Topography and high resolution di⁄raction studies in ...hera.ugr.es/doi/14995682.pdfTopography and high resolution di⁄raction studies in tetragonal lysozyme Fermı«n Otalora!,*,

*Corresponding author. Fax: #34 58 243384; e-mail:[email protected].

Journal of Crystal Growth 196 (1999) 546—558

Topography and high resolution diffraction studiesin tetragonal lysozyme

Fermı́n Otalora!,*, Juan Manuel Garcia-Ruiz!, José Antonio Gavira!, Bernard Capelle"! Instituto Andaluz de Ciencias de la Tierra. CSIC/Univ. Granada, Campus Fuentenueva (Facultad Ciencias), 18002 Granada, Spain" Laboratoire de Mine& ralogie-Cristallographie, Tour 16, 2e%me e& tage, 4, place Jussieu, Case 115, F-75252 Paris Cedex 05, France

Abstract

Two complementary approaches are used to enhance the usefulness of X-ray topographies obtained from proteincrystals. First, the use of thin plate-like crystals in conjunction with a high intensity, collimated and small source sizesynchrotron beam produces a large beneficial effect on the level of detail and contrast of topographies for thequantification of local misalignment in the crystal lattice. Second, the recording of topography series along the rockingcurve of a diffraction peak is proposed as a technique to combine the benefits of both rocking curves and topographiesand produce very detailed data (“rocking maps”) on the spatial distribution of lattice misalignments and mosaic spread(“local rocking curves”). The most important crystal features controlling the observed contrast are growth sectors andinter-sector boundaries, clearly observed in the topographies. Systems of parallel fringes are observed in many of thetopographies. Two alternative explanations for these fringes are discussed: (a) as moiré interference fringes or (b) asPendellösung fringes in a wedge shaped crystal volume. In both cases, growth sectors play a central role in the physics offringe generation. Many observations suggest the presence of a relatively large component of dynamical diffraction inthese crystals; the consequences of this new scenario are discussed. ( 1999 Elsevier Science B.V. All rights reserved.

PACS: 07.85.!m; 07.85.Qe; 87.64.Bx

Keywords: Protein; Crystal quality; Topography; Mosaicity; Synchrotron; Dynamical theory

1. Introduction

Two rather different approaches can be taken incharacterising the quality of protein crystals. First,

one might be interested in the resolution limit de-fined as some statistically meaningful limit in thesignal to noise statistics. This eminently practicalapproach is the best choice when the object of ourinterest is the protein structure, but in this case noinformation can be derived on the structure defectslimiting the resolution. “Defect” is used in this workas meaning “deviation from a perfect lattice” rather

0022-0248/99/$ — see front matter ( 1999 Elsevier Science B.V. All rights reserved.PII: S 0 0 2 2 - 0 2 4 8 ( 9 8 ) 0 0 8 4 9 - 5

than as individual defects like dislocations, va-cancies, stacking faults... A deeper understanding ofthe problem can be obtained by addressing thecharacterisation of the defective structure of crys-tals by using different X-ray diffraction techniques.These techniques usually produce results of lessdirect application for structural analysis, but aremuch more interesting than the resolution limit forquestions related to improving crystal growth tech-niques by understanding the influence of the manyphysicochemical parameters involved in crystalgrowth. The study of the physical properties ofprotein crystals can also benefit from these devel-opments. In some way, these studies can be con-sidered as a longer path (with many diversions)towards enhancing the resolution limit; neverthe-less, the continual enhancement of protein crystalgrowth techniques [1,2] and new results suggestingthe use of protein crystals as technological mate-rials [3] indicate that the crystallographic charac-terisation of protein crystals must be developedbeyond the useful resolution limit.

At present, studies on the quality of protein crys-tals concentrate on

f mosaic spread, by collecting rocking curves[4—8];

f mosaic block and defect distributions, by topo-graphy [5,9—12] and

f crystal defects and defect-generating surface pro-cesses, by atomic force microscopy [13—16].

