Computational simulation of protein-protein docking can expedite the process of molecular modeling and drug discovery. This paper reports on our new F2 Dock protocol which improves the state of the art in initial stage rigid body exhaustive docking search, scoring and ranking by introducing improvements in the shape-complementarity and electrostatics affinity functions, a new knowledge-based interface propensity term with FFT formulation, a set of novel knowledge-based filters and finally a solvation energy (GBSA) based reranking technique. Our algorithms are based on highly efficient data structures including the dynamic packing grids and octrees which significantly speed up the computations and also provide guaranteed bounds on approximation error.
The improved affinity functions show superior performance compared to their traditional counterparts in finding correct docking poses at higher ranks. We found that the new filters and the GBSA based reranking individually and in combination significantly improve the accuracy of docking predictions with only minor increase in computation time. We compared F2 Dock 2.0 with ZDock 3.0.2 and found improvements over it, specifically among 176 complexes in ZLab Benchmark 4.0, F2 Dock 2.0 finds a near-native solution as the top prediction for 22 complexes; where ZDock 3.0.2 does so for 13 complexes. F2 Dock 2.0 finds a near-native solution within the top 1000 predictions for 106 complexes as opposed to 104 complexes for ZDock 3.0.2. However, there are 17 and 15 complexes where F2 Dock 2.0 finds a solution but ZDock 3.0.2 does not and vice versa; which indicates that the two docking protocols can also complement each other.
The docking protocol has been implemented as a server with a graphical client (TexMol) which allows the user to manage multiple docking jobs, and visualize the docked poses and interfaces. Both the server and client are available for download. Server: http://www.cs.utexas.edu/~bajaj/cvc/software/f2dock.shtml. Client: http://www.cs.utexas.edu/~bajaj/cvc/software/f2dockclient.shtml.
Citation: Chowdhury R, Rasheed M, Keidel D, Moussalem M, Olson A, Sanner M, et al. (2013) Protein-Protein Docking with F2Dock 2.0 and GB-Rerank. PLoS ONE 8(3): e51307. doi:10.1371/journal.pone.0051307
Editor: Yang Zhang, University of Michigan, United States of America
Received: July 21, 2012; Accepted: October 31, 2012; Published: March 6, 2013
Copyright: © 2013 Chowdhury et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The research of C.B., R.C., M.M., and M.R. of University of Texas, was supported in part by National Science Foundation (NSF) grant CNS-0540033, and grants from the National Institutes of Health (NIH) R01-GM074258, R01-GM073087, R01-EB004873. The research of M.M. was additionally supported by an NSF Graduate Research Fellowship. The research of M.S. and A.O. of TSRI was supported in part by a subcontract on NIH grant R01-GM073087. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
The study of protein-protein interactions plays an important role in understanding the processes of life . Though advancements in X-ray crystallography and other imaging techniques have led to the extraction of near-atomic resolution information for numerous individual proteins; the creation, crystallization and imaging of macromolecular complexes, as extensively required for drug design, still remains a difficult task. Among the atomic structures of proteins deposited in the Protein Data Bank , only a very small percentage are complexes. Hence, the need for fast and robust computational approaches to reliably predict the structures of protein-protein complexes is growing. An important step towards understanding protein-protein interactions is protein-protein docking which can be defined as computationally finding the relative transformation and conformation of two proteins that results in a stable (energetically favorable) complex if one exists.
Given two rigid proteins and some characteristic (e.g., electron density) function(s) of the molecules, one can construct an appropriate representation of them and also define a correlation function based on cumulative overlap of the characteristic functions. Then it is possible to conduct a combinatorial search in a 6D parameter space of all possible relative translations and orientations of the two proteins to find the optimal. Hence in computational perspective, docking is a search over the space of possible orientation of two proteins to find the (set of) optimum(s) of a scoring function designed to mimic physico-chemical interaction of proteins.
The combinatorics of the search can be reduced by using coarse grids and rotational angles , and by using a-priori knowledge of suitable binding sites . For docking without prior knowledge about possible binding sites, exhaustive sampling is required to improve the probability of finding the global minimum energy configuration. In such cases, Fast Fourier Transforms has been used to speed up the cumulative scoring function computations –. Spherical Fourier correlation based approaches were presented in multiple studies –. However, if binding sites are known, or inferred based on some initial stage docking, then a finer resolution search involving local flexibility can be applied to improve the accuracy of the fit –.
Accuracy of docking predictions is dependent on the scoring model’s ability to distinguish between native and non-native poses. Docking based on structural (shape) complementarity alone has shown to be adequate for a range of proteins , , . To represent shape complementarity, a grid based double skin layer approach became the base of many variations and software, e.g., DOT , ZDOCK , PIPER , MolFit ,  and RDOCK . However, energy and bioinformatics based scoring terms have been shown to improve the accuracy of predictions and a combination of multiple scoring terms have become the norm in current docking software. For example, DOT 2.0  is based on van der Waals energy and Poisson-Boltzmann electrostatics, ZDock 3.0.2  uses pairwise shape complementarity, electrostatics, and pairwise potentials known as Interface Atomic Contact Energies (IFACE), PIPER  is based on shape complementarity and electrostatics using a Generalized Born (GB) type formulation, and uses a new class of structure-based potentials referred to as DARS (Decoys As the Reference State) where the potentials are derived from a large set of docking conformations as decoys. FRODOCK  is a recent spherical harmonics based docking tool that uses van der Waals, electrostatics and desolvation potential terms in its correlation function. Some docking or reranking techniques solely use coarse-grained potentials trained on large benchmark of decoys , . We leave the reader to consult the reviews – for further information.
In  we described a non-equispaced Fast Fourier Transform (NFFT) based rigid-body protein-protein docking algorithm for efficiently performing the initial docking search (based on shape and electrostatics complementarity). Compared to traditional grid based Fourier docking algorithms, the algorithm was shown to have lower computational complexity and memory requirement. The algorithm was extended in  to F2 Dock ( = Fast Fourier), which included an adaptive search phase (both translational and rotational) for faster execution.
In this paper we describe a new version (F2 Dock 2.0 ) which includes improved shape-complementarity and electrostatics functions as well as a new on-the-fly affinity function based on interface propensity and hydrophobicity. The current version uses uniform FFT, but exploits the sparsity of FFT grids for faster execution and restricts its search within a narrow band around the larger molecule. A clustering phase penalizes docking poses that are structurally similar to poses with better scores and a set of efficient on-the-fly filters penalize potential false positives based on Lennard-Jones potential, steric clashes, interface propensity, interface area, residue-residue contact preferences, antibody active sites, and glycine richness at the interface for enzymes. The filters are implemented using fast multipole type recursive spatial decomposition techniques , . A solvation energy based reranking program GB-rerank ,  has also been implemented using an approximation scheme which can be tuned for speed-accuracy trade-off. Both F2 Dock 2.0 and GB-rerank have been implemented as multithreaded programs for faster execution on multicore machines. Our molecular visualization software TeMol serves as a front-end to F2 Dock 2.0 in a client-server mode of execution . F2 Dock has been calibrated based on an extensive experimental study of the rigid-body complexes from Zlab benchmark 2.0  and tested on Zlab benchmark 4.0  (which includes the complexes from benchmark 2.0).
The paper is organized as follows. In the next section, we describe the latest version of F2 Dock 2.0. Results are presented and discussed in the next section, followed by conclusions and plans for future research.
