Finite temperature string method for the study of rare events

A method is presented for the study of rare events such as conformational changes arising in activated processes whose reaction coordinate is not known beforehand and for which the assumptions of transition state theory are invalid. The method samples the energy landscape adaptively and determines the isoprobability surfaces for the transition: by definition the trajectories initiated anywhere on one of these surfaces has equal probability to reach first one metastable set rather than the other. Upon weighting these surfaces by the equilibrium probability distribution, one obtains an effective transition pathway, i.e., a tube in configuration space inside which conformational changes occur with high probability, and the associated rate. The method is first validated on a simple two-dimensional example; then it is applied to a model of solid-solid transformation of a condensed system.     

The geometric minimum action method for computing minimum energy paths

An algorithm is proposed to calculate the minimum energy path (MEP). The algorithm is based on a variational formulation in which the MEP is characterized as the curve minimizing a certain functional. The algorithm performs this minimization using a preconditioned steepest-descent scheme with a reparametrization step to enforce a constraint on the curve parametrization.     

Adaptive importance sampling Monte Carlo simulation of rare transition events

We develop a general theoretical framework for the recently proposed importance sampling method for enhancing the efficiency of rare-event simulations [W. Cai, M. H. Kalos, M. de Koning, and V. V. Bulatov, Phys. Rev. E 66, 046703 (2002)], and discuss practical aspects of its application. We define the success/fail ensemble of all possible successful and failed transition paths of any duration and demonstrate that in this formulation the rare-event problem can be interpreted as a "hit-or-miss" Monte Carlo quadrature calculation of a path integral. The fact that the integrand contributes significantly only for a very tiny fraction of all possible paths then naturally leads to a "standard" importance sampling approach to Monte Carlo (MC) quadrature and the existence of an optimal importance function. In addition to showing that the approach is general and expected to be applicable beyond the realm of Markovian path simulations, for which the method was originally proposed, the formulation reveals a conceptual analogy with the variational MC (VMC) method. The search for the optimal importance function in the former is analogous to finding the ground-state wave function in the latter. In two model problems we discuss practical aspects of finding a suitable approximation for the optimal importance function. For this purpose we follow the strategy that is typically adopted in VMC calculations: the selection of a trial functional form for the optimal importance function, followed by the optimization of its adjustable parameters. The latter is accomplished by means of an adaptive optimization procedure based on a combination of steepest-descent and genetic algorithms.     

Simplified and improved string method for computing the minimum energy paths in barrier-crossing events

We present a simplified and improved version of the string method, originally proposed by E et al. [Phys. Rev. B 66, 052301 (2002)] for identifying the minimum energy paths in barrier-crossing events. In this new version, the step of projecting the potential force to the direction normal to the string is eliminated and the full potential force is used in the evolution of the string. This not only simplifies the numerical procedure, but also makes the method more stable and accurate. We discuss the algorithmic details of the improved string method, analyze its stability, accuracy and efficiency, and illustrate it via numerical examples. We also show how the string method can be combined with the climbing image technique for the accurate calculation of saddle points and we present another algorithm for the accurate calculation of the unstable directions at the saddle points.     

An efficient self-optimized sampling method for rare events in nonequilibrium systems

Rare events such as nucleation processes are of ubiquitous importance in real systems.The most popular method for nonequilibrium systems,forward flux sampling(FFS),samples rare events by using interfaces to partition the whole transition process into sequence of steps along an order parameter connecting the initial and final states.FFS usually suffers from two main difficulties:low computational efficiency due to bad interface locations and even being not applicable when trapping into unknown intermediate metastable states.In the present work,we propose an approach to overcome these difficulties,by self-adaptively locating the interfaces on the fly in an optimized manner.Contrary to the conventional FFS which set the interfaces with equal distance of the order parameter,our approach determines the interfaces with equal transition probability which is shown to satisfy the optimization condition.This is done by firstly running long local trajectories starting from the current interface i to get the conditional probability distribution Pc(>i|i),and then determining i+1by equaling Pc(i+1|i)to a give value p0.With these optimized interfaces,FFS can be run in a much more efficient way.In addition,our approach can conveniently find the intermediate metastable states by monitoring some special long trajectories that neither end at the initial state nor reach the next interface,the number of which will increase sharply from zero if such metastable states are encountered.We apply our approach to a two-state model system and a two-dimensional lattice gas Ising model.Our approach is shown to be much more efficient than the conventional FFS method without losing accuracy,and it can also well reproduce the two-step nucleation scenario of the Ising model with easy identification of the intermediate metastable state.     

