Accelerating and focusing protein-protein docking correlations using multi-dimensional rotational FFT generating functions

MOTIVATION: Predicting how proteins interact at the molecular level is a computationally intensive task. Many protein docking algorithms begin by using fast Fourier transform (FFT) correlation techniques to find putative rigid body docking orientations. Most such approaches use 3D Cartesian grids and are therefore limited to computing three dimensional (3D) translational correlations. However, translational FFTs can speed up the calculation in only three of the six rigid body degrees of freedom, and they cannot easily incorporate prior knowledge about a complex to focus and hence further accelerate the calculation. Furthemore, several groups have developed multi-term interaction potentials and others use multi-copy approaches to simulate protein flexibility, which both add to the computational cost of FFT-based docking algorithms. Hence there is a need to develop more powerful and more versatile FFT docking techniques. RESULTS: This article presents a closed-form 6D spherical polar Fourier correlation expression from which arbitrary multi-dimensional multi-property multi-resolution FFT correlations may be generated. The approach is demonstrated by calculating 1D, 3D and 5D rotational correlations of 3D shape and electrostatic expansions up to polynomial order L=30 on a 2 GB personal computer. As expected, 3D correlations are found to be considerably faster than 1D correlations but, surprisingly, 5D correlations are often slower than 3D correlations. Nonetheless, we show that 5D correlations will be advantageous when calculating multi-term knowledge-based interaction potentials. When docking the 84 complexes of the Protein Docking Benchmark, blind 3D shape plus electrostatic correlations take around 30 minutes on a contemporary personal computer and find acceptable solutions within the top 20 in 16 cases. Applying a simple angular constraint to focus the calculation around the receptor binding site produces acceptable solutions within the top 20 in 28 cases. Further constraining the search to the ligand binding site gives up to 48 solutions within the top 20, with calculation times of just a few minutes per complex. Hence the approach described provides a practical and fast tool for rigid body protein-protein docking, especially when prior knowledge about one or both binding sites is available.     

Optimized grid-based protein-protein docking as a global search tool followed by incorporating experimentally derivable restraints

Unbound protein docking, or the computational prediction of the structure of a protein complex from the structures of its separated components, is of importance but still challenging. A practical approach toward reliable results for unbound docking is to incorporate experimentally derived information with computation. To this end, truly systematic search of the global docking space is desirable. The fast Fourier transform (FFT) docking is a systematic search method with high computational efficiency. However, by using FFT to perform unbound docking, possible conformational changes upon binding must be treated implicitly. To better accommodate the implicit treatment of conformational flexibility, we develop a rational approach to optimize "softened" parameters for FFT docking. In connection with the increased "softness" of the parameters in this global search step, we use a revised rule to select candidate models from the search results. For complexes designated as of low and medium difficulty for unbound docking, these adaptations of the original FTDOCK program lead to substantial improvements of the global search results. Finally, we show that models resulted from FFT-based global search can be further filtered with restraints derivable from nuclear magnetic resonance (NMR) chemical shift perturbation or mutagenesis experiments, leading to a small set of models that can be feasibly refined and evaluated using computationally more expensive methods and that still include high-ranking near-native conformations.     

Exploring protein-protein interactions using the site-identification by ligand competitive saturation methodology

Protein docking methods are powerful computational tools to study protein-protein interactions (PPI). While a significant number of docking algorithms have been developed, they are usually based on rigid protein models or with limited considerations of protein flexibility and the desolvation effect is rarely considered in docking energy functions, which may lower the accuracy of the predictions. To address these issues, we introduce a PPI energy function based on the site-identification by ligand competitive saturation (SILCS) framework and utilize the fast Fourier transform (FFT) correlation approach. The free energy content of the SILCS FragMaps represent an alternative to traditional energy grids and they can be efficiently utilized to guide FFT-based protein docking. Application of the approach to eight diverse test cases, including seven from Protein Docking Benchmark 5.0, showed the PPI prediction using SILCS approach (SILCS-PPI) to be competitive with several commonly used protein docking methods indicating that the method has the ability to both qualitatively and quantitatively inform the prediction of PPI. Results show the utility of the SILCS-PPI docking approach for determination of probability distributions of PPI interactions over the surface of both partner proteins, allowing for identification of alternate binding poses. Such binding poses are confirmed by experimental crystal contacts in our test cases. While more computationally demanding than available PPI docking technologies, we anticipate that the SILCS-PPI docking approach will offer an alternative methodology for improved evaluation of PPIs that could be used in a variety of fields from systems biology to excipient design for biologics-based drugs.     

