Reaction-Diffusion Modelling of Pattern Formation.Charlotte E. Jupp*, Ruth E. Baker, Philip K. Maini.

Centre for Mathematical Biology, University of Oxford.Mathematical Institute, 24-29 St. Giles’, Oxford OX1 3LB.

* jupp@maths.ox.ac.uk

2. The Model

In his paper in 1952 [4], Turing suggested that chemicals can react and diffuse to produce patterns. Therefore, the formation of patterns in an organism can be modelled by a series of reaction-diffusion equations for the interacting species. If, on the removal of the diffusion terms, the species tend towards a linearly stable state, then, under certain conditions, patterns can evolve when the diffusion is re-introduced.

A non-dimensional model for the interaction of two species, u and v, was proposed by Schnakenberg [3] in 1979. In one spatial dimension, it is

.

,

2

22

2

22

x

vdvub

t

v

x

uvuua

t

u

represents the relative strength of the reaction terms,

d is the ratio of diffusion coefficients,

a and b are different ratios of the positive rate constants.

Neumann boundary conditions are applied on the domain [0,1].

4. Conditions needed for pattern formation

After analysis it was found that the following conditions need to be satisfied for diffusion driven instability to take place, i.e. for pattern formation to occur.

.2

4

1

,02

,

22

2

2

3

bababa

bdd

d

baba

bdd

abba

1. Introduction

Pattern formation is an important part of developmental biology. In the formation of an organism, cells differentiate, in the biological sense, according to where they are in the spatial organisation [2]. We have been looking at methods of modelling pattern formation, and how, by altering the parameters and initial conditions of the model, different pattern types can form.

3. Solution

The solution of the linearised system about the steady state has the form

,exp tikxV

U

v

u

where U, V are constants. Using the boundary conditions, it is shown that

,nk where n is an integer. k is the mode, and determines how many stripes form. e.g. k = 1 gives one stripe.

5. Results

Analysis showed that varying had the greatest effect upon the type of pattern formed with this model. By choosing appropriately, different modes, k, could be isolated separately, allowing for an increasing number of stripes to form. For example

30 115 250 450

Number Number of Stripesof Stripes

1 2 3 4

The other parameters took the values: a = 0.1, b = 1, d = 10.

These results agreed with computer simulations run using MATLAB, with random oscillatory initial conditions.

= 115. Two intersections at steady state, therefore 2 stripes.

= 450. Four intersections at steady state, therefore 4 stripes.

6. Future work

• Analyse the model in two spatial dimensions, to establish 2D patterns, such as spots.

• Look at other models, such as a Gierer-Meinhardt [1] system.

• Examine sequential pattern formation due to stimulation at one boundary.

References

[1] A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetik, 12:30-39, 1972.

[2] J. Murray. Mathematical Biology. Springer-Verlag, Berlin, Heidelberg, 2nd edition, 1989.

[3] J. Schnakenberg. Simple chemical reaction systems with limit cycle behaviour. J. Theor. Biol., 81:389-400, 1979.

[4] A. Turing. The chemical basis of morphogenesis. Phil. Trans. R. Soc. B., 237:37-72, 1952.

Acknowledgements.

This research was financially supported by a research studentship from Microsoft.