2.1. Anisotropy correction  

Where a crystal diffracts to different effective resolution limits along different directions in reciprocal space, the crystal is said to diffract anisotropically. The anisotropic variation in intensity restricts the sensitivity of MR functions, particularly maximum-likelihood MR functions. Before undertaking maximum-likelihood MR, it is thus important to computationally remove the anisotropic variation in intensity by applying an anisotropic B-factor correction. This can be thought of as up-weighting the observed structure factors (F _obs) in the direction of weak diffraction and/or down-weighting them in the direction of strong diffraction. Strictly, the correction is not applied directly to the F _obs, but via the reflection-wise normalization factors, Σ_N, that are used to calculate the E values (normalized structure factors) used for all calculations. The low F _obs/σ(F _obs) [E _obs/σ(E _obs)] ratio in the direction of weakest diffraction is accounted for by increasing the sigma of these reflections accordingly. The degree of anisotropy is measured as the difference between the B factors in the directions of strongest and weakest diffraction.

2.2. Model generation (ensembling)  

The coordinates of an MR model are converted to calculated structure factors for comparison with the observed data. In Phaser, this procedure (called ‘ensembling’, as it can be performed with a structurally aligned ‘ensemble’ of homologous models) uses the estimated r.m.s. deviation between the model and the target in the calculation of the structure factors. The initial estimate of the r.m.s. deviation is made via the formula of Chothia & Lesk (1986 ), which relates the r.m.s. deviation of C^α atoms to the fraction sequence identity (f _identity),

In Phaser the minimum r.m.s. deviation is increased to 0.8 Å, so for a fraction sequence identity of higher than 63% the r.m.s. deviation is 0.8 Å rather than the lower value given by the formula. A fraction sequence identity of 50% corresponds to an r.m.s. deviation of C^α atoms of 1.0 Å. In the limit of 0% sequence identity the formula would give a maximum r.m.s. deviation of C^α atoms of 2.6 Å. However, if the r.m.s. deviation estimated from fraction sequence identity is a severe underestimate of the true r.m.s. deviation, MR may fail. In Phaser, the correct MR solution may be rescued by manually entering an increased r.m.s. deviation estimate.

2.3. Rotation function  

In a ‘brute-force’ rotation function, the target function (which ‘scores’ the rotations) is calculated on a grid of orientations and the orientations with the highest score are selected for the next step in MR (which, in Phaser’s ‘automated MR’ mode, is a translation function). ‘Brute-force’ rotation functions are very slow when the target function is the maximum-likelihood rotation function (MLRF). A significant speed improvement is achieved in Phaser by the calculation of an approximation to the full MLRF, the likelihood-enhanced fast rotation function (LERF), via fast-Fourier transform (which is very fast). The LERF is the first term in the Taylor series expansion of the full MLRF and can be thought of as a scaled and variance-weighted version of the Patterson overlap function used in the traditional Crowther RF. The full MLRF contains many (an infinite number of) additional terms, the physical interpretation of which in terms of Patterson functions is more difficult. For example, part of the second term in the Taylor series expansion can be thought of as a ‘Patterson of a Patterson’, with the other part including cross-terms between symmetry-related models with different symmetry operations. For an intuitive interpretation of the full MLRF, it is far easier to consider a random walk of structure factors in reciprocal space rather than trying to find an interpretation in real (Patterson) space (see McCoy, 2004 ). The highest peaks from the fast but poorer scoring LERF are then re-scored with the full MLRF, which gives better discrimination of the correct orientation (Storoni et al., 2004 ). Apart from being more sensitive to the correct solution, the MLRF is also able to easily include knowledge of partial structure, so that MR components that have already been placed can be used to even further improve the sensitivity of the search under way. Inclusion of partial structure information in the rotation function has previously only been possible using Patterson subtraction techniques, i.e. using coefficients |F _o|² − |F _c|² (Nordman, 1994 ; Zhang & Matthews, 1994 ) or coefficients (|F _o| − |F _c|)² (Dauter et al., 1991 ), which suffer from the problem of achieving correct relative scaling between F _o and F _c and consequently have only ever been attempted in a few specialized cases.

2.4. Translation function  

The full maximum-likelihood translation function (MLTF) is the same function as the maximum-likelihood refinement function. As for the MLRF, the MLTF is slow to compute when used as the target function of a ‘brute-force’ search. A speed improvement is achieved in Phaser in the same way as for the MLRF, i.e. an approximation to the full MLTF, the likelihood-enhanced fast translation function (LETF) is calculated by fast Fourier transform and then the top peaks are re-scored with the full MLTF (McCoy et al., 2005 ). The MLTF also makes good use of partial structure information to enhance the signal from the search under way.

2.5. Packing function  

The packing of potential solutions is checked using a C^α clash test. Each C^α position is tested for the presence of any C^α from another model that is within 2 Å (a clash). The search includes other components in the asymmetric unit, their symmetry-related copies and symmetry-related copies of the model under consideration. Only potential solutions that have a number of clashes less than the user-specified number (default zero) are accepted for the next step (in Phaser’s ‘automated MR’ mode, refinement). The number of accepted clashes should be increased when the search model has low sequence identity with the target or has large flexible loops that could adopt an alternative conformation. However, it is best to edit the model so as to remove flexible loops and allow only a small number of clashes, as packing provides a very powerful constraint on the translation function.

