Efficient numerical reconstruction of protein folding kinetics with partial path sampling and pathlike variables

Kinetics from Metadynamics: Principles, Applications, and Outlook

Metadynamics is a popular enhanced sampling algorithm for computing the free energy landscape of rare events by using molecular dynamics simulation. Ten years ago, Tiwary and Parrinello introduced the infrequent metadynamics approach for calculating the kinetics of transitions across free energy barriers. Since then, metadynamics-based methods for obtaining rate constants have attracted significant attention in computational molecular science. Such methods have been applied to study a wide range of problems, including protein-ligand binding, protein folding, conformational transitions, chemical reactions, catalysis, and nucleation. Here, we review the principles of elucidating kinetics from metadynamics-like approaches, subsequent methodological developments in this area, and successful applications on chemical, biological, and material systems. We also highlight the challenges of reconstructing accurate kinetics from enhanced sampling simulations and the scope of future developments.

15 Years of molecular simulation of drug-binding kinetics

INTRODUCTION

Drug-binding kinetics has been increasingly recognized as an important factor to be considered in drug discovery. Long residence time could prolong the action of some drugs while produce toxicity on others. Early evaluation of the binding kinetics of drug candidates could reduce attrition rate late in the drug discovery process. Computational prediction of drug-binding kinetics is useful as compounds can be evaluated even before they are made. However, simulation of drug-binding kinetics is a challenging problem because of the long-time scale involved. Nevertheless, significant progress has been made.

AREAS COVERED

This review illustrates the rapid evolution of qualitative to quantitative molecular dynamics-based methods that have been developed over the last 15 years.

EXPERT OPINION

The development of new methods based on molecular dynamics simulations now enables computation of absolute association/dissociation rate constants. Cheaper methods capable of identifying candidates with fast or slow binding kinetics, or rank-ordering rate constants are also available. Together, these methods have generated useful insights into the molecular mechanisms of drug-binding kinetics, and the design of drug candidates with therapeutically favorable kinetics. Although predicting absolute rate constants is still expensive and challenging, rapid improvement is expected in the coming years with the continuing refinement of current technologies, development of new methodologies, and the utilization of machine learning.

Enhanced-Sampling Simulations for the Estimation of Ligand Binding Kinetics: Current Status and Perspective

The dissociation rate (k off) associated with ligand unbinding events from proteins is a parameter of fundamental importance in drug design. Here we review recent major advancements in molecular simulation methodologies for the prediction of k off. Next, we discuss the impact of the potential energy function models on the accuracy of calculated k off values. Finally, we provide a perspective from high-performance computing and machine learning which might help improve such predictions.

SGOOP-d: Estimating Kinetic Distances and Reaction Coordinate Dimensionality for Rare Event Systems from Biased/Unbiased Simulations

Understanding kinetics including reaction pathways and associated transition rates is an important yet difficult problem in numerous chemical and biological systems, especially in situations with multiple competing pathways. When these high-dimensional systems are projected on low-dimensional coordinates, which are often needed for enhanced sampling or for interpretation of simulations and experiments, one can end up losing the kinetic connectivity of the underlying high-dimensional landscape. Thus, in the low-dimensional projection, metastable states might appear closer or further than they actually are. To deal with this issue, in this work, we develop a formalism that learns a multidimensional yet minimally complex reaction coordinate (RC) for generic high-dimensional systems. When projected along this RC, all possible kinetically relevant pathways can be demarcated and the true high-dimensional connectivity is maintained. One of the defining attributes of our method lies in that it can work on long unbiased simulations as well as biased simulations often needed for rare event systems. We demonstrate the utility of the method by studying a range of model systems including conformational transitions in a small peptide Ace-Ala₃-Nme, where we show how two-dimensional and three-dimensional RCs found by our previously published spectral gap optimization method "SGOOP" [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. 2016, 113, 2839] can capture the kinetics for 23 and all 28 out of the 28 dominant state-to-state transitions, respectively.

Ligand-Binding Calculations with Metadynamics

All-atom molecular dynamics simulations can capture the dynamic degrees of freedom that characterize molecular recognition, the knowledge of which constitutes the cornerstone of rational approaches to drug design and optimization. In particular, enhanced sampling algorithms, such as metadynamics, are powerful tools to dramatically reduce the computational cost required for a mechanistic description of the binding process. Here, we describe the essential details characterizing these simulation strategies, focusing on the critical step of identifying suitable reaction coordinates, as well as on the different analysis algorithms to estimate binding affinity and residence times. We conclude with a survey of published applications that provides explicit examples of successful simulations for several targets.

