-
State uo = ($,ii) 6 . 3 . 1 Eigenvalues of fU(A,uo) 6 . 3 . 2 Conditions f o r u; to be negative 6 . 3 . 3 Conclusion Bifurcation Analysis 6 . 4 . 1 General framework 6 . 4 . 2 Application to the s-a system Numerical Methods and Results 6 . 5 . 1 One-dimensional case 6 . 5 . 2 Two-dimensional case
183 184 1 a6 186 1a 8 1a 8 191 196 198 20 1
xiii
TABLE OF CONTENTS Case of a surface Galerkin's method 6 . 6 - Mathematical Technical Details 6 . 6 . 1 Spectrum of f u ( X , u o ) 6 . 6 . 2 The principle of linearized stability 6 . 6 . 3 f is a twice continuously differentiable mapping 6 . 6 . 4 The range o f f is closed and of codimension 1 References 6.5.3 6.5.4
CHAPTER 7 7.1
- OSCILLATIONS AND WAVE FRONT PROPAGATION
- Papain System
Papain kinetics in a well-stirred solution 7 . 1 . 2 Papain kinetics in a membrane 7 . 1 . 3 Numerical methods 7 . 1 . 4 Numerical results 7 . 2 - Uricase System 7 . 3 - Wave Front Propagation in the s-a System References 7.1.1
214 221 222 223 225 232 234 236
241 242 242 244 245 246 248 252 258
This Page Intentionally Left Blank
CHAPTER 1 DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
In this chapter we present o u r basic biochemical model, governed by the equation: St
- sxx
+
us/(1
+
Isl) = 0,
t > 0, 0 < x < 1 ,
with boundary conditions s(0,t)
= c1
and s(1,t)
=
B
and initial condition s(x,O)
=
0.
Here u , a, and p are positive parameters and s = s(x,t) denotes the concentration of substrate at (x,t) in a one-dimensional membrane. The motivation for studying such immobilized enzyme systems will be Section 1.2 presents a brief introducthe subject of section 1 . 1 . tion to enzyme kinetics. Artificial enzymatically active enzyme membranes are described in Section 1.3. Their modeling by partial differential equations is given in Section 1.4. The numerical solution of these equations is indicated in Section 1.5, together with numerical results. Finally, Section 1.6 studies the existence and uniqueness of a positive solution for the evolution problem ( l . l ) , and the behavior of the concentration profile sC.,t) as t + + m . 1.1
Motivation for Studying Artificial Enzyme Membranes
Enzymes are molecules which catalyze the biochemical reactions in the metabolic pathways of living organisms, whence their importance. Most of the biochemical reactions are catalyzed by enzymes, each enzyme being highly specific f o r one type of reaction. But in living cells reaction interacts with diffusion because enzymes act within structured systems: either bound to cell organelles such as mitochondria o r linked to membrane structures where they catalyze meta-
2
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
bolic reactions taking a part i n the transfer of metabolites, or embedded within the cytoplasm, where the viscosity is high and the convection effects negligible. On the other hand, the complexity of living cells is such that it is difficult to model the whole system, where so many phenomena play a role. There are many enzyme reactions, many kinds of transport, by diffusion, electrical migration, convection or other active processes. Whence the idea of embedding enzyme molecules within artificial membranes in order to study the interaction of enzyme reaction and diffusion in a well defined context. Such membranes, where enzymes are physically confined and localized in a certain region of space with retention of their catalytic activities, and where enzymes can be used repeatedly and continuously, are called immobilized enzyme systems. 1.2
Enzyme Kinetics
Though enzyme reactions have been used for thousands of years in fermentations, it was not before the nineteenth century that the idea became clear that the biochemical reactions are not under the influence of some mysterious principle of Life, but are subject to the usual Law of Mass Action like other chemical reactions, and are catalyzed by enzymes. In 1 8 3 5 , Berzelius suggested that most of the biochemical reactions could occur under the influence of a new force which he calls "catalytic". In 1 8 7 8 , Kiihne coined the term "enzyme" for this catalytic force, from a greek word meaning "in yeast". In 1902, Brown suggested that the enzyme E combines with its substrate, S, to form an intermediate complex, ES, which then decomposes into the product, P , and E:
(1.2)
E + S
kl
k2 ES + E
+
P.
k2 This can be described in the "lock and key" theory, (Figure l . l ) , where the structures and shapes of enzyme and substrate molecules explain the high specificity of enzymes to substrate type observed in enzyme reactions. Other important facts can be explained by Brown's scheme and the "lock and key" theory: the possibility of inhibition of enzyme reactions by substances, called inhibitors, which can take the place of S on the enzyme molecule; the enzyme concentration may be small with respect to the substrate concentration, since enzyme
ENZYME KINETICS
3
molecules are released unchanged to be used again. In 1903, Henri [ I ] , and, ten years later, Michaelis and Menten [21, translated Brown's chemical scheme into a mathematical model, by using the Law of Mass Action:
[
d[ dt SI
-kl[EIIS]
= =
k-l[ESl,
+
-(k-l + kZ)[ESl
+
[Sl(O)
kl[EIISl,
= So,
[ESI(@)
=
0,
Here [El, [S], [ES] and [PI are the concentrations of free enzyme, free substrate, enzyme-substrate complex and product. The Henri-Michaelis-Menten derivation uses the so-called quasi equilibrium assumption, where the reaction E + S S E S is supposed to be alone and at equilibrium:
so that one finds:
with KS
=
k-l/kl and
VM
=
kZEo.
In 1913, Bodenstein points out that a less restrictive assumption than the quasi-equilibrium assumption is to suppose that d[ESl/dt = 0 (the so-called quasi-steady-state assumption). In 1925, Briggs and Haldane [31 use this assumption to write: 0 = -(k-l+k2)[ES3
+
kl[E3[Sl
=
-(k-,+k2)[ESl
+
kl(Eo-[ESl)[SI
which still yields ( 1 . 4 ) , but now the "Michaelis constant" KS is given by :
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
4
and the rate expression for S is found to b e ( b y using the first equation in (1.3) and the first equation in (1.4):
In 1967, Heineken, Tsuchiya, and Aris 141 justify this modeling b y using singular perturbations. Indeed, the equations' for S and ES alone may be written:
-kl(Eo
=
d[ES1 dt
=
- [ES])[Sl
-(k-l+kZ)[ES1
+
+
k-l[ES],
[ S ] ( O ) = So
kl(Eo - [ E S I ) t S ] ,
We nondimensionalize by setting:
s = [Sl KS,
c
=
[ESI(O) = 0
[ESI/Eo,
E
= Eo/KS.
Here and hereafter, KS is defined b y (1.5 Then the differential equations for s and c take the form: ds
- klKS ~(1-c))
= E ( ~ - ~ c
s ( 0 ) = so
(1.7) =
klKS(s - c(l+s))
c(0) = 0
If we take the stand that the large time behavior is the important one, we employ a larger time scale, E - I ,and define T = Et. Then rewriting (1.7) in terms of this new variable, we find: ds
k - l -~ k 1K s s(l-c),
=
s ( 0 ) = so
(1.8) E
dc a T
=
1 s( S -
k K
C(l+S)),
c(0)
=
0.
A naive formal approximation is found b y writing (1.9)
c
=
S ds l + s ' dT
-- -k2
.
E =
0 , to obtain:
ENZYME KINETICS
5
This is tantamount to making the quasi-steady-state hypothesis, since it gives: (1.10)
[ E S ] = Eo[S]/(KS+[SI)
and
=
-VMISl/(KS +[Sl)
=
-
d[Pl
Another naive formal approximation valid for moderate time is obtained by setting E = 0 in (1.7): ds 7L-F
=
(whence s(t) = s o )
O,
whence :
Thus we see that during a short interval of time, of the order of 1 / ~ ,s remains approximately constant while c rapidly varies from 0 to so/(l + s ) , The larger time behavior, which is usually the impor0 tant one in biological contexts is described by (1.9), or, with different units (1.lOj. Thus we have seen that the Briggs-Haldane quasisteady-state assumption is justified in the usual case where Eo is small with respect to KS. More details aboutthe singular perturbation analysis of enzyme kinetics may be found in ([51,[61). The same kind of quasi-steady-state assumption may be applied to the kinetics of inhibited reactions. A competitive inhibitor I competes with S for the active sites on the enzyme molecule, and we have the following scheme:
E + I
-
ki
k' -1
EI
with the corresponding equations:
6
DIFFUS ON AND REACTION IN AN ENZYME MEMBRANE
iv=-k
[ES](O)
=
0
[EIl(O)
=
0
[PI(O)
= 0
The quasi-steady-state assumption can be written, in this case, d[ES]/dt = d[EI]/dt = 0, whence [El = KSIESI/[S] and.[EIl = (KS/KI) [ES][I]/[S], where KI = (kll/ki) is the so-called inhibition constant of E for I. Putting these expressions of [El and LEI] in the last equation of (1.11), we obtain:
(1.12)
[ES]
=
EO
KS
s
KS [I1 + l + - - KI [Sl
-
Eo
KS(l
+
[SI [I1 ) I
+
[Sl
Thus we find the following rate expressions for S and P:
If we compare equations (1.10)and (1.12)- (1.13), we observe that a competitive inhibitor increases the apparent Michaelis constant of the enzyme for the substrate. It is worth noting the significance of these two constants VM and KS which appear in equations (1.9):
VM is the maximal value of the reaction rate. KS is the value of [ S l for which the value of this reaction rate is VM/2. The smaller the KS, the greedier the enzyme for its substrate. We see from (1.14) that VM is not an intrinsic constant of the enzyme, but is proportional to the initial quantity of free enzyme, E o , whereas KS is a constant characteristic of the enzyme, the so-called Michaelis constant. The net effect of adding an inhibitor to a solution of enzyme and substrate is to increase the apparent Mkchaelis constant, which becomes
ENZYME KINETICS
7
.
K S ( l + [I] ) KI
Up to now we have described the monoenzyme irreversible reaction ( l . Z ) , whose velocity term is given by ( 1 . 6 ) , eventually inhibited by a competitive inhibitor, in which case the velocity term is given by ( 1 . 3). In fact the rate expressions for enzyme reactions in a well-stirred solution vary greatly. For example we shall deal with substrate in hibited phenomena for which the rate expression is given by:
Here KSS is the inhibition constant of S for the enzyme. More than one.substrate can be involved substrate may be called a cosubstrate. duct can result from the reaction. For A = oxygen, E = uricase, P = allantoin, E (1.16)
S + A
+.
P
+
in the reaction. A second Similarly, more than one proexample, with S = uric acid, we can write:
other products
In the case of the uricase reaction ( . 1 6 ) , the rate expression may be taken as:
Here [A] is cosubstrate concentration and KA is the Michaelis constant of enzyme E for cosubstrate A. If [ A ] is small with respect to KA, we can approximate ( 1 . 1 7 ) by: (1.18)
T =
v
[A1
KS
+
[SI [Sl(l +
F) ss 1
If A is in excess, i.e. if [ A ] is large with respect t o KA, ( 7 . 7 7 ) reduces to ( 1 . 1 5 ) . In the following we shall choose KS as the unit of concentration, and use the dimensionless quantities:
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
8
(1.19)
s
=
[Sl/KS,
i
=
[Il/KS,
The rates of reaction ( 1 . 6 ) , then be written:
vM
(1.20)
r
=
(1.23)
r
= V
a
(1.13),
=
IAl/KS.
(l.lS),
(1.17),
and ( 1 . 1 8 )
can
1 + s '
-.-%K A + a KS
1 +
S
s + ks2
For more details on enzyme models, see [71, 1 8 1 , and, mainly [g]. 1.3
Enzyme Membranes
Enzyme kinetics described in the preceding section have been obtained by studying enzyme reactions in well-stirred solutions of the species involved (enzymes, substrates, products, inhibitors, activators). However, a s pointed out in Section 1 . 1 , in living organisms enzyme reaction is coupled to diffusion. Thus, after the first and necessary step of studying enzyme kinetics in an homogeneous phase, there is a need for a second step which would be enzyme kinetics in an heterogeneous phase 151. .In other terms, after studying "lumped" systems governed by ordinary differential equations, we now have to study "distributed" systems governed by partial differential equat ions. Our tool will be artificial enzyme membranes, more precisely membranes in which enzymes are insolubly embedded by one of several In particular the binding of enzyme molecules to inacmeans [ l o ] .
SIMPLE IRREVERSIBLE MONOEKZYMATIC REACTIONS
9
tive protein molecules under the action of glutaraldehyde produces a membrane [ 1 1 1. Without entering into details, we can say the following about the you make an homogeneous sorecipe for preparing a membrane [ I l l : lution of inactive protein (albumin for example), enzyme (glucose oxidase for example, and glutaraldehyde, then, as for a pancake, you pour the mixture on a flat surface (glass). You wait several hours, until the water has evaporated, and obtain a membrane where glutaraldehyde acts as a glue to bind the enzyme and albumin molecules. This membrane presents the aspect of a translucid film of thickness L = 5 0 p , with good mechanical properties. Figure 1.2 shows an electron microscope image of such a membrane. Remark the regular aspect and the absence of pores. Within this membrane, enzyme molecules are uniformly embedded. Due to their proteic environment, they exhibit an increased,stability and remain active for a longer time than in solution. Many enzymes can be immobilized ([lo], [121). 1.4
Simple Irreversible Monoenzymatic Reactions
For the simple biochemical model to be described in this section, we shall assume that an irreversible monoenzymatic reaction,such as ( l . Z ) , takes place. For example glucose oxidase (E) catalyzes the transformation of glucose (S) in gluconic acid (P). The rate expression, in solution, is given by (1.6). The membrane M separates two compartments, I and 1 1 , as depicted in Figure 1.3. Compartments I and I 1 are 5 to 10 cm. long and several centimeters in height. The compartments contain well-stirred solutions of S, of concentrations S 1 and Sz respectively. The membrane being initially empty of any substrate or product, the substrate moves in and is altered under the catalytic action of the enzyme. The membrane shape, with its slab geometry, is particularly convenient for an analysis of the phenomenon, inasmuch as it favors the direction transverse to the membrane. Letting S(x,t) denote the concentration of substrate at (x,t) in this one-dimensional me,g)rgng, ~8 have I. 131 : (1.25)
as -at (
as
at
as
diffusion
+ (d reaction
10
DIFFUS I O N A N D R E A C T I O N IN A N EN Z Y M E M E M B R A N E
n
w
-T"1-bk ES
Fig.
1.1
F i g . 1.2
SIMPLE IRREVERSIBLE MONOENZYMATIC REACTIONS
11
50P
H X Fig. 1 . 3 We can assume that the coefficient of diffusion DS is constant since the medium is homogeneous. Using Fick's law, we have:
as
(1.26)
'diffusion
(
=
DS
a2s ax.
*
The velocity due to reaction is given by:
as
(1.27) (
'reaction
=
-VM S / ( K M
+
S)
Thus the evolution of S in the membrane is given by:
Here L is the membrane thickness. If we introduce the non-dimensional variables: (1.29)
s
= S/KS,
x' = x/L,
t'
=
t/(L2/Ds),
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
12
substitute them into (1.28) and for convenience drop the primes, we obtain the dimensionless equation:
where the non-dimensional constant u is defined by:
(1.31)
u =
M' L2 DS
KS
u , square of the so-called Thiele modulus, is the ratio of L2/DS, characteristic time for diffusion, and Ks/VM, characteristic time for reaction, One may ask whether it is realistic to compare the coupling of reaction and diffusion within cell membranes, which are 7 5 to 100 x 1 0 - 4 p in thickness, and within artificial membranes of thickness S o p . In fact the single parameter u is of importance for the behavior of the membrane. For an artificial enzyme membrane, any desired value of u may be achieved by an appropriate choice of VM, which is at our disposal since it is proportional to the concentration Eo of enzyme added to inactive protein and glutaraldehyde when preparing the membrane.
For the simple biochemical model we have in mind, namely a glucose oxidase membrane, the parameter values are L = 5 10-3cm, DS = 5 cm'h-'(h = hour), VM = 3.66 10-3moles cm-3h-' , and KM = 1.3 lo-' moles whence u = 1.4. The characteristic time for diffusion h = 18 seconds. L2/DS = 5 Boundary conditions: if one assumes that the concentrations S 1 and S 2 are held fixed in compartments I and 11, one obtains "bath" or Dirichlet conditions: (1.32)
s(0,t)
=
a , s(1,t)
=
B,
t > 0,
where: a = S1/Ks and f3 = S2/Ks. Such boundary conditions are also appropriate if solution volumes in the compartments are large compared with membrane volume and if the experiment is of short duration. A configuration of interest is a membrane immersed in a solution of substrate. In this case both edges of the membrane are exposed to
SIMPLE IRREVERSIBLE MONOENZYMATIC REACTIONS
13
the same concentration a , concentration of substrate in the bulk solution, and we still have (1.32), with 6 = a. Initial conditions: the membrane being initially empty of any s u b strate, the initial condition is: (1.33)
s(x,O)
=
0,
O < X < l .
Product concentration: one can easily see that the product P concentration (again in dimensionless form) is given by:
1,t) = 0
Here p = [ P ] / K s , is the diffusion coefficient of P and it is asDP sumed that the membrane is initially empty of any product, and the adjacent compartments are large enough with respect to membrane volume for P concentration to remain 0 within them, although P may flow across the membrane edges. Thus S and P concentrations in an artificial membrane separating two compartments, and where a simple irreversible monoenzymatic reaction takes place, are governed by equations (1 .30) , (1 .32) , (1 .33) , (1.34) We mentioned that it was also the case for a membrane immersed in a solution of substrate.
.
Enzyme electrodes: Another configuration, important in many applications, also falls within this example: artificial membranes coupled with electrodes ( [ l o ] , [ 1 4 ] , [ 1 5 ] , [ 1 6 1 ) . A membrane i s attached to an electrode which is sensitive to the product P of the reaction. In Figure 1.4 for example, the substrate (urea) is transformed by the enzymatic layer (urease and protein) which covers the electrode bulb: as a result, monovalent cationsappear, which are detected by the electrode. When the electrode is in contact with a solution containing the substrate to be measured, S and P concentrations within the membrane are governed by equations:
14
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE sxx s(0,t)
+
oF(s)
= a,
=
p(0,t)
0 , Pt =
0,
Here and hereafter, st -- a s , sx
=, as
fi .
Due to the preax2 sence of the electrode, there is a zero-flux boundary condition at the impermeable wall x = 1 . The S and P profiles evolve towards a stable steady-state, as depicted in Figure 1.5, where substrate ( - - - ) and product ( - ) concentration profiles, solutions of (1.37) for u = 0.35 and a = 6 = 1, are represented at times t = 0.1,0.2, and 0.7. At time t = 0.7 the profiles have nearly attained the steady state. If P is a cation to which the electrode is sensitive, from the measurement of the electrode at steady state, one can get the concentration of P against the electrode and hence of S in bulk solution. Other electrodes, p02 electrodes, are sensitive to the oxygen pressure. Glucose electrodes can be made with such an electrode coupled with a membrane containing glucose oxidase. The reaction: (1.38)
glucose + O2
=
glucose oxidase
1-
sxx
=
gluconic acid + H202
consumes oxygen, s o that the oxygen pressure is less at the contact of the electrode (x = 1) than in solution (x = 0). A well-defined relation exists between the p02 level, at.x = 1 at steady state, and glucose concentration in the bulk solution. Due to the specificity of the enzyme, such an electrode is sensitive to glucose only. It is thus possible to prepare electrodes sensitive to one particular substrate (glucose, maltose, saccharose, lactose, etc.) and to continuously monitor the concentration of these substances in media like blood o r the broth of fermentations. Later is this chapter we shall be interested in the numerical and mathematical analysis of the systems introduced in this section, more precisely the membrane modeled by equations (1.30) , (1 .32) , (1 .33) , and (1.34), and the electrode modeled by equations (1.37). In each case p is defined by a linear parabolic equation. Thus we shall restrict ourselves to the equations giving s, the definition of p being straightforward once s is known. These equations are (1.1) and its
SIMPLE IRREVERSIBLE MONOENZYMATIC REACTIONS
ENZYME ELECTRODE SENSITIVE TO MONOVALENT
F i g . 1.4
1
U
Fig. 1.5
15
16
DIFFUSION AND REACTIOX IN AN ENZYME MEMBRANE
variant with a 0-flux boundary condition at x = 1 . 1.5
Numerical Solution
Because the concentration profiles within the membranes are not directly observable, biochemists are very interested by the numerical simulation of equations (1.1). Indeed the only quantities that biochemists can measure are -sx(O,t) and sx(l,t), fluxes of substrate entering into the membrane from the compartments, or, in the case of an electrode, S or P concentration along the electrode, namely s(1,t) or p(1,t). Fortunately, there exist simple and efficient numerical methods for solving equations like (l.l), the so-called explicit and implicit finite difference methods [ 1 7 ] . Although they are wellknown, we describe them here for the sake of completeness. They are very useful for solving, not only (l.l), but also similar reaction diffusion problems. In orderto show how simple they are we give, in Figures 1.6 and 1.7, listings of the corresponding Fortran programs. 1.5.1 - Explicit scheme: We divide the interval [0,11 into N equal intervals of length h = Ax = 1/N, and define a time step k = At. We call s n an approximat on of s(ih,nk). This approximation is defined by the following expl cit scheme:
It is easily seen that ”:s is explicitly known once the three values s s , and s:+~ are known. As s z = a, sp = 0 , 1 5 i - N-1, and 1-1’ s o = B , once can calculate the s 1i , 1 5 i 5 N - 1 , and so on: once
Sn
N
the s; are known at time level n , the level n + 1 (Figure 1.6), by formula:
5 ; ’ ’
may be calculated at time
17
NUMERICAL SOLUTION
1
2 3 4
5 6 7
8 9 10
it 12 13 14
15 16 17 18 10 20 21 22
23 24 25 26 27 211
Fig. 1.6 We may remark that if the s s are positive, so are the s : " ,
(1.41)
k < h2/(2
+
provided:
oh').
This stability condition ensures a good behavior of the explicit scheme. (In the case u = 0 i t reduces to the well known stability condition k/h2 < l / Z ) . F o r example, if N = 20, h = 0.05, u = 1 . 4 , the stability condition is k < 0.0013. We choose k = 0.001, This drawback of being constrained to choose small time increments may be overcome by using an implicit scheme. 1.5.2 - implicit schemes: There are many of them. totally implicit finite difference scheme:
Here we give the
18
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
(
s;
15 i
= 0,
5 N-1.
Here the's:'' are no more explicitly defined but, in order to find them at each time level n+l, we have to solve the set of N-1 nonlinear equations: u.
(1.43)
i
-
sn
uo = a ,
Uitl
UN
=
+
ui-l h2
- 2u.
U.
' + o + = o ,
I ~ ~ Z N - I ,
B.
A possible iterative method f o r solving (1.43) is the Newton method. Easier to implement is the following one: i/
take as starting values uo
ii/
once the um are known, define the umtl as the solution of the linear system:
iii/ Stop when
=
sn1'
max luytl - uyl < lLi5N-1
5 i 5 N-1.
1
E,
E
being some error tolerance.
In fact, in the program shown in Figure 1.7, it was decided to adopt n+ 1 i' = u 1' . 1 5 i 5 N-1, n 2 0 . Some words about the solution of the linear system (1.44).
It takes
NUMERICAL SOLUTION
19
only 7 lines of Fortran (lines 17-23 in Figure 1.7) to efficiently solve this tridiagonal system by Gauss elimination. Let us briefly recall the method [ 1 7 1 for the tridiagonal system: + b.u. -a.u. 1 1-1 1 1
C.U.
1 i+1
- di'
1 5 i
5 N-1
(1.45) us
uo = a ,
=
B
1 2
3 4 5 6 7
e
9
so
10
11
12
13 14
15 16 11
ie
19
20 21 22 23
100
150
24
25 26
27 25 29 30
NRITE (6,iiooj s CONTIWUE FORHAT(/' T I M E S T E P #',IU/) FOR~AT(1X,llE10~3)
200
1000
1100
STOP END
31
Fig. 1 .7 Equations (1.45) may be rewritten: (1.46)
ui
= E1 . u .1+1
This is evident for i
=
(1.47)
Fo
Eo = 0
and
0
+ Fi,
5 i 5 N-1
0 , with: =
a
This is true for i if it is also true for i-1.
Indeed, if u .
1-1
--
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
20
into (1.45), we obtain: E i - l u i + F i - l , then plugging this value of u. 1-1 -ai(Ei-,ui
+
F i - l ) + biui
-
C1 . Ui+1 .
- di’
di b.
a 1. F1-1 . a.E.
whence (1.46) with: (4.48)
E 1.
C. =
~b. - :.E. and 1
1 1-1
F 1. =
1
+
-
1
1-1
*
Thus (1.46) is proven by recurrence and the algorithm for solving (1.45) is the following: i/
ii/
using formulas (1.47) and (1.48), calculate the E i , F i , 1 5 i -< N-1 (triangularization), using formula (1.46), and knowing u N , calculate . . . , u 1 (backward substitution).
then
UN - 2 ,
Although this algorithm is slightly more complex than the former (31 lines of program in Figure 1.7 instead of 28 lines in Figure 1.6!) and more computer time consuming at each level of time (there may be several iterations with respect to m and within each of these iterations a certain amount of operations are required), however the overall advantage comes from the fact that we are no more constrained to small time steps k , but may choose any value for this increment of time. For example we may choose k = 0.01, which is forbidden in the explicit method. Of course with k very large we do not find the steady state solution in one step, but have to iterate several times. The successive profiles thus obtained do not have much to do with the transient evolution of the system, except that they represent closer and closer approximations of the steady state solution in what is in fact a fixed point algorithm to calculate this steady state solution.
Remark
1 . 1 - No-flux boundary condition sx(l,t) = 0: A slight modification of the numerical schemes described in (1.39) and (1.42) enables handling no-flux boundary conditions. In both cases we only have to write equation (1.39) (resp. (1.42)) for i = N with s:+~ = n sn+l = sn+l s ~ (resp. - ~ N+l N - l ) . In the last case trivial modifications
MATHEMATICAL METHODS
have to be made.
21
Since the last equation is:
and we have
we obtain:
1.5.3 - Numerical results: Application of either method provides, at each level of time, a profile of concentration as depicted in Figures 1.5 and 1.8. Figure 1.8 shows the evolution of the substrate concentration profile in an enzyme membrane shortly after introduction into a substrate solution. Here u = 1.3, and CY = 5 = 1 . The time increment is At = 0.001 and at time 700 At ( = 0.7) the concentration profile is nearly stationary. The existence of such concentration profiles can be shown by electron microscope images of the membrane (Figure 1.9) ( [ 1 8 ] , [ 1 9 ] , [ 2 0 ] ) . For other electron microscopy studies, see [ 2 1 ] and [ 2 2 ] . Figure 1.9 shows an electron microscope image of a membrane exhibiting a concentration profile: the membrane boundaries are highly contrasted and the density of the black points (due to insoluble product) is much lower in the middle. We may remark that the profiles found for the electrode system (1.37) for a given value of u are the same as the left-hand half parts of the profiles found for the system (1.1) with 5 = CY and u four times larger. 1.6
Mathematical Methods
For reaction-diffusion problems, there exist well-known methods for studying existence and uniqueness of a positive solution, and asympHere we shall present methods totic stability of a steady state [ 2 3 ] . based upon the existence of upper and lower solutions, as explained by Sattinger in [24] and [ 2 5 ] , Amann in [ 2 6 1 , Fife in [ 6 1 , and Pao' in [ 2 7 ] , although existence and uniqueness of a solution for problem (1.1) results classically from the monotonicity of the function:
22
DIFFUSION AND REACTION IY AN ENZYME MEMBRANE
0.5 Fig.
1.8
Fig.
1.9
MATHEMATICAL METHODS
23
s + s/(l + I s / ) 1 2 8 1 . In fact the notions and results introduced in this section, particularly theorem 1.1, will be useful for subsequent problems, more complex than the "model" case (1 .l) :
We shall show that the evolution problem (1.50) admits a positive solution and only one, a property we must expect because of the physical origin of this problem, and that the transient concentration profile tends, as t + m , towards a stationary profile, which is an unique solution of:
We shall essentially rely upon the notions of upper and lower solutions for problems (1.50) and (1.51). An upper solution for (1.50) is a function O(x,t) such that:
whereas a lower solution satisfies the reversed inequalities. Let us define: (1.53)
O(x)
=
a(1-x) + Bx.
It is easily seen that 0(x,t) = $(x) is an upper solution for the parabolic problem (1.50), while y(x,t) = 0 is a lower solution. As will follow from Proposition 1.1, this is enough to claim that there is an unique solution s(x,t) to (1 .SO), satisfying:
24
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
(1.54)
0 5 s(x,t) 5 $ J ( x ) .
Similarly, upper and lower solutions to (1.51) are functions O(x) and v(x) satisfying respectively:
and the reversed inequalities. Again $(x) and 0 are upper and lower solutions for (1.51), and, as will follow from Proposition 1.2, this is enough to ensure the existence of at least one solution s(x) for (1.51) such that:
Uniqueness of this stationary solution will result from the monotonicity of function F and Proposition 1.3. To begin with, let us state four propositions, which will be proven later in this chapter. Proposition 1 . 1 - Let 7 and 7 be a pair of lower and upper solutions such that 7 5 9 for the evolution problem: st
- s xx
+ U F ( S ) = 0,
s(0,t)
= a,
S(X,O)
=
s(1,t)
0 < x < 1,
t > 0,
= B,
so(x)
(So'
L2(0,l))
where F is Lipschitz-continuous. Then problem (1.57) has an unique solution s(x,t) such that:
Proposition 1.2 - Let 7 and 9 b e a p a i r of lower and upper solutions such that 7 5 9 for the stationary problem: (1.59)
i
-s"(x)
+
s(0) = a,
uF(s(x)) s(1)
= =
8,
0,
O < X < l ,
MATHEMATICAL METHODS
25
where F is Lipschitz-continuous. Then problem (1.59) has at least one solution s(x) such that: (1.60)
Y(x) 5 s(x) 5
?(XI,
O < X < l .
Proposition 1 . 3 - Moreover, if F is monotone increasing, then the solution of ( 1 . 5 9 ) is unique. Remark 1 . 2 (1.61)
Of course, in Proposition 1 . 1 , we must have:
s(x,O) 5 so(x) 5 g(x,O).
Remark 1 . 3 - F is said to be Lipschitz-continuous if there exists a constant L such that: (1.62)
IF(s1)
- F(s2)I 5
I,IS1
- 521
for every s l , s Z e R. Without l o s s of generality we can assume that L conveniently. Remark 1 . 4 -
= 1,
by choosing u
F is said to be monotone increasing if:
The way of applying Proposition 1 1 t o problem ( 1 . S O ) is quite apparent. The Lipschitz continuity of F is a consequence of I F ' ( 5 ) 1 5 1, Similarly Propositions 1 . 2 and 1 . 3 whence I F ( s , ) - F(s2)I 5 Is1 - s 2 apply obviously to problem ( 1 . 5 1 ) .
.
The relation between the transient profile s(.,t), solution of ( . S O ) , and the steady-state profile s e , solution of ( 1 . 5 1 ) , is given by the following proposition : Proposition 1 .4 - Let se be the solution of ( 1 . 5 1 ) . tion s of ( 1 . 5 0 ) satisfies the relation: se(x)
- re-"w(x)
5 s(x,t) 5 se(x)
+
Then the s lu-
re-Ptw(x),
o
< x < I,
for t large enough, where r is a non-negative constant and w is a
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
26
positive eigenfunction corresponding to the least eigenvalue of the eigenvalue problem: -w"(x)
=
0 < x < 1,
pw(x),
w ( 0 ) = w(1)
=
0
Remark 1.5 - Similar results hold when we have a no-flux boundary dondition at x = 1 . In this case we take, as an upper solution, the constant function p(x,t) 5 a. The remainder of this chapter is concerned with the statement and proof of Theorem 1.1, a most useful theorem due to Amann [ 2 6 ] , and the derivation of Propositions 1 . 1 - 1.4 from this theorem. Theorem 1 . 1 - Let (E,P) be an O.B.S. and let [?,PI be a nonempty order interval in E. Suppose that f : [ y , ? ] -t E is an increasing compact map such that 7 5 f(7) and f(p) 5 9. Then f possesses a minimal fixed point and a maximal fixed point 2 . Moreover,
x
-
x
=
lim fk(r)
and
k-
and the sequence (fk(7))
k ? = lim f
k-
(p),
is increasing and fk ( 9 ) ) is decreasing
Comments : Let us first explain the terms which occur in this statement. a real Banach space. A subset P of E is a cone if: P + P c P , R+PcP,
Pn(-P)
=
{O},
-
P
=
E is
P.
P induces an ordering in E, defined by: x 5 y* y - x E P . Endowed with this ordering, E is called an ordered Banach space (O.B.S.), denoted (E,P). The order interval [?,PI is the set of those z such that 7 5 x 2 7 . A map f of the O . B . S . (E,P) into the O.B.S. ( F , Q ) is said to be increasing if x 5 Y =+ f(x) 5 f(y).
X being a subset of E , f is a compact map of X into F if f is continuous and is compact.
MATHEMATICAL METHODS
27
-
x and R are called minimal and maximal fixed points of f in [ y , p l if every fixed point y of f in [y,p] satisfies 5 y 5 R.
x
- From the sequence (fk(y)) we can extract a subsequence converging towards 7 E E. As 7 5 fk(y) 5 9 , and as the cone P defining the ordering is closed, 7 5 7 5 7 . It is immediate that the whole sequence is converging towards Similarly the whole sequence (fk ( 9 ) ) converges towards an element R in [ y , ? ] which, since fk(y) 5 fk (p), satisfies 5 I. The fact that 2 is minimal follows from the consideration of a candidate fixed point x in [7,p].Since x = f(x), we can consider x as satisfying f(x) 5 x , and make the same analysis as above with the interval [y,xl instead Of so that < x. Similarly x 5 4. [y,?]. We shall find the same Proof of Theorem 1 .1
x,
x.
x
x,
x
Applications of Theorem 1 . 1 : Proof of F'r'rosition 1.1 - We are going to prove the existence and uniqueness of a positive solution for (1.57) on the time interval lO,T[ , T arbitrary, whence the property on the time interval ]O,+m[. x ]O,T[ , T > 0. We take E = L 2 (Q) , where Q = 30, [ v defined by: P = I V E L ~ ( Q )I v(x,t) 2 o a.e.1, f : u -f
vt - vXx
+
- UF
uv = ou
v(0,t)
= a,
V(X,O)
=
v(1,t)
=
u)
t
B
(we suppose so
so(x)
Let us check that the hypotheses of Theorem 1.1 i/
E
L' (0,1)). are satisfied
f is increasing: suppose u 1 2 u 2 and define vi by equations (1.64). The difference w = v 2 -.vl wt - w xx + uw = w(0,t) = w(1,t) w(x,O)
U ( U ~-
=
u l ) - o ( F ( u 2 ) - F(u,))
=
f(ui), i = 1,2 satisfies:
2
0
0,
= 0
where the inequality results from the Lipschitz continuity of F. maximum principle for parabolic problems implies that w 2 0. ii/
f is compact:
the map u
-L
v - 0 is continuous from L2(Q) to
The
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
28
W(O,T), where 0 = a(1-x) + Bx, W ( 0 , T ) = { w l w L2(0,T;V), ~ V' = H-'(R), D = ]0,1[, and the wt~L2(0,T;V')},V = injection from W ( 0 , T ) to L 2 ( Q ) is compact " 2 8 1 . Thus the map of f : u + v from E to E is compact.
Hi(D),
iii/
f(y) and f(7) 5 9 : in fact these inequalities are equivalent to the properties for 7 and 7 to be lower and upper solutions. Let us show for example that if 9 is an upper solution for problem (1.57), then f(9) 5 7 . Let v = f(9).
75 -
We have :
I
Subtracting (1.65) from (1.66) gives, for w
(1.67)
Wt
- wxx
+
=
9 - v,
uw 1. 0 ,
w(0,t) 2 0 , w(1,t) 1 0, W(X,O) 2 0 ,
whence, from the maximum principle for parabolic equations, w 2 0 , o r 9 - v 1. 0 , or, f(7) 5 9. Thus existence of at least one solution for (1.57) on [O,T] results In order to prove uniqueness on from application of Theorem 1 . 1 . the same time interval [O,T], we again use the Lipschitz continuity of F. Let s , and s 2 be two solutions, and w = s1 - s 2 . Then: Wt - wXx = U(F(s2) - F(sl)), w(0,t) = w(1,t) = 0 , w(x,O) = 0.
MATHEMATICAL METHODS
29
Hence : 1
Let:
$(t)
(wt
=
1
- wxx)wdx
1
=
1 2
=
1
(
w2dx)
+
1
widx
=
0
w2(x,t)dx.
0
We have successively:
(e But:
- 2ut $ ( t ) ) 5'
0 and the function t
$(t) 2 0 and $(O)
=
-+
0, therefore $(t)
e-2ut$(t) is decreasing : 0.
Remark 1.6 - Thus the solution s of problem (1.57) belongs to The only hypothesis on s o is that s o € L 2 ( 0 , 1 ) . In par$ + W(0,t). ticular we do not need to have s o ( 0 ) = CY or s o ( l ) = B . But on the other hand we know that as soon as t > 0 , the profile x -+ s ( x , t ) is smooth in the sense that s(.,t) E $ + HA(0,l) c C o ( [ O , l ] ) . Proof of Proposition 1 . 2 - We take E P = { V E Elv(x) > 0 a.e.1, 9 = a(1-x) by :
-v"
uv
+
uu
=
-
uF(u),
= L2(n), +
Bx,
7=
R = 10,1[, 0, f : u v defined -+
O < X < l ,
( 1 .69)
v(0) i/
=
v(1)
a,
c i s increasing: -(v2-v1)"
+
=
6.
suppose u 1 5 u z . u(v2-v1)
=
We have:
~ ( ~ 2 - u- ~o(F(uZ) )
together with v 2 - v 1 = 0 at x = 0 and x mum principle for elliptic problems [ 2 9 ] ,
=
- F(ul)) 2
0,
1 . Thus, from the maxiit follows that v 2 - v 1 2 0 .
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
30
ii/
f is continuous from E to E since u + v is continuous from E to H ’ ( f 2 ) and v + v is continuous from H ‘ ( f 2 ) to E.
iii/ f is compact from E to E since u + v i s continuous from E to H ’ ( R ) and v + v is compact from H ’ ( f 2 ) to E.
-
,iv/ y 2 f(y) and f(9) < 9 result from the hypothesis that 7 and are lower and upper solutions for problem ( 1 . 5 9 ) , as in the proof of Proposition 1.1. Proof of Proposition 1.3 - Theorem 1 . 1 does not guarantee uniqueness of a fixed point for the map f defined by (1.69). However, if F is a monotone increasing function, we can prove that s = 5 , s and 5 being the minimal and maximal fixed points of f in [ y , 9 ] . Let w = S - 3 . -w”
w(0)
- F(s)) 1. 0,
U(F(5)
= =
w(1)
=
since S 5 S
0.
Thus from the maximum principle for elliptic equations, w 2 0, and we have 7 2 3 , together with S 2 5 , hence S = 5 . 1.4 - This proof follows a method employed by Pao in 1 2 7 1 to analyze the behavior of immobilized enzyme systems as t + =. Let to: 0. We know that @ = s(*,to) E H ’ ( ~ ) , 2 = 1 0 , 1 [ , so that @ ~ C ~ ( [ o , l l ) Therefore: .
Let us define: (1.70)
7
se(x)
=
- pw(x)e
-11
(t-t,)
-
and let us check that, on the time interval [to,+m[, y is a lower solution for the problem:
i
st
(1.71)
-
s
s(0,t) xx =
+ uF(s) a,
=
s(1,t)
0 < x < 1,
0, =
B,
t > to,
MATHEMATICAL METHODS
31
We immediately have:
the first inequality following from pwlle-P(t-to)
-w” = uw
and
-s;
+
Yt - yxx = uF(se)
puwe -u(t-to)
-
s;
+
= 0.
Similarly, it is easy to check that: (1.72)
i.
