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electronic reprint Journal of

Applied Crystallography ISSN 0021-8898

Editor: Anke R. Pyzalla

Optimal side-chain packing in proteins and crystallographic refinement Swanand Gore and Tom Blundell

J. Appl. Cryst. (2008). 41, 319–328

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Many research topics in condensed matter research, materials science and the life sciences make use of crystallographic methods to study crystalline and non-crystalline matter with neutrons, X-rays and electrons. Articles published in the Journal of Applied Crystallography focus on these methods and their use in identifying structural and diffusioncontrolled phase transformations, structure–property relationships, structural changes of defects, interfaces and surfaces, etc. Developments of instrumentation and crystallographic apparatus, theory and interpretation, numerical analysis and other related subjects are also covered. The journal is the primary place where crystallographic computer program information is published.

Crystallography Journals Online is available from journals.iucr.org J. Appl. Cryst. (2008). 41, 319–328

Gore and Blundell · Optimal side-chain packing

research papers Journal of

Applied Crystallography

Optimal side-chain packing in proteins and crystallographic refinement

ISSN 0021-8898

Swanand Gore* and Tom Blundell Received 17 December 2007 Accepted 16 January 2008

# 2008 International Union of Crystallography Printed in Singapore – all rights reserved

Department of Biochemistry, University of Cambridge, 80 Tennis Court Road, Cambridge CB2 1GA, UK. Correspondence e-mail: [email protected]

Amino acid side chains often adopt one of a few distinct, physicochemically favourable conformational states called rotamers. Rotameric preferences and compact packing are sufﬁcient to estimate the conformations of most side chains, as demonstrated by approaches such as SCWRL in homology modelling, but such algorithms have not yet been applied to protein crystallographic reﬁnement. SCWRL’s combinatorial optimization algorithm was adapted for assigning side-chain rotameric states that maximize the electron density map occupation while minimizing steric clashes. Our program (OPSAX) was tested on ﬁve proteins by introducing error in main chains and comparing the subsequent CNS-only and CNS/OPSAX reﬁnements. The latter reﬁnement was also extended to multiconformer models. A sequence-assignment exercise examined whether CNS/OPSAX reﬁnement can discriminate between correct and incorrect assignments at various artiﬁcially lowered resolutions. The composite CNS/OPSAX reﬁnement yielded better Rfree values than CNS-only reﬁnement. A further drop in Rfree was observed with multiconformer reﬁnement. For complete main chains, the correct sequence could be discriminated efﬁciently in most cases even for a low observation-to-parameter ratio of 4, indicating that the OPSAX approach should ﬁnd useful applications in protein X-ray reﬁnement.

1. Introduction Amino acid side chains are vital to the folding, function and dynamics of proteins because they are the main ingredients of physicochemical diversity therein. Two salient features of side chains are their rotamericity and their compact packing in globular protein domains. Rotamericity is the tendency to be in one of a few, distinct, locally optimal conformations and was hypothesized from small-molecule analysis (Sasisekharan & Ponnuswamy, 1971). With the increasing size of the Protein Data Bank (PDB), this hypothesis was statistically conﬁrmed by many studies, leading to the compilation of rotamer libraries for use in homology modelling (Tuffery et al., 1997; De Maeyer et al., 1997; Dunbrack & Cohen, 1997; Lovell et al., 2000). Rotamer libraries describe a set of discrete states for side chains and their propensities, dependent on or regardless of the backbone conformation. Packing of side chains in globular protein domains is complementary, compact and like a jigsaw puzzle (Richards, 1973; Levitt et al., 1997). Taken together, packing and rotamericity can determine the conformations adopted by side chains. This has been demonstrated by side-chain placement programs (De Maeyer et al., 1997; Canutescu et al., 2003; Xu, 2005; Smith et al., 2007), which ﬁnd the optimal balance between rotamer propensity score, van der Waals energy and sometimes evolutionary information. These methods are capable of ﬁnding the global J. Appl. Cryst. (2008). 41, 319–328