Accelerating protein docking in ZDOCK using an advanced 3D convolution library

Computational prediction of the 3D structures of molecular interactions is a challenging area, often requiring significant computational resources to produce structural predictions with atomic-level accuracy. This can be particularly burdensome when modeling large sets of interactions, macromolecular assemblies, or interactions between flexible proteins. We previously developed a protein docking program, ZDOCK, which uses a fast Fourier transform to perform a 3D search of the spatial degrees of freedom between two molecules. By utilizing a pairwise statistical potential in the ZDOCK scoring function, there were notable gains in docking accuracy over previous versions, but this improvement in accuracy came at a substantial computational cost. In this study, we incorporated a recently developed 3D convolution library into ZDOCK, and additionally modified ZDOCK to dynamically orient the input proteins for more efficient convolution. These modifications resulted in an average of over 8.5-fold improvement in running time when tested on 176 cases in a newly released protein docking benchmark, as well as substantially less memory usage, with no loss in docking accuracy. We also applied these improvements to a previous version of ZDOCK that uses a simpler non-pairwise atomic potential, yielding an average speed improvement of over 5-fold on the docking benchmark, while maintaining predictive success. This permits the utilization of ZDOCK for more intensive tasks such as docking flexible molecules and modeling of interactomes, and can be run more readily by those with limited computational resources.     

Protein docking using spherical polar Fourier correlations

We present a new computational method of docking pairs of proteins by using spherical polar Fourier correlations to accelerate the search for candidate low-energy conformations. Interaction energies are estimated using a hydrophobic excluded volume model derived from the notion of "overlapping surface skins," augmented by a rigorous but "soft" model of electrostatic complementarity. This approach has several advantages over former three-dimensional grid-based fast Fourier transform (FFT) docking correlation methods even though there is no analogue to the FFT in a spherical polar representation. For example, a complete search over all six rigid-body degrees of freedom can be performed by rotating and translating only the initial expansion coefficients, many unfeasible orientations may be eliminated rapidly using only low-resolution terms, and the correlations are easily localized around known binding epitopes when this knowledge is available. Typical execution times on a single processor workstation range from 2 hours for a global search (5 x 10(8) trial orientations) to a few minutes for a local search (over 6 x 10(7) orientations). The method is illustrated with several domain dimer and enzyme-inhibitor complexes and 20 large antibody-antigen complexes, using both the bound and (when available) unbound subunits. The correct conformation of the complex is frequently identified when docking bound subunits, and a good docking orientation is ranked within the top 20 in 11 out of 18 cases when starting from unbound subunits. Proteins 2000;39:178-194.     

ClusPro: performance in CAPRI rounds 6-11 and the new server

ClusPro is the first fully automated, web-based program for docking protein structures. Users may upload the coordinate files of two protein structures through ClusPro's web interface, or enter the PDB codes of the respective structures. The server performs rigid body docking, energy screening, and clustering to produce models. The program output is a short list of putative complexes ranked according to their clustering properties. ClusPro has been participating in CAPRI since January 2003, submitting predictions within 24 h after a target becomes available. In Rounds 6-11, ClusPro generated acceptable submissions for Targets 22, 25, and 27. In general, acceptable models were obtained for the relatively easy targets without substantial conformational changes upon binding. We also describe the new version of ClusPro that incorporates our recently developed docking program PIPER. PIPER is based on the fast Fourier transform correlation approach, but the method is extended to use pairwise interaction potentials, thereby increasing the number of near-native docked structures.     

