stochastic cellular automata

Pergamon Nonlinear Analysis, Theory, Methods & Applications, Vol. 30, No. 3, pp. 1847-1858, 1997

Proc. 2nd World Congress of Nonlinear Analysts © 1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

Plh S0362-546X(96)00378-1

STOCHASTIC CELLULAR AUTOMATA

THOMAS FRICKE

Lehrstuhl fiir stochastische Prozesse, Institut ffir Physik, Humboldt-Universit£t, Invalidenstr. 110, D-10 115 Berlin,

email: t [email protected]

Keywords: stochastic processes, cellular automata, excitable media, global coupling, reaction-diffusion

1. INTRODUCTION

We describe a general algorithm for simulating large complex Markoff-processes. We have used a reaction-cell method in order to simulate arbitrary reactions, which can be used for any kind of reaction-diffusion-systems on arbitrary topologies. The events within a single cell are managed by an event handler which has been implemented independently of the system studied. The method is exact on the Markoff level including the correct treatment of finite numbers of molecules. To demonstrate its properties, we apply it on a very to a stochastic, three dimensional excitable system. Thus the dynamics of chemical reactious in homogeneous systems is well understood, they are easy to describe by a van ' t Hoff ansatz [1]. However, in inhomogeneous systems the transport mechanism has to be taken into account. We want to consider reaction-diffusion-systems (RDS) with a small number of molecules as they typically arise in biological systems on cellular level. There are only one or two single DNA or RNA molecules and a small number of messengers and regulator proteins.

RDS are known to be capable of a very complex behaviour like formation of spatial and temporal patterns. They show this great variety of effects, even if their discrete nature can be neglected and they are well described by the mean concentrations n~(~ . Population dynamics, which is often described by a formulation equivalent to chemical reactions, deals with much smaller concentrations, so the role of fluctuations is much more important than for chemical reactions.

2. REACTION-DIFFUSION-SYSTEMS

We want to denote different types of molecules E~ by a Greek index a, while the reactions are distinguished by the Latin letter i. The stoichiometric equations describe the chemical reaction of the i - th type unambiguously by the initial (or forward) f~ and final (or backward) b~ coefficients and the related reaction-rate A i

) i (2.1)

Ot

The common formulation of the mean-field equations for position-dependent mesoscopic concentrations n(x, t) leads to nonlinear coupled partial differential equations

-~n~, = D~,An~, + ~_, A i I I ( b ~ - f~) n 7 . (2.2)

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1848 Second World Congress of Nonlinear Analysts

However, in contrast to the simple derivations of the partial differential equations for the mean values, the formulation of the equations for the higher moments are much more complicated, even for simple systems.

The analytical treatment turns out to become very difficult, if diffusion and the discrete nature of the molecules together must be taken into account. Not only the cost of computation grows proportional to the volume, it has to be remarked that it also increases exponentially to the statistical moments taken into consideration.

Therefore, for small numbers of molecules within an w volume, the discrete nature of the molecules must be taken into consideration. The raw estimation for systems of N molecules leads to an estimation for the fluctuations of O(x/~) . Thus we need O ( N ) = 10,000 molecules to guarantee a relative error of percentage order if fluctuations are neglected.

Computer simulations of complex RDS focus on the mean-field-behaviour taking no notice of the fluctuations, whereas on the other hand stochastical simulations of extensive systems close to equilibrium are not well suited to dynamics.

In contrast, using a random-walk algorithm we may handle some 107 particles and are, further- more, able to study the crossover from the diffusion-controlled to the reaction controlled limit.

Following an analysis of the relevant time scales, the diffusion can be simulated on a coarse lattice of cells without affecting the exactness of the results. Our goal is to close the gap between solving the mean-field-equations and simulation techniques using cellular automata. Both techniques are included, but we want to emphasize, that standard algorithms may be more efficient to cover these limits.

As it has shown in [2] algorithms simulating Brownian motion adapt much better to steep gradients than standard techniques for partial differential equations.