Kinetics of Drug Binding and Residence Time

The kinetics of drug binding and unbinding is assuming an increasingly crucial role in the long, costly process of bringing a new medicine to patients. For example, the time a drug spends in contact with its biological target is known as residence time (the inverse of the kinetic constant of the drug-target unbinding, 1/ koff). Recent reports suggest that residence time could predict drug efficacy in vivo, perhaps even more effectively than conventional thermodynamic parameters (free energy, enthalpy, entropy). There are many experimental and computational methods for predicting drug-target residence time at an early stage of drug discovery programs. Here, we review and discuss the methodological approaches to estimating drug binding kinetics and residence time. We first introduce the theoretical background of drug binding kinetics from a physicochemical standpoint. We then analyze the recent literature in the field, starting from the experimental methodologies and applications thereof and moving to theoretical and computational approaches to the kinetics of drug binding and unbinding. We acknowledge the central role of molecular dynamics and related methods, which comprise a great number of the computational methods and applications reviewed here. However, we also consider kinetic Monte Carlo. We conclude with the outlook that drug (un)binding kinetics may soon become a go/no go step in the discovery and development of new medicines.

Defining an Optimal Metric for the Path Collective Variables

Path Collective Variables (PCVs) are a set of path-like variables that have been successfully used to investigate complex chemical and biological processes and compute their associated free energy surfaces and kinetics. Their current implementation relies on general, but at times inefficient, metrics (such as RMSD or DRMSD) to evaluate the distance between the instantaneous conformational state during the simulation and the reference coordinates defining the path. In this work, we present a new algorithm to construct optimal PCVs metrics as linear combinations of different CVs weighted through a spectral gap optimization procedure. The method was tested first on a simple model, trialanine peptide, in vacuo and then on a more complex path of an anticancer inhibitor binding to its pharmacological target. We also compared the results to those obtained with other path-based algorithms. We find that not only our proposed approach is able to automatically select relevant CVs for the PCVs metric but also that the resulting PCVs allow for reconstructing the associated free energy very efficiently. What is more, at difference with other path-based methods, our algorithm is able to explore nonlocally the reaction path space.

A Multiscale Simulation Approach to Modeling Drug-Protein Binding Kinetics

Drug-target binding kinetics has recently emerged as a sometimes critical determinant of in vivo efficacy and toxicity. Its rational optimization to improve potency or reduce side effects of drugs is, however, extremely difficult. Molecular simulations can play a crucial role in identifying features and properties of small ligands and their protein targets affecting the binding kinetics, but significant challenges include the long time scales involved in (un)binding events and the limited accuracy of empirical atomistic force fields (lacking, e.g., changes in electronic polarization). In an effort to overcome these hurdles, we propose a method that combines state-of-the-art enhanced sampling simulations and quantum mechanics/molecular mechanics (QM/MM) calculations at the BLYP/VDZ level to compute association free energy profiles and characterize the binding kinetics in terms of structure and dynamics of the transition state ensemble. We test our combined approach on the binding of the anticancer drug Imatinib to Src kinase, a well-characterized target for cancer therapy with a complex binding mechanism involving significant conformational changes. The results indicate significant changes in polarization along the binding pathways, which affect the predicted binding kinetics. This is likely to be of widespread importance in binding of ligands to protein targets.

Enriched Conformational Sampling of DNA and Proteins with a Hybrid Hamiltonian Derived from the Protein Data Bank

In this article, we present a method for the enhanced molecular dynamics simulation of protein and DNA systems called potential of mean force (PMF)-enriched sampling. The method uses partitions derived from the potentials of mean force, which we determined from DNA and protein structures in the Protein Data Bank (PDB). We define a partition function from a set of PDB-derived PMFs, which efficiently compensates for the error introduced by the assumption of a homogeneous partition function from the PDB datasets. The bias based on the PDB-derived partitions is added in the form of a hybrid Hamiltonian using a renormalization method, which adds the PMF-enriched gradient to the system depending on a linear weighting factor and the underlying force field. We validated the method using simulations of dialanine, the folding of TrpCage, and the conformational sampling of the Dickerson⁻Drew DNA dodecamer. Our results show the potential for the PMF-enriched simulation technique to enrich the conformational space of biomolecules along their order parameters, while we also observe a considerable speed increase in the sampling by factors ranging from 13.1 to 82. The novel method can effectively be combined with enhanced sampling or coarse-graining methods to enrich conformational sampling with a partition derived from the PDB.

