Turing Patterns : The Chemical Basis of Morphogenesis

Turing Patterns : The Chemical Basis of MorphogenesisWolfram Language Summer School, June 2018

Wim Poignon and Agnès Totschnig, Lycée Français de Berlin, Germany

Printed by Wolfram Mathematica Student Edition

Introduction to the problem

Can one general mathematical model explain the diversity of animal patterns in nature?

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Reaction

Chemical reactions of morphogens : activator and inhibitor

The patterns arise from the reaction of two morphogens stimulating the production of melanin.

Gierer-Meinhardt model

The evolution of the concentration over time can be described by the following mathematical model :

∂u

∂t= f(u, v) = s

u2

v1+ku2+bu - ru u .

∂v

∂t= g(u, v) = su2 - rv v .

◼ u and v are respectively the concentration of activator and inhibitor

◼ s is the source density

◼ bu is the small activator production

◼ ru and rv are the degradation rates

◼ ku2 is the saturation term

Approximation with Mathematica

We can solve the ODE by an approximation with the Euler method :

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ux,t+Δt = ux,t +Δt * f (ux,t, vx,t) .vx,t+Δt = vx,t +Δt * f (ux,t, vx,t) .

◼ Δt is the size of the step of the approximation

For initial conditions u0 = v0 = 1, we get the following evolution :

0 2000 4000 6000 8000 10 000Iterations

1

2

3

4

5

6Concentration

Activator

Inhibitor

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Diffusion

Intuitive Explanation

The process of diffusion is the spreading of molecules from areas with high concentration to areas with lower concentration.

Fick’s law of diffusion

The diffusion flux is proportional to the concentration gradient :

J(x, t) = -Du∂u(x,t)∂x

◼ J(x, t) is the diffusion flux

◼ Du is the diffusion coefficient

◼∂u(x,t)∂x

is the diffusion gradient

As we only consider diffusion, no mass is created or destroyed inside of each area, so we can apply the law of conservation of mass :

ⅆ

ⅆt ∫x0

x1u(x, t)ⅆx = J(x0, t) - J(x1, t)

Combining these two equations, we finally get the evolution of the concentration over time :

∂ u(x,t)∂ t

= Du*∂2u(x,t)∂ x2

Approximation with Mathematica

We can solve the PDE by an approximation with the explicit finite-difference method :

ux,t+Δt = Δt * [Du (ux-1,t -ux,t) +Du (ux+1,t -ux,t)] .vx,t+Δt = Δt * [Dv (vx-1,t - vx,t) +Dv (vx+1,t - vx,t)] .

For different diffusion coefficient, we get the following evolution :

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Animal Patterns : diffusion-driven instabilities

Structure of the algorithm

The concentration in the cells

size = 30;

initialisation[] := TableRandomVariateNormalDistribution[3.3,0.5],2,size;

concentration=initialisation[]

PlotTablePDFNormalDistribution[3.3,σ],x,{σ,{.5}}//Evaluate,{x,0,6},Filling→Axis, PlotRange

Out[ ]= {{3.58383, 2.94438}, {2.79789, 2.84424}, {4.05785, 3.28149},

{2.75419, 3.73823}, {2.25539, 3.38386}, {3.67961, 4.36445},

{3.67715, 2.22433}, {3.55133, 3.24847}, {3.62059, 3.25456}, {3.90991, 3.28188},

{3.8829, 2.90608}, {3.7748, 3.75322}, {2.48831, 3.83937}, {3.01641, 3.03226},

{2.84356, 2.98982}, {3.81287, 3.78253}, {4.15429, 3.00799}, {3.93625, 3.54328},

{2.90233, 2.68243}, {3.43307, 3.47585}, {2.99333, 2.95553}, {3.06619, 3.14006},

{2.49194, 3.98947}, {3.45877, 4.37862}, {3.26128, 2.04679}, {3.95125, 3.50162},

{3.63006, 3.48508}, {3.27772, 4.2492}, {2.66212, 2.95701}, {3.50316, 3.57598}}

Out[ ]=

1 2 3 4 5 6

0.2

0.4

0.6

0.8

1.0

The parameters

iterations = 2000;

Δt = 0.1;

Du = 0.002;

Dv=0.06;

ru=0.006;

rv=0.02;

s=0.006;

k=0;

bu=0.0004;

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Reaction of the inhibitors and the activators

In[ ]:= f[u_,v_]:=s(u^2/(v(1+k*u^2))+bu)-ru*u;

g[u_,v_]:=s*u^2-rv*v;

Diffusion of the inhibitors and activators

In[ ]:= DiffusionU[n_] :=Du*concentrationModn-1,size+1,1-2concentrationModn,size+1,1+concentration

DiffusionV[n_]:=Dv*concentrationModn-1,size+1,2-2concentrationModn,size+1,2+concentration

The evolution of the concentration over time

u[n_]:=concentrationModn,size+1,1+Δt*fconcentrationModn,size+1,1,concentration

v[n_]:=concentrationModn,size+1,2+Δt*gconcentrationModn,size+1,1,concentration

evolution=Tableconcentration=Table{u[n],v[n]},n,0,size-1,iterations;

Display

displayconcentration_List:=ListLinePlotTablen,concentration[[n,1]],n,1,size,

Tablen,concentration[[n,2]],n,1,size,

PlotRange→{6,0};

Results

In[ ]:=

importe

Import["simulation1.avi", "Animation"]

Out[ ]=

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Discussion

Encountered difficulties

◼ too many parameters to find stable state

◼ too long computation time due to slow convergence

Further research

◼ generalise to a two-dimensional space :

◼ study the influence of the shape and the size of the space

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Bibliography◼ A Two-Dimensional Numerical Study of Spatial Pattern Formation in Interacting Turing Systems, R.

A. Barrio, C. Varea, J. L. Aragon, P. K. Maini, 1999, Bulletin of Mathematical Biology

◼ La Formation de Motifs et l’Instabilité de Turing, Catherine Poissant, Département de mathématiques et de statistique, Université de Montréal, 2015, Bulletin de l’Association mathématique du Québec

◼ La mathématique du modèle, Université Francois-Rabelais, Tours

◼ Les motifs des pelages d’animaux, L.G. Vidiani

◼ Motifs des pelages d’animaux, C. Bejjani, N. Khattabi, R. Labib, École polytechnique de Montréal

◼ How the Leopard Gets Its Spots, James D. Murray, 1988, Scientific American

◼ The Algorithmic Beauty of Sea shells, H. Meinhardt, 2009, Springer

◼ Can Math Explain How Animals Get Their Patterns ? , MinuteEarth on YouTube : https://youtu.be/alH3yc6tX98

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Thank you for your attention!Do you have any question?

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Turing Patterns : The Chemical Basis of Morphogenesis

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