An exact expression to calculate the derivatives of position-dependent observables in molecular simulations with flexible constraints

In this work, we introduce an algorithm to compute the derivatives of physical observables along the constrained subspace when flexible constraints are imposed on the system (i.e., constraints in which the hard coordinates are fixed to configuration-dependent values). The presented scheme is exact, it does not contain any tunable parameter, and it only requires the calculation and inversion of a sub-block of the Hessian matrix of second derivatives of the function through which the constraints are defined. We also present a practical application to the case in which the sought observables are the Euclidean coordinates of complex molecular systems, and the function whose minimization defines the constraints is the potential energy. Finally, and in order to validate the method, which, as far as we are aware, is the first of its kind in the literature, we compare it to the natural and straightforward finite-differences approach in three molecules of biological relevance: methanol, N-methyl-acetamide and a tri-glycine peptideComment: 13 pages, 8 figures, published versio

Modeling Symmetric Macromolecular Structures in Rosetta3

Symmetric protein assemblies play important roles in many biochemical processes. However, the large size of such systems is challenging for traditional structure modeling methods. This paper describes the implementation of a general framework for modeling arbitrary symmetric systems in Rosetta3. We describe the various types of symmetries relevant to the study of protein structure that may be modeled using Rosetta's symmetric framework. We then describe how this symmetric framework is efficiently implemented within Rosetta, which restricts the conformational search space by sampling only symmetric degrees of freedom, and explicitly simulates only a subset of the interacting monomers. Finally, we describe structure prediction and design applications that utilize the Rosetta3 symmetric modeling capabilities, and provide a guide to running simulations on symmetric systems

An effective all-atom potential for proteins

We describe and test an implicit solvent all-atom potential for simulations of protein folding and aggregation. The potential is developed through studies of structural and thermodynamic properties of 17 peptides with diverse secondary structure. Results obtained using the final form of the potential are presented for all these peptides. The same model, with unchanged parameters, is furthermore applied to a heterodimeric coiled-coil system, a mixed alpha/beta protein and a three-helix-bundle protein, with very good results. The computational efficiency of the potential makes it possible to investigate the free-energy landscape of these 49--67-residue systems with high statistical accuracy, using only modest computational resources by today's standards

A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry

We present here a review of the fundamental topics of Hartree-Fock theory in Quantum Chemistry. From the molecular Hamiltonian, using and discussing the Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock equations for the electronic problem. Special emphasis is placed in the most relevant mathematical aspects of the theoretical derivation of the final equations, as well as in the results regarding the existence and uniqueness of their solutions. All Hartree-Fock versions with different spin restrictions are systematically extracted from the general case, thus providing a unifying framework. Then, the discretization of the one-electron orbitals space is reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition of the basic underlying concepts related to the construction and selection of Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we close the review with a section in which the most relevant modern developments (specially those related to the design of linear-scaling methods) are commented and linked to the issues discussed. The whole work is intentionally introductory and rather self-contained, so that it may be useful for non experts that aim to use quantum chemical methods in interdisciplinary applications. Moreover, much material that is found scattered in the literature has been put together here to facilitate comprehension and to serve as a handy reference.Comment: 64 pages, 3 figures, tMPH2e.cls style file, doublesp, mathbbol and subeqn package

Defining an Essence of Structure Determining Residue Contacts in Proteins

The network of native non-covalent residue contacts determines the three-dimensional structure of a protein. However, not all contacts are of equal structural significance, and little knowledge exists about a minimal, yet sufficient, subset required to define the global features of a protein. Characterisation of this “structural essence” has remained elusive so far: no algorithmic strategy has been devised to-date that could outperform a random selection in terms of 3D reconstruction accuracy (measured as the Ca RMSD). It is not only of theoretical interest (i.e., for design of advanced statistical potentials) to identify the number and nature of essential native contacts—such a subset of spatial constraints is very useful in a number of novel experimental methods (like EPR) which rely heavily on constraint-based protein modelling. To derive accurate three-dimensional models from distance constraints, we implemented a reconstruction pipeline using distance geometry. We selected a test-set of 12 protein structures from the four major SCOP fold classes and performed our reconstruction analysis. As a reference set, series of random subsets (ranging from 10% to 90% of native contacts) are generated for each protein, and the reconstruction accuracy is computed for each subset. We have developed a rational strategy, termed “cone-peeling” that combines sequence features and network descriptors to select minimal subsets that outperform the reference sets. We present, for the first time, a rational strategy to derive a structural essence of residue contacts and provide an estimate of the size of this minimal subset. Our algorithm computes sparse subsets capable of determining the tertiary structure at approximately 4.8 Å Ca RMSD with as little as 8% of the native contacts (Ca-Ca and Cb-Cb). At the same time, a randomly chosen subset of native contacts needs about twice as many contacts to reach the same level of accuracy. This “structural essence” opens new avenues in the fields of structure prediction, empirical potentials and docking

Hydrophilicity Matching – A Potential Prerequisite for the Formation of Protein-Protein Complexes in the Cell

