research papers
How to take advantage of
in `locked' rotation and translation functions^{a}Department of Biological Sciences, Columbia University, New York, NY 10027, USA
^{*}Correspondence email: tong@como.bio.columbia.edu
Many protein molecules form assemblies that obey pointgroup symmetry. These assemblies are often situated at general positions in the
such that the pointgroup symmetry of the assembly becomes (NCS) in the crystal. The presence of NCS places significant constraints on by the molecularreplacement method. The locked rotation and translation functions have been developed to take advantage of the presence of NCS in this which generally requires four steps. (i) The locked selfrotation function is used to determine the orientation of the NCS assembly in the crystal, relative to a predefined `standard' orientation of this NCS (ii) The locked crossrotation function is used to determine the orientation of one monomer of the assembly in the standard orientation. This calculation requires only the structure of the monomer as the search model. (iii) The locked translation function is used to determine the position of this monomer relative to the center of the assembly. Information obtained from steps (ii) and (iii) will produce a model of the entire assembly centered at the origin of the coordinate system. (iv) An ordinary translation function is used to determine the center of the assembly in the crystal using as the search model the structure of the entire assembly produced in step (iii). The locked rotation and translation functions simplify the structuredetermination process in the presence of NCS. Instead of searching for each monomer separately, the locked calculations search for a single rotation or translation. Moreover, the locked functions reduce the noise level in the calculation, owing to the averaging over the NCS elements, and increase the signals as all monomers of the assembly are taken into account at the same time.1. Introduction
Many proteins function as macromolecular assemblies. The monomers in such assemblies are often related to each other by pointgroup symmetry. For example, many protein homotetramers obey 222 pointgroup symmetry, while the protein capsid of icosahedral viruses possesses 532 pointgroup symmetry. When such assemblies are crystallized, the pointgroup symmetry of the assembly may superimpose with the
such that the assemblies are located at special positions in the However, it often happens that the assemblies are located at general positions in the In such cases, the pointgroup symmetry of the assembly exists within the asymmetric units of the crystal and thereby the symmetry of the assembly becomes (NCS) in the crystal.Traditionally, the individual molecules of the assembly are treated separately in the molecularreplacement (MR) calculation, with no assumption of or regard to the NCS of the assembly. However, the presence of NCS introduces significant constraints on structure solution by the molecularreplacement method. A correct solution from the MR calculation must obey the NCS of the assembly. Therefore, it is more appropriate in these cases to constrain the MR calculations such that any solution that is obtained will obey the NCS of the assembly. In other words, such MR calculations are locked to the NCS of the assembly, hence the name locked rotation and locked translation functions. The concept of locked selfrotation function was first proposed in 1972, in the study of the orientations of the tetramer of glyceraldehyde 3phosphate dehydrogenase (Rossmann et al., 1972).
The locked rotation and translation functions offer many advantages over the traditional MR calculations. First of all, a single rotation and translation can define the entire assembly, thereby simplifying the MR calculations. Traditional methods will need to define the orientation and position of each monomer of the NCS and this becomes extremely cumbersome in cases of high symmetry. More importantly, the locked MR calculations consider the contributions of the entire assembly at the same time. This should give rise to stronger signals in the calculation, especially for cases of high NCS. Not surprisingly, the locked rotation function (RF) has found the widest application in virus crystallography, owing to the high NCS that is often involved. However, the locked MR calculations should apply to most cases of NCS point groups.
In cases where the NCS does not belong to a
(improper symmetry), the application of the locked MR calculations becomes more difficult. For the locked RF calculations, the difficulty lies in the definition of the standard orientation of the assembly (see below). This requires knowledge of the orientations of the NCS axes relative to each other, which are generally not known beforehand with improper symmetry. In comparison, for a (proper symmetry) these relative orientations are fixed. In any event, if a standard orientation can be defined, the locked RF can be applied to cases where the NCS is not a On the other hand, the application of the locked translation function is limited to cases where the NCS is a For ease of discussion, here we will consider only cases where the NCS is a point group.2. The locked selfrotation function
When the crystal contains NCS, selfrotation functions (self RFs) are used to determine the orientations of the NCS elements in the crystal ).
Ordinary selfRF calculations make no assumptions about the NCS and determine the orientations of the NCS elements independently of each other. However, often the nature of the NCS is known beforehand. For example, a protein that migrates as a tetramer on gelfiltration columns may form a complex that obeys 222 pointgroup symmetry. Similarly, icosahedral viruses are expected to have 532 symmetry. With knowledge of the possible NCS the selfRF calculations can be locked to this giving rise to the locked self RF (Tong & Rossmann, 1990Three steps are involved in the calculation of a locked self RF.