minimum of a composite function consisting of the self-energy of rotameric states (rotameric propensity, conservation scores from homologous templates) and pairwise energies (van der Waals energy between close rotamers). They rely on techniques such as dead-end elimination, branch-and-bound and graph decomposition to reduce the search space and avoid futile energy calculations. Side-chain optimization programs are valuable in homology modelling because the main-chain conformers can be predicted from templates more accurately than side chains, in addition to the fact that side chains are densely packed. For the same reasons, they could also be useful in protein crystallography, for example, in the following scenarios. Firstly, reﬁnement of a model with an approximately known main chain can be automated with an optimal side-chain assignment procedure. Secondly, side-chain heterogeneity can be gauged by running the procedure several times and performing a multiconformer reﬁnement. Thirdly, hypotheses about sequence registry of a full or fragmented main chain can be quickly tested. Here we assess the value of optimal side-chain assignment in these crystallographic scenarios. We begin by describing the side-chain placement procedure (termed OPSAX, for optimal side-chain assignment for crystallography) and its implementation within the RapperTK framework (Gore & Blundell, 2007). Then we describe and compare the composite (CNS/OPSAX) reﬁnement protocol doi:10.1107/S0021889808001672

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Figure 1 A side-chain graph (left) is prepared by checking which side chains have clashing rotamers. This graph is decomposed into biconnected subgraphs and the articulation vertices are identiﬁed. The biconnected subgraphs can be represented as a graph (right); two subgraphs are connected if they share an articulation vertex. The biconnected subgraphs are ordered using a topological sort and solved in that order. The ﬁnal rotamer assignment is found by traversing the subgraphs in the reverse order.

against the CNS-only protocol (Bru¨nger et al., 1998) and show that the former generally outperforms the latter when reﬁnement statistics are compared. Multiconformer reﬁnement with two and four copies is performed to observe a further improvement in reﬁnement. We then ask whether the correct sequence registry can be efﬁciently discriminated with simulated low-resolution data sets for complete and fragmented main chains.

chosen because it seems to be large enough to allow averaging over the surrounding density while avoiding density that belongs to atoms not in the rotamer. The pairwise energy of rotamers of two different side chains is deﬁned as 1 or 0 depending on whether they sterically clash or not. The van der Waals radii of rotamer atoms are reduced by 50% while calculating the steric clashes, following previous work (DePristo et al., 2003). This is necessary because our experience shows that using the full radius of side-chain rotamers leads to excessive steric clashing, sometimes preventing allatom protein modelling. (3) Rotamers are further removed using dead-end elimination (Goldstein DEE; Goldstein, 1994). For a side chain, a set of neighbouring side chains which may sterically clash with it are identiﬁed (denoted Nbrs). Then two rotamers a; b of this side chain are compared by checking whether ða; iÞ ðb; iÞ þ

Nbrs P j6¼i

minc ða; i; c; jÞ ðb; i; c; jÞ > 0;

ð1Þ

where ða; iÞ is the self-energy of the rotamer a of residue i, and (a; i; c; j) is the pairwise energy of rotamer a of residue i and rotamer c of residue j. If this relation holds, then rotamer a is said to be dominated by rotamer b and is removed from further calculations. (4) Side-chain pairs with the possibility of steric clash are connected to form a side-chain graph (Fig. 1). A connected graph is biconnected when removal of any one vertex does not disconnect the graph. Articulation vertices are those that belong to more than one biconnected subgraph in a graph.

2. Side-chain reassignment procedure This procedure is inspired by SCWRL (Canutescu et al., 2003), a popular and very effective approach to optimal side-chain reassignment, which we ﬁnd difﬁcult to better in the absence of the use of evolutionary information. For clarity and completeness, we describe our implementation, which is a new module written in our conformationsampling software RapperTK. (1) We use rotamers from the Penultimate Rotamer Library (Lovell et al., 2000) in this work. For each side chain to be assigned, steric clashes of all of its rotamers with the rest of the system are checked. All clashing rotamers are removed. (2) The self-energy of a rotamer is the average value of the electron density in the 2Fo Fc map over grid ˚ of atoms in points that lie within 1 A ˚ the rotamer. A 1 A neighbourhood is

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Figure 2 Branch-and-bound in OPSAX. For each rotameric state of the articulation vertex of a biconnected subgraph with N + 1 nodes, there are N assignments to be decided in order to solve the component. This can be viewed as an N-level tree, each level corresponding to an assignment. A search of this tree can be pruned to make it more efﬁcient. The best assignment in the tree seen so far is remembered (Ebest ). At the current stage of a search, the best possible energy from the subtree below is computed and added to the present energy. If this sum is worse than Ebest then the subtree is not searched further.

