Modeling Approaches to Biological Problems Abstract: Modeling in biology requires the development of new techniques which reflect the uniqueness of biological problems. Discrete particle system models (cellular automata models) are especially well-suited for biological problems in which individual stochastic cellular or molecular interactions play an important role. While these models are flexible and straight-forward to implement, they are difficult to interpret analytically and are computationally intensive. Thus large-scale events are best described by continuum PDE models. Since biological problems are often characterized by a range of length scales, hybrid models that incorporate continuum and discrete elements can be very powerful. In this talk, the discrete LGCA and Potts models will be described with examples of these models applied to biological problems, typically with hybrid components.

Modeling Approaches to Biological Problems Audi Maria Byrne September 27th, 2007 USA Math and Stat Colloquium

Talk Outline Characteristics of Biological Problems Characteristics of Cellular Automata Lattice Gas Cellular Automata Biological Potts Model

Characteristics of Biological Problems

Characteristics of Biological Problems Purpose / function inherent in biological systems. Out of equilibrium. Large # of distinct components. Wide range of relevant length and time scales. Feedback interaction.

Perspectives in Biology Wolpertian organization: Positional coding of cell behaviors Behaviors are pre-mapped or communicated via long-range signals Specialized cells acts as “directors” to guide other cells Example: chemotaxis towards a signaling cell Typically addressed by continuum PDE model

Perspectives in Biology Self-organization: Local organization of cell behaviors Behaviors are relayed via direct cell-cell or cell-ECM contacts All cells are initially equivalent (and have no immediate information about position in tissue) Typically addressed by discrete cellular automata models

Example Wolpertian alignment: Self-organized alignment: A professor moves to a corner of the room and asks the class to turn towards that corner of the room. Self-organized alignment: The class is told to collectively face a new, random direction in the room.

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).

Characteristics of Cellular Automata

Origin & History of Cellular Automata Developed by Stanislaw Ulam, John von Neumann 1940’s Ulam’s lattice network model for crystal growth Von Neumann’s self-replicating systems Popularized by John Conway, 1970’s Game of Life Systematically investigated by Stephen Wolfram, 1980’s 1D “Elementary” Cellular Automata Wolfram’s “A New Kind of Science” in 2002

Properties, Definition of Cellular Automata Discretized Space A regular lattice of “nodes”, “sites”, or “cells” Discretized Time The lattice is a dynamical system updated with “time-steps”. Discretized States For Each Node E.g.; binary states

Properties, Definition of Cellular Automata Universal Rule for Updating Node States Applied to every node identically States at time t+1 are based on states at time t Neighborhood (local) Rule for Updating Node States New node states are determined by nearby states within the “interaction neighborhood” Rules may be deterministic or stochastic

Example of an Elementary CA Wolfram’s Rule #122 One-dimensional, only binary ‘0’ and ‘1’ Initial conditions: (t=0) Simple, Deterministic Update Rule: If exactly one neighbor is ‘1’, become ‘1’. If both neighbors are ‘1’, become ‘0’. States at t=1:

Example of an Elementary CA Wolfram’s Rule #122 As t increases 

Examples of Regular Lattices 1-D Triangular Square Hexagonal

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

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

Advantages of Cellular Automata Straight-forward implementation. Local rules are intuitive and easy to modify. Transversing length scales: Cells are a natural mesoscopic unit. With modern computational power, 10^3-10^5 cells can be modeled on a PC. Physical Mechanisms easily verified In silico response variables may be the same as for an in vivo experiment. Reflects intrinsic individuality of cells. Unbounded complexity potential.

Disadvantages of Cellular Automata Analytic analysis of results difficult. Description of ‘average’ behavior difficult. Require large computational resources. Arbitrary limitations Limited directions on a regular lattice Rigid body motion difficult Artifacts and discretization errors.

Example: Lattice Artifacts Round off errors: particles appear and disappear unnaturally. Overly regular structures. Unrealistic periodic behavior over time: “bouncing checkerboard behavior”.

Versatility of CA in Biology 1980-1995 Occular dominance in the visual cortex Swindale 1980 Tumor Growth Duchting & Vogelssaenger, 1983; Chodhury et al, 1991; Pjesevic & Jiang 2002 Microtubule Arrays Smith et al, 1984; Hammeroff et al, 1986 Animal Coat Markings Young 1984, Cocho et al, 1987 Cell sorting Bodenstein, 1986; Goel & Thompson 1988, Glazier & Graner 1993 Neural Networks Hoffman 1987 Nerve and muscle, cardiac function Kaplan et al 1988 Cell dispersion Othmer, Dunbar, Alt 1988 Predator Prey Models Dewdney 1988 Immunology Dayan et al, 1988; Sieburgh et al, 1990; DeBoer et al, 1991 Angiogenesis Stokes, 1989; Peirce & Skalak 2003 Cell Differentiation and Mitosis Nijhout et al 1986; Dawkins 1989 Plant Ecology Moloney et al 1991 Honey Bee Combs Camazine 1991 G-protein Activation Mahama et al 1994 Bacteria Growth Ben-Jacob et al 1994

