Part I: A discrete model of cell-cell adhesion Part II: Partial derivation of continuum equations...
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Transcript of Part I: A discrete model of cell-cell adhesion Part II: Partial derivation of continuum equations...
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
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
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
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.
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
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:
Interacting populations
• To consider cell sorting we look at interacting populations
• Adhesion will now include self-population adhesion and cross-population adhesion
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
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
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