Improving the efficiency and convergence of geometry optimization with the polarizable continuum model: new energy gradients and molecular surface tessellation

New equations are derived and implemented for efficient and accurate computation of solvation energy derivatives for the conductor-like polarizable continuum model (C-PCM) and the isotropic integral equation formalism polarizable continuum model (IEF-PCM). Two new molecular surface tessellation procedures GEPOL-RT and GEPOL-AS that generate near continuous potential energy surfaces are proposed for PCM geometry optimization. The combined use of these new techniques leads to efficient and convergent geometry optimizations with the PCMs.     

Effect of Protein Dimerization on Ion Conductivity of Gramicidin A Channel Studied Using Polarizable Force Field

In studies of ion channel systems, due to the huge computational cost of polarizable force fields, classical force fields remain the most widely used for a long time. In this work, we used the AMOEBA polarizable atomic multipole force field in enhanced sampling simulations of single-channel gramicidin A (gA) and double-channel gA systems and investigated its reliability in characterizing ion-transport properties of the gA ion channel under dimerization. The influence of gA dimerization on the permeation of potassium and sodium ions through the channel was described in terms of conductance, diffusion coefficient, and free energy profile. Results from the polarizable force field simulations show that the conductance of potassium and sodium ions passing through the single- and double-channel agrees well with experimental values. Further data analysis reveals that the molecular mechanism of protein dimerization affects the ion-transport properties of gA channels, i.e., protein dimerization accelerates the permeation of potassium and sodium ions passing through the double-channel by adjusting the environment around gA protein (the distribution of phospholipid head groups, ions outside the channel, and bulk water), rather than directly adjusting the conformation of gA protein.     

Key role of the polarization anisotropy of water in modeling classical polarizable force fields

We have evaluated the extent to which classical polarizable force fields, based either on the chemical potential equalization principle or on distributed polarizabilities in the framework of the Sum of Interactions Between Fragments Ab initio computed (SIBFA), can reproduce the ab initio polarization energy and the dipole moment of three distinct water oligomers: bifurcated chains, transverse hydrogen-bonded chains, and longitudinal hydrogen-bonded chains of helical shape. To analyze the many-body polarization effect, chains of different size, i.e., from 2 to 12 water monomers, have been considered. Although the dipole moment is a well-defined quantity in both classical polarizable models and quantum mechanical methods, polarization energy can be defined unequivocally only in the former type of approaches. In this study we have used the Kitaura-Morokuma (KM) procedure. Although the KM approach is on the one hand known to overestimate the polarization energy for strongly interacting molecules, on the other hand it can account for the many-body polarization effectively, whereas some other procedures do not. Our data show that, if off-centered lone pair polarizabilities are explicitly represented, classical polarizable force fields can afford a close agreement with the ab initio results, both in terms of polarization energy and in terms of dipole moment.     

Analytic gradient for second order Møller-Plesset perturbation theory with the polarizable continuum model based on the fragment molecular orbital method

A new energy expression is proposed for the fragment molecular orbital method interfaced with the polarizable continuum model (FMO/PCM). The solvation free energy is shown to be more accurate on a set of representative polypeptides with neutral and charged residues, in comparison to the original formulation at the same level of the many-body expansion of the electrostatic potential determining the apparent surface charges. The analytic first derivative of the energy with respect to nuclear coordinates is formulated at the second-order M?ller-Plesset (MP2) perturbation theory level combined with PCM, for which we derived coupled perturbed Hartree-Fock equations. The accuracy of the analytic gradient is demonstrated on test calculations in comparison to numeric gradient. Geometry optimization of the small Trp-cage protein (PDB: 1L2Y) is performed with FMO/PCM/6-31(+)G(d) at the MP2 and restricted Hartree-Fock with empirical dispersion (RHF/D). The root mean square deviations between the FMO optimized and NMR experimental structure are found to be 0.414 and 0.426 A? for RHF/D and MP2, respectively. The details of the hydrogen bond network in the Trp-cage protein are revealed.     

Conformational ensembles and sampled energy landscapes: Analysis and comparison

We present novel algorithms and software addressing four core problems in computational structural biology, namely analyzing a conformational ensemble, comparing two conformational ensembles, analyzing a sampled energy landscape, and comparing two sampled energy landscapes. Using recent developments in computational topology, graph theory, and combinatorial optimization, we make two notable contributions. First, we present a generic algorithm analyzing height fields. We then use this algorithm to perform density‐based clustering of conformations, and to analyze a sampled energy landscape in terms of basins and transitions between them. In both cases, topological persistence is used to manage (geometric) frustration. Second, we introduce two algorithms to compare transition graphs. The first is the classical earth mover distance metric which depends only on local minimum energy configurations along with their statistical weights, while the second incorporates topological constraints inherent to conformational transitions. Illustrations are provided on a simplified protein model (BLN69), whose frustrated potential energy landscape has been thoroughly studied. The software implementing our tools is also made available, and should prove valuable wherever conformational ensembles and energy landscapes are used. © 2015 Wiley Periodicals, Inc.     

