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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 |〉 ?
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Introduction – Expanding space
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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 – Expanding space
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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 – Expanding space
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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 – Expanding space
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Hallatschek, Hersen, Ramanathan, Nelson; PNAS (2007)
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Domain boundaries
domain boundaries grow perpendicular to surfaceXh is superdiffusive due to surface roughness
M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Domain boundaries
domain boundaries grow perpendicular to surfaceXh is superdiffusive due to surface roughness
M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Range expansion
fixed, finite size geometry: fixation (absorbing state in finite time)
range expansion: promotes diversity and segregation
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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 < ∞
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
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 < ∞
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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) ⟩
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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.
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Bacteria vs. yeast
Hallatschek, Hersen, Ramanathan, Nelson; PNAS (2007)
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Biology
ignore many biological details (shape, growth direction, etc.)
consider: correlations due to reproduction time
consider: overall geometry
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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.
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Surface – KPZ scaling
w(L, t) ∼
{t1/3, for t � L3/2
L1/2, for t � L3/2
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
Domain boundaries
domain boundaries grow perpendicular to surfaceXh is superdiffusive due to surface roughness
M(h) := 〈X 2h 〉 ≈ σ2h2γ , γ = 2/3
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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\
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
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.
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)
Somfai Ellak Skalainvarians dinamika tagulo terben ELTE, 2013.04.17.