coupling phenomena in nonlinear chemical...

Coupling Phenomena inNonlinear Chemical Kinetics

with Special Emphasis on the Light-Sensitive BZ Reaction

Henrik Skødt

August 2002

Dissertation submitted for obtainingthe Danish Ph.D. degree in chemistry.

Advisor Assoc. prof. Preben Graae SørensenChemistry Laboratory III, Dept. of Chemistry,University of Copenhagen

Evaluation Prof. Dr. rer. nat. habil. Stefan C. Muller,Comittee Otto-von-Guericke-Universitat, Magdeburg, Germany

Assoc. prof. Lars Folke Olsen, University of Southern DenmarkAssoc. prof. Finn Hynne, University of Copenhagen

Well I heard the bells ringing,I was thinking about winning

In this God forsaken place. . .

Well this is no New York street,and there’s no Bobby on the beat

And things ain’t just what they seem

And I’m taking in the Indian SummerAnd I’m soaking it up in my mind

And I’m pretending that it’s paradiseOn a golden autumn day, on a golden autumn day

-Van Morrison, Golden Autumn Day

Preface

This dissertation contains the bulk of my research during the past threeyears at West Virginia University (first year) and the University of Copen-hagen. The two main topics may therefore appear to have little in common,but both include some sort of coupling phenomenon and they both lead toconclusions regarding pattern formation in biological systems. The wordsystem is applied liberally throughout the dissertation and can mean any-thing from a system of differential equations to an actual oscillating chemicalor biological reaction, in a stirred or unstirred reactor. Hopefully the givencontext will reveal what is meant by system in each case.

The organisation of the chapters is such that there is a slow build-up witha more chemically oriented chapter 2, followed by chapters 3 and 4 with amore mathematical character. Especially chapter 3 can be useful for peopleunfamiliar with nonlinear dynamics. This is not to say that it will make iteasy to continue with the rest of the thesis, but it is an attempt to introduceall mathematical concepts encountered in later chapters. Chapters 5 and 6contain the main results of the research together with some experimentalresults shown in chapter 2. Finally I will try to make some conclusions anddraw perspectives for future work in chapter 7.

Acknowledgements

First of all I am thankful to Ken Showalter at WVU for getting me startedwith this Ph.D. work and for giving me an insight into research done theAmerican way. In his group at the time were also Michael Hildebrand andEugene Mihaliuk. Collaboration with Michael Hildebrand on the nonlocalcoupling project was very good, and the experiments would not have beenpossible without the aid of Eugene Mihaliuk. Especially in making theelectronics work and programming the software for the feedback he was likea wizard.

In Copenhagen a number of people have had a positive influence on theoutcome of this dissertation. Preben Graae Sørensen has provided excellentadvise, and his knowledge on geometrical optics and electronic componentsmade the building of the experimental setup much easier. I am also fortunateenough to be sharing an office with Sune Danø. He was always available forquestions regarding yeast, and always willingly discussed any subject I putforward. Discussing science with him is like playing tennis against a wall,in the sense that the shot you get back is always as least as good as the oneyou made yourself. Aage Nissen and Torben Hansen solved any technical

vi

problem related to the setup with great skill, and Mads Ipsen, Mads Madsen,and Jesper Schmidt Hansen all made useful remarks along the way.

Nuria, mi picharri, made the “God forsaken place” much less so, and isin general responsible for me trying to be a better person.

Henrik Skødt

Copenhagen, August 2002

Contents

Blues iii

Preface v

1 Introduction 1

2 Chemistry of the Light-Sensitive BZ Reaction 5

2.1 The Field-Koros-Noyes Mechanism . . . . . . . . . . . . . . . . 5

2.2 The Oregonator Model . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Dimensionless Model . . . . . . . . . . . . . . . . . . . 7

2.2.2 2D vs. 3D . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Mechanistic Details . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Light-Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Reaction-Diffusion Systems . . . . . . . . . . . . . . . . . . . . 11

2.5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . 12

2.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . 14

3 Differential Equations and Bifurcations 17

3.1 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Stationary States . . . . . . . . . . . . . . . . . . . . . 18

3.2 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Subspaces and Manifolds . . . . . . . . . . . . . . . . . 20

3.2.2 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . 20

3.3 Partial Differential Equations . . . . . . . . . . . . . . . . . . . 21

3.4 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.1 Hopf Bifurcations . . . . . . . . . . . . . . . . . . . . . 24

3.4.2 Excitable and Oscillatory . . . . . . . . . . . . . . . . . 25

3.4.3 Turing Bifurcations . . . . . . . . . . . . . . . . . . . . 26

4 Amplitude Equations 31

4.1 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Amplitude Equations . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Unfoldings of Amplitude Equations . . . . . . . . . . . . 33

4.2.2 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . 34

4.2.3 Pitchfork Bifurcation . . . . . . . . . . . . . . . . . . . 35

viii Contents

5 Nonlocal Coupling 375.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1.1 Excitable Conditions . . . . . . . . . . . . . . . . . . . . 405.1.2 Oscillatory Conditions . . . . . . . . . . . . . . . . . . . 41

5.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 465.4 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . 525.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6 Locally Coupled “BZ Cells” 576.1 A Single Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 2 by 2 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.2.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 626.3 Larger Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.3.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . 636.3.2 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 Flow-Distributed Structures . . . . . . . . . . . . . . . . . . . . 656.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Conclusions and Perspectives 69

A Appendix 73

B Specifications 77

Bibliography 79

1Introduction

Oscillations are found in many chemical reactions. Oscillations in the con-centrations of some of the participating species, that is. This may seem likean obvious statement to people that work with oscillating reactions everyday, but if one should mention oscillations to a “normal” chemist that iswhere the first confusion arises. Having regained his balance after being in-troduced to the concept of oscillations, the “normal” chemist can be thrownright back out of balance by mentioning that an oscillating reaction, whencoupled to diffusion, can produce chemical waves. The confusion usuallyincreases when supplying the information that two waves that meet will an-nihilate each other. This does not at all correspond to the popular imageof two mechanical waves meeting to produce constructive interference andthus a larger wave.

A single oscillator may oscillate in many different fashions. Like a cosine(or sine) function, where we speak of the phase relative to the x-axis whenmoving on the unit circle, we can speak of the phase (or angle) of a chemicaloscillator relative to the period and a given point in time. Observing asingle oscillator in an array of chemical oscillators, the nearest neighbourscan be leading and lacking the oscillator in phase respectively. If this isso for the whole array, then it will appear as if a wave is moving acrossthe array. If the are two waves in the array moving in opposite directions,they will, in the point where they “meet”, have the same phase. From thepoint of view of that oscillator it will see the same array of phases in bothdirections (assuming that all oscillators are identical and evenly spaced).Still, if one considers that same oscillator, the amplitude of its oscillationsdoes not change, which means that there is no increase in amplitude wheretwo waves meet. This type of wave is known as a phase wave. A way topicture this in a more colloquial situation could be to think of a football gamewhere the crowd has become bored. Typically they will start “initiating”waves around the stadium, and if for some reason two waves are running inopposite directions they will not achieve a larger “amplitude” where they

2 Introduction

meet.This is where the critical reader might ask why the chemical oscillators

should be arranged so that their phases match the description above. Partof the answer is: Diffusion. If there somewhere in a chemical solution (notnecessarily containing an oscillating reaction) should be a gradient in theconcentration of some species, diffusion will work to even out the differencein concentration. This is also the case for the array of oscillators (which isour picture of an oscillating reaction in an unstirred container), so diffusionwill work to even out the differences in the concentrations of the oscillatingspecies and hence the differences in phases. Since diffusion does not workinfinitely fast (a typical diffusion constant in aqueous solution is on the orderof 10−5 cm2/s) it will typically not be able to even out the phase differencesin oscillators completely, so that nearest neighbours will maintain a smallphase difference to allow for the propagation of waves.

Diffusion is a purely local coupling, and the fact that an unstirred os-cillating reaction can exhibit waves is old news [79]. For the most commonoscillating reactions in aqueous solution diffusion is also the only couplingbetween oscillators, so in order to obtain coupling between oscillators thatis non-local, i.e. anything that is not purely local, it has to somehow beimposed on the system. Considering surface chemical reactions [30] wherenonlocal coupling is an inherent part of the system, e.g. global couplingthrough the gas phase, nonlocal coupling leads to more complex and inter-esting behaviour than diffusion would have done alone. Thus it is naturalto take an interest in what similar coupling could achieve in oscillating re-actions in aqueous solution.

The Belousov-Zhabotinsky (BZ) reaction, which is the oxidation andbromination of malonic acid by bromate, catalysed by a metal-pair catalyst,has become the prototype oscillating reaction. When the catalyst is tris-bipyridyl ruthenium(II/III), the BZ reaction shows great sensitivity to light.Thus the state of the oscillating reaction can be altered by exposing it tolight. This has been utilised to make global coupling and periodic forcingof the unstirred system [73, 46]. Both types of coupling lead to interestingcluster patterns, and common for them is that they both use homogeneouslight for the feedback.

Returning to the purely local coupling for a moment, Alan Turing showedas early as 1952 that an activator-inhibitor system can show spatial struc-tures that are temporally stable if the inhibitor has longer diffusion-lengththan the activator [67]. Unfortunately, most chemical species have verysimilar diffusion constants in aqueous solution so the phenomenon is notobserved in purely aqueous solution. The spatial structures, known as Tur-ing structures, were first observed in 1990 [6] in the Chlorite-Iodide-MalonicAcid (CIMA) reaction in a polyacrylamide gel. In the BZ reaction a similaruse of gel does not lead to Turing structures.

Taking, again, inspiration from surface chemical reactions [25, 24], it

Introduction 3

leads to the idea of introducing a feedback to the light-sensitive BZ reactionwith different lengthscales built into the feedback. Such a feedback with twodifferent lengthscales, alternating between activation and inhibition, coupledby light to the BZ reaction is one of the main topics of this dissertation.It will be shown that with the right choice of nonlocal coupling, Turingstructures in the BZ reaction will be observed.

The motivation for Turing to do his original work was the fact thatfrom apparently homogeneous embryos, creatures could evolve with arms,legs, and head in the right positions. The nonlocal coupling used for theBZ reaction can also be constructed as a reaction-diffusion system with tworapidly relaxing species, and thus it provides an alternative mechanism forTuring structures in biological systems.

Biological relevance seems to have become a major issue in the nonlinearchemical research in recent years, but it need not have an instant directapplication to a biological problem to be of importance. One such exampleis the quenching method [29] which was first developed and used for theBZ reaction, but since was applied to oscillations in living yeast cells [9].Biological cells and their synchronisation coupling has also become a hottopic.

The second major topic of this thesis is concerned with the creation of“BZ cells” in fixed positions with respect to each other. These cells arecreated with dynamical properties very similar to the properties existing inyeast cells. The cells may be thought of as arrays of oscillators, coupling lo-cally through diffusion, interrupted by a short array of non-oscillators thatalso couple locally via diffusion. The local coupling between these cells,via the diffusion in the non-oscillatory part, is studied as well as the pat-terns they form. The coupling yields some surprising results and gives riseto some interesting questions regarding systems with built-in boundaries,and in particular cells’ ability to form stationary patterns when interactinglocally with other cells.

4 Introduction

2Chemistry of the

Light-SensitiveBZ Reaction

The mechanism of the BZ reaction has been studied intensively over theyears since the discovery by first Belousov and since Zhabotinsky [4, 80]. Ifone does not take any interest in the dynamics of the reaction, it can simplybe described as the oxidation and bromination of malonic acid (MA) bybromate, catalysed by metal ion couples. Those couples are most oftenCe3+/Ce4+, Ru(bpy)2+

3 /Ru(bpy)3+3 , and Fe(phen)2+

3 /Fe(phen)3+3 . There

are many different products of the reaction, some of whom still remain tobe identified.

2.1 The Field-Koros-Noyes Mechanism

The first serious attempt of modelling the mechanism of the BZ reaction wasdone by Field, Koros, and Noyes (FKN) in 1972 [11]. This mechanism (seetable 2.1) was able to account for many qualitative features of the reaction.The reactions (R1)-(R4) containing Br at different oxidation levels as wellas reactions (R5) and (R6) that include the autocatalytic growth of HBrO2

are generally regarded as being a correct account of the inorganic part ofthe reaction. Studies by Field and Forsterling [10] with a revision of therate constants indicate that the reaction rate of (R7) should be set to zero.Special attention should be given to the autocatalytic step, where HBrO2

catalyses its own formation. Adding (R5) + 2(R6),

BrO−3 + HBrO2 + 2Ce3+ + 3H+ 2HBrO2 + 2Ce4+ + H2O (A)

6 Chemistry of the Light-Sensitive BZ Reaction

(R1) HOBr + Br− + H+ Br2 + H2O(R2) HBrO2 + Br− + H+ → 2HOBr(R3) BrO−

3 + Br− + 2H+ → HBrO2 + HOBr(R4) 2HBrO2 → BrO−

3 + HOBr + H+

(R5) BrO−3 + HBrO2 + H+ 2BrO2· + H2O

(R6) BrO2· + Ce3+ + H+ HBrO2 + Ce4+

(R7) BrO2· + Ce4+ + H2O → BrO−3 + Ce3+ + 2H+

(R8) Br2 + MA → BrMA + Br− + H+

(R9) 6Ce4+ + MA + 2H2O → 6Ce3+ + HCOOH +2CO2 + 6H+

(R10) 4Ce4+ + BrMA + 2H2O → Br− + 4Ce3+ + HCOOH +2CO2 + 5H+

Table 2.1: The Field-Koros-Noyes mechanism. MA and BrMA are CH2(COOH)2 andBrCH(COOH)2 respectively.

we see that the autocatalysis is quadratic in HBrO2. Autocatalysis is knownto be an important premise for observing oscillations in a chemical reaction.

2.2 The Oregonator Model

The essential dynamics of the FKN mechanism are captured by the Oreg-onator model proposed in 1974 by Field and Noyes [12]. It includes threedynamical species, HBrO2, Br−, and Ce4+, represented by the symbols X,Y, and Z respectively. Bromate, represented by the symbol A, acts an ad-justable parameter. None of the reverse reactions are considered in themodel, and the H+-concentration is included in the rate constants. Theresulting reaction scheme can be viewed in table 2.2. Comparing the Oreg-onator model with the FKN scheme the origins of the different steps areobvious; e.g. the autocatalytic step, (O3), is clearly the equivalent of reac-tion (A). The last step in the Oregonator, (O5), incorporates the whole or-ganic subset of the FKN mechanism without specifying any organic species.The emphasis has been put on the fact that regeneration of Br− is the mostimportant feature of the organic subset, and the stoichiometric factor, f ,has been included to take into account that only some of the organic reac-tants react with Ce4+ to produce bromide. The factor, f , varies accordingto experimental conditions.

Applying mass action kinetics to the five reactions, (O1)-(O5), results inthe following expressions for the reaction velocities of the dynamical species

x = k1ay − k2xy + k3ax− 2k4x2 (2.2a)y = −k1ay − k2xy + fk5z (2.2b)z = 2k3ax− k5z (2.2c)

2.2 The Oregonator Model 7

(O1) A + Y k1−→ X

(O2) X + Y k2−→ P

(O3) A + X k3−→ 2X + 2Z

(O4) 2X k4−→ Q

(O5) Z k5−→ fY

Table 2.2: The Oregonator model. It is related to the FKN mechanism by the relations A= BrO−3 , X = HBrO2, Y = Br−, and Z = Ce4+. P and Q are arbitrary reaction products.

where k1 − k5 are the reaction rate constants, and lowercase letters denoteconcentrations of the corresponding uppercase letters, e.g. a = [A]. The dotsignifies differentiation with respect to time, t.

2.2.1 Dimensionless Model

It has certain, especially numerical, advantages to work with dimensionlessmodels. For the Oregonator model we shall follow the steps devised byTyson and Fife [69] to obtain a dimensionless model. For the variables fromeq. (2.2) we define the following transformations:

u =2k4

k3ax, v =

k4k5

(k3a)2z, w =

k2

k3ay, τ = k5 t (2.3)

Inserting these new variables into eq. (2.2) yields the dimensionless equations

εu = qw − uw + u− u2 (2.4a)v = u− v (2.4b)

ε′w = −qw − uw + 2fv (2.4c)

where the dot now signifies differentiation with respect to τ , and with thedimensionless constants defined from the rate constants and the bromateconcentration by:

ε =k5

k3a, ε′ =

2k4k5

k2k3a, q =

2k1k4

k2k3(2.5)

There is still a clear correspondence between the variables and the oscil-lating species, u corresponds to [HBrO2] etc., but now the oscillations takeplace on a “chemical” timescale and the amplitude of the oscillations of thedimensionless variables is several orders of magnitude larger compared tothe concentrations of the oscillating species. The hazard is that once youstart playing around with the dimensionless parameters and come acrosssome interesting behaviour in the simulations, they are not easily translatedback into rate constants (which are only constant at constant temperatureand constant H+-concentration).