=
se(x)
+
pw(x)<e - u (t-to)
is an upper solution for (1.71) on the time interval [to,+-[. In fact we even have s(x,t) 5 se(x) since se is an upper solution for problem (1.50). Therefore Proposition 1.4 follows from an application of Proposition 1 . 1 to problem (1.71) with 7 and 9 defined by (1.70) and (1.72), and:
Remark 1.7 - The main gochemical consequence of the above numerical and mathematical results is the existence of concentration profiles in enzyme membranes. It is very important to take into account this fact when explaining the behavior of biological membranes: interacting diffusion and reaction may cause substrate (resp. product) concentrations to be much lower (resp. higher) within membranes than at boundaries. And in any case, one cannot speak of a single internal concentration for each chemical species involved, but rather of a concentration profile. A modification of the environment causes the concentration profiles to undergo a fast transient evolution, during a period of the order of L2/DS, the diffusion characteristic time, and to approach exponentially a stable steady state.
DIFFUSION AND REACTION IN AN ENZYME MEMBRANE
32
References Henri, V., Lois G6n6rales de 1'Action des Diastases (Hermann, Paris, 1 9 0 3 ) . Michaelis, L. and Menten, M., Die Kinetik der Invertinwirkung, Biochem. Z 4 9 ( 1 9 1 3 ) 3 3 3 - 3 6 9 . Briggs, G.E. and Haldane, J.B.S., A note on the kinetics of enzyme action, Biochem. J. 1 9 ( 1 9 2 5 ) 3 3 8 - 3 3 9 . Heineken, F.G., Tsuchiya, H.M., and Aris, R., On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosci. 1 ( 1 9 6 7 ) 9 5 - 1 1 3 . Murray, J.D., Lectures on Nonlinear Differential-Equation Models in Biology (Clarendon Press, Oxford, 1 9 7 7 ) . Fife, P.C., Mathematical Aspects of Reacting* and Diffusing Systems (Springer-Verlag, Berlin, 1 9 7 9 ) . Banks, H.T., Modeling and Control in the Biomedical Sciences (Lecture Notes in Biomathematics N 0 6 , Springer Verlag, Berlin, 1975).
Walter, C., Contributions of enzyme models, in: Solomon. D.L. and Walter, C. (eds.), Mathematical Model: 'in Biological C S covery (Lecture Notes in Biomathematics N 3 , Springer-Ver ag > Berlin , 1 9 7 7 ) . Bernhard, S.A., The Structure and Function of Enzymes, (Benjamin, New York, 1 9 6 8 ) . Chibata, I., Immobilized Enzymes, Research and Development (Halsted Press, Wiley, New York, 1 9 7 8 ) . Broun G.. Thomas d.. Gellf G.. Domurado D.. Berionneau A . M . and Guillon C., New'methods for binding enkyme molecules into a water insoluble matrix : properties after insolubilization, Biotechnol. Bioeng., Vol. 1 5 ( 1 9 7 3 ) 3 5 9 - 3 7 5 . Thomas, D., Broun G., and Selegny E., Monoenzymatic model membranes : diffusion-reaction kinetics and phenomena, Biochimie, 5 4 ( 1 9 7 2 ) 2 2 9 - 2 4 4 . Thomas D., Bourdillon C., Broun G., and Kernevez J.P., Kinetic behavior of enzymes in artificial membranes. Inhibition and reversibility effects, Biochemistry, 1 3 ( 1 9 7 4 ) 2 9 9 5 3000.
Calvot, C., Berjonneau A.M., Gellf G., and Thomas D., Magnetic enzyme membranes as active elements of electrochemical sensors. Specific amino acid enzyme electrodes, FEBS Letters Vol. 5 9 ( 1 9 7 5 ) 2 5 8 - 2 6 2 . Cordonnier, M., Lawny F., Chapot D., and Thomas D., Magnetic enzyme membranes as active elements of electrochemical sensors. Lactose, saccharose, maltose bienzyme electrodes, FEBS Letters, Vol. 5 9 , N O 2 ( 1 9 7 5 ) 2 6 3 - 2 6 7 .
REFERENCES
33
L161 Durand, P., David A., and Thomas D., An enzyme electrode for acetylocholine, Biochimica et Biophysica Acta, 527 (1978) 277-281. [17l
Richtmyer, R.D. and Morton K.W., Difference Methods for Initial Value Problems (Wiley-Interscience, New York, 1957).
[181
Barbotin, J.N. and Thomas D., Electron microscopic and kinetic studies dealing with an artificial enzyme membrane, Application to a cytochemical model with the horseradish peroxidase-3,3l-diaminobenzidine system, J. Hystochem. Cytochem. Vol. 2 2 , N " l 1 (1974) 1048-1059.
[19J Barbotin, J.N., Electron microscopy as a tool for studying artificial enzyme membranes, in: Thomas, D. and Kernevez J.P. (eds.), Analysis and Control of Immobilized Enzyme Systems (North-Holland, Amsterdam, 1976). [201
Malpisce, Y., Sharan M., Barbotin J.N., Personne P., and Thomas D., A histochemical model.dealing with a glucose-oxidase-peroxidase immobilized bienzyme system: The influence of diffusion limitations on histochemical results (submitted for publication).
[211
Barbotin, J.N., Properties of lactate dehydrogenase immobilized in a lipid-protein matrix, FEBS Letters, Vol. 72, NO1 (1976) 93-97.
[221
Barbotin, J.N. and Thomasset, Immobilization of L-glutamate dehydrogenase into soluble cross-linked polymers - ADP effects and electron microscopy studies, Biochem. Biophys. Acta, 570 (1979) 1 1 - 2 1 .
[231
Auchmuty, J.F.G., Qualitative effects of diffusion in chemical systems, in: Lectures on Mathematics in the Life Science, Vol. 10 (The American Mathematical Society, Providence, 1978).
I241
Sattinger, D.H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. Jour. 21 (1972) 979-1000.
[251
Sattinger. D.H.. T o ~ i c sin Stability and Bifurcation Theory (Lectuge Notes in Mathematics N"309', Springer-Verlag, Berlin, 1973).
[261
Amann, H., Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, (preprint).
[271
Pao, C.V., Mathematical analysis of enzyme-substrate reaction diffusion in some biochemical systems, J. Nonlinear Analysis: Theory, Methods, and Applications (to appear).
[281
Lions, J.L., Quelques Methodes de Resolution des Problemes aux Limites Non-lineaires (Dunod, Paris, 1969, English translation by Le Van, 1971).
[291
Protter, M.H. and Weinberger H.F., Maximum Principles in Differential Equations (Prentice Hall, Englewood Cliffs, New Jersey, 1967).
This Page Intentionally Left Blank
CHAPTER 2 GLUCOSE PUMP
Suppose that two compartments I and 11, containing solutions of substrate at concentrations S 1 and S2 ( S 1 < S2), are separated by an inactive membrane, namely a membrane prepared as described in Chapter 1, but containing only inactive protein, without enzyme. There is no reaction within the membrane and the only phenomenon taking place in this system is a flow of substrate from compartment I1 to compartment I due to the diffusion of substrate across the membrane from regions at high concentration to regions at low concentration. Moreover, if concentrations S 1 and S z are not held fixed, their difference S2 - S 1 tends towards 0 as t -t a. Thomas et.al. ([1],[21,[3]) describe a membrane system able to transport glucose against the concentration gradient, namely from compartment I at low concentration S 1 to compartment I1 at high concentration S2 (Fig. 2.1). In this chapter we first describe this so-called "active transport" phenomenon (section Z.l), then model this membrane system by a pair of coupled partial differential equations governing S 1 and S2 concentrations (section 2 . 2 ) , then give numerical methods and results (section 2.3) and finally discuss mathematical aspects (section 2.4). 2.1
Description
The membrane is made of two enzyme layers sandwiched between two selective films (Fig. 2.2). The enzyme layers contain respectively hexokinase (El) and phosphatase (E2). Hexokinase catalyzes the phosphorylation of glucose (S) into glucose-6-phosphate (P), and phosphatase catalyzes the dephosphorylation of glucose-6-phosphate into glucose : E2 S + ATP ----$P + ADP, P ---)S + inorganic phosphate. In layer one S + E l + P + E l , while in layer two P + E 2 +. S + E 2 . The membrane is impregnated with ATP. The selective films are permeable to S, but impermeable to P. The membrane separates two compartments I and I 1 containing solutions of S at concentrations S 1 and S z (S, S z ) (Fig. 2.1). In the experiments reported by Thomas 35
GLUCOSE PUMP
36
the first (donor) compartment is large enough with respect to membrane volume for S concentration within it ( S 1 ) to be assumed constant, while the second (receptor) compartment is 1 5 0 times smaller, and S concentration within it ( S z ) is increasing (though S 2 2 S1 at the beginning of the experiment). Thus this unsymmetrical membrane is a model of a metabolic and spatial sequential enzyme system transporting glucose against the concentration gradient. This phenomgnon, called active transport can be explained by examining the S and P concentration profiles, obtained by numerical simulation S enters from compartment I into layer 1 where it is (Fig. 2.3). consumed, producing P. In layer 1 , the S profile is convex. P produced in layer 1 diffuses into the second layer where it is converted to s. When the S profile presents a hump in layer 2 as depicted in Fig. 2.3, S leaves the membrane into compartment TI where it is found at increasingly higher levels of concentration. This system behaves as a glucose pump where energy is provided by A T P .
* M
xi0
x=1/2
x=l
Fig. 2 . 1 - Double-layer membrane separating 2 solutions of substrate S at concentrations S 1 and S 2 in compartments I and 11. In laver 1 . containing enzyme E l . S + El P + El. In layer i, containing enzyme P + E2 S + Ez. -+
-+
ki,
DESCRIPTION
H E X
0 SOLUTION 1
K I N A S E
37
P H
0
S
P H A
SOLUTION 2
T A
S E
L A Y E R
Fig. 2.2 - Diagram of a double-layer membrane sandwiched between two selective layers.
Fig. 2.3 - S and P concentration profiles in the double layer membrane, indicated respectively by solid A for S ,ind the dashed line for P.
GLUCOSE PUMP
38
2.2
Modeling
The reaction S + E l + P + E l is assumed monoenzymatic i reversibl with competitive inhibition by the product P. Thus the BriggsHaldane reaction velocity in layer one is given by:
The reaction P + E2 +. S + E2 being monoenzymatic irreversible, we have in layer 2 the reaction velocity: v2
as
=
(
'reaction
VM2 p'(Kp
=
+
S and P concentrations within the two active layers are governed by
equations: (2.1)
as -
-
at
DS
fi + ax2
v
and
= 0
ap at
azp - D~ ax2
-
v = 0,
where the x-axis has been chosen perpendicular to the membrane surfaces, S and P have the same diffusion coefficients DS and D p , v = v 1 in layer one and v = -v2 in layer two, with VM = VM2 = VM. 1 The selective film's thickness in negligible with respect to the overall thickness L of the active layers. Let 8 denote the characteristic time for diffusion: 8 = L2/DS. If we introduce the nondimensional variables: s = S/KS, p = P/KS, x '
=
x/ L and t'
=
t/8,
substitute them into (2.1) and for convenience drop the primes, we get the dimensionless glucose pump equations: St
- s X x + F(*,s,p)
=
0,
Pt
- Pxx - F(.,S,P)
=
0,
0 < x < 1 , t > 0,
(2.2)
where the reaction term F is defined by:
MODEL ING
(2.4)
u
=
In ( 2 . 2 ) O < X < l ,
(VM/Ks)L2/Ds
and
39
A = KS/KI.
F(.,s,p) denotes the function (x,t) t > O .
-+
F(x,s(x,t),p(x,t)),
Initial conditions - The membrane being initially void of S and P , the initial conditions are: (2.5)
s(-,O)
=
p(*,O)
= 0,
where s ( * ,0) is the function x
+
s(x,O),
0 < x < 1.
Boundary conditions - Both walls of the membrane being impermeab1.eto P, appropriate boundary conditions for P are:
Here px(O,-) denotes the function
t
The boundary condition for S at x
= 0
(2.7)
s(O,.)
+
px(O,t),
t > 0.
is:
= a
where a = S,/Ks either is held fixed by some chemostat, or may be considered as a constant because the donor compartment is large enough, The boundary condition for S at x
=
1. is:
= S /K is the initial concentration in the receptor comwhere 2 s partment and:
GLUCOSE PUMP
40
(2.9)
c
=
"M vR
=
~
membrane volume receptor compartment volume
Condition (2.8) may be obtained by the following reasoning: the rate of increase of the total amount of S in the receptor compartment, vRst(l,t), equals the total amount of S leaving the membrane 'per unit of time, -Asx(l,t), A being the membrane area. As vM = thickness x area = 1 x A = A , equation ( 2 . 7 ) results, with c given hy (2.9).
In fact c < < 1 since the membrane volume vM is small compared to the receptor compartment volume vR. We may distinguish two epochs in the evolution of the system. In a first epoch, of small duration, the boundary value s(1,t) has no time to change appreciably, while the S and P profiles of concentration, s(.,t) and p(-,t), undergo a fast transition from initial state (2.5) towards a steady state satisfying: -s"(x)
+
F(x,s(x),P(x))
=
0, ~ ( 0 =) a , ~ ( 1 ) = B ,
(2.10)
-P"(x) - F(x,s(x),~(x)
0 , p'(0)
This fast transition obeys ( 2 . 2 ) , ( 2 . 5 ) , (2.11)
s(l,.)
=
=
p'(1)
(2.6),
=
0.
( 2 . 7 ) and
B.
Fig. 2.4 shows the S and P concentration profiles during this evolution, and the numerical simulations show that these profiles are unchanging after about t = 10 (remember that the unit of time, 8 = L 2 / D S , corresponds to 18 seconds if, as is the case, L = 5 10-3cm and DS = 5 10-3cm2h-'). Second epoch - For larger time behavior the boundary condition (2.11) for S is no more valid since the flow of S from x = 0 to x = 1 gives rise to an increase of S concentration in the receptor compartment, according to equations (2.8). For example, in Thomas' experiments, glucose concentration is observed in the receptor compartment during several hours. Thus, once a quasi-steady state has been attained through a fast transition, this quasi-steady state is going to change
MODELING
41
slowly during a long period, this evolution being governed by equations (2.2), (2.5), (2.6), ( 2 . 7 ) , and (2.8). The numerical simulations give the results shown in Fig. 2.5 and end with steady state
. .
prol'iles as hep'icieh 'in P'ig. tions :
Z.b
an> wYic\, are represenieh \by equa-
l ,s,p)
=
0,
=
0,
p'(0) = p'(1) = 0.
Note that when 2.12) is satisfied there is no more pumping action, since s ' ( 1 ) = 0
To summarize the above discussion, the modeling of the glucose pump system results in evolution problems governed by equations ( 2 . 2 ) , with initial conditions (2.5), supplemented with boundary conditions (2.6) for P, (2.7) for S at x = 0, and either (2.8) or (2.11) for S at x = 1 , and corresponding steady state problems (2.10) or (2.12).
Fig. 2.4 - Evolution of S (on the left) and right) concentration profiles for equations a = B = 1 and o = 700. Profiles - - - , correspond respectively to times t = 0.5, t
-*-,
-
P (on the
(2.1), with and = 1 , & t = 10.
GLUCOSE PUMP
42
Fig. 2.5 - Evolution of S (on the 1eft)and P (on the right) concentration profiles for equations ( 2 . 1 ) with B . C . ( 2 . 2 ) , with a = B = 1 , u = 700 and c = 0.005. Profiles ---- - & correspond respectively to times t = 0.05, t = 0.5, t = 5 , and t = 1 5 .
... .
9
130
100
50
/
1 r~______._
0
\ \
>.--.-
NUMERICAL SIMULATIONS
2.3
43
Numerical Simulations
The results of the numerical simulation of equations ( Z . Z ) , ( Z . S ) , ( 2 . 6 ) , (2.7), and (2.8) (resp. (2.11)) are given in Figures (2.4) and ( 2 . 5 ) . In the first case it appears that the profiles (s(*,t) and p(.,t)) tend increasingly towards a steady state (se,pe), solution of ( 2 . 1 0 ) , and, after t = 10, the approximations of (s(.,t), p(.,t)) vary no more with time. m = 10 for this system ! In the second case, on the contrary, it takes a longer time for the receptor compartment to attain a stationary value. This stationary value is about 130 times the concentration of S in the donor compartment (Fig. 2.6). These numerical simulations aid understanding why unicellular animals immersed in a solution of glucose at low concentration S 1 , continue to absorb glucose, even though glucose concentration within them, S 2 , is much higher: their membrane could be the seat of similar diffusion-reaction phenomena. There is little to say about the numerical methods used to obtain these curves in addition to what was said in Section 1.5 on finite difference methods. Again two schemes were employed, the explicit one and the totally implicit one, giving the same results. k stability condition for the explicit scheme being k = At
GLUCOSE PUMP
44
- n
sn+l N
k
2.4
sn+l - n+l N+1 'N-1 + c 2 h -
SN
=
0 , (k = At, h = AX),
Mathematical Analysis
The modeling of the glucose pump resulted in two models, one representing the fast transient evolution at the beginning.of the experiment, the other describing the slow variation of quasi-steady states towards an equilibrium. In this section we study the behavior of solutions in both cases. Namely, we prove the following proposition: Proposition 2.1 - Consider the problem: find functions s = s(x,t), p = p(x,t), 0 < x < 1, t > 0 , satisfying equations:
ini ial conditions (2. 4) s(x,O) = p(x,O)
=
0,
O < X < l
and boundary conditions
(2.15)
i
PX(l,t) = 0 ,
P,(O,t)
=
s(0,t)
= 0,
t > 0,
and either (2.16) s(1,t)
=
B,
t > O
or
I
st(l,t) + csx(l,t) = 0 ,
( 2 * 1 7 ) s(1,O)
=
0.
t > 0,
MATHEMATICAL ANALYSIS (resp. (2.17))
45
i/
Problem (2.13)-(2.16)
has at most one solution.
ii/
Problem (2.13)-(2.16) (resp. (2.17)) has at least one (hence exactly one) solution.
iii/
The solution of either problem is increasing with time.
The solution o f either problem tends increasingly, when t + - , iv/ towards a steady state, solution o f
i
-s"(x) + F(x,s(x),P(x)) (2.18) -P"(X) - F(x,s(x),p(x)) p'(0) = p'(1) = 0 , s(1)
0,
= = =
O < X < l
0, a,
and eithe (2.19) s ( 1 = B , or (2.20) s'(1) = 0. We are going to prove proposition 2.1 for problem one (boundary conditions (2.16) and (2,19)), then we shall indicate how to convert this proof to handle boundary conditions (2.17) and (2.20). We shall use tools which proved to be useful in Chapter 1, Amann theorem, and Maximum Principles for elliptic and parabolic problems: Amann theorem - E being an ordered Banach space, f an increasing, continuous and compact mapping of [7,91 E E into E such that y 5 f ( y ) and f ( p ) 5 7 , then there is at least one fixed point of f in [ Y , ? ] . Maximum principle for elliptic problems -w"(x) w(0)
Then:
+ GW(X)
0,
Suppose w satisfies:
o < x < 1 ,
2 0 and w(1) 2 0.
w(x) 2 0 ,
o < x < 1 .
GLUCOSE PUMP
46
Maximum principle for parabolic problems Wt
Then:
- wxx
+ ow
-
Suppose w satisfies:
0 < x < 1,
3 0,
w(x,t) 2 0 ,
t > 0,
O < x < l , t > O .
In order to handle problems involving no-flux boundary conditions, we add to our tool kit the: Maximum principle for elliptic problems with Neumann boundary condi tions - Suppose w satisfies:
i
-w"(x) + aw(x) 3 0 , -w'(O)
then:
O < X < l ,
2 0 and w'(1) 2 0 ,
w(x) 3 0 ,
O < X < l .
We have a similar property for parabolic problems. Proof of proposition 2.1 (2.19). i/
-
(with boundary conditions (2.16) and
Uniqueness - Let(s,,pl) and (s2,p2) be two solutions. difference (w,z) = (sl-s2,p1-p2)satisfy:
=
w(l,.)
= 0,
w(*,O) =
Z(*,O)
=
w(O,.)
ZX(O,')
=
ZX(l,')
=
Their
0,
0.
Multiplying the first equation by w, the second one by z, adding and integrating with respect to x on (O,l), we obtain:
where :
MATHEMATICAL ANALYSIS
47
by applying Cauchy-Schwarz inequality, Lipschitz continuity of F Let $(t) = lw(t)/’ + l$(t)/’. We have and ( a ? b ) ’ 5 Z(a’ + b ’ j . $‘(t) 5 4u$(t), $(O) = 0 , which, together with $(t) 2 0 , imply $(t) = 0. ii/
Existence - Existence of a solution on any bounded interval [O,Tl, together with uniqueness, implies existence and uniqueness of a solution on LO,+-[. Thus, let us prove existence of a solution for ( 2 . 1 3 ) - ( 2 . 1 6 ) on the time interval [O,T], T < m . For this purpose, we put the problem within the framework of theorem 1 .l, with E = L’(Q) x L ’ ( Q ) , (Q = l O , T [ x a , = 1 0 , 1 [ ) , P = I(u,v) I u(x,t) 2 0 , v(x,t) 0 a.e. (x,t)l, f : (u,v) + (s,p) defined by:
y
=
(0,O) and
q
=
$(x)
(2.23) r
=
(2.22)
+
9
=
(q,r) defined by:
ox(1-x)/Z,
-
m
+
a/4
(a/2)x2
m
+
(a/2)(1-x)2,
$(x) = o(1-x)
,
+
fix,
0 < x < 1/2, 1/2 < x < 1,
where m and a satisfy:
m 2
=
max q(x) 05x51
and
Let us check that the hypotheses of Amann theorem are satisfied
GLUCOSE PUMP
48 y
5 f(y) is obvious since, if f(y)= (s,p), s 2
f(?)
5 9
results from the fact that
0 and p E 0.
is an upper solution for pro-
~
blem ( 2 . 1 3 ) - ( 2 . 1 6 ) ,
in the sense that:
Indeed, we have, cn 0 < x
1/2,
and, on 1/2 < x < 1, qt - q,
t'
- rxx
+ ~(.,q,r) =
u
-
- ~ ( - , q , r )= -a +
u
+
> 0,
u T > -a l + r -
+
,
or(?) + r(l)
=
- a + a - 1 + m -> 0. The boundary and initial conditions for (q,r) to be an upper solution are trivially satisfied. Let f(?) = (s,p), defined by (2.21) with u = q and v = r . Subtracting these equations from (2.24) we obtain, for w = q - s and z = r - p,
I
wt - wxx
+
uw 2 0 , w ( O , . )
- zxx
+
uz 2 0 , - Z x ( O , . )
Zt
1. 0 , w(l,.)
20,
ZX(l,')
2 0 , w(.,O) L 0 , 2 0 , z(.,O) 2 0
whence w 0 and z 1. 0 from the Maximum Principle for parabolic problems. But (w,z) 1. 0 means p 2 f ( ? ) .
MATHEMATICAL ANALYSIS
49
f is increasing - Suppose ( u l , v l ) 5 (u2,v2) and let w = s 2 - s l , z = p2 - p l , where (si,pi) = f(ui,vi), i = 1 ,2. h'e have:
wt
-
wXx
+
ow
=
u(uZ-u1)
-
F ( * , u ~ , v ~ )+ F(-,u1,vl),
(2.25)
W(O,.)
= 0,
+
w(l,.)
=
0, w ( O , * )
uz = u ( v z - v 1 )
+
=
0,
F ( * , u ~ , v ~ )- F(a,u1,vl),
(2.26)
(ZX(O,')
=
ZX(l,')
=
0 , z(.,O)
=
0.
The second member in ( 2 . 2 5 ) can be written:
where the quan-tities within brackets are positive, the first one because of the Lipschitz continuity of F , the second one because F is a decreasing function of p. Thus, from the Maximum Principle for parabolic problems, w 0. Similarly, the second member in (2.26) is positive, hence z 0. Continuity and compactness of f - f, defined by equations (2.21), may be considered as built up from three mappings. The first one transforms ( u , v ) in (Q,R) and is continuous from E to E , as follows from Lipschitz continuity of F. The second one transforms (Q,R) in ( u , v ) , and is continuous from E to [ $ + W ( 0 , T ) l x W1(O,T), where @ = a ( l - X) + Bx,
Finally, the third one is the natural injection (s-$,p) from W ( 0 , T ) x W1(O,T) to E, and is compact.
+
(s-$,p)
iii/ t + (s(-,t),p(*,t)) is an increasing function of time - Let u(x,t) = s(x,t+?) and v(x,t) = p(x,t+~), where (s,p) is the solution of problem (2.13)-(2.16) and T > 0. u and v satisfy the same equations and boundary conditions as s and p, and initial conditions u ( x , o ) = S ( X , T ) > 0 , v ( x , O ) = P(X,T) > 0. Hence ( u , v ) is an upper
GLUCOSE PUMP
50
s o l u t i o n f o r problem ( 2 . 1 3 ) - ( 2 . 1 6 ) .
Then, r e p l a c i n g
t h e above argument, i t follows t h a t ( s , p ) s ( . , t ) 5 s ( - , t + . r ) and p ( . , t ) 5 p ( . , t + . r ) .
by ( u , v ) i n
5 ( u , v ) , w h i c h means t h a t
iv/ ( s ( * , t ) , p ( . , t ) ) t e n d s t o w a r d s a s t e a d y s t a t e when t + m : a s = ( q , r ) , f o r e v e r y x t h e f u n c t i o n s t +. s ( x , t ) (s(-,t),p(-,t)) 5 a n d t -+ p ( x , t ) t e n d t o w a r d s l i m i t s u ( x ) a n d v ( x ) . M o r e o v e r , f r o m t h e L e b e s g u e t h e o r e m , s ( . , t ) +. u i n L 2 ( n ) a n d p ( - , t ) v i n L2(Q], whence, by u s i n g t h e L i p s c h i t z c o n t i n u i t y o f F, F ( - , s ( . , t ) , p ( - , t ) ) F ( * , u , v ) i n L 2 ( n ) . L e t w and z b e t h e s o l u t i o n s o f : -+
-w” ( x )
,
-F
+
uz(x) = uv(x) + F ( x , u ( x ) , v ( x ) ) ,
(X , U
(x)
(x) )
=
,V
O < X < l ,
and (-z”(x)
We w i s h t o p r o v e t h a t ( w , z ) s(x,t).
=
(u,v).
O < X < l ,
Let us d e f i n e y ( x , t )
7
w(x) -
We h a v e :
where 0 in L2(n) as t
(2.28) f ( * , t )
+
since
= F(*,s(*,t),p(.,t))
f(.,t)
+ m,
-
F(*,u,v).
We n e e d t h e f o l l o w i n g Lemma: Lemma 2 . 1 L2(n) a s t +.
Let y s a t i s f y ( 2 . 2 7 )
-.
and ( 2 . 2 8 ) .
Then y ( . , t )
+.
0 in
-+
MATHEMATICAL ANALYSIS
51
___ Proof -
Let E > 0 be given, arbitrarily small. For t large enough, we have If(*,t)I < E, where 1 . 1 denotes the L 2 ( n ) - norm. If in (2.27) we multiply by y , and integrate with respect to x on (O,l), we obtain:
Applying PoincarC Inequality lyx(-,t)12 2 /y(*,t)l' and denoting $(t) = ly(.,t)I2, we have $'(t) + $(t) 5 E ~ which , implies $(t) 5 E ' + ($(O) - c2)e-t. Thus for t large enough $(t) 5 2 ~ ' . Applying lemma 2.1 to our problem, it results that s(.,t) + w as t + m , hence u = w. Similarly, p ( . , t ) -+ z as t m, and v E z . Thus u and v satisfy the steady state equations (2.18)-(2.19). The On the solutions of these equations are in H ' ( Q ) , hence in Co(E). other hand, for t > 0 , s(-,t) and p(-,t) belong to H ' ( R ) , hence to Co(a). By the Dini theoremL61 s(.,t) u in Co(E), and p(.,t) vin -+
-+
-+
co(a) .
-~ Remark 2.3 - The above argument demonstrates the existence of at least one steady-state, solution of (2.18)-(2.19). It is possible to prove existence of minimal and maximal solutions of (2.18)-(2.19) in [ ? , ? I , where 7 = (0,O) and = [p,ql as defined by (2.22) and (2.23), by applying Theorem 1 .1 to the situation where E =(L2(n))2 , P = {(u,v)lu(x) 2 0 and v(x) 2 0 a.e.1 and f : (u,v) (s,p) is defined by : -+
-s"
+ 5s = u u - F(*,u,v),
(2.29) s ( 0 ) = u , s(1) = 6 ,
and
I
-p"
+
up = uv + F(.,u,v),
(2.30)
p'(0)
=
p ' ( 1 ) = 0.
The details are similar to those found in the proofs of propositions
52
GLUCOSE PUMP
1.2 and 2.1. Let (2,p)and (2,g) be these minimal and maximal solutions. Whether they are the same or not is an open problem. Numerically, i t is possible to approximate (S,p) = X and ( ? , p ) = 2 by algorithms using the fact that fk(y) + and fk(p) -+ 4 (Theorem l.l), namely:
x
or, more precisely,
. . .
start either from s o = 0, p o = 0 , or from s o = q , p 0 = r , define sktl and p k + 1 as solutions of (2.29) and (2.30) with u and v = p k stop when s k and pk do not vary appreciably anymore.
=
.
-
s
k
-
At least for the values u 700, X 1 , and a = B = 1, there is strong numerical evidence that # 2 (Figure 2 . 7 ) . Anyway, even if
x
there are multiple steady states, (s(*,t),p(*,t)) converges towards the minimal solution of (2.18)-(2.19), x = (s,p), since (s,?)being an upper solution for the evolution problem (2.13)-(2.16), (s(*,t), p(.,t)) 5 (s,p) 5 (u,v). There was only one steady state for the simple biochemical system considered in Chapter 1 . Here, though i t seems probable that there exists only one steady state, we did not find any method to prove it. In subsequent chapters we shall deal with systems possessing multiple steady states. In fact, i t is the general situation: initial value problems, governed by equations such + G(y) = 0 , y(0) = yo, admit naturally only one evolution, as whereas the corresponding steady states, obeying G(y) = 0 , may be multiple.
2
We proved Proposition 2.1 in the case of boundary conditions (2.16) and (2.19) for s. In the case of boundary conditions (2.17) and (2.20), the reasoning is the same as above, with only two slight changes, relative to the definition of s and to the definition of an upper solution p. These modifications are explained in Remarks 2.2 and 2.3. Remark 2.4
-
Consider the problem:
MATHEMATICAL ANALYSIS
(2*31)po,t)
=
a,
st(l,t) + csx(l,t)
=
53
0,
Without entering into details let us say that the Galerkin method can be used to prove existence and uniqueness of a solution for ( 2 . 3 1 ) , the proof being based on the use of the variational formulation:
Remark 2 . 5 - An upper solution 9 for the case of "moving" boundary = (q,r) such that: values for s is q = a
+
(O/Z)X(2-X)
and r defined as in ( 2 . 2 3 ) , with:
Remark 2 . 6 - The biochemical interpretation of the above numerical ~and mathematical results is that two "scalar" phenomena, namely diffusion and reaction, can induce, when interacting, a "vectorial" phenomenon, more precisely an active transport of substrate.
54
GLUCOSE PUMP
r i g . L . I - x p p r o x i m a r i o n s o r S L X J rrom a D o v e L I J a n a rrom ~ e i o w I I (2) a f t e r 50 iterations. B o t h a p p r o x i m a t i o n s c o i n c i d e (3) after 7 5 0 iterations.
REFERENCES
55
References Broun, G., Thomas D., and Selegny E., Structured bienzymatical models formed by sequential enzymes bound into artificial supports: kctive glucose transport effect, J. Membrane Eiol. 8 (1972) 313-332. Thomas, D. and Broun, G., Some aspects of the regulation and transport in enzyme membrane models, Biochimie, Tome 55, NO8 (1973) 975-984. Thomas, D., Artificial enzyme membranes, transport, memory, and oscillatory phenomena, in: Thomas, D. and Kernevez ,J.P. (eds) Analysis and Control o f Immobilized Enzyme Systems (North Holland, Amsterdam, 1976). Amann, H., Nonlinear operators in ordered Banach spaces and some applications to nonlinear boundary value problems, (preprint)
.
Kernevez, J.P. and Thomas, D., Numerical analysis and control o f some biochemical systems, Appl. Math. Optimization, v o l . 1, NO3 (1975) 222-285. Dieudon6, J., Foundations of Modern Analysis (Academic Press, New Jersey, 1960).
This Page Intentionally Left Blank
CHAPTER 3 MULTIPLE STEADY STATES AND HYSTERESIS
In this chapter we shall be interested in the multiple steady states which may exist in systems involving diffusion and substrate inhibition kinetics. In Section 3.1 we shall consider the so-called zero-dimensional system, governed by equation:
where : (3.2)
F(s)
=
s/(l
+
Is1
+
ks’).
Equation (3.1) models an inactive membrane separating an enzymatically active compartment at concentration s from a reservoir at concentration s (Fig. 3.1). Here diffusion and reaction take place respectively within the membrane and the compartment. Though simple, this model can exhibit multiple steady states, governed by the algebraic equation: (3.3)
s o - s - PF(S) = 0
* S
Reservoir
Compartment \ L
Membrane
Fig. 3.1 57
MULTIPLE STEADY STATES AND HYSTERESIS
58
As a consequence we may have an hysteresis phenomenon when, p being
held fixed, so is allowed to vary, first increasingly, then decreasingly. A steady-state s that has been changed by a variation of s o may fail to return to its original value when the cause of the change is removed. In Section 3 . 2 diffusion and reaction take place in the same regions o f an artificial enzyme membrane, a one-dimensional system represented by equations: st
- s xx
+
uF(s)
= 0,
O < X < l , t > O ,
(3.4)
s(0,t)
=
s(1,t)
= so
for the transient evolution and - s " ( x ) + uF(s(x))
= 0,
O < X < l ,
(3.5) s ( 0 ) = s(1)
= so
for the stationary states. Depending upon the values of u and s o , the stationary problem (3.5) may admit multiple solutions. This is shown by numerical experiments, and can be demonstrated by elementary methods. We discuss the hysteretic behavior of these solutions when, u being fixed sufficiently large, s o varies. However it appears clearly that in more complex cases a mathematical analysis of similar situations is not within our grasp and that, even in the case of equation ( 3 . 5 ) , numerical methods should be used for exploring the family of solutions when some parameter, say s o , varies. By the family of solutions of ( 3 . 5 ) , we mean the subset Jt of R+ x B defined by: (A,u)~A
where B
=
C:([O,l])
-u"(x) =
+
uF(u(x)
+
A)
=
Cu(u~C~([O,ll,u(O))
O < X < l
0, = u(1)
=
01
.
Accordingly, Section 3.3 describes the continuation method of Kubicek
THE ZERO-DIMENSIONAL SYSTEM
59
[ l ] for problems where the steady-state u of the system depends on a parameter X through a relation of the form:
(3.6)
f(A,u)
= 0.
Here f is a continuously differentiable mapping of R+ x Rn into Rn. Section 3.4 is devoted to a mathematical analysis of the curve o f solutions for (3.5) when u is held fixed and s o varies. In addition to the familiar methods of upper and lower solutions, which enable us to show the existence of maximal and minimal solutions, we employ the Implicit Function Theorem, thus obtaining continuous curves o f solutions. In particular the turning points of the S-shaped curve of solutions (Figure 3 . 8 ) are studied by applying results of Crandall and Rabinowitz [ 2 3 . I n Section 3 . 5 we discuss the stability of the steady-state solutions
of (3.4), showing that the maximal and minimal solutions of (3.5) are stable and that the stability of the steady state solution is changing at turning points. Here again results of [ 2 ] are very helpful. Lastly, Section 3.6 argues the quasi-steady-state system (3.3) as a limit of the evolution system (3.1). 3.1
The Zero-dimensional System
Our model is a well-stirred, enzymatically active compartment separated from a well-stirred reservoir by a membrane (Figure 3.1). The membrane, in which all of the diffusive resistance to mass transfer is concentrated, is inactive, that is without enzyme. It is permeable to substrate, but not to enzyme. The enzyme is uricase, which catalyzes
(3.7)
the oxidation
o f
uric
acid a c c o r d i n s
to:
uricase uric acid + oxygen F a l l a n t o i n + other products
We can assume that oxygen undergoes zero order kinetics so that the only substrate to be considered is in our modelling uric acid. The compartment contains enzyme molecules which catalyze the transformation of this substrate with the rate expression: (3.8)
R(S)
=
VM S / ( K s
+
S(l
+
S/KSS)).
MULTIPLE STEADY STATES AND HYSTERESIS
60
In this commonly accepted rate law for substrate inhibition kinetics, S is the substrate (uric acid) concentration, while VM, KS, and KSS denote the familiar maximal rate, Michaelis, and inhibition constants. The reservoir contains substrate at a fixed concentration S o . If S is the substrate concentration in the compartment, the membrane transports substrate from the reservoir to the compartment with rate (Ds A/L)(So-S) where DS is the substrate diffusion coefficient, A the contact surface area and L the membrane thickness. Mass balance for S in the reactor is expressed by equation: Vr
=
(Ds A/L) ( S o
- S) - V r
where Vr is the reactor volume. efficient:
P
=
DS A/(L Vr)
R(S)
Let P denote the mass transfer co-
(Vm/Vr) DS/L2
=
where Vm is the membrane volume.
We have:
Let 0 be the characteristic time 0 = 1 / P . Taking 8 and KS as the natural units of time and concentration we obtain:
where : s = S / K ~ T, = t/e, 0
p
so =
s ~ / K ~
is the reaction-permeation modulus: P =
(VM/KS)O,
=
(Vr/Vm)LZ/DS,
+
ks2)
and (3.10)
F(s) = s / ( l
+
Is1
with k
=
Ks/Kss.
Henceforth, in this chapter F ( s ) will denote the rate expression (3.10). We have chosen to use the simple steady state form for both permeation and reaction in a system which is at least potentially
THE Z E R O - D I M E N S I O N A L
SYSTEM
61
non-stationary. A limit is thereby placed on the maximum frequency for which equation (3.9) will be a good model if s o is allowed to vary with time. This frequency is typically the order of magnitude P, since diffusion processes relax more slowly than most reactions In the stationary state, the time derivative in equation (3.9) van shes, and the mass balance equation reduces to the algebraic equation (3.11)
so
-
s
-
pF(s)
=
0.
Equation (3.11) is one of the simplest imaginable models in which the coupling of reaction and diffusion yields a phenomenon involving multiple steady states. In fact, it is also an open system, due to transport of substrate from outside into the compartment with a flux so - s. Without this transport the substrate concentration s would evolve according to the kinetic equation: (3.12)
ds =
-
p
F(s)
towards the unique equilibrium state s = 0. But equation (3.11) possesses multiple solutions, simply obtained by cutting the curve If we fix y = pF(s) by the straight line y = s o - s (Figure 3.2.a). p in (3.11) and wish to obtain s as s o varies, we can consider the functions:
see(Figures 3.3a and 3.3b) respectively. It is easily seen that if: p
-l/min F'(s), os,
there are 2 values s:: and s:::: of s such that: -ds0 ds - 1 + pF'(s)
=
0,
and for which so = s + p F ( s ) admits relative extrema. In that case we have the representations depicted in (Figures 3 . 3 ~and 3.3d) for s and y as functions of s o , y being the reaction rate of the system:
MULTIPLE STEADY STATES AND HYSTERESIS
62
(3.13)
y
= so
-
s = pF(s).
The plots given in Figure 3 . 3 show the existence, for sufficiently large values of p , o f a multiple steady state region in which three values of the state s are possible for a given external concentration The multiple stationary state region is bounded by a pair of so. concentrations s g and s::: for which the observed stationary state is a discontinuous function of s o if this parameter is allowed to vary quasi-statically, that is slowly enough for the system to undergo sequential steady states. There is a critically large value of p for the onset of this behavior, namely: p > -1/
min F'(s) os,
It is easy to understand the hysteresis phenomenon by observing the sequential stationary states when s o is allowed to vary slowly, first increasingly (points 1 to 4), then decreasingly (points 4 to 7) as indicated in Figures 3.2b and 3.4: (Figure 3.2) - Figure 3.2a shows that s is a solution of so - s = pF(s) if it is the abscissa of a point belonging equally to the curve y = p F ( s ) and the straight line y = s o - s. Figure 3.2b shows those points (s,y) obtained as s o varies, first increasingly (points 1-4), then decreasingly (points 4-7). Figure 3.4 shows the same states 1-7 on the path followed by the point ( s o , p , s ) on the catastrophe surface so - s - pF(s) = 0 when p is kept fixed and s o is allowed to vary as in Figure 3.2b. three types of steady states are possible, of which For s o € ] s : : : , s g [ two are stable. In one type of stable state (lying on the upper branch of the S-shaped curve in Figure 3 . 3 ~ ) the reaction is relatively slow and s is close to s o ; these will be designated reaction-controled states. In the other type of stable state (on the lower branch o f the S) the reaction is fast and s is limited by diffusion, and may be relatively far from s o ; these will be designated diffusioncontroled states. These notions of reaction and diffusion controled states were introduced by Hardt, Naparstek, Segel, and Caplan for the "papain system"[3].
