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Overview: A Sampling of Some Recent Computational Modeling Efforts MA354 – Computational Math Modeling

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Cellular Automata Models

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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

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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

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Versatility of CA in Biology 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

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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 Cellular Automata Approches to Biological Modeling Ermentrout and Edelstein-Keshet, J. theor. Biol, 1993 Versatility of CA in Biology

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Application: Modeling FRAP “Fluorescent Recovery After Photobleaching” Modeling the diffusion of fluorescent molecules and “photobleaching” a region of the lattice to look at fluorescence recovery. 1. Fluorescent molecules diffuse on the lattice. 2. All molecules in Region A are “photobleached” (state changes from ‘1’ to ‘0’). 3. Recovery: remaining flourescent molecules diffuse into Region A randomly.

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Application: Modeling FRAP Modeling the diffusion of fluorescent molecules and “photobleaching” a region of the lattice to look at fluorescence recovery.

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Ising Model: Ferromagnetism

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Extension: Cellular Potts Model

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Jiang and Pjesevic

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Differential Adhesion

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Limb Development Cellular Potts Model for cell-cell interactions (cell sorting into clusters that will become bones) Coupled with a reaction diffusion equation that instructs what shapes the clusters should be

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1. Model For Limb Chondrogenesis Reaction Diffusion Developmental Model Based On Reaction Diffusion and Cell-matrix Adhesion. Computational Model and Results “Interaction between reaction-diffusion and cell-matrix adhesion in a CA model for chondrogenic patterning: a prototype study for developmental modeling” Kiskowski, Alber, Thomas, Glazier, Bronstein and Newman, Dev. Biol., to appear.

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

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Bone formation occurs from the proximal to distal region. As the limb grows, the number of precartilage condensations increases. Bone formation is mediated by fibronectin, which links cells together. Limb Formation in vivo

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Computational Model Cells on a 2D circular spot on a square lattice -Simulation of simplified, in vitro model -quasi-3D Reaction-Diffusion Cell-Fibronectin Adhesion

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Activation: Inhibition: Up-regulation of Inhibitor: Inhibitor Decay: C A =Activator concentration C B =Inhibitor concentration n c = cell concentration Reaction (Occurs At Each Node Independently)

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Diffusion At each time-step, cells, activator molecules and inhibitor molecules diffuse by either: resting at their current node with probability p s (or ) moving right, up, left or down with probability (1- p s )/4. As the probability of resting p s increases, the diffusion rate of the particle decreases. Model Particles: Cells, Activator, Inhibitor, and Fibronectin

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Cell-Matrix Adhesion Cells produce fibronectin at threshold levels of activator. Fibronectin does not diffuse. Cells stick to fibronectin with probability p f and un-stick with probability 1-p f. Once stuck, cells do not diffuse during that time-step. Fibronectin With Stuck Cells

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Final Fibronectin Distribution Final Cell Distribution Reaction diffusion establishes pattern of activator peaks. Activator Peaks Inhibitor Peaks Fibronectin produced at activator peaks slows cell diffusion and cells cluster. Preliminary Results

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Limb Development

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Hybrid Models a hybrid model contains both discrete (for example, individual cells defined on a lattice) and continuous elements. These elements must be ‘coupled’ in some way so the model elements interact and exchange information

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Phototaxis during the Slug Stage of Dictyostelium discoideum: a Model Study Marée, Panfilov and Hogeweg Proceedings of the Royal Society of London. Series B. Biological sciences 266 (1999)

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Example: A Tumor Model Based on Diffusion and Growth with 2 Continuous Fields Model of Dormann & Deutsch, 2002 – Model components similar to that of Düchting and Vogelsaenger, 1983: Cell divisions based on cell cycle – Added stochastic transitions – Added cell density and nutrient dependence Two cell types: normal and fast growth Cell death (necrosis) based on cell cycle – Two continuous fields: Diffusing chemotactic field secreted by necrotic cells attracts cancer cells Diffusing nutrient field – 200x200 2D lattice – Results in layered tumor structure.

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(a) The tumor is cut in half and recovers. (b) Cell adhesion is lowered and tumor expands. (c) Necrosis rate is increased by 1000%, tumor survives.

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Prostate Tumorigenesis

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Paracrine Signaling Occurs when a cell or tissue produces a factor which acts upon an adjacent tissue.

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Mathematical modeling of epithelial-stromal interactions Modeling Goal How can we define epithelial and stromal cell rules that (1) are biologically motivated, (2) model correct proliferative behavior, (3) model correct invasive behavior? Method: Hypothesize a set of simplified biologically motivated rules and use computer simulations to check if they are sufficient to yield expected cell behaviors. Warning: If successful, we identify rules that are sufficient to explain experimental observations. Discourse between model predictions and further experiments are needed to further validate/refine the model.

