Return to Eden: How biologically relevant can onlattice models really be? 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. 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). Cellular automaton • • • • Pattern formation. Neural networks. Population biology. Tumour growth. See Ermentrout, G.B. and Edelstein-Keshet, L., Journal of Theoretical Biology 1993 Cellular Potts models • Immunology • Tumour growth •Metastasis •Developmental biology Cellular Potts Model of single ovarian cancer cell migrating through the mesothelial lining of the peritoneum. • • • • Position jump models Development Pattern formation Animal Movement Aggregation 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). 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. What’s best for… • …Parallelisation of code? – Can both on-lattice and off-lattice individualbased models be parallelised equally well? • …Boundary condition implementation? – Which type of model deals best with curved boundaries for example? Case Study 1: Position jump modelling of cell migration: Movement T- = A cell T+ Signal Sensing = A cell Some definitions Probability master equation Equivalent to PDE Local Signal Sensing Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Growth = A cell Exponential Domain Growth Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Domain length – Green star. Domain Growth PDE – Red. Average stochastic- Green. Individual Stochastic – Black. Density Dependent Domain Growth Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Domain length – Green star. Domain Growth PDE – Red. Average stochastic- Green. Individual Stochastic – Black. Incremental Domain Growth = A cell 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 Diffusion on the Voronoi domain partition Cell Density Profiles Individual model – Blue histograms. PDE – Red curve. Domain length – Green star. Domain Growth PDE – Red. Average stochastic- Green. Individual Stochastic – Black. Higher Dimensions PDE solution surface Individual based model – Square grid histogram Local sensing on a 50X50 square lattice Triangular Lattice PDE solution surface Individual based model – Traingle grid histogram Diffusion on a triangular lattice 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? Case Study 2: The Eden model 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: Eden A • A cell is added to any of the sites which neighbour the surface equiprobably. # surface neighbouring sites = 12 Example Eden A cluster Eden B • A cell is added to any of the edges of the surface equiprobably. # surface edges = 14 Example Eden B cluster 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 Example Eden C cluster Real Tumour Slices Images Courtesy of Kasia Bloch (Gray Institute for Radiation Oncology and Biology and the Centre for Mathematical Biology) Important properties • Growth rate • Morphology • Surface thickness • Genus (Holiness) Number of holes vs time Eden A Eden B All values are averaged over 50 repeats Eden C Surface scaling Surface scaling Universality Classes (UC) • By finding these coefficients we can classify these models into universality classes. • Some well-known universality classes are: Name Z EW ¼ ½ 2 KPZ 1/3 ½ 3/2 MH 3/8 3/2 4 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 growth Biophys. J., Elsevier, 2003, 85, 2948-2961 Eden universality • In strip geometry Eden is in KPZ. • But not so in radial clusters? • Why not? Anisotropy • Axial anisotropy cause problems. Eden A Eden B The three Eden models average over 50 repeats Eden C 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 clusters Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2007, 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 mitosis Math. Biosci., Elsevier, 1999. ** Drasdo, D. Coarse graining in simulated cell populations Advances in Complex Systems, Singapore: World Scientific, 2005. 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. 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? 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?