THE ZERO-DIMENSIONAL SYSTEM
63
a/ Solution of
h
Hysteresis
b ’
0
1 -2-3-3-4 1-7-6-5
3
S
Fig. 3.2
Fig. 3.3
MULTIPLE STEADY STATES AND HYSTERESIS
64
S
Fig. 3.4 A l l of the above is easily seen, as is the stability of the state s
with respect to the dynamical system ( 3 . 1 ) , which changes when 1 + p F ' ( s ) changes sign, that is at the turning points s = s:: and s = s:::: However, the behavior of this simple system extends to more complex systems. In particular an S-shaped curve as shown in Figure 3 . 3 ~will still represent the qualitative behavior or the distributed system studied in the next section.
.
3.2
The One-dimensional System
The distributed system to be discussed hereafter is more realistic than the previous one for representing the interaction of diffusion and reaction in living cells. In fact enzyme molecules are mostly located within cell membranes and organelles, that is in those regions where diffusive resistance to mass transfer is concentrated. The so-called one-dimensional system is an artificial enzyme membrane which was experimentally shown to exhibit an hysteretic phenomenon [4].
THE OKE-DIMENSIONAL SYSTEM
65
3.2.1 - Experimental s e t u p : [ 4 ] The enzyme is uricase, which catalyzes the oxidation of uric acid according to the stoechiometric equation (3.7). The artificial membrane was prepared by the following procedure: a cross-linking reaction occurred for 2 hours in a solution containing albumin, glutaraldehyde, and uricase. Then this solution was spread on a plane glass surface. After some hours a complete insolubilization occurred and a 50u-thick proteic membrane was produced. Once the membrane was produced, i t was immersed in a solution of uric acid. The reaction (3.7) needs oxygen as a cosubstrate. To reduce the mass transfer problems of oxygen, several pieces of the same membrane were immersed in a well stirred solution of substrate and put in continuous air bubbling. Though the oxygen concentration in the system is much lower than the uric acid concentration, zero order kinetics for oxygen exists, because the oxidation of uric acid to allantoin is the rate limiting step in the reaction chain. That is to say that the rate expression is given by (3.8) and depends only on S, uric acid concentration. A time gradient of uric acid was established in the bulk solution by a controled pumping (chemical stat). Thus the concentration of S at the membrane boundaries was changing slowly with time, either increasingly or decreasingly. This variation was slow enough for the membrane to undergo sequential steady states. While uric acid concentration in the bulk solution was thus allowed to vary quasi-sta. tically, the rate of substrate consumption was recorded spectrophotometrically. It was experimentally demonstrated that enzyme activity, that is the rate of substrate consumption, did present hysteresis when the substrate concentration in the bulk solution was changing.
3.2.2 - Mathematical modeling: Henceforth substrate will mean uric acid, the concentration of which within the membrane obeys:
-asat
D~
+
R(S)
=
0,
together with boundary conditions:
66
MULTIPLE STEADY STATES AND HYSTERESIS S = S0 at x = 0 and x = L ,
(L membrane thickness),
and initial conditions: S = 0 a t t = O o r S = So a t t = 0,
accord ng to whether the membrane is initially empty or full of substrate We take as new units KS for concentration, 6(= L2/DS) for time, for length, and define the dimensionless quantities s = S/KS, t' = t/e, and x ' = x / L . Dropping the primes for convenience, we obtain the following equations: st
- s xx
s(0,t)
=
+ uF(s)
s(1,t)
= 0,
0 < x < 1,
t > 0,
= so
given
s(x,O)
where : (3.15)
a
=
(VM/KS)B,
6 = L2/DS
and F ( s ) is given by (3.10). We are interested by the steady state, satisfying the 2-point boundary value problem:
i
-s"(x)
(3.16)
s(0)
+
uF(s(x)) = 0,
= s(1)
0 < x < 1,
= so.
If the parameter s o is allowed to vary with time, the evolution equations (3.14) are more appropriate than the steady state equations (3.16) for representing the system evolution. However, since the rate of change of s o is small with respect to the speed of the system when it evolves towards equilibrium for a given value of s o , i t is possible to make the so-called quasi-steady-state hypothesis, which consists in considering the system as well represented by the steady state equations (3.16), where time t is a parameter appearing through s o . The justification of such a quasi-steady-state hypothesis is given in Section 3.6 for the 0-dimensional case.
ONE-DIMENSIONAL SYSTEM
67
3.2.3 - Numerical simulations: Let us consider problems (3.14) and (3.16) for a given boundary value s o . It is fairly easy to check that both the evolution and stationary problems (3.14) and (3.16) admit 0 and s o as lower and upper solutions. Therefore, upon applying Propositions 1 . 1 and 1.2 we already know that the evolution problem (3.14) admits an unique positive solution satisfying 0 5 s(x,t) 5 s o , and the stationary problem (3.16) admits maximal and minimal solutions s and 2 , such that 0 5 s(x) 5 ?(x) 5 s o . Also, applying the same kind of arguments as in the proof of Proposition 2.1, we can show that if the evolution system starts from the initial condition s ( * , O ) = 0 (resp. s ( - , O ) = s o ) , then the profile s(*,t) tends increasingly (resp. decreasingly) towards the steady state S (resp. 2 ) .
A numerical simulation of (3.16) when
s o is a slowly varying function of time, first increasing, then decreasing, was performed according to the following procedure. We solved the initial boundary value problem (3.14) for a succession of values s o , starting at each step from the slightly perturbed steady state of the preceding step. Sequential stable steady state profiles were observed as indicated in Figure 3.5. (The profiles 1-2-3-4-5-6-7 represent s(x)/so as a function of x. Low profiles 1-2-3 correspond to increasing s o values, until a jump to the high profile 3 for s o = s g = 68. High profiles 4-5-6 correspond to decreasing s o values, until a jump to the low profile 6 for s o = s : : : : = 61. u = 1200, k = 0.1).
It can be checked that these steady states are the minimal o r maximal solutions of (3.16), which can be obtained by the following numerical scheme :
(i)
start from so(x) = s o when seeking the maximal solution, and so(x) = 0 for the minimal solution.
(ii)
define sm+’(x) from sm(x) by:
- _d_2 sm+l(x) + dx2
0
-
1
+
1
sm(x) + k(sm(x))’
(3.17) sm+l(0)
=
s m+ 1 (1)
= so
sm+l(x)
=
0 , 0 <x < 1,
68
MULTIPLE STEADY STATES AND HYSTERESIS
Fig. 3.5 and iterate until
max 05x51
Ism+’ (x)
-
sm(x)
1
<
E.
The justification of this algorithm for generating sequences of functions converging towards the extremal solutions of (3.16) will be given in the proof of Proposition 3.2. One obtains the same steady states either as limits of evolution profiles s(-,t) as t + m or as limits of the sequences sm(.) as m + m. The first important observation is that one may obtain two distinct stable steady states for the same boundary value s o . For example we can see in Figure 3.5 two profiles numbered 3, a low one and a high one, and which correspond to the same value s ; of the parameter s o . Similarly the two profiles numbered 6 correspond to the same value
THE ONE-DIMENSIONAL SYSTEM
69
s ,. " * ',. of so. Moreover, for each value s o E]s::::,s;[ there exist two profiles located respectively between the high profiles 3 and 6 and the low ones.
The second observation is that, for u large enough, we still have an hysteresis phenomenon (Figure 3.6 - Hysteresis loops which, for u = 800, 1000, and 1200, describe the activity versus s o . Here the activity y is:
I
I
I
50
60
I
70 0'
SUBSTRATE CONCENTRATION Fig. 3.6 This hysteresis phenomenon is more easily seen in Figure 3.6 than in Figure 3.5. The enzyme activity which is represented in Figure 3.6 is this quantity which can b e experimentally observed and which may be defined either as the total flux of substrate entering the membrane, y = s ' ( 1 ) - s ' ( 0 ) = 2 s ' ( 1 ) , or as the total quantity of substrate consumed within the membrane,
MULTIPLE STEADY STATES AND HYSTERESIS
70
both expressions being the same since, by integrating (3.16) on (0,l) one obtains: u
1 /o
F(s(x))dx
=
1
s"(x)dx
=
s'(1)
-
s'(0) =
2s'(l).
0
In Figure 3.6 we see two arcs, 1 and 2 , corresponding respectively to high and low activity. The corresponding steady states will be designated respectively diffusion-controled and reaction-controled. These two types of steady states appear clearly in Figure 3.5 where the quantity s(x)/so has been represented for several values of s 0' We distinguish "low profiles" 1,2,3, and "high profiles" 4,5,6, which are respectively diffusion-controled and reaction-controled. More precisely when s o increases from 0 to s;:: = 68 we have "low profiles',' f o r example 1,2,3. Then at so = s;:: there is a jump from the low profile 3 to the high profile 3. After, the larger the value of s o , the higher the profile. Now, if s decreases, the profile varies continuously through profiles 4,5,6. 0 In particular, for that value s o = s g : : for which there was a jump, the profile does not come back to the low profile 3, but remains a high profile until s o attains the value s ; = 61. For this value of s o , the profile is the high profile 6. But when s passes through S,; this profile falls down to the low profile 6. Afterwards, that is for s o < S,; when s continues decreasing, the steady state profiles will be the same as during the ascending phase. 3.2.4 - Theoretical justification: Although a more thorough analysis of the family of solutions will be performed in Section 3.4, we give here an elementary proof of the multiplicity of solutions for (3.16).
-Proposition 3.1 - For
u large enough and s o conveniently chosen, problem (3.16) admits multiple solutions.
Proof: Let
G be the indefinite integral:
G(s) =
1 0
F(u)du,
(GI =
F).
THE ONE-DIMENSIONAL SYSTEM
71
Any solution of (3.16) must satisfy the first integral obtained by multiplying in (3.16) by 2s'(x):
where C is a constant depending upon s o and the solution 0 , s is a convex function of x on [0,11, and, since s o , we have s " ( 0 ) = s"(1) = ZoG(so) - C , whence = m. A s the right (resp. left) half part of the s profile is obtained by solving the ordinary differen= 0 on the interval [1/2,11 (resp.) -s"(x) + uF(s(x)) Cauchy "initial" conditions s(1) = so,s'(l) = m (resp. s ( 0 ) = s o , s ' ( 0 ) = -m), the s profile is symmetric with respect to x = 1 / 2 . Moreover, if we restrict our considerations to the interval [1/2,13, we have s ' ( x ) 0 , whence:
Since s"(x) > s ( 0 ) = s(1) = s'(1) = - s ' ( O ) concentration tial equation [0,1/2]) with
ds = ( 2 o ) ' / ' ( G ( s ) dx
where 1-1
=
s(1/2),
(3.18)
( ~ u ) ' / * ( x-
-
G(p))'/'
o
The solution of (3. 6) is equivalent to the solution of (3.18) together with the condition requiring s(1) = s o :
In fact (3.16) has as many solutions as we can find several values of p satisfying (3.19),
(3.20)
or:
f(v,so) = ( 0 / 2 ) ' / '
where :
It can be shown that for s o large enough the representative curve of the function p +. f(p,so) has at least two relative extrema as curve
72
MULTIPLE STEADY STATES AND HYSTERESIS
C in Figure 3.7 so that there are at least 3 solutions to (3.20) if u is in a suitable interval. (Fig. 3.7 - Functions p + f ( p , s o ) for values of s o similar to those in Figure 3.5 and, for a given value of u , determination of the corresponding values of P . )
Fig. 3.7 Indeed, f ( p , s ) can be written:
whose derivative with respect to T,
=
-l/(G(so)
-
G(p))'/'
Let P be such that F'(P) < 0
LI
is f'(p,so)
=
T1
+
T2 where:
and
(P > 0). It is easily seen that, as I -< a, since F(S) - F ( C ) 5 0 for 5 2
so m, T 1 + 0 and T 2 -+ I , 0 < a. Then, by choosing s o sufficiently large f'(fi,so) > 0, so that the representative curve of f(-,so) possesses an arc where f is increasing, as curve C in Figure 3.7. On the other hand i t is easily seen m as P +. O + and f (LI , s o ) -+ 0 as LI +. s that f ( p , s o ) Thus for such a value s o , equation (3.20) admits at least 3 solutions for u suitably -+
-+
.
THE ONE-DIMENSIONAL SYSTEM
73
chosen. Finally, it is easy to see that once we have found a pair ( 3 , S o ) such that 3 = f(fi,g0) and f'(a,so) < 0 , then for every u > d it is possible to find s o > Z 0 such that u = f(G,so) and f'(p,so) < 0. The above proof is interesting not only because it is not always obvious whether the steady state is unique or not (see for example the glucose pump in Chapter Z ) , but also for the reason t.hat it introduces the family of curves p + f(p,so) as so varies, depicted in Figure 3.7, and which is useful for explaining the profiles described in Figure 3.5. The curves p + f(p,so) can be calculated for each value of s o , and, although we have not demonstrated mathematically that there are at most three solutions for (3.20), there is some numerical evidence for this assertion to be true. Thus if, in Figure 3.7, we follow by continuity the point common to the straight line i ~ . + ( u / Z ) ' / ' and to the curve p + f(p,so) as so varies, the abscissa of this point being p = s ( 1 / 2 ) , we find the s e quence of points numbered from 1 to 7, corresponding to the profiles in Figure 3.5 with the same numerical order. In particular we see that the two jumps 3-low -+ 3-high and 6-high -+ 6-low occur when the curve is tangent to the straight line. Alternatively we can represent for each value of s o the corresponding values of p and obtain the S-shaped curve of Figure 3.8, where the order of the steady states is the same as in Figures 3.5 and 3.7. (Figure 3.8 - Representation of p ( = s ( 1 / 2 ) ) as a function of s o ( = s ( 0 ) = s(1)). Solid (resp. dashed) curves are used for stable (resp. unstable) steady-states. The order of the steady states is the same as in Figures 3.5 and 3.7). Hysteresis may be observed for a given boundary value s o : the system can be in one of two possible stable steady states depending on its past history, Thus we have a simple model of a system with memory. The dashed curve did not appear in the numerical experiments of 53.2.3. However, we see in Figure 3.7 that whenever two distinct
MULTIPLE STEADY STATES AND HYSTERESIS
74
stable steady states exist, there is a third steady state lying between them. It will be the object of the next sections to investigate further the structure of the S-shaped curve of Figure 3.8, numerically in Section 3.3, mathematically in Sections 3.4 and 3.5.
Fig. 3.8 These studies will confirm the conclusion of the present section, namely that the curve of solutions is composed of three arcs: a first one, comprising the minimal solutions for 0 5 s o 5 s ; , which are diffusion controled, a second one, comprising the maximal solutions for s o < s : : : : , which are reaction controled, and a third one, comprising the unstable intermediate solutions between the two turning points. 3.3
Kubicek Method of Continuation
A finite difference discretization of (3.16) gives a system of n non-linear equations:
(3.21)
...... -S. 1- 1
+ 2si
-
S.
+ 2sN
+
P F ( s ~ )=
......
-s
N- 1
1+ 1
+
pF(si) so,
=
0
KUBICEK METHOD OF CONTINUATION where si is an approximation of s(ih), h = l / ( N + l ) , and p = oh2.
h being the space step
If u denotes the vector of concentrations u X = s o , problem ( 3 . 2 1 ) is of the form: (3.22)
f(X,u)
75
= [sl ,s2,.
. . ,sNIT and
= 0,
where f : R+ x RN -+ RN is a continuously differentiable mapping. More precisely, u stands for [u,, . . . ,uNIT and f(A,u) for [fl(X,ul ,...,uN), ...,fN(X,ul,...,%) I T . The derivatives fX(X,u) and fu(X,u) are respectively the vector:
and the matrix:
These equations involve the state of the system u, which is a vector of concentrations, and a parameter A , describing the environment. We are interested in determining how the state of the system varies as this parameter is altered. In general, there may be multiple solutions to the equations, and the number and stability of these solutions undergo changes at particular values of the parameter. We desire to comprehend the interrelation among multiple solutions which have been found in Section 3 . 2 . The numerical method of continuation we are going to present here is general and can be applied to any problem of the form ( 3 . 2 2 ) where f is a continuously differentiable ma,pping of R x RN into R N . Stability is relative to the evolution problem du/dt- + f(X,u) = 0. The special points where the number and stability of solutions may change are of two types, turning points and bifurcation points. In this section we shall deal only with continuation of curves of solutions with eventual turning points, the numerical exploration of bifurcating branches of solutions being treated in the next chapter. There is a well developed theory which permits the qualitative description of the behavior of solutions in the vicinity of a turning
MULTIPLE STEADY STATES AND HYSTERESIS
76
The results of this theory will be given in the next two
point [ 2 J sect ions.
The Kubicek method [ l J consists of parameterizing A and u by the arc length s along the curve of solutions. Let ( A ' ( s ) , u ' ( s ) ) be the tangent vector, where:
Along an arc the tangent vector varies continuously, and satisfies the condition: (3.23)
fA(S)
Al(s)
+ fu(s) ul(s)
= 0.
where: fX(s) = fA(A(s),u(s)) and fu(s) = fu(A(s),u(s)), and which follows from (3.22). The definition of arc length implies that the following normalization must be attached to (3.23): (3.24) Here
Xt2(s) +
1.1
IU'(S)~~
= 1.
denotes the euclidean norm in RN
If (A 0 ,u0 ) is some initial point on the curve of solutions, corresponding to, say, s = s o , then solving (3.22) is equivalent to finding the solution of the system of ordinary differential equations:
(3.25)
i
A'
=
A'(s),
A(so) = A 0 '
u'
=
u'(s),
u ( s o ) = u0 '
where A l ( s ) and u'(s) satisfy (3.23) and (3.24). In order to obtain (A' ( 5 ) , u l ( s ) ) at the point (A(s) , u ( s ) ) there are two cases: First if f;'(s)
exists, we solve:
Since we know that:
u'(s)
=
A'(s)y,
A ' ( s ) is determined by:
KUBICEK METHOD OF CONTINUATION
(3.26)
( 1 + Iy12)A”(s)
=
77
1.
There are two solutions to (3.26), which differ only according to the direction in which they trace the curve. We choose an initial direction at ( A 0 , u 0 ) by providing a vector of signs for A ‘ (0) and u f ( O ) , 1 5 i 5 N, and this direction is maintained until a turning point is encountered. Points where f-’(s) does not exist are of course isolated along the U curve of solutions, but near these points the algorithm described above loses its accuracy, due to the ill-conditioning of fU(s). The role played by X in the above treatment may be assumed by some other component ui ( 1 5 i 5 N), permitting the restatement of (3.21) which is less ill-conditioned. A strategy of full pivoting, applied to the N x ( N + l ) matrix [fA ful selects the new independent variable most effectively. The details as well as a Fortran code may be found in Kubicek [ l ] . Having obtained (A’(s),u’(s)) we can proceed to integrate (3.25) numerically, using the Adams-Bashforth formulas of maximal order 4, again following Kubicek. The step length and order of the integration formula are adjusted automatically. In particular, whenever the independent variable is changed, the order of the integration formula is reduced to 1 , and the step size is reduced by an empirically chosen factor (10). The numerical integration is a predictor step, proof the next point on the branch, viding an estimate (x(s+h),fi(s+h)) (A(s+h),u(s+h)), see Figure 3.9, (Scheme for tracing a curve of solutions in R x RN from known point (X(s) ,u(s)) by predictor step ( A ( s ) , u ( s ) ) +. (X(s+h) ,fi(s+h)), followed by corrector step (i(s+h), fi(s+h)) -+ (A(s+h) ,u(s+h)). This estimate is not without error, so it is taken as a starting point f o r a Newton iteration which refines the estimate up to some error tolerance. In case the independent variable is X, for example, the Newton method for variable u is:
( u o = O(s+h)
We use the number of Newton corrections as a basis for a second stage
MULTIPLE STEADY STATES AND HYSTERESIS
78
of control of step size in the predictor code. Whenever fewer iterations suffice to satisfy the error tolerance, size is increased by an empirically chosen factor (1.5), maximum step length, selected so as to produce curves o f density.
three or the step up to some adequate
UEB
R Fig. 3.9 The application of the Kubicek method to problem (3.21) with N = 20 gives the results depicted in Figure 3.10. Evidently, (O;O,...,0) is a solution to f(X,u) = 0 and we take it as a starting point o n the curve. Only slightly less evident: au
/ (O;O,...,0) > 0,
so the initial direction vector is obtained = (+l;+l,...,+ 1). The resulting collection of solution components is illustrated in Figure 3.10 where only the components corresponding to i = 1, 3, and 10 have been shown. Linearized stability was inferred from the change in sign of the determinant o f f U . There are two turning points on the curve, f o r s0 = s::0 and so = s:::: , at which this determinant changes does not change sign between the sign. Since the determinant of U turning points, we infer that the steady state is unstable all along the intermediate arc lying between these two turning points.
go
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
79
‘i max
‘i
(I
Fig. 3 . 1 0 (Fig. 3 . 1 0 - Components si o f the solution to ( 3 . 2 1 ) for i = 1 , 3 , and 10. The abscissa is so, the ordinate is si. si max = 3 0 . 6 0 f o r i = 1 , 2 8 . 9 2 for i = 3 , and 2 6 . 2 6 for i = 10. s i = 2 1 . 5 , s::::0 = 2 8 . 9 . u = 1200, h = 1 / 2 0 , p = 3 , and k = 1 ) . 3.4
Mathematical Analysis of the S-shaped Curve
We rewrite the problem ( 3 . 1 6 ) by calling A the parameter s o , and s ( x , A ) the solution of: -s”(x,A)
+
uF(s(x,A))
=
0,
O < X < l ,
(3.27) s(0,A) = s ( 1 , A ) = A ,
where the primes denote derivation with respect to x. Let us think o f s as the mapping A -+ s ( - , A ) where s ( - , A ) is the funcWe then write s(A) rather than s ( . , X ) , correspondtion x +. s(x,A).
80
MULTIPLE STEADY STATES AND HYSTERESIS
ing to s(x,A) via s(A)(x)
=
s(x,A).
Let u be such that problem (3.27) admits, for a range of conveniently chosen boundary values A , multiple solutions. We have shown, in the proof of Proposition 3.1, how to find such a u . Our aim in this section is to clarify the structure of the family of solutions of equations (3.27) when 0 is held fixed and A varies. It seems likely, from the discussion of Section 3.2 and the numerical results of section 3.3, that the points ( A , s ( A ) ) lie on an S-shaped curve. Indeed it is possible to demonstrate precise statements about the arcs of maximal and minimal solutions, and in particular to prove that they are continuous curves ending at turning points. Our proofs follow arguments of Crandall and Rabinowitz [ 2 , 5 , 6 1 , Brauner and Nicolaenko [ 7 1 , and Mignot and Puel [ E l . We shall first see what can be done by using upper and lower solution techniques. Then we shall add to our tool kit the Implicit Function Theorem, which is very useful for proving existence and uniqueness of a continuous curve of solutions passing through a point. Lastly we shall investigate the behavior of the curve of solutions near a turning point.
In the following c([o 1 1 ) will denote the Banach space of continuous functions u : [ o , ~ ] R equipped with the norm: -f
and Ck ([0,1]) the Banach space of k-times continuously differentiable functions u : [0,1] R under the norm: -f
It is easy to demonstrate the following result for problem (3.27): Proposition 3.2
(i)
-
For every X > 0 , there exists a maximal solution ? ( . , A ) minimal solution ?(.,A), satisfying: 0 ( ?(x,A)
(ii)
5 s^(x,A) 5 X ,
The mappings h
+.
S ( * , A ) and X
and a
O(X5l. +
?(*,A)
are monotone increasing
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
81
on LO,+-[.
(iii) For every A 2 0 ,
Proof
-
(i)
It is obvious that 0 is a lower solution and A an upper solution. Hence we can apply Proposition 1.2. Alternatively we can apply Theorem 1 .1 within the following framework.
E is the Banach space C([O,ll)
of continuous functions on [0,1], enmax)u(x)I. E is an ordered Banach o < x <1 space for the natural ordering7 5 uz u 1 ( x ) 5 u2(x) for every x ~ [ 0 , 1 ] . We consider the order interval [y,91 where y ( x ) : 0 and p(x) : A , and define f: [ ? , ? I + E by: u = f(v) if: dowed with the norm: 1Iu1I
=
c1
-u"
ouG(v) = 0 ,
+
u(0)
+
u(1)
=
= A,
where : (3.29)
G(v)
=
1/(1
+
v
+
kv').
We verify that f is a monotone increasing mapping. If v 1 '< v 2 and = u 2 - u , , we have, since G is monotone decreasing on R,,
w
-w"
+
owG(vZ)
=
ou,(G(vl)
-
G(v2))
2 0,
which, together with w(0) = w(1) = 0 , implies w > 0. We verify that y 5 f(7) and f(9) 5 9 . u = f(y) is defined by: (3.30)
-u"
+
uu
=
0, ~ ( 0 =) ~ ( 1 )
=
A.
The solution of (3.30) is: (3.31)
u
=
A[ch(u'/'(x
- 1/2)]/ch
0'/2/2
which at evidence satisfies u 2 0. u = f 9) is defined by: -u" + ouG(X) = 0 , u ( 0 ) = u(1) = A . Let w = A - u. We have -w"
+
MULTIPLE STEADY STATES AXD HYSTERESIS
82
o w G ( A ) = oF(A), w ( 0 ) = w ( 1 ) = 0.
From the Maximum Principle w 2 0 , hence u 5 A . Lastly we verify that compact. Let (v,) be a sequence in [ y , ? ] . The definition of un, - u “ + ounG(vn) = 0 , un(0) = un(l) = A , implies that, for every x~[o,l;,o 5 un(x) 5 A , 0 5 u;(x) 5 o h , lu;(x)l 5 aA, the last inequality resulting from the fact that there is an xo~[O,ll such that = 0 , and, for every x~[O,l], u;(xo)
f is
u;(x)
=
lXx
u:(S)dt.
0
The compactness of f The un remain in a bounded set of C 2 ( [ 0 , 1 ] ) . follows from the compactness of the natural embedding of C2([0,11) into C([O,l I). (ii) Let A 1 < A 2 . s ( * , A , ) is an upper solution for problem (3.27) Since the minimal solution of this problem eitlier in with A = A 1 . k interval [ O , A 1 ] or in interval [0,s(.,A2)] is the limit of f (0) as k + m, the minimal solution is the same in both intervals, so that Similarly S ( - , A , ) is a lower solution for pros(.,x,) 5 <(-,A2). blem (3.27) with A = h 2 , whose maximal solution is the same either in [ 0 , A 2 1 or in [ s ( - , A l ) , A , ] , implying ? ( * , A 1 ) 5 S(.,A,).
(iii) results from the fact that, according to the proof of Theorem < :(*,A), u being given by (3.31). 1 . 1 , 0 = 7 5 f(y) = u 5
...
-
s(A) and ?(A) may be the same. This is the case in particular when A is close to 0 or sufficiently large.
Proposition 3.3 following cases:
(i)
h : k-‘/‘,
Proof
-
We have :
The solution of problem (3.27) is unique in the
Let s 1 and s , be two solutions of (3.27), and w = s 1
-
s2
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
-w" + qw
w(0)
= 0,
w(1)
=
83
0
=
where :
- kslSZ)/[(l
q ='~(l
+
s1
+
ks;)(l
+
s2
ks;)l
+
In the case (i), since 0 < s1,s2 < A < k-'/2, we have q w = 0.
0 , hence
On the other hand, q > -u/(ks1s2) and, from ( 3 . 2 8 ) , s1,s2 2 X/ch(01/2/2 In the case (ii), q > - T ' , that is -och2(u'/2/2)/(kX2), Thus q q 2 -T' + a for some c1 > 0 , so that: 0 =
/' (-w"
+
lo1 wI2dx
qw)w dx 2
0
> a
+
( - a 2 + a)
/ 1 w2dx 0
1
j 0w2dx
(By applying Poincar6 Inequality w = 0.
1
j0
w"dx
1
low'dx).
> IT'
Thus
The families of minimal and maximal solutions can be a bit more precise.
?(*,A) (resp. A -+ ?(.,A)) Proposition 3.4 - The mapping A into C 3 ( [ 0,ll). (resp. right) - continuous from [(I,+@[ -+
proof
-
Let h
0
be any f i x e d v a l u e of h ( h
f a m i l y of :unctions
?(*,A)
d
is left
0)) and consider the
indexed by A w i t h 0 5 A
5
Ao,
Since t h i s
family is monotone increasing an3 bounded above by s(*,Xo), as X f X o , the ? ( * , A ) converge pointwise and in L2(0,1) towards some limit v 5. s(*,Aoj.It follows that F(s(* , A ) ) -+ F(v) in L2(0,1). Let u(X) = u ( * , h ) be the function u(9,X) = S(-,X) - X , satisfying -u"(x,X) =-oF(S(x,X)) , 0 < x < 1 , u ( 0 , X ) = u ( 1 , X ) = 0, and w the solution of -w" = -F(v), w(0) = w(1) = 0. Since F(s(.,X)) + F(v) in w in H2(0,1) n HA(0,l). But u ( * , A ) = ?(.,A) - h + L2(0,1), u ( - , X ) v - X in L2(0,1). Hence w = v - X and the limit v of s(.,X) as A f A. satisfies: -+
(3.22)
-vft+ uF(v)
=
0,
v(0)
=
v(1)
= Xo'
84
MULTIPLE STEADY STATES AND HYSTERESIS
On the other hand, since v satisWe already know that v 5 s(*,Xo). fies (3.22) and belongs to the order interval [O,Xo], we have ? ( . , A o ) < v. Therefore v = : ( * , A o ) . Lastly, since H2(0,1) is continuously n C’([O,ll), we a l s o have ?(*,A) + S(.,Ao) in C’([O,l]), +. F(s(*,Xo)) in C’([O,l]), whence ?(*,A) +. s ( * , X o ) in
continuity proof for X
+.
s ( * , A ) would be similar.
Proposition 3.4 is in agreement with Figures3.8 and 3.10 where it can be seen, in particular, that, as s + s : : from below, ? ( . , s o ) + 0 0 s(-,s:), whereas as s0 + s::0 from above, s ( . , s o ) does not converge towards ? ( . , s : ) . Similar remarks can be made for s o = s : : : : . In order to obtain further information about the curve of solutions, we need to employ the Implicit Function Theorem. For the sake of completeness we give below a version of this theorem which was stated by Crandall and Rabinowitz in [ 5 ] , and for the proof of which they refer to [91. Theorem 3.1-(The Implicit Function Theorem) - Let X, Y , Z be three Banach spaces, f a continuous mapping of an open subset A of X x Y into Z. Let the map y -+ f(x,y) of Ax = { y E Y : (X,Y)E A} into Z be differentiable in Ax for each x E X such that Ax # a , and assume the derivative of this map (denoted by f ) is continuous on A. Let Y ( x 0 , y o ) ~4 be such that f(xo,yo) = 0 , and f ( x o , y o ) is a linear homeoY morphism of Y onto Z. Then there are neighborhoods U of xo in X and V of yo in Y such that: (i)
U x V c A.
(ii)
There is exactly one function u:U for x E U .
+
V satisfying f(x,u(x))
= 0
(iii) The mapping u of (ii) is continuous. If, moreover, the mapping f is k-times continuously differentiable on A , then (iii) above may be replaced by:
(iv)
u is k-times continuously differentiable.
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
85
Due to its amazing scope and simplicity, the implicit function theorem is a very useful analytical tool. A nice introduction to the material involved in this theorem and its applications can be found in [ l o ] .
We begin with an application of the implicit function theorem to the following situation: X = R, Y = C;([O,l]), Banach space of twice continuously differentiable functions on [ 0 , 1 ] vanishing at x = 0 and x = 1, equipped with the norm llu112 , Z = C([O,ll), equipped with the norm IIuIIo , A = X x Y , f : (A,u) +. -(u + A ) " + o F ( u + A ) , x = 0 and y o = 0. It is easily seen that f(xo,yo) = 0 and that f (xo,yo) : Y w + -w" + uw is an homeomorphism of Y onto Z. Then there are neighborhoods U = I-€,+€[ of 0 in R and V of 0 in CG([O,ll) such that:
(i)
there is exactly one function u:U for A E U.
(ii)
the mapping u of (i) is continuous.
+
V satisfying € ( A , u ( X ) )
Moreover, since f (0,O) is non singular, fU(A,u(.,A)) for 0 5 X < E if 6 is chosen sufficiently small.
= 0
is non singular
Thus we have the:
-P-sition
3.5 - There is an interval LO,€[ of R and a neighborhood V of 0 in Ci([O,ll) such that:
there is exactly one function u:[O,c[
(i) (3.33)
for (ii)
-u"(* , A )
AE[O,E[
.
+oF(u(*,A)
+ A)
=
+.
V satisfying:
0
Moreover:
the mapping u of (i) is continuous,
(iii) the operator v + -v" + oF'(u(.,A) from c;([o,~]) into c([o,1]).
x,
+ X)v is invertible as a map
Proposition 3.6 - There is a number 0 < 5; < with respect to the following properties:
m,
which is maximal
MULTIPLE STEADY STATES AND HYSTERESIS
86
(i)
existence of a continuous function u:[0,5;[ which:
(ii)
for every A E [ O , X [ , the operator v + -v" + uF'(u(.,A) invertible as a map from Ci([O,l]) into C([O,ll).
+
c~([o,~]) for
+
A)v is
-
Moreover, A and u(A) are unique. Proof - We are interested by those intervals [O,a[ on which we can define a continuous function u:[O,a[ * Ci([O,l]) satisfying, for every A€[O,a[, (i)
f(A,u(A))
(ii)
fU(A,u(A))
=
0 and
is non singular.
Of course, for A small enough, such a function coincides with the function u provided by Proposition 3 . 5 , and, in particular the curve ( A , u ( A ) ) emanates for the point ( 0 , O ) . Moreover, if ul:[O,al[ +. Ci([O,l]) and u2:[0,a2[ +. Ci([O,l]) are two functions of A satisfying (i) and (ii) with a l 5 a2, then u l ( A ) = u,(A) for every Ae[O,alT. If not, since u 1 and u 2 are continuous functions of A , there exists a closed interval [O,bl with b < a l , such that u l ( A ) = u2(A) for every A=[O,b] but, whatever small q , and whatever the neighborhood V of ul(b) in Ci([O,l]), there exists at least one Ae[b,b+n[ for which u 1 (A) # u 2 ( A ) and u l ( A ) E V , u,cA) E V . Since the implicit function theorem,applied at xo = b , yo = ul(b), implies the existence of neighborhoods Ib-n,b+n[ of b and V of ul(b) such that there is an unique solution of f(A,u) = 0 satisfying (A,u) E ]b-q,b+n[ x V, there is a contradiction. Now it suffices to define
as the least upper bound of the numbers a such that on interval [O,a[ there exists a function u:[O,a[ +. Ci([O,l]) and the function satisfying (i) and (ii). It follows that u:[O,x[ +. Co([O,l]) satisfying (i).and (ii) are uniquely defined.
x
Let s(A) = s ( * , A ) = u ( * , A ) + by Proposition 3.6, and u(A)
A =
where u ( - , A ) is the function provided = 's(A) - A where ' s ( A ) is the
<(.,A)
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
87
minimal solution of problem (3.27). Before proving that s(A) = :(A) for X E [ O , x [ , we need the following: Lemma 3 . 1 - Suppose that Aoc [ O , x [ and satisfies s(Ao) = ? ( A o ) . Then there exists rl > 0 such that s(X) = ?(A) for every AE[A,,X~ + r l [ .
By the implicit function theorem there are neighborhoods 1A0-n,A0+77[ such that there is exactly one of A. in R and V of u(Xo) in C:([O,l]) function u : ]XO-q,AO+~[-+ V satisfying f(X,u(X)) = 0 for X E lXo-ri,Xo+q[. Moreover, since q can be chosen so that U ( A ) c V for A E [ X ~ , X ~ + ~ [ ,it This completes the proof follows that u(X) = U(A) for X E [Xo,Ao+q[. of the Lemma. Proposition 3 . 7 - s(X) = ?(A) s ( x ) is well defined by:
for A E [ O , X [ .
Moreover if
x
<
m
then
_Proof -
Since s ( 0 ) = ? ( O ) , Lemma 3.1 implies the existence of an inLet [ 0,b [ be its maximal extenterval [ O , q [ on which s(A) = ?(A). sion, that is let b be the least upper bound of those a such that s(A) = ?(A) for AE[O,a[. Suppose b < X. Since s(X) = ?(A) for ?(A) is left-continuous as a mapping of R+ into AE[O,b[ and S : X Thus, by Lemma 3 . 1 , s(X) = s(A) on C2([0,11), then s(b) = ?(b). [O,b+e[, E > 0, which contradicts the maximality of b. Hence b = 1. Moreover, if < m then s ( 1 ) is well defined since s(A) = :(A) + as A -+ X from below. -+
x
We now have all the elements for proving:
s(x)
MULTIPLE STEADY STATES AND HYSTERESIS
88
Proposition 3.8
-
s(Ao) f ? ( A o ) ,
- If there is at least one value X o of A such that then 5; < -.
By Proposition 3.7 s(X) = s(X) for every X E LO,+-[. By Proposition 3.3 s ( h ) = ?(A) for X large enough. Let ]Al,+-[ be the maximal extension of those intervals ]a,+-[ on which s(A) = ?(A). From the continuity of X + s(X) and the right conNow Of course X 1 2 X o . tinuity of X + ?(A), we have s ( X l ) = ? ( A 1 ) . consider the monotone increasing family Ir(A)IX < A l l , hounded above by ? ( h i ) * Proof
- Suppose 7
= + -,
This family converges pointwise and in L2(0,1) towards some element v 5 ?(Al). As in the proof of Proposition 3.4, it is easy to check that ?(A) converges in C2([0,1]) towards v, and v satisfies - v " ( x ) + oF(v(x)) = 0 for 0 < x < 1 , together with v(0) = v ( 1 ) = A 1 . Thus. s(A,) 5 v 5 S ( X l ) . But, since s(A1)= S ( A 1 ) , v = s ( A 1 ) . By the implicit function theorem there are neighborhoods ]X1-rl,A1+n[ of X 1 in R and V of s(X,) in C2([0,1]) such that there is exactly one function s:lX1-n,X1+n[ V satisfying f(X,s(X) - X) = 0 for A E lA1-n,X1+q[. Moreover we can choose n small enough for ?(A) to he in V for is X ~ I h ~ - r l , A ~ But l . this cannot occur because the interval [ A 1 , + " [ maximal. This completes the proof of the Proposition. -+
In order to obtain further information on the limit point ( x , s ( x ) ) , we shall use the two following theorems of Crandall and Rabinowitz 121, which give a description of the behavior of the curve of solutions for f(A,u) = 0 near this limit point. As Theorem 3.2 follows at once from the implicit function theorem, we shall give the details of its proof for the sake of completeness, whereas for Theorem 3.3, which is more complicated to demonstrate, we refer to the proof given in [2]. Theorem 3.2 - Let Y and Z be Banach spaces, ( X o , u o ) e R x Y and let f be a continuously differentiable mapping of an open neighborhood of (Xo,uo) into Z. Let f(ho,uo) = 0 , N(fU(Xo,uo)) = span { $ I ~ }be onedimensional and codim R(fU(Ao,uo)) = 1. Let fX(Xo,uo) e R(fU(Ao,uo)). If X is a complement of span {$11 in Y , then the solutions of f(X,u) = 0 near ( h o , u o ) form a curve ( A ( s ) , u ( s ) ) = ( A o + ~ ( s ) , uo + s $ ~+ x ( s ) ) where s (T(s) ,x(s)) E R x X is a continuously differentiable function near s = 0 and ~ ( 0 =) ~ ' ( 0 =) 0 , x ( 0 ) = x'(0) = 0. Moreover, if f is k-times continuously differentiable (analytic), so are T(S) , x ( s ) . -+
MATHEMATICAL ANALYSIS OF THE S-SHAPPD CURVE
89
Here dim and codim are abbreviations for dimension and codimension, respectively, (the codimension of a subspace W of Z is the dimension of Z / W ) , and N(T),R(T) denote the null space and range of a linear mapping T.
