Investigations of α-helix↔β-sheet transition pathways in a miniprotein using the finite-temperature string method

CHARMM at 45: Enhancements in Accessibility, Functionality, and Speed

Since its inception nearly a half century ago, CHARMM has been playing a central role in computational biochemistry and biophysics. Commensurate with the developments in experimental research and advances in computer hardware, the range of methods and applicability of CHARMM have also grown. This review summarizes major developments that occurred after 2009 when the last review of CHARMM was published. They include the following: new faster simulation engines, accessible user interfaces for convenient workflows, and a vast array of simulation and analysis methods that encompass quantum mechanical, atomistic, and coarse-grained levels, as well as extensive coverage of force fields. In addition to providing the current snapshot of the CHARMM development, this review may serve as a starting point for exploring relevant theories and computational methods for tackling contemporary and emerging problems in biomolecular systems. CHARMM is freely available for academic and nonprofit research at https://academiccharmm.org/program.

Theoretical Investigation into Polymorphic Transformation between β-HMX and δ-HMX by Finite Temperature String

Polymorphic transformation is important in chemical industries, in particular, in those involving explosive molecular crystals. However, due to simulating challenges in the rare event method and collective variables, understanding the transformation mechanism of molecular crystals with a complex structure at the molecular level is poor. In this work, with the constructed order parameters (OPs) and K-means clustering algorithm, the potential of mean force (PMF) along the minimum free-energy path connecting β-HMX and δ-HMX was calculated by the finite temperature string method in the collective variables (SMCV), the free-energy profile and nucleation kinetics were obtained by Markovian milestoning with Voronoi tessellations, and the temperature effect on nucleation was also clarified. The barriers of transformation were affected by the finite-size effects. The configuration with the lower potential barrier in the PMF corresponded to the critical nucleus. The time and free-energy barrier of the polymorphic transformation were reduced as the temperature increased, which was explained by the pre-exponential factor and nucleation rate. Thus, the polymorphic transformation of HMX could be controlled by the temperatures, as is consistent with previous experimental results. Finally, the HMX polymorph dependency of the impact sensitivity was discussed. This work provides an effective way to reveal the polymorphic transformation of the molecular crystal with a cyclic molecular structure, and further to prepare the desired explosive by controlling the transformation temperature.

Surface-Accelerated String Method for Locating Minimum Free Energy Paths

We present a surface-accelerated string method (SASM) to efficiently optimize low-dimensional reaction pathways from the sampling performed with expensive quantum mechanical/molecular mechanical (QM/MM) Hamiltonians. The SASM accelerates the convergence of the path using the aggregate sampling obtained from the current and previous string iterations, whereas approaches like the string method in collective variables (SMCV) or the modified string method in collective variables (MSMCV) update the path only from the sampling obtained from the current iteration. Furthermore, the SASM decouples the number of images used to perform sampling from the number of synthetic images used to represent the path. The path is optimized on the current best estimate of the free energy surface obtained from all available sampling, and the proposed set of new simulations is not restricted to being located along the optimized path. Instead, the umbrella potential placement is chosen to extend the range of the free energy surface and improve the quality of the free energy estimates near the path. In this manner, the SASM is shown to improve the exploration for a minimum free energy pathway in regions where the free energy surface is relatively flat. Furthermore, it improves the quality of the free energy profile when the string is discretized with too few images. We compare the SASM, SMCV, and MSMCV using 3 QM/MM applications: a ribozyme methyltransferase reaction using 2 reaction coordinates, the 2'-O-transphosphorylation reaction of Hammerhead ribozyme using 3 reaction coordinates, and a tautomeric reaction in B-DNA using 5 reaction coordinates. We show that SASM converges the paths using roughly 3 times less sampling than the SMCV and MSMCV methods. All three algorithms have been implemented in the FE-ToolKit package made freely available.