2.2.2 2D vs. 3D

The two parameters, ε and ε′, will typically have small values with ε′ as thesmallest of them. For realistic rate constants we will have ε/ε′ ≈ 500. Thefact that ε′ is usually very small can be utilised to simplify the dimensionlessOregonator even further. Assuming that ε′ is very small we see that therighthandside of eq. (2.4c) must be very close to zero. Setting it equal tozero we obtain

w =2fvu + q

(2.6)

Thus the equations (2.4) can be reduced to just two equations by insertingthe expression (2.6) for w

εu = u− u2 − 2fvu− q

u + q(2.7a)

v = u− v (2.7b)

Working with the reduced equations (2.7) has some advantages. One exam-ple, which can be found in the book by Gray and Scott [17] pp. 381, dealswith relaxation oscillations which can be analysed more easily by looking atthe nullclines of the reduced system. There are of course cases where the 2Dapproximation does not reproduce the results from the 3D system. Apartfrom stating the obvious in saying that ε′ needs to be very small, it is moreuseful to try to say something about how the oscillations should look in anexperiment in order to be able to use the reduced model (assuming that the3D model gives a good description of the system).

This investigation could be a whole research topic in its own right, butfor our purpose we just need to set up a guideline for when it is safe touse the reduced model. Consider figure 2.1. In (a) are shown the timeseries for a parameter set that will be used later on in this dissertation.The time series for the reduced model is shown as the solid line whereasthe dashed line corresponds to the full 3-dimensional system. In this caseand throughout the dissertation the relation ε = 500ε′ applies. Evidentlyboth the amplitude and period of the reduced model match those of thefull system to a very good approximation. In (b), however, the match isvery poor. Neither the period, nor the amplitude of the full system can bereproduced by the reduced model. These two examples are both clear-cutcases where it is easy to determine that the very relaxational oscillationscan be well modelled by the reduced model, whereas it is equally evidentthat the sinusoidal oscillations can not.

There are, of course, intermediate sets of parameter values that are not aseasy to categorise. Take the example in fig. 2.1 (c) and (d) for instance. Thetime series for the reduced and full model are shown in (c), and (d) shows thecorresponding phaseplot with u on the x-axis. From (c) we can tell that theoscillations have the same amplitude in y but have slightly differing periods,

2.3 Mechanistic Details 9

0 10 20 30 40 50τ

0.05

0.15

v

(c) 2D3D

0 5 10 15 20 25τ

0.05

0.15

v

(a) 2D3D

0 0.1 0.2u

0.05

0.15

v

(d) 2D3D

0 10 20 30 40 50τ

0.004

0.008

v

(b) 2D3D

Figure 2.1: (a) Relaxation oscillations with q = 0.005, ε = 0.025, ε′ = 0.00005, and f =0.95. (b) Sinusoidal oscillations with q = 0.00095, ε = 0.45, ε′ = 0.0009, and f = 0.79. (c)Time series and (d) phaseplot of the oscillations for the parameter values q = 0.00095, ε =0.2, ε′ = 0.0004, and f = 0.79. The initial conditions are the same for all plots, where thesolid line corresponds to the reduced 2D model and the dashed line corresponds to thefull 3-dimensional system.

and from (d) that the limit cycles are nearly identical (naturally the curveshown for the full model is only the projection of the limit cycle onto theu-v plane). We do not make any conclusions regarding the periods based onthis, since the biggest discrepancy between the Oregonator and experimentslies in the prediction of oscillation periods. Comparing (a) and (c) it shouldbe evident that, even though they can both be characterised as relaxationaloscillations with very sharp peaks, the rise in y is steeper in (a) than in (c).

The overall conclusion must be that, looking at experimental oscillationsand their shape, unless they are sinusoidal oscillations with small amplitude(compare the y-axes in fig. 2.1 (a) and (b)) it should be safe to use the re-duced model to achieve the same qualitative features. This is, as mentionedearlier, just a rule of thumb.

2.3 Mechanistic Details

Many (or perhaps most) people are perfectly happy with using just the FKNmechanism and Oregonator model for describing the BZ reaction, but there


are also a few that are concerned with that blank spot in the charts: Theorganic subset of the reaction. The first attempt at determining the organicproducts was done by looking at reaction products with Thin-Layer Chro-matography (TLC) and comparing with the most likely candidates amongstorganic acids [5]. The main conclusion was that bromomalonic acid, dibro-moacetic acid, and a third unknown product were produced in the reaction.The production of dibromoacetic acid suggests that some bromomalonic acidreacts without producing bromide. Earlier it was also known that CO2 isproduced in the reaction. This is seen by the development of bubbles in thereaction.

The natural substitute for TLC in determining reaction products is HighPerformance Liquid Chromatography (HPLC). A series of studies were madeusing this technique along with NMR. The first studies focused on the re-action of Ce4+ with malonic and bromomalonic acid [13, 61, 52]. FurtherHPLC studies [65, 51] investigated the role of UV radiation and the mecha-nism of CO2 formation. Another study was made of the bromination ofmalonic and bromomalonic acid [60], with the conclusion that brominationof bromomalonic acid is faster than bromination of malonic acid. This hasconsequences for the socalled Zhabotinsky method for producing bromoma-lonic acid [79] where bromide is added to an acidic solution of bromate andmalonic acid, since the products are not what they were assumed to be. Thevery same method is applied later in this dissertation, and for the sake ofreproducibility it is important to supply details of the synthetical procedure(see also [66], p. 415). The method of applying HPLC was extended to theinduction period of the BZ reaction in a batch reactor [20, 21] where furtherreaction channels were identified.

These initial studies, as one might call them, lead to the logical con-sequence of analysing the full BZ system with HPLC [22]. This study in-corporated all the previous results along with its own findings to proposethe Marburg-Budapest-Missoula (MBM) model. The result is a model thatquantitatively reproduces many features of the BZ reaction, but there arestill a few problems remaining with some of the organic compounds. Itis, however, impressive to see such an amount of work put into finding aquantitative model where a qualitatively good model has been around forthirty years. It should be noticed that this model incorporates reactionsof bromomalonic acid to yield different brominated organic species withoutproduction of bromide in those reactions. This is important in explainingcomplex transient oscillations in batch reactors (see [76] and [19] p. 90 ff.).

2.4 Light-Sensitivity

The light-sensitivity of the Ru(bpy)2+3 -catalysed BZ reaction is a well-studied

phenomenon [79, 14, 43] and has, as mentioned in the previous chapter, been

2.5 Reaction-Diffusion Systems 11

utilised for many different experiments [73, 35, 46, 49]. The mechanisms ofthe influence of light on both the organic and inorganic subsets have beeninvestigated by Kadar et al. and Amemiya et al. [34, 1]. The main conclu-sion is that the most important effect of illumination is the production ofbromide from the reaction of the excited Ru(II) complex with bromomalonicacid:

Ru(bpy)2+∗3 + 2BrMA −→ Ru(bpy)2+

3 + 2Br− + 2H+ + org.prod. (2.8)

For modelling purposes it is also important to note that the rate of bro-mide production is proportional to the light flux at constant concentrationof bromomalonic acid, and for practical purposes that the light-sensitivity(i.e. the bromide production rate) increases with increasing concentrationof bromomalonic acid. In the dimensionless Oregonator (2.4) these consid-erations give rise to adding the term ϕ to the bromide production rate togive the modified version:

εu = qw − uw + u− u2 (2.9a)v = u− v (2.9b)

ε′w = ϕ− qw − uw + 2fv (2.9c)

This is in accordance with the model proposed by Krug et al. [42], wherethe dimensionless light intensity, ϕ, is related to the light flux, Φ, via

ϕ =2k4

(k3a)2Φ.

The modified Oregonator can of course also be reduced to the 2-dimensionalversion:

εu = u− u2 − (ϕ + 2fv)u− q

u + q(2.10a)

v = u− v (2.10b)

This is the version of the Oregonator that will be used in chapter 5 on nonlo-cal coupling, whereas the full modified model (2.9) will be used in chapter 6regarding local coupling of sinusoidally oscillating cells, in accordance withthe guidelines set up in section 2.2.2.

2.5 Reaction-Diffusion Systems

In the introduction it was mentioned how oscillations coupled with diffusionmay produce spatial patterns, and in this section will be shown some ex-perimental examples of this. Included in the examples will be the first everexamples of small-amplitude waves in the Ru(bpy)3-catalysed BZ reaction.


1 2 3 3 4 5 6 7

Chemical system

8

Figure 2.2: The optical path of the perturbation light. The numbers indicate 1: Plane-convex lense, 2: Heat filter, 3:Coloured glass filter, 4: Plane-convex lense, 5: Lightmaskto be projected onto the chemical system, 6: 85 mm slide-projector objective, 7: Semi-transparent mirror, and 8: A CCD camera.

2.5.1 Experimental Setup

Experiments were performed in both Copenhagen and Morgantown usingdifferent setups, but applying the same principles. The chemical part of thesetups was identical in both cases.

A gel containing the reduced form of the catalyst, Ru(bpy)2+3 , was pre-

pared by acidifying a solution of 10% (w/w) Na2SiO3 and 2.0 mM Ru(bpy)2+3

with H2SO4 and casting a uniform 0.3 mm × 20 mm × 25 mm layer onto amicroscopic slide. A solution not containing any catalyst was prepared bymixing bromate, sulfuric acid, and malonic acid solutions and then adding,in an ice-bath, Br− solution dropwise to obtain the desired concentrationsof BrO−

3 , H2SO4, malonic acid, and bromomalonic acid (with the commentregarding BrMA and Br2MA in sec. 2.3 in mind). Immersing the gel in thecatalyst-free solution will then initiate wave activity which is confined to thegel due to the immobility of the catalyst in the gel. This way an open sys-tem can be maintained by continuously feeding fresh catalyst-free solutionto the reactor, e.g. a petri dish, containing the gel without disturbing thewave patterns.

In practice, in the actual experiments, the system could only be char-acterised as semi-open. The total volume of catalyst-free solution, ca. 230mL, was flown through the reactor and then recycled. This amount wasenough not to see any significant changes in the chemical dynamics duringthe course of one experiment. More pronounced was the effect of washingout catalyst from the gel, which limited the duration of one experimentto 4-5 hours. Another point of concern is decay of BrMA which is unsta-ble in aqueous solution. This was addressed by immersing the bulk of thecatalyst-free solution, stored in a plexi-glass tube, in an ice bath. The flowwould continuously replace a small part of this with solution from the reac-tor, thermostated at the operating temperature. The tube leading from the


20 40 60 80 100Position

60

80

100

Pixe

l val

ue

Figure 2.3: Enhanced experimental image of a pair of counter-rotating spirals to the left.To the right a profile of the waves emphasising the relaxational nature of the spirals.

plexi-glass tube to the reactor was lead through the thermostat bath. Theduration of the stay of the catalyst-free solution inside the thermostat waslong enough to reach the operating temperature before reaching the reactor.

The difference between the types of setup used lay in the applicationof perturbation light. As mentioned in the previous section light has aninfluence on the BZ reaction with the chosen catalyst. In figure 2.2 is illus-trated the principle of the optical part of the setup used in the Copenhagenexperiments. Starting from the left we have a pointlike (in principle atleast) lightsource. The beams of light are parallelised by the plane-convexlense. The parallel beams of light are lead through a series of different filtersbefore another plane-convex lense gathers them into a slide projector objec-tive. From the objective the beams are projected onto the semi-transparentmirror from which approx. half the beams are reflected onto the chemicalsystem, while the other half passes through the mirror. With the camera itis possible at the same time to look through the mirror and record picturesof the system. If some pattern to be projected onto the chemical systemis put immediately after the second plane-convex lense it can, by tuningthe distances between it and the objective and/or the distance from theobjective to the mirror, be focused on the chemical system. The projectionrelations, e.g. 1:1, 1:2 etc., can also be tuned by varying those distances.The advantage of this setup is that it is relatively easy to build and hasrelatively cheap components. The downside is that the projected patterncan not be made time-dependent. This can be achieved with a video pro-jector as the light-source, as was the case in the Morgantown experimentsdescribed in chapter 5.

Another light source for observing the gel with the camera was alsoneeded in these experiments, since an inhomogeneous pattern does not pro-vide a good background for grabbing images. The principle for the obser-vation light is the same as for the perturbation light as far as the secondplane-convex lense, with a 450nm interference filter between the two plane-


25 50 75Position

50

75

100

Pixe

l val

ue

Figure 2.4: Experimental image of a low-amplitude spiral. To the right the correspondingprofile compared to the profile of the relaxation wave. The profile of the low-amplitudewave is marked with circles.

convex lenses to achieve maximal contrast in the recorded images. Fromthere the light is gathered onto a fiber light-guide that ends up in a ring-light. That ring-light is placed between the mirror and the chemical system.It has a hole in the middle through which the camera may still grab images ofthe chemical system, and at the same time it provides a fairly homogeneouslight for taking pictures.

A synchronous motor mounted with a wheel with openings for the per-turbation light was placed between the mask and the objective, i.e. number5 and 6 in figure 2.2, to provide a timing device for grabbing images. Thiswas coupled to an electronic setup via an opto-sensor to provide the syn-chronisation between the camera and the shutting off of the perturbationlight. More detailed information about this part of the setup can be foundat: http://theochem.ki.ku.dk/~ hs/report.pdf

Details regarding chemicals and filters can be found in the specificationsin B on page 77.

2.5.2 Experimental Results

The setup was used to perform the coupling experiments described in chap-ter 6, but it can of course also be used to do normal experiments withformation of spirals. Having the perturbation light at your disposal is agreat advantage, though, since the spirals tend not to form spontaneouslyin the gel experiments. It is rather the norm that waves are generated atthe boundary of the gel and that they eventually will take over the entiremedium. One particular way to form a pair of counter-rotating spirals is tocut a wave front so that the free ends will start rotating to form spirals.

An example of this is shown in figure 2.3 where the experimental con-ditions are as described for the oscillatory case in chapter 5. To the left isshown an enhanced image of two counter-rotating spirals formed by cuttinga wave front with the perturbation light, and to the right is shown a profileof the wave train moving away from the spiral center to the left. Notice that


2000 2200 2400 2600time/secs

80

85

90

95

100

Pixe

l val

ue

Figure 2.5: Time series recordedin one point in the gel system.The oscillations have low ampli-tude. The near-sinusoidal form in-dicates proximity of a supercriticalHopf bifurcation.

the pixel values are the ones taken from the raw image that was enhanced toyield the picture to the left. The profile clearly emphasises the relaxationalnature of the waves, leading to the conclusion that modelling of these wavesmay be accomplished by using the 2-dimensional version of the Oregonator.

Experiments were also done with the conditions [BrO−3 ] = 0.0533 M,

[H2SO4] = 0.58 M, [MA] = 0.48 M, and [BrMA] = 0.02 M at 25◦ C. Herethe BrMA concentration is actually low enough compared to the MA con-centration to assume that all Br2 has reacted with MA to produce BrMA.A picture of a spiral formed under these conditions is shown in figure 2.4to the left. To the right is shown the corresponding profile together withthe profile of the relaxation waves shown earlier. Evidently the amplitudeis much smaller, and the wave peak appears to be much more symmetrical.This points to a sinusoidal oscillator dominated slightly by overtones of theharmonic oscillations as indicated by the narrowness of the peaks. Indeedit is possible, by reducing the bromate concentration, to make waves witheven smaller amplitudes and even more symmetrical peaks.

The operating point for the experiments in chapter 6 was without BrMAin the catalyst-free solution, and the concentrations were: [BrO−

3 ] = 0.013 M,[H2SO4] = 0.5 M, and [MA] = 0.5 M. Under these conditions the systemwas still light-sensitive, probably due to the low bromate concentration. Theoperating temperature was as in the other low-amplitude experiments 25 ◦

C. A sample time series at these conditions, recorded at a particular pointin the system is shown in figure 2.5. The amplitude here is about the samesize as in figure 2.4. This indicates that the operating point might be closeto a supercritical Hopf bifurcation. The standard way of testing this, as willbe described in chapter 4, is to make a plot of the square of the amplitudeas a function of the bifurcation parameter. This is not possible in this casedue to low signal-to-noise ratio.