3. SIMULATING MASTER EQUATIONS

We want to treat a chemical reaction as a Markoff-process, which is described by the time evolution of the probability p)? to be in state )( at time t following a stationary master-equation

dt ~, v

inflow into 3~ outflow from J(

in our case the discrete vector )( denoting the number of molecules [3]. The minimal-process-method simulates the master-equation as a random-walk in the space of

all possible configurations.

M i n i m a l - p r o c e s s - m e t h o d

1. Generate an exponentially distributed random number r with Pr(r) = W~7. e x p ( - W ~ , r), let t e-- t + r,

2. select )~r according to the probability Pr(.~ -+ _~') = W ~ _ ~ x , / W j I ,

3. go back to step 1.

Starting the random walk in state X, we only need to determine the t imeleaving )(, and the successor state _~, which is equivalent to draw the lifetime 7~ and the transition X --+ X ~ as random numbers.

Second World Congress of Nonlinear Analysts 1849

For both steps the knowledge of the flux out of X is completely sufficient. The lifetime 7-)7 is exponentially distr ibuted, while the selection of the transition requires the drawing of a random- number X~ proport ional to the transition rate W)?_~)?,. This property is due to discrete Markoff- processes, and we want to emphasize that it is ezact and not affected by any further assumptions. Wi th the sum

= Z (3.2)

the way to generate this random-walk and the sequence of states :V~(t) is given by the simple algorithm presented above.

4. CHEMICAL REACTIONS IN A HOMOGENEOUS VOLUME

For the present, we want to discuss arbi t rary chemical reactions in a small volume a~, for which the diffusion is so fast that any spatial inhomogeneity may be neglected. After that , we will introduce diffusion as a reaction-like process of molecules leaving the volume w. In this way, we want to describe the dynamics by which diffusion relates distances of an extensive length scale to a t ime scale.

According to (2.0) the number X~ of molecules -X~ changes from X~ to X~ + b~ - f~ each time the reaction of the i th type occurs. I.e. f~ molecules E~ are consumed and b i~ molecules E~ are produced. Forward and backward reactions shall be distinguished by their enumerations. If the reactions j and i are reciprocals of each other, the role of the coefficients is exchanged, i. e. J~ = b J and fJ = b x.

Since the number of molecules is an extensive quantity, the number of chemical reactions per unit t ime must be extensive, too. The rate of events, the reactivity A i of a chemical reaction is defined as

J'the reaction i occurs during'[ Aidt = Pr [ t he time interval [t, t + dt] j " (4.2)

According to van ' t Hoff [1], it is given by

X~! (4.3) Ai = ~i~ ~l-J (X~ -~i)! .d~"

where h i is an intensive constant which does not depend on X~. Note that the reactivities A depend on the s tate , i.e. A i = A ) , A = AX, the index .~ is usually suppressed. The fraction

X~,] Xc,(Xc~ - 1 ) . . . ( X , - f / + 1) (Xa - f~)! cog = w]~ (4.4)

takes into consideration that each individual molecule can only be consumed once. This is important ,

if the numbers X~ are small. For large Xa, this fraction turns into the more familiar form (x-~) G' ,

ignoring any power up to ( e ) " ' The van ' t Hoff approach suggests chemical reactions being Markoff-processes [3, 4], which may

be justified due to a stosszahlansatz for homogeneous concentrations. With these assumptions the sequence of states of a chemical system by the random-walk of the

vector .~ = X( t ) can be obtained by a minimal-process-method adapted to the dynamics of chemical reactions, which is based on the well known properties of discrete stochastic processes[5, 3, 4]. We summarise his main results according to our requirements. The total reactivity is the rate

A = ~ A i, (4.5)

all reactions i


which determines the probabili ty of a chemical reaction during the infinitesimM time interval [t, t+dt] according to

~'some chemical reaction changes the} Adt = Pr [.state Jf during It, t + dt] , " (4.6)

A is the inverse of tile average lifetime of 3~, therefore the random time rA until the next event occuring has the probabili ty density

Pr{rA, time to next reaction } = A e x p ( - A r h ) . (4.7)

