Full kinetics of CO entry, internal diffusion, and exit in myoglobin from transition-path theory simulations

Vibrational analyses of the reaction of oxymyoglobin with NO using a photolabile caged NO donor at cryogenic temperatures

The NO dioxygenation reaction catalyzed by heme-containing globin proteins is a crucial aerobic detoxification pathway. Accordingly, the second order reaction of NO with oxymyoglobin and oxyhemoglobin has been the focus of a large number of kinetic and spectroscopic studies. Stopped-flow and rapid-freeze-quench (RFQ) measurements have provided evidence for the formation of a Fe(III)-nitrato complex with millisecond lifetime prior to release of the nitrate product, but the temporal resolution of these techniques is insufficient for the characterization of precursor species. Most mechanistic models assume the formation of an initial Fe(III)-peroxynitrite species prior to homolytic cleavage of the OO bond and recombination of the resulting NO2 and Fe(IV)=O species. Here we report vibrational spectroscopy measurements for the reaction of oxymyoglobin with a photolabile caged NO donor at cryogenic temperatures. We show that this approach offers efficient formation and trapping of the Fe(III)-nitrato, enzyme-product, complex at 180 K. Resonance Raman spectra of the Fe(III)-nitrato complex trapped via RFQ in the liquid phase and photolabile NO release at cryogenic temperatures are indistinguishable, demonstrating the complementarity of these approaches. Caged NO is released by irradiation <180 K but diffusion into the heme pocket is fully inhibited. Our data provide no evidence for Fe(III)-peroxynitrite of Fe(IV)=O species, supporting low activation energies for the NO to nitrate conversion at the oxymyoglobin reaction site. Photorelease of NO at cryogenic temperatures allows monitoring of the reaction by transmittance FTIR which provides valuable quantitative information and promising prospects for the detection of protein sidechain reorganization events in NO-reacting metalloenzymes.

Markov State Models Underlying the N-Terminal Premodel of TOPK/PBK

Lymphokine-activated killer T-cell-originated protein kinase (TOPK) is a potential target for cancer therapy. To explore the micromechanism, we proposed the N-terminal premodel (NTPM) of the TOPK monomer via homology modeling and molecular dynamic simulations and analyzed the conformational dynamics by Markov state model analysis. The electronegative insert (ENI) motif of the NTPM can be opened with a small probability under wild type, regulated by the so-called "N-C" interaction zone consisting of the N-terminal head, the coil between β₃-strand and αC-helix, and the ENI motif. Glutamate substitution at threonine residue 9 or tyrosine residue 74 promotes the closed-open transition, revealing the details of phosphorylation. Allosteric effects induce functionally relevant structural changes, such as increased structural flexibility and active sites, which are thought to be necessary for further activation or binding. These findings provide rational structural templates for designing state-dependent inhibitors and give insight into the molecular regulatory mechanisms of TOPK monomers.

Nonadiabatic transition paths from quantum jump trajectories

We present a means of studying rare reactive pathways in open quantum systems using transition path theory and ensembles of quantum jump trajectories. This approach allows for the elucidation of reactive paths for dissipative, nonadiabatic dynamics when the system is embedded in a Markovian environment. We detail the dominant pathways and rates of thermally activated processes and the relaxation pathways and photoyields following vertical excitation in a minimal model of a conical intersection. We find that the geometry of the conical intersection affects the electronic character of the transition state as defined through a generalization of a committor function for a thermal barrier crossing event. Similarly, the geometry changes the mechanism of relaxation following a vertical excitation. Relaxation in models resulting from small diabatic coupling proceeds through pathways dominated by pure dephasing, while those with large diabatic coupling proceed through pathways limited by dissipation. The perspective introduced here for the nonadiabatic dynamics of open quantum systems generalizes classical notions of reactive paths to fundamentally quantum mechanical processes.

