mathematical models and biological insights · mathematical models and biological insights ruth...

Collective cell motility Mathematical models and biological insights

Ruth Baker @ruth_baker

Ruth Baker 2

Research focusTo understand the mechanisms driving collective cell motility, and their contributions to complex biological processes, such as those associated with development, disease and repair.

wound healing

embryo development

tumour growth

Goal: to interrogate multiplex quantitative data using validated and biologically realistic mathematical models.

Ruth Baker 3

Interdisciplinary methodology

Mathematical modelling computational simulations

Experiments: wildtype and perturbation

Data analysis and model testing guides

improves

guid

es

impr

oves

guidesim

proves

Integral in the cycle of predict - test - refine - predict

Ruth Baker

• Developing mathematical and computational models that can be used to test specific biological hypotheses.

• Efficient and accurate methods for computational simulation.

• Model coarse-graining / reduction to facilitate analysis.

• Extraction of useful (quantitative) summary statistics from experimental data.

• Inference of model parameters / model selection using quantitative data.

4

Theoretical contributions

Ruth Baker

• Developing mathematical and computational models that can be used to test specific biological hypotheses.

• Efficient and accurate methods for computational simulation.

• Model coarse-graining / reduction to facilitate analysis.

• Extraction of useful (quantitative) summary statistics from experimental data.

• Inference of model parameters / model selection using quantitative data.

5

Theoretical contributions

Ruth Baker

• Numerous examples of cell populations undergoing coordinated migration over long distances within developing embryos.

• What are the key driving mechanisms?

• Population heterogeneity.

• Cell-microenvironment interactions.

• Cell-cell interactions.

6

Collective motility in development

Ruth Baker

• Collaboration with Paul Kulesa at the Stowers Institute for Medical Research, Kansas City.

7

The neural crest as a model system

Kulesa et al. Developmental Biology (2010).

Ruth Baker

What is the role of population heterogeneity?

8

Ruth Baker

• Initial experiments:

• cells emerge without directionality;

• leading cells remain unaligned to the migratory route;

• trailing cells align to the route.

• Are the leaders creating a path towards the distal target sites?

9

Population heterogeneity

Kulesa et al. Developmental Biology. 2010

Ruth Baker

• Self-generated gradient of positional information.

• Cells at the front of the invading stream create and respond to a chemoattractant gradient.

• Do the cells further behind respond to the same chemoattractant gradient, or do they follow other cues?

10

Initial hypotheses

Kulesa et al. Developmental Biology. 2010

Ruth Baker

• At each time step: • insert new cells; • solve chemoattractant PDE; • grow tissue; • move cells.

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Hybrid agent-based - PDE model

A

CA0

1

cell-inducedgradient

B

chemotaxis(leaders)

C

leader-followerbehaviour

?

D

L

ttissuegrowth

migratorydomain

L(t)

t>tLF

t<tLF

F

E

G

H

I

FollowerLeaderKey

Neur

al tu

be

Dist

al ta

rget

Tissue growth (parameterised experimentally)

Ruth Baker

• Cells create, and move up, a gradient of a chemoattractant (for example, VEGF).

• Stream breaks apart as late-entering cells have no gradient to follow.

12

Self-generated positional information

chemoattractantconcentration

0 1

FollowerLeader

Ruth Baker 13

Hybrid agent-based - PDE model

A

CA0

1

cell-inducedgradient

B

chemotaxis(leaders)

C

leader-followerbehaviour

?

D

L

ttissuegrowth

migratorydomain

L(t)

t>tLF

t<tLF

F

E

G

H

I

FollowerLeaderKey

• What about population heterogeneity?

• Could later emerging cells play “follow my leader”?

Ruth Baker

• Fixed number of leader cells - create and respond to a gradient in chemoattractant.

• Trailing cells - obtain directional information from leading cells (“follow the leader”).

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Population heterogeneity


0 1

FollowerLeader


0 1

FollowerLeaderKey

Ruth Baker

• Distinct expression patterns.

• Leading cells - upregulated guidance factor receptors, MMPs, cadherins.

• Trailing cells - different cadherins.

15

Experimental verification

Ruth Baker

• What makes for efficient and / or robust migration?

16

A working model

Ruth Baker 17

Smaller leader fractions are optimal

McLennan, Schumacher et al. Development (2015)

• Simulate model with different, fixed leader fractions.

Ruth Baker

• Simulate model with different, fixed leader fractions.

18

Smaller leader fractions are optimal

x/µm

# c

ells / 5

0µ

m

0 100 200 300 400 500 600 700 8000

4

8

12

16 0.051

0.11

0.26

0.59

1

<fL>


Ruth Baker

• Simulate model with leaders peppered throughout the stream.

19

Leaders are better at the front


Ruth Baker

What is the role of cell-microenvironment interactions?