Because we have assumed that chemical reactions are Markoff-processes, this equation yields independently of the time when the last change occurred, which is a general s ta tement for any discrete Markoff-process. The probabili ty of the reaction of kind i is proportional to its contribution to A

A i Pr{Reaction i} = ~- . (4.8)

q r . is given by the probabili ty of all chemical reactions leading from .~ to The transit ion rate . .x-+g,

14~y,-~Z, = Y2~ Ai- (4.9) all reactions i with g , = . ~ _ f i + ~i

Note tha t different chemical reactions may change X by the same ~ = f3 _ ~'i because only the backward and forward stoichiometric coefficients f3 and ~'i together determine the reaction unambiguously.

For a rb i t ra ry chemical reactions it computes the random time-step ra until the next reaction by drawing an uniformly distr ibuted random number rnd E [0, 1). The reaction type i usually is selected by a simple loop, known as linear selection algorithm using the sum s for the integration. The variable t denotes the time, proceeding in exponential time steps. The order of reactions denoted by i is arbitrary. After the reaction has been carried out, the reaction reactivities A i and the total rate A = B"~ are computed again.

M i n i m a l p r o c e s s m e t h o d for c h e m i c a l s y s t e m s

I. rA +- --~ log(1 - rnd), t + - t + r A

2. r +-- rnd, s ~-- 0, i ' +--first reaction,

- I ' it 3. while s < rA, (a) i +-- ~', <--next reaction (b) add up s +-- s + A i

4. do reaction type i by .~ ~ Jf + 5~/, compute A i, and A e-- ~ i Ai,

5. go back to step 1.

Note the adapta t ion to the time scale A -1, which is responsible for the high flexibility of algorithm. Thus it is a fast and easy to use Monte-Car lo method for the dynamics of arbi t rary react ion-systems and has the main advantage over the mean-field description, tha t it takes into account internal fluctuations caused by the finite numbers of molecules. Because the random-walk imitates the sequence of s tates of the simulated Markoff-process, any correlations and higher moments may be measured as in a reM experiment.


5. REACTION-DIFFUSION-SYSTEMS

The molecules are treated as point-like particles with a finite interaction probability, implicitly assuming, that their Brownian motion is independent. Our simulation introduces a reaction-cell method. In general, for small diffusion-constants, the homogeneity condition of the van ' t Hoff ansatz cannot be fulfilled for the total d-dimensional volume fL Thus f~ is subdivided into small cells (cubes or squares etc.) of volume ¢0 = La---j = h d. We have to impose the condition, that the size of co may be justified by a local van' t Hoff ansatz. This prerequisite depends on the time-scale rR between two subsequent chemical reactions compared to the time scale rD of a molecule leaving a: by diffusion. The diffusion-time-scale has to be much faster than the reaction time-scale, i. e. ro ~< rR with rD = h 2 / D and 7R = 1/A.

In this section we want to introduce diffusion as a random-walk on the lattice of reaction-cells. The diffusion in an arbitrary, not necessarily Euclidean, topology may be regarded as a reaction-like step of a molecule .E~ from ff to one of its next neighbours r~,

A~,',rn -- - . = - (5.i)

For a diffusion-step of a molecule -Za the rate

1 ) ~ , ~ = Da~,~ ~ , (5.2)

with D~,~ denoting the local diffusion-constant, describes the probability

~a single molecule E~ jumps from ~7 to r?t} ~ , ~ d t -~ Pr [dur ing the time intervals [t, t + dr] " (5.3)

On a lattice built of volumes ~o = h d, we obtain in the limit t --~ oo the correct Brownish motion behaviour by the central-limit-theorem. The symbol (g) denotes the set of all 2d next neighbour cells of ft. Due to the symmetry of the rate to do a single step into any direction i.e. the rate of leaving ff by diffusion, is D ~ / h 2, with

),~ = ~ G~,~ = 2 d ) , ~ . (5.4)

In the isotropic and homogeneous case we omit the spatial indices ,~, = , ~ and the rate for stepping in a certain dimension gets , ~ / 2 d , while the total rate for leaving a cell is A~.

