Perspectives on Single Molecule Diffraction

Using the X-Ray Free Electron Laser

Gordon Webster A) and Rolf Hilgenfeld B)

A) Division of Experimental Medicine

Beth Israel Deaconess Medical Center

Harvard Institutes of Medicine

4 Blackfan Circle

Boston MA 02115, U.S.A.

gordon_webster@hms.harvard.edu

B) Department of Structural Biology and Crystallography

Institute of Molecular Biotechnology

Beutenbergstrasse 11

D-07745 Jena, Germany

hilgenfd@imb-jena.de

Correspondence to

Rolf Hilgenfeld

Department of Structural Biology and Crystallography

Institute of Molecular Biotechnology

Beutenbergstrasse 11

D-07745 Jena, Germany

phone ++49-3641-656061

fax ++49-3641-656062

e-mail hilgenfd@imb-jena.de

submitted 04 Feb 2002

accepted 01 Mar 2002

published 12 Mar 2002

keywords: Single-molecule diffraction, phase determination,

X-ray crystallography, radiation damage, free electron laser,

self-amplified spontaneous emission laser (SASE)

Abstract

A free electron laser capable of operating at the shortwavelengths and high energies characteristic of hard X-rays,could be a potentially powerful new tool for the determinationof the structures of single biological macromolecules,subcellular organelles, or whole cells at high resolution.Current methods using relatively low-intensity, incoherentX-ray sources depend upon the high signal gain achieved bythe coherent scattering of a large number of identicalmolecules from a crystalline sample. The prerequisitecrystallization of biological macromolecules is a majorbottleneck in this process and the discrete nature of theFourier transforms of crystalline materials also complicatesthe process of the reconstruction of the original molecularstructure from its corresponding diffraction pattern.

The ability to record detailed diffraction patterns fromsingle molecules would eliminate the need for crystallinesamples. Furthermore, the continuous molecular transformsof single molecules can be sampled finely enough to enablethe unrecorded phase information to be retrieved from the setof structural amplitudes without recourse to the additional, apriori structural information that is generally required for phaseassignment using sets of diffraction amplitudes fromcrystalline samples.

Currently, diffraction experiments with single biologicalmacromolecules or other non-crystalline material arehampered by the significant radiation damage to the samplesthat occurs as a result of the very large radiation doses thatare required. Model calculations indicate that this problemmay be solved by using the ultra-short (< 100 fs) pulses ofhard X-radiation that a new generation of X-ray free electronlasers should be capable of producing. In anticipation that thecurrent, rapid development of this technology will make thispotential a reality in the near future, the theoreticalfoundations of this new methodology are already being laid.

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Introduction

The past two years have witnessed the determination at highresolution of the structure of the bacterial ribosome [1-5], amajor achievement of X-ray crystallography believed to beunreachable by many only a decade ago. Where do we go fromhere? There can be little doubt that the next challenge forstructural biology is the elucidation of the structures of evenlarger molecular assemblies, subcellular organelles, andwhole cells at high resolution. Being barely amenable tocrystallization, these objects obviously require thedevelopment of entirely new and different approaches, if thisgoal is to be realized. Prospects for the structuraldetermination by X-ray diffraction of non-crystalline biologicalsamples, including single molecules, have gained momentumin the past few years with the parallel developments of newmethods for phase determination [6-9] and of high-intensityfree-electron lasers (FELs) [10-14]. Here we undertake tocritically review these promising developments.

Advantages and Disadvantages of BraggDiffraction from Crystals

The interaction of X-rays with matter is weak, and thescattering from a single molecule does not generally yieldstructural amplitudes of sufficient magnitude to be recordedusing current methods. Therefore, the coherent scatteringfrom many identical molecules localized in a spatially orderedcrystal lattice is exploited in X-ray crystallography, in order tobe able to record the diffracted, high-resolution intensities(Bragg reflections) on an X-ray sensitive detector. Thecorresponding gain in the signal-to-noise ratio resulting fromthis coherent scattering is a major advantage of this method.In spite of its successes however, X-ray crystallography alsohas a number of drawbacks. The requirement that the sampleto be studied must be obtained in the crystalline state hasthwarted many attempts at determining the structures ofinteresting biological macromolecules, especially for example,membrane proteins, which often prove difficult or impossibleto crystallize.

