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Computational strategies aimed at unveiling the thermodynamic and kinetic properties of G Protein-Coupled Receptor (GPCR) activation require extensive molecular dynamics simulations of the receptor embedded in an explicit lipid-water environment. A possible method for efficiently sampling the conformational space of such a complex system is metadynamics (MetaD) with path collective variables (CV). Here, we applied well-tempered MetaD with path CVs to one of the few GPCRs for which both inactive and fully active experimental structures are available, the μ-opioid receptor (MOR), and assessed the ability of this enhanced sampling method to estimate thermodynamic properties of receptor activation in line with those obtained by more computationally expensive adaptive sampling protocols. While n-body information theory (nBIT) analysis of these simulations confirmed that MetaD can efficiently characterize ligand-induced allosteric communication across the receptor, standard MetaD cannot be used directly to derive kinetic rates because transitions are accelerated by a bias potential. Applying the principle of Maximum Caliber (MaxCal) to the free-energy landscape of morphine-bound MOR reconstructed from MetaD, we obtained Markov State Models (MSMs) that yield kinetic rates of MOR activation in agreement with those obtained by adaptive sampling. Taken together, these results suggest that the MetaD-MaxCal combination creates an efficient strategy for estimating thermodynamic and kinetic properties of GPCR activation at an affordable computational cost.

We present a comprehensive benchmark study of the adsorption energy of a single water molecule on the (001) LiH surface using periodic coupled cluster and quantum Monte Carlo theories. We benchmark and compare different implementations of quantum chemical wave function based theories in order to verify the reliability of the predicted adsorption energies and the employed approximations. Furthermore we compare the predicted adsorption energies to those obtained employing widely used van der Waals density-functionals. Our findings show that quantum chemical approaches are becoming a robust and reliable tool for condensed phase electronic structure calculations, providing an additional tool that can also help in potentially improving currently available van der Waals density-functionals.

The probability density functions (PDFs) of the local measure of pressure as a function of the sampling volume are computed for a model Lennard-Jones (LJ) fluid using the Method of Planes (MOP) and Volume Averaging (VA) techniques. This builds on the study of Heyes, Dini, and Smith [J. Chem. Phys. 145, 104504 (2016)] which only considered the VA method for larger subvolumes. The focus here is typically on much smaller subvolumes than considered previously, which tend to the Irving-Kirkwood limit where the pressure tensor is defined at a point. The PDFs from the MOP and VA routes are compared for cubic subvolumes, V=ℓ3. Using very high grid-resolution and box-counting analysis, we also show that any measurement of pressure in a molecular system will fail to exactly capture the molecular configuration. This suggests that it is impossible to obtain the pressure in the Irving-Kirkwood limit using the commonly employed grid based averaging techniques. More importantly, below ℓ≈3 in LJ reduced units, the PDFs depart from Gaussian statistics, and for ℓ=1.0, a double peaked PDF is observed in the MOP but not VA pressure distributions. This departure from a Gaussian shape means that the average pressure is not the most representative or common value to arise. In addition to contributing to our understanding of local pressure formulas, this work shows a clear lower limit on the validity of simply taking the average value when coarse graining pressure from molecular (and colloidal) systems. EPSRC

The problem of predicting polymorphism in atomic and molecular crystals constitutes a significant challenge both experimentally and theoretically. From the theoretical viewpoint, polymorphism prediction falls into the general class of problems characterized by an underlying rough energy landscape, and consequently, free energy based enhanced sampling approaches can be brought to bear on the problem. In this paper, we build on a scheme previously introduced by two of the authors in which the lengths and angles of the supercell are targeted for enhanced sampling via temperature accelerated adiabatic free energy dynamics [T. Q. Yu and M. E. Tuckerman, Phys. Rev. Lett. 107, 015701 (2011)]. Here, that framework is expanded to include general order parameters that distinguish different crystalline arrangements as target collective variables for enhanced sampling. The resulting free energy surface, being of quite high dimension, is nontrivial to reconstruct, and we discuss one particular strategy for performing the free energy analysis. The method is applied to the study of polymorphism in xenon crystals at high pressure and temperature using the Steinhardt order parameters without and with the supercell included in the set of collective variables. The expected fcc and bcc structures are obtained, and when the supercell parameters are included as collective variables, we also find several new structures, including fcc states with hcp stacking faults. We also apply the new method to the solid-liquid phase transition in copper at 1300 K using the same Steinhardt order parameters. Our method is able to melt and refreeze the system repeatedly, and the free energy profile can be obtained with high efficiency.

Excited-state vibrational dynamics in molecules can be studied by an electronically off-resonant Raman process induced by a probe pulse with variable delay with respect to an actinic pulse. We establish the connection between several variants of the technique that involve either spontaneous or stimulated Raman detection and different pulse configurations. By using loop diagrams in the frequency domain, we show that all signals can be described as six wave mixing which depend on the same four point molecular correlation functions involving two transition dipoles and two polarizabilities and accompanied by a different gating. Simulations for the stochastic two-state-jump model illustrate the origin of the absorptive and dispersive features observed experimentally. © 2013 AIP Publishing LLC.

Effective coarse-grained representations of protein-protein interaction potentials are vital in the modeling of large scale systems. We develop a method to fit an arbitrary number of effective charges to approximate the electrostatic potential of a protein at a given pH in an ionic solution. We find that the effective charges can reproduce an input potential calculated from a high resolution Poisson-Boltzmann calculation. Since the effective charges used in this model are not constrained to the locations of the original charged groups, the extra degrees of freedom allows us to reproduce the field anisotropy with fewer charges. The fitting procedure uses a number of approximations in the charge magnitudes, initial conditions, and multipoles to speed convergence. The most significant gains are found by fitting the multipole moments of the effective charge potential to the moments of the original field. We show that the Yukawa potential is not only sufficient as a pairwise summation in reproducing the potential, but comes naturally from the linearized expansion of the Poisson-Boltzmann equation. We compute interaction energies and find excellent agreement to the original potential. From the effective charge model we compute the electrostatic contribution to the second virial coefficient.

