August 10, 2004 20:11 Proceedings Trim Size: 9in x 6in psb. Fast Geometric Algorithm for Rigid Protein Docking. Y. Wang. ¡. , P. K. Agarwal¢, P. Brown.
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Fast Geometric Algorithm for Rigid Protein Docking
Y. Wang, P. K. Agarwal, P. Brown , H. Edelsbrunner and J. Rudolph
Duke University, Durham, North Carolina
Abstract. In this paper, we present an efficient algorithm to generate a reasonably small but reliable set of coarse protein docking configurations by aligning meaningful features on protein surfaces. A protein is considered as a rigid body in our approach, and the output produced can serve as input data for other local improvement methods that allow protein flexibility. We demonstrate the performance of our algorithm by testing our algorithm on a diverse set of protein complexes from the Protein Data Bank.
1. Introduction Motivation. Highly organized transient or static assemblies of proteins control most cellular events. A better understanding of the protein-protein interactions involved in these assemblies would help elucidate how individual proteins form complexes and dynamically function together to generate the cell circuitry and its time-dependent responses to external stimuli. Protein structures determined at atomic resolution by X-ray crystallography, nuclear magnetic resonance, and increasingly by computer modeling provide one basis for the study of protein interactions. However, given the relative wealth of structural details for monomeric proteins compared to multimeric protein complexes, there exists a need for computational tools to predict protein docking. Prior work. Current research on protein-protein docking has been focused on either bound docking (the reassembly of known protein complexes), or unbound docking with fairly small protein conformational changes. Most approaches for unbound docking consist of two stages, with bound docking as a subroutine 19: (i) bound-docking stage, which produces a set of potential docking configurations
All authors are supported by NSF under grant CCR-00-86013. JR and HE are also supported by NIH under grant R01 GM61822-01. PA is also supported by NSF under grants EIA-01-31905 and CCR-02-04118 and by U.S.-Israel Binational Science Foundation. Department of Computer Science. Departments of Computer Science and Mathematics. Departments of Computer Science and Mathematics, and Raindrop Geomagic. Departments of Biochemistry and Chemistry.
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by considering only rigid motions; and (ii) refinement stage, which allows certain flexibility. The two essential components in both stages are: (i) a scoring function that can discriminate near-native docking configurations from incorrect ones; and (ii) a search algorithm to find (approximately) the best configuration under the scoring function exploited. In general, approaches for the bound-docking stage rely mainly on geometric complementarity. Many approaches discretize the search space of rigid motions, using methods such as exhaustive search3 , fast Fourier transform (FFT)6 11 15, and spherical harmonics16 17 . A second type of approach samples a small number of rigid motions non-uniformly by aligning points from molecular surfaces10 12 14 . The idea goes back to Connolly9, who also proposed to identify feature points by taking the minima and maxima of a certain function, widely known as the Connolly function. One representative here is by Fischer et al. 10 who use geometric hashing technique to align points computed by a variant of the Connolly function. The refinement stage is usually modeled as an energy-minimization problem, with the scoring functions focusing more on the thermodynamic aspects of the interaction. Even if we consider only sidechain flexibility, the problem is still quite complex as the dimension of the search space is high19. Recently, Vajda et al. have proposed a hierarchical, progressive refinement protocol 4 5 , which seems to reliably converge to a near-native docking configuration starting with initial configurations up to ˚ rmsd away. Little success has been achieved for including backbone conformational changes18. New work. Since each step in the refinement stage is quite costly, it is crucial to generate a small and reliable set of potential configurations in the bound-docking stage. On the other hand, even though only rigid motions are considered in this stage, the time complexity of current approaches is still not satisfactory, preventing us from experimenting with larger data sets. In this paper, we present an efficient algorithm for the bound-docking stage. We use geometric complementarity to guide us to search for a small set of rigid motions so that the two proteins fit loosely into each other. Such a set of potential configurations can be further refined independently, using both chemical information and flexibility 5 8 , to obtain more accurate docking predictions. We remark that for the case of unbound docking, it is especially important to consider coarse (not tight) fits between input components in order to provide enough tolerance for flexibility in the later refinement procedure. We describe our algorithm, called
, in Section 2. It relies on a novel approach to describe meaningful “features” on molecular surfaces, such as