If we have X~,~ molecules E , in cell if, the rate for the next diffusion step of any E , to r~ is

A~,~ = A~,~X~. (5.5)

This rate is the equivalent formulation to (4.2), treating a diffusion-step like a first order chemical reaction. If a diffusion step occurs, another cell r~ gets involved, because one E~ moves from g to

r~.. Thus a diffusion step is carried out by

1. X ~ <--- X ~ - 1, 2. X ~ +- X~a + 1. (5.7)

Notice that in our case A,,g~ depends only on the contents of if, from which a molecule hops into a neighbour N 6 (g), although the assigned diffusion-step changes the contents of the other cell rT~, too. Thus a diffusion-step changes the state of two cells, however, its rate depends only on the

contents of the first cell ~ triggering the event.


For the chemical reaction of type i we simply have to change the contents of ~ by 2~n +-- 2 ~ + ffi. Therefore, the probabili ty Qn of an event triggered by cell ~ is the sum of react ion-ra tes A~ and diffusion-rates A~n,~

i ~ ,ae(,~)

Because all react ion- and diffusion-steps are assigned to the cell ~, we may identify an arbi t rary step entirely by an integer i and the cell ~. We may ignore the additional index r5 for the selection mechanism. A reaction may be defined by an addition of its associated difference 5~, i.e. ) f e-- 2 + 5/, involving one or two cells. In section 6.1.1 below we suggest obvious generalisations of rates depending on the contents of the neighborhood of g. This way it is easy to introduce dynamics depending on gradients, like the movement of kinks and steps on surfaces.

6. TIMES SCALES OF REACTION- AND DIFFUSION-PROCESSES

K we want to fulfill the assumptions of a local van' t Hoff ansatz we have to satisfy the condition, that diffusion is much faster than any chemical reaction. More precisely spoken, this means, that the local reactions A~ and diffusion- A~,~a rates define different time scales, which have to be separated. Therefore we choose the size of our cells sufficiently small, for the condition

A / << Aa~,~, for any i , a (6.1)

to be satisfied. This can always be achieved, because reaction probabilities are extensive quantities A~ oc w = h d, while diffusion-rates increase with h -2. Thus )~a~r, o¢ w -2/d and A~a,~ ~ w 1-2/d,, the latt ice constant h always can be reduced to make the ratio

A~ ¢v2/d h2 = ( 6 . 2 ) Ac, ar~

arbi trar i ly small. For any practical purpose it is not possible to give an a priori est imation, so we recommend to choose ~: as large as possible rejecting the simulation runs if IOA i > A~ n t ~ '

7. SELECTING A SINGLE CELL BY THE METHOD OF LOGARITHMIC CLASSES

Although we have argued that there is no principal difference between the minimal-process-method in a homogeneous volume and on a lattice of reaction cells, a significant complication arises as the number of possible changes in each Markoff-step is an extensive quantity. If we make a straight- forward generalization of the Mgorithm using a linear selection s t rategy we run into the problem of selecting a single event from an extensive quantity of transitions. The total reactivity

Q= E ¢ . = EE¢

again determines the mean time step. However, this method suffers from the severe disadvantage tha t the sum computed by a loop

adds up an extensive number of reaction probabilities. Drawing a random cell due to its contribution Qa to Q, the loop consumes computing time proportional to the number of cells. This procedure is completely unsuitable for a single step of a computer algorithm, see figure l (a ) . If we do not have any addit ional information, we have on the average to walk the half length of the main loop.