The observables of multidimensional infrared spectroscopy may be calculated from nonlinear vibrational response functions. Fully quantum dynamical calculations of vibrational response functions are generally impractical, while completely classical calculations are qualitatively incorrect at long times. These challenges motivate the development of semiclassical approximations to quantum mechanics, which use classical mechanical information to reconstruct quantum effects. The mean-trajectory (MT) approximation is a semiclassical approach to quantum vibrational response functions employing classical trajectories linked by deterministic transitions representing the effects of the radiation-matter interaction. Previous application of the MT approximation to the third-order response function R(3)(t3, t2, t1) demonstrated that the method quantitatively describes the coherence dynamics of the t3 and t1 evolution times, but is qualitatively incorrect for the waiting-time t2 period. Here we develop an optimized version of the MT approximation by elucidating the connection between this semiclassical approach and the double-sided Feynman diagrams (2FD) that represent the quantum response. Establishing the direct connection between 2FD and semiclassical paths motivates a systematic derivation of an optimized MT approximation (OMT). The OMT uses classical mechanical inputs to accurately reproduce quantum dynamics associated with all three propagation times of the third-order vibrational response function.

We investigate the reversible diffusion-influenced reaction of an isolated pair in the presence of a non-Markovian generalization of the backreaction boundary condition in two space dimensions. Following earlier work by Agmon and Weiss, we consider residence time probability densities that decay slower than an exponential and that are characterized by a single parameter 0σ ≤ 1. We calculate an exact expression for a Green's function of the two-dimensional diffusion equation subject to a non-Markovian backreaction boundary condition that is valid for arbitrary σ and for all times. We use the obtained expression to derive the survival probability for the initially unbound pair and we calculate an exact expression for the probability S(t[line]*) that the initially bound particle is unbound. Finally, we obtain an approximate solution for long times. In particular, we show that the ultimate fate of the bound state is complete dissociation, as in the Markovian case. However, the limiting value is approached quite differently: Instead of a ~t(-1) decay, we obtain 1 - S(t[line]*) ~ t(-σ)ln t. The derived expressions should be relevant for a better understanding of reversible membrane-bound reactions in cell biology.

We investigate reversible diffusion-influenced reactions of an isolated pair in two dimensions. To this end, we employ convolution relations that permit deriving the survival probability of the reversible reaction directly in terms of the survival probability of the irreversible reaction. Furthermore, we make use of the mean reaction time approximation to write the irreversible survival probability in restricted spaces as a single exponential. In this way, we obtain exact and approximative expressions in the time domain for the reversible survival probability for three different two dimensional diffusion spaces: The infinite and restricted plane as well as the surface of a sphere. Our obtained results should prove useful in the context of membrane-bound reversible diffusion-influenced reactions in cell biology., Comment: 3 figures

We derive an exact Green's function of the diffusion equation for a pair of disk-shaped interacting particles in two dimensions subject to a backreaction boundary condition. Furthermore, we use the obtained function to calculate exact expressions for the survival probability and the time-dependent rate coefficient for the initially unbound pair and the survival probability of the bound state. The derived expressions will be of particular utility for the description of reversible membrane-bound reactions in cell biology.

We calculate the pair diffusion coefficient D(r) as a function of the distance r between two hard-sphere particles in a dense monodisperse suspension. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine D(r) from the propagators (Green's functions) of particle pairs obtained from discontinuous molecular dynamics simulations. At distances exceeding 3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamic theory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/r distance dependence of D(r) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated D(r) from the hydrodynamic models at short distances r reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry., 6 pages, 5 figures

The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized Fick-Jacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both Reguera-Rubi and Kalinay-Percus versions of the generalized Fick-Jacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period.

The weighted histogram analysis method (WHAM) is routinely used for computing free energies and expectations from multiple ensembles. Existing derivations of WHAM require observations to be discretized into a finite number of bins. Yet, WHAM formulas seem to hold even if the bin sizes are made arbitrarily small. The purpose of this article is to demonstrate both the validity and value of the multi-state Bennet acceptance ratio (MBAR) method seen as a binless extension of WHAM. We discuss two statistical arguments to derive the MBAR equations, in parallel to the self-consistency and maximum likelihood derivations already known for WHAM. We show that the binless method, like WHAM, can be used not only to estimate free energies and equilibrium expectations, but also to estimate equilibrium distributions. We also provide a number of useful results from the statistical literature, including the determination of MBAR estimators by minimization of a convex function. This leads to an approach to the computation of MBAR free energies by optimization algorithms, which can be more effective than existing algorithms. The advantages of MBAR are illustrated numerically for the calculation of absolute protein-ligand binding free energies by alchemical transformations with and without soft-core potentials. We show that binless statistical analysis can accurately treat sparsely distributed interaction energy samples as obtained from unmodified interaction potentials that cannot be properly analyzed using standard binning methods. This suggests that binless multi-state analysis of binding free energy simulations with unmodified potentials offers a straightforward alternative to the use of soft-core potentials for these alchemical transformations.

The Fourier transform Coulomb (FTC) method has been shown to be effective for the fast and accurate calculation of long-range Coulomb interactions between diffuse (low-energy cutoff) densities in quantum mechanical (QM) systems. In this work, we split the potential of a compact (high-energy cutoff) density into short-range and long-range components, similarly to how point charges are handled in the Ewald mesh methods in molecular mechanics simulations. With this linear scaling QM Ewald mesh method, the long-range potential of compact densities can be represented on the same grid as the diffuse densities that are treated by the FTC method. The new method is accurate and significantly reduces the amount of computational time on short-range interactions, especially when it is compared to the continuous fast multipole method.

