Quantum annealing: A new method for minimizing multidimensional functions

Computation based on molecular models is playing an increasingly important role in biology, biological chemistry, and biophysics. Since only a very limited number of properties of biomolecular systems is actually accessible to measurement by experimental means, computer simulation can complement experiment by providing not only averages, but also distributions and time series of any definable quantity, for example, conformational distributions or interactions between parts of systems. Present day biomolecular modeling is limited in its application by four main problems: 1) the force-field problem, 2) the search (sampling) problem, 3) the ensemble (sampling) problem, and 4) the experimental problem. These four problems are discussed and illustrated by practical examples. Perspectives are also outlined for pushing forward the limitations of biomolecular modeling.

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Biomolecular modeling: Goals, problems, perspectives.

A two-stage procedure for the determination of a united-residue potential designed for protein simulations is outlined. In the first stage, the long-range and local-interaction energy terms of the total energy of a polypeptide chain are determined by analyzing protein-crystal data and averaging the all-atom energy surfaces. In the second stage (described in the accompanying article), the relative weights of the energy terms are optimized so as to locate the native structures of selected test proteins as the lowest energy structures. The goal of the work in the present study is to parameterize physically reasonable functional forms of the potentials of mean force for side-chain interactions. The potentials are of both radial and anisotropic type. Radial potentials include the Lennard-Jones and the shifted Lennard-Jones potential (with the shift parameter independent of orientation). To treat the angular dependence of side-chain interactions, three functional forms of the potential that were designed previously to describe anisotropic systems are evaluated: Berne-Pechukas (dilated Lennard-Jones); Gay-Berne (shifted Lennard-Jones with orientation-dependent shift parameters); and Gay-Berne-Vorobjev (the same as the preceding one, but with one more set of variable parameters). These functional forms were used to parameterize, within a short-distance range, the potentials of mean force for side-chain pair interactions that are related by the Boltzmann principle to the pair correlation functions determined from protein-crystal data. Parameter determination was formulated as a generalized nonlinear least-squares problem with the target function being the weighted sum of squares of the differences between calculated and “experimental” (i.e., estimated from protein-crystal data) angular, radial-angular, and radial pair correlation functions, as well as contact free energies. A set of 195 high-resolution nonhomologous structures from the Protein Data Bank was used to calculate the “experimental” values. The contact free energies were scaled by the slope of the correlation line between side-chain hydrophobicities, calculated from the contact free energies, and those determined by Fauchere and Pliska from the partition coefficients of amino acids between water and n-octanol. The methylene group served to define the reference contact free energy corresponding to that between the glycine methylene groups of backbone residues. Statistical analysis of the goodness of fit revealed that the Gay-Berne-Vorobjev anisotropic potential fits best to the experimental radial and angular correlation functions and contact free energies and therefore represents the free-energy surface of side-chain-side-chain interactions most accurately. Thus, its choice for simulations of protein structure is probably the most appropriate. However, the use of simpler functional forms is recommended, if the speed of computations is an issue. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 849–873, 1997

A united-residue force field for off-lattice protein-structure simulations. I. Functional forms and parameters of long-range side-chain interaction potentials from protein crystal data

Methods to search for low-energy conformations, to generate a Boltzmann-weighted ensemble of configurations, or to generate classical-dynamical trajectories for molecular systems in the condensed liquid phase are briefly reviewed with an eye to application to biomolecular systems. After having chosen the degrees of freedom and method to generate molecular configurations, the efficiency of the search or sampling can be enhanced in various ways: (i) efficient calculation of the energy function and forces, (ii) application of a plethora of search enhancement techniques, (iii) use of a biasing potential energy term, and (iv) guiding the sampling using a reaction or transition pathway. The overview of the available methods should help the reader to choose the combination that is most suitable for the biomolecular system, degrees of freedom, interaction function, and molecular or thermodynamic properties of interest.

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On searching in, sampling of, and dynamically moving through conformational space of biomolecular systems: A review.

Based on the dipole model of peptide groups developed in our earlier work [Liwo et al., Prot. Sci.,2, 1697 (1993)], a cumulant expansion of the average free energy of the system of freely rotating peptide-group dipoles tethered to a fixed α-carbon trace is derived. A graphical approach is presented to find all nonvanishing terms in the cumulants. In particular, analytical expressions for three- and four-body (correlation) terms in the averaged interaction potential of united peptide groups are derived. These expressions are similar to the cooperative forces in hydrogen bonding introduced by Kolinski and Skolnick [J. Chem. Phys.,97, 9412 (1992)]. The cooperativity arises here naturally from the higher order terms in the power-series expansion (in the inverse of the temperature) for the average energy. Test calculations have shown that addition of the derived four-body term to the statistical united-residue potential of our earlier work [Liwo et al., J. Comput. Chem.,18, 849, 874 (1997)] greatly improves its performance in folding poly-l-alanine into an α-helix. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 259–276, 1998

United-residue force field for off-lattice protein-structure simulations: III. Origin of backbone hydrogen-bonding cooperativity in united-residue potentials

A self-consistent multitorsional field (SCMTF) method of global optimization of the conformational energy of an oligopeptide is proposed. The method is based on the idea that the maximum of the square of the ground-state wave function is very often close to the global minimum of the potential energy and may be relatively unaffected by the complexity of the potential, e.g., its many minima. Using a mean field approximation, a set of coupled Schroinger equations for the motion of the nuclei is solved iteratively in a dihedral-angle space, each equation depending on a single dihedral angle

Mean field theory as a tool for intramolecular conformational optimization. 1. Tests on terminally-blocked alanine and met-enkephalin

The self-consistent multitorsional field (SCMTF) method for global optimization of the conformational energy of polypeptides has been tested with the 20-residue membrane-bound portion of melittin. A new rapid method for calculating the effective potential is presented, together with an algorithm that accelerates convergence in the vicinity of the global minimum. The effective average potential energy for each dihedral angle was obtained by interpolating the Monte Carlo average ECEPP/2 (empirical conformational energy program for peptides) potential, calculated on an equally spaced wide grid supplemented by points drawn from the related probability density distribution

Mean field theory as a tool for intramolecular conformational optimization. 3. Test on melittin

Tests of the self-consistent multitorsional field (SCMTF) method for global optimization of the conformational energy of decaglycine (10-residue polyglycine) and icosalanine (20-residue poly-L-alanine) show that it is applicable for global minimization of larger oligopeptides than considered heretofore. The method is based on the fact that the maximum of the square of the ground-state wave function is very often close to the global minimum of the potential energy

H. A. Scheraga

Papers

Mean field theory as a tool for intramolecular conformational optimization. 1. Tests on terminally-blocked alanine and met-enkephalin

Mean field theory as a tool for intramolecular conformational optimization. 3. Test on melittin

Mean field theory as a tool for intramolecular conformational optimization. 2. Tests on the homopolypeptides decaglycine and icosalanine