(~)

(b)

(c)

10

8

6 O~ 4

Q~

l inear selec , r i , i

0 ' 0 10 20 3o 40elIO 60 7o 8o 9o loo

yon N e u m a n n re jec t ion 10 . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Q

6 c~N~ oc= ~ ~ re jected I~

0 ' 0 10 20 30 40ce1510~60 70 80 90 100

10 , m e t h o d of, logar i thmic c!asses ,

. . . . . . . . . . . . 3

6 Q~ Ce2 = c o n s t a n t z

4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F - 2

2 . . . . . . . . . . . . . . ~ z" 1

0 - ~ 10 20 30eI140~,c s n 50 60 70 80 90 100 r e a r r a n g e a

Fig. 1: Selection algorithms: (a) Linear s., the loop integrates Qa up to rQ, thus the cost is extensive, (b) Von Neumann rejection without improvement, the cost is the ratio of the area of the rectangle marked off by

the upper limit (~ to the area below Qa. A single peak may severely affect its efficiency, (c) s. by the method of logarithmic classes, the upper and lower limits for each class are denoted by dotted lines. The acceptance a > 0.5 is the ratio of the reordered Qa to its upper estimation 2 fl°°r (ld(Q,~)) + 1, we find the cost being

approximately constant.

1 8 5 4 S e c o n d W o r l d C o n g r e s s o f N o n l i n e a r A n a l y s t s

Even for a small latt ice of 104 -~ 100 × 100 cells this means an increase for the computing time compared to the algorithm without spatial resolution by an average factor of 5,000. It seems to be impossible to circumvent this problem by rearranging the cells or some kind of pre-sorting, because the selected reaction changes the cell-reactivity every time. A random selection according A. J. Walker 's [6] algorithm would be efficient only if the react ion-rates did not depend on time. The extensive quanti ty of cells to be considered is a principle obstacle.

These reflections show tha t l inear selection is total ly inefficient. Thus we have tried another a lgor i thm based on the yon Neumann rejection, which selects at first the reaction-cell and in a second step the reaction or diffusion in this cell.

The yon Neumann rejection requires an upper bound for the probabili ty to select a cell. Therefore we have to make the assumption, tha t

Qn _< Q for each cell g. (7.2)

The method does not depend on the number of cells, thus the problem of the dependency on the extensive number of cells does not arise. However, a more precise look reveals tha t the efficiency of the yon Neumann rejection strongly depends on the homogeneity of the RDS. This is demonstra ted in figure l (b) . The acceptance ratio a is the quotient of the area below Qa and the rectangle delimited by 0

E~Q~ a - - ^ , ( 7 . 3 )

QE~I and 1/a is proport ional to the number of runs through the loop needed to select a cell. Therefore the cost UrN R of the algorithm is proportional to the inverse of the acceptance ratio,

I OE~I C,r~R oc -- (7.4)

a E ~ Q ~

For very inhomogeneous systems, i.e. for systems dominated by a single peak, this method may slow down by an arbi t rary factor, which has been studied for the simulation of a biological system [7]. This disaster can be avoided, if we split up the cells into classes according to their dual order of magnitude. The acceptance ratio is improved to 75%.

If we want to handle all orders of magnitude of the cell reactivities Qz we have to implement a logarithmic classification scheme. We define the logarithmic class Lz as the set of all cells ff with a reactivity Q~ in the same order of dual magnitude. The symbol ld (x) = log 2 (x) denotes the logarithm with base 2, and the function floor(x) denotes the largest integer which is not greater than x

L~ = {~] fioor(ld(Q~)) = z}. (7.5)

The reactivity gz of a certain class is given by the sum of the reactivities of its elements

g~ = Z Q~" (7.6) ,~ELz

There is no principal restriction of the range of z

z - floor(ld(Qn)) E { - o c , . . . , - 2 , - 1 , 0, 1,2, 3 , . . .} . (7.7)

The class L _ ~ contains ~ll cells without any possibility of a reaction- or diffusion-step, above all the empty cells. Because L _ ~ cannot be selected, i.e g-o¢ = 0, there is no need to represent it in computer memory. For any practical application, z is restricted to a finite range Zmin <__ z < Zm~x.

