Sketch Outline Ising, bio-LGCA and bio-Potts models Potts model general description computational description Examples of ‘energies’ (specifying interactions)

Ising Model Lattice All nodes occupied with spins ±1 Local interaction LGCAPotts Particles in motion Lattice channels (spins) Two steps: interaction then transport Q spins At each node, spin can change according to interaction Hamiltonian Metropolis algorithm

LGCA model

Lattice Gas – Coulomb Interaction

Lattice Gas – log(1/r) Interaction

LGCA Examples: Ideal fluids and gases (Hardy, de Passis, Pomeau 1976) Rippling and aggregation in myxobacteria Application: large numbers of uniformly interacting particles Limitation: HPP model approximates the Navier-stokes equation at low velocities, but at higher velocities fails to conserve isotropy.

POTTS model

Lattice Energy The total energy of the system depends on cell- cell surface interaction. Surface interaction can also include terms for: Adhesion Volume constraints Chemical energy due to local chemical gradients

Jiang and Pjesevic

Metropolis Algorithm The lattice evolves by a Monte Carlo process. At each Monte Carlo step a random change is made at a lattice site  H is calculated probability of accepting this change is

Pott’s Model One cell is modeled as many lattice pixels A cell has area or volume Generalization of the Ising model since each cell has a unique “spin” Corresponds to a cell ID # At each time-step, a cell pixel may change spin Cell becomes part of another cell or becomes part of the extra-cellular matrix (ECM). Driving force is energy minimization.

Pott’s Model Examples: Sorting of biological cells via cell-cell adhesion (Glazier and Graner, 1993) Morphogenesis in Dicty (Maree et al, 2000) Zebrafish convergent extension (Zajac et al, 2002) Limb morphogenesis (Chaturvedi et al, 2004) Avascular tumor growth (Jiang et al., 2005)

Simple example: cell sorting 3 cell types “like” cell types have a lower surface energy than “unlike” cell types Like cells:  = 0 Unlike cells:  = 1

Simple example: cell sorting Random Initial Conditions

Simple example: cell sorting Simulation over 100 Time Steps

Biology… Proof of principle (self-organized verses directed) Transients Limiting cases

Examples of Interaction Neighborhoods von Neumann Moore Radial Biological Application: C- Signaling in Myxobacteria

Dale Kaiser Neighborhood for a Biological Application: C-Signaling in Myxobacteria

Example: Self-Organization Emerges from a Population of Equivalent Cells due to Local Individual Interactions Igoshin, O. A., Welch, R., Kaiser, D. & Oster, G. Waves and aggregation patterns in myxobacteria. Proc. Natl Acad. Sci. USA 101, 4256–4261 (2004).

Avascular Tumor Growth Model (Jiang and Pjesivac-Grbovic model) Hybrid model transversing three spatial scales Sub-cellular scale described by a Boolean network describing cell cycle Cellular scale described by a Potts model Tissue scale chemical diffusion described by PDEs

Avascular Tumor Growth Model (Jiang and Pjesivac-Grbovic model) Proliferating, quiescent, and necrotic cells Cell division based on cell cycle: proliferating cells increase cell volume until the cell is double the initual volume, then the cell divides Proliferating cells consume nutrients and excrete waste Cells become quiescent and necrotic depending upon nutrient levels PDE reaction-diffusion equations describe concentrations for oxygen, nutrients, and waste products

Lattice Energy S - cell identification number (1,2,3,…)  (S) - cell type (proliferating, quiescent, or necrotic) J  (S)  (S’) - coupling energy between cell types  (S) and  (S’) v - elasticity v s - cell Volume V s - target Volume C f - concentration of chemical (nutrient or waste) µ f - chemical potential

Chemical Diffusion in Tumor

Simulation After 440 Monte Carlo steps (from Jiang & Pjesivac-Grbovic)