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

Overview: A Sampling of Some Recent Computational

Modeling Efforts

MA354 – Computational Math Modeling

Cellular Automata Models

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

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

• 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 ModelingErmentrout and Edelstein-Keshet, J. theor. Biol, 1993

Versatility of CA in Biology 1995-2005

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.

Application: Modeling FRAP

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

Ising Model: Ferromagnetism

Extension: Cellular Potts Model

Jiang and Pjesevic

Differential Adhesion

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

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.

Chemical peaks occur in a system with an autocatalytic component (an activator) and a faster-diffusing inhibiting component (an inhibitor).

[Meinhardt, 1995]

Result: periodic peaksin stripes or spotsdescribed by complexBessel equations.

Reaction-Diffusion Systems

• Bone formation occurs from the proximal to distal region.

http://zygote.swarthmore.edu/limb4.html

• As the limb grows, the number of precartilage condensations increases.

• Bone formation is mediated by fibronectin, which links cells together.

Limb Formation in vivo

http://zygote/

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

Activation:

Inhibition:

Up-regulation of Inhibitor:

Inhibitor Decay:

CA=Activator

concentration

CB=Inhibitor

concentration

nc = cell

concentration

Reaction (Occurs At Each Node Independently)

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 ps increases, the diffusion rate of the particle decreases.

Model Particles:

Cells, Activator, Inhibitor, and Fibronectin

Cell-Matrix Adhesion• Cells produce fibronectin at threshold levels of activator.

• Fibronectin does not diffuse.

• Cells stick to fibronectin with probability pf and un-stick

with probability 1-pf.

• Once stuck, cells do not diffuse during that time-step.

FibronectinFibronectinWith Stuck

Cells

Final FibronectinDistribution

FinalCellDistribution

Reaction diffusion establishes pattern of activator peaks.

Activator Peaks

Inhibitor Peaks

Fibronectin produced at activator peaks slows cell diffusion and cells cluster.

Preliminary Results

Limb Development

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

Phototaxis during the Slug Stage of Dictyostelium discoideum: a Model StudyMarée, Panfilov and Hogeweg Proceedings of the Royal Society of London. Series B. Biological sciences 266 (1999) 1351-1360

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.

(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.

Prostate Tumorigenesis

Paracrine Signaling

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

Interactions

Epithelium

Stroma

Mathematical modeling of epithelial-stromal interactions

Modeling GoalHow 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.

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF

1

NormalStroma

AlteredStroma

NormalEpithelium

ProliferativeEpithelium

InvasiveEpithelium

HGF SDF

50% Altered Stroma Invasive Epithelium

21

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.

Hybrid Model

Continuous, PDE Component

•Morphogen production, diffusion and decay is modeled with the heat equation.

•Production rates k1, k2 (s-1)•Diffusion rates D1, D2 •Decay rates kd1, kd2

12

1111

11 , mDmkyxk

t

md

n

iii

s

22

2221

22 , mDmkyxk

t

md

a

iii

s

Morphogen Concentrations

Simulation Results

PIN

Invasion

Phase Diagram: Transitions Depend Weakly on Production Levels

‘Most Susceptible’ Epithelial Cells

The Prisoner’s Dilemma Applied to the Interaction of Black Flies

and Their Residents

Maria Byrne – Math & StatsJohn McCreadie – Biology

University of South Alabama

MAA Local Meeting University of West FloridaFriday, November 18th

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.)

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.


• Payoff Matrix


Silent Defects

SilentA: 1 month jailB: 1 month jail

A: freeB: 1 year jail

DefectA: 1 year jail

B: freeA: 3 months jailB: 3 months jail

Prisoner A

Pris

oner

B

• Payoff Matrix – From a Global Perspective


Silent Defects





Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”A: free

B: 1 year jail



Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail

“fair”1 year jail

“ not symmetric”



Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail



Defect1 year jail

“ not symmetric”A: 3 months jailB: 3 months jail

Prisoner A

Pris

oner

B



Silent Defects

Silent2 months jail



Defect1 year jail

“ not symmetric”6 months jail

“fair”

Prisoner A

Pris

oner

B

• Payoff Matrix – From Prisoner’s Perspective


Silent Defects





Prisoner A

Pris

oner

B

• Payoff Matrix – From Prisoner’s Perspective


Silent Defects





Prisoner A

Pris

oner

B

Either way – should defect!

• 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 Applied toInteraction 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.

Larval Black Fly

Host

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)

Spectrum of Host-Resident Interactions

Parasitic

Mutualistic

Spectrum of Host-Resident Interactions

Parasitic

Mutualistic

Commensalisti

c

Is the relationship parasitic or mutualistic?H

ost S

urvi

val (

%)

20

40

60

80

100

Commensalistic: Fungi have no effect on host survival.

McCreadie et al, 2005

FED


ost S

urvi

val (

%)

20

40

60

80

100


Hos

t Sur

viva

l (%

)

20

40

60

80

100

FED STARVED

Mutualistic: Fungi improve survival rate of flies in starvation conditions



ost S

urvi

val (

%)

20

40

60

80

100


Hos

t Sur

viva

l (%

)

20

40

60

80

100

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.


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

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 lx and the expected number of offspring at that age mx

(Brommer 2000, Roff 2008)

xx mlxF )(

Black Fly Fitness

• lx is the survival rate– Trichomycetes increase this in starvation conditions.

• mx 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.

xx mlxF )(

• Let f0 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

bE

bE

ERa

RfRF 0

Trichomycetes Benefit on FitnessVia Survival Term

Trichomycetes Benefit on FitnessVia Survival Term


• Model of Limiting Resource EWith N Microbes

bE

bE

ENkRa

NkRfNRF

1

10,

Trichomycetes Cost on FitnessVia Fertility Term


• Model of Fertility Cost of N Microbes

d

d

Nkc

Nkf

2

20 1

Net Result on Fitness

• Net effect on fitness (F(RE,N) compared to f0) depends upon amount of available resource E and number of trichomycetes N.

tfertilitybenefitsurvivalfNRF E cos, 0

d

d

bE

bE

ENkc

Nk

NkRa

NkRfNRF

2

2

1

10 1,


tfertilitybenefitsurvivalfNRF E cos, 0


d

d

bE

bE

ENkc

Nk

NkRa

NkRfNRF

2

2

1

10 1,


d

d

bE

bE

ENkc

Nk

NkRa

NkRfNRF

2

2

1

10 1,

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

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.

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.

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.


• Trichomycetes– Parasitic– Non-Parasitic

• Larval black flies– Tolerant– SemiTolerant (eject fungi during molt)– Intolerant


• Payoff Matrix

ResistantTo Fungi

Tolerant Until Molt

TolerantOf Fungi

Doesn’t Hijack

Ovaries

Fly: No benefit.T: No available host, extinction.

Fly: benefits in starvation conditions

Fly: benefits in starvation conditions

HijacksOvaries

Fly: No benefit.T: No available host, extinction.

T: benefits with extra

reproduction

T: benefits with extra

reproduction

Black Fly

Tric

hom

ycet

es


• Payoff Matrix

ResistantTo Fungi

Tolerant Until Molt

TolerantOf Fungi

Doesn’t Hijack

OvariesFly Defects Cooperative

HijacksOvaries

Fungi Defects

Black Fly

Tric

hom

ycet

es

Completely different types of models…

Network Models

“Scale Free Networks”

Bioinformatics (Protein Sequence Analysis)

Protein Folding

Protein folding video

http://www.youtube.com/watch?v=swEc_sUVz5I

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

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