 1. Introduction
 2. Theoretical principles
 3. Strength of the anomalous signal in MASC
 4. Experimental considerations
 5. Experimental testcase studies
 6. Extracting envelope structurefactor moduli, GU(h), from MASC data
 7. Phasing the moduli of envelope structure factors
 8. Conclusions and perspectives
 References
 1. Introduction
 2. Theoretical principles
 3. Strength of the anomalous signal in MASC
 4. Experimental considerations
 5. Experimental testcase studies
 6. Extracting envelope structurefactor moduli, GU(h), from MASC data
 7. Phasing the moduli of envelope structure factors
 8. Conclusions and perspectives
 References
research papers
Lowresolution phase information in multiplewavelength anomalous solvent contrast variation experiments
^{a}LURE, Bâtiment 209D, BP 34, Centre Université ParisSud, 91898 Orsay, France,^{b}Institut de Biologie Structurale, 41 Rue Jules Horowitz, 38027 Grenoble, France, and ^{c}Université d'EvryVald'Essonne, Boulevard François Mitterrand, 91025 Evry CEDEX, France
^{*}Correspondence email: shepard@lure.upsud.fr
The basic theory and principles of the multiplewavelength anomalous solventcontrast (MASC) method are introduced as a contrastvariation technique for generating lowresolution crystallographic phase information on the envelope of a macromolecule. Experimental techniques and practical considerations concerning the choice of anomalous scatterer, sample preparation and data acquisition are discussed. Test cases of crystals of three proteins of differing molecular weights from 14 kDa through to 173 kDa are illustrated. Methods for extracting the moduli of the anomalous structure factors from the MASC data are briefly discussed and the experimental results are compared with the known macromolecular envelopes. In all cases, the lowest resolution shells exhibit very large anomalous signals which diminish at higher resolution, as expected by theory. However, in each case the anomalous signal persists at high resolution, which is strong evidence for ordered sites of the anomalous scatterers. For the smaller two of these proteins the heavyatom parameters could be refined for some of these sites. Finally, a novel method for phasing the envelope structurefactor moduli is presented. This method takes into account the relatively low number of observations at low resolution and describes the macromolecular envelope with a small number of parameters by presuming that the envelope is a compact domain of known volume. The parameterized envelope is expressed as a linear combination of independent functions such as spherical harmonics. Phasing starts from solutions of a sphere in the
after positional from random trials and the parameters describing the envelope are then refined against the data of structurefactor moduli. The preliminary results using simulated data show that the method can be used to reconstruct the correct macromolecular envelope and is able to discriminate against some false solutions.Keywords: MASC.
1. Introduction
Contrast variation is a means of gleaning information from a scattering experiment on the form of a dispersed particle (a solute) by altering the level of its surrounding solvent. In experimental terms, this requires the measurement of scattering intensities of the particle in several different solvent compositions. The level of the solvent may be electron density which scatters Xrays, H/D isotope mixtures which scatter neutrons or any other physical density which interacts with the incident photons in a scattering experiment (for a review, see Williams et al., 1994). The difference between the solute and the solvent density levels is defined as the contrast (Stuhrmann & Kirste, 1965; Ibel & Stuhrmann, 1975).
This method is experiencing a renaissance in smallangle scattering studies on biological samples (e.g. Junemann et al., 1998; Stuhrmann & Nierhaus, 1996; Zhao et al., 1999). However, since 30–70% of the volume of macromolecular crystals constitutes solvent molecules, contrastvariation techniques can also be extended to macromolecular crystallography. Such was the case when Bragg & Perutz (1952) applied these methods to a haemoglobin crystal by altering the electronic density of the mother liquor and then observing changes in the intensities of Xray reflections at low resolution. The changes derived from a contrastvariation series provide information on the `solvent'accessible region of the crystal, a region which is closely related to the `negative' image of the macromolecular envelope. A number of contrastvariation experiments on macromolecular crystals have been demonstrated using either Xrays or neutrons to extract the lowresolution structures (Harrison, 1969; Jack et al., 1975; Moras et al., 1983; Roth et al., 1984; Bentley et al., 1984; Podjarny et al., 1987 and references cited therein; Carter et al., 1990; Badger, 1996) or to locate the lipid phase in membrane protein crystals (Roth et al., 1991; Timmins et al., 1992; PebayPeyroula et al., 1995; Timmins & PebayPeyroula, 1996; Penel et al., 1998; Pignol et al., 1998). These experiments highlight certain disadvantages in the contrastvariation methods used, notably in the Xray diffraction experiments, where there is a risk of lack of isomorphism when the unitcell parameters change after soaking the crystals in different mother liquors, and in the H/D isotopeexchange experiments, which suffer from the low of neutron sources. Although effects have been employed in smallangle scattering experiments at the Fe Kedge (Stuhrmann, 1980) and at the P Kedge (Hütsch, 1993), the possibility of using from the solvent in crystals has only been considered as a supplement to a standard contrastvariation series at a single wavelength by Wyckoff and others (Dumas, 1988; Crumley, 1989; Carter et al., 1990).