No confusion should be made between the parameter s in the statement of Theorem 3.2 and the solution of Problem 3.27. Let:
and f;
Proof -
= fu(Ao'Uo).
Define a function g by:
for ( s , T , x ) E R x R x X. Observe that the partial derivatives gT and gx are continuous in ( s , T , x ) . In addition:
and the assumptions of Theorem 3.2 imply that the Fr6chet derivative g(O,.r,x) at (T,x) = (O,O), which is the linear map: of the map ( T , x ) -f
(?,a)
-f
fl;?
+
f;a
of R x X into Z , is an isomorphism onto Z. The implicit function theorem implies the existence of functions T and x possessing the properties asserted in the theorem. In particular, since f is continuously differentiable, so are T(S) and x(s). More precisely the derivatives T ' ( s ) and x'(s) are given by:
where fA(s) and fU (s) are the values of fx and fU at ( A o + T(s), For s = 0 we obtain fiT'(0) + f{x'(O) + fA$l uo + s $ , + x(s)). Since f;$l = 0 , it follows that ~ ' ( 0=) 0 and x'(0) = 0. Theorem 3.3
- Let
0 be a simple eigenvalue of f;.
= 0.
There are continu-
MULTIPLE STEADY STATES AND HYSTERESIS
90
ous functions ~ ( s E ) R, w(s) E Y defined near s = 0 such that:
w(s)
-
$,
(iii) w(0)
=
$ 1 , ~ ( 0 =) 0
(ii)
E X
Then, near s = 0, < Z , @ l > p(s) and Let I EZ:: satisfy N(2) = R(f:). - A I ( s ) < 2,fi > have the same zeroes and, whenever A ' ( s ) # 0, the same sign. Moreover,
-c C ( [ O , l ] ) be given b y Proposition 3.9 - Let f:[O,+m[ x C : ( [ O , l l ) If u : [ O , X [ +. C : ( [ O , l l ) is defined a s af(A,u) = - u " + uF(u +A). bove, then:
(i)
lim u(A)
x4x
(ii) :f
=
=
exists and uo
uo
=
u(x).
fU(X,uo) has a null vector @ 1 which is positive on ] 0 , 1 [ .
(iii) the zeroes of f near ( x , u o ) form a smooth curve ( A ( s ) , G ( s ) ) for which ( A ( O ) , u ( O ) ) = ( x , u o ) , A ' ( 0 ) = 0 , X"(0) < 0 . (iv)
if p ( s ) is the least value of p f o r which the equation: -w"
oF'(A(s)
+
+
C~(S))W
=
pw,
w~C:([O,11),
has a non trivial solution, then p ( s ) and h ' ( s ) have the same sign for s # 0. More precisely: (3.35)
p(s)
=
s(
/
1
F"(s(X))@;dx) 0
/ (
/
1
@;dx) + O ( s 2 ) .
0
Proof
-
(i)
has already been shown in Proposition 3.7.
(ii)
Let r(X), 0 5 A 5
x be
the least number r such that the equation
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
-w"
+
oF'(s(h))w
91
rw
=
has a non trivial solution w ~C*([0,1]). Then r(X) is continuo u s on [O,XI. Since r(0) = IT' + u > 0, and fu(A,u(X)) is nonsingular on [ O , x [ , we have r(A) > 0 on [ O , X [ . Since r(X) = lim r(X) as X .f X, this implies that r ( X ) > 0. However is singular, so that r(X) 5 0. It follows that r(X) = 0. Since 0 is the least eigenvalue of ,:f the corresponding eigenvector $ 1 is of one sign on 10,1[ [ 1 1 ] .
f:
(iii) We verify that the assumptions of Theorem 3 . 2 are satisfied with:
f(X,U)
=
-u"
Clearly f(Xo,uo) R(f:)
= {WE
A).
+ UF(U +
=
0, N(fi)
I
C([O,lI)
=
span {01} is one-dimensional and
/ 1 $l(x)w(x)dx
= 01
0
is of codimension 1 ,
' 0
I
For X we can take:
x
=
Iw€C;([0,11)
1
[
$lwdx
=
01
and any element v o f Ci([O 1 1 ) will admit exactly one decomposition v = c $ l + w with w E X and:
All the assumptions of Theorem 3 . 2 clearly being satisfied, the solutions of f(X,u) = 0 near ( X , u ( X ) ) form a smooth curve (X(s),u(s)) for = (X,U(X)) and X ' ( 0 ) = 0. which (A(O),u(O))
MULTIPLE STEADY STATES AND HYSTERESIS
92
Calculation of A"(0) : fX(s)A'(s) fXX
X'2(s
+
fuu(s)G'2(s) At s
fu(s)G'(s)
+
)
twice derivating f (X(s) , u ( s ) ) =
0 , we obtain:
=
and
0,
2fhU(S)G' ( s ) h ' ( s )
+
f,(s)A"(s)
+
+ fu(s)G"(s) = 0.
= 0,
fj;WO)
+
fUU(O)Lq
+ f;
ti"(0)
= 0,
which implies that: fiA"(0)
+
fuu(0)@;
ER(f1).
If h " ( 0 ) is the first nonvanishing derivative of h ( s ) at s = 0 , then = 0. However we are unable to determine whether X"(0) # 0. A"(0) c 0 since X attains a local maximum at s
Although it has not been verified mathematically, there is strong numerical evidence that A"(0) # 0. Anyway, there lies a difficulty, already found by Crandall and Rabinowitz for the "chemical reaction There are limitations to a purely mathematical analysis problem" [ 2 ] . of such problems, since, already for this very simple case, we cannot decide whether at s = 0 the curve bends backwards or forwards. But fortunately a numerical analysis can aid and indicate the value of X"(0).
(iv)
0 being a simple eigenvalue of f;,
By this theorem,near s = 0,
we can apply Theorem 3.3 with:
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
93
lo $luF'(?((X))dx
1
1
$;dx p ( s ) and - A ' ( s )
have the same zeroes and, whenever A 1 ( s ) # 0, the same sign.
lo @luF'(?(x))dx
As:
1
< 0,
= 2$!(1)
then p ( s ) and A ' ( s ) have the same sign. It is possible to give a more precise result, since we know that:
lim s+o
=
X' ( s )
-
< 1,fi > / < 1 , $ 1 >
1
1
(Theorem 3.3)
and A"(0) =
-
F"(s(X
)$;dx/<
2,fi >
(3.36).
0
As
A ' ( s ) = sA"(0)
+
we obtain (3.35).
O(S*)
Piecing together the curves ( A , u ( X ) ) , 0 2 X 5
x, and
(A(s),G(s)):
Consider the continuous curve of points ( A , u ( X ) ) , 0 5 A 5 (X, linking the origin (0,O) to the limit point ( x , u o ) , and the smooth curve of points ( A ( s ) , C ( s ) ) passing through ( x , u o ) and provided by Proposition 3.9. Because of the local uniqueness of the curve passing through ((X,uo), an arc of the former must coincide with an arc of the latter, either the one obtained for the negative values of the .parameter s , or the one obtained for s z 0. The first alternative will be the good one if we succeed in proving that, at least for s sufficiently close to 0, the function s G ( s ) + A(s) is increasing. As: -f
G ( s ) + A(s)
=
uo
+ SG1
+
x(s) + 5; +
G ' ( s ) + X ' ( s ) = @ 1 + x'(s)
G ' ( 0 ) + A'(0) =
0,
'0
T(S),
+ ~ ' ( s ) ,
and
MULTIPLE STEADY STATES AND HYSTERESIS
94
Therefore u ( A ( s ) ) = S ( s ) for small negative s (as opposedto small positive s). Moreover for s < 0, since A ' ( s ) > 0 we have p ( s ) > 0, which we knew already, and for s > 0, since h f ( s ) < 0, we have p ( s ) < 0, which as will be seen in the next section, corresponds to a change in stability. Proposition 3.10 - The curve of solutions ( X , u ( X ) ) , h ~ [ O , x ] can , be continued uniquely and smoothly through ( X , u o ) . Moreover the least changes sign at s = 0. eigenvalue of f,(A(s),u(s)) far we justified the lower arc of the S-shaped curve and its continuation through the lower bending point. Arguing in exactly the same way as above we can prove the existence of a smooth curve of h ' > 0. The maximal solutions, on some maximal interval [A',+-[, essential steps are the following: So
(i)
For X large enough, there is exactly one solution to (3.27), and, since: s(x,A)
2
and F ' ( s )
+
h /ch(u'/*/2)
0 as s
.+
m,
we can choose h large enough for
F' (s(x,X)) to satisfy, whatever
E,
Therefore: 1
(-w"(x)
-
(i~' UE)
+
1'
uF'(s(x,X))w)wdx
1 >
wf2dx -
1 UE
w2dx 2
w2dx
0
for any w E C ; ( [ O , I I ) , implying that the least eigenvalue of It follows that we fU(X,u(X)) is positive (u(A) = s(X) - A). can apply the implicit function theorem and obtain a continuous arc of solutions ( A , u ( h ) ) . (ii)
the solutions ( X , s ( X ) ) ( A , S ( A ) ) 9
coincide with the maximal solutions
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
95
(iii) the limit point (X',?(A')) thus obtained is a bending point, that is the curve bends forwards and can be smoothly extended. What occurs between the two bending points is unknown. However we have the following theorem given by Rabinowitz in [ I 2 1 and by Leray and Schauder in E 1 3 1 : Theorem 3.4 - Let T:R x E +. E be a compact operator such that T(0,u) = 0 , V U E E . Let I denote the closure of the set of solutions for: u = T(h,u).
Let C be the component of I to which ( 0 , O ) belongs. Then C = C + u Cwhere C+ (resp. C-) is included in R+ x E (resp. R- x E) and is unbounded, and C* n C- = { (0,O)1 . We apply this theorem within the following framework:
E
= L2(0,1)
and T(X,u)
=
v
is defined by: (3.37)
-v"
UVG(U)
+
~(0) = ~ ( 1 ) = A,
= 0,
where : G(u)
= 1/(1
+
IuI
+
ku').
It is easy to check that the assumptions of Theorem 3 . 4 are satisfied: if
implies v
(i)
T(0,u)
(ii)
T is compact: if (X, u) lies in a bounded set of R have, by denoting w = v - X , I the norm in L 2 ( 0 , the norm in Hk(O,l),
=
0 :
A
= 0,
(3.37)
=
0
MULTIPLE STEADY STATES AND HYSTERESIS
96
1
5 j-
02x2
1
+ j-
llw//2
,
whence :
where C is some constant. The result follows from the compactness of the embedding of HA(0,l) in L'(o,~). Then the component C+ emanating from (0,O) is unbounded. This implies that C + must be connected with the curve of maximal solutions,somewhere between the two bending points. Thus we have the: Proposition 3.11 - The continuation of the curve of minimal solutions for (3.27) is connected to the curve of maximal solutions, through a smooth arc of solutions joining the two bending points.
-Remark
3.1 - We limit ourselves to one-dimensional problems. This is due to the fact that a biological membrane is a medium with the slab geometry where the only dimension to be taken into account corresponds to the direction transverse to the membrane. However if the immobilized enzyme system does not have the slab geometry but fills up the interior R of an n-dimensional domain (n 5 3) whose boundary I' is exposed to a fixed concentration of substrate so, this system is represented by equations: -As + o F ( s ) = 0 in n, (3.38)
s = s
on I'.
If we replace Ci([O,l]) by CiPa(E), the arguments just used carry over from the one-dimensional case to the 2 or 3-dimensional cases, with some slight modifications. In particular we have to use Schauder estimates, the Sobolev imbedding theorem and Lp estimates of AgmonDouglis-Niremberg [ 1 4 1 . However there is a notable difference with the one-dimensional case in that we are unable to directly find a pair (o,X) for which multiple solutions exist. Without entering into details, let us say that this difficulty has been overcome by Brauner and Nicolaenko [ 1 5 ] by using a regular perturbation argument. They
MATHEMATICAL ANALYSIS OF THE S-SHAPED CURVE
97
rewrite equation (3.38):
-au +
u/(s2 +
+ kB2uZ) = 0
BEU
in 0 ,
(3.39) u = l o n r by defining u = s / s o ,
E’
=
l/o,
B
= so€.
We have already made the remark that in order to obtain multiple steady states we must have u and so sufficiently large. Therefore the hypothesis u = l/c’ and s o = B / E seems reasonable, and in fact in the one-dimensional case, typical values for which there are multiple steady states are IJ = 1200 and s o = 64, corresponding to E E 1/32, 6 1 2 . Brauner and Nicolaenko define a new, simpler problem by setting E = 0 in (3.39), thus obtaining:
i
-AU
(3.40)
+
u = 1
i/(kfi2u) = 0 on
in a ,
r.
Then they prove the following: (i)
there exists a branch of maximal solutions f o r (3.40) as 6 varies, with a turning point T (Figure 3.11).
(ii)
for E small enough, the same is true f o r (3.39), with turning point T,,
(iii) there is at least one other turning point T 2 for (3.39).
Fig. 3.11
MULTIPLE STEADY STATES AND HYSTERESIS
98
(Fig. 3.1 1) is a graph of u(A) , A being any point in R . A complete treatment of point (i) can be found in [ 8 ] . Point (ii) uses arguments of Crandall and Rabinowitz [ 2 , page 1771. 3.5
Stability
In this section we study the stability of the equilibrium solutions for the one-dimensional case: ut
- uxx
+
r(u)
0,
=
O < X < l , t > O ,
(3.41) u(0,t)
=
u(1,t)
= 0,
t > 0.
Here u(x,t) = s(x,t) - so and r(u) = oF(so + u), and the evolution equation (3.41) must be supplemented with the initial condition: (3.42)
u(x,O)
=
uo(x),
O < X < l .
We say v is an equilibrium point of (3.41) if u(x,t) lution of (3.41). Namely: r(v(x))
-v"(x)
+
v(0)
v(1)
=
0,
=
v(x)
is a so-
0 < x < 1,
(3.43) =
= 0.
Equivalently, we say v is a stationary or steady state. Indeed, "stationary" is more appropriate than "steady", since stationary means "fixed in a position, not moving", whereas steady may mean "having a continuous movement", and may be applied, for example, to oscillations. However, when dealing with solutions of (3.43) as particular solutions of (3.41), we may and shall assume that these three expressions, equilibrium point, stationary state, and steady state have the same meaning. We say v is a stable equilibrium point of (3.41) provided that we can keep the solution of (3.41)-(3.42) as close to v as we desire for all t 2 0 by choosing uo sufficiently close to v. More precisely, for any E > 0 , there exists 6 > 0 such that any solution u of (3.41) with I l u ( 0 ) - v I / < 6 satisfies llu(t) - vI) < E for all t > 0. Here
STAB IL ITY
99
we write u(t) rather than u(.,t) for the function x 11 - 1 1 is the usual norm in HA(0,l).
+
u(x,t),
and
We say v is an asymptotically stable solution of (3.41) if it is v in H A ( 0 , I a s t - f m stable and we can also guarantee that u(t) by choosing I / u ( O ) sufficiently small. -f
vII
We say v is unstable if it is not stable. Specifically there exist E~ > 0 and {uon,n 2 l } with IIuon - v11+ 0 as n + m, but for all n , we can guarantee that there exists a time t such that the solution un(t) of (3.41) with initial condition uon satisfies:
In finite dimensions, that is u e R N and f:RN + RN it is well known that the solution v of f(u) = 0 is an asymptotically stable equilibrium point of du/dt t f(u) = 0 if the eigenvalues of fu(v) (the Jacobian matrix (afi/au.) evaluated at u = v) all have positive real I parts and is unstable if fu(v) has an eigenvalue with a negative real part. In infinite dimensions, as is the case for problem (3.41), analogues of this remain true. More precisely, consider the nonlinear problem (3.41) where the function r:R + R is continuously differentiable, its derivative being bounded. Suppose v is an equilibrium point of (3.41), such that: r(v
where
1.1
+
z)
(resp.
=
r(v) + r'(v)z
Il*jl
t
h(z)
) denotes the norm in L2(0,1)
(resp. HA(0,l)).
Moreover we assume u o to be in HA(0,'l). Let L be the "linearized operator" from C:([O,l]) L:w +-w" + r'(v)w.
into C(r0,ll)
100
MULTIPLE STEADY STATES AND HYSTERESIS
It is well known that the eigenvalue problem:
I
Lw = pw, w(0) = w(1)
O < X < l , =
0
possesses a sequence ui, i = 1,2,. . . , of real eigenvalues converging towards + We shall suppose that the p . are set in an increasing order. Then p, is a simple eigenvalue associated to an eigenfunction, w,, which is strictly positive in 1 0 , 1 [ . Then we have the:
-.
Theorem 3.5 (Principlesf linearized stability) sumptions on r , u0 ’ and v , we have the following:
With the above as-
(i)
(Existence and uniqueness) (3.41) has an unique solution on [O,+m[ with initial value u It is a continuous function u:[O,+-[ + L2(0,1) such that u ( 0 ) = u o and on l o , + - [ we have u(t) E H 2 (0,l) n HA(0,l).
(ii)
(Stability) If the spectrum of L lies in ]B,+-[ for some B > 0 , then v is asymptotically stable. More precisely, there exist 5 > 0 and M 2 1 such that if I / u o - v1I < 6 then on 0 5 t < the solution of (3.41) satisfies:
.
-
(iii) (Asymptotic behavior) Assume that 0 < p 1 i y < exists 5 > 0 so that IIuo 5 6 ,implies:
p2.
Then there
vII
u(t)
v + ~ ( u ~ ) e -+ pE(t) ~ ~
=
where :
Here K(v) = 0 and K(.) is a continuous map from a neighborhood of v in H’(0,l) into the one-dimensional space N(L - p I) = span {w,}, and: K(uo)
=
E1(uo
-
v)
+
O( /lug - vlI2) as uo
uhere E l is the projection of HA(0,l) onto span
-+
IW, 1 :
v,
STAB1 LITY
(iv)
(Instability) is unstable.
Assume
u1
< 0.
101
Then the equilibrium solution v
For the above statement we rely heavily on the work of D. Henry [161, to which we refer for more details, and in particular for the proof of similar results in a much more general framework. Coming back to the case of our substrate inhibited system, where f(u) = - u " + uF(u + s o ) , one has a linearized operator f ( v ) : w + -w" + uF'(v + s )w and its eigenvalues correspond in the above way to the stability of the equilibrium solution v, provided the assumptions of Theorem 3.5 hold. The only property iihich may not be evident is (3.44).
But:
< M for every 5 E R , we have: -
max Iz(x)l, C is some constant and the last inequaliO ~1 X ~ ty is a consequence of the continuous imbedding of H A ( 0 , l ) into
where:
Izlo
=
C"0,lI). This completes the verification of Theorem 3.5 assumptions. However we obtained precise results in Section 3.4 about the least eigenvalue of the linearized operator fU(v). Namely it is strictly positive on the upper and lower arcs of the S-shaped curve, those which correspond to maximal and minimal solutions, and it changes sign at turning points. Although a priori everything can happen between the two turning points, it results from the numerical exploration of the whole curve of solutions that between these turning points the least eigenvalue remains negative and no other eigenvalue crosses 0. Thus the situation is quite clear:
102
MULTIPLE STEADY STATES AND HYSTERESIS
the steady states on the upper and lower arcs are stable, whereas those on the intermediate curve are unstable. The only steady states having a physical existence are those which are extremal, either maximal or minimal: only these can be experimentally observed, in particular on the hysteresis loop. 3.6
Justification of the Quasi-steady-state Hypothesis
In Section 3.1 we modeled the 0-dimensional system by the evolution equation ( 3 . 9 ) , where s o was a time varying parameter, first increasing, then decreasing. Then we dropped the time deriva.tive, thus obtaining equation (3.11). The so-called quasi-steady-state hypothesis assumes the solution S(T) of the differential equation (3.9) to nearly satisfy the algebraic equation (3.11). The aim of this section is to make this statement more precise. The underlying assumption is that the diffusion-reaction process relaxes faster than s o is altered, namely that 8 is small with respect to T , T being the period of time during which s o increases from 0 to its maximum value. Thus it is appropriate to keep the original time scale t instead of T in (3.9), which gives:
where
E
=
Replacing
0 = E
1/P, t
= ET,
y(t)
= S(T)
and X(t)
= so(?),
by 0 in (3.45) gives:
We have seen in Section 3.1 that equation (3.46) may define multiple So we define z(t) as the minimal solution of: solutions y(t). (3.47)
A(t)
- z(t) - pF(z(t))
=
0,
O(t(T,
Our aim is to justify the replacement of equation (3.45) by equation (3.46). This will be achieved by Proposition 3.11. Since the s o l u tion y of (3.45) depends on E , which takes various values in the following, we shall denote it sometimes y(t,E) instead of y(t). When
JUSTIFICATION OF THE QUASI-STEADY-STATE HYPOTHESIS
103
takes sequential values cn, then yn will denote the function t Y(t,En).
E
-+
It is useful to consider the two points P = P(t,E) = (A(t) - Ey'(t,E), Here y'(t,E) denotes the deriy(t,E)) and Q = Q ( t )= (X(t),z(t)). vative of y with respect to t. Both points P and Q move on the curve OABCD (Figure 3.12) since their coordinates satisfy the equaThe path followed by the point Q tion of this curve, x = y + pF(y). as t varies from 0 to T is OACD, the last point D being attained at time T. T will denote the "jump time", at which the point Q jumps from A to C. Thus A ( T ) is the abscissa of the turning point A. Let rl be a strictly positive number, arbitrarily small. We have the following proposition:
Fig. 3.12 Proposition 3.11
- Let the €-system be defined by:
and the 0-system by:
(3*49)
i
z(t) is the minimal solution of z(t) + pF(z(t)) = A(t),
O z t z T ,
MULTIPLE STEADY STATES AND HYSTERESIS
104 where, (3.50)
A eH1(O,T),
Then when
E
(3.51)
y(t)
(3.52)
y
(3.54)
y
0,
-+
+
+
-f
Al(t) 3 0, A(0) = 0
z(t)
for every t #
z
in H'(O,T-Q) weak,
z
in L~(o,T) strong.
T
,
The proof of Proposition 3.11 will result from a series of Lemmas. ____ Lemma 3.2 -
If, at time to, y(to) 5 z(to), and if, interval (to,t,), both points P and Q remain on arc y(t) < z(t), that is P is on the left of Q , for to < same property holds if, instead of being on arc OA, and Q belong to arc BCD.
Proof:
w satisfies:
Let w = z - y.
(1
EW' + (
during the time then t < t l . The both points P
OA,
+ pF'(y +
@w))d@)w
= EZ'
0
(3.55) w(to)
3 0.
Since, when both points P and Q are on arc OA, (or BCD), 1
+
Lemma 3.3 -~
pF'(y
+
When t
ew) t
> 0
and
z' > 0 , then w > 0 .
]O,T[ , P is between 0 and Q.
Proof: We apply Lemma 3.2 with to = 0. As y(0) = z ( 0 ) = 0 , the only property t o check is that both points P and Q remain on arc OA during the time interval ( 0 , ~ ) .It is the case for Q by definition. P is on arc OA at least at the beginning, for t small enough, since the coordinates of P are continuous functions of t. During this period, from Lemma 3.2, y(t) < z(t) and P remains between 0 and Q.
JUSTIFICATION OF THE QUASI-STEADY-STATE HYPOTHESIS
105
On the other hand suppose P crosses Q on arc OA at some time t l < T . Since the function t y(t) - z(t) vanishes at t = t, by passing from negative to positive values, y'(tl) - z'(tl) is non-negative, and €y'(tl) = 0. Hence 0 < z'(tl) 5 y'(tl) = 0 , which is contradictory. -+
Just after the jump of Q from A to C , P is still somewhere between 0 and A. Then, since its coordinates are continuous functions of t , P moves continuously on the curve OABCD. As long as P remains on arc OABC, y(t) + pF(y(t)) 5 A(T), where A ( T ) is the abscissa of A. Thus if P E arc OABC and Q E arc CD, we have:
and P moves upwards. Lemma _~
3.4
- Let
t >
T.
For
E
small enough, P(~,E) t arc CD.
0 such that P(t,cn)E arc Proof: If not, there exists a sequence E~ (Figure 3.12). It OABC. Let I- = (t - ~ ) / 2and 6 = A(s + n) - A(T) < 8 < t, P(8,En) E arc OABC results from the above that, for T + with y(e,En) < y ( t , ~ ~ ) ,so that: -+
and
It follows that:
which is arbitrarily large. However this contradicts the fact that y(t,~,) - Y ( T + ~ , E ~ )is less than the ordinate of C. This completes the proof of the Lemma. Let be a positive number, arbitrarily small. We wish to compare the functions y(t,E) and z(t) on intervals ( O , T - ~ and ) (r+n,T), when E is sufficiently small for P(T+n,E) to belong to arc CD. As Q is on the right of P when P crosses C , Q remains on the right of P during
106
M U L T I P L E STEADY STATES AND
HYSTERESIS
the time interval (T,T). The reasoning is the same as in the proof of Lemma 3.3, by employing Lemma 3.2. Lemma 3.5 - The family of functions y(.,e) when E is small enough remains in a bounded set of H'(O,T-T~). (resp. H'(T+rl,T)). Proof: Let (to,t,) be either interval ( O , ~ - r l ) or (T+n,T). We have, upon multiplying (3.55) by w , integrating and using the fact that,
on (to,tl), 1
PF'(Y) 2 a > 0 ,
+
and the last inequality results from the well known formula: ab 5
4
a2
t
whatever a and b.
1 b2 2cY Thus, if
E
< 1,
It follows that: (3.56)
E'/'w
(3.57)
w
E
E
bounded set of Lm(to,tl)
bounded set of L2(to,tl).
Proof of Proposition 3.11 - Let (to,t ) be either ( O , T - T ~ ) o r ( T + Q , T ) 1 We can consider a sequence E +. 0 and such that: Y(.,En)
+.
Y(*,En)
+.
"A/'
u
Y'(*,En)
+.
5
in H'(to,tl)
weak,
in L2(to,tl)
strong and a.e
in Lm(to,tl)
weak::
JUSTIFICATION OF THE QUASI-STEADY-STATE HYPOTHESIS
which means t h a t :
and
For a n y f u n c t i o n
V E
L 2 ( t o , t , ) we h a v e :
Itt1
Av d t .
0
As n
+ m,
y,v
dt
and
It’
uv d t ,
+
L
L
t
F(yn)v d t
F(u)v d t ,
107
MULTIPLE STEADY STATES AND HYSTERESIS
108
so that:
[:’
(u + p F ( u )
-
X)v dt = 0
for every
V E
L2(to,t,),
whence : (3.58)
u
+
pF(u) = A.
But we know that y(t,En) 5 z(t). Thus :
and, as z is the minimal solution of (3.58), u(t)
=
z(t),
t €(to,tl).
(3.51) results from the fact that ri is arbitrarily small. (3.54) results from (3.51), the fact that y(t,En) 5 z(t), and the Lebesgue Theorem. 3.2 - The biochemical interpretation of the above numerical and mathematical results is that, even in very simple biochemical systems where diffusion and enzyme reaction interact, multiple stable steady-states may exist, together with an hysteresis phenomenon as some environmental parameter varies. I t has been suggested that such hysteresis effects in biochemical systems are adequate to account for short-term memory [ 1 7 1 , 1 8 , 1 9 1 .
Remark -
REFERENCES
109
References Kubicek, M., Dependence of solution of nonlinear systens on a parameter, ACM Transactions on Mathematical Software, Vol. 2 , # 1 (March 1 9 7 6 ) 9 8 - 1 0 7 . Crandall, M.G. and Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 5 2 ( 1 9 7 3 ) 1 6 1 - 1 8 0 . Hardt, S., Naparstek, A., Segel, L.A., and Caplan, S.R., Spatiotemporal structure formation and signal propagation in a homogeneous enzymatic membrane, in: Thomas, D . and Kernevez, J.P. (eds.), Analysis and Control of Immobilized Enzyme Systems (North-Holland, Amsterdam, 1 9 7 6 ) . Naparstek, A., Romette, J.L., Kernevez, J.P., and Thomas, D., Memory in enzyme membranes, Nature, 2 4 9 ( 1 9 7 4 ) 4 9 0 - 4 9 1 . Crandall, M.G. and Rabinowitz, P.H., Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 ( 1 9 7 1 ) 3 2 1 - 3 4 0 . Crandall, M.G. and Rabinowitz, P.H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 ( 1 9 7 5 ) 2 0 7 - 2 1 8 . Brauner, C.M. and Nicolaenko, B., Singular perturbation, multiple solutions, and hysteresis in a nonlinear problem, in: Singular Perturbation Bound. Layer Theory, Proc. Cong. Lyon 1 9 7 6 , Lecture Notes in Mathematics 5 9 4 (Springer, Berlin, 1 9 7 7 ) . Mignot, F. and Puel, J.P., Sur une classe de problsmes non-linCaires avec non lin6arit6 positive, croissante, convexe, Rapport N o 7 5 0 1 8 , Analyse NumGrique, Universit6 Paris 6 ( 1 9 7 8 ) . Dieudonng, J., Foundations of Modern Analysis (Academic Press, New Jersey, 1 9 6 0 ) . Crandall, M.G., An introduction to constructive aspects of bifurcation and the implicit function theorem, in: Rabinowitz (ed.) Applications of Bifurcation Theory (Academic Press, New York, 1977).
Krein. M.G. and Rutman. M.A.. Linear ooerators leaving invariant a cone in a Banach space, Trans. A.M.S:, Series 1 , 1 O U ( 1 9 6 2 ) 199-325.
Rabinowitz, P.H., Some aspects of nonlinear eigenvalue problems, Rocky Mountain Journal of Mathematics, Vol. 3;N02 (1973)161-202 Leray, J. and Schauder, J., Topologie et 6quations fonctionnelles, Ann. Sci. Ecole Norm. Sup. 3 , V o l . 5 1 ( 1 9 3 4 ) 4 5 - 7 8 . Sattinger, D.H., Topics in stability and bifurcation theory, Lecture Notes in Mathematics, NO309 (Springer Verlag, Berlin, 1973).
Brauner, C.M. and Nicolaenko, B., Perturbation singulisre, solutions multiDles et hvst&r$sis dans un nrobleme de biochimie. Cras S&rie 7 ( 1 9 7 6 ) . '
110
MULTIPLE STEADY STATES AND HYSTERESIS
[16] Henry, D., Geometric theory o f semilinear parabolic equations, Lecture Notes, University o f Kentucky (1976). 1171 Changeux, J.P. and Thiery, J., in: Jarnefelt J.(ed.), Regulatory Function o f Biological Membranes (Elsevier, New York, 1968). [181 Katchalsky, A. and Oplatka, A., Neurosciences Res. Symp., 1 (1966) 352-371. I191 Katchalsky, A. and Spangler, R., Quarterly Rev. Biophys., 1 (1968) 127-175.
CHAPTER 4 ASSEMBLAGES OF CELLS AND BIFURCATIONS
In our survey of the various phenomena induced by interacting diffusion and reaction in enzyme systems, we first observed the existence of concentration gradients in the profiles within the membranes. This phenomenon implies the necessity, when discussing membrane behavior, to take into account not only the external concentrations, but also the internal values which may be much smaller. Then we have seen how a "scalar" phenomenon, enzyme reaction, can cause a "vectorial" effect, glucose transport, in a glucose pump where transport against the concentration gradient is effected from an external reservoir at low concentration to an internal compartment at high concentration. Lastly, we studied a system endowed with an hysteretic behavior due to the existence of multiple steady states and able to act as a memory and store information. The goal of this chapter is, on one hand, to study a type of system interesting in itself, that is assemblages of cells, and on the other hand, to explain by a simple example a type of phenomenon that we shall encounter frequently in succeeding chapters, bifurcation. In particular, we shall study the solutions (s1,s2) of the system:
as the parameter s o varies. Here F ( s ) is, as in Chapter 3, the substrate inhibited rate expression: (4.2)
F(s)
= s / ( l + 151 + k s ' ) .
Equations (4.1) represent, in the case N the system:
111
=
2, the steady states o f
ASSEMBLAGES OF CELLS AND BIFURCATIONS
112
(4.3)
i
~
dsi dt
-
si-l
+
2s. - s .
s
+-
2sN ~
+
pF(si)
=
0,
2 5 i 5 N-1,
....
>:
_ -
~
+
pF(sx) = so
Here s l , s 2 , ..., sN are substrate concentrations in a linear array of cells containing a substrate inhibited enzyme, each cell being in contact with two neighbors, except for the two cells at the ends of the array which are in contact with well-stirred reservoirs at specified concentration s o (Figure 4.1). Such assemblages of cells with substrate inhibition kinetics have been studied by Bunow and Colton [ I ] , who so describe the biological interest of their model: "In the tissues of metazoan organisms, some cells have direct access to compartments or reservoirs which supply the substrates or remove the products; for other cells the access is diminished by the presence of neighboring cells through which these substances must pass in order to reach the interior of the tissue. The intervening cells may contain the same enzymes which catalyze reactions involving these substrates, thereby reducing the substrate concentration available to interior cells. These phenomena may be expected to change the apparent dependence of enzymatic reaction rates in an assemblage of cells upon the substrate concentration in the surrounding medium, in comparison either to the observed kinetics of single cells or to that of enzymes free in solution".
LINEAR ARRAY OF CELLS RESERVOIR
RESERVOIR
sN+l
\ IMEMtBRANES/ Fig. 4.1
INTRODUCTION An additional interest over prescribed ranges states can occur, some files of concentration
of Bunow and of reservoir of which are and reaction
113
Colton's work is to show that concentration so, multiple steady characterized by asymmetric prorates across the array.
Figure 4.2 shows the first component s 1 of the solution ( s 1 , s 2 ) of (4.1) as a function of the external concentration s o . In addition to a smooth curve of "symmetric" solutions, that is satisfying s 1 = s2, represented by the curve OABCDZ, we shall find branches of nonsymmetric solutions (sl # s2) emerging from this curve at two bifurcation points 5 and C (BEFC and BGHC).
Fig. 4.2 The outline for the remainder of this chapter is as Follows. Section 4.1 describes multiple cell assemblages. Because the analysis and exposition are simplest when there are only two cells, in Section 4.2 we restrict our attention to this case. There is a well developed theory which permits the qualitative description of the solutions for equations: (4.4)
f(X,u)
=
0
in the vicinity of a bifurcation point (A o , u o ) . In this chapter we restrict our analysis to mappings f : R+ x RN + R~ twice continuously differentiable with respect to both X and u in some neighborhood of the bifurcation point (ho,uo). In Section 4.3 W E set the framework
114
ASSEMBLAGES OF CELLS AND BIFURCATIONS
for further discussion, specify conditions for bifurcation to occur, prove the existence of bifurcations and study exchange of stability at bifurcation points. In Section 4.4 we describe numerical methods for computing bifurcation points and bifurcation branches. In both sections we rely upon the methods of H.B. Keller and his group "2,3,4,5,61.
Finally Section 4.5 deals with the case of slightly different boundary values:
Here E is a small "imperfection" the influence of which on the family of solutions is analyzed. More precisely, we explain why there is bifurcation breaking.
4.1
Multiple Cell Assemblages
Figure 4.1 shows an array of N identical cells, each containing an enzyme catalyzing a reaction inhibited by high substrate concentrations, with rate expression:
VM, KS, and KSS being the same in each cell. Furthermore, suppose that the cells communicate with one another by diffusion across their common boundary. Across the membrane separating cells i and i-1 the rate of substrate transport is PA(S.1 - 1 - Si), where P is the diffusive permeability and A is the contact surface area. Figure 4.1 shows a pair of reservoirs at the ends of the array, at specified concentrations S o and S N + l , which provide a continuous source of substrate. Mass balance for substrate in the ith cell is given by: dS.
(4.6)
V
-J-$ =
PA(Si-l
- Si)
+
PA(Si+l
- Si)
- VR(Si)
where V is the cell volume. By introducing a reduced substrate concentration s = S/KS and a reduced time T = t PA/V, equation (4.6) may be rewritten in dimensionless form:
TWO CELL
(4.7)
ds d.; -
=
s
~ - - 2s. ~1
+ S.
1+1
ASSEMBLAGE
- pF(si)
where F(s) is given by (4.2) with k (4.8)
M' V KS PA
p = - -
115
=
Ks/Kss
and:
*
The composition of the cells in the steady state is governed by:
)
2 < i 5 N-1
= O
1
In Section 4.2 we shall be in the case N = 2 and s ~ =+ s o~ . For the numerical experiments of Section 4.4 we shall take N 1. 2 and s ~ =+ ~ s
.
1.2
In Section 4.5 we shall have N Two Cell
=
2 but ss+l
=
so
+
E.
Assemblage
Consider the steady state equations:
as s varies. A first branch of solutions consists of "symmetric" profiles s 1 = s z obtained by replacing s 1 and sZby s into ( 4 . 1 0 ) , which gives :
Ke already encountered equation (4.11) in Section 3.1, and saw that, in the ( s o ' s ) plane, its solutions are located on an S-shaped curve. Thus Ke obtain a family of solutions of (4.10) parametrized by s , s
=
s + pF(s),
s1
=
S,
s2
= S .
As s varies from 0 to + m , the s 1 component is represented against The stability of so by the S-shaped curve OABCDZ of Figure 4 . 2 . these "symmetric" steady-states depends on the eigenvalues of the
ASSEMBLAGES OF CELLS A S D BIFURCATIONS
116
linearized operator:
s. These eigenvalues are p 1 = 1 + pF'(s) and u 2 = and we suppose p sufficiently large for p l and u 2 to be able to vanish for some values of s. Morepreciselypl = 0 when F ' ( s ) = - l / p , which occurs for s = t l and s = t2 (Figure 4.3), thus givingthe turning points A and D , and u 2 = 0 when F'(s) = - 3 / p , which occurs for s = b l and s = b2, corresponding to the bifurcation points B and C.
for s 1
= s2 =
3 + pF'(s),
Fig. 4.3 The stability analysis of the "symmetric" sta es is straightforward. They are stable when the leading eigenvalue p is positive, that is when 0 5 s < t l or s > t2, and unstable when t l < s < t 2 . Thus the middle states, which occur between A and D , in the region of negative slope, are unstable whereas the lower and upper states, on limbs OA and DZ of the sigmoid curve, are stable. It is a fairly easy exercise to check that points A and D are turning But we are mainpoints, by verifying the hypotheses of Theorem 3 . 2 . ly interested by the status o f points B and C, more precisely to know why branches of solutions emanate from these points and how they can be calculated. Before discussing mathematical and numerical aspects of bifurcations in Sections 4.3 and 4.4, let us change nota-
TWO CELL
ASSEMBLAGE
117
tions and specify the properties which will be used. Notations - It is more appealing to denote the variables by names reminiscent of their origin, for example so, s l , and s 2 for substrate concentrations. However, in order to cast problem ( 4 . 1 0 ) into a more general framework, we perform a dramatic change of variables:
A
= so,
u,
=
s l , u2 = s 2 .
Let u = ( u l , u 2 ) and f : R+ x R 2
+
R 2 be the mapping:
Furthermore s will designate in Sections 4.3 and 4.4, one of the (Ao,uo) two solutions b l and b2 of 3 + p F ' ( s o ) = 0 (Figure 4 . 3 ) . will be the corresponding bifurcation point (B or C) defined by X o = s + p F ( s o ) and u o l = u o 2 = s o . f{ = fu(Xo,uo) and fl = fA(Ao,uo) will be respectively the Jacobian matrix of f and the derivative of f with respect to A at ( A o , u o ) :
$
will be an eigenvector of :f
associated to the zero eigenvalue:
will be an eigenvector of fo:: the transpose of fi. here : f = fi, we shall have I):: = $:: (transpose of $)
In fact, since
$::
and range R(f:)
Properties - The null space Y(f: dim K(fo) N(fi)
=
=
codim R ( f $
span { $ I , N(f;::)
1,
=
=
span {$::I
of :f
=
[l
-11.
satisfy:
ASSEMBLAGES OF CELLS AND BIFURCATIONS
118
where I):: and v are understood to be respectively row and column vectors, and $::v the scalar product of these vectors. The vector fi is such that:
This is precisely the property which will permit us to recognize the turning points and bifurcation points as being different. At either type of point f; is singular, but at turning points $::fi# 0 (Theorem 3.2) whereas at bifurcation points $::fi = 0. Compare for example the calculated values of $::fi at turning points and at bifurcation points in Figure 4.4.
