Energetics and mechanism of the normal-to-amyloidogenic isomerization of β2-microglobulin: on-the-fly string method calculations

A Companion Guide to the String Method with Swarms of Trajectories: Characterization, Performance, and Pitfalls

The string method with swarms of trajectories (SMwST) is an algorithm that identifies a physically meaningful transition pathway─a one-dimensional curve, embedded within a high-dimensional space of selected collective variables. The SMwST algorithm leans on a series of short, unbiased molecular dynamics simulations spawned at different locations of the discretized path, from whence an average dynamic drift is determined to evolve the string toward an optimal pathway. However conceptually simple in both its theoretical formulation and practical implementation, the SMwST algorithm is computationally intensive and requires a careful choice of parameters for optimal cost-effectiveness in applications to challenging problems in chemistry and biology. In this contribution, the SMwST algorithm is presented in a self-contained manner, discussing with a critical eye its theoretical underpinnings, applicability, inherent limitations, and use in the context of path-following free-energy calculations and their possible extension to kinetics modeling. Through multiple simulations of a prototypical polypeptide, combining the search of the transition pathway and the computation of the potential of mean force along it, several practical aspects of the methodology are examined with the objective of optimizing the computational effort, yet without sacrificing accuracy. In light of the results reported here, we propose some general guidelines aimed at improving the efficiency and reliability of the computed pathways and free-energy profiles underlying the conformational transitions at hand.

String Method with Swarms-of-Trajectories, Mean Drifts, Lag Time, and Committor

The kinetics of a dynamical system comprising two metastable states is formulated in terms of a finite-time propagator in phase space (position and velocity) adapted to the underdamped Langevin equation. Dimensionality reduction to a subspace of collective variables yields familiar expressions for the propagator, committor, and steady-state flux. A quadratic expression for the steady-state flux between the two metastable states can serve as a robust variational principle to determine an optimal approximate committor expressed in terms of a set of collective variables. The theoretical formulation is exploited to clarify the foundation of the string method with swarms-of-trajectories, which relies on the mean drift of short trajectories to determine the optimal transition pathway. It is argued that the conditions for Markovity within a subspace of collective variables may not be satisfied with an arbitrary short time-step and that proper kinetic behaviors appear only when considering the effective propagator for longer lag times. The effective propagator with finite lag time is amenable to an eigenvalue-eigenvector spectral analysis, as elaborated previously in the context of position-based Markov models. The time-correlation functions calculated by swarms-of-trajectories along the string pathway constitutes a natural extension of these developments. The present formulation provides a powerful theoretical framework to characterize the optimal pathway between two metastable states of a system.

Proline isomerization effects in the amyloidogenic protein β2-microglobulin

The protein β₂-microglobulin can aggregate in insoluble amyloid fibrils. By relying on extensive sampling simulations, we study the Pro32 isomerization as a possible triggering factor leading to structural modifications in β₂-m.

Simulating ion channel activation mechanisms using swarms of trajectories

Atomic-level studies of protein activity represent a significant challenge as a result of the complexity of conformational changes occurring on wide-ranging timescales, often greatly exceeding that of even the longest simulations. A prime example is the elucidation of protein allosteric mechanisms, where localized perturbations transmit throughout a large macromolecule to generate a response signal. For example, the conversion of chemical to electrical signals during synaptic neurotransmission in the brain is achieved by specialized membrane proteins called pentameric ligand-gated ion channels. Here, the binding of a neurotransmitter results in a global conformational change to open an ion-conducting pore across the nerve cell membrane. X-ray crystallography has produced static structures of the open and closed states of the proton-gated GLIC pentameric ligand-gated ion channel protein, allowing for atomistic simulations that can uncover changes related to activation. We discuss a range of enhanced sampling approaches that could be used to explore activation mechanisms. In particular, we describe recent application of an atomistic string method, based on Roux's "swarms of trajectories" approach, to elucidate the sequence and interdependence of conformational changes during activation. We illustrate how this can be combined with transition analysis and Brownian dynamics to extract thermodynamic and kinetic information, leading to understanding of what controls ion channel function. © 2019 Wiley Periodicals, Inc.

Exploring Configuration Space and Path Space of Biomolecules Using Enhanced Sampling Techniques-Searching for Mechanism and Kinetics of Biomolecular Functions

To understand functions of biomolecules such as proteins, not only structures but their conformational change and kinetics need to be characterized, but its atomistic details are hard to obtain both experimentally and computationally. Here, we review our recent computational studies using novel enhanced sampling techniques for conformational sampling of biomolecules and calculations of their kinetics. For efficiently characterizing the free energy landscape of a biomolecule, we introduce the multiscale enhanced sampling method, which uses a combined system of atomistic and coarse-grained models. Based on the idea of Hamiltonian replica exchange, we can recover the statistical properties of the atomistic model without any biases. We next introduce the string method as a path search method to calculate the minimum free energy pathways along a multidimensional curve in high dimensional space. Finally we introduce novel methods to calculate kinetics of biomolecules based on the ideas of path sampling: one is the Onsager⁻Machlup action method, and the other is the weighted ensemble method. Some applications of the above methods to biomolecular systems are also discussed and illustrated.