Markovian Weighted Ensemble Milestoning (M-WEM): Long-Time Kinetics from Short Trajectories

We introduce a rare-event sampling scheme, named Markovian Weighted Ensemble Milestoning (M-WEM), which inlays a weighted ensemble framework within a Markovian milestoning theory to efficiently calculate thermodynamic and kinetic properties of long-time-scale biomolecular processes from short atomistic molecular dynamics simulations. M-WEM is tested on the Müller-Brown potential model, the conformational switching in alanine dipeptide, and the millisecond time-scale protein-ligand unbinding in a trypsin-benzamidine complex. Not only can M-WEM predict the kinetics of these processes with quantitative accuracy but it also allows for a scheme to reconstruct a multidimensional free-energy landscape along additional degrees of freedom, which are not part of the milestoning progress coordinate. For the ligand-receptor system, the experimental residence time, association and dissociation kinetics, and binding free energy could be reproduced using M-WEM within a simulation time of a few hundreds of nanoseconds, which is a fraction of the computational cost of other currently available methods, and close to 4 orders of magnitude less than the experimental residence time. Due to the high accuracy and low computational cost, the M-WEM approach can find potential applications in kinetics and free-energy-based computational drug design.

Water Dynamics in a Peptide-appended Pillar[5]arene Artificial Channel in Lipid and Biomimetic Membranes

Peptide-appended Pillar[5]arene (PAP) is an artificial water channel that can be incorporated into lipid and polymeric membranes to achieve high permeability and enhanced selectivity for angstrom-scale separations [Shen et al. Nat. Commun.9:2294 (2018)]. In comparison to commonly studied rigid carbon nanotubes, PAP channels are conformationally flexible, yet these channels allow a high water permeability [Y. Liu and H. Vashisth Phys. Chem. Chem. Phys.21:22711 (2019)]. Using molecular dynamics (MD) simulations, we study water dynamics in PAP channels embedded in biological (lipid) and biomimetic (block-copolymer) membranes to probe the effect of the membrane environment on water transport characteristics of PAP channels. We have resolved the free energy surface and local minima for water diffusion within the channel in each type of membrane. We find that water follows single file transport with low free-energy barriers in regions surroundings the central ring of the PAP channel and the single file diffusivity of water correlates with the number of hydrogen bonding sites within the channel, as is known for other sub-nm pore-size synthetic and biological water channels [Horner et al. Sci. Adv.1:e1400083 (2015)].

Collective variable-based enhanced sampling and machine learning

ABSTRACT

Collective variable-based enhanced sampling methods have been widely used to study thermodynamic properties of complex systems. Efficiency and accuracy of these enhanced sampling methods are affected by two factors: constructing appropriate collective variables for enhanced sampling and generating accurate free energy surfaces. Recently, many machine learning techniques have been developed to improve the quality of collective variables and the accuracy of free energy surfaces. Although machine learning has achieved great successes in improving enhanced sampling methods, there are still many challenges and open questions. In this perspective, we shall review recent developments on integrating machine learning techniques and collective variable-based enhanced sampling approaches. We also discuss challenges and future research directions including generating kinetic information, exploring high-dimensional free energy surfaces, and efficiently sampling all-atom configurations.

GRAPHIC ABSTRACT

Free energy and kinetics of cAMP permeation through connexin26 via applied voltage and milestoning

The connexin family is a diverse group of highly regulated wide-pore channels permeable to biological signaling molecules. Despite the critical roles of connexins in mediating selective molecular signaling in health and disease, the basis of molecular permeation through these pores remains unclear. Here, we report the thermodynamics and kinetics of binding and transport of a second messenger, adenosine-3',5'-cyclophosphate (cAMP), through a connexin26 hemichannel (Cx26). First, inward and outward fluxes of cAMP molecules solvated in KCl solution were obtained from 4 μs of ± 200 mV simulations. These fluxes data yielded a single-channel permeability of cAMP and cAMP/K⁺ permeability ratio consistent with experimentally measured values. The results from voltage simulations were then compared with the potential of mean force (PMF) and the mean first passage times (MFPTs) of a single cAMP without voltage, obtained from a total of 16.5 μs of Voronoi-tessellated Markovian milestoning simulations. Both the voltage simulations and the milestoning simulations revealed two cAMP-binding sites, for which the binding constants K_D and dissociation rates k_off were computed from PMF and MFPTs. The protein dipole inside the pore produces an asymmetric PMF, reflected in unequal cAMP MFPTs in each direction once within the pore. The free energy profiles under opposite voltages were derived from the milestoning PMF and revealed the interplay between voltage and channel polarity on the total free energy. In addition, we show how these factors influence the cAMP dipole vector during permeation, and how cAMP affects the local and nonlocal pore diameter in a position-dependent manner.