Harmonic Fourier beads method for studying rare events on rugged energy surfaces

We present a robust, distributable method for computing minimum free energy paths of large molecular systems with rugged energy landscapes. The method, which we call harmonic Fourier beads (HFB), exploits the Fourier representation of a path in an appropriate coordinate space and proceeds iteratively by evolving a discrete set of harmonically restrained path points-beads-to generate positions for the next path. The HFB method does not require explicit knowledge of the free energy to locate the path. To compute the free energy profile along the final path we employ an umbrella sampling method in two generalized dimensions. The proposed HFB method is anticipated to aid the study of rare events in biomolecular systems. Its utility is demonstrated with an application to conformational isomerization of the alanine dipeptide in gas phase.     

On the detection of rare,and moderately rare,nuclear events

Some of the more important developments in science and practical demands in commerce have been linked to attempts to detect rare events and rare contaminants, ranging from the early “counting” of solar neutrinos to the occurrence of dodder seeds in clover. For moderately rare events (≈5 to 50 counts) we consider limitations of the Poisson-normal approximation, together with the apparent problem of excessive false positives when a common expression is (mis-)used for detection decisions. For very rare events, rigorous approaches published more than half a century ago are applicable to such current problems as trace actinide contamination and nuclear treaty monitoring.     

PyRETIS 2: An improbability drive for rare events

The algorithmic development in the field of path sampling has made tremendous progress in recent years. Although the original transition path sampling method was mostly used as a qualitative tool to sample reaction paths, the more recent family of interface-based path sampling methods has paved the way for more quantitative rate calculation studies. Of the exact methods, the replica exchange transition interface sampling (RETIS) method is the most efficient, but rather difficult to implement. This has been the main motivation to develop the open-source Python-based computer library PyRETIS that was released in 2017. PyRETIS is designed to be easily interfaced with any molecular dynamics (MD) package using either classical or ab initio MD. In this study, we report on the principles and the software enhancements that are now included in PyRETIS 2, as well as the recent developments on the user interface, improvements of the efficiency via the implementation of new shooting moves, easier initialization procedures, analysis methods, and supported interfaced software. © 2019 The Authors. Journal of Computational Chemistry published by Wiley Periodicals, Inc.     

Improved transition path sampling methods for simulation of rare events

The free energy surfaces of a wide variety of systems encountered in physics, chemistry, and biology are characterized by the existence of deep minima separated by numerous barriers. One of the central aims of recent research in computational chemistry and physics has been to determine how transitions occur between deep local minima on rugged free energy landscapes, and transition path sampling (TPS) Monte-Carlo methods have emerged as an effective means for numerical investigation of such transitions. Many of the shortcomings of TPS-like approaches generally stem from their high computational demands. Two new algorithms are presented in this work that improve the efficiency of TPS simulations. The first algorithm uses biased shooting moves to render the sampling of reactive trajectories more efficient. The second algorithm is shown to substantially improve the accuracy of the transition state ensemble by introducing a subset of local transition path simulations in the transition state. The system considered in this work consists of a two-dimensional rough energy surface that is representative of numerous systems encountered in applications. When taken together, these algorithms provide gains in efficiency of over two orders of magnitude when compared to traditional TPS simulations.     

Sampling of rare events using hidden restraints

A method to enhance sampling of rare events is presented. It makes use of distance or dihedral-angle restraints to overcome an energy barrier separating two metastable states or to stabilize a transition state between the two metastable states. In order not to perturb these metastable end states themselves, a prefactor is introduced into the restraining energy function, which smoothly increases the weight of this function from zero to one at the transition state or on top of the separating energy barrier and then decreases the weight again to zero at the final state. The method is combined with multi-configurational thermodynamic integration and applied to two biomolecular systems, which were difficult to treat using standard thermodynamic integration. As first example the free energy difference of a cyclic alpha-aminoxy-hexapeptide-ion complex upon changing the ion from Cl- to Na+ was calculated. A large conformational rearrangement of the peptide was necessary to accommodate this change. Stabilizing the transition state by (hidden) restraints facilitates that. As a second example, the free energy difference between the 4C1 and the 1C4 conformation of beta-D-glucopyranoside was calculated. In unrestrained simulations the change from the 4C1 into the 1C4 conformation was never observed because of the high energy barrier separating the two states. Using (hidden) restraints, the transition from the 4C1 into the 1C4 state and back could be enforced without perturbing the end states. As comparison, for the same transitions the potential of mean force as obtained by using dihedral-angle constraints is provided.     

Sequential quadratic programming method for determining the minimum energy path

A new method, referred to as the sequential quadratic programming method, is presented for determining minimum energy paths. The method is based on minimizing the points representing the path in the subspace perpendicular to the tangent of the path while using a penalty term to prevent kinks from forming. Rather than taking one full step, the minimization is divided into a number of sequential steps on an approximate quadratic surface. The resulting method can efficiently determine the reaction mechanism, from which transition state can be easily identified and refined with other methods. To improve the resolution of the path close to the transition state, points are clustered close to this region with a reparametrization scheme. The usefulness of the algorithm is demonstrated for the Muller-Brown potential, amide hydrolysis, and an 89 atom cluster taken from the active site of 4-oxalocrotonate tautomerase for the reaction which catalyzes 2-oxo-4-hexenedioate to the intermediate 2-hydroxy-2,4-hexadienedioate.     