Flexible protein docking refinement using pose-dependent normal mode analysis

Modeling conformational changes in protein docking calculations is challenging. To make the calculations tractable, most current docking algorithms typically treat proteins as rigid bodies and use soft scoring functions that implicitly accommodate some degree of flexibility. Alternatively, ensembles of structures generated from molecular dynamics (MD) may be cross-docked. However, such combinatorial approaches can produce many thousands or even millions of docking poses, and require fast and sensitive scoring functions to distinguish them. Here, we present a novel approach called "EigenHex," which is based on normal mode analyses (NMAs) of a simple elastic network model of protein flexibility. We initially assume that the proteins to be docked are rigid, and we begin by performing conventional soft docking using the Hex polar Fourier correlation algorithm. We then apply a pose-dependent NMA to each of the top 1000 rigid body docking solutions, and we sample and re-score multiple perturbed docking conformations generated from linear combinations of up to 20 eigenvectors using a multi-threaded particle swarm optimization algorithm. When applied to the 63 "rigid body" targets of the Protein Docking Benchmark version 2.0, our results show that sampling and re-scoring from just one to three eigenvectors gives a modest but consistent improvement for these targets. Thus, pose-dependent NMA avoids the need to sample multiple eigenvectors and it offers a promising alternative to combinatorial cross-docking.     

A method for biomolecular structural recognition and docking allowing conformational flexibility.

In this work, we present an algorithm developed to handle biomolecular structural recognition problems, as part of an interdisciplinary research endeavor of the Computer Vision and Molecular Biology fields. A key problem in rational drug design and in biomolecular structural recognition is the generation of binding modes between two molecules, also known as molecular docking. Geometrical fitness is a necessary condition for molecular interaction. Hence, docking a ligand (e.g., a drug molecule or a protein molecule), to a protein receptor (e.g., enzyme), involves recognition of molecular surfaces. Conformational transitions by "hinge-bending" involves rotational movements of relatively rigid parts with respect to each other. The generation of docked binding modes between two associating molecules depends on their three dimensional structures (3-D) and their conformational flexibility. In comparison to the particular case of rigid-body docking, the computational difficulty grows considerably when taking into account the additional degrees of freedom intrinsic to the flexible molecular docking problem. Previous docking techniques have enabled hinge movements only within small ligands. Partial flexibility in the receptor molecule is enabled by a few techniques. Hinge-bending motions of protein receptors domains are not addressed by these methods, although these types of transitions are significant, e.g., in enzymes activity. Our approach allows hinge induced motions to exist in either the receptor or the ligand molecules of diverse sizes. We allow domains/subdomains/group of atoms movements in either of the associating molecules. We achieve this by adapting a technique developed in Computer Vision and Robotics for the efficient recognition of partially occluded articulated objects. These types of objects consist of rigid parts which are connected by rotary joints (hinges). Our method is based on an extension and generalization of the Hough transform and the Geometric Hashing paradigms for rigid object recognition. We show experimental results obtained by the successful application of the algorithm to cases of bound and unbound molecular complexes, yielding fast matching times. While the "correct" molecular conformations of the known complexes are obtained with small RMS distances, additional, predictive good-fitting binding modes are generated as well. We conclude by discussing the algorithm's implications and extensions, as well as its application to investigations of protein structures in Molecular Biology and recognition problems in Computer Vision.     

Hierarchical structure of cascade of primary and secondary periodicities in Fourier power spectrum of alphoid higher order repeats

Background  

Identification of approximate tandem repeats is an important task of broad significance and still remains a challenging problem of computational genomics. Often there is no single best approach to periodicity detection and a combination of different methods may improve the prediction accuracy. Discrete Fourier transform (DFT) has been extensively used to study primary periodicities in DNA sequences. Here we investigate the application of DFT method to identify and study alphoid higher order repeats.     

Mapping of antibody epitopes based on docking and homology modeling

Antibodies are key proteins produced by the immune system to target pathogen proteins termed antigens via specific binding to surface regions called epitopes. Given an antigen and the sequence of an antibody the knowledge of the epitope is critical for the discovery and development of antibody based therapeutics. In this work, we present a computational protocol that uses template-based modeling and docking to predict epitope residues. This protocol is implemented in three major steps. First, a template-based modeling approach is used to build the antibody structures. We tested several options, including generation of models using AlphaFold2. Second, each antibody model is docked to the antigen using the fast Fourier transform (FFT) based docking program PIPER. Attention is given to optimally selecting the docking energy parameters depending on the input data. In particular, the van der Waals energy terms are reduced for modeled antibodies relative to x-ray structures. Finally, ranking of antigen surface residues is produced. The ranking relies on the docking results, that is, how often the residue appears in the docking poses' interface, and also on the energy favorability of the docking pose in question. The method, called PIPER-Map, has been tested on a widely used antibody–antigen docking benchmark. The results show that PIPER-Map improves upon the existing epitope prediction methods. An interesting observation is that epitope prediction accuracy starting from antibody sequence alone does not significantly differ from that of starting from unbound (i.e., separately crystallized) antibody structure.     