Structural Mechanism of ω-Currents in a Mutated Kv7.2 Voltage Sensor Domain from Molecular Dynamics Simulations

Activation of voltage-gated ion channels is regulated by conformational changes of the voltage sensor domains (VSDs), four water- and ion-impermeable modules peripheral to the central, permeable pore domain. Anomalous currents, defined as ω-currents, have been recorded in response to mutations of residues on the VSD S4 helix and associated with ion fluxes through the VSDs. In humans, gene defects in the potassium channel Kv7.2 result in a broad range of epileptic disorders, from benign neonatal seizures to severe epileptic encephalopathies. Experimental evidence suggests that the R207Q mutation in S4, associated with peripheral nerve hyperexcitability, induces ω-currents at depolarized potentials, but the fine structural details are still elusive. In this work, we use atom-detailed molecular dynamics simulations and a refined model structure of the Kv7.2 VSD in the active conformation in a membrane/water environment to study the effect of R207Q and four additional mutations of proven clinical importance. Our results demonstrate that the R207Q mutant shows the most pronounced increase of hydration in the internal VSD cavity, a feature favoring the occurrence of ω-currents. Free energy and kinetics calculations of sodium permeation through the native and mutated VSD indicate as more favorable the formation of a cationic current in the latter. Overall, our simulations establish a mechanistic linkage between genetic variations and their physiological outcome, by providing a computational description that includes both thermodynamic and kinetic features of ion permeation associated with ω-currents.

Thermodynamics and Kinetics of Ion Permeation in Wild-Type and Mutated Open Active Conformation of the Human α7 Nicotinic Receptor

Molecular studies of human pentameric ligand-gated ion channels (LGICs) expressed in neurons and at neuromuscular junctions are of utmost importance in the development of therapeutic strategies for neurological disorders. We focus here on the nicotinic acetylcholine receptor nAChR-α7, a homopentameric channel widely expressed in the human brain, with a proven role in a wide spectrum of disorders including schizophrenia and Alzheimer’s disease. By exploiting an all-atom structural model of the full (transmembrane and extracellular) protein in the open, agonist-bound conformation we recently developed, we evaluate the free energy and the mean first passage time of single-ion permeation using molecular dynamics simulations and the milestoning method with Voronoi tessellation. The results for the wild-type channel provide the first available mapping of the potential of mean force in the full-length α7 nAChR, reveal its expected cationic nature, and are in good agreement with simulation data for other channels of the LGIC family and with experimental data on nAChRs. We then investigate the role of a specific mutation directly related to ion selectivity in LGICs, the E-1′ → A-1′ substitution at the cytoplasmatic selectivity filter. We find that the mutation strongly affects sodium and chloride permeation in opposite directions, leading to a complete inversion of selectivity, at variance with the limited experimental results available that classify this mutant as cationic. We thus provide structural determinants for the observed cationic-to-anionic inversion, revealing a key role of the protonation state of residue rings far from the mutation, in the proximity of the hydrophobic channel gate.

The Impact of Electron Correlation on Describing QM/MM Interactions in the Attendant Molecular Dynamics Simulations of CO in Myoglobin

The impact of the dispersion and electron correlation effects on describing quantum mechanics/molecular mechanics (QM/MM) interactions in QM/MM molecular dynamics (MD) simulations was explored by performing a series of up to 2 ns QM/MM MD simulations on the B states of the myoglobin-carbon monoxide (MbCO) system. The results indicate that both dispersion and electron correlations play significant roles in the simulation of the ratios of two B states (B₁/B₂), which suggests that the inclusion of the electron correlation effects is essential for accurately modeling the interactions between QM and MM subsystems. We found that the QM/MM interaction energies between the CO and the surroundings statistically present a linear correlation with the electric fields along the CO bond. This indicates that QM/MM interactions can be described by a simple physical model of a dipole with constant moment under the action of the electric fields. The treatment provides us with an accurate and effective approach to account for the electron correlation effects in QM/MM MD simulations.

Multiscale simulation approaches to modeling drug-protein binding

Simulations can provide detailed insight into the molecular processes involved in drug action, such as protein-ligand binding, and can therefore be a valuable tool for drug design and development. Processes with a large range of length and timescales may be involved, and understanding these different scales typically requires different types of simulation methodology. Ideally, simulations should be able to connect across scales, to analyze and predict how changes at one scale can influence another. Multiscale simulation methods, which combine different levels of treatment, are an emerging frontier with great potential in this area. Here we review multiscale frameworks of various types, and selected applications to biomolecular systems with a focus on drug-ligand binding.