All the evidence, although circumstancial, points to the operating pointbeing in the vicinity of a supercritical Hopf bifurcation. The starting pointfor trying to find this operating point were the conditions that, in a CSTR,


were close to a supercritical Hopf bifurcation [62]. To the knowledge of thisauthor, noone has ever done experiments with the Ru(bpy)3-catalysed BZreaction showing small-amplitude waves before.

The shape of the waves, and considering the shape of the time series,point towards modelling this operating point using the full 3-dimensionalOregonator (dimensionless or not) for obtaining satisfactory results.

3Differential Equations

and Bifurcations

This chapter provides a mathematical basis for proceeding with the subse-quent chapters. Only concepts encountered in those chapters are introducedand no proofs will be given for the results quoted.

3.1 Linear Systems

The time evolution of an oscillating chemical system is described by differ-ential equations in the concentrations of the participating chemical species.(A set of differential equations can in general be termed a dynamical systemeven though the variables may not have any chemical or physical meaning.)When one speaks of a linear system, it is a dynamical system that consists oflinear differential equations only. In general a linear system can be writtenas

x = A · x (3.1)

where x is a real n-dimensional vector and A is a real n×n matrix, the dotmeaning differentiation with respect to time, i.e. x = dx

dt . In two dimensions,eq. (3.1) appears as

x = ax + by

y = cx + dy

with

A =(

a bc d

)and x =

(xy

)

The solution to eq. (3.1) can be proven to be the same in the general n-dimensional case as if A and x were scalars, i.e.

x(t) = eAt · x0 (3.2)

18 Differential Equations and Bifurcations

where eA is defined by its Taylor series,

eA =∞∑

i=0

Ai

i!

and x0 is the initial condition for x(t) at t = 0. If n linearly independentsolutions to eq. (3.1) exist, it is possible to write any solution as a linearcombination of those. Assuming that A has n linearly independent eigen-vectors, e1, . . . , en, with corresponding eigenvalues λ1, . . . , λn, they are aconvenient choice, and the solution is

x(t) =n∑

i=1

aieλitei (3.3)

with the coefficients, ai, determined by the initial conditions.

3.1.1 Stationary States

The vector, x(t), describes the state of the system, and the state is sta-tionary when it no longer “moves”, i.e. when x = 0. For linear systemslike (3.1) there is only one stationary solution, the origin, and its stabilityis determined by the real part of the eigenvalues of A.

Consider the case where a2 = · · · = an = 0, corresponding to a pertur-bation of the stationary state along e1

x(t) = a1eλ1te1

If λ1 is real, its sign will determine if the motion of the system state willbe an exponential decay back towards the stationary state or away from it.From eq. (3.3) it can be seen that a random perturbation from the stationarystate will result in a motion that is a linear combination of motions alongeigenvectors. It is also clear that if just one of the eigenvalues, say λ1, ispositive then the result of the perturbation will be a motion away from thestationary state, provided a1 6= 0.

In the case of complex eigenvalues, λi = λi+1 = αi + iβi (a bar denotingcomplex conjugation) with corresponding eigenvector u + iv, the solutionsin real vector space can be chosen as

xi = eαit(cos(βit)u− sin(βit)v)xi+1 = eαit(cos(βit)u + sin(βit)v)

which are seen to be elliptical motions with exponentially varying ampli-tudes. The sign of the real part, αi, determines whether motion is towardsor away from the stationary state.

The stable subspace of the linear system (3.1) is spanned by the eigenvec-tors corresponding to eigenvalues with negative real part, and the unstable

3.2 Nonlinear Systems 19

subspace is spanned by the eigenvectors corresponding to eigenvalues withpositive real part. The center subspace is spanned by the eigenvectors corre-sponding to eigenvalues with zero real part and is neither stable nor unsta-ble. If just one eigenvector belongs to the unstable subspace, the stationarystate is unstable, but if all eigenvectors belong to the stable subspace, thestationary state is stable.

For two-dimensional systems with real eigenvalues of opposite sign theorigin is called a saddle-point, whereas systems with real eigenvalues of thesame sign have a stable or an unstable node at the origin. In the case of twocomplex conjugate eigenvalues the system has a focus at the origin, stabilityagain determined by the sign of the real part. If the real part is equal tozero the origin is called a center. These names are often also applied forsystems with more than 2 dimensions.

3.2 Nonlinear Systems

For oscillating chemical reactions a linear description is not sufficient, butthe concepts from the theory of linear systems can be recycled in the caseof nonlinear systems. For nonlinear systems (3.1) is replaced by the moregeneral expression

x = f(x) (3.4)

For chemical systems described by mass-action kinetics f(x) will consist ofpolynomials in the xi’s. The Oregonator is a good example of this.

Stationary states are characterised by x = f(xs) = 0 as for linear sys-tems, but are not necessarily located at the origin. In real oscillatory chem-ical systems the stationary states are not located at the origin. To make useof the theory in the previous section, we perform the coordinate transforma-tion u = x− xs of the stationary state to the origin. The time derivative ofu is then u = x = f(x). A Taylor expansion of u from the stationary state(remembering that u = x− xs and f(xs) = 0) yields

u = J · u +12!

F2xx · uu +

13!

F3xxx · uuu + · · · (3.5)

where

Jij =∂fi

∂xj

∣∣∣∣x=xs

F 2ijk =

∂2fi

∂xj∂xk

∣∣∣∣x=xs

F 3ijkl =

∂3fi

∂xj∂xk∂xl

∣∣∣∣x=xs

J is known as the Jacobian matrix of f . The Jacobian matrix provides uswith a first linear approximation of the system (3.4) close to the stationary


stateu ' J · u (3.6)

This equation is identical to eq. (3.1) for linear systems, although it onlyapplies in the neighbourhood around the stationary state. Thus station-ary states can be characterised in the same way for nonlinear systems asstationary states in linear systems, the eigenvalues of the Jacobian matrixdetermining the stability. Nonlinear systems can have several stationarystates, and for each stationary state there is a separate Jacobian matrixdetermining the stability.

Example Providing an example for later use, we can compute the Jacobianmatrix for a theoretical chemical reaction known as the Brusselator. Theequations are

c =(

xy

)f(c) =

(a− (b + 1)x + x2y

bx− x2y

)

i.e.

x = a− (b + 1)x + x2y (3.7a)y = bx− x2y (3.7b)

There is one possible stationary state which has the values (xs, ys) = (a, b/a),defined by the parameter values a and b. This provides us with the Jacobianmatrix

J =(

b− 1 a2

−b −a2

)(3.8)

3.2.1 Subspaces and Manifolds

Just as eq. (3.6) is an approximation of the general eq. (3.4) close to astationary state, so are the stable and unstable subspaces of (3.6) only anapproximation of the more general stable and unstable manifolds of (3.4).A manifold is a generalized surface of arbitrary dimension. In fact the stableand unstable subspaces in the stationary point are tangent to the stable andunstable manifolds respectively. Thus motion on the stable manifold will betowards the stationary state, while motion on the unstable manifold will beaway from the stationary state.

In the case of neutral stability the center manifold replaces the centersubspace, which is tangent to the center manifold in the stationary state.

3.2.2 Periodic Solutions

Nonlinear dynamical systems can also have periodic solutions, where afterone period the same state is achieved again. This is seen as oscillations of

3.3 Partial Differential Equations 21

(a) (b)

Figure 3.1: (a) Oscillations in the Ce4+-concentration and (b) the 2D projection of theassociated limit cycle on the [Ce4+]-[Br−] plane with a simulation of the unscaled Orego-nator.

concentrations in chemical systems, e.g. the BZ reaction, and can be ex-pressed mathematically as x(t) = x(t+T ), T being the period of oscillation.In phase space, e.g. the 3-dimensional concentration space in the case of theunscaled Oregonator, the solution must thus describe a closed curve. Thiscurve is called a limit cycle. Examples of a periodic solution of the Oreg-onator and the associated limit cycle can be seen in fig. 3.1. Limit cyclescan arise in different ways, e.g. a Hopf bifurcation which will be describedin section 3.4.

3.3 Partial Differential Equations

If one was to model a reaction-diffusion system like the one in section 2.5it is obvious that a spatial dependence of some sort has to be added on topof the time dependence investigated in the previous sections. If no stirringof the reacting species takes place, either because there is no stirring at allor because the reaction takes place inside a gel, then one simply adds adiffusion term to the reaction term in eq. (3.4) to obtain

∂c∂t

= f(c) + D∇2c (3.9)

where ∇2 is the Laplacian operator and D is the diffusion matrix, which isdiagonal for dilute solutions with the diffusion constants of the individualspecies in the diagonal. Equation (3.9) is a partial differential equation(PDE) with c depending on both space and time.

Since the Laplacian operator is linear, the linear part of eq. (3.9) isdifferent from the linear part of eq. (3.4), i.e. the linear part is

∂u∂t

' (J + D∇2

)u (3.10)

with u being the displacement from the homogeneous stationary state. Totest the stability of the stationary state one chooses the perturbation

u = eikr+λ(k)tu0


named a plane wave perturbation because it is sinusoidal in space. Thisleaves us, after letting the operators work, with the eigenvalue problem

(J− k2D

)u0 = λ(k)u0 (3.11)

i.e. the eigenvalues, λ(k), of the matrix (J− k2D) determine the linear sta-bility of the system with regards to homogeneous as well as inhomogeneousperturbations. The observant reader will have noticed by now that onlyone spatial dimension is considered, which simplifies matters greatly butdoes not influence the conclusions that might be reached regarding linearstability.

The eigenvalue of the linear problem now depends on k, the wavenumberof the plane wave, and that dependence is known as the dispersion relation.From the expression for the plane wave we see that if the wavelength isL then k must equal 2π/L. This means that the homogeneous case corre-sponding to an infinite wavelength results in k = 0, which in turn reduceseq. (3.11) back to the homogeneous case as expected.

Example The dimensionless Oregonator (2.4) may be used to produce spiralwaves if diffusion terms are added. With diffusion terms on the variables uand w but not on v, to resemble the experimental conditions described insection 2.5 as much as possible, the equations become

∂u∂τ

=1ε

{qw − uw + u + u2

}+∇2u

∂v∂τ

= u− v

∂w∂τ

=1ε′{−qw − uw + 2fv}+∇2w

with the dimensionless diffusion constants of u and w both set to 1. Thisimplies that the dimensionless spatial variable, ρ, depends on the real spacevector as ρ = r/

√Duk5. An example of a spiral wave is shown in figure 3.2.

Many other sets of parameter values could have been chosen to produce aspiral.

Textbooks concerned with partial differential equations with special rel-evance for chemical systems are hard to come by, but the best option foran introduction to the subject is the book by Mikhailov [50] which is quiteeasy to approach.

3.4 Bifurcations

In dynamical systems there may be a variable parameter, like the bromateconcentration in the Oregonator. When the value of this parameter is varied,

3.4 Bifurcations 23

Figure 3.2: Spiral produced us-ing the dimensionless Oregonatorwith added diffusion terms. Pa-rameter values are: ε = 0.45, ε′ =0.0009, q = 0.00095, f = 0.79.

a significant qualitative change in behaviour of the system may occur. If thatis the case, eq. (3.4) is more appropriately written

x = f(x, µ) (3.12)

where µ is a real, variable parameter. The value of µ where the systemchanges behaviour, µ0, is known as the bifurcation point. This could be achange in stability of a stationary state or periodic solution. If one was tolook for a mathematical definition in a textbook like [44] it is stated as:

The appearance of a topologically nonequivalent phase portraitunder variation of parameters is called a bifurcation.

As in many other situations it is useful to illustrate with an example.

Example Consider the real, one-dimensional system

x = f(x, µ) = µx− x3

For µ > 0 there are three stationary states, x = 0 and x = ±√µ. Forµ ≤ 0 there is only one stationary state, x = 0. To proceed, we calculate∂f∂x = µ− 3x2. We then insert the stationary state, x = 0, and get the

one-dimensional Jacobian matrix, ∂f∂x

∣∣∣x=0

= µ, which is identical to theeigenvalue. Thus it is obvious that the stationary state, x = 0, is stable forµ < 0 and unstable for µ > 0. Equivalently we get ∂f

∂x

∣∣∣x=±√µ

= −2µ for the

stationary states, x = ±√µ. Thus, they are both stable (since they onlyexist for µ > 0). The situation is illustrated in the bifurcation diagram infigure 3.3, where x is plotted as a function of µ. This explains the name


µ0

x Figure 3.3: Bifurcation diagramfor the pitchfork bifurcation. Thesolid lines are the stable station-ary states, the dashed line is un-stable.

pitchfork bifurcation for this type of bifurcations. In this example, where onestable stationary state changes to an unstable stationary state along withthe emergence of two new stable stationary states at µ = 0, the bifurcationpoint is µ0 = 0.

3.4.1 Hopf Bifurcations

Hopf bifurcations have been found in many different physical and chemi-cal, oscillating systems, e.g. the BZ reaction [41] and suspensions of yeastcells [9]. Thus the characteristics of the Hopf bifurcation deserve a littleattention.

Consider eq. (3.12). Assume that the system has a stationary state, xs,at the parameter value, µ0, with the two complex eigenvalues αi ± iβi ofthe Jacobian matrix. Assume also that the system is “well-behaved”. Thefollowing conditions apply for a Hopf bifurcation at the bifurcation point:

• The Jacobian matrix of (3.12) evaluated at (xs, µ0) has a pair of com-plex conjugated, purely imaginary eigenvalues, i.e. λi = λi+1 = iβi.The remaining n− 2 eigenvalues should have negative real part.

• The complex pair of eigenvalues crosses the imaginary axis with non-zero speed, i.e.

dαi(µ)dµ

∣∣∣∣µ=µ0

6= 0

This condition is known as the transversality condition.

• At the bifurcation point emerges a limit cycle, which has zero ampli-tude.

In practice, in real chemical systems, when observing oscillations in theconcentration of one of the participating species, it is impossible to determinewhether these conditions are met or not.

First of all it is necessary to distinguish between a supercritical Hopfbifurcation and a subcritical. In the case of a supercritical Hopf bifurcation,

3.4 Bifurcations 25

10 20t

0.8

1

1.2

1.4

x

Figure 3.4: Oscillations of thespecies x in the Brusselator. Pa-rameter values are a = 1, b = 2.1.

the system changes from having a stable focus to having an unstable focusencircled by a stable limit cycle. In the subcritical case, the change goesfrom unstable focus to stable focus encircled by an unstable limit cycle.Suppose that the bifurcation is supercritical, then it is possible to make thefollowing observations sufficiently close to the bifurcation point:

• The oscillations are nearly sinusoidal, cf. fig. 3.1, with a third ordercorrection term as the lowest nonlinear term.

• The square of the amplitude of the oscillations can be approximatedwith a straight line when plotted as a function of the bifurcation pa-rameter [41].

Both the last properties can be derived from the properties of the systemon the center manifold associated with the zero real part eigenvalue at thebifurcation point, and this will be treated in a little more detail in chapter 4.

Example To illuminate the subject further we proceed with another exam-ple. Recall that the Brusselator (3.7) has the Jacobian matrix (3.8) at thestationary state (xs, ys) = (a, b/a). It has the eigenvalues

λ1,2 =12

(b− 1− a2 ±

√(b− 1− a2)2 − 4a2

)

which at the bifurcation point b = 1 + a2 become

λ1,2 = ± i a

In figure 3.4 is shown the nearly sinusoidal oscillations for the parametervalues a = 1, b = 2.1.

3.4.2 Excitable and Oscillatory

It is possible for waves to propagate through a medium even if it is notoscillatory. The book by Gray and Scott [17] (amongst others) treats the


5 10τ

0.05

0.1

0.15

v

Figure 3.5: Exampledemonstrating the dif-ference between an ex-citable and an oscil-latory medium. Thesolid line shows theoscillations of the os-cillatory medium, thedashed line shows thesingle oscillation doneby the excitable sys-tem. Parameters arestated in the text.

subject of excitability for systems with a stable stationary state relativelyclose to a Hopf bifurcation.

If the system sits in the stationary state, a very small perturbation willonly lead to tiny damped oscillations back toward the stationary state. Alarger (but still relatively small) perturbation may lead to a very differentresponse. Then one might observe one large excursion toward the station-ary state, resembling the response observed in the nearby oscillatory regionin parameter space. An example of such an excitable response is shownin figure 3.5 (dashed line) for the 2D dimensionless Oregonator with theparameter values ε = 0.05, q = 0.005, and f = 1.15.

By changing the f -value very little to 1.10 the system is now in anoscillatory state, and starting with exactly the same initial conditions asbefore it is shown as the solid line in figure 3.5.