Forward flux sampling-type schemes for simulating rare events: efficiency analysis

We analyze the efficiency of several simulation methods which we have recently proposed for calculating rate constants for rare events in stochastic dynamical systems in or out of equilibrium. We derive analytical expressions for the computational cost of using these methods and for the statistical error in the final estimate of the rate constant for a given computational cost. These expressions can be used to determine which method to use for a given problem, to optimize the choice of parameters, and to evaluate the significance of the results obtained. We apply the expressions to the two-dimensional nonequilibrium rare event problem proposed by Maier and Stein [Phys. Rev. E 48, 931 (1993)]. For this problem, our analysis gives accurate quantitative predictions for the computational efficiency of the three methods.     

High throughput single molecule tracking for analysis of rare populations and events

High throughput single molecule tracking methods were developed to perform quantitative analyses of rare molecular populations. An optimization strategy for single molecule tracking at interfaces is described that allowed tracking of ~10(6) unique trajectories. These large statistical datasets were analyzed in order to identify and characterize distinct molecular populations based on their characteristic dynamic behavior (residence time or surface diffusion) and/or their spatial distribution. Cumulative (i.e. integrated) probability distributions were found to be several orders of magnitude more sensitive to rare populations than were raw probability distributions. Mapping using Accumulated Probe Trajectories (MAPT) was used to characterize molecular populations associated with rare surface heterogeneities. Importantly, large sample sizes were found to result in a dramatic enhancement in the ability to identify rare populations and to resolve their dynamic and spatial parameters.     

Optimizing the sampling and staging for simulations of rare events via forward flux sampling schemes

In this work, we present an adaptive algorithm to optimize the phase space sampling for simulations of rare events in complex systems via forward flux sampling (FFS) schemes. In FFS, interfaces are used to partition the phase space along an order parameter lambda connecting the initial and final regions of interest. Since the kinetic "bottleneck" regions along the order parameter are not usually known beforehand, an adaptive procedure is used that first finds these regions by estimating the rate constants associated with reaching subsequent interfaces; thereafter, the FFS simulation is reset to concentrate the sampling on those bottlenecks. The approach can optimize for either the number and position of the interfaces (i.e., optimized lambda phase staging) or the number M of fired trial runs per interface (i.e., the {M(i)} set) to minimize the statistical error in the rate constant estimation per simulation period. For example, the optimization of the lambda staging leads to a net constant flux of partial trajectories between interfaces and hence a constant flux of connected paths throughout the region between the two end states. The method is demonstrated for several test systems, including the folding of a lattice protein. It is shown that the proposed approach leads to an optimized lambda staging and {M(i)} set which increase the computational efficiency of the sampling algorithm.     

Atomistic simulations of rare events using gentlest ascent dynamics

The dynamics of complex systems often involve thermally activated barrier crossing events that allow these systems to move from one basin of attraction on the high dimensional energy surface to another. Such events are ubiquitous, but challenging to simulate using conventional simulation tools, such as molecular dynamics. Recently, E and Zhou [Nonlinearity 24(6), 1831 (2011)] proposed a set of dynamic equations, the gentlest ascent dynamics (GAD), to describe the escape of a system from a basin of attraction and proved that solutions of GAD converge to index-1 saddle points of the underlying energy. In this paper, we extend GAD to enable finite temperature simulations in which the system hops between different saddle points on the energy surface. An effective strategy to use GAD to sample an ensemble of low barrier saddle points located in the vicinity of a locally stable configuration on the high dimensional energy surface is proposed. The utility of the method is demonstrated by studying the low barrier saddle points associated with point defect activity on a surface. This is done for two representative systems, namely, (a) a surface vacancy and ad-atom pair and (b) a heptamer island on the (111) surface of copper.     

State-dependent doubly weighted stochastic simulation algorithm for automatic characterization of stochastic biochemical rare events

In recent years there has been substantial growth in the development of algorithms for characterizing rare events in stochastic biochemical systems. Two such algorithms, the state-dependent weighted stochastic simulation algorithm (swSSA) and the doubly weighted SSA (dwSSA) are extensions of the weighted SSA (wSSA) by H. Kuwahara and I. Mura [J. Chem. Phys. 129, 165101 (2008)]. The swSSA substantially reduces estimator variance by implementing system state-dependent importance sampling (IS) parameters, but lacks an automatic parameter identification strategy. In contrast, the dwSSA provides for the automatic determination of state-independent IS parameters, thus it is inefficient for systems whose states vary widely in time. We present a novel modification of the dwSSA--the state-dependent doubly weighted SSA (sdwSSA)--that combines the strengths of the swSSA and the dwSSA without inheriting their weaknesses. The sdwSSA automatically computes state-dependent IS parameters via the multilevel cross-entropy method. We apply the method to three examples: a reversible isomerization process, a yeast polarization model, and a lac operon model. Our results demonstrate that the sdwSSA offers substantial improvements over previous methods in terms of both accuracy and efficiency.     

