Transition paths in molecules at finite temperature

Abstract

In the zero temperature limit, it is well known that in systems evolving via Brownian dynamics, the most likely transition path between reactant and product may be found as a minimizer of the Freidlin-Wentzell action functional. An analog for finite temperature transitions is given by the Onsager-Machlup functional. The purpose of this work is to investigate properties of OnsagerMachlup minimizers. We study transition paths for thermally activated molecules governed by the Langevin equation in the overdamped limit of Brownian dynamics. Using gradient descent in pathspace, we minimize the Onsager-Machlup functional for a range of model problems in one and two dimensions and then for some simple atomic models including Lennard-Jones seven-atom and 38-atom clusters, as well as for a model of vacancy diffusion in a planar crystal. Our results demonstrate interesting effects, which can occur at nonzero temperature, showing transition paths that could not be predicted on the basis of the zero temperature limit. However the results also demonstrate unphysical features associated with such Onsager-Machlup minimizers. As there is a growing literature that addresses transition path sampling by related techniques, these insights add a potentially useful perspective into the interpretation of this body of work.

A modified PATH algorithm rapidly generates transition states comparable to those found by other well established algorithms

“…In summary: We confirm and amplify observations summarized by Pinski and Stuart 33 that minimizing the Onsager-Machlup (OM) functional requires optimized estimates of both the time taken for the transition and the energy difference between initial and final states, and that as such paths can involve non-physical features, they must be treated with caution, and hence validated by other types of information, which here include comparing different algorithms and molecular systems.These difficulties notwithstanding, distinct algorithms including PATH 18 and ANMPathway 8 produce quite similar transition-state structures to that generated using the temperature-dependent string method, 31 which can be considered a “gold standard.”The transition state structure obtained by modifying the PATH algorithm to eliminate non-physical invariant portions of the trajectory coincides closely with the saddle point in the free energy surface simulated using Discrete Molecular Dynamics (DMD) 34,42

Cooperative repacking of aromatic side-chains is a common feature of transition state structures for domain rearrangements in three unrelated protein systems.

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“…We confirm and amplify observations summarized by Pinski and Stuart 33 that minimizing the Onsager-Machlup (OM) functional requires optimized estimates of both the time taken for the transition and the energy difference between initial and final states, and that as such paths can involve non-physical features, they must be treated with caution, and hence validated by other types of information, which here include comparing different algorithms and molecular systems.…”

“…1, it can be seen that if either force constant, k l or k r , the relative energy difference between the two wells, (Δ E ), or t f are changed, then the minimum action path that the system will take would be different. This means that for different values of Δ E and t f , and as noted previously, 33 there are different minimum action paths between the given equilibrium states, each defined by a different $\overline{t}$. As previously mentioned, since $\overline{t}$ uniquely identifies each path, when plotted against different values of Δ E and t f , it gives rise to the surface that we call the convergence surface (Fig.…”

“…The essential difference between the two approaches is that entropic changes play no role at zero temperature. Pinski and Stuart 33 showed that the effects of temperature are of little significance, provided there was no energy difference between initial and final states, but that significant energy differences between states introduced comparable changes in the transition states obtained at different temperatures when minimizing the Freidlin-Wentzell action functional. The relevance of free-energy differences between initial and final conformational states emphasizes the value of minimizing the Onsager-Machlup action functional as a useful approximation to the computationally intensive string method.…”

Linear response theory and optimal control for a molecular system under non-equilibrium conditions

“…Then, using Euler's method for (6), it follows that (9) is basically the likelihood ratio between the time-discrete path densities of (6b) and (6a). Another route to the same result is by using the Onsager-Machlup functional [15] for…”