Lessons Learned from 50 Years of Hemoglobin Research: Unstirred and Cell-Free Layers, Electrostatics, Baseball Gloves, and Molten Globules

Significance: Over the past 50 years, the mechanisms for O₂ storage and transport have been determined quantitatively on distance scales from millimeters to tenths of nanometers and timescales from seconds to picoseconds. Recent Advances: In this review, I have described four key conclusions from work done by my group and our close colleagues. (i) O₂ uptake by mammalian red cells is limited by diffusion through unstirred water layers adjacent to the cell surface and across cell-free layers adjacent to vessel walls. (ii) In most vertebrates, hemoglobins (Hbs) and myoglobins (Mbs), the distal histidine at the E7 helical position donates a strong hydrogen bond to bound O₂, which selectively enhances O₂ affinity, prevents carbon monoxide poisoning, and markedly slows autoxidation. (iii) O₂ binding to mammalian Hbs and Mbs occurs by migration of the ligand through a channel created by upward rotation of the His(E7) side chain, capture in the empty space of the distal pocket, and then coordination with the ferroprotoporphyrin IX (heme) iron atom. (iv) The assembly of Mbs and Hbs occurs by formation of molten globule intermediates, in which the N- and C-terminal helices have almost fully formed secondary structures, but the heme pockets are disordered and followed by high-affinity binding of heme. Critical Issues: These conclusions indicate that there are often compromises between O₂ transport function, holoprotein stability, and the efficiency of assembly. Future Directions: However, the biochemical mechanisms underlying these conclusions provide the framework for understanding globin evolution in greater detail and for engineering more efficient and stable globins.

Mapping saddles and minima on free energy surfaces using multiple climbing strings

Locating saddle points on free energy surfaces is key in characterizing multistate transition events in complicated molecular-scale systems. Because these saddle points represent transition states, determining minimum free energy pathways to these saddles and measuring their free energies relative to their connected minima are further necessary, for instance, to estimate transition rates. In this work, we propose a new multistring version of the climbing string method in collective variables to locate all saddles and corresponding pathways on free energy surfaces. The method uses dynamic strings to locate saddles and static strings to keep a history of prior strings converged to saddles. Interaction of the dynamic strings with the static strings is used to avoid the convergence to already-identified saddles. Additionally, because the strings approximate curves in collective-variable space, and we can measure free energy along each curve, identification of any saddle's two connected minima is guaranteed. We demonstrate this method to map the network of stationary points in the 2D and 4D free energy surfaces of alanine dipeptide and alanine tripeptide, respectively.

Diffusion network of CO in FeFe-Hydrogenase

FeFe-hydrogenase is an efficient enzyme to produce H₂ under optimal conditions. However, the activity of this enzyme is highly sensitive to the presence of inhibitory gases CO and O₂ that cause irreversible damage to the active site. Therefore, a detailed knowledge of the diffusion pathways of these inhibitory gases is necessary to develop strategies for designing novel enzymes that are tolerant to these gases. In this work, we studied the diffusion pathways of CO in the CpI FeFe-hydrogenase from Clostridium pasteurianum. Specifically, we used several enhanced sampling and free-energy simulation methods to reconstruct a three-dimensional free-energy surface for CO diffusion which revealed 45 free-energy minima forming an interconnected network of pathways. We discovered multiple pathways of minimal free-energy as diffusion portals for CO and found that previously suggested hydrophobic pathways are not thermodynamically favorable for CO diffusion. We also observed that the global minimum in the free-energy surface is located in the vicinity of the active-site metal cluster, the H-cluster, which suggests a high-affinity for CO near the active site. Among 19 potential residues that we propose as candidates for future mutagenesis studies, 11 residues are shared with residues that have been previously proposed to increase the tolerance of this enzyme for O₂. We hypothesize that these shared candidate residues are potentially useful for designing new variants of this enzyme that are tolerant to both inhibitory gases.

