UC Davis Genome Center and Bioinformatics Program, Department of Applied Science, University of California, One Shields Avenue, Davis, CA 95616, USA
1 To whom correspondence should be addressed. E-mail: duan{at}ucdavis.edu
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Abstract |
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Keywords: antibodyantigen/enzyme inhibitor/intermolecular distance/proteinprotein docking
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Introduction |
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One of the greatest challenges in proteinprotein docking is to reduce the number of candidates from typically hundreds of thousands to hundreds while still retaining a reasonable number of near-native complex structures. Part of the challenge can be attributed to the flexibility of ligands and receptors, especially at the interface (Betts and Sternberg, 1999), and the rough docking energy landscape. This problem is further aggravated by the marginal stability of proteinprotein complexes (typically near 10 kcal/mol) (Gray et al., 2003
). Although protein flexibility can be modeled by Gaussian network (Bahar et al., 1997
) and other methods (Jacobs et al., 2001
), it is usually infeasible to introduce flexibility at the early stage of docking. Therefore, reduced side chain representations (Heifetz and Eisenstein, 2003
; Li et al., 2003b
; Zacharias, 2003
) and soft potentials (Fernandez-Recio et al., 2002
) have been applied to smooth the energy landscape in rigid-body docking. Along this line, alternative approaches such as the use of multiple copies of receptors or ligands combined with more efficient search algorithms could also be contemplated.
With the aim of predicting proteinprotein interactions, comprehensive studies have been conducted to characterize proteinprotein interfaces (Nooren and Thornton, 2003a; Ofran and Rost, 2003
). Some of the features of the proteinprotein interface identified in these studies include the number of surface patches, contact area, polarity, flatness and residue conservation (Nooren and Thornton, 2003b
). Interfaces smaller than 1000 Å2 usually consist of only a single patch with a core and a rim (Chakrabarti and Janin, 2002
). The composition of core residues is distinctive from rim and non-interacting surfaces. Interfaces are less planar for enzymeinhibitor complexes than other types of interfaces. Hydrophobic residues are more abundant for larger interfaces and polar residues are more abundant for smaller interfaces. Hot spots (Ma et al., 2001
) are interface residues whose mutations have a significant effect on the binding free energies. Most hot spots comprise tryptophan, tyrosine or arginine residues, whereas serine, threonine and leucine residues are least likely to be hot spots (Bogan and Thorn, 1998
). Interface residues tend to be more structurally conserved than non-interface residues (Ma et al., 2003
). For the enzymeinhibitor interface, the conservation is significantly higher at the enzyme interface than the inhibitor interface (Bradford and Westhead, 2003
). Residue pairing also has distinct preferences (Glaser et al., 2001
), favoring hydrophobichydrophobic and polarpolar pairing.
Universal scoring functions have been dominant in the field of proteinprotein docking. However, different classes of complexes may have different binding mechanisms. It was noted that protease inhibitors interact predominantly through main chain to main chain interactions whereas sidechain interactions play a dominant role in antibodyantigen complexes (Jackson, 1999). Hence, our strategy is to develop different procedures based on the specific features for a class of complexes (Li et al., 2003a
). Among the structurally determined complexes, enzymeinhibitor and antibodyantigen are the two dominant classes. For antibodyantigen complexes, complementarity-determining regions (CDR) of antibodies are conserved both sequentially and structurally. This feature has been widely used to limit the search space for antibodyantigen complexes (Chothia and Lesk, 1987
) and was shown to enhance greatly the predictive power of docking (Schneidman-Duhovny et al., 2003
). In addition to the CDR restriction, antibody interfaces are also abundant in hot spot residues tryptophan and tyrosine. As for the enzymeinhibitor class, most complexes share a common geometric feature at the interface, which is a good packing comprised of a convex inhibitor surface and a concave enzyme surface (activity pocket). Although this is a well-known geometric feature, it has not been well utilized in guiding the docking of enzymeinhibitor complexes.
