模拟退火方法研究线性肽构象与环化的关系

采用蒙特卡罗模拟退火方法对一系列线性五肽进行了构象分析，探索了线性肽的优势构象与其环化产率之间的关系．构象分析所得到的预测结果与实验结果吻合．     

通用模拟退火用于稳健多元分析校正

模拟退火是一种全局优化算法，具有跨越局部最优点的机制，最小一乘是一种较常用的最小二乘更为稳健的优化准则，更适用于可能偏离正态分布的实际数据集，本文探讨了用最小一乘为准则并利用模拟退火方法同时测定多组分体系的可能性。应用于2－3组分药物体系分析，获得了满意的结果，本文还探讨了改变步长提高模拟退火算法优化精度的方法。     

A comparative study of the simulated-annealing and Monte Carlo-with-minimization approaches to the minimum-energy structures of polypeptides: [Met]-enkephalin

A comparison of two methods for surmounting the multiple-minima problem, Simulated Annealing (SA) and Monte Carlo with Minimization (MCM), is presented with applications to [Met]-enkephalin in the absence and in the presence of water. SA explores a continuous space of internal variables, while MCM explores a discrete space consisting of the local energy minima on that space. Starting from random conformations chosen from the whole conformational space in both cases, it is found that, while SA converges to low-energy structures significantly faster than MCM, the former does not converge to a unique minimum whereas the latter does. Furthermore, the behavior of the RMS deviations with respect to the apparent global minimum (for enkephalin in the absence of water) shows no correlation with the observed overall energy decrease in the case of SA, whereas such a correlation is quite evident with MCM; this implies that, even though the potential energy decreases in the annealing process, the Monte Carlo SA trajectory does not proceed towards the global minimum. Possible reasons for these differences between the two methods are discussed. It is concluded that, while SA presents attractive prospects for possibly improving or refining given structures, it must be considered inferior to MCM, at least in problems where little or no structural information is available for the molecule of interest.     

Stochastic molecular optimization using generalized simulated annealing

We propose a stochastic optimization technique based on a generalized simulated annealing (GSA) method for mapping minima points of molecular conformational energy surfaces. The energy maps are obtained by coupling a classical molecular force field (THOR package) with a GSA procedure. Unlike the usual molecular dynamics (MD) method, the method proposed in this study is force independent; that is, we obtain the optimized conformation without calculating the force, and only potential energy is involved. Therefore, we do not need to know the conformational energy gradient to arrive at equilibrium conformations. Its utility in molecular mechanics is illustrated by applying it to examples of simple molecules (H₂O and H₂O₃) and to polypeptides. The results obtained for H₂O and H₂O₃ using Tsallis thermostatistics suggest that the GSA approach is faster than the other two conventional methods (Boltzmann and Cauchy machines). The results for polypeptides show that pentalanine does not form a stable α-helix structure, probably because the number of hydrogen bonds is insufficient to maintain the helical array. On the contrary, the icoalanine molecule forms an α-helix structure. We obtain this structure simulating all Φ, Ψ pairs using only a few steps, as compared with conventional methods. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 647–657, 1998     

Global optimization of binary Lennard-Jones clusters using three perturbation operators

Global optimization of binary Lennard-Jones clusters is a challenging problem in computational chemistry. The difficulty lies in not only that there are enormous local minima on the potential energy surface but also that we must determine both the coordinate position and the atom type for each atom and thus have to deal with both continuous and combinatorial optimization. This paper presents a heuristic algorithm (denoted by 3OP) which makes extensive use of three perturbation operators. With these operators, the proposed 3OP algorithm can efficiently move from a poor local minimum to another better local minimum and detect the global minimum through a sequence of local minima with decreasing energy. The proposed 3OP algorithm has been evaluated on a set of 96 × 6 instances with up to 100 atoms. We have found most putative global minima listed in the Cambridge Cluster Database as well as discovering 12 new global minima missed in previous research.     

Semiglobal simplex optimization and its application to determining the preferred solvation sites of proteins

The classical simplex method is extended into the Semiglobal Simplex (SGS) algorithm. Although SGS does not guarantee finding the global minimum, it affords a much more thorough exploration of the local minima than any traditional minimization method. The basic idea of SGS is to perform a local minimization in each step of the simplex algorithm, and thus, similarly to the Convex Global Underestimator (CGU) method, the search is carried out on a surface spanned by local minima. The SGS and CGU methods are compared by minimizing a set of test functions of increasing complexity, each with a known global minimum and many local minima. Although CGU delivers substantially better success rates in simple problems, the two methods become comparable as the complexity of the problems increases. Because SGS is generally faster than CGU, it is the method of choice for solving optimization problems in which function evaluation is computationally inexpensive and the search region is large. The extreme simplicity of the method is also a factor. The SGS method is applied here to the problem of finding the most preferred (i.e., minimum free energy) solvation sites on a streptavidin monomer. It is shown that the SGS method locates the same lowest free energy positions as an exhaustive multistart Simplex search of the protein surface, with less than one-tenth the number of minizations. The combination of the two methods, i.e.. multistart simplex and SGS, provides a reliable procedure for predicting all potential solvation sites of a protein.     

