Thermodynamics of Gradient Driven Transport: Application to Single-Particle Tracking

Single-particle tracking techniques allow measurements of the trajectories of individual particles (and sometimes single molecules) as they move. Here, we show, in accord with an equality derived by Bier et a], (Phys. Rev. E 1999, 59 6422-6432), that even in the presence of gradients of pressure, temperature, electric field strength, and so forth, the ratio of the probabilities for forward and backward trajectories between any two points is given by the exponential of the difference between the particle's free energy at the start and end points of the trajectory, that is, P(x(a) -> x(b), Delta t) = P(x(b) -> x(a), Delta t) exp(-Delta G/k(B)T), where Delta G = G(x(b)) - G(x(a)). Thus, experimental approaches based on single-particle tracking can be used to map the free-energy landscape for transport of a particle without reference to whether the overall system is, or is not, in thermodynamic equilibrium.