Reference energy extremal optimization: a stochastic search algorithm applied to computational protein design

We adapt a combinatorial optimization algorithm, extremal optimization (EO), for the search problem in computational protein design. This algorithm takes advantage of the knowledge of local energy information and systematically improves on the residues that have high local energies. Power-law probability distributions are used to select the backbone sites to be improved on and the rotamer choices to be changed to. We compare this method with simulated annealing (SA) and motivate and present an improved method, which we call reference energy extremal optimization (REEO). REEO uses reference energies to convert a problem with a structured local-energy profile to one with more random profile, and extremal optimization proves to be extremely efficient for the latter problem. We show in detail the large improvement we have achieved using REEO as compared to simulated annealing and discuss a number of other heuristics we have attempted to date.     

CHARMM fluctuating charge force field for proteins: II protein/solvent properties from molecular dynamics simulations using a nonadditive electrostatic model

A fluctuating charge (FQ) force field is applied to molecular dynamics simulations for six small proteins in explicit polarizable solvent represented by the TIP4P-FQ potential. The proteins include 1FSV, 1ENH, 1PGB, 1VII, 1H8K, and 1CRN, representing both helical and beta-sheet secondary structural elements. Constant pressure and temperature (NPT) molecular dynamics simulations are performed on time scales of several nanoseconds, the longest simulations yet reported using explicitly polarizable all-atom empirical potentials (for both solvent and protein) in the condensed phase. In terms of structure, the FQ force field allows deviations from native structure up to 2.5 A (with a range of 1.0 to 2.5 A). This is commensurate to the performance of the CHARMM22 nonpolarizable model and other currently existing polarizable models. Importantly, secondary structural elements maintain native structure in general to within 1 A (both helix and beta-strands), again in good agreement with the nonpolarizable case. In qualitative agreement with QM/MM ab initio dynamics on crambin (Liu et al. Proteins 2001, 44, 484), there is a sequence dependence of average condensed phase atomic charge for all proteins, a dependence one would anticipate considering the differing chemical environments around individual atoms; this is a subtle quantum mechanical feature captured in the FQ model but absent in current state-of-the-art nonpolarizable models. Furthermore, there is a mutual polarization of solvent and protein in the condensed phase. Solvent dipole moment distributions within the first and second solvation shells around the protein display a shift towards higher dipole moments (increases on the order of 0.2-0.3 Debye) relative to the bulk; protein polarization is manifested via the enhanced condensed phase charges of typical polar atoms such as backbone carbonyl oxygens, amide nitrogens, and amide hydrogens. Finally, to enlarge the sample set of proteins, gas-phase minimizations and 1 ps constant temperature simulations are performed on various-sized proteins to compare to earlier work by Kaminsky et al. (J Comp Chem 2002, 23, 1515). The present work establishes the feasibility of applying a fully polarizable force field for protein simulations and demonstrates the approach employed in extending the CHARMM force field to include these effects.     

Polarizable empirical force field for alkanes based on the classical Drude oscillator model

Recent extensions of potential energy functions used in empirical force field calculations have involved the inclusion of electronic polarizability. To properly include this extension into a potential energy function it is necessary to systematically and rigorously optimize the associated parameters based on model compounds for which extensive experimental data are available. In the present work, optimization of parameters for alkanes in a polarizable empirical force field based on a classical Drude oscillator is presented. Emphasis is placed on the development of parameters for CH3, CH2, and CH moieties that are directly transferable to long chain alkanes, as required for lipids and other biomolecules. It is shown that a variety of quantum mechanical and experimental target data are reproduced by the polarizable model. Notable is the proper treatment of the dielectric constant of pure alkanes by the polarizable force field, a property essential for the accurate treatment of, for example, hydrophobic solvation in lipid bilayers. The present alkane force field will act as the basis for the aliphatic moieties in an extensive empirical force field for biomolecules that includes the explicit treatment of electronic polarizability.     

Self-consistent treatment of the local dielectric permittivity and electrostatic potential in solution for polarizable macromolecular force fields

A self-consistent method is presented for the calculation of the local dielectric permittivity and electrostatic potential generated by a solute of arbitrary shape and charge distribution in a polar and polarizable liquid. The structure and dynamics behavior of the liquid at the solute∕liquid interface determine the spatial variations of the density and the dielectric response. Emphasis here is on the treatment of the interface. The method is an extension of conventional methods used in continuum protein electrostatics, and can be used to estimate changes in the static dielectric response of the liquid as it adapts to charge redistribution within the solute. This is most relevant in the context of polarizable force fields, during electron structure optimization in quantum chemical calculations, or upon charge transfer. The method is computationally efficient and well suited for code parallelization, and can be used for on-the-fly calculations of the local permittivity in dynamics simulations of systems with large and heterogeneous charge distributions, such as proteins, nucleic acids, and polyelectrolytes. Numerical calculation of the system free energy is discussed for the general case of a liquid with field-dependent dielectric response.     

