Sk alainvari ans dinamika t agul o t erbenglu.elte.hu/~statfiz/eloadasok/2013-04-17-Somfai.pdfcan be...

Skalainvarians dinamikatagulo terben

Somfai EllakUniversity of Warwick, Wigner FK SZFI

Adnan AliStefan GrosskinskyRobin Ball

University of Warwick

Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.

Introduction

Scale invariant trajectories – random walk, (fractional) Brownianmotion

dX ∼ dtγ (eg. γ = 1/2)evetually slower than ballistic

in 1D: all trajectories meet (with probability 1)

Our question:

what if space expands faster than 〈|dX |〉 ?


Introduction – Expanding space


Cosmology

physicsforme.wordpress.com

Thin sheets

Klein, Efrati, Sharon, Science (2007)

Growing substrate

stanford.edu/group/brainsinsilicon

Indirect: domain walls

Hallatschek, Hersen, Ramanathan, Nelson; PNAS (2007)

Introduction – Genetic drift and range expansion



Domain boundaries

domain boundaries grow perpendicular to surfaceXh is superdiffusive due to surface roughness

M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3


Range expansion

fixed, finite size geometry: fixation (absorbing state in finite time)

range expansion: promotes diversity and segregation


Mapping

understand radial growthwith help of same dynamics in fixed size rectangular space

polar-like coordinate transformation:linear: 0 ≤ Xh < L, 0 ≤ h <∞radial: 0 ≤ Yr < 2πr , r0 ≤ r <∞, 2πr0 = L


Mapping

polar-like coordinate transformation:linear: 0 ≤ Xh < L, 0 ≤ h <∞radial: 0 ≤ Yr < 2πr , r0 ≤ r <∞, 2πr0 = L

Yr =r

r0Xh

increment:

dYr =dr

r0Xh +

r

r0dXh = Yr

dr

r+ dYr ,

r

r0dXh = dYr

preserving local structure:

dXh ∼ (dh)γ , dYr ∼ (dr)γγ = 1/2 diffusive fluctuationsγ = 2/3 KPZ domain boundaries

dh

dr=

(dXh

dYr

)1/γ

=( r0r

)1/γ


Mapping

integrating differential equation:

h(r) =

{r0

γ1−γ

(1− (r0/r)

1−γγ

), γ 6= 1

r0 ln(r/r0) , γ = 1

local interaction:{ r0rYr

}dist.={Xh(r)

}for all r ≥ r0

properties:h(r) ≈ r − r0 for r close to r0hγ(∞) = limr→∞ h(r) = γ

1−γ r0 < ∞


1 2 3 4 5 60

0.5

1

1.5

r′ = r/r0

h′ =

h/r

0

γ=2/3

γ=1/3

γ=1/2

h1/3

(∞)/r0

h1/2

(∞)/r0

γ=1

Results

number of domains:

101

102

103

100

101

h, r−r0

⟨ N ⟩

× ⟨ NF(h) ⟩

° ⟨ NR(r) ⟩

101

101

h, h(r)

⟨ N ⟩

° ⟨ NR(r) ⟩

h1/2

(∞)

× ⟨ NF(h) ⟩


Results

Different processes:

Levy flight:jump size distribution: P(Xh+ − Xh = x) ∼ C |x |−(1+α) (α > 0)Markovianγ = max{1/α, 1/2}

fractional Brownian motion:covariances: 〈Xh+∆hXh〉 ∼ (h + ∆h)2γ + h2γ − (∆h)2γ

non-Markovian


Results

Levy flight (γ = max{1/α, 1/2}) + absorption:number of surviving trajectories N,

mean square distance: DF (h)2 =N(h)∑i=1

(X

(i+1)h − X

(i)h

)2

100

101

100

101

h

⟨ N ⟩

Levy(a)

α=3/2

h1/2

(∞)

h2/3

(∞)

× ⟨ NF(h) ⟩

° ⟨ NR(r) ⟩

α=1

α=5/2

100

101

102

103

104

h

⟨ D2 ⟩

Levy(b)

h1/2

(∞)

h2/3

(∞)° ⟨ (r

0D

R(r)/r)2 ⟩

× ⟨ D2F(h) ⟩

α=3/2α=5/2

α=1


Results

fractional Brownian motion:number of surviving trajectories N,

mean square distance: DF (h)2 =N(h)∑i=1

(X

(i+1)h − X

(i)h

)2

100

101

102

101

h

⟨ N ⟩

h1/2

(∞)h

2/3(∞)γ=2/3

γ=1/2

(c)fBm

× ⟨ NF(h) ⟩

° ⟨ NR(r) ⟩

h1/3

(∞)

