A cell contains copies of a protein

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Probability, statistics and stochastic methods

Error, Chance and the Machine, Ch Random processes in continuous time

Problem

A cell contains N=5N=5 copies of a protein. New copies are made at a constant rate b=5 s−1b = 5\,\mathrm{s^{-1}} (birth), and each copy is degraded independently at rate k=1 s−1k = 1\,\mathrm{s^{-1}}, so the death rate is kNkN and the total exit rate from the state is RN=b+kNR_N = b + kN. Two uniform random numbers on [0,1)[0,1) are drawn for one step of the stochastic simulation algorithm: u1=0.3u_1 = 0.3 for the waiting time and u2=0.6u_2 = 0.6 for the choice of jump, with birth listed first in the running sum. ((a)) Find the waiting time tw=−ln⁡(1−u1)/RNt_{\rm w} = -\ln(1-u_1)/R_N to the next event. ((b)) Decide which event occurs and state the new molecule count NN.

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