Part I: A discrete model of cell-cell adhesion Part II: Partial derivation of continuum equations from the discrete model Part III: A new continuum model

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Part I A discrete model of cell-cell
adhesionPart II Partial derivation of
continuum equations from the discrete modelPart
III A new continuum model
Continuum Modelling of Cell-Cell Adhesion
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What is cell-cell adhesion?

• Cells bind to each other through cell adhesion
  molecules
• This is important for tissue stability
• Embryonic cells adhere selectively and sort out
  forming tissues and organs
• Altered adhesion properties are thought to be
  important in tissue breakdown during tumour
  invasion

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Part I A Discrete Model of Cell-Cell Adhesion
Stephen Turner Western General
Hospital Jonathan Sherratt Mathematics, Heriot
Watt David Cameron Clinical oncology, Western
General Hospital, Edinburgh
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A discrete model of cell movement

• The extended Potts model is a discrete model of
  biological cell movement which we apply to
  modelling cancer invasion.
• Each cell is represented as a group of squares on
  a lattice
• Cell movement occurs via rearrangements that tend
  to reduce overall energy

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Discrete model The Potts Lattice
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Discrete model The Potts Lattice
The cells are adhesive
The cells are elastic
So the total energy is
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Discrete model Energy minimisation
Copy the parameters for a lattice point inside
one cell into a neighbouring cell. This will give
rise to a change in total energy DE.
If DE is negative, accept it. If it is positive,
accept it with Boltzmann-weighted probability
If DElt0
If DEgt0
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Cancer Invasion
Right carcinoma of the uterine cervix, just
beginning to invade (at green arrow) Left
corresponding healthy tissue
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Potts model simulation of cancer invasion
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Maximum Invasion Distance
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Part II Partial Derivation of Continuum
Equations from the Discrete Model
Stephen Turner - Western General
Hospital Jonathan Sherratt - Mathematics,
Heriot-Watt University Kevin Painter
Mathematics, Heriot-Watt University Nick Savill
Biology, University of Edinburgh
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Single Cell in One Dimension
What is the effective diffusion coefficient of
the centre of the cell?
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From a discrete to a continuous model
Set T T -a, a constant, so
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From a discrete to a continuous model
If we set ni-1n(x-h), ni1n(xh), and tlt,
then take the limit
then we obtain the diffusion equation
where DaD, a constant. So we have used a
knowledge of the transition probabilites for
individual cells on the lattice to derive a
macroscopic quantity (the diffusion coefficient).
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The diffusion coefficient of Potts modelled cells
If we set PL probability of being at length L,
probability of moving to the right while at
length L,
where the summation is over all possible values
of cell length.
PL is related to the difference between the
energy at this length, EL and the minimum energy,
Emin
where Z is a partition function which ensures
normalisation.
The probability of a cell of length L moving to
the right is given by
Where DEL is the change in energy associated with
this move.
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If we assume that the cells are non-interacting,
so T T , and remembering our result from the
derivation of the diffusion equation, where DT
, we can say
We can test this formula by performing a
numerical experiment.
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Comparison of theory and experiment
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Conclusions

• We have derived a formula for the effective
  diffusion coefficient
• But it is a complicated expression
• Moreover derivation of a directed movement term
  due to adhesion is much more difficult
• So develop a new continuum model

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Part III A New Continuum Model of Cell-Cell
Adhesion

• Nicola J. Armstrong
• Kevin Painter
• Jonathan A. Sherratt
• Department of Mathematics,
• Heriot-Watt University

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Aggregation and cell sorting

• (a) After 5 hours
• (b) After 19 hours
• (c) After 2 days

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Derivation of the model

• Assume
• No cell birth or death
• Movement due to random motion and adhesion
• Mass conservation gt

• where
• u(x,t) cell density
• J flux due to diffusion and adhesion

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• Diffusive flux

• where D is a positive constant
• Adhesive flux

• where
• F total force due to breaking and forming
  adhesive bonds
• f constant related to viscosity
• R sensing radius of cells

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• Force on cells at x exerted by cells a distance
  x0 away depends on
• cell density at xx0
• distance x0
• direction of force depends on position x0
  relative to x

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R The sensing radius of cells
In 1D
x
x R
x - R
Range over which cells can detect surroundings
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w(x0)

• w(x0) is an odd function
• for simplicity we assume

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Modelling one cell population
Dimensionless equations

• Assume g(u) u
• Expect aggregation of disassociated cells
• Stability analysis and PDE approximation suggest
  aggregating behaviour is possible
• critical in determining model behaviour

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Numerical results
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Aggregation in Two Dimensions
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Interacting populations

• To consider cell sorting we look at interacting
  populations
• Adhesion will now include self-population
  adhesion and cross-population adhesion

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• Initially we assume linear functions

• This simplifies the adhesion terms to

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Numerical Results

• (a) C 0, Su gt Sv
• (b) Su gt C gt Sv

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g(u,v)

• Linear form of g(u,v) unrealistic
• Steep aggregations with progressive coarsening
• Biologically likely that there exists a density
  limit beyond which cells will no longer aggregate
• Introduce a limiting form of g(u,v) to account
  for this

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Numerical Results with Limiting g(u,v)

• C 0, Su gt Sv

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Experimental cell sorting results

• A Mixing C gt (Su Sv) / 2
• B Engulfment Su gt C gt Sv
• C Partial engulfment C lt Su and C lt Sv
• D Complete sorting C 0

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Numerical results A
( C gt (Su Sv) / 2 )
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Numerical Results - B
( Su gt C gt Sv )
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Numerical Results - C
( C lt Su and C lt Sv )
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Numerical Results - D
( C 0 )
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Experimental results and numerical model results
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Future work

• Cell-cell adhesion is important in areas such as
  developmental biology and tumour invasion
• Largely ignored until now due to difficulties in
  modelling
• Many possible areas for application
• Current model has no kinetics
• May be some interesting behaviour if kinetics
  were included
• Cell-cell adhesion is a three dimensional
  phenomenon
• Could be an argument for extending the model to 3D