Reaction Coordinate-Free Approach to Recovering Kinetics from Potential-Scaled Simulations: Application of Kramers’ Rate Theory

Ligand Gaussian Accelerated Molecular Dynamics 3 (LiGaMD3): Improved Calculations of Binding Thermodynamics and Kinetics of Both Small Molecules and Flexible Peptides

Binding thermodynamics and kinetics play critical roles in drug design. However, it has proven challenging to efficiently predict ligand binding thermodynamics and kinetics of small molecules and flexible peptides using conventional molecular dynamics (cMD), due to limited simulation time scales. Based on our previously developed ligand Gaussian accelerated molecular dynamics (LiGaMD) method, we present a new approach, termed "LiGaMD3″, in which we introduce triple boosts into three individual energy terms that play important roles in small-molecule/peptide dissociation, rebinding, and system conformational changes to improve the sampling efficiency of small-molecule/peptide interactions with target proteins. To validate the performance of LiGaMD3, MDM2 bound by a small molecule (Nutlin 3) and two highly flexible peptides (PMI and P53) were chosen as the model systems. LiGaMD3 could efficiently capture repetitive small-molecule/peptide dissociation and binding events within 2 μs simulations. The predicted binding kinetic constant rates and free energies from LiGaMD3 were in agreement with the available experimental values and previous simulation results. Therefore, LiGaMD3 provides a more general and efficient approach to capture dissociation and binding of both small-molecule ligands and flexible peptides, allowing for accurate prediction of their binding thermodynamics and kinetics.

Dynamical Reweighting for Biased Rare Event Simulations

Dynamical reweighting techniques aim to recover the correct molecular dynamics from a simulation at a modified potential energy surface. They are important for unbiasing enhanced sampling simulations of molecular rare events. Here, we review the theoretical frameworks of dynamical reweighting for modified potentials. Based on an overview of kinetic models with increasing level of detail, we discuss techniques to reweight two-state dynamics, multistate dynamics, and path integrals. We explore the natural link to transition path sampling and how the effect of nonequilibrium forces can be reweighted. We end by providing an outlook on how dynamical reweighting integrates with techniques for optimizing collective variables and with modern potential energy surfaces.

Ligand Gaussian accelerated Molecular Dynamics 3 (LiGaMD3): Improved Calculations of Binding Thermodynamics and Kinetics of Both Small Molecules and Flexible Peptides

Binding thermodynamics and kinetics play critical roles in drug design. However, it has proven challenging to efficiently predict ligand binding thermodynamics and kinetics of small molecules and flexible peptides using conventional Molecular Dynamics (cMD), due to limited simulation timescales. Based on our previously developed Ligand Gaussian accelerated Molecular Dynamics (LiGaMD) method, we present a new approach, termed "LiGaMD3", in which we introduce triple boosts into three individual energy terms that play important roles in small-molecule/peptide dissociation, rebinding and system conformational changes to improve the sampling efficiency of small-molecule/peptide interactions with target proteins. To validate the performance of LiGaMD3, MDM2 bound by a small molecule (Nutlin 3) and two highly flexible peptides (PMI and P53) were chosen as model systems. LiGaMD3 could efficiently capture repetitive small-molecule/peptide dissociation and binding events within 2 microsecond simulations. The predicted binding kinetic constant rates and free energies from LiGaMD3 agreed with available experimental values and previous simulation results. Therefore, LiGaMD3 provides a more general and efficient approach to capture dissociation and binding of both small-molecule ligand and flexible peptides, allowing for accurate prediction of their binding thermodynamics and kinetics.