Considering again the excitable response in an array of identical points,it is not hard to imagine how the reponse shown in fig. 3.5 in one pointmay cause the neighbouring points to do the same due to the influence ofdiffusion. This is because only a small perturbation away from the stationarypoint may initiate the excitable response. Observing this excitable responsemoving along the array will in any practical situation be almost impossibleto distinguish from the propagating waves in an oscillatory system, thusmaking it hard to tell whether the system is excitable or oscillatory. The bestoption to tell the difference in an experimental situation is to see whetherthe system displays homogeneous oscillations or not.

3.4.3 Turing Bifurcations

The Turing bifurcation takes its name after the famous mathematician AlanM. Turing, famous for conceiving the idea of the Turing machine, a hypo-thetical machine for doing calculations that he “invented” in the 1930s, as

3.4 Bifurcations 27

well as for inventing the bombes that the allies used for breaking the Ger-man Enigma codes during WWII. In the nonlinear dynamics community heis more famous for predicting the appearance of stationary patterns in anactivator-inhibitor system with unequal diffusivities.

At this point it might be suitable to make a small footnote. When Turingpredicted the stationary patterns in 1952 [67] he did so for a two-componentactivator-inhibitor system with the inhibitor having faster diffusion thanthe activator. For conservatists this means that only systems with theseexact properties can be said to posess Turing structures. It is reminiscent ofanother oddity regarding nonlinear dynamics. Amongst physicists it is com-monly accepted that only physicists can be said to do nonlinear dynamics,whereas the nonlinear chemical kinetics done by chemists do not fall intothat category . . .

To clarify what this author means by a Turing bifurcation and hence theappearing Turing structures we return to the concept of dispersion relationsfrom section 3.3. From eq. (3.11) we know that the linear eigenvalues ofa partial differential equation depend on the wavenumber, k, of the planewave perturbation performed to test the stability of the system. Thus it isnot at all inconceivable that diffusion may destabilise an otherwise stablestationary state for a given wavenumber. For a Turing bifurcation to occur,the following conditions regarding one of the linear eigenvalues need to bemet for some wavenumber k0 6= 0:

λ(k0) = 0,dλ

dk

∣∣∣∣k=k0

= 0 (3.13)

Another requirement is that the imaginary part of the eigenvalue is equal tozero on either side of the bifurcation point and not just at the bifurcationpoint.

The requirement is thus that a single real eigenvalue becomes positive atthe bifurcation point, and that it happens for a single wavenumber differentfrom zero. It should be emphasised that any bifurcation with these propertieswill be considered a Turing bifurcation. The k0 for which the eigenvaluefirst becomes positive will be the spatial mode to grow first, but as for theoscillations originating from the Hopf bifurcation it will not grow indefinitelyas nonlinear terms will take over. More about the amplitude of the stablespatial waves, the Turing structures, originating from a Turing bifurcationin section 4.2.3.

Example The Brusselator has been studied extensively [57, 78, 45] regard-ing Turing instabilities. Including diffusion terms it is written

x = a− (b + 1)x + x2y + dd2xdr2

(3.14a)

y = bx− x2y + d qd2ydr2

(3.14b)


0.5 1k

-2

-1

0

1

λ

q = 3.0q = 4.0q = 5.0q = 4.0

Figure 3.6: Dispersion relationsfor the Brusselator. Parametervalues are a = 2, b = 4, and d = 2with q being varied. For q = 3 thereal part of λ (dotted line) is neg-ative for all values of k. At q = 4the real part of λ (dashed line) isequal to zero for one k-value, andat q = 5 the real part of λ (dot-dashed line) is positive for a wholerange of k-values. The solid lineshows the imaginary part of λ forq = 4.

where q is the proportion between the diffusion constants of the inhibitor,y, and the activator, x. Applying eqs. (3.11) and (3.13) to eq. (3.14) resultsin a k0 given by

k0 =

√(b− 1)q − a2

2qd

requiring that (b − 1)q − a2 > 0. By insertion of the expression for k0 intothe equation λ(k) = 0 one eventually finds that

q =(

a√b− 1

)2

in the bifurcation point. Choosing the parameter values a = 2, b = 4,and d = 2 the bifurcation is expected to occur for q = 4 at k0 = 1/

√2 as

shown in fig. 3.6. For q = 3 the real part of the eigenvalue is negative forall values of k, but for q = 4 it is zero at k = 1/

√2 and for q = 5 it is

positive for a whole range of k-values. Also shown is the imaginary partof the eigenvalue for q = 4 to emphasise that the bifurcating eigenvalueis real in the neighbourhood around k0. Since b = 4 < a2 + 1 = 5 weexpect the eigenvalue to be negative for k = 0 according to the example insec. 3.4.1, which is also the case. Had b been equal to 5 we would observea simultaneous transition of eigenvalues from negative to positive for bothk = 0 (Hopf bifurcation) and for k > 0 (Turing bifurcation). This is knownas a codimension-2 bifurcation point since two different instabilities occurat the same point in parameter space. In this case we are talking about asocalled Turing-Hopf bifurcation.

The stationary structure that evolves from the Turing bifurcation isshown in fig. 3.7 for the species x. It resembles the time-periodic oscil-lations in fig. 3.4 but now the periodicity is in space and the structure isstable in time. Taking the comparison even further one might suspect thatthis structure originates from a supercritical bifurcation, and it does show a

3.4 Bifurcations 29

50 100 150 200 250position

1

1.5

2

2.5

3

x

Figure 3.7: The stationary pat-tern evolving for the Brussela-tor when doing a simulation withthe parameter values a = 2, b =4, d = 2, and q = 5. The lengthof the system is 6.284 and the 1-dimensional grid has 256 points.

linear dependence of the square of the amplitude on the distance from thebifurcation point. More about this in section 4.2.3.

4Amplitude Equations

For every kind of local bifurcation exists a generic equation which close tothe bifurcation point of a given system, in most cases, is topologically equiv-alent to that system with as few nonlinear terms as possible. This genericequation is called the normal form for the corresponding bifurcation. Thecoefficients of the normal form can conveniently be calculated by applyingthe theory of amplitude equations. Two systems, both in the vicinity ofthe same bifurcation, can then be compared directly. The results of thischapter are taken from the Ph. D. thesis by Mads Ipsen [33] and furtherreading about normal forms and amplitude equations can be found in thatthesis and its references. A particularly systematic treatment of amplitudeequations can be found in [31]. It avoids explicit treatment of normal formsand has results for different common bifurcations, but is somewhat harderto approach.

4.1 Normal Forms

We consider a dynamical system in n dimensions

x = f(x) (4.1)

with 0 as a stationary solution of the system. We seek a transformation inthe bifurcation point of x

x = z + hr(z) (4.2)

so that the expression for z contains as few nonlinear terms as possible. Tofigure out which terms should be included we need to consider the eigen-values, λ1, . . . , λn, of the Jacobian matrix of the system (4.1). They areincluded in the key quantity

n∑

j=1

mjλj − λi (4.3)

32 Amplitude Equations

with i = 1, . . . , n and∑n

j=1 mj = r ≥ 2. The eigenvalues, λ1, . . . , λn, aresaid to be resonant of order r if the quantity (4.3) is equal to zero for somecombination of m’s and λ’s. If a system is resonant of order r it means thatthat system will have terms of order r in the expression for z. If it is notpossible to find a resonant term for a given order, that order can be excludedin the expression for z.

Examples Here will be presented the normal forms up to third order forthe pitchfork and Hopf bifurcations without any further derivation. For thepitchfork bifurcation the normal form is

z = g3z3 (4.4)

with z being a real variable and z = zu, u being the eigenvector correspond-ing to the bifurcating eigenvalue. For the Hopf bifurcation the normal formis

z = iω0z + g3z |z|2 (4.5)

with iω0 as the bifurcating eigenvalue in the bifurcation point and z be-ing a complex variable and z = zu + zu, u and u being the eigenvectorscorresponding to the bifurcating eigenvalues.

4.2 Amplitude Equations

The normal forms are not very useful without knowledge of the coefficientvalues for the system in question, and normal form theory does providea method to determine those coefficients, although the procedure is longand tedious. So fortunately there exists another method to determine thecoefficients, namely by looking at the amplitude equations. The theory isbased on the transformation (4.2) with x on the center manifold and z inthe center subspace. The transformation is uniquely determined when ittransforms (4.1) into the normal form. As a basis for the center subspacewe choose the eigenvectors corresponding to the bifurcating eigenvalues, e1,. . . , enc , with nc being the dimension of the center manifold. The motionin the center subspace is then written

z(t) =nc∑

n=1

Ai(t)ei (4.6)

The Ai’s are the amplitudes and the amplitude equation is

Ai = fi(A1, . . . , Anc) (4.7)

We may write the map h(z) as a Taylor expansion in the amplitudes to adesired order

h(z) =∑m

AmΦm (4.8)

4.2 Amplitude Equations 33

where m = (m1, . . . , mnc) with∑nc

i=1 mi = r ≥ 2, Am =∏

i Amii , and

the n-dimensional vector Φm is determined from the normal form. Equa-tions (4.6) and (4.8) can then be inserted into the transformation (4.2) anddifferentiated with respect to time. This yields, when put equal to eq. (3.5)

nc∑

i=1

Ai

[ei +

∑m

∂Ψm

∂Ai

]= J ·

[nc∑

i=1

Aiei +∑m

Ψm

]+

(4.9)

12Fxx ·

[nc∑

i=1

Aiei +∑m

Ψm

]2

+13!

Fxxx ·[

nc∑

i=1

Aiei +∑m

Ψm

]3

+ · · ·

with Ψm = AmΦm. This equation can be solved recursively for the coeffi-cients of the amplitude equation and the vectors Φm for consecutive ordersstarting with the lowest order r = 2, increasing to the desired order.

4.2.1 Unfoldings of Amplitude Equations

In real systems it is not usually desirable to operate the system at thebifurcation point, since the oscillations (in the case of a Hopf bifurcation)are infinitely small there. To be able to observe oscillations one needs tomove away from the bifurcation point. For this purpose the dynamicalsystem (4.1) needs to be extended to

x = f(x,µ) (4.10)

assuming that the bifurcation has codimension-one, and that it occurs atµ = 0. The Jacobian matrix, J, depends on the bifurcation parameter and sodo the bifurcating eigenvalue, λi, and associated right and left eigenvectors,ei and e∗i . At µ = 0 it is possible to write the expansions

J(µ) = J0 + J1µ + J2µ2 + · · · (4.11a)

λi(µ) = λi0 + λi1µ + λi2µ2 + · · · (4.11b)

ei(µ) = ei0 + ei1µ + ei2µ2 + · · · (4.11c)

e∗i (µ) = e∗i0 + e∗i1µ + e∗i2µ2 + · · · (4.11d)

It is of particular physical relevance to consider the dependence of the bi-furcating eigenvalue on µ, and λi1 can be found by solving λi(µ) = e∗i (µ) ·J(µ) · ei(µ) to the first order

λi1 = e∗i0 · J1 · ei0 (4.12)

where J1 is the derivative of f(x, µ) with respect to µ.


4.2.2 Hopf Bifurcation

For a Hopf bifurcation the normal form including the unfolding is written

z = (iω0 + σ1µ)z + g3z |z|2 (4.13)

The vectors, Φm, are in this case also calculated according to eq. (4.9)yielding the normal form coefficients

g3 = e∗1 · Fxx(e1,Φ110) + e∗1 · Fxx(e1,Φ200) +12e∗1 · Fxxx(e1, e1, e1) (4.14a)

σ1 = e∗1 · Fxµ · e1 + e∗1 · Fxx(e1,Φ001) (4.14b)

where a bar signifies complex conjugation. A full table of expressions forthe vectors Φm for this kind of bifurcation can be found in [31] table II.Equation (4.13) is also known as the Stuart-Landau equation.

It can be very instructive to write eq. (4.13) in polar coordinates, i.e. forz = reiθ

r = σr1µr + grr

3 (4.15a)θ = ω0 + σi

1µ + gir2 (4.15b)

where g3 = gr+igi and σ1 = σr1+iσi

1. In the following we assume Re(

dλidµ

)=

σr1 > 0, i.e. that the real parts of the complex conjugate eigenvalues become

positive when increasing µ above 0. Since r can not be negative it has two

stationary states, rs = 0 and rs =√

σr1µ−gr

. First we consider drdr = σr

1µ+3grr2s

for rs = 0, i.e. drdr = σr

1µ. This means that it is stable for µ < 0 and unstable

for µ > 0 as expected. The stationary state rs =√

σr1µ−gr

exists only for

−µ gr > 0. For this state drdr = 2grr

2s , and we see that its stability depends

on the sign of gr. For gr < 0 it is stable and for gr > 0 it is unstable.For gr < 0 this state exists only for µ > 0 and for gr > 0 only for µ < 0.Figure 4.1 descibes the two different cases, gr < 0 and gr > 0, with solidlines indicating stable states and dashed lines indicating unstable states.The case where gr < 0 is called a supercritical Hopf bifurcation, whereas thethe bifurcation is called subcritical for gr > 0. If σr

1 < 0 the dependence ofthe criticality on gr does not change, and in the bifurcation plots in fig. 4.1we would have to make the transformation µ → −µ, i.e. make a reflectionof the plots in the y-axis.

It should be noted that in the supercritical case where rs =√

σr1µ−gr

a plotof the square of the amplitude vs. the bifurcation parameter should resultin a straight line. This is the most convenient method to experimentallyverify whether an observed Hopf bifurcation is supercritical or not.

4.2 Amplitude Equations 35

-0.5 0 0.5 1µ

-0.5

0

0.5

1

r

(a)

s

-0.5 0 0.5 1µ

-0.5

0

0.5

1

rs

(b)

s

Figure 4.1: Supercritical (a) and subcritical (b) Hopf bifurcation for the two cases gr < 0and gr > 0 respectively. Solid lines indicate stable states and dashed lines indicate unstablestates.

The Complex Ginzburg-Landau Equation

If one adds a diffusion term to the Stuart-Landau equation it becomes

z = (iω0 + σ1µ)z + g3z |z|2 + d∇2z (4.16)

Changing variables, w = eiω0tz, we arrive at the Complex Ginzburg-LandauEquation (CGLE)

w = σ1µw + g3w |w|2 + d∇2w (4.17)

where the diffusion term, d, is found as d = e∗i ·D·ei, with D being the diffu-sion matrix belonging to the system (4.10). Like the Stuart-Landau equationis the generic equation for systems close to a Hopf bifurcation, the CGLE isthe generic equation for unstirred systems close to a Hopf bifurcation and itcan be used to explain observed instabilities in experiments [53, 54, 81, 82].For a review of some of the properties of the CGLE, especially in the contextof chemical reactions, see reference [32]

4.2.3 Pitchfork Bifurcation

For the pitchfork bifurcation the normal form including the unfolding is

z = σ1µz + g3z3 (4.18)

The vectors, Φm, are calculated according to eq. (4.9) and the normal formcoefficients are then

g3 = e∗1 · Fxx(e1,Φ20) +16e∗1 · Fxxx(e1, e1, e1) (4.19a)

σ1 = e∗1 · Fxµ · e1 + e∗1 · Fxx(e1,Φ01) (4.19b)


A full table of expressions for the vectors Φm for this bifurcation can befound in [31] table I.

If one was to derive the amplitude equation for the Turing bifurcation(including unfolding) the expression is identical with the expression for thepitchfork bifurcation. This applies when looking exclusively at the modethat first becomes unstable at the bifurcation point [77]. It means that thesquare of that amplitude of that mode is expected to grow linearly with thebifurcation parameter when g3 < 0, i.e. in the supercritical case. This wasthe case in the example with the Brusselator, and we thus conclude that ithas a supercritical Turing bifurcation.

Unfortunately there are no formulas readily available to calculate the co-efficients, since the Laplacian operator adds to the complexity of the deriva-tion of the formulas, and one has to take discretisation due to finite systemsize into account.

5Nonlocal Coupling

This chapter is, as the title indicates, concerned with nonlocal coupling.By nonlocal is meant something that is neither local, i.e. like a diffusionalcoupling, nor global, i.e. depending on the whole system at once. It is rathersomething that depends on the system state within a certain distance whichdepending on several lengthscales.

In this study the light-sensitivity described in section 2.4 is utilised. Ithas been used before for different kinds of external forcing of the BZ reac-tion [63, 18, 55, 35, 49], including periodic forcing [46, 47] and global couplingexperiments [73, 74]. There is also a study linking the two phenomena [75].

