10/28/2015

Chris Miles

QUESTION:

Does a general theory exist describing the formation of patterns from non-patterns?

MORPHOGENESIS

• biological process of organism developing its shape

• all cells start from single precursor

• incredibly complex!

KEY COMPONENTS

local activation
long range inhibition

reaction = stuff interacting

diffusion = stuff moving around

Turing instability: diffusion driven

without diffusion: no pattern (stable)

diffusion: pattern

WAVE FORMATION

diffusion driven pattern!

MATHEMATICAL FORMATION

\frac{\partial u}{\partial t} = f(u,v) + D_u \frac{\partial^2 u}{\partial x^2}
$\frac{\partial u}{\partial t} = f(u,v) + D_u \frac{\partial^2 u}{\partial x^2}$

rate of change of chemical concentration

reactions

diffusion

\frac{\partial v}{\partial t} = g(u,v) + D_u \frac{\partial^2 v}{\partial x^2}
$\frac{\partial v}{\partial t} = g(u,v) + D_u \frac{\partial^2 v}{\partial x^2}$
u(x,t)
$u(x,t)$
v(x,t)
$v(x,t)$

= concentration of morpohgen at position x, time t

GIERER-MEINHARDT

\frac{\partial u}{\partial t} = \frac{ u^2}{v} - bu + D_u \frac{\partial^2u}{\partial x^2},
$\frac{\partial u}{\partial t} = \frac{ u^2}{v} - bu + D_u \frac{\partial^2u}{\partial x^2},$

used to describe sea-shell patterns (1972)

\frac{\partial v}{\partial t} =u^2 - v + D_v \frac{\partial^2v}{\partial x^2}.
$\frac{\partial v}{\partial t} =u^2 - v + D_v \frac{\partial^2v}{\partial x^2}.$

With no diffusion:

\frac{\partial u}{\partial t} = \frac{ u^2}{v} - bu,
$\frac{\partial u}{\partial t} = \frac{ u^2}{v} - bu,$
\frac{\partial v}{\partial t} =u^2 - v.
$\frac{\partial v}{\partial t} =u^2 - v.$

0= \frac{ u^2}{v} - bu,
$0= \frac{ u^2}{v} - bu,$
0=u^2 - v.
$0=u^2 - v.$
(u_0, v_0) = (1/b, 1/b^2)
$(u_0, v_0) = (1/b, 1/b^2)$
D_u, D_v =0
$D_u, D_v =0$

LINEARIZATION

(u_0, v_0) = (1/b, 1/b^2)
$(u_0, v_0) = (1/b, 1/b^2)$

Perturb slightly from our steady state:

u(x,t) = u_0 + \tilde{u},
$u(x,t) = u_0 + \tilde{u},$
v(x,t) = v_0 + \tilde{v}
$v(x,t) = v_0 + \tilde{v}$

f(u,v) = f(u_0, v_0) + \tilde{u}\frac{\partial f(u_0,v_0)}{\partial u} + \tilde{v}\frac{\partial f(u_0,v_0)}{\partial v} + ...
$f(u,v) = f(u_0, v_0) + \tilde{u}\frac{\partial f(u_0,v_0)}{\partial u} + \tilde{v}\frac{\partial f(u_0,v_0)}{\partial v} + ...$
\frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} = \left(D\frac{\partial}{\partial x^2} + J \right) \binom{\tilde{u}}{\tilde{v}},
$\frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} = \left(D\frac{\partial}{\partial x^2} + J \right) \binom{\tilde{u}}{\tilde{v}},$

J = \left( \begin{array}{cc} \partial f / \partial u & \partial f/ \partial v \\ \partial g/\partial u & \partial g / \partial v \end{array} \right)
$J = \left( \begin{array}{cc} \partial f / \partial u & \partial f/ \partial v \\ \partial g/\partial u & \partial g / \partial v \end{array} \right)$
D = \left( \begin{array}{cc} D_u & 0 \\ 0 & D_v \end{array} \right)
$D = \left( \begin{array}{cc} D_u & 0 \\ 0 & D_v \end{array} \right)$

diffusion driven = stable with no diffusion

\left.J\right|_{(1/b, 1/b^2)} = \left. \left(\begin{array}{cc} -b + \frac{2u}{v} & - \frac{u^2}{v^2} \\ 2u & -1 \end{array} \right) \right|_{(1/b, 1/b^2)} = \left(\begin{array}{cc} b & - b^2 \\ \frac{2}{b} & -1 \end{array} \right).
$\left.J\right|_{(1/b, 1/b^2)} = \left. \left(\begin{array}{cc} -b + \frac{2u}{v} & - \frac{u^2}{v^2} \\ 2u & -1 \end{array} \right) \right|_{(1/b, 1/b^2)} = \left(\begin{array}{cc} b & - b^2 \\ \frac{2}{b} & -1 \end{array} \right).$
\frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} =J \binom{\tilde{u}}{\tilde{v}},
$\frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} =J \binom{\tilde{u}}{\tilde{v}},$

for Gierer-Meinhardt

result from ODE theory: solutions look like

\binom{\tilde{u}}{\tilde{v}} = e^{\lambda_1 t} a(x)+ e^{\lambda_2 t} b(x)
$\binom{\tilde{u}}{\tilde{v}} = e^{\lambda_1 t} a(x)+ e^{\lambda_2 t} b(x)$

are eigenvalues of

\lambda_1, \lambda_2
$\lambda_1, \lambda_2$
J
$J$
D_u, D_v =0
$D_u, D_v =0$

fact:

e^{\lambda t} \to 0
$e^{\lambda t} \to 0$
Re (\lambda) < 0
$Re (\lambda) < 0$

if

thus, this boils down to:

tr(J) = b-1 < 0
$tr(J) = b-1 < 0$
det(J) = b >0
$det(J) = b >0$
0 < b < 1
$0 < b < 1$

