The Impact of Protonation on Early Translocation of Anthrax Lethal Factor: Kinetics from Molecular Dynamics Simulations and Milestoning Theory

ScMiles2: A Script to Conduct and Analyze Milestoning Trajectories for Long Time Dynamics

Milestoning is a theory and an algorithm that computes kinetics and thermodynamics at long time scales. It is based on partitioning the (phase) space into cells and running a large number of short trajectories between the boundaries of the cells. The termination points of the trajectories are analyzed with the Milestoning theory to obtain kinetic and thermodynamic information. Managing the tens to hundreds of thousands of Milestoning trajectories is a challenge, which we handle with a python script, ScMiles. Here, we introduce a new version of the python script ScMiles2 to conduct Milestoning simulations. Major enhancements are: (i) post analysis of Milestoning trajectories to obtain the free energy, mean first passage time, the committor function, and exit times; (ii) similar to (i) but the post analysis is for a single long trajectory; (iii) we support the use of the GROMACS software in addition to NAMD; (iv) a restart option; (v) the automated finding, sampling, and launching trajectories from new milestones that are found on the fly; and (vi) support Milestoning calculations with several coarse variables and for complex reaction coordinates. We also evaluate the simulation parameters and suggest new algorithmic features to enhance the rate of convergence of observables. We propose the use of an iteration-averaged kinetic matrix for a rapid approach to asymptotic values. Illustrations are provided for small systems and one large example.

Design of Peptides for Membrane Insertion: The Critical Role of Charge Separation

A physical understanding of membrane permeation and translocation by small, positively charged molecules can illuminate cell penetrating peptide mechanisms of entry and inform drug design. We have previously investigated the permeation of the doubly charged peptide WKW and proposed a defect-assisted permeation mechanism where a small molecule with +2 charge can achieve a metastable state spanning the bilayer by forming a membrane defect with charges stabilized by phospholipid phosphate groups. Here, we investigate the membrane permeation of two doubly charged peptides, WWK and WWWK, with charges separated by different lengths. Through complementary experiments and molecular dynamics simulations, we show that membrane permeation was an order of magnitude more favorable when charges were separated by an ∼2-3 Å greater distance on WWWK compared to WWK. These results agree with the previously proposed defect-assisted permeation mechanism, where a greater distance between positive charges would require a less extreme membrane defect to stabilize the membrane-spanning metastable state. We discuss the implications of these results in understanding the membrane permeation of cell-penetrating peptides and other small, positively charged membrane permeants.

Peptide Permeation across a Phosphocholine Membrane: An Atomically Detailed Mechanism Determined through Simulations and Supported by Experimentation

Cell-penetrating peptides (CPPs) facilitate translocation across biological membranes and are of significant biological and medical interest. Several CPPs can permeate into specific cells and organelles. We examine the incorporation and translocation of a novel anticancer CPP in a dioleoylphosphatidylcholine (DOPC) lipid bilayer membrane. The peptide, NAF-1^44-67, is a short fragment of a transmembrane protein, consisting of hydrophobic N-terminal and charged C-terminal segments. Experiments using fluorescently labeled NAF-1^44-67 in ∼100 nm DOPC vesicles and atomically detailed simulations conducted with Milestoning support a model in which a significant barrier for peptide-membrane entry is found at the interface between the aqueous solution and membrane. The initial step is the insertion of the N-terminal segment and the hydrophobic helix into the membrane, passing the hydrophilic head groups. Both experiments and simulations suggest that the free energy difference in the first step of the permeation mechanism in which the hydrophobic helix crosses the phospholipid head groups is -0.4 kcal mol^-1 slightly favoring motion into the membrane. Milestoning calculations of the mean first passage time and the committor function underscore the existence of an early polar barrier followed by a diffusive barrierless motion in the lipid tail region. Permeation events are coupled to membrane fluctuations that are examined in detail. Our study opens the way to investigate in atomistic resolution the molecular mechanism, kinetics, and thermodynamics of CPP permeation to diverse membranes.