An additional problem arises from the fact that thescattered diffraction amplitudes are generally recorded on a"photographic" medium in which their electromagnetic energyis transformed into some form of quantifiable signal (e.g. adischarge of electrons in a charge-coupled detector orphosphorescence in the visible spectrum in the case of animaging-plate system). Since the sampled energy of eachdiffraction maximum is directly related only to its intensity,which in turn, is proportional to the square of its amplitude,none of the phase information necessary for the direct Fourierreconstruction of the sample's molecular structure from itscorresponding diffraction pattern is recorded. This "phase

problem" in X-ray crystallography is further complicated by thediscrete nature of the molecular transforms of crystallinesamples. The strong and discrete diffraction maxima thatoccur at the Bragg angles facilitate the recording of thecrystal's diffraction pattern, but at the same time impose alimit on the sampling rate of its Fourier transform with theresult that the molecular transform cannot be sampled finelyenough for the phases to be retrieved without additionalinformation [8,15].

Diffraction from Non-crystalline Specimens

The phase problem is somewhat different for non-crystallinespecimens than for crystals. The diffraction pattern of a singlemolecule, if measurable, would be continuous, in contrast tothe discrete diffraction pattern obtained from a crystal [16].From basic sampling theory (as based on Shannon'stheorem), it is known that the accurate reconstruction of anycontinuous function from a sampled set of discrete datapoints requires that the function itself be sampled at afrequency greater than the so-called Nyquist frequency [6-8].Below the Nyquist frequency, the reconstructive processsuffers from information loss that manifests itself in aliasingerrors. In other words, the Nyquist frequency is the minimumsampling frequency required for the correct inversion of theFourier transform when its amplitudes and phases are known[8]. In the case of diffraction from a non-crystalline specimen,the Nyquist frequency, fN, is defined as the inverse of the size,s, of the specimen (fN = 1/s).

In the reconstruction of a three-dimensional electrondensity distribution from its corresponding moleculartransform (diffraction pattern), the phase problem consists ofsolving a series of independent equations of the form

F( , , )(

k k k x,y,z ex y z

x

l

y

m

z

ni kx

x

==

−

=

−

=

−⋅∑ ∑ ∑

0

1

0

1

0

12ρ( ) π

l y

y

m zz

nk k

x y zk l k m k n

+ ⋅ + ⋅

= − = − = −

)

,..., ,..., ,...,0 1 0 1 0 1

(1)

Where F( , , )k k kx y z is the magnitude of the moleculartransform and ρ( , , )x y z corresponds to the electron density ateach point in the sample. Since the electron density, undermost conditions, is real rather than complex, when samplingthe discrete, three-dimensional array (l,m,n) at the Nyquistfrequency, the number of unknown variables is lmn whereasthe number of independent equations is lmn/2 due to thecentrosymmetry of the diffraction pattern. This explains whythis sampling limitation, in the case of crystals, prevents thereconstruction of the original electron density from thesampled transform without the use of additional information.In the case of a single molecule however, the continuousmolecular transform can be sampled at greater than theNyquist frequency, increasing the number of independent

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equations for the same number of unknown variables. It hasbeen shown that phase retrieval from an oversampleddiffraction pattern is possible at a sampling of twice theNyquist frequency fN [6]. If the 2 fN x 2 fN array resulting fromthe oversampling procedure is subjected to the reverse fastFourier transform (FFT) at this sampling, an array is obtainedin real space which corresponds to the specimen surroundedby a large region containing no electron density (Fig. 1c).

Miao and Sayre [7,8] have developed an iterativeprocedure for phasing oversampled diffraction patterns. Thealgorithm alternates iteratively between reciprocal (diffraction)space and real space (Fig. 1) and is reminiscent of the phaseoptimization procedures involving density modification (realspace) and statistical phase combination (reciprocal space)that are currently used in macromolecular crystallography[17]. Initially, the diffraction pattern is combined with randomphases. In the resulting real space image, a boundary isdefined, separating the electron density region and thezero-density region (which is due to the oversampling, asexplained above). Since determination of a correct envelopesurrounding the specimen is experimentally difficult, the initialboundary is chosen generously, assuming the form of a loose

support (Fig. 1c). Constraints are then applied in real space:Since the electron density of the specimen is usually real andpositive, all negative density inside the loose support is set tozero. In addition, all density outside the loose support is drivento zero by the algorithm. By applying the FFT to the modifieddensity, a new set of phases is obtained. Over many iterationsof the algorithm, these phases will be statistically combinedwith the original measured amplitudes and converted againthrough the reverse FFT to yield a new electron density. Thisprocedure has been shown to converge to the correct phaseset and corresponding image over the course of severalhundred to several thousand iterations [7,8].

