Kinetic Diagram Analysis: A Python Library for Calculating Steady-State Observables of Kinetic Systems Analytically.

Kinetic diagrams are commonly used to represent biochemical systems in order to study phenomena such as free energy transduction and ion selectivity. While numerical methods are commonly used to analyze such kinetic networks, the diagram method by King, Altman and Hill makes it possible to construct exact algebraic expressions for steady-state observables in terms of the rate constants of the kinetic diagram. However, manually obtaining these expressions becomes infeasible for models of even modest complexity as the number of the required intermediate diagrams grows with the factorial of the number of states in the diagram. We developed Kinetic Diagram Analysis (KDA), a Python library that programmatically generates the relevant diagrams and expressions from a user-defined kinetic diagram. KDA outputs symbolic expressions for state probabilities and cycle fluxes at steady-state that can be symbolically manipulated and evaluated to quantify macroscopic system observables. We demonstrate the KDA approach for examples drawn from the biophysics of active secondary transmembrane transporters. For a generic 6-state antiporter model, we show how the introduction of a single leakage transition reduces transport efficiency by quantifying substrate turnover. We apply KDA to a real-world example, the 8-state free exchange model of the small multidrug resistance transporter EmrE of Hussey et al (J General Physiology 152 (2020), e201912437), where a change in transporter phenotype is achieved by biasing two different subsets of kinetic rates: alternating access and substrate unbinding rates. KDA is made available as open source software under the GNU General Public License version 3.

Chloride binding and cholesterol effects on high frequency complex nonlinear capacitance (cNLC) in the mouse outer hair cell: experiment and molecular dynamics.

The function of prestin (SLC26a5), an anion transport family member, has evolved to enhance auditory sensitivity and frequency selectivity by providing mechanical feedback via outer hair cells (OHC) into the organ of Corti. The frequency extent of this boost is governed by the voltage-dependent kinetics of the protein's charge movements, otherwise known as nonlinear capacitance (NLC) that we measure in membrane patches under voltage clamp. Here we extend our previous studies on guinea pig OHCs by studying the frequency response of NLC in the mouse OHC, a species with higher frequency auditory needs. We find that the characteristic frequency cut-off (F is ) for the mouse surpasses that of the guinea pig, being 27 kHz vs. 19 kHz, respectively; nevertheless, each shows significant activity in the ultrasonic range. We also evaluate the influence of anion binding on prestin frequency response. Several single point mutations within the chloride binding pocket of prestin (e.g., S396E, S398E) lack anion influence. In agreement, we show absence of anion binding through molecular dynamics (MD) simulations. NLC F is in the S396E knock-in mouse remains the same as controls, indicating that high frequency activity is likely governed by viscoelastic loads within the membrane characterized by stretched-exponential frequency roll-off. Accordingly, treatment with MßCD, which removes membrane cholesterol, possibly from prestin itself, and can alter membrane fluidity, augments NLC F is out to 39 kHz. Although interactions between membrane lipid and prestin have been suggested from structural studies to arise at their interfacial boundaries within the membrane, our MD simulations suggest that phospholipids can insert within transmembrane domains of prestin during voltage perturbation. Such novel lipid-protein interactions could account for our observed changes in the phase of prestin's voltage-sensor charge movements across frequency. We hypothesize that because prestin tertiary structures of all species studied to-date are indistinguishable, it is likely that any special auditory requirements of individual species for cochlear amplification have evolved to capitalize on prestin performance by modifying, not the protein itself, but the external loads on the protein, including those within the membrane and organ of Corti. Significance: Prestin is believed to provide cochlear amplification in mammals that possess a wide range of frequency sensitivities, yet its tertiary structure is indistinguishable among those species studied. We find that prestin kinetics is faster in mice than in guinea pigs, mice showing higher frequency auditory capabilities. Chloride binding is not influential, but membrane lipids/viscosity is. We suggest that the evolution of prestin's species performance involves modifications of impinging loads, not the protein itself.