In most of the mentioned studies homogeneous light has been applied inthe feedback, thus leaving no opportunity to introduce different lengthscalesinto the system. To be able to observe Turing patterns it is necessary to havedifferent lengthscales, classically observed as different diffusion lengths inactivator-inhibitor systems as mentioned in section 3.4.3. Different diffusionlengths are only possible in experiments not taking place in purely aqueoussolution, but the gel in the BZ reaction does not solve that problem sinceit is not the activator that is immobilised. The first experimental finding ofTuring structures in the BZ reaction [26], part of this study, was done byapplying inhomogeneous light to the BZ reaction with different lengthscales.Subsequently Turing structures have also been discovered in the BZ systemdispersed in an AOT microemulsion [71].

Taking inspiration from surface chemical reactions where nonlocal inter-actions are an inherent part of the system, it was necessary to choose a wayto introduce the nonlocality. Belintsev et al. showed that by choosing aGaussian kernel and introducing it into a one-component, linear chemicalsystem, a Turing instability was possible [3, 2]. In their example an effec-tive long-range repulsive interaction resulted from adiabatic elimination ofa fast-relaxing inhibitor species. This idea is applied in this study to showhow a 4-species model may have a Turing instability.

Choosing a specific kernel for the nonlocal feedback via the sensitivity

38 Nonlocal Coupling

Computer

CCD Camera

VideoProjector

Mirror

Reactor

Gel

Figure 5.1: Schematic drawing of thesetup. The CCD camera records apicture of the gel through the semi-transparent mirror while the video pro-jector is projecting a homogeneous back-ground light. The recorded image isthen transferred from the camera to thecomputer which in turn calculates thefeedback image to send to the video pro-jector. The video projector projects thefeedback light on the gel via the mirror,interrupted very shortly for the record-ing of a new image by the camera. Thewhole circle is completed in 2 seconds.

to light and the corresponding term in the 2D dimensionless Oregonatorit is predicted with a linear stability analysis that it should be possible toproduce Turing structures in the BZ reaction. It is also predicted that thesystem will contain codimension-2 points. The predictions are confirmed bydoing experiments and numerical simulations.

5.1 Experiments

The experiments for the nonlocal coupling, the Morgantown experiments,were carried out with the light-sensitive BZ reaction, as mentioned earlier.The use of gel and solution in the petri dish was as described in chapter 2with a total volume of ca. 150 mL. As the lightsource was used a modifiedvideo projector both for recording images and perturbing the reaction. Thisenables a time-dependent, inhomogeneous distribution of light across thesurface of the system.

Prior to each experiment, the image projected by the video projector(consult fig. 5.1 for references to the experimental setup) was adjusted ateach pixel by an iterative algorithm to ensure a spatially uniform illumina-tion field, upon which all subsequent projected images was based. The localconcentration of oxidised catalyst was recorded with the CCD-camera, andthe recorded image was divided into an array of 100 × 128 square cells. In allof the experiments, the lateral size of each cell was approx. 5 times smallerthan the spiral wave length. The feedback occurred via an illumination field(2.0 cm × 2.5 cm) projected from the video projector onto the face of the gelmedium through a 460 nm interference filter. The nonlocal feedback at eachgrid point was computed according to the recorded image, with the result-ing projected image updated at 2 s intervals (which is significantly smallerthan the oscillation period). Hence, bromide ions were locally produced tomodify the local excitability as described in chapter 2.

The intensity, I(r, t), of the projected illumination field was computed

5.1 Experiments 39

Figure 5.2: Result for r1 = 2r2 = 14 sqs, with the system excitable in the absence offeedback. The two images were recorded 30 minutes apart and we observe that there aretwo types of domains. One contains a steady state, the other contains waves and spirals.By comparing the two images it is clear that the boundary between the domains does notmove significantly within 30 minutes.

once every cycle as

I(r) =Imax

2{1− tanh [χV (r)]} , (5.1)

where Imax (ca. 80 mW/cm2) is the maximum light intensity. The parameterχ characterises the width of the interface in which I increases from zero toImax, and in the experiments χ was chosen such that in practice tanh[χV (r)]= H(z), where H(z) = 1 for z > 0 and H(z) = −1 for z < 0. The nonlocalpotential, V (r) =

∑w(r−r′) v(r′), represents the coarse grained convolution

of the observed concentration of oxidised catalyst, v (cf. fig. 5.2 where whitemeans large v and black means v = 0), with the kernel w(r) representing an“effective binary potential” [23, 3] which vanishes for distances exceeding acharacteristic interaction radius. Here, we choose

w(r) = − 1√πr2

1

exp(−r2

r21

)+

1√πr2

2

exp(−r2

r22

), (5.2)

with r1 > r2, i.e., w(r) is positive for short distances r and negative forlarger distances. In the calculation of the potential, V (r), the summationover r′ extends over the entire array of square cells, and in section 5.1.2 theissue of applying different boundary conditions is addressed. The charac-teristic radii, r1 and r2, can be chosen independently from the remainingexperimental parameters. In these experiments the condition, r1 = 2r2, wasobeyed throughout.

Two different sets of conditions on the chemical composition of thecatalyst-free solution were applied. One set corresponded to an excitablesystem in the absence of feedback, while the other set of conditions meantthat the system was oscillatory in the absence of feedback.


Figure 5.3: Result for r1 = 2r2 = 28 sqs, with the system excitable in the absence offeedback. The two images were recorded 30 minutes apart and as in fig. 5.2 we observespiral domains on a steady state background. The domain sizes have increased with theincrease in the radii as expected.

5.1.1 Excitable Conditions

The composition of the catalyst-free solution was: [BrO−3 ] = 0.28M, [H2SO4]

= 0.4M, [MA] = 0.175M, [BrMA] = 0.15M 1. For r1 = 14 in units of squarecells (sqs) the results shown in fig. 5.2 were obtained by applying the feed-back. The two images were recorded 30 minutes apart and we observe thatthere are two types of domains. One contains a stationary state, the othercontains spiral waves. By comparing the two images it is clear that theboundary between the domains does not move significantly within 30 min-utes. We therefore conclude that the domain boundaries are stable but atthe same time there is wave activity within the spiral domains, as can beseen by the difference in the position of the wave fronts. Before recordingthe first image came a transient period where the domain boundaries weremoving around until they found the positions shown in the images.

For r1 = 28 sqs a similar result was obtained, as shown in fig. 5.3. Asbefore the images are separated by 30 minutes and here domains of spiralwaves on a stationary state background with stationary domain boundariesare also observed. The only difference is in the sizes of the domains, whichhave increased as expected from increasing the radii, r1 and r2. If the radiiare decreased to or slightly below the size of one wavelength, what appearsto be, with the contrasting capability of the camera in use, a stationarystate throughout the system is observed. This is most likely due to the factthat the domain sizes set by the radii are no longer large enough fit a spiral.What actually happens in the experiment is that the waves present are

1Bearing in mind the discussion from sec. 2.3 it should be mentioned that a 1M Br−

solution was pumped at 1 mL/min. into the solution containing the remaining speciesto produce Br2 and subsequently BrMA. The given concentration is assuming that onlyBrMA and no Br2MA was produced

5.1 Experiments 41

Figure 5.4: Result for r1 = 2r2 = 14 sqs, with the system oscillatory in the absence offeedback. The two images were recorded 30 minutes apart and as in fig. 5.2 we observespiral domains on a steady state background with roughly equal domain sizes.

driven to the edge, resulting in the stationary state. With the whole systemcovered with a stationary state, the light feedback is half the maximum lightintensity throughout the system. This prevents any waves from reappearing.

5.1.2 Oscillatory Conditions

The composition of the catalyst-free solution was: [BrO−3 ] = 0.5M, [H2SO4]

= 0.25M, [MA] = 0.125M, [BrMA] = 0.125M. For this composition thefeedback with r1 = 14 sqs was also applied. The result can be viewed infig. 5.4. Again the images have been recorded 30 minutes apart and as inthe excitable case stable spiral domains on a stationary background wereobserved. For comparison, different radii were also tried in the oscillatorycase. This time the radii were decreased so that r1 = 8 sqs. The result canbe viewed in fig. 5.5. We observe that the domain sizes decrease as expectedand that the system can just barely fit the spirals into the domains.

To illustrate the connection between the feedback and the system statefig. 5.6 has been included. The image projected from the video projector isshown on the left and the corresponding image of the system on the right.Evidently the domains containing a stationary state correspond to the partsof the feedback with high light intensity (the white areas) and the partsof the feedback with no light intensity (the black areas) correspond to thespiral domains. Outside the region of interest, i.e. the region for which thefeedback is calculated, the feedback is set to the maximum light intensityof the video projector, thus killing all wave activity outside the region ofinterest by making the medium non-excitable. This procedure is necessarydue to the fact that casting the gel on the microscope slide can not be doneto fit exactly the measures of the region of interest. From the recorded imagewe know that the areas with high feedback intensity are black, correspondingto a low concentration of oxidised catalyst. This is also the case for the part


Figure 5.5: Result for r1 = 2r2 = 8 sqs, with the system oscillatory in the absence offeedback. The two images were recorded 30 minutes apart and again we observe spiraldomains on a steady state background, this time with very small domain sizes just largeenough to contain the spirals.

of the gel outside the region of interest, so in calculating the potential, V (r),the concentration of oxidised catalyst has been set to zero outside the regionof interest, thus applying the socalled Dirichlet boundary conditions. Theseboundary conditions were applied during most of the experiments, but tomake sure that the boundary conditions did not have any significant effecton the obtained results the experiment with r1 = 14 sqs was repeated forperiodic and no-flux boundaries. A comparison of the three different setsof boundary conditions can be seen in fig. 5.7 where the first panel showsthe resulting domains with Dirichlet boundaries and the last two panelsshow the resulting domains for periodic and no-flux boundaries respectively.Qualitatively the results are the same, the only difference being the locationof the domains. In the Dirichlet case the spiral domains tend to cluster nearthe boundaries of the medium whereas in the other two cases they tend tocluster away from the medium boundaries. This is not surprising consideringthe nature of the boundary conditions.

The radii can be reduced below one wavelength to r1 = 4 sqs. Usingexcitable conditions only a stationary state can be observed throughoutthe medium. Under oscillatory conditions, however, complex behavior isfound, with travelling wave fragments that interact and split as they reacha critical size (see fig. 5.8). In this case the domain sizes set by the nonlocalpotential are clearly too small to contain spirals. Qualitatively, we alsoobserve the same result in this case when changing the boundary conditionsof the potential.

If the radii are increased slightly to r1 = 6 sqs a mixture of the two kindsof behaviour previously observed is found. In fig. 5.9 is displayed a series ofimages recorded 400 seconds apart. Considering the upper right corner ofthe system we find that this part of the system seems to contain some kindof stationary structure, whereas the remaining part of the system behaves

5.2 The Model 43

Figure 5.6: Comparison of the result from fig. 5.5 with the image fed back from theprojector onto the system. Areas with high light intensity (white) correspond to thesteady state domains whereas areas with low light intensity (black) correspond to thespiral domains.

in a manner similar to r1 = 4. This behaviour could be due to the factthat r1 is larger than one wavelength and r2 is smaller than one wavelength.Slightly different results can be obtained by decreasing the maximum lightintensity, which simulations indicate is equivalent to decreasing the updatetime of the feedback. This was not possible in the experiments due to thelimitation set by the computer in computing the feedback. For Imax ≈ 75mW/cm2 and r1 = 10 sqs the result can be viewed in fig. 5.10. The imageswere recorded 10 seconds apart, and again the spiral domains can be found.This time, however, we observe waves propagating through the stationarydomains. These waves have a larger wavelength than the waves in the spiraldomains due to the different excitability of these domains. This effect isparticularly pronounced in the upper left corner.

5.2 The Model

To simulate wave propagation in the light-sensitive BZ-medium we considerthe 2D dimensionless Oregonator model, including the additional term ϕ =ϕ(r, t) taking into account the additional bromide production that is inducedby the external illumination of the system (sec. 2.4, eq. (2.10)):

∂u∂t

= ∇2u +1ε

{u− u2 − [2fv + ϕ(r, t)]

u− q

u + q

},

(5.3)∂v∂t

= u− v,

where the variables u and v correspond to the concentrations of the autocat-alytic species HBrO2 and the oxidised form of the catalyst, respectively. The


Figure 5.7: Result for r1 = 2r2 = 14 sqs in the oscillatory system with different boundaryconditions. From left to right the boundary conditions are: Dirichlet, periodic, and no-flux. Qualitatively, the results do not differ.

absence of a diffusion term for v is supposed to model the immobilisation ofthe catalyst in the gel.

With reference to the discussion in chapter 2 it may be noted that thetype of oscillations for the parameter values used in the numerical simula-tions of eq. (5.3) (sec. 5.4) are highly relaxational. This justifies the use ofthe 2-dimensional version of the Oregonator. The linear stability analysis ofthe system is also simplified by choosing the 2D version. At the end of theday, the important thing is whether or not the simulations with the chosenmodel have any predictive power regarding the experiments.

Naturally we choose the feedback term, ϕ, as in the experiment

ϕ(r, t) =K

2[1− tanh [χV (r, t)]] , (5.4)

with ϕ = I KImax

.The nonlocal potential, V (r), is modeled as:

V (r) =∫

w(r− r′) v(r′) dr′, (5.5)

where the integration extends over the entire medium and w(r) is the sameas in eq. (5.2). This replaces the coarse-grained potential we used in theexperiments for practical reasons.

Moreover, it should be noted that the system of equations (5.3) togetherwith the definitions eqs. (5.5) and (5.4)) can be interpreted as a limitingcase of a more general four-variable reaction-diffusion system, if in the two-dimensional system instead of eq. (5.3) we choose

w(r) = − 12r1

exp(−|r|

r1

)+

12r2

exp(−|r|

r2

). (5.6)

in 1 spatial dimension.

5.2 The Model 45

Figure 5.8: Result for r1 = 2r2 = 4 sqs in the oscillatory system. The four images wererecorded 2 seconds apart and we observe spiral breakup and fragments floating aroundthe system.

Then, considering the system of reaction-diffusion equations

∂u∂t

= ∇2u +1ε

{u− u2 −

[2fv +

K

2[1− tanh [χ(−w1 + w2)]]

]u− q

u + q

},

∂v∂t

= u− v,

(5.7)∂w1

∂t= τ−1

1

[v − w1 + r2

1∇2w1

],

∂w2

∂t= τ−1

2

[v − w2 + r2

2∇2w2

],

we obtain in the limit τ1, τ2 → 0 (c.f [3, 2]), i.e. when the species w1 andw2 relax much faster than v,

w2(r, t)− w1(r, t) = V (r, t), (5.8)

and hence, the system of eqs. (5.7) reduces to eqs. (5.3).Considering those equations we see that, depending on the nature of

the feedback, ϕ, we can go in different directions with the dynamics of the


Figure 5.9: Result for r1 = 2r2 = 6 sqs in the oscillatory system. The four images wererecorded 400 seconds apart and we observe fragments floating around some parts of thesystem, whereas the upper right corner seems to contain some kind of stationary structure.

light-sensitive BZ reaction diffusion system. Known examples of feedbackare periodic forcing [55, 46, 47] and global coupling [73, 74], that both showstanding wave behaviour when applied to the BZ reaction. Global coupling,as well as periodic forcing [75], results in standing cluster patterns withno intrinsic wavelength and thus can not be characterised as true standingwaves. Common to the two kinds of feedback is that none of them are spa-tially specific, i.e. in neither of the two is the feedback different in differentparts of the spatially extended system. Designing the feedback as describedin this section we obtain the possibility of spatial instabilities.

5.3 Linear Stability Analysis

The stationary uniform states of the system eqs. (5.3) satisfy u=v=u0 whereu0 is obtained as a solution of the cubic equation

u3 − (1− q − 2f) u2 − (q + 2qf −K/2) u− qK/2 = 0, (5.9)

since ϕ = K/2 in the uniform stationary state. For typical parameter values,this equation has a single solution. It can be either stable or unstable withrespect to uniform perturbations.

5.3 Linear Stability Analysis 47

Figure 5.10: Result for r1 = 2r2 = 10 sqs in the oscillatory system with Imax ≈ 75mW/cm2. The four images were recorded 10 seconds apart and we observe waves withlarger wavelengths propagating through the “steady state” domains.