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.

Anthrax toxin translocation complex reveals insight into the lethal factor unfolding and refolding mechanism

Translocation is essential to the anthrax toxin mechanism. Protective antigen (PA), the binding component of this AB toxin, forms an oligomeric pore that translocates lethal factor (LF) or edema factor, the active components of the toxin, into the cell. Structural details of the translocation process have remained elusive despite their biological importance. To overcome the technical challenges of studying translocation intermediates, we developed a method to immobilize, transition, and stabilize anthrax toxin to mimic important physiological steps in the intoxication process. Here, we report a cryoEM snapshot of PA_pore translocating the N-terminal domain of LF (LF_N). The resulting 3.3 Å structure of the complex shows density of partially unfolded LF_N near the canonical PA_pore binding site. Interestingly, we also observe density consistent with an α helix emerging from the 100 Å β barrel channel suggesting LF secondary structural elements begin to refold in the pore channel. We conclude the anthrax toxin β barrel aids in efficient folding of its enzymatic payload prior to channel exit. Our hypothesized refolding mechanism has broader implications for pore length of other protein translocating toxins.

Computer Simulations of the Dissociation Mechanism of Gleevec from Abl Kinase with Milestoning

Gleevec (a.k.a., imatinib) is an important anticancer (e.g., chronic myeloid leukemia) chemotherapeutic drug due to its inhibitory interaction with the Abl kinase. Here, we use atomically detailed simulations within the Milestoning framework to study the molecular dissociation mechanism of Gleevec from Abl kinase. We compute the dissociation free energy profile, the mean first passage time for unbinding, and explore the transition state ensemble of conformations. The milestones form a multidimensional network with average connectivity of about 2.93, which is significantly higher than the connectivity for a one-dimensional reaction coordinate. The free energy barrier for Gleevec dissociation is estimated to be ∼10 kcal/mol, and the exit time is ∼55 ms. We examined the transition state conformations using both, the committor and transition function. We show that near the transition state the highly conserved salt bridge K217 and E286 is transiently broken. Together with the calculated free energy profile, these calculations can advance the understanding of the molecular interaction mechanisms between Gleevec and Abl kinase and play a role in future drug design and optimization studies.

Length Dependent Folding Kinetics of Alanine-Based Helical Peptides from Optimal Dimensionality Reduction

We present a computer simulation study of helix folding in alanine homopeptides (ALA)n of length n = 5, 8, 15, and 21 residues. Based on multi-microsecond molecular dynamics simulations at room temperature, we found helix populations and relaxation times increasing from about 6% and ~2 ns for ALA5 to about 60% and ~500 ns for ALA21, and folding free energies decreasing linearly with the increasing number of residues. The helix folding was analyzed with the Optimal Dimensionality Reduction method, yielding coarse-grained kinetic models that provided a detailed representation of the folding process. The shorter peptides, ALA5 and ALA8, tended to convert directly from coil to helix, while ALA15 and ALA21 traveled through several intermediates. Coarse-grained aggregate states representing the helix, coil, and intermediates were heterogeneous, encompassing multiple peptide conformations. The folding involved multiple pathways and interesting intermediate states were present on the folding paths, with partially formed helices, turns, and compact coils. Statistically, helix initiation was favored at both termini, and the helix was most stable in the central region. Importantly, we found the presence of underlying universal local dynamics in helical peptides with correlated transitions for neighboring hydrogen bonds. Overall, the structural and dynamical parameters extracted from the trajectories are in good agreement with experimental observables, providing microscopic insights into the complex helix folding kinetics.

