Application of Empirical and Quantum-chemical Computational Methods in the Determination of the Free Conformational Energy of Substituents in 1,3-Dioxanes

The advantages and disadvantages of empirical and quantum-chemical methods for the determination of the free conformational energy of methyl and phenyl substituents at the C₍₄₎ and C₍₅₎ atoms of the ring in the molecules of 1,3-dioxanes are analyzed.     

Ab initio prediction of helical segments in polypeptides

An ab initio method has been developed to predict helix formation for polypeptides. The approach relies on the systematic analysis of overlapping oligopeptides to determine the helical propensity for individual residues. Detailed atomistic level modeling, including entropic contributions, and solvation/ionization energies calculated through the solution of the Poisson-Boltzmann equation, is utilized. The calculation of probabilities for helix formation is based on the generation of ensembles of low energy conformers. The approach, which is easily amenable to parallelization, is shown to perform very well for several benchmark polypeptide systems, including the bovine pancreatic trypsin inhibitor, the immunoglobulin binding domain of protein G, the chymotrypsin inhibitor 2, the R69 N-terminal domain of phage 434 repressor, and the wheat germ agglutinin.     

Conformational free energy surface of the linear DPDPE peptide: cost of pre-organization for disulfide bond formation

We have calculated the free energy differences between four conformers of the linear form of the opioid pentapeptide DPDPE in aqueous solution. The conformers are Cyc, representing the structure adopted by the linear peptide prior to disulfide bond formation, β _C and β _E , two slightly different β-turns previously identified in unconstrained molecular dynamics simulations, and Ext, an extended structure. Our simulations indicate that β _E is the most stable of the studied conformers of linear DPDPE in aqueous solution, with β _C , Cyc and Ext having free energies higher by 2.3, 6.3, and 28.2 kcal/mol, respectively. The free energy differences of 4.0 kcal/mol between β _C and Cyc, and 6.3 kcal/mol between β _E and Cyc, reflect the cost of pre-organizing the linear peptide into a conformation conducive for disulfide bond formation. Such a conformational change is a pre-requisite for the chemical reaction of S–S bond formation to proceed. The relatively low population of the cyclic-like structure agrees qualitatively with observed lower potency and different receptor specificity of the linear form relative to the cyclic peptide, and with previous unconstrained simulation results. Free energy component analysis indicates that the moderate stability difference of 4.0–6.3 kcal/mol between the β-turns and the cyclic-like structure results from cancellation of two large opposing effects. In accord with intuition, the relaxed β-turns have conformational strain 43–45 kcal/mol lower than the Cyc structure. However, the cyclic-like conformer interacts with water about 39 kcal/mol strongly than the open β-turns. Our simulations are the first application of the recently developed multidimensional conformational free energy thermodynamic integration (CFTI) protocol to a solvated system, with fast convergence of the free energy obtained by fixing all flexible dihedrals. Additionally, the availability of the CFTI multidimensional free energy gradient leads to a new decomposition scheme, giving the contribution of each fixed dihedral to the overall free energy change and providing additional insight into the microscopic mechanisms of the studied processes. Received: 20 April 1998 / Accepted: 9 September 1998 / Published online: 7 December 1998     

Some practical aspects of free energy calculations from molecular dynamics simulation

In this work, we address two critical aspects of calculation of the free energy differences in molecular systems from molecular simulations. The first aspect involves checking whether the calculated free energy difference depends significantly on the extent of perturbation used for accomplishment of a given transformation. The second aspect of interest is to verify if the sampling errors in calculating the free energy differences between the wild-type molecule and a mutated one in its free state and in a complex are similar, or not, for a finite-length dynamic simulation. The reliability of the free energy estimates obtained from molecular simulations using thermodynamic cycles depends in part on this fact. For investigating these aspects, we use a self-transformation scheme in which a transformation of a part of a molecular system into itself is considered. We perform MD simulations of DNA fragments in which a part of a specific base is subjected to such a self-transformation. Results indicate that the estimated free energy differences do not depend significantly on the extent of perturbation used to achieve the transformation. Interestingly, the variation in the cumulative free energy difference, ΔA, with the coupling parameter, λ, depends significantly on the extent of perturbation. We examine the physical basis of the observed nature of the variation of the accumulated free energy difference, ΔA, against the λ value in the case of a self-transformation. In a thermodynamic cycle, the sampling errors due to the finite-length simulation for the molecular system are found to be similar to each other for the two perturbations (free and in a complex) justifying the use of such approach in calculating ΔΔA in molecular complexes. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 877–885, 1999     