Studying the Collective Functional Response of a Receptor in Alchemical Ligand Binding Free Energy Simulations with Accelerated Solvation Layer Dynamics

Ligand binding free energy simulations (LB-FES) that involve sampling of protein functional conformations have been longstanding challenges in research on molecular recognition. Particularly, modeling of the conformational transition pathway and design of the heuristic biasing mechanism are severe bottlenecks for the existing enhanced configurational sampling (ECS) methods. Inspired by the key role of hydration in regulating conformational dynamics of macromolecules, this report proposes a novel ECS approach that facilitates binding-associated structural dynamics by accelerated hydration transitions in combination with the λ-exchange of free energy perturbation (FEP). Two challenging protein-ligand binding processes involving large configurational transitions of the receptor are studied, with hydration transitions at binding sites accelerated by Hamiltonian-simulated annealing of the hydration layer. Without the need for pathway analysis or ad hoc barrier flattening potential, LB-FES were performed with FEP/λ-exchange molecular dynamics simulation at a minor overhead for annealing of the hydration layer. The LB-FES studies showed that the accelerated rehydration significantly enhances the collective conformational transitions of the receptor, and convergence of binding affinity calculations is obtained at a sweet-spot simulation time scale. Alchemical LB-FES with the proposed ECS strategy is free from the effort of trial and error for the setup and realizes efficient on-the-fly sampling for the collective functional response of the receptor and bound water and therefore presents a practical approach to high-throughput screening in drug discovery.

Mechanism of Ligand Binding to Theophylline RNA Aptamer

Studying RNA-ligand interactions and quantifying their binding thermodynamics and kinetics are of particular relevance in the field of drug discovery. Here, we combined biochemical binding assays and accelerated molecular simulations to investigate ligand binding and dissociation in RNA using the theophylline-binding RNA as a model system. All-atom simulations using a Ligand Gaussian accelerated Molecular Dynamics method (LiGaMD) have captured repetitive binding and dissociation of theophylline and caffeine to RNA. Theophylline's binding free energy and kinetic rate constants align with our experimental data, while caffeine's binding affinity is over 10,000 times weaker, and its kinetics could not be determined. LiGaMD simulations allowed us to identify distinct low-energy conformations and multiple ligand binding pathways to RNA. Simulations revealed a "conformational selection" mechanism for ligand binding to the flexible RNA aptamer, which provides important mechanistic insights into ligand binding to the theophylline-binding model. Our findings suggest that compound docking using a structural ensemble of representative RNA conformations would be necessary for structure-based drug design of flexible RNA.

Ligand Gaussian Accelerated Molecular Dynamics 2 (LiGaMD2): Improved Calculations of Ligand Binding Thermodynamics and Kinetics with Closed Protein Pocket

Ligand binding thermodynamics and kinetics are critical parameters for drug design. However, it has proven challenging to efficiently predict ligand binding thermodynamics and kinetics from molecular simulations due to limited simulation timescales. Protein dynamics, especially in the ligand binding pocket, often plays an important role in ligand binding. Based on our previously developed Ligand Gaussian accelerated molecular dynamics (LiGaMD), here we present LiGaMD2 in which a selective boost potential was applied to both the ligand and protein residues in the binding pocket to improve sampling of ligand binding and dissociation. To validate the performance of LiGaMD2, the T4 lysozyme (T4L) mutants with open and closed pockets bound by different ligands were chosen as model systems. LiGaMD2 could efficiently capture repetitive ligand dissociation and binding within microsecond simulations of all T4L systems. The obtained ligand binding kinetic rates and free energies agreed well with available experimental values and previous modeling results. Therefore, LiGaMD2 provides an improved approach to sample opening of closed protein pockets for ligand dissociation and binding, thereby allowing for efficient calculations of ligand binding thermodynamics and kinetics.

Structural insights into Nirmatrelvir (PF-07321332)-3C-like SARS-CoV-2 protease complexation: a ligand Gaussian accelerated molecular dynamics study

Coronavirus 3C-like protease (3CLpro) is found in SARS-CoV-2 virus, which causes COVID-19. 3CLpro controls virus replication and is a major target for target-based antiviral discovery. As reported by Pfizer, Nirmatrelvir (PF-07321332) is a competitive protein inhibitor and a clinical candidate for orally delivered medication. However, the binding mechanisms between Nirmatrelvir and 3CLpro complex structures remain unknown. This study incorporated ligand Gaussian accelerated molecular dynamics, the one-dimensional and two-dimensional potential of mean force, normal molecular dynamics, and Kramers' rate theory to determine the binding and dissociation rate constants (k_off and k_on) associated with the binding of the 3CLpro protein to the Nirmatrelvir inhibitor. The proposed approach addresses the challenges in designing small-molecule antiviral drugs.

