Voronoi binding site models: Calculation of binding modes and influence of drug binding data accuracy

A new and accurate method for calculating the geometrically allowed modes of binding of a ligand molecule to a Voronoi site model is reported. It is shown that the feasibility of the binding of a group of atoms to a Voronoi site reduces to a simple set of linear and quadratic inequalities and quadratic equalities which can be solved by minimization of a simple function. Newton's numerical method of solution coupled to a line search proved to be successful. Moreover, we have developed efficient molecular and site data bases to discard quickly infeasible binding modes without time-consuming numerical calculation. The method is tested with a data set consisting of the binding constants for a series of biphenyls binding to prealbumin. After determination of the conformation space of the molecules and proposal of a Voronoi site geometry, the geometrically feasible modes are calculated and the energy interaction parameters determined to fit the observed binding energies to the site within experimental error ranges. We actually allowed these ranges to vary in order to study the influence of their broadness on the site geometry and found that as they increase, one can first model the receptor as a three-region site then as a single region site, but never as a two-region site.     

The multiscale coarse-graining method. VIII. Multiresolution hierarchical basis functions and basis function selection in the construction of coarse-grained force fields

The multiscale coarse-graining (MS-CG) method is a method for determining the effective potential energy function for a coarse-grained (CG) model of a molecular system using data obtained from molecular dynamics simulation of the corresponding atomically detailed model. The coarse-grained potential obtained using the MS-CG method is a variational approximation for the exact many-body potential of mean force for the coarse-grained sites. Here we propose a new numerical algorithm with noise suppression capabilities and enhanced numerical stability for the solution of the MS-CG variational problem. The new method, which is a variant of the elastic net method [Friedman et al., Ann. Appl. Stat. 1, 302 (2007)], allows us to construct a large basis set, and for each value of a so-called "penalty parameter" the method automatically chooses a subset of the basis that is most important for representing the MS-CG potential. The size of the subset increases as the penalty parameter is decreased. The appropriate value to choose for the penalty parameter is the one that gives a basis set that is large enough to fit the data in the simulation data set without fitting the noise. This procedure provides regularization to mitigate potential numerical problems in the associated linear least squares calculation, and it provides a way to avoid fitting statistical error. We also develop new basis functions that are similar to multiresolution Haar functions and that have the differentiability properties that are appropriate for representing CG potentials. We demonstrate the feasibility of the combined use of the elastic net method and the multiresolution basis functions by performing a variational calculation of the CG potential for a relatively simple system. We develop a method to choose the appropriate value of the penalty parameter to give the optimal basis set. The combined effect of the new basis functions and the regularization provided by the elastic net method opens the possibility of using very large basis sets for complicated CG systems with many interaction potentials without encountering numerical problems in the variational calculation.     

Comparison of methods for outlier detection and their effects on the classification results for a particular data base

One of the drawbacks for using linear discriminant analysis (LDA) is the presence of outliers. Some methods of detecting outliers are compared and applied to a particular data base. When multivariate methods (multinormal distribution procedure and Hawkins' procedure) were applied, the two subsets produced did not differ greatly. Assumptions needed for the application of LDA were evaluated for each subset. Classification ability, feature selection and prediction ability were considered for each subset. Results for each subset were quite different. Hawkins' procedure seems the better method for detecting outliers.     

A powerful truncated Newton method for potential energy minimization

With advances in computer architecture and software, Newton methods are becoming not only feasible for large-scale nonlinear optimization problems, but also reliable, fast and efficient. Truncated Newton methods, in particular, are emerging as a versatile subclass. In this article we present a truncated Newton algorithm specifically developed for potential energy minimization. The method is globally convergent with local quadratic convergence. Its key ingredients are: (1) approximation of the Newton direction far away from local minima, (2) solution of the Newton equation iteratively by the linear Conjugate Gradient method, and (3) preconditioning of the Newton equation by the analytic second-derivative components of the “local” chemical interactions: bond length, bond angle and torsional potentials. Relaxation of the required accuracy of the Newton search direction diverts the minimization search away from regions where the function is nonconvex and towards physically interesting regions. The preconditioning strategy significantly accelerates the iterative solution for the Newton search direction, and therefore reduces the computation time for each iteration. With algorithmic variations, the truncated Newton method can be formulated so that storage and computational requirements are comparable to those of the nonlinear Conjugate Gradient method. As the convergence rate of nonlinear Conjugate Gradient methods is linear and performance less predictable, the application of the truncated Newton code to potential energy functions is promising.     

