Bioinformatics Advance Access originally published online on October 12, 2006
Bioinformatics 2007 23(4):421-426; doi:10.1093/bioinformatics/btl524
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Inherent limitations in protein–protein docking procedures
1 Department of Biological Chemistry, Weizmann Institute of Science Rehovot, Israel
2 Department of Chemical Research Support, Weizmann Institute of Science Rehovot, Israel
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONREFERENCES |
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Motivation: The limited success rate of protein–protein docking procedures is generally attributed to structure differences between the bound and unbound states of the molecules. Herein we analyze a large dataset of protein–protein docking results and identify additional parameters that affect the performance of docking procedures.
Results: We find that the distinction between nearly correct models (NCMs) and decoys depends on the size of the interface to be predicted thus setting a limit to the prediction ability of docking procedures, particularly those in which the geometric complementarity descriptor is dominant. The geometric complementarity score in grid-based docking carries a large statistical error which further reduces the distinction between NCMs and decoys. We propose a method for correcting the statistical error and show that the distinction is improved when the docking models are ranked by statistically equivalent scores.
Availability: MolFit can be downloaded from our website http://www.weizmann.ac.il/Chemical_Research_Support/molfit
Contact: miriam.eisenstein@weizmann.ac.il
Supplementary information: Supplementary data are available at Bioinformatics online.
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONREFERENCES |
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The past decade witnessed a considerable upsurge of in silico docking procedures aiming to predict the structures of protein–protein complexes given the structures of the component molecules (docking). Such prediction methods try to form correct or nearly correct models (NCMs) and to distinguish them from false models. Different configurations of the docked molecules are formed and evaluated using a scoring function that includes several more or less discriminative descriptors (Eisenstein et al., 2004; Halperin et al., 2002; Sternberg et al., 1998). The performance of the different docking procedures is very encouraging (Mendez et al., 2003, 2005) yet it is limited, making it important to study the origins of the difficulties that some prediction procedures encounter. Naturally the high dimensionality of the docking problem and the need to resort to approximations in the depiction of the molecular structures and/or the evaluation of the different docking configurations has a dominant effect on the performance of docking tests; but is this the only limiting factor?
The docking procedure developed in our group, MolFit, performs a comprehensive scan in the rotation-translation space. The molecules are represented by three-dimensional (3D) grids that carry information on the shape and the chemical character of the molecular surfaces. The grids are correlated using fast Fourier transformations (FFTs), producing a matrix of complementarity scores for N3 relative translations, where N is the dimension of the grid. Iteration of the procedure for a large number of relative orientations of the molecules completes the comprehensive stepwise scanning of the rotation-translation space (Katchalski-Katzir et al., 1992).
Most of the configurations tested in the MolFit docking scan are erroneous, producing many false predictions or decoys. We statistically analyzed the distributions of the complementarity scores in attempt to identify parameters that affect our ability to distinguish between correct and false docking models. We concentrated in this study on the basic test of bound docking; this eliminates errors due to the structural differences between bound and unbound molecules and highlights additional parameters that affect the results of docking tests. We found that, as expected, the shape complementarity scores calculated by MolFit for NCMs are linearly related to the size of the interface, whereas the electrostatic and hydrophobic complementarity scores are not. The mean score in the rotation-translation scan, which mostly reflects the quality of the decoys, shows little variation; hence, the distinction between correct and false docking models depends on the size of the predicted interface. This is related to the dominant role of the geometric descriptor and is therefore relevant to practically all docking procedures.
The difficulty to distinguish correct models from false ones is further aggravated by errors introduced by the discrete 3D grid representation and the correlation procedure employed in MolFit. The digitization of an image or an object introduces two kinds of errors (Ben-Tzvi and Weizman, 1996): Sampling errors originate from the ‘division’ of the object into a finite number of smaller pieces (pixels or voxels in 2D and 3D grids, respectively); they cause loss of some of the object's details and the addition of artificial details, thereby deforming the object. It is possible to overcome sampling errors by reducing the voxel size (increased grid resolution) as infinitely small voxels reproduce the real shape of the object. The second kind of error is known as the ‘quantization error’. It originates from the need to describe a continuous function by discrete values at grid points. Most studies that measure the extent of grid-based errors deal with the quantization problem (Luty et al., 1995; Wu et al., 2003). This error is however negligible in MolFit, at least for the dominant geometric descriptor, because we can safely assume that a given conformer of a molecule has a well-defined (although not necessarily correct) shape that is not a continuous function.