Let and be two proteins with and atoms respectively. Without loss of generality, we assume that , i.e., is the larger of the two proteins. We refer to and as “receptor” and “ligand”, respectively.
Figure 1 gives a high level overview of the algorithm. The rest of this section details the various aspects of the algorithm.
F2 Dock 2.0 performs exhaustive 6D search in discretized rotational and translational space where it computes a score for each sampled orientation of the ligand with respect to a stationary receptor. The scoring function is a weighted combination of shape complementarity, electrostatics and interface propensity based affinity terms (refer to Section 2.3 for details). The top few orientations (poses) of the ligand are kept in a priority queue. Then top several thousand poses from the queue are clustered based on the distance between the geometric centers of different poses of . All but the best scoring pose of a cluster is penalized by reducing the score. The resulting reordered list is then passed through several soft filters in order to further penalize potential false positives. Finally, as a separate post-processing step, the ranked docking poses are re-scored and reranked based on the change in solvation energy caused by each pose.
2.1 Overall Strategy
First, F2 Dock 2.0 performs exhaustive search in 6D space of relative configuration of with respect to . We use a discrete and uniform sampling of 3D rotational space and then use FFT to score a discrete 3D translational space. Given rotational samples and translational grid, F2 Dock 2.0 computes scores. However, only a constant multiple of scores and their corresponding poses are retained for the next step. Let us denote this set as . A particular pose is expressed as a tuple where is the translation, is the rotation and is the corresponding score.
We apply a very simple clustering scheme based on proximity of the poses in to reshuffle the order such that the top few poses are dissimilar to each other. Though this step does not affect the overall ratio of true and false positives, it increases the chance of finding at least one near-native solution at the top of the order. It is important because the next stage of filtering is only performed on the top few (2000 by default) poses. Let this reduced list be called .
The filters are designed based on knowledge-based scoring potentials, which are described in Section 2.5, to update the scores of the poses of , reorder them and output them as final predictions from F2 Dock 2.0. Some filters are defined only for specific types of proteins like Antibodies or Enzymes.
The results from F2 Dock 2.0, or a subset of it, can optionally be reranked using a solvation energy (generalized Boltzman model) based reranker called GB-rerank which generally improves ranks of near native solutions.
2.2 Exhaustive Search Using FFT
2.2.1 Rotational sampling.
The rotation space is sampled using uniformly distributed Euler angles as in , , . For each sampling interval the sample set is equivalent to a set of points uniformly distributed on a projective sphere such that the angular distance  between any two points in the set is at most . This approach provides a much better distribution of samples than sampling each angle () separately and requires fewer samples for the same .
2.2.2 Translational sampling and scoring.
FFT-based scoring of the translational grid (see, e.g., , ) involves two forward (one each for molecules and ) and one inverse FFT computations. Since the forward FFT of the stationary molecule can be precomputed, in practice, only one forward (involving molecule ) and one inverse FFT must be computed for each rotation. Current version of F2 Dock 2.0 uses uniform FFT but exploits the sparsity of the input and the output grids for faster computation as follows.
- - Forward FFT (sparsity of the input grid): The input grids are large enough so that they can at least contain the following two spheres side-by-side: the smallest sphere enclosing molecule , and another sphere having the same radius as the largest distance from the geometric center of molecule to an atom of . Hence, when discretized to such a grid molecule will occupy only a fraction of the grid points around the grid center. Thus many grid planes will remain completely empty (i.e., have zero values only). 3D FFT is sped up by ignoring recursive calls that compute 2D FFT’s of such empty planes.
- - Inverse FFT (sparsity of the output grid): In the output translational grid we need to score only the grid points that are within a band around the stationary molecule such that if the geometric center of molecule lies outside the band the two molecules can never touch. Note that the number of gridpoints in the band is as opposed to . This band can be approximated during the initial precomputation phase by running a sphere of radius (defined in previous paragraph) over the surface of which can be done using a single call to FFT. This sparseness of the output grid is exploited to speed up inverse FFT.
2.2.3 Cost of FFT-based affinity function computations.
For any rotation the FFT-based scoring takes time, where is the size of the FFT grid. Hence, for rotations the total running time is . We use the publicly available FFTW package  for our FFT-based scoring.
2.3 FFT-based Affinity Functions
F2 Dock 2.0 uses three affinity functions during exhaustive search over the rotational and translational space. The score is defined as a weighted sum of the shape complementarity, electrostatics and interface propensity scores. Using a new approach to skin-core definition and weighting, the shape complementarity term has been greatly improved from the original version (F2 Dock ), and the interface propensity is a novel addition defined using statistical residue potentials.
2.3.1 Shape complementarity (SC).
The original version of F2 Dock  used the traditional double skin layer approach for shape complementarity . Two skin regions are defined (Figure 2): a grown skin region around , and the surface skin of , which consists of the surface atoms of . The atoms of and the inner atoms of form core regions.
(Left) Traditional double skin-layer approach for shape complementarity, (Right) Improved approach with curvature-based weighting of skin atoms and depth dependent weighting of core atoms of molecule , and depth dependent weighting of the atoms of .
A good docking pose of and will have large skin-skin overlap and small core-core overlap, and in order to identify such poses constant positive imaginary weights are assigned to the core atoms and constant positive real weights to skin atoms/pseudo-atoms. An integral of the superposition of the molecules has two real contributions: the core-core overlaps contribute negatively and the skin-skin overlaps contribute positively. Hence the real part can be used to rank docking poses based on shape complementarity. The magnitude of the imaginary part of the integral due to skin-core clashes (caused by pseudo-atom vs atom overlaps) is not desirable and assigned a smaller negative weight in the accumulated score.
Improved double skin approach.
The current version (F2 Dock 2.0 ) uses an improved double skin layer approach which differs from the traditional approach in four ways. First, the receptor skin layer does not touch the receptor van der Waals surface and the radius of skin atoms are different. This is based on our observation of the gap between the VDW surfaces of the receptors and ligands in Zlab benchmark 2.0 . Second, the weights assigned to receptor skin atoms are computed based on the curvature of the skin around that atom. Such weighting encourages convex-concave and concave-convex interfaces as opposed to large flat interfaces. Third, the core atoms of molecule are assigned weights using an increasing function of depth (distance of the atom center from the surface of ). Such weightings discourage deeper core-core overlaps more strongly. And fourth, since in the traditional approach the ligand skin is defined using its surface atoms, the skin thickness varies and can be too thin in some areas. Therefore, we use a double layer of ligand atoms as its skin. Refer to supplemental materials (Supplement S1) for in depth discussion on the skin-core definition as well as FFT based correlations.
2.3.2 Electrostatics (E).
In the previous version (F2 Dock ) , we defined the electrostatics affinity function similar to the simplified model for electrostatics described by Gabb et. al. , which allows efficient FFT-based computation. The first protein’s electric potential is computed and matched against the charges in the other. In this version (F2 Dock 2.0 ), we replace point charges with a Gaussian to reduce discretization errors on the grid (See Supplement S1 for details).
2.3.3 Interface propensity (IP) and hydrophobicity (HP).
F2 Dock 2.0 scores the interfaces between molecules and using the per-residue interface propensity values computed in  which are based on relative frequencies of residues in the interfaces of a set of 63 protein-protein complexes . Let denote the natural logarithm of the interface propensity value of a residue . The values for the 20 amino acid residues lie between −0.38 (ASP) and 0.83 (TRP). A residue with higher value is likely to occur more frequently in a protein-protein interface than one with a lower value.
