Post on 17-Feb-2017
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Collective motion of interacting random walkers
Catherine Penington1, Kerry Landman2, Barry Hughes2
1School of Mathematical Sciences,Queensland University of Technology
2Department of Mathematics and Statistics,University of Melbourne
13th March, 2015
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction
‘Movement rules’
Mean �eld assumptions
∂C
∂t= D0∇ •
(D(C )∇C
)where C is the local average occupancy
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Three topics
Three different classes of discrete model:
Monomer agents, general movement rules
Collective motion of dimers (agents occupying two sites)
Agents of any length, moving any distance
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Section 2
Building macroscale models from microscaleprobabilistic models
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction
‘Movement rules’
Mean �eld assumptions
∂C
∂t= D0∇ •
(D(C )∇C
)where C is the local average occupancy
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Examples
Simple exclusion process
∂C
∂t= D0∇2C
Myopic exclusion process
∂C
∂t= D0∇ •
[(1− 5C 4 +
8
3
(C − C 4
1− C
))∇C
]
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Examples
Simple exclusion process
∂C
∂t= D0∇2C
Myopic exclusion process
∂C
∂t= D0∇ •
[(1− 5C 4 +
8
3
(C − C 4
1− C
))∇C
]
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Probabilistic Model
Assume agents occupy single sites
Agents move from one lattice siteto one of its neighbours
Discrete time and discrete spacesimulation method
Not necessarily exclusion
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Lattice Structure
Regular lattice with bondsof unit length
Dimension d = 1, 2 or 3
N (v) ={nearest-neighbour sites}N = |N (v)|Choose coordinate system sothat 0 is a lattice site
Figure: For site v shown in blue,N (v) is the set of sites shown in red
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agent Movement
*0
Q sequential independent randomchoices of agent each time step
Movement probability P
Tn(ex |0) depends on occupancy ofinfluence region MM is set of m sites not necessarilycoinciding with N (v)
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Transition Probabilities
v′ = v + ex
*v
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Transition Probabilities
v′ = v + Arex
*
v
Tn(v′|v) = F(〈Cn
(v + Arw1)〉, . . . ,
〈Cn(v + Arwm)〉)
P(site is occupied) = averageoccupancy of site
〈Cn+1(v)〉 − 〈Cn(v)〉= − P
∑v′∈N (v) Tn(v′|v)〈Cn(v)〉 out
+ P∑
v′∈N (v) Tn(v|v′)〈Cn(v′)〉 in
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Transition Probabilities
v′ = v + Arex
*
v
Tn(v′|v) = F(〈Cn
(v + Arw1)〉, . . . ,
〈Cn(v + Arwm)〉)
P(site is occupied) = averageoccupancy of site
〈Cn+1(v)〉 − 〈Cn(v)〉= − P
∑v′∈N (v) Tn(v′|v)〈Cn(v)〉 out
+ P∑
v′∈N (v) Tn(v|v′)〈Cn(v′)〉 in
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Transition Probabilities
v′ = v + Arex
*
v
Tn(v′|v) = F(〈Cn
(v + Arw1)〉, . . . ,
〈Cn(v + Arwm)〉)
P(site is occupied) = averageoccupancy of site
〈Cn+1(v)〉 − 〈Cn(v)〉= − P
∑v′∈N (v) Tn(v′|v)〈Cn(v)〉 out
+ P∑
v′∈N (v) Tn(v|v′)〈Cn(v′)〉 in
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Continuum Limit
Small time step τ , t = nτ
Small distance ∆, x = ∆v
〈Cn(v)〉 = C (x, t) ∈ [0, 1] local average occupancy
Taylor series in ∆, τ
〈Cn(v + z)〉 = C + ∆z •∇C +∆2
2(z •∇)2C + o(∆2)
τ∂C
∂t+ o(τ) = P
[H0(C ) + H1(C )∆ + H2(C )∆2 + o(∆2)
]
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Symmetries and Consequences
N−1∑r=0
Arex = 0
N−1∑r=0
a(r)i a
(r)j =
N
dδij
Arex =(a
(r)1 , . . . , a
(r)d
)
N−1∑r=0
(Arex •∇)2C =N
d∇2C
N−1∑r=0
(Arex •∇C )2 =N
d(∇C •∇C )
N−1∑r=0
(Arex •∇C )(Arwk •∇C )
= (ex •wk)N
d(∇C •∇C )
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Diffusion Equation for average occupancy
