Cluster expansion models for flexible‐backbone protein energetics

Protein structure prediction and design often involve discrete modeling of side‐chain conformations on structural templates. Introducing backbone flexibility into such models has proven important in many different applications. Backbone flexibility improves model accuracy and provides access to larger sequence spaces in computational design, although at a cost in complexity and time. Here, we show that the influence of backbone flexibility on protein conformational energetics can be treated implicitly, at the level of sequence, using the technique of cluster expansion. Cluster expansion provides a way to convert structure‐based energies into functions of sequence alone. It leads to dramatic speed‐ups in energy evaluation and provides a convenient functional form for the analysis and optimization of sequence‐structure relationships. We show that it can be applied effectively to flexible‐backbone structural models using four proteins: α‐helical coiled‐coil dimers and trimers, zinc fingers, and Bcl‐x_L/peptide complexes. For each of these, low errors for the sequence‐based models when compared with structure‐based evaluations show that this new way of treating backbone flexibility has considerable promise, particularly for protein design. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

基于氨基酸模糊聚类分析的跨膜区域预测

跨膜蛋白在进化过程中,序列保守性较差,即使是同源蛋白序列的一致性程度也较低,因而在跨膜区预测算法中,通过序列的一致性程度来选取训练集并不能有效地消除预测结果对训练集的过度适应性．本文提出了一种基于氨基酸模糊聚类分析的预测算法,通过氨基酸在各个区域分布的相似性程度进行模糊聚类,从而根据一类氨基酸的分布特性而不是各个氨基酸的分布特性进行跨膜区预测．结果表明,该方法能在一定程度上消除训练集的选取对测试结果的影响,提高跨膜蛋白拓扑结构预测的准确度,特别是提高对目前知之甚少的跨膜蛋白的预测准确度．     

A universal state-selective approach to multireference coupled-cluster non-iterative corrections

A new form of the asymmetric energy functional for multireference coupled cluster (MRCC) theories is discussed from the point of view of an energy expansion in a quasidegenerate situation. The resulting expansion for the exact electronic energy can be used to define the non-iterative corrections to approximate MRCC approaches. In particular, we show that in the proposed framework the essential part of dynamic correlation can be encapsulated in the so-called correlation Hamiltonian, which in analogy to the effective Hamiltonian, is defined in the model space (M(0)). The proper parametrization of the exact/trial wavefunctions leads to the cancellation of the overlap-type numerators and to a connected form of the correlation Hamiltonian and size-extensive energies. Within this parametrization, when the trial wavefunctions are determined without invoking a specific form of the MRCC sufficiency conditions, the ensuing correction can be universally applied to any type of the approximate MRCC method. The analogies with other MRCC triples corrections to MRCC theories with singles and doubles (MRCCSD) are outlined. In particular, we discuss the approach, which in analogy to the Λ-Mk-MRCCSD(T) method [F. A. Evangelista, E. Prochnow, J. Gauss, H. F. Schaefer III, J. Chem. Phys. 132, 074107 (2010)], introduces an approximate form of the triply-excited clusters into the effective and correlation Hamiltonians. Since the discussed corrections can be calculated as a sum of independent reference-related contributions, possible parallel algorithms are also outlined.     

Surface induced reactions of cluster ions

First results are presented from a new apparatus, consisting of a supersonic beam for generating neutral clusters, a variable energy electron gun for ionizing the clusters, and a tandem mass spectrometer set-up for studying surface induced reactions of mass and energy selected cluster ions. Rare gas cluster ions, fragment ions from SF₆, benzene ions and benzene cluster ions have been investigated so far. Cluster ion dissociation, intracluster ion molecule reactions and surface reactions with adsorbed hydrocarbons have been shown to be important reaction channels for these ion-surface collision at energies ranging from a few eV to 500 eV. The surface induced fragmentation spectrum is demonstrated to be a useful tool for probing binding energy and structure of cluster ions.     