Powerful simulated-annealing algorithm locates global minimum of protein-folding potentials from multiple starting conformations

Protein-folding potentials, designed with the explicit goal that the global energy minimum correspond to crystallographically observed conformations of protein molecules, may offer great promise toward calculating native protein structures. Achieving this promise, however, depends on finding an effective means of dealing with the multiple-minimum problem inherent in such potentials. In this study, a protein-folding-potential test system has been developed that exhibits the properties of general protein-folding potentials yet has a unique well-defined global energy minimum corresponding to the crystallographically determined conformation of the test molecule. A simulated-annealing algorithm is developed that locates the global minimum of this potential in four of eight test runs from random starting conformations. Exploration of the energy-conformation surface of the potential indicates that it contains the numerous local minima typical of protein-folding potentials and that the global minimum is not easily located by conventional minimization procedures. When the annealing algorithm is applied to a previously developed actual folding potential to analyze the conformation of avian pancreatic polypeptide, a new conformer is located that is lower in energy than any conformer located in previous studies using a variety of minimization techniques.     

A line integral reaction path approximation for large systems via nonlinear constrained optimization: application to alanine dipeptide and the beta hairpin of protein G

A variation of the line integral method of Elber with self-avoiding walk has been implemented using a state of the art nonlinear constrained optimization procedure. The new implementation appears to be robust in finding approximate reaction paths for small and large systems. Exact transition states and intermediates for the resulting paths can easily be pinpointed with subsequent application of the conjugate peak refinement method [S. Fischer and M. Karplus, Chem. Phys. Lett. 194, 252 (1992)] and unconstrained minimization, respectively. Unlike previous implementations utilizing a penalty function approach, the present implementation generates an exact solution of the underlying problem. Most importantly, this formulation does not require an initial guess for the path, which makes it particularly useful for studying complex molecular rearrangements. The method has been applied to conformational rearrangements of the alanine dipeptide in the gas phase and in water, and folding of the beta hairpin of protein G in water. In the latter case a procedure was developed to systematically sample the potential energy surface underlying folding and reconstruct folding pathways within the nearest-neighbor hopping approximation.     

Linear and nonlinear excitations in a polyethylene crystal, 1. Vibrational modes and linear equations

Linear excitations in a polyethylene crystal are investigated by a new analytical method from the “first principles”. Elastic moduli are calculated from the parameters of molecular potentials of interaction. High anisotropy allows to derive comparatively simple approximate equations describing the dynamics of the crystal. Approximate expressions for the low-frequency branches of vibrational spectra are obtained. The role of internal degrees of freedom is revealed. Approximate expressions governing the dynamics of all degrees of freedom considered are obtained in analytic form suitable for nonlinear generalizations.     

Prediction of stability changes upon single-site mutations using database-derived potentials

One of the purposes of studying protein stability changes upon mutations is to get information about the dominating interactions that drive folding and stabilise the native structure. With this in mind, we present a method that predicts folding free-energy variations caused by point mutations using combinations of two types of database-derived potentials, i.e. backbone torsion-angle potentials and distance potentials, describing local and non-local interactions along the chain, respectively. The method is applied to evaluate the folding free-energy changes of 344 single-site mutations introduced in six different proteins and a synthetic peptide. We found that the relative importance of local versus non-local interactions along the chain is essentially a function of the solvent accessibility of the mutated residues. For the subset of totally buried residues, the optimal potential is the sum of a distance potential and a torsion potential weighted by a factor of 0.4. This combination yields a correlation coefficient between measured and computed changes in folding free energy of 0.80. For mutations of partially buried residues, the best potential is the sum of a torsion potential and a distance potential weighted by 0.7. For fully accessible residues, the torsion potentials taken alone perform best, reaching correlation coefficients of 0.87 on all but 10 mutations; the excluded mutations seem to modify the backbone structure or to involve interactions that are atypical for the surface. These results show that the relative weight of non-local interactions along the sequence decreases as the solvent accessibility of the mutated residue increases, and vanishes at the protein surface. On the contrary, the weight of local interactions increases with solvent accessibility. The latter interactions are nevertheless never negligible, even for the most buried residues. Received: 20 May 1998 / Accepted: 3 September 1998 / Published online: 7 December 1998     