Let and denote the set of atoms in the interface of in this docking pose that have positive and negative values, respectively. Also let , for . We assign an interface propensity score to the pose:where . We also penalize very small interfaces by setting to a negative value when is below a user-defined threshold.
See Suppplement S1 for details on the FFT based formulation and parameter selections.
2.4 Proximity Clustering
We consider all docking poses in sorted by score, and penalize a pose if there are similar poses with higher score. Suppose is a docking pose with score that is currently being considered. For a given , let be the number of docking poses with score at least and with geometric center within distance from that of . Then we penalize as follows.
- If , then the score of the pose is reduced by ,
- otherwise, if , then the score of the pose is reduced by ,
- otherwise, if , then the score of the pose is reduced by .
The objective of this proximity based penalty is to increase diversity at the top of the order which increases the possibility of getting at least one near native solution at the top of the order. But it also penalizes all true positives (except one). As a result, F2 Dock 2.0 tends to get a near native solution at high ranks for many complexes, but the total number of near native solutions for any particular complex is not high.
2.4.1 Cost of clustering.
We use our dynamic packing grid data structure  to speed-up this computation, and the overall time required for this step is (with high probability), where is the number of poses originally in .
To penalize potential false positives and thus improve the ranks of correct solutions, F2 Dock 2.0 uses several filters based on the Lennard-Jones potential, the number of steric clashes, interface area, interface propensity, residue-residue contact preferences, antibody active sites, and the frequency of glycine residues at the interface for enzymes.
Only the interface regions of the two molecules at a given pose contribute to the terms used in the filters. We have developed an octree-based hierarchical spatial decomposition technique  and the Dynamic Packing Grids data structure  for efficiently locating the interface regions and for computing local interactions. Since the overall algorithm for computing each term is similar and only varies in the exact type of local interactions we compute, we present the algorithm only once (in our discussion of the interface propensity filter).
2.5.1 Interface propensity filter.
This filter computes the interface propensity of the interfaces of the molecules at a given pose and penalizes or rewards the pose based on empirically determined thresholds.
We sample and weigh quadrature/integration points from the surface of each molecule as described in , . The sampling can be considered a decomposition of the surface into small patches, where each quadrature point is representative of a patch . The weight of a quadrature point is the same as the area of the corresponding patch. Each quadrature point is also labeled with the average of the interface propensity values of the atoms near the point. Let, denote the interface propensity label of a quadrature point. The interface propensity contribution of a quadrature point is defined as if is on the interface, and otherwise.
We approximate the interface propensity as , where, (defined in Section 2.3.3). Here and are the sum of the negative interface propensity contributions of the quadrature points of two molecules; and and are the sum of the positive contributions. We reward docking poses with large values, and penalize a pose if its is below a threshold.
The crucial step in the approximation is identifying quadrature points that are on the interface. We store the quadrature points into a DPG data structure, and we also store them in an octree. The octree is a hierarchical and adaptive subdivision of space such that a node of the tree represents a regular cube in 3-space, A node is split if it contains more than a user-defined number of quadrature points in it. Given a specific pose, we trace the two octrees starting from the roots to identify the leaves that are close to each other. Then for each pair of neighboring octree-cells, we use DPG to identify quadrature points in one leaf which have a neighbor in the other leaf. The overall cost of the algorithm is with high probability, where and are the number of atoms in and . However, in practice it runs even faster and approaches , where is the number of quadrature patches on the interface.
2.5.2 Residue-Residue contact filter.
Contact preferences derived from a non-redundant set of 621 protein-protein interfaces of known high resolution structures  are used to penalize potential false positives. Two residues are considered to be in contact if the distance between their atoms ( for Gly) is less than 6 Å. In , log-normalized contact preferences for each pair of amino acid types are reported. Positive values of indicate that residues and prefer to form contacts, negative values indicate the opposite.
Given a docking pose , we identify all residue-residue contacts at the interface of the two molecules using a fast algorithm similar to the one used in Section 2.5.1, and compute the sum of all positive and negative values denoted by and , respectively. Then we penalize the pose if the ratio of and is outside a user-specified range.
2.5.3 Lennard-Jones filter.
We approximate the Lennard-Jones () potential between molecules and as follows: where is the distance between atoms and , and constants and depend on the atom types. The well depths and equivalence contact distances of homogeneous pairs are taken from the Amber force field , . Poses with positive potential are penalized. However, we allow soft clashes in the cases of unbound-(un)bound docking by reducing the values by a constant factor which effectively reduces the inter-atomic clash distances ( values).
2.5.4 Clash filter.
Two atoms and with van der Waals radii and , respectively, are considered to be clashing provided the distance between their centers is smaller than a threshold. F2 Dock 2.0 counts the total number of clashes between molecules and and penalizes if , where is a user-defined constant.
2.5.5 Interface area filter.
This filter penalizes a docking pose if the interface area is outside the range of areas derived empirically from known native interfaces. We define the interface area as the sum of the weights of the quadrature points on the interface, where the weights and the interface is defined the same way as in the interface propensity filter.
2.5.6 Glycine filter.
Enzyme active sites are rich in Glycines, particularly G-X-Y and Y-X-G oligopeptides, where and are polar and non-polar residues, respectively, and is glycine . The and residues are typically small in size and low in polarity, and the frequency of those two types of oligopeptides is significantly higher in enzyme active regions than in other parts of the enzyme molecule. Therefore, enzyme surface oligopeptides with these properties are marked and for a given docking pose, the number of these motifs occurring at the interface are counted. If this count is below a user-specified threshold , the pose is penalized. Conversely, poses with higher G-X-Y/Y-X-G frequency at the interface are rewarded by adding this (weighted) count to the total score.
2.5.7 Antibody-Antigen contact filter.
As reported in  (available at http://www.bioinf.org.uk/abs/allContacts.html) based on a set of 26 known antibody-antigen complexes, each of the following three regions in an antibody will make at least one antigen contact (burial by at least 1 Å change in solvent accessibility): CDR-L1 or CDR-H1, CDR-L3 and CDR-H3.
Given a potential antibody-antigen docking pose, F2 Dock 2.0 computes , and , the number of antigen atoms that are in the close neighborhood of any atom in the antibody regions CDR-L1/CDR-H1, CDR-L3 and CDR-H3, respectively. The CDR (Complementarity Determining Region) loops are identified using the method described in . F2 Dock 2.0 penalizes poses if the computed values are outside the ranges observed in the native antibody-antigen interfaces in our training set.
2.5.8 Cost of filtering.
Using our algorithm described in  based on octrees  and our Dynamic Packing Grid (DPG) data structure , the scores for each filter can be evaluated in w.h.p. (for an input of size , an event E occurs w.h.p. (with high probability) if, for any and independent of , .) time and space. Assuming that each filter is applied on at most configurations, the total time taken by all filters is (w.h.p.). Assuming , where is the number of samples in the rotations space, the running time reduces to (w.h.p.).
2.6 Solvation Energy Based Reranking with GB-rerank
GB-rerank approximates the change in solvation energy of a complex and reranks the list of top docking poses produced by F2 Dock 2.0 based on the resulting values. In order to approximate , GB-rerank precomputes the values for molecules and , and then computes for each docking pose.
The solvation energy consists of the energy to form cavity in the solvent (), the solute-solvent van der Waals interaction energy (), and the electrostatic potential energy change due to the solvation (also known as the polarization energy, ) –.
