Sergei V. Krivov

Free energy disconnectivity graphs: Application to peptide models

Disconnectivity graphs are widely used for understanding the multidimensional potential energy surfaces (PES) of complex systems. Since entropic contributions to the free energy can be important, particularly for polypeptide chains and other polymers, conclusions concerning the equilibrium properties and kinetics of the system based on potential energy disconnectivity graphs (PE DG) can be misleading. We present an approach for constructing free energy surfaces (FES) and free energy disconnectivity graphs (FE DG) and give examples of their applications to peptides. They show that the FES and FE DG can differ significantly from the PES and PE DG.

Diffusive reaction dynamics on invariant free energy profiles

A fundamental problem in the analysis of protein folding and other complex reactions in which the entropy plays an important role is the determination of the activation free energy from experimental measurements or computer simulations. This article shows how to combine minimum-cut-based free energy profiles (FC), obtained from equilibrium molecular dynamics simulations, with conventional histogram-based free energy profiles (FH) to extract the coordinate-dependent diffusion coefficient on the FC (i.e., the method determines free energies and a diffusive preexponential factor along an appropriate reaction coordinate). The FC, in contrast to the FH, is shown to be invariant with respect to arbitrary transformations of the reaction coordinate, which makes possible partition of configuration space into basins in an invariant way. A “natural coordinate,” for which FH and FC differ by a multiplicative constant (constant diffusion coefficient), is introduced. The approach is illustrated by a model one-dimensional system, the alanine dipeptide, and the folding reaction of a double β-hairpin miniprotein. It is shown how the results can be used to test whether the putative reaction coordinate is a good reaction coordinate.

One-Dimensional Barrier-Preserving Free-Energy Projections of β-sheet Miniprotein: New Insights into the Folding Process

The conformational space of a 20-residue three-stranded antiparallel beta-sheet peptide (double hairpin) was sampled by equilibrium folding/unfolding molecular dynamics simulations for a total of 20 micros. The resulting one-dimensional free-energy profiles (FEPs) provide a detailed description of the free-energy basins and barriers for the folding reaction. The similarity of the FEPs obtained using the probability of folding before unfolding (pfold) or the mean first passage time supports the robustness of the procedure. The folded state and the most populated free-energy basins in the denatured state are described by the one-dimensional FEPs, which avoid the overlap of states present in the usual one- or two-dimensional projections. Within the denatured state, a basin with fluctuating helical conformations and a heterogeneous entropic state are populated near the melting temperature at about 11% and 33%, respectively. Folding pathways from the helical basin or enthalpic traps (with only one of the two hairpins formed) reach the native state through the entropic state, which is on-pathway and is separated by a low barrier from the folded state. A simplified equilibrium kinetic network based on the FEPs shows the complexity of the folding reaction and indicates, as augmented by additional analyses, that the basins in the denatured state are connected primarily by the native state. The overall folding kinetics shows single-exponential behavior because barriers between the non-native basins and the folded state have similar heights.

The Free Energy Landscape Analysis of Protein (FIP35) Folding Dynamics

A fundamental problem in the analysis of protein folding and other complex reactions is the determination of the reaction free energy landscape. The current experimental techniques lack the necessary spatial and temporal resolution to construct such landscapes. The properties of the landscapes can be probed only indirectly. Simulation, assuming that it reproduces the experimental dynamics, can provide the necessary spatial and temporal resolution. It is, arguably, the only way for direct rigorous construction of the quantitatively accurate free energy landscapes. Here, such landscape is constructed from the equilibrium folding simulation of FIP35 protein reported by Shaw et al. Science 2010, 330, 341-346. For the dynamics to be accurately described as diffusion on the free energy landscape, the choice of reaction coordinates is crucial. The reaction coordinate used here is such that the dynamics projected on it is diffusive, so the description is consistent and accurate. The obtained landscape suggests an alternative interpretation of the simulation, markedly different from that of Shaw et al. In particular, FIP35 is not an incipient downhill folder, it folds via a populated on-pathway intermediate separated by high free energy barriers; the high free energy barriers rather than landscape roughness are a major determinant of the rates for conformational transitions; the preexponential factor of folding kinetics 1/k(0) ∼ 10 ns rather than 1 μs.

Importance of Metastable States in the Free Energy Landscapes of Polypeptide Chains

We show that the interplay between excluded volume effects, hydrophobicity, and hydrogen bonding in a tubelike representation of a polypeptide chain gives rise to free energy landscapes that, in addition to a clear global minimum, are characterized by the general presence of a small number of metastable minima, which correspond to common structural motifs observed in proteins. The complexity of the landscape increases only moderately with the length of the chain. Analysis of the temperature dependence of these landscapes reveals that the stability of specific metastable states is maximal at a temperature close to the midpoint of folding. These mestastable states are therefore likely to be of particular significance in determining the generic tendency of proteins to aggregate into potentially pathogenic agents.