Finite temperature string by K-means clustering sampling with order parameters as collective variables for molecular crystals: application to polymorphic transformation between β-CL-20 and ε-CL-20

Polymorphic transformation of molecular crystals is a fundamental phase transition process, and it is important practically in the chemical, material, biopharmaceutical, and energy storage industries. However, understanding of the transformation mechanism at the molecular level is poor due to the extreme simulating challenges in enhanced sampling and formulating order parameters (OPs) as the collective variables that can distinguish polymorphs with quite similar and complicated structures so as to describe the reaction coordinate. In this work, two kinds of OPs for CL-20 were constructed by the bond distances, bond orientations and relative orientations. A K-means clustering algorithm based on the Euclidean distance and sample weight was used to smooth the initial finite temperature string (FTS), and the minimum free energy path connecting β-CL-20 and ε-CL-20 was sketched by the string method in collective variables, and the free energy profile along the path and the nucleation kinetics were obtained by Markovian milestoning with Voronoi tessellations. In comparison with the average-based sampling, the K-means clustering algorithm provided an improved convergence rate of FTS. The simulation of transformation was independent of OP types but was affected greatly by finite-size effects. A surface-mediated local nucleation mechanism was confirmed and the configuration located at the shoulder of potential of mean force, rather than overall maximum, was confirmed to be the critical nucleus formed by the cooperative effect of the intermolecular interactions. This work provides an effective way to explore the polymorphic transformation of caged molecular crystals at the molecular level.

Free Energy Simulations of Receptor-Binding Domain Opening of the SARS-CoV-2 Spike Indicate a Barrierless Transition with Slow Conformational Motions

Infection by sarbecoviruses begins with the attachment of the homotrimeric viral "spike" protein to the angiotensin-converting enzyme 2 receptor on the surface of mammalian cells. This requires one or more receptor-binding domains (RBDs) to be in the open (up) position. Here, we present the results of long molecular dynamics simulations with umbrella sampling (US) to compute a one-dimensional free energy profile of RBD opening/closing and the associated transition times. After ≃3.58μs of simulation time per US window (∼229 μs in total), which was required to approach trajectory decorrelation, the computed free energy profile was found to be without large barriers. This suggests that the RBD diffuses between the open and closed positions without significant energetic hindrance. This interpretation appears consistent with experiments but is at odds with some previous simulations. Modeling the RBD motion as diffusive dynamics along the computed free energy profile, we find that the overall time required for the transition is only about 2 μs, which is 5 orders of magnitude shorter than experimentally measured transition times. We speculate that the most likely reason for the transition time mismatch is our use of very short glycans, which was required to make the simulations performed here feasible. Despite the long simulation times, the final free energy profile is not fully converged with statistical errors of ≃1.16 kcal/mol, which were found to be consistent with the slow time decay in the autocorrelation of the conformational motions of the protein. The simulation lengths that would be required to obtain fully converged results remain unknown, but the present calculations would benefit from at least an order-of-magnitude extension.

Molecular Simulations of Conformational Transitions within the Insulin Receptor Kinase Reveal Consensus Features in a Multistep Activation Pathway

Modulating the transitions between active and inactive conformations of protein kinases is the primary means of regulating their catalytic activity, achieved by phosphorylation of the activation loop (A-loop). To elucidate the mechanism of this conformational activation, we applied the string method to determine the conformational transition path of insulin receptor kinase between the active and inactive conformations and the corresponding free-energy profiles with and without A-loop phosphorylation. The conformational change was found to proceed in three sequential steps: first, the flipping of the DFG motif of the active site; second, rotation of the A-loop; finally, the inward movement of the αC helix. The main energetic bottleneck corresponds to the conformational change in the A-loop, while changes in the DFG motif and αC helix occur before and after A-loop conformational change, respectively. In accordance with this, two intermediate states are identified, the first state just after the DFG flipping and the second state after the A-loop rotation. These intermediates exhibit structural features characteristic of the corresponding inactive and active conformations of other protein kinases. To understand the impact of A-loop phosphorylation on kinase conformation, the free energies of A-loop phosphorylation were determined at several states along the conformational transition path using the free-energy perturbation simulations. The calculated free energies reveal that while the unphosphorylated kinase interconverts between the inactive and active conformations, A-loop phosphorylation restricts access to the inactive conformation, thereby increasing the active conformation population. Overall, this study suggests a consensus mechanism of conformational activation between different protein kinases.