Among these techniques, AFM produces verydetailed information but is restricted to small re-gions on the crystal surface, providing no informa-tion on the crystal volume. The two X-raytechniques, on the other hand, produce com-plementary information on the crystal volume.Rocking curves contain statistical information onthe angular space, the intensity I(h#*h) recordedat a given angle close to the Bragg angle h beingproportional to the crystal volume whose misalign-ment is *h. Topographic images contain informa-tion in real space on the distribution of the parts ofthe crystal volume whose misalignment is *h. Inboth cases a gaussian-like “experimental function”accounting for the bandwidth of the radiation, thedivergence of the beam and other experimental

factors is convoluted to the measured distributions.The complementary character of topographies androcking curves is just part of the landscape of possi-bilities that high resolution and imaging X-ray dif-fraction techniques offer. In the field of proteins,new X-ray techniques, like reciprocal space map-ping [17], are currently being developed to provideadditional information on the protein crystal lat-tice. In this paper we present our latest develop-ments in exploiting topography, rocking curves andtheir relations to gain a better insight into thequality of protein crystals.

In Fig. 1, an imaginary diffraction experimentis presented to introduce some central conceptsinvolved in this work. Fig. 1a shows a locally de-fective crystal structure; clear disks indicate thetheoretical (“perfect”) position of the lattice nodes,while dark disks show the actual position of nodesin this crystal volume. In this situation, the lattice islocally misaligned by an angle that is continuouslychanging throughout the volume. This distributionof local misalignment can be represented by a greyvalue map, with the colour of each point indicatingthe local angular misalignment, as in the bottompart of Fig. 1a. Obviously, if the width of the ex-perimental function is small, not all of the crystalvolume will be simultaneously in the Bragg condi-tion and, for each *h angle, different parts of thecrystal will diffract. In Fig. 1b, a two-dimensionalcrystal, represented by one such misalignment mapis shown. Let us assume that this crystal is illumi-nated by a collimated, monochromatic X-ray beamand that the Bragg condition is fulfilled, producinga diffracted beam in the vertical direction. By plac-ing a high resolution film (or any other imagingdetector) in the direction of the diffracted beam,a diffraction image of the crystal is obtained wherethe parts of the crystal that fulfil the Bragg condi-tion appear projected in the direction of the diffrac-ted beam; this is a topography of the crystal(Fig. 1c). In this example we obtain one-dimen-sional topographies, but in real experiments two-dimensional topographies are recorded which arethe projection of a three-dimensional volume in thedirection of the diffracted beam. By acquiringa series of such topographies around the Braggangle at different *h values and stacking them,a map is obtained (Fig. 1d) that contains all the

F. Otalora et al. / Journal of Crystal Growth 196 (1999) 546–558 547

Fig. 1. Schematic view of the techniques used. (a) Definition of the misalignment inside a crystal volume (close to a crystal defect).Representation of the misalignment field by grey value maps. (b) Hypothetical diffraction experiment using a nonperfect two-dimensional crystal. The crystal is represented by the rectangular grey value map encoding local misalignment values. The Braggcondition is assumed to be fulfilled producing a diffracted beam in the vertical direction. (c) Integration of crystal volumes along thediffracted beam direction to produce topographies (top). Distribution of crystal volumes fulfilling the Bragg condition for two different (hand h#*h) angles and the corresponding topographies obtained (bottom). In these images, the grey value corresponds to scatteredintensity in the direction of the diffracted beam, not to the misalignment angle. (d) Space/angle map produced by stacking topographiesat different h values close to the Bragg angle (grey value represents scattered intensity). Horizontal sections are single topographies,vertical sections are local rocking curves and the integration over the horizontal direction is the global rocking curve of the crystal.

information on the local lattice misalignments inthe crystal. This map shows the relation betweenthe different X-ray techniques presently used toassess crystal quality: each horizontal section ofthis map is a single topography and the integrationover the horizontal direction is the global rockingcurve of the crystal, which could be recovered bysetting a scintillation counter (or any other pointdetector) in the position of the film and rocking thecrystal around h. This is just a unifying view of

topography and rocking curves, but new and stillunexplored possibilities appear when working withvertical sections. Each vertical section of the map inFig. 1d is a “local rocking curve”, which is therocking curve we would obtain from the differentialvolume of the crystal corresponding to a linethrough the crystal in the direction of the diffractedbeam and ending at the given point (Fig. 1c top).All the features along this line are integrated atthis single data point of the topography. This