Protein-Protein Interaction-Gaussian Accelerated Molecular Dynamics (PPI-GaMD): Characterization of Protein Binding Thermodynamics and Kinetics

Protein-protein interactions (PPIs) play key roles in many fundamental biological processes such as cellular signaling and immune responses. However, it has proven challenging to simulate repetitive protein association and dissociation in order to calculate binding free energies and kinetics of PPIs due to long biological timescales and complex protein dynamics. To address this challenge, we have developed a new computational approach to all-atom simulations of PPIs based on a robust Gaussian accelerated molecular dynamics (GaMD) technique. The method, termed "PPI-GaMD", selectively boosts interaction potential energy between protein partners to facilitate their slow dissociation. Meanwhile, another boost potential is applied to the remaining potential energy of the entire system to effectively model the protein's flexibility and rebinding. PPI-GaMD has been demonstrated on a model system of the ribonuclease barnase interactions with its inhibitor barstar. Six independent 2 μs PPI-GaMD simulations have captured repetitive barstar dissociation and rebinding events, which enable calculations of the protein binding thermodynamics and kinetics simultaneously. The calculated binding free energies and kinetic rate constants agree well with the experimental data. Furthermore, PPI-GaMD simulations have provided mechanistic insights into barstar binding to barnase, which involves long-range electrostatic interactions and multiple binding pathways, being consistent with previous experimental and computational findings of this model system. In summary, PPI-GaMD provides a highly efficient and easy-to-use approach for binding free energy and kinetics calculations of PPIs.

Peptide Gaussian accelerated molecular dynamics (Pep-GaMD): Enhanced sampling and free energy and kinetics calculations of peptide binding

Peptides mediate up to 40% of known protein-protein interactions in higher eukaryotes and play an important role in cellular signaling. However, it is challenging to simulate both binding and unbinding of peptides and calculate peptide binding free energies through conventional molecular dynamics, due to long biological timescales and extremely high flexibility of the peptides. Based on the Gaussian accelerated molecular dynamics (GaMD) enhanced sampling technique, we have developed a new computational method "Pep-GaMD," which selectively boosts essential potential energy of the peptide in order to effectively model its high flexibility. In addition, another boost potential is applied to the remaining potential energy of the entire system in a dual-boost algorithm. Pep-GaMD has been demonstrated on binding of three model peptides to the SH3 domains. Independent 1 µs dual-boost Pep-GaMD simulations have captured repetitive peptide dissociation and binding events, which enable us to calculate peptide binding thermodynamics and kinetics. The calculated binding free energies and kinetic rate constants agreed very well with available experimental data. Furthermore, the all-atom Pep-GaMD simulations have provided important insights into the mechanism of peptide binding to proteins that involves long-range electrostatic interactions and mainly conformational selection. In summary, Pep-GaMD provides a highly efficient, easy-to-use approach for unconstrained enhanced sampling and calculations of peptide binding free energies and kinetics.