N − 1 peaks in the ordinary self RF, a single peak is sought in the locked self RF. It must be stressed, however, that this rotation in the locked self RF is a general rotation. For example, for the 222 the rotation [E] can have any κ value (in polar angles). The locked selfRF calculation in this case cannot be limited to the κ = 180° plane, in contrast to the ordinary self RF where κ would normally be fixed at 180°. As the rotation in the locked self RF is a general one, it is generally better to carry out the calculations in Eulerian angles. This also makes it easier to define the unique region of the rotation space (see below).Another major advantage of the locked self RF is that it reduces the noise in the calculation owing to the averaging of the ordinary RF values (2). It can be expected statistically that the noise level in the RF will be reduced by a factor of (N − 1)^{1/2} by the averaging process and this has been shown to be roughly correct based on actual calculations (Tong & Rossmann, 1990). Therefore, for icosahedral viruses roughly an eightfold noise reduction can be achieved with the locked self RF.
The symmetry of the locked self RF is generally rather complicated. It depends on the E] is applied to this standard orientation. If [H] is a 90° rotation around the Z axis, applying the rotation [E][H] to the standard orientation should produce the same orientation of the NCS as applying the rotation [E]. This is owing to the fact that the rotation [H] only swaps the X and Y axes, but does not cause a net change to the standard orientation. Similarly, a 120° rotation around the [111] direction will not change the standard orientation either, as it only causes a cyclic permutation of the twofold axes. Therefore, for 222 pointgroup symmetry, the locked self RF appears to have at least 432 symmetry, as the collection of [H] matrices have 432 symmetry (Tong & Rossmann, 1997). The unique region of rotation space in this case can be defined to cover the regions 0–90° for all three Eulerian angles. More generally, if rotation [H] satisfies the condition [H][I_{n}][H]^{−1} = [I_{m}], applying [E] and [E][H] to the standard orientation will produce the same results. These two rotations are related by the symmetry of the locked self RF. Occasionally, additional symmetry of the locked self RF can be generated by the In practice, the locked selfRF calculations can be rather fast. One should generally cover a large region of rotation space and then classify the resulting solutions based on the orientations of the NCS that they produce.
and the NCS and also depends on the definition of the standard orientation. To illustrate this symmetry, the 222 is used here as an example. Assume that the standard orientation is defined such that the twofold axes are parallel to the Cartesian coordinate axes and that a rotation [3. The locked crossrotation function
The locked self RF has found the widest use so far in macromolecular crystallography, especially for icosahedral viruses, to determine the orientation of the NCS pointgroup symmetry elements in the crystal
The locked crossrotation function (locked cross RF) and the locked translation function can be used to solve the structure of the crystal when the atomic model of only the monomer of the NCS assembly is available. For example, the structure of the monomer may have been determined in a different crystal form, by NMR or other methods, but it is not known how the monomers are arranged in the NCS assembly. Alternatively, it may be possible that the NCS assembly has undergone a reorganization, for example owing to ligand binding, leading to large changes in the relative orientation and position of the monomers in the assembly. In such a case, it is more appropriate to determine the structure of the new assembly with the model of the monomer.With traditional MR methods, the individual monomers of the assembly are treated essentially independently in such a
The orientation and position of one monomer is determined first, followed by the determination of the second and additional monomers. This procedure is not only tedious, it also suffers from having low signals in the calculation, especially for locating the first monomer when the NCS is high. For example, with an assembly obeying 222 symmetry, the first monomer will only account for 25% of the diffracting power of the crystal and this will reduce the signals in both the ordinary RF and TF calculations to locate this monomer.Similar to the locked self RF, one can take advantage of the presence of NCS in such a ).
The entire NCS assembly is considered in the locked calculations, which should increase the signal and reduce the noise. Overall, four steps are needed in the calculations that utilize the NCS (Fig. 1

For the locked cross RF, assume [F] is a rotation that makes the orientation of the monomer search model the same as one of the monomers of the assembly in the standard orientation; the orientations of all the monomers in the crystal is then given by (Tong & Rossmann, 1997),
In other words, [ρ_{n}] represents the (cross) rotational relationship between the monomer search model and the monomers of the assembly in the crystal. Therefore, an ordinary cross RF value R_{n} can be calculated for each of the rotations [ρ_{n}] and the locked cross RF value is defined as the average
Like the ordinary cross RF, the rotation [F] is completely general and can assume any value. The symmetry of the locked cross RF depends on the symmetry of the NCS and the definition of the standard orientation. It is however independent of the as the rotation [F] relates the orientation of the search model to the NCS assembly in a specific crystallographic with its orientation defined by the rotation [E]. The unique region of the rotation space for the locked cross RF can be derived from the fact that rotations [F] and [I_{n}][F] will produce the same set of rotational relationships between the search model and the crystal. Therefore, the unique region of the locked cross RF can be the same as that of an ordinary cross RF between a P1 crystal and a crystal with spacegroup symmetry that is equivalent to the NCS For example, with a 222 tetramer, the unique region of the locked cross RF can be the same as that of an ordinary cross RF between space groups P1 and P222, which has already been defined (Rao et al., 1980). For this correspondence to work, however, the NCS standard orientation must be defined in the same way as that in the equivalent For example, for 422 pointgroup symmetry, the standard orientation must be defined such that the fourfold axis is along the Cartesian Z axis and one of the twofold axes is along the Cartesian X axis.