FTIR,FT-Raman spectra and quantum chemical studies of nebivolol

Fourier transform infrared and Raman spectra of nebivolol have been recorded. The structure, conformational stability, geometry optimisation, and vibrational wave numbers have been investigated. Satisfactory vibrational assignments were made for the stable conformer of the molecule using Restricted Hartree–Fock (RHF) and density functional theory (DFT) calculation (B3LYP) with the 6-31G(d,p) basis set. Comparison of the observed fundamental vibrational wave numbers of the molecule and calculated results by RHF and DFT methods indicates that B3LYP is superior for molecular vibrational problems. Comparison of the simulated spectra with the experimental spectra provides important information about the ability of the computational method to describe the vibrational modes. The RHF and DFT-based NMR calculation procedure was also done. It was used to assign the ¹³C NMR chemical shift of nebivolol.     

Molecular modeling of the interaction between heparan sulfate and cellular growth factors: bringing pieces together

Heparan sulfate is a polysaccharide belonging to the glycaminoglycan family. It interacts with numerous proteins of the extracellular matrix, in particular cellular growth factors. The number of experimental protein-heparin sulfate complexes obtained by crystallography or nuclear magnetic resonance is limited. Alternatively, computational approaches can be employed. Generally, they restrain the conformation of the glycosidic rings and linkages in order to reduce the complexity of the problem. Modeling the interaction between protein and heparan sulfate is indeed challenging because of the large size of the fragment needed for a strong binding, the flexibility brought by the glycosidic rings and linkages and the high density of negative charges. We propose a two-step method based on molecular docking and molecular dynamics simulation. Molecular docking allows exploring the positioning of a rigid heparin sulfate fragment on the protein surface. Molecular dynamics refine selected docking models by explicitly representing solvent molecules and not restraining the polysaccharide backbone. The interaction of a hexamer of heparin sulfate was studied in interaction with fibroblast growth factor 2 and stromal cell-derived factor 1α. This approach shed light on the plasticity of the growth factors interacting with heparan sulfate. This approach can be extended to the study of other protein/glycosaminoglycan complexes.     

Examination of shape complementarity in docking of unbound proteins.

Here we carry out an examination of shape complementarity as a criterion in protein-protein docking and binding. Specifically, we examine the quality of shape complementarity as a critical determinant not only in the docking of 26 protein-protein "bound" complexed cases, but in particular, of 19 "unbound" protein-protein cases, where the structures have been determined separately. In all cases, entire molecular surfaces are utilized in the docking, with no consideration of the location of the active site, or of particular residues/atoms in either the receptor or the ligand that participate in the binding. To evaluate the goodness of the strictly geometry-based shape complementarity in the docking process as compared to the main favorable and unfavorable energy components, we study systematically a potential correlation between each of these components and the root mean square deviation (RMSD) of the "unbound" protein-protein cases. Specifically, we examine the non-polar buried surface area, polar buried surface area, buried surface area relating to groups bearing unsatisfied buried charges, and the number of hydrogen bonds in all docked protein-protein interfaces. For these cases, where the two proteins have been crystallized separately, and where entire molecular surfaces are considered without a predefinition of the binding site, no correlation is observed. None of these parameters appears to consistently improve on shape complementarity in the docking of unbound molecules. These findings argue that simplicity in the docking process, utilizing geometrical shape criteria may capture many of the essential features in protein-protein docking. In particular, they further reinforce the long held notion of the importance of molecular surface shape complementarity in the binding, and hence in docking. This is particularly interesting in light of the fact that the structures of the docked pairs have been determined separately, allowing side chains on the surface of the proteins to move relatively freely. This study has been enabled by our efficient, computer vision-based docking algorithms. The fast CPU matching times, on the order of minutes on a PC, allow such large-scale docking experiments of large molecules, which may not be feasible by other techniques. Proteins 1999;36:307-317.     

