research papers
– historical background
^{a}Department of Biological Sciences, Purdue University, West Lafayette, IN 479071392, USA
^{*}Correspondence email: mgr@indiana.bio.purdue.edu
A review is given of the mathematical procedures required for a molecularreplacement
These apply equally to the more frequently encountered situations where a known homologous structure can be used as a search model and to phase determination in the presence of (NCS). In general, the former represents improper NCS between two different unit cells, whereas the latter occurs when there is proper NCS within one unit cell.Keywords: molecular replacement.
1. Introduction
Although the concept of ) on what is now known as the molecularreplacement (MR) method, the first recorded use of the term `molecular replacement' was, as far as I can remember, in the title of my book on this subject (Rossmann, 1972), published ten years later. The summary that I wrote for this book in 1972 remains relevant and will serve as a component of the following text. I have indicated the original text by using italics. Numerous other reviews on MR have also been written (Lawrence, 1991), including some by myself (Rossmann, 1990, 1995) and by Eddy Arnold and myself (Rossmann & Arnold, 1993). In addition, there have been at least two Daresbury Study Weekends (Machin, 1985; Dodson et al., 1992) devoted to MR. Furthermore, the independent development by Walter Hoppe and colleagues of the Faltmolekülmethode should not be forgotten (Hoppe, 1957a,b).
(NCS) was introduced in the first paper (Rossmann & Blow, 1962Undoubtedly, the most significant change in emphasis since the 1972 publication (Rossmann, 1972) is as a result of the now very large PDB database. Whereas in 1972 most structural studies had to employ NCS for ab initio phase determinations, today the most frequent use of MR is to solve a structure when that of an analogous molecule or domain is already known. Indeed, it is to be anticipated that with the high level of funding for structural genomics, that there will soon be available systematic search methods that compare an unknown structure with a representative set of known protein folds.
My intent in coining the description `molecular replacement' was to cover all methods that utilize NCS whether within or between crystal forms. However, comments by Phil Evans and Gerard Bricogne during the 2001 CCP4 meeting indicated that they preferred to limit the term `molecular replacement' to the case where an unknown structure is to be solved with a known search molecule. As the computational procedures have much in common in all uses of NCS, I will here continue to employ the term MR in the manner I had first intended (Appendix A).
2. Genesis of molecular replacement
By the summer of 1960 only two proteins had been solved: myoglobin at 2.0 Å resolution and hemoglobin at 5.5 Å resolution. A number of other proteins were being studied, but none had given up its secrets. The principal difficulty lay in the preparation of good singlesite heavyatom derivatives. It was necessary either to make some advance in the alchemy used in the preparation of the derivatives or to find an alternative method if the two respiratory proteins were not to stand unchallenged for many years. Some handle was needed which was a natural property of these large molecules and which did not depend on some artificial device as is implied in the isomorphous replacement technique.
David Blow and I had already shown that much could be performed with only one heavyatom derivative (Blow & Rossmann, 1961), but the enormity of the impasse was still considerable. While discussing these problems with colleagues, at the time of the International Union of Crystallography held in Cambridge in 1960, it occurred to me that both the ability of proteins to crystallize in different space groups and their frequent property of being made up of identically folded polypeptide chains might form a basis for an alternative process in the solution of the phase problem.
Many larger protein molecules are made of identical, or closely similar, subunits. Plausible causes for such aggregates of the protein parts of virus structures were set out by Crick & Watson, 1956). Changes in the amino acid sequence of functionally similar proteins produce only minor structural alterations indicating that the evolution of divergent functions can occur without large changes in structure.