∂C
∂t= D0∇ •
(D(C )∇C
)where
D0 =P
2dlim
∆,τ→0
∆2
τ
and
D(C ) = N
[F + C
m∑k=1
(1− 2ex •wk)∂F
∂yk
]Tn(v′|v) = F
(〈Cn
(v + Arw1)〉, . . . , 〈Cn(v + Arwm)〉
)
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Diffusion Equation for average occupancy
‘Movement rules’
Formula
∂C
∂t= D0∇ •
(D(C )∇C
)Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Comparison to simulations
100 200
20t=0
x
y
100 200
20t=200
x
y
100 200
20t=0
x
y
100 200
20t=200
x
y
100 200
20t=0
x
y
100 200
20t=200
x
y
115
135
px
<x>
100 200t
f(K) = 1
(a)
(b)
(c)
(d)
100 150
1
t=200
x
C
<C>
100 150
1
t=200
x100 150
1
t=200
x
C
<C>
C
<C>
0 0 0
f(K) = eK
f(K) = e3K
50 50
C 10
D(C)
1.2
-1.2
C 10
D(C)
1.2
-1.2
C 10
D(C)
1.2
-1.2
(e)
115
135
px
<x>
100 200t
115
135
px
<x>
100 200t
t
C 10
D(C)
1.2
-1.2
(e)
100 200
20t=0
x
y
100 200
20t=200
x
y
100 200
20t=0
x
y
100 200
20t=200
x
y
100 200
20t=0
x
y
100 200
20t=200
x
y
115
135
px
<x>
100 200t
f(K) = 1
(a)
(b)
(c)
(d)
100 150
1
t=200
x
C
<C>
100 150
1
t=200
x100 150
1
t=200
x
C
<C>
C
<C>
0 0 0
f(K) = eK
f(K) = e3K
50 50
C 10
D(C)
1.2
-1.2
C 10
D(C)
1.2
-1.2
C 10
D(C)
1.2
-1.2
(e)
115
135
px
<x>
100 200t
115
135
px
<x>
100 200t
100 200
20t=0
x
y
100 200
20t=200
x
y
100 200
20t=0
x
y
100 200
20t=200
x
y
100 200
20t=0
x
y
100 200
20t=200
x
y
115
135
px
<x>
100 200t
f(K) = 1
(a)
(b)
(c)
(d)
100 150
1
t=200
x
C
<C>
100 150
1
t=200
x100 150
1
t=200
x
C
<C>
C
<C>
0 0 0
f(K) = eK
f(K) = e3K
50 50
C 10
D(C)
1.2
-1.2
C 10
D(C)
1.2
-1.2
C 10
D(C)
1.2
-1.2
(e)
115
135
px
<x>
100 200t
115
135
px
<x>
100 200t
Simulation results in bluePDE results in red
Reference: Penington, C. J., Hughes, B. D., & Landman, K. A. (2011).
Building macroscale models from microscale probabilistic models: a general
probabilistic approach for nonlinear diffusion and multispecies phenomena.
Physical Review E, 84(4), 041120.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Section 3
Collective motion of dimers
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction: collective motion of monomers
‘Movement rules’
Mean �eld assumptions
∂C
∂t= D0∇ •
(D(C )∇C
)where C is the local average occupancy
Reference: Penington et al., Building macroscale models from microscale probabilistic models, Phys. Rev. E (2011)
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction: collective motion of dimers
‘Movement rules’
Mean �eld assumptions
∂C
∂t= D0∇ •
(D(C )∇C
)where C is the local average occupancy
Reference: Simpson et al., Models of collective cell spreading with variable cell aspect ratio, Phys. Rev. E (2011)
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in one dimension
Agents occupy two sites on a one-dimensional lattice.
Agents move in either direction with equal probability.
If an agent already occupies the site, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in one dimension
Agents occupy two sites on a one-dimensional lattice.
Agents move in either direction with equal probability.
If an agent already occupies the site, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in one dimension
Agents occupy two sites on a one-dimensional lattice.
Agents move in either direction with equal probability.
If an agent already occupies the site, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in one dimension
Agents occupy two sites on a one-dimensional lattice.
Agents move in either direction with equal probability.
If an agent already occupies the site, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in one dimension
Agents occupy two sites on a one-dimensional lattice.
Agents move in either direction with equal probability.
If an agent already occupies the site, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in one dimension
Agents occupy two sites on a one-dimensional lattice.
