A colleague proposes a motor made from a bead diffusing in a static
PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Molecular machines, transport and bioenergetics · Molecular machines, transport and bioenergetics
Forces, Machines and Signals, Ch Enzymes and molecular motors
Problem
A colleague proposes a motor made from a bead diffusing in a static, periodic, asymmetric sawtooth potential of period L=8nm (a short, steep face and a long, shallow face), with the whole system held at a single uniform temperature and no fuel (no ATP or other free-energy source) supplied. Every well of the sawtooth has the same depth, and the barrier separating adjacent wells has height ΔU=6kBT measured from the bottom of each well. The colleague reasons: "Because the potential is asymmetric, hopping over the short, steep face must be easier than hopping back over the long, shallow face, so the barrier seen when hopping forward is lower than the barrier seen when hopping backward. Using attempt frequency ν=1×107s−1, I get a forward rate using an apparently lower barrier of 4kBT, kf=νe−4=1.8×105s−1, but a backward rate kb=νe−6=2.5×104s−1, giving a large steady drift."
((a)) Is the colleague's assignment of different barrier heights to the forward and backward hops correct?
((b)) State the correct long-time mean drift velocity of the bead, and explain the error using the idea of detailed balance in a periodic potential at one temperature.