A rigorous formalism for estimating noncovalent binding free energies and thermodynamic expectations from calculations in which receptor configurations are sampled independently from the ligand is derived. Due to this separation, receptor configurations only need to be sampled once, facilitating the use of binding free energy calculations in virtual screening. Demonstrative calculations on a host-guest system yield good agreement with previous free energy calculations and isothermal titration calorimetry measurements. Implicit ligand theory provides guidance on how to improve existing molecular docking algorithms and insight into the concepts of induced fit and conformational selection in noncovalent macromolecular recognition., Comment: 21 pages, 2 figures, 4 tables, 3 supplemental figures, 2 supplemental tables

We study two-dimensional diffusion in a channel formed by periodic overlapping circles. Periodic variation of the channel width leads to the slowdown of diffusion along the channel axis. There are several approximate approaches, which allow one to analyze the slowdown. We use these approaches to derive five expressions for the effective diffusion coefficient of a point Brownian particle in the channel. To check the accuracy of the expressions we compare their predictions with the effective diffusion coefficient obtained from Brownian dynamics simulations.

A direct numerical simulation (DNS) procedure is employed to study the thermal motion of a nanoparticle in an incompressible Newtonian stationary fluid medium with the generalized Langevin approach. We consider both the Markovian (white noise) and non-Markovian (Ornstein-Uhlenbeck noise and Mittag-Leffler noise) processes. Initial locations of the particle are at various distances from the bounding wall to delineate wall effects. At thermal equilibrium, the numerical results are validated by comparing the calculated translational and rotational temperatures of the particle with those obtained from the equipartition theorem. The nature of the hydrodynamic interactions is verified by comparing the velocity autocorrelation functions and mean square displacements with analytical results. Numerical predictions of wall interactions with the particle in terms of mean square displacements are compared with analytical results. In the non-Markovian Langevin approach, an appropriate choice of colored noise is required to satisfy the power-law decay in the velocity autocorrelation function at long times. The results obtained by using non-Markovian Mittag-Leffler noise simultaneously satisfy the equipartition theorem and the long-time behavior of the hydrodynamic correlations for a range of memory correlation times. The Ornstein-Uhlenbeck process does not provide the appropriate hydrodynamic correlations. Comparing our DNS results to the solution of an one-dimensional generalized Langevin equation, it is observed that where the thermostat adheres to the equipartition theorem, the characteristic memory time in the noise is consistent with the inherent time scale of the memory kernel. The performance of the thermostat with respect to equilibrium and dynamic properties for various noise schemes is discussed.

Ionic solutions are mixtures of interacting anions and cations. They hardly resemble dilute gases of uncharged noninteracting point particles described in elementary textbooks. Biological and electrochemical solutions have many components that interact strongly as they flow in concentrated environments near electrodes, ion channels, or active sites of enzymes. Interactions in concentrated environments help determine the characteristic properties of electrodes, enzymes, and ion channels. Flows are driven by a combination of electrical and chemical potentials that depend on the charges, concentrations, and sizes of all ions, not just the same type of ion. We use a variational method EnVarA (energy variational analysis) that combines Hamilton’s least action and Rayleigh’s dissipation principles to create a variational field theory that includes flow, friction, and complex structure with physical boundary conditions. EnVarA optimizes both the action integral functional of classical mechanics and the dissipation functional. These functionals can include entropy and dissipation as well as potential energy. The stationary point of the action is determined with respect to the trajectory of particles. The stationary point of the dissipation is determined with respect to rate functions (such as velocity). Both variations are written in one Eulerian (laboratory) framework. In variational analysis, an “extra layer” of mathematics is used to derive partial differential equations. Energies and dissipations of different components are combined in EnVarA and Euler–Lagrange equations are then derived. These partial differential equations are the unique consequence of the contributions of individual components. The form and parameters of the partial differential equations are determined by algebra without additional physical content or assumptions. The partial differential equations of mixtures automatically combine physical properties of individual (unmixed) components. If a new component is added to the energy or dissipation, the Euler–Lagrange equations change form and interaction terms appear without additional adjustable parameters. EnVarA has previously been used to compute properties of liquid crystals, polymer fluids, and electrorheological fluids containing solid balls and charged oil droplets that fission and fuse. Here we apply EnVarA to the primitive model of electrolytes in which ions are spheres in a frictional dielectric. The resulting Euler–Lagrange equations include electrostatics and diffusion and friction. They are a time dependent generalization of the Poisson–Nernst–Planck equations of semiconductors, electrochemistry, and molecular biophysics. They include the finite diameter of ions. The EnVarA treatment is applied to ions next to a charged wall, where layering is observed. Applied to an ion channel, EnVarA calculates a quick transient pile-up of electric charge, transient and steady flow through the channel, stationary “binding” in the channel, and the eventual accumulation of salts in “unstirred layers” near channels. EnVarA treats electrolytes in a unified way as complex rather than simple fluids. Ad hoc descriptions of interactions and flow have been used in many areas of science to deal with the nonideal properties of electrolytes. It seems likely that the variational treatment can simplify, unify, and perhaps derive and improve those descriptions.

Understanding energy landscapes is a major challenge in chemistry and biology. Although a wide variety of methods have been invented and applied to this problem, very little is understood about the actual mathematical structures underlying such landscapes. Perhaps the most general assumption is the idea that energy landscapes are low-dimensional manifolds embedded in high-dimensional Euclidean space. While this is a very mild assumption, we have discovered an example of an energy landscape which is nonmanifold, demonstrating previously unknown mathematical complexity. The example occurs in the energy landscape of cyclo-octane, which was found to have the structure of a reducible algebraic variety, composed of the union of a sphere and a Klein bottle, intersecting in two rings.

We present a powerful replica exchange method, particularly suited to first-order phase transitions associated with the backbending in the statistical temperature, by merging an optimally designed generalized ensemble sampling with replica exchanges. The key ingredients of our method are parametrized effective sampling weights, smoothly joining ordered and disordered phases with a succession of unimodal energy distributions by transforming unstable or metastable energy states of canonical ensembles into stable ones. The inverse mapping between the sampling weight and the effective temperature provides a systematic way to design the effective sampling weights and determine a dynamic range of relevant parameters. Illustrative simulations on Potts spins with varying system size and simulation conditions demonstrate a comprehensive sampling for phase-coexistent states with a dramatic acceleration of tunneling transitions. A significant improvement over the power-law slowing down of mean tunneling times with increasing system size is obtained, and the underlying mechanism for accelerated tunneling is discussed.

