collective motion of interacting random walkers

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



Introduction

‘Movement rules’

Mean �eld assumptions

∂C

∂t= D0∇ •

(D(C )∇C

)where C is the local average occupancy




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




Section 2

Building macroscale models from microscaleprobabilistic models




Introduction



∂C

∂t= D0∇ •

(D(C )∇C





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

]




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




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




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)




Transition Probabilities

v′ = v + ex

*v




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




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)

]




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 )




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)〉

)




Diffusion Equation for average occupancy


Formula

∂C

∂t= D0∇ •

(D(C )∇C

)Penington et al. Collective motion of interacting random walkers



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

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100 200

20t=0

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100 200

20t=200

x

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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)

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20t=0

x

y

100 200

20t=200

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y

100 200

20t=0

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<x>

100 200t

f(K) = 1

(a)

(b)

(c)

(d)

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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

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<x>

100 200t

115

135

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<x>

100 200t

100 200

20t=0

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100 200

20t=200

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y

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y

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y

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y

100 200

20t=200

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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.




Section 3

Collective motion of dimers




Introduction: collective motion of monomers



∂C

∂t= D0∇ •

(D(C )∇C


Reference: Penington et al., Building macroscale models from microscale probabilistic models, Phys. Rev. E (2011)




Introduction: collective motion of dimers



∂C

∂t= D0∇ •

(D(C )∇C


Reference: Simpson et al., Models of collective cell spreading with variable cell aspect ratio, Phys. Rev. E (2011)




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.




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.




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.




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)

}.




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.




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.




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.




Model in two dimensions




Approximation

The probability of positionu being occupied isindependent of theoccupancy of any positionthat does not overlap it.

00

0.35


Co

nd

itio

na

l p

rob

ab

ility

Full density

Actual values in red on upper line.

Approximation is lower line.




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)],




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 ).




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.




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




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.




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




Section 4

Beyond 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




Introduction

Lattice spacing = size of agents = movement distance




Introduction

?

∂C

∂t= D0∇ •

(D(C )∇C





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.





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 .




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)

)}.






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)

)}.






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)

)}.




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.




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)




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.





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.




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 .




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.




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).




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).




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


Cond

ition

al p

roba

bilit

y

(a)

1

L=2




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




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.




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)




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




Summary

Careful probability arguments required

Off lattice model




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.




Thank you

Kerry Landman and Barry Hughes, my supervisors


collective motion of interacting random walkers

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