Anomalous scattering effects used at multiple wavelengths, however, can offer an excellent way of generating contrast variation by tuning the Xray wavelength about an externally to the sample. Consequently, isomorphism is conserved. The possibility of exploiting at multiple wavelengths for contrastvariation experiments was originally suggested by Bricogne (1993), but its full potential was realised and demonstrated in the theoretical formalism and preliminary results presented by Fourme et al. (1995). Here, we will present the principles, experimental aspects and recent advances that have been made in what is known as the multiplewavelength anomalous solvent contrast method or MASC.
of the anomalous scatterer, which is dispersed in the solvent channels of the macromolecular crystal. This method avoids the difficulties associated with the lack of isomorphism often encountered in Xray chemical contrastvariation experiments because the changes in Xray wavelength are physical changes applied2. Theoretical principles
The theoretical principles of MASC (Fourme et al., 1995) start at the basic principles of contrast variation, where the macromolecular crystal (volume V) is assumed to be separated into two phases: the domain U occupied by the macromolecule and the domain V − U occupied by the solvent, which is in a liquidlike state of rapid exchange (Fig. 1a). The domain U containing the macromolecule is presumed to be ordered and this also includes the shell of ordered solvent molecules, whereas the domain V − U is presumed to be the solvent region, which is completely disordered (Fig. 1b). It will be seen later that this model is oversimplified because the solvent region overlaps with the ordered zone.
A formalism derived by Bricogne (unpublished work) and demonstrated in Carter et al. (1990) separates the diffraction effects of the ordered and disordered domains by defining G_{U}(h) as the Fourier transform of the indicator function χ_{U}(h), which is 1 inside the domain U and 0 elsewhere (Fig. 1c). Likewise, G_{V−U}(h) is the FT of the complementary indicator function χ_{V−U}(h) which corresponds to the disordered solvent zones (Fig. 1d). It should be noted that G_{U}(h) = G_{V−U}(h) for all h ≠ 0. The overall F(h) can then be written as two components, one as a term for the ordered structure domain, F_{P}(h), and the other as the term for the disordered solvent domains, ρ_{s}G_{V−U}(h) or ρ_{s}G_{U}(h) for h ≠ 0, where ρ_{s} is the constant electron density of this flat and featureless domain,
Note that F_{P}(h) is the Fourier transform of the protein in a vacuum (Fig. 1e) and may be rewritten in terms of the average electron density inside the domain U, 〈ρ_{P}〉, and the internal fluctuations from this average (Fig. 1g), 〈ρ_{P}〉 – ρ_{P}(r), such that their Fourier transforms become 〈ρ_{P}〉G_{U}(h) and Δ(h), respectively. Substituting this into the overall expression one obtains
The term 〈ρ_{P}〉 −ρ_{s} is defined as the contrast as illustrated in Fig. 1(f) (Stuhrmann & Kirste, 1965). When 〈ρ_{P}〉 = ρ_{s} the system is said to be at the contrast matching point, and only the Fourier transform of the internal electron density, Δ(h), contributes to the overall F(h).
In order to account for anomalous scatterers included in the solvent, we use the seminal idea of Karle (1980) where the structure factors of an anomalously scattering atom, A, are separated into wavelengthindependent () and wavelengthdependent (^{λ} + i^{λ}) parts,
We then consider that the ^{λ}f is constant to a firstorder approximation with respect to scattering angle at low resolution. The density of the anomalous scatterers in the solvent can be treated as a complex quantity, ^{λ}ρ_{sA}, which is dependent upon wavelength and can be separated into wavelengthindependent and wavelengthdependent parts, i.e.
The total electronic density of the solvent, ^{λ}ρ_{s}, becomes a function of wavelength and separable into wavelengthindependent and wavelengthdependent parts,
Note that the term °ρ_{s} includes the normal scattering part of the anomalous scatterer. Thus, one obtains for the overall ^{λ}F(h),
The terms between the first set of brackets represent the wavelengthindependent part of the overall F(h). It includes the envelope, the contrast and the fluctuation terms. The second set of brackets is wavelengthdependent and incorporates the envelope and the anomalous structure factors of A, ^{λ} and ^{λ}. Note that the wavelengthdependent contribution is subtracted from the normal scattering part, indicating that the anomalous and dispersive structure factors of A are applied to the Fourier transform of the indicator function of the solventaccessible domain −G_{U}(h). The overall contrast becomes 〈ρ_{p}〉 − ^{λ}ρ_{s} and contrast variation is generated by tuning the wavelength to different values of ^{λ} and ^{λ}.
denoted °By defining Γ(h) = −°ρ_{sA}G_{U}(h), one generates a expression of the overall similar to the starting point used for the algebraic MAD method (Hendrickson, 1985), where Γ(h) replaces the normal scattering component of the A, °F_{A}(h),
The substitution of Γ(h) for °F_{A}(h) has an obvious physical meaning. The anomalous A, which is a set of a few punctual and ordered scatterers in a MAD experiment, is exchanged for an extended uniform electron density in a MASC experiment (see Fig. 1h). The separation of the effects of the anomalous A (and hence the Fourier transform of the solventaccessible volume) from the overall diffraction effects can be applied using a set of equations analogous to the MADLSQ equations, where they are solved for °F_{T}(h), Γ(h) and the phase difference between °F_{T}(h) and Γ(h), Δφ = (φ_{T} − φ_{Γ}), i.e.