Electrostatic effects in nanosystems are understood via a physical picture built on their multiscale character and the distinct behavior of mobile ions versus charge groups fixed to the nanostructure. The Poisson-Boltzmann equation is nondimensionalized to introduce a factor lambda that measures the density of mobile ion charge versus that due to fixed charges; the diffusive smearing and volume exclusion effects of the former tend to diminish its value relative to that from the fixed charges. We introduce the ratio sigma of the average nearest-neighbor atom distance to the characteristic size of the features of the nanostructure of interest (e.g., a viral capsomer). We show that a unified treatment (i.e., lambda proportional to sigma) and a perturbation expansion around sigma=0 yields, through analytic continuation, an approximation to the electrostatic potential of high accuracy and computational efficiency. The approach was analyzed via Padé approximants and demonstrated on viral system electrostatics; it can be generalized to accommodate extended Poisson-Boltzmann models, and has wider applicability to nonequilibrium electrodiffusion and many-particle quantum systems.

Polymer fluids are modeled with dissipative particle dynamics (DPD) as undiluted bead-spring chains and their solutions. The models are assessed by investigating their steady shear-rate properties. Non-Newtonian viscosity and normal stress coefficients, for shear rates from the lower to the upper Newtonian regimes, are calculated from both plane Couette and plane Poiseuille flows. The latter is realized as reverse Poiseuille flow (RPF) generated from two Poiseuille flows driven by uniform body forces in opposite directions along two-halves of a computational domain. Periodic boundary conditions ensure the RPF wall velocity to be zero without density fluctuations. In overlapping shear-rate regimes the RPF properties are confirmed to be in good agreement with those calculated from plane Couette flow with Lees–Edwards periodic boundary conditions (LECs), the standard virtual rheometer for steady shear-rate properties. The concentration and the temperature dependence of the properties of the model fluids are shown to satisfy the principles of concentration and temperature superposition commonly employed in the empirical correlation of real polymer-fluid properties. The thermodynamic validity of the equation of state is found to be a crucial factor for the achievement of time-temperature superposition. With these models, RPF is demonstrated to be an accurate and convenient virtual rheometer for the acquisition of steady shear-rate rheological properties. It complements, confirms, and extends the results obtained with the standard LEC configuration, and it can be used with the output from other particle-based methods, including molecular dynamics, Brownian dynamics, smooth particle hydrodynamics, and the lattice Boltzmann method.

Classical density functional theory (DFT) of fluids is a valuable tool to analyze inhomogeneous fluids. However, few numerical solution algorithms for three-dimensional systems exist. Here we present an efficient numerical scheme for fluids of charged, hard spheres that uses $\mathcal{O}(N\log N)$ operations and $\mathcal{O}(N)$ memory, where $N$ is the number of grid points. This system-size scaling is significant because of the very large $N$ required for three-dimensional systems. The algorithm uses fast Fourier transforms (FFT) to evaluate the convolutions of the DFT Euler-Lagrange equations and Picard (iterative substitution) iteration with line search to solve the equations. The pros and cons of this FFT/Picard technique are compared to those of alternative solution methods that use real-space integration of the convolutions instead of FFTs and Newton iteration instead of Picard. For the hard-sphere DFT we use Fundamental Measure Theory. For the electrostatic DFT we present two algorithms. One is for the \textquotedblleft bulk-fluid\textquotedblright functional of Rosenfeld [Y. Rosenfeld. \textit{J. Chem. Phys.} 98, 8126 (1993)] that uses $\mathcal{O}(N\log N)$ operations. The other is for the \textquotedblleft reference fluid density\textquotedblright (RFD) functional [D. Gillespie et al., J. Phys.: Condens. Matter 14, 12129 (2002)]. This functional is significantly more accurate than the bulk-fluid functional, but the RFD algorithm requires $\mathcal{O}(N^{2})$ operations., 23 pages, 4 figures

We report a single-copy tempering method for simulating large complex systems. In a generalized ensemble, the method uses runtime estimate of the thermal average energy computed from a novel integral identity to guide a continuous temperature-space random walk. We first validated the method in a two-dimensional Ising model and a Lennard-Jones liquid system. It was then applied to folding of three small proteins, trpzip2, trp-cage, and villin headpiece in explicit solvent. Within 0.5~1 microsecond, all three systems were folded into atomic accuracy: the alpha carbon root mean square deviations of the best folded conformations from the native states were 0.2 A, 0.4 A, and 0.4 A, for trpzip2, trp-cage, and villin headpiece, respectively., 60 pages, 13 figures

International audience; The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE solver. While both methods are not guaranteed to converge, numerical evidences suggest that they do and that their convergence is also superlinear. Both variants are significantly faster than the solver based on the exact Jacobian, with a much smaller memory footprint. All three methods have been implemented in a new code named AQUASOL, which is freely available.

We propose a method for incorporating nuclear quantum effects in transition path sampling studies of systems that consist of a few degrees of freedom that must be treated quantum mechanically, while the rest are classical-like. We used the normal mode centroid method to describe the quantum subsystem, which is a method that is not CPU intensive but still reasonably accurate. We applied this mixed centroid/classical transition path sampling method to a model system that has nontrivial quantum behavior, and showed that it can capture the correct quantum dynamical features.

Hybrid Monte Carlo (HMC) is a rigorous sampling method that uses molecular dynamics (MD) as a global Monte Carlo move. The acceptance rate of HMC decays exponentially with system size. The shadow hybrid Monte Carlo (SHMC) was previously introduced to reduce this performance degradation by sampling instead from the shadow Hamiltonian defined for MD when using a symplectic integrator. SHMC’s performance is limited by the need to generate momenta for the MD step from a nonseparable shadow Hamiltonian. We introduce the separable shadow Hamiltonian hybrid Monte Carlo (S2HMC) method based on a formulation of the leapfrog/Verlet integrator that corresponds to a separable shadow Hamiltonian, which allows efficient generation of momenta. S2HMC gives the acceptance rate of a fourth order integrator at the cost of a second-order integrator. Through numerical experiments we show that S2HMC consistently gives a speedup greater than two over HMC for systems with more than 4000 atoms for the same variance. By comparison, SHMC gave a maximum speedup of only 1.6 over HMC. S2HMC has the additional advantage of not requiring any user parameters beyond those of HMC. S2HMC is available in the program PROTOMOL 2.1. A Python version, adequate for didactic purposes, is also in MDL (http://mdlab.sourceforge.net/s2hmc).