Calculations of the free energy of interaction of the c-Fos-c-Jun coiled coil: effects of the solvation model and the inclusion of polarization effects

The leucine zipper region of activator protein-1 (AP-1) comprises the c-Jun and c-Fos proteins and constitutes a well-known coiled coil protein-protein interaction motif. We have used molecular dynamics (MD) simulations in conjunction with the molecular mechanics/Poisson-Boltzmann generalized-Born surface area [MM/PB(GB)SA] methods to predict the free energy of interaction of these proteins. In particular, the influence of the choice of solvation model, protein force field, and water potential on the stability and dynamic properties of the c-Fos-c-Jun complex were investigated. Use of the AMBER polarizable force field ff02 in combination with the polarizable POL3 water potential was found to result in increased stability of the c-Fos-c-Jun complex. MM/PB(GB)SA calculations revealed that MD simulations using the POL3 water potential give the lowest predicted free energies of interaction compared to other nonpolarizable water potentials. In addition, the calculated absolute free energy of binding was predicted to be closest to the experimental value using the MM/GBSA method with independent MD simulation trajectories using the POL3 water potential and the polarizable ff02 force field, while all other binding affinities were overestimated.     

A molecule on the surface or inside a spherical nanoparticle: the computation of the interaction energy in terms of the dielectric continuum model

The electrostatic interaction energy between a charged or polar molecule and a spherical polarizable nanoparticle is studied within the advanced dielectric continuum model proposed previously. The molecule can be located either inside or outside the nanoparticle or in the vicinity of its boundary surface. The nanoparticle and its environment are considered as a polarizable medium and described in terms of a nonuniform dielectric continuum approximation with a position-dependent dielectric permittivity function e(r) \varepsilon (r) , where r is the position vector. A special construction of this function accounts for the proper treatment of sophisticated boundary effects. Test computations are performed for a number of sample molecules.     

Calculating absorption shifts for retinal proteins: computational challenges

Rhodopsins can modulate the optical properties of their chromophores over a wide range of wavelengths. The mechanism for this spectral tuning is based on the response of the retinal chromophore to external stress and the interaction with the charged, polar, and polarizable amino acids of the protein environment and is connected to its large change in dipole moment upon excitation, its large electronic polarizability, and its structural flexibility. In this work, we investigate the accuracy of computational approaches for modeling changes in absorption energies with respect to changes in geometry and applied external electric fields. We illustrate the high sensitivity of absorption energies on the ground-state structure of retinal, which varies significantly with the computational method used for geometry optimization. The response to external fields, in particular to point charges which model the protein environment in combined quantum mechanical/molecular mechanical (QM/MM) applications, is a crucial feature, which is not properly represented by previously used methods, such as time-dependent density functional theory (TDDFT), complete active space self-consistent field (CASSCF), and Hartree-Fock (HF) or semiempirical configuration interaction singles (CIS). This is discussed in detail for bacteriorhodopsin (bR), a protein which blue-shifts retinal gas-phase excitation energy by about 0.5 eV. As a result of this study, we propose a procedure which combines structure optimization or molecular dynamics simulation using DFT methods with a semiempirical or ab initio multireference configuration interaction treatment of the excitation energies. Using a conventional QM/MM point charge representation of the protein environment, we obtain an absorption energy for bR of 2.34 eV. This result is already close to the experimental value of 2.18 eV, even without considering the effects of protein polarization, differential dispersion, and conformational sampling.     

萃取溶剂选择的计算机辅助分子设计中的数学方法

萃取精馏是分离沸点相近或具有恒沸组成混合物的一种重要方法。选择最优的萃取剂是提高生产能力和降低能耗的根本途径。萃取剂选择的计算机辅助分子设计（CAMD）是基于性质估算的方法即通过CAMD来生成一组具有期望性质的分子，然后再对候选分子进行筛选。目前，CAMD法的研究主要可分为生成一验证、数学优化和组合优化。生成一验证法通常使用启发式的策略，是基于知识的，无法克服基团的组合爆炸问题，也不能保证结果的最优性；数学优化方法，包括混合整数非线性规划（MINLP）和混合整数线性规划（MILP），可用于表达非线性或线性的结构．性质关系，但准确表达结构-性质关系还有困难；组合优化法是使用遗传算法或模拟退火算法、禁忌搜索等的方法，理论上可克服以上问题。本文评述了以上三种方法在萃取精馏溶剂选择领域中的用途、原理及其工业化所面临的主要问题。     