where
3. Strength of the anomalous signal in MASC
The strength of the anomalous signal in a MASC experiment can be estimated as in a MAD experiment by measuring differences between Bijvoet pairs (anomalous or ^{λ} contribution) and wavelength pairs (dispersive or Δ contribution). Intuitively, the magnitude of the anomalous signal in a MASC experiment is expected to vary considerably with resolution, being very large in the lowest resolution shells and then diminishing rapidly with increasing resolution. One also expects the anomalous signal to be directly proportional to the concentration of the anomalous scatterer in the solventaccessible volume. Furthermore, the signal will be maximized at the point of contrast matching. By making a certain number of approximations (globular protein, Porod's law, smooth interface etc), it is possible to derive expressions for and calculate the expected anomalous and dispersive ratios (Fourme et al., 1995), but to be concise only the final expressions will be given here. Thus, for anomalous and dispersive differences, respectively, one obtains^{1}
and
M_{w} is the molecular weight and f_{eff} is the r.m.s. of the structurefactor moduli of the atoms in the macromolecule (f_{eff} = 6.7 electrons for proteins at s = 0). Obviously, the anomalous signal is dependent on the molar concentration of the anomalous scatterer, [A], and the magnitudes of Δ and ^{λ}, but the strongest effect on the anomalous signal is a result of the resolution, s = 2sinθ/λ, diminishing as a function of 1/s^{2} and exp(−Bs^{2}/4). In the term exp(−Bs^{2}/4), B is a pseudotemperature factor which models the combined effects of the smearing of the envelope boundary as well as the temperature factor of the solvent. However, even with a perfectly sharp interface (i.e. B = 0 Å^{2}), the anomalous signal still drops away as 1/s^{2} owing to Porod's law (Porod, 1951). The loose dependence upon the molecular weight arises from the assumptions of a globular protein and Porod's law. Hypothetical MASC signals are plotted against resolution for different molecular weights in Fig. 2. As is clearly illustrated, one expects very large signals in the lowest resolution shells that will diminish sharply with increasing resolution. Thus, to obtain a measurable anomalous signal (>0.020) out to 10 Å resolution, either multimolar quantities of a Kedge scatterer or molar quantities of an Ledge scatterer are required.
4. Experimental considerations
4.1. The experimental setup
Since a MASC experiment is analogous to a MAD experiment but at low resolution, the data collection should ideally be carried out at Xray wavelengths near an ^{λ} and ^{λ} vary abruptly. Thus, a MASC experiment requires, in addition to the usual equipment of a crystallography experiment, a tuneable source of Xrays with a narrow bandpass (Δλ/λ ≃ 10^{−4}), an Xray fluorescence detector to determine precisely the wavelengths of ^{λ}_{max} and ^{λ}_{max}, and an experimental setup designed to collect reflections at the lowest possible resolution. This last aspect is not as trivial as it first appears (also see Evans et al., 2000), since smallangle scattering from different components of the setup can severely add to the background and deteriorate the quality of the data, especially if the solvent electron density is at the contrast matching where the lowresolution reflections are weakest. The Xray background around the beamstop originates from several sources, in particular from the slits, the air path and the windows of the He cone, as well as the sample and its holder. The mounting of a lowresolution beamstop requires special attention and a schematic layout is shown in Fig. 3. Typically, a MASC experiment requires the mounting of a small beamstop just in front of the detector entrance window inside the He cone. However, a very small beamstop which is only slightly larger than the beam size and is accurately placed just behind the crystal will minimize considerably the background arising from the air scatter by cutting down on the length of the air path travelled by the Xray beam. If a long crystaltodetector distance is used, then it is important that the beam divergence is low or that the beam is carefully focused onto the detector rather than the crystal. This helps to avoid an enlargement or elongation of the reflection profiles.
of the anomalous scatterer where the4.2. Choice of the anomalous scatterer and preparation of crystals
A variety of anomalous scatterers may be used in a MASC experiment and the most suitable ones will depend on the crystallization conditions of the macromolecule. Analogues of the precipitating agent are good choices, as such compounds are less likely to perturb the crystalline lattice (e.g. selenate for sulfate, bromide for chloride, tribromoacetate for acetate etc). In general, the MASC compound used should have a high solubility in the mother liquor, should not influence the pH and should be inert to avoid binding to and denaturing the protein. Crystals may be prepared either by cocrystallization with the MASC compound or by soaking techniques. In the latter case, special attention is often needed to prevent crystals from cracking owing to the osmotic shock upon addition of the high concentration of the MASC compound. Small amounts of the MASC compound can be added progressively to the mother liquor over a period of several hours, or vapourdiffusion techniques may be used to slowly increase the of the mother liquor before attaining the desired concentration. Dialysis buttons could also be used, although their rate of exchange is often too fast for many crystals unless the reservoirs are incremented in a stepwise fashion. Another factor to consider is that the macromolecular crystals of interest may dissolve at high ionic strengths and this could be understood as a form of `dehydration' of the crystal, which suffers from overcompetition with the MASC compound for water molecules. Finally, it can be difficult to find a MASC compound suitable for those crystals which grow under low conditions, such as polyethylene glycol precipitating agents. One possibility is to use neutral polar compounds, such as aurothioglucose; however, these compounds remain to be examined in detail.
4.3. and absorption effects
The combination of working with high concentrations of an anomalous scatterer in the crystal and with Xray wavelengths at one of its absorption edges will provoke two drawbacks: severe absorption of the Bragg reflection intensities of the crystal and an increased background from μ. Fluorescence is the emission of a photon after the absorption of an incident photon with an energy greater than that of the effects can be very strong in MASC experiments because of the high concentrations of anomalous scatterers in the crystal. The is at a maximum at Xray wavelengths where ^{λ} is a maximum. The effects can be very detrimental at the sharp whiteline resonances, where the emitted radiation floods out the diffraction image (see Fig. 4). This is especially true for K edges, which have higher fluorescence yields than L edges. Remedies include increasing the crystaltodetector distance, collecting images with small rotation slices or maximizing the dispersive differences by collecting diffraction data at the inflection point and then at two other wavelengths as remote as possible from the However, perhaps the best solution is to switch the anomalous scatterer from a Kedge element to an Ledge element. For Ledge whiteline resonances, ^{λ} can be over 20 electrons, so that only a fraction of the concentration is needed to generate an anomalous signal; near 1 Å the yield can be halved (Kortright, 1986).
