Return to Eden: How biologically relevant can on-lattice models really be?

1. Return to Eden:How biologically relevant can on-lattice models really be?

2. Outline • What sorts of on-lattice models are there? • What do/can we model on-lattice? • Pros. • Cons. • Two case studies • Position jump modelling of cell migration. • Models for tumour growth.

3. Types of on lattice model • Cellular automaton. • Exclusion processes. • Game of life. • Cellular Potts model. • Lattice gas automaton. • Lattice-Boltzmann. • Ising model. • Position jump models (on lattice).

4. Cellular automaton • Pattern formation. • Neural networks. • Population biology. • Tumour growth. See Ermentrout, G.B. and Edelstein-Keshet, L., Journal of Theoretical Biology 1993

5. Cellular Potts models • Metastasis • Developmental biology • Immunology • Tumour growth Cellular Potts Model of single ovarian cancer cell migrating through the mesothelial lining of the peritoneum.

6. Position jump models • Development • Pattern formation • Animal Movement • Aggregation

7. Advantages • Simple to formulate and adapt. • Easy to explain to biologists. • Can capture phenomenological details. • Mathematically and computationally tractable. • Makes multiscale description possible (i.e. can often derive PDEs).

8. Problems with on-lattice models • Geometry - Cells aren’t squares! • Hard to convince biologists. • Changing lattices are difficult to deal with (i.e. how to implement cell birth/death). • Inherent anisotropy. • Artificial noise effects.

9. What’s best for… • …Parallelisation of code? • Can both on-lattice and off-lattice individual-based models be parallelised equally well? • …Boundary condition implementation? • Which type of model deals best with curved boundaries for example?

10. Case Study 1:Position jump modelling of cell migration:Movement T- T+ = A cell

11. Signal Sensing = A cell

12. Some definitions

13. Probability master equation

14. Equivalent to PDE

15. Local Signal Sensing Cell Density Profiles Individual model – Blue histograms. PDE – Red curve.

16. Growth = A cell

17. Exponential Domain Growth Domain Growth PDE – Red. Average stochastic- Green. Individual Stochastic – Black. Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Domain length – Green star.

18. Density Dependent Domain Growth Domain Growth PDE – Red. Average stochastic- Green. Individual Stochastic – Black. Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Domain length – Green star.

19. Incremental Domain Growth = A cell

20. Connecting to a PDE • In order to connect the PDE with the cell density we had to enforce a Voronoi domain partition. Interval Centred Domain Partition Vornoi Domain Partition

21. Diffusion on the Voronoi domain partition Domain Growth PDE – Red. Average stochastic- Green. Individual Stochastic – Black. Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Domain length – Green star.

22. Higher Dimensions PDE solution surface Individual based model – Square grid histogram Local sensing on a 50X50 square lattice

23. Triangular Lattice PDE solution surface Individual based model – Traingle grid histogram Diffusion on a triangular lattice

24. Growth in two-dimensions? • Circular or square domain to make PDEs tractable. • Apical growth? • How much can lattice sites push each other out of the way? • Can we make on lattice models replicate real biological dynamics, at least qualitatively?

25. Case Study 2:The Eden model

26. The Eden model • Produces roughly circular growth (especially for large clusters) • Start of with an initial “cell” configuration or a single seed. • Square cells are added one at a time to the edges of the cluster in one of three ways:

27. Eden A • A cell is added to any of the sites which neighbour the surface equiprobably. # surface neighbouring sites = 12

28. Example Eden A cluster

29. Eden B • A cell is added to any of the edges of the surface equiprobably. # surface edges = 14

30. Example Eden B cluster

31. Eden C • A surface cell is chosen equiprobably and one of its edges chosen equiprobably to have a cell added to it. # surface cells = 8

32. Example Eden C cluster

33. Real Tumour Slices Images Courtesy of Kasia Bloch (Gray Institute for Radiation Oncology and Biology and the Centre for Mathematical Biology)

34. Important properties • Growth rate • Morphology • Surface thickness • Genus (Holiness)

35. Number of holes vs time Eden A Eden B Eden C All values are averaged over 50 repeats

36. Surface scaling

37. Surface scaling

38. Universality Classes (UC) • By finding these coefficients we can classify these models into universality classes. • Some well-known universality classes are:

39. Tumour universality class • Brú et al*. found a universality class for tumours. • They placed tumours in the MH universality class. *Brú, A.; Albertos, S.; Luis Subiza, J.; Garcia-Asenjo, J. & Brú, I.The universal dynamics of tumor growthBiophys. J., Elsevier, 2003, 85, 2948-2961

40. Eden universality • In strip geometry Eden is in KPZ. • But not so in radial clusters? • Why not?

41. Anisotropy • Axial anisotropy cause problems. Eden A Eden B Eden C The three Eden models average over 50 repeats

42. Anisotropy correction • Even model C exhibits a 2% axial anisotropy. • But Paiva & Ferreira* have found a way to correct for this. • Once corrected and surface thickness determined in the proper way it was found the radial Eden clusters fall into the KPZ UC. *Paiva, L. & Ferreira Jr, S.Universality class of isotropic on-lattice Eden clustersJournal of Physics A: Mathematical and Theoretical, IOP Publishing, 2007,

43. Mitosis • Off-lattice Eden model – Ho and Wang*. • Isotropic but no use to us as it’s off lattice. • On lattice with limited pushing range – Drasdo**. • Limited range of pushing. • Anisotropic. *Ho, P. & Wang, C.Cluster growth by mitosisMath. Biosci., Elsevier, 1999. ** Drasdo, D.Coarse graining in simulated cell populationsAdvances in Complex Systems, Singapore: World Scientific, 2005.

44. Adapted mitosis model • Division in 8 neighbouring directions. • No limit as to how far we can push other cells. • Isotropic? Tentative yes. • Universality class? Too early to say.

45. Summary • Lattice model examples. • Pros and cons. • Position jump case study. • Cluster growth case study. • Lattice models can be compared to real-world phenomena (e.g. universality classes, genus). • But how realistic are they?

46. Discussion points • Will on-lattice models continue to be of use in the future? • Will on lattice models ever be as realistic as off-lattice models? • Why use a lattice model when an off-lattice model works just as well (and vice versa)? • Do lattice models have a role in communicating our modelling ideas to biologists?