Agents move in either direction with equal probability.
If an agent already occupies the site, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Occupancy probabilities
Indicator function
γn(i) =
1 if site i is occupied by the right side
of an agent after n time-steps,
0 otherwise.
At maximum density, half of the sites are occupied by the rightside of an agent.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Changes in (right side) occupancy
There are three ways occupancy of site i can change:
An agent at i moves away,
An agent at i + 1 moves to i ,
An agent at i − 1 moves to i .
P(γn+1(i) = 1 | γn(i + s) = 1) =P
2P(γn(i − s) = 0 | γn(i + s) = 1)
where s = ±1 and agents move with probability P.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Changes in (right side) occupancy
P(γn+1(i) = 1)− P(γn(i) = 1)
=P
2
{P(γn(i − 1) = 0 | γn(i + 1) = 1)P(γn(i + 1) = 1)
+ P(γn(i + 1) = 0 | γn(i − 1) = 1)P(γn(i − 1) = 1)
− P(γn(i + 2) = 0 | γn(i) = 1)P(γn(i) = 1)
− P(γn(i − 2) = 0 | γn(i) = 1)P(γn(i) = 1)
}.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Approximation
The probability of site jbeing (right side) occupiedis independent of the(right side) occupancy ofsites j ± 2
00
1
Probability position occupied
Co
nd
itio
na
l p
rob
ab
ility
Full density
Actual values in red on upper line.
Approximation is lower line.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Approximation
The probability of site jbeing (right side) occupiedis independent of the(right side) occupancy ofsites j ± 2
00
1
Probability position occupied
Co
nd
itio
na
l p
rob
ab
ility
Full density
Actual values in red on upper line.
Approximation is lower line.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Master equations for right side occupancy
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2
{−rn(i)
∑s=±1
[1− rn(i + 2s)
]+
∑s=±1
rn(i + s)[1− rn(i − s)
]}.
If x = i∆, t = nτ and R is the continuous local average occupancyby the right side of agents,
∂R
∂t= D
(1)0
∂
∂x
[(1 + 2R)
∂R
∂x
],
where D(1)0 =
P
2lim
∆,τ→0
∆2
τ.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Master equations for right side occupancy
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2
{−rn(i)
∑s=±1
[1− rn(i + 2s)
]+
∑s=±1
rn(i + s)[1− rn(i − s)
]}.
If x = i∆, t = nτ and R is the continuous local average occupancyby the right side of agents,
∂R
∂t= D
(1)0
∂
∂x
[(1 + 2R)
∂R
∂x
],
where D(1)0 =
P
2lim
∆,τ→0
∆2
τ.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Simulations
50 100 150 200 2500
0.05
0.1
0.15
0.2
0.25
0.3Half density
x
R
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
Full density
x
R
PDE solutions are shown in red
Simulation results are shown in black
Results are shown for times t = 100, t = 300 and t = 500. Simulation resultsare averaged over 10,000 simulations.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Model in two dimensions
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Approximation
The probability of positionu being occupied isindependent of theoccupancy of any positionthat does not overlap it.
00
0.35
Probability position occupied
Co
nd
itio
na
l p
rob
ab
ility
Full density
Actual values in red on upper line.
Approximation is lower line.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Approximation
The probability of positionu being occupied isindependent of theoccupancy of any positionthat does not overlap it.
00
0.35
Probability position occupied
Co
nd
itio
na
l p
rob
ab
ility
Full density
Actual values in red on upper line.
Approximation is lower line.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Partial Differential Equations
(x , y) = ∆(i , j) and t = nτ ,
H is local average occupancy by horizontal agents,
V is local average occupancy by vertical agents,
τ∂H
∂t+ o(τ) = P[QH
0 (H,V ) + ∆QH1 (H,V ) + ∆2QH
2 (H,V ) + o(∆2)],
τ∂V
∂t+ o(τ) = P[QV
0 (H,V ) + ∆QV1 (H,V ) + ∆2QV
2 (H,V ) + o(∆2)],
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Partial Differential Equations
(x , y) = ∆(i , j) and t = nτ ,
H is local average occupancy by horizontal agents,
V is local average occupancy by vertical agents,
τ∂H
∂t+ o(τ) = P[QH
0 (H,V ) + ∆2QH2 (H,V ) + o(∆2)],
τ∂V
∂t+ o(τ) = P[QV
0 (H,V ) + ∆2QV2 (H,V ) + o(∆2)],
QH1 (H,V ) = QV
1 (H,V ) = 0,
QH0 (H,V ) = −QV
0 (H,V ) =2
3(V − H)(1− 2H − 2V ).