Versatility of CA in Biology 1995-2005 Population dynamics Janecky & Lawniczak 1995 Reaction diffusion Chen, Dawson, Doolen 1995 Actin Filaments Besseau & Geraud-Guille 1995 Animal Herds Mogilner & Edelstein-Keshet 1996 Shell pigmentation Kusch & Markus 1996 Alignment Cook, Deutsch, Mogilner 1997 Fruiting Body Formation of Dicty Maree & Hogeweg 2000 Convergent Extension Zebrafish Zajac, Jones, Glazier 2002 Fruiting Body Formation Myxobacteria Alber, Jiang, Kiskowski, 2004 Limb Chondrogenesis Kiskowski et al, 2004; Chaturvedi et al, 2004 T-cell Synapse Formation Casal, Sumen, Reddy, Alber, Lee, 2005 Avascular Tumor Growth Jiang, Pjesivac-Grbovic, Cantrell, Freyer, 2005 Tumor-Induced Angiogenesis Bauer, Jackson, Jiang, 2007 Cellular Automata Approaches to Biological Modeling Ermentrout and Edelstein-Keshet, J. theor. Biol, 1993

Types of CA Models Alternative Names Lattice Gases (LGCA) Bio-LGCA Pott’s Model Hybrid Models Alternative Names Cell-based Discrete-state Discrete particle systems Lattice-based Monte Carlo

Hybrid Models While CA are fully discrete, a hybrid model may contain continuous elements. Coupled map lattices: continuous states E.g., finite difference and finite element methods Interacting particle systems: time is continuous Coupled differential equations: continuous state and time Very commonly, a CA interacts with a continuously-defined field (chemotactic gradient, etc)

LGCA

LGCA Cells/particles are modeled as points. The cell state corresponds to the cell “channel”, or orientation. Important exclusion rule: only one cell per channel (all cells at the same node have different orientations) Computationally efficient updating with a 2-step transition rule at every time-step: (1) Interaction step: Cells are assigned new velocities. (e.g., collisions, alignment, birth, death.) (2) Transport step: All cells are transported simultaneously along velocity channels.

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

Biological LGCA Coined by Edelstein-Keshet and Ermentrout, 1990 May relax exclusion principle or include novel modeling elements

Example 1: Diffusion Albert Einstein (1905) Treated molecules as a statistical ensemble Deduced that diffusion could be the result of numerous collisions Random Memory-less Einstein-Stokes Equation D=kβT/(6πηr) D = diffusion constant r = particle radius η = viscosity of the fluid kβ = Boltzmann constant T = temperature

Analytic Description: The Heat Equation Describes the mean concentration of particles or the probability density function: Molecules are treated as a statistical ensemble. www.amath.unc.edu/SCC/showcase.html

Discrete description of cell diffusion Diffusion of a cell is modeled by moving the cell in a random direction at each time-step. For each cell, choose a random number between 0 and 4: 0 => rest 1 => right 2 => up 3 => left 4 => down

Diffusion Simulation

Gaussian Blur

Example 2: Reaction Diffusion Chemical peaks occur in a system with an autocatalytic component (an activator) and a faster-diffusing inhibiting component (an inhibitor). Result: periodic peaks in stripes or spots described by complex Bessel equations. [Meinhardt, Bioessays, 1995]

Reaction Diffusion: Reproduces many biological patterns Figure: J.D. Murray, Mathematical Biology [D.A. Young, 1984] Figure: L.Wolpert, Principles of Dev. Biol. [ Also: Meinhardt, Klinger, Fowler, 1980s] Figure:Painter, Miani and Othmer, 1999 [ Kondo and Asai, 1995] Figure: L.Wolpert, Principles of Dev. Biol. [Gierer and Meinhardt, 1971] Figure: Klausmeier,1999 Also: [Hardenburg et al, 2001]

PDE Model: FitzHugh-Nagumo Eqn For positive constants du, dv, τ, σ and λ, with f(u)=λu -u3 - κ which describes how an action potential travels through a nerve.

Fitzhugh Nagumo Simulation http://cnls.lanl.gov/~aric/Simulations/Simulations.html

Discrete model: Reaction occurs at each node independently after diffusion step. Activation: CA=Activator concentration CB=Inhibitor nc = cell Up-regulation of Inhibitor: Inhibition: Inhibitor Decay:

Application: chondrogenesis in the chick limb Activator Peaks Inhibitor Peaks Final Cell Distribution Kiskowski, Alber, Thomas, Glazier, Bronstein and Newman

Application: chondrogenesis in the chick limb Activator Peaks Inhibitor Peaks Final Cell Distribution Kiskowski, Alber, Thomas, Glazier, Bronstein and Newman

POTTS model

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.

Jiang and Pjesevic

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)

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

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

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

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 vs - cell Volume Vs - target Volume Cf - concentration of chemical (nutrient or waste) µf - chemical potential

Chemical Diffusion in Tumor

Growth of Tumor - 12 Hours (from Jiang & Pjesivac-Grbovic)

Nutrient Diffusion: PDE model (from Jiang & Pjesivac-Grbovic)

Waste Distribution

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

8.5 Days

Number of Cells over Time (from Jiang & Pjesivac-Grbovic)

Growth in Number of Cells versus O2 and Nutrient

Funding slide: NIH Mitchell Cancer Institute Thanks very much!