CryoEM structures of anion exchanger 1 capture multiple states of inward- and outward-facing conformations

Anion exchanger 1 (AE1, band 3) is a major membrane protein of red blood cells and plays a key role in acid-base homeostasis, urine acidification, red blood cell shape regulation, and removal of carbon dioxide during respiration. Though structures of the transmembrane domain (TMD) of three SLC4 transporters, including AE1, have been resolved previously in their outward-facing (OF) state, no mammalian SLC4 structure has been reported in the inward-facing (IF) conformation. Here we present the cryoEM structures of full-length bovine AE1 with its TMD captured in both IF and OF conformations. Remarkably, both IF-IF homodimers and IF-OF heterodimers were detected. The IF structures feature downward movement in the core domain with significant unexpected elongation of TM11. Molecular modeling and structure guided mutagenesis confirmed the functional significance of residues involved in TM11 elongation. Our data provide direct evidence for an elevator-like mechanism of ion transport by an SLC4 family member.

Finite temperature string method with umbrella sampling using path collective variables: application to secondary structure change in a protein

The transition of an α-helix to a β-sheet in proteins is among the most complex conformational changes seen in biomolecular systems. Due to long time scales involved in the transition, it is challenging to study such protein conformational changes using direct molecular dynamics simulations. This limitation is typically overcome using an indirect approach wherein one computes the free energy landscape associated with the transition. Computation of free energy landscapes, however, requires a suitable set of collective variables that describe the transition. In this work, we demonstrate the use of path collective variables [D. Branduardi, F. L. Gervasio and M. Parrinello, J. Chem. Phys., 2007, 126, 054103] and combine it with the finite temperature string (FTS) method [E. Weinan, W. Ren and E. Vanden-Eijnden, J. Phys. Chem. B, 2005, 109, 6688-6693] to determine the molecular mechanisms involved during the structural transition of the mini G-protein from an α-helix to a β-hairpin. The transition from the α-helix proceeds via unfolding of the terminal residues, giving rise to a β-turn unfolded intermediate to eventually form the β-hairpin. Our proposed algorithm uses umbrella sampling simulations to simulate images along the string and the weighted histogram analysis to compute the free energy along the computed transition path. This work demonstrates that the string method in combination with path collective variables can be exploited to study complex protein conformational changes such as a complete change in the secondary structure.

Hierarchical Markov State Model Building to Describe Molecular Processes

Markov state models can describe ensembles of pathways via kinetic networks but are difficult to create when large free-energy barriers limit unbiased sampling. Chain-of-states simulations allow sampling over large free-energy barriers but are often constructed using a single pathway that is unlikely to thermodynamically average over orthogonal degrees of freedom in complex systems. Here, we combine the advantages of these two approaches in the form of a Markov state model of Markov state models, which we call a Hierarchical Markov state model. In this approach, independent Markov models are constructed in regions of configuration space that are locally well sampled but are separated by large free-energy barriers from other regions. A string method is used to construct an ensemble of pathways connecting the states of these different local Markov models, and the rate through each pathway is then estimated. These rates are then combined with the rate information from the local Markov models in a master equation to predict global rates, fluxes, and populations. By applying this hierarchical approach to tractable systems, a toy potential and dipeptides, we demonstrate that it is more accurate than the conventional single-pathway description. The advantages of this approach are that it (i) is more realistic than the conventional chain-of-states approach, as an ensemble of pathways rather than a single pathway is used to describe processes in high-dimensional systems, and (ii) it resolves the issue of poor sampling in Markov State model building when large free-energy barriers are present. The divide-and-conquer strategy inherent to this approach should make this procedure straightforward to apply to more complex systems.