The definition of the locked cross RF presented here (Tong & Rossmann, 1997) is different from the original one (Tong & Rossmann, 1990), where the rotation [F] relates the orientation of the monomer search model and a monomer of the NCS assembly in the actual orientation in the crystal. While both definitions are functionally correct, the new definition is preferred as it greatly simplifies the understanding of the symmetry of the locked cross RF.
4. The locked translation function
Once the orientation of one monomer of the NCS assembly is defined by the locked cross RF, the orientations of all the monomers of the assembly are defined (3). The next step in the locked MR calculation is to determine how the monomers are positioned in the NCS assembly with the locked TF (Tong, 1996). Ordinary TF calculations are based on comparisons of intermolecular vectors, where the molecules are related by the In contrast, the locked TF calculations are based on vectors among molecules that are related by the NCS. The locked TF does not take into account the of the crystal.
For the locked TF, the center of the NCS assembly is placed (arbitrarily) at the origin of the coordinate system. The rotation [F] that brings the monomer search model into the same orientation as one of the monomers of the NCS assembly in the standard orientation is determined from the locked cross RF. If V_{0} is the translation vector that places this monomer in the same position as the monomer in the NCS assembly in the standard orientation, the entire assembly is defined by (Tong, 1996)
where X_{j}^{0} is the atomic coordinates of the jth atom in the monomer search model. The atomic coordinates of the entire assembly in the crystal centered at the origin, is given by
where [α] is the deorthogonalization matrix (Rossmann & Blow, 1962). The calculated structure factors based on this single NCS assembly in the crystal ignoring the is then
where
The locked TF is based on the overlap between the intermolecular vectors within this NCS assembly and the observed , 1996),
(Tong, 1993where F_{h}^{o} is the observed structurefactor amplitude. A constant term has been omitted in (9) (Tong, 1996).
The equation for the locked TF (9) bears remarkable resemblance to that for the ordinary Patterson correlation translation function (Harada et al., 1981; Tong, 1993), with the interchange of the crystallographic (T_{n}) and NCS ([θ_{n}]) parameters (Tong, 1996). The evaluation of the locked TF is however more complicated. The fast Fourier transform (FFT) method cannot be applied to (9) directly, as the [θ_{n}] matrices are generally nonintegral. Direct summation can be used to evaluate (9), but it would take too much time for most cases. In practice, N(N − 1)/2 FFTs of the form
are calculated first and the locked TF values are then obtained by interpolating among these transforms. In selecting solutions from the locked TF, the packing of the monomers in the NCS assembly is also examined to remove those solutions that cause serious steric clashes among the monomers.
There is no inherent symmetry in the locked TF. The unique region of the locked TF is generally a sphere or spherical shell centered at the origin of the coordinate system, if the monomer search model has been positioned such that its center is at the origin. If the monomer search model is not centered at the origin, the unique region of the locked TF will depend on both the rotation [F] and the position of the center. Therefore, the center of the monomer search model should be placed at the origin for all locked TF calculations. The radius of the sphere is determined by the distance between the center of the monomer and the center of the NCS assembly, which can be affected both by the size of the monomer and by the packing of the monomers in the assembly. Alternatively, the unique region of the locked TF can be defined as a cube centered at the origin. In special cases, the unique region of the locked TF can be limited to two dimensions. For example, if the NCS has sixfold symmetry and the standard orientation is defined such that the sixfold is along the Z axis, only the XY plane needs to be covered in the locked TF calculations.
5. An example
All the locked MR calculations described here are supported in the GLRF program (Tong & Rossmann, 1990, 1997), which is available freely to academic users as part of the Replace program package (Tong, 1993). To illustrate the concept and the application of the locked MR method, the structure solution of a new crystal form of the human malic enzyme is presented here as an example (Yang & Tong, 2000). Malic enzymes are tetrameric in solution and the tetramers obey 222 pointgroup symmetry (Bhargava et al., 1999). The tetramer interface undergoes large reorganizations depending on whether transitionstate analog inhibitors are bound to the enzyme (Xu et al., 1999; Yang et al., 2000). For the example here, the monomer of the enzyme was used as the search model to solve the structure of the enzyme in a different crystal form. This new crystal belongs to the P2_{1}, with a tetramer of the enzyme in the (Bhargava et al. , 1999).