The stability of the uniform stationary states is tested by adding planewave perturbations in one dimension. Substituting u=u0 +δu exp(λkt+ ikr)and v=u0 + δv exp(λkt + ikr) into eqs. (5.3) and linearising, we arrive atthe eigenvalue problem

(J(k)− λkI)(

δuδv

)= 0, (5.10)

that determines the linear growth rates λk as a function of the dimension-less wavenumber k (I denotes the identity matrix). This is the eigenvalueproblem revisited from eq. (3.11) in a slightly different form. The elementsof the dimensionless 2× 2 linearisation matrix J(k) are given by

J11(k) = ε−1

[1− 2u0 − 2q

(u0 + q)2(2fu0 + K/2)

]− k2,

J12(k) = −2f

ε

u0 − q

u0 + q+ Ck, (5.11)

J21(k) = 1, J22(k) = −1,


where

Ck = −Kχ

2ε

u0 − q

u0 + q

{exp

[−r21k

2

4

]− exp

[−r22k

2

4

]}, (5.12)

if the effective binary potential is defined by eq. (5.2), and

Ck =Kχ

2ε

u0 − q

u0 + q

[r21k

2

1 + r21k

2− r2

2k2

1 + r22k

2

], (5.13)

if w(r) is given by eq. (5.6). Note that in both cases Ck is positive if and onlyif the condition r1 > r2 is satisfied. Moreover, note that Ck tends to infinityas χ → ∞, and that the results are obtained for an indefinite medium.Derivations of eqs. (5.12) and (5.13) can be found in the appendix.

The eigenvalues λ±k are hence given by the equation

λ±k =J11(k)− 1

2±

[(J11(k)− 1)2

4+ J11(k) + J12(k)

] 12

. (5.14)

They can be either real or complex. It is easy to show, however, thatunstable modes with nonzero wave numbers have always real growth rates.The situation is similar to the classic Turing instability demonstrated forthe Brusselator in chapter 3. The dispersion Re

[λ+

k

]has a single maximum

at a wave number k0 which changes its sign at the instability, determinedby the conditions (3.13). Note that here, these conditions correspond to theequations

Bk ≡ J11(k) + J12(k) = 0,dBk

dk2= 0, (5.15)

which can be satisfied if r1 > r2 and the parameter combination Kχ islarge enough. In general, eqs. (5.15) can be solved only numerically for theinstability boundaries and the wave number k0. In order to obtain somefurther insight into the dependence of k0 on the parameters, it is instructiveto analyse the limiting cases ρ → 0 and ρ → 1, where ρ ≡ r2/r1. Assumingthat r2 → 0 with r1 remaining finite, i.e. ρ → 0, we obtain the followingexpression for k0 from eq. (5.15)

k20 = − 4

r21

ln(

4Keff r2

1

), (5.16)

where the notation Keff = (2 ε)−1Kχ(u0− q)/(u0 + q) has been introduced.The instability boundary is determined by the equation

Bk=0 = −Keff +4r21

[1− ln

(4

Keff r21

)]. (5.17)

In this limit a Turing instability is hence possible providing Keffr21 > 4.

Note also that in this case k0 → 0, i.e. the critical wave length diverges, as


0 1k

-1

-0.5

0

λk

+ Figure 5.11: Dispersion relationsRe[λ+

k ] vs k for the reduced uni-form steady state in eq. (5.9).The parameters are ε = 0.05, f= 0.95, q = 0.005, r1 = 2r2 = 4,K = 0.1, and χ = 1620 (dashedline), χ = 1820 (solid line), χ =2020 (dot-dashed line).

r1 → ∞. This is not surprising since w(r) → 0 in that case, meaning thatϕ becomes homogeneous in space and thus does not leave the possibility ofspatial instabilities.

In the other limit, i.e. assuming that ρ → 1, one arrives at

Keff (1− ρ) = 8 exp(

r21k

20

4

) (4 r2

1 − r41k

20

)−1. (5.18)

for the instability boundary. In this limit, the critical wavelength is alsoexpected to diverge as r1 →∞, since this also means that w(r) → 0. Hencethe solution

k20 =

Bk=0

2−

[B2

k=0

4− 4Bk=0

r21

]1/2

, (5.19)

for k20 is preferred from the equation k4

0 − Bk=0k20 + 4Bk=0/r2

1 = 0. Itshould be noticed that the condition Bk=0 > 0 applies for the existence ofthe solutions for k2

0. This is a severe constraint on the possible choices ofparameter values. In the limit r1 →∞ eq. (5.18) reduces to Keff (1− ρ) =−Bk=0e/2.

The overall conclusion on this analysis of the limits is that r2 should notbe too close to r1 to be able to observe a Turing bifurcation, and r1 shouldnot become too large either. Then it is possible by choosing a sufficientlylarge value of Keff , e.g. by increasing χ, to cross the instability boundary.

Typical dispersion relations Re[λ+

k

]vs. k, as obtained numerically from

solving eq. (5.14), are shown in Fig. 5.11 for K = 0.1: For χ = 1620 thedispersion already has developed a single maximum, however all growth ratesare still negative (dashed line). The solid line shows the critical situation forχ = 1820, where the maximum of λ+

k changes its sign, whereas for χ = 2020perturbations grow for a whole interval of wavenumbers (dot-dashed line).We note that, at the critical value of χ, Ck is predicted to have a maximum atk2 = ln(4)/3 ≈ 0.462 (see [3]), which agrees well with the observed maximumat k = 0.67 since the term −k2 in J11 does not, with the given parameter


0 0.25 0.5f

0

500

1000

χ a

0 0.25 0.5f

0

200

400

χb

0 0.5 1f

0

500

1000

χ c Figure 5.12: Bifurcation diagrams in thef − χ plane for the parameter valuesε = 0.05, q=0.005, and (a) K = 0.1,(b) K = 0.05, (c) K = 0.02. The verti-cal, dashed lines denote Hopf bifurcationsof the individual states, the black circlescorrespond to the codimension-2 points.The symmetry breaking instabilities aremarked by the solid and dot-dashed lines(see also text).

values, contribute significantly in the determination of the k-value for whichdBk/dk2 = 0.

Figure 5.11 shows the dispersion relations for r1 = 2r2 = 4. If largerradii are chosen, the k0 where λk has a maximum will shift to lower values,meaning that the spatial instability will have longer wavelength. This is inagreement with intuition and the experimental results from section 5.1.

Figure 5.12 displays the instability boundaries of the homogeneous sta-tionary state in the parameter plane (f , χ) for a relatively large value of thecoupling constant [K = 0.1 in (a)], intermediate coupling [K = 0.05 in (b)],and rather weak coupling [K = 0.02 in (c)]. For K = 0.1 and arbitrary fthere is a homogeneous steady state solution (the reduced state) of eqs. (5.3)with low concentration of the oxidised catalyst which is always stable withrespect to homogeneous perturbations. For f < 0.2175 it coexists with an-other steady state solution with relatively high concentration (the oxidizedstate) which is also stable with respect to homogeneous perturbations. Thelatter becomes unstable in a Hopf-bifurcation at f = 0.2175 (dashed line)and disappears at f = 0.296 in a saddle-node bifurcation. Both stationarystates become unstable with respect to inhomogeneous perturbations if χexceeds a threshold. The upper dot-dashed line in Fig. 5.12 (a) marks thesymmetry breaking instability for the reduced steady state, while the lowersolid line corresponds to the oxidised steady state. Note that for the latterstate a codimension-2 situation can be observed, where it simultaneouslyundergoes a Hopf-bifurcation with respect to homogeneous perturbations


0 0.5k

-0.5

0

0.5

λk+

Figure 5.13: Dispersion rela-tions for r1 = 2r2 = 4 nearthe codimension-2 point in fig.5.12 (a). The solid line showsRe[λ+

k ] in the critical situationwith χ = 17.24 where the homo-geneous steady state simultane-ously becomes unstable with re-spect to both homogeneous andnon-homogeneous perturbations.The dotted line shows Im[λ+

k ] forthe same χ-value. The dashedand dot-dashed lines show Re[λ+

k ]for χ = 17.15 and χ = 17.40 re-spectively.

and a Turing bifurcation (black dot).Fig. 5.13 shows dispersion relations in the vicinity of the codimension-2

point shown in Fig. 5.12 (a). The solid line shows Re[λ+

k

]in the critical

situation where the homogeneous stationary state simultaneously becomesunstable with respect to homogeneous perturbations (with nonzero Im

[λ+

k

],

see dotted line) and perturbations with k = k0 6= 0 (with vanishing Im[λ+

k

]).

This situation is similar to the classical Turing-Hopf instability observed innumerical simulations of reaction-diffusion systems (c.f. [78, 45, 48]).

For low values of K [fig. 5.12 (c)], there exists at most a single station-ary state which is stable with respect to homogeneous perturbations. AtK = 0.02 it becomes unstable in Hopf-bifurcations at f = 0.7435 resp. atf = 0.2605, and hence we find two codimension-2 situations in this case atthe intersections between the symmetry breaking instability (solid line) andthe Hopf bifurcations (dashed lines). This single state can be considered a“mixture” of the reduced and oxidised states in the previous case, since atlow f -values it has a relatively high concentration of oxidised catalyst, andat high f it has a relatively low concentration of oxidised catalyst.

For intermediate values of K [fig. 5.12 (b)], there are two steady statescoexisting in a certain interval, with neither of them being stable for all f -values. With K = 0.05 there exists an oxidised steady state which is stablewith respect to homogeneous perturbations for f < 0.2445. This steadystate loses its stability in a Hopf bifurcation at f = 0.2445 and at f = 0.374it disappears in a saddle-node bifurcation. The reduced steady state is sta-ble with respect to homogeneous perturbations for f > 0.151 and disappearsat f = 0.148. The solid line marks the symmetry breaking instability forthe oxidised steady state, the dot-dashed line marks the symmetry breakinginstability for the reduced steady state. The dashed lines signify Hopf bifur-cations (one for each steady state) and the black dots mark the Turing-Hopfpoints at the intersections with the symmetry breaking instabilities.


By applying the formulas described in chapter 4 it is possible to deter-mine the nature of the Hopf bifurcations. All of the shown Hopf bifurcationsin fig. 5.12, except for the one in (c) at f = 0.7435, are subcritical. Unfortu-nately there is no similar tool available for easily characterising the Turinginstabilities, but simulations with eq. (5.3) indicate that they are subcritical.

If Ck in the eigenvalue equation (5.10) is replaced with the one givenby (5.13), it produces analogous results to the ones presented here. Re-turning to eqs. (5.7) we note that if in the expression for ∂u/∂t we replaceK2 [1− tanh[χ(−w1 +w2)]] with kw(w1−w2), which is perhaps more realisticin chemical kinetics, the linear stability analysis is analogous with Kχ/2replaced by kw in eq. (5.13). This also explains the necessity for the tanhterm in eq. (5.1), adding the extra possibility of adjusting the value of χ,since a linear dependency on V would require an unrealistically high maxi-mum light intensity, Imax, in the experiments to be able to observe the samephenomena.

Holding on to the 4-dimensional system for a moment, kw does not nec-essarily suffer from the same limitations as Imax, meaning that a Turinginstability may in fact be possible for the system (5.7).

5.4 Numerical simulations

The main emphasis of this chapter is on the experiments, but a few numericalresults will also be shown to demonstrate similarities with the experiments,rather than doing an exhaustive investigation of all corners of parameterspace looking for exotic behaviour. The simulations are largely due to theeffort of Michael Hildebrand.

The oscillations under normal experimental conditions for the BZ reac-tion are more similar to the ones found when using relatively high f -valuesin simulations. As the operating point for simulations the parameter valuesq = 0.005, ε = 0.05, f = 0.95, χ = 22000, and K = 0.1 have been chosen.This corresponds to a point outside the displayed part of figure 5.12 (a), buttaking the high χ-value into consideration it should be possible to observesome sort of Turing pattern in the simulations. For an f -value this highthere exists only the reduced stationary state which is stable with respectto homogeneous perturbations, but it should be noted that this applies forthe homogeneous steady state corresponding to ϕ = K/2. If ϕ = 0 withthe parameter values remaining unchanged, there exists also only a reducedstationary state, but now it is unstable with respect to homogeneous per-turbations.

The simulation results for the given parameter values are shown in fig-ure 5.14 with additional details regarding numerical technicalities describedin the figure caption. The boundary conditions on the potential, V , areDirichlet boundary conditions for all simulations, i.e. setting v = 0 out-

5.4 Numerical simulations 53

t

x

d

a b

e

hg

c

f

i

Figure 5.14: Two-dimensional evolution of the oxidised catalyst concentration, v(x, y),(increasing from black to white) in eqs. (5.3) for a relatively large f . The time evolutionin the one-dimensional cross-sections indicated by the dashed white lines in (a), (d), and(g) is shown in (c), (f), and (i) during time T ; successive snapshots are separated by ∆t,and the nonlocal potential, V is updated at intervals of tup. The parameters q = 0.005,ε = 0.05, f = 0.95, χ = 22000, and K = 0.1, were used. Other parameters are (a)-(f):r1 = 2r2 = 50, system size L = 256, ∆t = T = 31.3, (a)-(c) tup = 0.03125, and (d)-(f)tup = 0.625; (g)-(i): r1 = 2r2 = 4, L = 50, ∆t = T = 62.5, and tup = 0.0016. The latticespacing was dx = 0.25 and the time step dt = 1.6 × 10−4. V was calculated on a coarsegrained lattice with spacing 4 dx.

side the system, whereas the boundary conditions on the variables, u andv, are no-flux. The simulation result in fig. 5.14 (a)-(c) is generated withexactly the same parameters as in (d)-(f) except that the update time onthe potential, V , has been decreased. This means that in (a)-(c), instead ofobserving a domain containing a stationary state and a spiral domain as in(d)-(f), one observes waves going through the “stationary” domain with alarger wavelength than in the spiral domain. The domain boundaries appearto be stable, though. In both cases the spiral domains cluster around theboundaries of the system as a result of the Dirichlet boundary conditions. Ifthe characteristic radii, r1 and r2 greatly reduced as in (g)-(h), spirals breakup and wave fragments float around the system.

Comparing with the experiments (a)-(d) corresponds to fig. 5.10, (d)-(f)


t

xa b

e

g

d

h

c

f

i

Figure 5.15: Two-dimensional evolution of the oxidised catalyst concentration, v(x, y), ineqs. (5.3) for relatively small f , q = 0.005, ε = 0.05, and K = 0.1. Notations and numericalparameters are as in fig. 5.14, with other parameters (a)-(c): f = 0.125, r1 = 2r2 = 4,χ = 2500, L = 50, ∆t = 305, tup = 1.6 × 10−4, and T = 101.6; (d)-(f): r1 = 2r2 = 4,χ = 2000, f = 0.185, L = 50, ∆t = 62.5, tup = 0.0016, and T = 62.5; (g)-(i): f = 0.185,r1 = 2r2 = 25, χ = 2400, L = 200, ∆t = 62.5, tup = 0.0016, and T = 31.3.

corresponds to fig. 5.4, and (g)-(h) corresponds to fig. 5.8. If f is increasedfurther to move into the region that is excitable in the absence of feedback,the simulations show pure Turing structures with very low amplitude atsmall radii. This is in accordance with the fact that in the experiments aseemingly homogeneous stationary state is observed throughout the system,when applying the feedback to the excitable case at small radii.

The advantage of doing simulations is that all parameter regions arereadily accessible. Hence also the part of figure 5.12 (a) that lies to the leftof the codimension-2 point. Figure 5.15 shows three different simulations inthat parameter region. For very low f and small radii [(a)-(c)] a stationaryspatially periodic structure with large amplitude is formed. A “true” Turingstructure one might call it. Moving closer to the codimension-2 point [(d)-(f)], but otherwise with unchanged parameters, the domains now containhomogeneous oscillations. Increasing the radii [(g)-(i)] will, as in the case ofhigher f -values, allows for spiral formation inside the domains.

The general conclusion is that as long as K is large enough there is a

5.5 Discussion 55

whole zoo of different patterns in the system, with the common feature thatthey are all the result of Turing instability.

5.5 Discussion

In the previous sections it has been demonstrated that applying an alter-nating nonlocal feedback to the light-sensitive BZ reaction will result instationary domain patterns containing spiral waves or travelling wave frag-ments. Which pattern is selected depends on the characteristic lengthscales,the radii r1 and r2, of the feedback. It was also shown that the domain sizesdepend on the value of those radii. The spiral domains are the result ofapplying the feedback both under oscillatory and excitable conditions, andcan be considered the fundamental result of applying the feedback to theBZ reaction in the most easily accessible parameter regime.

Different boundary conditions did not result in any significant changes,and thus we are lead to conclude that boundary effects play a small or norole in the phenomena observed. Quoting [6] on the topic of Turing patterns:

The patterns are characterized by an intrinsic wavelength whichdoes not depend on the geometrical parameters...

This is consistent with figure 5.7.Decreasing the maximum light intensity in the experiment resulted in

similar spiral domains, but this time with waves of longer wavelength movingthrough the steady state domains.

