Stochastic models, patterns formation and diffusion

Transcription

1 Stochastic models, patterns formation and diffusion Duccio Fanelli Francesca Di Patti, Tommaso Biancalani Dipartimento di Energetica, Università degli Studi di Firenze CSDC Centro Interdipartimentale per lo Studio delle Dinamiche Complesse INFN Istituto Nazionale di Fisica Nucleare, Sezione di Firenze CNISM Sezione di Firenze Bari September 2011

2 Patterns formation Investigating the dynamical evolution of an ensemble made of microscopic entities in mutual interaction constitutes a rich and fascinating problem, of paramount importance and cross-disciplinary interest. Complex microscopic interactions can eventually yield to macroscopically organized patterns. Temporal and spatial order manifests as an emerging property of the system dynamics

3 The Belousov-Zhabotinsky reaction. Highlighting the peculiarities: First system to display self-organization Regular oscillations between homogeneous states. Experimental evidence

4 The Belousov-Zhabotinsky reaction. Highlighting the peculiarities: First system to display self-organization Regular oscillations between homogeneous states. Experimental evidence Belousov e Zhabotinsky Med. Publ. Moscow (1959) - Biofizika (1964)

5 The Belousov-Zhabotinsky reaction. Highlighting the peculiarities: First system to display self-organization Regular oscillations between homogeneous states. Experimental evidence Spatially organized patterns develop (Turing instability) - Vanag e Epstein Phys. Rev. Lett. (2001)

6 The theoretical frameworks Model the dynamics of the population involved (family of homologous chemicals) From the microscopic picture... Assign the microscopic rules of interactions Deterministic formulation (continuum limit hypothesis) Differential equations No fluctuations allowed Discrete, many particles model Stochastic model (respecting the intimate discreteness) Stochastic processes Statistical, finite sizes fluctuations

7 Two messages I. Finite size corrections do matter: macroscopic order, both in time and space, can emerge as mediated by the microscopic disorder (inherent granularity and stochasticity) II. Bottom-up modeling returns mathematical formulation justified from first principles, as opposed to any heuristically proposed scheme.

8 A classical model: the a-spatial Brusselator Microscopic formulation a X X b Y 2X + Y c 3X X d Prigogine e Glandsdorff (1971) Two species are involved: X e Y Paradigmatic model of self-organized dynamics Population: ensemble made of individual entities Mutual interactions are specified via = chemical equations a, b, c, d = reaction constants

9 The deterministic (continuum) picture The law of mass action From chemical equations to ordinary differential equations: X λ = φ = λφ

10 The deterministic (continuum) picture The law of mass action From chemical equations to ordinary differential equations: X λ = φ = λφ The a-spatial Brusselator Continuum concentration { φ = φ(t) species X ψ = ψ(t) species Y Equations { φ = a (b + d)φ + cφ 2 ψ ψ = bφ cφ 2 ψ Φ t, Ψ t 6 a = d = c = 1 b =

11 Including space: the spatial Brusselator. Spatial model Add "by hand" the diffusive transport. { t φ = a (b + d)φ + cφ 2 ψ + µ 2 φ t ψ = bφ cφ 2 ψ + δ 2 ψ { φ = φ( r, t) ψ = ψ( r, t)

12 Including space: the spatial Brusselator. Spatial model Add "by hand" the diffusive transport. { t φ = a (b + d)φ + cφ 2 ψ + µ 2 φ t ψ = bφ cφ 2 ψ + δ 2 ψ Turing instabilities. Yang, Zhabotinsky, Epstein, Phys. Rev. Lett. (2004) 10 c { φ = φ( r, t) ψ = ψ( r, t) = 15 a = d = Μ = b

13 On the stochastic (individual based) approach. Chemical equations E i a Xi, X i b Yi, 2X i + Y i c 3Xi, X i d Ei, Assumptions Consider a spatially extended system composed of Ω cells. i identifies the cell where the molecules are located. E stand for the empties and impose a finite carrying capacity in each micro-cell.

14 The molecules are however allowed to migrate between adjacent cells, which in turn implies an effective spatial coupling imputed to the microscopic molecular diffusion. i and j are neighbors cells X i + E j µ Ei + X j, Y i + E j δ Ei + Y j.

15 The molecules are however allowed to migrate between adjacent cells, which in turn implies an effective spatial coupling imputed to the microscopic molecular diffusion. i and j are neighbors cells X i + E j µ Ei + X j, Y i + E j δ Ei + Y j. Define: n i number of molecule X in cell i m i number of molecule Y in cell i N number of molecules that can be host in any cell

16 The transition probabilities

17 The transition probabilities (1) Reactions T (n i + 1, m i n i, m i ) = a N n i m i, NΩ T (n i 1, m i + 1 n i, m i ) = b n i NΩ, T (n i + 1, m i 1 n i, m i ) = c n i 2 m i N 3 Ω, T (n i 1, m i n i, m i ) = d n i NΩ, (1) (2) Diffusion T (n i 1, n j + 1 n i, n j ) = µ n i N N n j m j, (2) NΩz

18 Step operators Master equation (ME) E ±1 i f (n) = f (n 1,..., n i ± 1,..., n k ) Ω [ t P(n, m, t) = (ɛ X i 1) T (n i + 1, m i ) i + (ɛ + X i 1) T (n i 1, m i, ) + (ɛ + X i ɛ Y,i 1) T (n i 1, m i + 1 ) + (ɛ X i ɛ + Y i 1) T (n i + 1, m i 1 ) + [ (ɛ + X i ɛ X j 1) T (n i 1, n j + 1 )+ j i + (ɛ + Y i ɛ Y j 1) T (m i 1, m j + 1 ) ]] P(n, m, t)

19 New variables n i N = φ i(t) + ξ i, N m i N = ψ i(t) + η i, (3) N φ i (t) and ψ i (t) are the mean field deterministic variables. ξ i and η i stand for the stochastic contribution. Here 1/ N plays the role of a small parameter and paves the way to a perturbative analysis of the master equation (the van Kampen system size expansion).

