Constrained global optimization for estimating molecular structure from atomic distances

Common enzymological experiments allow free energy profile determination

The determination of a complete set of rate constants [free energy profiles (FEPs)] for a complex kinetic mechanism is challenging. Enzymologists have devised a variety of informative steady-state kinetic experiments (e.g., Michaelis-Menten kinetics, viscosity dependence of kinetic parameters, kinetic isotope effects, etc.) that each provide distinct information regarding a particular kinetic system. A simple method for combining steady-state experiments in a single analysis is presented here, which allows microscopic rate constants and intrinsic kinetic isotope effects to be determined. It is first shown that Michaelis-Menten kinetic parameters (kcat and Km values), kinetic isotope efffets, solvent viscosity effects, and intermediate partitioning measurements are sufficient to define the rate constants for a reversible uni-uni mechanism with an intermediate, EZ, between the ES and EP complexes. Global optimization provides the framework for combining the independent experimental measurements, and the search for rate constants is performed using algorithms implemented in the biochemical software COPASI. This method is applied to the determination of FEPs for both alanine racemase and triosephosphate isomerase. The FEPs obtained from global optimization agree with those in the literature, with important exceptions. The method opens the door to routine and large-scale determination of FEPs for enzymes.

SYNTHESIS AND GIAO NMR CALCULATIONS FOR TWO DIASTEREOISOMERS OF 2′-ACETYLOXY-2′-PHENYLSPIRO[INDENO[1,2-b]QUINOXALIN-11,1′-CYCLOPROPANE]

Two diastereoisomers of 2′-acetyloxy-2′-phenylspiro[indeno[1,2-b]quinoxalin-11,1′-cyclopropane] were synthesized and their 1 H NMR spectra were recorded. Their chemical structures were fully optimized at B3LYP/6-311+G(d,p) level of theory using the Gaussian 03W program package. The 1 H NMR chemical shifts were calculated for geometry-optimized structures of the diastereoisomers with the gauge independent atomic orbital (GIAO) and B3LYP method with the 6-311+G(d,p), 6-311++G(d), 6-31++G(d,p) and 6-31+G(d) basis sets. The computational results were then compared with the experimental values and the structures associated with each spectrum were assigned.

Determining protein structures from NOESY distance constraints by semidefinite programming

Contemporary practical methods for protein nuclear magnetic resonance (NMR) structure determination use molecular dynamics coupled with a simulated annealing schedule. The objective of these methods is to minimize the error of deviating from the nuclear overhauser effect (NOE) distance constraints. However, the corresponding objective function is highly nonconvex and, consequently, difficult to optimize. Euclidean distance matrix (EDM) methods based on semidefinite programming (SDP) provide a natural framework for these problems. However, the high complexity of SDP solvers and the often noisy distance constraints provide major challenges to this approach. The main contribution of this article is a new SDP formulation for the EDM approach that overcomes these two difficulties. We model the protein as a set of intersecting two- and three-dimensional cliques. Then, we adapt and extend a technique called semidefinite facial reduction to reduce the SDP problem size to approximately one quarter of the size of the original problem. The reduced SDP problem can be solved approximately 100 times faster, and it is also more resistant to numerical problems from erroneous and inexact distance bounds.

Generating properly weighted ensemble of conformations of proteins from sparse or indirect distance constraints

Inferring three-dimensional structural information of biomacromolecules such as proteins from limited experimental data is an important and challenging task. Nuclear Overhauser effect measurements based on nucleic magnetic resonance, disulfide linking, and electron paramagnetic resonance labeling studies can all provide useful partial distance constraint characteristic of the conformations of proteins. In this study, we describe a general approach for reconstructing conformations of biomolecules that are consistent with given distance constraints. Such constraints can be in the form of upper bounds and lower bounds of distances between residue pairs, contact maps based on specific contact distance cutoff values, or indirect distance constraints such as experimental phi-value measurement. Our approach is based on the framework of sequential Monte Carlo method, a chain growth-based method. We have developed a novel growth potential function to guide the generation of conformations that satisfy given distance constraints. This potential function incorporates not only the distance information of current residue during growth but also the distance information of future residues by introducing global distance upper bounds between residue pairs and the placement of reference points. To obtain protein conformations from indirect distance constraints in the form of experimental phi-values, we first generate properly weighted contact maps satisfying phi-value constraints, we then generate conformations from these contact maps. We show that our approach can faithfully generate conformations that satisfy the given constraints, which approach the native structures when distance constraints for all residue pairs are given.

