Self-Organisation in Reaction-Diffusion Systems

Peter Rashkov

University of Exeter Biosciences

22.1.2015

“ . . . Later in the book we discuss in considerable detail variouspossible mechanisms for generating spatial patterns, includingreaction-diffusion systems. I firmly believe that the process here ismechanistic and not genetic.”

— J. D. Murray, Mathematical Biology

Overview

Morphogenesis

Hair follicle spacing in miceAnalytic propertiesLimiting formTuring space

Outlook & Literature

Mathematical models of morphogenesis

I chemical gradients influence fate of surrounding tissue

I cell differentiation

I tissue and organ formation – epidermal appendages (hairs,feathers)

I mathematical model for the chemical basis of morphogenesis– Turing (1952)

I activator-inhibitor model – Gierer and Meinhardt (1972)

I various models for generation of spatial patterns (spatialheterogeneity)

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Turing’s model

I reaction kinetic system

(u, v)t = F (u, v), u, v ∈ R, F : R2 → R2

I steady state (u, v) : F (u, v) = 0

I asymptotically stable to spatially homogeneous perturbations

I reaction-diffusion system

(u, v)t = D∇2(u, v) + F (u, v)

I unstable to spatially heterogeneous perturbations

I D 6= I – diffusion rates for u, v must be different

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Gierer and Meinhardt’s model

ut − du∆u =up

vq+ σ1(x)− µuu in [0,T )× Ω

vt − dv∆v =ur

v s+ σ2(x)− µvv in [0,T )× Ω

(1)

I Ω is a bounded smooth domain in Rn, and T ∈ (0,∞]

I initial data u(0, ·), v(0, ·) ∈ L∞(Ω)

I homogeneous Neumann boundary conditions

n · ∇u = n · ∇v = 0 in ∂Ω

I p = r = 2, q = 1, s = 0, Turing space (µu, µv )

I vast literature on existence and uniqueness of solutions fordifferent p, q, r , s, σi

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Gierer and Meinhardt’s model

(Loading movie...)

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Model for hair follicle localisation in mice

wild-type mouse vs. DKK-transgenetic mutant mouse

5 / 16Sick, Reinker, Timmer & Schlake, Science, 314, 2006.

Model for hair follicle spacing in mice

[Sick et al., 2006]

I WNT signalling pathway in hair follicle localisation

I DKK has inhibitory effect on WNT

I GM kinetics with saturation, Turing instability

I identical production term for activator u & inhibitor v

ut − du∇2u =ρu

v + γ· u2

1 + κu2− µuu in [0,T )× Ω

vt − dv∇2v =ρv

v + γ· u2

1 + κu2− µvv in [0,T )× Ω

(2)

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Reaction-kinetic system

I GM type without source terms

I normalise ρu = ρv = 1, µu := µu

ρu, µv := µv

ρv

ut = u2

(v+γ)(1+κu2) − µuu

vt = u2

(v+γ)(1+κu2) − µvv

ut = 0

vt = 0

v

uO

µv ≤ µ2u

µv > µ2u

vt = 0

E

E ′

−γ

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Analytical properties

I solution (u, v) of (2) has global existence in L∞ on (0,∞]× Ω forstrictly positive initial data in L∞

I (0, 0) is the unique homogeneous steady state for µv < µ2u

(corresponds to values used by Sick et al. (2006))

I it can be shown analytically that u = 0, v = 0 is local attractor forthe model system (2)

I Turing instability is never possible at (0, 0) no matter what du, dvare!

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Analytical properties

I solution (u, v) of (2) has global existence in L∞ on (0,∞]× Ω forstrictly positive initial data in L∞

I (0, 0) is the unique homogeneous steady state for µv < µ2u

(corresponds to values used by Sick et al. (2006))

I it can be shown analytically that u = 0, v = 0 is local attractor forthe model system (2)

I Turing instability is never possible at (0, 0) no matter what du, dvare!

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Analytical properties

I solution (u, v) of (2) has global existence in L∞ on (0,∞]× Ω forstrictly positive initial data in L∞

I (0, 0) is the unique homogeneous steady state for µv < µ2u

(corresponds to values used by Sick et al. (2006))

I it can be shown analytically that u = 0, v = 0 is local attractor forthe model system (2)

I Turing instability is never possible at (0, 0) no matter what du, dvare!

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Analytical properties

I solution (u, v) of (2) has global existence in L∞ on (0,∞]× Ω forstrictly positive initial data in L∞

I (0, 0) is the unique homogeneous steady state for µv < µ2u

(corresponds to values used by Sick et al. (2006))

I it can be shown analytically that u = 0, v = 0 is local attractor forthe model system (2)

I Turing instability is never possible at (0, 0) no matter what du, dvare!

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Limiting form of model system

ut − du∇2u =u2

v(1 + κu2)− µuu in [0,T )× Ω

vt − dv∇2v =u2

v(1 + κu2)− µvv in [0,T )× Ω

(3)

I γ = 0, du = 1, dv = d – approximation when γ ≈ 0

I solution (u, v) of (3) has global existence in L∞ on(0,∞]× Ω for strictly positive initial data in L∞

I a-priori bounds on stationary solutions in t

I find Turing space

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Modified model, phase portrait

ut = 0

vt = 0

v

uO

µv ≤ µ2u

µv > µ2u

vt = 0

E

I µ2u ≥ µv : no homogeneous steady state for (3)I O is a singularity for the reaction kinetic systemI when µu > µv , for appropriately chosen initial data u0, v0 the

solutions of (3)

u, v → 0 uniformly on Ω, t →∞

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Turing space for fixed Ω, d , κ

µv

µu0

µ v=µ2 uI

I

V

C

IVh1(µu, µv) = 0

II

IIIh 2(µ

u, µ

v)=0

subregion V – Turing instability, subregion C – collapsing solutionssubregion IV – far-from-equilibrium stationary solutions

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Summary

I “modified” GM – uniform boundedness, global existence

I conditions for pattern formation in Sick et al. (2006) not relevant

I patterns observed are not due to Turing bifurcation, butconvergence to a far-from-equilibrium solution

I sensitivity to initial conditions because of collapsing solutions

u

0 20 40 60 80 100

0

20

40

60

80

100

0

0.5

1

1.5

2

2.5

v

0 20 40 60 80 100

0

20

40

60

80

100

0

0.5

1

1.5

2

Figure: Asymmetric pattern converging to some non-trivial solutionbranch stemming from some (unknown) steady state

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Literature

I P. Rashkov.Remarks on pattern formation in a model for hair follicle spacing.(under review)

I P. Rashkov.Regular and discontinuous solutions in a reaction-diffusion model forhair follicle spacing.Biomath J. 3(2): 1411111, 2014.

I S. Sick, S. Reinker, J. Timmer, and T. Schlake.WNT and DKK determine hair follicle spacing through areaction-diffusion mechanism.

Science 314(5804): 1447–1450, 2006.

Thanks to : S. Dahlke, B. Schmitt (Marburg), J. Sherratt(Heriott-Watt), A. Marciniak-Czochra (Heidelberg)

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