Estimating transition path times and shapes from single-molecule photon trajectories: A simulation analysis

Abstract

In a two-state molecular system, transition paths comprise the portions of trajectories during which the system transits from one stable state to the other. Because of their low population, it is essentially impossible to obtain information on transition paths from experiments on a large sample of molecules. However, single-molecule experiments such as laser optical tweezers or Förster resonance energy transfer (FRET) spectroscopy have allowed transition-path durations to be estimated. Here, we use molecular simulations to test the methodology for obtaining information on transition paths in single-molecule FRET by generating photon trajectories from the distance trajectories obtained in the simulation. Encouragingly, we find that this maximum likelihood analysis yields transition-path times within a factor of 2–4 of the values estimated using a good coordinate for folding, but tends to systematically underestimate them. The underestimation can be attributed partly to the fact that the large changes in the end–end distance occur mostly early in a folding trajectory. However, even if the transfer efficiency is a good reaction coordinate for folding, the assumption that the transition-path shape is a step function still leads to an underestimation of the transition-path time as defined here. We find that allowing more flexibility in the form of the transition path model allows more accurate transition-path times to be extracted and points the way toward further improvements in methods for estimating transition-path time and transition-path shape.

Protein Folding Dynamics as Diffusion on a Free Energy Surface: Rate Equation Terms, Transition Paths, and Analysis of Single-Molecule Photon Trajectories

“…As the barrier decreases, however, the estimated and actual TPT diverge, with the Δlog L 2si‑2s maximum underestimating the TPT by up to 2.5-fold for the lowest barriers (Figure ). This result is consistent with the findings of Taumoefolau and Best on surfaces projected from coarse-grained simulations, which had barriers on the lower end of the range we explore here . Notably, the Δlog L 2si‑2s value at maximum follows the opposite trend, with the marginal 0.4 k B T barrier producing a maximum at ∼240 (statistical significance of 1–10 –240 ), whereas it is only ∼7 (significance of 1–10 –7 ) for the 8.1 k B T barrier (magenta scale in Figure ).…”

“…We then investigated the performance of various MLA-PT approaches for extracting transition path information. Our results show that the original analysis (Δlog L 2si‑2s ) underestimates the TPT, particularly for barriers smaller than 4 k B T , consistently with recent results from a similar study based on coarse-grained molecular simulations . The major factor behind such underestimation is that the two-state model shifts the two minima closer together in response to the population at the barrier, which makes the estimated transition path shorter than the real one.…”

“…We note that these issues arise from the use of discrete kinetic models. Hence, we conclude that an MLA procedure that directly uses a model of diffusion on a simple 1D free surface is preferable, as it has been proposed before by us and more recently by others . This is particularly true if the shape of the idealized 1D surface (barrier height and asymmetry) can be controlled with specific parameters, as with the Landau model we introduce here.…”

“…The lifetime of the virtual intermediate in the three-state model represents the TPT; hence, the comparison between the three-state likelihood as a function of the intermediate lifetime (1/2 k S ) and the likelihood of the two-state model (Δlog L 2si‑2s ) provides an estimate of whether the TPT is resolved and of its value (the maximum along the Δlog L 2si‑2s curve). This methodology has been recently tested by Taumoefolau and Best using coarse-grained molecular simulations of protein folding to explore fundamental issues such as the significance of using projected surfaces and E as reaction coordinate . Here we perform a similar test but using photon trajectories generated from the diffusive stochastic kinetics on 1D free energy surfaces with E as reaction coordinate.…”

“…That study showed that the method could extract the barrier height and diffusion coefficient when the shape of the free energy surface is predefined . Very recently, Taumoefolau and Best have used coarse-grained folding simulations to assess the suitability of the FRET efficiency ( E ) as reaction coordinate and of the two-/three-state MLA-PT procedure to estimate the TPT . In that case, the authors compared the results from the simulated photon trajectories with the TPT obtained by projecting the molecular trajectories as a function of the fraction of native contacts ( Q ), an order parameter that is generally accepted as a suitable folding reaction coordinate. ,, …”