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The Simplified string method in the Calculation of the Minimum Energy Path

Computational Mathematics and Modeling 2025年 1-8页

作者： Poluyan, S.V. Ershov, N.M. Institute of System Analysis and Management Dubna State University Universitetskaya Ul. 19 Moscow region Dubna 141982 Russian Federation Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State University Leninskiye Gory 1–52 Moscow 119991 Russian Federation

The problem of determining a minimum energy path is a common aspect in modeling various processes within physical and chemical kinetics. Often, this path corresponds to the transition of the modeled molecular system from one state to another. In this paper, we present a formulation of the problem related to identifying such a path, along with one approach for its determination – the simplified string method. We implement the method using numerical differentiation and investigate the impact of parameters on the formation of paths across different problems © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.

关键词： Energy potential Minimum energy path string method

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A new method for finding the minimum free energy pathway of ions and small molecule transportation through protein based on 3D-RISM theory and the string method

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CHEMICAL PHYSICS LETTERS 2018年 699卷 22-27页

作者： Yoshida, Norio Kyushu Univ Grad Sch Sci Dept Chem Nishi Ku 744 Motooka Fukuoka 8190395 Japan

A new method for finding the minimum free energy pathway (MFEP) of ions and small molecule transportation through a protein based on the three-dimensional reference interaction site model (3D-RISM) theory combined with the string method has been proposed. The 3D-RISM theory produces the distribution function, or the potential of mean force (PMF), for transporting substances around the given protein structures. By applying the string method to the PMF surface, one can readily determine the MFEP on the PMF surface. The method has been applied to consider the Na+ conduction pathway of channelrhodopsin as an example. (C) 2018 Elsevier B.V. All rights reserved.

关键词： 3D-RISM string method Ion channel Molecular transportation

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STABILITY AND CONVERGENCE OF THE string method FOR COMPUTING MINIMUM ENERGY PATHS

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MULTISCALE MODELING & SIMULATION 2019年第2期17卷 873-898页

作者： Van Koten, Brian Luskin, Mitchell Univ Massachusetts Amherst MA 01003 USA Univ Minnesota Sch Math Minneapolis MN 55455 USA

We analyze the convergence of the string method of E, Ren, and Vanden-Eijnden [J. Chem. Phys., 126 (2007), 164103] to a minimum energy path. Under some assumptions relating to the critical points on the minimum energy path, we show that the string method initialized in a neighborhood of the minimum energy path converges to an arbitrarily small neighborhood of the minimum energy path as the number of images is increased.

关键词： string method minimum energy path convergence stability

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The string method as a Dynamical System

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JOURNAL OF NONLINEAR SCIENCE 2011年第2期21卷 193-230页

作者： Cameron, Maria Kohn, Robert V. Vanden-Eijnden, Eric NYU Courant Inst Math Sci Dept Math New York NY 10012 USA

This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts. We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP.

关键词： string method Curve evolution Limit set Morse index Convergence Critical point Heteroclinic trajectory

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Finite temperature string method for the study of rare events

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JOURNAL OF PHYSICAL CHEMISTRY B 2005年第14期109卷 6688-6693页

作者： Weinan, E Ren, WQ Vanden-Eijnden, E Princeton Univ Dept Math Princeton NJ 08544 USA NYU Courant Inst New York NY 10012 USA

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.

关键词： .high probability rare metastable reach energy landscape Finite Temperature Phys Chem VANDEN-EIJNDEN transition state conformational energy barrier reaction coordinate beforehand Configuration Space string method isoprobability effective transition

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Numerical study of metastability due to tunneling: The quantum string method

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PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS 2007年第2期379卷 491-502页

作者： Qian, Tiezheng Ren, Weiqing Shi, Jing E, Weinan Sheng, Ping Hong Kong Univ Sci & Technol Dept Math Kowloon Hong Kong Peoples R China NYU Courant Inst Math Sci New York NY 10012 USA Princeton Univ Program Appl & Computat Math Princeton NJ 08544 USA Princeton Univ Dept Math Princeton NJ 08544 USA Princeton Univ PACM Princeton NJ 08544 USA Hong Kong Univ Sci & Technol Dept Phys Kowloon Hong Kong Peoples R China

We generalize the string method, originally designed for the study of thermally activated rare events, to the calculation of quantum tunneling rates. This generalization is based on the formal analogy between quantum mechanics and statistical mechanics in the path-integral formalism. The quantum string method first locates the minimal action path (MAP), which is a smooth curve connecting two minima of the imaginary-time action in the space of imaginary-time trajectories. From the MAP, the saddle point of the action (called "the bounce") associated with the exponential factor for barrier tunneling probability is obtained and the pre-exponential factor (the ratio of determinants) for the tunneling rate evaluated using stochastic simulation. The quantum string method is implemented to calculate the bounce and rate of tunneling for the Mueller potential in two dimensions. The quantum problem is much more difficult than the thermally activated barrier crossing problem for the same potential. The model calculations show the string method to be an efficient numerical tool for the study of barrier tunneling in higher dimension, from the determination of the saddle point to the computation of the pre-exponential factor. (C) 2007 Elsevier B.V. All rights reserved.

