Stochastic models, patterns formation and diffusion

Stochastic models, patterns formation anddiffusion

Duccio FanelliFrancesca Di Patti, Tommaso Biancalani

Dipartimento di Energetica, Università degli Studi di FirenzeCSDC – Centro Interdipartimentale per lo Studio delle Dinamiche Complesse

INFN – Istituto Nazionale di Fisica Nucleare, Sezione di FirenzeCNISM – Sezione di Firenze

BariSeptember 2011

D. Fanelli Stochastic models, patterns formation and diffusion

Patterns formation

Investigating the dynamical evolution of an ensemble made ofmicroscopic entities in mutual interaction constitutes a rich andfascinating problem, of paramount importance andcross-disciplinary interest.

Complex microscopic interactions can eventually yield tomacroscopically organized patterns.Temporal and spatial order manifests as an emergingproperty of the system dynamics


The Belousov-Zhabotinsky reaction.

Highlighting the peculiarities:

First system to display self-organizationRegular oscillations between homogeneousstates.

Experimental evidence






Belousov e ZhabotinskyMed. Publ. Moscow (1959) - Biofizika (1964)






Spatially organized patterns develop (Turinginstability) - Vanag e Epstein

Phys. Rev. Lett. (2001)


The theoretical frameworks

Model the dynamics of the population involved(family of homologous chemicals)

From the microscopic picture ...

Assign the microscopicrules of interactions

Discrete, many particlesmodel

Deterministicformulation (continuum

limit hypothesis)Differential equationsNo fluctuations allowed

Stochastic model(respecting the intimate

discreteness)Stochastic processesStatistical, finite sizesfluctuations


Two messages

I. Finite size corrections do matter: macroscopicorder, both in time and space, can emerge as

mediated by the microscopic disorder (inherentgranularity and stochasticity)

II. Bottom-up modeling returns mathematicalformulation justified from first principles, as

opposed to any heuristically proposed scheme.


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 YParadigmatic model of self-organizeddynamics

Population: ensemble made of individual entitiesMutual interactions are specified via =⇒ chemicalequationsa,b, c,d =⇒ reaction constants


The deterministic (continuum) picture

The law of mass actionFrom chemical equations to ordinary differential equations:

X λ→ ∅ =⇒ φ = −λφ

The a-spatial BrusselatorContinuum concentrationφ = φ(t) species Xψ = ψ(t) species Y

Equationsφ = a− (b + d)φ+ cφ2ψ

ψ = bφ− cφ2ψ

a = d = c = 1 b = 3

5 10 15 20t

1

2

3

4

5

6ΦHtL, ΨHtL


The deterministic (continuum) picture

The law of mass actionFrom chemical equations to ordinary differential equations:

X λ→ ∅ =⇒ φ = −λφ

The a-spatial BrusselatorContinuum concentrationφ = φ(t) species Xψ = ψ(t) species Y

Equationsφ = a− (b + d)φ+ cφ2ψ

ψ = bφ− cφ2ψ

a = d = c = 1 b = 3

5 10 15 20t

1

2

3

4

5

6ΦHtL, ΨHtL


Including space: the spatial Brusselator.

Spatial modelAdd "by hand" the diffusive transport.

∂tφ = a− (b + d)φ+ cφ2ψ + µ ∇2φ

∂tψ = bφ− cφ2ψ + δ ∇2ψ

φ = φ(~r , t)ψ = ψ(~r , t)

Turing instabilities.

Yang, Zhabotinsky, Epstein,Phys. Rev. Lett. (2004)

∆ = 15

a = d = Μ = 1

2 4 6 8 10b

2

4

6

8

10c


Including space: the spatial Brusselator.

Spatial modelAdd "by hand" the diffusive transport.

∂tφ = a− (b + d)φ+ cφ2ψ + µ ∇2φ

∂tψ = bφ− cφ2ψ + δ ∇2ψ

φ = φ(~r , t)ψ = ψ(~r , t)

Turing instabilities.

Yang, Zhabotinsky, Epstein,Phys. Rev. Lett. (2004)

∆ = 15

a = d = Μ = 1

2 4 6 8 10b

2

4

6

8

10c


On the stochastic (individual based) approach.

