lab session: numerical simulations of sponateous …1- di usion2- di usion+transport3-...

1- Diffusion 2- Diffusion+transport 3- Self-similarity 4- Cluster formation

Lab session: numerical simulations of sponateouspolarization

Emeric Bouin & Vincent CalvezCNRS, ENS Lyon, France

CIMPA, Hammamet, March 2012


Spontaneous cell polarization: the 1D case

The Hawkins-Voituriez model for spontaneous polarization in 1Dreads as follows:

∂tρ(t, x) = ∂xxρ(t, x) + ρ(t, 0)∂xρ(t, x) , t > 0 , x ∈ (0,+∞) .

The equation is a transport-diffusion equation.

Numerical issues:

• blow-up,

• non-trivial steady states,

• self-similar decay.


Behaviour of solutions

TheoremThere is a nice and simple dichotomy:

• If M < 1, the solution is global in time. It converges towardsa (unique) self-similar profile GM :

limt→+∞

∥∥∥∥ρ(t, x)− 1√t

GM

(x√t

)∥∥∥∥L1

= 0 .

• If M = 1 there is a one-parameter family of steady states:να(x) = α exp(−αx).Convergence holds + the first moment is conserved:α−1 =

∫x>0 ρ

0(x) dx.

• If M > 1 and ∀x > 0 ∂xρ0(x) ≤ 0, the solution blows up in

finite time.


Contents

1- Diffusion equation

2- The diffusion-transport equation

3- The diffusion-transport equation in self-similar variables

4- Cluster formation


The diffusion equation in the half-line

We begin with the diffusion equation

∂tρ(t, x) = ∂xxρ(t, x) , x > 0

+ Neumann boundary condition at x = 0

We rewrite the equation in ”divergence” form,

∂tρ(t, x) + ∂xF = 0 ,

The flux F is given by Fick’s law: F = −∂xρ.

The boundary condition at {x = 0} is F (0) = 0 (no-flux boundarycondition).


Time-Space discretization

We discretize the function ρ(t, x) on a regular grid

[0 : ∆t : T ]× [0 : ∆x : L] ,

• ∆t is the time step,

• T is the final time of computation,

• ∆x is the space step,

• L is the length of the numerical interval.


Strategy for computing ρ(t, x) numerically

We replace the equation ∂tρ(t, x) + ∂xF = 0 with the numericaldiscretization

ρ(t + ∆t, x)− ρ(t, x)

∆t+

F (t, x + ∆x/2)− F (t, x −∆x/2)

∆x= 0 .

(1)We introduce

• ρni = ρ(n∆t, i∆x)

• F ni± 1

2

= F (n∆t, x ±∆x/2).

The equation (1) rewrites at (t, x) = (tn, xi ),

ρn+1i − ρn

i

∆t+

F ni+ 1

2

− F ni− 1

2

∆x= 0


Boundary conditions

The boundary condition F (0) = 0 reads

F 12

= 0 .

For i = 1 the equation (1) reads

ρn+11 − ρn

1

∆t+

F n1+ 1

2

− 0

∆x= 0

We have similarly for i = Nx

ρn+1Nx − ρ

nNx

∆t+

0− F nNx− 1

2

∆x= 0


The diffusion equation

We have F = −∂xρ, therefore

F (t, x + ∆x/2) = −ρ(t, x + ∆x)− ρ(t, x)

∆x

The scheme writes finally

ρn+1i − ρn

i

∆t+

1

∆x

(−ρni+1 − ρn

i

∆x+ρni − ρn

i−1

∆x

)= 0 ,

Remark. It coincides with the classical finite difference scheme forthe heat equation

ρn+1i =

(1− 2

∆t

∆x2

)ρni +

∆t

∆x2ρni+1 +

∆t

∆x2ρni−1 ,


Explicit scheme for the diffusion equation

• Left boundary condition (zero-flux) when i = 1

ρn+1i − ρn

i

∆t+

1

∆x

(−ρni+1 − ρn

i

∆x+ 0

)= 0 ,

• Right boundary condition (zero-flux) when i = Nx

ρn+1i − ρn

i

∆t+

1

∆x

(0 +

ρni − ρn

i−1

∆x

)= 0 ,

• Flux of the density when 2 ≤ i ≤ Nx − 1

ρn+1i − ρn

i

∆t+

1

∆x

(−ρni+1 − ρn

i

∆x+ρni − ρn

i−1

∆x

)= 0 ,


Structure of the scilab code 1/4

Space and time grid of resolution.

%% Space discretisation

L = 10;dx = 0.1;x = [0:dx:L];Nx = length(x);

%% Time discretisation

T = 10;dt = dx^2/4;t = [0:dt:T];Nt = length(t);



Initial data ρ0: we start with a gaussian

%% Initial condition for the density of molecules rho_0

sigma = 1;rho0 = exp(-x.^2/sigma^2);

Z = sum(rho0*dx);rho0 = (M/Z)*rho0;

rho1 = rho0;



Time loop to update the value of the density ρ at each time step n

for n = 1:Nt

rho0 = rho1;

...

end


Structure of the scilab code 4/4Space loop to update the value of the density ρ at each point ofthe grid i

for n = 1:Nt

rho0 = rho1;

for i = 1:Nx

if i == 1 %% Left boundary conditionrho1(i) = ...

elseif i == Nx %% Right boundary conditionrho1(i) = ...

elserho1(i) = ...

end

endend


The maximum principleRecall the numerical scheme for the diffusion equation

ρn+1i =

(1− 2

∆t

∆x2

)ρni +

∆t

∆x2ρni+1 +

∆t

∆x2ρni−1 ,

Observation. ρn+1i is a linear combination of ρn

i−1, ρni and ρn

i+1.