Second World Congress of Nonlinear Analysts 1 8 5 5

The intersection of two different classes is empty Lz A L~, = ~, z ~ z', thus the total reactivity, is expressed by the sum

f i Zinax

Q = ~ Q~ = G = ~_, G. (7.8) ~t z~.~- ~ oo Z~Zmi n

The inequality

2 ~ _< Qz < 2 ~+1, g E L. (7.9)

guarantees, that the reaction-rates are relatively homogeneous within a certain class. The rates of two members of any class do not differ by more than a factor of 2

max Q~ ~ E L . min Q~, < 2. (7.10)

g ' E L ~

This is an ideal starting point for the yon Neumann rejection. We therefore decided to implement an event handler, which is able to select a reaction-cell according to its reaction-rate by avon Neumann rejection step within the class. The number of classes is small, i.e. of O(10), thus the choice of a class is based on a linear selection algorithm.

8. SELECTING A REACTION

The problem has been split into three qualitatively different steps:

1. select a class L, with probability G/Q,

2. select a cell ff E Lz with probability Q~/G,

3. select a reaction-step i within cell g with probability Q i / Q z.

We stress again, that the index i represents both reaction and diffusion transitions. The last two steps are selected according their conditional probabilities

G Pr{select class z) = - -

Pr{select cell ~ ] class z has been selected}

Pr{select reaction i I cell ~ has been selected}

therefore the probability of selecting a reaction in a cell

Pr{select reaction i in cell ~} -

Q, (8.1)

Qn - ¢ , (8.2)

_ Q~ Q , (8.3)

Q G Q ~ Q (8,4)

has been maintained correctly. The algorithm choosing a class is a linear selection. However, it is not necessary to order the

sequence of classes ( z i , . . . , zj) in a naive way with zi < zi+l or zi > zi+l for all i. Moreover, it has a favourable effect, if the classes are sorted with respect to their reactivity, i.e gz~ >_ G;+~. To speed up the selection, the classes of the highest probabilities to be selected are successively moved to the head of the loop by a bubble-sort. This is the most efficient algorithm to select a reaction in a class according to its rate G~, because the classes with the smallest probabilities to be drawn least are


moved to the tail. As the number of logarithmic classes is small, the selection can be performed by standard linear selection.

After z has been selected, it is easy to propose a cell ~7 by drawing a uniform random number u E {0 , . . . , ~,~ - 1}. We have implemented each class Lz as an array F~[0 . . . . , u~ - 1] of t,z =] L~ ] elements, each describing the state of a single cell. The following subroutine does the yon Neumann re ection of a cell in the previous selected class z.

S e l e c t i n g a cel l ff in c lass L~ by y o n N e u r n a n n r e j e c t i o n

1. propose u +- ttoor(u~rnd), ~ e-- Fz[u],

2. draw a uniform random number r e- rnd,

3. if Qn > r2 z+l then return if, else go back to step 1.

Because of the inequality (7.8) we know lower and upper limits of the reactivity, 2 * _< Q~ < 2 z+a. Therefore, it is guaranteed, that the probability for a rejection is less than 0.5. If we assume that the rates in each class are distributed uniformly, we get an acceptance ratio a = 0.75.

In the cell ff the local reaction is chosen by a linear selection method, because the number of possibilities usually is small O(1) . . .O(10) . The changes in all cells and classes involved have to be registered by bookkeeping steps which require the computation of the reaction-rates Q~, the local reactivity of a cell Qa, its class reactivity gf and the global reactivity Q. For a diffusion-step concerning a further cell rfi the bookkeeping has to be done for these cells, too.

9. THE RINZEL KELLER MODEL WITH GLOBAL COUPLING

The Rinzel-Keller model describes the activator-inhibitor dynamics by a piece-wise linear function. With global inhibition it has been demonstrated in one and two dimensions, that the system shows localized excitations propagating like macroscopic particles, "moving spots" [8]. We extend this simulations to three dimensions overcoming the arising numerical problems of deterministic algorithms by a random-walk approach. The simulations are performed by stochastic methods, activator and inhibitor are represented by different types of random walkers, performing birth and death processes according to their multiplication and extinction rates, which are the same as in the deterministic approach. This methods have been shown to be very efficient simulating large and complex reaction- diffusion systems [2, 7, 9]. This way we introduce noise into the system, which on the one hand, can be seen as the source of new physical effects. On the other hand, if noise is undesired, its disturbing effects can be eliminated using a sufficiently large number of random-walkers.

