Efficient calculation of rate constants: downhill versus uphill sampling.

The classical transition state theory (TST), together with the notion of transmission coefficient, provides a useful tool for calculation of rate constants for rare events. However, in complex biomolecular reactions, such as protein folding, it is difficult to find a good reaction coordinate, so the transition state is ill-defined. In this case, other approaches are more popular, such as the transition interface sampling (TIS) and the forward flux sampling (FFS). Here, we show that the algorithms developed in the frames of TIS and FFS can be successfully applied, after a modification, for calculation of the transmission coefficient. The new procedure (which we call "downhill sampling") is more efficient in comparison with the traditional TIS and FFS ("uphill sampling") even if the reaction coordinate is bad. We also propose a new computational scheme that combines the advantages of TST, TIS, and FFS.

Calculation of the "absolute" free energy of a ß-hairpin in an all-atom force field.

We propose a new approach to calculate the conformational free energy of a macromolecule in a compact stable state in implicit solvent. The free energy is evaluated with respect to an artificial reference system without internal interactions that is confined to a small well-defined multidimensional volume of a regular shape occupying approximately the same part of the conformational space as the macromolecule of interest. We present a practical implementation of our method, successfully apply it to a ß-hairpin in all-atom representation, verify the results with direct parallel tempering simulations, and discuss the possibilities of further improvements.

Structure simulation with calculated NMR parameters - integrating COSMOS into the CCPN framework.

The Collaborative Computing Project for NMR (CCPN) has build a software framework consisting of the CCPN data model (with APIs) for NMR related data, the CcpNmr Analysis program and additional tools like CcpNmr FormatConverter. The open architecture allows for the integration of external software to extend the abilities of the CCPN framework with additional calculation methods. Recently, we have carried out the first steps for integrating our software Computer Simulation of Molecular Structures (COSMOS) into the CCPN framework. The COSMOS-NMR force field unites quantum chemical routines for the calculation of molecular properties with a molecular mechanics force field yielding the relative molecular energies. COSMOS-NMR allows introducing NMR parameters as constraints into molecular mechanics calculations. The resulting infrastructure will be made available for the NMR community. As a first application we have tested the evaluation of calculated protein structures using COSMOS-derived 13C Cα and Cß chemical shifts. In this paper we give an overview of the methodology and a roadmap for future developments and applications.

Transition network based on equilibrium sampling: a new method for extracting kinetic information from Monte Carlo simulations of protein folding.

We propose a new type of transition network for modeling of protein dynamics. The nodes of the network correspond to the conformations taken from random sampling of equilibrium ensemble available, e.g., by Monte Carlo simulations. Although this approach does not provide absolute values of folding/unfolding rates, it allows identification of reaction pathways, transition state ensemble, and, eventually, intermediates. The new method is verified by a comparison with direct molecular dynamic simulations performed for a coarse-grained Go-like model of proteins. As an illustrative example, we analyze kinetics of formation of a small ß-hairpin (Trp zipper 1) in the all-atom representation.

Derivatives of molecular surface area and volume: simple and exact analytical formulas.

The computational effort of biomolecular simulations can be significantly reduced by means of implicit solvent models in which the energy generally contains a correction depending on the surface area and/or the volume of the molecule. In this article, we present simple derivation of exact, easy-to-use analytical formulas for these quantities and their derivatives with respect to atomic coordinates. In addition, we provide an efficient, linear-scaling algorithm for the construction of the power diagram required for practical implementation of these formulas. Our approach is implemented in a C++ header-only template library.

Modelling proteins: conformational sampling and reconstruction of folding kinetics.

In the last decades biomolecular simulation has made tremendous inroads to help elucidate biomolecular processes in-silico. Despite enormous advances in molecular dynamics techniques and the available computational power, many problems involve long time scales and large-scale molecular rearrangements that are still difficult to sample adequately. In this review we therefore summarise recent efforts to fundamentally improve this situation by decoupling the sampling of the energy landscape from the description of the kinetics of the process. Recent years have seen the emergence of many advanced sampling techniques, which permit efficient characterisation of the relevant family of molecular conformations by dispensing with the details of the short-term kinetics of the process. Because these methods generate thermodynamic information at best, they must be complemented by techniques to reconstruct the kinetics of the process using the ensemble of relevant conformations. Here we review recent advances for both types of methods and discuss their perspectives to permit efficient and accurate modelling of large-scale conformational changes in biomolecules. This article is part of a Special Issue entitled: Protein Dynamics: Experimental and Computational Approaches.

A method for the calculation of rate constants from stochastic transition paths.

We present a novel and efficient method for computation of rate constants in the systems where two metastable states are separated by a high free energy barrier. Our approach is based on the thermodynamic integration applied to the grand canonical ensemble of the stochastic transition paths. As illustrated on a multidimensional model system, the required computational costs depend only weakly on the barrier height, which provides a speedup of orders of magnitude in comparison to direct simulations.

Filamentous biopolymers on surfaces: atomic force microscopy images compared with Brownian dynamics simulation of filament deposition.

