On the improvement of free-energy calculation from steered molecular dynamics simulations using adaptive stochastic perturbation protocols

Computed Three-Dimensional Atomic Force Microscopy Images of Biopolymers Using the Jarzynski Equality

Three-dimensional atomic force microscopy (3D-AFM) has resolved three-dimensional distributions of solvent molecules at solid-liquid interfaces at the subnanometer scale. This method is now being extended to the imaging of biopolymer assemblies such as chromosomes or proteins in cells, with the expectation of being able to resolve their three-dimensional structures. Here, we have developed a computational method to simulate 3D-AFM images of biopolymers by using the Jarzynski equality. It is found that some parts of the fiber structure of biopolymers are indeed resolved in the 3D-AFM image. The dependency of 3D-AFM images on the vertical scanning velocity is investigated, and optimum scanning velocities are found. It is also clarified that forces in nonequilibrium processes are measured in 3D-AFM measurements when the dynamics of polymers are slower than the scanning of the probe.

Improving the Potential of Mean Force and Nonequilibrium Pulling Simulations by Simultaneous Alchemical Modifications

We present an approach combining alchemical modifications and physical-pathway methods to calculate absolute binding free energies. The employed physical-pathway method is either a stratified umbrella sampling to calculate a potential of mean force or nonequilibrium pulling. We devised two basic approaches: the simultaneous approach (S-approach), where, along the physical unbinding pathway, an alchemical transformation of ligand-protein interactions is installed and deinstalled, and the prior-plus-simultaneous approach (PPS-approach), where, prior to the physical-pathway simulation, an alchemical transformation of ligand-protein interactions is installed in the binding site and deinstalled during the physical-pathway simulation. Using a mutant of T4 lysozyme with a benzene ligand as an example, we show that installation and deinstallation of soft-core interactions concurrent with physical ligand unbinding (S-approach) allow successful potential of mean force calculations and nonequilibrium pulling simulations despite the problems posed by the occluded nature of the lysozyme binding pocket. Good agreement between the potential of the mean-force-based S-approach and double decoupling simulations as well as a remarkable efficiency and accuracy of the nonequilibrium-pulling-based S-approach is found. The latter turned out to be more compute-efficient than the potential of mean force calculation by approximately 70%. Furthermore, we illustrate the merits of reducing ligand-protein interactions prior to potential of mean force calculations using the murine double minute homologue protein MDM2 with a p53-derived peptide ligand (PPS-approach). Here, the problem of breaking strong interactions in the binding pocket is transferred to a prior alchemical transformation that reduces the free-energy barrier between the bound and unbound state in the potential of mean force. Besides, disentangling physical ligand displacement from the deinstallation of ligand-protein interactions was seen to allow a more uniform sampling of distance histograms in the umbrella sampling. In the future, physical ligand unbinding combined with simultaneous alchemical modifications may prove useful in the calculation of protein-protein binding free energies, where sampling problems posed by multiple, possibly sticky interactions and potential steric clashes can thus be reduced.

On the calculation of free energies over Hamiltonian and order parameters via perturbation and thermodynamic integration

In this work, complementary formulas are presented to compute free-energy differences via perturbation (FEP) methods and thermodynamic integration (TI). These formulas are derived by selecting only the most statistically significant data from the information extractable from the simulated points involved. On the one hand, commonly used FEP techniques based on overlap sampling leverage the full information contained in the overlapping macrostate probability distributions. On the other hand, conventional TI methods only use information on the first moments of those distributions, as embodied by the first derivatives of the free energy. Since the accuracy of simulation data degrades considerably for high-order moments (for FEP) or free-energy derivatives (for TI), it is proposed to consider, consistently for both methods, data up to second-order moments/derivatives. This provides a compromise between the limiting strategies embodied by common FEP and TI and leads to simple, optimized expressions to evaluate free-energy differences. The proposed formulas are validated with an analytically solvable harmonic Hamiltonian (for assessing systematic errors), an atomistic system (for computing the potential of mean force with coordinate-dependent order parameters), and a binary-component coarse-grained model (for tracing a solid-liquid phase diagram in an ensemble sampled through alchemical transformations). It is shown that the proposed FEP and TI formulas are straightforward to implement, perform similarly well, and allow robust estimation of free-energy differences even when the spacing of successive points does not guarantee them to have proper overlapping in phase space.