Adaptive enhanced sampling with a path-variable for the simulation of protein folding and aggregation

In this article, we present a novel adaptive enhanced sampling molecular dynamics (MD) method for the accelerated simulation of protein folding and aggregation. We introduce a path-variable L based on the un-biased momenta p and displacements dq for the definition of the bias s applied to the system and derive 3 algorithms: general adaptive bias MD, adaptive path-sampling, and a hybrid method which combines the first 2 methodologies. Through the analysis of the correlations between the bias and the un-biased gradient in the system, we find that the hybrid methodology leads to an improved force correlation and acceleration in the sampling of the phase space. We apply our method on SPC/E water, where we find a conservation of the average water structure. We then use our method to sample dialanine and the folding of TrpCage, where we find a good agreement with simulation data reported in the literature. Finally, we apply our methodologies on the initial stages of aggregation of a hexamer of Alzheimer's amyloid β fragment 25-35 (Aβ 25-35) and find that transitions within the hexameric aggregate are dominated by entropic barriers, while we speculate that especially the conformation entropy plays a major role in the formation of the fibril as a rate limiting factor.

Multiscale Methods in Drug Design Bridge Chemical and Biological Complexity in the Search for Cures

Drug action is inherently multiscale: it connects molecular interactions to emergent properties at cellular and larger scales. Simulation techniques at each of these different scales are already central to drug design and development, but methods capable of connecting across these scales will extend understanding of complex mechanisms and the ability to predict biological effects. Improved algorithms, ever-more-powerful computing architectures and the accelerating growth of rich datasets are driving advances in multiscale modeling methods capable of bridging chemical and biological complexity from the atom to the cell. Particularly exciting is the development of highly detailed, structure-based, physical simulations of biochemical systems, which are now able to access experimentally relevant timescales for large systems and, at the same time, achieve unprecedented accuracy. In this Perspective, we discuss how emerging data-rich, physics-based multiscale approaches are of the cusp of realizing long-promised impact in the discovery, design and development of novel therapeutics. We highlight emerging methods and applications in this growing field, and outline how different scales can be combined in practical modelling and simulation strategies.

An adaptive bias - hybrid MD/kMC algorithm for protein folding and aggregation

In this paper, we present a novel hybrid Molecular Dynamics/kinetic Monte Carlo (MD/kMC) algorithm and apply it to protein folding and aggregation in explicit solvent. The new algorithm uses a dynamical definition of biases throughout the MD component of the simulation, normalized in relation to the unbiased forces. The algorithm guarantees sampling of the underlying ensemble in dependency of one average linear coupling factor 〈α〉_τ. We test the validity of the kinetics in simulations of dialanine and compare dihedral transition kinetics with long-time MD-simulations. We find that for low 〈α〉_τ values, kinetics are in good quantitative agreement. In folding simulations of TrpCage and TrpZip4 in explicit solvent, we also find good quantitative agreement with experimental results and prior MD/kMC simulations. Finally, we apply our algorithm to study growth of the Alzheimer Amyloid Aβ 16-22 fibril by monomer addition. We observe two possible binding modes, one at the extremity of the fibril (elongation) and one on the surface of the fibril (lateral growth), on timescales ranging from ns to 8 μs.

Metadynamics Enhanced Markov Modeling of Protein Dynamics

Enhanced sampling techniques represent a versatile approach to account for rare conformational transitions in biomolecules. A particularly promising strategy is to combine massive parallel computing of short molecular dynamics (MD) trajectories (to sample the free energy landscape of the system) with Markov state modeling (to rebuild the kinetics from the sampled data). To obtain well-distributed initial structures for the short trajectories, it is proposed to employ metadynamics MD, which quickly sweeps through the entire free energy landscape of interest. Being only used to generate initial conformations, the implementation of metadynamics can be simple and fast. The conformational dynamics of helical peptide Aib₉ is adopted to discuss various technical issues of the approach, including metadynamics settings, minimal number and length of short MD trajectories, and the validation of the resulting Markov models. Using metadynamics to launch some thousands of nanosecond trajectories, several Markov state models are constructed that reveal that previous unbiased MD simulations of in total 16 μs length cannot provide correct equilibrium populations or qualitative features of the pathway distribution of the short peptide.