Little can be done to minimize severe absorption effects except to optimize the crystal size such that the reflection intensities are maximized with respect to the crystal volume and its5. Experimental testcase studies
To date, MASC data have been collected and analysed on crystals of three proteins of differing molecular weights (14, 54 and 173 kDa) and using a variety of anomalous scatterers (see Table 1). All of the cases are known crystal structures, which have allowed the experimental results to be compared with their known envelope structurefactor moduli and phases (Ramin et al., 1999). In each of the experiments, the Xray diffraction data were recorded at the wavelengths corresponding to the ^{λ}_{max} and ^{λ}_{max}, which were determined from the spectra from a solution of the anomalous scatterer, as well as for at least one wavelength remote from the A small beamstop (∼2.5 mm) was mounted and aligned just in front of the entrance window of the detector. Where possible, the crystallographic axes were aligned so that Bijvoet pairs could be measured on the same image. The full details of these experiments have been reported in Ramin et al. (1999) and here we will present each case study briefly.

5.1. Hen eggwhite lysozyme cocrystallized in YbCl_{3}
As a first test case, hen eggwhite lysozyme (HEWL) crystals were grown directly from YbCl_{3} solutions to obtain robust HEWL crystals, which normally crystallize in NaCl, and to exploit the whiteline structure of the Yb L_{III}edge. Xray diffraction data were collected for the first ever MASC experiment in 1993 on station D23 at LUREDCI (Kahn et al., 1986) at the Yb L_{III}edge from a single crystal of HEWL containing 0.8 M YbCl_{3} and then later with 0.5 M YbCl_{3} crystals on station DW21b at LUREDCI. In both experiments, the results confirmed a large anomalous signal at low resolution as expected by theory (see Fig. 5a). The internal agreement between true equivalent reflections is within ∼1–3%, implying that the anomalous signal is reproducible and not an artefact of either absorption effects, data processing or reflections hidden behind the beamstop. In all cases, the anomalous signal extends well beyond 10 Å resolution, indicating that Yb^{3+} ions have bound to the protein in the The metal sites have been identified from phased anomalous difference Fourier maps (Fig. 6a) and their number and occupancies vary for each crystal. In one case using SHARP (de La Fortelle & Bricogne, 1997), as many as six metal sites could be refined (Ramin et al., 1999). It is important to note that the data in the lowest resolution shells are rather incomplete as the crystals were aligned along the crystallographic fourfold axis. This is a consequence of the combination of the large blind zone formed by the curvature of the at the wavelength of the Yb L_{III} edge (1.3862 Å) and the relatively small unitcell parameters of HEWL.
5.2. Outer membrane meningitidis protein (P64k) soaked in (NH_{4})_{2}SeO_{4}
One of the proteins under study in our laboratory is a 54 kDa domain of the outer membrane protein (P64k) from Neisseria meningitidis (Li de la Sierra et al., 1994, 1997) that crystallizes from ammonium sulfate solutions and for which the mother liquor can be substituted with multimolar concentrations of ammonium selenate via simple soaking techniques. The selenate anion is isostructural with sulfate and has a sharp whiteline feature at the Se Kedge (∼0.9796 Å). Crystals of P64k withstand 3.5 M (NH_{4})_{2}SeO_{4} solutions which bring the solvent electronic density equal to the average protein electronic density, i.e. the contrast matching point. MASC data were collected at the Se Kedge on the TROIKA station at the ESRF. Severe effects were observed because of the sharp whiteline resonance of the selenate anion coupled with its high concentration and the fine Xray bandpass from the monochromator (Fig. 4). The Xray data in the lowest resolution shells are virtually complete because the curve of the is relatively flat at ∼1 Å wavelength, so much so that Friedel pairs often appeared on the same image. Despite the difficulties arising from fluorescence, the anomalous signals are large in the lowresolution shells but do not completely disappear at higher resolution (Fig. 5b); phased anomalous difference Fourier maps (Fig. 6b) confirmed the existence of at least one selenate site bound in a pocket of the macromolecule. Anomalous difference Patterson maps recovered this site which could be refined using SHARP, but no other sites could be located (Ramin et al., 1999).
5.3. Xylose isomerase soaked in (NH_{4})_{2}SeO_{4}
Xylose isomerase (XI) also crystallizes from ammonium sulfate solutions but as a large tetramer of 173.2 kDa in the et al., 1988). This represents a fairly large macromolecular structure on the scale of those typically solved by the MAD method. Crystals were soaked in 2 M (NH_{4})_{2}SeO_{4} solutions and MASC data were collected at the Se Kedge on the TROIKA station in the same fashion as for the P64k crystals. Evidence for several selenate anion sites have been found in the anomalous signal and in the phased anomalous difference Fourier maps of XI (Figs. 5c and 6c). All of the sites are at or near the macromolecular boundary; however, attempts to refine any of these have not proved fruitful (Ramin et al., 1999).