A multifunnel energy landscape encodes the competing α-helix and β-hairpin conformations for a designed peptide

Depending on the amino acid sequence, as well as the local environment, some peptides have the capability to fold into multiple secondary structures. Conformational switching between such structures is a key element of protein folding and aggregation. Specifically, understanding the molecular mechanism underlying the transition from an α-helix to a β-hairpin is critical because it is thought to be a harbinger of amyloid assembly. In this study, we explore the energy landscape for an 18-residue peptide (DP5), designed by Araki and Tamura to exhibit equal propensities for the α-helical and β-hairpin forms. We find that the degeneracy is encoded in the multifunnel nature of the underlying free energy landscape. In agreement with experiment, we also observe that mutation of tyrosine at position 12 to serine shifts the equilibrium in favor of the α-helix conformation, by altering the landscape topography. The transition from the α-helix to the β-hairpin is a complex stepwise process, and occurs via collapsed coil-like intermediates. Our findings suggest that even a single mutation can tune the emergent features of the landscape, providing an efficient route to protein design. Interestingly, the transition pathways for the conformational switch seem to be minimally perturbed upon mutation, suggesting that there could be universal microscopic features that are conserved among different switch-competent protein sequences.

Accelerated computation of free energy profile at ab initio quantum mechanical/molecular mechanical accuracy via a semi-empirical reference potential. II. Recalibrating semi-empirical parameters with force matching

An efficient and accurate reference potential simulation protocol is proposed for producing ab initio quantum mechanical/molecular mechanical (AI-QM/MM) quality free energy profiles for chemical reactions in a solvent or macromolecular environment. This protocol involves three stages: (a) using force matching to recalibrate a semi-empirical quantum mechanical (SE-QM) Hamiltonian for the specific reaction under study; (b) employing the recalibrated SE-QM Hamiltonian (in combination with molecular mechanical force fields) as the reference potential to drive umbrella samplings along the reaction pathway; and (c) computing AI-QM/MM energy values for collected configurations from the sampling and performing weighted thermodynamic perturbation to acquire an AI-QM/MM corrected reaction free energy profile. For three model reactions (identity SN2 reaction, Menshutkin reaction, and glycine proton transfer reaction) in aqueous solution and one enzyme reaction (Claisen arrangement in chorismate mutase), our simulations using recalibrated PM3 SE-QM Hamiltonians well reproduced QM/MM free energy profiles at the B3LYP/6-31G* level of theory all within 1 kcal mol-1 with a 20 to 45 fold reduction in the computer time.

The Renormalization Group and Its Applications to Generating Coarse-Grained Models of Large Biological Molecular Systems

Understanding the dynamics of biomolecules is the key to understanding their biological activities. Computational methods ranging from all-atom molecular dynamics simulations to coarse-grained normal-mode analyses based on simplified elastic networks provide a general framework to studying these dynamics. Despite recent successes in studying very large systems with up to a 100,000,000 atoms, those methods are currently limited to studying small- to medium-sized molecular systems due to computational limitations. One solution to circumvent these limitations is to reduce the size of the system under study. In this paper, we argue that coarse-graining, the standard approach to such size reduction, must define a hierarchy of models of decreasing sizes that are consistent with each other, i.e., that each model contains the information of the dynamics of its predecessor. We propose a new method, Decimate, for generating such a hierarchy within the context of elastic networks for normal-mode analysis. This method is based on the concept of the renormalization group developed in statistical physics. We highlight the details of its implementation, with a special focus on its scalability to large systems of up to millions of atoms. We illustrate its application on two large systems, the capsid of a virus and the ribosome translation complex. We show that highly decimated representations of those systems, containing down to 1% of their original number of atoms, still capture qualitatively and quantitatively their dynamics. Decimate is available as an OpenSource resource.

The allosteric switching mechanism in bacteriophage MS2

We use all-atom simulations to elucidate the mechanisms underlying conformational switching and allostery within the coat protein of the bacteriophage MS2. Assembly of most icosahedral virus capsids requires that the capsid protein adopts different conformations at precise locations within the capsid. It has been shown that a 19 nucleotide stem loop (TR) from the MS2 genome acts as an allosteric effector, guiding conformational switching of the coat protein during capsid assembly. Since the principal conformational changes occur far from the TR binding site, it is important to understand the molecular mechanism underlying this allosteric communication. To this end, we use all-atom simulations with explicit water combined with a path sampling technique to sample the MS2 coat protein conformational transition, in the presence and absence of TR-binding. The calculations find that TR binding strongly alters the transition free energy profile, leading to a switch in the favored conformation. We discuss changes in molecular interactions responsible for this shift. We then identify networks of amino acids with correlated motions to reveal the mechanism by which effects of TR binding span the protein. We find that TR binding strongly affects residues located at the 5-fold and quasi-sixfold interfaces in the assembled capsid, suggesting a mechanism by which the TR binding could direct formation of the native capsid geometry. The analysis predicts amino acids whose substitution by mutagenesis could alter populations of the conformational substates or their transition rates.