Here we explore the applicability of intermolecular distance on the evaluation of packing and subsequently the ranking of docking candidates. We adopted a three-step approach: (a) global search by rigid-body docking using FTDOCK; (b) filtering by intermolecular distance; additional filters CDR and physical energy were applied to antibodyantigen docking; (c) ranking by a composite scoring function including geometry, energy and informatics terms. The distance between the residues at the ligand interface and the center of the receptor is defined as the intermolecular distance. The geometry terms include the intermolecular distance and the interface size as approximated by the number of residues at the interface. There are three energy terms. The physical energy includes the soft electrostatics and van der Waals potentials. The statistical potential at residue level is represented by residue-pair (RP) score (Moont et al., 1999), the default scoring function of the docking suite 3D-DOCK (Gabb et al., 1997
). The third energy term is the statistical potential at atom level, i.e. atom contact energy (ACE). The informatics terms include the CDR score and the score for the enrichment of Trp and Tyr. The informatics terms only apply to the antibodyantigen complexes. A subset of
100 highest ranked candidates for each target formed the prediction set and the performance of the scoring function was evaluated based on the quality of the prediction set.
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Methods |
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FTDOCK and RP score are part of the 3D-DOCK suite (Gabb et al., 1997) which was obtained from the Sternberg group (http://www.bmm.icnet.uk/docking). The RP score is based on statistical analysis of residue pairing preferences of a non-redundant protein complex database (Glaser et al., 2001
). To avoid unnecessary sampling, the non-CDR domains of the antibodies were removed. The coordinates of receptors and ligands were preprocessed to remove hydrogen atoms and alternative atom coordinates by the preprocess-pdb.perl utility program in 3D-DOCK. The setup parameters for the FTDOCK runs included a 0.7 Å grid unit, 1.3 Å surface thickness and 9° rotation angle step size. Receptors (enzyme or antibody) were set as the static moiety and ligands (inhibitor or antigen) were set as the mobile moiety. The only exception was the target 1DFJ, for which the static moiety was the inhibitor because of the much larger size of the inhibitor (456 residues) compared with the enzyme (124 residues). There were three independent runs for each target; the starting coordinates for the second and third runs were generated by the randomspin utility program. For the enzymeinhibitor targets, 10 000 candidates were saved in each run, resulting in a total of 30 000 candidates for each target. The number of saved candidates was doubled for the antibodyantigen targets to account for their larger system sizes. Each of the three runs was analyzed separately throughout this work. Following the completion of each FTDOCK run, the residue-pair potential for each candidate was calculated using the rpscore program and normalized to 0100 for each run.
The quality of the docking candidates was evaluated by the ligand root mean square deviation (r.m.s.d.) against the native complex. We followed the standard procedure where the receptors were aligned first and then the r.m.s.d. of the ligand C atoms was calculated (Mendez et al., 2003
). A candidate with ligand r.m.s.d. <10 Å is defined as a hit. The build utility program was used to regenerate candidate structures for r.m.s.d. evaluation and other calculations. If any atom of a residue is within 5 Å of any atom on the other protein, this residue is defined as an interface residue. This is the standard cutoff used in CAPRI. The intermolecular distances include the average and minimum distances (ADIS and MDIS) between the ligand interface and the receptor center (defined as center of geometry). These two distances were normalized to 0100 for each run.
The charges and radii for calculating the interaction energy between the receptor and ligand were taken from a recently revised AMBER force field (Duan et al., 2003). Only electrostatics and van der Waals terms were included in the interaction energy. To reduce the noise caused by atom overlap in rigid-body docking, smooth energy functions were adopted. The electrostatic energy terms were truncated when atoms were closer than their van der Waals distances. Furthermore, only the attractive London dispersion energy terms were considered and the repulsive terms were ignored. These terms were combined as interaction energy and normalized to 0100 for each run. The parameters for calculating atomic contact energy were taken from Zhang et al. (1997)
. The contact radius was set to 6.0 Å according to the original work. The contact energy between the receptor and the ligand was calculated for each candidate. The ACE energy was also normalized to 0100 for each run.
For enzymeinhibitor complexes, the only filter was ADIS with a cutoff of 50. Following the filtering, a composite scoring function was applied to rank the remaining candidates. The scoring function SENZ consists of two components: energy terms and geometry terms.
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For antibodyantigen complexes, the filtering consists of two steps. First, each residue of the antibody was classified into CDR residue and non-CDR residue. For each candidate, the percentage of CDR residues at the antibody interface was calculated and the candidate is retained if this percentage is above 60%. The first filter is to ensure that most of the antibodyantigen interaction involves the CDR residues. In the second filtering step, the physical interaction energy filter was also applied in addition to the distance filter and the cutoff was also set to 50. After filtering, a composite scoring function was applied to rank the remaining candidates. The scoring function SANT consists of three components: energy terms, geometry terms and informatics terms.
