Modeling molecular kinetics with Milestoning

Kinetic Network in Milestoning: Clustering, Reduction, and Transition Path Analysis

We present a reduction of the Milestoning (ReM) algorithm to analyze the high-dimensional Milestoning kinetic network. The algorithm reduces the Milestoning network to low dimensions but preserves essential kinetic information, such as local residence time, exit time, and mean first passage time between any two states. This is achieved in three steps. First, nodes (milestones) in the high-dimensional Milestoning network are grouped into clusters based on the metastability identified by an auxiliary continuous-time Markov chain. Our clustering method is applicable not only to time-reversible networks but also to nonreversible networks generated from practical simulations with statistical fluctuations. Second, a reduced network is established via network transformation, containing only the core sets of clusters as nodes. Finally, transition pathways are analyzed in the reduced network based on the transition path theory. The algorithm is illustrated using a toy model and a solvated alanine dipeptide in two and four dihedral angles.

Comparison of Accuracy and Efficiency of Milestoning Variants: Introducing Buffer Milestoning

The Milestoning algorithm is a method for long-time molecular dynamics simulations. It enables the sampling of rare events. The precise calculations of observables depend on accurately determining the first hitting point distribution (FHPD) for each milestone. There is no analytical expression for FHPD, which is estimated numerically. Several variants of Milestoning offer approximations to the FHPD. Here, we examine in detail the FHPD of an exact calculation and Milestoning variants. We also introduce a new version of the Milestoning algorithm, buffer Milestoning, with a comparable cost to conventional Milestoning but higher accuracy. We use the mean first passage time and the free energy to assess the simulation quality, and we compare the accuracy and efficiency of buffer Milestoning to exact calculations, conventional Milestoning, local-passage-time-weighted Milestoning, Markovian Milestoning with Voronoi tessellation, and exact Milestoning. Conventional Milestoning requires milestone decorrelation. If this condition is not satisfied, it is the least accurate approach of all the techniques we examined. We conclude that for a small increase in cost compared to conventional Milestoning, buffer Milestoning provides accurate results for a range of problems, including more correlated milestones and is, therefore, versatile compared to other variants. Local-passage-time-weighted Milestoning provides accuracy similar to that of buffer Milestoning but at an increased simulation cost. Markovian Milestoning with Voronoi tessellation is the most accurate compared with other approximations, but it is less stable for high barriers and more expensive.

Approximating First Hitting Point Distribution in Milestoning for Rare Event Kinetics

Milestoning is an efficient method for rare event kinetics calculation using short trajectory parallelization. Mean first passage time (MFPT) is the key kinetic output of Milestoning, whose accuracy crucially depends on the initial distribution of the short trajectory ensemble. The true initial distribution, i.e., the first hitting point distribution (FHPD), has no analytic expression in the general case. Here, we introduce two algorithms, local passage time weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to accurately and efficiently approximate FHPD for systems at equilibrium condition. Starting from sampling the Boltzmann distribution on milestones, we calculate the proper weighting factor for the short trajectory ensemble. The methods are tested on two model examples for illustration purpose. Both methods improve significantly over the widely used classical Milestoning (CM) method in terms of the accuracy of MFPT. In particular, BI-M covers the directional Milestoning method as a special case in deterministic Hamiltonian dynamics. LPT-M is especially advantageous in terms of computational costs and robustness with respect to the increasing number of intermediate milestones. Furthermore, a locally iterative correction algorithm for nonequilibrium stationary FHPD is developed for exact MFPT calculation, which can be combined with LPT-M/BI-M and is much cheaper than the exact Milestoning (ExM) method.

Visualization of Molecular Permeation into a Multi-compartment Phospholipid Vesicle

Passive permeation of small molecules into vesicles with multiple compartments is a critical event in many chemical and biological processes. We consider the translocation of the peptide NAF-144-67 labeled with a fluorescent fluorescein dye across membranes of rhodamine-labeled 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) into liposomes with internal vesicles. Time-resolved microscopy revealed a sequential absorbance of the peptide in both the outer and inner micrometer vesicles that developed over a time period of minutes to hours, illustrating the spatial and temporal progress of the permeation. There is minimal perturbation of the membrane structure and no evidence for pore formation. On the basis of molecular dynamics simulations of NAF-144-67, we extended a local defect model to migration processes that include multiple compartments. The model captures the long residence time of the peptide within the membrane and the rate of permeation through the liposome and its internal compartments. Imaging experiments confirm the semi-quantitative description of the permeation of the model by activated diffusion and open the way for studies of more complex systems.

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.

Stochastic Model of Solvent Exchange in the First Coordination Shell of Aqua Ions

Ion microsolvation is a basic, yet fundamental, process of ionic solutions underlying many relevant phenomena in either biological or nanotechnological applications, such as solvent reorganization energy, ion transport, catalytic activity, and so on. As a consequence, it is a topic of extensive investigations by various experimental techniques, ranging from X-ray diffraction to NMR relaxation and from calorimetry to vibrational spectroscopy, and theoretical approaches, especially those based on molecular dynamics (MD) simulations. The conventional microscopic view of ion solvation is usually provided by a "static" cluster model representing the first ion-solvent coordination shell. Despite the merits of such a simple model, however, ion coordination in solution should be better regarded as a complex population of dynamically interchanging molecular configurations. Such a more comprehensive view is more subtle to characterize and often elusive to standard approaches. In this work, we report on an effective computational strategy aiming at providing a detailed picture of solvent coordination and exchange around aqua ions, thus including the main structural, thermodynamic, and dynamic properties of ion microsolvation, such as the most probable first-shell complex structures, the corresponding free energies, the interchanging energy barriers, and the solvent-exchange rates. Assuming the solvent coordination number as an effective reaction coordinate and combining MD simulations with enhanced sampling and master-equation approaches, we propose a stochastic model suitable for properly describing, at the same time, the thermodynamics and kinetics of ion-water coordination. The model is successfully tested toward various divalent ions (Ca²⁺, Zn²⁺, Hg²⁺, and Cd²⁺) in aqueous solution, considering also the case of a high ionic concentration. Results show a very good agreement with those issuing from brute-force MD simulations, when available, and support the reliable prediction of rare ion-water complexes and slow water exchange rates not easily accessible to usual computational methods.

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.

Impact of the Protonation State of Phosphatidylinositol 4,5-Bisphosphate (PIP2) on the Binding Kinetics and Thermodynamics to Transient Receptor Potential Vanilloid (TRPV5): A Milestoning Study

The binding of phosphatidylinositol 4,5-bisphosphate (PIP2) to the ion channel transient receptor potential vanilloid 5 (TRPV5) is critical for its function. We use atomically detailed simulations and the milestoning theory to compute the free energy profile and the kinetics of PIP2 binding to TRPV5. We estimate the rate of binding and the impact of the protonation state on the process. Several channel residues are identified as influential in the association event and will be interesting targets for mutation analysis. Our simulations reveal that PIP2 binds to TRPV5 in an unprotonated state and is protonated in the membrane. The switch between the protonation state of PIP2 is modeled as a diabatic transition and occurs about halfway through the reaction.