Comparison of umbrella sampling and steered molecular dynamics methods for computing free energy profiles of aromatic substrates through phospholipid bilayers

Understanding the permeation of molecules through lipid membranes is fundamental for predicting the cellular uptake of solutes and drug delivery mechanisms. In molecular simulations, the usual approach is to compute the free energy (FE) profile of a molecule across a model lipid bilayer, which can then be used to estimate the permeability of the molecule. Umbrella Sampling (US), which involves carrying out a series of biased simulations along a defined reaction coordinate (usually the bilayer normal direction), is a popular method for the computation of such FE profiles. However, US can be challenging to implement because the results are dependent on the strength of the biasing potential and the spacing of windows along the reaction coordinate, which, in practice, are usually optimized by an inefficient trial and error approach. The Steered Molecular Dynamics implementation of the Jarzynski Equality (JE-SMD) has been identified as an alternative to equilibrium sampling methods for measuring the FE change across a reaction coordinate. In the JE-SMD approach, equilibrium FE values are evaluated from the average of rapid non-equilibrium trajectories, thus avoiding the practical issues that come with US. Here, we use three different corrections of the JE-SMD method to calculate the FE change for the translocation of two aromatic substrates, phenylalanine and toluene, across a lipid bilayer and compare the accuracy and computational efficiency of these approaches to the results obtained using US. We show evidence that when computing the free energy profile, the JE-SMD approach suffers from insufficient sampling convergence of the bilayer environment and is dependent on the characteristic of the aromatic substrate itself. We deduce that, despite its drawbacks, US remains the more viable approach of the two for computing the FE profile.

Recognition of Potential COVID-19 Drug Treatments through the Study of Existing Protein-Drug and Protein-Protein Structures: An Analysis of Kinetically Active Residues

We report the results of our in silico study of approved drugs as potential treatments for COVID-19. The study is based on the analysis of normal modes of proteins. The drugs studied include chloroquine, ivermectin, remdesivir, sofosbuvir, boceprevir, and α-difluoromethylornithine (DMFO). We applied the tools we developed and standard tools used in the structural biology community. Our results indicate that small molecules selectively bind to stable, kinetically active residues and residues adjoining them on the surface of proteins and inside protein pockets, and that some prefer hydrophobic sites over other active sites. Our approach is not restricted to viruses and can facilitate rational drug design, as well as improve our understanding of molecular interactions, in general.

Constructing a multidimensional free energy surface like a spider weaving a web

Complete free energy surface in the collective variable space provides important information of the reaction mechanisms of the molecules. But, sufficient sampling in the collective variable space is not easy. The space expands quickly with the number of the collective variables. To solve the problem, many methods utilize artificial biasing potentials to flatten out the original free energy surface of the molecule in the simulation. Their performances are sensitive to the definitions of the biasing potentials. Fast-growing biasing potential accelerates the sampling speed but decreases the accuracy of the free energy result. Slow-growing biasing potential gives an optimized result but needs more simulation time. In this article, we propose an alternative method. It adds the biasing potential to a representative point of the molecule in the collective variable space to improve the conformational sampling. And the free energy surface is calculated from the free energy gradient in the constrained simulation, not given by the negative of the biasing potential as previous methods. So the presented method does not require the biasing potential to remove all the barriers and basins on the free energy surface exactly. Practical applications show that the method in this work is able to produce the accurate free energy surfaces for different molecules in a short time period. The free energy errors are small in the cases of various biasing potentials. © 2017 Wiley Periodicals, Inc.

Fast exploration of an optimal path on the multidimensional free energy surface

In a reaction, determination of an optimal path with a high reaction rate (or a low free energy barrier) is important for the study of the reaction mechanism. This is a complicated problem that involves lots of degrees of freedom. For simple models, one can build an initial path in the collective variable space by the interpolation method first and then update the whole path constantly in the optimization. However, such interpolation method could be risky in the high dimensional space for large molecules. On the path, steric clashes between neighboring atoms could cause extremely high energy barriers and thus fail the optimization. Moreover, performing simulations for all the snapshots on the path is also time-consuming. In this paper, we build and optimize the path by a growing method on the free energy surface. The method grows a path from the reactant and extends its length in the collective variable space step by step. The growing direction is determined by both the free energy gradient at the end of the path and the direction vector pointing at the product. With fewer snapshots on the path, this strategy can let the path avoid the high energy states in the growing process and save the precious simulation time at each iteration step. Applications show that the presented method is efficient enough to produce optimal paths on either the two-dimensional or the twelve-dimensional free energy surfaces of different small molecules.

