Notes on Modeling with Discrete Particle Systems Audi Byrne July 28 th, 2004 Kenworthy Lab Meeting Deutsch et al.

Presentation Outline I.Modeling Context in Biological Applications Validation and Purpose of a Model Continuous verses Discrete Models II. Discrete Particle Systems Detailed How-To: Cell Diffusion III. Characteristics of Discrete Particle Systems Self-Organization Non-Trivial Emergent Behavior Artifacts

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 Biological applications: Cells/Molecules = density field. 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

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 diameter 1 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 matrix 1 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.