Non-Markovian modeling of protein folding

Abstract

We extract the folding free energy landscape and the time-dependent friction function, the two ingredients of the generalized Langevin equation (GLE), from explicit-water molecular dynamics (MD) simulations of the α-helix forming polypeptide alanine9 for a one-dimensional reaction coordinate based on the sum of the native H-bond distances. Folding and unfolding times from numerical integration of the GLE agree accurately with MD results, which demonstrate the robustness of our GLE-based non-Markovian model. In contrast, Markovian models do not accurately describe the peptide kinetics and in particular, cannot reproduce the folding and unfolding kinetics simultaneously, even if a spatially dependent friction profile is used. Analysis of the GLE demonstrates that memory effects in the friction significantly speed up peptide folding and unfolding kinetics, as predicted by the Grote–Hynes theory, and are the cause of anomalous diffusion in configuration space. Our methods are applicable to any reaction coordinate and in principle, also to experimental trajectories from single-molecule experiments. Our results demonstrate that a consistent description of protein-folding dynamics must account for memory friction effects.

Generalized Langevin Equation with a Non-Linear Potential of Mean Force and Non-Linear Memory Friction From a Hybrid Projection Scheme

“…( 48), and Γ app (t), defined within the approximate GLE in eq. ( 28) and obtained via a Volterra scheme [20,37], are pronounced and already suggest that non-linear friction effects, not captured by Γ app (t), are present. A closer look at the results in fig.…”

“…While for various applications the approximate GLE has been demonstrated to reproduce the full system dynamics very accurately [20,45], it is difficult to check for realistic systems whether the ad-hoc assumption Γ Z (t − s, r s , p s ) ≈ Γ app (t − s) is in fact valid. This is one motivation for our hybrid projection scheme, since it allows to derive all parameters of the exact GLE from trajectory data and thereby to access the validity of the approximate GLE explicitly.…”

“…When dealing with the kinetics of such systems, one is typically interested in the dynamics of a low-dimensional reaction coordinate, which is, however, generally influenced by the entire system [1]. Examples include the motion of a particle in a liquid [2][3][4][5][6][7], vibrational modes of a molecule in the gas phase or in a liquid [8][9][10][11], chemical or associative reactions between molecules [12][13][14][15][16] and protein folding [17][18][19][20]. To predict the dynamics of the reaction coordinate, one in principle has to solve the equation of motion of the underlying many-body system, which is typically analytically impossible and is only numerically possible for small systems and over short times.…”

“…The applications of the approximate GLE range from non-Markovian rate theory [41][42][43], over protein folding dynamics [17][18][19][20] to molecular diffusion and conformational dynamics [6,7,37,44]. Methods to derive memory functions from trajectory data for non-linear systems within the framework of the approximate GLE have been introduced and it was demonstrated that the resulting GLE correctly describes the multi-scale fractal dynamics of protein folding [20] and the vibrational spectra of molecules in non-linear bond-length and bondangle potentials [45]. Although widely used, the validity of the approximate GLE in the presence of a non-linear potential is subject to ongoing discussions [46,47].…”

Position-dependent memory kernel in generalized Langevin equations: theory and numerical estimation

“…As we will see, along the formal derivation, the kernel K is proven to satisfy some integrodifferential equations of Volterra type. As a natural continuation, our second contribution is the implementation of a numerical scheme based on these equations to estimate the kernel and parametrize GLE models from all-atom simulations, thus adaptating the inverse Volterra method [14,[25][26][27][28][29] to the position-dependent kernel settings.…”