A Simple and Accurate Method To Calculate Free Energy Profiles and Reaction Rates from Restrained Molecular Simulations of Diffusive Processes

A method is developed to obtain simultaneously free energy profiles and diffusion constants from restrained molecular simulations in diffusive systems. The method is based on low-order expansions of the free energy and diffusivity as functions of the reaction coordinate. These expansions lead to simple analytical relationships between simulation statistics and model parameters. The method is tested on 1D and 2D model systems; its accuracy is found to be comparable to or better than that of the existing alternatives, which are briefly discussed. An important aspect of the method is that the free energy is constructed by integrating its derivatives, which can be computed without need for overlapping sampling windows. The implementation of the method in any molecular simulation program that supports external umbrella potentials (e.g., CHARMM) requires modification of only a few lines of code. As a demonstration of its applicability to realistic biomolecular systems, the method is applied to model the α-helix ↔ β-sheet transition in a 16-residue peptide in implicit solvent, with the reaction coordinate provided by the string method. Possible modifications of the method are briefly discussed; they include generalization to multidimensional reaction coordinates [in the spirit of the model of Ermak and McCammon (Ermak, D. L.; McCammon, J. A. J. Chem. Phys. 1978, 69, 1352-1360)], a higher-order expansion of the free energy surface, applicability in nonequilibrium systems, and a simple test for Markovianity. In view of the small overhead of the method relative to standard umbrella sampling, we suggest its routine application in the cases where umbrella potential simulations are appropriate.

Principles and Overview of Sampling Methods for Modeling Macromolecular Structure and Dynamics

Investigation of macromolecular structure and dynamics is fundamental to understanding how macromolecules carry out their functions in the cell. Significant advances have been made toward this end in silico, with a growing number of computational methods proposed yearly to study and simulate various aspects of macromolecular structure and dynamics. This review aims to provide an overview of recent advances, focusing primarily on methods proposed for exploring the structure space of macromolecules in isolation and in assemblies for the purpose of characterizing equilibrium structure and dynamics. In addition to surveying recent applications that showcase current capabilities of computational methods, this review highlights state-of-the-art algorithmic techniques proposed to overcome challenges posed in silico by the disparate spatial and time scales accessed by dynamic macromolecules. This review is not meant to be exhaustive, as such an endeavor is impossible, but rather aims to balance breadth and depth of strategies for modeling macromolecular structure and dynamics for a broad audience of novices and experts.

Synthesis and Dynamics of Nanosized Phenylene-Ethynylene-Butadiynylene Rotaxanes and the Role of Shape Persistence

Phenylacetylene‐based [2]rotaxanes were synthesized by a covalent‐template approach by aminolysis of the corresponding prerotaxanes. The wheel and the bulky stoppers are made of phenylene–ethynylene–butadiynylene macrocycles of the same size. The stoppers are large enough to enable the synthesis and purification of the rotaxane. However, the wheel unthreads from the axle at elevated temperatures. The deslipping kinetics and the activation parameters were determined. We described theoretically the unthreading by state‐of‐the‐art DFT‐based molecular‐mechanics models and a string method for the simulation of rare events. This approach enabled us to characterize in detail the unthreading mechanism, which involves the folding of the stopper during its passage through the wheel opening, a process that defies intuitive geometrical considerations. The conformational and energetic features of the transition allowed us to infer the molecular residues controlling the disassembly timescale.