Energetics and conformational pathways of functional rotation in the multidrug transporter AcrB

The multidrug transporter AcrB transports a broad range of drugs out of the cell by means of the proton-motive force. The asymmetric crystal structure of trimeric AcrB suggests a functionally rotating mechanism for drug transport. Despite various supportive forms of evidence from biochemical and simulation studies for this mechanism, the link between the functional rotation and proton translocation across the membrane remains elusive. Here, calculating the minimum free energy pathway of the functional rotation for the complete AcrB trimer, we describe the structural and energetic basis behind the coupling between the functional rotation and the proton translocation at atomic resolution. Free energy calculations show that protonation of Asp408 in the transmembrane portion of the drug-bound protomer drives the functional rotation. The conformational pathway identifies vertical shear motions among several transmembrane helices, which regulate alternate access of water in the transmembrane as well as peristaltic motions that pump drugs in the periplasm.

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.

Extended Phase-Space Methods for Enhanced Sampling in Molecular Simulations: A Review

Molecular Dynamics simulations are a powerful approach to study biomolecular conformational changes or protein-ligand, protein-protein, and protein-DNA/RNA interactions. Straightforward applications, however, are often hampered by incomplete sampling, since in a typical simulated trajectory the system will spend most of its time trapped by high energy barriers in restricted regions of the configuration space. Over the years, several techniques have been designed to overcome this problem and enhance space sampling. Here, we review a class of methods that rely on the idea of extending the set of dynamical variables of the system by adding extra ones associated to functions describing the process under study. In particular, we illustrate the Temperature Accelerated Molecular Dynamics (TAMD), Logarithmic Mean Force Dynamics (LogMFD), and Multiscale Enhanced Sampling (MSES) algorithms. We also discuss combinations with techniques for searching reaction paths. We show the advantages presented by this approach and how it allows to quickly sample important regions of the free-energy landscape via automatic exploration.

Probing the influence of citrate-capped gold nanoparticles on an amyloidogenic protein

Nanoparticles (NPs) are known to exhibit distinct physical and chemical properties compared with the same materials in bulk form. NPs have been repeatedly reported to interact with proteins, and this interaction can be exploited to affect processes undergone by proteins, such as fibrillogenesis. Fibrillation is common to many proteins, and in living organisms, it causes tissue-specific or systemic amyloid diseases. The nature of NPs and their surface chemistry is crucial in assessing their affinity for proteins and their effects on them. Here we present the first detailed structural characterization and molecular mechanics model of the interaction between a fibrillogenic protein, β2-microglobulin, and a NP, 5 nm hydrophilic citrate-capped gold nanoparticles. NMR measurements and simulations at multiple levels (enhanced sampling molecular dynamics, Brownian dynamics, and Poisson-Boltzmann electrostatics) explain the origin of the observed protein perturbations mostly localized at the amino-terminal region. Experiments show that the protein-NP interaction is weak in the physiological-like, conditions and do not induce protein fibrillation. Simulations reproduce these findings and reveal instead the role of the citrate in destabilizing the lower pH protonated form of β2-microglobulin. The results offer possible strategies for controlling the desired effect of NPs on the conformational changes of the proteins, which have significant roles in the fibrillation process.

A simulated intermediate state for folding and aggregation provides insights into ΔN6 β2-microglobulin amyloidogenic behavior

A major component of ex vivo amyloid plaques of patients with dialysis-related amyloidosis (DRA) is a cleaved variant of β₂-microglobulin (ΔN6) lacking the first six N-terminal residues. Here we perform a computational study on ΔN6, which provides clues to understand the amyloidogenicity of the full-length β₂-microglobulin. Contrary to the wild-type form, ΔN6 is able to efficiently nucleate fibrillogenesis in vitro at physiological pH. This behavior is enhanced by a mild acidification of the medium such as that occurring in the synovial fluid of DRA patients. Results reported in this work, based on molecular simulations, indicate that deletion of the N-terminal hexapeptide triggers the formation of an intermediate state for folding and aggregation with an unstructured strand A and a native-like core. Strand A plays a pivotal role in aggregation by acting as a sticky hook in dimer assembly. This study further predicts that the detachment of strand A from the core is maximized at pH 6.2 resulting into higher aggregation efficiency. The structural mapping of the dimerization interface suggests that Tyr10, His13, Phe30 and His84 are hot-spot residues in ΔN6 amyloidogenesis.