Many-body optimization using an ab initio monte carlo method

Advances in computing power have made it possible to study solvated molecules using ab initio quantum chemistry. Inclusion of discrete solvent molecules is required to determine geometric information about solute/solvent clusters. Monte Carlo methods are well suited to finding minima in many-body systems, and ab initio methods are applicable to the widest range of systems. A first principles Monte Carlo (FPMC) method was developed to find minima in many-body systems, and emphasis was placed on implementing moves that increase the likelihood of finding minimum energy structures. Partial optimization and molecular interchange moves aid in finding minima and overcome the incomplete sampling that is unavoidable when using ab initio methods. FPMC was validated by studying the boron trifluoride-water system, and then the method was used to examine the methyl carbenium ion in water to demonstrate its application to solvation problems.     

Toward absolute density of states calculations for proteins

The density of states (DOS), which gives the number of conformations with a particular energy E, is a prerequisite in computing most thermodynamic quantities and in elucidating important biological processes such as the mechanism of protein folding. However, current methods for computing DOS of large systems such as proteins generally yield only the ratios of microstate counts for different energies, which could yield absolute conformation counts if the total number of conformations in phase space is known, thus motivating this work. Here, the total number of energy minima of 50-mer polyalanine, whose size corresponds to naturally occurring small proteins, was estimated under an all-atom potential energy function based on the cumulative distribution function (CDF) of conformational differences to be approximately 10(38). This estimate can place any DOS function, such as the Gaussian DOS distribution in the random energy model, on an absolute scale. Comparing the absolute conformational counts from a Gaussian DOS function with those from the CDF derived from quenched molecular dynamics ensembles shows that the former are far greater than the latter, indicating far fewer low-energy minima actually exist. In addition to showing how CDF and relative DOS calculations can yield absolute DOS for a discrete system, we also show how they can yield absolute DOS for continuous variable systems to a specified atomic variance. In the context of protein folding, knowing this phase-space "volume" of conformations in a DOS function, as well as characteristic transition times, constrains the set of possible folding mechanisms.     

神经网络方法用于分辨3种化学物质

神经网络方法用于分辨３种化学物质冯伟,胡上序（浙江大学化工系,杭州,３１００２７）关键词模式分类,神经网络,模拟退火,遗传算法,传感器阵列传感器阵列技术是利用传感器阵列所提供的交叉敏和模式识别及微机处理技术,来提高传感器的选择性和传感器的测量精度［１...     

Potential energy and free energy landscapes

Familiar concepts for small molecules may require reinterpretation for larger systems. For example, rearrangements between geometrical isomers are usually considered in terms of transitions between the corresponding local minima on the underlying potential energy surface, V. However, transitions between bulk phases such as solid and liquid, or between the denatured and native states of a protein, are normally addressed in terms of free energy minima. To reestablish a connection with the potential energy surface we must think in terms of representative samples of local minima of V, from which a free energy surface is projected by averaging over most of the coordinates. The present contribution outlines how this connection can be developed into a tool for quantitative calculations. In particular, stepping between the local minima of V provides powerful methods for locating the global potential energy minimum, and for calculating global thermodynamic properties. When the transition states that link local minima are also sampled we can exploit statistical rate theory to obtain insight into global dynamics and rare events. Visualizing the potential energy landscape helps to explain how the network of local minima and transition states determines properties such as heat capacity features, which signify transitions between free energy minima. The organization of the landscape also reveals how certain systems can reliably locate particular structures on the experimental time scale from among an exponentially large number of local minima. Such directed searches not only enable proteins to overcome Levinthal's paradox but may also underlie the formation of "magic numbers" in molecular beams, the self-assembly of macromolecular structures, and crystallization.     

Computer optimization of small atomic clusters

We have successfully identified stable configurations of both rare-gas and NaCl clusters with a new optimization procedure. An initial cluster configuration is prepared in a so-called shoot-and-stay process. Its total energy is then minimized with respect to the atomic coordinates. To prevent the system from being locked in local minima, the step size of each move is chosen as the width of the energy well at a higher level. As the system evolves, the global minimum is contained in the volume bounded by the decreasing value of step sizes. We have also carried out the optimization of NaCl clusters by the simulated annealing technique, for comparison. The results show that for such heterogeneous systems, the latter method cannot always find the global minimum, because of large energy gaps between different catchment regions in phase space.     

