sb3-hw13-2016 - JHU 580.429 SB3HW13 Stochastic dynamics...

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JHU 580.429 SB3HW13: Stochastic dynamics, fluctuation-dissipation, stability. 1. Stochastic protein dynamics. Consider a stochastic system in which particular protein in a cell has copy number n . Starting in state n , the transition rate for n n + 1 is β and the transition rate for n n - 1 is α n . The steady-state probability that the system is in state n is P n . (a) Use detailed balance to obtain the steady-state relationship between P n and P n + 1 , and then derive a closed form expression for the probability distribution P n . (b) Provide a closed form expression for the generating function ˜ P ( φ ) = n = 0 e - φ n P n ; ex- press the mean h n i and the variance h n 2 i - h n i 2 in terms of derivatives of ˜ P ( φ ) ; and evaluate the mean and variance in terms of model parameters. (c) Suppose you want to run a stochastic simulation. Starting in state n , what states can be reached? Define the random variable τ as the waiting time before the first transition. What is the mean time h τ i for this first transition? What is the probability distribution p ( τ ) for the first transition?
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  • Fall '15
  • Dynamics, Stability, Stochastic, Stochastic Systems, Probability theory, Stability Analysis, Fluctuation, SBE3, Stochastic Dynamics, Fluctuation-dissipation, Dissipation, pn, symmetric fixed point

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