The first two terms are often modeled as and , where and are the position and center of atom , is the solvent pressure, is the molecular volume, is the solvent accessible surface area of atom and is its solvation parameter, is the bulk density, and is the van der Waals dispersive component of the interaction between atom and the solvent , . The last term, , can be approximated using the Generalized Born theory , whereby:(1)where are the atomic charges and is the distance between atoms and , is the effective Born radius of atom and .
The algorithms for rapidly approximating these terms have been presented in .
2.7 Dataset Preparation
F2Dock takes two PDB files as inputs. First the PDB files are processed by PDB2PQR  where missing atoms such as Hydrogens are added, the protein is optimized for hydrogen bonding, and charge and radius parameters are assigned using the AMBER force-field available in PDB2PQR. If the given PDB has missing residues or too many residues with missing backbone atoms, then our curation process fails and F2Dock cannot be used without manually curating the PDB or using other curation software.
Then pseudo-atoms are added above the surface of the receptor (i.e., stationary molecule), and surface atoms of the ligand (i.e., moving molecule) are detected. These atoms are marked as skin atoms, and the rest as core atoms.
2.8 Parameter Selection Based on Complex Type
F2 Dock 2.0 has several free parameters in its pipeline. We can broadly classify the parameters into several groups. For parameters like the charge and radii of atoms, or the hydrophobicity and interface propensity of residues etc., we either use well-established parameters (for example, from the AMBER  force field) or derive from previously published results (for example, interface propensity values from ). Some parameters are internal to a scoring function for example the distance dependent dielectric for electrostatics, or the thickness of the skin used in shape complementarity. These parameters are trained using manual parameter sweeps based on a small number (4–5 per complex type) of complexes. However, we produced multiple configurations for each complex and chose the set of parameters which maximizes the corresponding individual scoring term for the near native poses. A similar strategy was used for selecting the thresholds used to penalize poses during filtering. Finally, there are the parameters that govern the weights assigned to different scoring terms when they are combined as well as the weights (or percentages) by which poses are penalized. These parameters are the most difficult to train as the scoring terms are not independent and the relative influence of a term might vary for different complexes. These parameters were trained based on the 60 complexes from Zlab’s protein-protein benchmark 2.0  as follows.
The complexes in the benchmark are categorized into four main types: Antibody-Antigen (A) and Antibody-bound Antigen (AB), Enzyme-Inhibitor/Enzyme-Substrate (E), and other (O) types. We identify that the classification is not only functional, but it also has significant effect on scoring function design since different scoring terms bear different level of significance for different categories of complexes. For example, it is known that binding interfaces of Enzymes are rich in Glycines, which lead us to design a filter based on Glycine richness and it is applied only for Enzyme type of complexes. For each class of complexes (9 Antibody-Antigen, 9 Antibody-bound Antigen, 21 Enzyme-Inhibitor/Enzyme-Substrate and 21 Others), we train the weight parameters separately. The objective for the training is to improve the ranks of near-native solutions for as many complexes as possible. We performed parameter sweeps for each of the weights that combines the FFT based scores based on the above objective for each of the categories. Then we examined the effect of applying each of the filter, one at a time, and controlled its penalty to improve the results.
We do realize that our manual scheme has its drawbacks, specially since it does not sufficiently cover the entire space of possible values for the parameters. We are actively trying to use machine learning schemes to train the parameters in a more robust way. However, so far our attempt of using quadratic programming and random forest learning based on thousands of negative and positive examples based on this benchmark have failed to produce a set of parameters which outperform the manually calibrated set of parameters.
Default values of all the parameters for different types of complexes can be found in the user manual for F2 Dock 2.0 downloadable from our website (link given in the abstract).
2.8.1 Automated detection of complex types.
Since F2 Dock 2.0’s parameters are optimized separately for antibody-antigens and enzyme-inhibitors/enzyme-substrates, and a general set of parameters are used for all other types of complexes, the user only needs to specify the complex type to ensure the set of optimized parameters are applied. If the type is unknown, F2 Dock 2.0 tries to determine which set of parameters to use as follows. If F2 Dock 2.0 locates the six CDR loops (L1, L2, L3, H1, H2 and H3) in the protein sequence using the algorithm in , it identifies it as an antibody and uses the corresponding parameter set. Otherwise, if neither molecule is identified as an antibody and at least one of the molecules has at least 200 residues and at least 8% of its surface residues are Glycines then F2 Dock 2.0 uses the enzyme complex parameter set. Finally, if both tests fail, a set of parameters for the general case is used. Among the complexes in the Zlab benchmark 2.0, F2 Dock 2.0 fails to identify only one antibody (1KXQ) and three enzymes (1AY7, 1UDI and 2MTA). See Supplement S1 for details.
We present the results of our experiments to explore the contribution of the new scoring terms and filters available in F2 Dock 2.0 as well as the solvation energy based re-ranker GB-rerank on prediction accuracy. These experiments are carried out on the set of complexes in Zlab’s benchmark 2.0  which contains 60 complexes. Then we run F2 Dock 2.0 with the best set of parameters on the complexes in the Zlab benchmark 4.0 , and compare the performance with ZDock 3.0.2 . The complexes in both the benchmarks are categorized into rigid-body (easy), medium and difficult (flexible) based on the RMSD between the bound and unbound states of the proteins. They are also categorized into four main types: Antibody-Antigen (A) and Antibody-bound Antigen (AB), Enzyme-Inhibitor/Enzyme-Substrate (E), and other (O) types. As mentioned before, F2 Dock 2.0 uses different set of parameters for the different categories and we have also compared our results for each category separately.
3.9 Evaluation Criteria
F2 Dock 2.0’s search leaves the receptor stationary and searches over the orientations of the ligand. Hence, to evaluate the accuracy of a predicted pose, we compute the deviation between the predicted position of the ligand and its correct position as the root mean squared distance (RMSD) of the interface atoms. Note that correct position of the ligand for unbound test cases can be approximated by aligning the unbound components to their bound counterparts. The unbound ligands in the ZLab benchmarks are provided after alignment with bound counterparts and hence can be used as the approximate truth without further manipulations. We assume that an atom is on the interface if the distance between its center and the center of any atom on the other molecule is less than 10Å. We define as the set of all backbone atoms of the ligand which are on the interface when the ligand is placed in its native pose w.r.t the receptor (to find the native pose for an unbound case, we simply align the unbound receptor and unbound ligand to their bound counterparts). If the position of ligand atom is in the native pose and in a predicted pose , then the interface RMSD is computed as . A predicted solution is considered a hit provided its IRMSD value is at most 5Å.
In the remaining text and supplement S1, we refer to the hit with the lowest RMSD as the ‘best’ hit and the hit with the highest rank as the ‘top’ hit. In most of our results, we compare protocols based on the rank of the ‘top’ hit. Given a set of complexes , and a protocol , we define as the set of complexes such that for each complex , the top hit lies within the range . Clearly, for a given a higher is better. Hence, to compare the accuracy of two protocols and , we can simply compare and for different . In general we use a few specific values for ([1,1], , , , , , , [1,100], [1,500] and [1,1000]). We are specially interested in the first few ranges which shows off the accuracy of the scoring model, and the last range which shows off the applicability of the model over a broad range of complexes.
Two residues and are considered to be in contact if the distance between the centers of any atom and any atom is less than a threshold. The set of residue-residue contacts for the native pose of the receptor and ligand are defined as the native contacts . For a given predicted pose, we compute the set of residue-residue contacts for that pose as . The set of native contacts for that pose is hence defined as . Now, we define another metric based on native contacts as . We follow the well known CAPRI criteria that uses a combination of and to classify predictions as high, medium, acceptable and incorrect.