The replica exchange statistical temperature Monte Carlo algorithm (RESTMC) is presented, extending the single-replica STMC algorithm [J. Kim, J. E. Straub, and T. Keyes, Phys. Rev. Lett. 97, 050601 (2006)] to alleviate the slow convergence of the conventional temperature replica exchange method (t-REM) with increasing system size. In contrast to the Gibbs–Boltzmann sampling at a specific temperature characteristic of the standard t-REM, RESTMC samples a range of temperatures in each replica and achieves a flat energy sampling employing the generalized sampling weight, which is automatically determined via the dynamic modification of the replica-dependent statistical temperature. Faster weight determination, through the dynamic update of the statistical temperature, and the flat energy sampling, maximizing energy overlaps between neighboring replicas, lead to a considerable acceleration in the convergence of simulations even while employing significantly fewer replicas. The performance of RESTMC is demonstrated and quantitatively compared with that of the conventional t-REM under varying simulation conditions for Lennard-Jones 19, 31, and 55 atomic clusters, exhibiting single- and double-funneled energy landscapes.

Studying the Brownian motion of a system driven by an external control from one macroscopic state to another macroscopic state, this paper presents the derivation of a nonlinear fluctuation-dissipation theorem (FDT). The new FDT relates the nonequilibrium work to the equilibrium free-energy difference in a very simple manner. It is valid wherever the Brownian dynamics is applicable. It recovers the well-known Crooks fluctuation theorem (CFT) within the quasiequilibrium regime where the dissipative work is nearly zero. It will also be shown that the CFT’s fundamental assumption of microscopic reversibility is not obeyed in experiments such as mechanically unfolding biological molecules, in which the external driving forces depend on the system’s coordinates.

The Jarzynski equality and the fluctuation theorem relate equilibrium free energy differences to non-equilibrium measurements of the work. These relations extend to single-molecule experiments that have probed the finite-time thermodynamics of proteins and nucleic acids. The effects of experimental error and instrument noise have not previously been considered. Here, we present a Bayesian formalism for estimating free-energy changes from non-equilibrium work measurements that compensates for instrument noise and combines data from multiple driving protocols. We reanalyze a recent set of experiments in which a single RNA hairpin is unfolded and refolded using optical tweezers at three different rates. Interestingly, the fastest and farthest-from-equilibrium measurements contain the least instrumental noise, and therefore provide a more accurate estimate of the free energies than a few slow, more noisy, near-equilibrium measurements. The methods we propose here will extend the scope of single-molecule experiments; they can be used in the analysis of data from measurements with AFM, optical, and magnetic tweezers., 8 pages

Coarse-grained (CG) models provide a computationally efficient method for rapidly investigating the long time- and length-scale processes that play a critical role in many important biological and soft matter processes. Recently, Izvekov and Voth introduced a new multiscale coarse-graining (MS-CG) method [J. Phys. Chem. B 109, 2469 (2005); J. Chem. Phys. 123, 134105 (2005)] for determining the effective interactions between CG sites using information from simulations of atomically detailed models. The present work develops a formal statistical mechanical framework for the MS-CG method and demonstrates that the variational principle underlying the method may, in principle, be employed to determine the many-body potential of mean force (PMF) that governs the equilibrium distribution of positions of the CG sites for the MS-CG models. A CG model that employs such a PMF as a “potential energy function” will generate an equilibrium probability distribution of CG sites that is consistent with the atomically detailed model from which the PMF is derived. Consequently, the MS-CG method provides a formal multiscale bridge rigorously connecting the equilibrium ensembles generated with atomistic and CG models. The variational principle also suggests a class of practical algorithms for calculating approximations to this many-body PMF that are optimal. These algorithms use computer simulation data from the atomically detailed model. Finally, important generalizations of the MS-CG method are introduced for treating systems with rigid intramolecular constraints and for developing CG models whose equilibrium momentum distribution is consistent with that of an atomically detailed model.

The multiscale coarse-graining MS-CG method S. Izvekov and G.A. Voth, J. Phys. Chem. B 109, 2469 2005; J. Chem. Phys. 123, 134105 2005 employs a variational principle to determine an interaction potential for a CG model from simulations of an atomically detailed model of the same system. The companion paper proved that, if no restrictions regarding the form of the CG interaction potential are introduced and if the equilibrium distribution of the atomistic model has been adequately sampled, then the MS-CG variational principle determines the exact many-body potential of mean force PMF governing the equilibrium distribution of CG sites generated by the atomistic model. In practice, though, CG force fields are not completely flexible, but only include particular types of interactions between CG sites, e.g., nonbonded forces between pairs of sites. If the CG force field depends linearly on the force field parameters, then the vector valued functions that relate the CG forces to these parameters determine a set of basis vectors that span a vector subspace of CG force fields. The companion paper introduced a distance metric for the vector space of CG force fields and proved that the MS-CG variational principle determines the CG force force field that is within that vector subspace and that is closest to the force field determined by the many-body PMF. The present paper applies the MS-CG variational principle for parametrizing molecular CG force fields and derives a linear least squares problem for the parameter set determining the optimal approximation to this many-body PMF. Linear systems of equations for these CG force field parameters are derived and analyzed in terms of equilibrium structural correlation functions. Numerical calculations for a one-site CG model of methanol and a molecular CG model of the EMIM + /NO 3 ionic liquid are provided to illustrate the method. © 2008 American

Next-generation molecular force fields deliver accurate descriptions of non-covalent interactions by employing more elaborate functional forms than their predecessors. Much work has been dedicated to improving the description of the electrostatic potential (ESP) generated by these force fields. A common approach to improving the ESP is by augmenting the point charges on each center with higher-order multipole moments. The resulting anisotropy greatly improves the directionality of the non-covalent bonding, with a concomitant increase in computational cost. In this work, we develop an efficient strategy for enumerating multipole interactions, by casting an efficient spherical harmonic based approach within a particle mesh Ewald (PME) framework. Although the derivation involves lengthy algebra, the final expressions are relatively compact, yielding an approach that can efficiently handle both finite and periodic systems without imposing any approximations beyond PME. Forces and torques are readily obtained, making our method well suited to modern molecular dynamics simulations.

