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Animal Coat Patterns |
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.
Let C be the vector of morphogen concentrations. The vector equation giving the spatial and temporal dynamics of morphogen concentrations is:
where F(C) is a nonlinear vector function and D is diagonal matrix.
The equations can be put into a form such that the diagonal matrix D
is of the form
D = diag(1,d). Thus d represents the relative magnitude of
the diffusion coefficient of one morphogen compared to the other.
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:
Let C* be a solution to F(C)=0. C* is a solution which is uniform over space and thus ∇^{2}C* = 0. Now consider a linearization of the equation about the point C*. Thus
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. The existence of γ and its relation to the scale of the system requires further analysis.^{1}
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:
This system is stable if all of the roots of the determinant equation
have negative real parts. In general, the product of all the roots of this equation when A is n×n is equal to γ^{n}det[A] and the sum of the roots is equal to γtr(A).
For a two-component system the equation for the eigenvalues is:
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.
Now let us go back to the equation that includes spatial
variation and consider a separation of variables; i.e.,
c(r,t)= W(r)T(t) where W(r) is a scalar-valued function and
T(t) is a
vector function. Thus,
In order to proceed further it is necessary to limit W to functions which are solutions to the boundary value problem:
Now the system of equation is
For the solution for T(t) to be of the form exp(λt)T, we must have
The λ's are the eigenvalues of the matrix M = γA - k^{2}D. For spatial instability one of the eigenvalues for some value of k^{2} must have a positive real part. For k^{2} = 0 none of the eigenvalues has a real part.
For a two-component system the equation satisfied by the eigenvalues is:
In order for one of the solutions of the quadratic equation to have a positive real part for some value of k it is necessary that either:
Since -tr(M) = -γtr(A) + k^{2}tr(D) and tr(A) was assumed to be negative for steady state stability there is no way that the coefficient of λ, -tr(M), can be negative. Therefore the only way that at least one root can have a positive real value is that det(M) be negative for some value of k^{2}.
The graph shows the case that det(M) is negative for some value of k^{2}. For this case to hold there must be a value of d such that for some value of k^{2} det(M)=0 and the derivative of det(M) with respect to k^{2} is also zero. The value of d such that det(M) is tangent to the k^{2} axis is called the critical value of d, d_{c}.
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 as shown on the accompanying reproductions from J.D. Murray's book.
Murray's method of solution makes use of solutions to the Helmholtz equation,
He presents the patterns that occur on the special domains shaped as hexagons, rhombuses and triangles. The results shown in the following are evocative of types of patterns that occur on reptiles.
The most striking of Murray's simulations are those for a domain shaped like an animal skin with four legs, a head and a tail. He examines what the equations give for the junction of a horizontal stripe pattern on a leg with a vertical stripe pattern on the torso. The results are consistent with the pattern of zebra stripes at the junction of leg and torso.
Proto-types of Patterns of Animal Tails
Source: J.D. Murray, Mathematical Biology
Simulations of Animal Coat Patterns
Source: J.D. Murray, Mathematical Biology
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