Gradient tabu search

This paper presents a modification of the tabu search called gradient tabu search (GTS). It uses analytical gradients for a fast minimization to the next local minimum and analytical diagonal elements of the Hessian to escape local minima. For an efficient blocking of already visited areas tabu regions and tabu directions are introduced into the tabu list (TL). Trials with various well-known test functions indicate that the GTS is a very promising approach to determine local and global minima of differentiable functions. Possible application areas could be optimization routines for force field parameters or conformational searches for large molecules.     

Simulated annealing to locate various stationary points in semiempirical methods

In this work, an algorithm was developed to study the potential energy surfaces in the coordinate spaces of molecules by a nonlocal way, in contrast to classic energy minimizers as the BFGS or the DFP method. This algorithm, based on the specificities of semiempirical methods, mixes simulated annealing and local searches to reduce computation costs. By this technique, the global energy minimum can be localized. Moreover, local minima that are close in energy to the global minimum are also obtained. If the search is not only for minima but for all stationary points (minima, saddle points…), then the energy is replaced by the gradient norm, which reaches its minimum values at stationary points. The annealing process is stopped before having accurately reached the global minimum and generates a list of geometries whose energies (respectively, whose gradients) are optimized by local minimizers. This list of geometries is shortened from the nearly equivalent geometries by a dynamic single-clustering analysis. The energy/gradient local minimizers act on the clustered list to produce a set of minima/stationary points. A targeted search of these points and reduction of the costs are reached by the way of several penalty functions. They eliminate—without energy calculation—most of the points generated by random walks on the potential energy surface. These penalty functions (on the total moment of inertia or on interatomic distances) are specific to the class of problem studied. They account for the nonrupture of either specific chemical bonds or rings in cyclic molecules, they assure that molecular systems are kept bonded, and they avoid the collapsing of atoms. © 1992 John Wiley & Sons, Inc.     

Ab initio studies of conformational properties of dimethyl diphosphate dianion and its complex with magnesium

The conformational properties of the diphosphate linkage have been studied with ab initio methods using the dimethyl diphosphate dianion (1) and magnesium dimethyl diphosphate (2) as models. The ab initio energy and geometry of the conformers around the P-O bonds have been determined at the self-consistent-field (SCF) using the 6-31G* and the tzp basis sets; whereas, the 6-31G* basis set alone has been used for 2. In addition, the adiabatic connection method (ACM) of density functional theory (DFT) using the dzvp basis set has been employed for 1. The optimization of all possible staggered conformers assumed for the four P-O bonds, led to nine minima for 1. In agreement with the general anomeric effect, the sc conformation about the P-O bonds is clearly preferred over the ap one. Vibrational frequencies were calculated at the SCF level using the 6-31G* basis set and used to evaluate zero-point energies, thermal energies, and entropies for all minima of 1. The effect of zero-point energies and thermal energies is quite small. However, the effect of entropies, mainly resulting from a multiplicity contribution, changes the stability of the conformers. For each minimum of 1, up to six different arrangements of the Mg²⁺ were used to determine minima of 2. This procedure led to 21 distinct minima. The presence of the magnesium counter-ion appeared to completely change the structure and relative energy of the conformers. The preferred structures of the complex exhibit the (sc, ap) orientation around the two central P–O bonds and an arrangement in which the magnesium cation is coordinated by three phosphoryl oxygen atoms. The results of this work clearly demonstrate that interactions with the metal counter-ion can induce conformational changes in the overall 3D-shape adopted by molecules containing diphosphate linkages. The PM3 and MNDO quantum semi-empirical methods and molecular mechanics methods using the CVFF force field were tested and large differences in the minimum structures, as well as in the conformational energies between these and ab initio methods, are discussed.     

Monte Carlo temperature basin paving with effective fragment potential: an efficient and fast method for finding low-energy structures of water clusters (H2O)20 and (H2O)25

