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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.