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research papers Such biconnected subgraphs and articulation vertices are identiﬁed by using the Boost Graph Library (Siek et al., 2002). The biconnected subgraphs and articulation points can be viewed as vertices and edges in a complementary graph, which is then topologically sorted to order the biconnected subgraphs. These subgraphs are then solved in that order. (5) A biconnected subgraph has an articulation vertex assigned to it. Solving a subgraph means ﬁnding the best rotamer assignment for the subgraph for each rotamer of the articulation vertex. A biconnected subgraph is solved with a branch-and-bound technique. In principle, the number of possible rotamer assignments in a subgraph grows exponentially with the number of vertices. This can be viewed as a tree (Fig. 2), but most of the leaves of the tree are not feasible due to steric clashes. In order to avoid visiting all possible rotamer assignments for the components, at each stage of the search, the best possible energy from unassigned nodes in the component is calculated in the context of currently assigned nodes. This is the bounding energy: P PP mina;b ði; a; k; bÞ: ð2Þ Ebound ¼ mina ði; aÞ þ i>j

DePristo et al. (2004). In CNS-only reﬁnement, the trace is reﬁned with ten rounds of CNS, each round consisting of three cycles of molecular dynamics/simulated annealing (MDSA) starting from 5000 K. At the start of CNS/OPSAX composite reﬁnement, C and side-chain atoms are removed and deposited structure factors are phased with the main chain. Side chains are reassigned within this map after calculating the

i>j kj

If Ebound þ Ej > Ebest then the current subtree of search is abandoned as it is not possible to obtain a solution better than that already known from that subtree. When a subgraph is solved, the internal energy of each rotamer of its articulation vertex is replaced by the minimum energy of the subgraph with articulation vertex in that rotameric state and the corresponding subgraph rotamer assignment is remembered. (6) After all subgraphs have been solved in this way, the biconnected subgraphs are visited in reverse order. For each articulation vertex assigned to the subgraph, the least-energy rotamer is now known. This process is used to assign rotamers to all vertices; this is the global minimum assignment for this energy function. (7) Unconnected side chains are assigned the least-energy rotamer.

Figure 3 Single-conformer reﬁnement ﬂowchart. The difference between the CNSonly (left) and CNS/OPSAX (right) protocols is that the ill-ﬁtting side chains are identiﬁed and reassigned in the latter.

3. Single- and multiconformer refinement 3.1. Refinement protocols

The goal of this exercise is to compare the CNS-only and CNS/OPSAX reﬁnement procedures to assess the value of the side-chain placement procedure (Fig. 3). The deposited structure is perturbed by carrying out an all-atom C trace (DePristo et al., 2003; Gore & Blundell, 2007) on the structure. C tracing is a procedure that samples a protein chain sequentially from N to C termini using a genetic algorithm cum branch-and-bound (GABB) technique. It employs a residue-speciﬁc ﬁne-grained ’– library and a rotamer library for sampling the main chain and side chains, respectively. Every C atom is restrained positionally within a sphere centred on the given C coordinates and having a radius called the C restraint radius or the trace radius. Side chains can also be positionally restrained in this procedure, but that has not been done here. After tracing, B factors of all protein atoms are reset to 30, a typical value of B factor, similar to J. Appl. Cryst. (2008). 41, 319–328

Figure 4

Multiconformer reﬁnement. 2Fo Fc maps are calculated using the composite model and equal partial occupancies. A composite model at the ith stage is split into single conformers. Ill-ﬁtting side chains in single conformers are identiﬁed and reassigned using the composite map. Rebuilt conformers are combined into a multiconformer model and reﬁned with CNS in three cycles of MDSA starting at 5000 K. Gore and Blundell