Reciprocal adjustment of approximate coupled cluster and configuration interaction approaches

The linear version of the externally corrected coupled cluster method with singles and doubles (ecLCCSD), the recently proposed coupled cluster corrections to the multireference configuration interaction (ccMRCI) energies, and the so‐called self‐consistent, size‐consistent [(SC)²] approaches, which are designed to correct for the dynamic correlation effects and the size inconsistency of the MRCI energies, are analyzed and compared using several illustrative examples, including the dissociation of a triple‐zeta (TZ) model of the N₂ molecule. It is emphasized that the exponential cluster ansatz for the wave function is the basis of all these approaches, and appropriate cluster analysis of the MRCI wave function is the key step for both ecLCCSD and ccMRCI. The contributions from the orthogonal complement of the MRCI space, which can be generated by relying on such a cluster analysis, are responsible for a substantial part of the missing correlation energy. The ecLCCSD approach seems to represent a particularly attractive alternative to other highly accurate methods for the calculation of the ground‐state energy in the presence of quasidegeneracy, both due to its efficiency and affordability. It may in fact be regarded as a simple alternative to the iterative reduced multireference (RMR) CCSD method. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 77: 693–703, 2000     

Electronic structure and stability of binary and ternary aluminum‐bismuth‐nitrogen nanoclusters

The electronic structure and stability in binary and ternary aluminum‐bismuth‐nitrogen nanoclusters up to six atoms are studied using density functional theory (DFT). The lowest energy geometries were obtained by sampling the geometrical space with a Monte Carlo method and geometry optimizations, at DFT level, with M06L functional. The clusters stability is analyzed using formation and fragmentation energies. Our results show that a high concentration of nitrogen presents a tendency to form nitrogen clusters. highest occupied molecular orbital‐lowest unoccupied molecular orbital gaps show the well‐known oscillation as the number of atoms is increased. Bonding between Al, Bi, and N has mainly a π character. Bismuth and aluminum atoms tend to promote high multiplicity states in small clusters. These new binary and ternary materials provide a potential new field in optoelectronics and high energetic material compounds. © 2014 Wiley Periodicals, Inc.     

Size‐guided multi‐seed heuristic method for geometry optimization of clusters: Application to benzene clusters

Since searching for the global minimum on the potential energy surface of a cluster is very difficult, many geometry optimization methods have been proposed, in which initial geometries are randomly generated and subsequently improved with different algorithms. In this study, a size‐guided multi‐seed heuristic method is developed and applied to benzene clusters. It produces initial configurations of the cluster with n molecules from the lowest‐energy configurations of the cluster with n − 1 molecules (seeds). The initial geometries are further optimized with the geometrical perturbations previously used for molecular clusters. These steps are repeated until the size n satisfies a predefined one. The method locates putative global minima of benzene clusters with up to 65 molecules. The performance of the method is discussed using the computational cost, rates to locate the global minima, and energies of initial geometries. © 2018 Wiley Periodicals, Inc.     

Interaction energies of large clusters from many-body expansion

In the canonical supermolecular approach, calculations of interaction energies for molecular clusters involve a calculation of the whole cluster, which becomes expensive as the cluster size increases. We propose a novel approach to this task by demonstrating that interaction energies of such clusters can be constructed from those of small subclusters with a much lower computational cost by applying progressively lower-level methods for subsequent terms in the many-body expansion. The efficiency of such "stratified approximation" many-body approach (SAMBA) is due to the rapid convergence of the many-body expansion for typical molecular clusters. The method has been applied to water clusters (H(2)O)(n), n = 6, 16, 24. For the hexamer, the best results that can be obtained with current computational resources in the canonical supermolecular method were reproduced to within about one tenth of the uncertainty of the canonical approach while using 24 times less computer time in the many-body expansion calculations. For (H(2)O)(24), SAMBA is particularly beneficial and we report interaction energies with accuracy that is currently impossible to obtain with the canonical supermolecular approach. Moreover, our results were computed using two orders of magnitude smaller computer resources than used in the previous best calculations for this system. We also show that the basis-set superposition errors should be removed in calculations for large clusters.     