(Rey5.4. General comments
The anomalous signal for all test cases follows the expected trend, being very large at lowest resolution and decreasing rapidly with increasing resolution. At higher resolution, the Bijvoet ratios for all protein crystals remains significantly higher than zero and this suggests ordered sites of the anomalous scatterers bound to the macromolecule. The existence of these sites appears to be more general than expected and, although their relative occupancies may be low, it opens up a potential of phasing to higher resolution. Indeed, this has recently been shown in Dauter & Dauter (1999) and in Dauter et al. (2000). This effectively converts a MASC experiment into a MAD experiment. Other MASC compounds are under consideration, especially where Kedge elements can be replaced by Ledge elements to reduce the background owing to fluorescence as well as to prevent binding to the macromolecule. From this last point of view, nondetergent sulfobetains (NDSB), which are neutral commonly used as solubilizing agents to help the crystallization of macromolecules, are of special interest. Girard et al. (1999) have tested tetragonal crystals of HEWL cocrystallized with a monobrominated sulfobetain at a concentration of 0.72 M. As verified by a Fourier difference map, this anomalous contrast agent does not bind to the macromolecule and as predicted the MASC signal was observed at low resolution whereas no anomalous signal could be detected at higher resolution.
6. Extracting envelope structurefactor moduli, G_{U}(h), from MASC data
Extraction of the envelope structurefactor moduli, G_{U}(h), from MASC data can be performed in two ways. The first is to treat the MASC data as a MAD data by determining the anomalous °F_{A}(h), with the program MADLSQ (Hendrickson, 1985). The second method is to consider the MASC data as a chemical contrastvariation series, ^{i}F_{obs}(h), using a modified version of the program GFROMF (Carter & Bricogne, 1987). Both methods give satisfactory results for data to at least 20 Å resolution.
6.1. MADLSQ
The program MADLSQ was originally designed for MAD data and solves the set of algebraic MADLSQ equations by nonlinear leastsquares for °F_{T}, °F_{A} and the phase difference Δφ_{T−A}.^{2} These equations can be applied to MASC data by substituting Γ(h) = −ρ_{sA}G_{U}(h) for °F_{A}(h) and the phase difference for Δφ_{T−G}. The program has the ability to refine or fix the values of ^{λ} and ^{λ} of the different wavelengths. Comparisons of the G_{U}(h) from MADLSQ with the G_{U}(h) calculated from the threedimensional coordinates in the PDB are shown in Figs. 7(a)–7(c) for the three different protein crystals (R factors of ∼16–33%; see Table 1). Note the sharp asymptotic decrease in G_{U}(h) with increasing resolution. The agreement between model and experiment deteriorates beyond 10–20 Å resolution for several reasons: (i) the relative magnitudes of G_{U}(h) are small, (ii) the absorption effects are more pronounced at higher diffracting angles and (iii) the possibility of ordered sites contributing to the extracted from the MADLSQ equations [i.e. Γ(h) is more precisely defined as Γ(h) + °F_{A}(h)].
6.2. GFROMF
In chemical contrastvariation studies the diffraction data are a contrast series of ^{i}F_{obs}(h) for i = 1, …, N, where i corresponds to a different solventdensity level ^{i}ρ_{s}. To extend this to a multiplewavelength case, we simply substitute in for the contrast series ^{λi}F_{obs}(h), where λ_{i} = λ_{1}, …, λ_{N} and the solvent density becomes ^{λi}ρ_{s}. The same formalism is used to describe the overall in terms of the Fourier transforms of the envelope [G_{U}(h)] and the internal density fluctuations [Δ(h)]. If X(h) and Y(h) are the real and imaginary components of Δ(h) relative to G_{U}(h), one has
The GFROMF (Carter & Bricogne, 1987) program carries out the nonlinear leastsquares of G_{U}(h), X(h) and Y(h) from scaled data summed over all contrasts series and minimizes the function
where σ_{obs}(h) is the standard deviation of ^{i}F_{obs}(h). X(h) and Y(h) effectively represent the magnitude and the phase difference between G_{U}(h) and Δ(h). In practice, a scale factor between the different data sets, K_{i}, should be refined for all but one contrast or wavelength.
The original program was modified to incorporate
contributions such that,Trials on simulated MASC data of kallikrein (52 kDa) at three different contrasts of selenate and three wavelengths per contrast returned exact values of G_{U}(h), X(h) and Y(h) of the simulated observed data (Ramin, 1999). With experimental data, the results gave R factors of 27–45% for the testcase crystals (see Table 1). This level of agreement is satisfactory considering that many of the parameters are left unrefined. In particular, the values of ^{λi} and ^{λi} utilized were derived from previous runs of MADLSQ and theoretical values of the contrast were used rather than allowing them to refine. The scale factors between different wavelengths (^{λi}K) were set to unity since the data were already set on a common scale. In principle, all of these parameters should be refined in the GFROMF scheme, even though the number of observations in the lower resolution shells is not large. What is certain is that prior precise knowledge of the values of the contrasts and the factors ^{λi} and ^{λi} is important to extract G_{U}(h) values of satisfactory quality.
6.3. Separation of anomalous signals from punctual sites and the disordered solvent
Separation of the anomalous signal from punctual sites (MAD) and from the disordered region of the solvent (MASC) can at least be partially accomplished by delimiting the Xray diffraction data into the very lowest resolution shell, say below 20 Å resolution, where the MASC effects are strongest, or into the higher resolution shells, say above 5 Å resolution, where the MAD effects are strongest. This however is not a deconvolution, since even in the lowest resolution shells, which are dominated by the enormous MASC signals, the MAD signals are present albeit weak. The same is also true for MASC signals in the higher resolution shells. At moderate resolution, roughly between 20 and 5 Å resolution, MAD and MASC signals are mixed in nonnegligible proportions. In order to deconvolute punctual sites and the disordered solvent region, additional information is necessary either in the form of another contrast series or as structural information on the punctual sites or of the solvent region. As such, a model for the punctual sites may be determined from the highresolution anomalous data and these sites could then be used to extract out a model for the solvent at lower resolution. Both models could then in principle be refined against the entire data set of Γ(h) + °F_{A}(h).