Evaluating the dynamics and electrostatic interactions of folded proteins in implicit solvents

Three implicit solvent models, namely GBMVII, FACTS, and SCPISM, were evaluated for their abilities to emulate an explicit solvent environment by comparing the simulated conformational ensembles, dynamics, and electrostatic interactions of the Src SH2 domain and the Lyn kinase domain. This assessment in terms of structural features in folded proteins expands upon the use of hydration energy as a metric for comparison. All-against-all rms coordinate deviation, average positional fluctuations, and ion-pair distance distribution were used to compare the implicit solvent models with the TIP3P explicit solvent model. Our study shows that the Src SH2 domains solvated with TIP3P, GBMVII, and FACTS sample similar global conformations. Additionally, the Src SH2 ion-pair distance distributions of solvent-exposed side chains corresponding to TIP3P, GBMVII, and FACTS do not differ substantially, indicating that GBMVII and FACTS are capable of modeling these electrostatic interactions. The ion-pair distance distributions of SCPISM are distinct from others, demonstrating that these electrostatic interactions are not adequately reproduced with the SCPISM model. On the other hand, for the Lyn kinase domain, a non-globular protein with bilobal structure and a large concavity on the surface, implicit solvent does not accurately model solvation to faithfully reproduce partially buried electrostatic interactions and lobe-lobe conformations. Our work reveals that local structure and dynamics of small, globular proteins are modeled well using FACTS and GBMVII. Nonetheless, global conformations and electrostatic interactions in concavities of multi-lobal proteins resulting from simulations with implicit solvent models do not match those obtained from explicit water simulations.

Molecular simulation of homogeneous nucleation of crystals of an ionic liquid from the melt

The homogeneous nucleation of crystals of the ionic liquid [dmim(+)][Cl(-)] from its supercooled liquid phase in the bulk (P = 1 bar, T = 340 K, representing a supercooling of 58 K) was studied using molecular simulations. The string method in collective variables [Maragliano et al., J. Chem. Phys. 125, 024106 (2006)] was used in combination with Markovian milestoning with Voronoi tessellations [Maragliano et al., J. Chem. Theory Comput. 5, 2589-2594 (2009)] and order parameters for molecular crystals [E. E. Santiso and B. L. Trout, J. Chem. Phys. 134, 064109 (2011)] to sketch a minimum free energy path connecting the supercooled liquid and the monoclinic crystal phases, and to determine the free energy and the rates involved in the homogeneous nucleation process. The physical significance of the configurations found along this minimum free energy path is discussed with the help of calculations based on classical nucleation theory and with additional simulation results obtained for a larger system. Our results indicate that, at a supercooling of 58 K, the liquid has to overcome a free energy barrier of the order of 60 kcal/mol and to form a critical nucleus with an average size of about 3.6 nm, before it reaches the thermodynamically stable crystal phase. A simulated homogeneous nucleation rate of 5.0 × 10(10) cm(-3) s(-1) was obtained for our system, which is in reasonable agreement with experimental and simulation rates for homogeneous nucleation of ice at similar degrees of supercooling. This study represents our first step in a series of studies aimed at understanding the nucleation and growth of crystals of organic salts near surfaces and inside nanopores.

The Energy Difference between the Triply-Bridged and All-Terminal Structures of Co₄(CO)₁₂, Rh₄(CO)₁₂, and Ir₄(CO)₁₂: A Difficult Test for Conventional Density Functional Methods

The M4(CO)12 molecules Co4(CO)12, Rh4(CO)12, and Ir4(CO)12 have two low-lying structures, the all-terminal structure with Td symmetry and the triply bridged structure with C3v symmetry. A total of 45 density functional theory (DFT) methods have been used to predict structures and vibrational frequencies for Co4(CO)12, Rh4(CO)12, and Ir4(CO)12. The different DFT methods show a broad range of energy differences ΔE = E(Td) - E(C3v). For Rh4(CO)12, none of the 45 DFT predictions is within 11 kcal/mol of the 2005 experimental value of 5.1 ± 0.6 kcal/mol reported by Allian and Garland (Dalton Trans. 2005, 1957 - 1965). For the challenging Ir4(CO)12 molecule, 21 DFT methods predict the correct Td structure, while 24 DFT methods predict the C3v structure to lie lower in energy. This research reveals many peculiar problems in the computation of the vibrational frequencies for the all-terminal structure.