Drift-diffusion (DrDiff) framework determines kinetics and thermodynamics of two-state folding trajectory and tunes diffusion models

Abstract

The stochastic drift-diffusion (DrDiff) theory is an approach used to characterize the dynamical properties of simulation data. With new features in transition times analyses, the framework characterized the thermodynamic free-energy profile [F(Q)], the folding time (τf), and transition path time (τTP) by determining the coordinate-dependent drift-velocity [v(Q)] and diffusion [D(Q)] coefficients from trajectory time traces. In order to explore the DrDiff approach and to tune it with two other methods (Bayesian analysis and fep1D algorithm), a numerical integration of the Langevin equation with known D(Q) and F(Q) was performed and the inputted coefficients were recovered with success by the diffusion models. DrDiff was also applied to investigate the prion protein (PrP) kinetics and thermodynamics by analyzing folding/unfolding simulations. The protein structure-based model, the well-known Go¯-model, was employed in a coarse-grained Cα level to generate long constant-temperature time series. PrP was chosen due to recent experimental single-molecule studies in D and τTP that stressed the importance and the difficulty of probing these quantities and the rare transition state events related to prion misfolding and aggregation. The PrP thermodynamic double-well F(Q) profile, the “X” shape of τf(T), and the linear shape of τTP(T) were predicted with v(Q) and D(Q) obtained by the DrDiff algorithm. With the advance of single-molecule techniques, the DrDiff framework might be a useful ally for determining kinetic and thermodynamic properties by analyzing time observables of biomolecular systems. The code is freely available at https://github.com/ronaldolab/DrDiff.

Extension-Dependent Drift Velocity and Diffusion (DrDiff) Directly Reconstructs the Folding Free Energy Landscape of Atomic Force Microscopy Experiments

“…21 Local roughness has typically different energy barrier distributions in the energy landscape, resulting in different time scales for each position q, and thus, diffusion is also time-dependent. 22,23 The determination of a constant D or a position-dependent D(q) remains an experimental 10,15,24 and theoretical 16,17,23 challenge to overcome, despite its fundamental role to the biomolecule kinetics. Also, there are many discrepancies in the literature regarding the position dependency of D determined theoretically: some were found to vary a great deal 11,25 and some D(q) was roughly constant 8,26 all along the coordinate.…”

“…Comparisons of state-of-the-art single-molecule experiments also yielded conflicting results: a position-dependent D that was affected by experimental setups 15,27 or D roughly remained constant. 28 Nevertheless, there is an attempt to investigate the differences resulting from some theoretical/computational methodologies that characterize D(q), 16,17 Experimentally the measured extension (q) contains the sum of the probe, polymer linkers, and molecule (x) extensions, which are hard to decouple. 29 The measured q(t) has a "hidden" coordinate x that is the intrinsic biomolecular dynamics, and thus, the diffusion coefficient of the molecule and the pulling setup can differ.…”

“…Two measured trajectories with high resolution in time and space of different biomolecules were investigated by this framework: one is an HIV RNA hairpin, 32 and the another is three amino acids (aa) of the bacteriorhodopsin membrane protein (BR). 33 The reconstruction of F(q) might be possible by using the drift-diffusion (DrDiff) framework 16,17 as one simplified approach. The stochastic drift-diffusion theory 16,17,34 states that thermodynamic quantities can be reconstructed with the local position-dependent kinetic quantities D(q) and v(q) coefficients.…”

“…The method is more detailed in the Supporting Information. The free-energy profile is reconstructed from the equilibrium solution of the Fokker−Planck equation 16,17,34 by…”

“…32,36,42,43 Figure 2D shows the free-energy profile as a function of the measured extension pulling dynamics [F(q)] with the inverse Boltzmann method before (in cyan) and after (in red) deconvolution of the signal shown in Figure 2A. 32 Figure 2D also shows F reconstructed by the DrDiff methodology 16,17 by employing eq 1 (in green) with the characterized measured extension-dependent D (Figure 2B) and v (Figure 2C). In Figure 2D, the comparison is shown between the energy barrier and the two minima in F obtained by the inverse Boltzmann method, after the deconvolution of the trajectory signal, and the DrDiff algorithm, with a good agreement among them, although the well curvatures were slightly distinct.…”