7. Phasing the moduli of envelope structure factors
7.1. The at low resolution
The problem of directly phasing the structurefactor moduli of macromolecular envelopes is rather unusual in crystallography since it does not adhere to the characteristics normally found in smallmolecule or protein crystallography. For example, envelopes do not have atomic or point scatterer character and thus the principles of `atomicity' do not apply. Furthermore, by definition the density of an envelope of a macromolecule is anything but randomly distributed throughout the i.e. a continuous single volume described by a closed single surface). For an assembly of macromolecules this approximation still holds true, but whereas in solution these compact domains are well separated, in a crystal they are adjoining owing to the crystal packing contacts made between macromolecules in the crystal lattice.
quite the contrary, its distribution is binary and can be considered – in a first approximation – to be confined to a `compact' domain (An important problem is that there is no prior information on the scale factor between the observed diffraction data (structurefactor moduli of macromolecular envelope) and the calculated data (from the envelope model). In smallmolecule or macromolecule crystallography, the scale factors used are based upon Wilson statistics (Wilson, 1942), which assumes a random distribution of point scatterers in the However, Wilson statistics can only be successfully applied to diffraction data at moderate to high resolution; at resolutions below 3.5 Å Wilson statistics breaks down because the diffraction process is then dominated by the diffraction of secondary structures. The determination of the scale factor is further complicated by the large `dynamic range' of values for the structurefactor moduli of macromolecular envelopes, which can vary over at least two orders of magnitude in the resolution range extending from 100 to 10 Å and are very large in the lowest resolution shells, diminishing sharply with increasing resolution.
Another aspect which should be kept in mind is that at low resolution there is a relatively small number of observations; in fact, the number of observations is a small fraction of the number of nonH atoms in the
For example, lysozyme, a 14 kDa protein, consists of approximately 1500 nonH atoms, but its tetragonal crystal form has only 202 independent observations to 7.5 Å resolution.It should now begin to be clear to the reader that `traditional' ^{3} (Sayre, 1952) is complicated by the limited number of observations at low resolution and their large since this equation is a selfconvolution process requiring the products of F(k)F(h − k) be summed over all k, which should cover as much of as possible.
used in solving the for smallmolecule structures are unsuitable in the determination of macromolecular envelopes. Even the application of Sayre's equationUnless constraints to the model of the envelope are applied, knowledge of only the moduli of the envelope structure factors would yield an infinite number of solutions to the
Our approach is to constrain the solution in real space by imposing a constant electron density inside the volume delimited by the envelope of a single molecule that is reproduced by the crystal symmetry.7.2. Method and theory
7.2.1. Parameterization of the macromolecular envelope^{4}
Since there are a small number of observations at low resolution, the envelope of the macromolecule must be described by a small number of parameters. We start with the hypothesis that the envelope delimits a mathematically compact domain U_{0} in which the indicator function is unity inside the domain and zero outside the domain (Fig. 8). The boundary of the domain is unknown. The proportion of the cell volume occupied by this domain is presumed to be known. The parameters describing the frontier of the domain U_{0} are refined in order to improve the agreement between calculated and observed structure factors, G_{cal}(h) and G_{obs}(h), respectively. For reasons of convenience, spherical coordinates (r, θ, φ) are used to describe the macromolecular envelope (Fig. 8a). The origin C of this coordinate system is set arbitrarily inside the molecule. The surface is described by the value R(θ, φ), taken as the distance from the origin to the boundary of the domain for given angular variables (θ, φ)
It should be noted that R(θ, φ) is a function limited to a single value for each (θ, φ) and as a consequence certain concave features of a surface cannot be described by this method (see Fig. 8b). R(θ, φ) is expressed as a linear combination of a set of functions f_{k}(θ, φ)
and the refined parameters thus become the coefficients a_{k}. In principle, any set of linearly independent functions may be used to describe the envelope. Here, we develop the use of spherical harmonics Y_{l,m}(θ,φ) as the set of linear functions which describe the envelope, because this set of functions is orthonormal and allows one to control the `order of detail' which increases with the spherical harmonic order. Mathematical methods employing spherical harmonics are well developed. Its use in smallangle scattering has been introduced by Stuhrmann (1970), for which extensive use of the orthonormal properties of spherical harmonics can be made because the scattering pattern results from an average over all possible orientations of the envelope. In our application, where the orientation of the macromolecule is not time averaged, no specific simplification occurs from the orthonormal properties of the spherical harmonics during the computation of structure factors.
7.2.2. Expression for G_{U}(h) and its derivatives for a single molecule
An expression for G_{U}(h) and its derivatives for a single macromolecule can be derived in spherical coordinates. The Fourier integrals are estimated numerically taking into account the peculiar structure of the indicator function χ_{U}(r). The volume integral G_{U}(h) of a single macromolecule is transformed into a surface integral to minimize the number of integration points and to improve precision,
where is the unit vector normal to the surface S and pointing toward the exterior of the volume enclosed by S. From R(θ, φ) one can easily deduce dS in terms of polar coordinates (see Fig. 8c for definitions of , _{θ} and _{φ}). Since
one obtains
In order to refine the model, the derivatives of G_{U}(h) should be computed with respect to the parameters a_{k}. From the expression of G_{U}(h) in polar coordinates,
one can deduce derivatives of G_{U}(h) for the parameters a_{k},
G_{U}(h) and its derivatives [G_{U}(h)/a_{k}] are calculated by numerical integration using the same sample points (θ, φ) in angular space.