MetREx: A protein design approach for the exploration of sequence‐reactivity relationships in metalloenzymes

Metalloenzymes represent a particular challenge for any rational (re)design approach because the modeling of reaction events at their metallic cofactors requires time‐consuming quantum mechanical calculations, which cannot easily be reconciled with the fast, knowledge‐based approaches commonly applied in protein design studies. Here, an approach for the exploration of sequence‐reactivity relationships in metalloenzymes is presented (MetREx) that consists of force field‐based screening of mutants that lie energetically between a wild‐type sequence and the global minimum energy conformation and which should, therefore, be compatible with a given protein fold. Mutant candidates are subsequently evaluated with a fast and approximate quantum mechanical/molecular mechanical‐like procedure that models the influence of the protein environment on the active site by taking partial charges and van der Waals repulsions into account. The feasibility of the procedure is demonstrated for the active site of [FeFe] hydrogenase from Desulfovibrio desulfuricans. The method described allows for the identification of mutants with altered properties, such as inhibitor‐coordination energies, and the understanding of the robustness of enzymatic reaction steps with respect to variations in sequence space. © 2015 Wiley Periodicals, Inc.     

Ionized cluster beam technique for thin film deposition

The ionized cluster beam (ICB) technique can be classified as an ion-assisted technique for film formation, and it has the feature of transferring low energy and equivalently high current beams. The clusters can be created by condensation of supersaturated vapour atoms produced by an adiabatic expansion process. These clusters are large size macro-aggregates of 100–2,000 atoms formed by pure expansion of vapourized solid state materials. The clusters are first partially ionized by an electron impact, then the kinetic energy can be added to the ionized clusters. The ICB has unique capabilities of film deposition due to cluster properties and its low energy ion beam transport in a range from thermal energy to a few hundred eV, with the ability to use the effective influence from the ions without space charge problems. This allows high quality deposition and epitaxy of materials at low temperature onto a wide variety of substrates and even permits the formation of thin film materials not previously possible.     

Single determinant N‐representability and the kernel energy method applied to water clusters

The Kernel energy method (KEM) is a quantum chemical calculation method that has been shown to provide accurate energies for large molecules. KEM performs calculations on subsets of a molecule (called kernels) and so the computational difficulty of KEM calculations scales more softly than full molecule methods. Although KEM provides accurate energies those energies are not required to satisfy the variational theorem. In this article, KEM is extended to provide a full molecule single‐determinant N‐representable one‐body density matrix. A kernel expansion for the one‐body density matrix analogous to the kernel expansion for energy is defined. This matrix is converted to a normalized projector by an algorithm due to Clinton. The resulting single‐determinant N‐representable density matrix maps to a quantum mechanically valid wavefunction which satisfies the variational theorem. The process is demonstrated on clusters of three to twenty water molecules. The resulting energies are more accurate than the straightforward KEM energy results and all violations of the variational theorem are resolved. The N‐representability studied in this article is applicable to the study of quantum crystallography. © 2017 Wiley Periodicals, Inc.     

An efficient algorithm for multistate protein design based on FASTER

Most of the methods that have been developed for computational protein design involve the selection of side‐chain conformations in the context of a single, fixed main‐chain structure. In contrast, multistate design (MSD) methods allow sequence selection to be driven by the energetic contributions of multiple structural or chemical states simultaneously. This methodology is expected to be useful when the design target is an ensemble of related states rather than a single structure, or when a protein sequence must assume several distinct conformations to function. MSD can also be used with explicit negative design to suggest sequences with altered structural, binding, or catalytic specificity. We report implementation details of an efficient multistate design optimization algorithm based on FASTER (MSD‐FASTER). We subjected the algorithm to a battery of computational tests and found it to be generally applicable to various multistate design problems; designs with a large number of states and many designed positions are completely feasible. A direct comparison of MSD‐FASTER and multistate design Monte Carlo indicated that MSD‐FASTER discovers low‐energy sequences much more consistently. MSD‐FASTER likely performs better because amino acid substitutions are chosen on an energetic basis rather than randomly, and because multiple substitutions are applied together. Through its greater efficiency, MSD‐FASTER should allow protein designers to test experimentally better‐scoring sequences, and thus accelerate progress in the development of improved scoring functions and models for computational protein design. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Heuristic energy landscape paving for protein folding problem in the three-dimensional HP lattice model

