Comparing search strategies for finding global optima on energy landscapes

Monte Carlo Sampling with Hierarchical Move Sets: POSH Monte Carlo

We present a new Monte Carlo method for sampling rugged energy landscapes that allows for efficient transitions across sparsely distributed local basins. The trial move consists of two steps. The first step is a large initial trial move, and the second step is a Monte Carlo trajectory generated using smaller trial moves. To maintain detailed balance, a reverse transition probability is estimated along a path that differs from the forward path. Since the forward and reverse transitions are different, we name the algorithm POSH (port out, starboard home) Monte Carlo. The process obeys detailed balance to the extent that the transition probabilities are correctly estimated. There is an optimal range of performance for a given energy landscape, which depends on how sparsely the low energy states of the system are distributed. For simple model systems, adequate precision is obtained over a large range of inner steps settings. Side chain sampling of residues in the binding region of progesterone antibody 1dba are studied, and show that significant improvement over a comparable standard protocol can be obtained using POSH sampling. To compare with experimental data, the phosphopeptide Ace-Gly-Ser-pSer-Ser-Nma is also studied, and the resulting NMR observables compare well with experiment. For the biomolecular systems studied, we show that POSH sampling generates precise distributions using the number of inner steps set up to 20.

Energy-directed tree search: an efficient systematic algorithm for finding the lowest energy conformation of molecules

We present a new systematic algorithm, energy-directed tree search (EDTS), for exploring the conformational space of molecules. The algorithm has been designed to reliably locate the global minimum (or, in the worst case, a structure within 4 kJ mol(-1) of this species) at a fraction of the cost of a full conformational search, and in this way extend the range of chemical systems for which accurate thermochemistry can be studied. The algorithm is inspired by the build-up approach but is performed on the original molecule as a whole, and objectively determines the combinations of torsional angles to optimise using a learning process. The algorithm was tested for a set of 22 large molecules, including open- and closed-shell species, stable structures and transition structures, and neutral and charged species, incorporating a range of functional groups (such as phenyl rings, esters, thioesters and phosphines), and covering polymers, peptides, drugs, and natural products. For most of the species studied the global minimum energy structure was obtained; for the rest the EDTS algorithm found conformations whose total electronic energies are within chemical accuracy from the true global minima. When the conformational space is searched at a resolution of 120 degrees , the cost of the EDTS algorithm (in its worst-case scenario) scales as 2(N) for large N (where N is the number of rotatable bonds), compared with 3(N) for the corresponding systematic search.

Grow to Fit Molecular Dynamics (G2FMD): an ab initio method for protein side-chain assignment and refinement

The rough energy landscapes and tight packing of protein interiors are two of the critical factors that have prevented the wide application of physics-based models in protein side-chain assignment and protein structure prediction in general. Complementing the rotamer-based methods, we propose an ab initio method that utilizes molecular mechanics simulations for protein side-chain assignment and refinement. By reducing the side-chain size, a smooth energy landscape was obtained owing to the increased distances between the side chains. The side chains then gradually grow back during molecular dynamics simulations while adjusting to their surrounding driven by the interaction energies. The method overcomes the barriers due to tight packing that limit conformational sampling of physics-based models. A key feature of this approach is that the resulting structures are free from steric collisions and allow the application of all-atom models in the subsequent refinement. Tests on a small set of proteins showed nearly 100% accuracy on both chi1 and chi2 of buried residues and 94% of them were within 20 degrees from the native conformation, 79% were within 10 degrees and 42% were within 5 degrees . However, the accuracy decreased when exposed side chains were involved. Further improvement and application of the method and the possible reasons that affect the accuracy on the exposed side chains are discussed.

Protein decoy assembly using short fragments under geometric constraints

A small set of protein fragments can represent adequately all known local protein structure. This set of fragments, along with a construction scheme that assembles these fragments into structures, defines a discrete (relatively small) conformation space, which approximates protein structures accurately. We generate protein decoys by sampling geometrically valid structures from this conformation space, biased by the secondary structure prediction for the protein. Unlike other methods, secondary structure prediction is the only protein-specific information used for generating the decoys. Nevertheless, these decoys are qualitatively similar to those found by others. The method works well for all-alpha proteins, and shows promising results for alpha and beta proteins.