Mechanism of substrate binding and transport in BASS transporters.

The bile acid sodium symporter (BASS) family transports a wide array of molecules across membranes, including bile acids in humans, and small metabolites in plants. These transporters, many of which are sodium-coupled, have been shown to use an elevator mechanism of transport, but exactly how substrate binding is coupled to sodium ion binding and transport is not clear. Here, we solve the crystal structure at 2.3 Å of a transporter from Neisseria meningitidis (ASBTNM) in complex with pantoate, a potential substrate of ASBTNM. The BASS family is characterised by two helices that cross-over in the centre of the protein in an arrangement that is intricately held together by two sodium ions. We observe that the pantoate binds, specifically, between the N-termini of two of the opposing helices in this cross-over region. During molecular dynamics simulations the pantoate remains in this position when sodium ions are present but is more mobile in their absence. Comparison of structures in the presence and absence of pantoate demonstrates that pantoate elicits a conformational change in one of the cross-over helices. This modifies the interface between the two domains that move relative to one another to elicit the elevator mechanism. These results have implications, not only for ASBTNM but for the BASS family as a whole and indeed other transporters that work through the elevator mechanism.

Energy coupling and stoichiometry of Zn2+/H+ antiport by the prokaryotic cation diffusion facilitator YiiP.

YiiP from Shewanella oneidensis is a prokaryotic Zn2+/H+ antiporter that serves as a model for the Cation Diffusion Facilitator (CDF) superfamily, members of which are generally responsible for homeostasis of transition metal ions. Previous studies of YiiP as well as related CDF transporters have established a homodimeric architecture and the presence of three distinct Zn2+ binding sites named A, B, and C. In this study, we use cryo-EM, microscale thermophoresis and molecular dynamics simulations to address the structural and functional roles of individual sites as well as the interplay between Zn2+ binding and protonation. Structural studies indicate that site C in the cytoplasmic domain is primarily responsible for stabilizing the dimer and that site B at the cytoplasmic membrane surface controls the structural transition from an inward facing conformation to an occluded conformation. Binding data show that intramembrane site A, which is directly responsible for transport, has a dramatic pH dependence consistent with coupling to the proton motive force. A comprehensive thermodynamic model encompassing Zn2+ binding and protonation states of individual residues indicates a transport stoichiometry of 1 Zn2+ to 2-3 H+ depending on the external pH. This stoichiometry would be favorable in a physiological context, allowing the cell to use the proton gradient as well as the membrane potential to drive the export of Zn2+.

Thermodynamically consistent determination of free energies and rates in kinetic cycle models.

Kinetic and thermodynamic models of biological systems are commonly used to connect microscopic features to system function in a bottom-up multiscale approach. The parameters of such models-free energy differences for equilibrium properties and in general rates for equilibrium and out-of-equilibrium observables-have to be measured by different experiments or calculated from multiple computer simulations. All such parameters necessarily come with uncertainties so that when they are naively combined in a full model of the process of interest, they will generally violate fundamental statistical mechanical equalities, namely detailed balance and an equality of forward/backward rate products in cycles due to Hill. If left uncorrected, such models can produce arbitrary outputs that are physically inconsistent. Here, we develop a maximum likelihood approach (named multibind) based on the so-called potential graph to combine kinetic or thermodynamic measurements to yield state-resolved models that are thermodynamically consistent while being most consistent with the provided data and their uncertainties. We demonstrate the approach with two theoretical models, a generic two-proton binding site and a simplified model of a sodium/proton antiporter. We also describe an algorithm to use the multibind approach to solve the inverse problem of determining microscopic quantities from macroscopic measurements and, as an example, we predict the microscopic pKa values and protonation states of a small organic molecule from 1D NMR data. The multibind approach is applicable to any thermodynamic or kinetic model that describes a system as transitions between well-defined states with associated free energy differences or rates between these states. A Python package multibind, which implements the approach described here, is made publicly available under the MIT Open Source license.