The emerging complexity of large macromolecules has led to challenges in their full scale theoretical description and computer simulation. Multiscale multiphysics and multidomain models have been introduced to reduce the number of degrees of freedom while maintaining modeling accuracy and achieving computational efficiency. A total energy functional is constructed to put energies for polar and nonpolar solvation, chemical potential, fluid flow, molecular mechanics, and elastic dynamics on an equal footing. The variational principle is utilized to derive coupled governing equations for the above mentioned multiphysical descriptions. Among these governing equations is the Poisson-Boltzmann equation which describes continuum electrostatics with atomic charges. The present work introduces the theory of continuum elasticity with atomic rigidity (CEWAR). The essence of CEWAR is to formulate the shear modulus as a continuous function of atomic rigidity. As a result, the dynamics complexity of a macromolecular system is separated from its static complexity so that the more time-consuming dynamics is handled with continuum elasticity theory, while the less time-consuming static analysis is pursued with atomic approaches. We propose a simple method, flexibility-rigidity index (FRI), to analyze macromolecular flexibility and rigidity in atomic detail. The construction of FRI relies on the fundamental assumption that protein functions, such as flexibility, rigidity, and energy, are entirely determined by the structure of the protein and its environment, although the structure is in turn determined by all the interactions. As such, the FRI measures the topological connectivity of protein atoms or residues and characterizes the geometric compactness of the protein structure. As a consequence, the FRI does not resort to the interaction Hamiltonian and bypasses matrix diagonalization, which underpins most other flexibility analysis methods. FRI's computational complexity is of \documentclass[12pt]{minimal}\begin{document}${\cal O}(N^2)$\end{document}O(N2) at most, where N is the number of atoms or residues, in contrast to \documentclass[12pt]{minimal}\begin{document}${\cal O}(N^3)$\end{document}O(N3) for Hamiltonian based methods. We demonstrate that the proposed FRI gives rise to accurate prediction of protein B-Factor for a set of 263 proteins. We show that a parameter free FRI is able to achieve about 95% accuracy of the parameter optimized FRI. An interpolation algorithm is developed to construct continuous atomic flexibility functions for visualization and use with CEWAR.

Stochastic models of reaction-diffusion systems are important for the study of biochemical reaction networks where species are present in low copy numbers or if reactions are highly diffusion limited. In living cells many such systems include reactions and transport on one-dimensional structures, such as DNA and microtubules. The cytoskeleton is a dynamic structure where individual fibers move, grow, and shrink. In this paper we present a simulation algorithm that combines single molecule simulations in three-dimensional space with single molecule simulations on one-dimensional structures of arbitrary shape. Molecules diffuse and react with each other in space, they associate with and dissociate from one-dimensional structures as well as diffuse and react with each other on the one-dimensional structure. A general curve embedded in space can be approximated by a piecewise linear curve to arbitrary accuracy. The resulting algorithm is hence very flexible. Molecules bound to a curve can move by pure diffusion or via active transport, and the curve can move in space as well as grow and shrink. The flexibility and accuracy of the algorithm is demonstrated in five numerical examples.

A simple approximate formula is derived for the rate constant that describes steady-state flux of diffusing particles through a cluster of perfectly absorbing disks on the otherwise reflecting flat wall, assuming that the disk centers occupy neighboring sites of a square lattice. A distinctive feature of trapping by a disk cluster is that disks located at the cluster periphery shield the disks in the center of the cluster. This competition of the disks for diffusing particles makes it impossible to find an exact analytical solution for the rate constant in the general case. To derive the approximate formula, we use a recently suggested approach [A. M. Berezhkovskii, L. Dagdug, V. A. Lizunov, J. Zimmerberg, and S. M. Bezrukov, J. Chem. Phys. 136, 211102 (2012)] doi: 10.1063/1.4726015., which is based on the replacement of the disk cluster by an effective uniform partially absorbing spot. The formula shows how the rate constant depends on the size and shape of the cluster. To check the accuracy of the formula, we compare its predictions with the values of the rate constant obtained from Brownian dynamics simulations. The comparison made for 18 clusters of various shapes and sizes shows good agreement between the theoretical predictions and numerical results.

The mean lifetime of a particle diffusing in a cylindrical cavity with a circular absorbing spot on the cavity wall is studied analytically as a function of the spot radius, its location on the wall, the particle initial position, and the cavity shape determined by its length and radius. When the spot radius tends to zero our formulas for the mean lifetime reduce to the result given by the solution of the narrow escape problem, according to which the mean lifetime is proportional to the ratio of the cavity volume to the spot radius and is independent of the cavity shape, the spot location on the cavity wall, and the particle starting point, assuming that this point is not too close to the spot. When the spot radius is not small enough, the asymptotic narrow escape formula for the mean lifetime fails, and one should use more general formulas derived in the present study. To check the accuracy and to establish the range of applicability of the formulas, we compare our theoretical predictions with the results of Brownian dynamics simulations.