Why Does the Linear Driving Force Model for Adsorption Kinetics Work?

The Linear Driving Force (LDF) model for gas adsorption kinetics is frequently and successfully used for analysis of adsorption column dynamic data and for adsorptive process designs because it is simple, analytic, and physically consistent. Yet, there is a substantial difference in the characteristics of isothermal batch uptake curves on adsorbent particles by the LDF and the more rigorous Fickian Diffusion (FD) model. It is demonstrated by using simple model systems that the characteristics of the adsorption kinetics at the single pore or the adsorbent particle level are lost in (a) evaluating overall uptake on a heterogeneous porous solid, (b) calculating breakthrough curves from a packed adsorbent column, and (c) establishing the efficiency of separation by an adsorptive process due to repeated averaging of the base kinetic property. That is why the LDF model works in practice.     

A self-consistent Ornstein-Zernike approximation for a fluid with a screened power series interaction

We present a thermodynamically self-consistent Ornstein-Zernike approximation (SCOZA) for a fluid of spherical particles with a pair potential given by a hard-core repulsion and screened power series (SPS) tails. We take advantage of the known analytic properties of the solution of the Ornstein-Zernike equation for the case in which the direct correlation function outside the repulsive core is given by the SPS tails [M. Yasutomi, J. Phys.: Condens. Matter 13, L255 (2001)]: c(r)=∑(n=1) (N)exp(-z(n)r)∑(τ=-1) (L(n) )K((n,τ))z(n) (τ+1)r(τ) r>1. The analytic properties are rewritten so as to be optimally suited to the numerical computations. The SCOZA is known to provide very good overall thermodynamics, remarkably accurate critical point, and coexistence curve. In this paper, we present some numerical results for parameters in c(r) which are chosen to fit the Lennard-Jones potential. We show that both the energy and the compressibility paths lead to the same thermodynamics with high accuracy due to the thermodynamic consistency condition that has been enforced. The present method will be applicable to fluids with a large variety of smooth, realistic isotropic potentials where the pair potentials can be fitted by the SPS tails. The fitting procedure is superior to that by multi-Yukawa tails which is the only method presented so far.     

Communication: Hybrid ensembles for improved force matching

Force matching is a method for parameterizing empirical potentials in which the empirical parameters are fitted to a reference potential energy surface (PES). Typically, training data are sampled from a canonical ensemble generated with either the empirical potential or the reference PES. In this Communication, we show that sampling from either ensemble risks excluding critical regions of configuration space, leading to fitted potentials that deviate significantly from the reference PES. We present a hybrid ensemble which combines the Boltzmann probabilities of both potential surfaces into the fitting procedure, and we demonstrate that this technique improves the quality and stability of empirical potentials.     

Simulation of the kinetics of complex heterogeneous catalytic reactions under diffusion limitations

A method is suggested for determining kinetic parameters of arbitrary complex heterogeneous catalytic reactions under conditions such that the reaction rate is significantly affected by heat and mass transfer to and in the catalyst pellets. Linear relationships are established between the concentrations of the key and dependent reactants (diffusion stoichiometry relationships) and between temperature and the concentrations of the key components. A solution procedure is suggested for the problems involved in the determination of kinetic parameters for reactions proceeding under diffusion limitations. The procedure is illustrated by analysis of data earlier reported for steam methane reforming over a nickel catalyst.     