Ligand Gaussian Accelerated Molecular Dynamics (LiGaMD): Characterization of Ligand Binding Thermodynamics and Kinetics

Calculations of ligand binding free energies and kinetic rates are important for drug design. However, such tasks have proven challenging in computational chemistry and biophysics. To address this challenge, we have developed a new computational method, ligand Gaussian accelerated molecular dynamics (LiGaMD), which selectively boosts the ligand nonbonded interaction potential energy based on the Gaussian accelerated molecular dynamics (GaMD) enhanced sampling technique. Another boost potential could be applied to the remaining potential energy of the entire system in a dual-boost algorithm (LiGaMD_Dual) to facilitate ligand binding. LiGaMD has been demonstrated on host-guest and protein-ligand binding model systems. Repetitive guest binding and unbinding in the β-cyclodextrin host were observed in hundreds-of-nanosecond LiGaMD_Dual simulations. The calculated guest binding free energies agreed excellently with experimental data with <1.0 kcal/mol errors. Compared with converged microsecond-time scale conventional molecular dynamics simulations, the sampling errors of LiGaMD_Dual simulations were also <1.0 kcal/mol. Accelerations of ligand kinetic rate constants in LiGaMD simulations were properly estimated using Kramers' rate theory. Furthermore, LiGaMD allowed us to capture repetitive dissociation and binding of the benzamidine inhibitor in trypsin within 1 μs simulations. The calculated ligand binding free energy and kinetic rate constants compared well with the experimental data. In summary, LiGaMD provides a powerful enhanced sampling approach for characterizing ligand binding thermodynamics and kinetics simultaneously, which is expected to facilitate computer-aided drug design.

Long-time methods for molecular dynamics simulations: Markov State Models and Milestoning

Molecular dynamics (MD) studies of biomolecules require the ability to simulate complex biochemical systems with an increasingly larger number of particles and for longer time scales, a problem that cannot be overcome by computational hardware advances alone. A main problem springs from the intrinsically high-dimensional and complex nature of the underlying free energy landscape of most systems, and from the necessity to sample accurately such landscapes for identifying kinetic and thermodynamic states in the configurations space, and for accurate calculations of both free energy differences and of the corresponding transition rates between states. Here, we review and present applications of two increasingly popular methods that allow long-time MD simulations of biomolecular systems that can open a broad spectrum of new studies. A first approach, Markov State Models (MSMs), relies on identifying a set of configuration states in which the system resides sufficiently long to relax and loose the memory of previous transitions, and on using simulations for mapping the underlying complex energy landscape and for extracting accurate thermodynamic and kinetic information. The Markovian independence of the underlying transition probabilities creates the opportunity to increase the sampling efficiency by using sets of appropriately initialized short simulations rather than typically long MD trajectories, which also enhances sampling. This allows MSM-based studies to unveil bio-molecular mechanisms and to estimate free energy barriers with high accuracy, in a manner that is both systematic and relatively automatic, which accounts for their increasing popularity. The second approach presented, Milestoning, targets accurate studies of the ensemble of pathways connecting specific end-states (e.g., reactants and products) in a similarly systematic, accurate and highly automatic manner. Applications presented range from studies of conformational dynamics and binding of amyloid-forming peptides, cell-penetrating peptides and the DFG-flip dynamics in Abl kinase. As highlighted by the increasing number of studies using both methods, we anticipate that they will open new avenues for the investigation of systematic sampling of reactions pathways and mechanisms occurring on longer time scales than currently accessible by purely computational hardware developments.

Dynamical reweighting methods for Markov models

Conformational dynamics is essential to biomolecular processes. Markov State Models (MSMs) are widely used to elucidate dynamic properties of molecular systems from unbiased Molecular Dynamics (MD). However, the implementation of reweighting schemes for MSMs to analyze biased simulations is still at an early stage of development. Several dynamical reweighing approaches have been proposed, which can be classified as approaches based on (i) Kramers rate theory, (ii) rescaling of the probability density flux, (iii) reweighting by formulating a likelihood function, (iv) path reweighting. We present the state-of-the-art and discuss the methodological differences of these methods, their limitations and recent applications.