In order to guarantee the maximum principle, it is necessary toimpose a condition between ∆t and ∆x :

2∆t

∆x2< 1 .

CFL condition. In this case, ρn+1i is a convex combination of

ρni−1, ρ

ni and ρn

i+1.

Therefore,

∀i = 1 . . .Nx minjρ0j ≤ ρn

i ≤ maxjρ0j


Conservation of mass

The conservation of mass is automatic when the numerical schemeis written with the flux formulation

ρn+1i − ρn

i

∆t+

F ni+ 1

2

− F ni− 1

2

∆x= 0

To see this, simply sum up the equation:

1

∆t

Nx∑i=1

ρn+1i ∆x − 1

∆t

Nx∑i=1

ρni ∆x + F n

Nx+ 12− F n

12

= 0

The no-flux boundary condition F n12

= F nNx+ 1

2

= 0 guarantees

∑i

ρn+1i ∆x =

∑i

ρni ∆x =

∑i

ρ0i ∆x .


Contents






The diffusion-transport equation

We switch to the one-dimensional polarization equation.

∂tρ(t, x)− ∂x (µ(t)ρ(t, x)) = ∂xxρ(t, x)

+ Neumann boundary condition at x = 0

+ nonlinear coupling via µ(t) = ρ(t, 0).

Again, we rewrite the equation in ”divergence” form,

∂tρ(t, x) + ∂xF = 0 ,

The flux F is given by: F = −∂xρ− µρ.

The boundary condition at {x = 0} is F (0) = 0 (no-flux boundarycondition).

We assume that the speed µ(t) is nonnegative µ(t) ≥ 0.


Strategy for computing ρ(t, x) numerically

We replace the equation ∂tρ(t, x) + ∂xF = 0 with the numericaldiscretization

ρn+1i − ρn

i

∆t+

F ni+ 1

2

− F ni− 1

2

∆x= 0

F ni+ 1

2= −

ρni+1 − ρn

i

∆x︸︷︷︸Fick ′slaw

+ µn ρni+ 1

2︸︷︷︸transport

.

Important question. How to interpolate the value

ρni+ 1

2= ρ(t, x + ∆x/2) ?


The correct choice for ρni+ 1

2

for transport only

Recall the key assumption µn ≥ 0.

Choice 1. ρni+ 1

2

= ρni

ρn+1i =

(1 + µn ∆t

∆x

)ρni − µn ∆t

∆xρni−1 ,

Choice 2. ρni+ 1

2

= ρni+1 [Upwind scheme]

ρn+1i =

(1− µn ∆t

∆x

)ρni + µn ∆t

∆xρni+1 ,

Choice 3. ρni+ 1

2

= 12 (ρn

i + ρni+1) [Centered scheme]

ρn+1i = ρn

i + µn ∆t

2∆xρni+1 − µn ∆t

2∆xρni−1 ,

Only the second choice preserves the maximum principle, under thecondition

µn ∆t

∆x< 1 .


Combination of the two fluxes

We eventually get

ρn+1i − ρn

i

∆t+

1

∆x

(−ρni+1 − ρn

i

∆x− µnρn

i+1 +ρni − ρn

i−1

∆x+ µnρn

i

)= 0 ,

with suitable boundary conditions for i = 1 and i = Nx .


Contents






Self-similar rescaling

Rewrite the density ρ in self-similar variables,

ρ(t, x) =1√

1 + 2tu

(log√

1 + 2t,x√

1 + 2t

)The equation for u(τ, y) is the following,

∂τu(τ, y)−∂y

(yu(τ, y)+v(τ)u(τ, y)

)= ∂yy u(τ, y) , v(τ) = u(τ, 0) .

We get an additional drift term −∂y (yu(τ, y)) due to the changeof frame (t, x)→ (τ, y).


Long-time asymptotics

In the sub-critical regime, M < 1, the solution u(τ, y) convergestowards a unique stationary state G , given by

G (y) = α exp

(−αy − y 2

2

).

The parameter α is fixed by the conservation of mass∫y>0

G (y) dy =

∫y>0

u(τ, y) dy =

∫x>0

ρ(t, x) dx = M .

Exercise. Perform the numerical simulations for u(τ, y), solutionto the polarization equation in self-similar variables (the onlydifference is the additional drift term). Compare the long-timebehaviour with the expected value for G .


Contents






A one-dimensional model for cluster formation

The simplest model for bacterial chemotaxis is the followingcoupled system for the cell density ρ(t, x) and the chemicalconcentration S(t, x).{

∂tρ(t, x) + ∂x (ρ(t, x) u(t, x)) = ∂xxρ(t, x) , x ∈ R

−∂xxS(t, x) = ρ(t, x)

u(t, x) = −∫

v∈Vvφ (v∂xS(t, x)) dv

For simplicity we set V = (−1, 1).

The function φ depends on individual features of the bacteria (e.g.the way they react to the chemical signal).


Cluster formation

Exercise. Perform the numerical simulations for the couple (ρ, S)for different choices of function φ:

• φ(Y ) = −Y , i.e. the Keller-Segel model,

• φ(Y ) = −sign (Y ), i.e. a step function: it is a good model forchemotaxis in bacteria populations.


References

V. Calvez, N. Meunier and R. Voituriez, C. R. Math. Acad. Sci. Paris(2010)

J. Saragosti, V. Calvez, N. Bournaveas, A. Buguin, P. Silberzan, B.

Perthame, PLoS Comput Biol (2010)

lab session: numerical simulations of sponateous …1- di usion2- di usion+transport3-...

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