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There were a few changes compared with the scoring function for enzymeinhibitor docking. SMDIS was removed because antibodies do not have reaction pockets like enzymes do and a score reflecting the size of the ligand interface (SNL) was added. The informatics terms (Sinfo) are specific for the antibodyantigen class, which favor the candidates with a higher percentage of CDR residues (SCDR) and more hot spot residues Tyr and Trp (Shot) at the antibody interface. The signs for the informatics terms were reversed to be consistent with the whole scoring scheme. The number of hot spot residues was scaled up 10-fold to achieve a similar variation range as for other terms. The percentage of CDR residues was scaled up 4-fold because this is the most prominent feature for antibodyantigen interaction.
The prediction set of each target was composed of 33 highest ranked candidates from each of the three runs, making a total of 99 candidates in the prediction set. When the component score or individual terms were examined, the ranking was based on the specific score studied rather than the composite score. The performance was evaluated mainly by the number of hits in the prediction set. In addition, the lowest r.m.s.d. of the candidates in the prediction set and the highest rank of the hits were also presented for evaluation.
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Results |
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Generating candidate pool by FTDOCK
Among the 22 enzymeinhibitor targets studied here, 16 started from unbound-unbound structures and the other six started from unbound-bound structures (see Methods). There were three rigid-body docking runs for each target, starting from different orientations of both the receptor and the ligand. In each of the three runs, 10 000 candidates were saved. The quality of each candidate was evaluated by the ligand r.m.s.d. (C) of the candidate using the native complex as reference (see Methods). The number of hits is summarized in Table I (group definition will be described later in this section). Here we define a hit as a candidate with a ligand r.m.s.d. below 10 Å. The number of hits among the 30 000 candidates of each target ranged from 13 to 534 with a median value of 46. Most (17) of the 22 targets had fewer than 100 hits. The median hits-to-candidates ratio was 1:652, making it very challenging to pick up hits with limited selections.
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Here we define the best hit of a target as the candidate with the lowest r.m.s.d. The lowest r.m.s.d. among the 22 targets ranged from 1.07 to 6.14 Å with a median value of 2.45 Å. The best hits of the two complexes with the smallest ligands (1PPE and 1TAB) are of very high quality (1.07 and 1.18 Å). Out of 22 targets, 20 complexes have the lowest r.m.s.d. <5.0 Å and only two complexes have the lowest r.m.s.d. >5 Å (6.14 Å for 1DFJ and 5.44 Å for 1BRS). The low quality of the best hit for 1DFJ can be attributed to the size (large ligand and receptor). Interestingly, 1BRS is a receptor that is half the size of most complexes. However, its best hit is only 5.44 Å and is considered a low-quality hit. For the six unbound/bound targets, the quality of the best hits was higher, with r.m.s.d. ranging from 1.07 to 2.46 Å. This suggests that accounting for the conformational changes will likely be helpful, but it is beyond the scope of the current study, which is based on rigid-body docking approaches and is meant to examine the issue of scoring function.
We further examined each complex structure and found that there is a great variety among these enzymeinhibitor complexes. The classification based on interface characteristics is shown in Table I. The representative structures are shown in Figure 1. There are four major groups among these 22 complexes. The first group has 11 members with the interface comprised of a pseudo-loop of the ligand and a concave surface of the receptor. A pseudo-loop is defined here as a loop with one free end, compared with the classic loop in which both ends are attached to internal secondary structures. A representative structure of the first group is 1CGI, shown in Figure 1. Three members of the first group (1AVW, 2SIC and 4HTC) do not strictly conform to the definition of the group, but they all have at least one free end involved in the binding. The second group has four members with the interface comprised of an expanded loop of the ligand and a concave surface of the receptor. A representative structure of the second group is 1CSE (Figure 1). The third group has three members with the interface comprised of a ß-strand of the ligand and a cleft surface of the receptor. A representative structure of the third group is 1UDI. The remaining four complexes do not belong to these three groups. The interface of 1BRS consists of an -helix of the ligand and a ß-sheet of the receptor. 1DFJ has a large circular inhibitor formed by internal repeats. 1FSS and 1MAH both have large globular receptors over 500 residues. We put these four complexes in the fourth group. Overall, for the fourth group, the number of hits is less and the quality of the best hits is lower.