Accelerating Rare Dissociative Processes in Biomolecules Using Selectively Scaled MD Simulations

Molecular dynamics (MD) simulations can be a powerful tool for modeling complex dissociative processes such as ligand unbinding. However, many biologically relevant dissociative processes occur on timescales that far exceed the timescales of typical MD simulations. Here, we implement and apply an enhanced sampling method in which specific energy terms in the potential energy function are selectively "scaled" to accelerate dissociative events during simulations. Using ligand unbinding as an example of a complex dissociative process, we selectively scaled up ligand-water interactions in an attempt to increase the rate of ligand unbinding. After applying our selectively scaled MD (ssMD) approach to several cyclin-dependent kinase-inhibitor complexes, we discovered that we could accelerate ligand unbinding, thereby allowing, in some cases, unbinding events to occur within as little as 2 ns. Moreover, we found that we could make realistic estimates of the initial unbinding times (τ_unbind^sim) as well as the accompanying change in free energy (ΔG^sim) of the inhibitors from our ssMD simulation data. To accomplish this, we employed a previously described Kramers'-based rate extrapolation method and a newly described free energy extrapolation method. Because our ssMD approach is general, it should find utility as an easy-to-deploy, enhanced sampling method for modeling other dissociative processes.

Conformational analysis of replica exchange MD: Temperature-dependent Markov networks for FF amyloid peptides

Recent molecular modeling methods using Markovian descriptions of conformational states of biomolecular systems have led to powerful analysis frameworks that can accurately describe their complex dynamical behavior. In conjunction with enhanced sampling methods, such as replica exchange molecular dynamics (REMD), these frameworks allow the systematic and accurate extraction of transition probabilities between the corresponding states, in the case of Markov state models, and of statistically-optimized transition rates, in the case of the corresponding coarse master equations. However, applying automatically such methods to large molecular dynamics (MD) simulations, with explicit water molecules, remains limited both by the initial ability to identify good candidates for the underlying Markovian states and by the necessity to do so using good collective variables as reaction coordinates that allow the correct counting of inter-state transitions at various lag times. Here, we show that, in cases when representative molecular conformations can be identified for the corresponding Markovian states, and thus their corresponding collective evolution of atomic positions can be calculated along MD trajectories, one can use them to build a new type of simple collective variable, which can be particularly useful in both the correct state assignment and in the subsequent accurate counting of inter-state transition probabilities. In the case of the ubiquitously used root-mean-square deviation (RMSD) of atomic positions, we introduce the relative RMSD (RelRMSD) measure as a good reaction coordinate candidate. We apply this method to the analysis of REMD trajectories of amyloid-forming diphenylalanine (FF) peptides-a system with important nanotechnology and biomedical applications due to its self-assembling and piezoelectric properties-illustrating the use of RelRMSD in extracting its temperature-dependent intrinsic kinetics, without a priori assumptions on the functional form (e.g., Arrhenius or not) of the underlying conformational transition rates. The RelRMSD analysis enables as well a more objective assessment of the convergence of the REMD simulations. This type of collective variable may be generalized to other observables that could accurately capture conformational differences between the underlying Markov states (e.g., distance RMSD, the fraction of native contacts, etc.).

Acceleration of biomolecular kinetics in Gaussian accelerated molecular dynamics

Recent studies demonstrated that Gaussian accelerated molecular dynamics (GaMD) is a robust computational technique, which provides simultaneous unconstrained enhanced sampling and free energy calculations of biomolecules. However, the exact acceleration of biomolecular dynamics or speedup of kinetic rates in GaMD simulations and, more broadly, in enhanced sampling methods, remains a challenging task to be determined. Here, the GaMD acceleration is examined using alanine dipeptide in explicit solvent as a biomolecular model system. Relative to long conventional molecular dynamics simulation, GaMD simulations exhibited ∼36-67 times speedup for sampling of the backbone dihedral transitions. The acceleration depended on level of the GaMD boost potential. Furthermore, Kramers' rate theory was applied to estimate GaMD acceleration using simulation-derived diffusion coefficients, curvatures and barriers of free energy profiles. In most cases, the calculations also showed significant speedup of dihedral transitions in GaMD, although the GaMD acceleration factors tended to be underestimated by ∼3-96 fold. Because greater boost potential can be applied in GaMD simulations of systems with increased sizes, which potentially leads to higher acceleration, it is subject to future studies on accelerating the dynamics and recovering kinetic rates of larger biomolecules such as proteins and protein-protein/nucleic acid complexes.