548 F. Otalora et al. / Journal of Crystal Growth 196 (1999) 546–558

integration of features is a reasonable explanationfor the usual scarceness of contrast in topographiesobtained from protein crystals: sharp features arenot usually seen in these topographies, probablybecause the misalignments in the crystal lattice ofproteins comes mainly from continuous deforma-tions of the lattice rather than from sharp featureslike dislocations or other defects. Each such localrocking curve has its own width and Bragg angleand the sum of all these provides the global rockingcurve.

2. Materials and methods

Putting together all these ideas, we designed anexperiment to collect series of topographies aroundthe Bragg angle using thin, plate-like protein crys-tals as samples. The gel acupuncture technique[18—21] was used to produce the crystals because ofits shaping capabilities [1,2]: plate-like crystalswere grown simply by using flat X-ray capillaries(4.0]0.1 mm cross section) as the growth chamber.Solutions of 100 mg/ml lysozyme (L-6876 Sigmawithout further purification) were prepared at pH4.5 (50 mM acetate buffer). The precipitating agentsolution was prepared using 20% w/v NaCl in thesame buffer. Topographies were collected fromcrystals inside the same capillaries where they weregrown, to minimise crystal damage duringmounting and handling. The reflections used tocollect topographies were selected among those al-most perpendicular to the crystal plate to minimisethe path of the diffracted beam inside the crystalvolume and, consequently, minimise the integra-tion of features in the resulting image. The thinnessof the samples imposed the use of a very highbrilliance X-ray source; for these experiments, weused the Topography and High Resolution Beam-line (ID19) of the European Synchrotron RadiationFacility in Grenoble [22,23], a high intensity wig-gler beamline with small source size, which favourshigh spatial and angular resolution. A shortwavelength (0.688 A_ ) was selected. Series of topo-graphies were collected on the same film (KodakSR) by slightly displacing the film holder betweenconsecutive rotation steps around h (typically 10~3degrees per step). Exposure times of between 1 and

3 s were needed. After developing the film, eachsingle topography was digitised using a high resolu-tion video camera mounted in a microscope andconnected to a frame grabber operated by a com-puter. Standard image analysis procedures wereused to align and stack these sub-images makinga three-dimensional (x, y, h) map equivalent to thetwo-dimensional one illustrated in Fig. 1d.

3. Results

Fig. 2 shows part of one of the recorded series.The level of detail and contrast of the topographiesobtained with this experimental set-up is muchbetter than that in any previous topography experi-ment using protein crystals. This is explained easily:most of the intensity in the four topographiescomes from the dark fringe travelling from bottomto top as h increases during the rocking of thecrystal. This moving fringe is due to a continuousbending of the crystal lattice, something rathercommon in protein crystals. If the crystal thicknessintegrated to produce these images were much lar-ger than the 100 lm used here, the possibility ofhaving other crystal volumes in the Bragg condi-tion above or below (in the direction of the diffrac-ted beam) the volume producing the fringe wouldbe much higher, thus hiding this feature.

The most important contribution to the defectstructure of tetragonal lysozyme crystals comesfrom growth sectors and inter-sector boundaries. Inthe topographies obtained, most of the contrast notattributable to continuous bending can be inter-preted in terms of growth sectors and inter-sectorboundaries. Fig. 3 shows a complete series of topo-graphies acquired at 0.0025° steps. The boundariesbetween growth sectors can be clearly seen as sharpchanges in the grey level in images 3a and 3e and ascontrast (illuminated/not illuminated) between dif-ferent sectors in almost all the images, especiallyin 3d where the outline of a 1 1 0 growth sectornot at its Bragg angle (not scattering) is veryclearly shown. The whole series can be decomposedinto separate sub-series for each of the growthsectors present in the crystal: the 1 1 0 sector on theright is at its Bragg position in Fig. 3a, the intensityscattered by this sector then decreasing through