A de novo designed protein protein interface

As an approach to both explore the physical/chemical parameters that drive molecular self-assembly and to generate novel protein oligomers, we have developed a procedure to generate protein dimers from monomeric proteins using computational protein docking and amino acid sequence design. A fast Fourier transform-based docking algorithm was used to generate a model for a dimeric version of the 56-amino-acid beta1 domain of streptococcal protein G. Computational amino acid sequence design of 24 residues at the dimer interface resulted in a heterodimer comprised of 12-fold and eightfold variants of the wild-type protein. The designed proteins were expressed, purified, and characterized using analytical ultracentrifugation and heteronuclear NMR techniques. Although the measured dissociation constant was modest ( approximately 300 microM), 2D-[(1)H,(15)N]-HSQC NMR spectra of one of the designed proteins in the absence and presence of its binding partner showed clear evidence of specific dimer formation.     

Automatic identification and representation of protein binding sites for molecular docking.

Molecular docking is a popular way to screen for novel drug compounds. The method involves aligning small molecules to a protein structure and estimating their binding affinity. To do this rapidly for tens of thousands of molecules requires an effective representation of the binding region of the target protein. This paper presents an algorithm for representing a protein's binding site in a way that is specifically suited to molecular docking applications. Initially the protein's surface is coated with a collection of molecular fragments that could potentially interact with the protein. Each fragment, or probe, serves as a potential alignment point for atoms in a ligand, and is scored to represent that probe's affinity for the protein. Probes are then clustered by accumulating their affinities, where high affinity clusters are identified as being the "stickiest" portions of the protein surface. The stickiest cluster is used as a computational binding "pocket" for docking. This method of site identification was tested on a number of ligand-protein complexes; in each case the pocket constructed by the algorithm coincided with the known ligand binding site. Successful docking experiments demonstrated the effectiveness of the probe representation.     

Flexible protein-flexible ligand docking with disrupted velocity simulated annealing

By docking flexible balanol to a rigid model of protein kinase A (PKA), we found that a new simulated annealing protocol termed disrupted velocity simulated annealing (DIVE-SA) outperformed the replica-exchange method and the traditional simulated annealing method in identifying the correct docking pose. In this protocol, the atomic velocities were reassigned periodically to encourage the system to sample a large conformational space. We also found that scaling potential energy surface to reduce structural transition barriers could further facilitate docking. The DIVE-SA method was then evaluated on its ability to perform flexible ligand-flexible protein docking of three ligands (balanol, a balanol analog, and ATP) to PKA. To reduce computational time and to avoid possible unphysical structural changes resulting from the use of nonoptimal force fields, a soft restrain was applied to keep the root-mean-square-deviation (RMSD) between instantaneous protein structures and a chosen reference structure small. Because the restrain was applied to the overall RMSD rather than to individual atoms, a protein could still experience relatively large conformational changes during docking. To examine the impact of applying such a restrain on docking, we constructed two semi-flexible protein models by choosing two different crystal structures as reference. Both the balanol analog and ATP were able to dock to either one of these semi-flexible protein models. On the other hand, balanol could only dock well to one of them. Further analysis indicated that the restrain on the glycine-rich loop was too strong, preventing it to adjust its structure to accommodate balanol in the binding pocket of PKA. Removing the restrain on the glycine-rich loop resulted in much better docking poses. This finding demonstrates the important role that the flexibility of the glycine-rich loop play in accepting different ligands and should profitably not be restrained in molecular docking so that more diverse ligands can be studied.     

Density functional theory,restricted Hartree–Fock simulations and vibrational spectroscopic studies of nicorandil

Fourier transform infrared and Raman spectra of nicorandil have been recorded. The structure, conformational stability, geometry optimisation and vibrational frequencies have been investigated. Complete vibrational assignments were made for the stable conformer of the molecule using restricted Hartree–Fock (RHF) and density functional theory (DFT) calculations (B3LYP) with the 6-31G(d,p) basis set. Comparison of the observed fundamental vibrational frequencies of the molecule and calculated results by RHF and DFT methods indicates that B3LYP is superior for molecular vibrational problems. The thermodynamic functions of the title molecule were also performed using the RHF and DFT methods. Natural bond order analysis of the title molecule was also carried out. Comparison of the simulated spectra with the experimental spectra provides important information about the ability of the computational method to describe the vibration modes.     