Nanomechanical properties of filamentous biopolymers, such as the persistence length, may be determined from two-dimensional images of molecules immobilized on surfaces. For a single filament in solution, two principal adsorption scenarios are possible. Both scenarios depend primarily on the interaction strength between the filament and the support: i) For interactions in the range of the thermal energy, the filament can freely equilibrate on the surface during adsorption; ii) For interactions much stronger than the thermal energy, the filament will be captured by the surface without having equilibrated. Such a 'trapping' mechanism leads to more condensed filament images and hence to a smaller value for the apparent persistence length. To understand the capture mechanism in more detail we have performed Brownian dynamics simulations of relatively short filaments by taking the two extreme scenarios into account. We then compared these 'ideal' adsorption scenarios with observed images of immobilized vimentin intermediate filaments on different surfaces. We found a good agreement between the contours of the deposited vimentin filaments on mica ('ideal' trapping) and on glass ('ideal' equilibrated) with our simulations. Based on these data, we have developed a strategy to reliably extract the persistence length of short worm-like chain fragments or network forming filaments with unknown polymer-surface interactions.

Probing hot spots on protein-protein interfaces with all-atom free-energy simulation.

Modulation of protein-protein interactions by competitive small-molecule binding emerges as a promising avenue for drug discovery. Hot spots, i.e., amino acids with important contributions to the overall interaction energy, provide useful targets within these interfaces. To avoid time-consuming mutagenesis experiments, computational alanine screening has been developed for the prediction of hot spots based on existing structural information. Here we use the all-atom free-energy force field PFF02 to identify important amino acid residues in the complexes of the chemokine interleukin-8 (CXCL8) and an N-terminal peptide of its cognate receptor CXCR1, and of ERBIN, a molecular marker of the basolateral membrane in epithelial cells, in complex with the ERBIN-binding domain of tyrosin kinase ERBB2. The results of our analysis agree with available experimental functional assays, indicating that this approach is suitable for computational alanine screening and may help to identify competitive peptides as starting points for the development of inhibitors of protein-protein interactions for pharmaceutically relevant targets.

Template-free protein structure prediction and quality assessment with an all-atom free-energy model.

Biophysical forcefields have contributed less than originally anticipated to recent progress in protein structure prediction. Here, we have investigated the selectivity of a recently developed all-atom free-energy forcefield for protein structure prediction and quality assessment (QA). Using a heuristic method, but excluding homology, we generated decoy-sets for all targets of the CASP7 protein structure prediction assessment with <150 amino acids. The decoys in each set were then ranked by energy in short relaxation simulations and the best low-energy cluster was submitted as a prediction. For four of nine template-free targets, this approach generated high-ranking predictions within the top 10 models submitted in CASP7 for the respective targets. For these targets, our de-novo predictions had an average GDT_S score of 42.81, significantly above the average of all groups. The refinement protocol has difficulty for oligomeric targets and when no near-native decoys are generated in the decoy library. For targets with high-quality decoy sets the refinement approach was highly selective. Motivated by this observation, we rescored all server submissions up to 200 amino acids using a similar refinement protocol, but using no clustering, in a QA exercise. We found an excellent correlation between the best server models and those with the lowest energy in the forcefield. The free-energy refinement protocol may thus be an efficient tool for relative QA and protein structure prediction.

Facilitated diffusion of DNA-binding proteins: Simulation of large systems.

The recently introduced method of excess collisions to estimate reaction times of protein-DNA systems in the presence of facilitated diffusion ("sliding") requires a cell of full system size. This bottleneck is avoided with a modification, by which a set of empirical parameters is calibrated using numerical simulations of a small test system. Once this is done, reaction times for systems of arbitrary dimensions are derived by extrapolation. It is shown that at physiological sliding lengths a test system of the order of 100 nm radius suffices to extract accurate reaction times for realistic cell dimensions. The achieved speedup, when compared to explicit simulations of the reaction process, is increasing in third order of the extrapolated radius of the cell.

Facilitated diffusion of DNA-binding proteins: efficient simulation with the method of excess collisions.

In this paper, a new method to efficiently simulate diffusion controlled second order chemical reactions is derived and applied to site-specific DNA-binding proteins. The protein enters a spherical cell and propagates via two competing modes, a free diffusion and a DNA-sliding mode, to search for its specific binding site in the center of the cell. There is no need for a straightforward simulation of this process. Instead, an alternative and exact approach is shown to be essentially faster than explicit random walk simulations. The speed-up of this novel simulation technique is rapidly growing with system size.

Facilitated diffusion of DNA-binding proteins.

The diffusion-controlled limit of reaction times for site-specific DNA-binding proteins is derived from first principles. We follow the generally accepted concept that a protein propagates via two competitive modes, a three-dimensional diffusion in space and a one-dimensional sliding along the DNA. However, our theoretical treatment of the problem is new. The accuracy of our analytical model is verified by numerical simulations. The results confirm that the unspecific binding of protein to DNA, combined with sliding, is capable to reduce the reaction times significantly.