3. Defining NCS
Crystallographic and noncrystallographic symmetry are differentiated by the property that an operator applies throughout the whole infinite crystal for the former, whereas a noncrystallographic symmetry element relates only to a localized volume within a crystal. For instance, in Fig. 1 there are noncrystallographic twofold axes in the plane of the paper valid only in the immediate vicinity of each line. However, the crystallographic twofold axes perpendicular to the plane of the paper apply to every point within the crystal. Two types of can occur. A `proper' element is independent of the sense of rotation. An example of this kind is a fivefold axis in a virus. Clearly a rotation of the crystal either left or right by onefifth of a revolution will leave all parts of a given virus coat (but not the whole crystal) in equivalent positions. Improper rotation axes are found if two molecules are arbitrarily oriented in the same or are in two entirely different crystal lattices. It follows that any noncrystallographic operator that possesses an element of translation must be improper. Because proteins consist only of Lamino acids and have only Dribose rings, NCS in crystals of biological macromolecules consists exclusively of rotation axes (with or without translational components).
4. Utilization of NCS
Molecular replacement utilizes the similarity or identity of structure in different parts of the crystallographic asymmetric unit caused by a repetition of the same subunit structure in the formation of a whole molecule. Molecular replacement may also utilize the relationship between differing crystal forms of the same or similar molecules. Three principal stages in determining a structure in the presence of NCS are as follows.

5. Early results and problems
It was fortunate that, by the time I had formulated the general concepts of molecular replacement, I had also established a remarkable collaboration with David Blow. His natural caution and skepticism caused me to put rigor into my derivations; his willing ear and astute criticisms gave me sufficient encouragement to proceed. Encouragement came also from Max Perutz who first suggested to Dorothy Hodgkin that the rotation function might determine the relative orientation of the two crystallographically independent molecules in rhombohedral insulin. The peak in the ensuing rotation function of R3 insulin (Fig. 2) was most pronounced (Rossmann & Blow, 1962) and was the beginning of a happy collaboration on the application of the molecular replacement method to insulin.
There were, however, formidable objections to the solution of the translation problem and possibly even greater obstacles in the solution of the phase problem. Frances Crick was active in pointing out that the amount of translation required to superimpose two identical objects, after they had been similarly oriented, would depend on the position of the axis of rotation. How then could there be a unique solution of the translation problem at all? For a time this appeared to be an insurmountable hurdle, but gradually, as we considered the specific translation problem of R3 insulin, the difficulties became less formidable. Nevertheless, some of the biggest unsolved problems (even today) still lie in the determination of equivalent origins in differing crystal forms of equivalent molecular species; that is in the solution of the translation problem in the presence of improper noncrystallographic symmetry, in the absence of a known search model.
Frances Crick and Max Perutz had in addition serious objections to the third stage, the solution of the phase problem. It had been pointed out that the information presented by identical subunits which were different crystallographically was similar to that obtained by sampling the reciprocal lattice at nonintegral lattice points. This is readily understood when it is shown (see Appendix B) that the rotation, represented by x′ = [C]x, of the position vector x to x′ in real space corresponds to the rotation of the reciprocallattice vector h to h′ given by h′ = [C^{T}]h. Thus, the integral reciprocallattice point h will become a nonintegral reciprocallattice point h′ at the same resolution. The situation was indeed similar to the hemoglobin shrinkage stages investigated by Perutz (1952) (Fig. 3). When the sign of the molecular transform in the centric h0l zone changes, then a discontinuity results in the plot of its magnitude. Since various shrinkage stages permitted the sampling of the transform's magnitude at points far closer than its anticipated minimum wave length, it became possible to determine the position of these discontinuities and hence the sign of the transform at every point. Frances Crick pointed out that, in the more general noncentric case, there would be no discontinuities in the function , as might be present separately in the real, A, or imaginary, B, parts. That is, even if the magnitude of a transform of a molecule is known at every point in space, Crick argued, the structure of the molecule could not be determined. These arguments were powerful, indeed sufficiently so, that I found myself working alone for some time.
In hindsight, it is easy to see that the use of the hemoglobin shrinkage stages to phase (sign) determination is an example of MR in the presence of multiple crystal forms. Thus, Max Perutz should be credited with the first successful application of the MR method.
The first ideas on how to use NCS for phase determination were based on equating the electron density, ρ, at points in the which were equivalent chemically. Thus, if the noncrystallographic relationship is expressed as
then
Each relationship can be expressed as a Fourier summation where the unknowns are the phase angles, α_{h}.