Fig. 2. Four topographies from one of the recorded series. The four pictures are close to the Bragg angle and are 4 mm wide.

images 3b and 3c. The 1 1 0 sector on the rightscatters the maximum intensity at image 3e andagain the intensity decreases through images 3d, 3cand 3b to the lower h values and image 3f to thehigher. The two 1 0 1 sectors are at their Braggangle in 3b (the top sector) and 3e (the bottom one),both scattering some intensity in the previous andfollowing images. To be precise, what we called“1 0 1 sector” is really a group of 2 or more 1 0 1sectors with almost the same misalignment. Theeffect of growth sectors on the contrast of topo-graphies is so important that it can even be used tocompute approximate orientations for the crystalslices as shown in Fig. 3h.

As stated in the introduction, by collecting seriesof topographies at different Bragg angles and stack-

ing them, a local misalignment map is obtainedfrom which local rocking curves can be computed.Using the series illustrated in Fig. 2, we computedsuch a set of local rocking curves and fitted them togaussian functions. Fig. 4 shows a two-part “rock-ing map” composed by mapping the width of thefitted gaussians (i.e. the local mosaicity) and theposition of their maximum (i.e. the local Braggangle). For the first time, these maps show clearlythe quality, misalignment and curvature distribu-tions inside the crystal volume. In this crystal,a continuously bent, low mosaicity zone exists inthe centre of the crystal while high mosaicity zonesappear close to the six vertices. These high mosaic-ity areas are related to highly defective zones in thecase of the two vertices between the 1 0 1 faces


Fig. 3. (a)—(f) Series of topographies acquired from the same reflection at different h values. Between two consecutive pictures, the crystalwas rotated by 0.0025°. (g) Schematic view of the growth morphology of the tetragonal lysozyme crystals used in the experiment. Thetwo different growth sectors are illustrated. Eight symmetry-equivalent 1 0 1 sectors and four 1 1 0 sectors exist. (h) Example of the use oftopographies to approximately orient the 0.1 mm wide crystal slices grown inside the flat capillaries. Three of the four 1 0 1 sectors areclose to the Bragg angle; one corner between two 1 0 1 and one 1 1 0 sectors is observed.

(particularly the one at the bottom that corres-ponds to a cracked zone seen in Fig. 6c) and to thebehaviour of boundaries between growth sectors inthe case of the four high mosaicity zones close tothe 1 0 1/1 1 0 vertices, as we will see later. Thismap shows large differences in local mosaicity witha factor of 8 variation across the crystal. This unex-pected range of variation can be explained by thehigh spatial resolution achieved in the map shownin Fig. 4. Even measuring local rocking curves, allthe crystal volume under the measurement windowcontributes to the rocking curve, which is aweighted average of very wide and very narrowpeaks coming, respectively, from very high and verylow quality crystal lattice zones included within themeasurement window. As the size of the measure-ment window for computing local rocking curves isreduced, the differences between extreme mosaicity

values increases because high and low quality zonesof the crystal are isolated in a single window (i.e.a single width value). Fig. 4 has been computedusing a window of 16 square pixel, small enough tomake evident the large differences in local mosaicspread between different parts of the crystal.

By adding all the local rocking curves, the globalrocking curve is recovered (Fig. 5) having a full-width at half-maximum of 10 arcsec. This value isoverestimated because the experimental width hasnot been deconvolved and because the height of thecentral maximum is underestimated due to theexistence of several saturated zones in the topo-graphies corresponding to the maximum.

The most striking feature of the topographiesobtained during this study is the presence, in al-most all the images, of sets of parallel fringes. Fig. 6shows two examples from the same topography.


Fig. 4. Mapping of the spatial distribution of the local Bragg Angle (top) and local mosaic spread (bottom) for the crystal illustrated inFig. 2. The larger mosaic spread of the six corners and the curvature of the central part of the crystal are clearly observed. The units ofboth scales arcsec.