Mapping the Substrate Recognition Pathway in Cytochrome P450

Cytochrome P450s are ubiquitous metalloenzymes involved in the metabolism and detoxification of foreign components via catalysis of the hydroxylation reactions of a vast array of organic substrates. However, the mechanism underlying the pharmaceutically critical process of substrate access to the catalytic center of cytochrome P450 is a long-standing puzzle, further complicated by the crystallographic evidence of a closed catalytic center in both substrate-free and substrate-bound cytochrome P450. Here, we address a crucial question whether the conformational heterogeneity prevalent in cytochrome P450 translates to heterogeneous pathways for substrate access to the catalytic center of these metalloenzymes. By atomistically capturing the full process of spontaneous substrate association from bulk solvent to the occluded catalytic center of an archetypal system P450cam in multi-microsecond-long continuous unbiased molecular dynamics simulations, we here demonstrate that the substrate recognition in P450cam always occurs through a single well-defined dominant pathway. The simulated final bound pose resulting from these unguided simulations is in striking resemblance with the crystallographic bound pose. Each individual binding trajectory reveals that the substrate, initially placed at random locations in bulk solvent, spontaneously lands on a single key channel on the protein-surface of P450cam and resides there for an uncharacteristically long period, before correctly identifying the occluded target-binding cavity. Surprisingly, the passage of substrate to the closed catalytic center is not accompanied by any large-scale opening in protein. Rather, the unbiased simulated trajectories (∼57 μs) and underlying Markov state model, in combination with free-energy analysis, unequivocally show that the substrate recognition process in P450cam needs a substrate-induced side-chain displacement coupled with a complex array of dynamical interconversions of multiple metastable substrate conformations. Further, the work reconciles multiple precedent experimental and theoretical observations on P450cam and establishes a comprehensive view of substrate-recognition in cytochrome P450 that only occurs via substrate-induced structural rearrangements.

Quantitative Ranking of Ligand Binding Kinetics with a Multiscale Milestoning Simulation Approach

Efficient prediction and ranking of small molecule binders by their kinetic ( kon and koff) and thermodynamic ( Δ G) properties can be a valuable metric for drug lead optimization, as these quantities are often indicators of in vivo efficacy. We have previously described a hybrid molecular dynamics, Brownian dynamics, and milestoning model, Simulation Enabled Estimation of Kinetic Rates (SEEKR), that can predict kon's, koff's, and Δ G's. Here we demonstrate the effectiveness of this approach for ranking a series of seven small molecule compounds for the model system, β-cyclodextrin, based on predicted kon's and koff's. We compare our results using SEEKR to experimentally determined rates as well as rates calculated using long time scale molecular dynamics simulations and show that SEEKR can effectively rank the compounds by koff and Δ G with reduced computational cost. We also provide a discussion of convergence properties and sensitivities of calculations with SEEKR to establish "best practices" for its future use.

Atomic resolution mechanism of ligand binding to a solvent inaccessible cavity in T4 lysozyme

Ligand binding sites in proteins are often localized to deeply buried cavities, inaccessible to bulk solvent. Yet, in many cases binding of cognate ligands occurs rapidly. An intriguing system is presented by the L99A cavity mutant of T4 Lysozyme (T4L L99A) that rapidly binds benzene (~106 M-1s-1). Although the protein has long served as a model system for protein thermodynamics and crystal structures of both free and benzene-bound T4L L99A are available, the kinetic pathways by which benzene reaches its solvent-inaccessible binding cavity remain elusive. The current work, using extensive molecular dynamics simulation, achieves this by capturing the complete process of spontaneous recognition of benzene by T4L L99A at atomistic resolution. A series of multi-microsecond unbiased molecular dynamics simulation trajectories unequivocally reveal how benzene, starting in bulk solvent, diffuses to the protein and spontaneously reaches the solvent inaccessible cavity of T4L L99A. The simulated and high-resolution X-ray derived bound structures are in excellent agreement. A robust four-state Markov model, developed using cumulative 60 μs trajectories, identifies and quantifies multiple ligand binding pathways with low activation barriers. Interestingly, none of these identified binding pathways required large conformational changes for ligand access to the buried cavity. Rather, these involve transient but crucial opening of a channel to the cavity via subtle displacements in the positions of key helices (helix4/helix6, helix7/helix9) leading to rapid binding. Free energy simulations further elucidate that these channel-opening events would have been unfavorable in wild type T4L. Taken together and via integrating with results from experiments, these simulations provide unprecedented mechanistic insights into the complete ligand recognition process in a buried cavity. By illustrating the power of subtle helix movements in opening up multiple pathways for ligand access, this work offers an alternate view of ligand recognition in a solvent-inaccessible cavity, contrary to the common perception of a single dominant pathway for ligand binding.