Combining a polarizable force-field and a coarse-grained polarizable solvent model. II. Accounting for hydrophobic effects

A revised and improved version of our efficient polarizable force-field/coarse grained solvent combined approach (Masella, Borgis, and Cuniasse, J. Comput. Chem. 2008, 29, 1707) is described. The polarizable pseudo-particle solvent model represents the macroscopic solvent polarization by induced dipoles placed on mobile pseudo-particles. In this study, we propose a new formulation of the energy term handling the nonelectrostatic interactions among the pseudo-particles. This term is now able to reproduce the energetic and structural response of liquid water due to the presence of a hydrophobic spherical cavity. Accordingly, the parameters of the energy term handling the nonpolar solute/solvent interactions have been refined to reproduce the free-solvation energy of small solutes, based on a standard thermodynamic integration scheme. The reliability of this new approach has been checked for the properties of solvated methane and of the solvated methane dimer, as well as by performing 10 × 20 ns molecular dynamics (MD) trajectories for three solvated proteins. A long-time stability of the protein structures along the trajectories is observed. Moreover, our method still provides a measure of the protein solvation thermodynamic at the same accuracy as standard Poisson-Boltzman continuum methods. These results show the relevance of our approach and its applicability to massively coupled MD schemes to accurately and intensively explore solvated macromolecule potential energy surfaces.     

Lineshapes in carbon 1s photoelectron spectra of methanol clusters

A general protocol for theoretical modeling of inner-shell photoelectron spectra of molecular clusters is presented and applied to C1s spectra of oligomers and medium-sized clusters of methanol. The protocol employs molecular dynamics for obtaining cluster geometries and a polarizable force field for computing site-specific chemical shifts in ionization energy and linewidth. Comparisons to spectra computed from first-principle theories are used to establish the accuracy of the proposed force field approach. The model is used to analyze the C1s photoelectron spectrum of medium-sized clusters in terms of surface and bulk contributions. By treating the surface-to-bulk ratio as an adjustable parameter, satisfactory fits are obtained to experimental C1s spectra of a beam of methanol clusters.     

Ultrafast de novo docking combining pharmacophores and combinatorics

We report on a successful de novo design approach which relies on the combination of multi-million compound combinatorial docking under receptor-based pharmacophore constraints. Inspired by a rationale by A.R. Leach et al., we document on the unification of two steps into one for ligand assembly. In the original work, fragments known to bind in protein active sites were connected forming novel ligand compounds by means of generic skeleton linkers and following a combinatorial approach. In our approach, the knowledge of fragments binding to the protein has been expressed in terms of a receptor-based pharmacophore definition. The combinatorial linking step is performed in situ during docking, starting from combinatorial libraries. Three sample scenarios growing in size and complexity (combinatorial libraries of 1 million, 1.3 million, and 22.4 million compounds) have been created to illustrate the method. Docking could be accomplished between minutes and several hours depending on the outset; the results were throughout promising. Technically, a module compatibility between FlexX(C) and FlexX-Pharm has been established. The background is explained, and the crucial points from an information scientist's perspective are highlighted.     

针对蛋白质复合物Other类型的打分函数

在不同类型复合物结合界面的物理化学特征不同的基础上, 针对较难预测的Other 类型复合物设计出特异性打分函数, 用于在对接过程中挑选出有效结构. 该函数由原子接触能(EACE)、范德华和静电相互作用能组成,通过多元线性回归方法获得各项的权重系数. 对来自CAPRI benchmark1 中17 个Other 类复合物例子进行打分测试. 结果表明,组合打分能够刻画出Other 类型复合物单体间相互作用的特征, 反映出复合物形成前后的能量变化, 具备一定的从众多样本中筛选出有效结构的能力. 相对于残基成对势(RP), 该组合打分获得了更高的打分成功率. 对CAPRI 第八轮竞赛中两个结构预测模型进行打分排序, 该组合打分也体现出强于RP 的鉴别有效结合模式潜力.     

Computer simulation of electron transfer processes across the electrode|electrolyte interface: a treatment of solvent and electrode polarizability

A polarizable molecular dynamics model for adiabatic electron transfer across the electrode|electrolyte interface is presented. The electronic polarizability of the water and of the metal electrode is accounted for by a dynamical fluctuating charge algorithm, image charges, and the Ewald summation adapted for a conducting interface. The effects of the solvent electronic polarizability are studied by computing the diabatic and adiabatic free energy curves for both polarizable and non-polarizable water models. This represents the first effort to compute the adiabatic free energy curves from simulation for a fully polarizable electrochemical system.