DIFFUSION DRIVEN INSTABILITY

D_u, D_v \neq 0
$D_u, D_v \neq 0$
u(x,t) = u_0 + \tilde{u},
$u(x,t) = u_0 + \tilde{u},$
v(x,t) = v_0 + \tilde{v}
$v(x,t) = v_0 + \tilde{v}$
\frac{\partial \tilde{u}}{\partial t} = b\tilde{u} - b^2 \tilde{v}+ D_u \frac{\partial^2\tilde{u}}{\partial x^2},
$\frac{\partial \tilde{u}}{\partial t} = b\tilde{u} - b^2 \tilde{v}+ D_u \frac{\partial^2\tilde{u}}{\partial x^2},$
\frac{\partial \tilde{v}}{\partial t} = \frac{2}{b} \tilde{u} - \tilde{v} + D_v \frac{\partial^2\tilde{v}}{\partial x^2}.
$\frac{\partial \tilde{v}}{\partial t} = \frac{2}{b} \tilde{u} - \tilde{v} + D_v \frac{\partial^2\tilde{v}}{\partial x^2}.$
\binom{\tilde{u}}{\tilde{v}} =\binom{ A(t)e^{iqx} }{B(t)e^{iqx}}
$\binom{\tilde{u}}{\tilde{v}} =\binom{ A(t)e^{iqx} }{B(t)e^{iqx}}$

Look for solutions of the form:

e^{iqx} = \cos(qx) + i \sin(qx)
$e^{iqx} = \cos(qx) + i \sin(qx)$

Euler's identity

waves!

\frac{\partial}{\partial t} \binom{\tilde{u}}{\tilde{v}} = \left( \begin{array}{cc} b - q^2D_u & -b^2 \\ \frac{2}{b} & -1-q^2D_v \end{array} \right) \binom{\tilde{u}}{\tilde{v}}
$\frac{\partial}{\partial t} \binom{\tilde{u}}{\tilde{v}} = \left( \begin{array}{cc} b - q^2D_u & -b^2 \\ \frac{2}{b} & -1-q^2D_v \end{array} \right) \binom{\tilde{u}}{\tilde{v}}$

DIFFUSION DRIVEN INSTABILITY

\frac{\partial}{\partial t} \binom{\tilde{u}}{\tilde{v}} = \left( \begin{array}{cc} b - q^2D_u & -b^2 \\ \frac{2}{b} & -1-q^2D_v \end{array} \right) \binom{\tilde{u}}{\tilde{v}}
$\frac{\partial}{\partial t} \binom{\tilde{u}}{\tilde{v}} = \left( \begin{array}{cc} b - q^2D_u & -b^2 \\ \frac{2}{b} & -1-q^2D_v \end{array} \right) \binom{\tilde{u}}{\tilde{v}}$

want INSTABILITY

tr(J) = b-1-q^2(D_u+D_v) <0
$tr(J) = b-1-q^2(D_u+D_v) <0$
b<1
$b<1$
det(J) = (b-D_u q^2)(-1 - D_v q^2) < 0
$det(J) = (b-D_u q^2)(-1 - D_v q^2) < 0$

always true since

DOMAIN DEPENDENCE

What is q?

If our domain is [0,L], solutions look like:

\sum_{k} A_k e^{\lambda q^2 t} \cos (qx), \qquad q = \frac{n\pi}{L}
$\sum_{k} A_k e^{\lambda q^2 t} \cos (qx), \qquad q = \frac{n\pi}{L}$

(from Fourier)

critical domain size!

OTHER REACTIONS

\frac{\partial u}{\partial t} = D_u \Delta u - uv^2 +F(1-u),
$\frac{\partial u}{\partial t} = D_u \Delta u - uv^2 +F(1-u),$
\frac{\partial v}{\partial t} = D_v \Delta u +uv^2-(F+c)v.
$\frac{\partial v}{\partial t} = D_v \Delta u +uv^2-(F+c)v.$

Gray-Scott:

\Delta u = \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
$\Delta u = \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}$

OTHER DOMAINS

https://github.com/gollygang/ready

REFERENCES

1. S. Kondo and T. Miura, “Reaction-diffusion model as a framework for understanding biological pattern formation,” Science, vol. 329, no. 5999, pp. 1616–1620, (2010).
2. http://www.theshapeofmath.com/princeton/dynsys
3.  P.K. Maini, The impact of Turing's work on pattern formation in biology, Mathematics Today, 40(4), 140-141 (2004).
4. Theoretical aspects of pattern formation and neuronal development, Hans Meinhardt

this presentation is on my website: http://chrismil.es

By Chris Miles

HOW THE ZEBRA GOT ITS STRIPES: MATHEMATICAL PATTERN FORMATION

A presentation for the undergraduate mathematics colloquium at the University of Utah on mathematical pattern formation and Turing instabilities.

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