2.6. Refinement  

Rotation-function and translation-function searches are on a grid of orientations and positions. However, the best orientation and position need not (and in general will not) lie exactly on this grid. In addition, for the rotation function, the true orientation may be a shoulder of a peak in the rotation search rather than exactly where the peak maximum indicates. Therefore, a rigid-body refinement is performed to optimize the orientation and position of a model. This can greatly improve the likelihood score for a given solution from marginally above the noise level to a solution with a significant signal-to-noise ratio.

3.1. BETA–BLIP  

The case of β-lactamase (BETA)–β-lactamase inhibitor (BLIP) has been used repeatedly as a test case for Phaser (Storoni et al., 2004 ; McCoy et al., 2005 ) because the original structure solution by MR using AMoRe (Navaza, 1994 ) was difficult even though good models were available (the structures of both components had already been solved in isolation; Strynadka et al., 1996 ; Fig. 4 a). The difficult part of the MR solution was placing BLIP.

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Figure 4

(a) Structure of the β-lactamase (BETA)–β-lactamase inhibitor (BLIP) complex. BETA is in blue and BLIP is in yellow. (b) Structure of the ROP four-helix bundle structure. The asymmetric unit is shown in red and crystallographically related molecules are shown in white. Together, they form two four-helix bundles. The search model was a 26-residue polyalanine helix. (c) The 15 molecules in the asymmetric unit for the V_κ antibody fibre. The molecules form a continuous fibre along the 6₄ axis in the crystals (space group P6₄22). (d) The AP2 complex of four proteins. The α subunit (a superhelix of α-helices) is shown in red, the β2 subunit in blue (a similar superhelix of helices), the σ2 subunit in cyan (mixed α-helix/β-sheet structure) and the μ2 subunit in magenta (which consists of an N-­terminal domain structurally homologous to the σ2 subunit and a larger C-terminal mixed α-­helix/β-sheet structure).

The command script for the solution of BETA–BLIP using the ‘automated MR’ mode of Phaser is shown in Appendix A1. The search order is given as BETA and then BLIP. This is because BETA, with 62% of the molecular weight, would be expected to have the highest fraction scattering (and indeed it does, as the B factors for BETA and BLIP are comparable). Phaser rapidly produces a correct solution for the complex. This previously difficult structure solution becomes trivial because of two algorithms implemented in Phaser. The first is the anisotropy correction; there is significant anisotropy in the data (the maximum B-factor difference in different directions is 32 Ų). The second is the improved rotation-function target in MLRF, particularly in that the solution for BETA can be used to find the correct rotation-function solution for BLIP. Using the traditional Crowther (1972 ) fast rotation function, the Z score for the correct BLIP placement is 3.8 and the top Z score of 4.4 corresponds to an incorrect placement. Using MLRF and the prior knowledge about the placement of BETA, the correct placement of BLIP has a Z score of 6.5 and is the highest score in the search. (These results are for data that have had the anisotropy correction applied, to illustrate the improvement given by the MLRF alone.)

This example is illustrative of the case where one component of the asymmetric unit is easy to find in isolation and another is difficult or impossible to find. Knowledge of the partial structure of the component that is easy to find, introduced using the maximum-likelihood algorithms, enables the complex to be easily built up by addition.

3.2. ROP four-helix bundle  

The A31P mutant of ROP forms a helix–turn–helix motif that homodimerizes to form a four-helix bundle. The asymmetric unit contains two copies of the helix–turn–helix motif (Glykos & Kokkinidis, 2003 ; Fig. 4 b). The structure was originally solved with a 26-residue polyalanine single helix as the model and an extremely computationally intensive 23-dimensional Monte-Carlo search implemented in the program Queen of Spades (Glykos & Kokkinidis, 2001 ).

The command script for the solution of the ROP four-helix bundle using the ‘automated MR’ mode of Phaser is shown in Appendix A2. The r.m.s. deviation for ROP is given as 1.0 Å, since the sequence identity of the polyalanine helix is not a valid estimation of the r.m.s. deviation between the polyalanine helix and the backbone atoms of the ROP structure. The r.m.s. deviation value of 1.0 Å is a reasonable guess. Phaser produces eight solutions after a tree search with hundreds of branches, especially in the search for the second of the four helices. The eight solutions are nearly identical, differing only in the registration of the model to the structure helices (i.e. the C^α residues slip up or down the helix). All eight solutions generate a complete structure after model building with ARP/wARP (Perrakis et al., 2001 ). Although many potential solutions are stored in the intermediate stages of the search, the search itself is not computationally intensive.

