1. Monte Carlo methods
10/20/11

2. Deterministic vs. stochastic modeling
ODE: same trajectory every time
Stochastic ODE: noise adds uncertainty to trajectory
Stochastic process: each trajectory is unique

3. Although Monte Carlo methods are today computational, they originated pre-computer age
“Monte Carlo” was code term for war-time work on neutron fission
Random number generators = roll of the dice at a casino

4. Very early application: calculation of integrals
Laplace, Kelvin, Buffon, Gosset cited examples of how accumulation of random trials could be used in calculations of p, time integrals of kinetic energy, statistical distributions, etc.
Consider:

5. Ratio of area of circle to square = p/4 = 0.785398
Suppose you set a round pan placed inside a square pan out in the rain. Over time, you would expect that the number of raindrops that fall inside the round pan versus inside the round+square pans to approach p/4.
This would only hold true if you collected a lot of raindrops though!
Large sampling # -> convergence to definite integral
Numerically, this can be demonstrated using a random number generator to determine x,y coordinates of raindrops

6. Movement of particle through medium
For each time step, there is movement with arbitrary directionality based upon diffusivity (can be on a lattice or free coordinates)
At new location, there is a probability of encountering another molecule (based upon density, size)
Based upon intent of simulation, there can be energy transfer upon collision, molecular complex formation, conservation of kinetic energy, etc.
New time step initiated, process repeated.

7. Example: chemical reaction
Puskar et al, Phys Review E, 2007.

8. A + B  C k+ k- The ODEs for the deterministic description:
The equations used for MC simulation:

10. Molecular crowding effects
No inert particles
Moving inert particles
Stationary inert particles
4000 particles:
3000 stationary (black)
1000 moving (gray)
“trapping effect” results, with higher reaction rates in heterogeneous regions
Both high number and excluded volume effects come into play

11. example: influence of low #’s on receptor:ligand binding

12. Motivation: most VEGF experiments are performed in the picomolar range
< 1 ligand/mm3 of fluid!
Are deterministic models valid at these levels?
What is the variability of receptor-ligand binding around the mean?
Could result in localized responses, population heterogeneity among cells

13. 3 types of simulations performed
1) deterministic PDE model
2) MC simulations in 3D space with receptors immobilized on 2D surface
analogous to PDE simulations
3) MC simulations w/o diffusion description, just stochasticity of kinetics
analogous to ODE simulations

14. Deterministic model
Receptors assumed to be uniform over the entire cell surface
Ligand assumed to be uniform over planes parallel to surface
Ligand gradients only in y-direction
Ligand can be released, internalization possible at some temperatures

15. MC simulations w/diffusion http://www.mcell.cnl.salk.edu/
Receptor placement on membrane lattice is randomized
Ligand placement is initialized
106 time steps run with binding probabilities = 0
Binding parameters set to new values (same as 1st model), repeat simulations run. Diffusion coefficient also same as 1st model
Fixed time step of 2 x 10-4 s used

16. MC simulations w/o diffusion (ODE-like)
Total # of species (R, V, RV) tracked
Reaction rates a function of # molecules
The time to next reaction is stochastic:
# of R, V, RV and p1 p2 are updated, repeat

17. Comparing PDE model to MC model w/diffusion
Changing ligand conc Changing membrane patch size

18. Comparing PDE model to MC model w/diffusion
Variability decreases with increased receptor density

19. Comparing PDE model to MC model w/o diffusion

20. Comparing MC model w/diffusion to MC model w/o diffusion

21. Overall, the two MC modeling approaches agreed quite well
The MC models also approximated the deterministic model for most cases (deviation occurred when smaller area to calculate fractional occupancy was averaged)
Variability does exist, could affect population averaged interpretation of VEGF responses

22. Other applications
MC modeling is very popular for neuron simulations

23. Glutamate release as a function of transporter density and time
Franks et al, Biophys J, 2002.

24. Some final thoughts on MC modeling
MC is a term used broadly, describes many different types of random number usage within models – be sure you know what is being done!
Randomization of parameters
Interaction probabilities
Diffusive movement
MC modeling is only as good as the random number generator being applied
This is less of an issue these days, but be aware of potential bias
Often, time steps are arbitrary in choice
Test features of MC model (e.g. # of molecules) to determine whether computational cost is necessary for appropriate description