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A simple measurement that captures the concaveconvex features of enzymeinhibitor complexes is ADIS, the average distance between residues at the ligand interface and the geometric center of the receptor. ADIS was normalized to a score between 0 and 100 in each FTDOCK run. In the filtering process, ADIS was the only filter applied to the enzymeinhibitor targets. To avoid over-fitting to this data set, we chose to use a rather coarse cutoff. Candidates with normalized ADIS score >50 were filtered out. A summary of the filtering is shown in Table I. The total number of the remaining candidates ranged from 3953 to 23 344 with a median of 13 163. The percentage of the hits retained was >90% for all targets except for 4HTC, where a long tail of the inhibitor is involved in the interaction. For half of the 22 targets, there was no loss of hits after filtering. This shows that filtering reduced the candidate space to 43.9% (median) of the original set while retaining nearly all of the hits. Additionally, the lowest r.m.s.d. stayed the same for 21 of the 22 targets and there was a slight increase of the lowest r.m.s.d. from 2.46 to 2.73 Å for the target 4HTC.
Ranking by a composite scoring function
The composite scoring function SENZ included energy terms (Sener) and geometry terms (Sgeo). The energy terms included physical energy, atomic contact energy and residue-pair potential. The geometry terms included ADIS, MDIS and the size of the receptor interface. To evaluate the ranking ability of SENZ, 33 highest ranked candidates were selected from each of the three runs, forming the prediction set of 99 candidates. The performance by this scoring function is shown in Table II. Within the prediction set, 288 hits were retained for the 22 targets. The median number of hits was 11.5. According to the lowest r.m.s.d., the quality of the best hits in the prediction set remains the same as or similar to the best hits in the filtered set for most cases. The only target with significant increase of r.m.s.d. was 1ACB (the lowest r.m.s.d. increased from 2.83 to 4.56 Å). The best ranks of the hits were between 1 and 26 for the 22 targets and half of the best ranks were 1.
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RP score is the default and only score in 3D-DOCK. When compared with other individual terms used in this study, its performance was less desirable than any of the geometric terms. Since it missed nine out of the 22 targets, the ranking ability of RP score itself appears to be unsatisfactory for enzymeinhibitor complexes. However, it works exceptionally well for six targets in group 1, where at least 10 hits were retained in the prediction set. The biased performance by RP score could be caused by the possible bias inherent in the training database from which the residue-pair potential was derived. It is also possible that the preferential pair interactions are more conserved in the first group.
A study of the enrichment of hits was also conducted. Among all the candidates in the filtered set, we chose 33, 100, 150, 200, 250 and 300 highest ranked candidates from each of the three runs based on the composite score, component scores or the individual scores and the number of hits was examined within the subsets of 99, 300, 450, 600, 750 and 900 selected candidates. The median number of hits within different selection sets is shown in Figure 3. For the composite score SENZ, the number of hits increased from 11.5 to 30.5 when the selection size increased from 99 to 900. The increase rate was the highest in the first segment when the selection size changed from 99 to 300, the number of hits almost doubled. It is interesting that while the number of selected hits was significantly lower by the individual energy terms (lower panel in Figure 3) compared with the geometry terms (middle panel in Figure 3), the component score Sener performed slightly better than Sgeo when the selection size was >300 (upper panel in Figure 3).
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In the antibodyantigen class of the same benchmark set, there are four targets with single domain antibody-formed complex with enzyme antigen, 1KXQ, 1KXT, 1KXV and 1MEL. We reassigned those targets to the enzymeinhibitor class and conducted the test following the same procedure as described above. A major challenge is to predict the multiple binding modes of the same enzyme pancreatic -amylase with different antibody VHH domains in the three complexes 1KXQ, 1KXT and 1KXV. As shown in Figure 4, the VHH domains bind to the major pocket, a minor groove and a flat surface of the pancreatic
-amylase in the three complexes. These are three of the seven targets in the first and second round of CAPRI experiments. The only correctly predicted binding mode was 1KXQ by any participating groups. In our test, FTDOCK generated 39, 12 and 26 hits in the candidate pool for 1KXQ, 1KXT and 1KXV, respectively. In the prediction set, the number of hits was 18, 1 and 2, respectively. The best rank of hits was 2, 25 and 22. The lowest r.m.s.d. was 1.57, 1.87 and 6.83 Å. Although the scoring function is apparently biased to the binding mode of classical enzymeinhibitor interaction as in 1KXQ, it is encouraging that the scoring function can correctly identify all three binding modes with simple rigid-body docking. For the other target 1MEL, 31 hits were generated in the candidate pool by FTDOCK. There were eight hits within the prediction set. The lowest r.m.s.d. was 4.76 Å and the best rank was 2.