3.10 Analyzing the Improvements due to New Affinity Functions and Filters
3.10.1 Effectiveness of the new skin-core definition.
We have compared the new improved double skin approach to the traditional approach (used in F2 Dock ) in terms of their prediction accuracy on the rigid-body complexes of the Zlab Benchmark 4.0. In these tests only the shape complementarity term was used, and hence the results are not as accurate as the default combination of scoring and filtering terms can produce.
In Figure 3(a), we clearly notice the improvement offered by the floating skin approach over the traditional which validates our idea that a softer definition of skin is better for unbound docking. However, the traditional skin approach performs slightly better for the bound-bound (re-docking) test cases (Figure 3(b)). Figure 3(c) shows that as a result of the improved skin definition, F2 Dock 2.0’s shape complementarity function outperforms DOT and ZDock on the rigid complexes from Zlab benchmark 2.0 (bound-bound).
(a–b) Comparison of the performance of F2 Dock 2.0’s shape complementarity function with traditional skin and the new floating skin approach, in terms of the number of complexes for which the top hit is within the ranges mentioned in the X-axis. (a) On the rigid-body unbound-(un)bound complexes from Zlab Benchmark 4.0. (b) On the rigid-body bound-bound complexes from Zlab Benchmark 4.0. (c) Comparison of the shape complementarity functions of DOT, ZDock 2.1 and F2 Dock 2.0 on the rigid-body bound-bound complexes from Zlab benchmark 2.0.
3.10.2 Effects of various filters on quality of solutions.
Figure 4(top) shows how the number of test cases (rigid-body test cases from Zlab benchmark 2.0 ) with at least one hit in top 1, top 10, top 50, top 100, top 500 and top 1000 changes as various affinity functions and filters in F2 Dock 2.0 are applied. The filters are applied to the top 2000 predictions after using the FFT based affinity terms and clustering. In this experiment, we have specified the complex type (A/AB, E and O) for each test case. Clearly, each of the filters (except interface area filter) individually improves the ranks of the top solution, and the best outcome is generated when the default combination of filters are used. For example, after the FFT based scoring, we get a hit at rank 1 for 10 complexes, but after filtering it improves to 17. Since the antibody and enzyme filters do not apply to all types of complexes, we compare their effect only on the particular type of complexes. For example, Figure 4(bottom) shows the effectiveness of the enzyme filter.
(top) Improvements in the rank of the top hit (of rigid-body test cases from Zlab benchmark 4.0) as various affinity functions and filters in F2 Dock 2.0 are activated one after another. (bottom) Improvements in the rank of the top hit for the Enzyme type of complexes from Zlab benchmark 4.0.
The series of plots in Figure 5 shows a detailed breakdown of the effect of different scores/filters for each complex separately. On the X-axis, we list the complexes and the Y-axis shows the change of the rank of the top hit. In the figures, an improvement is defined as producing the top hit at a better rank. We use the results of using just shape complementarity as the base case and analyze the relative improvements as more and more terms are added.
(a) Lennard-Jones Filter (LJ), clash filter (CL) and proximity clustering (PC) are activated after shape complementarity (SC), (b) electrostatics & charge complementarity (EL) after SC+LJ+CL+PC, (c) interface propensity (IP) after SC+LJ+CL+PC+EL. (d) interface propensity filter (PF) after SC+LJ+CL+PC+EL+IP, (e) residue-residue contact filter (RC) after SC+LJ+CL+PC+EL+IP+PF, and (f) antibody contact filter (AF) or glycine filter (GF) after SC+LJ+CL+PC+EL+IP+PF+RC.
When we activate Lennard-Jones filter, clash filter and proximity clustering after shape complementarity we get hits for 4 new test cases, and the rank of the top hit improves for 15 more (see Figure 5(a)). However, we also lose hits in top 1000 for 3 test cases, and the rank of the top hit degrades for one test case. Overall, the application of these filters and clustering seem largely beneficial. The best results are obtained for enzyme-inhibitor/enzyme-substrate complexes, as for more than 50% of these complexes rank of the top hit improves.
When electrostatics is turned on we get hits in top 1000 for 9 test cases for which we did not have a single hit before, and for 14 other cases rank of the top hit improve (see Figure 5(b)). However, we lose hits 1 test case, and for 4 others rank of the top hit degrades.
The FFT-based interface propensity scoring is activated next which improves the rank of the top hit for 30 test cases (i.e., for around 50% of all cases) among which 7 cases did not have a single hit before (see Figure 5(c)). Among these 7 cases with new first hits 5 are antibody-antigen or antigen-bound antibody complexes, and none are enzyme-inhibitor or enzyme-substrate.
The interface propensity filter is turned on next. It improves the rank of the top hit for 25 complexes, and degrades for 5 (see Figure 5(d)). For 3 test cases we did not have a single hit in top 1000 before among which 2 are antibody-antigens.
The residue-residue contact filter which is activated next improves the rank of the top hit for 27 test cases, and degrades for none (see Figure 5(e)). The enzyme-inhibitor and enzyme-substrate complexes seem to have benefited the least from this filter.
Next we apply the antibody contact filer and the Glycine filter. The antibody contact filter improves the rank of the top hit for 9 antibody-antigen and antigen-bound antibody test cases, and degrades for 3, while the Glycine filter slightly improves the same for 4 enzyme-inhibitor/enzyme-substrate complexes (see Figure 5(f)).
More comparisons with respect to the RMSD of the best hit, the total number of hits, and the lowest RMSD are provided in Supplement S1.
3.10.3 Effects of post-processing with GB-rerank.
Figure 6 shows the impact of applying GB-rerank (after the initial docking phase) on the rigid-body test cases from Zlab benchmark 2.0 . GB-rerank improves the ranks of the top hit for 9 antibody-antigen and antigen-bound antibody complexes, and 10 complexes of type “other” (see Figure 6).
The plot shows the change of the rank for the first hit. A positive change indicates that the reranker improves the result. For most complexes, specially for complexes where a knowledge-based based filter (Antibody or Enzyme) could not be applied, GB-rerank improves the rank of top hit compared to the results produced by F2 Dock 2.0 (for the rigid-body test cases from Zlab benchmark 2.0 ).
The post-processor is least effective on enzyme-inhibitor/enzyme-substrate complexes since the enzyme filter has already improved the ranks quite well. On the other hand, for the ‘other’ complexes, GB-rerank produces the most significant improvements, since specific filters cannot be applied in these cases. Hence if the complex is known to be Enzyme, then GB-rerank should not be applied.
3.10.4 Performance of F2 Dock 2.0 with and without user-specified complex type.
Figure 7 compares the performance of F2 Dock 2.0 with and without user-specified complex types on Zlab’s protein-protein docking benchmark 2.0. When no complex type is specified F2 Dock 2.0 tries to identify antibody-antigen complexes by locating the CDR loop regions of the antibody. Among the 17 such complexes in our experiments 16 are correctly identified by F2 Dock 2.0. It fails to identify 1KXQ which is an antibody-antigen complex from a Camelid (camels, llamas, etc.) . Camelids produce functional antibodies that do not have light chains and CH1 domains, and so F2 Dock 2.0’s antibody detection system fails to identify such antibodies. Hence for 1KXQ the set of parameter values optimized for complexes of “other” type is applied, and the result is only slightly worse than what is obtained with the parameter set optimized for antibody-antigen complexes. F2 Dock 2.0 fails to select the correct parameter set for the following three enzyme-inhibitor/enzyme-substrate complexes among the 21 included in the experiments: 1AY7, 1UDI and 2MTA. While for 1UDI and 1AY7 F2 Dock 2.0 is still able to get a hit in the top 100 and top 500, respectively, it fails to get any hit in the top 1000 for 2MTA. For all other complexes the results remain the same except for 1WEJ for which we get slightly different results in the two set of experiments due to the non-determinism (arising from multiple concurrent threads) that exists in the proximity clustering phase.