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protrusions and cavities, using a succinct set of pairs of points computed from the elevation function as defined by Agarwal et al.1 . We then align such pairs and evaluate the resulting alignments by a simple and rapid scoring function. Compared to other similar approaches10 14 that align the so-called feature points, our algorithm inspects orders of magnitude fewer transformations by aligning only meaningful features to produce a reliable set of potential docking positions. In Section 3, we demonstrate the performance of our approach by testing a set of 25 bound protein-complexes obtained from the Protein Data Bank 2 . We also show with the local improvement prothat by combining our algorithm 8 cedure described in , we are able to efficiently find accurate near-native docking positions for all but one case without any false positives . Additionally, we have tested our algorithm on the unbound protein docking benchmark 7, demonstrating that can generate useful protein poses that can serve as input for refinement methods that take into account protein flexibility. We conclude and discuss future work in Section 4. 2. Methods
and Assume that we are given two proteins !#"$ &% (' *)+,.- ) &/)' , where each (resp. ) represents " 103254 - )607284 % an atom centered at (resp. ) with van der Waals radius (resp. /) for produces a docking configuration ( > ). At a high level, we first construct a set A1B (resp. ADC ) of features based on elevation function as defined in1 , each consisting of two points along with a nor mal direction, to characterize the molecular surface of protein (resp. ). These two sets of features, A B and A C , are the inputs to our coarse alignment algorithm which then outputs E , a list of possible configurations sorted by their scores. Below, we start with describing the scoring function that we use. Scoring function. A good scoring function should produce higher scores for (near) native configurations than for non-native ones. We design our scoring function to describe the geometric complementarity between and : Let
q
TLd FGIHJLKMONQP#R S T U8HVWVWXWYQXWHZ[MONQP\R S^]!` ac _ kmb l T dTLd b
degfeihjP if fnodpP if P otherwise
A false positive is a configuration with high score but far from native configuration.
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";
- )
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where , and is some prefixed constant that we re fer to as contact-threshold. The scoring function between and involves two ; // Sort > by 0213 ;
) 5 +/6)879.,. ; /
is valid
Figure 2. The coarse alignment algorithm.
good fits between the input proteins should have some features aligned, such as a protrusion from one surface fitting inside a cave from the other. Hence if we align all “features” from to those from , we will cover potentially good fits. Here features are defined as in Figure 1 (b). As we wish to align only important features, we preprocess A B and A C by removing features with small elevation value or short length. We now explain steps (1), (2) and (3) from the above algorithm. ' @? 3? ' Function . The? function A -A ' ?6 computes CB ,Bj ' a rigid motion > that aligns the pair of points onto ? ? . In particular, assume that and are the normals associated with and respectively. Let
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D ]FE HG L K I' J K L K K P
and
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E MG L O IN J O QL P O O
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AA
To obtain the transformation, we first translate the of segment A2A onto B[ B A 2Amidpoint that of segment . Next, we rotate segment so that (i) segment lies B B % % on the line passing through ; and (ii) coincides with . See Figure 3 for AA an illustration. Note that there is ambiguity in (i) as vector I'J can either be in
Figure 3. First move the midpoints (two empty dots) together. Then rotate so that
coincide. Last, rotate 7 w.r.t. so that coincides with .
and
B jB
the same direction or in opposite direction as vector INJ . As such, the function .' in fact returns two distinct transformations, although for simplicity, we present as if there were only one.
?
5'
?
Function . Obviously, if and are fairly “dissimilar”, then they will not align well with each other in any good configuration. By dissimilar we mean that either we are trying to align a protrusion (resp. cave) from one B B surface with a protrusion (resp. cave) from the other, or, the length of is too @? 3? ' A A different from that of . Function returns false if either of above two conditions hold.
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?=
?=
'
'
Compute 9;: and . By definition, 9;: and ' can be computed trivially in time, where and are the number of atoms ' ' in and respectively. We compute them in time by building a hierarchical data structure for and .
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3. Results In this section, we first provide a detailed study on one protein complex. We next test our docking algorithm on a diverse data set of 25 bound protein complexes chosen from the Protein Data Bank 2 . Finally, we provide results for testing the unbound protein benchmark7.