As many equations of this type can be written as there are unknown phases to be determined. Each expresses the equality of electron density at a different pair of points x and x′. The problem is then one of solving this nonlinear system of equations for the unknown phases. From here it was only a small step to employ a continuous integration over the volume of one molecule (see Appendix B). Yet no success was encountered in the solution of the equations.
The cause of failure was not in the physical content of the equations, but in the method of their solution. All attempts so far had been based on separating real and imaginary parts and then solving these two families of equations separately for cosa and sina. Solutions arrived at in this way did not in general satisfy the condition cos^{2}α + sin^{2}α = 1.
6. Eventual success in the use of NCS for phase determination
Details of the early attempts to solve the resultant MR equations (Main & Rossmann, 1966; Crowther, 1967, 1969; Main, 1967) are recorded in my book, but are no longer important for us today. A major breakthrough came with the solution of the structure of glyceraldehyde3phosphate dehydrogenase (Buehner et al., 1973) in which the 222 NCS of the molecule was used to improve the MIR phasing by using realspace averaging. The equivalence of realspace averaging to the reciprocalspace solution of the MR equations had been hinted at by Main & Rossmann (1966), Colman (1974) and Bricogne (1974) (see Appendix B). The principal advantage of realspace averaging was that it was easier to program and the results were easy to appreciate and understand. Reciprocalspace phase determination in the presence of NCS was revisited by Tong & Rossmann (1995). Although provides insight to possible problems and can be represented in a simple and elegant mathematical form, experience in imposing NCS in real space makes it likely that most or all future phase determination or using NCS and solvent flattening will be by realspace density averaging.
Phase extension using NCS remained a highly contentious subject. However, in 1984, Hol and colleagues (Gaykema et al., 1984) extended phases from 4.0 to 3.2 Å resolution in their of hemocyanin. Soon afterwards, encouraged by Hol's results, we used realspace averaging to extend phases from 6.0 to 3.5 Å resolution in our of human rhinovirus 14 (Rossmann et al., 1985) and since then quite a number of virus structures have been solved by phase extension from about 20 Å. It therefore took 25 years (from 1960 to 1985) to see the complete success of the proposal I made at the time of the International Union of Crystallography meeting in Cambridge in 1960.
7. Babinet inversions and other problems encountered in MR
If very low resolution data is missing from a data set and phase determination has been initiated with a lowresolution model (e.g. with a cryoEM structure), then there is a danger of obtaining a Babinet inverted phase solution where the phase solution is α + π instead of α. This means that positive density is negative and vice versa. The cause of such an inversion can be readily understood by examination of Fig. 3. If the lowresolution terms were absent from the (00l) reflections, then it would be easy to see how the continuous hemoglobin Fourier transform could be followed while interpreting all negative regions as positive and positive regions as negative. A similar situation could have occurred in the initial phasing of the lowresolution terms in the of southern bean mosaic virus (Fig. 4). The correct assumption that the virus particle radius is 150 Å shows that the lowest recorded resolution data, between 64 and 51 Å, have a phase of 0, whereas an assumption of a radius of 200 Å would make these phases have a value of π. Yet either assumption would have given an almost equally good fit to the experimentally observed data.
A related problem occurred in the structure solution of the bacterial phage φ29 (Simpson et al., 2000). Here, the NCS 12fold rotation axis was roughly perpendicular to the a*b* reciprocallattice planes. Thus, the process of electrondensity averaging (see Appendix B) only generated interpolations between the reciprocallattice points within each l = 0, 1, 2,… reciprocallattice plane, with little interaction between planes. Hence, the Babinet solution and handedness of the phases of each plane were mostly independent of each other, although consistent within a plane. As a result, the electron density could not be interpreted until a better overall phasing start had been obtained from an isomorphous derivative.
8. Conclusions
I had imagined that it might be difficult for the inexperienced crystallographer to use MR without a sufficient mathematical foundation. However, I am delighted that there are now a number of easytouse programming packages that allow even the most inexperienced student to use MR without tears. Foremost among these programs are AMoRe by Navaza (1994), DM (Cowtan, 1994), CNS (Brunger et al., 1998) and Patsol (Tong, 1993). These automatic procedures are likely to be of progressively more help in solving structures as the number of depositions in the Protein Data Bank (PDB) steadily increases. In addition, the use of MR for NCS electrondensity averaging is much better understood and will allow more ab initio determinations as the complexity of structures being determined continues to increase.