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Fifteen antibodyantigen targets were included in this study (Table III). The antigens were lysozyme in the first seven targets. In the FTDOCK step, 60 000 candidates were saved for each target. The increased size of candidate pool compared with the enzymeinhibitor docking was due to the significantly lower quality of candidates. The number of hits ranged from 9 to 98 with a median of 27, which was much fewer than the hits generated in the enzymeinhibitor docking with only half of the candidates. The hits-to-candidates ratio was 1:2222 in the candidate pool. This made it much more challenging than the enzymeinhibitor docking. The lowest r.m.s.d. ranged from 0.82 to 6.10 Å with a median of 2.7 Å, comparable to the lowest r.m.s.d. in enzymeinhibitor docking.
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The ranking performance of the composite scoring function SANT and its component scores Sgeo, Sener and Sinfo is presented in Table IV. Although the component scores only had correct prediction on 78 targets, the composite score correctly predicted 13 of the 15 targets, among which three of the highest ranks are 1. It is intriguing that most of the lowest r.m.s.d. candidates were missed by SANT.
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Discussion |
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Chen et al. developed ZDOCK and tested it on the same enzymeinhibitor set (Chen et al., 2003a) examined in the current study. Several scoring schemes were tested in the work of Chen et al. Among them, the best performance was by a combination of pairwise shape complementarity score, desolvation energy and electrostatics. For four of the enzymeinhibitor complexes, 2PTC, 1CSE, 2KAI and 2SNI, the first hit was ranked 193 or lower. In the worst case, only one hit was in the top 2000 for 2SNI and it was ranked 1262. The performance on the antibodyantigen targets was much less desirable. Only four targets (1AHW, 1BQL, 1NCA and 1MEL) had hits ranked among the top 100. Gray et al. also conducted a comprehensive test on the same complex set (Gray et al., 2003
). The energy terms used in their study included van der Waals, electrostatics, solvation, hydrogen bonding, residue pair probability and rotamer probability terms. In the global docking test, no hits were found among the top 25 clusters for three enzymeinhibitor targets, 1DFJ, 1BRC and 2PTC. For the antibodyantigen targets, no hits were found among the top 25 clusters for six targets, 1DQJ, 1WEJ, 1EO8, 1IAI, 1NCA and 2VIR. In summary, there are 34 incorrect predictions for the enzymeinhibitor targets and
6 incorrect predictions for the antibodyantigen targets. As a comparison, all the enzymeinhibitor targets were correctly predicted and only two antibodyantigen targets were incorrectly predicted in the current work.
It should be noted that comparison with other studies is usually indirect. Different procedures and criteria are used by different groups. For example, the rotation step size was 6° in the ZDOCK global search and 5° in Gray et al.'s work (Gray et al., 2003). The rotation step size was 9° in this work. When compared with 12° rotation (default in FTDOCK), candidate quality was improved at a 9° rotation step (data not shown). Unfortunately, rotation steps smaller than 9° are not allowed in FTDOCK without major modification of the program code. As a result, the candidate quality is expected to be slightly lower in this study than that in ZDOCK and in the work of Gray et al. Additionally, interface r.m.s.d. was used as a criterion for hits in ZDOCK. Both the ligand r.m.s.d. and interface r.m.s.d. are used in CAPRI evaluation. Since interface r.m.s.d. is sensitive to side chain orientation on the interface and geometric features are not very sensitive to side chain orientation, we decided to use ligand C
r.m.s.d. as criteria for hits. The highest populated clusters instead of individual candidates were used for ranking in Gray et al.'s work. Assuming that each cluster contains 20 members, the top 25 clusters would correspond to 500 individual candidates in the prediction set. Since a significant number of hits was found among the top 99 selections for most complexes in this work, we decided that it is not necessary to do clustering.