A new paradigm for atomically detailed simulations of kinetics in biophysical systems

The kinetics of biochemical and biophysical events determined the course of life processes and attracted considerable interest and research. For example, modeling of biological networks and cellular responses relies on the availability of information on rate coefficients. Atomically detailed simulations hold the promise of supplementing experimental data to obtain a more complete kinetic picture. However, simulations at biological time scales are challenging. Typical computer resources are insufficient to provide the ensemble of trajectories at the correct length that is required for straightforward calculations of time scales. In the last years, new technologies emerged that make atomically detailed simulations of rate coefficients possible. Instead of computing complete trajectories from reactants to products, these approaches launch a large number of short trajectories at different positions. Since the trajectories are short, they are computed trivially in parallel on modern computer architecture. The starting and termination positions of the short trajectories are chosen, following statistical mechanics theory, to enhance efficiency. These trajectories are analyzed. The analysis produces accurate estimates of time scales as long as hours. The theory of Milestoning that exploits the use of short trajectories is discussed, and several applications are described.

Fast Dynamic Docking Guided by Adaptive Electrostatic Bias: The MD-Binding Approach

Engineering chemical entities to modify how pharmaceutical targets function, as it is done in drug design, requires a good understanding of molecular recognition and binding. In this context, the limitations of statically describing bimolecular recognition, as done in docking/scoring, call for insightful and efficient dynamical investigations. On the experimental side, the characterization of dynamical binding processes is still in its infancy. Thus, computer simulations, particularly molecular dynamics (MD), are compelled to play a prominent role, allowing a deeper comprehension of the binding process and its causes and thus a more informed compound selection, making more significant the computational contribution to drug discovery (Carlson, H. A. Curr. Opin. Chem. Biol. 2002, 6, 447-452). Unfortunately, MD-based approaches cannot yet describe complex events without incurring prohibitive time and computational costs. Here, we present a new method for fully and dynamically simulating drug-target-complex formations, tested against a real world and pharmaceutically relevant benchmark set. The method, based on an adaptive, electrostatics-inspired bias, envisions a campaign of trivially parallel short MD simulations and a strategy to identify a near native binding pose from the sampled configurations. At an affordable computational cost, this method provided predictions of good accuracy also when the starting protein conformation was different from that of the crystal complex, a known hurdle for traditional molecular docking (Lexa, K. W.; Carlson, H. A. Q. Rev. Biophys. 2012, 45, 301-343). Moreover, along the observed binding routes, it identified some key features also found by much more computationally expensive plain-MD simulations. Overall, this methodology represents significant progress in the description of binding phenomena.

SEEKR: Simulation Enabled Estimation of Kinetic Rates, A Computational Tool to Estimate Molecular Kinetics and Its Application to Trypsin-Benzamidine Binding

We present the Simulation Enabled Estimation of Kinetic Rates (SEEKR) package, a suite of open-source scripts and tools designed to enable researchers to perform multiscale computation of the kinetics of molecular binding, unbinding, and transport using a combination of molecular dynamics, Brownian dynamics, and milestoning theory. To demonstrate its utility, we compute the kon, koff, and ΔGbind for the protein trypsin with its noncovalent binder, benzamidine, and examine the kinetics and other results generated in the context of the new software, and compare our findings to previous studies performed on the same system. We compute a kon estimate of (2.1 ± 0.3) × 107 M-1 s-1, a koff estimate of 83 ± 14 s-1, and a ΔGbind of -7.4 ± 0.1 kcal·mol-1, all of which compare closely to the experimentally measured values of 2.9 × 107 M-1 s-1, 600 ± 300 s-1, and -6.71 ± 0.05 kcal·mol-1, respectively.

Ligand diffusion in proteins via enhanced sampling in molecular dynamics

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

Rate Constants and Mechanisms of Protein-Ligand Binding

Whereas protein-ligand binding affinities have long-established prominence, binding rate constants and binding mechanisms have gained increasing attention in recent years. Both new computational methods and new experimental techniques have been developed to characterize the latter properties. It is now realized that binding mechanisms, like binding rate constants, can and should be quantitatively determined. In this review, we summarize studies and synthesize ideas on several topics in the hope of providing a coherent picture of and physical insight into binding kinetics. The topics include microscopic formulation of the kinetic problem and its reduction to simple rate equations; computation of binding rate constants; quantitative determination of binding mechanisms; and elucidation of physical factors that control binding rate constants and mechanisms.