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Partial Differential Equations
(x , y) = ∆(i , j) and t = nτ ,
H is local average occupancy by horizontal agents,
V is local average occupancy by vertical agents,
τ∂H
∂t+ o(τ) = P[QH
0 (H,V ) + ∆2QH2 (H,V ) + o(∆2)],
τ∂V
∂t+ o(τ) = P[QV
0 (H,V ) + ∆2QV2 (H,V ) + o(∆2)],
QH1 (H,V ) = QV
1 (H,V ) = 0,
QH0 (H,V ) = −QV
0 (H,V ) =2
3(V − H)(1− 2H − 2V ).
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Total agent occupancy and orientation imbalance
Total agent occupancy T = H + VOrientation imbalance S = H − V
QT0 (T ,S) = 0,
QS0 (T ,S) = −4
3S (1− 2T ).
∂S
∂t≈ − 4
3τS(1− 2T ).
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
t
Normalised orientation balanceversus time. The blue arrowshows increasing density.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Total agent occupancy and orientation imbalance
Total agent occupancy T = H + VOrientation imbalance S = H − V
QT0 (T ,S) = 0,
QS0 (T ,S) = −4
3S (1− 2T ).
∂S
∂t≈ − 4
3τS(1− 2T ).
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
t
Normalised orientation balanceversus time. The blue arrowshows increasing density.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Total agent occupancy and orientation imbalance
Total agent occupancy T = H + VOrientation imbalance S = H − V
QT0 (T ,S) = 0,
QS0 (T ,S) = −4
3S (1− 2T ).
∂S
∂t≈ − 4
3τS(1− 2T ).
0 10 20 30 40 50 60 70 80 90 1000.2
0
0.2
0.4
0.6
0.8
1
1.2
t
Normalised orientation balanceversus time. The blue arrowshows increasing density.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Diffusion equation
∂T
∂t= D
(2)0 ∇ •
[(1 + 2T )∇T
],
where
D(2)0 =
P
6lim
∆,τ→0
∆2
τ.
80 100 120 140 160 180 200 2200
0.05
0.1
0.15
0.2
0.25
80 100 120 140 160 180 200 2200
0.1
0.2
0.3
0.4
0.5
T T
x x
2/5 density Full density
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Allowing overlaps
Allowed
Not allowed
∂R
∂t= D
(1)0
∂
∂x
[(1 + 7R2
) ∂R∂x
],
where D(1)0 =
P
2lim
∆,τ→0
∆2
τ.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Simulations
50 100 150 200 2500
0.05
0.1
0.15
0.2
0.25
0.3
0.35Half density
R
x50 100 150 200 250
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7Full density
R
x
Onedimensionalmodel withoverlaps
80 100 120 140 160 180 200 2200
0.1
0.2
0.3
0.4
0.5
80 100 120 140 160 180 200 2200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
T T
x x
3/4 density Full density
Twodimensionalmodel withoverlaps
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Section 4
Beyond monomers and nearest neighbour steps
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction
Reference: pulpbits.com, cells under a
microscope
28 Bulletin of Mathematical Biology (2006) 68: 25–52
Fig. 1 Neuronal cell distribution from a circular explant from Ward et al. (2003). Images of theexplant before and after placement of an aggregate; the edge of the aggregate indicated by ablack line. (a) Control experiment at 24 h, just before non-Slit secreting aggregate placement.(b) Control experiment at 48 h. (c) Slit experiment at 24 h, just before Slit-secreting aggregateplacement. (d) Slit experiment at 48 h. The photographs are reproduced with the permission ofthe Society of Neuroscience. Copyright 2003 Society of Neuroscience.
are difficult to follow. From this, we see that the distinction between repellents andinhibitors is far from clear.
In this paper, modelling and simulation techniques are applied to unravellingdirectional guidance and motility regulation. Using the Ward et al. (2003) experi-ments as a guide, we simulate cell migration from an explant in the presence andabsence of a signalling molecule. In Section 2, we consider population-level con-tinuum models based on mass conservation equations. These equations are recastinto transition probabilities governing individual cell motility. We consider vari-ous strategies whereby a cell senses a signalling molecule and discuss the motilityregulation and/or directional guidance effects. In Section 3 we use our models tosimulate the Ward et al. (2003) experiments. Different cell-sensing models giverise to differences in population-level distributions and individual cell motility andturning. We compare the simulation results and determine the motility and repul-sive effects. Finally, we discuss our findings in relation to the results of Ward et al.(2003) and assess the difficulties and limitations in deducing cell migration rulesfrom time-lapse imaging and/or simulation realisations.