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research papers C atom positions. This is followed by similar CNS reﬁnement as a CNS-only protocol. In subsequent CNS/OPSAX rounds, all side chains are ﬁrst scored by the correlation coefﬁcient ˚ of between Fc and 2Fo Fc maps on grid points around 1 A the side-chain atoms. Only the side chains with a correlation

lower than 0.9 are reassigned. Whenever a rotamer is reassigned, its B factors are set to 30; otherwise they are copied from the previous round. In both protocols, structures with the best Rfree are considered for further analysis. The CNS/OPSAX protocol is extended to multiconformer models with two and four copies. In the two-copy case, two C traces are generated and stripped of their side chains and C atoms. They are assigned a partial occupancy of 0.5 and deposited structure factors are phased with this model. C and side-chain atoms are built for both traces with this map. Allatom models with partial occupancy are reﬁned with CNS. In subsequent rounds, in a way similar to single-conformer models, each copy is assessed to detect ill-ﬁtting side chains and rebuilt independently of others, but subjected to CNS reﬁnement together as a multiconformer model. This is depicted in Fig. 4. A similar protocol is replicated for the fourcopy case. Single and multiconformer reﬁnements are carried out over a range of C trace radii. Reﬁnement with each radius is repeated ﬁve times. The data set used in this work is shown in Table 1.

Figure 5 The improvement in Rfree calculated with the side-chain placement protocol as compared with a CNS-only protocol for (a) 9ilb, (b) 1aac, (c) 1kx8, (d ) 1g35 and (e) 8cho. Each CNS run consists of two cycles of MDSA starting at 5000 K with isotropic B factor reﬁnement and a maximum likelihood target.

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research papers 3.2. Single-conformer refinement

Fig. 5 shows the mean Rfree value plotted against the trace radius for all proteins. For the single conformer case, we observe that the Rfree values are consistently better in CNS/ OPSAX reﬁnement than in CNS-only reﬁnement over all trace radii, with the improvements more noticeable in 1aac,

1g35 and 8cho. Fig. 6 shows the comparison of rotameric states between the deposited structure and single-conformer models. The 1 and 1;2 values of the structure and a model are compared by deﬁning 1 to be similar for two corresponding residues if they are within 40 of one another, and by deﬁning 1;2 to be similar if both 1 and 2 for two corresponding residues are within 40 of one another. Clearly, the rotamer accuracies depend on the quality of the main chain. Fig. 6 shows that the CNS/OPSAX procedure models rotamers similar to those in the deposited structure when the resolution is high, but this is not so as the resolution decreases; when compared with rotamer statistics of the CNS-only protocol, the rotamer statistics of CNS/OPSAX are better for 1aac and 1g35, comparable for 1kx8, and worse for 9ilb and 8cho. This is perhaps expected because the rotamericity of low-resolution structures deposited in the PDB tends to be lower than that of the high-resolution structures, and OPSAX uses only the rotameric side chains in the reassignment procedure. Fig. 7 shows the reduction in rotamer accuracies as resolution drops. The HIV protease structure (1g35) does not follow this trend, but it can be explained by the existence of a large ligand (inhibitor AHA024). In the CNS/OPSAX

Figure 6

Side-chain (1 , 1;2 ) accuracy statistics for CNS/OPSAX and CNS-only protocols for (a) 9ilb, (b) 1aac, (c) 1kx8, (d ) 1g35 and (e) 8cho. J. Appl. Cryst. (2008). 41, 319–328

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research papers Table 1 The ﬁve-protein set used in this work. #AA denotes the number of amino acid residues. CDPI is the Cruickshank DPI value [Murshudov & Dodson, 1997, equation (3)]. The number and percentage of loop residues are also listed. PDB id 1aac 1g35 8cho 9ilb 1kx8

Name

#AA

No. (%) of loop AA

Amicyanin HIV protease 5-3-Ketosteroid isomerase Interleukin- Chemosensory protein A6

105 198 174

64 (61) 95 (48) 40 (23)

1.31 1.80 1.47

21199 19059 7439

0.002 0.022 0.114

153 109

82 (54) 20 (18)