A Minimal, Adaptive Binning Scheme for Weighted Ensemble Simulations

A promising approach for simulating rare events with rigorous kinetics is the weighted ensemble path sampling strategy. One challenge of this strategy is the division of configurational space into bins for sampling. Here we present a minimal adaptive binning (MAB) scheme for the automated, adaptive placement of bins along a progress coordinate within the framework of the weighted ensemble strategy. Results reveal that the MAB binning scheme, despite its simplicity, is more efficient than a manual, fixed binning scheme in generating transitions over large free energy barriers, generating a diversity of pathways, estimating rate constants, and sampling conformations. The scheme is general and extensible to any rare-events sampling strategy that employs progress coordinates.

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.

Kinetics and free energy of ligand dissociation using weighted ensemble milestoning

We consider the recently developed weighted ensemble milestoning (WEM) scheme [D. Ray and I. Andricioaei, J. Chem. Phys. 152, 234114 (2020)] and test its capability of simulating ligand-receptor dissociation dynamics. We performed WEM simulations on the following host-guest systems: Na⁺/Cl^- ion pair and 4-hydroxy-2-butanone ligand with FK506 binding protein. As a proof of principle, we show that the WEM formalism reproduces the Na⁺/Cl^- ion pair dissociation timescale and the free energy profile obtained from long conventional MD simulation. To increase the accuracy of WEM calculations applied to kinetics and thermodynamics in protein-ligand binding, we introduced a modified WEM scheme called weighted ensemble milestoning with restraint release (WEM-RR), which can increase the number of starting points per milestone without adding additional computational cost. WEM-RR calculations obtained a ligand residence time and binding free energy in agreement with experimental and previous computational results. Moreover, using the milestoning framework, the binding time and rate constants, dissociation constants, and committor probabilities could also be calculated at a low computational cost. We also present an analytical approach for estimating the association rate constant (k_on) when binding is primarily diffusion driven. We show that the WEM method can efficiently calculate multiple experimental observables describing ligand-receptor binding/unbinding and is a promising candidate for computer-aided inhibitor design.

Exploring the Reaction Mechanism of HIV Reverse Transcriptase with a Nucleotide Substrate

Enzymatic reactions consist of several steps: (i) a weak binding event of the substrate to the enzyme, (ii) an induced fit or a protein conformational transition upon ligand binding, (iii) the chemical reaction, and (iv) the release of the product. Here we focus on step iii of the reaction of a DNA polymerase, HIV RT, with a nucleotide. We determine the rate and the free energy profile for the addition of a nucleotide to a DNA strand using a combination of a QM/MM model, the string method, and exact Milestoning. The barrier height and the time scale of the reaction are consistent with experiment. We show that the observables (free energies and mean first passage time) converge rapidly, as a function of the Milestoning iteration number. We also consider the substitution of an oxygen of the incoming nucleotide by a nonbridging sulfur atom and its impact on the enzymatic reaction. This substitution has been suggested in the past as a tool to examine the influence of the chemical step on the overall rate. Our joint computational and experimental study suggests that the impact of the substitution is small. Computationally, the differences between the two are within the estimated error bars. Experiments suggest a small difference. Finally, we examine step i, the weak binding of the nucleotide to the protein surface. We suggest that this step has only a small contribution to the selectivity of the enzyme. Comments are made on the impact of these steps on the overall mechanism.

The transition between active and inactive conformations of Abl kinase studied by rock climbing and Milestoning

BACKGROUND

Kinases are a family of enzymes that catalyze the transfer of the ɤ-phosphate group from ATP to a protein's residue. Malfunctioning kinases are involved in many health problems such as cardiovascular diseases, diabetes, and cancer. Kinases transitions between multiple conformations of inactive to active forms attracted considerable interest.

METHOD

A reaction coordinate is computed for the transition between the active to inactive conformation in Abl kinase with a focus on the DFG-in to DFG-out flip. The method of Rock Climbing is used to construct a path locally, which is subsequently optimized using a functional of the entire path. The discrete coordinate sets along the reaction path are used in a Milestoning calculation of the free energy landscape and the rate of the transition.