In spite of the obvious advantages of the oversampling, itcould be argued that due to their non-linearity, a uniquesolution to a set of independent equations of the form shownin equation 1 cannot be guaranteed even for the casedescribed above, where they outnumber the unknownvariables (i.e. that more than one homometric set of electrondensity distributions ρ( , , )x y z may give rise to the same set ofFourier magnitudes F( , , )k k kx y z ). In practice however, it hasbeen shown that the occurrence of more than onehomometric set for a real three-dimensional electron densitydistribution that satisfies sensible stereochemical criteria isextremely unlikely [18]. By contrast, in the case of diffractionfrom crystalline samples, the unfavorable ratio ofmeasurements to unknown variables effectively precludes thesuccessful convergence of such an algorithm unlessadditional a priori information about the structure is included.Even if such information is available, the vast size of the set ofpossible real space models may severely limit the efficacy of asearch algorithm and more sophisticated approaches arenecessary such as the evolutionary computational algorithmfor ab initio phasing recently described by the authors [19].

The phasing procedure developed by Sayre and hiscolleagues [7,8] has been shown to work not only intheoretical model calculations but also experimentally. Usingsoft X-rays (λ = 1.7 nm) at a synchrotron source, Miao et al.[7,20] were able to record a diffraction pattern from acollection of gold dots deposited on a solid membrane, and toreconstruct the image at ~75 nm resolution by using thedescribed phasing algorithm. The method has also beenshown to be applicable not only to non-crystalline materialsbut also to nanocrystals that are sufficiently small to generatea continuous diffraction pattern [8,15].

The Radiation Damage Problem

In view of the apparent power of the oversampling techniquein retrieving the phases directly from the diffraction pattern ofboth nanocrystals and single molecules, it might seemsurprising that the method was only recently developed forand applied to the phasing of continuous X-ray diffractionpatterns from non-crystalline samples. The reason for this liesin its one major disadvantage, the high radiation dose that

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Perspectives on Single Molecule DiffractionUsing the X-Ray Free Electron Laser

65

Fig. 1. Iterative algorithm for phase determination by

oversampling (from reference [8], with permission of the

IURr). (a) Oversampled Fourier transform combining

diffraction amplitudes with a current phase set. At the outset

of the procedure, random phases are chosen. (b) Electron

density obtained by applying the reverse FFT to (a). (c)

Density modification: electron density outside the loose

support (envelope, S) is set to zero, and so is negative

density inside the support. (d) New oversampled Fourier

transform obtained by applying the FFT to (c).

needs to be applied to the specimen under study. Sayre &Chapman [21] have calculated that the X-ray dose required toobtain an image of a non-crystalline specimen rises as theeighth power of the diffracting resolution. Thus, radiationdoses would be required which are far beyond what biologicalsamples can tolerate. It has been demonstrated that aradiation dose of 105 Gray (1 Gray = 100 rad) leads toimmediate morphological changes in living cells [22] andchemical fixation of wet biological specimens or drying themimproves the tolerance to X-rays by a maximum of only twoorders of magnitude [23,24]. By cryo-cooling the specimensto the temperature of liquid nitrogen, doses of up to 1010 Graycan be applied without causing morphological damage [25].This would correspond to a maximum achievable resolution (inthree dimensions) of ~10 nm [8]. Thus, the method is notsuitable for structural studies on single molecules usingconventional radiation sources. However, recent modelcalculations by Hajdu and colleagues [26,27] have suggestedthat radiation tolerance could be substantially enhanced (byup to five orders of magnitude) with hard X-rays at very highdose rates and very short exposure times. These authors haveanalyzed the various types of interactions of X-rays withbiological materials and simulated its effects on the structuralintegrity of T4 lysozyme. They concluded that the removal ofelectrons from the protein, primarily through the photoelectriceffect and through Auger emission, would lead to a build-up ofpositive charge that would eventually lead to the explosion ofthe molecule. However, with very high X-ray dose rates andpulses with durations of a few femtoseconds, the calculationsalso suggested that the experiments may provide usefulstructural information before radiation damage destroys thesample. Radiation with such properties would be availablefrom the free electron X-ray lasers which are being developedat DESY, Hamburg (TESLA – TeV-Energy SuperconductingLinear Accelerator) [10-12], and Stanford, California (LCLS –Linac Coherent Light Source) [13,14], although in the currentplans, the radiation pulses would have durations of ~100 –200 fs.

