BIOL/EBME/MATH/PHOL/SYBB 319/419 ∼∼∼ Biological Stochastic Processes – Spring 2020 ∼∼∼ Homework # 5 ∼∼∼ Due before the end of term (Monday April 27, 5 pm). You may work these exercises by hand, or with assistance from symbolic solvers such as Maple or Math- ematica, or numerical solvers such as Matlab or XPP/AUTO (except as indicated otherwise in individual exercises). Please indicate the methods employed. I encourage you to discuss these problems with other students, but you should prepare your own solutions in the end. Questions marked with an asterisk (*) are optional except for students in MATH 419. Please turn in all written components together on paper; please combine all electronic components (figures) into a single PDF file with each figure clearly marked to indicate which problem it corresponds to, and upload these to Canvas. Please upload any executable code to canvas as well. Random Velocity Process Random velocity processes provide models for the random movement of cells during chemotaxis, the growth and shrinkage of tubulin, actin, and other monomers, and the random gating of ion channels. 1. (20 pts) Question from RVP lecture: for what value of c is V = 0? Interpret this “critical concentra- tion”. Run simulations illustrating the behavior above and below the critical concentration. 2. *(10 pts) Compare and contrast the critical concentration in the preceding model to the critical concen- tration for the growth of fTSZ (filamentous temperature sensitive protein Z) in Miraldi, Thomas, and Romberg.“Allosteric models for cooperative polymerization of linear polymers.” Biophysical journal 95.5 (2008): 2470-2486. Reaction-Diffusion Equations Questions related to material on reaction-diffusion equations. 3. (20 pts) The following two calculations establish a correspondence between a master equation (discrete- space, continuous-time) representation of diffusion and the standard diffusion equation (continuous- space, continuous-time). (a) (10 pts) A single particle diffuses along an infinite one-dimensional domain −∞ < X(t) < ∞. Suppose at time t0 the particle has probability density ρ(x, t0) satisfying ρ(x, t0) ≥ 0 and∫∞ −∞ ρ(x, t0) dx = 1, and that for t ≥ t0 the density satisfies the diffusion equation, ∂ρ(x, t) ∂t = D ∂2ρ(x, t) ∂x2 . (1) Assume also that for any integer p, and any t ≥ t0, we have lim |x|→∞ ∣∣ρ(x, t)xp∣∣ = 0, and (2) lim |x|→∞ ∣∣∣∣∂ρ(x, t)∂x xp ∣∣∣∣ = 0. (3) Recall that the mean of X(t) is X(t) = ∫ ∞ −∞ ρ(x, t)x dx 1 and the variance of X(t) is σ2X(t) = ∫ ∞ −∞ ρ(x, t)x2 dx− [X(t)]2 . Show that the rate of change of the variance satisfies dσ2X(t) dt = 2D. (4) Hint: use integration by parts. (b) (10 pts) Suppose a particle moves by making discrete jumps of ±∆x at exponentially distributed intervals, so that X → X + ∆x, at rate k X → X −∆x, at rate k. (5) Suppose we start a single particle at x = 0 at time t = 0. Let σ2X be the variance in the position X(t). Set up the discrete-state, continuous-time master equation for the particle’s position, and show that at t = 0, dσ2X(t) dt = 2k(∆x)2. (6) Together, parts (a-b) show that choosing k = D/(∆x)2 matches the 2nd moment of the diffusion process. 4. (40 pts) Here we study the location at which a reaction A + B → C occurs. In the matlab simula- tion code rdme1D a plus b to c.m1 we solve numerically for the probability distribution of a pair of molecules (one of type A, one of type B) released at two ends of a 1D interval, that react with rate k/∆x when they are in the same subvolume of size ∆x. Otherwise, they diffuse independently with diffusion constant D. The following questions ask you to discuss the behavior of the system under two limits: fast-diffusion and slow reaction (D k) and vice-versa (D k). The computer simulations should give you intuition, but your answers should go beyond the simulations if possible. (a) Describe the conditional distribution of A and B, in the limit that k is very fast and D is slow. (Here the “conditional distribution” means the joint probability distribution for the location of A and B, given that they have not yet reacted.) (b) Describe the long-time distribution of C, in the limit that k is very fast and D is slow. In the simulation, C does not diffuse, so the location distribution of C at the end of the simulation is the same as the distribution of the location at which the reaction occurred. (c) Describe the conditional distribution of A and B, in the limit that k is slow and D is fast. (d) Describe the long-time distribution of C, in the limit that k is slow and D is fast. 5. (20 pts) Get a version of the Lotka-Volterra equations working in 1D and 2D, using your choice of simulation software.2 Document your efforts and results. Extra credit if you can numerically demonstrate the presence of stochastic traveling wave solutions. 1Available on canvas. Similar situations are available in the URDME distribution (linked from canvas – see URDME’s “annihilation” example) and (probably) also StochSS. 2This should be possible in URDME, in StochSS, in MCell, and also in R-code examples accompanying Wilkinson’s textbook. 2