The first step in a locked MR calculation is to determine the orientation of the NCS axes. For this example, the ordinary self RF clearly showed the orientations of the NCS twofold axes, demonstrating the 222 symmetry of the tetramer (Bhargava et al., 1999). For the locked self RF, the standard orientation of the was defined such that the three twofold axes are parallel to the Cartesian coordinate axes. The calculation covered the region 0–90° for each Eulerian angle with a grid interval of 3°. An ordinary self RF was calculated first with the fast rotation function (Crowther, 1972) using reflection data between 10 and 3.5 Å resolution. The radius of integration was 35 Å. The locked selfRF values were then obtained by interpolating in the ordinary selfRF map. The entire calculation took roughly 4 min of CPU time on an SGI O2 R10000 workstation.
The highest peak in the locked self RF stands out from the rest of the peaks, suggesting that it is likely to be the correct solution (Table 1). However, peaks 2–4 also have reasonably high locked selfRF values (Table 1). The orientations of the NCS elements corresponding to each of the top four peaks in the locked selfRF map were then plotted in a stereographic projection and compared with the ordinary self RF (Fig. 2). It clearly shows that the top peak in the locked self RF is the correct solution. However, peaks 2–4 are erroneous, owing to accidental overlap of one of the twofold axes with the correct orientation. Such noises in the locked self RF are expected to be more serious when the NCS is low. When the NCS is high, for example for icosahedral viruses, the background noise is reduced significantly by the averaging and the correct solution is essentially the only peak in the locked self RF (Tong & Rossmann, 1990). In addition, once two noncollinear NCS axes are matched by a rotation, the entire NCS is matched. Therefore, when the NCS is low it may be important to cross check the solution from the locked self RF with the results from the ordinary self RF.

With the knowledge that the top peak in the locked self RF is correct, a fine search was then carried out using 1° intervals around the rotational parameters of this top peak. This produced more accurate parameters for the rotation [E], at 32, 34, 24°.
The locked cross RF was then calculated to determine the orientation of the monomer in the NCS assembly. As with the ordinary cross RF, the monomer search model was placed in a large P1 cell with dimensions of a = b = c = 100 Å and structure factors to 3.5 Å resolution were calculated for this artificial crystal. An ordinary cross RF was calculated with the fast RF (Crowther, 1972), using reflection data between 10 and 3.5 Å resolution and a radius of integration of 35 Å; the locked cross RF values were obtained by interpolating in this map. The entire calculation took roughly 6 min CPU time, covering the region 0–180° in θ_{1} and θ_{3}, and 0–90° in θ_{2}, with 3° grid intervals. The locked cross RF contained one significant peak whose height was about twice that of the second peak in the function (Table 1). This clearly demonstrated that the correct orientation of the monomer has been found. More accurate parameters for the rotation were obtained from a subsequent fine search, with 1° intervals in the three Euler angles.
With the knowledge of the orientation of the NCS assembly (rotation [E]) and the orientation of the monomer in this assembly (rotation [F]), the locked TF was then calculated to determine the position of this monomer relative to the center of the NCS. Reflection data between 10 and 3.5 Å resolution were used in the calculation. The monomer search model was centered at the origin of the coordinate system. The search region was defined as a spherical shell with an inside radius of 15 Å and an outside radius of 40 Å, as it is known from the structures of the other tetramers of this enzyme that the center of the monomer is about 35 Å from the center of the tetramer. The grid interval along the three axes was 1 Å. The calculation took about 6 min CPU time. There was only one significant peak in the locked TF (Table 1) and placing the monomer at this position also gives rise to reasonable packing of the monomers in the NCS assembly. Therefore, this is likely the correct solution from the locked TF. A fine search was then carried out using 0.5 Å intervals to obtain more accurate parameters for the position of the monomer.
At the completion of the locked TF calculation, the GLRF program outputs the atomic model for the entire NCS assembly in the standard orientation and centered at the origin. This model for the NCS assembly was then used in an ordinary TF calculation to determine the center of the NCS assembly in the crystal. The TF program of the Replace package was used in this example (Tong, 1993), using reflection data between 10 and 4 Å resolution. It clearly revealed the location of the NCS assembly in the crystal (Table 1). The top peak has significantly better (CC) and Rfactor values. In addition, there are few steric clashes among crystallographically related molecules based on this solution (Tong, 1993). This confirms that the locked MR calculations successfully determined the structure of this new crystal form of human malic enzyme.
Acknowledgements
This research is supported by a grant from the National Science Foundation (DBI9876668).
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