Mechanism of substrate binding and transport in BASS transporters.

The Bile Acid Sodium Symporter (BASS) family transports a wide array of molecules across membranes, including bile acids in humans, and small metabolites in plants. These transporters, many of which are sodium-coupled, have been shown to use an elevator mechanism of transport, but exactly how substrate binding is coupled to sodium ion binding and transport is not clear. Here we solve the crystal structure at 2.3 Å of a transporter from Neisseria Meningitidis (ASBTNM) in complex with pantoate, a potential substrate of ASBTNM. The BASS family is characterised by two helices that cross-over in the centre of the protein in an arrangement that is intricately held together by two sodium ions. We observe that the pantoate binds, specifically, between the N-termini of two of the opposing helices in this cross-over region. During molecular dynamics simulations the pantoate remains in this position when sodium ions are present but is more mobile in their absence. Comparison of structures in the presence and absence of pantoate demonstrates that pantoate elicits a conformational change in one of the cross-over helices. This modifies the interface between the two domains that move relative to one another to elicit the elevator mechanism. These results have implications, not only for ASBTNM but for the BASS family as a whole and indeed other transporters that work through the elevator mechanism.

Thermodynamically consistent determination of free energies and rates in kinetic cycle models.

Kinetic and thermodynamic models of biological systems are commonly used to connect microscopic features to system function in a bottom-up multiscale approach. The parameters of such models-free energy differences for equilibrium properties and in general rates for equilibrium and out-of-equilibrium observables-have to be measured by different experiments or calculated from multiple computer simulations. All such parameters necessarily come with uncertainties so that when they are naively combined in a full model of the process of interest, they will generally violate fundamental statistical mechanical equalities, namely detailed balance and an equality of forward/backward rate products in cycles due to T. Hill. If left uncorrected, such models can produce arbitrary outputs that are physically inconsistent. Here we develop a maximum likelihood approach (named multibind ) based on the so-called potential graph to combine kinetic or thermodynamic measurements to yield state resolved models that are thermodynamically consistent while being most consistent with the provided data and their uncertainties. We demonstrate the approach with two theoretical models, a generic two-proton binding site and a simplified model of a sodium/proton antiporter. We also describe an algorithm to use the multibind approach to solve the inverse problem of determining microscopic quantities from macroscopic measurements and as an example we predict the microscopic p K a s and protonation states of a small organic molecule from 1D NMR data. The multibind approach is applicable to any thermodynamic or kinetic model that describes a system as transitions between well-defined states with associated free energy differences or rates between these states. A Python package multibind , which implements the approach described here, is made publicly available under the MIT Open Source license. WHY IT MATTERS: The increase in computational efficiency and rapid advances in methodology for quantitative free energy and rate calculations has allowed for the construction of increasingly complex thermodynamic or kinetic "bottom-up" models of chemical and biological processes. These multi-scale models serve as a framework for analyzing aspects of cellular function in terms of microscopic, molecular properties and provide an opportunity to connect molecular mechanisms to cellular function. The underlying model parameters-free energy differences or rates-are constrained by thermodynamic identities over cycles of states but these identities are not necessarily obeyed during model construction, thus potentially leading to inconsistent models. We address these inconsistencies through the use of a maximum likelihood approach for free energies and rates to adjust the model parameters in such a way that they are maximally consistent with the input parameters and exactly fulfill the thermodynamic cycle constraints. This approach enables formulation of thermodynamically consistent multi-scale models from simulated or experimental measurements.

Energy Coupling and Stoichiometry of Zn2+/H+ Antiport by the Cation Diffusion Facilitator YiiP.

