Lab sesson: numercal smulatons of sponateous polarzaton Emerc Boun & Vncent Calvez CNRS, ENS Lyon, France CIMPA, Hammamet, March 2012
Spontaneous cell polarzaton: the 1D case The Hawkns-Voturez model for spontaneous polarzaton n 1D reads as follows: t ρ(t, x) = xx ρ(t, x) + ρ(t, 0) x ρ(t, x), t > 0, x (0, + ). The equaton s a transport-dffuson equaton. Numercal ssues: blow-up, non-trval steady states, self-smlar decay.
Behavour of solutons Theorem There s a nce and smple dchotomy: If M < 1, the soluton s global n tme. It converges towards a (unque) self-smlar profle G M : lm ρ(t, x) 1 ( ) x t L G M = 0. t 1 t + If M = 1 there s a one-parameter famly of steady states: ν α (x) = α exp( αx). Convergence holds + the frst moment s conserved: α 1 = x>0 ρ0 (x) dx. If M > 1 and x > 0 x ρ 0 (x) 0, the soluton blows up n fnte tme.
Contents 1- Dffuson equaton 2- The dffuson-transport equaton 3- The dffuson-transport equaton n self-smlar varables 4- Cluster formaton
Contents 1- Dffuson equaton 2- The dffuson-transport equaton 3- The dffuson-transport equaton n self-smlar varables 4- Cluster formaton
The dffuson equaton n the half-lne We begn wth the dffuson equaton t ρ(t, x) = xx ρ(t, x), x > 0 + Neumann boundary condton at x = 0 We rewrte the equaton n dvergence form, t ρ(t, x) + x F = 0, The flux F s gven by Fck s law: F = x ρ. The boundary condton at {x = 0} s F (0) = 0 (no-flux boundary condton).
Tme-Space dscretzaton We dscretze the functon ρ(t, x) on a regular grd [0 : t : T ] [0 : x : L], t s the tme step, T s the fnal tme of computaton, x s the space step, L s the length of the numercal nterval.
Strategy for computng ρ(t, x) numercally We replace the equaton t ρ(t, x) + x F = 0 wth the numercal dscretzaton ρ(t + t, x) ρ(t, x) t We ntroduce + F (t, x + x/2) F (t, x x/2) x ρ n = ρ(n t, x) F n = F (n t, x ± x/2). ± 1 2 The equaton (1) rewrtes at (t, x) = (t n, x ), = 0. (1) ρ n t + F n F n + 1 1 2 2 x = 0
Boundary condtons The boundary condton F (0) = 0 reads F 1 2 For = 1 the equaton (1) reads We have smlarly for = Nx = 0. 1 ρ n F n 0 1 1+ + 1 2 = 0 t x Nx ρ n Nx t + 0 F n Nx 1 2 x = 0
We have F = x ρ, therefore The dffuson equaton ρ(t, x + x) ρ(t, x) F (t, x + x/2) = x The scheme wrtes fnally ρ n t + 1 ( ρn +1 ρn x x + ρn ρ n ) 1 = 0, x Remark. It concdes wth the classcal fnte dfference scheme for the heat equaton ( = 1 2 t ) x 2 ρ n + t x 2 ρn +1 + t x 2 ρn 1,
Explct scheme for the dffuson equaton Left boundary condton (zero-flux) when = 1 ρ n t + 1 ( ρn +1 ρn x x ) + 0 = 0, Rght boundary condton (zero-flux) when = Nx ρ n t + 1 ( 0 + ρn ρ n ) 1 = 0, x x Flux of the densty when 2 Nx 1 ρ n t + 1 ( ρn +1 ρn x x + ρn ρ n ) 1 = 0, x
Explct scheme for the dffuson equaton Left boundary condton (zero-flux) when = 1 ρ n t + 1 ( ρn +1 ρn x x ) + 0 = 0, Rght boundary condton (zero-flux) when = Nx ρ n t + 1 ( 0 + ρn ρ n ) 1 = 0, x x Flux of the densty when 2 Nx 1 ρ n t + 1 ( ρn +1 ρn x x + ρn ρ n ) 1 = 0, x
Explct scheme for the dffuson equaton Left boundary condton (zero-flux) when = 1 ρ n t + 1 ( ρn +1 ρn x x ) + 0 = 0, Rght boundary condton (zero-flux) when = Nx ρ n t + 1 ( 0 + ρn ρ n ) 1 = 0, x x Flux of the densty when 2 Nx 1 ρ n t + 1 ( ρn +1 ρn x x + ρn ρ n ) 1 = 0, x
Structure of the sclab code 1/4 Space and tme grd of resoluton. %% Space dscretsaton L = 10; dx = 0.1; x = [0:dx:L]; Nx = length(x); %% Tme dscretsaton T = 10; dt = dx^2/4; t = [0:dt:T]; Nt = length(t);
Structure of the sclab code 2/4 Intal data ρ 0 : we start wth a gaussan %% Intal condton for the densty of molecules rho_0 sgma = 1; rho0 = exp(-x.^2/sgma^2); Z = sum(rho0*dx); rho0 = (M/Z)*rho0; rho1 = rho0;
Structure of the sclab code 3/4 Tme loop to update the value of the densty ρ at each tme step n for n = 1:Nt rho0 = rho1;... end
Structure of the sclab code 4/4 Space loop to update the value of the densty ρ at each pont of the grd for n = 1:Nt end end rho0 = rho1; for = 1:Nx f == 1 %% Left boundary condton rho1() =... elsef == Nx %% Rght boundary condton rho1() =... else rho1() =... end
The maxmum prncple Recall the numercal scheme for the dffuson equaton ( = 1 2 t ) x 2 ρ n + t x 2 ρn +1 + t x 2 ρn 1, Observaton. s a lnear combnaton of ρ n 1, ρn and ρ n +1. In order to guarantee the maxmum prncple, t s necessary to mpose a condton between t and x : 2 t x 2 < 1. CFL condton. In ths case, s a convex combnaton of ρ n 1, ρn and ρ n +1. Therefore, = 1... Nx mn ρ 0 j ρ n max j j ρ 0 j
Conservaton of mass The conservaton of mass s automatc when the numercal scheme s wrtten wth the flux formulaton F n F n + 1 1 2 2 ρ n + t x To see ths, smply sum up the equaton: 1 t Nx =1 x 1 t Nx =1 = 0 ρ n x + F n Nx+ 1 2 F n 1 2 = 0 The no-flux boundary condton F n 1 = F n Nx+ 1 2 2 x = ρ n x = = 0 guarantees ρ 0 x.