Temperature evolution of Trp-cage folding pathways: An analysis by dividing the probability flux field into stream tubes

Owing to its small size and very fast folding rate, the Trp-cage miniprotein has become a benchmark system to study protein folding. Two folding pathways were found to be characteristic of this protein: pathway I, in which the hydrophobic collapse precedes the formation of α-helix, and pathway II, in which the events occur in the reverse order. At the same time, the relative contribution of these pathways at different temperatures as well as the nature of transition from one pathway to the other remain unclear. To gain insight into this issue, we employ a recently proposed hydrodynamic description of protein folding, in which the process of folding is considered as a motion of a "folding fluid" (Chekmarev et al., Phys. Rev. Lett. 100(1), 018107 2008). Using molecular dynamics simulations, we determine the field of probability fluxes of transitions in a space of collective variables and divide it into stream tubes. Each tube contains a definite fraction of the total folding flow and can be associated with a certain pathway. Specifically, three temperatures were considered, T = 285K, T = 315K, and T = 325K. We have found that as the temperature increases, the contribution of pathway I, which is approximately 90% of the total folding flow at T = 285K, decreases to approximately 10% at T = 325K, i.e., pathway II becomes dominant. At T = 315K, both pathways contribute approximately equally. All these temperatures are found below the calculated melting point, which suggests that the Trp-cage folding mechanism is determined by kinetic factors rather than thermodynamics.

An Efficient Metadynamics-Based Protocol To Model the Binding Affinity and the Transition State Ensemble of G-Protein-Coupled Receptor Ligands

A generally applicable metadynamics scheme for predicting the free energy profile of ligand binding to G-protein-coupled receptors (GPCRs) is described. A common and effective collective variable (CV) has been defined using the ideally placed and highly conserved Trp6.48 as a reference point for ligand-GPCR distance measurement and the common orientation of GPCRs in the cell membrane. Using this single CV together with well-tempered multiple-walker metadynamics with a funnel-like boundary allows an efficient exploration of the entire ligand binding path from the extracellular medium to the orthosteric binding site, including vestibule and intermediate sites. The protocol can be used with X-ray structures or high-quality homology models (based on a high-quality template and after thorough refinement) for the receptor and is universally applicable to agonists, antagonists, and partial and reverse agonists. The root-mean-square error (RMSE) in predicted binding free energies for 12 diverse ligands in five receptors (a total of 23 data points) is surprisingly small (less than 1 kcal mol^-1). The RMSEs for simulations that use receptor X-ray structures and homology models are very similar.

Protein States as Symmetry Transitions in the Correlation Matrices

Over the last few years, there has been significant progress in the knowledge on protein folding. However, some aspects of protein folding still need further attention. One of these is the exact relationship between the folded and unfolded states and the differences between them. Whereas the folded state is well known, at least from a structural point of view (just think of the thousands of structures in online databases), the unfolded state is more elusive. Also, these are dynamic states of matter, and this aspect cannot be overlooked. Molecular dynamics-derived correlation matrices are an invaluable source of information on the protein dynamics. Here, bulk eigenvalue spectra of the correlation matrices obtained from the Trp-cage dynamics in the folded and unfolded states have been analyzed. The associated modes represent localized vibrations and are significantly affected by the fine details of the structure and interactions. Therefore, these bulk modes can be used as probes of the protein local dynamics in different states. The results of these analyses show that the correlation matrices describing the folded and unfolded dynamics belong to different symmetry classes. This finding provides new support to the phase-transition models of protein folding.

Crystal Nucleation in Liquids: Open Questions and Future Challenges in Molecular Dynamics Simulations

The nucleation of crystals in liquids is one of nature's most ubiquitous phenomena, playing an important role in areas such as climate change and the production of drugs. As the early stages of nucleation involve exceedingly small time and length scales, atomistic computer simulations can provide unique insights into the microscopic aspects of crystallization. In this review, we take stock of the numerous molecular dynamics simulations that, in the past few decades, have unraveled crucial aspects of crystal nucleation in liquids. We put into context the theoretical framework of classical nucleation theory and the state-of-the-art computational methods by reviewing simulations of such processes as ice nucleation and the crystallization of molecules in solutions. We shall see that molecular dynamics simulations have provided key insights into diverse nucleation scenarios, ranging from colloidal particles to natural gas hydrates, and that, as a result, the general applicability of classical nucleation theory has been repeatedly called into question. We have attempted to identify the most pressing open questions in the field. We believe that, by improving (i) existing interatomic potentials and (ii) currently available enhanced sampling methods, the community can move toward accurate investigations of realistic systems of practical interest, thus bringing simulations a step closer to experiments.