When complex type is not specified in the input, F2 Dock 2.0’s performance does not change significantly. In most cases, it can automatically detect the complex-type and apply the correct set of parameters. Tests are based on rigid body cases from Zlab’s Protein-protein docking Benchmark 2.0.
3.11 Comparison with ZDock
In this section we compare the performance of F2 Dock 2.0 and ZDock 3.0.2 ,  on the complexes from Zlab benchmark 4.0 . We acquired the executable for ZDock 3.0.2 from their website and ran it following the steps specified in the accompanying instructions and used the PDB files downloaded from ZLab’s website without any modification. F2 Dock 2.0 used the same set of PDBs after performing the preprocessing we mentioned in Section 2.7. Note that ZDock 3.0.2 also applies their own preprocessing which is part of the mark_sur script provided with the executable. Both programs used 15° rotational sampling. F2 Dock 2.0 used user-specified complex types.
In Figure 8, we show a summary of the performances in terms of the number of complexes where each protocol found at least one hit in different ranges (see the X-axis). Note that having a higher Y-axis value for any instance shows that the corresponding protocol is successful on complexes than the other. In Figure 8(a) we compare the performances over the entire Zlab benchmark 4.0 containing 176 complexes. We find that for each of the ranges except one, F2 Dock 2.0 performs better than ZDock 3.0.2. F2 Dock 2.0 is specially impressive since it gets a hit at rank 1 for 22 of the complexes (which is 1/8th of the dataset) as opposed to 13 found by ZDock 3.0.2. Overall both ZDock 3.0.2 and F2 Dock 2.0 finds at least one solution for about the same number of complexes, 104 and 106 respectively.
(a) On all 176 complexes from Zlab Benchmark 4.0 , (b) On 25 antibody-antigen and antigen-bound antibody complexes, (c) On 52 enzyme-inhibitor and enzyme-substrate complexes, and (d) on the 99 other type of complexes.
Figures 8(b)–(d) compares F2 Dock 2.0 and ZDock 3.0.2 using the same metrics but considers each type of complex separately. For antibodies there is not much to choose between the two protocols. For other types F2 Dock 2.0 is successful for a lower number of complexes, and is comparable only at relatively high ranks. However, for Enzymes, F2 Dock 2.0 completely outperforms ZDock 3.0.2 across the board.
Based on these results, we can clearly see that F2 Dock 2.0 produces much more reliable predictions for Enzymes, but there is not much difference for antibodies and other type of complexes. But Tables 1, 2, 3, and 4 show that even for antibodies and other types F2 Dock 2.0 provides significant contributions since the two protocols are often successful for different complexes and hence compliment each other. For example, among the antibodies, F2 Dock 2.0 finds a solution for 1QFW and 1I9R for which ZDock 3.0.2 does not find any solutions, on the other hand ZDock 3.0.2 finds a solution for 1NSN where F2 Dock 2.0 fails. Similarly among the other complexes, only F2 Dock 2.0 is successful for 1J2J, 2A5T, 2A9K, 2HQS, 3BP8, 1K5D, 1R6Q, 2Z0E, 3CPH and 1ATN. Hence, it is advisable to use both of these protocols specially for other type of complexes to increase the possibility of finding a correct solution.
Next, we compare the rate of success of the two protocols. Let us assume that the total number of hits (counting multiple hits found for a complex) found within a range across all the complexes be . Now we define the rate of success as which measures how quickly a protocol finds its hits. A protocol with a higher ratio has higher true positive rate near the top of the list. If we plot this function, we expect to see a curve which rises sharply and then gradually flattens and converges to . In Figure 9, we see that F2 Dock 2.0 has consistently better success rate than ZDock 3.0.2.
On the 176 complexes from ZLab’s benchmark 4.0. Rate of success is defined as the percentage of the hits found within the top ranks, where is the corresponding value of the X-axis. Clearly F2 Dock 2.0 has a better ratio.
A closer look at Tables 1, 2, 3, and 4 shows that the RMSDs of the predictions by F2 Dock 2.0 is poorer than ZDock 3.0.2 in more occasions than it is better. This is due to our softer skin approach which rewards docking poses which have slightly larger gap between them, and our stringent clash and VDW filters which discard ligand poses which comes too close. This is beneficial for unbound complexes with larger conformational change, but prevents ligands of rigid (easy cases in the benchmark) from getting as close as they could be placed. The result clearly shows that ZDock 3.0.2 gets better RMSDs for rigid cases, and F2 Dock 2.0 is better for non-rigid cases. At this point, it should be mentioned that F2 Dock 2.0 is designed solely as a initial stage docking tool, which can quickly perform exhaustive search and return good leads at high ranks. Hence the poses it finds are generally acceptable or medium quality as defined in the criteria used in the CAPRI  challenge (tables summarizing F2 Dock 2.0’s performance using the CAPRI criteria can be found in Supplement S1). Local refinements (rigid body or flexible) can then be performed on a small number of top solutions to further improve their RMSDs and minimize the energies. There are a host of such tools available including ROSETTA , Amber , FireDock  etc.
We conclude this section with the observation that F2 Dock 2.0 shows better overall performance, with significant improvement for Enzymes. For other type of complexes the performance is comparable and sometimes complementary.
3.12 Running Times
To evaluate the average running times and the relative consumption by each scoring term/filter we performed a set of experiments run on a 3 GHz dual-core (i.e., 4 cores) AMD Opteron 2222 processor with 4 GB RAM. On average, the FFT phase took around 23 minutes or 35% of the total running time, the interface propensity filter took 20%, GB-rerank accounted for around 42%, and the remaining 3% is spent on the other filters. GB-rerank and interface propensity filter take longer to compute than other filters, since the computation is based on surface quadrature points, whose number is a constant multiple of the number of atoms. Figure 10 shows how the different components of F2 Dock 2.0 and GB-rerank contribute to the total running time of the docking and reranking process on the rigid-body test cases from Zlab benchmark 2.0 . Overall, about 30% time is taken up by the FFT based affinity functions, 30% is taken up the the filters (mostly the interface propensity filter), and around 40% by the GB-rerank.
(a) Average running time of each affinity function and filter of F2 Dock 2.0. GB-rerank consumes a major portion of the time (42%), the FFT phase takes about 30% time and the rest is taken by the filters and clustering. The labels in the figure are actual time in minutes. (b) Running times of F2 Dock 2.0 on the rigid-body test cases from Zlab benchmark 2.0  showing percentage of running time due to each affinity function and filter of F2 Dock 2.0 for each complex.
F2 Dock 2.0 leverages from the embarrassingly parallel nature of the computation using multithreaded computations on multi-core machines. Note that each of the FFT computations are independent of each other and can be run in parallel. Scores for each of filter terms for each of the poses in can also be computed in parallel. Specifically, given cores and tasks the simplest strategy is to distribute tasks to each cores. But this approach often leads to unbalanced exploitation of the cores if the tasks given to different cores take different amount of time to complete. For example, the running times of the filters are proportional to the size of the interface which varies between different poses. So our technique initially sends only one task to each core and maintains a queue of remaining tasks, and then whenever a core is done with its task, it gets another one from the queue. This scheduling ensures that every core is exploited equally and hence the overall completion time is quicker.