Fig. 6. Two fringe systems from the same topography. (a) and (b) are enlargements of the complete picture shown in (c) (where thecrystal outline has been added for clarity). (a) “Y-shaped” fringes close to one of the 1 1 0 growth sectors. The stem of the Y is the endof the moving fringe shown in Fig. 2 while the two branches are associated with the boundary between the 1 1 0 sector and the con-tiguous 1 0 1 sectors. (b) Complex fringe system depicting a strain field at the joining point of two cracks. A well-developed fringesystem is observed at the top. Two fringe systems coexist at the bottom, one having the same appearance as the one at the top andrepresented in this image by a single fringe and the other perpendicular to this one and showing many fringes, which are more tightlyspaced.

Fig. 5. Global rocking curve of the crystal illustrated in Fig. 2.This rocking curve was computed by adding all the local rockingcurves mapped in Fig. 4. The height of the maximum is under-estimated due to the overexposure of several areas in the topo-graphies close to the global Bragg angle.

The “Y-shaped” fringe system in picture 6a, cha-nges its position with h and is associated with theboundaries between the 1 1 0 and 1 0 1 growth sec-tors (see the whole topography in Fig. 6c). The stemof the darkest “Y-shaped” fringe in both zonesseems to be the end of the moving fringe shown inFig. 2, which suggests that the global curvature ofthe crystal can somehow be controlled by the distri-bution of growth sector misalignment. The slowdisplacement of these fringes and their presence inmany consecutive topographies in the series is theorigin of the high mosaic spread measured close tothe 1 0 1/1 1 0 vertices. The fringe system illustratedin Fig. 6b is associated with the joining point oftwo cracks (clearly seen in Fig. 6c) and depictsa stress field in this area produced by the twocracks. The bottom part of the picture shows the


same effect at the opposite side of the cracks witha curved single fringe having the same aspect asthose above and another system of fringes perpen-dicular to the first one and more tightly spaced.

4. Discussion and conclusions

The enhancement in the level of detail and con-trast observed in these images is partly due to thecharacteristics of the ID19 beamline. This X-raysource is very well suited for topography experi-ments due to their small source size (0.2]0.1 mm2H]») and divergence (0.3]0.1 mrad2 H]»FWHM), which favours high spatial and angularresolution. In addition, short wavelengths can beused while still preserving a high incident intensity.However, comparison of the acquired images withother topographies obtained at this beamline usingthick lysozyme crystals [24] clearly shows that thelargest contribution to the quality of the topo-graphies comes from the use of thin proteincrystals, which allow the observation of local mis-alignments without 3D masking effects. The con-trast observed in the topographies supports theinitial hypothesis on the prevalence of continuousbending and smooth local contrast as the predomi-nant features in the defect distribution of lysozymecrystals. Sharp contrast features indicating singledefects were not observed. The absence of thesedefect records seems to be commonplace in proteincrystal topography except in experiments with sud-den changes in the growth conditions [12].

The most important features controlling the de-fect structure of tetragonal lysozyme crystals arethe growth sectors and the boundaries betweengrowth sectors. They are so clearly seen in thetopographies as to allow the study of separate rock-ing sub-series for each individual growth sector andthe determination of the approximate orientationof the crystal slices used as samples. In addition,misaligned growth sectors produce wedge-shapedvolumes giving rise to Pendellösung fringes orMoiré interference. Misalignment between growthsectors could produce some degree of stress insidethe crystal lattice, relieved in the form of large scalecontinuous bending of the lattice, which could bethe factor responsible for the large moving fringe

illustrated in Fig. 2. This result is suggested by theevolution of the “Y-shaped” fringes close to the1 1 0 growth sectors, the stem of which is the end ofthe moving fringe. The presence of these growthsectors has been suggested previously [25] but noclear evidence of their ubiquity and their impor-tance in controlling the internal lattice distortionhas been presented. This importance might be dueto the role of inter-sector boundaries as the site ofaccumulation of stress or defect clusters, relievingthe small unit cell volume differences produced bythe selective incorporation of impurities within dif-ferent crystal faces [26]. The misalignment of differ-ent growth sectors could even be the result of thestress field produced by these changes in volumebetween different growth sectors.

