A molecular motor protein cycles through three conformational states,
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Biophysics & Medical Physics
Error, Chance and the Machine, Ch Random walks and Markov chains
Problem
A molecular motor protein cycles through three conformational states, 1→2→3→1, driven by ATP hydrolysis. Once every catalytic attempt the motor either advances to the next state in the cycle or fails and remains in its current state; there is no step backwards round the cycle. The advance probabilities per attempt are P12=0.30, P23=0.40, P31=0.20, with the remaining probability in each row a self-loop (a failed attempt).
((a)) Write down the 3×3 transition matrix and find the stationary distribution π.
((b)) Check detailed balance between each pair of communicating states. Where it fails, compute the net probability current Jij=πiPij−πjPji for each of the three links of the cycle.
((c)) Explain physically why detailed balance fails here, and state, with justification, whether the chain still converges to a single stationary distribution from every starting distribution.