YiiP is a prokaryotic Zn2+/H+ antiporter that serves as a model for the Cation Diffusion Facilitator (CDF) superfamily, members of which are generally responsible for homeostasis of transition metal ions. Previous studies of YiiP as well as related CDF transporters have established a homodimeric architecture and the presence of three distinct Zn2+ binding sites named A, B, and C. In this study, we use cryo-EM, microscale thermophoresis and molecular dynamics simulations to address the structural and functional roles of individual sites as well as the interplay between Zn2+ binding and protonation. Structural studies indicate that site C in the cytoplasmic domain is primarily responsible for stabilizing the dimer and that site B at the cytoplasmic membrane surface controls the structural transition from an inward facing conformation to an occluded conformation. Binding data show that intramembrane site A, which is directly responsible for transport, has a dramatic pH dependence consistent with coupling to the proton motive force. A comprehensive thermodynamic model encompassing Zn2+ binding and protonation states of individual residues indicates a transport stoichiometry of 1 Zn2+ to 2-3 H+ depending on the external pH. This stoichiometry would be favorable in a physiological context, allowing the cell to use the proton gradient as well as the membrane potential to drive the export of Zn2+.

Crystal structure of the Na+/H+ antiporter NhaA at active pH reveals the mechanistic basis for pH sensing.

The strict exchange of protons for sodium ions across cell membranes by Na+/H+ exchangers is a fundamental mechanism for cell homeostasis. At active pH, Na+/H+ exchange can be modelled as competition between H+ and Na+ to an ion-binding site, harbouring either one or two aspartic-acid residues. Nevertheless, extensive analysis on the model Na+/H+ antiporter NhaA from Escherichia coli, has shown that residues on the cytoplasmic surface, termed the pH sensor, shifts the pH at which NhaA becomes active. It was unclear how to incorporate the pH senor model into an alternating-access mechanism based on the NhaA structure at inactive pH 4. Here, we report the crystal structure of NhaA at active pH 6.5, and to an improved resolution of 2.2 Å. We show that at pH 6.5, residues in the pH sensor rearrange to form new salt-bridge interactions involving key histidine residues that widen the inward-facing cavity. What we now refer to as a pH gate, triggers a conformational change that enables water and Na+ to access the ion-binding site, as supported by molecular dynamics (MD) simulations. Our work highlights a unique, channel-like switch prior to substrate translocation in a secondary-active transporter.

General principles of secondary active transporter function.

Transport of ions and small molecules across the cell membrane against electrochemical gradients is catalyzed by integral membrane proteins that use a source of free energy to drive the energetically uphill flux of the transported substrate. Secondary active transporters couple the spontaneous influx of a "driving" ion such as Na+ or H+ to the flux of the substrate. The thermodynamics of such cyclical non-equilibrium systems are well understood, and recent work has focused on the molecular mechanism of secondary active transport. The fact that these transporters change their conformation between an inward-facing and outward-facing conformation in a cyclical fashion, called the alternating access model, is broadly recognized as the molecular framework in which to describe transporter function. However, only with the advent of high resolution crystal structures and detailed computer simulations, it has become possible to recognize common molecular-level principles between disparate transporter families. Inverted repeat symmetry in secondary active transporters has shed light onto how protein structures can encode a bi-stable two-state system. Based on structural data, three broad classes of alternating access transitions have been described as rocker-switch, rocking-bundle, and elevator mechanisms. More detailed analysis indicates that transporters can be understood as gated pores with at least two coupled gates. These gates are not just a convenient cartoon element to illustrate a putative mechanism but map to distinct parts of the transporter protein. Enumerating all distinct gate states naturally includes occluded states in the alternating access picture and also suggests what kind of protein conformations might be observable. By connecting the possible conformational states and ion/substrate bound states in a kinetic model, a unified picture emerges in which the symporter, antiporter, and uniporter functions are extremes in a continuum of functionality. As usual with biological systems, few principles and rules are absolute and exceptions are discussed as well as how biological complexity may be integrated in quantitative kinetic models that may provide a bridge from the structure to function.

Structure, mechanism and lipid-mediated remodeling of the mammalian Na+/H+ exchanger NHA2.

