Monte Carlo methods 10/20/11

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

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

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:

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

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.

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

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

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

example: influence of low #’s on receptor:ligand binding http://www.rosenthallab.com/gallery/list.php?pageNum_rsDisplay=14&totalRows_rsDisplay=72

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

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

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

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

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

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

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

Comparing PDE model to MC model w/o diffusion

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

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

Other applications MC modeling is very popular for neuron simulations

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

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