Dialysis-related amyloidosis (DRA) is a conformational disease that affects individuals undergoing long-term haemodialysis. In DRA the progressive accumulation of protein human β₂-microglobulin (Hβ₂m) in the osteoarticular system, followed by its assembly into amyloid fibrils, eventually leads to tissue erosion and destruction. Disclosing the aggregation mechanism of Hβ₂m under physiologically relevant conditions represents a major challenge due to the inability of the protein to efficiently nucleate fibrillogenesis in vitro at physiological pH. On the other hand, ΔN6, a truncated variant of Hβ₂m, which is also a major component of ex vivo amyloid deposits extracted from DRA patients, is able to efficiently form amyloid fibrils de novo in physiological conditions. This amyloidogenic behavior is dramatically enhanced in a slightly more acidic pH (6.2) compatible with the mild acidification that occurs in the synovial fluid of DRA patients. In this work, an innovative three-stage methodological approach, relying on an array of molecular simulations, spanning different levels of resolution is used to investigate the initial stage of the de novo aggregation mechanism of ΔN6 in a physiologically relevant pH range. We identify an intermediate state for folding and aggregation, whose potential to dimerize is enhanced at pH 6.2. Our results provide rationalizations for previous experimental observations and new insights into the molecular basis of DRA.

Exploring the conformational transitions of biomolecular systems using a simple two-state anisotropic network model

Biomolecular conformational transitions are essential to biological functions. Most experimental methods report on the long-lived functional states of biomolecules, but information about the transition pathways between these stable states is generally scarce. Such transitions involve short-lived conformational states that are difficult to detect experimentally. For this reason, computational methods are needed to produce plausible hypothetical transition pathways that can then be probed experimentally. Here we propose a simple and computationally efficient method, called ANMPathway, for constructing a physically reasonable pathway between two endpoints of a conformational transition. We adopt a coarse-grained representation of the protein and construct a two-state potential by combining two elastic network models (ENMs) representative of the experimental structures resolved for the endpoints. The two-state potential has a cusp hypersurface in the configuration space where the energies from both the ENMs are equal. We first search for the minimum energy structure on the cusp hypersurface and then treat it as the transition state. The continuous pathway is subsequently constructed by following the steepest descent energy minimization trajectories starting from the transition state on each side of the cusp hypersurface. Application to several systems of broad biological interest such as adenylate kinase, ATP-driven calcium pump SERCA, leucine transporter and glutamate transporter shows that ANMPathway yields results in good agreement with those from other similar methods and with data obtained from all-atom molecular dynamics simulations, in support of the utility of this simple and efficient approach. Notably the method provides experimentally testable predictions, including the formation of non-native contacts during the transition which we were able to detect in two of the systems we studied. An open-access web server has been created to deliver ANMPathway results.

Many biomolecules are like tiny molecular machines that need to change their shapes and visit many states to perform their biological functions. For a complete molecular understanding of a biological process, one needs to have information on the relevant stable states of the system in question, as well as the pathways by which the system travels from one state to another. We report here an efficient computational method that uses the knowledge of experimental structures of a pair of stable states in order to construct an energetically favoravle pathway between them. We adopt a simple representation of the molecular system by replacing the atoms with beads connected by springs and constructing an energy function with two minima around the end-states. We searched for the structure with highest energy that the system is most likely to visit during the transition and created two paths starting from this structure and proceeding toward the end-states. The combined result of these two paths is the minimum energy pathway between the two stable states. We apply this method to study important structural changes in one enzyme and three large proteins that transport small molecules and ions across the cell membrane.

Comparison between Mean Forces and Swarms-of-Trajectories String Methods

The original formulation of the string method in collective variable space is compared with a recent variant called string method with swarms-of-trajectories. The assumptions made in the original method are revisited and the significance of the minimum free energy path (MFEP) is discussed in the context of reactive events. These assumptions are compared to those made in the string method with swarms-of-trajectories, and shown to be equivalent in a certain regime: in particular an expression for the path identified by the swarms-of-trajectories method is given and shown to be closely related to the MFEP. Finally, the algorithmic aspects of both methods are compared.

Toward an Automatic Determination of Enzymatic Reaction Mechanisms and Their Activation Free Energies

We present a combination of the string method and a path collective variable for the exploration of the free energy surface associated to a chemical reaction in condensed environments. The on-the-fly string method is employed to find the minimum free energy paths on a multidimensional free energy surface defined in terms of interatomic distances, which is a convenient selection to study bond forming/breaking processes. Once the paths have been determined, a reaction coordinate is defined as a measure of the advance of the system along these paths. This reaction coordinate can be then used to trace the reaction Potential of Mean Force from which the activation free energy can be obtained. This combination of methodologies has been here applied to the study, by means of Quantum Mechanics/Molecular Mechanics simulations, of the reaction catalyzed by guanidinoacetate methyltransferase. This enzyme catalyzes the methylation of guanidinoacetate by S-adenosyl-l-methionine, a reaction that involves a methyl transfer and a proton transfer and for which different reaction mechanisms have been proposed.