Kinetics of antigen binding to antibody microspots: strong limitation by mass transport to the surface.

It is well documented that diffusion has generally a strong effect on the binding kinetics in the microtiter plate immunoassays. However, a systematic quantitative experimental evaluation of the microspot kinetics is still missing in the literature. Our work aims at filling this important gap of knowledge on the example of antigen binding to antibody microspots. A mathematical model was derived within the framework of two-compartment model and applied to the quantitative analysis of the experimental data obtained for typical antibody microspot assays. A strong mass-transport dependence of the antigen-antibody microspot kinetics was identified to be one of the main restrictions of this new technology. The binding reactions are slowed down in the microspot immunoassays by several orders of magnitude as compared with the corresponding well-stirred bulk reactions. The task to relax the mass-transport limitations should thus be one of the most important issues in designing the antibody microarrays. These limitations notwithstanding, the detection range of more than five orders of magnitude and the high sensitivity in the low femtomolar range were experimentally achieved in our study, demonstrating thus an enormous potential of this highly capable technology.

Kinetics of protein binding in solid-phase immunoassays: theory.

In a solid-phase immunoassay, binding between an antigen and its specific antibody takes place at the boundary of a liquid and a solid phase. One of the reactants (receptor) is immobilized on a surface. The other reactant (ligand) is initially free in solution. We present a theory describing the kinetics of immunochemical reaction in such a system. A single essential restriction of the theory is the assumption that the reaction conditions are uniform along the binding surface. In general, the reaction rate as a function of time can be obtained numerically as a solution of a nonlinear integral equation. For some special cases, analytical solutions are available. Various immunoassay geometries are considered, in particular, the case when the reaction is carried out on a microspot.

Modeling of intramolecular reactions of polymers: an efficient method based on Brownian dynamics simulations.

By the traditional approach to the Brownian dynamics simulations of intrachain reactions of polymers, the initial chain conformation is sampled from the equilibrium distribution. A dynamic trajectory is carried out until a "collision" of the reactive groups takes place, i.e., the distance between their centers becomes less that a certain reaction radius. The average length of the trajectory is equal to the mean time tauF of a diffusion-controlled reaction. In this work we propose another computational scheme. The trajectory begins at the instant of collision and is carried out until the chain is relaxed. The length of the trajectory has the order of the relaxation time taurel of the distance between the reactive groups. For polymer systems with taurel << tauF, this scheme allows the computation of tauF with considerable gain in computational time. Using the present approach, we calculated the mean time of DNA cyclization for the molecule length in the range from 50 to 500 nm.

Kinetics of site-site interactions in supercoiled DNA with bent sequences.

A curved DNA segment is known to adopt a preferred end loop localization in superhelical (sc) DNA and thus may organize the overall conformation of the molecule. Through this process it influences the probability of site juxtaposition. We addressed the effect of a curvature on site-site interactions quantitatively by measuring the kinetics of cross-linking of two biotinylated positions in scDNA by streptavidin. The DNA was biotinylated at either symmetric or asymmetric positions with respect to a curved insert via triplex-forming oligonucleotides (TFOs) modified with biotin. We used a quench-flow device to mix the DNA with the protein and scanning force microscopy to quantify the reaction products. As a measure of the interaction probability, rate constants of cross-linking and local concentrations j(M) of one biotinylated site in the vicinity of the other were determined and compared to Monte Carlo simulations for corresponding DNAs. In good agreement with the simulations, a j(M) value of 1.74 microM between two sites 500bp apart was measured for an scDNA without curvature. When a curvature was centered between the sites, the interaction probability increased about twofold over the DNA without curvature, significantly less than expected from the simulations. However, the relative differences of the interaction probabilities due to varied biotin positions with respect to the curvature agreed quantitatively with the theory.

Computational analysis of the chiral action of type II DNA topoisomerases.

It was found recently that bacterial type II DNA topoisomerase, topo IV, is much more efficient in relaxing (+) DNA supercoiling than (-) supercoiling. This means that the DNA-enzyme complex is chiral. This chirality can appear upon binding the first segment that participates in the strand passing reaction (G segment) or only after the second segment (T segment) joins the complex. The former possibility is analyzed here. We assume that upon binding the enzyme, the G segment forms a part of left-handed helical turn. This model is an extension of the hairpin model introduced earlier to explain simplification of DNA topology by these enzymes. Using statistical-mechanical simulation of DNA properties, we estimated different consequences of the model: (1) relative rates of relaxation of (+) and (-) supercoiling by the enzyme; (2) the distribution of positions of the G segment in supercoiled molecules; (3) steady-state distribution of knots in circular molecules created by the topoisomerase; (4) the variance of topoisomer distribution created by the enzyme; (5) the effect of (+) and (-) supercoiling on the binding topo II with G segment. The simulation results are capable of explaining nearly all available experimental data, at least semiquantitatively. A few predictions obtained in the model analysis can be tested experimentally.