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Altered Stroma Normal Epithelium Proliferative Epithelium Invasive Epithelium HGF 1

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Normal Stroma Altered Stroma Normal Epithelium Proliferative Epithelium Invasive Epithelium HGF SDF 50% Altered Stroma Invasive Epithelium 2 1

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Hybrid Model Discrete, Cell-based Component Cells are modeled as discrete, individual entities in 2D space. Stromal and epithelial cells: 5 cell types. Stromal cells are ‘normal’ or ‘altered’. Epithelial cells are ‘normal’, ‘proliferative’ or ‘invasive’. Different stromal types secrete different morphogens. Epithelial cells progress sequentially from normal to proliferative to invasive if there are threshold levels of the required morphogen.

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Hybrid Model Continuous, PDE Component Morphogen production, diffusion and decay is modeled with the heat equation. Production rates k 1, k 2 (s -1 ) Diffusion rates D 1, D 2 Decay rates k d1, k d2

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Morphogen Concentrations

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Simulation Results PIN Invasion

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Phase Diagram: Transitions Depend Weakly on Production Levels

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‘Most Susceptible’ Epithelial Cells

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The Prisoner’s Dilemma Applied to the Interaction of Black Flies and Their Residents Maria Byrne – Math & Stats John McCreadie – Biology University of South Alabama MAA Local Meeting University of West Florida Friday, November 18 th

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Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) Game Theory: Analysis of decisions made by rational agents in a hypothetical situation with fixed rules (game) where each agent has options that affect themselves and the group (different payoffs). When will cooperative or altruistic behavior be the winning strategy? (Verses uncooperative or ‘cheating’ behavior.)

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Prisoner’s Dilemma (Axelrod, 1984) Prisoner’s Dilemma – Two Prisoners – Police do not have enough evidence for a conviction. Prisoner Options (Silence, Defection) – The prisoners can stay silent, in which case they will be sentenced for 1 month on a minor charge. – A prisoner can inform on the other prisoner (defect) in which case that prisoner goes free and the other serves a year in jail. – If both prisoners defect, they both serve 3 months in jail. Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)

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Payoff Matrix Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect 1 year jail “ not symmetric” A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect 1 year jail “ not symmetric” 6 months jail “fair” Prisoner A Prisoner B

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Payoff Matrix – From a Global Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent 2 months jail “fair” 1 year jail “ not symmetric” Defect 1 year jail “ not symmetric” 6 months jail “fair” Prisoner A Prisoner B

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Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B

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Payoff Matrix – From Prisoner’s Perspective Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950) SilentDefects Silent A: 1 month jail B: 1 month jail A: free B: 1 year jail Defect A: 1 year jail B: free A: 3 months jail B: 3 months jail Prisoner A Prisoner B Either way – should defect!

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Conundrum: Rational strategy is for both prisoners is to defect, however this leads to an outcome where the outcome is worse than if they had cooperated! Solution: Extended game play. The players gain information over time regarding whether the other is trustworthy, and have motive to cooperate so the other will too. Extended game play Evolutionary Timescales Prisoner’s Dilemma (Melvin Dresher and Merrill Flood, 1950)

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Prisoner’s Dilemma Applied to Interaction of Black Flies and their Resident Fungi How to characterize their relationship? – Black flies spend larval stage in moving water, where they may encounter fungi that take residence in their gut. – The fungi (Harpella Zygomycota, “trichomycetes”) require black fly larvae for reproduction, so for them the interaction is beneficial. – If the fungi benefit the black fly, the relationship is mutualistic; if the fungi harm the black fly the relationship is parasitic.

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Larval Black Fly Host

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Prisoner’s Dilemma Applied to Interaction of Black Flies and their Resident Fungi How to characterize their relationship? – Black flies spend larval stage in moving water, where they may encounter fungi that take residence in their gut. – The fungi (Harpella Zygomycota, “trichomycetes”) require black fly larvae for reproduction, so for them the interaction is beneficial. – If the fungi benefit the black fly, the relationship is mutualistic; if the fungi harm the black fly the relationship is parasitic.

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Trichomycetes group of cosmopolitan filamentous fungi obligate endosymbiotes in the guts of arthropods 300 species world wide 35 species in black flies trichospores (water column)

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Prisoner’s Dilemma Applied to Interaction of Black Flies and their Resident Fungi How to characterize their relationship? – Black flies spend larval stage in moving water, where they may encounter fungi that take residence in their gut. – The fungi (Harpella Zygomycota, “trichomycetes”) require black fly larvae for reproduction, so for them the interaction is beneficial. – If the fungi benefit the black fly, the relationship is mutualistic; if the fungi harm the black fly the relationship is parasitic.