Grid-based docking is prone to another kind of error originating from the stepwise sampling of the rotation-translation space. Thus, the vector connecting the centers of mass of the docked molecules (V) for any tested docking configuration consists of integer voxel translations along three perpendicular axes. In contrast, the components of V in the correct structure are rarely integer multiples of the voxel size; hence, the predicted V can only be an approximation of the experimental distance between the molecules even when the relative orientation is exact. Again, small grid intervals are likely to provide better estimates of the correct distance between the molecules.
In view of the abundance of grid-based docking procedures (reviewed by Eisenstein et al., 2004) we set out to determine the extent of the scoring error introduced by the use of a grid. We designed a computational tool that reproduces a given docking model, rotates it to new, randomly selected orientations relative to the grid and re-evaluates the complementarity score. We found that the distribution of scores for a given model is generally wide and that the average score (AV) and standard deviation (SD) of these distributions behave differently for NCMs and for false predictions. We then re-ranked the docking models employing statistically equivalent scores (such as AV or AV + n x SD). This led to higher ranking of the NCMs and enriched the fraction of NCMs amongst the high ranking predictions. The analyses of NCMs and decoys provide new insight regarding the difficulties that docking procedures encounter and point out new plausible directions in protein–protein docking.
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2 METHODS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONREFERENCES |
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2.1 MolFit rotation-translation scans
Bound docking scans were performed for the 84 systems in benchmark V2.0 (Mintseris et al., 2005). All the water molecules were omitted. Hetero-groups within the proteins were retained. Only the Fv domains of antibodies were used but docking to the face that interacts with the Fc domain in an intact antibody was impeded (Heifetz et al., 2002). Missing side chains, but not missing residues, were completed in all the structures. We followed the chain specifications in the benchmark; in some cases however we used a dimer (1bj1), a trimer (1kac) or a tetramer (1k4c) instead of a monomer, assuming that there is no data limiting the interaction to only one subunit within the oligomer.
Geometric and geometric–hydrophobic [residues+ representation; Berchanski et al. (2004)] docking scans were performed for all 84 systems. Geometric–electrostatic docking scans (Heifetz et al., 2002) were performed for 75 of the systems in the database; we excluded systems in which residues are missing from within the polypeptide chain of either molecule and the electrostatic potential may therefore be miscalculated. Electrostatic potentials were computed using the program Delphi (Honig and Nicholls, 1995) as implemented in the InsightII package (Accelrys Inc., San-Diego, CA), using the PARSE set of parameters (Sitkoff, 1994) to which we added the necessary hetero-groups. A grid interval of 0.8 Å was employed in the Delphi computations except for the largest molecules, for which larger grid intervals (between 0.8 and 1.1 Å) had to be used. Geometric–electrostatic–hydrophobic (GEH) complementarity scores were obtained by intersecting the lists of predictions from geometric, geometric–electrostatic and geometric–hydrophobic docking scans, as described previously (Berchanski et al., 2004). The predictions were compared to all the symmetry-related native interactions and NCMs were defined as the models with root mean square deviation (RMSD) <3 Å from the experimental structure, calculated for all the common C
atoms in the complex; the ligand-RMSDs (RMSD calculated only for the moving molecule, which is usually the smaller of the docked molecules) for the highest ranking NCMs were <4.2 Å.
We employed fixed translational and rotational grid intervals of 1.05 Å and 12°, respectively, in all the MolFit scans unless otherwise stated. The two highest scoring models for each relative orientation were saved. As a result of this selection the distribution of the 17 520 scores in each docking scan follows an extreme value distribution (Levitt and Gerstein, 1998) as demonstrated in Figure 1. Estimates for the mean score (µ) and standard deviation (
) were calculated for each distribution and E-values (the probability of obtaining the same or higher scores) were calculated for each model.