The spatial direct method with gradient-based diffusion is an accelerated stochastic reaction-diffusion simulation algorithm that treats diffusive transfers between neighboring subvolumes based on concentration gradients. This recent method achieved a marked improvement in simulation speed and reduction in the number of time-steps required to complete a simulation run, compared with the exact algorithm, by sampling only the net diffusion events, instead of sampling all diffusion events. Although the spatial direct method with gradient-based diffusion gives accurate means of simulation ensembles, its gradient-based diffusion strategy results in reduced fluctuations in populations of diffusive species. In this paper, we present a new improved algorithm that is able to anticipate all possible microscopic fluctuations due to diffusive transfers in the system and incorporate this information to retain the same degree of fluctuations in populations of diffusing species as the exact algorithm. The new algorithm also provides a capability to set the desired level of fluctuation per diffusing species, which facilitates adjusting the balance between the degree of exactness in simulation results and the simulation speed. We present numerical results that illustrate the recovery of fluctuations together with the accuracy and efficiency of the new algorithm.

The displacement of perturbed water upon binding is believed to play a critical role in the thermodynamics of biomolecular recognition, but it is nontrivial to unambiguously define and answer questions about this process. We address this issue by introducing grid inhomogeneous solvation theory (GIST), which discretizes the equations of inhomogeneous solvation theory (IST) onto a three-dimensional grid situated in the region of interest around a solute molecule or complex. Snapshots from explicit solvent simulations are used to estimate localized solvation entropies, energies, and free energies associated with the grid boxes, or voxels, and properly summing these thermodynamic quantities over voxels yields information about hydration thermodynamics. GIST thus provides a smoothly varying representation of water properties as a function of position, rather than focusing on hydration sites where solvent is present at high density. It therefore accounts for full or partial displacement of water from sites that are highly occupied by water, as well as for partly occupied and water-depleted regions around the solute. GIST can also provide a well-defined estimate of the solvation free energy and therefore enables a rigorous end-states analysis of binding. For example, one may not only use a first GIST calculation to project the thermodynamic consequences of displacing water from the surface of a receptor by a ligand, but also account, in a second GIST calculation, for the thermodynamics of subsequent solvent reorganization around the bound complex. In the present study, a first GIST analysis of the molecular host cucurbit[7]uril is found to yield a rich picture of hydration structure and thermodynamics in and around this miniature receptor. One of the most striking results is the observation of a toroidal region of high water density at the center of the host's nonpolar cavity. Despite its high density, the water in this toroidal region is disfavored energetically and entropically, and hence may contribute to the known ability of this small receptor to bind guest molecules with unusually high affinities. Interestingly, the toroidal region of high water density persists even when all partial charges of the receptor are set to zero. Thus, localized regions of high solvent density can be generated in a binding site without strong, attractive solute-solvent interactions.

We investigate transport of point Brownian particles in a tube formed by identical periodic compartments of varying diameter, focusing on the effects due to the compartment asymmetry. The paper contains two parts. First, we study the force-dependent mobility of the particle. The mobility is a symmetric non-monotonic function of the driving force, F, when the compartment is symmetric. Compartment asymmetry gives rise to an asymmetric force-dependent mobility, which remains non-monotonic when the compartment asymmetry is not too high. The F-dependence of the mobility becomes monotonic in tubes formed by highly asymmetric compartments. The transition of the F-dependence of the mobility from non-monotonic to monotonic behavior results in important consequences for the particle motion under the action of a time-periodic force with zero mean, which are discussed in the second part of the paper: In a tube formed by moderately asymmetric compartments, the particle under the action of such a force moves with an effective drift velocity that vanishes at small and large values of the force amplitude having a maximum in between. In a tube formed by highly asymmetric compartments, the effective drift velocity monotonically increases with the amplitude of the driving force and becomes unboundedly large as the amplitude tends to infinity.

As a key process of many biological reactions such as biological energy transduction or human sensory systems, proton transport has attracted much research attention in biological, biophysical, and mathematical fields. A quantum dynamics in continuum framework has been proposed to study proton permeation through membrane proteins in our earlier work and the present work focuses on the generalized correlation of protons with their environment. Being complementary to electrostatic potentials, generalized correlations consist of proton-proton, proton-ion, proton-protein, and proton-water interactions. In our approach, protons are treated as quantum particles while other components of generalized correlations are described classically and in different levels of approximations upon simulation feasibility and difficulty. Specifically, the membrane protein is modeled as a group of discrete atoms, while ion densities are approximated by Boltzmann distributions, and water molecules are represented as a dielectric continuum. These proton-environment interactions are formulated as convolutions between number densities of species and their corresponding interaction kernels, in which parameters are obtained from experimental data. In the present formulation, generalized correlations are important components in the total Hamiltonian of protons, and thus is seamlessly embedded in the multiscale/multiphysics total variational model of the system. It takes care of non-electrostatic interactions, including the finite size effect, the geometry confinement induced channel barriers, dehydration and hydrogen bond effects, etc. The variational principle or the Euler-Lagrange equation is utilized to minimize the total energy functional, which includes the total Hamiltonian of protons, and obtain a new version of generalized Laplace-Beltrami equation, generalized Poisson-Boltzmann equation and generalized Kohn-Sham equation. A set of numerical algorithms, such as the matched interface and boundary method, the Dirichlet to Neumann mapping, Gummel iteration, and Krylov space techniques, is employed to improve the accuracy, efficiency, and robustness of model simulations. Finally, comparisons between the present model predictions and experimental data of current-voltage curves, as well as current-concentration curves of the Gramicidin A channel, verify our new model.

We study the search of a small round hole in the wall of a spherical cavity by a diffusing particle, which can reversibly bind to the cavity wall and diffuse on the surface being in the bound state. There are two channels for the particle first passage to the hole, through the bulk, and through the surface. We propose a coarse-grained model of the search process and use it to derive simple approximate formulas for the mean time required for the particle to reach the hole for the first time and for the probability of the first passage to the hole through the bulk channel. This is done for two distributions of the particle starting point: (1) Uniform distribution over the cavity volume and (2) uniform distribution over the cavity wall. We check the accuracy of the approximate formulas by comparing their predictions with the corresponding quantities found by solving the mixed bulk-surface diffusion problem numerically by the finite difference method. The comparison shows excellent agreement between the analytical and numerical results.