2.28 2.80

9535 5096

0.036 0.261

Resolution ˚) (A

No. of reﬂections

CDPI ˚) (A

reﬁnement protocol, the ligand is copied from the deposited structure at the beginning and then copied from the previous to the present iteration. This indirectly informs CNS about incorrectly placed rotamers. With the exception of this outlier, lowering of mean rotamer accuracy and an increase in its standard deviation is a clear tendency and indicates that lower-resolution data accommodate greater variability. The improvement in the CNS reﬁnement as a result of using the OPSAX procedure is primarily due to the fact that CNS alone rarely relocates bulky side chains into better density. Because of the gradient-driven nature of CNS, relocation into better density is likely only if there is a continuous favourable gradient towards a correct solution, but this is sometimes not the case because of intermediate regions of low density. OPSAX relocates such side chains whenever permitted by rotamericity and excluded volume restraints. An ill-ﬁtting side chain will be moved out of a patch of density if there is another side chain which ﬁts better. This process enriches the model with more well placed rotamers and leads to better reﬁnement. However, the best Rfree statistics do not indicate native-like reﬁnement, but rather show a dependence on the quality of the main chain provided at the start of reﬁnement. This is due to the fact that OPSAX does not modify incorrectly placed parts of main chain. For this reason, it is not often that reﬁnement statistics become comparable to those of the native, but in the rare instances when they do, the corresponding models can indicate the structural heterogeneity. For instance, in the 1kx8 case (see Fig. 8), we observed that three of the 15 models generated had comparable Rfree values (0.287, 0.287, 0.284) to the native (0.283), yet the differences between the deposited structure and the models were different for each model. This example suggests that OPSAX can be a quick way to identify the regions of structural heterogeneity. 3.3. Multiconformer refinement

Previous work (e.g. Wilson & Brunger, 2000) has identiﬁed two ways to describe structural heterogeneity better than with a single-conformer isotropic B factor model: a singleconformer anisotropic B factor model or a multiconformer isotropic B factor model with partial occupancies on each

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Figure 7

Accuracies of (a) 1 and (b) 1;2 as a function of resolution. The accuracy generally drops with resolution and the sizes of the error bars increase, indicating greater variability accommodated by low-resolution data. Note that the increasing order of resolution is 1aac, 8cho, 1g35, 9ilb and 1kx8. The bump for 1g35 is due to a large ligand.

conformer. The previous section suggested that a collection of single-conformer isotropic B factor models generated with the CNS/OPSAX protocol can give clues about structural heterogeneity. This encouraged us to extend the protocol to multiconformer reﬁnement for deriving an ensemble. Multiconformer reﬁnement is generally expected to yield better reﬁnement statistics than its single-conformer counterpart because the reﬁnement process has more parameters to ﬁt to the experimental data. Fig. 5 shows that this is true in all ﬁve cases for the two-copy reﬁnement, but contrary to expectations, four-copy reﬁnement is not consistently better than two-copy reﬁnement. This can be explained as follows: (a) all conformers have different and erroneous main chains, (b) owing to the nature of the derivative calculation procedure, errors in a conformation are not corrected as vigorously as in a single-conformer case, (c) over-ﬁtting may occur as a result of there being too many parameters, and (d ) the relationship between multiple conformers and their combined density is not as clear as a single conformer and its density.

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research papers In spite of these difﬁculties, sometimes the reﬁnement yields Rfree values slightly lower than the native model and the resulting ensemble may be considered as good an interpretation of the diffraction data as the deposited model. We ﬁnd four such cases, two each for 1kx8 and 1aac; the two cases for both proteins consist of a two- and a four-conformer reﬁnement (Fig. 9). Both ensembles show variability in the same regions, lending credibility to the interpretations. The fourconformer ensemble shows more variation than the twoconformer ensemble. An interesting case of concerted conformational change is observed in the four-conformer 1kx8 case (Fig. 10). Residues Tyr98, Trp81, Leu73 and His72 are systematically different in one of the copies, while the other copies are very similar to each other. It is unclear whether this is an artefact of the CNS/OPSAX protocol or a genuine

concerted movement. In any case, the observed ensemble reﬁnements suggest that the protocol developed here is useful to derive multiconformer interpretations.

