HOW THE ZEBRA GOT ITS STRIPES: THE MATHEMATICS OF PATTERN FORMATION
Undergraduate Mathematics Colloquium
10/28/2015
Chris Miles
PATTERNS
QUESTION:
Does a general theory exist describing the formation of patterns from non-patterns?
ALAN TURING
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
MORPHOGENS
WAVE FORMATION
diffusion driven pattern!
MATHEMATICAL FORMATION
\frac{\partial u}{\partial t} = f(u,v) + D_u \frac{\partial^2 u}{\partial x^2}
∂t∂u=f(u,v)+Du∂x2∂2u
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}
∂t∂v=g(u,v)+Du∂x2∂2v
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},
∂t∂u=vu2−bu+Du∂x2∂2u,
used to describe sea-shell patterns (1972)
\frac{\partial v}{\partial t} =u^2 - v + D_v \frac{\partial^2v}{\partial x^2}.
∂t∂v=u2−v+Dv∂x2∂2v.
STEADY STATE
With no diffusion:
\frac{\partial u}{\partial t} = \frac{ u^2}{v}
- bu,
∂t∂u=vu2−bu,
\frac{\partial v}{\partial t} =u^2 - v.
∂t∂v=u2−v.
Steady state:
0= \frac{ u^2}{v}
- bu,
0=vu2−bu,
0=u^2 - v.
0=u2−v.
(u_0, v_0) = (1/b, 1/b^2)
(u0,v0)=(1/b,1/b2)
D_u, D_v =0
Du,Dv=0
LINEARIZATION
(u_0, v_0) = (1/b, 1/b^2)
(u0,v0)=(1/b,1/b2)
Perturb slightly from our steady state:
u(x,t) = u_0 + \tilde{u},
u(x,t)=u0+u~,
v(x,t) = v_0 + \tilde{v}
v(x,t)=v0+v~
Taylor expand around steady state:
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(u0,v0)+u~∂u∂f(u0,v0)+v~∂v∂f(u0,v0)+...
\frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} = \left(D\frac{\partial}{\partial x^2} + J \right) \binom{\tilde{u}}{\tilde{v}},
∂t∂(v~u~)=(D∂x2∂+J)(v~u~),
Only take leading order terms:
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=(∂f/∂u∂g/∂u∂f/∂v∂g/∂v)
D = \left( \begin{array}{cc} D_u & 0 \\ 0 & D_v \end{array} \right)
D=(Du00Dv)
STABLE STEADY STATE
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).
J∣(1/b,1/b2)=(−b+v2u2u−v2u2−1)(1/b,1/b2)=(bb2−b2−1).
\frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} =J \binom{\tilde{u}}{\tilde{v}},
∂t∂(v~u~)=J(v~u~),
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)
(v~u~)=eλ1ta(x)+eλ2tb(x)
are eigenvalues of
\lambda_1, \lambda_2
λ1,λ2
J
J
D_u, D_v =0
Du,Dv=0
STABLE STEADY STATE
fact:
e^{\lambda t} \to 0
eλt→0
Re (\lambda) < 0
Re(λ)<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
Du,Dv=0
u(x,t) = u_0 + \tilde{u},
u(x,t)=u0+u~,
v(x,t) = v_0 + \tilde{v}
v(x,t)=v0+v~
\frac{\partial \tilde{u}}{\partial t} = b\tilde{u} - b^2 \tilde{v}+ D_u \frac{\partial^2\tilde{u}}{\partial x^2},
∂t∂u~=bu~−b2v~+Du∂x2∂2u~,
\frac{\partial \tilde{v}}{\partial t} = \frac{2}{b} \tilde{u} - \tilde{v} +
D_v \frac{\partial^2\tilde{v}}{\partial x^2}.
∂t∂v~=b2u~−v~+Dv∂x2∂2v~.
\binom{\tilde{u}}{\tilde{v}} =\binom{ A(t)e^{iqx} }{B(t)e^{iqx}}
(v~u~)=(B(t)eiqxA(t)eiqx)
Look for solutions of the form:
e^{iqx} = \cos(qx) + i \sin(qx)
eiqx=cos(qx)+isin(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}}
∂t∂(v~u~)=(b−q2Dub2−b2−1−q2Dv)(v~u~)
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}}
∂t∂(v~u~)=(b−q2Dub2−b2−1−q2Dv)(v~u~)
want INSTABILITY
tr(J) = b-1-q^2(D_u+D_v) <0
tr(J)=b−1−q2(Du+Dv)<0
b<1
b<1
det(J) = (b-D_u q^2)(-1 - D_v q^2) < 0
det(J)=(b−Duq2)(−1−Dvq2)<0
always true since
PATTERN FORMATION
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}
∑kAkeλq2tcos(qx),q=Lnπ
(from Fourier)
critical domain size!
INSUFFICIENT L
OTHER REACTIONS
\frac{\partial u}{\partial t} = D_u \Delta u - uv^2 +F(1-u),
∂t∂u=DuΔu−uv2+F(1−u),
\frac{\partial v}{\partial t} = D_v \Delta u +uv^2-(F+c)v.
∂t∂v=DvΔu+uv2−(F+c)v.
Gray-Scott:
\Delta u = \nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}
Δu=∇2u=∂x2∂2u+∂y2∂2u
OTHER DOMAINS
CURRENT RESEARCH
REFERENCES
- 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).
- http://www.theshapeofmath.com/princeton/dynsys
- P.K. Maini, “The impact of Turing's work on pattern formation in biology,” Mathematics Today, 40(4), 140-141 (2004).
- “Theoretical aspects of pattern formation and neuronal development”, Hans Meinhardt
this presentation is on my website: http://chrismil.es
HOW THE ZEBRA GOT ITS STRIPES: THE MATHEMATICS OF PATTERN FORMATION Undergraduate Mathematics Colloquium 10/28/2015 Chris Miles