1. 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

2. 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

3. 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

4. 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

5. Discrete model: The Potts Lattice

6. The cells are elastic: So the total energy is: The cells are adhesive:

7. Discrete model: Energy minimisation 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:

8. Cancer Invasion Right – carcinoma of the uterine cervix, just beginning to invade (at green arrow) Left – corresponding healthy tissue

9. Potts model simulation of cancer invasion

10. Maximum Invasion Distance

11. 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

12. Single Cell in One Dimension What is the effective diffusion coefficient of the centre of the cell?

13. From a discrete to a continuous model Set T + =T - = , a constant, so:

14. From a discrete to a continuous model If we set n i-1 =n(x-h), n i+1 =n(x+h), and t= , then take the limit: then we obtain the diffusion equation 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).

15. The diffusion coefficient of Potts modelled cells P L is related to the difference between the energy at this length, E L and the minimum energy, E min : where Z is a partition function which ensures normalisation. If we set P L = probability of being at length L, The probability of a cell of length L moving to the right is given by: Where  E L is the change in energy associated with this move. = probability of moving to the right while at length L, where the summation is over all possible values of cell length.

16. 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.

17. Comparison of theory and experiment

18. 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

19. Part III: A New Continuum Model of Cell-Cell Adhesion Nicola J. Armstrong Kevin Painter Jonathan A. Sherratt Department of Mathematics, Heriot-Watt University

20. 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

21. 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

22. 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

23. Force on cells at x exerted by cells a distance x 0 away depends on 1.cell density at x+x 0 2.distance x 0 3.direction of force depends on position x 0 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

24. R – The sensing radius of cells Cell R In 1D xx + Rx - R Range over which cells can detect surroundings

25.  (x 0 )  (x 0 ) is an odd function –for simplicity we assume

26. 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:

27. Numerical results

28. Aggregation in Two Dimensions

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

31. Initially we assume linear functions This simplifies the adhesion terms to

32. Numerical Results (a) C = 0, S u > S v (b) S u > C > S v

33. 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

34. Numerical Results with Limiting g(u,v) C = 0, S u > S v

35. Experimental cell sorting results A: MixingC > (S u + S v ) / 2 B: EngulfmentS u > C > S v C: Partial engulfmentC < S u and C < S v D: Complete sortingC = 0

36. Numerical results – A ( C > (S u + S v ) / 2 )

37. Numerical Results - B ( S u > C > S v )

38. Numerical Results - C ( C < S u and C < S v )

39. Numerical Results - D ( C = 0 )

40. Experimental results and numerical model results

41. 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