Free energy perturbation approach to the critical assessment of selective cyclooxygenase-2 inhibitors

The discovery of selective cyclooxygenase-2 (COX-2) inhibitors represents a major achievement of the efforts over the past few decades to develop therapeutic treatments for inflammation. To gain insights into designing new COX-2-selective inhibitors, we address the energetic and structural basis for the selective inhibition of COX isozymes by means of a combined computational protocol involving docking experiment, force field design for the heme prothetic group, and free energy perturbation (FEP) simulation. We consider both COX-2- and COX-1-selective inhibitors taking the V523I mutant of COX-2 to be a relevant structural model for COX-1 as confirmed by a variety of experimental and theoretical evidences. For all COX-2-selective inhibitors under consideration, we find that free energies of binding become less favorable as the receptor changes from COX-2 to COX-1, due to the weakening and/or loss of hydrogen bond and hydrophobic interactions that stabilize the inhibitors in the COX-2 active site. On the other hand, COX-1-selective oxicam inhibitors gain extra stabilization energy with the change of residue 523 from valine to isoleucine because of the formations of new hydrogen bonds in the enzyme-inhibitor complexes. The utility of the combined computational approach, as a valuable tool for in silico screening of COX-2-selective inhibitors, is further exemplified by identifying the physicochemical origins of the enantiospecific selective inhibition of COX-2 by -substituted indomethacin ethanolamide inhibitors.     

MC(JBW): Simple but smart Monte Carlo algorithm for free energy simulations of multiconformational molecules

Many of the most common molecular simulation methods, including Monte Carlo (MC) and molecular or stochastic dynamics (MD or SD), have significant difficulties in sampling the space of molecular potential energy surfaces characterized by multiple conformational minima and significant energy barriers. In such cases improved sampling can be obtained by special techniques that lower such barriers or somehow direct search steps toward different low energy regions of space. We recently described a hybrid MC/SD algorithm [MC(JBW)/SD] incorporating such a technique that directed MC moves of selected torsion and bond angles toward known low energy regions of conformational space. Exploration of other degrees of freedom was left to the SD part of the hybrid algorithm. In the work described here, we develop a related but simpler simulation algorithm that uses only MC to sample all degrees of freedom (e.g., stretch, bend, and torsion). We term this algorithm MC(JBW). Using simulations on various model potential energy surfaces and on simple molecular systems (n-pentane, n-butane, and cyclohexane), MC(JBW) is shown to generate ensembles of states that are indistinguishable from the canonical ensembles generated by classical Metropolis MC in the limit of very long simulations. We further demonstrate the utility of MC(JBW) by evaluating the room temperature free energy differences between conformers of various substituted cyclohexanes and the larger ring hydrocarbons cycloheptane, cyclooctane, cyclononane, and cyclodecane. The results compare favorably with available experimental data and results from previously reported MC(JBW)/SD conformational free energy calculations. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1736–1745, 1998     

Free energy in vertically aligned liquid crystal mode

Vertical alignment (VA) is a widely applied operation mode for liquid crystal displays. To achieve optimum brightness, the electrode of VA is often patterned with fish-bone fine slits to generate fringe field, so the negative liquid crystal aligns along the fine slits when the electric field is applied. VA is usually simply modelled by the bend geometry along the cell gap. However, defects, domain boundaries and periodical splay induced by the fine slits also exist in real pixels and disturb the liquid crystal alignment. Polymer-stabilised VA test cells with various fine slit pitches which lead to various strength of fringe field were fabricated to observe the deformation of liquid crystal. Then the models of liquid crystal deformation nearby the defects and in the fine slit area were proposed to calculate the electromagnetic (f_EM) and elastic free energy (f_elastic). The results show that the key factor to regulate f_EM and f_elastic is the pitch of the fine slits, and the optimum liquid crystal alignment is obtained when f_EM and f_elastic are equal. The models are useful for further investigation on the dynamics of liquid crystal alignment and applications in industrial products.     