Chemical equations

Eia→ Xi ,

Xib→ Yi ,

2Xi + Yic→ 3Xi ,

Xid→ Ei ,

AssumptionsConsider a spatially extendedsystem composed of Ω cells.i identifies the cell where themolecules are located.E stand for the empties andimpose a finite carryingcapacity in each micro-cell.


The molecules are however allowed to migratebetween adjacent cells, which in turn impliesan effective spatial coupling imputed to the

microscopic molecular diffusion.

i and j are neighbors cells

Xi + Ejµ→ Ei + Xj ,

Yi + Ejδ→ Ei + Yj .


The molecules are however allowed to migratebetween adjacent cells, which in turn impliesan effective spatial coupling imputed to the

microscopic molecular diffusion.

i and j are neighbors cells

Xi + Ejµ→ Ei + Xj ,

Yi + Ejδ→ Ei + Yj .

Define:ni → number of molecule X in cell imi → number of molecule Y in cell iN → number of molecules that can behost in any cell


The transition probabilities

(1) Reactions

T (ni + 1,mi |ni ,mi) = aN − ni −mi

NΩ,

T (ni − 1,mi + 1|ni ,mi) = bni

NΩ,

T (ni + 1,mi − 1|ni ,mi) = cni

2mi

N3Ω,

T (ni − 1,mi |ni ,mi) = dni

NΩ,

(1)

(2) Diffusion

T (ni − 1,nj + 1|ni ,nj) = µni

NN − nj −mj

NΩz, (2)


The transition probabilities

(1) Reactions

T (ni + 1,mi |ni ,mi) = aN − ni −mi

NΩ,

T (ni − 1,mi + 1|ni ,mi) = bni

NΩ,

T (ni + 1,mi − 1|ni ,mi) = cni

2mi

N3Ω,

T (ni − 1,mi |ni ,mi) = dni

NΩ,

(1)

(2) Diffusion

T (ni − 1,nj + 1|ni ,nj) = µni

NN − nj −mj

NΩz, (2)


Step–operators

E±1i f (n) = f (n1, . . . ,ni ± 1, . . . ,nk )

Master equation (ME)

∂

∂tP(n,m, t) =

Ω∑i

[(ε−Xi

− 1) T (ni + 1,mi |·)

+ (ε+Xi− 1) T (ni − 1,mi , |·) + (ε+Xi

ε−Y ,i − 1) T (ni − 1,mi + 1|·)

+ (ε−Xiε+Yi− 1) T (ni + 1,mi − 1|·)

+∑j∈i

[(ε+Xi

ε−Xj− 1) T (ni − 1,nj + 1|·)+

+ (ε+Yiε−Yj− 1) T (mi − 1,mj + 1|·)

]]P(n,m, t)


New variablesni

N= φi(t) +

ξi√N,

mi

N= ψi(t) +

ηi√N, (3)

φi(t) and ψi(t) are the mean field deterministic variables. ξi andηi stand for the stochastic contribution. Here 1/

√N plays the

role of a small parameter and paves the way to a perturbativeanalysis of the master equation (the van Kampen system sizeexpansion).


Expanding the left–hand side of the Master equation

New probability

P(n, t) → Π(ξ, t)

ddτ

P(n, t) =∂

∂τΠ(ξ, t)− 1√

N

k∑i=1

∂Π(ξ, t)∂ξi

dφi

dτ

E±1i = 1± 1√

N∂

∂ξi+

12N

∂2

∂ξ2i

+ . . .


The leading order: N0

The mean-field equations

φ = a(1− φ− ψ)− (d + b)φ+ cφ2ψ + µ[∇2φ+ φ∇2ψ − ψ∇2φ

]ψ = bφ− cφ2ψ + δ

[∇2ψ + ψ∇2φ− φ∇2ψ

]Surprising facts:

Homogeneous fixed points: no oscillations exist!Cross diffusion terms emerge from the microscopicformulation of the problem φ∇2ψ − ψ∇2φ

In the diluted limit, normal diffusion is recoveredCross-terms manifest under crowding conditions


What is the impact of crowding on the region of deterministicTuring-like order?

