A bacterium's flagellar motor is observed once every and is always found in one of…

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 walks and Markov chains

Problem

A bacterium's flagellar motor is observed once every Δt=10 ms\Delta t = 10\,\mathrm{ms} and is always found in one of three states: running (R), tumbling (T) or paused (P). From the observations: if running, it stays running with probability 0.70.7, switches to tumbling with probability 0.20.2, and pauses with probability 0.10.1; if tumbling, it switches to running with probability 0.60.6, stays tumbling with probability 0.30.3, and pauses with probability 0.10.1; if paused, it resumes running with probability 0.50.5, starts tumbling with probability 0.10.1, and stays paused with probability 0.40.4. The state occupied at successive observations forms a Markov chain with no memory of earlier states. ((a)) Write the row-stochastic transition matrix P\mathsf{P} in the order (R, T, P), and check that each row sums to 11. ((b)) The motor is observed running at time 00, so p0=(1,0,0)\mathbf{p}_0 = (1,0,0). Find p2=p0P2\mathbf{p}_2 = \mathbf{p}_0\mathsf{P}^2, the distribution over states after two observation intervals (20 ms20\,\mathrm{ms}).

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