Basic ingredients of free energy calculations: A review

Methods to compute free energy differences between different states of a molecular system are reviewed with the aim of identifying their basic ingredients and their utility when applied in practice to biomolecular systems. A free energy calculation is comprised of three basic components: (i) a suitable model or Hamiltonian, (ii) a sampling protocol with which one can generate a representative ensemble of molecular configurations, and (iii) an estimator of the free energy difference itself. Alternative sampling protocols can be distinguished according to whether one or more states are to be sampled. In cases where only a single state is considered, six alternative techniques could be distinguished: (i) changing the dynamics, (ii) deforming the energy surface, (iii) extending the dimensionality, (iv) perturbing the forces, (v) reducing the number of degrees of freedom, and (vi) multi‐copy approaches. In cases where multiple states are to be sampled, the three primary techniques are staging, importance sampling, and adiabatic decoupling. Estimators of the free energy can be classified as global methods that either count the number of times a given state is sampled or use energy differences. Or, they can be classified as local methods that either make use of the force or are based on transition probabilities. Finally, this overview of the available techniques and how they can be best used in a practical context is aimed at helping the reader choose the most appropriate combination of approaches for the biomolecular system, Hamiltonian and free energy difference of interest. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

Computer simulation on the energy transfer processes in allophycocyanins

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自由能计算方法及其在生物大分子体系中的适用性问题

本文总结了计算科学领域中被广泛使用的自由能计算方法,详述了各种自由能计算方法的基本原理、操作过程以及所具有的特点,并在最后对各种方法在生物大分子体系中的适用性做了概括,以期使读者在自由能相关研究工作中,根据自身研究体系和研究目的,选择合适的计算方法开展研究工作.     

Graph‐theoretical identification of dissociation pathways on free energy landscapes of biomolecular interaction

Biomolecular association and dissociation reactions take place on complicated interaction free energy landscapes that are still very hard to characterize computationally. For large enough distances, though, it often suffices to consider the six relative translational and rotational degrees of freedom of the two particles treated as rigid bodies. Here, we computed the six‐dimensional free energy surface of a dimer of water‐soluble alpha‐helices by scanning these six degrees of freedom in about one million grid points. In each point, the relative free energy difference was computed as the sum of the polar and nonpolar solvation free energies of the helix dimer and of the intermolecular coulombic interaction energy. The Dijkstra graph algorithm was then applied to search for the lowest cost dissociation pathways based on a weighted, directed graph, where the vertices represent the grid points, the edges connect the grid points and their neighbors, and the weights are the reaction costs between adjacent pairs of grid points. As an example, the configuration of the bound state was chosen as the source node, and the eight corners of the translational cube were chosen as the destination nodes. With the strong electrostatic interaction of the two helices giving rise to a clearly funnel‐shaped energy landscape, the eight lowest‐energy cost pathways coming from different orientations converge into a well‐defined pathway for association. We believe that the methodology presented here will prove useful for identifying low‐energy association and dissociation pathways in future studies of complicated free energy landscapes for biomolecular interaction. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2010     

A physically meaningful method for the comparison of potential energy functions

In the study of the conformational behavior of complex systems, such as proteins, several related statistical measures are commonly used to compare two different potential energy functions. Among them, the Pearson's correlation coefficient r has no units and allows only semiquantitative statements to be made. Those that do have units of energy and whose value may be compared to a physically relevant scale, such as the root-mean-square deviation (RMSD), the mean error of the energies (ER), the standard deviation of the error (SDER) or the mean absolute error (AER), overestimate the distance between potentials. Moreover, their precise statistical meaning is far from clear. In this article, a new measure of the distance between potential energy functions is defined that overcomes the aforementioned difficulties. In addition, its precise physical meaning is discussed, the important issue of its additivity is investigated, and some possible applications are proposed. Finally, two of these applications are illustrated with practical examples: the study of the van der Waals energy, as implemented in CHARMM, in the Trp-Cage protein (PDB code 1L2Y) and the comparison of different levels of the theory in the ab initio study of the Ramachandran map of the model peptide HCO-L-Ala-NH2.     