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2.2 Estimating the scoring error caused by the grid representation
The scoring error was estimated as follows: (1) A docking model was reproduced based on the rotations and translations put out by MolFit; (2) the model was rotated as a whole to a new, randomly selected orientation; hence, the relative position of the two molecules in the model did not change but they were disposed differently on the grid; (3) the complementarity score was re-evaluated for the translation closest to the translation in the original model and for 26 additional translations (±1 step along each principal axis), and the highest score was retained. Steps (2) and (3) were repeated M times producing M score values for the model formed in the first step. Notably, the scores were calculated using Boolean operations and not FFT because FFT is faster when scores for the whole 3D translation matrix are calculated but not for a small 3 x 3 x 3 sub-matrix.
2.3 Buried surface area
Buried surface areas (BSAs) in the experimentally determined structures of the complexes were calculated by subtracting the surface area of the complex from the sum of the surface areas of the individual molecules. We used the program NACCESS (Hubbard and Thornton, 1993) and a probe radius of 1.4 Å in these computations.
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3 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONREFERENCES |
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3.1 Geometric complementarity and distinction between correct and false models
The geometric score of MolFit reflects the level of surface complementarity for a given relative position of the docked molecules, penalized for interpenetrations. We therefore expect that in bound docking the geometric scores of the NCMs (SNCM) will be highly correlated to the BSA in the native structure of the complex. Figure 2 shows that such high correlation exists, with a correlation coefficient R2 = 0.903 (for 84 systems). Moreover, the correlation coefficient is independent of the type of the complex; thus, the subgroups of enzyme–inhibitor or antibody–antigen complexes behave in the same way as all other systems (Fig. 2). Importantly, the complementarity scores have arbitrary units; they are matched for all the systems tested here because we used the same grid interval in the computations.
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Figure 2 also shows that µ is only weakly dependent on the BSA (R2 = 0.288). Most of the relative orientations tested in a scan are erroneous therefore µ and
characterize the decoys. These are not randomly selected decoys; rather they are the best decoys selected by our scoring function and therefore the values of µ are substantial and for complexes that have very small interfaces (less than 
900 Å2) SNCM and µ are similar, making the NCMs and the decoys indistinguishable. The values are independent of the BSA; they are consistently 8–10% of µ.
The success of a docking experiment depends on the score of the NCM and also on the statistical significance of the score, measured by its E-value, and by the gap between SNCM and the score of the top ranking false prediction (
Sc). In geometric docking both measures show high dependency on the BSA, with R2 = 0.716 for log(ENCM) and R2 = 0.757 for Sc.
3.1.1 The effect of the grid interval and the penalty for interpenetration parameters
MolFit employs a very small number of adjustable parameters, namely, the translational and rotational grid intervals, the thickness of the surface layer and the penalty for interpenetration (Katchalski-Katzir et al., 1992). We repeated the geometric docking scans employing a smaller grid interval of 0.8 Å or different interpenetration penalties (–7.5, –15 and –30) in order to test the effect of these parameters on the BSA dependency of the scores. These computations were performed for a selected subset of 14 systems (1ktz, 1wej, 1qa9, 1acb, 1hia, 1iqd, 1mah, 1f51, 1dfj, 1i2m, 1f34, 1kxp, 1n2c, 1bgx), which display a wide range of BSAs (between 989 and 5814 Å2).
The BSA dependency of SNCM persists in computations with the modified parameters. Employing a smaller grid interval results in an increase in both SNCM and µ (Fig. 3). The change in µ is however smaller than the change in SNCM and therefore the BSA dependency regression lines for SNCM and µ meet at BSA = 765 Å2 for a grid interval of 0.8 Å compared with 870 Å2 for a grid interval of 1.05 Å (calculated for the same selection of systems), indicating that the smaller grid interval extends the range of distinction between NCMs and decoys by 100 Å2. The ratios between the SNCM values obtained with the two grid intervals are very similar to the corresponding ratios for µ and (2.1 on the average) and therefore the ENCM values are barely affected by the change in grid interval (RMSD = 0.76 between the two sets of logE values). Hence, the smaller grid interval does not improve the statistical distinction of the NCMs.
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The BSA dependency of SNCM is stronger when the interpenetration penalty
is less negative (higher R2 values and larger slopes of the regression lines; the grid interval in these computations was fixed at 1.05 Å). This feature however does not improve the distinction of the NCMs because µ increases correspondingly (see Supplementary Figure 1). Judging by the meeting points of the BSA dependency regression lines for SNCM and µ, at 950, 870 and 958 Å2 for = –7.5, –15 and –30, respectively, we conclude that the previously determined value ( = –15) is adequate. We did not test the effect of the rotational grid interval at this point because this parameter will affect the number of decoys but not their distribution; by docking disassembled structures we made sure that the exact orientation is included in the scan. The thickness of the surface layer was optimized for unbound docking; this parameter is likely to influence the distinction between NCMs and decoys but not the dependence on the BSA and therefore it was not modified in this study.