The Na+/H+ exchanger SLC9B2, also known as NHA2, correlates with the long-sought-after Na+/Li+ exchanger linked to the pathogenesis of diabetes mellitus and essential hypertension in humans. Despite the functional importance of NHA2, structural information and the molecular basis for its ion-exchange mechanism have been lacking. Here we report the cryo-EM structures of bison NHA2 in detergent and in nanodiscs, at 3.0 and 3.5 Å resolution, respectively. The bison NHA2 structure, together with solid-state membrane-based electrophysiology, establishes the molecular basis for electroneutral ion exchange. NHA2 consists of 14 transmembrane (TM) segments, rather than the 13 TMs previously observed in mammalian Na+/H+ exchangers (NHEs) and related bacterial antiporters. The additional N-terminal helix in NHA2 forms a unique homodimer interface with a large intracellular gap between the protomers, which closes in the presence of phosphoinositol lipids. We propose that the additional N-terminal helix has evolved as a lipid-mediated remodeling switch for the regulation of NHA2 activity.

Precise force-field-based calculations of octanol-water partition coefficients for the SAMPL7 molecules.

We predicted water-octanol partition coefficients for the molecules in the SAMPL7 challenge with explicit solvent classical molecular dynamics (MD) simulations. Water hydration free energies and octanol solvation free energies were calculated with a windowed alchemical free energy approach. Three commonly used force fields (AMBER GAFF, CHARMM CGenFF, OPLS-AA) were tested. Special emphasis was placed on converging all simulations, using a criterion developed for the SAMPL6 challenge. In aggregate, over 1000 [Formula: see text]s of simulations were performed, with some free energy windows remaining not fully converged even after 1 [Formula: see text]s of simulation time. Nevertheless, the amount of sampling produced [Formula: see text] estimates with a precision of 0.1 log units or better for converged simulations. Despite being probably as fully sampled as can expected and is feasible, the agreement with experiment remained modest for all force fields, with no force field performing better than 1.6 in root mean squared error. Overall, our results indicate that a large amount of sampling is necessary to produce precise [Formula: see text] predictions for the SAMPL7 compounds and that high precision does not necessarily lead to high accuracy. Thus, fundamental problems remain to be solved for physics-based [Formula: see text] predictions.

Zinc binding alters the conformational dynamics and drives the transport cycle of the cation diffusion facilitator YiiP.

YiiP is a secondary transporter that couples Zn2+ transport to the proton motive force. Structural studies of YiiP from prokaryotes and Znt8 from humans have revealed three different Zn2+ sites and a conserved homodimeric architecture. These structures define the inward-facing and outward-facing states that characterize the archetypal alternating access mechanism of transport. To study the effects of Zn2+ binding on the conformational transition, we use cryo-EM together with molecular dynamics simulation to compare structures of YiiP from Shewanella oneidensis in the presence and absence of Zn2+. To enable single-particle cryo-EM, we used a phage-display library to develop a Fab antibody fragment with high affinity for YiiP, thus producing a YiiP/Fab complex. To perform MD simulations, we developed a nonbonded dummy model for Zn2+ and validated its performance with known Zn2+-binding proteins. Using these tools, we find that, in the presence of Zn2+, YiiP adopts an inward-facing conformation consistent with that previously seen in tubular crystals. After removal of Zn2+ with high-affinity chelators, YiiP exhibits enhanced flexibility and adopts a novel conformation that appears to be intermediate between inward-facing and outward-facing states. This conformation involves closure of a hydrophobic gate that has been postulated to control access to the primary transport site. Comparison of several independent cryo-EM maps suggests that the transition from the inward-facing state is controlled by occupancy of a secondary Zn2+ site at the cytoplasmic membrane interface. This work enhances our understanding of individual Zn2+ binding sites and their role in the conformational dynamics that govern the transport cycle.

Evidence that specific interactions play a role in the cholesterol sensitivity of G protein-coupled receptors.