How Does Solvation Layer Mobility Affect Protein Structural Dynamics?

Solvation is critical for protein structural dynamics. Spectroscopic studies have indicated relationships between protein and solvent dynamics, and rates of gas binding to heme proteins in aqueous solution were previously observed to depend inversely on solution viscosity. In this work, the solvent-compatible enzyme Candida antarctica lipase B, which functions in aqueous and organic solvents, was modeled using molecular dynamics simulations. Data was obtained for the enzyme in acetonitrile, cyclohexane, n-butanol, and tert-butanol, in addition to water. Protein dynamics and solvation shell dynamics are characterized regionally: for each α-helix, β-sheet, and loop or connector region. Correlations are seen between solvent mobility and protein flexibility. So, does local viscosity explain the relationship between protein structural dynamics and solvation layer dynamics? Halle and Davidovic presented a cogent analysis of data describing the global hydrodynamics of a protein (tumbling in solution) that fits a model in which the protein's interfacial viscosity is higher than that of bulk water's, due to retarded water dynamics in the hydration layer (measured in NMR τ2 reorientation times). Numerous experiments have shown coupling between protein and solvation layer dynamics in site-specific measurements. Our data provides spatially-resolved characterization of solvent shell dynamics, showing correlations between regional solvation layer dynamics and protein dynamics in both aqueous and organic solvents. Correlations between protein flexibility and inverse solvent viscosity (1/η) are considered across several protein regions and for a rather disparate collection of solvents. It is seen that the correlation is consistently higher when local solvent shell dynamics are considered, rather than bulk viscosity. Protein flexibility is seen to correlate best with either the local interfacial viscosity or the ratio of the mobility of an organic solvent in a regional solvation layer relative to hydration dynamics around the same region. Results provide insight into the function of aqueous proteins, while also suggesting a framework for interpreting and predicting enzyme structural dynamics in non-aqueous solvents, based on the mobility of solvents within the solvation layer. We suggest that Kramers' theory may be used in future work to model protein conformational transitions in different solvents by incorporating local viscosity effects.

Peptide dimerization-dissociation rates from replica exchange molecular dynamics

We show how accurate rates of formation and dissociation of peptide dimers can be calculated using direct transition counting (DTC) from replica-exchange molecular dynamics (REMD) simulations. First, continuous trajectories corresponding to system replicas evolving at different temperatures are used to assign conformational states. Second, we analyze the entire REMD data to calculate the corresponding rates at each temperature directly from the number of transition counts. Finally, we compare the kinetics extracted directly, using the DTC method, with indirect estimations based on trajectory likelihood maximization using short-time propagators and on decay rates of state autocorrelation functions. For systems with relatively low-dimensional intrinsic conformational dynamics, the DTC method is simple to implement and leads to accurate temperature-dependent rates. We apply the DTC rate-extraction method to all-atom REMD simulations of dimerization of amyloid-forming NNQQ tetrapetides in explicit water. In an assessment of the REMD sampling efficiency with respect to standard MD, we find a gain of more than a factor of two at the lowest temperature.

Kinetics for Drug Discovery: an industry-driven effort to target drug residence time

A considerable number of approved drugs show non-equilibrium binding characteristics, emphasizing the potential role of drug residence times for in vivo efficacy. Therefore, a detailed understanding of the kinetics of association and dissociation of a target-ligand complex might provide crucial insight into the molecular mechanism-of-action of a compound. This deeper understanding will help to improve decision making in drug discovery, thus leading to a better selection of interesting compounds to be profiled further. In this review, we highlight the contributions of the Kinetics for Drug Discovery (K4DD) Consortium, which targets major open questions related to binding kinetics in an industry-driven public-private partnership.