On Reaction Coordinate Optimality.

The following question is addressed: how to establish that a constructed reaction coordinate is optimal, i.e., that it provides an accurate description of dynamics. It is shown that the reaction coordinate is optimal if its cut free energy profile, determined using length-weighted transitions, is constant, i.e., it is position and sampling interval independent. The observation leads to a number of interesting results. In particular, the equilibrium flux between two boundary states can be computed exactly as diffusion on a free energy profile associated with the coordinate. The mean square displacement, for the trajectory projected onto the coordinate, grows linear with time. That for the same trajectory projected onto a suboptimal coordinate grows slower than linear with time. The results are illustrated on a number of model systems, Sierpinski gasket, FIP35 protein, and beta3s peptide.

Numerical construction of the p(fold) (committor) reaction coordinate for a Markov process.

To simplify the description of a complex multidimensional dynamical process, one often projects it onto a single reaction coordinate. In protein folding studies, the folding probability p(fold) is an optimal reaction coordinate which preserves many important properties of the dynamics. The construction of the coordinate is difficult. Here, an efficient numerical approach to construct the p(fold) reaction coordinate for a Markov process (satisfying the detailed balance) is described. The coordinate is obtained by optimizing parameters of a chosen functional form to make a generalized cut-based free energy profile the highest. The approach is illustrated by constructing the p(fold) reaction coordinate for the equilibrium folding simulation of FIP35 protein reported by Shaw et al. (Science 2010, 330, 341-346).

Is Protein Folding Sub-Diffusive?

Protein folding dynamics is often described as diffusion on a free energy surface considered as a function of one or few reaction coordinates. However, a growing number of experiments and models show that, when projected onto a reaction coordinate, protein dynamics is sub-diffusive. This raises the question as to whether the conventionally used diffusive description of the dynamics is adequate. Here, we numerically construct the optimum reaction coordinate for a long equilibrium folding trajectory of a Go model of a -repressor protein. The trajectory projected onto this coordinate exhibits diffusive dynamics, while the dynamics of the same trajectory projected onto a sub-optimal reaction coordinate is sub-diffusive. We show that the higher the (cut-based) free energy profile for the putative reaction coordinate, the more diffusive the dynamics become when projected on this coordinate. The results suggest that whether the projected dynamics is diffusive or sub-diffusive depends on the chosen reaction coordinate. Protein folding can be described as diffusion on the free energy surface as function of the optimum reaction coordinate. And conversely, the conventional reaction coordinates, even though they might be based on physical intuition, are often sub-optimal and, hence, show sub-diffusive dynamics.

Analysis of the Free-Energy Surface of Proteins from Reversible Folding Simulations

Computer generated trajectories can, in principle, reveal the folding pathways of a protein at atomic resolution and possibly suggest general and simple rules for predicting the folded structure of a given sequence. While such reversible folding trajectories can only be determined ab initio using all-atom transferable force-fields for a few small proteins, they can be determined for a large number of proteins using coarse-grained and structure-based force-fields, in which a known folded structure is by construction the absolute energy and free-energy minimum. Here we use a model of the fast folding helical λ-repressor protein to generate trajectories in which native and non-native states are in equilibrium and transitions are accurately sampled. Yet, representation of the free-energy surface, which underlies the thermodynamic and dynamic properties of the protein model, from such a trajectory remains a challenge. Projections over one or a small number of arbitrarily chosen progress variables often hide the most important features of such surfaces. The results unequivocally show that an unprojected representation of the free-energy surface provides important and unbiased information and allows a simple and meaningful description of many-dimensional, heterogeneous trajectories, providing new insight into the possible mechanisms of fast-folding proteins.

Free-Energy Landscapes of Proteins in the Presence and Absence of Force

The equilibrium properties of the fourth immunoglobulin domain of filamin from Dictyostelium discoideum (ddFLN4) in the absence and presence of a small force (0-6 pN) pulling the termini apart is characterized through atomistic numerical simulation. The equilibrium free-energy landscape of ddFLN4 is found to change in a complex fashion that cannot be described in terms of one-dimensional projections as usually done in the interpretation of mechanical (un)folding experiments. Nonequilibrium unfolding simulations reveal that the major unfolding intermediate corresponds to a marginally populated state at equilibrium that only appears when a force larger than 4 pN is applied. Finally, we show that if the free-energy difference between states is taken to be linear in the applied force, the proportionality coefficient is not the difference in the end-to-end distance between pair of states as generally assumed even though the data can be reasonably fitted. The present results suggest that mechanical unfolding experiments may reveal states that are not accessible in the absence of force. Thus, special care should be taken when trying to interpret both equilibrium and nonequilibrium mechanical (un)folding experiments in light of the (un)folding properties in the absence of a force.