Comparison of implicit solvent models for the simulation of protein-surface interactions

Empirical force field-based molecular simulations can provide valuable atomistic-level insights into protein-surface interactions in aqueous solution. While the implicit treatment of solvation effects is desired as a means of improving simulation efficiency, existing implicit solvent models were primarily developed for the simulation of peptide or protein behavior in solution alone, and thus may not be appropriate for protein interactions with synthetic material surfaces. The objective of this research was to calculate the change in free energy as a function of surface-separation distance for peptide-surface interactions using different empirical force field-based implicit solvation models (ACE, ASP, EEF1, and RDIE with the CHARMM 19 force field), and to compare these results with the same calculations conducted using density functional theory (DFT) combined with the self-consistent reaction field (SCRF) implicit solvation model. These comparisons show that distinctly different types of behavior are predicted with each implicit solvation method, with ACE providing the best overall agreement with DFT/SCRF calculations. These results also identify areas where ACE is in need of improvement for this application and provide a basis for subsequent parameter refinement.     

低自由能固体表面的制备及其应用*

低自由能表面具有一些独特的性能,在工业和日常生活中具有广泛的应用,本文就低自由能表面的制备及其发展概况进行综述,共引用文献83篇。     

Free energy calculations using flexible-constrained, hard-constrained and non-constrained molecular dynamics simulations.

A comparison of different treatments of bond-stretching interactions in molecular dynamics simulation is presented. Relative free energies from simulations using rigid bonds maintained with the SHAKE algorithm, using partially rigid bonds maintained with a recently introduced flexible constraints algorithm, and using fully flexible bonds are compared in a multi-configurational thermodynamic integration calculation of changing liquid water into liquid methanol. The formula for the free energy change due to a changing flexible constraint in a flexible constraint simulation is derived. To allow for a more direct comparison between these three methods, three different pairs of models for water and methanol were used: a flexible model (simulated without constraints and with flexible constraints), a rigid model (simulated with standard hard constraints), and an alternative flexible model (simulated with flexible constraints and standard hard constraints) in which the ideal or constrained bond lengths correspond to the average bond lengths obtained from a short simulation of the unconstrained flexible model. The particular treatment of the bonds induces differences of up to 2 % in the liquid densities, whereas (excess) free energy differences of up to 5.7 (4.3) kJ mol(-1) are observed. These values are smaller than the differences observed between the three different pairs of methanol/water models: up to 5 % in density and up to 8.5 kJ mol(-1) in (excess) free energy.     

Conformational Composition of 2,5-Disubstituted 1,3,2-Dioxaborinanes

We have used ¹H NMR spectra and also MM+ and AM1 calculation methods to show that the conformational equilibrium of 2,5-disubstituted 1,3,2-dioxaborinane molecules, including two sofa forms, is shifted toward the equatorial conformer. We have established the values of G ⁰ for a number of substituents on the C₍₅₎ ring atom.     

Sampling potential energy surface of glycyl glycine peptide: Comparison of Metropolis Monte Carlo and stochastic dynamics

A comparative study was carried out to test the efficiency with which Metropolis Monte Carlo (MC) and stochastic dynamics (SD) sample the potential energy surface of the N-acetyl glycyl glycine methylamide peptide as defined by the united atom AMBER* force field. Boltzmann-weighted ensembles were generated with variations of all internal degrees of freedom (i.e., stretch, bend, and torsion) for a single N-acetyl glycyl glycine methylamide molecule at 300 K by 10⁸-step MC and 100-ns SD simulations. As expected, both methods gave the same final energetic results. However, convergence was found to be ∼10 times faster with MC than with SD as measured by comparisons of the populations of all symmetrically equivalent conformers. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1294–1299, 1998     

Free energy perturbation and molecular dynamics calculations of copper binding to azurin

Free energy perturbation/molecular dynamics simulations have been carried out on copper/azurin systems calculating the binding affinities of copper (II) ion to azurin either in the native or in the unfolded state. In order to test the validity of the strategy adopted for the calculations and to establish what force field is suitable for these kinds of calculations, three different force fields, AMBER, CVFF, and CFF, have been alternatively used for the calculations and the results have been compared with experimental data obtained by spectroscopic titrations of copper (II)/azurin solutions and denaturation experiments. Our findings have pointed out that only CFF gives satisfactory results, thus providing a reliable tool for copper binding simulations in copper protein.