As progressively more structures are solved by MR, the danger of major errors increases owing to bias introduced by the initial phasing model. This is a significant problem when phasing is initiated from an assumed structure as is necessary both when using a homologous model or even with `ab initio' phasing. Notwithstanding the term `ab initio', it is minimally necessary to assume a very low resolution model (such as a solid sphere) to initiate phasing. It will be essential to observe that the MR procedure has produced anticipated structural features (e.g. a Trp where there was a Gly in the phasing model) that are consistent with information not present in the phasing model.
APPENDIX A
The equivalence of different MR procedures
The rotation function R(θ_{1}θ_{2}θ_{3}) can be defined as a match between two Patterson functions (Rossmann & Blow, 1962), such that
where R(θ_{1}θ_{2}θ_{3}) represents the rotation function when the Patterson P_{1}(x) is rotated by the angles θ_{1}, θ_{2}, θ_{3} from an arbitrarily defined original orientation and then superimposed onto another Patterson P_{2}(x′). The superimposed Pattersons are then integrated over the volume U. A large value of R indicates a good match between the two Pattersons. It can readily be shown (Rossmann & Blow, 1962) that expression (1) is equivalent to the corresponding reciprocalspace expression
where and are the Patterson Fourier coefficients for the Patterson P_{1} and P_{2}, respectively. G_{hp} is a diffraction function of the form
(see the body of this paper for the definition of [C]).
Two types of MR situations were mentioned in this paper. The first is when a homologous phasing structure is available. In that case, the first Patterson P_{1} (with Fourier coefficients ) would be those of the known molecule and the second Patterson P_{2} (with coefficients ) would be those of the unknown molecule placed in an arbitrary generally of P1. A `crossrotation function' (equation 2 above) then determines the orientation of the known molecule with respect to the unknown molecule in each of the asymmetric units of the crystal under investigation.
The second type of MR situation utilizes the NCS relating different copies of the unknown molecule (or unknown subunits within a larger molecule). In that case, a `selfrotation function' (equation 2 above) determines these relationships by comparing the Patterson P_{1} (with coefficients ) with itself (the same coefficients now identified by ).
Thus, the first step in the MR process for either of these two types of situations requires an identical computational procedure. Similar arguments apply to the more difficult translational problem.
In the third stage of MR, a known structure is used for initiating phasing. The known structure can be a homologous molecule or it can be a simple assumption about the nature of the molecule, such that it is a roughly spherical object. The latter is a minimal assumption when NCS is used for `ab initio' phasing. Subsequently, these phases are usually refined using NCS electrondensity averaging (if available), plus solvent flattening (NCS implies that part of the cell has no NCS and must, therefore, be solvent flattened) or by only solvent flattening. Thus, again all the MR applications discussed in this paper require identical calculations, with the case of the NCS redundancy of 1 being a special situation. Once atomic positions can be picked (possibly after phase extension during the densitymodification procedure), normal crystallographic can take over.
APPENDIX B
The equivalence of real and reciprocalspace phase determination in the presence of NCS restraints
It will be assumed that the electron density ρ(x) at the NCSrelated positions x_{n}(n = 1,N) should be equal. Hence, the averaged electron density at x is given by
The NCS relating the different densities is given by
where [C_{n}] is the rotation matrix relating the nth density to the reference density and d_{n} is the corresponding translational component with respect to an arbitrarily selected origin. By replacing the electron density ρ(x_{n}) by its corresponding Fourier summation, it is seen that
where h is the Now, recomputing structure factors using the averaged density and assuming zero density outside the molecular envelopes, we have for reflection p
where U_{n} bounds the volume containing the nth molecular subunit. By substitution of (6) into (7), it follows that
If we now define
where U is the sum of the volumes bounded by U_{n} (n = 1, 2, …, N) and define
or
where = and corresponds to the rotation of p in equivalent to [C_{n}] in real space. Thus, is the at the nonintegral reciprocallattice point corresponding to the rotation of p by the nth NCS element. Hence, F_{p} represents the complex averaging of structure factors at the N noncrystallographically equivalent positions in reciprocal space.