The origin of the observed fringe systems is stillunclear although the possibilities seem to be limitedto (a) Moiré fringes due to the superposition in thedirection of the diffracted beam of different crystalvolumes having slightly different misalignment orlattice parameters [27] or (b) Pendellösung fringesdue to the presence of wedge-shaped crystal vol-umes [28].

To give an idea of what this means in terms ofmisalignment or d-spacing differences, Fig. 7ashows the spacing between consecutive fringes inthe topography previously shown in Fig. 3. Thespacing between consecutive fringes is maximumbetween the first (darkest) and the second fringe,decreasing in this direction. An inter-fringe spacingof e corresponds to the interference of two diffrac-ted beams making an angle *h such that

tan(*h)"je

with j being the wavelength. These two interferingdiffracted beams can be produced by two crystalblocks misaligned by *h (provided that *h is small-er than the experimental width) or by two crystalblocks with slightly different d

h k lspacing such that

*dh k l

"dh k lA

sin(h)sin(h#*h)

!1B.Fig. 7c and Fig. 7d show the two alternative inter-pretations of fringe spacing. Block misalignmentsbetween 1 and 2 arcsec or d

h k ldifferences in the


Fig. 7. Inter-fringe spacing and its possible interpretation in terms of Moiré interference. (a) Grey level through a section perpendicularto the fringe system from a topography of the series illustrated in Fig. 3. (b) Inter-fringe spacing; the fringes are numbered starting fromthe most intense. Misalignment between blocks (c) and difference (]105) in d-spacing in A_ (d) that could explain the inter-fringe spacingshown in Fig. 7b.

order of 10~5 A_ (uniform enough over the volumeshowing fringes) could explain the observed fringepattern in terms of Moiré interference.

An alternative explanation for these fringes isthat they are Pendellösung fringes produced bya wedge-shaped crystal. The reflectivity of a crystal

in the Laue case (no total reflection) of the dynam-ical theory of diffraction is (in the simplest, symmet-rical, case)

Rh k l

"Ds

)D

sin(2h)¼(A)


Fig. 8. (a) The reflectivity function for a given reflection as a function of the A (normalised thickness) parameter. (b) Schematicexplanation of the observed fringes as Pendellösung fringes. Two crystal blocks (I and II) corresponding to slightly misoriented growthsectors are assumed to make up a crystal slice such as those used in the experiment. If crystal quality is sufficient to observe a dynamicalbehaviour and the h angle is set such that block I fulfils the Bragg condition and block II does not (as in Fig. 3), the diffraction physicscorresponds to the Laue case, and the wedge-shaped (I) block will have a variable reflectivity associated with the thickness gradient asshown in Fig. 8a.

[28], where s)and ¼(A), called the Waller integral,

are defined as

s)"

j2p»

#

F), ¼(A)"

p

2P2A

0

J0(u) du with A"

pt

K,

where J0

is the Bessel function of rank 0, t is thecrystal thickness and K is the Pendellösung (extinc-tion) length.

K"1

Cr0

»0

jDFD,

where C is the polarisation factor, r0

the classicalelectron radius, »

0the volume of the unit cell, j the

wavelength and DFD the structure factor amplitude.The reflectivity value R

h k las a function of A is

shown in Fig. 8a. It is clear that, for crystals in theLaue case, a continuous variation of the crystalthickness will produce intensity fringes as the reflec-tivity changes. As seen in Fig. 3, the misalignmentbetween growth sectors can be large enough to

produce two contiguous sectors, one in the Braggangle and the other completely out of the diffrac-tion condition. In this situation, if the boundarybetween sectors is not perpendicular to the crystalplate, the scattering volume has the wedge shaperequired to produce these fringes (Fig. 8b).