γ=1/3

100

101

101

102

h

⟨ D2 ⟩

γ=2/3γ=1/2

h2/3

(∞)

(d) fBm

h1/2

(∞)

× ⟨ DF2(h) ⟩

° ⟨ (r0D

R(r)/r)2⟩

h1/3

(∞)

γ=1/3


Results – finite particle size

fixed size d in expanding space corresponds todecreasing size r0

r(h)d in fixed space

100

101

102

103

h

⟨ N ⟩

Bm 2+1d=0.3d=0.5

° ⟨ NR ⟩

+ ⟨ NF,uncorrected

⟩

h1/2

(∞)

(e)

× ⟨ NF ⟩

d=1


Results – branching-coalescing Brownian motionbranching rate RR in expanding space vs.branching rate RF in fixed space:

RR

RF=

∆R(dr)/dr

∆F (dh)/dh=

dh

dr=( r0r

)1/γ

102

100

101

r

⟨ N⟩/r

° ⟨ NR(r) ⟩/r

(f)

branching−coalescing Bm

× ⟨ NF(h) ⟩/r(h)

RR=0.1

RR=1

RR=5


Conclusions – mapping

locally scale invariant growth processesmapped from homogenously expanding space to fixed space

can be used eg. to handle 2D radial growth,asymptotic state in radial growth corresponds to finite time in fixed space

can be extended to include:

I finite particle size,I branching-coalescing,I higher dimensions etc.


Bacteria vs. yeast



Biology

ignore many biological details (shape, growth direction, etc.)

consider: correlations due to reproduction time

consider: overall geometry


Bacteria (E. Coli)

emc.maricopa.edu

Yiest (S. cerevisiae)

visualphotos.com

Model

off-lattice Eden growth model

one-parameter family, δ ∈ [0, 1]: reproduction time T with delay 1− δdistribution: T ∼ 1− δ + Exp(1/δ)normalized average: 〈T 〉 = 1variation coefficient: σ(T )/〈T 〉 = δ


Surface – KPZ scaling

on large scale:

∂th(x , t) = v0 + ν∂2xh +

λ

2(∂xh)2 + Dη(x , t)

scaling:x → x ′ = bx , t → t ′ = bz t, h→ h′ = bαhstatistical scale invariance: h(x , t) ∼ h′(x ′, t ′)Family-Vicsek scaling: w(L, t) :=

√〈(h − 〈h〉x)2〉x ∼ Lαf (t/Lz)

ahol f (u) ∼

{uβ, if u � 1

const, if u � 1

exponents: α + z = 2, z = α/β1D: α = 1/2, β = 1/3, z = 3/2.


Surface – KPZ scaling

w(L, t) ∼

{t1/3, for t � L3/2

L1/2, for t � L3/2


Domain boundaries

domain boundaries grow perpendicular to surfaceXh is superdiffusive due to surface roughness

M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3


Correlations

Partial synchronization leads to intrinsic vertical correlations.

N(t) growth events with height ∆hi : hN(t) =∑N(t)

i=1 ∆hi

var[hN(t)] = 〈∆hi 〉2var[N(t)] + 〈N(t)〉var[∆hi ]

var[hN(t)] = t〈∆hi 〉2(δ2 + ε2)!

= O(1)where correlation coefficient due to geometric effects:ε =

√var[∆hi ]/〈∆hi 〉

intrinsic vertical correlation: τ ∼ 1δ2+ε2

lateral correlation length: ξ‖(t) ∼ (t/τ)1/z

mean square displacementM(h) := 〈[X (h)− X (0)]2〉 ≈ σ2

δh2γ ∼ ξ2

‖(h)

γ = 2/3 and σ2δ ∝ (δ2 + ε2)4/3


Correlations

Prefactor of mean square displacement: σ2δ ∝ (δ2 + ε2)4/3

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.15

0.2

0.25

0.3

0.35

0.4

δ

σ2 δ

Data

0.2765(δ2+0.3958)4/3


S. cerevisiae (yeast)

Replace δ-family with realistic reproduction times.

Tr

Tm+Tr

T H∆=0.248L

T H∆=0.190L

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

2.5

t�XT\

pdf

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60.08

0.1

0.12

0.14

0.16

0.18

0.2

δσ δ2

δ−familyMDM and D


Conclusions – partially synchronized growth

extension of off-lattice Eden modelreproduction time has variation coefficient δ

stays in KPZ universality class

changes in patterns are due to changing prefactors – quantified

works for realistic reproduction times (S. cerevisiae)


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