Conformational Selection and Induced Fit Mechanisms in the Binding of an Anticancer Drug to the c-Src Kinase

Understanding the conformational changes associated with the binding of small ligands to their biological targets is a fascinating and meaningful question in chemistry, biology and drug discovery. One of the most studied and important is the so-called "DFG-flip" of tyrosine kinases. The conserved three amino-acid DFG motif undergoes an "in to out" movement resulting in a particular inactive conformation to which "type II" kinase inhibitors, such as the anti-cancer drug Imatinib, bind. Despite many studies, the details of this prototypical conformational change are still debated. Here we combine various NMR experiments and surface plasmon resonance with enhanced sampling molecular dynamics simulations to shed light into the conformational dynamics associated with the binding of Imatinib to the proto-oncogene c-Src. We find that both conformational selection and induced fit play a role in the binding mechanism, reconciling opposing views held in the literature. Moreover, an external binding pose and local unfolding (cracking) of the aG helix are observed.

Adaptive free energy sampling in multidimensional collective variable space using boxed molecular dynamics

The past decade has seen the development of a new class of rare event methods in which molecular configuration space is divided into a set of boundaries/interfaces, and then short trajectories are run between boundaries. For all these methods, an important concern is how to generate boundaries. In this paper, we outline an algorithm for adaptively generating boundaries along a free energy surface in multi-dimensional collective variable (CV) space, building on the boxed molecular dynamics (BXD) rare event algorithm. BXD is a simple technique for accelerating the simulation of rare events and free energy sampling which has proven useful for calculating kinetics and free energy profiles in reactive and non-reactive molecular dynamics (MD) simulations across a range of systems, in both NVT and NVE ensembles. Two key developments outlined in this paper make it possible to automate BXD, and to adaptively map free energy and kinetics in complex systems. First, we have generalized BXD to multidimensional CV space. Using strategies from rigid-body dynamics, we have derived a simple and general velocity-reflection procedure that conserves energy for arbitrary collective variable definitions in multiple dimensions, and show that it is straightforward to apply BXD to sampling in multidimensional CV space so long as the Cartesian gradients ∇CV are available. Second, we have modified BXD to undertake on-the-fly statistical analysis during a trajectory, harnessing the information content latent in the dynamics to automatically determine boundary locations. Such automation not only makes BXD considerably easier to use; it also guarantees optimal boundaries, speeding up convergence. We have tested the multidimensional adaptive BXD procedure by calculating the potential of mean force for a chemical reaction recently investigated using both experimental and computational approaches – i.e., F + CD₃CN → DF + D₂CN in both the gas phase and a strongly coupled explicit CD₃CN solvent. The results obtained using multidimensional adaptive BXD agree well with previously published experimental and computational results, providing good evidence for its reliability.

Correlation Analysis of Trp-Cage Dynamics in Folded and Unfolded States

A fundamental and still debated problem is how folded structures of proteins are related to their unfolded state. Besides the classical view, in which a large number of conformations characterize the unfolded state while the folded one is dominated by a single structure, recently a reassessment of the denatured state has been suggested. A growing amount of evidence indicates that not only the folded but also the unfolded state is at least partially organized. Here, we try to answer the question of how different protein dynamics is in folded and unfolded states by performing all-atom molecular dynamics simulations on the model protein Trp-cage. Random matrix theory inspired analysis of the correlation matrices has been carried out. The spectra of these correlation matrices show that the low rank modes of Trp-cage dynamics are outside of the limit expected for a random system both in folded and in unfolded conditions. These findings shed light on the nature of the unfolded state of the proteins, suggesting that it is much less random than previously thought.