4. Sequence assignment

Typical protein crystallographic reﬁnement begins with approximate identiﬁcation of a complete or fragmented main chain, followed by sequence identiﬁcation and detailed allatom iterative reﬁnement. At medium and low resolutions, sequence assignment on an approximate main chain becomes difﬁcult because electron density contours may not contain much shape information and side-chain densities may be absent or intermerged. Pattern matching techniques, exploitation of knowledge of the structures of homologues and trialand-error all play a part in addressing this problem. In theory, all possible assignments can be made and reﬁnement carried out to ﬁnd the correct one, but it is easy to see that even ﬁve mainchain fragments create a combinatorial explosion which cannot be addressed with brute force. Here we investigate whether an efﬁcient testing of a sequence assignment is possible at low resolution when the main chain is known only approximately. The same ﬁve proteins used in previous sections are used here, but with artiﬁcially lowered resolution such that the observation-to-parameter ratio drops to three. The observation-toparameter ratio is calculated as the ratio of the number of observed X-ray reﬂections to the number of protein atoms. Resolution is lowered by gradually removing the high-resolution reﬂections till the desired observations˚ to-parameter ratio is obtained. On a 2 A C-traced main chain, 21 different sequences are assigned and reﬁned. The 21 sequences consist of the correct sequence and the correct one offset by up to ten residues in either direction. The terminal region left vacant by offsetting is assigned a random sequence. The CNS reﬁnement carried out here is shorter than the previous protocol: a round consists of two cycles of MDSA starting at 3000 K. After initial sequence assignment, ﬁve rounds of such CNS/OPSAX reﬁnement are carried out. A sequence discrimination exercise consists of 21 sequence trials Figure 8 and it is repeated 25 times to obtain a Structural heterogeneity identiﬁed for 1kx8. (a) The residue-wise all-atom r.m.s. deviation in the statistical estimate of the success rate in three models, with Rfree comparable to the deposited value. (b) The deposited model as green sticks and the three models as lines. Heterogeneity occurs at different residues in different models. identiﬁcation of the correct sequence. J. Appl. Cryst. (2008). 41, 319–328

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Figure 9 Two-conformer (left) and four-conformer (right) ensemble interpretations of 1aac (top) and 1kx8 (bottom), having a lower Rfree value than the deposited structure.

Figure 11 (a) Success rates of registry recognition and (b) average time required for reﬁning a main chain for a registry, for various X-ray observation-toparameter ratios. Observations are removed gradually, those in higherresolution shells before those in lower-resolution shells. All main chains ˚ restraints. For a main chain, 20 incorrect were generated with 2 A registries are tried in addition to the correct one. Each CNS run consists of two cycles of MDSA starting at 3000 K with isotropic B factor reﬁnement with a standard residual target.

Figure 10 Conformational variability in the 1kx8 multiconformer reﬁned model with four copies. The Rfree values are comparable for the multiconformer and deposited models. Tyr98, Trp81, Leu73 and His72 are signiﬁcantly different in one of the copies (sticks) than the other three (lines) and the deposited model (red lines). This can be interpreted as the concerted motion occurring rarely.

An exercise is successful if Rfree for the correct sequence is lower than those of the other 20 incorrect sequences. Fig. 11 shows the success rate (i.e. discriminating power) in identifying correct sequences as a function of the X-ray observation-to-parameter ratio. For even a small X-ray observation-to-parameter ratio of 4, the discriminating power is more than 80% for four of the ﬁve proteins. In other words, in four of ﬁve attempts, the correct sequence reﬁnes to a smaller Rfree than other sequences. Thus the CNS/OPSAX protocol is highly effective in identifying a correct sequence.

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As expected, the discriminatory power increases with increasing X-ray observation-to-parameter ratio. The time required to reﬁne a combination of sequence and main chain for an X-ray observation-to-parameter ratio of 4 is around 20 min, i.e. around 7 h for 21 sequence/main-chain combinations, suggesting that the protocol is also efﬁcient. Investigation of the 1aac case where the protocol does not work well revealed that too many X-ray observations were removed in the artiﬁcial lowering of resolution, and the protocol worked satisfactorily when more data were used, e.g. the correct registry was identiﬁed 60 and 80% of the time for X-ray observation-to-parameter ratios of 10 and 20, respectively. We attempted to extend this exercise further to fragmented main chains. We assumed that secondary structure elements were approximately known and we repeated the same procedure as before to estimate the discriminatory power. Loop residues were not included in the model during any round of reﬁnement. Our results conﬁrm our expectation that

