stochastic models, patterns formation and diffusion
TRANSCRIPT
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
D. Fanelli Stochastic models, patterns formation and diffusion
The Belousov-Zhabotinsky reaction.
Highlighting the peculiarities:
First system to display self-organizationRegular oscillations between homogeneousstates.
Experimental evidence
D. Fanelli Stochastic models, patterns formation and diffusion
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)
D. Fanelli Stochastic models, patterns formation and diffusion
The Belousov-Zhabotinsky reaction.
Highlighting the peculiarities:
First system to display self-organizationRegular oscillations between homogeneousstates.
Experimental evidence
Spatially organized patterns develop (Turinginstability) - Vanag e Epstein
Phys. Rev. Lett. (2001)
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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.
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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.
D. Fanelli Stochastic models, patterns formation and diffusion
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 .
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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)
D. Fanelli Stochastic models, patterns formation and diffusion
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)
D. Fanelli Stochastic models, patterns formation and diffusion
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)
D. Fanelli Stochastic models, patterns formation and diffusion
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).
D. Fanelli Stochastic models, patterns formation and diffusion
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
+ . . .
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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)
D. Fanelli Stochastic models, patterns formation and diffusion
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)
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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"
D. Fanelli Stochastic models, patterns formation and diffusion
Deterministic vs. stochastic simulations inside the region ofTuring order
D. Fanelli Stochastic models, patterns formation and diffusion
Deterministic vs. stochastic simulations inside the region ofTuring order
D. Fanelli Stochastic models, patterns formation and diffusion
Deterministic vs. stochastic simulations outside the region ofTuring order
D. Fanelli Stochastic models, patterns formation and diffusion
Deterministic vs. stochastic simulations outside the region ofTuring order
D. Fanelli Stochastic models, patterns formation and diffusion
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.
D. Fanelli Stochastic models, patterns formation and diffusion
Back to crowding and cross diffusion
Cross diffusive terms emerge as follows the assumption offinite spatial resources: ink drops evolution
Snapshots from the experiments
D. Fanelli Stochastic models, patterns formation and diffusion
Back to crowding and cross diffusion
Cross diffusive terms emerge as follows the assumption offinite spatial resources: ink drops evolution
Snapshots from the experiments
D. Fanelli Stochastic models, patterns formation and diffusion
Comparing theory prediction and the experimentoutcome
Time evolution of the radii of a drop along and perpendicularlyto the direction of collision
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
D. Fanelli Stochastic models, patterns formation and diffusion
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
ns = number of molecules of typeXs
k∑i=1
ni + nE = N
nE = N −k∑
i=1
ni
n ≡ (n1, . . . ,nk )
D. Fanelli Stochastic models, patterns formation and diffusion
Order mediated by finite size effects
I. Analytical power spectra.
D. Fanelli Stochastic models, patterns formation and diffusion