Determining low-energy structures of large water clusters is a challenge for any optimization algorithm. In this work, we have developed a new Monte Carlo (MC)-based method, temperature basin paving (TBP), which is related to the well-known basin hopping method. In the TBP method, the Boltzmann weight factor used in MC methods is dynamically modified based on the history of the simulation. The states that are visited more are given a lower probability by increasing their temperatures and vice versa. This allows faster escapes from the states frequently visited in the simulation. We have used the TBP method to find a large number of low-energy minima of water clusters of size 20 and 25. We have found structures energetically same to the global minimum structures known for these two clusters. We have compared the efficiency of this method to the basin-hopping method and found that it can locate the minima faster. Statistical efficiency of the new method has been investigated by running a large number of trajectories. The new method can locate low-energy structures of both the clusters faster than some of the reported algorithms for water clusters and can switch between high energy and low-energy structures multiple times in a simulation illustrating its efficiency. The large number of minima obtained from the simulations is used to get both general and specific features of the minima. The distribution of minima for these two clusters based on the similarity of their oxygen frames shows that the (H(2)O)(20) can have different variety of structures, but for (H(2)O)(25), low-energy structures are mostly cagelike. Several (H(2)O)(25) structures are found with similar energy but with different cage architectures. Noncage structures of (H(2)O)(25) are also found but they are 6-7 kcal/mol higher in energy from the global minimum. The TBP method is likely to play an important role for exploring the complex energy landscape of large molecules.     

Kinetics of cooperative protein folding involving two separate conformational families

The master equation that describes the kinetics of protein folding is solved numerically for a portion of Staphylococcal Protein A by a Laplace transformation. The calculations are carried out with 50 local-minimum conformations belonging to two conformational families. The master equation allows for transitions among all the 50 conformations in the evolution toward the final folded equilibrium distribution of conformations. It is concluded that the native protein folds in a fast cooperative process. The global energy minimum of a native protein can be reached after a sufficiently long folding time regardless of the initial state and the existence of a large number of local energy minima. Conformations representing non-native states of the protein can transform to the native state even if they do not belong to the native conformational family. Given a starting conformation, the protein molecule can fold to its final conformation through different paths. Finally, when the folding reaches the equilibrium distribution, the protein molecule adopts a set of conformations in which the global minimum has the largest average probability.     

Determination of an empirical energy function for protein conformational analysis by energy embedding

It is quite easy to propose an empirical potential for conformational analysis such that given crystal structures lie near local minima. What is much more difficult, is to devise a function such that the native structure lies near a relatively deep local minimum, at least in some neighborhood of the native in conformation space. An algorithm is presented for finding such a potential acting on proteins where each amino acid residue is represented by a single point. When the given structure is either an α-helical, β-strand, or hairpin bend segment of pancreatic trypsin inhibitor, the resulting potential function in each case possesses a deep minimum within 0.10 Å of the native conformation. The improved energy embedding algorithm locates a marginally better minimum in each case only 0.1–1.3 Å away from the respective native state. In other words, this potential function guides a conformational search toward structures very close to the native over a wide range of conformation space.     

Exploring the similarities between potential smoothing and simulated annealing

Simulated annealing and potential function smoothing are two widely used approaches for global energy optimization of molecular systems. Potential smoothing as implemented in the diffusion equation method has been applied to study partitioning of the potential energy surface (PES) for N‐Acetyl‐Ala‐Ala‐N‐Methylamide (CDAP) and the clustering of conformations on deformed surfaces. A deformable version of the united‐atom OPLS force field is described, and used to locate all local minima and conformational transition states on the CDAP surface. It is shown that the smoothing process clusters conformations in a manner consistent with the inherent structure of the undeformed PES. Smoothing deforms the original surface in three ways: structural shifting of individual minima, merging of adjacent minima, and energy crossings between unrelated minima. A master equation approach and explicit molecular dynamics trajectories are used to uncover similar features in the equilibrium probability distribution of CDAP minima as a function of temperature. Qualitative and quantitative correlations between the simulated annealing and potential smoothing approaches to enhanced conformational sampling are established. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 531–552, 2000     

Identification of hits and lead structure candidates with limited resources by adaptive optimization

Three stochastic optimization algorithms (Simulated Annealing (SA), Evolution Strategy (ES), and Particle Swarm Optimization (PSO)) and a Random Search were assessed for their ability to generate small activity-enriched subsets of molecular compound libraries. The optimization algorithms were employed to perform an "intelligent" iterative sampling of library molecules avoiding the biological testing of the full library. This study was performed to find a suitable optimization algorithm along with suitable parametrization. Particularly, the optimal number of iterations and population size were of interest. Optimizations were performed with limited resources as the maximal number of compound evaluations was restricted to 300. Results show that all three optimization algorithms are able to produce comparably good results, clearly outperforming a Random Search. While ES was able to come up with good solutions after a few optimization cycles, SA favored high numbers of iterations and was therefore less suited for library design. We introduce PSOs as an alternative approach to focused library design. PSO was able to produce high quality solutions while exhibiting marked autoadaptivity. Its implicit step size control makes it a straightforward out-of-the-box optimization algorithm. We further demonstrate that a nearest neighbor algorithm can successfully be applied to map from continuous search space to discrete chemical space.