Ensemble docking of multiple protein structures: considering protein structural variations in molecular docking

One approach to incorporate protein flexibility in molecular docking is the use of an ensemble consisting of multiple protein structures. Sequentially docking each ligand into a large number of protein structures is computationally too expensive to allow large-scale database screening. It is challenging to achieve a good balance between docking accuracy and computational efficiency. In this work, we have developed a fast, novel docking algorithm utilizing multiple protein structures, referred to as ensemble docking, to account for protein structural variations. The algorithm can simultaneously dock a ligand into an ensemble of protein structures and automatically select an optimal protein structure that best fits the ligand by optimizing both ligand coordinates and the conformational variable m, where m represents the m-th structure in the protein ensemble. The docking algorithm was validated on 10 protein ensembles containing 105 crystal structures and 87 ligands in terms of binding mode and energy score predictions. A success rate of 93% was obtained with the criterion of root-mean-square deviation <2.5 A if the top five orientations for each ligand were considered, comparable to that of sequential docking in which scores for individual docking are merged into one list by re-ranking, and significantly better than that of single rigid-receptor docking (75% on average). Similar trends were also observed in binding score predictions and enrichment tests of virtual database screening. The ensemble docking algorithm is computationally efficient, with a computational time comparable to that for docking a ligand into a single protein structure. In contrast, the computational time for the sequential docking method increases linearly with the number of protein structures in the ensemble. The algorithm was further evaluated using a more realistic ensemble in which the corresponding bound protein structures of inhibitors were excluded. The results show that ensemble docking successfully predicts the binding modes of the inhibitors, and discriminates the inhibitors from a set of noninhibitors with similar chemical properties. Although multiple experimental structures were used in the present work, our algorithm can be easily applied to multiple protein conformations generated by computational methods, and helps improve the efficiency of other existing multiple protein structure(MPS)-based methods to accommodate protein flexibility.     

Computational insights into the O<Subscript>2</Subscript>-evolving complex of photosystem II

Mechanistic investigations of the water-splitting reaction of the oxygen-evolving complex (OEC) of photosystem II (PSII) are fundamentally informed by structural studies. Many physical techniques have provided important insights into the OEC structure and function, including X-ray diffraction (XRD) and extended X-ray absorption fine structure (EXAFS) spectroscopy as well as mass spectrometry (MS), electron paramagnetic resonance (EPR) spectroscopy, and Fourier transform infrared spectroscopy applied in conjunction with mutagenesis studies. However, experimental studies have yet to yield consensus as to the exact configuration of the catalytic metal cluster and its ligation scheme. Computational modeling studies, including density functional (DFT) theory combined with quantum mechanics/molecular mechanics (QM/MM) hybrid methods for explicitly including the influence of the surrounding protein, have proposed chemically satisfactory models of the fully ligated OEC within PSII that are maximally consistent with experimental results. The inorganic core of these models is similar to the crystallographic model upon which they were based, but comprises important modifications due to structural refinement, hydration, and proteinaceous ligation which improve agreement with a wide range of experimental data. The computational models are useful for rationalizing spectroscopic and crystallographic results and for building a complete structure-based mechanism of water-splitting in PSII as described by the intermediate oxidation states of the OEC. This review summarizes these recent advances in QM/MM modeling of PSII within the context of recent experimental studies.     

On the characterization and selection of diverse conformational ensembles with applications to flexible docking

To address challenging flexible docking problems, a number of docking algorithms pregenerate large collections of candidate conformers. To remove the redundancy from such ensembles, a central problem in this context is to report a selection of conformers maximizing some geometric diversity criterion. We make three contributions to this problem. First, we resort to geometric optimization so as to report selections maximizing the molecular volume or molecular surface area (MSA) of the selection. Greedy strategies are developed, together with approximation bounds. Second, to assess the efficacy of our algorithms, we investigate two conformer ensembles corresponding to a flexible loop of four protein complexes. By focusing on the MSA of the selection, we show that our strategy matches the MSA of standard selection methods, but resorting to a number of conformers between one and two orders of magnitude smaller. This observation is qualitatively explained using the Betti numbers of the union of balls of the selection. Finally, we replace the conformer selection problem in the context of multiple-copy flexible docking. On the aforementioned systems, we show that using the loops selected by our strategy can improve the result of the docking process.