The paper deals with a reaction-diffusion problem that arises in developmental biology when describing the formation of the concentration profiles of signaling molecules, called morphogens, which control gene expression and, hence, cell differentiation. The mean local accumulation time, which is the mean time required to reach the steady state at a fixed point of a patterned tissue, is an important characteristic of the formation process. We show that this time is a sum of two times, the conditional mean first-passage time from the source to the observation point and the mean local accumulation time in the situation when the source is localized at the observation point.

Basis set superposition error (BSSE) is a significant contributor to errors in quantum-based energy functions, especially for large chemical systems with many molecular contacts such as folded proteins and protein-ligand complexes. While the counterpoise method has become a standard procedure for correcting intermolecular BSSE, most current approaches to correcting intramolecular BSSE are simply fragment-based analogues of the counterpoise method which require many (two times the number of fragments) additional quantum calculations in their application. We propose that magnitudes of both forms of BSSE can be quickly estimated by dividing a system into interacting fragments, estimating each fragment's contribution to the overall BSSE with a simple statistical model, and then propagating these errors throughout the entire system. Such a method requires no additional quantum calculations, but rather only an analysis of the system's interacting fragments. The method is described herein and is applied to a protein-ligand system, a small helical protein, and a set of native and decoy protein folds.

When there is a separation of time scales, an effective description of the dynamics of the slow variables can be obtained by adiabatic elimination of fast ones. For example, for anisotropic Langevin dynamics in two dimensions, the conventional procedure leads to a Langevin equation for the slow coordinate that involves the potential of the mean force. The friction constant along this coordinate remains unchanged. Here, we show that a more accurate, but still Markovian, description of the slow dynamics can be obtained by using position-dependent friction that is related to the time integral of the autocorrelation function of the difference between the actual force and the mean force by a Kirkwood-like formula. The result is generalized to many dimensions, where the slow or reaction coordinate is an arbitrary function of the Cartesian coordinates. When the fast variables are effectively one-dimensional, the additional friction along the slow coordinate can be expressed in closed form for an arbitrary potential. For a cylindrically symmetric channel of varying cross section with winding centerline, our analytical expression immediately yields the multidimensional version of the Zwanzig-Bradley formula for the position-dependent diffusion coefficient.

Effective mobility and diffusion coefficient of a particle in a tube with identical periodic dead ends characterize the motion on large time scale, when the particle displacement significantly exceeds the tube period. We derive formulas that show how these transport coefficients depend on the driving force and the geometric parameters of the system. Numerical tests show that values of the transport coefficients obtained from Brownian dynamics simulations are in excellent agreement with our theoretical predictions.

The recently proposed mixed quantum-classical method is extended to applications at finite temperatures. The method is designed to treat complex systems consisting of a low-dimensional quantum part (the primary system) coupled to a dissipative bath described classically. The method is based on a formalism showing how to systematically correct the approximate zeroth-order evolution rule. The corrections are defined in terms of the total quantum Hamiltonian and are taken to the classical limit by introducing the frozen Gaussian approximation for the bath degrees of freedom. The evolution of the primary system is governed by the corrected propagator yielding the exact quantum dynamics. The method has been tested on a standard model system describing proton transfer in a condensed-phase environment: a symmetric double-well potential bilinearly coupled to a bath of harmonic oscillators. Flux correlation functions and thermal rate constants have been calculated at two different temperatures for a range of coupling strengths. The results have been compared to the fully quantum simulations of Topaler and Makri [J. Chem. Phys. 101, 7500 (1994)] with the real path integral method.

We study the effect of the driving force on brownian motion of a point particle in a tube formed by identical spherical compartments, which create periodic entropy potential for the motion along the tube axis. The focus is on (i) the effective mobility and diffusion coefficient of the particle as functions of the driving force, (ii) localization of the particle in the central part of the tube induced by the driving force, and (iii) transit time of the particle between the openings connecting neighboring compartments. Some of the results at very small and large driving force are obtained analytically, while the majority of the results are obtained from brownian dynamics simulations.

The coherent third order optical response of molecular aggregates with fluctuating frequencies, couplings, and transition dipole moments is studied. We derived stochastic nonlinear exciton equations (SNEEs) by combining the quasiparticle picture of excitons with the path integral over stochastic bath paths described by the stochastic Liouville equations. Coherent two-dimensional (2D) spectra are calculated for a tetramer model system whose transition dipole orientations undergo two-state stochastic jumps on an arbitrary timescale. Correspondence between domains of ordered dipoles, which determine the exciton coherence length and the absorption peaks, is established. Signatures of domain coherence length fluctuations are observed in the cross peak dynamics of the 2D spectra in specific pulse polarization configurations.

Reduction of three-dimensional (3D) description of diffusion in a tube of variable cross section to an approximate one-dimensional (1D) description has been studied in detail previously only in tubes of slowly varying diameter. Here we discuss an effective 1D description in the opposite limiting case when the tube diameter changes abruptly, i.e., in a tube composed of any number of cylindrical sections of different diameters. The key step of our approach is an approximate description of the particle transitions between the wide and narrow parts of the tube as trapping by partially absorbing boundaries with appropriately chosen trapping rates. Boundary homogenization is used to determine the trapping rate for transitions from the wide part of the tube to the narrow one. This trapping rate is then used in combination with the condition of detailed balance to find the trapping rate for transitions in the opposite direction, from the narrow part of the tube to the wide one. Comparison with numerical solution of the 3D diffusion equation allows us to test the approximate 1D description and to establish the conditions of its applicability. We find that suggested 1D description works quite well when the wide part of the tube is not too short, whereas the length of the narrow part can be arbitrary. Taking advantage of this description in the problem of escape of diffusing particle from a cylindrical cavity through a cylindrical tunnel we can lift restricting assumptions accepted in earlier theories: We can consider the particle motion in the tunnel and in the cavity on an equal footing, i.e., we can relax the assumption of fast intracavity relaxation used in all earlier theories. As a consequence, the dependence of the escape kinetics on the particle initial position in the system can be analyzed. Moreover, using the 1D description we can analyze the escape kinetics at an arbitrary tunnel radius, whereas all earlier theories are based on the assumption that the tunnel is narrow.