We have developed an enhanced version (F2 Dock 2.0 ) of our protein-protein docking program F2 Dock 2.0 with improved scoring functions, complete with dynamic clustering and filtering and generalized Born based solvation energetic reranking. The on-the-fly FFT-based scoring function is a weighted combination of shape-complementarity, Coulombic electrostatics complementarity, and interface propensity terms. The on-the-fly docking also includes filters based on Lennard-Jones potential, steric clashes, residue-residue contact statistics and an extremely fast approximation of solvation energy using a newly developed fast multipole type implementation with octree data structures. Our implementation results and numerous tests show that each of these terms and filters significantly improves the accuracy of docking predictions. Our use of highly efficient data structures including the dynamic packing grids for near constant time neighborhood search and near-far distance clustering using octrees, significantly speed up the computations for each of the ‘on-the-fly’ scoring and filtering terms. GB-rerank ’s solvation energy based post-processing suite is also optimized using these efficient data structures with the best tradeoffs of docking accuracy vs. speed. The entire software is highly parallel and can be run efficiently on multicores and clusters of multicores (e.g., many modern supercomputers). We have also developed a GUI based interface (TexMol) for easily preparing and running a docking process and interactively visualize, compare different solutions along with several relevant statistics including interface area, residue contacts, binding energy etc.
4.13 Future Work
Currently F2 Dock 2.0 addresses flexibility by simply performing a ‘soft’ complementarity with the goal of identifying a near-native solution at a higher rank. The solutions can be optimized further by side chain refinement near the binding site as well as applying small rigid-body perturbations which moves all atoms of the ligand. There are several software including AMBER , ROSETTA , SCRWL  etc. which can be used to achieve this objective. We are currently working on an algorithm for dead-end elimination with better complexities compared to SCRWL. The automatic assignment of all the docking parameters remains a very active area of research for us; we are currently pursuing a semi-supervised computational learning algorithm over the space of different protein families which will further improve the performance of F2 Dock 2.0 and GB-rerank. We are exploring ways to further improve the speed of F2 Dock 2.0 specially using GPU level parallelism. We are also actively working on extending F2 Dock 2.0 to Protein-RNA and RNA-RNA docking. Finally, a web service supporting similar features as TexMol is under construction.
The molecular visualizations shown in this paper were all produced via the latest version of our in house program TexMol .
Conceived and designed the experiments: CB RC AO MR MS. Performed the experiments: RC DK MM MR. Analyzed the data: CB RC DK MM AO MR MS. Contributed reagents/materials/analysis tools: CB RC MR MS. Wrote the paper: CB RC MR.
- 1. Fruton JS (1999) Proteins, Enzymes, Genes: The Interplay of Chemistry and Biology. Yale Uni-versity Press.
- 2. Berman HM, Westbrook J, Feng Z, Gilliland G, Bhat TN, et al. (2000) The protein data bank. Nucleic Acids Res 28: 235–42.
- 3. Katchalski-Katzir E, Shariv I, Eisenstein M, Friesem AA, Aalo C, et al. (1992) Molecular sur-face recognition: determination of geometric fit between proteins and their ligands by correlation techniques. Proceedings of the National Academy of Sciences of the United States of America 89: 2195–2199.
- 4. Gabb HA, Jackson RM, Sternberg MJE (1997) Modelling protein docking using shape complemen-tarity, electrostatics and biochemical information. Journal of Molecular Biology 272: 106–120.
- 5. Ritchie DW, Kemp GJ (2000) Protein docking using spherical polar Fourier correlations. Proteins: Structure, Function, and Genetics 39: 178–194.
- 6. Ritchie DW (1998) Parametric Protein Shape Recognition. Phd thesis, Departments of Computer Science & Molecular and Cell Biology, University of Aberdeen, King’s College, Aberdeen, UK.
- 7. Duncan BS, Olson AJ (1993) Approximation and characterization of molecular surfaces. Biopoly-mers 33: 219–229.
- 8. Max NL, Getzoff ED (1988) Spherical harmonic molecular surfaces. IEEE Computer Graphics & Applications 8: 42–50.
- 9. Kovacs JA, Chacon P, Cong Y, Metwally E, Wriggers W (2003) Fast rotational matching of rigid bodies by fast fourier transform acceleration of five degrees of freedom. Acta Crystallographica, Biological Crystallography D59: 1371–1376.
- 10. Das R, Andre I, Shen Y, Wu Y, Lemak A, et al. (2009) Simultaneous prediction of protein folding and docking at high resolution. Proceedings of the National Academy of Sciences of the United States of America 106: 18978–18983.
- 11. Dominguez C, Boelens R, Bonvin AMJJ (2003) Haddock: a protein-protein docking approach based on biochemical or biophysical information. Journal of the American Chemical Society 125: 1731–1737.
- 12. Huang W, Liu H (2012) Optimized grid-based protein-protein docking as a global search tool followed by incorporating experimentally derivable restraints. Proteins 80: 691–702.
- 13. Chen R, Weng Z (2003) A novel shape complementarity scoring function for protein-protein dock-ing. Proteins: Structure, Function, and Genetics 51: 397–408.
- 14. Gu S, Koehl P, Hass J, Amenta N (2012) Surface-histogram: A new shape descriptor for protein-protein docking. Proteins: Structure, Function, and Bioinformatics 80: 221–238.
- 15. Mandell JG, Roberts VA, Pique ME, Kotlovyi V, Mitchell JC, et al. (2001) Protein docking using continuum electrostatics and geometric fit. Protein Engineering 14: 105–113.
- 16. Chen R, Weng Z (2002) Docking unbound proteins using shape complementarity, desolvation, and electrostatics. Proteins: Structure, Function, and Genetics 47: 281–294.
- 17. Kozakov D, Brenke R, Comeau S, Vajda S (2006) PIPER: An FFT-based protein docking program with pairwise potentials. Proteins: Structure, Function, and Bioinformatics 65: 392–406.
- 18. Ben-Zeev E, Eisenstein M (2003) Weighted geometric docking: incorporating external information in the rotation-translation scan. Proteins: Structure, Function, and Bioinformatics 52: 24–27.
- 19. Ben-Zeev E, Berchanski A, Heifetz A, Shapira B, Eisenstein M (2003) Prediction of the unknown: Inspiring experience with the CAPRI experiment. Proteins: Structure, Function, and Bioinfor- matics 52: 41–46.
- 20. Li L, Chen R, Weng Z (2003) Rdock: Refinement of rigid-body protein docking predictions. Pro-teins: Structure, Function, and Genetics 53: 693–707.
- 21. Mintseris J, Pierce B, Wiehe K, Anderson R, Chen R, et al. (2007) Integrating statistical pair potentials into protein complex prediction. Proteins: Structure, Function, and Bioinformatics 69: 511–520.
- 22. Garzon J, Lopez-Blanco J, Pons C, Kovacs J, Abagyan R, et al. (2009) FRODOCK: a new approach for fast rotational protein-protein docking. Bioinformatics 25: 2544–2551.
- 23. Ravikant DVS, Elber R (2010) Pie - efficient filters and coarse grained potentials for unbound protein-protein docking. Proteins 78: 400–419.
