Pattern Formation

Patrick Lucey

• Why do some animals have spotted bodies and striped tails, but there are no animals with striped bodies and spotted tails?

• All of these questions now have a mathematical answer. The model in question describes the way in which two different chemical products react and are propagated on the skin : one which colors the skin, and one which does not color it; or more precisely, one which stimulates the production of melanin (coloring the skin) and one which inhibits this production.

Alan Turing• In the quest for understanding

biological growth, it was Alan Turing who first demonstrated how a simple model system of coupled reaction-diffusion equations could give rise to spatial patterns in chemical concentrations through a process of chemical instability.

• Humans have long had a fascination with the coat patterns of animals. Alan Turing was the first to articulate an explanation of how the patterns of animals like leopards, jaguars and zebras are determined. Turing asserted that the patterns can arise as a result of instabilities in the diffusion of morphogenetic chemicals in the animals' skins during the embryonic stage of development.

Turing’s Interest

• Turing worked from 1952 until his death in 1954 on mathematical biology, specifically morphogenesis. He published one paper on the subject called "The Chemical Basis of Morphogenesis" in 1952, putting forth the Turing hypothesis of pattern formation His central interest in the field was understanding Fibonacci phyllotaxis , the existence of Fibonacci numbers in plant structures. He used reaction-diffusion equations which are now central to the field of pattern formation.

• Morphogenesis : a study concerned with the shapes of tissues, organs and entire organisms and the positions of the various specialized cell types

The Model

• Let C be the vector of morphogen concentrations. The vector equation giving the spatial and temporal dynamics of morphogen concentrations is:

• ∂C/∂t = F(C) + D 2C∇

• where F(C) is a nonlinear vector function and D is diagonal matrix.

• Turing said in effect that if D=0 and C goes to the steady state solution of F(C*)=0, then under some circumstance the introduction of nonzero D can create spatial variations that produce patterns.

• For the problem to be properly formulated there must be boundary and initial conditions specified. Let B be the domain of the problem and let ∂B be the boundary of the domain. Then the boundary and intial conditions are:

• n· C = 0 on ∂B∇C(r,0) = G(r).

• Let C* be a solution to F(C)=0. C* is a solution which is uniform over space and thus 2C* = 0. Now consider a ∇linearization of the equation about the point C*. Thus

• ∂C/∂t = F(C) + D 2C∇∂C*/∂t=F(C*)+D 2C*∇∂c/∂t = γAc+ D 2c∇

• where c = C - C* and γA = [∂F/∂C], where γ is a nondimensional scale parameter that is proportional to the square of the linear dimension of a one dimensional system and the area of a two dimensional system.

• In the absence of spatial variation in c the behavior of the system for |c| small is the same as the solution to the system:

• ∂z/∂t = γAz

• This system is stable if all of the roots of the determinant equation

• det[λI - γA] = 0

• For a two-component system the equation for the eigenvalues is:

• Λ^2 - γtr(A)λ + γ^2det(A)= 0 The condition that the real parts of both roots be negative is that the det(A) be positive and the trace of A be negative. These conditions are equivalent to both roots having the same sign and the sum of the roots being negative.

On an Interesting Note

• For a periodic solution in an essentially linear domain, as for example in a snake skin, coloration of all regions above a threshold level would result in stripes. In more rectangular domains there could also be stripes, but there can also be spots.

A Visual