Multiscale implementation of infinite-swap replica exchange molecular dynamics

Replica exchange molecular dynamics (REMD) is a popular method to accelerate conformational sampling of complex molecular systems. The idea is to run several replicas of the system in parallel at different temperatures that are swapped periodically. These swaps are typically attempted every few MD steps and accepted or rejected according to a Metropolis-Hastings criterion. This guarantees that the joint distribution of the composite system of replicas is the normalized sum of the symmetrized product of the canonical distributions of these replicas at the different temperatures. Here we propose a different implementation of REMD in which (i) the swaps obey a continuous-time Markov jump process implemented via Gillespie's stochastic simulation algorithm (SSA), which also samples exactly the aforementioned joint distribution and has the advantage of being rejection free, and (ii) this REMD-SSA is combined with the heterogeneous multiscale method to accelerate the rate of the swaps and reach the so-called infinite-swap limit that is known to optimize sampling efficiency. The method is easy to implement and can be trivially parallelized. Here we illustrate its accuracy and efficiency on the examples of alanine dipeptide in vacuum and C-terminal β-hairpin of protein G in explicit solvent. In this latter example, our results indicate that the landscape of the protein is a triple funnel with two folded structures and one misfolded structure that are stabilized by H-bonds.

Kinetics of O2 Entry and Exit in Monomeric Sarcosine Oxidase via Markovian Milestoning Molecular Dynamics

The flavoenzyme monomeric sarcosine oxidase (MSOX) catalyzes a complex set of reactions currently lacking a consensus mechanism. A key question that arises in weighing competing mechanistic models of MSOX function is to what extent ingress of O2 from the solvent (and its egress after an unsuccessful oxidation attempt) limits the overall catalytic rate. To address this question, we have applied to the MSOX/O2 system the relatively new simulation method of Markovian milestoning molecular dynamics simulations, which, as we recently showed [ Yu et al. J. Am. Chem. Soc. 2015 , 137 , 3041 ], accurately predicted the entry and exit kinetics of CO in myoglobin. We show that the mechanism of O2 entry and exit, in terms of which possible solvent-to-active-site channels contribute to the flow of O2, is sensitive to the presence of the substrate-mimicking competitive inhibitor 2-furoate in the substrate site. The second-order O2 entry rate constants were computed to be 8.1 × 10(6) and 3.1 × 10(6) M(-1) s(-1) for bound and apo MSOX, respectively, both of which moderately exceed the experimentally determined second-order rate constant of (2.83 ± 0.07) × 10(5) M(-1) s(-1) for flavin oxidation by O2 in MSOX. This suggests that the rate of flavin oxidation by O2 is likely not strongly limited by diffusion from the solvent to the active site. The first-order exit rate constants were computed to be 10(7) s(-1) and 7.2 × 10(6) s(-1) for the apo and bound states, respectively. The predicted faster entry and slower exit of O2 for the bound state indicate a longer residence time within MSOX, increasing the likelihood of collisions with the flavin isoalloxazine ring, a step required for reduction of molecular O2 and subsequent reoxidation of the flavin. This is also indirectly supported by previous experimental evidence favoring the so-called modified ping-pong mechanism, the distinguishing feature of which is an intermediate complex involving O2, the flavin, and the oxidized substrate simultaneously in the cavity. These findings demonstrate the utility of the Markovian milestoning approach in contributing new understanding of complicated enyzmatic function.

Effect of Outer-Sphere Side Chain Substitutions on the Fate of the trans Iron-Nitrosyl Dimer in Heme/Nonheme Engineered Myoglobins (Fe(B)Mbs): Insights into the Mechanism of Denitrifying NO Reductases