Reference: A.Q. Cai, K.A.
Landman, B.D. Hughes, Bull.
of Math. Bio. 2006
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction
Lattice spacing = size of agents = movement distance
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Introduction
?
∂C
∂t= D0∇ •
(D(C )∇C
)where C is the local average occupancy
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Monomer agents moving d sites
N agents each occupy one site on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Monomer agents moving d sites
N agents each occupy one site on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Monomer agents moving d sites
N agents each occupy one site on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Monomer agents moving d sites
N agents each occupy one site on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Monomer agents moving d sites
N agents each occupy one site on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Monomer agents moving d sites
N agents each occupy one site on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Occupancy probabilities
Indicator function
γn(i) =
{1 if site i is occupied by an agent after n time-steps,
0 otherwise.
There are four ways occupancy of site i can change:
An agent at i moves d sites to the left,
An agent at i moves d sites to the right,
An agent at i + d moves to i ,
An agent at i − d moves to i .
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Mean-field approximation
The probability that a site j is occupied is independent of theoccupancy of its neighbours.
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2N
{(rn(i + d)− rn(i)
) d−1∏s=1
(1− rn(i + s)
)+(rn(i − d)− rn(i)
) d−1∏s=1
(1− rn(i − s)
)}.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Mean-field approximation
The probability that a site j is occupied is independent of theoccupancy of its neighbours.
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2N
{(rn(i + d)− rn(i)
) d−1∏s=1
(1− rn(i + s)
)+(rn(i − d)− rn(i)
) d−1∏s=1
(1− rn(i − s)
)}.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Mean-field approximation
The probability that a site j is occupied is independent of theoccupancy of its neighbours.
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2N
{(rn(i + d)− rn(i)
) d−1∏s=1
(1− rn(i + s)
)+(rn(i − d)− rn(i)
) d−1∏s=1
(1− rn(i − s)
)}.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Mean-field approximation
The probability that a site j is occupied is independent of theoccupancy of its neighbours.
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2N
{(rn(i + d)−rn(i)
) d−1∏s=1
(1− rn(i + s)
)+(rn(i − d)−rn(i)
) d−1∏s=1
(1− rn(i − s)
)}.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Mean-field approximation
The probability that a site j is occupied is independent of theoccupancy of its neighbours.
rn(i) := P(γn(i) = 1),
rn+1(i)− rn(i) =P
2N
{(rn(i + d)− rn(i)
)d−1∏s=1
(1− rn(i + s)
)+(rn(i − d)− rn(i)
)d−1∏s=1
(1− rn(i − s)
)}.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Diffusion equation
If x = i∆, t = nτ and C is the continuous local average occupancyof agents,
∂C
∂t= D0
∂
∂x
(d2(1− C )d−1∂C
∂x
),
where
D0 =P
2lim
∆,τ→0
∆2
τ.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Simulations
PDE solutions are shownin red
Simulation results areshown in blue
Results are shown for timest = 100, t = 300 and t = 500.Simulation results are averagedover 10,000 simulations.
0 50 100 150 200 250 300 350 4000
0.1
0.2
0.3
0.4
0.5
0.6
x
C d=2
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
d=3C
x
(a)
(b)
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agents with length L > 1
N agents each occupy L sites on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agents with length L > 1
N agents each occupy L sites on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agents with length L > 1
N agents each occupy L sites on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agents with length L > 1
N agents each occupy L sites on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agents with length L > 1
N agents each occupy L sites on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Agents with length L > 1
N agents each occupy L sites on a one-dimensional lattice.
Agents move d sites in either direction with equal probability.
If an agent already occupies the any site the agent moves toor through, the move is aborted.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Occupancy probabilities
Indicator function
γn(i) =
1 if site i is occupied by the right-most end
of an agent after n time-steps,
0 otherwise.
At maximum density, 1/L of the sites are occupied by the rightside of an agent.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Changes in (right-end) occupancy
There are four ways the (right-end) occupancy of site i canchange:
An agent at i moves d sites to the left,
An agent at i moves d sites to the right,
An agent at i + d moves to i ,
An agent at i − d moves to i .