The protein folding problem, i.e., the prediction of the tertiary structures of protein molecules from their amino acid sequences is one of the most important problems in computational biology and biochemistry. However, the extremely difficult optimization problem arising from energy function is a key challenge in protein folding simulation. The energy landscape paving (ELP) method has already been applied very successfully to off-lattice protein models and other optimization problems with complex energy landscape in continuous space. By improving the ELP method, and subsequently incorporating the neighborhood strategy with the pull-move set into the improved ELP method, a heuristic ELP algorithm is proposed to find low-energy conformations of 3D HP lattice model proteins in the discrete space. The algorithm is tested on three sets of 3D HP benchmark instances consisting 31 sequences. For eleven sequences with 27 monomers, the proposed method explores the conformation surfaces more efficiently than other methods, and finds new lower energies in several cases. For ten 48-monomer sequences, we find the lowest energies so far. With the achieved results, the algorithm converges rapidly and efficiently. For all ten 64-monomer sequences, the algorithm finds lower energies within comparable computation times than previous methods. Numeric results show that the heuristic ELP method is a competitive tool for protein folding simulation in 3D lattice model. To the best of our knowledge, this is the first application of ELP to the 3D discrete space.     

Force fields for metallic clusters and nanoparticles

Atomic force fields for simulating copper, silver, and gold clusters and nanoparticles are developed. Potential energy functions are obtained for both monatomic and binary metallic systems using an embedded atom method. Many cluster configurations of varying size and shape are used to constrain the parametrization for each system. Binding energies for these training clusters were computed using density functional theory (DFT) with the Perdew-Wang exchange-correlation functional in the generalized gradients approximation. Extensive testing shows that the many-body potentials are able to reproduce the DFT energies for most of the structures that were included in the training set. The force fields were used to calculate surface energies, bulk structures, and thermodynamic properties. The results are in good agreement with the DFT values and consistent with the available experimental data.     

Bare Clusters Derived from Protein Templates: Au25+, Au38+ and Au102+

A discrete sequence of bare gold clusters of well‐defined nuclearity, namely Au₂₅⁺, Au₃₈⁺ and Au₁₀₂⁺, formed in a process that starts from gold‐bound adducts of the protein lysozyme, were detected in the gas phase. It is proposed that subsequent to laser desorption ionization, gold clusters form in the gas phase, with the protein serving as a confining growth environment that provides an effective reservoir for dissipation of the cluster aggregation and stabilization energy. First‐principles calculations reveal that the growing gold clusters can be electronically stabilized in the protein environment, achieving electronic closed‐shell structures as a result of bonding interactions with the protein. Calculations for a cluster with 38 gold atoms reveal that gold interaction with the protein results in breaking of the disulfide bonds of the cystine units, and that the binding of the cysteine residues to the cluster depletes the number of delocalized electrons in the cluster, resulting in opening of a super‐atom electronic gap. This shell‐closure stabilization mechanism confers enhanced stability to the gold clusters. Once formed as stable magic number aggregates in the protein growth medium, the gold clusters become detached from the protein template and are observed as bare Au_n⁺ (n=25, 38, and 102) clusters.     

A New Many-Body Expansion Scheme for Atomic Clusters: Application to Nitrogen Clusters

Although the many-body expansion (MBE) approach is widely applied to estimate the energy of large systems containing weak interactions, it is inapplicable to calculating the energies of covalent or metal clusters. In this work, we propose an interaction many-body expansion (IMBE) to calculate the energy of atomic clusters containing covalent bonds. In this approach, the energy of a system is expressed as the sum of the energy of atoms and the interaction energy between the atom and its surrounding atoms. The IMBE method is first applied to calculate the energies of nitrogen clusters, in which the interatomic interactions are truncated to four-body terms. The results show that the IMBE approach could significantly reduce the energy error for nitrogen clusters compared with the traditional MBE method. The weak size and structure dependence of the IMBE error with respect to DFT calculations indicates the IMBE method has good potential application in estimating energy of large covalent systems.     