3.2 The effect of electrostatic and hydrophobic complementarity
The hydrophobic scores of the NCMs (the difference between geometric–hydrophobic and geometric scores) depend weakly on the BSA (R2 = 0.312) and the corresponding electrostatic scores (the difference between geometric–electrostatic and geometric scores) do not depend on the BSA (R2 = 0.130; see Supplementary Figure 2). However, the BSA dependency of the GEH scores (see Supplementary Figure 3) is close to that of the geometric scores (R2 = 0.840 and 0.862, respectively; for 75 systems), reflecting the dominance of the geometric term in the MolFit scoring function. Importantly, µ is affected considerably less than SNCM by the electrostatic and hydrophobic complementarity terms and therefore the BSA dependency regression lines for SNCM and µ meet at BSA = 783 Å2 for GEH scores compared with BSA = 883 Å2 for the geometric scores (for 75 systems). In addition, inclusion of the electrostatic and hydrophobic complementarity terms reduces ENCM and increases the Sc values in almost every case (by 81 score units on the average); hence, the GEH scores provide a considerable improvement in the potential ability of MolFit to distinguish between NCMs and decoys.
3.3 Ranking by statistically equivalent scores
Estimates of the scoring error due to the grid orientation (for grid interval of 1.05 Å) were obtained as described in Section 2. Initially, we calculated the complementarity scores for 1000 different orientations on the grid for each, the highest ranking NCM and the highest ranking false prediction from the geometric MolFit scans for systems 1ay7 and 1ewy (in both systems the interfaces are small and the NCMs are not ranked at the top). The normal-like distributions of the 1000 scores are very wide (275 score units on the average) and the MolFit scan scores are often found away from the center of the distribution (Fig. 4), highlighting the considerable scoring error. Table 1 shows that the AVs calculated for subgroups of 200 or 100 orientations provide good approximations of the AV obtained for the 1000 orientations (deviations of 0.3% and 0.5–0.6% for subgroups of 200 and 100 orientations, respectively).
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We selected 20 systems (1ktz, 1de4, 1wej, 1ay7, 1fc2, 1mlc, 1ewy, 1fsk, 1ijk, 1hia, 1nca, 1kxq, 1bvn, 1f51, 1dfj, 1wq1, 1f34, 1ibr, 1n2c, 1bgx), which display a wide range of BSAs, preferring systems for which an NCM was not ranked 1 in the MolFit scan, and computed AV and SD for the 1000 top ranking MolFit predictions for each system (with M set to 100). The BSA dependency of the AV values for NCMs is very similar to the BSA dependency of the MolFit scores (R2 = 0.982 and 0.976, respectively, for 20 systems). However, the distribution of the 1000 AV values for each system differs considerably from the distribution of the corresponding MolFit scores, suggesting significant shuffling of the predictions. The average of the differences
= (MolFit score) – AV is positive for all 20 systems tested. This deviation from normality is related to the mode of selection of decoys in the MolFit scan: the highest scoring decoys are likely to be enriched with models whose scores exceed AV. Indeed, for 19 out of the 20 systems the average for the NCMs is smaller than the average for the decoys (data not shown). The SDs are generally larger for NCMs than for the decoys.
Next, we re-ranked the 1000 predictions in each system by statistically equivalent scores (Table 2). The AV values rank the NCMs consistently higher than in the MolFit scan (where possible); re-ranking by AV + n * SD (n = 1, 2) had a small effect that depended on the system. In addition, the Sc values calculated from the AV scores are generally higher than the corresponding Sc for MolFit scores, emphasizing the improved distinction between NCMs and decoys. Another important feature of the ranking by AV is the enrichment of NCMs among the high ranks. Thus, the number of NCMs ranking 1–50 is larger for 18 of the 20 systems (1mlc and 1hia excluded) and the fraction of NCMs ranking 1–50 increases by 20% on the average compared with the MolFit ranking (comparing only the NCMs ranked 1–1000).