20

Ruth Baker 21

Phenotypic plasticity

McLennan, Schumacher et al. Development (2015) McLennan, Dyson et al. Development (2012)

Ruth Baker 22

Characterising switching behaviour

cell starts as a leader

cell switches to following

time

signa

l sen

sed }~ time-scale

of switching behaviour

cell switches to leading upper

lower

no chemoattractan

McLennan, Schumacher et al. Developmental Biology (2015)

time (min)0 30 60 90

rela

tive

exp

ress

ion

0

1

2

3

genesmean

time (min)90 120 150 180

rela

tive

exp

ress

ion

0

1

2

3

genesmean

time (min)0 30 60 90

rela

tive

exp

ress

ion

0

1

2

3

genesmean

time (min)90 120 150 180

rela

tive

exp

ress

ion

0

1

2

3

genesmean

Ruth Baker 23

Efficiency and variability

McLennan, Schumacher et al. Developmental Biology (2015)

relative switch time τL->F

/τF->L

0 2 4 6 8 10 12 14 16 18 20

mig

ratio

n e

ffic

iency

, µ

(-)

, µ

/σ (

--)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τF->L

1 min 2 min 4 min 8 min12 min16 min24 min32 min40 min48 min56 min

Ruth Baker

• Cell-induced positional information gradients provide a viable mechanism for collective cell migration in the embryo.

• Population heterogeneity can facilitate migration over long distances.

• Interactions with the local microenvironment are sufficient to give rise to this heterogeneity.

24

Conclusions

Ruth Baker

• Assumed that cell-cell interactions are required for leader-follower behaviour, but what do they actually look like?

• How do cells sense and respond to the chemoattractant?

• Modelling questions

• To what extent can we parameterise these kinds of models using available quantitative data?

• Can analysis of coarse-grained models elucidate classes of potential invasion mechanisms?

25

Outstanding questions

Ruth Baker

Coarse-grained models of collective cell motility

What is the role of cell shape?

26

Ruth Baker

• Often it is useful to derive simplified models that are analytically tractable.

• Continuum diffusion approximations are widely used to represent collective cell motility.

• Most models incorporate simple linear diffusion:

• Whilst others incorporate nonlinear diffusion terms, e.g.

27

Continuum diffusion approximations

@C

@t= D0r2C

@C

@t= D0r · [D(C)rC] D(C) = Cn, n > 0

Ruth Baker

• Arguments for the use of a nonlinear diffusion model:

• solutions can have interfaces beyond which the density is zero;

• can represent e.g. population pressure, contact-mediated motility.

• Arguments supporting the use of a linear diffusion model:

• no guidance available to suggest how best to choose function form of diffusion coefficient;

• simple to solve / analyse models.

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What is appropriate?

Ruth Baker 29

What is appropriate?

(a) (b)cell

dens

ity (c

ells

cm

-2) 1

04

time

position (cm) cell

dens

ity (c

ells

cm

-2) 1

04

time

position (cm)

time

linear diffusion degenerate nonlinear diffusion

Sengers et al., Experimental characterisation and computational modelling of two-dimensional cell spreading

for skeletal regeneration. J. R. Soc. Interface (2007).

Ruth Baker

• Use an on-lattice exclusion process approach to explore what “sensible” continuum approximations might look like.

• Simple means to incorporate excluded volume interactions.

• Can use standard techniques to produce numerical realisations, write down conservation statements, derive corresponding partial differential equations.

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A method to tackle this problem

Ruth Baker 31

The simplest case

• On-lattice model (spacing ), volume excluding, agent-based.

• Initial conditions - each site population uniformly at random with given probability.

• Events:

• motility, rate

• proliferation, rate

Pm

Pp

� ! 0� ! 0

Baker and Simpson, Phys. Rev. E (2010).

Ruth Baker

• At the population level, evolution of the cell density is described by a simple, linear diffusion equation.

32

The simplest case

@C

@t= D0r2C

Ruth Baker

• Why is the spreading of certain cell populations best described by a linear diffusion mechanism?

• On the other hand, why is the spreading of other cell populations best described by a nonlinear diffusion model?

33

Questions

Nonlinear diffusion effects play an important role in describing the spreading of cell populations when we consider the effects of varying the cell aspect ratio together with volume exclusion.

Simpson, Baker and McCue, PRE (2011) and Baker and Simpson, Physica A (2012)

Ruth Baker

• Simple to simulate from the individual-level model, but can we derive a corresponding population-level model?