Fig. 4.4 4.3
Nonisolated Solutions and Bifurcation
Consider the equation: (4.13)
f(X,u) = 0
NONISOLATED SOLUTIONS AND BIFURCATIONS
119
where f : R+ x RN +. RN is a twice continuously differentiable mapping, We deliberately restrict ourselves to RN in this Chapter and the next because the states u of the systems therein considered lie in finite dimensional spaces, and also because RN is the usual framework for numerical approximations of infinite dimensional problems such as encountered later. However, since most of the concepts and results of this section are extendable to mappings f : R+ x B +. B where B is a real Banach space, RN will be denoted B. Keener [ 7 1 gives the following definitions of branching and bifurcation points which, although somewhat restrictive, encompass all of the phenomena we discuss in this book. Definition 4.1 - Suppose ( A o , u o ) is a solution of (4.13). Then (Ao,uo) is a branching point of (4.13) if in every open neighborhood of (Ao,uo) there exist distinct solutions ( A 1 , u l ) and (Az,u2) of (4.13) with A 1 = A 2 . It isthecase of points A,B,C,D,E,F,G,H in Figure 4.2. However there isa difference of nature between turning points A,D,E,F,G and H and bifurcation points B and C. For a branching point to be a bifurcation pojnt,we need an additional property. Definition 4.2 - A branching point is a bifurcation point if, furthermore, every open neighborhood of (Ao,uo) contains at least one solution of (4.13) for A > A. and also for A < A o . Definition 4.3 - By a smooth branch or arc r of solutions for (4.13) we mean a twice continuously differentiable mapping r : s +. (A(s), u ( s ) ) of a compact interval [ a , ~c]R into R+ x B, satisfying f(A(s), u ( s ) ) = 0. In other words it is a smooth one parameter family of solutions for (4.13), (A(s) , u ( s ) ) , s describing the interval [ a , B ] . Let:
be the Jacobian matrix of f, and :f
=
fU(Ao,uo),
If :f is nonsingular, the implicit function theorem implies that there are neighborhoods U of A. in R+ and V of u o in B such that there is
ASSEMBLAGES OF CELLS AND BIFURCATIONS
120
exactly one function u : U + V satisfying f(X,u(A)) = 0 for X E U . Moreover,the mapping f being twice continuously differentiable on R t x B, u is twice continuously differentiable and the solution (Ao,uo) can be continued into exactly one smooth branch of solutions for (4.13).
Definition 4.4 - A solution pair ( X o , u o ) at which fi is non singular (resp. singular) is called an isolated (resp. a nonisolated) solution of (4.13). Of course a necessary condition for a solution ( A o,uo) of (4.13) to be a branching solution is that i t be a nonisolated solution. Suppose that (Ao,uo) is a nonisolated solution of (4.13), that fi has null space N(fi) spanned by @ , and that N(fi::), the null space of the transpose fo::of fi, is spanned by$::. Let:
The existence of a nonisolated solution is not sufficient to guarantee branching. However there are two important possibilities. Possibility 1 (4.14)
$::fi
-
If:
# 0
theorem 3.2 tells us that the solutions of f(X,u) = 0 near (ho,uc) form a curve ( A ( s ) , u ( s ) ) = ( A o + -r(s),u0 + s @ + x ( s ) ) , where ~ ( 0 =) ~ ' ( 0=) 0 , x(0) = x'(0) = 0 and @ : : x ( s )= 0. Moreover, since f is twice continuously differentiable, so is X and the calculation of A " ( O ) , resulting in formula (3.36), shows that:
Here f:U@2 is a shortened notation for fiu@@, where f:uvw tor of components:
is the vec-
Thus if we suppose that (4.14) holds, ( A o , u ) is a turning point
NONISOLATED SOLUTIONS AND BIFURCATIONS
121
(also referred to as a limit point o r a bending point) provided: (4.16)
@::f;U$2
#
0.
Depending upon cases (X"(0) < 0 or X"(0) > 0), there is no solution to the nonlinear problem f(X,u) = 0 either for A > X o or A < A o . In the particular case A"(0) = 0, then it is easy to see that ( A o , u o ) is not a branching point o r is a turning point according to whether the first nonvanishing derivative A(p)(0) is of odd o r even order.
- Therefore in order that
Possibility 2 cat ion, (4.17)
@::fi
( A o , u o ) be a point of bifur-
= 0
must hold. We will see that under an additional condition which is in general satisfied, (4.17), together with the above assumptions on f;, ensures the existence of switching branches at (Ao,uo). We rely upon [ 2 ] , [ 3 1 , and [ 5 1 for analyzing this situation. We first specify the structure of the tangents to the branches at a bifurcation point. Theorem 4.1
- Suppose:
(4.18)
f(Ao,uo)
= 0
(4.19)
dim N(fi)
=
(4.20)
fi
(4.21)
N(f;)
Let
be the unique element of B such that:
$
(4.22)
f;$"
E
codim R(f;)
= 1,
R(fi), =
+
span { $ I , N(fA::)
fi
= 0, I )::$
We define a , b, c by:
=
= 0.
span
{@::}.
ASSEMBLAGES OF CELLS AND BIFURCATIONS
122
Let r : s + ( X ( s ) , u ( s ) ) be any smooth branch of (4.13) through the point ( X o , u o ) , say with X ( s o ) = Ao,u(so) = u o . Then the tangent to
r
at ( A o , u o ) is of the form:
where the scalars (ao,al) satisfy the Bifurcation Equation: (4.25)
aa;
2baoal + ca; = 0
+
Proof: (4.17) implies the existence of an unique element fying (4.22). Differentiating the identity f(A(s),u(s)) it follows that:
fU,(S)U'2(S)
where:
fu(s)
=
+
2fUA(S)A' (s)u'(s)
+
f,(S)A"(S)
+
fu(s)u"(s)
+
B satis0 twice,
+ o t
=
f,,(sjA'2(s)
= 0
...
f,(u(s),A(s)),
We get at s = s o : (4.26)
f{u'(so)
(4.27)
fiu"(so)= -[fo uuU'~(S,)
+
where:
f{ = fU(so)
f:h'(s,)
= 0
+
2f,"AA'(so)u'(so)
+
f~AX12(so)]
f;[x'ysoj =
fu(Ao,uo),
... .
It results from (4.22) and (4.26) that fi(u'(so) - X'(s0)$,) = 0, that is u'(so) - A ' ( S ~ ) + ~ EN(f:), and there exist two scalars a. and a l for which (4.24) h o l d s . Since both f{u"(so) and f:A"(s,) are in R(f{) the same must be true of the bracketed term of (4.27). Thus Ql[ . . . I = 0 must hold for this term and so using (4.24) i t follows that the two scalars a. and a 1 must satisfy the quadratic equation ( 4 . 2 5 ) , where a, b, and c are
NONISOLATED SOLUTIONS AND BIFURCATIONS
123
given by (4.23) and this completes the proof. The quadratic form (4.25) and its role in bifurcation theory was first pointed out in [ 2 1 . Note that the eigenvectors I$ and I):,: the vector @ o and the coefficients a, b , and c will be the same for any branch r passing through ( A o , u o ) , and are defined by quantities involving the single function f and In case there is a second its derivatives at that point ( X o , u o ) . smooth branch of (4.13), r ' : t + (p(t),w(t)), passing through the then it follows from point (Xo,uo), say with ~ ( 0 =) X o and w ( 0 ) = u 0' theorem 4.1 that the tangent to r ' at (Ao,uo) is of the form:
where the scalars Bo and B 1 satisfy the same quadratic equation as a and a l , namely the bifurcation equation:
Now we specify the condition under which the roots of the bifurcation equation : (4.28)
"5;
-+
2bSoC1
+
cc:
=
0
are distinct. As (4.28) is a homogeneous quadratic equation, it admits rays of solutions. Its roots are defined up to a scalar multiple, and two roots ( a 0 , a 1 ) and ( B o , B 1 ) are said to be distinct when:
Lemma 4.1 provided: (4.30)
-
The roots of the bifurcation equation (4.28) are distinct
+"(fiUu'(so)
+
fiXX'(so))@
# 0
when
X'(so) # 0
and
Proof:
We already know that the quadratic (4,28).admits a nontrivial
ASSEMBLAGES OF CELLS AND BIFURCATIONS
124 root ( a o , a l ) . aal
+
The other root is distinct provided: bao # 0
when
a.
# 0
when
a.
= 0.
and b # O
The proof of the Lemma is completed by verifying that the left-hand sides in (4.30) and (4.31) are precisely a a l + bclo and b. Under the assumptions of Theorem 4.1, together with (4.30) and (4.31), our goal is to prove the existence of a second curve of solutions emanating from (Ao,uo) and distinct from r . Let r ' : t +(U(t),w(t)) Then Theorem 4.1 b e such a curve, say with ~ ( 0 =) A. and w(0) = p 0' tells us that we necessarily have:
where ( B O , B 1 ) is the second root of the bifurcation equation (4.28). We just noticed in Lemma 4 . 1 that (4.30) and (4.31) are precisely conditions ensuring that the pair of roots (ao,al) and ( B o , B 1 ) are distinct. Thus we must have:
=
B,,
$J%(t)
= 0
together with some normalization, say: (4.34)
U'2(t) + /w'(t)1 2
In particular, at t B;
+
lBolDo
=
+
=
1.
0, the scaling of B,
and B 1 is specified by:
B1@l* = 1
We demonstrate the existence of
r ' using implicit function theoretic
NONISOLATED SOLUTIONS AND BIFURCATIONS
125
techniques, following [ 5 1 , where the analysis is performed under much more general conditions. Application of the Implicit Function Theorem: following: We define the function g : R x R x B x R
We see immediately that g(O;Bo,B1,vo) lution of:
+
= 0
The framework is the
B x R x R by:
where vo is the unique s o -
The claim about existence and uniqueness of a solution to ( 4 . 3 5 ) follows from: Lemma 4 . 2 Proof:
I)::[.
The bracketed term in ( 4 . 3 5 )
.,]
= aB;
+
2bB0B1
+
cB;
=
is in R(f;).
0
by definition of ( B o , B 1 ) . We now investigate the derivative of g with respect to (SO,cl,v) at (O;BO,B,
,vo), applied to ( f o , E l ,O):
126
ASSEMBLAGES OF CELLS AND BIFURCATIONS
where :
We verify the following lemma: Lemma 4.3
-
~ ~ ~ o , S 1 , v ~ ~ ~ ; B o(Eo,El,~l , B 1 , ~ =o ~O*
go
=
E,
= 0
and O = 0.
Proof: For the first component in (4.36) to vanish, i t is necessary that the bracketed term be in R(f:), whence I$::[ 1 = 0, that is:
...
Observe that the quadratic equation (4.28), the roots of which are ( a o , a l ) and ( B o , B l ) , can be written:
with condition (4.29), expressing that the roots are distinct Thus we can write, instead of (4.37):
-
(a1B0
and, as a,Bo
-
aoB,)(B1io
aoBl f 0 ,
Setting io = A B o and i , ferential, we must have: +
whence
A
=
lBo@o
0, and
+
-
= 0
B l i o - Boil = =
el
0.
A B 1 into the second component of the dif-
B1@12)
to =
BoE1)
= 0
= 0.
Finally O:f = 0 together with $::O = 0 imply O = 0. And this completes the proof of the lemma. Thus g
(0;B ,B , v ) is non singular. We see from the defini(Co,S,,v) 0 1 0 tion of g that this function is continuous in all arguments and continuously differentiable with respect to S o , S l , and v. Thus the
NONISOLATED SOLUTIONS AND BIFURCATIONS
127
Implicit Function Theorem insures the existence of an unique continuous (co(t) , E l (t) ,v(t)) satisfying, for all t near 0, g(t;co(t), cl(t),v(t)) = 0 with initial values: Co(0) = B o ,
C,(O)
=
B1,
and v(0)
=
vo.
This in turn implies the existence of a continuous curve:
of solutions of f ( A , u ) = 0, given by (4.32), (4.33), and (4.34). Summing up the above discussion, we have the following statement:
- Suppose a smooth arc r : (X(s) , u ( s ) ) of solutions of (4.13) passes through a point ( h o , u o ) , say with X(so) = X and u ( s o ) = u o , and that the following properties hold: (4.19), (4.20), (4.21), (4.30), and (4.31). Let 0, be defined by ( 4 . 2 2 ) , a , b, c by (4.23), (Bo,B1) by the bifurcation equation (4.28). Then there exists a second continuous branch of solutions of (4.13), I ? ' , passing through (Xo,uo). r ' has nontangential intersectionwith r , andnear (Xo,uo) is of the form (4.32), (4.33), (4.34).
Theorem 4.2 -_____
Both cases (4.30) and (4.31) are depicted in Figure 4.5, where the branch (X(s),u(s)) is called the primary branch, whereas the bifurcating branches are termed secondary. As will be seen below, at those bifurcation points such as A where X ' ( s o ) # 0 the determinant of the Jacobian matrix fu(X(s),u(s)) changes sign, whereas at those, such as B, where X'(so) = 0, it does not change sign. This difference between both situations is important for numerical methods since it is precisely through this change of sign of the determinant that a point of bifurcation can be detected. Change of stability - A rapid formal calculation shows the behavior which vanishes at s = s as of that eigenvalue y ( s ) of fu(X(s),u(s)) the point ( X ( s ) , u ( s ) ) crosses the bifurcation point:
i
fu(X(s),u(s))
(4.38)
w(s) = 4 $::Z(s)
=
w(s)
+ z(s)
0.
=
Y(S)W(S)
128
ASSEMBLAGES OF CELLS ASD BIFURCATIONS
U
I
A Fig. 4.5
If we assume the functions y ( s ) and z ( s ) to be continuously differentiable in some neighborhood of s o , we obtain by differentiating (4.38) with respect t o s at s = s o :
Acting upon the first equation in (4.39) with $:: and replacing by a0@, + a l $ , A ' ( s o ) by cto, and w(so) by $ we find:
u l ( s
)
Hence :
We distinguished two possibilities in Theorem 4.2. Either a0 # 0 and aal + bcxo f 0, or 0: = 0 and b 0. It f o l l o w s from (4.40) that y ' ( s o ) # 0 in the first case, whereas in the second case, since a = 0 , y ' f & , ) = 0 . Thus we have the following result:
+
Proposition 4.1 - Under the assumptions of Theorem 4.2, let y ( s ) be the eigenvalue of f u ( X ( s ) , u ( s ) ) which vanishes at s = s o . Then: Y'(S0)
# 0
when A'(s0)
# 0
NONISOLATED SOLUTIONS AND BIFURCATIONS
129
(there is a stability change through s o ) Y'(S0)
= o
when A ' ( s o )
=
0
Thus although det(fu(X(s),u(s)) vanishes at both points A and B in Figure 4.5, its sign changes at A but not at B (in the general case y " ( s o ) # 0 there is no change in stability through B). In particular if all the eigenvalues of fu(h(s),u(s)) are positive before crossing A , that is if the state u ( s ) is stable, then after crossing the least eigenvalue is negative, and the state u ( s ) is unstable. There is a change of stability when ( A ( s ) , u ( s ) ) crosses A. On the other hand the number of negative eigenvalues is the same before and after crossing B and the stability is the same on both sides of B. Much more can be said about the behavior of the eigenvalues which vanish at the bifurcation point as the points ( h ( s ) , u ( s ) ) and (p(t), w(t)) pass through this point ( X o , u o ) , namely we have the phenomenon of:
Exchange of stability: We describe two possibilities, which encompass all of the situations we shall be faced with in our subsequent analyses of stationary states.
___-__ Possibility
1: A ' ( s o ) # 0 and ~ ' ( 0#) 0 (Point A in Figure 4.5) Formula (4.40) applies not only to the eigenvalue y ( s ) which vanishes as ( A ( s ) , u ( s ) ) crosses ( X o , u o ) , but also to the eigenvalue 6(t) of of r ' vanishes at t = 0, as the point (p(t),w(t))
Thus y'(so)s'(0)
=
(aB,
+
bBO) (aa,
+
bao)/($"$)2.
It is easy to check that: yt(so)61(0)
=
- aoB0(b2 - ac)/($;:$)
ASSEMBLAGES OF CELLS AND BIFURCATIONS
130
Since the bifurcation equation possesses distinct real roots we have b z - ac > 0. Now in order to move along the two branches r and r ' in the same direction, we choose c1 ( = A' (so)) and Po ( = ~ ' ( 0 )to ) be of the same sign, say positive. With this choice we see that: y'(so)Sf(O)
< 0.
Hence the eigenvalues y ( s ) and 6(t) cross the origin in opposite directions. In particular, if the state u(s) loses stab lity as the point ( X ( s ) , u(s)) on arc r crosses (Ao,uo), then y ' ( s o ) < 0 and S ' ( 0 ) > 0 , implying that the point (p(t),w(t)) on arc r ' gains stability. More generally, since the Jacobian matrices fU(A s ) ,u(s)) and fu(~(t) ,w(t)) coincide at (Xo,uo), the number of strictly negative eigenvalues of the first one is transferred to the second one when crossing (A 0'
uo)
*
Possibility 2 : A ' ( s o ) # 0 and ~ ' ( 0=) 0 , ~ " ( 0 ) f 0
(points B and C in Figure 4.6). of the following result: 1im t+O
-t"
( t ) Y ' (so) 6 (t) X ' ( s o )
=
We refer to [51 and [ 8 1 for a proof
1,
which implies that the full (resp. dashed) lines in Figure 4.6 correspond to the same even (resp. odd) number of negative values for fU (A
,u)
Application to the two-cellassemblage of Section 4.2: We already found a "trivial" branch of "symmetric" states, OABCDZ (Figure 4.2) , with the parametrization:
on which points B and C satisfy the assumptions of theorem 4.2, as is easily checked. I n particular A ' ( s o ) = 1 + pF'(so) = - 2 , and $::(f:Uu'(so) + f:AA'(so))$ = - 2 , so that the bifurcation equation has
NUMERICAL ANALYSIS OF BIFURCATIONS
131
distinct roots
det f, > 0 det fu< 0
------
Fig. 4.6 Lastly, O 0 = Thus the conc usion of theorem 4.2 holds, namely B and C are bifurcation points and the tangent to the bifurcating curve at B o r C is given by: lJ (0) = B o ,
w'(0) = B O O O
+
B1@
where (BO,B1) satisfies the bifurcation equation: bBoB1
0.
=
The two possible solutions are ( 0 , l ) and (1,O) (up to some multiplicative factor). As ci0 = A'(s0) # 0 , necessarily P o = 0 and the bifurcation is one-sided. 4.4
Numerical Analysis of Bifurcations (Keller [ 2 , 3 1 )
We have seen insection 3.3 how to trace a smooth arc f(A,u)
=
0, X ER',
u
E
R
r
of solutions for:
N
by using a continuation method such as Kubicek method, which consists in parametrizing by the arc length s along the arc r f(A(s),u(s))
=
0
and
X"(s)
+
IU'(S)~~
=
1,
ASSEMBLAGES OF CELLS AND BIFURCATIONS
132
and solving a system of ordinary differential equations:
with initial conditions: X(0) =
xo,
u ( 0 ) = uo
where ( X o , u o ) is an isolated starting point. Such a method, applied to problem:
(4.41)
I
2 u , - uz -u,
+
+
PF(U,)
Z U ~+ p F ( u Z
- x = o - x
=
0,
with starting point (0,O) provides the arc O.4BCDZ of Figure 4.2. However the bifurcation points B and C and the bifurcating branches are ignored by such an algorithm. It is therefore desirable to enhance the method by providing the following possibilities:
(i) location of branching points (ii) distinction of bifurcation points from turning points (iii) computation of points on bifurcation branches in the vicinity of a bifurcation point, in order to treat them as starting points in subsequent applications of a continuation method. These additions to the standard program of Kubicek (or any method of continuation) are easily done by following closely the results of the previous section. (i) Determining theposition of branching points: At each point ( X ( s ) , u ( s ) ) along the curve, we evaluate the sign of the determinant When the determinant changes sign between succesdet(fu(X(s),u(s)). sive points of the arc, we employ a bisection search to refine the position of the singular point up to an error tolerance. We suppose that this process has produced a (nearly) singular point ( X o , u ) corresponding to the arc coordinate s = s
.
NUMERICAL ANALYSIS OF BIFURCATIONS
133
(ii) Distinction Of bifurcation points from turning points: l-he point (Ao,uo) may possibly be either a turning point or a bifurcation point. We have seen in Section 4.3 that generically the following holds. Either:
and (Xo,uo) is a turning point, or:
and (Xo,uo) is a bifurcation point. Figure 4.4 shows the values of $::fi at the branching points of equations (4.10), and i t appears that bifurcation points have a value of $;:fi which is very small (O(10-lo)), whereas for turning points +::fi = 1.41. (In fact fi, as may be directly checked!) at the branching
In order to calculate $::fi we need to calculate point ( A o , u o ) .
_______ Calculation of
@ and 9::: It is done most efficiently in the followThe triangularization of matrix fo:: with partial pivoting way [ 9 ] . ing yields a lower triangular matrix L with ones on the diagonal and an upper triangular matrix U with 0 at the last diagonal term. Since L is nonsingular, solving LU+ = 0 is equivalent to Ui$ = 0 , that is:
whose solution is calculated from +N stitution.
=
1 for example by backward sub-
For calculating @:: remark that if P is the permutation matrix corresponding to the partial pivoting, the solution of f;::@ = 0 satisfies U::L::PJI = 0 and is obtained by solving successively: u::x = 0
( j x = (0,O, . . . )0,l));
L::y
=
x,
P $ = y
134
ASSEMBLAGES OF CELLS AND BIFURCATIONS
(iii) Computation of points on bifurcation branches in the vicinity
of a bifurcation point: --__ __-____ r l , (X'(so),u'(so)) are
Since we are currently on a particular branch known up to a multiplicative factor. We use the upper and lower bounds on (Ao,uo) obtained in the bisection search to estimate ( A ' ( s o ) , u ' ( s o ) ) (Fig. 4.7) :
where E is arclength between points M and P but will not appear in the final formulas. Since u ' ( s ) =
+ al$
with a0
=
A'(so) and I)::$o
= 0, we
obtain:
In particular
Now we have all the elements to calculate the coefficients a, h , c of the bifurcation equation:
The vectors f o vw and f o v can be obtained analytically from f directuu Uh ly, or approximated by finite differences:
AS we already know one solution (a ,a ) to the bifurcation equation, 0 1 the other ( B 0 , B 1 ) can be obtained simply:
NUMERICAL ANALYSIS OF BIFURCATIONS
The multiplicative uncertainty in a.o,a.l disappears in a l / a o fact:
135
.
In
Normalizing:
whence :
The tangent (A;(to)
,u;(to))
to the bifurcating curve is thus determined:
A pair of points, one on either side of the bifurcation point, and lying in the bifurcated branch are obtained by taking a short step in For problem (4.41) the length of this the direction t (A;(to),u~(to)). step was selected empirically to be 0.1 but it can be as small as 0.001. Evidently this estimate is subject to error, so a Newton correction procedure is applied until an error tolerance is satisfied. Determination of the connected component of solutions: We are now in a position to attempt to trace out the collection of branches which are linked to an initially known solution, ( A o , u o ) . Starting from (Ao,uo) we trace r l , the "trivial" branch, noting the occurrence of each bifurcation or turning point. A s they are found, a pair of points on the putative bifurcated branch r 2 are obtained. The direction vectors to the bifurcated branch are simply the signs of the differences:
136
ASSEMBLAGES OF CELLS AND BIFURCATIONS
where M , P denote the direction taken along r 2 . These provide the starting points and direction vectors for the "primary" bifurcation branches. The process is continued until either the connected component is exhausted, or the state variables or the parameter A exceed preset limits. Application to problem (4.41) (two-cell assemblage): The case N = 2 with p = 100, k = 1 already illustrates the capacities of the algorithm and the complexity of the criteria. Figure 4.4 shows s , ( = u l ) as a function of s o ( = A). There are two turning points and two bifurcation points on the trivial branch. The bifurcations are both unilateral, and lead to stable solutions after a turning point. In fact the stability of the solutions on arcs E F and GH has to be checked separately since the only information we possess a priori is that the determinant of fU(A,u) changes sign as the point ( A , u ) crosses, say E , so that either 2 or 0 eigenvalues are negative on arc E F . For N = 3 there is another pair of bifurcation points on the trivial solution, and another closed primary bifurcation branch. There are no secondary bifurcation points, only numerous turning points, and each branch has a pair of arcs of stable solutions. In conclusion: We have at our disposal, with the Kubicek method and Keller method for bifurcations, an efficient algorithm for the numerical exploration of a connected component of stationary solutions.
4.5
Imperfections
In equations (4.1) the substrate concentrations in both reservoirs were assumed to be the same. It is a highly idealized situation because it is practically impossible to impose exactly the same concentration in two distinct reservoirs, and the following equations, more realistic, hold instead of (4.1):
(-s1 where
E
+
2sz
+
p F ( s 2 ) = so
+
E
is supposed to be a small imperfection or lack of homogeneity,
IMPERFECTIONS
137
What happens to the "trivial" solution and to the bifurcated branch of solutions in the neighborhood of a bifurcation point as the magnitude of the imperfection deviates from zero has been studied by J.P. Keener and H.B. Keller for a very broad class of problems [ l o ] . It is the phenomenon of perturbed bifurcation. In fact, by adding a slight imperfection, the whole nature of the set of solutions is changed. The aim of this section is to show on the trivial example (4.42) what happens and try to explain it. Figure 4.8 shows the set of solutions for (4.42) when E = 0 (heavier lines), with the "trivial" branch of "symmetric" solutions OABCDZ and the bifurcated loop BEFCHGB. It shows also the set of solutions for (4.42) when E # 0 (lighter lines), with a disconnected locus of solutions which forms an "isola" beside the rest of the set of solutions. This locus has a visual appearance which suggests, correctly, that it is composed of symmetrical and asymmetrical segments BC and BEFC. Each segment of the disconnected locus corresponds in its stability and symmetry to the analogue segment of the connected locus immediately adjacent. BC and BEFC disappear from the main locus, and only on this branch remain BGHC, OAB, and CDZ.
As E is increased, the closed locus becomes smaller and increasingly separate from the main locus, ultimately disappearing for large values of E [I]. First explanation of perturbed bifurcation 1 1 1 1 : On one hand we consider solutions of (4.1) such that s o = s + pF(s), s, = s , and s 2 = s , s being close to a solution b of 3 + pF'(b) = 0. Thus the point of coordinates (so,s,s) lies on the arc AD of symmetric solutions, close to one of the two bifurcation points B and C. (Fig. 4.8). On the other hand, for the same value s o , we consider nearby solutions of (4.42), (so,s1,s2). Let u 1 = s l - s and u 2 = s z - s . Since s + pF(s) = s o , we have: 2ul - u2
whence :
+
p(F(s+ul)
- F(s))
=
0
138
ASSEMBLAGES OF C E L L S AKD B I F U R C A T I O N S
U E
Fig. 4.7
i mper f ec t ion
SO F i g . 4.8
IMPERFECTIONS
139
Using :
in ( 4 . 4 3 ) , then putting the resulting expression of uz into the second equation of ( 4 . 4 3 ) , we obtain at last: (4.44)
E
=
-ZpF"(b) (s-b)ul
- p ( l F"' (b) + 3
+
p(pF"'(b)
5 F"*(b))u;
- F"'(b))
(s-b)'u,
- $2F"2(b)(s-b:hlf
+ O(4).
Thus, up to order tWo, we have: (4.15)
E
=
-2pF"(b) ( s - b ) u ,
iihich is the equation of an hyperbolic paraboloid, of the form z = kxy with z = E , k = -2pF"(b), x = s-b, and y = u l . This explains the hyperbolic form of the curves of solutions for [4.42) near B and L.
The net result of adding an imperfection is thus of breaking the bifurcation, which appears only in the idealized case E = 0. Second explanation ___---_
of perturbed bifurcation 1121:
Consider the equa-
tion: (3.46) f(A,T,U) = 0
bhere f : R x R x B B is a twice continuously differentiable mapping satisfying, a t the point ( X o , ~ o , ~ o ) , -f
ASSEMBLAGES OF CELLS AND BIFURCATIONS
140
$XfO
= 0
A
$XfO
=
T
d # 0.
Here B = R N for example, and f o = f(Ao,~o,uo),... pose that the bifurcation equation: a57
2b5,S1
+
cst
+
admits distinct roots. a
=
$::fouu 4 , b
c
=
$::(f;u$o@o
=
.
Moreover we sup-
0
=
Here: $::(f;U$o
Zf;
+
fAA)$
+
Q0
and
fix)
+
where $o is the solut on of f;$o
+
fl
= 0,
$ ::@ 0
=
0.
First remark that if: X 1 = {x,
E
B ; $::xl
01
=
then any X E B can be uniquely decomposed into x Let x = c $ + x 1 with x l c X l a n d 5 = $::x.
=
C $ + x 1 with x l ~ X 1 .
If we denote P the projection of B on @ parallel to X,, Px = 5 $ = ( $ ; : x ) @ and ( 1 - P ) x = x - Px, then equation x = 0 in B is equivalent to: (1-P)x Px
=
=
0
in
X1
and
in span{$}.
0
Now let (A,T,u) be a solution to (4.46). u - uo =
E$
+
Then if we decompose:
with v e x 1 ,
v
equation (4.46) can be rewritten: (4.47) (l-P)f(A,T,uo
+
€I$
+
v)
=
0
in X 1
and
IMPERFECTIONS
(4.48) Pf(A,T,uo
E$
+
+
v)
=
in span{$}
0
141
.
Equation (4.47) defines v as a function of ( E , A , T ) , as is easily seen by applying the implicit function theorem to the mapping:
Indeed :
and (1-P)f;O
=
0
with
6t
X1
implies 0 = 0. Thus there exists an unique C'smooth v ( E , X , T ) E X1 satisfying (4.47) near (O,Ao,~,;O). Replacing v by this function in (4.48) yields : (4.49) g(E,A,T)
=
$"f(A,T,Uo
+
€$
+
V(E,X,T))
=
0.
This is analogous to the so-called bifurcation equation of LiapunovS chmidt procedure. Expanding about ( E , A , T) = (0,A o , T ~ yields ) : g(c,A,T)
= d(T-To)
+
1 Z[ae2
+
2b€(A-Ao)
=
- 2 d ( - r - ~ ~+) h.o.t.'s.
+ C ( A - A ~ ) ~ ]+
h.o.t.'s
whence (4.50)
aE2
+
2b€(A-Ao) + c(A-A0)'
In fact we have no more bifurcation, but perturbed bifurcation, since the curves of solutions to (4.46) near ( A o , ~ o , ~ o ) are given by:
where v is the solution of (4.47) and E , A , T are related by ( 4 . 5 0 ) , and thus homotopic, for given T , to the two distinct "hyperbolic" curves described by (€,A) in the ( € , A ) - plane.
142
ASSEMBLAGES O F C E L L S AND BIFURCATIONS
Application to the 2-cells assemblage with imperfection: Problem (4.42) can be rewritten f(X,T,u) = 0 where X = s o , T = E , u = ( u , , u 2 ) = ( s 1 , s 2 ) and f(A,T,u) = ( Z u , - uz t pF(ul) - A , - u , t 2u2 + pF(u2) X - T). Moreover, as seen at the end of Section 4.3, a = c = 0 and b = - pF"(bi), bi,(i = 1,2), satisfying 3 t pF'(b.) = 0. Since d = Ji::fo = 1, the bifurcation equation reduces to: T =
pF"(bi)~(X-Ao)
+
h.0.t.'~.
which is to be compared to (4.44). The information brought to us by both methods is about the qualitative behavior of the solution curves near (Xo,~o,uo). In order to know their global behavior numerical methods of continuation such as those previously exposed have to be employed. interpretation of the above numerical Remark 4.1 - The &ochemical and mathematical results is that, in assemblages of cells where an enzyme reaction occurs, new non-symmetric stable steady-states may exist when the cells are separated by diffusion barriers. We shall develop this idea further in the next two chapters, where pattern formation in the so-called S-A system will result from sequential bifurcations and will be compared to morphogenesis in developing embryos.
REFERENCES
143
References Bunow B. and Colton C.K., Substrate inhibition kinetics in assemblages of cells, Biosystems , 5 (1975). Keller, H.B., Numerical solution of bifurcation and nonlinear eigenvalue problems, 359-384: Rabinowitz, P.H. (ed.), Applications of Bifurcation Theory (Academic Press, New York, 1977). Keller, H.B., Constructive methods for bifurcation and nonlinear eigenvalue problems, Proceedings of a meeting in IRIA, 1 - 1 1 (1977). Keller, H.B., Global homotopies and Newton methods, 73-94: C. de Boor and G. Golub (eds.), Recent Advances in Numerical Analysis (Academic Press, New York, 1978). Decker, D . W . , Topics in bifurcation theory, Ph.D. Thesis, California Institute of Technology, Pasadena, California (1978) 152 pp. + V . Szeto, R., The flow between rotating coaxial disks, Ph.D. Thesis California Institute of Technology, Pasadena, California (1978). Keener, J.P., Sec ondary bifurca t ion and mult iple eigenvalues, S.I.A.M. J. Appl. Math. Vol. 7 , N O 2 (1979) 330-349. Crandall, M.G. and Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal. 52 (1973) 161-180. Keller, H.B., Personal communication. Keener, H.P. and Keller, H.B., Perturbed bifurcation theory, Arch. Rat. Mech. Anal., Vol. 50 (1973) 159-175. Iooss, G., Personal communication Keller, H.B., Two new bifurcation phenomena, IRIA Research Report NO369 (1979) 1-22. Ray, W.H., Bifurcation phenomena n chemically reacting systems, 285-315: Rabinowitz, P.H. (ed.), Applications of Bifurcation Theory (Academic Press, New York, 1977).
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CHAPTER 5 THE URICASE SYSTEM AND PATTERN FORMATION
Pattern formation is probably one of the most surprising phenomena which occur in reacting and diffusing systems. Its biological interest is to suggest an explanation for morphogenesis in developing organisms. This chapter and the next one are devoted to the study of pattern formation in enz.yme systems. By pattern formation we mean that stable spatially non-uniform concentration profiles supersede spatially uniform concentration profiles which become unstable as a bifurcation parameter crosses a critical value. The systems under consideration are open systems where an external reservoir supplies a flow of substrates, the substrates diffuse with distinct diffusion coefficients and they are consumed in an enzyme reaction. In the absence of diffusion constraints in the reactive medium, an equilibrium state settles with the same concentration values in all the cells. This so-called trivial steady state is stable, and remains stable as long as relatively small diffusion constraints are imposed to the substrates within the reactive medium. However this uniform distribution of diffusible reacting substrates loses stability as the diffusion constraints cross a critical value, and a stable patterned distribution of the substrates emerge. This is the so-called Turing effect [l], also known as "diffusioninduced ins tab il i ty" or "diffusive instab il ity"
.
For example Figure 5.1 shows concentration patterns corresponding to successive stable steady states, as A varies, for the system:
145
146
THE URICASE SYSTEM AKD PATTERS FORMATION s t - As
+
X[R(s,a) - ( s o - s ) l
at - BAa
+
R(s,a)
pas/(l
=
A[R(s,a)
-
+ s +
=
in f?
0
a(ao-a)l
=
0
ks')
+ no-flux boundary conditions,
where f? is an elliptical domain:
A= 0.6
A=0.3
Fig
5.1
Such distributed systems will be studied in the next chapter and, since the analysis is simpler and easily extendable, vie restrict this chapter to the so-called 0-dimensional case, which is an assemblage of N cells communicating by diffusion and supplied with substrates from an external reservoir. This "zero-dimensional" system
INTRODUCTIOK
147
is governed by the equations: dS1 dt
+
s1 - s2
dsi
af- - s 1. - 1
+
A[palF(sl) -(so-sl)l = 0
+ 2si - s .
1+ 1
+
A[paiF(si)
-d -s ~ + s h + A[pa F ( s dt ss-l N N
-
-(sO-sN)]
(s0 - s 1. ) l = 0 (25i5N-1) =
0
and
d t + B(-ai-l
(5.2)
__
t
2a. - a. ) 1 l+l
t
A oaiF(si) - a(ao-ai)]
=
0
(Zzi5N-1 )
-
a(ao-aN)l
=
0
where s . (resp. a.) denotes the substrate S (resp. cosubstrate A) concentrat:on in thelith cell (1 5 i 5 N), s o and a . are the concentrations of S and .A in the reservoir, a, 6 , p are positive parameters and : F s)
(5.3)
=
s/(l
+
s
+
ksZ)
We explain the modeling of the S-cells assemblage in Section 5.1. In particular we define the trivial steady-state s = . . . = sN = S and - aN = Z, S and Z being the one solution of: a = 1
...,
The stability of this trivial steady state is analyzed in Section 5.2. For an appropriate choice of the parameter values a , B , p , s o , and a o , the spectrum of the linearized operator consists only of real eigenvalues. \Ye take X as the bifurcation parameter, its physical meaning being that:
148
THE URICASE SYSTEM AND PATTERN FORlrlATION
is the ratio of OD and BT which are respectively the characteristic times of diffusion between cells and of transport of S from the reservoir to the cells. The main result of Section 5.2 is the existence of K - 1 bounded intervals In (1 5 n 5 N - 1 ) such that: (ij
if A belongs to at least one of these intervals, then the trivial steady-state (5,a) is unstable, and
(iij if A is exterior to all these intervals, then (<,iT)
is stable.
Then we see in Section 5.3 that a stable pattern (or structure) is triggered off by random disturbances as X crosses a critical value separating the regions of stability and instability, that is the trivial steady state transmits its stability to an emerging non-uniform steady state. Finally the numerical techniques described in the two preceding chapters are employed in Section 5.4 to yield a complete panorama of the system behavior. There are two reasons for studying this "zero-dimensional" case before n-dimensional cases (n = 1 , 2 or 3) in the next chapter. The first one is that the analysis is simpler. The second is that cell assemblages are good models for pattern formation in biology. Already in 1952 Turing [ I ] studied stationary waves on a ring of cells in order to explain the regeneration of the tentacles of Hydra and the appearance of the whorls of leaves of certain plants such as Woodruff. Since then, this pioneering paper is inspiring the research of those, biologists, biochemists, or biomathematicians, i
PRESENTATION OF THE URICASE SYSTEM 5.1
149
Presentation of the Uricase System
5 . 1 . 1 - Case of a single cell: We first consider a single cell or compartment containing an homogeneous solution of uricase, uric acid, and oxygen. Uricase is an enzyme which catalyzes the consumption of uric acid and oxygen according to the stoichiometric scheme:
uric acid
+
uricase oxygen.-*allantoin
t
other products.
Uric acid and oxygen will be called respectively the substrate S and the cosubstrate A , and their concentrations will be denoted [S] and [A]. Their reaction rate was already discussed in Section 1.2 and given in formula (1.18) :
Many reactions have similar rate expressions. For example Figure 5 . 2 shows the measured activity (i.e. rate expression) of the reaction:
ATP
PFK F ~ -products P
t
where PFK (phosphofructokinase) catalyzes the consumption of ATP (substrate S) and F6P (cosubstrate A). As for rate expression (5.6), we see that if [ A ] is held fixed the activity as a function of IS] is substrate inhibited, whereas if [S] is held fixed, it is an increasing function of [A], thus cosubstrate activated. However we restrict o u r attention to the precise and simple rate expression given in (5.6). Thus the S and A concentrations in a single cell evolve according to the equations:
where : F(s)
=
s/(l
+
1st
+
ks2)
and : s = [Sl/KS, a = [Al/KS, B R = K A / V M ,
k
=
Ks/~ss.