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Figure 12 Success rates of registry recognition in the fragmented main-chain case. Only the residues in secondary structure elements are used in the reﬁnement. Discrimination of the correct sequence is correlated to the fraction of the secondary structure residues.

this case is more challenging than the complete main chain (Fig. 12). For a low-resolution scenario, the success rate is better than 50% only for 1kx8. Even when all reﬂections are used, only 1kx8 and 8cho have satisfactory success rates. This observation is consistent with the secondary structure content: both 1kx8 and 8cho have nearly 80% of the residues in secondary structure compared with around 50% for the others (Table 1). This suggests that in the fragmented main-chain case, the packing restraint is lost because of the exclusion of side chains and main chains of loop residues, and the packing restraint is important to drive reﬁnement to a nonrandom Rfree . Perhaps rotameric propensities would be helpful here to compensate for loss of packing, and this approach needs to be further investigated.

5. Conclusions One of the motivations for this work was to experiment with orderless sampling. In Rapper’s GABB algorithm (DePristo et al., 2003), there is a ﬁxed N-to-C order on sampling. In RapperTK, this order can be suitably changed but not eliminated. Indeed the existence of sampling order is essential to keep the sampling tractable. However, some scenarios where sets of conformational samples of residues are independent of one another allow orderless sampling. Side-chain sampling is a classic example of this. The limitations of this approach are computational in nature; for instance, we noted that it takes a long time to solve large biconnected components, as observed by Canutescu et al. (2003). The availability of an orderless sampling algorithm makes RapperTK a more useful software package and complements the GABB algorithm, extending its applicability. OPSAX can be extended beyond amino acid side chains, to entities such as ﬂexible ligands or main-chain fragments adopting a limited set of conformations. We have shown that orderless sampling as implemented in OPSAX is useful for protein crystallography: (a) side-chain J. Appl. Cryst. (2008). 41, 319–328

optimization can be used in conjunction with CNS to carry out automated reﬁnement starting from approximate main chains; (b) multiconformer reﬁnement improves reﬁnement statistics and may help in speculating concerted conformational changes; (c) efﬁcient discrimination of a correct sequence assignment is possible at low resolution with an approximate main chain. Where Rfree of a single- or multiconformer model is comparable to Rfree of the deposited structure, our approach is useful to assess the structural heterogeneity inherent in the structure afforded by the diffraction data. Single-conformer reﬁnements lead to collections and multiconformer reﬁnements lead to ensembles. A simple application of the CNS/ OPSAX protocol might be to obtain both collections and ensembles. Another interesting avenue would be to detect rotameric heterogeneity and build multiple side chains instead of one. Efﬁcient discrimination of a correct sequence, given an approximate main chain and low-resolution data, is a promising result. However, we also realize that the same procedure was not so promising for fragmented main chains owing to loss of packing. Further work will need to ﬁnd ways to simulate the packing effect that missing loops would provide, e.g. building missing loops ﬁrst before sequence assignment, use of rotameric propensities in the energy function, more realistic treatment of van der Waals interactions etc. From the computational perspective, the OPSAX algorithm can be improved by identifying articulation edges and components, especially when the biconnected subgraphs are large. Large subgraphs can be divided into more manageable subgraphs this way and many interesting scenarios can be addressed, which would otherwise be computationally prohibitive, e.g. taking into account long-range correlations between side-chain rotamers such as NMR NOEs and electrostatic energy. To conclude, this work demonstrates that optimal side-chain assignment is useful for crystallographic reﬁnement and points to ways to adapt it better to multiconformer and low-resolution applications. This approach and the applications described here will be made freely available online in the near future. We thank Dr Nick Furnham and Anjum Karmali for helpful discussions. SG thanks Cambridge Commonwealth Trust and Universities UK for funding his PhD studentship, and Mr Anand Chitipothu for technical help.

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Optimal side-chain packing

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J. Appl. Cryst. (2008). 41, 319–328

electronic reprint Bis(tetraethylammonium)