1. 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.

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

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

4. Characteristics of Biological Problems

5. 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.

6. 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

7. 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

8. 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.

9. 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).

10. Characteristics of Cellular Automata

11. 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

12. 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

13. 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

14. 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:

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

16. Examples of Regular Lattices
1-D
Triangular
Square
Hexagonal

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

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

19. 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.

20. 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.

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

22. 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

23. 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

24. 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

25. 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)

26. LGCA

27. 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.

28. 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.

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

30. 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

31. Analytic Description: The Heat Equation
Describes the mean concentration of particles or the probability density function:
Molecules are treated as a statistical ensemble.

32. 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

33. Diffusion Simulation

34. Gaussian Blur

35. 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]

36. 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]

37. 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.

38. Fitzhugh Nagumo Simulation

39. 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:

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

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

42. POTTS model

43. 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.

44. Jiang and Pjesevic

45. 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)

46. 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

47. 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

48. 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

49. Simple example: cell sorting
Random Initial Conditions

50. Simple example: cell sorting
Simulation over 100 Time Steps

51. 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

52. 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

53. 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

54. Chemical Diffusion in Tumor

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

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

57. Waste Distribution

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

59. 8.5 Days

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

61. Growth in Number of Cells versus O2 and Nutrient

62. Funding slide:
NIH
Mitchell Cancer Institute
Thanks very much!