RESULTS

The estimated transition times are between a few milliseconds and seconds, consistent with simulations of the kinetics and with indirect experimental data. The activation requires the transient dissociation of the salt bridge between Lys271 and Glu286. The salt bridge reforms once the DFG motif is stabilized by a locked conformation of Phe382. About ten residues are identified that contribute significantly to the process and are included as part of the reaction space.

CONCLUSIONS

The transition from DFG-in to DFG-out in Abl kinase was simulated using atomic resolution of a fully solvated protein yielding detailed description of the kinetics and the mechanism of the DFG flip. The results are consistent with other computational methods that simulate the kinetics and with some indirect experimental measurements.

GENERAL SIGNIFICANCE

The activation of kinases includes a conformational transition of the DFG motif that is important for enzyme activity but is not accessible to conventional Molecular Dynamics. We propose a detailed mechanism for the transition, at a timescale longer than conventional MD, using a combination of reaction path and Milestoning algorithms. The mechanism includes local structural adjustments near the binding site as well as collective interactions with more remote residues.

ScMile: A Script to Investigate Kinetics with Short Time Molecular Dynamics Trajectories and the Milestoning Theory

Studies of complex and rare events in condensed phase systems continue to attract considerable attention. Milestoning is a useful theory and algorithm to investigate the long-time dynamics of activated molecular events. It is based on launching a large number of short trajectories and statistical analysis of the outcome. The implementation of the theory in a computer script is described that enables more efficient Milestoning calculation, reducing user time and errors, and automating a significant fraction of the algorithm. The script exploits a molecular dynamics engine, which at present is NAMD, to run the short trajectories. However, since the script is external to the engine, the script can be easily adapted to different molecular dynamics codes. The outcomes of the short trajectories are analyzed to obtain a kinetic and thermodynamic description of the entire process. While many examples of Milestoning were published in the past, we provide two simple examples (a conformational transition of alanine dipeptide in a vacuum and aqueous solution) to illustrate the use of the script.

Defect-Assisted Permeation Through a Phospholipid Membrane: Experimental and Computational Study of the Peptide WKW

We investigate membrane permeation by the peptide WKW that is amidated at its C-terminus and therefore carries a positive charge of +2. To facilitate an efficient calculation, we introduce a novel set of simple coarse variables that measure permeation depth and membrane distortion. The phospholipid head groups shift toward the center of the membrane, following the permeating peptide, and create a defect that assists permeation. The Milestoning algorithm was used in the new coarse space to compute the free-energy profile and the mean first passage time. The barrier was lower than expected from a simple continuum estimate. This behavior is consistent with the known behavior of positively charged cell-penetrating peptides, and is explained by a detailed mechanism of defect formation and propagation revealed by the simulations.

Ion Permeation through a Phospholipid Membrane: Transition State, Path Splitting, and Calculation of Permeability

We investigate the thermodynamics and kinetics of the permeation of a potassium ion through a phospholipid membrane. We illustrate that the conventional reaction coordinate (the position of the ion along the normal to the membrane plane) is insufficient to capture essential elements of the process. It is necessary to add coarse variables that measure membrane distortion. New coarse variables are suggested, and a two-dimensional coarse-space is proposed to describe the permeation. We illustrate path splitting and two transition states of comparable barrier heights. The alternative pathways differ by the extent of water solvation of the ion-phosphate pairs. The permeation process cannot be described by a local one-dimensional reaction coordinate, and a network formulation is more appropriate. We use Milestoning with Voronoi tessellation in two dimensions to quantify the equilibrium and rate of the permeation of the positively charged ion. The permeation coefficient is computed and compared favorably to experiment.