III

III

IV

2 4 6 8 10b

20

40

60

80

100c

0.5 1.0 1.5 2.0 2.5k

-6

-4

-2

ΛHkL



III

III

IV

2 4 6 8 10b

20

40

60

80

100c



III

III

IV

2 4 6 8 10b

20

40

60

80

100c


The next–to–leading order N1/2: Fokker Planck equation

∂Π

∂τ= −

∑p

∂

∂ξp

[Ap(ξ)Π

]+

12

∑l,p

Blp∂2Π

∂ξl∂ξp,

where A and B are matrices whose entries depend on thechemical parameters of the model.

The Fokker-Planck equation allows us to explicitly quantify therole played by finite size corrections (demographic noise)


The emergence of regular structures as mediated bythe stochastic component

Pk (ω) =< |ξk (ω)|2 >=C1 + Bk ,11ω

2

(ω2 − Ω2k ,0)

2 + Γ2kω

2(4)


Stochastic vs. deterministic order I

Region of stochastic order

Existence of alocalized peak in the(spatial) powerspectrumLarger than expected!Distinct regions foreach of the twospecies!Stochastic TuringPatterns (see alsoButler-Goldenfeld)

2 4 6 8 10b

20

40

60

80

100c


Stochastic vs. deterministic order II

Order is found for identical diffusion amount, µ = δ.

0.00.5

1.01.5 2.0Ω

0

2

4

k

1234

1 2 3 4 5k

0.5

1.0

1.5

2.0PX!Ω"0, k"


Deterministic vs. stochastic simulations inside the region ofTuring order


Deterministic vs. stochastic simulations inside the region ofTuring order


Deterministic vs. stochastic simulations outside the region ofTuring order


Deterministic vs. stochastic simulations outside the region ofTuring order


Summing up...

Importance of grounding the model on a solid microscopicdescription.Unexpected and unconventional mean-field equationsarise (see crowding).Temporal and spatial order can emerge as mediated by thestochastic component of the dynamics (demographicnoise).Patterns may arise also when not predicted within therealm of the classical Turing paradigm.


Back to crowding and cross diffusion

Cross diffusive terms emerge as follows the assumption offinite spatial resources: ink drops evolution

Snapshots from the experiments


Back to crowding and cross diffusion

Cross diffusive terms emerge as follows the assumption offinite spatial resources: ink drops evolution

Snapshots from the experiments


Comparing theory prediction and the experimentoutcome

Time evolution of the radii of a drop along and perpendicularlyto the direction of collision


On the origin of spatial order...

Position of the peak in Pk (ω = 0) vs. maximum of λ(k)

4.0 4.5 5.0 5.5 6.0 6.5 7.0b

0.2

0.4

0.6

0.8

1.0

1.2

1.4

kmax

b = 6

b = 7.2

0.5 1.0 1.5 2.0 2.5k

-3

-2

-1

ΛHkL


Beyond the Brusselator: protocells dynamics

Autocatalytic reaction

Xs + Xs+1rs+1−→ 2Xs+1

with Xk+1 ≡ X1

Inward diffusion

E αs−→ Xs

Outward diffusion

Xsβs−→ E




Xs + Xs+1rs+1−→ 2Xs+1

with Xk+1 ≡ X1

Inward diffusion

E αs−→ Xs

Outward diffusion

Xsβs−→ E




Xs + Xs+1rs+1−→ 2Xs+1

with Xk+1 ≡ X1

Inward diffusion

E αs−→ Xs

Outward diffusion

Xsβs−→ E




Xs + Xs+1rs+1−→ 2Xs+1

with Xk+1 ≡ X1

Inward diffusion

E αs−→ Xs

Outward diffusion

Xsβs−→ E

ns = number of molecules of typeXs

k∑i=1

ni + nE = N

nE = N −k∑

i=1

ni

n ≡ (n1, . . . ,nk )


Order mediated by finite size effects

I. Analytical power spectra.


Order mediated by finite size effects

II. Numerical (Gillespie) power spectra.


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Stochastic models, patterns formation and diffusion