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.
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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.
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Collective motility in development
Ruth Baker
• Collaboration with Paul Kulesa at the Stowers Institute for Medical Research, Kansas City.
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The neural crest as a model system
Kulesa et al. Developmental Biology (2010).
Ruth Baker
What is the role of population heterogeneity?
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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?
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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?
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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.
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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
chemoattractantconcentration
0 1
FollowerLeader
chemoattractantconcentration
0 1
FollowerLeaderKey
Ruth Baker
• Distinct expression patterns.
• Leading cells - upregulated guidance factor receptors, MMPs, cadherins.
• Trailing cells - different cadherins.
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Experimental verification
Ruth Baker
• What makes for efficient and / or robust migration?
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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.
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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>
McLennan, Schumacher et al. Development (2015)
Ruth Baker
• Simulate model with leaders peppered throughout the stream.
19
Leaders are better at the front
McLennan, Schumacher et al. Development (2015)
Ruth Baker
What is the role of cell-microenvironment interactions?
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Phenotypic plasticity
McLennan, Schumacher et al. Development (2015) McLennan, Dyson et al. Development (2012)
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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
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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.
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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?
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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.
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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
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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.
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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?
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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?
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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
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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)
Physical System Mathematical Idealization
2 Δ
Δ
2 Δ
Δi i+1 i
Individual Agents Approach (IA)
Physical System Mathematical Idealization
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.
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Comparing the different models
200 500 8001
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y t = 0
200 500 8001
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y t = 500
200 500 8001
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y t = 1000
200 500 8000.0
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1.0IAISSL<C>
x
dens
ity
200 500 8000.0
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1.0IAISSL<C>
x
dens
ity
200 500 8001
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xy t = 0
200 500 8001
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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
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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
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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.
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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?
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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)
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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
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