A case study. We take the protein complex barnase/barstar (pdb-id 1brs, chain and ), as an example. The two chains have 864 and 693 atoms respectively. We use the msms program (also available as part of the VMD software13) to generate a triangulation of the molecular surface for each input chain. The resulting triangulations have and !#"$ vertices respectively. In the left of Table 1, we show the number of features generated from the four different types of max-
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A D
index-2
index-3
index-4
1044 828
696 510
156 154
index-2
index-3
index-4
279 224
402 307
78 83
A D
ima of the elevation function: A feature is of index if it comes from a -legged maximum. In the right, we show the number of features whose length is at least ˚ : these features are the inputs to our coarse alignment algorithm. We note that significantly higher percentage of index-3 features are longer than ˚ , compared to those of index-4 and especially index-2 features. Given the two sets of features that are longer than ˚ and whose elevation ! are greater than ! , algorithm
generates a family E of valid configurations whose score is higher than ( ! ! valid ones whose score is greater than ). The running time is around 3 minutes on a single processor PIII 1GHz machine. Each configuration in E corresponds to a transformation > for chain . We compute the rmsd (root mean squared deviation) between the ' centers of all atoms from chain and those of > , and use it to measure how good our predictions are: the smaller the rmsd of a configuration is, the closer it is to the native docking position. The top ranked configuration in E has a rmsd of , and 6 out of the top 10 configurations have a rmsd smaller than " ˚ . Next consider only a subset E , the top 100 ranked configurations from E . We refine each coarse alignment in E by applying the local improvement heuristic8 and re-rank E afterwards based on the new scores. The results as shown in Figure 4 demonstrate that generates multiple useful poses for chains A and D that can be refined to yield a near-native final configuration.
After LocalImprove rank 1 2 3 4 5 6 7 8 9 10
score 359 338 328 314 311 310 307 281 251 213
rmsd 0.54 0.80 0.72 0.80 0.91 0.78 1.50 1.47 2.09 39.96
Before LocalImprove rank 12 5 1 4 2 59 3 11 14 76
coll. 24 48 23 49 39 12 29 18 16 29
rmsd 3.23 2.42 1.59 3.57 1.70 2.84 2.32 3.07 3.00 39.39
Figure 4. Top 10 ranked configurations after local improvements (only those with 4 H; kept), and the corresponding ranking beforehand. coll. means the collision number.
are
More bound protein complexes. We extend our experiments on a diverse set of 25 bound protein complexes obtained from the protein data bank. For each complex, we generate a set of features for each input chain as in the case of 1brs, the size of which is usually roughly the same as the number of atoms of that given these inputs. In Table 5, we show chain. Next, we perform
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rank
coll.
rmsd
time
1A22 (A, B) 1BI8 (A, B) 1BRS (A, D) 1BUH (A, B) 1BXI (B, A) 1CHO (E, I) 1CSE (E, I) 1DFJ (I, E) 1F47 (B, A) 1FC2 (D, C) 1FIN (A, B) 1FS1 (B, A) 1JAT (A, B) 1JLT (A, B) 1MCT (A, I) 1MEE (A, I) 1STF (E, I) 1TEC (E, I) 1TGS (Z, I) 1TX4 (A, B) 2PTC (E, I) 3HLA (A, B) 3SGB (E, I) 3YGS (C, P) 4SGB (E, I)
2 12 1 5 3 1 2 78 15 5 11 1 522 8 1 1 1 9 1 2 1 1 1 6 10
23 43 11 14 34 14 22 11 1 49 44 29 20 23 27 23 43 54 46 4 18 19 38 7 33
2.75 2.48 1.52 1.85 2.54 2.71 2.21 3.09 1.49 4.13 3.70 1.62 1.20 3.64 3.49 1.33 1.18 3.07 2.61 3.35 4.55 1.87 3.21 1.07 2.33
20 26 3 2 8 3 9 27 1 6 41 5 9 10 3 9 8 7 6 14 6 16 5 6 4
Figure 5. For each protein, the rank of the first configuration with low rmsd (smaller than ˚ ). Coll. represents collision number. The last column of the table shows the running time of algorithm 4 in minutes.