An algebraic method that includes Gibbs minimization for performing phase equilibrium calculations for any number of components or phases

The most widely used technique for performing phase equilibria calculations is the K-value method (equality of chemical potentials). This paper proposes a more efficient algorithm to achieve the results that includes Gibbs minimization when we know the number of phases. Using the orthogonal derivatives, the tangent plane equation and mass balances, it is possible to reduce the Gibbs minimization procedure to the task of finding the solution of a system of non-linear equations. Such an operation is easier and faster than finding tangents or areas, and appears to converge as fast as the K-value method. Examples illustrate application of the new technique to two and three phases in equilibrium for binary and ternary mixtures.     

Electrochemical studies and square wave adsorptive stripping voltammetry of the antidepressant fluoxetine

The electrochemical reduction of the antidepressant drug fluoxetine was investigated by cyclic, linear sweep, differential pulse and square wave voltammetry using a hanging mercury drop electrode in alkaline buffer solution in water and in a water/acetonitrile mixed solvent. Cyclic voltammograms in aqueous solution showed very strong adsorption of fluoxetine on the electrode with formation of a compact film. The effect of addition of different percentages of acetonitrile on the voltammetric response was evaluated. It is shown that acetonitrile protects the electrode surface, thus preventing the adsorption of fluoxetine as a compact film, although reduction occurs at more negative potentials. Adsorption was used to accumulate the drug onto the electrode surface. The adsorbed species were measured voltammetrically by reduction at -1.3 V in an aqueous 0.05 M Ringer buffer, pH 12, 20% acetonitrile v/v. Linear calibration graphs were obtained in the range 0.52-5.2 M. The quantification of fluoxetine in pharmacological formulations existing in the market was performed using adsorptive square wave cathodic stripping voltammetry. and compared with data from UV spectrophotometry. The method is simple and not time-consuming. A comparative high performance liquid chromatography assay with UV detection was performed. Recovery data for both methods are reported.     

The rigorous computation of the molecular electric potential

A method is presented for the rigorous computation of the electric potential of molecules of arbitrary shape, under the assumption of continuous linear dielectric media. The computational technique involves finding the distribution of induced polarization charge on the molecular surface, and proceeds by an application of the method of boundary elements. The surface, which separates the molecular interior (of low dielectric constant) from the highly polar solvent, is given a piece-wise analytic representation as a collection of curvilinear elements. Given a set of internal fixed charges, the distribution of polarization-charge is found as a continuous function over the surface elements, and the electric potential (including all polarization effects) is then easily computed at any point. The method is applied to a spherical interface, and to several small molecules of biological interest, including a hexapeptide. The resulting potentials show good convergence in all cases. The future application of the method to macromolecules is discussed.     

Monte carlo calculations of second virial coefficients of chain molecules

Monte Carlo calculations have been performed for different types of chain molecules whose units interact through Lennard-Jones potentials. From the averaged Mayer function, we have evaluated the intermolecular two-body cluster integral, obtaining results for second virial coefficients. We have investigated the following points: a) the site modelization of alkanes by comparison of our results with gas phase data of different linear and branched alkanes and their mixtures. b) the prediction of interpenetration factors for flexible linear and star polymer chains in a good solvent (or excluded volume conditions). c) the determination of the theta point for a model of flexible polymer chains and the comparison of data for finite chains with theoretical predictions.     

Time-dependent rate coefficients for diffusion-influenced reactions with centrosymmetric potentials

Simple closed-form expressions are presented for the time-dependent rate coefficients of diffusion-influenced reactions in the presence of spherically symmetric potentials. For diffusion-controlled contact reactions, our expression reproduces the first two terms in both the short- and long-time expansions of the rate coefficient. At intermediate times, agreement with numerical results for the Debye-Hückel potential is found to be within a few percent for a wide range of parameters. For diffusion-influenced contact reactions (described by the radiation boundary condition), the agreement is even better. When the reactivity depends on the distance between the reactants (e.g., exponentially), our analytic result is less accurate, because it reproduces the two terms in the long-time expansion only to the linear order of the reciprocal of the diffusion coefficient. Our results should prove useful in the analysis of experimental data for diffusion-influenced reactions with centrosymmetric interaction potentials.