7.2.3. Application of crystallographic symmetry
One can compute G_{U}(h) for the entire by defining G_{0}(h) as the inverse Fourier transform of the indicator function (r) for a single molecule occupying the domain U_{0} centred at C. The unitcell origin O is taken as the origin for the calculation and we define = r_{C}, u = , = r (r = r_{C} + u). Then, G_{0}(h) for a single molecule becomes
The indicator function can be rotated and translated for any g, of the such that χ_{U} (t_{g} + r) = χ_{U}(r) and then G_{U}(h) for the entire develops to
and finally,
More simply put, the overall G_{U}(h) for the entire is defined as the sum of the G_{0}(h) for each single molecule in the after applying the corresponding rotations and translations of the crystallographic symmetries in can also be applied in an analogous way.
7.3. The program NVLOP: procedure and criteria
The program NVLOP (Kahn, in preparation) determines a parameterized envelope using spherical harmonics given a set of G_{obs}(h). A sphere is used as a starting model and the volume occupied by this sphere is considered to be known. The centre of the sphere is determined from a few random trials followed by a positional minimizing the residual R_{ref},
where m(h) is the number of reflections equivalent to h by symmetry and K is a scale factor. After the initial sphere positioning, the clusters of these solutions are identified and the using the linear combination of spherical harmonics is started. The refined parameters are the coefficients a_{k}, the coordinates of the centre C of one molecule and the scale factor K. The only constraint is the constant volume imposed onto the molecular envelope. Thus, during the process, the component along the gradient of the volume of the vector describing the variations of the set of parameters is forced to vanish. Attempts to introduce a global thermal factor as a refined parameter were unsuccessful and led to abnormally high values for the thermal parameter coupled with very high values for the coefficients a_{k} of highest order.
The program requires the maximum order of the spherical harmonics describing the envelope to be set for the NVLOP includes the possibility of introducing a starting model from a predefined set of parameters (the centre C, the scale factor K and the coefficients a_{k}). The procedure is performed using a leastsquares normal matrix. This procedure is also rather insensitive to missing data.
This order can be incremented in steps and the can be restrained to the evenorder spherical harmonics which are centrosymmetric. This is of particular importance, because the oddorder (noncentrosymmetric) spherical harmonics are strongly coupled to shifts in the centre of mass. Subsequently, eigenvalue filters have been incorporated and are very important when refining the oddorder spherical harmonics. Another attribute ofAfter the last cycle of i.e. overlapping and nonoverlapping pixels are equally weighted). To make the comparison of preexisting maps easier, the program can output all of the maps using different possible origins.
the parameterized envelope in the might overlap with its symmetryrelated counterparts. The overlapping grid points are set to unity and play the same role as nonoverlapping pixels in the calculation of the final map (7.3.1. Simulated data
A simulated set, G_{sim}(h), of G_{obs}(h) for hen eggwhite tetragonal lysozyme has been generated to a resolution of 7.5 Å using the following procedure.
7.3.2. Initial sphere positioning and envelope refinement
The best results of the envelopedetermination process were obtained using a volume significantly higher than the actual value (75% instead of 67.3%). 200 random trials of initial sphere positioning were undertaken. (It should be remembered that the initial sphere positioning fixes the hand and the origin of the final solution.) These trials converged into 11 different clusters of solutions defined as those solutions which are separated by no more than 0.005 in their fractional coordinates or by ∼0.4 Å. This was found to be the coarsest spacing possible before neighbouring spherical solutions would refine into nonidentical spherical harmonic solutions. The resulting refined coordinates X_{c}, Y_{c}, Z_{c} of the sphere centre and the corresponding value for R_{ref} as well as the number N_{freq} of trials corresponding to equivalent solutions are reported in Table 2. Each of these 11 cluster solutions have been used as a starting model for subsequent envelope Spherical harmonics up to the order l = 4 (29 refined parameters) and up to l = 5 (40 refined parameters) were successively tested. These refinements converged to 11 different solutions. Results for these solutions are summarized in Tables 3 and 4. Sections z = 0 of the indicator functions for these 11 solutions and the corresponding section for the model are shown in Fig. 9.



7.3.3. General comments on the phasing method
Several solutions with a high proportion of grid points correctly set have been found. Nevertheless, these results address the problem of selecting the correct solutions from a few indicators. First of all, it should be noted that the correct solution corresponds to the lowest residual R_{ref} before overlap removal using spherical harmonics up to the order l = 4 and its residual is ranked second lowest after removal of the overlaps. In both refinements using spherical harmonics of orders l = 4 and l = 5, solutions 9–11 can be eliminated because their number of overlapping grid points are too high and consequently their residuals after removal are high. Similarly, solution 6 can be eliminated according to its high residual after overlap removal. When compared with the known model, the remaining seven solutions refined using spherical harmonics of orders l = 5 have more than 70% of grid points correctly set.
When higher orders of spherical harmonics are employed (i.e. above l = 6, the equivalent of 53 refinable parameters), the diverges. This gives an idea of the practical upper limit of the number of spherical harmonic orders which can be effectively refined, at least for this crystal form of lysozyme. In theory, the upper limit of the number of spherical harmonic orders would be restricted by the number of independent observations. For the above case of lysozyme with 202 reflections at 7.5 Å resolution, this leads to an upper limit of l = 14 for a 1:1 ratio of parameters to observations.