关键词： string method barrier tunneling rate of decay quantum metastability

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Kinetics of phase transitions in two dimensional Ising models studied with the string method

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JOURNAL OF MATHEMATICAL CHEMISTRY 2009年第1期45卷 188-222页

作者： Venturoli, Maddalena Vanden-Eijnden, Eric Ciccotti, Giovanni New York Univ Courant Inst Math Sci New York NY 10012 USA Univ Roma La Sapienza Dipartimento Fis I-00185 Rome Italy Univ Roma La Sapienza Unita Roma 1 CNISM I-00185 Rome Italy

The kinetics of phase transitions in the two dimensional Ising model under different conditions is studied using the string method. The key idea is to work in collective variables, consisting of block of spins, which allow for a continuous approximation of the collective variable state-space. The string method computes the minimum free energy path (MFEP) in this collective variable space, which is shown to explain the mechanism of the phase transformation (in particular, an approximation of its committor function, its free energy and its transition state). In this paper the theoretical background of the technique as well as its computational aspects are discussed in details. The string method is then used to analyze phase transition in the Ising model with imposed boundary conditions and in a periodic system under an external field of increasing magnitude. In each case, the mechanism of the phase transformation is elucidated.

关键词： Minimum free energy path string method Sampling Phase transition Ising model

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Optimization-Based string method for Finding Minimum Energy Path

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Communications in Computational Physics 2013年第7期14卷 265-275页

作者： Amit Samanta Weinan E Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Department of Mathematics and Program in Applied and Computational Mathematics Princeton UniversityPrincetonNew JerseyUSA Beijing International Center for Mathematical Research Peking UniversityBeijingChina

We present an efficient algorithm for calculating the minimum energy path(MEP)and energy barriers between local minima on a multidimensional potential energy surface(PES).Such paths play a central role in the understanding of transition pathways between metastable *** method relies on the original formulation of the string method[***.B,66,052301(2002)],*** evolve a smooth curve along a direction normal to the *** algorithm works by performing minimization steps on hyperplanes normal to the *** the problem of finding MEP on the PES is remodeled as a set of constrained minimization *** provides the flexibility of using minimization algorithms faster than the steepest descent method used in the simplified string method[***.,126(16),164103(2007)].At the same time,it provides a more direct analog of the finite temperature string *** applicability of the algorithm is demonstrated using various examples.

关键词： Rare events string method minimum energy path dislocation nucleation

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string method Study on the Proton Transfer Process of a simplified model system from CcO D-channel

String Method Study on the Proton Transfer Process of a simp...

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第十五届全国化学动力学会议

作者：刘润泽 Keli Han Dalian Institute of Chemical Physics

The cytochrome c oxidase d-channel is responsible for the uploading of all pumping protons through the membrane. However, the water wire in the channel is often blocked by the two Asn residues near the entrance. The proton loading process could become complex because the rotation of Asn is coupled to the water wire reformation. We proposed a possible mechanism of loading a proton using Asn residue as an intermediate. Thus we build a simplified model system to investigate this process. Three residues Asp, Asn, and Asp are solvated in a water droplet and the proton transfer process goes upward through the rotation of Asn. The whole proton transfer process involves more than three collective variables to describe, so string method is the ideal approach for this study. Both SMCV1 and FTSM2 methods are carried out to calculate the minimum free energy path and the free energy profile with the contribution of entropy. The results are further validated by umbrella sampling results and higher DFT level method by FEP. Committor analysis shows that both O-H distances and rotation dihedrals are required to describe the process, and string method is capable of calculating such complex transfer process involves a "hard" degree of freedom（O-H-O bond distances） combined with a "soft" process（residue group rotation）. Also our model reveals that the proton transfer through Asn is possible in biological systems and could be a potential competing mechanics of proton loading in d-channel of Cc O. Further study is needed to perform in the real Cc O d-channel.

关键词： Proton transfer SCC-DFTB string method SMCV FTSM

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Single-Ended Transition State Finding with the Growing string method

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JOURNAL OF COMPUTATIONAL CHEMISTRY 2015年第9期36卷 601-611页

作者： Zimmerman, Paul M. Univ Michigan Dept Chem Ann Arbor MI 48109 USA

Reaction path finding and transition state (TS) searching are important tasks in computational chemistry. methods that seek to optimize an evenly distributed set of structures to represent a chemical reaction path are known as double-ended string methods. Such methods can be highly reliable because the endpoints of the string are fixed, which effectively lowers the dimensionality of the reaction path search. string methods, however, require that the reactant and product structures are known beforehand, which limits their ability for systematic exploration of reactive steps. In this article, a single-ended growing string method (GSM) is introduced which allows for reaction path searches starting from a single structure. The method works by sequentially adding nodes along coordinates that drive bonds, angles, and/or torsions to a desired reactive outcome. After the string is grown and an approximate reaction path through the TS is found, string optimization commences and the exact TS is located along with the reaction path. Fast convergence of the string is achieved through use of internal coordinates and eigenvector optimization schemes combined with Hessian estimates. Comparison to the double-ended GSM shows that single-ended method can be even more computationally efficient than the already rapid double-ended method. Examples, including transition metal reactivity and a systematic, automated search for unknown reactivity, demonstrate the efficacy of the new method. This automated reaction search is able to find 165 reaction paths from 333 searches for the reaction of NH3BH3 and (LiH)(4), all without guidance from user intuition. (c) 2015 Wiley Periodicals, Inc.

关键词： transition state single-ended string method

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