THE URICASE SYSTEM AND PATTERN FORMATION
150
> 4I>
-
I
Fig. 5.2
Of the two parameters which play a role in equations ( 5 . 7 ) , one, k , is determined by the kinetic properties of the enzyme reaction, whereas the other, OR, which is the reaction characteristic time, depends upon VM which is proportional to the enzyme concentration in the cell, and thus is at our disposal. The system represented by equations (5.7) has a rather poor behavior, both concentrations s and a converging to 0 as t + +-. In order to keep the activity of the system from becoming smaller and smaller, we can supply substrate and cosubstrate coming from an external reservoir through a membrane (Fig. 5.3.a:. The reservoir contains a well-stirred solution of S and A at fixed concentrations s o and ao, and the membrane is inactive, that is without enzyme, and only plays the role of transporting S and A from the reservoir to the reac-
PRESENTATION OF THE URICASE SYSTEM
151
tive medium with the rates: ds
(
so-s
'transport = D B - L'
(
da
'transport
C' R'
and
a0-a C ' R'
= D A - - L'
where DS (resp. DA) is the S ( r e s p . A) diffusion coefficient in the membrane, L ' the membrane thickness, C ' the contact area and VR the volume of the reacting solution in the cell. We define the characteristic time for transport of S from the reservoir to the cell to be:
where V' is the membrane volume, and a to be the ratio of A and S diffusion coefficients:
N cells Fig. 5.3 Then the equations governing the evolution of s and a in a s ngle cell fed from an external reservoir are: (5.8)
ds dt
-
- -
ds (=)reaction
R
aF(s)
+
1
T
(so
-
s)
THE URICASE SYSTEM AND PATTERN FORMATION
152
and (5.8) continued: da +
-
transport
R
aF(s)
+
-% (a -a).
'T
5 . 1 . 2 - The trivial steady state ( $ , a ) : The eventual steady states of the one-cell system (5.8) satisfy the algebraic equations:
paF(s)
- (so-s) = 0
paF(s)
-
(5.9) a(ao-a)
=
0
where :
It is fairly easy to express a as a function of s for each equation: a
=
(so-s)/[~F(s)1
a
=
aao/[a+pF(s)l
and to plot these two functions in the (s,a) phase-plane. Figure 5.4 shows these two curves represented respectively with dots and stars. These two curves admit at least one point of intersection, and we can choose the parameter values s o , ao, p and c1 so that: (5.11)
the system of equations (5.9) admits exactly one solution ( S , H ) and,
(5.12)
the steady-state system (5.8)
(?,a)
is a stable node for the dynamical
Let: (5.13)
R(s,a)
=
paF(s).
The property that ( ? , a ) is a stable node is equivalent to asserting that the two eigenvalues of the Jacobian matrix:
PRESEKTATION OF THE URICASE SYSTEM
1 +
Es
153
a'
(5.14)
are strictly positive.
- -- a~ Rs as
(?,a)
In ( 5 . 1 4 ) , =
gs and ga denote:
SF'(?),
=
aR aa
(S,B)
=
pF(S).
a
aC
a
S
Fig. 5.4 In particular we must have, for the trace tr(0) and the determinant det(0) of the Jacobian matrix: (5.15)
tr(0)
=
1
+
gs
+ N
+
ga
> 0
and
det(0)
=
a
+
Ea
+
'a
> 0.
Again the behavior of s and a is not very interesting; this time they converge towards s" and 5 as t ++m. 5.1.3 - N cells: Hohever the situation becomes much more interesting when, instead of only one cell, the system consists of a linear array of h' identical cells communicating by diffusion through membranes separating two neighboring cells, as pictured in Figure 5.3.b. By identical xe mean that, to simplify, each cell is characterized by the
154
THE URICASE SYSTEM AND PATTERN FORMATION
same characteristic times B R and BT, is fed from the same reservoir at concentrations s and ao, and is separated from its neighbors by inactive membranes having the same thickness L , the same contact area 1, thus the same volume V , and the same diffusion coefficients DS and DA. Then the governing equations for the concentrations si and a. in the ith cell ( 2 5 i 5 N-1) are: dsi dt
+
(5.16) dai +
\
where :
BD
=
1
eD
6
2si
-
1 s ~ + ~+ ]1 a.F(s.) - (so - s . ) R 1 'T
[-ai-l + 2ai
-
ai+l] +
[-5.
1-1
+
3 2 3,Q D ~ C nS v =
=
1 a.F(s.1 R 1
- 2 'T
(ao
-
=
0
a,) = 0
DA/Ds, and for i = 1 (resp. N) the
bracketed terms have to be replaced by s, - s 2 and a l - a2 (resp. sN - sh.- 1 and aN - aN-1)' Of course the system (5.16) admits the trivial solution s 1 = s 2 = = s = and a l = a = = aN = 2 , corresponding to the N cells functioning at the same levels of concentration as the previous But, even when the single cell at the stable steady state ( 2 , ; ) . parameter values s a ci and p are such that (5.11) and (5.12) hold, 0' 0' it is possible that ( ? , a ) be an unstable steady-state for the dynamica1 system (5.16). The simple fact of placing diffusion barriers destabilizes the steady-state ( ? , 2 ) . In order to study more closely this phenomenon we write down the governing equations of the uricase system in the slightly more abstract setting:
...
...
s
ds
+
As + X[R(s,a)
+
BAa + A[R[s,a)
- (so-s)]
=
0
(5.17) da dT where:
T =
- a(ao-a)]
t/eD, s = [sl sz
R(s,a) = [R(sl,al)
...
...
sNl T ,
R(sN,a,)l
0.
=
a
=
T,
[a, a2
...
aN]T ,
so - s = [so-s,
...
and A is the tridiagonal matrix the non-zero terms of which are:
so-sN1T ,
STABILITY OF THE TRIVIAL STEADY STATE
(5.18)
a l l = aNN
=
1,
a.. 11
=
2 , 2
155
5 i 5 N-1, ai,i-l = a i - l , i =
-1,
25i:N. For example in the case N = 5, which will be the subject of numerical simulations in Section 5.5,
(5.19)
A=I t
01-1
I
O
I
2 1-1 1 0
I
I
I
I
I
O 1-1
I
1-1
1
4
I
A is the N x N matrix of diffusion of S between neighboring cells. The bifurcation parameter A is the ratio:
of the characteristic times f o r intercellular diffusion and reservoircell diffusion. Finally the equilibrium solutions of (5.17) are given by the system of 2N nonlinear equations:
(5.20)
\..
+ A
R(s,a)
- (so-s)l
= 0
+
BAa All the other parameters being kept fixed, we wish to study the stability of the trivial steady state ( T , 1 ) as A varies. 5.2
Stability of the Trivial Steady State
We are in the well known case of an ordinary differential system: du
J J
+
f ( X , u ) = 0 , ~ ( t )E R 2N ,
possessing a solution u
independent of t and
A.
This equilibrium
156
THE URICASE SYSTEM AND PATTERN FORMATION
solution is stable when all the eigenvalues of the Jacobian matrix fU@,uo) have positive real part, and unstable when at least one of the eigenvalues has negative real part. The Jacobian matrix is:
where :
- -Rs
a~ (T,Z) as
Ra
and
(%,a).
=
We want to solve the eigenvalue problem: f u ( L u o ) @ = u@.
This problem can be written, if AX + h(Es+l)X
+
XkaY
$ =
[X
Y] 'I' ,
uX
=
(5.21)
ARsX
+
BAY + X(fia+a)Y
=
uY.
Now we introduce the N eigenvectors wn of A:
Since A is symmetric the corresponding eigenvalues vn are real. It is obvious that p o = 0 is an eigenvalue of A , associated with the eigenvector wo = [ l 1 . . . 1 1 T . Since A is semi-positive definite the other eigenvalues wn are strictly positive. Moreover the N eigenvectors w n constitute an orthogonal basis of R'Y , Thus X and Y admit the decomposition:
x =
N- 1
C
n=O
N- 1
Snwn
and
Y
=
Z
n=O
rlnwn
and the system (5.21) can be written:
N- 1
c
n=O and
[LInCn
+
X(iis+l)Sn
- USn
+
Aiiallnlwn
=
0
STABILITY OF THE TRIVIAL STEADY STATE
N- 1
c qs,
+
n=O
f3Pn?ln +
A(ia+4?ln -
uilnlwn
=
157
0.
(cn,nn) satisfying:
Thus the problem is to find N pairs
(5.22)
Bun
+
X(iia+a)
-
at least one of these pairs being different from (0,O). For ( 5 . 2 2 ) to admit a nontrivial solution we must have a vanishing determinant for the matrix of ( 5 . 2 2 ) , that is u must satisfy: (5.23)
uz
- tr(n)u
det(n)
+
=
0
where :
Thus for each value of n (0 5 n 5 N-1) one obtains two eigenvalues, un and u:, of fU(h,uo), solutions of the quadratic equation ( 5 . 2 3 ) ,
where MA is defined by the first equation of ( 5 . 2 2 ) with u and 'I, =: :M n'
+ X(ES+l)
-
u:
+
XfiaM:
=
=
cr:,
En=
0.
The essential observation, due t o G. Joly [31, is that, under the hypotheses: (5.24)
1 < a < B
and k s < 0, these eigenvalues are
real.
Indeed one verifies that:
1
THE URICASE SYSTEM AND PATTERN FORMATION
158
{(B-l)pn Suppose u,
+ '/:;[A
+(-fis)1/2J2+
A(a-1))
.
At least one of these eigenvalues is positive since:
< u:.
(using the fact that pn 1. 0 and the first inequality in (5.15)). Thus u, < 0 if and only if det(n) < 0. But det(n) = A2T(un/A) where T(z) is defined by:
Thus we are led to assume that this quadratic expression possesses two positive roots z ' and z":
T(z)
=
0 <
B(z-z')(z-z"),
Z' < Z"
so that:
Necessary and sufficient conditions for T to admit positive roots are:
[fia
+ ci
ciES
+
B(is+l)
-
fia +
B(iis+1)12
+ ci
> 0
iia
+ c1
> -4gisia
< 0
We shall see below how to find parameter values in order to satisfy these conditions, but already we can see that the third condition in (5.25) implies Rs < 0. Concluding, we have the: Proposition 5.1 - Under the hypotheses(5.11), (5.15), ( 5 . 2 4 ) , and (S.25), the trivial steady state ( 2 , s ) is stable provided:
STABILITY OF THE TRIVIAL STEADY STATE
A @
159
N- 1
u Tn
n= 1
and is unstable as soon as:
A
N- 1 E
u
n= 1
In.
Choice of the parameters: Parameters k and 6 are given. It is not obvious that the other parameters s o , ao, a and p can be chosen so that ( 5 . 1 1 ) , ( 5 . 1 5 ) , ( 5 . 2 4 ) , and ( 5 . 2 5 ) hold. It is possible to obtain such a set of parameters by the following procedure. ( 5 . 2 5 ) is equivalent to the fact that the polynomial T can be written: T(z) = B(z-z') (z-z"),
0 < z' <
2"
which itself is equivalent to: 6(kS+l) +
iia
+
a = -B(Z'+Z"),
a(ks+l)
t
iia
=
BZ'Z"
or
..
As a by-product we see that we must have a < We know that: R(s,a) = paF(s) with F(s)
= s/(l
in order that Ra > 0.
+ s
+
ks').
a Now take values for z ' , z " , ? , s such that 0 < z ' < z " , 1 < c1 < B and F ' ( S ) < 0 . From the second equation in ( 5 . 2 6 ) we know the value of Ea = pF(S), whence p , and from the first one the value of GS = pZF'(S), whence 8. By using equations ( 5 . 4 ) we obtain s o and a Then we check whether the first condition in ( 5 . 1 5 ) is satisfied and at last by considering the representative curves of:
.
a
=
(so-s)/(pF(s))
and
a = aao/(a
we readily see whether ( 5 . 1 1 ) holds.
+
pF(s))
THE URICASE SYSTEM AND PATTERN FORMATION
160
Bifurcation from the Trivial Steady State
5.3
Let f : R+ x RZN
-+
RZN be the mapping:
I
As + A(R(s,a)
f(X,u)
where u
=
[I]
=
BAa
.
+
-
A(R(s,a)
We shall denote u o
(so-s))
- a(ao-a)) =
[:]
I
the trivial steady state
and A. one of the endpoints of the interval In (1 5 n 5 N-1) : X o = pn/z" or pn/z'. We verify that ( A o , u o ) is a bifurcation point on the branch of trivial solutions [ ( A , u o ) , X E R'].
where :
is that eigenvector of fu associated to u, = 0 and @:: is the corresponding eigenvector of fi::, defined by:
(iii)
(iv)
fi
=
[
R(s,5)
-
R(5,H)
- a(ao-5)
(so-?)
I=[ 1
]ER(f:)
=
[w:
.
The roots o f the bifurcation equation are distinct.
Indeed if we parametrize the trivial branch by:
N-w"'] n n
BIFURCATION FROM THE TRIVIAL STEADY STATE X(t)
=
t , u(t)
=
161
uo
a first root of the bifurcation equation, ( a o , a , ) , is determined by:
A'(Xo)
= a.
u ' ( h o ) = ao$o
1,
=
But @ o = 0 since fi = 0. Thus a 1 the bifurcation equation we have: aa;
+
and we obtain c
Zbaoal =
+
ca;
=
=
0.
+
al$ =
Since a.
0
and a, must satisfy
0
0 , which is easily found directly, since:
The bifurcation equation can be written: ;'a
+
2bSOS1 = 0
and the second root (6, ed b f 0. But:
=
a, B 1 = -2b) is distinct from ( 1 , O ) provid-
a n d , from ( 5 . 2 2 ) with X
=
Ao,o
=
o n = 0 , En = 1 , and 'ln =
where I is the N x N unit matrix.
Mn,
THE URICASE SYSTEM AND PATTERN FORMATION
162
Since:
-
-
B(Rs+l) + Ra
+
~1
=
-B(z'
+
z"),
Thus if the roots z ' and z " of the quadratic expression T are distinct, so are the roots of the bifurcation equation. Moreover the bifurcation will be one-sided if a = 0 , which depends upon whether:
or not, since, as is easily checked:
Once we have verified that (Ao,uo) is a bifurcation point, we can exa mine further the behavior of the steady state as A crosses A. and the trivial steady state uo loses stability. For example in the case N = 5 (Figure 5 . 5 ) which will be completely explored in the next section, this occurs at points A,B,C, and H. At points A and B a, changes sign, it is negative between A and B, and positive outside. At point C u i changes sign, and at H u i changes sign. But i t results from the discussion of Section 4.3 that the bifurcating branches inherit the stability of the trivial branch. Voreover if the tangent at (Xo,uo) to the bifurcating branch is defined by: A;
= Po = 0,
u;
=
B1@,
it f o l l o w s that, at least in the vicinity of the bifurcation point, is roughly proportional to:
u - u
Thus the vectors s
-
S and a - 2 are roughly proportional to w n , that
NUMERICAL EXPLORATION OF THE STEADY STATES
163
is out of all the components of the "thermal noise" which cause deviation of the state from the trivial state as the latter becomes unstable, only one is amplified, the one corresponding to x The system acts as a filter which selects, in tile vicinity of .A and B , only the mode w,, in the vicinity of C w 2 and in the vicinity of H w4.
.
Due to the ever present small fluctuations in the concentrations of S and A in the cells, there is no chance for the trivial state ( 2 , s ) to have a physical existence as h crosses one of the critical values X o , and the fluctuations trigger off a new stable steady state characterized by unequal concentrations in the cells.
Fig. 5 . 5
5.4
Xumerical Exploration of the Steady States [ 4 ]
It is evident that although the previous analysis gives us some infor-
164
THE URICASE SYSTEM AND PATTERN FORMATION
mation about the bifurcations from the curve of trivial steady states, however i t is purely local and restricted to the behavior of the bifurcated branches near the bifurcation points. For example, we know for a given point (X,uo) on the trivial branch, how many negative eigenvalues lie in the spectrum of fU(A,uo). It is equal to the number of intervals In to which X belongs. As this number of negative eigenvalues is transferred from the trivial branch to the bifurcating branch, we thus know the kind of stability of the steady states on that branch, at least in the vicinity of the bifurcation point. But far from the bifurcation the behavior of the branch is unknown. It is the reason why it is necessary, if we want to know more about the whole connected set of steady state solutions to f(A,uj = 0 , to employ the numerical techniques described in the two previous chapters. Recall that all we need is a subprogram providing f ( X , u ) and fu(X,u) when (A,uj is given. We will now give the conditions and results of the numerical experi. = 7 9 . 2 , CY = 1.45, i? = 5 , and ments. We have taken s o = 1 0 2 . 5 , a p = 13. There exists a "trivial" solution for all values of A , at, approximately, s = S = 7.84 and a = Z = 1 3 . 9 . This defines the trivial branch. Figure 5.5 shows for N = 5 the entire connected component issued from (O,uoj, more precisely the projection of the solution onto the ( A , s Z ) plane for lack of any better means of visualizing a curve in an 11-dimensional space. We hasten to point out that the solution would look quite different, had we used ( A , s 3 ) for example. The vertical coordinate is s 2 . Stable The horizontal coordinate is A . solutions are shown using solid lines, while unstable solutions are indicated by dashed lines. Also indicated on the Figure are the numbers of negative eigenvalues of f ( A , U ) . u o Starting from X = 0 ue have indicated the trivial branch as a straight line on which appear the bifurcation points A,B,C,D,E,F,G,and H corresponding to the eigenvectors w 1 (A and B), w2 (C and F) ,w3 (D and G), and w4 (E and H). The bifurcation points are indicated by 1 . 1 .
At each of these points there is a primary bifurcating branch of solutions. The first one has a relatively simple structure, being a closed loop passing through A and B. The bifurcation is unilateral
NUMERICAL EXPLORATION OF THE STEADY STATES
165
and there is an exchange of stability. Between B and C the trivial state is stable. At C there is another bifurcation. Again the branch forms a closed loop, intersecting the trivial branch at E, F, and H. There are 4 secondary bifurcation points, I, J , K, L on this closed loop. The bifurcation is forward at I so that the resulting bifurcation branch is stable. On the contrary, that at J is reverse, and the secondary branch is unstable. I and J are connected to distinct bifurcation branches. There is a third forward bifurcation at D. It becoming negative. Since the trivial solution is corresponds to already unstable with u i < 0 , the bifurcating branch is unstable with one negative eigenvalue in the spectrum of the Jacobian matrix. Extending this branch in either direction there is a turning point beyond which the solutions are again stable. It is this branch at which the secondary bifurcation at I is attached. The branch emanating from J belongs to none of the primary bifurcations. Following it from J we encounter a turning point on both sides of the bifurcation at J , beyond which the solutions are stable for a considerable interval of A. Finally, at another turning point, the branch describes only unstable solutions, closing at the tertiary bifurcation point (in this sense) K. The branch of solutions CI encounters a secondary bifurcation point L which lies on a second bifurcation branch "belonging" to w3, that is bifurcating at G from the trivial steady state as u 3 crosses 0. This branch contains only unstable solutions.
ui
The continuation of the curve CIL coincides with the branch which bifurcates at H from the trivial branch. H is one of the points on the trivial branch where u i changes sign. The other one is E but, before attaining it, the curve CILHKFE encounters first the bifurcation point K, which is a secondary bifurcation point in this sense, then the bifurcation point F. F is one of the points on the trivial branch where ui changes sign, the other being C. .Thus we see that i t was impossible to forecast the global behavior of the branch bifurcating from C , neither that of any bifurcating branch. However the numerical exploration of the whole connected set of solutions emanating from the trivial branch gives us a complete picture of the behavior of these solutions. Evidently, the response of the system to a variation of X in the interval (0,lO) depends upon the sense of variation. Here again if we follow by continuity the solutions of f(X,u) = 0 as X varies, as long as possible, then jumping when necessary from a
166
THE URICASE SYSTEM AND PATTERN FORMATION
stable branch to another one, we observe an hysteresis phenomenon when X first increases from 0 to 10, then decreases from 10 to 0. Lastly we can see the efficiency of the numerical method if we consider that the distance between the points E and F is of the order of 0.01, whereas the distance between points 0 and H is of the order of 10. The algorithm described the curve enlarged in the insert, automatically reducing the step length in the vicinity of the points E and F. The results obtained for S = 5 and the matrix (5.19) a’re represented in Figure 5.5. Figure 5.6 shows the results when the matrix (5.19) is replaced by the matrix:
(5.27)
AN
The advantage of matrix (5.27) is that its eigenvalues and correspond ing eigenvectors are known:
and that the matrix: (N-1)
’ AN
d 2 with no-flux boundary is a finite difference approximation of - dx conditions at x = 0 and x = 1 , when the interval is divided into K - 1 intervals.
’
Remark 5.1 - The biological interpretation of the above numerical and mathematical results is that assemblages of cells, in which enzyme reaction occurs and which communicate by diffusion with one another and with an external medium, can show multiple stable steady states. There are two ways for such a system to pass from one kind
NUMERICAL EXPLORATION OF THE STEADY STATES
167
of steady state to another. Either through a stability breaking bifurcation, where the stability of the previous steady state is transferred to the new one, or at a turning point, in the neighborhood of which a slight change of an environmental parameter may cause the system to jump to an entirely different kind of steady state, thus triggering dramatic modifications.
Fig. 5.6
THE URICASE SYSTEM AND PATTERN FORMATION
168 References
Turing, A.M., The chemical basis of morphogenesis, Phil. Trans. Roy. S O C . London, Vol. B 237 ( 1 9 5 2 1 3 7 - 7 2 . Bunow, B. and Colton, C.K., Multiple steady states in cellular arrays with hydrogen-ion activation kinetics, in: Thomas, D. and Kernevez, J.P. (eds.), Analysis and Control of Immobilized Enzyme Systems (North-Holland, Amsterdam, 1976). J o l y , G., Th6se d'Etat, Compibgne, 1980.
Bunow, B. and Kernevez, J.P., Numerical exploration of bifurcating branches of solutions to reaction-diffusion equations describing the kinetics of immobilized enzymes, (preprint).
CHAPTER 6 PATTERN FORMATION IN A MONOENZYME MEMBRANE
In this chapter we deal v th pattern formation in the s-a system, governed by the equations - (so-s)] = 0 in Q *t - A s + X[R(s,a at - gAa + X[R(s,a) - cr(ao-a)l = 0 (6.1)
R(s,a)
=
pas/(l
I s 1 + ks')
+
with zero-flux boundary conditions, where 0 is a bounded domain in Rn (n = 1 , Z , or 3), more precisely an uricase membrane, and the conditions on the parameters s o , ao, a , p , and B are the same as in the preceding chapter. In particular the following holds: there is exactly one solution, ( S , 5 ) , tem, R(S,2)
-
(so-:)
R(S,H)
-
a(a -2)
to the algebraic sys-
0
=
=
0
( S , 2 ) is a stable node for the dynamical system:
ds
R(s,a)
-
(so-s)
; da l i + R(s,a)
-
a(ao-a) = 0
+
=
0
plain that ( $ , a ) is a trivial steady-state solution of (6.1). By steady state solution of (6.1) we mean a solution to the station169
PATTERN FORMATION IN A MONOENZYME MEMBRANE
170
I
ary problem:
(6.6)
-As + A[R(s,a)
-Baa
+ A[R(s,a)
- (so-s)]
-
=
a(ao-a)l
in R
0 =
0
with zero-flux boundary conditions.
The important fact is that, as in the preceding chapter, this equilibrium solution ( ? , a ) may become unstable as A crosses a critical value. The stability analysis of ( ? , a ) is formally the same as in the preceding chapter, the only difference being that the .states (s(t), a(t)) of the system are no more belonging to a finite, but an infinite dimensional vector space.
As a consequence, the spectrum of the linearized operator consists of an infinite number of eigenvalues. Spatially non-uniform steadystate solutions inherit the stability of the trivial branch as A crosses those critical values at which an eigenvalue of this spectrum changes sign. These bifurcating solutions can be obtained by numerical methods, yielding spatially non-uniform stable steadystates even far from the bifurcation points. The phenomena of diffusion limitation combined with the regularity properties inherent in enzymes being universal characteristics of living systems, it seems reasonable to suppose that phenomena such as pattern formation in the s-a system might well play a role in many biological processes. More precisely, while it is doubtless correct that there is a great deal more to biological pattern formation than just chemical instabilities in reaction-diffusion systems, and by no means do we suppose that uricase has any role to play in the determination of embryonic morphology in any organism, however we feel that an extensive exploration of the types of patterns to be found in the s-a system may provide a basis for studying this problem. The origin of biological form is a problem which has challenged embryologists since the discipline was founded. Every biological organism starts life as a single cell. According to molecular biology, every detail of every phase of development is coded in the genetic material of the first cell, and embryological development procedes like clockwork. But he genetic program, by itself, is insufficient to specify all of the elements of development, and it is still a
INTRODUCTION
171
problem how the genetic information induces the formation of a pattern of differentiated cells. Some additional information is provided by the interaction of developing cells with their environment. A particular engaging suggestion is that part of this additional information is positional information, and that the information is coded in the form of concentration gradients before the actual differentiation may be observed. In a later stage, differentiation occurs according to the assigned positional information. How does this non-homogeneous positional information is established
is the essential problem. cal consequence.
The later development is only a biochemi-
One simple form of this hypothesis is very ancient in embryology: the organizing center. It is supposed that a small nucleus of cells produce a product which diffuses away, perhaps consumed by surrounding cells. The resulting field of concentration measures distance away from the organizing center. A more sophisticated version of the hypothesis was proposed by Turing [ l ] who pointed out that the interaction of diffusive transport and non-linear chemical kinetics of chemical substances, called morphogens, can lead to instability of their concentration field, with the result that the initially homogeneous system develops a finally non-homogeneous distribution. Since its proposal by Turing, many workers have attempted to explain morphogenesis using reaction-diffusion models. We give, in chronological order, references to this work [ 2 - 4 7 ] bhich are not exhaustive. Further referencc; can be found in the books [ 3 0 , 3 5 , 36, 4 3 1 . Recently, Kauffman, Shymko, and Trabert [ 4 0 1 have made a reactiondiffusion model of the sequential formation of compartmental boundaries in the embryonic differentiation of the wing imaginal disk of the fruitfly Drosophila melanogaster. Since the s-a system (6.1) falls within the framework of their model, we thus have a strong motivation for studying pattern formation in an artificial uricase membrane, that is the succession of stable spatially non uniform steady states, solutions of (6.6), as the bifurcation parameter A varies. The plan of the Chapter is as follows.
Pattern formation in fruitfly
172
PATTERN FORMATION IN A MONOENZYME MEMBRANE
Drosophila imaginal disks is explained in Section 6.1. GarciaBellido observations [48-511 and Kauffman's theory [ 4 0 1 are presented as a motivation for further study of the s-a system. The s-a system is described in Section 6.2 and its modeling yields the evolution equations (6.1) (in shortened form du/dt + f ( h , u ) = 0) and the stationary equations (6.6) (in shortened form f(X,u) = 0), a particular solution of which is a spatially uniform steady state ( T , i i ) satisfying (6.2), ( 6 . 3 ) , and (6.3). The linear stability analysis of this equilibrium solution u o = ( S , Z ) is performed in Section 6.3, where conditions for the eigenvalues of the Jacobian fu(X,uo) to be real are given, together with conditions for an eigenvalue to be negative. Finally we can completely analyse the stability of the trivial steadystate u o , mainly because this steady state is spatially uniform. Section 6.4 deals with the bifurcation analysis at those points ( X o , u ) where an eigenvalue of f (A , u ) crosses 0. The problem is u o o set in a more general framework than in the preceding chapters, namely the mapping: f : R + x X + Y involves Banach spaces X and Y. The s-a system fits this general framework and patterned solutions emanate from these bifurcation points (Xo,uo). How to calculate these spatially non-uniform steady states and the numerical results so obtained are indicated in Section 6.5, for the one dimensional case as well as for two-dimensional cases or diffusion reaction on a surface (involving the LaplaceBeltrami operator). Finally mathematical technical details are gi\,en in Section 6.6. In particular the spectrum of fU(A,uo) is carefully studied and a general linearized stability theorem is given, justifying the study presented in Section 6.3, and the properties of the Jacobian operator fu(Ao,uo) at the bifurcation point (Xo,uo) are shown justifying the analysis presented in Section 6.4. 6.1
Pattern Formation in Fruitfly Drosophila Imaginal Disks
Fruitfly Drosophila has been extensively studied by zoologists and the experimental embryology of this insect is one of the best knokn. The adult organs of the fly develop out of a number of approximately planar amorphous blocks of cells, called imaginal disks. There are
PATTERN FORMATION IN FRUITFLY DROSOPHILA IMAGINAL DISKS
173
the following pairs of disks, among others: wing, first leg, second leg, third leg, haltere, eye-antenna, and the single bilaterally fused genital disk. The disks differ from one another in their size and shape. As development procedes, the disks enlarge, while more or less retaining their original shape. We restrict ourselves to the wing disk for our explanations. 6.1.1 - Garcia-Bellido observations: The experiments of GarciaBellido consist in determining the ultimate position in the wing of cells in the wing disk which are marked by genetic intervention at various times during maturation. Garcia-Bellido and coworkers observed the formation of compartments, defined by the anatomic location of the progeny of early embryonic cells labelled genetically by X-irradiation. There are a series of compartmental lines which form, defined by the observation that descendants of primordial cells never cross the lines. The lines are formed sequentially, and successively subdivide the disk into progressively smaller and anatomically more specialized regions. The five observed compartmental boundaries are shown schematically in Figure 6.1 : (1 ) anterior-posterior : ( 2 ) dorsal -ventral : (3) wingthorax : (4) scutum-scutellum and postscutum-postscutellum : ( 5 ) proximal-distal wing. During metamorphosis, the wing disk folds along the dorsal-ventral line ( Z ) , and apposes dorsal and ventral thorax, while the wing everts. The u-ing imaginal disk is not the only one on which sequential compartmental lines were observed by GarciaBellido and coworkers. Figure 6.2 shows the schematic compartmental lines on the leg, genital, and haltere. 6.1.2 - The model of Kauffman, Shymko, and Trabert for morphogenesis Kauffman, Trabert, and Shymko [40] elaborated Turing's hypothesis into a qualitative model for the detailed development of the wing imaginal disk of the fruitfly Drosophila. They proposed a model involving spontaneous development of concentration patterns from the interaction of reaction and diffusion within the disk. They observed that the position of the compartmental lines on the wing imaginal disk was reminiscent of the nodal lines of the first few eigenfunctions of the Laplacian operator on the disk surface with Neumann boundary conditions: in Drosophila [ 4 0 ] :
174
PATTERN FORMATION IN A MONOENZYME MEMBRANE
VENTRAL
b2v:E Fig. 6.1
LEG
ANTENNA
GENITAL Fig. 6.2
PATTERN FORMATION IN FRUITFLY DROSOPHILA IMAGINAL DISKS
awn - 0 -
175
on an
Here - A is the Laplacian operator on the disk 0 an 2 av is the (outward) normal derivative at its boundary an, un are the eigenvalues, wn the corresponding eigenvectors, n = 0 , 1 , 2 , For n = 0 , un = 0 and w z 1 . The nodal lines of wn are those lines along which w = 0. This apparently arcane similarity was the basis of suggesting that the compartmental lines were formed by cells responding to a concentration field which developed as a result of an instability in a reaction diffusion system:
... .
1'
(6.8)
=
F(X,Y)
+
D, V 2 X
=
G(X,Y)
+
D 2 v2y
with no flux boundary conditions "Such a model can have the property that its spatially uniform steady state, X = Xo, Y = Y 0 [defined by the simultaneous solution of F(Xo,Yo) = 0 , G(Xo,Yo) = 01 is stable to all spatially distributed perturbations except those perturbations whose spatial wavelengths fall in a narrow range around some specific, characteristic wavelength, lo ." In a linear approximation, the form of the concentration instability would resemble the eigenfunction of a Laplacinn, with high concentrations on one side of the nodal line, and low on the other. Which eigenfunction is selected for amplification depends on parameter values. Since the imaginal disk is growing the .parameters are changing, and i t is reasonable that a sequence of successive patterns should appear. The particular sequence of patterns to be observed depends upon the shape of the domain. In the model, as the size and shape of the disk vary , distinct chemical patterns resembling the eigenfunctions of the Laplacian emerge from the spatially homogeneous steady state and decay back to i t .
176
PATTERN FORMATION IN A MONOENZYME MEMBRANE
Since a critical feature of compartmental boundaries is that they arise in a well defined sequence as the imaginal disk grows in size, the concentration field (X,Y) would pass, as the imaginal disk enlarges, through a sequence of stable non-uniform steady states, inducing one commitment (for example anterior) in cells where X (or Y) concentration is above some threshold level, and the alternate commitment (posterior) in cells below threshold. This reaction-diffusion system, acting throughout development, generates a sequence of differently shaped chemical patterns. 6.1.3 - Sprey's observations [ 5 2 - 5 3 1 : Sprey observed the distribution of the enzyme aldehyde oxidase (AO) in the imaginal discs of some diptera, including Drosophila melanogaster. An illustration of his observations is given in Figure 6.3, which shows a clear-cut pattern of the enzyme in the left wing imaginal disc of Drosophila melanogaster. More generally Sprey observed the existence of a straight and sharp boundary between the AO-positive and AO-negative (i.e. with and without aldehyde oxidase) areas in the discs of all AO-positive flies. Interestingly, this boundary in Drosophila melanogaster coincides with a part of the borderline between the anterior and posterior developmental compartments described by Garcia-Bellido et al. [ 4 9 , 5 1 ] for the wing disc and by Steiner [ 5 4 ] for the leg disc. Also due to Sprey [ 5 5 ] are the photographs of Figure 6.4 and the drawings of Figure 6.5, which show actual imaginal discs. More precisely, from top to bottom, we have a Calliphora leg disc, a Calliphora wing disc and a Calliphora wing disc. It appears at once that the leg disc is not a flat disc! In fact i t is a simplification to consider an imaginal disc as a planar surface. The surface of an actual imaginal disc is more complicated, involving invaginations, thick layers and thin layers of cells, etc
...
Fig. 6.3
PATTERh’ FORMATION IN F R U I T F L Y D R O S O P H I L A IMAGIKAL DISKS
Fig. 6.4
1-7
PATTERN FORMATION IN A MONOENZYME VEMBRANE
178
.part of thin peripodial .1-5 dorsal segments
.
vent ra I w ingb Iade
.dorsa I wingblade .median cross ridg .thorax forming pa
:a terior p o s t e r A WING DISC .thorax forming part .median cross ridge Owing forming part
Fig. 6.5
DESCRIPTION OF THE S-A SYSTEM 6.2
179
Description of the S-A System
From the preceding section it results that reaction-diffusion could well play a non-negligible role in pattern formation. Kauffman's theory accounts for sequential compartment formation in Drosophila. Even if in the precise case of insect imaginal discs the morphogens X and Y occuring in (6.8) are hypothetic and unknown, there are various developmental systems in which the existence of morphogenetic The fact that gradients is well established experimentally [ 5 6 - 6 2 1 . imaginal discs are not planar surfaces is not very important since, as will result from our analysis, as soon as a given biochemical system shows pattern formation in the 0-dimensional case studied in the preceding chapter, i t is able to show patterns also in various configurations, in the 1 or 2-dimensional cases, or on non planar surfaces. T h u s i t is interesting to study, in a well defined context, pattern
formation due to diffusive instabilities in a biochemical system. Again, artificial enzyme membranes are a good tool to achieve this goal. 6.2.1 - g d e l i n g of the S-A system: Figure 6.6 shows an active layer, separated from a bulk solution by an unstirred layer. e l and e2 are the active and unstirred layers thicknesses. The cross section of these cylinders will not be in general circular, but will be taken to be a region shaped like the wing imaginal disc of Figure 6.1 for example. The active layer is an artificial enzymatically active membrane. The enzyme is immobilized i>,ithinthe membrane. Its kinetics are those of the enzyme uricase, which catalyzes an irreversible reaction involving the two substrates oxygen and uric acid, here denoted by A and S. In an appropriate regime of concentrations, an empirical expression for the reaction rate is:
where S is the concentration of the substrate uric acid, and A the concentration of the co-substrate oxygen, while VM, KA, KS, and KSS are all constants characteristic of the enzyme defined by kinetic
PATTERN FORMATION IN A MONOENZYME MEMBRANE
180
study in free solution.
=2
t
Fig. 6.6
L is some length related to the size of the membrane (diameter of its cross section for example, i.e. maximum distance between two points of this cross section). We suppose that el is very small compared to L, so that transport in the narrow dimension is very rapid. Within a cross section of the active layer, however, diffusion is slow enough to permit significant concentration gradients to b e established. The modeling assumes that in the active layer the concentrations S and A depend only upon x and y I coordinates in a Cartesian frame taken in a cross section. The bulk solution is a nutrient reservoir containing the two substrates at fixed concentrations S o and Ao. We suppose that eZ is very small compared to L. A Fick expression for transport through the thickness of the unstirred layer is appropriate because the layer is assumed to be so thin that the concentration profile in that direction is always in a stationary state with respect to the reservoir concentrations. The equations of material balance state the condition of conservation f o r each of the substrates in this system:
-a s-
- DS AS
- Jr
+
D&(So-S)/(ele2)
:t
=
DA AA
- Jr
+
Di(Ao-A)/(e,ez)
at
(6.9)
-
DESCRIPTION OF THE S-A SYSTEM
181
where DS and DA (resp. DA and DA) are substrate (resp. cosubstrate) diffusion coefficients in the active and inactive layers. We now nondimensionalize (6.9) by writing:
i
s
BD
a = A/KS,
S/KS,
=
BT
L2/DS,
=
P = eT/eR, A =
ti
=
e 1 e 2 /D’ S’
B = DL/DA
DA/Ds,
ci =
=
t/ e D ,
a .
so = So/KS,
‘R
= KA/V?I
Ao/KS,
=
’
,
B D / BT ’
XI =
X/L, y ’
=
Y/L.
\ If we substitute the non-dimensional variables t ’ , x ’ , y ’ into (6.9) and for convenience drop the primes we get:
(6.11)
_ as at
AS +
aat _ a
PAa
~ [ ~ ( s , a )- ( s 0 - s ) l = +
A[R(s,a)
-
a(ao-a)l
+
ks’),
o = 0
where : R(s,a)
(6.12)
= pas/(l
+
s
(k = Ks/Kss).
This biochemical system occurs in a bounded domain R with no flux of reactants S and A through the boundaries: and
as - 0 av
(6.13)
aa -av
on
afi
av
. the outward normal derivative on the boundary afi of R where a is In other words: (6.14)
VS-U
=
0
and
Va-w
=
0
where Vs is the gradient of s and v is the outwa d unit normal. Note that here 0 5 x,y 5 1 and that the domain s ze L has been incorporated into the parameter A . Thus an isotropic growth in the cross section of this system, the
182
PATTERN FORMATION IN A MOKOENZYME MEMBRANE
shape of this cross section remaining geometrically similar, will appear in our model as an increase of the parameter A , the other parameters being unchanged and in particular the scaled domain R being the same. Note also the similarity with the model of the preceding chapter. The only difference between equations (5.17) and equations (6.1) is that A has been replaced by - A . As in Chapter 5 , three characteristic times play a fundamental role: eD, time of diffusion within the active layer, eT, time of transport of S from the reservoir to the active layer, and BR, reaction time, If a dimension of R is so narrow that transport in this dimension is very rapid, the system reduces to a one-dimensional immobilized uricase ribbon, governed by the equations: st
- sxx
X[R(s,a)
+
at - Baxx
+
s,(O,t)
sx(l,t)
=
- ( s o - s ) ] = 0,
A[R(s,a)
R(s,a) = oas/(l
+
=
- a(ao-a)]
ax(O,t)
=
0 < x < 1, t > 0,
= 0,
ax(l,t) = 0 ,
s + ks2).
Such a model corresponds to situations where there exists a privileged direction. Up to now we said nothing about the initial conditions which are necessary for the problem (6.1) to be well posed. In fact any regime of concentrations is conceivable, but we shall be mainly interested by the "trivial" (time and space independent, to be studied next) solution subject to a small, random perturbation. Equations 6.2.2 - The spatially uniform steady-state u o = ( 2 , Z ) : (6.1) always possess at least one trivial solution s = 2 , a = 2 which is constant in both time and space. More precisely from the boundary conditions i t is clear that there will exist a time and space independent solution ( ? , a ) if: s
(6.16)
0
-
S
-
plF(S)
=
a(ao - 2 ) - plF(S)
0 = 0
LINEAR STABILITY ANALYSIS
183
There may be multiple solutions to these equations. We choose the parameters ( s o , ao, a and p ) so as to avoid this possibility, and to have a set of nulclines for the space-independent form of differential equations (6.5) similar to that shown in Figure (5.4). Moreover we choose these parameters for ( S , 1 ) to be a stable node for the dynamical system (6.5). A convenient set of parameters, already adopted in Chapter 5 , is : s
102.5,
=
0
a .
79.2, p = 13, and a = 1.35, for which
=
(6.17)
T
=
8 and 1
14.