A binding event between two proteins typically consists of a diffusional search of binding partners for one another, followed by a specific recognition of the compatible binding sites resulting in the formation of the complex. However, it is unclear how binding partners find each other in the context of the crowded, constantly fluctuating, and interaction-rich cellular environment. Here we examine the non-specific component of protein-protein interactions, which refers to those physicochemical properties of the binding partners that are independent of the exact details of their binding sites, but which can affect their localization or diffusional search for one another. We show that, for a large set of high-resolution experimental 3D structures of binary, transient protein complexes taken from the DOCKGROUND database, the binding partners display a surprising, statistically significant similarity in terms of their total hydration free energies normalized by a size-dependent variable. We hypothesize that colocalization of binding partners, even within individual cellular compartments such as the cytoplasm, may be influenced by their relative hydrophilicity, potentially in response to local hydrophilic gradients

Functional Analysis and Molecular Dynamics Simulation of LOX-1 K167N Polymorphism Reveal Alteration of Receptor Activity

The human lectin-like oxidized low density lipoprotein receptor 1 LOX-1, encoded by the ORL1 gene, is the major scavenger receptor for oxidized low density lipoprotein in endothelial cells. Here we report on the functional effects of a coding SNP, c.501G>C, which produces a single amino acid change (K>N at codon 167). Our study was aimed at elucidating whether the c.501G>C polymorphism changes the binding affinity of LOX-1 receptor altering its function. The presence of p.K167N mutation reduces ox-LDL binding and uptake. Ox-LDL activated extracellular signal-regulated kinases 1 and 2 (ERK 1/2) is inhibited. Furthermore, ox-LDL induced biosynthesis of LOX-1 receptors is dependent on the p.K167N variation. In human macrophages, derived from c.501G>C heterozygous individuals, the ox-LDL induced LOX-1 46 kDa band is markedly lower than in induced macrophages derived from c.501G>C controls. Investigation of p.K167N mutation through molecular dynamics simulation and electrostatic analysis suggests that the ox-LDL binding may be attributed to the coupling between the electrostatic potential distribution and the asymmetric flexibility of the basic spine residues. The N/N-LOX-1 mutant has either interrupted electrostatic potential and asymmetric fluctuations of the basic spine arginines

Predicting Target DNA Sequences of DNA-Binding Proteins Based on Unbound Structures

DNA-binding proteins such as transcription factors use DNA-binding domains (DBDs) to bind to specific sequences in the genome to initiate many important biological functions. Accurate prediction of such target sequences, often represented by position weight matrices (PWMs), is an important step to understand many biological processes. Recent studies have shown that knowledge-based potential functions can be applied on protein-DNA co-crystallized structures to generate PWMs that are considerably consistent with experimental data. However, this success has not been extended to DNA-binding proteins lacking co-crystallized structures. This study aims at investigating the possibility of predicting the DNA sequences bound by DNA-binding proteins from the proteins' unbound structures (structures of the unbound state). Given an unbound query protein and a template complex, the proposed method first employs structure alignment to generate synthetic protein-DNA complexes for the query protein. Once a complex is available, an atomic-level knowledge-based potential function is employed to predict PWMs characterizing the sequences to which the query protein can bind. The evaluation of the proposed method is based on seven DNA-binding proteins, which have structures of both DNA-bound and unbound forms for prediction as well as annotated PWMs for validation. Since this work is the first attempt to predict target sequences of DNA-binding proteins from their unbound structures, three types of structural variations that presumably influence the prediction accuracy were examined and discussed. Based on the analyses conducted in this study, the conformational change of proteins upon binding DNA was shown to be the key factor. This study sheds light on the challenge of predicting the target DNA sequences of a protein lacking co-crystallized structures, which encourages more efforts on the structure alignment-based approaches in addition to docking- and homology modeling-based approaches for generating synthetic complexes

A Ligand Channel through the G Protein Coupled Receptor Opsin

The G protein coupled receptor rhodopsin contains a pocket within its seven-transmembrane helix (TM) structure, which bears the inactivating 11-cis-retinal bound by a protonated Schiff-base to Lys296 in TM7. Light-induced 11-cis-/all-trans-isomerization leads to the Schiff-base deprotonated active Meta II intermediate. With Meta II decay, the Schiff-base bond is hydrolyzed, all-trans-retinal is released from the pocket, and the apoprotein opsin reloaded with new 11-cis-retinal. The crystal structure of opsin in its active Ops* conformation provides the basis for computational modeling of retinal release and uptake. The ligand-free 7TM bundle of opsin opens into the hydrophobic membrane layer through openings A (between TM1 and 7), and B (between TM5 and 6), respectively. Using skeleton search and molecular docking, we find a continuous channel through the protein that connects these two openings and comprises in its central part the retinal binding pocket. The channel traverses the receptor over a distance of ca. 70 Å and is between 11.6 and 3.2 Å wide. Both openings are lined with aromatic residues, while the central part is highly polar. Four constrictions within the channel are so narrow that they must stretch to allow passage of the retinal β-ionone-ring. Constrictions are at openings A and B, respectively, and at Trp265 and Lys296 within the retinal pocket. The lysine enforces a 90° elbow-like kink in the channel which limits retinal passage. With a favorable Lys side chain conformation, 11-cis-retinal can take the turn, whereas passage of the all-trans isomer would require more global conformational changes. We discuss possible scenarios for the uptake of 11-cis- and release of all-trans-retinal. If the uptake gate of 11-cis-retinal is assigned to opening B, all-trans is likely to leave through the same gate. The unidirectional passage proposed previously requires uptake of 11-cis-retinal through A and release of photolyzed all-trans-retinal through B