Notes on Modelingwith

Discrete Particle Systems

Audi Byrne

July 28th, 2004

Kenworthy Lab Meeting

Deutsch et al.

Presentation Outline

I. Modeling Context in Biological Applications

Validation and Purpose of a ModelContinuous verses Discrete Models

II. Discrete Particle Systems

Detailed How-To: Cell Diffusion

III. Characteristics of Discrete Particle Systems

Self-OrganizationNon-Trivial Emergent BehaviorArtifacts

Modeling in Biological Applications

• Models = extreme simplifications

• Model validation:– capturing relevant behavior– new predictions are empirically confirmed

• Model value:– New understanding of known phenomena– New phenomena motivating further expts

Modeling Approaches

• Continuous Approaches (PDEs)

• Discrete Approaches (lattices)

Continuous Models

• E.g., “PDE”s

• Typically describe “fields” and long-range effects

• Large-scale events– Diffusion: Fick’s Law– Fluids: Navier-Stokes Equation

• Good models in bio for growth and population dynamics, biofilms.

Continuous Models

http://math.uc.edu/~srdjan/movie2.gif

Biological applications:

Cells/Molecules = density field.

http://www.eng.vt.edu/fluids/msc/gallery/gall.htm

Rotating Vortices

Discrete Models

• E.g., cellular automata.• Typically describe micro-scale events and short-range

interactions• “Local rules” define particle behavior• Space is discrete => space is a grid.• Time is discrete => “simulations” and “timesteps” • Good models when a small number of elements can

have a large, stochastic effect on entire system.

Discrete Particle Systems

Cells = Independent Agents

Cell behavior defined by arbitrary

local rules

Discrete Particle Systems

How-To Example: Diffusion

Example: Diffusion

1. Space is a matrix corresponding to a square lattice:

Example: Diffusion

2. Cells are “occupied nodes” where matrix values are non-zero.

Example: Diffusion

3. Different cells can be modeled as different matrix values.

Example: Diffusion

5. Diffusion of a cell is modeled by moving the cell in a random direction at each time-step.

Choose a random number between 0 and 4:

0 => rest

1 => right

2 => up

3 => left

4 => down

Example: Diffusion

4. Cells move by updating the lattice. Ex: Moving Right

Cell Diffusion

Slower diffusion is modeled by adding an increased

probability that the cell rests during a timestep.

Fast: P(resting)=0 Slow: P(resting)=.9

Modeling FRAP

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

1. Fluorescent molecules are added at random to a lattice (‘1’s added to a matrix)

2. Assumption: flourescence at a node occurs wherever there is a flourescent molecule at a node

3. Molecules are allowed to diffuse and total flourescence is a region A is measured

4. All molecules in A are photobleached (state changes from ‘1’ to ‘0’)

5. Remaining flourescent molecules will diffuse into A.

Modeling FRAP

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

Some Characteristics of Discrete Particle Systems

1. Self-Organization

2. Emergent Properties

3. Artifacts

Directed Pattern Formation

Wolpertian point of view:

Cells are organized by external signals; there is a pacemaker or director cell.

1. Self-Organization

Self-organization point of view: Cells are self-organized so there is no

need for a special director cell.

Self-Organization

Alber, Jiang, Kiskowski“A model for rippling and aggregation in myxobacteria”Physica D.

2. Emergent Behaviors

• There is no limit on the possible outcomes.

• There is no faster way to predict the outcome of a simulation than to run the simulation itself.

• Example: tail-following in myxobacteria

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

C-Signaling

Myxobacteria C-signal only when they transfer C-factor via their cell poles. Aggregation is controlled by interactions between their head and another cell tail.

Stream Formation

Orbit Formation

Orbit Formation

Stream and Orbit Dynamics

Lattice Artifacts

• Round off errors.

• Overly regular structures.

• Unrealistic periodic behavior over time: “bouncing checkerboard behavior”.

Defining Spatial and Temporal Scales

Spatial scale:

(1) Using minimum particle distance.Ex: SA is 5nm in diameter1 node = 5nm

(2) Using average particle distance.Ex: 100 limb bud cells are found along 1.4mm, though most of this space is extra-cellular matrix1 node = 1.4/100 mm

Defining Spatial and Temporal Scales

Temporal scale:

(1) Spatial scale combined with known diffusion rates often describe temporal scale.

(2) Comparing time-evolution of pattern in simulation with that of experiment.

(3) Intrinsic temporal scale: cell or molecule timer.

Modeling a Particle Timer

Timer: During flourescence, a flourophore is excited for L timesteps before releasing its energy.

(1) An unexcited flourophore is represented by a lattice state “1”.

(2) When excited, the florophore is assigned the state “L”.

(3) At every timestep, if the flourophore is excited (state>1), then the state is decreased by 1.

Modeling a Particle Timer

States 2,3,…L represent a timer for the excited state. If the experimental excitement time of a flourophore is 10 ns, then one simulation time-step corresponds to 10/L ns.