=
We have already seen in Chapter 5 that, provided a > 1, (S,?i) stable node for the dynamical system (6.5) if and only if:
(6.18)
i
det(0)
-
where Rs 6.3
=
aR as
=
cigs
(?,a)
+
a
+
ga
= p?iF'(S)
is a
> 0,
and
-
a~ (5,s) = p F ( S ) . Ra -- aa
Linear Stability Analys s of the Spatially Uniform Steady State u o = (S,1)
Thus ( ? , H ) is a stable equilibrium point for the dynamical system (6.5). Since (;,a) is also an equilibrium point for the dynamical system (6.1), the question is: is this equilibrium point stable ? The answer is: not always. More precisely, with an appropriate choice of the other parameters, it is possible that ( S , H ) be stable or not according to the value of the parameter A. As in the 0-dimensional case studied in Chapter 5, diffusion can be stability breaking. In order to investigate more closely the n-dimensional case (n = 1,2 or 3) we reformulate it under the form: (6.19) where:
du u
+
=
f(A,u) = 0 [s
a] T ,
du/dt = [ s t
atiT
and:
PATTERN FORMATION IN A MONOENZYME MEMBRANE
184
(6.20)
f(A,u)
=
I
-
As + A[R(s,a)
-
BAa
+
A[R(s,a)
- (so-S)l - a(ao-a)l
I
The interest o f formulation (6.19) is that i t is formally the same as in Chapter 5 , with A instead of - A . Evidently the essential difference is that now f is a mapping: f : R+xX-+Y where X and Y are infinite dimensional Hilbert spaces. cisely: (6.21)
Y
(6.22)
X = V x V
=
L2(Q) x L2(Q)
More pre-
and where
However, as in the finite dimensional case, an equilibrium point u o f (6.19) is stable if the eigenvalues of the Jacobian:
lie on the right-hand side of the imaginary axis in the complex plane, and is unstable if at least one of these eigenvalues lie in the left-hand side. The reason for the stability o f an equilibrium point u o of (6.19) to be governed by the spectrum of f ( X , u o ) and for this spectrum to be composed of eigenvalues will be given in Section 6.6. We restrict ourselves here to the study of these eigenvalues in the case of the trivial steady-state of the s-a system, u o = (?,2), which satisfies: (6.24)
f(A,uo)
=
0
for every
X
E
R+
Eigenvalues u and corresponding 6.3.1 - Eigenvalues o f fU(A,uo): eigenvectors 0 of fu(A,u ) are non trivial solutions ( I J , ~ )to the eigenproblem: (6.25)
fu(X,uo)O
= u0
LINEAR STABILITY ANALYSIS
185
J.A. Boa [ 1 2 , 2 3 ] studied a similar problem in the case of the Bruxellator. The procedure to study problem ( 6 . 2 5 ) will be formally the same as in the 0-dimensional case. First we define the eigenpairs (pn,wn) of - A on Q subject to Neumann boundary conditions on an:
i
-Awn
(6.26)
=
n
unwn,
=
...
0, 1 , 2 ,
and we decompose:
@ =
m
[:I
n=O ‘nwn m
1 n=O
=
nnwn
Hereinafter we assume the wn to be normalized: wi(x)dx
=
1.
Since : -A + X(jis+l)
Xii,
hi i s
-gA
+
h(iia+cc)
m
=
I2o
{rpn
0 is an eigenvector associated to the eigenvalue u if and only if there is, for at least one value of n , a non trivial solution to:
PATTERN FORMATION IN A MONOENZYME MEMBRANE
186
Therefore, exactly like in the 0-dimensional case, to each eigenvalue They are p n of - A correspond two eigenvalues u - and : IC of fU(A,O). the roots of the dispersion equation: (6.27)
uz
-
tr(n)u
det(n)
+
= 0
where tr(n) and det(n) are given in Section 5.2. Exactly like in the preceding chapter, these roots are real, provided Rs < 0 , ci > 1 , Associated eigenvectors @ - and @: are: B > I . Let u, < 0 : .
(6.29)
un
+
X(Rs+l)
Moreover, since X -
- u:
+
:A
+
=
a'M n = 0.
tr(n)
> 0 , at least
> 0.
6.3.2 - Conditions for u- to be negative: Following the same lines as in Chapter 5 , we arrive at the conclusion that:
where z ' and z" are the positive roots of the quadratic expression T defined in Section 5.2. Conditions for T to admit two positive roots are expressed in ( 5 . 2 5 ) . 6.3.3 - Conclusion: Under the hypotheses (6.2), ( 6 . 1 8 ) , ( 5 . 2 4 ) , and ( 5 . 2 5 ) , ( ? , a ) is a stable steady state of the dynamical system (6.1) provided X does not belong to any of the closed intervals T n , and is unstable if X belongs to at least one of these intervals.
It is worth pointing out that, in the one-dimensional case for example, we have : (6.30)
1-1,
=
n2n2,
w
=
cos nTix
and that, for n large enough, the intervals In overlap. ?lore precisely, as long as pn/z' < pn+,/z", there is, between In and In+,, an interval of stability for ( 2 , F i ) . This condition can be written:
LINEAR STABILITY ANALYSIS
(6.31)
n < ~/((Z"/Z')'/~
-
187
1).
But, for n larger than this value, the intervals I n overlap. This is shown in Figure 6.7, where interval In is denoted 1cn,dn[, and where it can be seenthat for n 2 2,the intervals In overlap.
Fig. 6.7 Figure 6.7 also shows that at the points cn and dn spatially non uniform solutions bifurcate from the trivial branch, represented by a straight line. These bifurcated profiles are characterized by n nodes (i.e. the profile of s intersects n times the horizontal s = S ) if they bifurcate at cn or dn (n = 1 , 2 , . . . ) . The object of the next section is to explain this kind of behavior. Finally note that the reason for the linearized stability analysis to be so simple is that we linearize about a space independent steady state. Thus the partial derivatives gs and Ea are constant, and the eigenvalues and eigenvectors of fU(X,uo) are easily related to those of - A .
PATTERN FORMATION IN A MONOENZYME MEMBRANE
188
6.4
Bifurcation Analysis
Before showing why bifurcation occurs in the s-a system we present a general framework within which it is practical to deal with bifurcation, at both the mathematical and numerical points of view. This framework is due to H.B. Keller [ 6 3 , 6 4 ] and described by D . W . Decker in [ 6 5 1 . Then we explain how this structure fits for the s-a system and indicate why bifurcation occurs from the "trivial" branch of the s-a system as A crosses the endpoints of intervals In. We do not intend to enter into mathematical details in this justification, and delay technicalities until Section 6 . 6 . 6.4.1 (6.32)
- General framework: Consider the equation: f(h,u)
=
0
where f is a twice continuously Fr6chet differentiable mapping of R + x X into Y. Here X and Y are real Banach spaces, with X c Y, the embedding being continuous. We suppose that (Ao,uo) is a solution of ( 6 . 3 2 ) possessing the following properties: (i)
the Fr6chet derivative :f = fu(Ao,uo) and its adjoint f:: a one-dimensional null space: a)
N(f:)
=
have
span {@I, Q EX;
(6.33)
Here Y:: denotes the dual space of Y. Further we assume the structure of the zero eigenvalue of :f such that : (6.33)
c) +::I$
to be
# 0
This is equivalent to demanding that this eigenvalue be simple, that is :
BIFURCATION ANALYSIS
(ii) (6.33)
We require the range of :f d)
R(f:)
to be closed and of codimension 1:
is closed, codim (R(f:))
It results from the fact that R(f:) (6.34)
a)
R(f;)
b)
Y
Y1
=
=
189
{ y E Y : $::y
=
1.
is closed that: = 0)
Thus : (6.34)
span {Q)@Yl.
=
On the other hand we have the decomposition: (6.34)
c)
X = span {$l@Xl
d)
X1 =
where : (6.34)
{XE
X : $::x
= 0).
(iii) Another requirement is that: (6.35)
$::fi = 0 ,
i.e.
fl
E
R(f:).
As a consequence there exists an unique (6.36)
f:Oo
+
f i = 0,
$::@
@
0
E
X such that:
= 0.
(iv) Finally we assume the roots of the bifurcation equation t o be distinct. More precisely, we recall that the bifurcation equation is : (6.37) where :
"5;
+ 2bS0C1
+
cs; = 0
PATTERN FORMATION I N A MONOENZYME MEMBRANE
190
and that the root ( 5 , , c 1 ) a5,
+
bCo # 0
is said to be isolated if:
if
5, f 0
if
5,
(6.39)
b f O
= 0.
It is easy to check the equivalence between the existence of distinct roots (i.e. distinct rays [(kC0,k<,); k ER]) and the property to be isolated for any of these roots. Then we have the following theorem, which is a particular case of a more general result proven by Decker [65] and Keller [641 in the case of multiple bifurcation (i.e. when dim N(f:) = codim R(fo) = m > 1). k
Theorem 6.1 (Keller and Decker) - Let f ( h , u ) be a C map (k 2 2) from R x X into Y. Let (Xo,uo) be a solution o f ( 6 . 3 2 ) for which fi satisfies ( 6 . 3 3 ) and fiER(f;). Let be an isolated root of the Bifurcation Equation,(6.37). Let v o be the unique solution of: a) f:vo
=
- [f:uOoOo
+Zf:XOoS:
+
f~x~A’I,
(6.40)
b ) v ” E X,
where :
and O o is defined in ( 6 . 3 6 ) .
Then f o r some Ic(E),~O(E),v(E)I
where :
E~
> 0 there exist unique Ck-‘ smooth functions
such that for all
I E ~<
E
~
:
BIFURCATION ANALYSIS
191
Note that the geometric meaning of 5, and 5 is that: (6.42)
X ' ( 0 ) = 5,
and
~ ' ( 0=) c o o o
+
50.
6.4.2 - Qplication to the s-a system: We apply theorem 6.1 to the mapping f defined in (6.20) and the pair ( X o , u o ) where u o = (2,a) is the trivial steady state of the s-a system and A an endpoint (cn or dn) of an interval In (n = 1,2,. . . ) . We review the hypotheses of this theorem. The first one, about the C 2 smoothness of f in both arguments, will be verified in Section 6.6. Property (6.33)a) is the very definition of A o . More precisely, for X = X there is an integer n E { 1 , 2 , . . . I such that u, = 0 , and the associated eigenvector $ - = 4 , defined by:
spans N ( f ; ) . Property (6.33) b) : Since:
each eigenvalue 0: of fU(X,uo) is also an eigenvalue of f;(h,uo), with the associated eigenvector:
where N'
is defined by:
In particular for h = X o , un- = 0 and N(f{::)
=
span { + I where:
192
PATTERN FORMATION IN A MONOENZYME MEMBRANE
-P r o p e r t y
( 6 . 3 3 ) c ) : S i n c e Y = L 2 ( n ) x L 2 ( n ) , Y:: c a n be i d e n t i f i e d w i t h Y a n d , s i n c e t h e s c a l a r p r o d u c t o f two e l e m e n t s u = ( u , , v , ) and v
=
(vl , v 2 ) of Y i s : (u,v)
=
In
[ u l ( x ) v l (XI
+
u2(x)v2(x)ldx
we h a v e :
From t h e d e f i n i t i o n s o f = so t h a t :
“Mi
RsNi,
Mi
i n ( 6 . 4 3 ) and
Ni
i n (6.46) it r e s u l t s t h a t
E q u a l i t y (1) u s e s t h e f a c t t h a t t h e determinant of t h e m a t r i x :
v a n i s h e s , e q u a l i t y ( 2 ) u s e s t h e d e f i n i t i o n of M- i n ( 6 . 4 3 ) , and n e q u a l i t i e s ( 3 ) a n d ( 4 ) come f r o m t h e d e f i n i t i o n o f t r ( n ) a n d t h e f a c t that tr(n) =
0:
+
u,
=
u+
.
BIFURCATION ANALYSIS
193
Thus zero is a simple eigenvalue of fi. Property (6.33) d) will be proven in Section 6.6. Property (6.35) results at once from the fact that:
and implies that the vector $ o defined by (6.36) is 0 , (6.47)
= 0.
C$o
Isolation of the roots of the bifurcation equation: the coefficients of the bifurcation equation are: (6.48)
a
=
$::f{u$C$
, b
$::fO 0 Uh
=
A s a consequence
and c = 0
Thus the bifurcation equation is: (6.49)
aE;; + ZbcOcl = 0
a root of which is 5, = 1 , 6, = 0. Since 5 # 0 , the condition for this root to be isolated is aS, + bSo = b # 0 , which is realized since, like in Chapter 5 , it can be shown that:
b = (
? Bpn ( 2
widx) BPn
+
' - z ")
-.
Ao(Ra+")
where z ' and z " are the (distinct) roots of the quadratic expression T defined in Section 5.2. This first root [ t o = 1, 5 , = 01 clearly corresponds to the trivial branch parametrized by A . Thus the second root 15, = a, 6, = -2bl corresponds to a bifurcating branch. More precisely:
___-Proposition v(E))
6.1: There exist unique C o smooth functions ( S ( E ) , such that for all / E I < E ~ :
S O ( ~ ) ,
PATTERN FORMATION IN A MONOENZYME MEMBRANE
194
f(X(E),U(E))
0,
=
where :
and : (6.50)
b) E ( 0 )
So(0) = a, v ( 0 ) = 0
= -2b,
Proof: All the hypotheses of Theorem 6.1 are verified, except the C 2 smoothness of f and the properties that the range of f o is closed and of codimension 1, which are verified in Section 6.6. Moreover, since $ = 0 and fix = 0 , we have: (6.51)
v o = 0.
It is easy to verify that, if u:: = ( u l , u 2 ) E Y::, v = ( v l , v ? ) E~ X and G then f&vw = (6ssv1w1 + isa(v,w2 + v2x1) + $aav2w2)[i] w = ( W ~ , W ? ) ~X, so that: u::fouuvw
/n(kssvlwl
=
+
...) (u,
+
u2)dx.
Therefore: a
$::fouu $I$
=
I / i l w~(x)dxl(l
=
+
Ni)(iss
and the bifurcation is one sided (i.e. X ' ( 0 ) (6.52)
/Rwi(x)dx
0
=
=
+
2Miksa
a
= 0)
+
Mn2kaa)
if:
.
In the case of a 2 - or 3-dimensional domain R with arbitrary shape, there is no reason for (6.52) to hold. However, in the 1-dimensional case ( 6 . 1 5 ) , where (see (6.30)) wn(x) = cos nnx,
[
1
wi(x)dx
=
0
A l s o when R has an axis of symmetry, say 0 case, suppose w is an eigenfunction:
-Aw(x,y)
=
pw(x,y),
2 av =
- y in the 2-dimensional
0 , / f i w 2 d ~= 1 ,
BIFURCATION AXALYSIS and consider v(x,y)
=
195
w(-x,y):
which, together with the boundary condition and the normalization: /nv2dx = 1, implies that v(x,y) = - C w ( x , y ) , i.e. w ( - x , y ) is the possibility for w to satisfy w(-x,y)
=
- + w ( x , y ) , so that there
=
-w(x,y).
In that case:
and (6.52) holds. The case Q = ] 0 , 1 [ is not unrealistic. It corresponds to the modeling of an ellipsoid with large axis ratio when one is interested in the first eigenmodes. One can thus explain the differentiation of regions in an insect egg by a pattern of concentrations in the privileged direction [40]. More generally, the onset of polarity [ 6 6 1 can be explained this way. The establishment of a pattern of the form: s(x) =
s
a(x) = B
+ ECOS
+ E
7Tx +
M; cos
O(E2)
~ T X+
0 < x < 1
O(E')
which corresponds, according to (6.50), to a value of A in the vicinity of .A = c , or d,, has been invoked to explain head-tail distinction. Of course a one-sided bifurcation at (cl,uo) for,example, gives rise to a stable pattern only if i t is forvard. If it is reverse the bifurcating states will be unstable, at least until a turning point is encountered. In order to see whether the bifurcation is forward or reverse we need the sign of A " ( 0 ) . But calculation of X"(0) is tedious and the result complicated. Thus we are one more time led to the conclusion that the best way to see what actually occurs is to employ numerical methods.
196 6.5
PATTERN FORMATION I N A MONOENZYME MEMBRANE Numerical Methods and Results
We now propose to obtain the patterns which appear sequentially as A increases. Recall that increasing A is equivalent to growth without deformation of the domain. Two kinds of numerical methods are at our disposal for obtaining the solutions of (6.6) as A varies.
(i) numerical methods for continuation and bifurcation of solutions as already employed in the 0-dimensional case in Chapter 5 ; they apply to finite difference or finite element discretizations of problem (6.6). Unfortunately these methods become very time consuming as the number of degrees of freedom increases. (ii) methods in which the steady states are obtained by solving the evolution equations (6.1) until one estimates that the system has attained a stable steady state. However these methods do not permit obtaining unstable steady-states. Since, anyway, our aim is to obtain sequential stable steady states of (6.1) as A increases, we will describe methods of the second type and their results. Since monotonically increasing values of A correspond to a spatial domain enlarging and remaining geometrically similar to the shape of R , a numerical experiment appropriate for following pattern formation in the s-a system is the following: We find the numerical solution of the initial boundary value problem (6.1) for a succession of values A and use at each step the steady state of the preceding step as an initial condition. If the step in A is small, this procedure is analogous to following the evolution of the solution on a growing domain, Everytime A is changed, the preceding steady-state is triggered off by random disturbances in order to have the whole range of spatial wavelengths in the Fourier expansion of the initial condition, In fact these environmental conditions, so that the system may undergo its sequential alterations, seem to be a good simulation of the ever present "thermal" noise in Nature. All the numerical experiments to be described hereinafter were per-
197
NUMERICAL METHODS AKD RESULTS formed with the following values of the parameters: (6.53)
s
=
102.5, a .
=
79.5, a. = 1.45,
p =
13, k
=
0.1, and B = 5,
satisfying the conditions (6.2), (6.18), (5.24), and ( 5 . 2 5 ) for pattern formation to occur. We (i.e. B. Bunow, M.C. Duban, G. Joly, and the Author) used the finite difference method in the 1-dimensional case, and the finite element method in the 2-dimensional case (either on a planar disc or on a more general surface). We discretize with respect to time according to the following scheme, in order to isolate the nonlinearities in the right-hand-side:
k being a time increment, and sn and an being approximations of
s
and a at time nk. Then we discretize (6.54) with respect to space, by the finite difference method in the 1-dimensional case and by the finite element method in the 2-dimensional case. The interest of the scheme (6.54) is that both equations are linear equations of the form: (-PA
+ r)u = f
(6.55) with no-flux boundary conditions where
and r are strictly positive constants and the linear operator r is the same at each time level. Thus for every value of X we only have to perform the triangularization of the matrix coming from the discretization once. -PA
p
+
We calculate sn and an for n = l,Z,,.. until we estimate that a steady state has been reached.
198
PATTERN FORMATION IN A MONOENZYME MEMBRANE
6.5.1 - _ One-dimensional _ _ _ _ _case: - ~ A finite difference approximation of: -pu"(x)
+
ru(x)
=
f(x),
o < x < 1 ,
(6.56) u ' ( 0 ) = u'(1) = 0
is given by:
where: h = 1/N is a space increment and u . is an approximation of u(ih). We have indicated in Chapter 1 how to solve such a linear system. This algorithm (Gauss elimination for a tridiagonal matrix) For every value of X involves coefficients Ei and Fi (0 5 i 5 N). we have to perform the calculation of the coefficients Ei only once. Results are shown in Figures 6.7, 6.8, 6.9, and 6.10. Figure 6.7 (communicated by M . C . Duban) schematically shows the non uniform steady states (N.U.S.S.) as X 'monotonically increases (top), then decreases (bottom). The abscissa is X and the ordinate, not indicated, is some quantity related to the steady state, s ( 1 / 2 ) or s'(1) for example.
In the figure, heavy lines are used for stable steady states obtained by a continuous increase in X (top) and a continuous decrease of X (bottom). Dashed lines are used for unstable solutions. Recall that in the interval (cn,dn), ( 5 , s ) is unstable, at least because u; < 0. Thus the uniform steady-state (U.S.S.) is stable for 0 < X < c 1 and d , < A < c 2 , and unstable elsewhere. Continuous increases in X produce a family of solutions which are generally smoothly related one to the next, although there are some discontinuous jumps. The numbers n = 1, 2 , , . . , 10 indicate to which interval (cn,dn) the branch of solutions "belongs", and also the number of nodes in the concentration profiles. Since s and a profiles l o o k the same, we have represented the s profiles only, for n = 1 , 2 , 3, and 8 . Starting from X = 0 , s = S , a = Z , the U.S.S. is stable until
NUMERICAL METHODS AND RESULTS
199
A = c l . Then a polar pattern arises, corresponding to n = 1 , one node, and with the concentrations of s above 3 in one half of the spatial domain, and below S in the other half. It is this kind of pattern which is sometimes invoked to explain head-tail differentiation. A rather uncommon feature in bifurcation diagrams is that this pattern vanishes as A crosses d,, and the system comes back to the trivial steady-state, which is stable again for d, < X < c2. For the particular parameter values adopted here the intervals (cn, dn) overlap for n 2 2. But it is possible to choose these values so that the intervals (cn,dn) overlap only for n 2 n o z 2. As an example for: (6.58)
a
1.45, k
=
=
0.1,
s
=
99.2, . a
=
76.4,
p = 12.2,
and 8,
=
5
we have : (6.59)
z'
=
0.45, z "
=
0.74, and no
=
4.
Coming back to Figure 6.7, for X > c 2 the U.S.S. is no more stable, and we observe a sequence of more and more complex stable patterns with 2, then 3 , ..., 8 , nodes. At evidence the transition from n to n+l nodes is not continuous and there are jumps from each kind of pattern to the following.
...
Now if A is allowed to decrease, the system passes through stable steady states which are not necessarily the same as in the ascending phase (Figure 6.7, bottom). There is an hysteresis phenomenon, due to the existence of multiple stable steady states, Figure 6.8 shows the profile of s for n = 5, i.e. with n nodes. Note the large amplitude of the deviation from 3 . Figures =and 6.10 (communicated by M.C. Duban) 'show the s - patterns during the ascending phase for A = 20, 79, 120, 250, and 400. Remark that, although we are far from the bifurcation point and the deviations from S are large, the profiles still resemble cosine curves s = 3 + A cos n T x (n = 1, 2, 3, 4, and 5).
PATTERN FORMATION IN A MONOENZYME i’IEhIBRANE
S
15
10
5
C Fig. 6.8
:
1
0
1 Flg. 6.9
x
NUMERICAL METHODS AND RESULTS
201
Fig. 6.10 6.5.2 - Two-dimensional case: To solve (6.55) on a planar disk R we discretize with respect to space by the finite element method. Let us recall briefly how the finite element method works in this case. We subdivide the spatial domain R into M elements Rm (lLmiM) on each of which any function u(x,y) is approximated by a polynomial expression. Figures 6.11-6.15 show such a subdivision in quadrilateral elements of spatial domains having the shape respectively of a wing disk, an eye-antenna disk, a leg disk, a genital disk, and an ellipse. The elements are 8 nodes quadrilaterals, on which piecewise biquadratic expressions are determined by their values u i at nodes j (1 < i < 8) and are of the form:
where wi(x,y) is the i-th “basis function”: it equals 1 at node i, 0 at other nodes, and is different from zero only in the interior of the elements, R m , of which node i is a member.
202
PATTERN FORMATION IN A MONOENZYME MEMBRANE
Depending on the complexity of the domain, from 25 to 35 elements suffice to describe it, with between 100 and 150 total degrees of freedom in all. Without entering into details, for which we refer to Bathe and Wilson [ 6 7 ] , and Strang and Fix [ 6 8 ] , let us say that an equation like (6.55) is approximated by : (uK
+
rM)U = MF
where K is the "stiffness" matrix, (6.60)
K.. 1J
//fi(Vw.)
=
J
T (Vwi)dxdy ,
M is the "mass" matrix,
= ( u 1 , u 2 , . . . , u ~ is ) the ~ vector of (unknown) degrees of freedom and F = (fl,f2, . . . , fW)T is the vector of (known) nodal values of f.
u
Figure
6.11 shows the discretization which was employed for the study of the wing imaginal disk. 33 elements with a total of 122 nodal points were used. For those who are familiar with finite difference methods for partial differential equations, this discretization seems very coarse. However, the smoothness of the basis functions in the finite element method is such that a coarse discretization is satisfactory as long as the maximum spatial frequency in the exact solution is low enough. Coming back to (6.54)we have to solve, at each step of time, in order to calculate sn+l and an+' from sn and an, two linear systems:
where: K1 = K K2
=
pK
+
(X+l/k)M + (aX+l/k)M
NUMERICAL METHODS AND RESULTS
203
while:
( F ? ) ~ = x[s0 -
n n R(s~,~~)I
For every value of A we only have to assemble the two matrices K and and to perform the triangularizations of K 1 and K 2 once andfor all time steps. For every value of A we take as starting values so,ao, the steady state obtained for the preceding value of X , slightly perturbed by random noise, and calculate sn and an for n = 1 , 2 , . . until we estimate that a steady state has been reached again. M
.
Results - We have already shown in Figure 5.1 a selection of the solutions obtained in this manner on an ellipse. Since both s and a show the same concentration pattern, only s needs to be represented. Here as well as in the subsequent figures, darker plotting represent larger values. These patterns, obtained by G. Joly [ 4 6 1 , are the stable steady states, as X increases, for X = 0.3, 0 . 6 , O . ? , 2 . , 3.5, and 5 , (top to bottom, right to left).
Fig. 6.11
204
PATTERS FORMATION IN A MONOENZYME MEMBRANE
Fig. 6 . 1 2
Fig. 6.13
NUMERICAL METHODS AND RESULTS
Fig. 6.14
Fig. 6.15
205
206
PATTERN FORMATION IN A MONOENZYME MEMBRANE
FQures 6.16 and 6.17 show sequential patterns of s concentration on --------the wing disk for A = 1, 8 , 10, 14, 18, 26, 32, and 36 (Fig. 6.16), and X = 42, 46, 62, 70, 72, and 90 (Fig. 6.17). Successive patterns are displayed across the top row from left to right, then across the next row, and so on. The resulting patterns become more and more complex as X increases. Figures 6.18 and 6.19 show similarly sequential patterning on an eye antenna disk shaped domain, as X increases; the patterns corresponding to X = 2, 20, 28 (top row), 3 2 , 40, 52 (next row), 76, 96, 112 (last row of Figure 6.18), and 128, 148, 152, 164, 188, and.200 (Figure 6.19) have been represented. I n t h e s e examples [wing and e y e - a n t e n n a d i s k s ) the increment for X was 1 , and it was observed that, as X is increased, Zhe s o i u c : 1 u n ~ vary continuously through the sequence, with sometimes a jump between two values of A , exactly as in the 1-dimensional case.
Finally we present three figures communicated by B. Bunow [ 6 9 ] . -~ Figure 6.20 shows the discretization grid for the finite element method on the wing imaginal disk. Thirty quadrilateral elements with eight points were employed, for a total of 115 nodal points. This mesh was sufficiently fine to represent accurately a l l the solutions obtained on the wing disk. Deformations of the disk were introduced by applying an analytic distortion function to the coordinates of the nodal points, leaving their numbering and connectivity unchanged.
Figure 6.21 shows a sequence of solutions to equations (6.6) on wing disk. As the disk is enlarged, the solutions vary continuously through the sequence and are similar to those displayed in Figures 6.16 and 6.17. FQure _-__ 6.22 shows a sequence of solutions to equations (6.6) on an ellipse with the same parameter values.
Note,in particular, a deformed pattern of stripes. Stripe patterns have been implicated in blastoderm differentiation [ 7 0 , 3 3 ] . On other domains, the same parameter values can produce quite different patterns, including herring-bones, bull's eyes, chevron arrays, and bifurcating patterns resembling leaf veination.
KUMERICAL METHODS AND RESULTS
. . . ..
Fig. 6.16
207
208
P A T T E R N FORMATION I N A MONOENZYME MEMBRANE
Fig. 6.17
NUMERICAL METHODS AND RESULTS
Fig. 6.18
209
210
PATTERN FORMATION IN
4 410hOENZYME MEMBRANE
Fig. 6.19
NUMERICAL METHODS AND RESULTS
21 1
4
37
109
F i g . 6.20
38
39
40
41
21 2
P A T T E R h FORMATION IK A MONOENZYME MEMBRANE
W
NUMERICAL METHODS AND R E S U L T S
21 3
N N
iD
w
i
L
PATTERK FORMATION IN .$ MONOENZYME MEMBRANE
214
6 . 5 . 3 - Case o f a s u r f a c e : The s - a s y s t e m more g e n e r a l l y c a n r e p r e s e n t d i f f u s i o n a n d enzyme r e a c t i o n on a s u r f a c e S s e p a r a t e d f r o m a n e x t e r n a l medium by a b o u n d a r y l a y e r . The L a p l a c e - B e l t r a m i o p e r a t o r A B i s t h e n a p p r o p r i a t e t o r e p r e s e n t d i f f u s i o n on t h e s u r f a c e , and t h e
s - a system equations a r e :
st
-
ABs
at
-
BABa
X[R(s,a)
+
-
(so-s)]
= 0
(6.62) A[R(s,a)
+
-
a(ao-a)]
=
0
w i t h no boundary c o n d i t i o n s i f t h e s u r f a c e i s c l o s e d .
Let us r e c a l l
t h e d e f i n i t i o n of t h e Laplace-Beltrami o p e r a t o r : l o c a l l y t h e s u r f a c e S ( t h i n k o f a cucumber s u r f a c e ! ) c a n be mapped i n t o t h e i n t e r i o r o f - 1 < r , s < + 1 ( F i g u r e 6 . 2 3 ) and x,y,z c o o r d i n a t e s on S
a square:
can be expressed a s : (6.63)
x = x(u,,u2),
y = y(u1,u2),
=
z ( u 1 ,u21
+
2a12duldu2
Define : (6.64)
dZ2
=
(6.65)
a
alla22
=
dxZ
dy2 + d z 2
+
-
= alldu;
a;2
-1
+
az2du:,
NUMERICAL METHODS AND RESULTS
215
Then : (6.67)
dS = all2 du,du2
(6.68)
(6.69)
s =+1
6 =+1 2,. .:5 . l 8 * 6:
3" "7 '4 t s =-1
r =-1-l
Fig. 6.24 It is no more difficult to solve numerically the's-a system on a cucumber surface than on a planar surface by using the finite element method. However we have in mind the surface of an egg rather than the surface of a cucumber. Let us give some details on the calculation of the "stiffness" and "mass" matrices K and M in the case of a surface (a planar domain being a particular case). The surface is divided into curved 8-nodes quadrilaterals (see Figure 6.24). The nodes are defined by their coordinates (xi,yi,zi for node number i of the quadrilateral, 1 ' i58). We call Hi (r,s) the "shape functions":
216
PATTERN FORMATION I N A MONOENZYME MEMBRANE
(6.70)
H1(r,s)
=
1 -(l+r) (l+s) 4
1 1 - -H 2 5 (r,s) - -H 2 8(r,s)
H2(r,s)
=
1 -(l-r)(l+s) 4
1 (r,s) - -H 1 - -H 2 6 2 5(
H3(r,s)
=
1 -(l-r)(l-s) 4
1 - -H 2 7(
H4(r,s)
=
1 -(l+r)(l-s) 4
1 1 - 7H8(r,s) - -H 2 7 (r,s)
H,.(r,s)
=
Z(l-r2) 1 (l+s)
H6(r,s)
=
1 -(l-s2) (1-r) 2
H7(r,s)
=
- ( l - r 2 ) (1-s)
H8(r,s)
=
1 -(l-s2) (l+r) 2
r , ~ )
1 r , ~ -) -H (r,s) 2 6
1 2
The part of the surface delineated by the quadrilateral is approximated by the surface S(m) whose parametric representation is:
x (6.71)
i
8
=
1
i= 1
xiHi(r,s),y
8
=
2
i=l
yiHi(r,s),
8
z =
1
i= 1
ziHi(r,S)
-1 5 r , s 5 +l.
The elementary mass and stiffness matrices are:
where dS(m), B(m),
D ( m ) , and H(m)
are defined below.
To each pair (r,s) E [-1,+1] x [ - 1 , + 1 ] corresponds a point (x,y,z) on S(m) by equations (6.71) and we call: (6.73)
Him)(x,y,z)
=
Hi(r,s)
1 :.
i 5 8.
Thus Hjm) is a function of the point (x,y,z) on the surface S ( m ) . H(m) denotes the vector:
NUMERICAL METHODS AND RESULTS
H(m)
[Hl(m) ,Hi"),
=
. ..,
Him)].
Similarly we define:
. . . aHg ar (6.74)
2 as
...
H(m) and B(m) are 1 x 8 and 2 x 8 matrices defined on S L r n ) . Now, following equations (6.64)-(6.67) we define:
al,dr2 + 2a12drds + a22ds2
=
+ '
2'
all
=
V1 , a 1 2
+. [ iR1 v1
=
1
=
2'
V1V2, a Z 2 = V2
+
aHi aH. x. 1 ar , z y i ,
1 2 . 1
2 ar aH.
1
Then we define: lla22 - a;2 (6.76) aZ2/a,
a''
=
al,/a, a''
=
- a lz / a
21 7
21 8
(6.77)
PATTERN FORMATION IN A MONOENZYME MEMBRANE
dS(m)
=
a1l2drd ,
(6.78)
(6.79)
(6.80)
Using this method and the program SSPACE (subspace iteration method) o f Bathe and Wilson [ 6 7 ] , the first eigenvalues and eigenfunctions of the Laplace Beltrami operator - A B on the surface of an elongated ellipsoid with the form of an egg can be obtained (Bunow et.al. [ 6 9 1 ) . The first one corresponds to the onset of polarity (distinction headtail), the second one to the dorsal-ventral separation and the third one to the left-right differentiation (Figure 6.25). Figure 6.26, provided by B. Bunow, shows the sequence of eigenfunctions on the surface of a Drosophila egg at the cellular blastoderm stage. The left and right surfaces of the egg have been projected onto the sagittal plane. The first eigenfunction is shown at the bottom of the first two columns, From bottom to top of these two columns we have the first four eigenfunctions each column corresponding to a side of the egg. Across the 3rd and 4th columns, from bottom to top, we have the next five eigenfunctions. It results from our previous analysis of bifurcation from the uniform state in the s-a system that it is possible to find patterns similar to these first eigenfunctions when solving the nonlinear equations (6.62) instead of the eigenvalue problem. Surfaces where the diffusion coefficients are varying (because of a layer of cells of variable size for instance) model more closely the shape of the real imaginal discs (Figure 6.27). It is thus possible to solve the s-a system equations on such surfaces and study the sequence of stable patterns.
NUMERICAL METHODS AND R E S U L T S
TAIL
219
aA
a
DORSAL PART
VENTRAL PART
SIDE
Fig. 6.25
THIN CELLS LAYER
THICK 'CELLS LAYER Fig. 6.27
220
P A T T E R N FORMATION I N A MONOENZY!4E
MEMBRANE
I
F i g . 6.26
S U M E R I C A L METHODS AND R E S U L T S
221
If the diffusion coefficients are varying on a surface, we only have to multiply the entries ail of the matrix D(m) in (6.78) by an appropriate coefficient. This does not complicate the calculation of the stiffness matrix K. Thus the finite element method is a very efficient tool which allows us to solve the s-a system on any kind of surface S, planar or not, hrith any field of diffusion coefficients. Moreover i t enables us to calculate the first few eigenvalues and eigenfunctions: (6.81)
-
V(DVwi)
O c i c M
= p . ~ . 1
1’
on S with eventually no-flux boundary conditions if S is not closed. Here D = D(x) may be a variable coefficient. This knowledge of the first few eigenvalues and eigenfunctions is the basis of a method which saves computer time, and that we are going to describe now, namely Galerkin’s method. 6.5.4 - Galerkin’s method: Ke explain Galerkin’s method for the evolution problem (6.11 on the wing disk of Figure 6.11. Here we pick out the first I.! eigenvalues and eigenfunctions defined by (6.81). Of course ive work with time and space discretizations, but we prefer explaining the method on the original problem (6.1). We define approximations s), and ah! of s and a by: (6.82)
sM(t) and ahl(t)
t
span
{wo,wl, . . . , w,)
>lore precisely we restrict ourselves to the problem of finding the 2 ( > 1 + 1 ) functions ci(t) and qi(t), 0 2 i 5 M, such that:
/ nIs;
- Ash1
/n{a;
- @AaM+ X [ R ( s M , a M )
+
X[R(sM,aM)
- (so-sM)l)widx
=
0 O
-
a(ao-ahl)l?widx = 0
~
i
~
M
PATTERN FORMATION IN A MONOENZYME MEMBRANE
222
where of course the eigenfunctions wi are functions of the spatial variable x ,
Equations (6.83) are in fact a system of 2 ( M + 1 ) equations:
ordinary differential
Here we have used the normalization condition: (6.85)
R ~ i w j d x= 6 . .
11
.
This method has been employed by G. Joiy [ 7 i ] . Taking M = 5 , she finds, for A increasing continuously from 0 to 200, the same stable patterns with 2(M+1) = 1 2 degrees of freedom than with 2 x 122 = 244 degrees of freedom in the previous method ( 2 deOf grees of freedom si and ai for each of the 122 nodal points). course a preliminary calculation of the eigenpairs ( ~ 1. , w1 . ) is necessary, but they can be calculated once and for all for a lot of applications involving the solution of the evolution equations as previously described or the continuation of solutions as in the preceding chapter. For details we refer to [ 7 1 1 . 6.6
Mathematical Technical Details
We delayed until now the justification of some important mathematical properties. In the present Section we essentially investigate the properties of the operator:
MATHEMATICAL TECHNICAL DETAILS
-A
(6.86)
+
A(gs+l)
223
ARa
fU(A,uo) =
-
- @ A + A(Ra+a)
We first show that its spectrum is only composed of eigenvalues with finite multiplicities. Then we give a linarized stability theorem for the proof of which we refer to [ 7 2 ] , and apply it to the stability analysis of the trivial state ( $ , Z ) . Finally we prove that f is a twice continously differentiable mapping, the range of which is closed and of codimension 1 . 6.6.1 - Spectrum of fU(A,uo): We generalize here some of the methods employed by Meurant and Saut [ 3 4 ] in the study of bifurcation and stability in the Bruxellator. We are within the framework of Section 6.3, with X and Y defined by (6.21)-(6.23) and f : R+ x X + Y defined by (6.20). The spatial domain n is supposed to be an open bounded ,.) the scalar product in Y set in Rn (n = 1 , 2, or 3 ) . We denote (or in L2(Q), no ambiguity being possible), and 1 . 1 the corresponding We define the operators norm. Recall u o is the trivial state ( ? , a ) . L and L 1 as: ( a
(6.87)
Lo
=
[iA
@
-6A
1
and
L, =
1
iis+1 RS
I
Ra+a iia
and (6.88)
LA
=
Lo
+
AL,
=
fU(A,uo).
Definition
6.1 - Let T be a linear operator from D(T) = X c Y into Y. The "numerical range" O(T) of T is the set of those z E Z! such that:
z
__ Example T
=
=
UEX,
(TU,U),
6.1 - Let U L A , We have:
=
IUI
=
1.
( u , v ) E X , X defined in (6.22) and (6.23), and
PATTERN FORMATION IN A MONOENZYME MEMBRANE
224
where the functions u and v may be complex-valued and,
Definition 6.2 - A linear operator T is said to be "sectorial", with -_--____
O(T) is included in
vertex y and half-angle 0 if the numerical range the complex sector: (6.89)
I: Rez 1. y ,
Example 6.2
-
Since
Re(LhU,U)
=
IIm z l 5 tg 0 (Rez - y).
Ks
< 0 and
[u12
+
ia
B[v12
> 0 , we have:
+ h(is+l)
1uI2
+
h(Ra+a)lv12
and
Thus :
Lemma6.1 - LA
i s a sectorial operator with vertex y h = 2Es -
ka
and
half-angle n/4.
~Lemma 6.2 -
If 6 2 y h , then (LA
Proof - Since LO' : Y is compact and so is A
Since Re(iLh
-
+
=
&I)-' is compact.
+
Y
s compact and L, : Y XL Lo' + 6L01 : Y + Y.
yXI)U,U) 3 0 and 6
+
-f
Y is bounded, L I L o l
y h > 0 we have :
MATHEMATICAL TECHNICAL DETAILS
Re((LA
+
GI)U,Uj 2 (6+yX)IU/* and LA
+
In other words LOU + ALIU + 6U = 0 j U = 0 , or -U admits the single solution U = 0 .