Analysis of the corresponding Oregonator equations including the nonlo-cal feedback term told us that the system contains both Turing bifurcationsand Turing-Hopf bifurcations. We therefore expected to see spatially peri-odic structures in the numerical simulations along with stable domains thatare somehow modulated in time. This was exactly the case, and the resultsare furthermore in excellent qualitative agreement with the ones obtainedexperimentally.

The spiral domains are obtained in both simulations and experiments aswell as the travelling wave fragments. The same goes for the long-wavelengthwaves going through the steady state domains although they are obtained inslightly different ways. In the simulations it was accomplished by decreas-ing the time interval between successive updates of the nonlocal potential,whereas in experiments, since the smallest obtainable interval of 2 secondswas probably too long, it was accomplished by decreasing the maximumlight intensity.

Based on this agreement between numerical simulations and actual ex-periments it is concluded that applying an alternating nonlocal feedbackto the light-sensitive BZ reaction results in symmetry breaking instabilitiesto yield Turing patterns that had not previously been observed in this re-action. The numerical simulations also predict that if experimental states


corresponding to low f -values can be achieved with sufficient image-contrastthen we should be able to observe “true” Turing structures in the experi-ments. So far, though, it has not been possible to produce such a state withsufficient contrast to reproduce the low-f results experimentally.

Along with the results obtained by applying periodic forcing and globalcoupling to the BZ reaction this study is another example of how one can,via coupling with light, design the properties of a dynamical system thatdoes not produce a given instability autonomously. Thus it is very conceiv-able that applying a different feedback to the system could also yield otherpreviously unobserved instabilities, e.g. a wave bifurcation [25].

Regarding the discussion of the relevance of Turing structures in bio-logical systems, eq. (5.7) tells us that the kind of feedback applied to theBZ reaction may also result from a pure reaction-diffusion system with fast-relaxing species. We notice that, as in the classical Turing case, this systemalso possesses different diffusion lengths, but that the difference is not asso-ciated with a classical activator-inhibitor system. It is one more example ofhow stationary structures may arise in dynamical systems.

6Locally Coupled

“BZ Cells”

A variety of biological phenomena can be described by assemblies of coupledoscillatory cells. Examples are beta cells in the pancreas of mice [16], neuralcells in the visual cortex [68] or other parts of the brain, and yeast cells ina CSTR [9].

Modelling efforts of synchronisation have so far concentrated mainly onrather abstract models that can not easily be related to something physi-cal. Another common feature is that synchronisation has been modelled forpoint-like objects with zero length and width.

Suspensions of living yeast cells in open systems have been shown tohave oscillations arising from a supercritical Hopf bifurcation [9]. The mod-elling of the coupled yeast cells in the stirred suspension was attemptedby modelling each cell as a Stuart-Landau oscillator [8]. This is very rea-sonable since the Stuart-Landau equation is the normal form for the Hopfbifurcation. The normal form parameters were obtained from quenchingexperiments with the yeast cells [9]. Analytical results of that study wereobtained in the limit of zero extracellular volume, and still the cells weretreated as being homogeneous. It makes sense, though, to assume thatthe cells are always very close to each other with thorough stirring of thesuspension.

Real cells have an internal, inhomogeneous distribution of metabolites,and models should take this into consideration. A layer of living yeast cellshas successfully been cast inside a gel [7], and it raises the question of howto model cells in fixed positions relative to each other. Attempting to modelcells located inside living tissue would also imply fixed positions of the cells.

In this study such a system is emulated by using the light-sensitive BZsystem so that there is a mask of light on the system, i.e. so that there are

58 Locally Coupled “BZ Cells”

Parameter Intracellular Extracellularε 0.45 0.45ε′ 0.0009 0.0009q 0.00095 0.00095f 0.79 0.79ϕ 0 0.001

Table 6.1: Parameter valuesused to create the intracellu-lar and extracellular media.

square areas unexposed to light, the “cells”, surrounded by areas exposedto light, the “extracellular medium”. The parameters chosen for the mod-elling study mean that the BZ system will be close to a supercritical Hopfbifurcation, just like the yeast cells. This approach takes the distribution ofmetabolites across the cell into account.

The BZ system was chosen for this study because both experiments andsimulations were well-known and relatively easy to manipulate. Indeed, theversion of the Oregonator used for numerical simulations is adapted as muchas possible to the gel experiments with the setup described in chapter 2.

6.1 A Single Cell

To be properly introduced to the topic of this chapter it is relevant to con-sider the properties of a single cell. The cell is in simulations constructed byapplying two different sets of parameter values (see table 6.1) of the 3D di-mensionless Oregonator, including diffusion as in the example in section 3.3,

∂u∂τ

=1ε

{qw − uw + u + u2

}+∇2u

∂v∂τ

= u− v (6.1)

∂w∂τ

=1ε′{ϕ(r)− qw − uw + 2fv}+∇2w

to different locations in the system. The only difference between the two setsof parameters is that in the area termed the extracellular medium, the valueof ϕ has been set to 0.001. This means that this area has a stable stationarystate (a stable focus). The area termed the intracellular medium has astable limit cycle with sinusoidal oscillations of the three variables. Thispoint in parameter space is just on the unstable side of a supercritical Hopfbifurcation, the same generic operating point used in the yeast experiments.The points where the intracellular and extracellular media meet are referredto as the cell-boundaries.

The simulations in the first three sections of this chapter are all done ongrids with a gridsize of 0.24 on the dimensionless lengthscale of the Oregona-tor (ρ = r/

√Duk5), and again the diffusion term for v has been omitted to

be as close as possible to experimental conditions in the gel experiments with

6.1 A Single Cell 59

0.0015

0.0085

v

20 40 60 80Position

0.004

0.006

v

Figure 6.1: Properties of a single cell. To the left is shown a snapshot at a certain timeinstant with the colourscale as indicated, to the right is shown profiles through the middleat different instants as well.

the light-sensitive BZ reaction. The dimensions of cells and other distancesare given in number of gridpoints with that size. In figure 6.1 to the left isshown an image from a simulation done on a 100 × 100 grid. The central70 × 70 points compose the intracellular medium whereas the remainingpoints compose the extracellular medium. The colourscale is as indicated,and this is the colourscale applied throughout this chapter. This image cor-responds to the black line in the plot on the right, which is a cross-sectiondown through the middle. The red and blue lines show the cross-section atother instants. The feature to be noticed here is that there is a continuous,inhomogeneous distribution of v across the system with the gradients im-posed at the cell-boundaries. The extracellular medium, although having astable stationary state for the given parameter values, oscillates with a verysmall amplitude driven by the oscillations of the intracellular medium.

The amplitude, and frequency of the oscillations of the intracellularmedium depends on the size of the cell as well as the size of the extracellularmedium. If the cell becomes as small as the central 40 × 40 points it is nolonger able to sustain oscillations (in a 100 × 100 system). This size alsodepends on the thickness of the extracellular medium, but it demonstratesthat cells can not be made arbitrarily small.

To estimate whether the characteristics of this cell are desirable, it mightbe relevant to compare with some characteristics of yeast cells. It is well-known that yeast cells come in different sizes and that they may not alloscillate with the same individual frequencies. A result from a full-scalemodel of glycolysis in yeast [27] shows that even though the extracellularacetaldehyde (ACA) in a suspension of living yeast cells does not oscillateautonomously, intracellular oscillations in ACA may drive the oscillationsof extracellular ACA. This is known to cause the synchronisation of yeastcells [58] Metabolites will most likely have different concentrations inside andoutside cells (only uncharged metabolites diffuse through the cell walls),and hence there must be a gradient across the cell wall. It is generally


5 10 15τ

0.004

0.006

v

Figure 6.2: Synchronisation in anti-phase of 2 by 2 cells. To the left is shown a snapshotof the cells, to the right is shown the time series of the centres of the four cells. Thecolourscale is shown in fig. 6.1.

believed that the yeast cells are too small to contain propagating waves,so the cells constructed here should not contain any internally propagatingwaves either [28]. This will not be case for the cell sizes chosen here. Largercells, though, may have waves propagating inside them [56], but this studywill focus on the beforementioned type of cells.

6.2 2 by 2 Cells

Considering what may be achieved with an experimental setup like the onein chapter 2 and to get a first glimpse at the coupling and synchronisationproperties of the constructed cells, we start by looking at an array of 2 by2 cells. The main result observed is that the four cells synchronise eitherin-phase or anti-phase, but in either case the cells end up with the sameperiod. In figure 6.2 is shown the result for four 70 × 70 cells with no-fluxboundary conditions. The four cells are not surrounded by any additionalinactive medium apart from 3 gridpoints of extracellular medium to createa cell-boundary. The cells are separated by extracellular medium with adistance of 6. To the left is shown a snapshot of four cells synchronising inanti-phase. The image also indicates the possibility of spiral formation inlarger systems. To the right is shown the resulting time series for the centresof the four cells.

Whether in-phase or anti-phase synchronisation is observed depends onthe initial condition, and unless the cells are leading and lacking their neigh-bours in phase in the manner depicted in figure 6.2 in-phase synchronisationis favoured.

With a thick boundary of extracellular medium the results stay the same,but there are indications, though, already in the results with 2 by 2 cells,that the gradient induced by a thick extracellular medium results in an

6.2 2 by 2 Cells 61

25 50 75τ

0.004

0.006

v

Figure 6.3: Mixing of two setsof cells initially in the oppositephase of each other. The averageof v in the centre of the 4 cells isshown as a time series.

unsymmetrical distribution of v inside the cells. A thick boundary of light-suppressed areas is also necessary in the actual experiments to keep waves,generated outside the area of interest in the gel, away from the cells.

The cell-cell distance is in this case not very important, since the cellswill eventually synchronise for even very large cell-cell distances. The syn-chronisation is of course faster for shorter cell-cell distances. For largersystems containing more cells the distance becomes important.

The synchronisation can also be viewed as a consequence of the factthat the extracellular medium actually oscillates in these simulations. Theintracellular oscillations drive the extracellular oscillations, and with shorterdistance between cells (higher cell density) the amplitude of the extracellularoscillations is larger. Comparing with yeast, it is similar to acetaldehyde(ACA) synchronising yeast cells in stirred suspensions. This is because theintracellular ACA oscillations drive the extracellular ACA oscillations tomediate the synchronisation [58].

An important thing to notice is that if the four cells have different sizesand hence different autonomous frequencies, the in-phase synchronisationstill takes place. The differences in frequencies are not large, but this is notexpected to be the case for yeast, or other types of cells, either.

If one was to consider how to best use this concept for making cells tocompare with the experiments of stirred suspensions of living yeast cells,one would have to make sure that the cells on average always are relativelyclose to all the remaining cells. A way to do this for 4 cells is to imposeperiodic boundary conditions on the system with the cell-end-of-system dis-tance being half the cell-cell distance. With these conditions the cells canonly synchronise in-phase, and in figure 6.3 is shown the average of v in thecentres of the four cells when two cells in opposite corners of the system arestarted in the opposite phase of the other two cells. The cells have in thiscase a size of 50 × 50 and a cell-cell distance of 2. Larger distances meanlonger time before full synchronisation. Comparing with figure 1 in [58] andeven earlier experiments [15] there is a nice agreement between it and the


1000 2000 3000 4000 5000time/secs

80

90

100

110

Pixe

l Val

ueFigure 6.4: Enhanced experimental image of synchronisation in-phase to the left. To theright are shown the time series of the centres of the four cells. Initially they are out ofphase, but eventually they synchronise in-phase.

“mixing” response in fig. 6.3. When measuring the NADH oscillations inyeast cells it is also a measurement of the average NADH level of all thecells.

6.2.1 Experiments

Experiments with the setup described in chapter 2 confirm the simulationresults with 2 by 2 cells. A pattern to produce the same mask of light used inthe simulations was placed right after number 5 in figure 2.2, and a snapshottaken after approximately 5500 seconds of the system being exposed to themask of light is shown to the left in figure 6.4. The image has been enhanced,demonstrating the relatively low contrast, and it shows the four cells allhaving high concentration of oxidised catalyst. To the right is shown thecorresponding time series for what are approximately the centres of the fourcells. Initially they are out of phase, but eventually they synchronise in-phase (after approximately 4000 s). Anti-phase synchronisation has so farnot been observed in the experiments, but this is to be expected. Only initialconditions with cell-oscillation phases in a particular order should result inanti-phase synchronisation in the simulations.

6.3 Larger Systems

In this section the results for larger systems with the possibility of actualpattern formation will be shown. Due to limited computational capacitythese systems can not be made infinitely large, in fact there will not beshown results for more than 8 by 8 cells. This is enough, though, to showthe general trend where one very important feature dominates. The gradientimposed at the boundary of the outer cells with the extracellular medium is

6.3 Larger Systems 63

Figure 6.5: Large antispiral and wave sink formed in an array of 8 × 8 cells. Parametervalues were the same for both simulations, the only difference were the initial conditions.The colourscale is shown in fig. 6.1.

a very important factor in the pattern formation, and therefore the thicknessof the extracellular layer between the outer cells and the boundaries of thesystem has been made very large in the results shown here. This is merelyto emphasise the effect.

6.3.1 Initial Conditions

The main results for the larger systems can, as it was the case for the small2 by 2 systems, be divided into two categories: Inwardly moving spirals,termed antispirals by Vanag and Epstein [70], and wave sinks. Again thepattern that is selected depends on the initial conditions.

In figure 6.5 is shown an antispiral and a wave sink. The antispiral wasinitiated by letting the four quadrants of the system be approximately 90degrees out of phase with the neighbouring quadrants in terms of cell oscil-lations. The quadrants do not have the same initial conditions throughout,but rather a narrow distribution of oscillation phases. The wave sink wasinitiated by letting all the cells be oscillating almost in phase, having onlya narrow distribution of oscillation phases compared to the average oscilla-tion phase. Still the final result is a wave sink with far from homogeneousconditions. The name wave sink, as opposed to wave source, means thatthe waves move in the inward direction. The dimensions of the results infigure 6.5 are a gridsize of 0.24 as before, a cell size of 70 × 70, a cell-celldistance of 6, and an extracellular boundary of 49 outside the cells.

This behaviour can be compared to what the response would be with ϕ= 0 (in eq. (6.1)) throughout the system, i.e. no light feedback whatsoever.The initial conditions that were used to produce the spiral would result ina plane wave (not even a full wavelength with the given dimensions of the


Figure 6.6: Two examples of different “boundary conditions”. To the left a plane wavein a system with periodic boundary conditions, demonstrating the wave propagation inlarger systems. To the right a spiral in a system of cells with different cell sizes.

system) moving from one side to the other. The initial conditions used toproduce the wave sink would just result in homogeneous oscillations of thewhole system. The spirals normally observed in the BZ system (see fig. 3.2)are, as in most excitable systems, spiralling outwards. This does not dependon whether they are relaxation type waves or small-amplitude waves.

6.3.2 Boundaries

It seems like the grid of ϕ = 0.001 imposed on the system has a signifi-cant effect. It actually imposes a whole new set of boundary conditions onthe system, i.e. the requirement that two regions with different dynamicalproperties have a continuous distribution of variables across the boundaries.

In figure 6.5 a large edge of extracellular medium was used. The perhapsmost striking feature of the images at a first glance is the squareness of theantispiral and wave sink. This is certainly an effect of the relatively largeedge of inactive medium compared to the system as a whole, and one mightask what the effect of reducing the boundaries would be. Qualitatively thereis no effect and the observed patterns are the same, but the squareness ofthe patterns is reduced. At the same time the wavelength of the spiral andwave sink are increased, so ideally one should make simulations with largerdimensions. Unfortunately this is not an option with available computerpower. The significance of the large edge is demonstrated by the fact thathaving that edge without cells in the middle, i.e. ϕ is only different fromzero at the edge of the system, then the resulting pattern is a wave sink(regardles of initial conditions).

Using periodic boundary conditions for the system with the outer edgebeing one half cell-cell distance provides an insight into the properties of

6.4 Flow-Distributed Structures 65

systems with larger dimensions. In figure 6.6 to the left is shown the result ofa simulation run with periodic boundary conditions. Cell size is 70 × 70 andcell-cell distance is 6. Initial conditions were such that the fourth quadrantwas started approximately 180 degrees out of phase with the remainingsystem. This results in a plane wave moving indefinitely through the system.One should remember that the wave fronts originating from a spiral or wavesource/sink are merely plave waves sufficiently far from the center. Thisleads to the conclusion that the same phenomena observed here would beobserved in larger systems as well.