Future development
A considerable challenge is to account for the conformational changes during docking. Indeed, some complexes undergo large conformational changes upon binding. They are categorized into the difficult class in the benchmark set compiled by Chen et al. (2003b). This class provides the greatest challenge for docking. Although this is beyond the scope of the present study, we tried to apply the scoring scheme to assess the limitation. One example is complex 1BTH, which is an enzymeinhibitor complex in the difficult class. Although its main structural features resemble those complexes in group one, the ß-loops at the receptor interface changed from open to close state upon binding. No hits were found among the top 25 clusters in Gray et al.'s work (Gray et al., 2003
); the lowest ligand r.m.s.d. in the top 10 clusters was 16.96 Å. In our test, FTDOCK generated 21 hits in the candidate pool. The best hit had a ligand r.m.s.d. of 7.08 Å. Among the 99 selections in the prediction set, there were 12 candidates with r.m.s.d. <15 Å; the lowest r.m.s.d. was 11.84 Å. Although it did not survive the 10 Å cutoff, it is a reasonable starting point for further refinement.
Ultimately, proteinprotein interactions are governed by free energy (Fernandez-Recio et al., 2004). This is why most of the scoring functions for proteinprotein docking employ atomic-level interaction energy terms. However, there are a few pitfalls in energy-based approaches. First, energy at the atomic level is sensitive to small perturbations. Hence incorporation of flexibility becomes necessary in the early stage of docking. This makes the global search computationally intensive. Second, entropy is an important part of free energy. Although certain solvation terms have entropy partially included in the parameterization process, it is extremely difficult to evaluate the complete entropic contribution to binding. Third, the accuracy of physical force fields is still uncertain despite decades of effort. Small systematic errors can lead to large errors exceeding binding energy when thousands of atoms are involved. These pitfalls of atomic interaction energy make it necessary to utilize information at coarse-grain level. The information does not have to be universal to all proteinprotein complexes, provided that it is a general feature for certain groups of complexes. For example, restriction to CDR regions has been a common protocol for antibodyantigen docking. Interface residue conservation (Bradford and Westhead, 2003
) has also been suggested as a filter for docking candidates. Intermolecular distance has been shown in this work to be of utility in selecting good candidates. Further improvement will require introducing flexibility. This will comprise the next stage of the docking procedure.
Conclusions
We investigated the application of a new packing factor, namely intermolecular distance, to the docking of proteinprotein complexes. A new selection strategy was developed on 22 enzymeinhibitor targets and 15 antibodyantigen targets from a proteinprotein docking benchmark set. A pool of docking candidates was generated by FTDOCK. This was followed by filtering with the intermolecular distance and ranking by the composite scoring function. This strategy succeeded on all of the 22 enzymeinhibitor targets and 13 of the 15 antibodyantigen targets. Most interestingly, this selection strategy succeeded on four test targets, within which three complexes have the same receptor yet different binding modes.
Although the three-step procedure was adopted in the docking of both enzymeinhibitor and antibodyantigen complexes, there are variations in each of the three steps. In the global sampling step by FTDOCK, 30 000 candidates were saved for the enzymeinhibitor targets, whereas 60 000 candidates were saved for the antibodyantigen targets. The increased size for the latter case was due to the much lower quality of the candidate pool. In the filtering step, only the ADIS filter was applied for the enzymeinhibitor targets, whereas two more filters (CDR and physical energy) were added for the antibodyantigen targets. CDR information has been routinely integrated in antibodyantigen docking, either in the sampling step or in the filtering step. The physical energy filter was added because of the good correlation between this energy and candidate quality. In the final ranking step, the composite score SENZ consists of geometry terms and energy terms, while the composite score SANT includes additional informatics terms. All these variations were due to the significantly greater difficulty of antibodyantigen docking, as been shown in many previous docking studies.
Intermolecular distance was applied in two of the three steps in this docking study. In the filtering step, it was able to reduce greatly the number of candidates without significant loss of hits. In the ranking step, the addition of the intermolecular distance was shown to be helpful in giving the hits higher ranks. The contribution of intermolecular distance to the filtering and ranking was manifested in this work for the docking of both the enzymeinhibitor complexes and the antibodyantigen complexes. More work is under way to evaluate its application to other proteinprotein complexes.
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Acknowledgments |
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Received August 15, 2004; revised October 22, 2004; accepted December 28, 2004.
Edited by Bruce Tidor
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