- 24. Liu S, Vakser IA (2011) DECK: Distance and environment-dependent, coarse-grained, knowledge- based potentials for protein-protein docking. BMC Bioinformatics 12.
- 25. Ritchie D (2008) Recent progress and future directions in protein-protein docking. Current Protein and Peptide Science 9: 1–15.
- 26. Pierce B, Weng Z (2007) Structure prediction of protein complexes. Computational Methods for Protein Structure Prediction and Modeling : 109–134.
- 27. Gray J (2006) High-resolution protein-protein docking. Current opinion in structural biology 16: 183–193.
- 28. Bonvin A (2006) Flexible protein-protein docking. Current opinion in structural biology 16: 194–200.
- 29. Halperin I, Ma B, Wolfson H, Nussinov R (2002) Principles of docking: an overview of search algorithms and a guide to scoring functions. Proteins: Structure, Function, and Bioinformatics 47: 409–443.
- 30. Castrillon-Candas J, Siddavanahalli V, Bajaj C (2005) Nonequispaced fourier transforms for protein-protein docking. ICES Report 05–44, The University of Texas at Austin.
- 31. Bajaj C, Chowdhury RA, Siddavanahalli V (2011) F2Dock: Fast Fourier Protein-Protein Docking. IEEE/ACM Transactions on Computational Biology and Bioinformatics 8: 45–58.
- 32. Chowdhury RA, Bajaj C (2010) Algorithms for faster molecular energetics, forces and interfaces. ICES report 10–32, Institute for Computational Engineering & Science, The University of Texas at Austin.
- 33. Bajaj C, Chowdhury RA, Rasheed M (2010) A dynamic data structure for flexible molecular maintenance and informatics. Bioinformatics 27: 55–62.
- 34. Bajaj C (2010) ZhaoW (2010) Fast molecular solvation energetics and forces computation. SIAM Journal on Scientific Computing 31: 4524–4552.
- 35. Bajaj C, Djeu P, Siddavanahalli V, Thane A (2004) TexMol: Interactive visual exploration of large flexible multi-component molecular complexes. In: Proc. of the Annual IEEE Visualization Conference. pp. 243–250.
- 36. Mintseris J, Wiehe K, Pierce B, Anderson R, Chen R, et al. (2003) Protein-protein docking bench-mark 2.0: an update. Proteins 60: 214–6.
- 37. Hwang H, Vreven T, Janin J, Weng Z (2010) Protein-protein docking benchmark version 4.0. Proteins: Structure, Function, and Bioinformatics 78: 3111–3114.
- 38. Mitchell JC (2008) Sampling rotation groups by successive orthogonal images. SIAM Journal on Scientific Computing 30: 525–547.
- 39. Lattman EE (1972) Optimal sampling of the rotation function. Acta Crystallographica Section B 28: 1065–1068.
- 40. Frigo M, Johnson SG (2005) The design and implementation of FFTW3. Proceedings of the IEEE Invited paper, Special Issue on Program Generation, Optimization, and Platform Adaptation 93: 216–231.
- 41. Wang H (1991) Grid-search molecular accessible surface algorithm for solving the protein docking problem. Journal of Computational Chemistry 12: 746–750.
- 42. Jones S, Thornton JM (1997) Analysis of protein-protein interaction sites using surface patches. Journal of Molecular Biology 272: 121–132.
- 43. Jones S, Thornton JM (1996) Principles of protein-protein interactions. Proceedings of the National Academy of Sciences of the United States of America 93: 13–20.
- 44. Glaser F, Steinberg DM, Vakser IA, Ben-Tal N (2001) Residue frequencies and pairing preferences at protein-protein interfaces. Proteins: Structure, Function, and Genetics 43: 89–102.
- 45. Weiner SJ, Kollman PA, Case DA, Singh UC, Ghio C, et al. (1984) A new force field for molecular mechanical simulation of nucleic acids and proteins. Journal of the American Chemical Society 106: 765–784.
- 46. Morris GM, Goodsell DS, Huey R, Olson AJ (1996) Distributed automated docking of flexible ligands to proteins: Parallel applications of AutoDock 2.4. Journal of Computer-Aided Molecular Design 10: 293–304.
- 47. Yan B, Sun Y (1997) Glycine residues provide exibility for enzyme active sites. Journal of Bio-logical Chemistry 272: 3190.
- 48. MacCallum R, Martin A, Thornton J (1996) Antibody-antigen interactions: contact analysis and binding site topography. Journal of Molecular Biology 262: 732–745.
- 49. Martin A (2010) Protein sequence and structure analysis of antibody variable domains. Antibody engineering: 33–51.
- 50. Jackins CL, Tanimoto SL (1980) Oct-trees and their use in representing three-dimensional objects. Computer Graphics and Image Processing 14: 249–270.
- 51. Eisenberg D, Mclachlan A (1986) Solvation energy in protein folding and binding. Nature 319: 199–203.
- 52. Gilson M, Davis M, Luty B, McCammon J (1993) Computation of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation. Journal of Physical Chemistry 97: 3591–3600.
- 53. Hermann RB (1972) Theory of hydrophobic bonding. ii. correlation of hydrocarbon solubility in water with solvent cavity surface area. Journal of Physical Chemistry 76: 2754–2759.
- 54. Sharp K (1991) Incorporating solvent and ion screening into molecular dynamics using the finite-difference poisson-boltzmann method. Journal of Computational Chemistry 12: 454–468.
- 55. Simonson T, Bruenger AT (1994) Solvation free energies estimated from macroscopic continuum theory: An accuracy assessment. Journal of Physical Chemistry 98: 4683–4694.
- 56. Wagoner J, Baker N (2006) Assessing implicit models for nonpolar mean solvation forces: The importance of dispersion and volume terms. Proceesings of the National Academy of Sciences of the USA 103: 8331–8336.
- 57. Still WC, Tempczyk A, Hawley RC, Hendrickson T (1990) Semianalytical treatment of solvation for molecular mechanics and dynamics. Journal of the American Chemical Society 112: 6127–6129.
- 58. Dolinsky TJ, Nielsen JE, McCammon JA, Baker NA (2004) PDB2PQR: An automated pipeline for the setup, execution, and analysis of Poisson-Boltzmann electrostatics calculations. Nucleic Acids Research 32: w665–w667.
- 59. Duan Y, Wu C, Chowdhury S, Lee M, Xiong G, et al. (2003) A point-charge force field for molecular mechanics simulations of proteins based on condensed-phase quantum mechanical calculations. Journal of Computational Chemistry 24: 1999–2012.
- 60. Desmyter A, Spinelli S, Payan F, Lauwereys M, Wyns L, et al. (2002) Three camelid vhh domains in complex with porcine pancreatic alpha-amylase. inhibition and versatility of binding topology. Journal of Biological Chemistry 277: 23645–23650.
- 61. Pierce BG, Hourai Y, Weng Z (2011) Accelerating protein docking in ZDOCK using an advanced 3d convolution library. PLoS ONE 6: e24657.
- 62. Janin J, Henrick K, Moult J, Eyck LT, Sternberg MJE, et al. (2003) CAPRI: A Critical Assessment of PRedicted Interactions. Proteins 52: 2–9.
- 63. Andrusier N, Nussinov R, Wolfson H (2007) FireDock: fast interaction refinement in molecular docking. Proteins 69: 139–159.
- 64. Canutescu A, Shelenkov A, Dunbrack Jr R (2003) A graph-theory algorithm for rapid protein side-chain prediction. Protein Science 12: 2001–2014.