a low-rmsd configuration for each protein complex, as well as its rank (by score) ˚ ! in the set of configurations output by our algorithm, with parameters o and . Note that in all but one case, we have at least one configuration with low rmsd among the top 100 configurations. The last column shows the running time for algorithm
, which does not include the time to compute the molecular surfaces and the elevation function. For each protein complex, we next perform the local improvement heuristic 8 on the top configurations, and re-rank them based on new scores. The results are illustrated in Table 6 where we choose (other than for the case . In all but of 1JAT). We consider only those configurations with one case, the top-ranked (or the No. 2 ranked in one case; 1FC2) configuration after local improvement has a small rmsd. In the only remaining case, we can obtain a small rmsd for top-ranked configuration if we relax . In other words, in 23 out of these 25 test complexes, our coarse alignment algorithm, when combined with the local search heuristics from8, can predict an accurate near-native docking configurations without false positives.
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pdbid
rank
1A22 1BI8 1BRS 1BUH 1BXI 1CHO 1CSE 1DFJ 1F47 1FC2 1FIN 1FS1 1JAT 1JLT 1MCT 1MEE 1STF 1TEC 1TGS 1TX4 2PTC 3HLA 3SGB 3YGS 4SGB
2 62 2 5 16 1 23 78 15 5 34 2 522 3 84 1 1 10 2 80 1 1 1 6 10
Before Improve score coll. rmsd 270 211 389 209 217 243 273 178 129 356 382 309 168 232 233 373 408 362 227 243 265 246 320 216 260
23 10 37 14 21 14 36 11 1 49 54 27 21 14 34 23 43 51 13 25 18 19 38 7 33
2.75 30.0 3.01 1.85 5.59 2.71 2.57 3.09 1.49 4.13 9.94 1.59 1.20 6.17 3.57 1.33 1.18 4.51 2.71 4.34 4.55 1.97 3.21 1.03 2.33
After Improve rank score rmsd 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
475 234 349 256 289 305 317 220 221 200 413 326 288 310 322 372 314 304 348 355 314 416 257 209 266
1.08 29.88 1.07 0.61 0.63 0.99 0.82 1.28 0.56 1.33 0.61 0.89 0.87 1.77 0.32 0.57 0.79 1.28 0.44 0.36 0.66 0.70 2.24 0.85 2.50
Figure 6. For each protein, column 2 – 5 shows the first configuration with low rmsd after locally improve the top coarse alignments and re-ranking, with coll. means the collision number. Columns 6 – 9 show the corresponding configuration before local improvements and re-ranking.
on the proteinUnbound protein benchmark. We further tested
protein docking benchmark provided in7 . We omitted the seven complexes classified as difficult in7 because they have significantly different conformations in the unbound vs. bound structures. We also omitted complexes 1IAI, 1WQ1 and 2PCC due to difficulties in generating molecular surfaces of required quality. Of the 49 remaining complexes, 25 are so-called bound-unbound cases where one of the components is rigid. For each complex, we fix one chain as , which is the rigid chain for the bound-unbound cases, or the receptor for the unbound-unbound cases. We generate E , a set of the potential positions for the other component, . 0 For each transformation > E , we measure the rmsd between the interface ' atoms from and those from > , and refer to it as rmsd . As the unbound structures (i.e, the input to our algorithm) provided in the benchmark are superimposed onto their crystalized correspondances, this value is close to the rmsd measured ' between > and the crystalized structure for . Now take E , the top 2000
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C-id
Top 2000 rmsd
hits
rank
3.70 5.51 4.66 1.60 3.04 2.35 3.15 6.44 7.65 2.78 5.27 7.95 6.55 4.55 4.04 6.34 4.13 1.41 3.78 4.50 1.42 5.94 9.38 1.95 4.59 3.71 6.27 6.98 2.31 6.49 3.47 5.99 4.52 2.48 2.21 1.75 7.18 1.97 3.46 1.08 4.06 2.75 2.91 5.95 1.52 2.48 2.83 5.02
20 8 35 7 5 27 7 2 2 4 18 3 26 32 27 10 10 8 3 3 5 2 1 5 7 7 3 11 1 8 18 2 12 7 8 7 7 4 19 11 8 2 40 3 8 3 3 2
3,951 4,698 1,629 426 695 92 15,271 1,433 10,721 1,561 543 8,268 2,560 4,983 76 4,894 37 1 48 1,124 6 396 2,781 1,189 710 6949 4,659 10,388 11 11,783 14,185 1,495 153 321 73 621 14,202 12 115 1 4,243 1136 19,167 3950 1 25,307 3,260 617 10,132
1ACB 1AVW 1BRC 1BRS 1CGI 1CHO 1CSE 1DFJ 1FSS 1MAH 1TGS 1UGH 2KAI 2PTC 2SIC 2SNI 1PPE 1STF 1TAB 1UDI 2TEC 4HTC 1AHW 1BVK 1DQJ 1MLC 1WEJ 1BQL 1EO8 1FBI 1JHL 1KXQ 1KXT 1KXV 1MEL 1NCA 1NMB 1QFU 2JEL 2VIR 1AVZ 1L0Y 2MTA 1A0O 1ATN 1GLA 1IGC 1SPB 2BTF
All outputs rmsd
1.75 5.42 4.66 1.60 3.04 2.35 2.74 6.44 5.15 2.78 5.27 7.16 3.41 4.16 4.04 4.58 4.13 1.41 3.78 4.50 1.42 5.94 4.37 1.95 4.59 3.32 5.86 4.39 2.31 2.30 2.61 5.99 4.52 2.48 2.21 1.75 2.72 1.97 3.46 1.08 3.52 2.75 2.07 4.35 1.52 2.82 2.06 2.83 3.28