Ligand diffusion in proteins via enhanced sampling in molecular dynamics

Computational simulations in biophysics describe the dynamics and functions of biological macromolecules at the atomic level. Among motions particularly important for life are the transport processes in heterogeneous media. The process of ligand diffusion inside proteins is an example of a complex rare event that can be modeled using molecular dynamics simulations. The study of physical interactions between a ligand and its biological target is of paramount importance for the design of novel drugs and enzymes. Unfortunately, the process of ligand diffusion is difficult to study experimentally. The need for identifying the ligand egress pathways and understanding how ligands migrate through protein tunnels has spurred the development of several methodological approaches to this problem. The complex topology of protein channels and the transient nature of the ligand passage pose difficulties in the modeling of the ligand entry/escape pathways by canonical molecular dynamics simulations. In this review, we report a methodology involving a reconstruction of the ligand diffusion reaction coordinates and the free-energy profiles along these reaction coordinates using enhanced sampling of conformational space. We illustrate the above methods on several ligand-protein systems, including cytochromes and G-protein-coupled receptors. The methods are general and may be adopted to other transport processes in living matter.

Unbinding pathway energy of glyphosate from the EPSPs enzyme binding site characterized by Steered Molecular Dynamics and Potential of Mean Force

The quantification of herbicides in the environment, like glyphosate, is extremely important to prevent contamination. Nanobiosensors stands out in the quantization process, because of the high selectivity, sensitivity and short response time of the method. In order to emulate the detection of glyphosate using a specific nanobiossensor through an Atomic Force Microscope (AFM), this work carried out Steered Molecular Dynamics simulations (SMD) in which the herbicide was unbinded from the active site of the enzyme 5- enolpyruvylshikimate 3 phosphate synthase (EPSPS) along three different directions.After the simulations, Potential of Mean Force calculations were carried, from a cumulant expansion of Jarzynski's equation to obtain the profile of free energy of interaction between the herbicide and the active site of the enzyme in the presence of shikimate-3 substrate phosphate (S3P). The set of values for external work, had a Gaussian distribution. The PMF values ranged according to the directions of the unbindong pahway of each simulation, displaying energy values of 10.7, 14.7 and 19.5KJmol-1. The results provide a theoretical support in order to assist the construction of a specific nanobiossensor to quantify the glyphosate herbicide.

Unraveling the Conformational Landscape of Ligand Binding to Glucose/Galactose-Binding Protein by Paramagnetic NMR and MD Simulations

Protein dynamics related to function can nowadays be structurally well characterized (i.e., instances obtained by high resolution structures), but they are still ill-defined energetically, and the energy landscapes are only accessible computationally. This is the case for glucose-galactose binding protein (GGBP), where the crystal structures of the apo and holo states provide structural information for the domain rearrangement upon ligand binding, while the time scale and the energetic determinants for such concerted dynamics have been so far elusive. Here, we use GGBP as a paradigm to define a functional conformational landscape, both structurally and energetically, by using an innovative combination of paramagnetic NMR experiments and MD simulations. Anisotropic NMR parameters induced by self-alignment of paramagnetic metal ions was used to characterize the ensemble of conformations adopted by the protein in solution while the rate of interconversion between conformations was elucidated by long molecular dynamics simulation on two states of GGBP, the closed-liganded (holo_cl) and open-unloaded (apo_op) states. Our results demonstrate that, in its apo state, the protein coexists between open-like (68%) and closed-like (32%) conformations, with an exchange rate around 25 ns. Despite such conformational heterogeneity, the presence of the ligand is the ultimate driving force to unbalance the equilibrium toward the holo_cl form, in a mechanism largely governed by a conformational selection mechanism.