The possibility of having dynamical effects on thediffraction of protein crystals gives rise to manynew and interesting questions. The most trivial ofthese questions is the correct relation betweenstructure factors and intensities: structure factorsare proportional to the square root of the inte-grated intensity in the framework of the kinemati-cal theory but to the integrated intensity in thedynamical theory. Therefore, the correctness of thecalculations used to obtain the structure factor dataused for structure determination could be compro-mised. Recent studies on primary extinction in pro-tein crystals [29] show that dynamical extinctioncorrections can be 13—15% for the strongest reflec-tions. Dynamical effects may also control the lattice


Table 1Classification of X-ray diffraction techniques, a unified view

Space dimension Rocking angles Technique

x y z a1

a1

I I I I F Integrated intensity Volume integrating techniquesI I I S F Rocking curveI I I S S Reciprocal space mapping

S S I F F Topography Imaging techniquesS S F F F Section topographyS S I S F Rocking maps

S S S F F Stereoscopic topography Three-dimensional techniquesS S S S F Tomographic topography

Note: F N “Fix” meaning that data are collected for only one value of this parameter; I N “Integrated” meaning that data for a rangeof this parameter is added in one single intensity value; S N “Scanned” meaning that actual intensity values are sampled at severalvalues of the parameter.

features observable by topography: in the low ab-sorption case, the predominant contrast mecha-nism is the so-called “direct image”, produced byan additional diffracted intensity coming from dis-torted lattice areas close to each crystal defect, likethe one shown in Fig. 1a. Direct images are onlyseen in crystals that are thicker than about K/3.Below this thickness there is not enough differencebetween the intensity scattered by the “distorted”region and the “perfect” environment, and defectscannot be observed [30]. This could be the ex-planation for the ubiquitous absence of defect re-cords in protein crystal topographies. These resultshighlight the need to have an idea of the Pendel-lösung length for the crystals we are working with.All the values involved in the computation of K areeasily available (C+1 and j+1 A_ for usual ex-periments in synchrotron sources, r

0"2.82]

10~15 m and »0"2.335]105 A_ 3 for tetragonal

lysozyme) except the structure factor, which cannotbe computed exactly due to the large amount oftotally or partially disordered water molecules in-side the unit cell. Estimating [31] an average valueDFD+596 for lysozyme (DFD+1800 for the strongestreflections commonly used in topography), a valueof 1540 lm (500 lm for the strongest reflections) isfound for K. This value is much larger than theusual value (1—100 lm) found for small molecule

crystal. This means that the protein crystals com-monly used in diffraction experiments, measuringbetween 100 and 2000 lm are within the limits ofthe kinematical and the dynamical theories.

The new concepts “rocking maps” and “localrocking curves” have been introduced and theirusefulness demonstrated. These constructs appearnaturally through a unified view of rocking curvemeasurement and topography studies that is, infact, part of a vastly larger landscape (see Table 1):in terms of information content, during a diffrac-tion experiment, we are working with the threedimensions of space and some angles. Several tech-niques integrate the spatial information whilescanning or keeping the remaining parameters con-stant; among these techniques, we found the integ-rated intensity measurements (for example by thescreenless oscillation method) used for structureanalysis. By scanning one angular parameter rock-ing curves are recorded. Reciprocal space mappingscan be obtained by simultaneously scanning twoangular dimensions. Most existing X-ray dif-fraction imaging techniques like topography andsection topography, as well as new ones such asrocking maps, work by scanning two spatial para-meters. In the near future, we will start workingwith techniques that scan the three spatial dimen-sions, like stereoscopic topography [32] and new


ones such as tomographic topography. Neverthe-less, many possible and interesting combinationsare lacking from Table 1, especially if a sixth di-mension, namely the time during which crystalgrows, is added as we did in our study of theevolution of mosaic spread during protein crystalgrowth [33]. In short, why stop at rocking curvesand topography rather than making the most of thepotential of X-ray diffraction experiments?

Acknowledgements

Thanks are due to the staff of the ID19 beamline,José Baruchel, Jurgen Härtwig, José Espeso andAngel Mazuelas, for their kind help during theexperiments. This work has been performed withfinancial support from the Spanish Ministerio deEducacio&n y Cultura project PB-0220. Travel andaccommodation expenses during experiments weremet by the ESRF and MEC.

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