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3.4 Comparison with other docking methods
Shape complementarity terms (geometric descriptor) are essential in every docking procedure. We therefore compared the BSA dependency of the MolFit geometric complementarity score and the PatchDock (Schneidman-Duhovny et al., 2003) score. We chose PatchDock because the depiction of the molecules in this algorithm and the scoring of the docking models are different from those in MolFit (not a grid-based method); moreover, the PatchDock server is very fast. PatchDock was run for the 20 systems listed above, employing the default clustering radius. The number of predictions per system varied from 954 to 16 976. NCMs were determined in the same way as for the MolFit models. PatchDock did not identify NCMs for two of the systems (1ktz and 1de4), which were therefore omitted from the analysis. We found that the PatchDock scores for the NCMs are highly correlated with the BSA (R2 = 0.945 for 18 systems) as do the MolFit scores. The values of µ, estimated from plots of the distribution of scores for each system, show considerable dependency on BSA (R2 = 0.852) but with a much smaller slope, resulting in a similar limiting BSA as for the MolFit geometric scans.
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4 DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS4 DISCUSSIONREFERENCES |
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Protein–protein docking is a difficult challenge; the difficulty is mostly attributed to the differences between the conformations of the bound and unbound molecules, which increase the dimensionality of the problem. Herein we show that additional factors may affect the performance of some docking procedures. We concentrate on a basic test in docking, namely, the reassembly of disassembled complexes, and apply our docking program MolFit to a large dataset. We find that the gap between the score of the highest ranking NCM and the mean score for the scan depends on the size of the interface to be predicted. Unfortunately the sizes of biologically relevant interfaces are not correlated with the sizes of the interacting molecules (Chakrabarti and Janin, 2002). Taken together, these relations indicate that small biologically relevant interfaces are indistinguishable from decoy interfaces. The limiting BSA based on the GEH scores is
780 Å2, considerably below the standard interface size in experimental complexes, 1600 Å2 ± 400 (Bahadur et al., 2004; Chakrabarti et al., 2002). The error introduced by the grid may elevate this estimate for the limiting BSA, whereas using a thinner surface layer is likely to improve the distinction between NCMs and decoys in bound docking. Notably, the distinction between biologically relevant interfaces and biologically irrelevant crystal contacts (that may be considered as decoys) using various interface properties such as interface area, polar/non-polar composition, number of hydrogen bonds and residue propensity is not clear cut as the distributions of these properties for the two types of interfaces overlap considerably (Bahadur et al., 2004). Grid-related errors can in principle be reduced by employing smaller grid intervals. Our tests indicate that reducing the grid interval from 1.05 to 0.8 Å extends the range of distinction between NCMs and decoys by 100 Å2 but it does not improve the statistical significance of the NCMs. Docking of small ligands to proteins using different grid intervals (0.25–2 Å) showed that only very high-resolution grid representations (<0.4 Å) produced good predictions (Blom and Sygusch, 1997). Working with such high resolutions is currently unrealistic in protein–protein docking because of the memory load and computation time. We present in this study a general method for correcting grid-related errors, to be used when the resolution is limited. Here it consists of calculating the correlation values for a single docking solution positioned randomly on the grid M times. The average of the M tests provides better discrimination between NCMs and decoys.
The scoring function in MolFit, as in all other docking procedures, is designed to represent the properties of biologically relevant interfaces. Scoring functions consist of various descriptors but shape complementarity is always included, either as an empirical descriptor or as a physical energy term (e.g. van der Waals energy); hence, the BSA dependency described here is likely to be general. Moreover, this dependency is likely to persist in unbound docking because it is not related to the conformations of the molecules. However, larger interfaces often undergo larger conformational changes upon docking (Conte et al., 1999) that may negate the favorable correlation between the docking scores and the BSAs. During the scan stage the same scoring function is used for the selection of NCMs and decoys thereby limiting the ability of the docking procedure to discriminate between them. However, once a selection of models (NCMs and decoys) is obtained, better distinction can be achieved by employing a different set of descriptors that are not related to the size of the interface being predicted.
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Acknowledgments |
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We thank Prof Ephraim Katchalski-Katzir for valuable discussion and support. This study was supported by the Kimmelman center for Biomolecular Structure and Assembly and by the Mazer center for Structural Biology.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Alex Bateman
Received on August 9, 2006; revised on October 1, 2006; accepted on October 6, 2006
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