Free Energy Landscapes, Diffusion Coefficients, and Kinetic Rates from Transition Paths

Abstract

We address the problem of constructing accurate mathematical models of the dynamics of complex systems projected on a collective variable. To this aim we introduce a conceptually simple yet effective algorithm for estimating the parameters of Langevin and Fokker–Planck equations from a set of short, possibly out-of-equilibrium molecular dynamics trajectories, obtained for instance from transition path sampling or as relaxation from high free-energy configurations. The approach maximizes the model likelihood based on any explicit expression of the short-time propagator, hence it can be applied to different evolution equations. We demonstrate the numerical efficiency and robustness of the algorithm on model systems, and we apply it to reconstruct the projected dynamics of pairs of C60 and C240 fullerene molecules in explicit water. Our methodology allows reconstructing the accurate thermodynamics and kinetics of activated processes, namely free energy landscapes, diffusion coefficients, and kinetic rates. Compared to existing enhanced sampling methods, we directly exploit short unbiased trajectories at a competitive computational cost.

Optimal Reaction Coordinates and Kinetic Rates from the Projected Dynamics of Transition Paths

“…The phase-space information used for RC optimization is represented by MD trajectories spanning transitions between the reactant and product regions of configuration space. We adopt TPS as a source of input data for several reasons: ,, (i) ergodic trajectories (spontaneously spanning the transitions) are unfeasible in the presence of free-energy barriers ≫ k B T , while TPS has an affordable cost for many systems; (ii) TPS trajectories are free from biasing forces and, at statistical convergence, faithfully reproduce ergodic trajectories; (iii) recent numerical evidence indicate that ∼100 TPS-like MD trajectories projected on a suitable CV are sufficient to reconstruct accurate free-energies and rates by means of overdamped Langevin models …”

“…A newly proposed RC is accepted or rejected based on a Metropolis criterion aimed at minimizing the kinetic rate estimated from a maximum-likelihood Langevin model. The latter is optimized following the method in ref . For a sufficiently small time interval τ, the transition probability density p (propagator) between points q and q′ in the CV-space can be approximated as p ( q ′ , t + τ | q , t ) ≈ 1 2 π μ e − false( q ′ − q − ϕ false) 2 / 2 μ ϕ = a τ + 1 2 ( a a ′ + D a ″ ) τ 2 , goodbreak0em1em⁣ μ = 2 D τ + ( a D ′ + 2 a ′ D + D D ″ ) τ 2 where the prime indicates d / dq , a = − βDF′ + D′ is the drift in eq , and the approximation includes terms up to order τ 2 .…”

“…The method we present exploits these principles: putative CVs are generated in a Monte Carlo scheme, and the kinetic rates of associated effective low-dimensional models are estimated and systematically minimized, leading to optimal RCs and accurate kinetic predictions. An important ingredient of our approach is the automatic construction of stochastic (Langevin) models of CV evolution: for this purpose, we adopt the maximum likelihood algorithm proposed in ref , based on short TPS-like MD trajectories. We remark that such a task is nontrivial from a numerical viewpoint, and alternative approaches exist. − …”

Formation of membrane invaginations by curvature-inducing peripheral proteins: free energy profiles, kinetics, and membrane-mediated effects

“…( 3) to obtain free energies. Kinetics: Given that the free energy profile as a function of the reaction coordinate λ is available, one simple approach to estimating the kinetics of spontaneous processes is to consider a diffusion process on this onedimensional energy landscape [27]. To estimate the timescales of slow processes in such a description, we take two different approaches.…”

Coordinate-Dependent Drift-Diffusion Reveals the Kinetic Intermediate Traps of Top7-Based Proteins

“…The resulting projection will suffer from local conformational landscapes of different local environments from other projected coordinates . Consequently, the local scape time that is represented by diffusion is commonly position-dependent ( D ( Q )). − Folding and transition path times can take D ( Q ) into account in the Kramers’ original theory to deliver the underlying kinetics with improved accuracy. − In addition, the coordinate-dependent drift-velocity coefficient ( v ( Q )), which is proportional to the gradient of the effective diffusive potential of the system, can be used to recover the one-dimensional free energy landscape ( F ( Q )). , An outcome of the diffusion theory is that D ( Q ) incorporates the multiple diffusions over local energetic barriers or microstates of F ( Q ). If present, the local kinetic traps might be uncovered by the position-dependent D and v .…”