This example is illustrative of the case where there are multiple components of the asymmetric unit and a poor signal for each component in isolation. The correct solution for placing the first few components was only found after placing the last component. After the search for first few components, there were a large number of branches on the tree and it was not apparent that a solution would eventually be found. However, the ‘tree search with pruning’ strategy, when left to run to completion, found the correct solution with the placement of the last model, when the signal of the correct solution finally became significant.

3.3. V_κ antibody fibre  

Unlike the previous two examples, other MR software had not solved this example of an aggregation-prone antibody variable domain of the kappa subgroup (V_κ) prior to structure solution with Phaser (James et al., 2006 ). Structures of antibody domains are of course well known; it was the association of the domains in the aggregate that was of interest in this structure. The aggregation-prone V_κ antibody domain crystallized in space group P6₄22 in a unit cell such that the Matthews coefficient (Matthews, 1966 ) indicated that there were most probably 18 molecules in the asymmetric unit. The search model had 98% sequence identity to the target structure, i.e. three point mutations, which give it the tendency to aggregate.

The first command script for the solution of the V_κ antibody domain using the ‘automated MR’ mode of Phaser is shown in Appendix A 3.1. In this step, only one of the potential 18 molecules in the asymmetric unit was searched for, the aim being to determine whether or not there was any signal in the search for a single molecule. Rather surprisingly, this step produced two placements with much higher Z scores than all the others. In this case, the signal from the single component was significant because subsets of V_κ domains had similar orientations and so there was a signal for these orientations in the rotation function. Using this clear rotation-function signal, the top Z score from the translation function was 19.2 and the second was 16.0. This indicated that the problem was solvable by short-cutting the automated MR job. The structure was therefore solved by manually editing the output ‘solution’ files from Phaser and checking the packing of the resulting solutions with Phaser’s ‘packing’ mode, as described below.

Since the first step produced two clear solutions, the two solutions were combined into a single solution by editing the ‘solution’ file as described in Appendix A 3.2. However, before proceeding to the searches for more molecules, it was necessary to check the packing of the two components in the solution by running Phaser’s ‘packing’ mode (Appendix A 3.3) and only accepting the subset of placements that had no clashes. The packing test showed that the two placements packed with no clashes and thus no placements needed to be deleted. This ‘solution’ was then used as the background of the search for the next molecule (Appendix A 3.4). Three more rounds of searching, manual editing of the solution files and checking of the packing gave a solution with 15 molecules in the asymmetric unit (Fig. 4 c). The noncrystallographic symmetry of the 15 molecules and the crystallographic symmetry along the 6₄ axis form a continuous chain of V_κ antibody domains, showing that the aggregation assembly is a fibre structure.

This example is illustrative of the case where there are multiple copies of the same component in the asymmetric unit and there is a signal from the search for individual components. The ‘tree search with pruning’ strategy is suboptimal in this case because the tree has multiple branches, each with a subset of the complete solution. The solutions only converge onto one branch (solution) with the placement of the last component on each of the branches. In this case the optimal search strategy is to add multiple components at each search step (rather than branching at each search step), but this search strategy must currently be performed semi-manually as described above. If there is no signal from the search for individual components, it is necessary to perform the search using the full tree search with pruning strategy as described in test case 2 (ROP four-helix bundle).

I thank my co-authors of Phaser, Randy Read, Laurent Storoni and Ralf Grosse-Kunstleve, and collaborators in the Phenix consortium. I am also grateful to Michael James and Natalie Strynadka for supplying the data for the BETA–BLIP test case, Nicolas Glykos and coworkers for making the ROP four-helix bundle test case available, Leo James and my other collaborators on the V_κ antibody fibre and David Owen, my collaborator on AP2.

A1. BETA–BLIP  

The script for running the ‘automated MR’ mode of Phaser to obtain a solution for the BETA–BLIP complex test case is shown below.

A2. ROP four-helix bundle  

The script for running the ‘automated MR’ mode of Phaser to obtain a solution for the ROP four-helix bundle test case is shown below.

A3. V_κ antibody domain  

Solution of the V_K antibody domain was achieved by short-circuiting the automated MR mode of Phaser as described in the text.

A3.1. Round 1  

The script for running the ‘automated MR’ mode of Phaser to search for one copy of the V_κ domain in the asymmetric unit is shown below. Two placements with high Z scores were found from this search. The ‘solution’ file output by this script has the name round1.sol.

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A3.2. Solution file from the ‘round 1’ script  

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A3.3. Edited ‘solution’ file from the ‘round 1’ script  

The two solutions given in the round1.sol file were combined into a single solution by editing the file to remove the second SOLUTION SET line; in Phaser, SOLUTION SET commands delineate separate solutions.

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A3.4. Packing  

The packing of the two placements in the edited solution file was checked using the script below, which uses Phaser’s @ pre-processor command to include the data from the edited round1.sol file.

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A3.5. Round 2  

The script for running the ‘automated MR’ mode of Phaser to search for one copy of the V_K domain in the asymmetric unit in the presence of the two molecules found in the round 1 search is shown below. The solutions found from this run of Phaser were added to the solution set as described and the search continued until no more molecules could be found.

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