G protein-coupled receptors (GPCRs) are known to be modulated by membrane cholesterol levels, but whether or not the effects are caused by specific receptor-cholesterol interactions or cholesterol's general effects on the membrane is not well-understood. We performed coarse-grained molecular dynamics (CGMD) simulations coupled with structural bioinformatics approaches on the ß2-adrenergic receptor (ß2AR) and the cholecystokinin (CCK) receptor subfamily. The ß2AR has been shown to be sensitive to membrane cholesterol and cholesterol molecules have been clearly resolved in numerous ß2AR crystal structures. The two CCK receptors are highly homologous and preserve similar cholesterol recognition motifs but despite their homology, CCK1R shows functional sensitivity to membrane cholesterol while CCK2R does not. Our results offer new insights into how cholesterol modulates GPCR function by showing cholesterol interactions with ß2AR that agree with previously published data; additionally, we observe differential and specific cholesterol binding in the CCK receptor subfamily while revealing a previously unreported Cholesterol Recognition Amino-acid Consensus (CRAC) sequence that is also conserved across 38% of class A GPCRs. A thermal denaturation assay (LCP-Tm) shows that mutation of a conserved CRAC sequence on TM7 of the ß2AR affects cholesterol stabilization of the receptor in a lipid bilayer. The results of this study provide a better understanding of receptor-cholesterol interactions that can contribute to novel and improved therapeutics for a variety of diseases.

Structure and elevator mechanism of the mammalian sodium/proton exchanger NHE9.

Na+ /H+ exchangers (NHEs) are ancient membrane-bound nanomachines that work to regulate intracellular pH, sodium levels and cell volume. NHE activities contribute to the control of the cell cycle, cell proliferation, cell migration and vesicle trafficking. NHE dysfunction has been linked to many diseases, and they are targets of pharmaceutical drugs. Despite their fundamental importance to cell homeostasis and human physiology, structural information for the mammalian NHE was lacking. Here, we report the cryogenic electron microscopy structure of NHE isoform 9 (SLC9A9) from Equus caballus at 3.2 Å resolution, an endosomal isoform highly expressed in the brain and associated with autism spectrum (ASD) and attention deficit hyperactivity (ADHD) disorders. Despite low sequence identity, the NHE9 architecture and ion-binding site are remarkably similar to distantly related bacterial Na+ /H+  antiporters with 13 transmembrane segments. Collectively, we reveal the conserved architecture of the NHE ion-binding site, their elevator-like structural transitions, the functional implications of autism disease mutations and the role of phosphoinositide lipids to promote homodimerization that, together, have important physiological ramifications.

Alternative proton-binding site and long-distance coupling in Escherichia coli sodium-proton antiporter NhaA.

Escherichia coli NhaA is a prototypical sodium-proton antiporter responsible for maintaining cellular ion and volume homeostasis by exchanging two protons for one sodium ion; despite two decades of research, the transport mechanism of NhaA remains poorly understood. Recent crystal structure and computational studies suggested Lys300 as a second proton-binding site; however, functional measurements of several K300 mutants demonstrated electrogenic transport, thereby casting doubt on the role of Lys300. To address the controversy, we carried out state-of-the-art continuous constant pH molecular dynamics simulations of NhaA mutants K300A, K300R, K300Q/D163N, and K300Q/D163N/D133A. Simulations suggested that K300 mutants maintain the electrogenic transport by utilizing an alternative proton-binding residue Asp133. Surprisingly, while Asp133 is solely responsible for binding the second proton in K300R, Asp133 and Asp163 jointly bind the second proton in K300A, and Asp133 and Asp164 jointly bind two protons in K300Q/D163N. Intriguingly, the coupling between Asp133 and Asp163 or Asp164 is enabled through the proton-coupled hydrogen-bonding network at the flexible intersection of two disrupted helices. These data resolve the controversy and highlight the intricacy of the compensatory transport mechanism of NhaA mutants. Alternative proton-binding site and proton sharing between distant aspartates may represent important general mechanisms of proton-coupled transport in secondary active transporters.