size 14,426 23,565 12,770 11,607 10,135 11,815 21,068 35,231 25,609 25,402 11,383 14,656 13,478 13,929 20,065 15,830 7,660 15,082 8,296 21,133 21,134 14,032 32,919 24,611 28, 694 29,747 18,194 23,308 45,512 26,036 32,091 37,218 39,240 46,368 17,741 49,600 42,066 47,693 34,072 40,813 7,895 34,044 36,903 9,113 50,729 33,879 25,303 13,728 33,480
Figure 7. Protein-protein benchmark. C-id means complex-id.
ranked configurations from E . The results are shown in Figure 3, where we show in column 2 the smallest rmsd generated from E , and in column 3 the number of configurations from E with a rmsd smaller than ˚ . Columns 4-5 provide information (rank and rmsd ) of the configuration in E with the smallest rmsd , and the size of E is shown in column 6. Our results demonstrate a number of favorable characteristics for a coarse rigid body protein docking algorithm. First, within the relatively small set of ! top-scoring configurations ( E ), 38/49 complexes yield a configuration below ˚ rmsd . All but one complex yield a configuration below the ˚ cut-off needed as input for the hierarchical, progressive refinement protocol in4 5 . The fact that most complexes generate multiple hits increases the probability that a local refinement will not be trapped in a local minimum and instead find a correct solution. Second, within all the configurations generated ( E , at most 50,000), 47/49 complexes yield a configuration below ˚ , typically within ˚ , in the top 10,000 scores. All 49 generate at least one configuration below
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at most ! scores. How these coarse alignments re-rank to yield high scoring solutions with low rmsd remains to be investigated. We also remark that it is possible to futher reduce the size of E by clustering similar configurations 10. 4. Discussion We have presented in this paper an efficient alignment-based algorithm to compute a set of coarse configurations given two rigid input proteins. We have shown in Section 3 that when combined with the local improvement heuristics from 8 , our algorithm can predict an accurate near-native docking configurations for 23 out of 25 test bound protein complexes without false positives. When tested on the unbound protein docking benchmark7, our algorithm is able to rapidly produce a relatively short list of potential configurations which can be inputs to other local improvement emthods that allow protein flexibility and thus have the potential to solve the unbound protein docking problem. Comparisons. Current approaches for the bound docking stage differ in how they sample the search space and how they evaluate the docking score. FFT-type methods discretize the search space in a rather uniform way. They produce more accurate predictions for recombining known complexes, but at a much higher computational cost. For the case of unbound docking, it is possible to run those algorithms at a low resolution to provide some tolerance for flexibility. However, if the resolution is too low, there is a danger of missing good alignments completely; while if the resolution is high, too many configurations will be generated and the selection of a small set of good candidates for the refinement stage is not a trivial problem. Alignment-based methods tend to sample the search space in a more selective manner guided by shape complementarity. Such earlier methods align feature points residing at protrusions and cavities. In particular, given two sets of such points representing and , they align all possible pairs of points from one set with all possible pairs from the other to generate potential rigid motions. Since the number of feature pairs is of the same order as that of feature points, these 4 algorithms inspect significantly more transformations than ours ( vs. ) due to meaningless ones and duplicates. In contrast, by aligning features computed from the elevation function, we sample the transformational space in a much sparser manner than previous approaches, focusing only at potentially good docking lo; ! cations. The size of the output of our algorithm is much smaller (with configurations without clustering for 1BRS), and as they are generated by fitting features, we expect to capture reasonable configurations unless proteins undergo dramatic conformational changes. We also comment that it is possible to further improve the speed of our algorithm by the geometric hashing technique as in 10 .