34

Elongated cells

200 500 8001

25

x

y t = 0

200 500 8001

25

x

y t = 500

200 500 8001

25

x

y t = 1000

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8001

25

x

y t = 0

200 500 8001

25

x

y t = 500

200 500 8001

25

x

y t = 1000

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8001

25

x

y t = 0

200 500 8001

25

x

y t = 500

200 500 8001

25

x

y t = 1000

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity(a) (b) (c)

(i)(h)(g)

(f)(e)(d)

Ruth Baker 35

Modelling options

i i+1 i+2 i+3i-1i-2i-3 i+4 i+5i-4

2 Δ

Δ

Δ

ΔIndividual Sites Approach (IS)

Physical System Mathematical Idealization

i i i+1i+1

Stretched Lattice Approach (SL)


2 Δ

Δ

2 Δ

Δi i+1 i

Individual Agents Approach (IA)


2 Δ

Δ

2 Δ

Δi i+1 i i+1

j

j+1

j+2

@C

@t= D0

@

@x

✓DIS(C)

@C

@x

◆, DIS(C) = L2CL�1

@C

@t= D0

@

@x

✓DSL(C)

@C

@x

◆, DSL(C) = L

@C

@t= D0

@

@x

✓DIA(C)

@C

@x

◆, DIA(C) = 1 + 2(L� 1)C

Ruth Baker

• Compare performance of each population-level model against ensemble-averaged discrete data.

• Validity affected by the mean-field approximation underlying the model derivation.

36

Comparing the different models

200 500 8001

25

x

y t = 0

200 500 8001

25

x

y t = 500

200 500 8001

25

x

y t = 1000

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8001

25

xy t = 0

200 500 8001

25

x

y t = 500

200 500 8001

25

x

y t = 1000

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

200 500 8001

25

x

y t = 0

200 500 8001

25

x

y t = 500

200 500 8001

25

x

y t = 1000

200 500 8000.0

0.5

1.0IAISSL<C>

xde

nsity

200 500 8000.0

0.5

1.0IAISSL<C>

x

dens

ity

(a) (b) (c)

(i)(h)(g)

(f)(e)(d)

Ruth Baker

• Compare performance of each population-level model against ensemble-averaged discrete data.

• Validity affected by the mean-field approximation underlying the model derivation.

37

Comparing the different models

1 500 10001

25

xy t = 0

1 500 10001

25

x

y t = 500

1 500 10001

25

x

y t = 1000

1 500 10000.0

0.5

1.0IAISSL<C>

x

dens

ity

1 500 10000.0

0.5

1.0IAISSL<C>

x

dens

ity

1 500 10001

25

x

y t = 0

1 500 10001

25

x

y t = 500

1 500 10001

25

x

y t = 1000

1 500 10000.0

0.5

1.0IAISSL<C>

x

dens

ity

1 500 10000.0

0.5

1.0IAISSL<C>

x

dens

ity

1 500 10001

25

x

y t = 0

1 500 10001

25

x

y t = 500

1 500 10001

25

x

y t = 1000

1 500 10000.0

0.5

1.0IAISSL<C>

x

dens

ity

1 500 10000.0

0.5

1.0IAISSL<C>

xde

nsity

(a) (b) (c)

(i)(h)(g)

(f)(e)(d)

Ruth Baker

• Individual agents model well able to capture invasion speed.

• Wave profile not not well captured.

38

Cell invasion profiles

-4 -3 -2 -10.0

0.1

0.2

0.3

0.4

0.5

log10(Pp)

wave

spe

ed

IAISSL<C>

0 1000 2000 30000.0

0.5

1.0IAISSL<C>

x

dens

ity

0 1000 20000.0

0.5

1.0IAISSL<C>

x

dens

ity

(a) (b) (c)

-4 -3 -2 -10.0

0.1

0.2

0.3

0.4

0.5

log10(Pp)

wave

spe

ed

IAISSL<C>

0 1000 2000 30000.0

0.5

1.0IAISSL<C>

x

dens

ity

0 1000 20000.0

0.5

1.0IAISSL<C>

x

dens

ity

(a) (b) (c)

Ruth Baker

• Cell shape / aspect ratio, together with volume exclusion, provides a motivation for the use of nonlinear diffusion coefficients for collective cell motility.

• We can motivate a range of possible forms for the diffusion coefficient.

• Outstanding questions:

• How to correct the mean-field approximation. Can we use e.g. a moment dynamics description?

• Can we retain a notion of “agent-size” in the coarse-grained models?

39

Conclusions

Ruth Baker

• Louise Dyson (Warwick)

• Linus Schumacher (Imperial College)

• Mat Simpson (Queensland University of Technology)

• Philip Maini and David Kay (Oxford)

• Paul Kulesa and Rebecca McLennan (Stowers Institute)

40

AcknowledgementsBaker Group

Quantitative Approaches to Developmental Biology www.iamruthbaker.com

Daniel Wilson

Louise Dyson

Fergus Cooper

Casper Beentjes

David Warne (QUT)

Rasa Giniunaite

Andrew Parker

Jonathan Harrison

Bartosz Bartmanski

Linus Schumacher Paul Kulesa

Maxandre Jacqueline

Mat Simpson Rebecca McLennan

mathematical models and biological insights · mathematical models and biological insights ruth...

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