Nonparametric variational optimization of reaction coordinates

State of the art realistic simulations of complex atomic processes commonly produce trajectories of large size, making the development of automated analysis tools very important. A popular approach aimed at extracting dynamical information consists of projecting these trajectories into optimally selected reaction coordinates or collective variables. For equilibrium dynamics between any two boundary states, the committor function also known as the folding probability in protein folding studies is often considered as the optimal coordinate. To determine it, one selects a functional form with many parameters and trains it on the trajectories using various criteria. A major problem with such an approach is that a poor initial choice of the functional form may lead to sub-optimal results. Here, we describe an approach which allows one to optimize the reaction coordinate without selecting its functional form and thus avoiding this source of error

Fep1d: A Script for the Analysis of Reaction Coordinates

The dynamics of complex systems with many degrees of freedom can be analyzed by projecting it onto one or few coordinates (collective variables). The dynamics is often described then as diffusion on a free energy landscape associated with the coordinates. Fep1d is a script for the analysis of such one-dimensional coordinates. The script allows one to construct conventional and cut-based free energy profiles, to assess the optimality of a reaction coordinate, to inspect whether the dynamics projected on the coordinate is diffusive, to transform (rescale) the reaction coordinate to more convenient ones, and to compute such quantities as the mean first passage time, the transition path times, the coordinate dependent diffusion coefficient, and so forth. Here, we describe the implemented functionality together with the underlying theoretical framework

Optimal Reaction Coordinates

The dynamic behavior of complex systems with many degrees of freedom is often analyzed by projection onto one or a few reaction coordinates. The dynamics is then described in a simple and intuitive way as diffusion on the associated free energy pro le. In order to use such a picture for a quantitative description of the dynamics one needs to select the coordinate in an optimal way so as to minimize non-Markovian effects due to the projection. For equilibrium dynamics between two boundary states (e.g., a reaction) the optimal coordinate is known as the committor or the pfold coordinate in protein folding studies. While the dynamics projected on the committor is not Markovian, many important quantities of the original multidimensional dynamics on an arbitrarily complex landscape can be computed exactly. Here we summarize the derivation of this result, discuss different approaches to determine and validate the committor coordinate and present three illustrative applications: protein folding, the game of chess, and patient recovery dynamics after kidney transplant

Binding and Docking Interactions of NO, CO and O2 in Heme Proteins as Probed by Density Functional Theory

Dynamics and reactivity in heme proteins include direct and indirect interactions of the ligands/substrates like CO, NO and O2 with the environment. Direct electrostatic interactions result from amino acid side chains in the inner cavities and/or metal coordination in the active site, whereas indirect interactions result by ligands in the same coordination sphere. Interactions play a crucial role in stabilizing transition states in catalysis or altering ligation chemistry. We have probed, by Density Functional Theory (DFT), the perturbation degree in the stretching vibrational frequencies of CO, NO and O2 molecules in the presence of electrostatic interactions or hydrogen bonds, under conditions simulating the inner cavities. Moreover, we have studied the vibrational characteristics of the heme bound form of the CO and NO ligands by altering the chemistry of the proximal to the heme ligand. CO, NO and O2 molecules are highly polarizable exerting vibrational shifts up to 80, 200 and 120 cm−1, respectively, compared to the non-interacting ligand. The importance of Density Functional Theory (DFT) methodology in the investigation of the heme-ligand-protein interactions is also addressed

Structural Interpretation of Metastable States in Myoglobin–NO

Nitric oxide binding and unbinding from myoglobin (Mb) is central to the function of the protein. By using reactive molecular dynamics (MD) simulations, the dynamics following NO dissociation were characterized in both time and space. Ligand rebinding can be described by two processes on the 10 ps and 100 ps timescale, which agrees with recent optical and X-ray absorption experiments. Explicitly including the iron out-of-plane (Fe-oop) coordinate is essential for a meaningful interpretation of the data. The proposed existence of an "Fe-oop/NO-bound" state is confirmed and assigned to NO at a distance of approximately 3 Å away from the iron atom. However, calculated XANES spectra suggest that it is diffcult to distinguish between NO close to the heme-Fe and positions further away in the primary site. Another elusive state, with Fe-ON coordination, was not observed experimentally because it is masked by the energetically more favorable but dissociative (4) A state in this region, which makes the Fe-ON local minimum unobservable in wild-type Mb. However, suitable active-site mutations may stabilize this state

High-resolution free energy landscape analysis of protein folding

The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics.</jats:p

High-Resolution Free-Energy Landscape Analysis of α-Helical Protein Folding: HP35 and Its Double Mutant.

The free-energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. Here, we construct the optimal coordinate and the associated free-energy profile for all-helical proteins HP35 and its norleucine (Nle/Nle) double mutant, based on realistic equilibrium folding simulations [Piana et al. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 17845]. From the obtained profiles, we directly determine such basic properties of folding dynamics as the configurations of the minima and transition states (TS), the formation of secondary structure and hydrophobic core during the folding process, the value of the pre-exponential factor and its relation to the transition path times, the relation between the autocorrelation times in TS and minima. We also present an investigation of the accuracy of the pre-exponential factor estimation based on the transition-path times. Four different estimations of the pre-exponential factor for both proteins give k 0 (-1) values of approximately a few tens of nanoseconds. Our analysis gives detailed information about folding of the proteins and can serve as a rigorous common language for extensive comparison between experiment and simulation

Single biomolecules -in silico, in vitro and in vivo High-resolution free energy landscape analysis of protein folding

Abstract The free energy landscape can provide a quantitative description of folding dynamics, if determined as a function of an optimally chosen reaction coordinate. The profile together with the optimal coordinate allows one to directly determine such basic properties of folding dynamics as the configurations of the minima and transition states, the heights of the barriers, the value of the pre-exponential factor and its relation to the transition path times. In the present study, we review the framework, in particular, the approach to determine such an optimal coordinate, and its application to the analysis of simulated protein folding dynamics