A hydrodynamic view of the first-passage folding of Trp-cage miniprotein

We study folding of Trp-cage miniprotein in the conditions when the native state of the protein is stable and unfolding events are improbable, which corresponds to physiological conditions. Using molecular dynamics simulations with an implicit solvent model, an ensemble of folding trajectories from unfolded (practically extended) states of the protein to the native state was generated. To get insight into the folding kinetics, the free energy surface and kinetic network projected on this surface were constructed. This, "conventional" analysis of the folding reaction was followed by a recently proposed hydrodynamic description of protein folding (Chekmarev et al. in Phys Rev Lett 100(1):018107, 2008), in which the process of the first-passage folding is viewed as a stationary flow of a folding "fluid" from the unfolded to native state. This approach is conceptually different from the previously used approaches and thus allows an alternative view of the folding dynamics and kinetics of Trp-cage, the conclusions about which are very diverse. In agreement with most previous studies, we observed two characteristic folding pathways: in one pathway (I), the collapse of the hydrophobic core precedes the formation of the [Formula: see text]-helix, and in the other pathway (II), these events occur in the reverse order. We found that although pathway II is complicated by a repeated partial protein unfolding, it contributes to the total folding flow as little as ≈10%, so that the folding kinetics remain essentially single-exponential.

New Insights into the Molecular Mechanism of E-Cadherin-Mediated Cell Adhesion by Free Energy Calculations

Three-dimensional domain swapping is an important mode of protein association leading to the formation of stable dimers. Monomers associating via this mechanism mutually exchange a domain to form a homodimer. Classical cadherins, an increasingly important target for anticancer therapy, use domain swapping to mediate cell adhesion. However, despite its importance, the molecular mechanism of domain swapping is still debated. Here, we study the conformational changes that lead to activation and dimerization via domain swapping of E-cadherin. Using state-of-the-art enhanced sampling atomistic simulations, we reconstruct its conformational free energy landscape, obtaining the free energy profile connecting the inactive and active form. Our simulations predict that the E-cadherin monomer populates the open and closed forms almost equally, which is in agreement with the proposed "selected fit" mechanism in which monomers in an active conformational state bind to form a homodimer, analogous to the conformational selection mechanism often observed in ligand-target binding. Moreover, we find that the open state population is increased in the presence of calcium ions at the extracellular boundary, suggesting their possible role as allosteric activators of the conformational change.

Investigating drug-target association and dissociation mechanisms using metadynamics-based algorithms

CONSPECTUS: This Account highlights recent advances and discusses major challenges in the field of drug-target recognition, binding, and unbinding studied using metadynamics-based approaches, with particular emphasis on their role in structure-based design. Computational chemistry has significantly contributed to drug design and optimization in an extremely broad range of areas, including prediction of target druggability and drug likeness, de novo design, fragment screening, ligand docking, estimation of binding affinity, and modulation of ADMET (absorption, distribution, metabolism, excretion, toxicity) properties. Computationally driven drug discovery must continuously adapt to keep pace with the evolving knowledge of the factors that modulate the pharmacological action of drugs. There is thus an urgent need for novel computational approaches that integrate the vast amount of complex information currently available for small (bio)organic compounds, biologically relevant targets and their complexes, while also accounting accurately for the thermodynamics and kinetics of drug-target association, the intrinsic dynamical behavior of biomolecular systems, and the complexity of protein-protein networks. Understanding the mechanism of drug binding to and unbinding from biological targets is fundamental for optimizing lead compounds and designing novel biologically active ones. One major challenge is the accurate description of the conformational complexity prior to and upon formation of drug-target complexes. Recently, enhanced sampling methods, including metadynamics and related approaches, have been successfully applied to investigate complex mechanisms of drugs binding to flexible targets. Metadynamics is a family of enhanced sampling techniques aimed at enhancing the rare events and reconstructing the underlying free energy landscape as a function of a set of order parameters, usually referred to as collective variables. Studies of drug binding mechanisms have predicted the most probable association and dissociation pathways and the related binding free energy profile. In addition, the availability of an efficient open-source implementation, running on cost-effective GPU (i.e., graphical processor unit) architectures, has considerably decreased the learning curve and the computational costs of the methods, and increased their adoption by the community. Here, we review the recent contributions of metadynamics and other enhanced sampling methods to the field of drug-target recognition and binding. We discuss how metadynamics has been used to search for transition states, to predict binding and unbinding paths, to treat conformational flexibility, and to compute free energy profiles. We highlight the importance of such predictions in drug discovery. Major challenges in the field and possible solutions will finally be discussed.