225
61 is injective.
AU
=
(ALIL;’+ 6 L0’)U
=
Since - 1 is not an eigenvalue of the compact operator A , it belongs to its resolvent set. Thus (I+A)-’ is hounded. It follows that:
is compact as the product of a bounded operator and a compact operator. Lemma 6.3
-
The resolvent of L A is compact.
Proof ___
It is well known [ 7 3 , 7 4 1 that if T is a closed operator in a Banach space Y such that the resolvent (
+
f(u)
=
stability: Let u o be an equilibrium
0.
Suppose;
which means that the real part of the elements of the spectrum of f u ( u o ) , u (fu(uoj), is strictly positive. Can we assert that u o is stable? The answer is yes, provided fu (uo ) is an m-sectorial operator. Before stating more precisely this property, we must define what we mean by m-sectorial operators, and also the norm I I - I / C 1under which
226
PATTERN FORMATION IN A MONOENZYME MEMBRANE
Ilu(t)
0 as t
-
+
-, provided
IIu(0)
-
u O I l a < E.
Definition 6.3 - A linear operator T in a Banach space Y is said to be m-accretive if, for Rea > 0 , (T
+
aI)-’eL(Y)
I / (T
and
+ aI)-’ll
5 1/Rea
Here L(Y) is the space of bounded linear operators in Y and the norm in L ( Y )
.
II-II
is
g i n i t i o n 6.4 - T is said to be quasi-m-accretive if, for some real c , T + CI is m-accretive. Definition 6.5 - We call T an m-sectorial operator if it is sectorial and quasi-m-accretive.
Define:
Example 6.3 -
a(u,v)
=
i=,
\
au av dx + l 0 u v dx for u and v e H ’ ( Q ) , axi axi
and consider those u such that there exists a function f that:
Let A : D ( A )
+
L 2 ( 0 ) be the linear mapping defined by
( A u , v ) = a(u,v)
Au
E
=
L 2 ( 0 ) such
f:
VveH’(Sl).
It is well-known [ 7 5 ] that the set of those u such that the mapping v +. a(u,v) is continuous for the topology induced on H ’ (0) by L 2 (0),is: D(A)
=
l u I u ~ H ’ ( 0 ) and
AueL2(n)},
and, if an is smooth enough, D(A)
=
{UIUE
H2(n)
and
- -- 0) av
It is also well- known that, if u eD(A),
=
v.
then Au
=
-Au + u , and,
MATHEMATICAL TECHNICAL DETAILS
227
endowed with the graph norm, D(A) is a Banach space:
The operator A is m-accretive. Indeed, if Reo > 0 and f e L2(n), the equation Au + u u = f admits an unique solution, satisfying: a(u,v)
+
whence, f o r v =
a(u,v
= (f,v)
VVE H’(Q),
u,
[u12+ ( l + o )IUI
=
(f,U).
Taking the real part, we have: [ u ] ’ + (l+Reu)lulZ
=
Re(f,u) 5
If1 1 ~ 1
and in part cular: < IUI -
Ifl/Rea
whence :
Example 6.4 - Consider the operator L o defined in (6.87), with D(Lo) = V x V , and consider the equation ( L o + cI + o I ) U = G , where c E R + , Rea > 0, U = (u,v) and G = (g,h)E Y. This equation splits into:
and it results from example 6.3 that 1uI 5 Ig//Reo and IvI 5 / h / / R e u , whence / u I 2 + I v / ’ 5 ( / g ~ 2 + / h / 2 ) / ( R e u ) 2that , is IU/ 5 IGl/Reo, SO that: ll(L0+CI+ 01)-Ill L (Y) < l/Reo, Therefore
Lo
is quasi-m-accretive.
Y
=
L 2 ( 0 ) x L’(0).
Moreover it is clear that L
is
PATTERN FORMATION I N A MOSOEKZYME MEMBRANE
228
s e c t o r i a l w i t h v e r t e x 0 and h a l f - a n g l e any 0
E
( 0 , ~ / 2 ) . Thus
Lo-&
m-sectorial. ______ I f T i s an m - s e c t o r i a l o p e r a t o r i n a H i l b e r t space Y , then it i s poss i b l e t o d e f i n e B a n a c h s p a c e s Y" ( a 2 0 ) . F o r a g e n e r a l d e f i n i t i o n F o r t h e p a r t i c u l a r c a s e we o f t h e s e s p a c e s we r e f e r t o [ 7 2 , 7 3 , 7 4 1 . a r e i n t e r e s t e d i n , w h e r e Y = L 2 ( Q ) x L 2 ( n ) a n d T i s t h e o p e r a t o r Lo o f e x a m p l e 6 . 4 , i t s u f f i c e s t o s a y t h a t Y"
can be d e f i n e d a s f o l l o w s :
Let ( p . , w . ) b e t h e e i g e n p a i r s o f t h e o p e r a t o r A o f example 6 . 3 : 1
1
-Aw.
+
= V.W.
W.
1 1'
,
- 0
i = 0,1,2
,...
,
I f u and v e L 2 ( E ) , t h e y c a n be I n r i t t e n : m
u
The p a i r U
Y"
1 uiwi
=
i =O =
m
and
v =
1
viwi,
i=O
( u , v ) will b e s a i d t o b e l o n g t o Y"
if:
i s a Banach s p a c e w i t h t h e norm:
I n p a r t i c u l a r Y o = Y a n d Y' = D ( L o ) = X . We c o n s i d e r now t h e f o l l o w i n g situation. L e t A b e a n m s e c t o r i a l o p e r a t o r i n a H i l b e r t s p a c e Y , and l e t r : U + Y w h e r e U i s a n e i g h b o r h o o d i n 'Y (some a < 1 ) o f u o . We say u i s an e q u i l i b r i u m p o i n t of t h e dynamical system:
(6.90) i f uo
E
$
+
Au
=
r(u)
D ( A ) a n d Auo = r (u,)
,
An e q u i l i b r i u m s o l u t i o n u o i s s t a b l e ( i n Y") i f , f o r a n y E > 0 , t h e r e e x i s t s 6 > 0 such t h a t any s o l u t i o n u of ( 6 . 9 0 ) w i t h Ilu(0) - uoIIcl< 6
MATHEMATICAL TECHNICAL DETAILS
229
satisfies llu(t) - uojIc( < E for all t > 0. We say u o is asymptotically stable if it is stable and we can also guarantee that u(t) + u o in 'Y as t +. m by choosing I l u ( 0 ) - uolIcl sufficiently small. We say u o is unstable if it is not stable. We assume f is locally Lipschitzian on U. More precisely, if u , E U , there exists a neighborhood U , c U of u , such that for x e U 1 , v s U 1 ,
for some constant L > 0. Suppose : r(uo+z) = r(uO)
+
r'(uo)z
+
g(z)
where r'(uo) is a bounded linear map from Y'
to Y , and:
Let L be the "linearized operator" from X to Y:
The following results have been obtained in a more general framework in [ 7 2 ] , to which we refer for the proofs. Theorem 6.2 - With the above assumptions on r , u o , and u l , we have the following:
(i)
(Existence and uniqueness) (6.90) has an unique solution on [@,+a[ with initial value u , E Ya. It is a continuous function u : [0,m[ + Y such that u ( @ ) = u 1 and on ] O , m [ we have u(t) E D (A) .
(ii)
(Stability) If the spectrum of L lies in {Rez > 6 1 for some More precisely, there exist 6 > 0 and M 3 1 such that if / / u , - u o l l a < 6 , then on 05 t < the solution of (6.90) satisfies: B > 0 , then uo is asymptotically stable.
230
PATTERN FORMATION IN A MONOENZYME MEMBRANE
(Asymptotic behavior) Assume that u(L) c { B } u {Rez > 6 ' 3 , where B is a simple positive eigenvalue of L and a ' > 6 > 0 . Then for any y in B < y < min(B',ZB), there exists 6 > 0 so that I I U , - ~ o l I " < 6 implies:
(iii)
u(t)
=
uo
+
K(u,)e-"
t
E(t)
where :
Here K ( u o ) = 0 and K(.) is a continuous map from a neighborhood of u o in Ya into the one-dimensional space N(L - @I), and, if E, is the associated projection onto N ( L - BI), K ( u l ) = E1(ul-uo) + 0( I / u l
(iv)
-
uolIi)
as
u1
+
u
.
(Instability) Assume a ( L ) n {Rez < 03 is a nonempty spectral set. Then the equilibrium solution uo is unstable.
Application to the stability analysis of the trivial state ( 5 , a ) . We verify the hypotheses of Theorem 6.2. Here A and r ' ( u o ) are the operators Lo and AL1 defined in (6.87),, while Y and D ( L o ) = X are the spaces defined in (6.21), (6.22), and (6.23). (i)
We know that Lo is m-sectorial.
(ii)
L, is a bounded linear map from Y" to Y , for some
CY in 0 5 a < 1 . Indeed L , is bounded from Y to Y , and it is well known that Y" is continuously imbedded into Y B if a > B , whence:
(iii) r is Lipschitzian on Ya. We apply again the continuous imbedding of Y" into Y for a 2 0 , and obtain:
for every x and y in Y".
231
MATHEMATICAL TECHNICAL DETAILS
(iv)
We finally have to verify the property (6.91) for:
The quantity within braces can be rewritten:
Q
=
1 [Rs s ( 5 1 ) ~ '
+
ZRsa(5f)uv
+
Raa(M)v21
~
where M(x) is a point: M(X)
=
(s
+
e(x)u(x),
z
+
o
e(x)v(x)),
< e(x) < 1.
Since Rss, Rsa, and Raa are bounded functions, we have:
But we know [72,p.35] that we have the continuous inclusion, for O < a < l .
Therefore we have, if
< c1 < 1 ,
But [ 7 2 , p . j 5 ] we have the continuous inclusion for 0 <
ci
< 1
Thus, combining the conclusions of Theorem 6.2 with the discussion of Section 6.3, we obtain the following proposition, where (cn,dn) (pn/z", pn/z') denotes, for n = l , Z , . . . , the intervals defined in this section:
__ Proposition
6.2 -
-
Suppose max(l/Z,n/4)
<
ct
< 1.
Then:
=
PATTERN FORMATION IN A MONOENZYME MEMBRANE
232
(i)
(6.1) has an unique solution on [O,m[ with initial value It is a continuous function (s,a) : 0,- [ , a(0) = a l and on l o , + - [ we have
(sl,al)E ya.
u1
+.
Y
m
(ii)
[cn,dn], then (?,a) is asymptotically
n=
stable. if:
More precisely there exist 6 > 0 and c > 0 such that
then on 0 5 t 5
m
the solution of (6.1) satisf
where fi is the least eigenvalue of L A .
u n= m
(iii) Assume A €
1
]cn,dn[.
Then the equilibrium solution (?,a)
is
unstable.
- The reasonings of this section can be easily extended to the stability analysis of any stationary solution v of ( 6 . 1 ) , even with spatially non uniform concentrations: The spectrum of fu(v) ( = Lo + r ' ( v ) ) is only composed of isolated eigenvalues, and the stability of the steady state depends upon the sign of the real part of the least eigenvalue. Remark 6.1
6.6.3 - f is a twice continuously differentiable mapping: For the mapping f : X +. Y defined by (6.20)-(6.23), the first few derivatives are : fu(h,u)
:
X
+.
Y, fA(A,u)
fuA(A,u) : X x R
+.
Y,
:
and
R
+
: X x X
Y , fUU(A,u)
fAA(A,u) : R x R
+.
Y
defined by: ARa -,BA
+
A(Ra+a)
+.
Y,
MATHEMATICAL TECHNICAL DETAILS
where:
Rs
=
(s,a),
233
...,
+
and f A X
= 0.
Here Rss for example denotes the mapping: Rss(s(*),a(.))
: H2(n)
x H2(n)
+
L2(n)
which associates, to
each pair of functions (v,y) in H 2 ( n ) x H 2 ( n ) , the function Rss(s,a) Rxx(s(x),a(x))v(x)y(x), x E n . This mapping is continuous. vy : x Indeed, since H 2 ( Q ) is continuously imbedded into Co(B) for n < 4, we have : -f
and,since IR,,(C,n)l < c, , whence:
is bounded for 5 E R and 0 5 q 5 ao, IR(s(x),a(x))l
where c , c , , c2 are various constants. Thus RsS(s,a)e L(X,L2(Q)), space of bounded linear operators from X = V x V into L 2 ( n ) . We now have to show the continuity of the L(X,L*(Q)) associating to each pair of functions mapping R s s : X -f
234
PATTERN FORMATION IN A MONOENZYME MEMBRANE
This property is an immediate con(s,a) in X the operator Rss(s,a). sequence of the Lipschitz continuity of the function Rss, which implies that:
whence :
6.6.4 - The range of fU- ( A , u o ) is closed and of codimension 1 . The following definitions and theorem can be found, together with the proof, in [ 7 6 , 7 7 1 . Definition 6.5 - Let X,Y be two Banach spaces. T E L(X,Y) (space of bounded linear operators from X into Y) is called a Fredholm operator if: (i)
N(T)
(ii)
Y/T(X)
is finite dimensional is finite dimensional
T is said to be Fredholm of type (p,q) if N(T) p-dimensional (resp. q-dimensional) .
(resp. Y/T(X))
is
Let F(X,Y) denote the set of Fredholm operators from X into Y. Definition 6.6
ind(T)
=
-
dim N(T)
Theorem 6.3
The index function ind : F(X,Y)
-+
Z is defined by
- dim Y/T(X).
-
(i)
Let X,Y be two Banach spaces and T E F(X,Y). in Y.
(ii)
I f T E F(X,Y) and K EK(X,Y) (space of compact linear operators from X into Y), then T + K EF(X,Y) and ind (T+K) = ind (T).
Then T(X)
is closed
MATHEMATICAL TECHNICAL DETAILS
235
defined in (6.86) - Here the spaces X and Since fU(A,uo) = T + K where Y are those defined in (6.21)-(6.23).
Application to fu(h,uo)
all we have to check is that T E F(X,Y) with index 0 and K E K(X,Y). Since - A + 1 (resp. - g A + a ) is an isomorphism of V onto L 2 ( 0 ) , the former property is immediate. Since the imbedding of X into Y is compact and K eL(Y), K is compact as the product of a bounded op'erator and a compact operator. We thus have: Proposition 6.3 - The operator fU(A,uo) defined in (6.86) is Fredholm with index 0. Its range is closed. If A is one of the endpoints of intervals (cn,dn), n = 1 , 2 , . . , , then: dim N(fU(A,uo))
=
codim R(fU(A,uo))
=
d
where d is the dimension of the null space of -A-unI. Remark 6.2 - In all the examples encountered we had d = 1 . But d could be greater than 1 . For example on the surface of a sphere the eigenfunctions are the spherical harmonics and the null-space associated to 1-1, is 2n + 1-dimensional. Conclusion - Diffusive instability is a somewhat unexpected phenomenon since it is commonly accepted that diffusion has a smoothing action and a tendancy to make uniform the values in a field of temperatures or of concentrations. The possible occurrence of diffusion driven instability in morphogenesis justifies the study of systems like the s-a system, which is only an example in a large class of 2-species systems involving diffusion, enzyme reaction, and transport from the outside. Pattern formation in fluids is known since a long time, with Bdnard and Taylor phenomena. In ecological systems it is called patchiness, and we refer to Okubo's book [781 and the references therein for a detailed study of this phenomenon. Finally, for discussions about the relations between pattern formation by dynamic systems and pattern recognition, we refer to [791.
PATTERN FORMATION IN A MONOENZYME MEMBRANE
236
References Turing, A.M., The chemical basis of morphogenesis, Phil. Trans. Roy. SOC. London, Vol. B 237 ( 1 9 5 2 ) 3 7 - 7 2 . Gmitro, J.I. and Scriven, L.E., in: A Physicochemical Basis for Pattern & Rhythm,in: Warren, K.B. (ed.), Intracellular Transpor (Academic Press, New York, 1 9 6 6 ) . Prigogine, I. and Nicolis, G., On symmetry breaking instabilit e s in dissipative systems, J . Chem. Phys., 4 6 ( 1 9 6 7 ) 3 5 4 2 - 3 5 5 0 . Prigogine I. and Lefever R., Symmetry breaking instabilities in dissipative systems 11, J. Chem. Phys. 4 8 ( 1 9 6 8 ) 1 6 9 5 - 1 7 0 0 . Prigogine I., Lefever R., Goldbeter A . , and Herschkowitz-Kauhan M., Synrmetry breaking instabilities in biological systems, Nature, 2 2 3 (1969) 913-916. Goodwin, B.C. and Cohen, M.H., A phase shift model for the spatial and temporal organisation of developing systems, J. Theor. Biol., 25 ( 1 9 6 9 ) 4 9 - 1 0 7 . Othmer, H.G. and Scriven, L.E., Interactions of reaction and diffusion in open systems, I. & E. C. Fund. 8 ( 1 9 6 9 ) 3 0 3 - 3 1 3 . Glansdorff,P. and Prigogine, I., Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley Interscience, New York, 1971).
Othmer, H.G. and Scriven, L.E., Instability and dynamic pattern in cellular networks, J. Theor. Biol. ( 1 9 7 1 ) 3 2 , 5 0 7 - 5 3 7 . Gierer, A. and Meinhardt, H., A theory of biological pattern formation, Kybernetika (Prague) 1 2 ( 1 9 7 2 ) 3 0 - 3 9 . Herschkowitz-Kaufmann, M. and Kicolis, G., Localized spatial structures and nonlinear chemical waves in dissipative systems, J . Chem. Phys. 5 6 ( 1 9 7 2 ) 1 8 9 0 - 1 8 9 5 . Boa, J.A., A model biochemical reaction, Ph.D. Thesis, Caltech, Pasadena (1 9 7 4 ) . Gierer, A. and Meinhardt, H., Biological pattern formation involving lateral inhibition: Levin, S.A. (ed.), Some Mathematical Questions in Biology 6 (The American Mathematical Society, Providence, 1 9 7 4 ) . Meinhardt, H. and Gierer, . 4 . , Applications of a theory of biological pattern formation based on lateral inhibition, J . Cell. Sci., 1 5 ( 1 9 7 4 ) 3 2 1 - 3 4 6 . Nicolis, G. and Auchmuty, J.F.G., Dissipative structures, catastrophes and pattern formation: A bifurcation analysis, Proc. Nat. Acad. Sci. 7 1 ( 1 9 7 4 ) 2 7 4 8 - 2 7 5 1 . Wolpert, L., Positional information and the development of pattern and form: Cowan, J.D. (ed.), Some Mathematical Questions in Biology 5 (The American Mathematical Society, Providence, 1974).
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Maginu, K., Reaction-diffusion equations describing morphogenesis I : Waveform stability of stationary wave solutions in a one-dimensional model, Math. Biosciences 27 (1975) 17-98.
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Boa, J.A. and Cohen, D.S., Bifurcation of localized disturbances in a model biochemical reaction, SIAM J . Appl. Math., Vol. 30, NO1 (1976) 123-135.
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Goodwin, B.C., On some relationships between embryogenesis and cognition, Theoria to Theory, 10 (1976) 33-44.
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Goodwin, B.C., Analytical Physiology of Cells and Developing Organisms (Academic Press, New York, 1976)
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Babloyantz, A., Mathematical models for morphogenesis: Levin, S.A (ed.), Mathematical Models in Biological Discovery (Lecture Notes in Biomathematics N013, Springer-Verlag, Berlin, 1977).
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CHAPTER 7 OSCILLATIONS AND WAVE FRONT PROPAGATION
In Chapters I1 to VI we mainly dealt with stationary solutions of enzyme distributed systems, our goal being to present, in Chapters V and VI, spatial structurations in the s-a system. In the present chapter we describe other types of structurations, namely spatiotemporal structurations such as oscillations and wave front propagation. I n Section 7.1 we talk about oscillations in the so-called papain system, in Section 7.2 we give an account of oscillations in a two substrates system catalyzed by uricase, and in Section 7 . 3 we show that the s-a system of Chapter 6 may propagate waves, for parameter values suited to this purpose. We only give numerical results, and omit both mathematical and numerical aspects including Hopf bifurcation and continuation of periodic solutions. Indeed these subjects would necessitate another book! From the biological point of view, sustained oscillations have been experimentally observed and established beyond doubt for glycolysis [1,2,4]. Models to represent the observed oscillations have been worked out [7,2]. A model taking explicitly into account the allosteric effects which is free from phenomenological factors has been constructed [ 1 4 ] . Circadian rhythms exhibit approximately 24-hour periodicity. General accounts of oscillatory phenomena are given in [3,16]. 1291 is devoted more specifically to oscillatory enzymatic reactions. We have listed from [ I ] to 1391 in chronological order, a sample of the experimental and theoretical work on biological, biochemical, and chemical oscillations. The two "stars" are the glycolytic oscillation and the Belousov-Zhabotinskii oscillation However the latter is not a biochemical oscillation , but occurs during the cerium ion catalyzed oxidation of malonic acid by bromate ion. This reaction requires elaborate kinetics to account for its obserbed behavior [271.
.
241
OSCILLATIONS AND WAVE FRONT PROPAGATION
242
Much simpler, from the kinetic point of view, is the papain membrane [15, 19, 2 2 , 29, 3 2 1 , which is described in Section 1 . From the mathematical point of view the appearance of oscillations is usually related to Hopf bifurcation [ 4 0 - 4 2 1 . References [ 4 3 - 6 9 1 , listed in chronological order, present some of the work done in the field of nonlinear wave propagation in diffusing and reacting systems. Here the outstanding models are Hodgkin-Huxley theory and the Fitzhugh-Nagumo equations. However a mechanism of transmission of a solitary front quite different from that in HodgkinOur model of Huxley nerve action potentials is presented in [ 6 8 ] . Section 7.3 is very close to this mechanism. 7.1
Papain System
The aim of this section is to present the evolution of the concentration profiles of a substrate and its product within an artificial enzyme membrane. 7 . 1 . 1 - Papain kinetics in a well-stirred solution: The enzyme E is papain, the substrate S is benzoyl-L-argenine ethyl ester, BAEE, and t h e product is hydrogen-ion, H+. The reaction scheme is: k k2 k3 1 ES + ES' + P 1 -+ E + P 1 + P2 E + S
k- 1 where ES is the enzyme-substrate complex, ES' is an acyl-enzyme complex, and P 1 and P 2 are the alcohol and acid produced by the splitting of the ester substrate 1 1 1 . In solution we have
i$
;tf + k l ES - k-,C = 0
dS
+
k, ES - k - C
- kgC'
= 0
1 + kZ)C = 0 = 0,
-dP2 dt
k3C'
=
0
PAPAIN SYSTEM
243
where S, E , C , C ' , and P 2 are the substrate, enzyme, enzyme-substrate complex, acyl-enzyme complex, and product concentrations (mole Making the quasi-steady-state assumption we obtain:
which, together with: E
C
+
+
C'
=
Eo,
gives the rate expression
((7.1)
R
=
k2 where:
KS
=
R
=
dS - dt . '
k2k3Eo + k3 + k3KSS-'
(kl + k2)/kl
The pH dependence of the reaction is given by the relations:
K~
= 5.45
x
k2
=
6 4 . 5 / ( 1 + 107'29H + 1 0 - 1 1 ' 4 9H
k3
=
?0.2/(1
mole cm-3
f
106'92 H)
-'
)
pec - 1 sec - 1
where 13 is the hydrogen ion concentration (mole [ 11. Note that H = P2. Incidentally let us recall that the pH is defined. as: p~
=
-log 10 I O ~ H .
I t is easy to see that for S held fixed the dependence of R upon H is of the form:
OSCILLATIONS AND WAVE FRONT PROPAGATION
244
R = a
H 1 + bH
+
cH2
Thus papain has an autocatalytic character in hydrogen-ion concentration, and is inhibited by excessive H. For H held fixed, the dependence of R upon S is Michaelian, that is of the form:
Thus H and S in the papain system play the role of S and A in the S-A system. We may consider [ 3 2 ] that equilibrium between the combination of hydrogen and hydroxyl ions (concentration OH) to form water, and the reverse ionization o f water, is instantaneous as compared with the enzyme reaction rate:
7.1.2 - Papain kinetics in a membrane: A membrane (thickness L) bearing papain is formed on a glass pH electrode and immersed in a bath of substrate. [ 2 2 ] . The conservation equations governing the system may be formulated as:
- DSSxx + R(S,H) At - DHAxx - R(S,H) St
(7.4)
1:;
A
=
H
= 0, =
O < X < L , t > O ,
0,
- KW/H,
t) ,t)
= So, =
A(0,t)
Ax(l,t)
=
= Ao,
0
where R(S,H is given by ( 7 . 1 ) , together with (7.2), and DS and DH are the diffusion coefficients of S and H within the membrane. The equation for A = H - OH is obtained by using (7.3) and subtracting the equation for OH- from the equation for :'H
PAPAIN SYSTEM
245
Here we assume the same diffusivity for hydrogen and hydroxyl ions, and take into account the rate of water formation. S 0 is the bath substrate concentration, and A. = Po Po is the bath H+ concentration, more precisely:
- Kw /Po where
p 0 = 1 0 - l ~ mole
Thus the external pH in the bath is kept at pH
= 10.
For L = 10-2cm, S = 5.5*10-7 mole ~ m - E~o =, mole ~ m - DH ~ ,= 2 2 1 6.4 x cm sec", and DS = 1 .3 x cm sec , an oscillation in time of the measured pH inside the membrane spontaneously occurs [22]. The period of oscillation is around 1/2 minute. This membrane There is qualitative agreehas been studied theoretically [ 1 5 , 1 9 1 . ment between the computer simulation and the experimental results. 7.1.3 - Numerical methods: Approximate solutions to (7.4) can be obtained by using either explicit or implicit finite difference schemes. Whether the step length is taken to be L/20 or L/100, both methods give similar results. It is somewhat unexpected and rather a good surprise that even the simple explicit method with a reasonable time step give good results, because, as will be seen, the values of H range over 6 decades, from to 10-7 . Due to this variation of H over 6 decades, special care must be exercised for the solution of the equation:
where A is given. formula: H
=
Instead of calculating the positive root by the
0.5 (A + (1 + 4 . 1 0 - 2 0 ) 1 / 2 1 ,
it is preferable to employ fixed point algorithms such as:
H ~ =+ A ~+ I O - ~ O / H ~if A > 0 ,
or
246
OSCILLATIOSS AND WAVE FRONT PROPAGATION
7.1.4 - Numerical results: The numerical simulations aid to understand the experimental observation that H(L,t) oscillates in time between and Suppose that we start with a uniform concentration of H+ within the and a memmembrane, the same as in the bath, say H(x,O) = brane empty of substrate, S(x,O) = 0. S molecules diffuse in from the bath, react, and liberate H+ at each spatial location. While the S concentrations increase, thus enhancing the reaction, more and more H+ is produced, and, since the reaction is autocatalytic, there is soon a burst of H+ concentrations, which pass quickly from about 1 0 - l ~ to mole cm - 3 . Figures 7.1.a and b show respectively the evolution of S and H+ concentration profiles during this stage where S decreases from profiles 1 to 3 and H increases from profiles 1 to 3 . At the end there is a pH 1 10) steep front connecting the low H+ concentrations (H = and the high H+ concentrations (H 1 lom7, pH = 4). This steep front arises near the impermeable boundary and propagates towards the bath. Then the reaction is inhibited by high H+ levels. Another reason for the reaction rate to be slow is that the substrate level within the membrane is now very low: a huge quantity of S has been consumed to feed the reaction and it has not been compensated by the S ingoing flux, because S is not diffusing quick enough. On the other hand H+ diffuses 5 times quicker, and, due to the steep gradient of H+ concentration near the bath, there is a large rate of hydrogen ion exit. This causes the sharp front to move back, from the left to the impermeable plane x = L on the right (profiles 3 to 5 in Figure 7.1.d). Then the boundary value H(L,t) drops from l o - ' to (profile 6), which corresponds to the experimentally observed variation of pH from 4 to 10. While H+ is flushed out, the S profile terminates its descent, and begins swelling again, since substrate consumption is low as compared to the rate of substrate entry. When the S level is high enough so that an explosion of H+ can be fed again, the cycle is repeated:
PAPAIN SYSTEM
247
explosion of H+ profile, exhausting S , then propagation of a steep front of H+ tou-ards the right and decrease of H+ concentrations while S increases.
Fig. 7.1 That oscillating front that moves back and forth periodically across the membrane has been experimentally observed by Graves [ 5 1 , by using a laser spectrophotometer double beam, a 10 u 2 laser beam allowing the measurement of the optical density in the thickness of the membrane itself. the osciIf the substrate boundary value S o is decreased to 4 llation ceases. Starting with S(x,O) = 0 and P(x,O) = as above we observe an explosive increase of the H+ level and a huge substrate consumption. Then the product concentrations decrease and both substrate and hydrogen-ion profiles tend towards equilibrium profiles, with low H+ concentrations. If we increase the boundary substrate concentration to 6 we obtain the same behavior, except that the and S o = H+ equilibrium level is higher. In both cases ( S o = 4 6 the substrate level is insufficient to feed a second explo-
OSCILLATIONS AND WAVE FRONT PROPAGATION
248
sion. Thus it appears that the oscillations occur only for S o in a 7 6 10- )(all other parameters the narrow range contained in (4 10 same). Also one can view these oscillations as occuring between diffusion controlled and reaction controlled states, the diffusion controlled states being those where pH = 4 (high H+ level) and the reaction controlled states being those at pH = 10 (low H+ level).
-',
__ In conclusion
the papain system is interesting for the following
reasons : (i)
i t is a simple and well defined enzyme system, where truly biochemical oscillations occur.
(ii)
by tuning the substrate boundary value S o for example, it is possible t o act upon these oscillations. Indeed S o is a candidate "bifurcation" parameter for further mathematical or numerical study.
(iii) the back and forth oscillation of the steep front looks like a wave-front propagation. 7.2
Uricase System
Consider the equations:
- sxx + oaF(s) = 0 , at - aaXX + oaF(s) = 0 ,
O < X < l , t > O ,
st
s(0,t)
= so,
sx(l,t)
=
a(0,t)
ax(l,t)
=
=
ao,
0,
F ( s ) = s / ( l + s + ks').
The reader should recognize at once the modeling of a membrane coated along an electrode (at x. = 1 , where no-flux boundary conditions occur) and immersed into a bath (at x = 0 where bath boundary conditions s(0,t) = so and a(0,t) = a . occur). Two species, in fact two substrates, S and A , diffuse in from the bath and react, liberating a product. S + A
E +.
P
URICASE SYSTEM
249
Here E is uricase, S and A are respectively uric acid and oxygen, and P is allantoin plus other products. Thus it is the same enzyme reaction as in the s-a system. The difference is that here the feeding from the outside occurs by entry of S and A from the bath at x = 0. It is one of the simplest enzyme systems which can oscillate. A numerical study of this system shows that all concentrations s(x,t) and a(x,t) oscillate in phase for the following values of the parameters: so = 100, a .
(7.6)
=
500, c1
= 0.2,
k
=
0.1, and
4 < u < 8.5.
Indeed one can say why oscillations occur. For the values (7.6) of the parameters the system (7.5) possesses a
"trivial" steady state ( S , Z ) ,
(
S(0) = so,
S'(1)
=
defined by:
0.
By the same arguments as in the preceding chapter, the stability of ( $ , a ) depends upon the spectrum of the linearized operator:
L
02 ( a ) F ' ( S ( * ) )
-a-
d2 + u F ( T ( * ) : dx2
from {uEH2(0,1)lu(0) = 0 , u'(1) = 01' to L 2 ( 0 , 1 ) * , which is only composed of eigenvalues. The first few eigenvalues of L u were numerically approximated by M.C. Duban [ 3 3 1 who found that the real part of the spectrum is positive for u l o , , u 2 ] where u l = 4 and - 8 . 5 , and at least one pair of complex conjugate eigenvalues u2 has a negative real part if u E ( u , , u 2 ) . As u crosses u 1 or u Z the real part of the two complex conjugate
250
OSCILLATIONS AND WAVE FRONT PROPAGATION
leading eigenvalues crosses zero.
We thus have a Hopf bifurcation,
and it was verified numerically [ 3 3 1 that the system (7.5) oscillates for every u E ( u l , u 2 ) . Figure 7.2 shows in the complex plane the two complex conjugate leading eigenvalues of L as u varies.
0 Fig. 7.2 It is not evident whether the Hopf bifurcation at u = u 1 for example is forward or backward. One could have a forward bifurcation of stable oscillations ( s . o . ) from a branch of stable steady states (s.s.s.) which become unstable ( u . s . s . ) , as indicated schematically in Figure 7.3 (top); or a backward bifurcation of unstable periodic solutions, followed by a turning point and a branch of stable periodic solutions, as indicated in Figure 7.3 (bottom). Which eventuali-
URICASE SYSTEM
251
ty actually occurs cannot be recognized by simply solving the evolution equations (7.5) in the vicinity of u l . In both cases one passes, when solving equations (7.5) for successive values of u , from a stable steady state to a stable periodic solution.
Fig. 7.3 Thus a numerical continuation of the curve of periodic solutions after the Hopf bifurcation would be helpful to provide more information about what actually occurs in the vicinity of u 1 and u 2 and between u , and u 2 . Unfortunately numerical methods for detecting the bifurcation of periodic solutions are more complicated than for detecting the bifurcation of stationary solutions. This is mainly due to the fact that Hopf bifurcations do not correspond to the vanishing of the Jacobian determinant, but only to the vanishing of the real part of two complex conjugate eigenvalues. Moreover, the following of a curve of periodic solutions, after discretization with respect to space and time, involves much more degrees of freedom than for a curve of stationary solutions. For example for the problem ( 7 . 5 ) , if we divide the space interval (0,1) into 20 equal intervals and the time interval (O,T), where T is the period, into 100 intervals, the "state" of the system is a vector in R 2000, .
OSCILLATIONS AND WAVE FRONT PROPAGATION
25 2
7.3
Wave Front Propagation in the s - a System
The system is governed by equations:
(7.8)
st
-
at
- Baxx
sxx
sx(O,t) R(s,a)
= =
-
X[R(s,a)
+ +
X[R(s,a)
sx(l,t) pas/(l
- a(ao-a)l
=
+
= 0,
(so-s)]
ax(O,t)
t > 0,
= 0,
ax(l,t)
=
0 < x < 1,
=
0,
s + ks’).
Here the parameters s o y a o y a , p and k are such that, in the (s,a) phase plane (Figure 7.4) the isocline curves r l and r 2 : R(s,a)
-
(so-s)
0
=
(T,)
and R(s,a)
- a(ao-a)
=
0
(r2)
intersect at a point (%,a) which is a stable steady state for the dynamical system:
af
+
X[R(s,a)
-
a(ao-a)]
= 0.
da
s
so
S
Fig. 7.4 Moreover, (T,Z) lies slightly on the right of the superior turning point of the first isocline r , , as indicated in Figure 7.4.
WAVE FRONT PROPAGATION IN THE s-a SYSTEM
253
An appropriate set of parameters for this to occur is: (7.10)
k = 1,
for which S
= 22
p =
0.9,
and 5
=
s0
=
39.2,
a.
=
784, and a
=
0.05,
440 [ 6 4 1 .
It is well known that a particle, subject to the dynamics (7.9), and displaced from the equilibrium point ( S , Z ) , can describe a large trajectory like the one in Figure 7.5 before coming back to the resting position ( S , g ) . In fact the direction of its movement is ds da which can change only by crossdetermined by the signs of ; i i and ,iI; ing the isocline curves, and are, for example, positive in the region near ( 0 , O ) .
Fig. 7.5 Let us now turn our attention to the distributed system (7.8). We choose 6 = 1 and X E (20,100), for which values, together with ( 7 . 1 0 ) , the uniform steady state ( S , g ) is stable. For this distributed system, the particle has to be replaced by a "snake" in the phase plane. At each time t the state of the system is a curve [(s(x,t), a(x,t)), 0 < x < 11, which we shall call a "snake". The system being
OSCILLATIONS AND WAVE FRONT PROPAGATION
254
initially at rest ( s ( x , O ) = S , a(x,O) = a ) , during a small interval of time (0 < t < 0.1) s(0,t) is forced to be low: s(0,t) = 0.01 for instance. This can he the result of a strong reaction of disappearance of S localized in space and time and represents a signal. For t > 0.1 this excitation is suppressed and the boundary condition is sx(O,t) = 0. The effect of this excitation on the system (7.8) can be decomposed in four phases and is in strong opposition with the behavior of a similar system without enzyme activity. We describe here the results of numerical experiments done with the set of parameters (7.10) and B = 1, X = 2 0 . Stage 1 (0 < t < 0.18) At the beginning, s(0,t) is constrained to be small, and the phase plane portrait is given by the full line 0.12 in Figure 7 . 5 . The corresponding S profile of concentration undergoes an abrupt change across a wave front which rapidly moves from x = 0 to x = 1 . Figure 7 . 6 shows these successive profiles, from t = 0.02 to t = 0.18, when the profile has become flat.
2:
WAVE FRONT PROPAGATION IN THE s-a SYSTEM
255
After this first stage, all the points [ ( s ( x ) , a ( x ) ) , 0 < x < 1 1 are roughly represented by a single point, which corresponds to uniform S and A concentration profiles, the S profile being very low. The signal has been transmitted from x = 0 to x = 1, and during the following phases the system is coming back to its resting state ( ? , a ) . The subsequent evolution of the system is similar to the recovery of the stable steady state (5,Z) by the dynamical system (7.9), the S and A profiles remaining uniform. The larger the A , the faster the recovery. (In the first stage too, the larger the A , the faster the wave front propagation). With h = 20, the respect ve durations of the 4 stages are approximately: T, = 0.18, T 2
=
1.1,
T3
=
0.3, and T4 = 2.9
Stage 2 (0.18 < t < 1.3) The representative point moves slowly southward. F ( s , a ) - (so-s) is slightly negative, very close to zero, and F(s,a) - a ( a o - a ) is frankly positive, so that the representative point moves downward, nearly ds = 0 . The S concentration is still very low, on the isocline curve dt whereas that of A decreases from 390 to 130. When the representative point arrives in the vicinity of the first isocline inferior turning point, it moves faster and faster, crosses the second isocline and enters the next stage. Stage 3 (1.3 < t < 1.6) da ds . Sow dt i s much larger than dt , and the representative point moves rapidly eastward. It crosses the first isocline curve and the trajectory enters the region where ds < 0, < 0. Stage 4 (1.6 < t < 4.5) The point follows s l o w l y a path very close to the S isocline, and tends toward the equilibrium point ( 5 , Z ) . Figure 7 . 7 compares this wave front propagation with the propagation of the same signal in a system without reaction: St
- sxx + X(s - 2 )
=
0,
O < X < l , t > O ,
OSCILLATIONS AND WAVE FRONT PROPAGATION
256
s(O,t).= 0.01 for 0 < t < 0.1, sx(O,t) = 0 for t > 0 . 1 , sx(l,t)
2:
=
0, s(x,O) =
s
t = 0.5
-__-----DIFFUSION ONLY -DIFFUSION AND REACTION
1(
Fig. 7 . 7 We see from the responses s(1,t) of both systems that diffusion alone is unable to transmit from one end to another a wave front created by an excitation limited in time. There are analogies between the behavior of our system and propagation of nervous impulse. I n both cases there is a threshold, signal propagation, and, in the recovery phase, an overshoot. However our model does not seem to be in competition with well established models of nerve impulse propagation, but maybe could aid to understand the dissemination of information in systems like slime molds [ 2 4 , 6 5 ] . Finally, Figure 7.8 shows the response s(1,t) of the distributed system (7.8) to a perturbation at x = 0 of duration to = 0.05, for X = 30.
WAVE FRONT PROPAGATION IN THE s-a SYSTEM
257
30
Fig. 7.8 Remark 7.1 - The biochemical interpretation of the above numerical results is that oscillations as well as wave front propagation can occur in immobilized enzyme systems, as a consequence of competing diffusion and enzyme reaction. In particular it is not necessary, for oscillations to occur, that several enzyme reactions be implied, with a possible feedback of the last product on the first reaction. A single reaction is sufficient, provided it is coupled to diffusion Similarly the coupling of diffusion and enzyme reaction may induce the production of propagating wave fronts, much faster than purely diffusing chemical species.
OSCILLATIONS AND WAVE FRONT PROPAGATION
258
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