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Acknowledgement. The authors would like to thank Vicky Choi for the local
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improvement software. References 1. P. K. Agarwal, H. Edelsbrunner, J. Harer, and Y. Wang. Extreme elevation on a 2manifold. In ACM Symposium on Computational Geometry, 2004. 2. H. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. Bhat, H. Weissig, I. Shinkdyalov, and P. E. Bourne. The protein data bank. Nucleic Acid Res., 28:235–242, 2000. 3. S. Bespamyatnikh, V. Choi, H. Edelsbrunner, and J. Rudolph. Accurate protein docking by shape complementarity alone, 2004. 4. C. J. Camacho, D. W. Gatchell, S. R. Kimura, and S. Vajda. Scoring docked conformations generated by rigid-body protein-protein docking. Proteins, 40:525–537, 2000. 5. C. J. Camacho and S. Vajda. Protein docking along smooth association pathways. Proc. Natl. Acad. Sci., 98:10636–10641, 2001. 6. R. Chen, L. Li, and Z. Weng. ZDOCK: An initial-stage protein docking algorithm. Proteins, 52(1):80–87, 2003. 7. R. Chen, J. Mintseris, J. Janin, and Z. Weng. A protein-protein docking benchmark. Proteins, 52:88–91, 2003. 8. V. Choi, P. K. Agarwal, H. Edelsbrunner, and J. Rudolph. Local search heuristic for rigid protein docking. In 4th Workshop on Algorithms in Bioinformatics, 2004. 9. M. L. Connolly. Shape complementarity at the hemo-globin albl subunit interface. Biopolymers, 25:1229–1247, 1986. 10. D. Fischer, S. L. Lin, H. Wolfson, and R. Nussinov. A geometry-based suite of molecular docking processes. J. Mol. Biol., 248:459–477, 1995. 11. H. A. Gabb, R. M. Jackson, and M. J. Sternberg. Modeling protein docking using shape complementarity, electrostatics and biochemical information. J. Mol. Biol., 272(1):106–120, 1997. 12. B. B. Goldman and W. T. Wipke. QSD: quadratic shape descriptors. 2. Molecular docking using quadratic shape descriptors (QSDock). Proteins, 38:79–94, 2000. 13. W. Humphrey, A. Dalke, and K. Schulten. VMD– Visual Molecular Dynamics. J. Molec. Graphics, 15:33 – 38, 1996. 14. H. Lenhof. An algorithm for the protein docking problem. Bioinformatics: From Nucleic Acids and Proteins to Cell Metabolism, pages 125–139, 1995. 15. G. Moont and M. J. E. Sternberg. Modelling protein-protein and protein-dna docking. Bioinformatics – From Genomes to Drugs, 1:361–404, 2001. 16. D. W. Ritchie. Evaluation of protein docking predictions using hex 3.1 in capri rounds 1 and 2. Proteins, 52(1):98–106, 2003. 17. D. W. Ritchie and G. J. L. Kemp. Protein docking using spherical poloar fourier correlations. Proteins, 39:178–194, 2000. 18. B. Sandak, R. Nussinov, and H. J. Wolfson. A method for biomolecular structural recognition and docking allowing conformational flexibility. J. Comput. Biol., 5:631– 654, 1998. 19. G. R. Smith and M. J. E. Sternberg. Prediction of protein-protein interactions by docking methods. Current Opinion in Structural Biology, 12:29–35, 2002.