20 Expanding the left hand side of the Master equation New probability P(n, t) Π(ξ, t) d 1 P(n, t) = Π(ξ, t) dτ τ N k i=1 Π(ξ, t) dφ i ξ i dτ E ±1 i = 1 ± 1 N ξ i + 1 2N 2 ξ 2 i +...

21 The leading order: N 0 The mean-field equations φ = a(1 φ ψ) (d + b)φ + cφ 2 ψ + µ [ 2 φ + φ 2 ψ ψ 2 φ ] ψ = bφ cφ 2 ψ + δ [ 2 ψ + ψ 2 φ φ 2 ψ ] Surprising facts: Homogeneous fixed points: no oscillations exist! Cross diffusion terms emerge from the microscopic formulation of the problem φ 2 ψ ψ 2 φ In the diluted limit, normal diffusion is recovered Cross-terms manifest under crowding conditions

22 What is the impact of crowding on the region of deterministic Turing-like order? III I II IV

23 What is the impact of crowding on the region of deterministic Turing-like order? III I II IV 20 b

24 What is the impact of crowding on the region of deterministic Turing-like order? III I II IV 20 b

25 The next to leading order N 1/2 : Fokker Planck equation Π τ = p [ ] A p (ξ)π + 1 ξ p 2 l,p B lp 2 Π, ξ l ξ p where A and B are matrices whose entries depend on the chemical parameters of the model. The Fokker-Planck equation allows us to explicitly quantify the role played by finite size corrections (demographic noise)

26 The emergence of regular structures as mediated by the stochastic component P k (ω) =< ξ k (ω) 2 >= C 1 + B k,11 ω 2 (ω 2 Ω 2 k,0 )2 + Γ 2 k ω2 (4)

27 Stochastic vs. deterministic order I Region of stochastic order Existence of a localized peak in the (spatial) power spectrum Larger than expected! Distinct regions for each of the two species! Stochastic Turing Patterns (see also Butler-Goldenfeld) c b

28 Stochastic vs. deterministic order II Order is found for identical diffusion amount, µ = δ Ω P X Ω 0, k k k

29 Deterministic vs. stochastic simulations inside the region of Turing order

30 Deterministic vs. stochastic simulations inside the region of Turing order

31 Deterministic vs. stochastic simulations outside the region of Turing order

32 Deterministic vs. stochastic simulations outside the region of Turing order

33 Summing up... Importance of grounding the model on a solid microscopic description. Unexpected and unconventional mean-field equations arise (see crowding). Temporal and spatial order can emerge as mediated by the stochastic component of the dynamics (demographic noise). Patterns may arise also when not predicted within the realm of the classical Turing paradigm.

34 Back to crowding and cross diffusion Cross diffusive terms emerge as follows the assumption of finite spatial resources: ink drops evolution Snapshots from the experiments

35 Back to crowding and cross diffusion Cross diffusive terms emerge as follows the assumption of finite spatial resources: ink drops evolution Snapshots from the experiments

36 Comparing theory prediction and the experiment outcome Time evolution of the radii of a drop along and perpendicularly to the direction of collision

37 On the origin of spatial order... Position of the peak in P k (ω = 0) vs. maximum of λ(k) b = 6 b = 7.2

38 Beyond the Brusselator: protocells dynamics Autocatalytic reaction rs+1 Xs + Xs+1 2Xs+1 with Xk +1 X1 Inward diffusion α s E Xs Outward diffusion βs Xs E D. Fanelli Stochastic models, patterns formation and diffusion

39 Beyond the Brusselator: protocells dynamics Autocatalytic reaction rs+1 Xs + Xs+1 2Xs+1 with Xk +1 X1 Inward diffusion α s E Xs Outward diffusion βs Xs E D. Fanelli Stochastic models, patterns formation and diffusion

40 Beyond the Brusselator: protocells dynamics Autocatalytic reaction rs+1 Xs + Xs+1 2Xs+1 with Xk +1 X1 Inward diffusion α s E Xs Outward diffusion βs Xs E D. Fanelli Stochastic models, patterns formation and diffusion

41 Beyond the Brusselator: protocells dynamics Autocatalytic reaction r s+1 X s + X s+1 2Xs+1 n s = number of molecules of type X s with X k+1 X 1 Inward diffusion E αs X s Outward diffusion X s β s E k n i + n E = N i=1 n E = N k i=1 n i n (n 1,..., n k )

42 Order mediated by finite size effects I. Analytical power spectra.

43 Order mediated by finite size effects II. Numerical (Gillespie) power spectra.