Using surface envelopes to constrain molecular modeling

Molecular density information (as measured by electron microscopic reconstructions or crystallographic density maps) can be a powerful source of information for molecular modeling. Molecular density constrains models by specifying where atoms should and should not be. Low-resolution density information can often be obtained relatively quickly, and there is a need for methods that use it effectively. We have previously described a method for scoring molecular models with surface envelopes to discriminate between plausible and implausible fits. We showed that we could successfully filter out models with the wrong shape based on this discrimination power. Ideally, however, surface information should be used during the modeling process to constrain the conformations that are sampled. In this paper, we describe an extension of our method for using shape information during computational modeling. We use the envelope scoring metric as part of an objective function in a global optimization that also optimizes distances and angles while avoiding collisions. We systematically tested surface representations of proteins (using all nonhydrogen heavy atoms) with different abundance of distance information and showed that the root mean square deviation (RMSD) of models built with envelope information is consistently improved, particularly in data sets with relatively small sets of short-range distances.

Untying the FIV frameshifting pseudoknot structure by MS3D

The structure of the putative feline immunodeficiency virus (FIV) ribosomal frameshifting pseudoknot (PK) has been investigated by a mass spectrometric three-dimensional (MS3D) approach, which involves the application of established solvent-accessibility probes and chemical crosslinkers with detection by electrospray ionization (ESI) Fourier transform mass spectrometry (FTMS). Regardless of their size, probed substrates can be treated with ribonucleases and analyzed by ESI-FTMS to obtain the correct position of chemically modified nucleotides. Protection maps and distance information can be utilized to generate 3D models using the constraint satisfaction algorithm provided by MC-SYM and the energy minimization modules included in CNS. Control experiments were performed on a mutant of mouse mammary tumor virus pseudoknot (VPK), for which an NMR structure is available. Comparison between the MS3D model and the high-resolution structure provided a approximately 3A root-mean-square deviation calculated from all the atoms present in double-stranded regions. Applied to FIV-PK, the MS3D approach confirmed that the selected sequence could fold into an actual pseudoknot, supporting the sequence alignment predictions. Characteristic features of H-type pseudoknots were recognized immediately, but a putative A13-U30 pair was not observed at the stem junction, making FIV-PK resemble VPK more closely than the initially suggested simian retrovirus type-1 pseudoknot. In our model, the unpaired U30 protrudes into the medium, while the hinging A13 assumes a stacked conformation that enables the stems to form a approximately 60 degrees bend and relieve the strain caused by a short loop 1. The model provided the basis to explain the different alkylation patterns observed in the absence and presence of Mg(2+), suggesting the possible formation of a specific metal-binding site between loop 1 and stem 2. This instance illustrates how the MS3D model of FIV-PK can be utilized effectively to generate hypotheses and support functional observations in the absence of a high-resolution structure.

Using surface envelopes for discrimination of molecular models

Shape information about macromolecules is increasingly available but is difficult to use in modeling efforts. We demonstrate that shape information alone can often distinguish structural models of biological macromolecules. By using a data structure called a surface envelope (SE) to represent the shape of the molecule, we propose a method that generates a fitness score for the shape of a particular molecular model. This score correlates well with root mean squared deviation (RMSD) of the model to the known test structures and can be used to filter models in decoy sets. The scoring method requires both alignment of the model to the SE in three-dimensional space and assessment of the degree to which atoms in the model fill the SE. Alignment combines a hybrid algorithm using principal components and a previously published iterated closest point algorithm. We test our method against models generated from random atom perturbation from crystal structures, published decoy sets used in structure prediction, and models created from the trajectories of atoms in molecular modeling runs. We also test our alignment algorithm against experimental electron microscopic data from rice dwarf virus. The alignment performance is reliable, and we show a high correlation between model RMSD and score function. This correlation is stronger for molecular models with greater oblong character (as measured by the ratio of largest to smallest principal component).

Cluster distance geometry of polypeptide chains

Distance geometry has been a broadly useful tool for dealing with conformational calculations. Customarily each atom is represented as a point, constraints on the distances between some atoms are obtained from experimental or theoretical sources, and then a random sampling of conformations can be calculated that are consistent with the constraints. Although these methods can be applied to small proteins having on the order of 1000 atoms, for some purposes it is advantageous to view the problem at lower resolution. Here distance geometry is generalized to deal with distances between sets of points. In the end, much of the same techniques produce a sampling of different configurations of these sets of points subject to distance constraints, but now the radii of gyration of the different sets play an important role. A simple example is given of how the packing constraints for polypeptide chains combine with loose distance constraints to give good calculated protein conformers at a very low resolution.