Semiglobal simplex optimization and its application to determining the preferred solvation sites of proteins

The classical simplex method is extended into the Semiglobal Simplex (SGS) algorithm. Although SGS does not guarantee finding the global minimum, it affords a much more thorough exploration of the local minima than any traditional minimization method. The basic idea of SGS is to perform a local minimization in each step of the simplex algorithm, and thus, similarly to the Convex Global Underestimator (CGU) method, the search is carried out on a surface spanned by local minima. The SGS and CGU methods are compared by minimizing a set of test functions of increasing complexity, each with a known global minimum and many local minima. Although CGU delivers substantially better success rates in simple problems, the two methods become comparable as the complexity of the problems increases. Because SGS is generally faster than CGU, it is the method of choice for solving optimization problems in which function evaluation is computationally inexpensive and the search region is large. The extreme simplicity of the method is also a factor. The SGS method is applied here to the problem of finding the most preferred (i.e., minimum free energy) solvation sites on a streptavidin monomer. It is shown that the SGS method locates the same lowest free energy positions as an exhaustive multistart Simplex search of the protein surface, with less than one-tenth the number of minizations. The combination of the two methods, i.e.. multistart simplex and SGS, provides a reliable procedure for predicting all potential solvation sites of a protein.

Prediction of protein-protein interactions by docking methods

Recently, developments have been made in predicting the structure of docked complexes when the coordinates of the components are known. The process generally consists of a stage during which the components are combined rigidly and then a refinement stage. Several rapid new algorithms have been introduced in the rigid docking problem and promising refinement techniques have been developed, based on modified molecular mechanics force fields and empirical measures of desolvation, combined with minimisations that switch on the short-range interactions gradually. There has also been progress in developing a benchmark set of targets for docking and a blind trial, similar to the trials of protein structure prediction, has taken place.

Exploratory studies of ab initio protein structure prediction: multiple copy simulated annealing, AMBER energy functions, and a generalized born/solvent accessibility solvation model

A theoretical and computational approach to ab initio structure prediction for polypeptides in water is described and applied to selected amino acid sequences for testing and preliminary validation. The method builds systematically on the extensive efforts applied to parameterization of molecular dynamics (MD) force fields, employs an empirically well-validated continuum dielectric model for solvation, and an eminently parallelizable approach to conformational search. The effective free energy of polypeptide chains is estimated from AMBER united atom potential functions, with internal degrees of freedom for both backbone and amino acid side chains explicitly treated. The hydration free energy of each structure is determined using the Generalized Born/Solvent Accessibility (GBSA) method, modified and reparameterized to include atom types consistent with the AMBER force field. The conformational search procedure employs a multiple copy, Monte Carlo simulated annealing (MCSA) protocol in full torsion angle space, applied iteratively on sets of structures of progressively lower free energy until a prediction of a structure with lowest effective free energy is obtained. Calibration tests for the effective energy function and search algorithm are performed on the alanine dipeptide, selected protein crystal structures, and united atom decoys on barnase, crambin, and six examples from the Rosetta set. Specific demonstration cases of the method are provided for the 8-mer sequence of Ala residues, a 12-residue peptide with longer side chains QLLKKLLQQLKQ, a de novo designed 16 residue peptide of sequence (AAQAA)3Y, a 15-residue sequence with a beta sheet motif, GEWTWDATKTFTVTE, and a 36 residue small protein, Villin headpiece. The Ala 8-mer readily formed an alpha-helix. An alpha-helix structure was predicted for the 16-mer, consistent with observed results from IR and CD spectroscopy and with the pattern in psi/straight phi angles of known protein structures. The predicted structure for the 12-mer, composed of a mix of helix and less regular elements of secondary structure, lies 2.65 A RMS from the observed crystal structure. Structure prediction for the 8-mer beta-motif resulted in form 4.50 A RMS from the crystal geometry. For Villin, the predicted native form is very close to the crystal structure, RMS values of 3.5 A (including sidechains), and 1.01 A (main chain only). The methodology permits a detailed analysis of the molecular forces which dominate various segments of the predicted folding trajectory. Analysis of the results in terms of internal torsional, electrostatic and van der Waals and the electrostatic and non-electrostatic contributions to hydration, including the hydrophobic effect, is presented.

RNA folding energy landscapes

Using a statistical mechanical treatment, we study RNA folding energy landscapes. We first validate the theory by showing that, for the RNA molecules we tested having only secondary structures, this treatment (i) predicts about the same native structures as the Zuker method, and (ii) qualitatively predicts the melting curve peaks and shoulders seen in experiments. We then predict thermodynamic folding intermediates. For one hairpin sequence, unfolding is a simple unzipping process. But for another sequence, unfolding is more complex. It involves multiple stable intermediates and a rezipping into a completely non-native conformation before unfolding. The principle that emerges, for which there is growing experimental support, is that although protein folding tends to involve highly cooperative two-state thermodynamic transitions, without detectable intermediates, the folding of RNA secondary structures may involve rugged landscapes, often with more complex intermediate states.