Impact of different potentials on the structures and energies of clusters

Cluster studies have attracted much interest in the past decades because of their extraordinary properties. To describe the interaction between atoms or molecules and predict the energies and structures, potential functions are developed. However, different potentials generally produce different structures and energies for a cluster. To study the effect of potentials on the structure of a cluster, He clusters in the size range of 13-140 are investigated by Lennard-Jones (LJ), Pirani, and Hartree-Fock-dispersion individual damping (HFD-ID) potential with dynamic lattice searching (DLS) method. Potential function curves, cluster structures, bonds, and energies of the global minima are compared. The results show that cluster energies decrease with the values of the potential functions, the differences between structures depend upon the disagreements of the potentials, and the preferable motif of a cluster changes from icosahedron to decahedron with the increase of the derivative of the short-range part of the potentials.     

An ab initio study of water clusters in gas phase and bulk aqueous media: (H2O)n,n=1–12

Geometries of several clusters of water molecules including single minimum energy structures of n‐mers (n=1–5), several hexamers and two structures of each of heptamer to decamer derived from hexamer cage and hexamer prism were optimized. One structural form of each of 11‐mer and 12‐mer were also studied. The geometry optimization calculations were performed at the RHF/6‐311G* level for all the cases and at the MP2/6‐311++G** level for some selected cases. The optimized cluster geometries were used to calculate total energies of the clusters in gas phase employing the B3LYP density functional method and the 6‐311G* basis set. Frequency analysis was carried out in all the cases to ensure that the optimized geometries corresponded to total energy minima. Zero‐point and thermal free energy corrections were applied for comparison of energies of certain hexamers. The optimized cluster geometries were used to solvate the clusters in bulk water using the polarized continuum model (PCM) of the self‐consistent reaction field (SCRF) theory, the 6‐311G* basis set, and the B3LYP density functional method. For the cases for which MP2/6‐311++G** geometry optimization was performed, solvation calculations in water were also carried out using the B3LYP density functional method, the 6‐311++G** basis set, and the PCM model of SCRF theory, besides the corresponding gas‐phase calculations. It is found that the cage form of water hexamer cluster is most stable in gas phase among the different hexamers, which is in agreement with the earlier theoretical and experimental results. Further, use of a newly defined relative population index (RPI) in terms of successive total energy differences per water molecule for different cluster sizes suggests that stabilities of trimers, hexamers, and nonamers in gas phase and those of hexamers and nonamers in bulk water would be favored while those of pentamer and decamer in both the phases would be relatively disfavored. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 90–104, 2001     

Solidification of small p-H2 clusters at zero temperature

We have determined the ground-state energies of para-H(2) clusters at zero temperature using the diffusion Monte Carlo method. The liquid or solid character of each cluster is investigated by restricting the phase through the use of proper importance sampling. Our results show inhomogeneous crystallization of clusters, with alternating behavior between liquid and solid phases up to N = 55. From there on, all clusters are solid. The ground-state energies in the range N = 13-75 are established, and the stable phase of each cluster is determined. In spite of the small differences observed between the energy of liquid and solid clusters, the corresponding density profiles are significantly different, a feature that can help to solve ambiguities in the determination of the specific phase of H(2) clusters.     

Magic numbers, excitation levels, and other properties of small neutral 4He clusters (N < or = 50)

The ground-state energies and the radial and pair distribution functions of neutral 4He clusters are systematically calculated by the diffusion Monte Carlo method in steps of one 4He atom from 3 to 50 atoms. In addition the chemical potential and the low-lying excitation levels of each cluster are determined with high precision. These calculations reveal that the "magic numbers" observed in experimental 4He cluster size distributions, measured for free jet gas expansions by nondestructive matter-wave diffraction, are not caused by enhanced stabilities. Instead they are explained in terms of an enhanced growth due to sharp peaks in the equilibrium concentrations in the early part of the expansion. These peaks appear at cluster sizes which can just accommodate one more additional stable excitation. The good agreement with experiment provides not only experimental confirmation of the energy level and the chemical potential calculations, but also evidence for a new mechanism which can lead to magic numbers in cluster size distributions. By accounting for the falloff of the radial density distributions at the surface and a size-dependent surface tension, the energy levels are demonstrated to be consistent with a modified Rayleigh model of surface excitations. The compressibility coefficient of these small clusters is found to be one order of magnitude smaller than the bulk compressibility.