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Changes in (right-end) occupancy
There are four ways the (right-end) occupancy of site i canchange:
An agent at i moves d sites to the left,
An agent at i moves d sites to the right,
An agent at i + d moves to i ,
An agent at i − d moves to i .
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Three possible states
With monomer agents, lattice sites have two possible states:vacant or occupied.
With longer agents, there are now three possibilities: vacant ofany agent, occupied by the right end of an agent and occupiedby another part of an agent.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Three possible states
With monomer agents, lattice sites have two possible states:vacant or occupied.
With longer agents, there are now three possibilities: vacant ofany agent, occupied by the right end of an agent and occupiedby another part of an agent.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
A first approximation
If we have no knowledge of the surroundings, a lattice site isequally likely to be occupied by any part of an agent.
P (site j occupied by right end of an agent) = P(γn(j) = 1),
P (site j occupied by left end of an agent) ≈ P(γn(j) = 1),
P (site j occupied by any other part of an agent) ≈ P(γn(j) = 1),
P (site j is completely vacant) ≈ 1− LP(γn(j) = 1).
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
A first approximation
If we have no knowledge of the surroundings, a lattice site isequally likely to be occupied by any part of an agent.
P (site j occupied by right end of an agent) = P(γn(j) = 1),
P (site j occupied by left end of an agent) ≈ P(γn(j) = 1),
P (site j occupied by any other part of an agent) ≈ P(γn(j) = 1),
P (site j is completely vacant) ≈ 1− LP(γn(j) = 1).
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
A second approximation
If there is an agent with its right side at site j + L, the onlypossible part of an agent at site j is the right-most end.
The relative probabilities that site j is vacant or occupied by theright side of an agent remain the same.
P(γn(j) = 1 | γn(j + L) = 1) ≈ P(γn(j) = 1)
1− (L− 1)P(γn(j) = 1).
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
A second approximation
If there is an agent with its right side at site j + L, the onlypossible part of an agent at site j is the right-most end.
The relative probabilities that site j is vacant or occupied by theright side of an agent remain the same.
P(γn(j) = 1 | γn(j + L) = 1) ≈ P(γn(j) = 1)
1− (L− 1)P(γn(j) = 1).
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Accuracy
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability position occupied
Cond
ition
al p
roba
bilit
y
(a)
1
L=2
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Accuracy 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability position occupiedCo
nditi
onal
pro
babi
lity
(a)
0 0.1 0.2Probability position occupied
Cond
ition
al p
roba
bilit
y
0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) L=5
L=2
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Diffusion equation
If x = i∆, t = nτ and C is the continuous local average occupancyof agents,
∂C
∂t= D0∇ •
[D(C )∇C
],
where
D(C ) = d2 (1− LC )d−1(1− (L− 1)C
)d+1
(1 + L(L− 1)C 2
),
and
D0 =P
2Nlim
∆,τ→0
∆2
τ.
Note that D(C ) is not a polynomial.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Simulations
PDE solutions are shown inred
Simulation results are shownin blue
Results are shown for times t = 100,t = 300 and t = 500. Simulationresults are averaged over 10,000simulations.
0 50 100 150 200 250 300 350 4000
0.05
0.1
0.15
0.2
0.25
0.3
x
C d=2, L=2
0 100 200 300 400 500 600 700 8000
0.05
0.1
0.15
0.2
0.25
0.3
x
C d=4, L=2
(a)
(b)
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Summary
Careful probability arguments required0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Probability position occupiedCo
nditi
onal
pro
babi
lity
(a)
0 0.1 0.2Probability position occupied
Cond
ition
al p
roba
bilit
y
0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(b) L=5
L=2
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Summary
Careful probability arguments required
Off lattice model
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Summary
Careful probability arguments required
Off lattice model
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Summary
Careful probability arguments required
Off lattice model
Reference: Penington, C. J., Hughes, B. D., & Landman, K. A. (2014).Interacting motile agents: Taking a mean-field approach beyond monomers andnearest-neighbor steps. Physical review E, 89(3), 032714.
Penington et al. Collective motion of interacting random walkers
IntroductionBuilding macroscale models from microscale probabilistic models
Collective motion of dimersBeyond monomers and nearest neighbour steps
Thank you
Kerry Landman and Barry Hughes, my supervisors
Penington et al. Collective motion of interacting random walkers