Contents 1- Dffuson equaton 2- The dffuson-transport equaton 3- The dffuson-transport equaton n self-smlar varables 4- Cluster formaton
The dffuson-transport equaton We swtch to the one-dmensonal polarzaton equaton. t ρ(t, x) x (µ(t)ρ(t, x)) = xx ρ(t, x) + Neumann boundary condton at x = 0 + nonlnear couplng va µ(t) = ρ(t, 0). Agan, we rewrte the equaton n dvergence form, t ρ(t, x) + x F = 0, The flux F s gven by: F = x ρ µρ. The boundary condton at {x = 0} s F (0) = 0 (no-flux boundary condton). We assume that the speed µ(t) s nonnegatve µ(t) 0.
Strategy for computng ρ(t, x) numercally We replace the equaton t ρ(t, x) + x F = 0 wth the numercal dscretzaton ρ n F n F n + + 1 1 2 2 = 0 t x F n + 1 2 = ρn +1 ρn x }{{} Fck slaw + µ n ρ n. + 1 2 }{{} transport Important queston. How to nterpolate the value ρ n + 1 2 = ρ(t, x + x/2)?
The correct choce for ρ n + 1 2 Recall the key assumpton µ n 0. Choce 1. ρ n + 1 2 Choce 2. ρ n + 1 2 Choce 3. ρ n + 1 2 = ρ n = = for transport only ( 1 + µ n t ) ρ n µ n t x x ρn 1, = ρ n +1 [Upwnd scheme] ( 1 µ n t x ) ρ n + µ n t x ρn +1, = 1 2 (ρn + ρ n +1 ) [Centered scheme] = ρ n + µ n t 2 x ρn +1 µ n t 2 x ρn 1, Only the second choce preserves the maxmum prncple, under the condton µ n t x < 1.
Combnaton of the two fluxes We eventually get ρ n t + 1 ( ρn +1 ρn x x µ n ρ n +1 + ρn ρ n 1 x ) + µ n ρ n = 0, wth sutable boundary condtons for = 1 and = Nx.
Contents 1- Dffuson equaton 2- The dffuson-transport equaton 3- The dffuson-transport equaton n self-smlar varables 4- Cluster formaton
Self-smlar rescalng Rewrte the densty ρ n self-smlar varables, ( 1 ρ(t, x) = u log ) x 1 + 2t, 1 + 2t 1 + 2t The equaton for u(τ, y) s the followng, ) τ u(τ, y) y (yu(τ, y)+v(τ)u(τ, y) = yy u(τ, y), v(τ) = u(τ, 0). We get an addtonal drft term y (yu(τ, y)) due to the change of frame (t, x) (τ, y).
Long-tme asymptotcs In the sub-crtcal regme, M < 1, the soluton u(τ, y) converges towards a unque statonary state G, gven by G(y) = α exp ( αy y 2 ). 2 The parameter α s fxed by the conservaton of mass G(y) dy = u(τ, y) dy = ρ(t, x) dx = M. y>0 y>0 x>0 Exercse. Perform the numercal smulatons for u(τ, y), soluton to the polarzaton equaton n self-smlar varables (the only dfference s the addtonal drft term). Compare the long-tme behavour wth the expected value for G.
Contents 1- Dffuson equaton 2- The dffuson-transport equaton 3- The dffuson-transport equaton n self-smlar varables 4- Cluster formaton
A one-dmensonal model for cluster formaton The smplest model for bacteral chemotaxs s the followng coupled system for the cell densty ρ(t, x) and the chemcal concentraton S(t, x). { t ρ(t, x) + x (ρ(t, x) u(t, x)) = xx ρ(t, x), x R xx S(t, x) = ρ(t, x) u(t, x) = vφ (v x S(t, x)) dv v V For smplcty we set V = ( 1, 1). The functon φ depends on ndvdual features of the bactera (e.g. the way they react to the chemcal sgnal).
Cluster formaton Exercse. Perform the numercal smulatons for the couple (ρ, S) for dfferent choces of functon φ: φ(y ) = Y,.e. the Keller-Segel model, φ(y ) = sgn (Y ),.e. a step functon: t s a good model for chemotaxs n bactera populatons.
References V. Calvez, N. Meuner and R. Voturez, C. R. Math. Acad. Sc. Pars (2010) J. Saragost, V. Calvez, N. Bournaveas, A. Bugun, P. Slberzan, B. Perthame, PLoS Comput Bol (2010)