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Spectrum of Host-Resident Interactions Parasitic Mutualistic

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Spectrum of Host-Resident Interactions Parasitic Mutualistic Commensalistic

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Is the relationship parasitic or mutualistic? Host Survival (%) Commensalistic: Fungi have no effect on host survival. McCreadie et al, 2005 FED

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Is the relationship parasitic or mutualistic? Host Survival (%) Commensalistic: Fungi have no effect on host survival. Host Survival (%) FED STARVED Mutualistic: Fungi improve survival rate of flies in starvation conditions McCreadie et al, 2005

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Is the relationship parasitic or mutualistic? Host Survival (%) Commensalistic: Fungi have no effect on host survival. Host Survival (%) FED STARVED Mutualistic: Fungi improve survival rate of flies in starvation conditions Parasitic: Some species invade larval germ tissue and ‘hijack’ the ovaries of the female adult fly. McCreadie et al, 2005

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Conceptual Framework The trichomycete-simuliid relationship changes with environmental factors. A model is developed to explore movement of the relationship along the P-C-M axis depending upon the number of fungi and host food supply. M P C

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Developing a Model of Cost-Benefit of Fungi on Simuliid Fitness Fitness: A measure of the reproductive success of an individual allele, organism or species, depending on the context. Formal definition: the fitness F at age x is sum of the products of the relative rate of survival to a certain age l x and the expected number of offspring at that age m x (Brommer 2000, Roff 2008)

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Black Fly Fitness l x is the survival rate – Trichomycetes increase this in starvation conditions. m x is the reproductive rate – Trichomycetes decrease this. There is a fitness trade-off, where trichomycetes exert a benefit for one term and a cost for the other.

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Let f 0 be the mean adult fitness (reproductive number) of an individual fly when resource E is not limiting, in the absence of parasitism. Model of Limiting Resource E Trichomycetes Benefit on Fitness Via Survival Term

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Let f 0 be the mean adult fitness (reproductive number) of an individual fly when resource E is not limiting, in the absence of parasitism. Model of Limiting Resource E With N Microbes

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Trichomycetes Cost on Fitness Via Fertility Term Let f 0 be the mean adult fitness (reproductive number) of an individual fly when resource E is not limiting, in the absence of parasitism. Model of Fertility Cost of N Microbes

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Net Result on Fitness Net effect on fitness (F(R E,N) compared to f 0 ) depends upon amount of available resource E and number of trichomycetes N.

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Net Result on Fitness

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In practice, commensalism would be a band because some minimal difference is needed before beneficial or parasitic effects would be detectable.

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Summary So Far We have developed a cost-benefit fitness model that shows quantitatively how a host- resident relationship can vary from parasitic to mutualistic depending upon environmental factors.

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Back to the Prisoner’s Dilemma Combine our cost-benefit model for fitness with an evolutionary model where different fly and trichomycetes types compete for survival. Will the flies and trichomycetes defect or cooperate over time? Species ‘choose’ options by increasing their frequency in the dynamic, time-evolved population model according to the immediate fitness of that option.

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Fly and Trichomycetes “Options” Trichomycetes do not have to be parasitic. – Only some species invade larval germ tissue of the fly. Larval black flies eject trichomycetes when they molt. – Will consider the hypothetical cases that some species of larvae may retain trichomycetes and some species may be resistant to residence.

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Fly and Trichomycetes “Options” Trichomycetes – Parasitic – Non-Parasitic Larval black flies – Tolerant – SemiTolerant (eject fungi during molt) – Intolerant

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Fly and Trichomycetes “Options” Payoff Matrix Resistant To Fungi Tolerant Until Molt Tolerant Of Fungi Doesn’t Hijack Ovaries Fly: No benefit. T: No available host, extinction. Fly: benefits in starvation conditions Hijacks Ovaries Fly: No benefit. T: No available host, extinction. T: benefits with extra reproduction Black Fly Trichomycetes

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Fly and Trichomycetes “Options” Payoff Matrix Resistant To Fungi Tolerant Until Molt Tolerant Of Fungi Doesn’t Hijack Ovaries Fly DefectsCooperative Hijacks Ovaries Fungi Defects Black Fly Trichomycetes

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Completely different types of models…

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Network Models

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“Scale Free Networks”

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Bioinformatics (Protein Sequence Analysis)

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Protein Folding Protein folding video

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