The preliminary results of this phasing method are very encouraging. However, these results need to be confirmed on simulated data from other macromolecules (globular proteins as well as DNA, RNA etc.) and on experimentally obtained envelope structurefactor moduli. This latter point raises the concern of the effect of ordered sites of the anomalous scatterer, since such sites begin to `pollute' the experimental moduli of envelope structure factors in the higher resolution shells and where these values are generally weak.
The program gives a fair estimation of the absolute scale factor, without any prior knowledge except for the solvent content of the crystal. In addition, e.g. heterodimers), unless the entire complex was included into a single envelope.
can be easily implemented for cases containing several molecules per but the method is not adapted to cases involving mixtures or complexes of more than one macromolecule (Improvements can be made in both the method and algorithm, most notably in the treatment of the overlapping regions between symmetryrelated envelopes. A particular cause for concern is the loss in discrimination between refined solutions after the overlapping regions have been forced to unity. Work on a more robust minimization algorithm will be needed, especially when one considers that the effect of ordered sites of the
is as yet uncertain and may prove detrimental to resolving the envelope of the macromolecule. The difficulties associated with possible ordered sites of the anomalous scatterer could in principle be resolved by implementing a maximumentropy procedure using a loworder spherical harmonic envelope as a starting point.8. Conclusions and perspectives
There are three principal advantages to generating contrast variation in the MASC method: (i) the contrast is applied externally to a singlecrystal ensuring isomorphism, (ii) the scattering factor supplies extra phase information owing to the breakdown of Friedel's law and (iii) it uses the intense synchrotronradiation Xray sources which are much brighter than neutron sources. However, the disadvantages are the effects arising from absorption and fluorescence, and the weaker contrast between data sets compared with H/D
The risk of anomalous scatterers binding to ordered sites will contribute to the anomalous signal and contaminate the extraction of the structurefactor moduli of the envelope in the intermediate and higher resolution shells. This is where the disordered solvent effects are mixed in with the ordered solvent effects and thus MASC signals are combined with MAD signals.In principle, ordered anomalous sites could help to phase the protein structure to higher resolution, as has been shown recently by Dauter et al. (2000). The attempts to directly phase with the ordered anomalous sites found in the anomalous difference Patterson maps of HEWL and P64k test cases were hindered by the moderate resolution of the data (3.9–4.1 Å) because of the long crystaltodetector distances required. Ideally, the multiplewavelength data should be collected in two runs, one to collect the lowresolution data (MASC signal) and another to collect highresolution data (MAD signal). Such a strategy is complicated by absorption and fluorescence effects washing out the typically weak reflections at high resolution and the risk of generating many lowoccupancy sites which would render the determination of their location difficult either from Patterson maps or via by using programs such as ShakeandBake (Miller & Weeks, 1998) and SHELXD (Sheldrick, 1998). However, if a few well occupied sites could be generated and found then other sites may be localized using differencemap techniques.
The presence of ordered anomalous scatterers suggests that the biphasic model of the e.g. termini), an intermediate zone exists between the macromolecule and the bulk solvent where the electron density shows some order but is still in exchange with the bulk solvent. In a MASC experiment, the anomalous may not necessarily be restricted to the disordered zone of the bulk solvent, although it can permeate into this intermediate zone. Indeed, the anomalous scatterers may not be evenly distributed throughout the bulksolvent zone. One would expect ions to be absent near a macromolecular surface which has a like charge. How an anomalous scatterer is distributed about the crystal will depend on its physical chemistry character (charge, size etc.), the bulksolvent medium and the surface of the macromolecule crystallized.
in a macromolecular crystal needs to be revised. In fact, the macromolecule and solvent domains are not identical to the ordered and disordered domains in the Since all macromolecular crystals contain solvent shells of ordered molecules and crystals often contain disordered segments of the macromolecule (We have demonstrated that the envelope moduli can be phased by treating the envelope as a compact domain of known volume and then expressing it as a linear combination of a set of functions, such as spherical harmonics. This allows the macromolecular envelope to be expressed as a small number of parameters, given that there are relatively few independent observations. The preliminary results presented here are encouraging, but further research is necessary to optimize this phasing step and other methods may also prove to be successful (e.g. Andersson & Hovmöller, 1996; Badger, 1996; Harris, 1995; Lunin et al., 1995; Urzhumtsev et al., 1996, Subbiah, 1991, 1993). As such, it turns the MASC method into a complete technique of envelope determination.
Footnotes
^{1}Where s = 2sinθ/λ, ^{λ}ΔF(±h) = ^{λ}F(+h) − ^{λ}F(−h)/〈^{λ}F(±h)〉, 〈^{λ}F(±h)〉 = ^{λ}F(+h) + ^{λ}F(−h)/2, ^{Δλ}ΔF(h) = ^{λi}F(h) − ^{λj}F(h)/〈^{Δλ}F(h)〉, 〈^{Δλ}F(h)〉 =[^{λi}F(h) + ^{λj}F(h)]/2 and Δ = ^{λi} − ^{λj}.
^{2}°F_{T} is the moduli of normal scattering for the total structure, °F_{A} is the moduli of normal scattering for the anomalous and Δφ_{T−A} is the phase difference between °F_{T} and °F_{A}.
^{3}In principle, Sayre's equation is valid for envelope structure factors because the square of the indicator function, equal to zero or unity, is virtually identical to itself.
^{4}A complete mathematical treatment will be submitted for publication in the near future (Kahn, in preparation).
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