Denitrifying NO reductases are transmembrane protein complexes that utilize a heme/nonheme diiron center at their active sites to reduce two NO molecules to the innocuous gas N2O. Fe(B)Mb proteins, with their nonheme iron sites engineered into the heme distal pocket of sperm whale myoglobin, are attractive models for studying the molecular details of the NO reduction reaction. Spectroscopic and structural studies of Fe(B)Mb constructs have confirmed that they reproduce the metal coordination spheres observed at the active site of the cytochrome c-dependent NO reductase from Pseudomonas aeruginosa. Exposure of Fe(B)Mb to excess NO, as examined by analytical and spectroscopic techniques, results primarily in the formation of a five-coordinate heme-nitrosyl complex without N2O production. However, substitution of the outer-sphere residue Ile107 with a glutamic acid (i.e., I107E) decreases the formation rate of the five-coordinate heme-nitrosyl complex and allows for the substoichiometric production of N2O. Here, we aim to better characterize the formation of the five-coordinate heme-nitrosyl complex and to explain why the level of N2O production increases with the I107E substitution. We follow the formation of the five-coordinate heme-nitrosyl inhibitory complex through the sequential exposure of Fe(B)Mb to different NO isotopomers using rapid-freeze-quench resonance Raman spectroscopy. The data show that the complex is formed by the displacement of the proximal histidine by a new NO molecule after the weakening of the Fe(II)-His bond in the intermediate six-coordinate low-spin (6cLS) heme-nitrosyl complex. These results lead us to explore diatomic migration within the scaffold of myoglobin and whether substitutions at residue 107 can be sufficient to control access to the proximal heme cavities. Results on a new Fe(B)Mb construct with an I107F substitution (Fe(B)Mb3) show an increased rate for the formation of the five-coordinate low-spin heme-nitrosyl complex without N2O production. Taken together, our results suggest that production of N2O from the [6cLS heme {FeNO}(7)/{Fe(B)NO}(7)] trans iron-nitrosyl dimer intermediate requires a proton transfer event facilitated by an outer-sphere residue such as E107 in Fe(B)Mb2 and E280 in P. aeruginosa cNOR.

Benchmarking Rapid TLES Simulations of Gas Diffusion in Proteins: Mapping O2 Migration and Escape in Myoglobin as a Case Study

Standard molecular dynamics (MD) simulations of gas diffusion consume considerable computational time and resources even for small proteins. To combat this, temperature-controlled locally enhanced sampling (TLES) examines multiple diffusion trajectories per simulation by accommodating multiple noninteracting copies of a gas molecule that diffuse independently, while the protein and water molecules experience an average interaction from all copies. Furthermore, gas migration within a protein matrix can be accelerated without altering protein dynamics by increasing the effective temperature of the TLES copies. These features of TLES enable rapid simulations of gas diffusion within a protein matrix at significantly reduced (∼98%) computational cost. However, the results of TLES and standard MD simulations have not been systematically compared, which limits the adoption of the TLES approach. We address this drawback here by benchmarking TLES against standard MD in the simulation of O2 diffusion in myoglobin (Mb) as a case study since this model system has been extensively characterized. We find that 2 ns TLES and 108 ns standard simulations map the same network of diffusion tunnels in Mb and uncover the same docking sites, barriers, and escape portals. We further discuss the influence of simulation time as well as the number of independent simulations on the O2 population density within the diffusion tunnels and on the sampling of Mb's conformational space as revealed by principal component analysis. Overall, our comprehensive benchmarking reveals that TLES is an appropriate and robust tool for the rapid mapping of gas diffusion in proteins when the kinetic data provided by standard MD are not required. Furthermore, TLES provides explicit ligand diffusion pathways, unlike most rapid methods.

Understanding the kinetics of ligand binding to globins with molecular dynamics simulations: the necessity of multiple state models

Residue conformational changes and internal cavity migration processes play a key role in regulating the kinetics of ligand migration and binding events in globins. Molecular dynamics simulations have demonstrated their value in the study of these processes in different haemoglobins, but derivation of kinetic data demands the use of more complex techniques like enhanced sampling molecular dynamics methods. This review discusses the different methodologies that are currently applied to study the ligand migration process in globins and highlight those specially developed to derive kinetic data.

Multiscale Estimation of Binding Kinetics Using Brownian Dynamics, Molecular Dynamics and Milestoning

The kinetic rate constants of binding were estimated for four biochemically relevant molecular systems by a method that uses milestoning theory to combine Brownian dynamics simulations with more detailed molecular dynamics simulations. The rate constants found using this method agreed well with experimentally and theoretically obtained values. We predicted the association rate of a small charged molecule toward both a charged and an uncharged spherical receptor and verified the estimated value with Smoluchowski theory. We also calculated the kon rate constant for superoxide dismutase with its natural substrate, O2-, in a validation of a previous experiment using similar methods but with a number of important improvements. We also calculated the kon for a new system: the N-terminal domain of Troponin C with its natural substrate Ca2+. The kon calculated for the latter two systems closely resemble experimentally obtained values. This novel multiscale approach is computationally cheaper and more parallelizable when compared to other methods of similar accuracy. We anticipate that this methodology will be useful for predicting kinetic rate constants and for understanding the process of binding between a small molecule and a protein receptor.