Maximum Flux Transition Paths of Conformational Change.

Given two metastable states A and B of a biomolecular system, the problem is to calculate the likely paths of the transition from A to B. Such a calculation is more informative and more manageable if done for a reduced set of collective variables chosen so that paths cluster in collective variable space. The computational task becomes that of computing the "center" of such a cluster. A good way to define the center employs the concept of a committor, whose value at a point in collective variable space is the probability that a trajectory at that point will reach B before A. The committor "foliates" the transition region into a set of isocommittors. The maximum flux transition path is defined as a path that crosses each isocommittor at a point which (locally) has the highest crossing rate of distinct reactive trajectories. This path is based on the same principle as the minimum resistance path of Berkowitz et al (1983), but it has two advantages: (i) the path is invariant with respect to a change of coordinates in collective variable space and (ii) the differential equations that define the path are simpler. It is argued that such a path is nearer to an ideal path than others that have been proposed with the possible exception of the finite-temperature string method path. To make the calculation tractable, three approximations are introduced, yielding a path that is the solution of a nonsingular two-point boundary-value problem. For such a problem, one can construct a simple and robust algorithm. One such algorithm and its performance is discussed.