Abstract
Recent advances in experimental measurements of transition path time distributions have raised intriguing theoretical questions. The present interpretation of the experimental data indicates a small value of the fitted transition path barrier height as compared to the barrier height of the unfolded to folded transition. Secondly, as shown in this paper, it is essential to analyse the experimental data using absorbing boundary conditions at the end points used to determine the transition paths. Such an analysis reveals long time tails that have thus far eluded quantitative theoretical interpretation. Is this due to uncertainty in the experimental data or does it call for a rethinking of the theoretical interpretation? A detailed study of the transition path time distribution using a diffusive model leads to the following conclusions. a. The present experimental data is not accurate enough to discern between functional forms of bell shaped free energy barriers. b. Long time tails indicate the possible existence of a "trap" in the transition path region. c. The "trap" may be considered as a well in the free energy surface. d. The long time tail is quite sensitive to the form of the trap so that future measurements of the long time tail as a function of the location of the end points of the transition path may make it possible to not only determine the well depth but also to distinguish between different functional forms for the free energy surface. e. Introduction of a well along the transition path leads to good fits with the experimental data provided that the transition path barrier height is ∼3kBT, substantially higher than the estimates of ∼1kBT based on bell shaped functions. The results presented here negate the need of introducing multi-dimensional effects, free energy barrier asymmetry, sub-diffusive memory kernels or systematic ruggedness to explain the experimentally measured data.
Highlights
Understanding the folding and unfolding dynamics of biomolecules of proteins and nucleic acids from the perspective of both theory[1,2,3,4,5] and experiment[6,7,8,9,10,11,12,13,14,15] has been an important aim in recent years, especially in view of the experimental results
We find that such a model accounts for the long time tails, it suggests that the transition path free energy barrier measured in the experiments is of the order of B3kBT, while the transition path well depth is similar
We find that the measured data is sufficient for extracting the well depth of this intermediate state, though one cannot distinguish the functional form of the free energy, both cusped and piece-wise quadratic forms give a good fit to the experimental data
Summary
The first such measurements of the detailed transition path time distribution[11] were explained in terms of a one dimensional diffusion equation in which the shape of the barrier was assumed to be parabolic. A few previous experiments have indicated the possibility of ‘‘traps’’ mediating between the unfolded and folded states.[36,37] The experimentally measured long time tails suggest that perhaps there exists a trap in the transition region which prevents a rapid crossing With this in mind, we consider the effect of introducing a well along the free energy surface separated by barriers from the folded and unfolded states. We consider the effect of introducing a well along the free energy surface separated by barriers from the folded and unfolded states We find that such a model accounts for the long time tails, it suggests that the transition path free energy barrier measured in the experiments is of the order of B3kBT, while the transition path well depth is similar. We conclude with a discussion of the results, suggesting that this present work indicates how experimental data may be further used to elucidate more information on the transition path dynamics, while at the same time, noting the limitations and some remaining open theoretical questions
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