Why Does RNA Collapse? The Importance of Water in a Simulation Study of Helix-Junction-Helix Systems

Using computer simulations, we consider the balance of thermodynamic forces that collapse RNA. A model helix-junction-helix (HJH) construct is used to investigate the transition from an extended to a collapsed conformation. Conventional Molecular Dynamics and Milestoning Simulations are used to study the free energy profile of the process for two ion concentrations. We illustrate that HJH folds to a collapsed state with two types of counterions (Mg²⁺ and K⁺). By dissecting the free energy landscape into energetic and entropic contributions, we illustrate that the electrostatic forces between the RNA and the mobile ions do not drive the RNA to a collapsed state. Instead, entropy gains from water expulsion near the neighborhood of the RNA provide the stabilization free energy that tilt HJH into more compact structures. Further simulations of a three-helix hammerhead ribozyme show a similar behavior and support the idea of collapse due to increased gain in water entropy.

Probing Translocation in Mutants of the Anthrax Channel: Atomically Detailed Simulations with Milestoning

Anthrax toxin consists of a cation channel and two protein factors. Translocation of the anthrax protein factors from endosomal to the cytosolic compartment is a complex process which utilizes the cation channel. An atomically detailed understanding of the function of the anthrax translocation machinery is incomplete. We report atomically detailed simulations of the lethal factor and channel mutants. Kinetic and thermodynamic properties of early events in the translocation process are computed within the Milestoning theory and algorithm. Several mutants of the channel illustrate that long-range electrostatic interactions provide the dominant driving force for translocation. No external energy input is required because the lower pH in the endosome relative to the cytosol drives the initial translocation process forward. Channel mutants with variable sizes cause smaller effects on translocation events relative to charge manipulations. Comparison with available experimental data is provided.

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.

Nanoporous Hydrogels for the Observation of Anthrax Exotoxin Translocation Dynamics

The ability to observe lethal anthrax exotoxins translocating through size-constricting nanopores in vitro, combined with detailed sequence and structural data, has aided in elucidated mechanisms of exotoxin cell entry and toxicity. However, due to limited observations of anthrax exotoxins translocating through protective antigen nanopores in vitro and the instability of protective antigen-functionalized suspended lipid bilayers, questions remain regarding the native mechanisms of cell entry. Nanoporous hydrogel membranes offer a robust tool for studying protein translocation with ensemble measurements that complement conventional single-molecule translocation measurements. Here, we utilize nanoporous hydrogel membranes to assess the translocation of full-length anthrax lethal and edema factors through nanopores similar in diameter to protective antigen translocons. We find that, relative to globular serum and other proteins that do not translocate natively through nanopores, anthrax exotoxins demonstrate significantly reduced barriers to pore entry. Computed free-energy barriers to the unfolding of proteins and the dissociation of macromolecular complexes are generally found to coincide with translocation. Finally, a nanopore-blocking strategy is developed that utilizes nonspecific synthetic peptide constructs and effectively prevents LF translocation within the nanoporous hydrogel.

Pyrophosphate Release in the Protein HIV Reverse Transcriptase

Enzymatic reactions usually occur in several steps: a step of substrate binding to the surface of the protein, a step of protein reorganization around the substrate and conduction of a chemical reaction, and a step of product release. The release of inorganic phosphate-PPi-from the matrix of the protein HIV reverse transcriptase is investigated computationally. Atomically detailed simulations with explicit solvent are analyzed to obtain the free energy profile, mean first passage time, and detailed molecular mechanisms of PPi escape. A challenge for the computations is of time scales. The experimental time scale of the process of interest is in milliseconds, and straightforward molecular dynamics simulations are in sub-microseconds. To overcome the time scale gap, we use the algorithm of Milestoning along a reaction coordinate to compute the overall free energy profile and rate. The methods of locally enhanced sampling and steered molecular dynamics determine plausible reaction coordinates. The observed molecular mechanism couples the transfer of the PPi to positively charged lysine side chains that are found on the exit pathway and to an exiting magnesium ion. In accord with experimental findings, the release rate is comparable to the chemical step, allowing for variations in substrate (DNA or RNA template) in which the release becomes rate determining.