Another way to change the imposed boundary conditions is to make cellswith different sizes. This is perhaps also more reasonable if a comparisonwith yeast cells is sought for, since the yeast cells do not all have the samesize or autonomous oscillation frequency. For 2 by 2 cells it was observed thatcells with slightly different sizes would synchronise, and for larger systemsdifferent cell sizes does not change the observed patterns either. In figure 6.6to the right is shown a spiral formed for cells having the dimensions 60 ×60, 60 × 70, or 70 × 70. The cell-cell distance is still 6 and the inactive edgehas a thickness of 25.

So far results have only been shown for cell-cell distances of 6. Decreasingthe distance does not make any qualitative changes, but increasing it does.Running simulations with larger cell-cell distance still results in cell-cellinteractions in the sense that a cell responds to the state of its neighbours,but it does not result in any ordered pattern like an antispiral or wave sink.One might say that the diffusional coupling between cells is too slow tomediate the ordering of the individual cells that is necessary to form thespirals.

A point worth making regarding the results shown in this section forlarger systems, is that in none of the results for the parameters investigated,is there a coupling between cells resulting in homogeneous oscillations. Thismeans that the average oscillations of the system are even smaller than thealready small oscillations of the individual cells. Hence one would not expectto be able to observe macroscopic oscillations in a layer of yeast cells basedon these results, i.e. measuring the fluorescence of a layer of yeast cellswould not result in the observation of any oscillations.

6.4 Flow-Distributed Structures

The parameter region explored in the previous sections is by far not the onlyone where the cell-concept can be applied.

The concept of flow-distributed structures was brought to life experi-mentally by Kærn and Menzinger [37]. It was used to explain bands ofgene expression in growing chick embryos [38] by equalling the flow with thegrowth by looking at different coordinate systems. Analyses and simula-


Parameter Intracellular Extracellularε 0.05 0.05ε′ 0.0001 0.0001q 0.005 0.005f 0.15-0.325 0.265ϕ 0-0.03 0.5

Table 6.2: Conditions usedfor creating the cells in thegrowing domain.

tions show that a flow through an oscillatory system gives rise to stationarystructures [59, 39]. The biological community has not yet embraced thesefindings, one of the objections being that the simulations were done for sys-tems that have absolutely nothing to do with biology. Also it was hardto swallow that a flow of “cells” should be equal to stationary cells withgrowth at the end. One of the objections has been countered by letting partof a reactor containing the (numerical) chlorine dioxide iodide malonic acid(CDIMA) reaction be exposed to light. In the CDIMA reaction the light hasa suppressing effect on oscillations as it does in the BZ reaction. The partof the reactor unexposed to light was then allowed to grow by continuallymoving the light boundary, thus creating a growing oscillatory system [40].

Another objection one might have had could be that a system of cellswith growth at the end was modelled without cells. The systems that areable to display these are systems that are already Turing unstable (thewavelength is determined by the forcing flow) and the systems that containbistable states [39]. Knowing from chapter 5 that the Oregonator is bistableat low f -values for ϕ > 0 and that cells could be made with the conceptillustrated in the previous sections, it was decided to attempt to make flow-distributed structures in BZ cells.

Devoting an entire thesis to the subject of flow-distributed structures,Mads Kærn lists the key features of somitogenesis in a chick embryo [36]:

• Axial growth of a structure comprised of cells.

• Each cell contains a genetic oscillator.

• Cells are added with a periodically recurrent phase.

• The period of oscillation increases farther from the boundary.

These features were all incorporated into simulations with eq. (6.1) andthe parameters listed in table 6.2. The result is shown in figure 6.7 in aspace-time plot. Space is on the y-axis, time is on the x-axis. The lengthof the system is 110 on the dimensionless lengthscale of the Oregonator,with a gridsize of 0.05, cells are 80 gridpoints long and cell-cell distance is 6gridpoints.

The extracellular medium had constant conditions throughout, as shownin table 6.2, and the twelve cells originally present at the top of the system

6.5 Discussion 67

0.7

v

0

Figure 6.7: Space-time plot of a growing cell-domain with cells added periodically at thebottom, and cells at the top entering bistable states. Space is on the y-axis, time is onthe x-axis. The greyscale for v is as indicated.

were all in oscillatory states with f ranging equidistantly from 0.265 at thetop cell to 0.325 at the bottom cell. The light-parameter, ϕ was equal to 0inside all the cells. With this distribution the “oldest” cell at the top hasa longer period than the “youngest” cell at the bottom. Moving forward intime, i.e. to the right in the figure, a cell is added at the bottom at τ = 6.56with the parameters of the “youngest” cell, i.e. f = 0.325 and ϕ = 0,simultaneously with the ageing of the cells higher up. Most prominent isthe change in the uppermost cell that changes to f = 0.15 and ϕ = 0.03,meaning that it becomes bistable. The second cell from the top now hasthe parameters that the top cell had before. After this, cells are added atthe bottom with a steady period of 4.31 whereas ageing is a little slowerwith a period of 6.14. The two periods should, ideally speaking, be thesame according to [64], but it was not possible to find parameter valueswhere this was true simultaneously with having the alternating pattern asthe result. It does show, however, that if the cells end up in a bistablestate, it is possible to get an alternating stationary pattern based on a clockfrequency. The example shown here fulfill all the requirements listed above,and figure 6.7 does resemble the figure in [64] showing a picture of how thesomites are believed to develop, except for the small discrepancy regardingthe frequencies.

6.5 Discussion

The topic of flow-distributed structures was chosen as an example of howthe cell-concept may be applied to other cases than the sinusoidal oscilla-tions, because it has been studied extensively in recent times and becauseit somehow needs to address the issue of including cells and the boundaries


between them. There are of course many other directions to take this idea,but that would take a very long time to explore.

The patterns of the sinusoidally oscillating cells studied here are the re-sult of one dominating feature, the gradients induced at the cell-boundaries.It was argued that these gradients most likely also exist in biological cells,and to resemble yeast cells as much as possible, an operating point close toa supercritical Hopf bifurcation was chosen. One may ask the relevance ofapplying a model of an essentially inorganic reaction to explain biochemicalphenomena, but one should rather focus on the dynamic aspects of the con-structed cells and ask whether the dynamics are reasonable. In [8] it wasargued why applying the Stuart-Landau equation for the individual cells inthe study of synchronisation in the stirred suspension of yeast cells is rea-sonable, and the same argument applies in this case. In fact, one could eventake the step completely and use the complex Ginzburg-Landau equationwith different parameters for different parts of the system. The advantageof using the Oregonator in this study was the straightforward comparisonbetween modelling and experiments. The experiments done with 2 by 2cells show the same behaviour as the simulations, and it is concluded, if alarger number of cells could have been achieved in the experiments, thatthey would also have shown the same behaviour as in the simulations. It isthus clear that the behaviour shown is this chapter is also present in realsystems and not just numerical systems.

In another real system, experiments with inwardly rotating spirals andwave sinks have been reported for the BZ reaction in the water-in-oil AOTmicroemulsion [72, 70], but not in “normal” BZ experiments or other oscil-lating chemical systems, and characteristic for those experiments is also thepresence of BZ reagents in two different states (the water and oil phases) sur-rounding each other, as well as the presence of a boundary-imposed gradient.Modelling of the experiments was done with different diffusion coefficientsfor the two phases, which definitely is something to consider when trying toapply this cell-concept for modelling real biological cells. There may alsobe other transporting mechanisms than diffusion involved which could alsobe incorporated into the cells. The important thing is to apply parametervalues that yield the correct dynamical behaviour the cells.

7Conclusions and

Perspectives

Different coupling phenomena have been presented in this dissertation, bothlocal and nonlocal. Both types of coupling have changed the properties ofthe systems they were applied to. Biological relevance was mentioned inthe introduction, and both the nonlocal and the local “cell” coupling can beargued to be of some relevance to biological systems.

The results obtained from applying nonlocal coupling showed very goodqualitative agreement between experiments and simulations. They both dis-played formation of stationary spiral domains. Analysis of the simulationmodel revealed that the nonlocal coupling should result in a Turing insta-bility, which turned out to be true. The analysis was not extended intothe slightly nonlinear regime, but judging from the simulations it must besubcritical. Due to the fine agreement between experiments and simulationsone may conclude that the stationary structures in the experiments are alsotrue Turing structures. This is confirmed by the fact that the spatial wave-length is insensitive to the boundary conditions on the nonlocal feedback.The wavelength of the Turing structures was predicted to increase with in-creasing radii by the linear stability analysis, and this was also the case.Because the Turing bifurcation is subcritical it is not possible to predictwhich wavenumber is selected by doing a linear analysis, but intuitively italso seems reasonable that increasing the radii increases the wavelength.

This was, as mentioned, the first example of Turing patterns in the BZreaction. With the use of the video projector the light feedback becamepart of the dynamical system rather than something that was imposed onit. Hence the word “coupling” in nonlocal coupling. Something similar canbe said about global coupling, which might be considered a special caseof nonlocal coupling. Together with periodic forcing they are examples of

70 Conclusions and Perspectives

how one can utilise the light-sensitivity of the BZ reaction to “design” thedynamical properties of the system.

The example with the 4-dimensional reaction-diffusion system with tworapidly relaxing species showed how this “design” may be a built-in featureof a biological system. The number of reacting species does not have to berestricted to four to be able to observe stationary structures, but at leasttwo must relax rapidly with different lengthscales to create the alternatingfeedback. A non-alternating feedback will not result in stationary structures.

The cells presented in chapter 6 may not be directly related to anybiological system in particular, but as it was argued they do have manyfeatures in common with yeast cells. It is not that hard to imagine acetalde-hyde oscillating in a yeast cell with profiles similar to the ones shown infigure 6.1, and certainly extracellular acetaldehyde oscillations work to me-diate cell synchronisation. To compare directly with experimental results,simulations with 2 by 2 cells were performed. They showed two kinds of syn-chronisation, in-phase and anti-phase, with the former being observed formost initial conditions. This was confirmed with the observation of in-phasesynchronisation in the experiments. Another comparison with experimentswas the “mixing” simulation of two sets of cells opposite in phase, yieldinga similar response to that shown in experiments.

Simulations with larger systems yielded the surprising results of antispi-rals and wave sinks. Surprising because the “normal” BZ system does notshow this kind of behaviour at all. The simulation with periodic boundaryconditions, yielding a plane wave solution, indicated that the results ob-tained here are also valid for even larger systems. A natural consequenceof the comparison with the properties of yeast cells was to repeat the sim-ulations with cells of different sizes and oscillation frequencies. This didnot yield different results, indicating that they are robust with respect tovariations in size and frequency.

Experiments with yeast cells in fixed positions are a realistic possibil-ity [7], and the results presented in this dissertation give a hint of what toexpect from such experiments when successfully executed. Trying to observeNADH-oscillations as a collective phenomenon of all the cells in a layer ofgel will, if the results presented here are indicative, not be successful. Onewould have to be able to distinguish the individual cells, e.g. in a fluores-cence microscope.

In the experiments with the BZ reaction in an AOT-microemulsion, theoil phase may be considered the extracellular medium that does not oscil-late but works to couple the “cells”, i.e. the aqueous phase. The “cells”diffuse with respect to each other on top of the diffusion of species insidethe “cells” [70, 71, 72]. The citation marks are supposed to indicate thatthey are only cells in the sense that they have similar features to the cellsproduced with the mask of light. The AOT system has different phases withdifferent concentrations of the reacting species of the BZ reaction, creating

Conclusions and Perspectives 71

the same type of gradients that the light does. The real similarity withthis study, though, lies in the fact that the BZ-AOT system also displaysantispirals and wave sinks. Knowing that it also displays stationary struc-tures under slightly different conditions, it seems relevant to ask if stationarystructures is an inherent part of the properties of oscillating cells. If so, thismight be another explanation why seemingly homogeneous embryos evolveto have arms and legs in the right positions.

Certainly, there is no explanation available today that contains the ul-timate truth in this regard. This dissertation presents, including the flow-distributed structures, three suggestions that may all contribute toward abetter understanding of stationary structures in nature.

72 Conclusions and Perspectives

AAppendix

Derivation of eqs. (5.12) and (5.13)

The analysis in one spatial dimension of eq. (5.3) begins with the insertionof the plane wave perturbations

u = u0 + δu exp(λkt + ikr)v = u0 + δv exp(λkt + ikr)

into the equation. This yields, after evaluation of the differentials and a fewre-arrangements:

0 = −λkδu− k2δu +1ε ξ

{u0 + δuξ − u2

0 − (δu)2ξ2 − 2u0δuξ

− [2f(u0 + δvξ) + ϕ]u0 − q + δuξ

u0 + q + δuξ

}

0 = −λkδv + (δu− δv)

with ξ = exp(λkt + ikr). Linearising the two equations in the usual mannerone ends up with the eigenvalue equation (5.10) with the elements

J11(k) = ε−1

[1− 2u0 − 2q

(u0 + q)2(2fu0 + ϕ|δu=δv=0)

]− k2,

J12(k) = −1ε

u0 − q

u0 + q

[2f +

1ξ

∂ϕ

∂(δv)

∣∣∣∣δu=δv=0

],

J21(k) = 1, J22(k) = −1.

of J(k). Choosing ϕ as

ϕ(v, r, t) =K

2[1− tanh (χV (v, r, t))]

74 Appendix

the derivative is∂ϕ

∂(δv)

∣∣∣∣δu=δv=0

= −χK

2∂V

∂(δv)

∣∣∣∣δu=δv=0

assuming V |δu=δv=0 = 0. In one spatial dimension eq. (5.5) becomes

V (r) =∫ ∞

−∞w(r − r′) v(r′) dr′,

and still assuming that the choice of w means that V |δu=δv=0 = 0 it becomes

V (r) =∫ ∞

−∞w(r − r′) δv exp(λkt + ikr′) dr′

= δv eλkt

∫ ∞

−∞w(r − r′) eikr′ dr′

= δv ξ

∫ ∞

−∞w(y) e−ikydy, y = r − r′

The derivative of ϕ can then be evaluated to

∂ϕ

∂(δv)

∣∣∣∣δu=δv=0

= −χK

2ξ

∫ ∞

−∞w(y) e−ikydy,

meaning that the elements of J(k) evaluate to

J11(k) = ε−1

[1− 2u0 − 2q

(u0 + q)2

(2fu0 +

K

2

)]− k2,

J12(k) = −1ε

u0 − q

u0 + q

[2f − χK

2

∫ ∞

−∞w(y) e−ikydy

],

J21(k) = 1, J22(k) = −1.

Choosing the kernel, w(r), as in eq. (5.2), the evalutation of the integral isstraightforward, and the result is

∫ ∞

−∞w(y) e−ikydy = − exp

(−k2r21

4

)+ exp

(−k2r22

4

),

leading to the result in eq. (5.12) by looking up the integral in standardmathematical tables. This kernel also satisfies the beforementioned assump-tions on V . Choosing the kernel as (5.6) one is faced with an evaluation ofthe type

∫ ∞

−∞w(|y|) e−ikydy =

∫ 0

−∞w(−y) e−ikydy +

∫ ∞

0w(y) e−ikydy

=∫ ∞

0w(y) eikydy +

∫ ∞

0w(y) e−ikydy

= 2∫ ∞

0w(y) cos(y)dy

Appendix 75

which again, after applying standard mathematical tables for evaluation ofthe integral, leads to the result

∫ ∞

−∞w(y) e−ikydy = −

[r21k

2

1 + r21k

2− r2

2k2

1 + r22k

2

]

and subsequently to eq. (5.13).

76 Appendix

BSpecifications

This chapter contains some technical specifications regarding the experi-ments done with the BZ reaction.

Chemicals

The chemicals used in the preparation of gel and catalyst-free solution arelisted by manufacturer and purity.

• NaBrO3, Fluka, purum

• Malonic Acid, Aldrich, 99 %

• H2SO4, Baker, 95 % - 98 %

• NaBr, Merck, extra pure

• Ru(bpy)3Cl2 · 6H2O, Fluka, no purity indication

• sodium trisilicate, Fluka, purum

All chemicals, except Ru(bpy)3Cl2, were used without further processing.The impurities in malonic acid can have a great effect on the nature ofthe oscillations in the BZ reaction, but it has been found that the malonicacid from Aldrich produced reproducible results compared to malonic acidthat had been recrystallised twice. The Ru(bpy)3Cl2 has to be dissolvedand passed through an ion-exchange column saturated with concentratedH2SO4 to yield Ru(bpy)3SO4, because the chloride ions would have a hugeimpact on the BZ reaction.

Filters

The filters that were used for the cell experiments were:

• A 450nm interference filter from Linos for the observation light.

78 Specifications

• Two coloured glass filters, BG 23 and BG 38, also from Linos, for theperturbation light.

• In front of those filters were placed heat filters, KG1, from Linos.

The light source in these experiments was a 100 W, 12 V lamp from Osram.

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