We consider a three dimensional model without inhibitor diffusion, where the activator u produces inhibitor v with the rate # u. The activator and inhibitor decay with exponentially distributed average lifetimes ru and vv. For the production of the activator its concentration needs to exceed a threshold a which is increased by the total number of activator Su introducing a global coupling with a constant

parameter So. The inhibition is assumed to be piece-wise linear - q v 0(u). However, for a random-walk approach

it is necessary to avoid negative concentrations, thus we have to cut off the inhibition by the function 0(u) if there is no activator present. The global coupling S~ prevents the system from growing into the entire volume.

Ou _ gO u - a - - q v O ( u ) - - - + D A u Ot r.

Second World Congress of Nonlinear Analysts

f

) 0 0 0 ~ . ~ 50( ~2000

1857

J

r ' - - . J

• K 201

Fig. 2: Trajectories of five spots with So = 500, 1000, 2000, 5000, 10000. Three views in x, y, z-direction. The simulations start at the same point, x ---- (0, 0~ 0) running until ~ ---- 4.

= f i , (u , v) + D A u , (9.1) Ov v Ot -- # u - - - v Tv

= fo (u , v) , (9.2)

S,, = f d d x u ( x ) . (9.3) J

10. THE MACROSCOPIC DESCRIPTION OF A SPOT

For the analysis of the behaviour of the spot as a macroscopic object we consider the spots' coordinates x and velocity v. The coordinates are given by the center of concentration of the activator and the inhibitor component.

If we assume, that the absolute value of the velocity is nearly constant, we can solve the auto- correlation function problem and the compute the diffusion

(x2(t) ~ = 2 (v 2) t = 6 D t (10.1) % , - - /


where ;~ is the au tocor re la t ion cons t an t of the velocity. To co mp u t e ~ f rom

v ( t ) . v ( t ' ) = v . v ' = vo~e - ~ ' = ~(t) (10.2)

we have to solve the diffusion equat ion on the velocity sphere of fixed radius vo and ob ta in finally

v~ v~ (10.3) D - 3 ~ -- 6 D s

REFERENCES

1. VAN KAMPEN N. G., Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam 1981

2. SCHIMANSKY GEIER L.,MIETH M., ROSI~ H., MALCHOW H., Structure formation by active Brownian particles, Phys. Lett. A 207, 140 (1995)

3. G]LLESPIE D. T., A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. of Cornp. Phys., 22, 403 (1976)

4. GILLESPIE D. T., Monte Carlo simulation of random walks with resident time dependent transition ratesJ, of Comp. Phys, 28, 435 (1978)

5. KARLIN S., TAYLOR H. M., A First Course in Stochastic Processes, Academic Press, New York 1975

6. KNUTH D. E., The Art of Computer Programming, Voh 2,Addison Wesley 1973

7. FRICKE T., SCHNAKENBERG J., Monte Carlo Simulation of an inhomogeneous reaction-diffusion system in the biophysics of receptor cells, Z. Phys. B. 83, 277 (1991)

8. KR1SCHER K., M1KHAILOV A..S., Bifurcation to traveling spots in Reaction Diffusion Systems, Phys. Rev. Left. 73, 23 (1994)

9. FRICKE T., WENDT. D, The Markoff automaton - a new algorithm for simulating the time-evolution of large stochastic dynamic systems, Int. Y. Mod. Phys. C, 2,277 (1995)

10. FEISTEL R., EBELING W., Evolution of Complex Systems, Kluwer Dordrecht, 1989

11. WENDT D., FRICKE T., SCHNAKENBERG J., Stochastic Simulation of the Reaction-Diffusion System A + B -+inert in Integer and Fractal Dimensions, Z. Phys. B. 96, 541 (1995)

12. The animations and several related stuff is available in the World Wide Web http ://summa. physik .hu-berlin .de/ thomas/spots3d

stochastic cellular automata

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