Tuning the Attempt Frequency of Protein Folding Dynamics via Transition-State Rigidification: Application to Trp-Cage

The attempt frequency or prefactor (k0) of the transition-state rate equation of protein folding kinetics has been estimated to be on the order of 10(6) s(-1), which is many orders of magnitude smaller than that of chemical reactions. Herein we use the mini-protein Trp-cage to show that it is possible to significantly increase the value of k0 for a protein folding reaction by rigidifying the transition state. This is achieved by reducing the conformational flexibility of a key structural element (i.e., an α-helix) formed in the transition state via photoisomerization of an azobenzene cross-linker. We find that this strategy not only decreases the folding time of the Trp-cage peptide by more than an order of magnitude (to ∼100 ns at 25°C) but also exposes parallel folding pathways, allowing us to provide, to the best of our knowledge, the first quantitative assessment of the curvature of the transition-state free-energy surface of a protein.

Sampling the equilibrium kinetic network of Trp-cage in explicit solvent

We employed the single replica multiple state transition interface sampling (MSTIS) approach to sample the kinetic (un)folding network of Trp-cage mini-protein in explicit water. Cluster analysis yielded 14 important metastable states in the network. The MSTIS simulation thus resulted in a full 14 × 14 rate matrix. Analysis of the kinetic rate matrix indicates the presence of a near native intermediate state characterized by a fully formed alpha helix, a slightly disordered proline tail, a broken salt-bridge, and a rotated arginine residue. This intermediate was also found in recent IR experiments. Moreover, the predicted rate constants and timescales are in agreement with previous experiments and simulations.

A hybrid MD-kMC algorithm for folding proteins in explicit solvent

We present a novel hybrid MD-kMC algorithm that is capable of efficiently folding proteins in explicit solvent. We apply this algorithm to the folding of a small protein, Trp-Cage. Different kMC move sets that capture different possible rate limiting steps are implemented. The first uses secondary structure formation as a relevant rate event (a combination of dihedral rotations and hydrogen-bonding formation and breakage). The second uses tertiary structure formation events through formation of contacts via translational moves. Both methods fold the protein, but via different mechanisms and with different folding kinetics. The first method leads to folding via a structured helical state, with kinetics fit by a single exponential. The second method leads to folding via a collapsed loop, with kinetics poorly fit by single or double exponentials. In both cases, folding times are faster than experimentally reported values, The secondary and tertiary move sets are integrated in a third MD-kMC implementation, which now leads to folding of the protein via both pathways, with single and double-exponential fits to the rates, and to folding rates in good agreement with experimental values. The competition between secondary and tertiary structure leads to a longer search for the helix-rich intermediate in the case of the first pathway, and to the emergence of a kinetically trapped long-lived molten-globule collapsed state in the case of the second pathway. The algorithm presented not only captures experimentally observed folding intermediates and kinetics, but yields insights into the relative roles of local and global interactions in determining folding mechanisms and rates.

From metadynamics to dynamics

Metadynamics is a commonly used and successful enhanced sampling method. By the introduction of a history dependent bias which depends on a restricted number of collective variables it can explore complex free energy surfaces characterized by several metastable states separated by large free energy barriers. Here we extend its scope by introducing a simple yet powerful method for calculating the rates of transition between different metastable states. The method does not rely on a previous knowledge of the transition states or reaction coordinates, as long as collective variables are known that can distinguish between the various stable minima in free energy space. We demonstrate that our method recovers the correct escape rates out of these stable states and also preserves the correct sequence of state-to-state transitions, with minimal extra computational effort needed over ordinary metadynamics. We apply the formalism to three different problems and in each case find excellent agreement with the results of long unbiased molecular dynamics runs.

Assessment of local friction in protein folding dynamics using a helix cross-linker

Internal friction arising from local steric hindrance and/or the excluded volume effect plays an important role in controlling not only the dynamics of protein folding but also conformational transitions occurring within the native state potential well. However, experimental assessment of such local friction is difficult because it does not manifest itself as an independent experimental observable. Herein, we demonstrate, using the miniprotein trp-cage as a testbed, that it is possible to selectively increase the local mass density in a protein and hence the magnitude of local friction, thus making its effect directly measurable via folding kinetic studies. Specifically, we show that when a helix cross-linker, m-xylene, is placed near the most congested region of the trp-cage it leads to a significant decrease in both the folding rate (by a factor of 3.8) and unfolding rate (by a factor of 2.5 at 35 °C) but has little effect on protein stability. Thus, these results, in conjunction with those obtained with another cross-linked trp-cage and two uncross-linked variants, demonstrate the feasibility of using a nonperturbing cross-linker to help quantify the effect of internal friction. In addition, we estimate that a m-xylene cross-linker could lead to an increase in the roughness of the folding energy landscape by as much as 0.4-1.0k(B)T.