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

Continuum Modelling of

Cell-Cell Adhesion

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

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

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

Discrete model: The Potts Lattice

Discrete model: The Potts Lattice

ji ji

adh jiijJJE

, ',')()( ''

The cells are elastic:

2)( Tel VvE

So the total energy is:

2

'')()( )('' T

ij jitot VvJJE

jiij

The cells are adhesive:

Discrete model: Energy minimisation

)/exp(

1)( ''

E

p jiij

If E<0

If E>0

Copy the parameters for a lattice point inside one cell into a neighbouring cell. This will give rise to a change in total energy E.

If E is negative, accept it. If it is positive, accept it with Boltzmann-weighted probability:

Cancer Invasion

Right – carcinoma of the uterine cervix, just beginning

to invade (at green arrow)

Left – corresponding healthy tissue

Potts model simulation of cancer invasion

Maximum Invasion Distance

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

Single Cell in One Dimension

What is the effective diffusion coefficient of

the centre of the cell?

From a discrete to a continuous model

iiiiiiiii nTnTnTnTt

n

1111

Set T +=T -=, a constant, so:

iiii nnnt

n211

From a discrete to a continuous model

If we set ni-1=n(x-h), ni+1=n(x+h), and t=, then take the limit:

Dh )lim( 20h

then we obtain the diffusion equation

2

2*

x

nD

n

where D*=D, 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).

iiii nnnt

n211

The diffusion coefficient of Potts modelled cells

PL is related to the difference between the energy at this length, EL and the minimum energy, Emin :

minexp

1 EE

ZP LL

where Z is a partition function which ensures normalisation.

If we set PL = probability of being at length L,

L

RLLPPT

The probability of a cell of length L moving to the right is given by:

LR

L

EP exp1

4

1

Where EL is the change in energy associated with this move.

RLP = probability of moving to the right while at length L,

where the summation is over all possible values of cell length.

L

LL EEE

ZD

minexpexp1

4

1

If we assume that the cells are non-interacting, so T += T – , and

remembering our result from the derivation of the diffusion equation, where D=T +, we can say

We can test this formula by performing a numerical experiment.

Comparison of theory and experiment

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

Part III: A New Continuum Model of Cell-Cell Adhesion

Nicola J. ArmstrongKevin Painter

Jonathan A. Sherratt

Department of Mathematics,Heriot-Watt University

Armstrong, P.B. 1971. Wilhelm Roux' Archiv 168, 125-141

Aggregation and cell sorting

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

Derivation of the model

• Assume– No cell birth or death– Movement due to random motion and adhesion

• Mass conservation =>

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

• Diffusive flux

where D is a positive constant

• Adhesive flux

– where • F = total force due to breaking and forming

adhesive bonds = constant related to viscosity• R = sensing radius of cells

• Force on cells at x exerted by cells a distance x0

away depends on1. cell density at x+x0

2. distance x0

3. direction of force depends on position x0 relative to x

• Total force = sum of all forces acting on cells at x• If cells detect forces over the range

x – R < x < x + R then

R – The sensing radius of cells

Cell

R

In 1D

x x + Rx - R

Range over which cells can detect surroundings

(x0)

(x0) is an odd function– for simplicity we assume

Modelling one cell population

• Assume g(u) = u• Expect aggregation of disassociated cells• Stability analysis and PDE approximation

suggest aggregating behaviour is possible• critical in determining model behaviour

Dimensionless equations:

Numerical results

Aggregation in Two Dimensions

Interacting populations

• To consider cell sorting we look at interacting populations

• Adhesion will now include self-population adhesion and cross-population adhesion

• Initially we assume linear functions

• This simplifies the adhesion terms to

Numerical Results

• (a) C = 0, Su > Sv

• (b) Su > C > Sv

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

Numerical Results with Limiting g(u,v)

• C = 0, Su > Sv

Experimental cell sorting results

• A: Mixing C > (Su + Sv) / 2

• B: Engulfment Su > C > Sv

• C: Partial engulfment C < Su and C < Sv

• D: Complete sorting C = 0

Numerical results – A

( C > (Su + Sv) / 2 )

Numerical Results - B

( Su > C > Sv )

Numerical Results - C

( C < Su and C < Sv )

Numerical Results - D

( C = 0 )

Experimental results and numerical model results

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