Review L15 Turing-Gierer-Meinhardt models Local excitation, global inhibition 22r+hq a+D a +d a concentration activator concentration inhibitor variables time X posItion basal activator synthesis rate Ka, k;: rate constant for synthesis constants decay rates (parameters) D.D. diffusion constants
Review L15 Turing-Gierer-Meinhardt models Local excitation, global inhibition 2 2 2 2 2 2 x i k a i D t i x a a D i a r k t a i i i a a a a ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ γ γ a: concentration activator i: concentration inhibitor t: time x: position variables ra: basal activator synthesis rate ka, ki: rate constant for synthesis γa,γi : decay rates Da, Di: diffusion constants constants (parameters) 1
aa tk rat =Ka i+ d choose normalize dimensionless 4 variables variable oA 1+R A-A+2c2 Q +p homogeneous solution =R+1 0/as=0/at=0 =(R+1)2
2 2 2 2 2 2 x i k a i D t i x a a D i a r k t a i i i a a a a ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ γ γ choose dimensionless variable normalize 4 variables ( ) 2 2 2 2 2 2 1 s I Q A I P τ I s A A I A R τ A ∂ ∂ = − + ∂ ∂ ∂ ∂ = + − + ∂ ∂ 2 ( 1) 1 = + = + I R A R homogeneous solution ∂ / ∂s = ∂ / ∂t = 0 2
homogeneous solution 0/as=0/ot=0 A
homogeneous solution ∂ / ∂s = ∂ / ∂t = 0 A 3 A s I I s
stability of homogeneous solution eRA Ra R R trace o 2AO Q」L2(R+1)Q R-1 0 inhomogeneous A(S,r)=A+A(S,7) solution: /(S,7)=I+/(S)
stability of homogeneous solution ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + − + − + − = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − − − R Q Q R R R R A Q Q I RA I RA 2 ( 1 ) 1 ( 1 ) 1 2 1 2 2 2 2 trace 0 0 1 1 > < + − Q Q R R ( , ) '( , ) ( , ) '( , ) τ τ τ τ I s I I s A s A A s = + inhomogeneous = + solution: 4
inhomogeneous solution A(S,)=A+A(S,) A /(S,)=I+/(S,7) s,τ
inhomogeneous solution 5 A s s A I I’(s,τ) ( , ) '( , ) ( , ) '( , ) τ τ τ τ I s I I s A s A A s = + = + I
OA R-1 R (S,)=A+A(S,) + R+1(1+R)2a /(S,z)=I+/(S,) 2Q(1+R)A-Q/+P A'(S,)=A(T)cos() trial solution I(s, t=I(o)cos(
2 2 2 2 2 ' 2 (1 ) ' ' ' ' ' (1 ) ' 1 ' 1 s I Q R A QI P I s A I R R A R A R ∂ ∂ = + − + ∂ ∂ ∂ ∂ + + − + − = ∂ ∂ τ τ ( , ) '( , ) ( , ) '( , ) τ τ τ τ I s I I s A s A A s = + = + ( ) cos( ) ˆ '( , ) ( ) cos( ) ˆ '( , ) l l s I s I s A s A τ τ τ τ = = trial solution: 6
A'(s,D=A(Tcos( A I'(s, t)=l(T)cos( (S,τ (S,)=A+A(S,) (S,)=1+/(S,t)
7 ( ) cos( ) ˆ '( , ) ( ) cos( ) ˆ '( , ) l l s I s I s A s A τ τ τ τ = = A I s s A I I’(s,τ) ( , ) '( , ) ( , ) '( , ) τ τ τ τ I s I I s A s A A s = + = +
dA(R-11 R s. T TCOS dr\R+12)(1+R) "(S,z)=()cos( d P =2Q(1+R)A-Q+ dt R-11 P 2OR stability 0+ R+12人c2)1+R inhomogeneous P(R-1 solution Q R+1 Q、R-1 P R+l
I P Q R A Q ddI I R R A R R d dA ˆ ˆ 2 (1 ) ˆ ˆ (1 ) ˆ 1 1 1 ˆ 2 2 2 ⎟⎠⎞ ⎜⎝⎛ = + − + + ⎟ − ⎠⎞ ⎜⎝⎛ − +− = l l τ τ ( ) cos( ) ˆ '( , ) ( ) cos( ) ˆ '( , ) l l s I s I s A s A τ τ τ τ = = 0 1 1 1 0 1 1 2 1 1 2 2 2 2 ⎟ + ⎟ +⎠⎞ ⎜⎝⎛⎟ + ⎠⎞ ⎜⎝⎛ − +− − l l l l R P R Q R P QR Q RR stability inhomogeneous solution 1 1 + − > R R P Q 8
R-1 homogeneous stability R+1 inhomogeneous stability Q、R-1 P R+l (S,τ
1 1 + − > R R homogeneous stability: Q inhomogeneous stability: 1 1 + − > R R P Q s I I’(s,τ) I 9
homogeneously stable I relaxes back to previous value after small uniform disturbance △A△△ inhomogeneously stable I' relaxes back to after small spatial disturbance
s I I homogeneously stable: I relaxes back to previous value after small uniform disturbance s I I I’ relaxes back to after small spatial disturbance inhomogeneously stable: I 10