If we simplify (11) by putting
then
These are the `molecularreplacement equations' defined by Main & Rossmann (1966), whose coefficients are equivalent to the [H] matrix of Crowther (1967, 1969). Here, the complex coefficients a_{hp} are dependent only on a knowledge of the orientation, position and extent of the NCS elements. The molecularreplacement equations are exact other than the assumptions that the NCS holds to within the resolution limits of the available data and that the solvent regions of the cell can be approximated by a constant level of electron density.
Substitution of currently available approximate phases on the righthand side of (13) will produce an improved set of phases on the lefthand side in a process that is entirely equivalent and the same as the realspace averaging and backtransformation procedure (Arnold et al., 1987). Approximation enters in as far as many terms must be neglected in setting up the molecularreplacement equations because they are deemed too small in magnitude to matter. The same approximation occurs in calculating a rotation function (Rossmann & Blow, 1962). In real space, the approximations relate to linear (Bricogne, 1974) or nonlinear (Nordman, 1980) interpolation to obtain the value of electron density at nonintegral grid points. The elegance of merely substituting phases in a set of complex linear equations is selfapparent (at least to me!) and is also highly suitable for rapid arithmetic in parallel processing computers.
Just as is the case for the G function in its application to the rotation function, so here also the largest coefficient a_{hp} will be between terms of about the same resolution. Thus, the interactions represented by (13) will be significant only in a relatively thin shell at the same resolution as that of the F_{p}. If we approximate the molecular envelope to be spherical with a radius R and if it is, say, in a with cell dimensions 4R (about eight particles in the cell), then the argument H·R of G is given by (n/4R)R, where n is the number of reciprocallattice points represented by the difference p − h[C^{T}]. Now, G for a spherical envelope, becomes zero for the first time when its argument is 0.7. Thus, the thickness of a shell required to include the larger interactions in the molecularreplacement equations (13) has a halfwidth when n/4 = 0.7 or n = 2.8, which is about three reciprocallattice units. The interpolation for each value of within a radius of three reciprocallattice points around is equivalent in real space to the interpolation required to find the electron density at a nonintegral grid point.
Omission of a significant coefficient F_{h} on the righthand side of (13) will cause an error in the value of F_{p}. It is clearly more prudent to include an estimate of that value if there is no observed value available. This can be obtained by the calculations of F_{h} from the molecularreplacement equation when p = h. This process is identical to the inclusion of F_{calc} values obtained by backtransformation of an averaged electrondensity map in the computation of a new and improved map. Rayment (1983) and Arnold et al. (1987) have shown that such a procedure leads to a truer and faster phase determination. Furthermore, it is now seen that if there were no observed amplitudes, then the value of F_{p} on substituting all the F_{calc} values on the right would not change from the previous F_{calc} value. Thus, absence of some observed amplitudes slows down convergence, but does not entirely stop progress towards a phase solution.
Similarly, omission of structure factors outside the current resolution limit leads to a decrease of satisfaction of the molecularreplacement equations. An equation for a F_{p} at the exact limit of resolution will have half its terms missing and cannot give a good estimate of F_{p}. The lack of agreement between observed and calculated amplitudes at the limit of resolution is thus easily appreciated. Since the sum of the terms on the righthand side is essentially the sum of a random set of vectors, omission of some terms will cause an overall reduction of calculated F_{p} values. Hence, calculated structure factors will require progressively further upscaling as they approach the limit of current resolution, as indeed is found in practice (Arnold et al., 1987; Luo et al., 1989).
Acknowledgements
I would like to thank my numerous coworkers who have joined me for longer or shorter periods of times while employing the MR method in various different ways. We have always learned more with each new experience. I also thank Cheryl Towell and Sharon Wilder for help in preparing this manuscript. These studies have been supported by grants from the NSF and NIH.
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