Flashcards

B5.1

Ben Tidswell Part B uni English 30 cards

Suppose we have a system of q chemical reactions with propensity functions α_i(t), i=1,...,q. State the Gillespie SSA

Cards in this deck (30)

1Suppose we have a system of q chemical reactions with propensity functions α_i(t), i=1,...,q. State the Gillespie SSA
2Draw a table giving the form of the basic chemical reactions in a space with volume ν up to third order and their propensity functions α(t)
3Suppose we have a system of q chemical reactions with propensity functions α_i(t), i=1,...,q. Describe how to find the chemical master equation
4Derive a PDE for p(x,t) for a simple one dimensional velocity-jump process with position X(t) and velocity V(t)=±s and turning frequency λ
5Give the computational definition of the SDE X(t+dt)=X(t)+f(X(t),t)dt+g(X(t),t)dW
6State the Fokker-Planck equation for the SDE X(t+dt)=X(t)+f(X(t),t)dt+g(X(t),t)dW
7State the probability flux for the Fokker-Planck equation for the SDE X(t+dt)=X(t)+f(X(t),t)dt+g(X(t),t)dW given X(s)=y for some s≤t
8Suppose X satisfies the SDE X(t+dt)=X(t)+f(X(t))dt+g(X(t))dW with 2 stable fixed points x_s1,x_s2 and an unstable fixed point x_u with x_s1
9State the Kolmogorov Backward equation for the SDE X(t+dt)=X(t)+f(X(t),t)dt+g(X(t),t)dW satisfying the Fokker-Planck equation given X(s)=y for some s≤t
10Let p_n(t) be the probability that there are n molecules of A at time t in a system. Define the probability generating function G(x,t) and state the mean M(t) and the variance V(t) in terms of G(x,t)
11State the quotient rule
12State the stationary distribution for p(x,t) satisfying the Kolmogorov backward equation with the usual f(x,t), g(x,t) notation
13State the Chapman-Kolmogorov equation and use it to derive the Fokker-Planck equation
14Under what coordinates does the Fokker-Planck equation reduce to a diffusion equation
15Formulate a chemical reaction model for a compartment-based diffusion model and explain how you would find the CME
16State a simple degradation SSA for a species A degrading at rate k
17Suppose a molecule satisfies 𝗫(0)=𝘆 and moves under diffusion with diffusion constant D. State a differential equation for τ(𝘆), the average time for the molecule to be absorbed given that it started at 𝘆
18State the Laplacian in polar and spherical coordinates
19State the del operator in polar and spherical coordinates
20Suppose we have the following system for the position, velocity, and acceleration of a diffusing particle: X(t+dt)=X(t)+V(t)dt V(t+dt)=V(t)+U(t)dt U(t+dt)=U(t)-(α_1V(t)+α_2U(t))dt+α_3dW Derive the generalized Langevin equation and generalized fluctuation-dissipation theorem
21State the value of the sum of the first n terms of a geometric series with first term a and constant ratio r
22State the equilibrium velocity autocorrelation function formulae
23Define the Laplace transform and draw a table of common ones
24State the generalized fluctuation-dissipation theorem
25Derive the formula for D=lim_(t->∞)d/dt
26If X~N(a_x,b_x), Y~N(a_y,b_y) are independent, what is the distribution of X+Y
27What equations can we derive for the compartment based diffusion model with K boxes of length h and diffusion constant D. Also state the chemical master equation in this situation
28State the probability mass function given the probability generating function
29State the f and d terms in the Fokker-Planck equation for a system of reactions with propensity functions α_i and changes in biomass v_i
30State the stationary distribution for a Fokker-Planck equation