Substrate Specificity of OXA-48 after ß5-ß6 Loop Replacement.

OXA-48 carbapenemase has rapidly spread in many countries worldwide with several OXA-48-variants being described, differing by a few amino acid (AA) substitutions or deletions, mostly in the ß5-ß6 loop. While single AA substitutions have only a minor impact on OXA-48 hydrolytic profiles, others with 4 AA deletions result in loss of carbapenem hydrolysis and gain of expanded-spectrum cephalosporin (ESC) hydrolysis. We have replaced the ß5-ß6 loop of OXA-48 with that of OXA-18, a clavulanic-acid inhibited oxacillinase capable of hydrolyzing ESCs but not carbapenems. The hybrid enzyme OXA-48Loop18 was able to hydrolyze ESCs and carbapenems (although with a lower kcat), even though the ß5-ß6 loop was longer and its sequence quite different from that of OXA-48. The kinetic parameters of OXA-48Loop18 were in agreement with the MIC values. X-ray crystallography and molecular modeling suggest that the conformation of the grafted loop allows the binding of bulkier substrates, unlike that of the native loop, expanding the hydrolytic profile. This seems to be due not only to differences in AA sequence, but also to the backbone conformation the loop can adopt. Finally, our results provide further experimental evidence for the role of the ß5-ß6 loop in substrate selectivity of OXA-48-like enzymes and additional details on the structure-function relationship of ß-lactamases, demonstrating how localized changes in these proteins can alter or expand their function, highlighting their plasticity.

Prediction of octanol-water partition coefficients for the SAMPL6-[Formula: see text] molecules using molecular dynamics simulations with OPLS-AA, AMBER and CHARMM force fields.

All-atom molecular dynamics simulations with stratified alchemical free energy calculations were used to predict the octanol-water partition coefficient [Formula: see text] of eleven small molecules as part of the SAMPL6-[Formula: see text] blind prediction challenge using four different force field parametrizations: standard OPLS-AA with transferable charges, OPLS-AA with non-transferable CM1A charges, AMBER/GAFF, and CHARMM/CGenFF. Octanol parameters for OPLS-AA, GAFF and CHARMM were validated by comparing the density as a function of temperature, the chemical potential, and the hydration free energy to experimental values. The partition coefficients were calculated from the solvation free energy for the compounds in water and pure ("dry") octanol or "wet" octanol with 27 mol% water dissolved. Absolute solvation free energies were computed by thermodynamic integration (TI) and the multistate Bennett acceptance ratio with uncorrelated samples from data generated by an established protocol using 5-ns windowed alchemical free energy perturbation (FEP) calculations with the Gromacs molecular dynamics package. Equilibration of sets of FEP simulations was quantified by a new measure of convergence based on the analysis of forward and time-reversed trajectories. The accuracy of the [Formula: see text] predictions was assessed by descriptive statistical measures such as the root mean square error (RMSE) of the data set compared to the experimental values. Discarding the first 1 ns of each 5-ns window as an equilibration phase had a large effect on the GAFF data, where it improved the RMSE by up to 0.8 log units, while the effect for other data sets was smaller or marginally worsened the agreement. Overall, CGenFF gave the best prediction with RMSE 1.2 log units, although for only eight molecules because the current CGenFF workflow for Gromacs does not generate files for certain halogen-containing compounds. Over all eleven compounds, GAFF gave an RMSE of 1.5. The effect of using a mixed water/octanol solvent slightly decreased the accuracy for CGenFF and GAFF and slightly increased it for OPLS-AA. The GAFF and OPLS-AA results displayed a systematic error where molecules were too hydrophobic whereas CGenFF appeared to be more balanced, at least on this small data set.

Correction: Path Similarity Analysis: A Method for Quantifying Macromolecular Pathways.

[This corrects the article DOI: 10.1371/journal.pcbi.1004568.].