Repetitive spectral subtraction method for the spectrophotometric determination of rare earth elements

Summary A new multiwavelength data-analysis method for the determination of multicomponent mixtures, the repetitive spectral subtraction method (RSSM), is proposed. In RSSM, spectral data are obtained at evenly spaced wavelengths in the range of interest, and the analysis of the spectrum is performed as repetitive determination-subtraction cycle. In the determination step, the series of data in the narrow wavelength range is used, which is specified as to include the most characteristic or the largest peak for each component. Assuming a linear background, the target component, which can be a small group of components, is simultaneously determined together with co-existing components. In the next subtraction step, the calculated contribution of the component is subtracted from the spectral data in the whole measured wavelength range, and the component is assigned as determined and is excluded from the following analysis. This determination-subtraction cycle is repeated until all the components are determined.With RSSM, a mixture of many components can be precisely determined by the use of most information contained in the spectrum. The accuracy of determination can be improved by automatic estimation of the background and the gradual decrease of interferences. In order to demonstrate the usefulness of RSSM, the 8 rare earth elements (Pr, Nd, Sm, Eu, Dy, Ho, Er, Tm) in the model mixtures were determined by spectrophotometry, and the results were compared with those of the usual correction factor method and the derivative method. Furthermore, the 8 rare earth elements were determined in ores, monazite and xenotime.

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Spektralphotometrische Bestimmung von Seltenen Erden mit Hilfe eines Verfahrens der wiederholten Subtraktion

     

An extraction-photometric method for the determination of rare earths using mixed ligands

Summary A new procedure has been developed for the photometric determination of rare-earth metal ions with 4-(2-pyridyl azo)resorcinol (PAR), antipyrine and perchlorate and extraction into nitrobenzene of the mixed ligand complex formed at pH 6.5. Optimum conditions for the photometric determination, Beer's law range, sensitivity of the colour reaction, photometric precision and the interference of various foreign ions are reported. The method is simple, sensitive and fairly accurate. The composition of the mixed-ligand complexes and their conditional extraction constants have been determined.

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Eine Methode zur Extraktion und photometrischen Bestimmung Seltener Erden mit gemischten Liganden

Zusammenfassung Ein neues Verfahren zur photometrischen Bestimmung Seltener Erden mit 4-(2-Pyridylazo)~resorcin (PAR), Antipyrin und Perchlorat nach Extraktion der Komplexverbindung bei pH 6,5 mit Nitrobenzol wurde beschrieben. Die optimalen Arbeitsbedingungen, das Gültigkeitsgebiet des Beerschen Gesetzes, die Empfindlichkeit der Farbreaktion, die photometrische Genauigkeit und die Störung durch verschiedene Fremdionen wurden be- schrieben. Das Verfahren ist einfach, empfindlich und genau. Die Zusammensetzung der Komplexverbindungen und deren jeweilige Extraktionskonstanten wurden ermittelt.

     

A rapid carrier-free separation method for divalent rare earths

A separation of carrier-free divalent rare earths by electrolytic reduction and amalgamation is described. The separation is carried out in a water-jacketed cell, 1 cm diameter by 10 cm long, fitted with a stopcock at the bottom. The cell utilizes a Pt anode and Hg cathode. The electrolyte used is potassium citrate-rare earth acetate at a concentration of 10 mg rare earth oxide per ml. The decontamination factor observed for Lu in a typical separation for Yb is ≥10⁴. Sm, Eu, and Yb are all amalgamated if present in the sample. For the present study, the parameters have been evaluated primarily for the case of Yb. A current of 100 mA is applied to 0.5 ml of electrolyte containing 1–1000 μg of Yb(III) for 5 min. The amalgam is decomposed and the Yb recovered by shaking for 1 min with 6M HCl. The overall time required is about 10 min and the yield is reproducible at 84–85%. Under these conditions, the electrolyte may contain up to 3.5 mg of Yb although at 2 mg the yield begins to decrease. The effects of current time, density and electrolyte concentration are discussed. This method has the advantage of being rapid and carrier-free, and the separated rare earth is in a form suitable for further chemical separation if required. Based on work performed under the auspices of the U.S. Atomic Energy Commission.