The Free Electron Laser

In theory, the free electron laser (FEL) is able to produceintense, coherent radiation from any part of theelectromagnetic spectrum by passing a beam of relativisticelectrons from an accelerator through the transverse, periodicmagnetic field produced by an undulator device of the kindalready in use in many synchrotron radiation facilities [28].Each electron radiates electromagnetic energy proportional tothe changes in its momentum that occur as it undergoes theeffects of the undulator’s alternating magnetic field. Theincoherent radiation produced in this manner is already widelyused by structural biologists at existing synchrotron radiationsources and its intensity is proportional to the number ofelectrons per unit wavelength of the undulator. Under the right

conditions however, an efficient resonant exchange of energybetween the electromagnetic field of the electrons in thebeam and that of the undulator can cause stimulatedemission as electrons form coherent bunches while passingthrough the undulator field. In fact, a beat wave known as aponderomotive wave is set up by the interference of the twofields. This ponderomotive wave has the same frequency asthe induced radiation, but its wave number is the sum of thewave numbers of the radiation and undulator electromagneticfields. Thus it travels slower than the speed of light, and if it ismatched to the velocity of the accelerated beam of electrons,the electrons are able to gain energy from it when theradiation wavelength λ , the axial velocity of the electron beamc z and the undulator period λ0 satisfy the resonancecondition

λ λ λ β= −0 ( )z (2)

where c is the speed of light. Under resonant conditions, theintensity of the emitted radiation now becomes proportional tothe square of the number of electrons, offering potential gainsof orders of magnitude in intensity over the current incoherentradiation sources that use undulator devices. From equation 2

it is clear that the FEL can be continuously tuned by alteringthe kinetic energy of the relativistic electrons and that thediscrete energy levels characteristic of conventional lasers donot apply to the FEL, in which electron energies occur as acontinuum. Free electron lasers also have a further andsignificant advantage over conventional laser sources in thatthe lasing medium is transparent at all wavelengths andtherefore cannot be damaged by the intense electromagneticfields that are produced. Additionally, in an FEL, waste energyis dissipated at velocities close to the speed of light (108 m/s),compared with the acoustic velocities (103 m/s) with which itis carried away in a conventional laser. As a result, freeelectron lasers are capable of producing extremely high peakpowers over a very broad range of wavelengths, and peakpowers in the gigawatt range have already been demonstrated[29]. Unlike conventional lasers, free electron lasers do notrequire mirrors if the gain of the laser can be achieved in asingle pass through a long undulator device. Most of the freeelectron lasers under current development fall into this classof so-called self-amplified spontaneous emission (SASE)lasers.

Laser amplification in a SASE free electron laser proceedsin three stages as the electron beam passes through theundulator [28,30]. The initial low gain stage occurs near thestart of the undulator where the radiation amplification isroughly proportional to the cube of the distance traveled bythe electron beam. In the second stage of exponential gain,the radiation power increases by a factor of e (the base ofnatural logarithms) over the gain length of the undulator. Theundulator itself must be longer than several gain lengths inorder to achieve this exponential gain. The third, nonlinear

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stage occurs once electrons become completely bunched inthe ponderomotive wave and saturation occurs when thenumber of electrons gaining energy from the wave is inequilibrium with the number of electrons losing energy to it.The construction of an X-ray free electron laser (XFEL) requiresthe design of undulators of lengths of 10 m or more which canoperate at the very low spatial tolerances (combined withperiodic focusing) required to maintain the precisesynchronization of the electron beam and the inducedelectromagnetic field over this length. Although the undulatoris the most prominent component of the FEL, the quality of theelectron source becomes increasingly critical as the design ofthe FEL shifts towards ever shorter photon wavelengths. Inparticular, for an X-ray free electron laser, the normalizedemittance of the electron source will be a major factor indetermining the saturation power of the laser at high gain. Inspite of these challenges, recent technological advances havemade the construction of an XFEL altogether feasible and theX-FEL devices at DESY and at Stanford are currently alreadybeyond the planning phase [10-14]. The TESLA Test Facility(TTF) at DESY has recently generated 800 µs radiation pulsesin the UV range of radiation. While it is still a long way to go tothe realization of the X-FEL, the proof-of-principle has beenachieved. Once completed, both TESLA and LCLS will produceX-ray pulses with durations of < 100 fs, wavelengths around0.1 – 0.15 nm, and with a peak brilliance 10 - 11 orders ofmagnitude higher than that of radiation currently produced bythe best third-generation synchrotrons (ESRF, APS, SPring-8).

Sample handling – another challenge

With individual molecules and other ultra-small samples,standard procedures for sample positioning will no longer beapplicable. Any sample holder within the beam path would beimaged. Therefore, novel container-free methods based onspraying techniques similar to those used in massspectrometry are being developed to solve this problem. It hasbeen shown that viruses and large assemblies such asribosomes retain their three-dimensional structures, includingtheir characteristic patterns of hydration, under theseconditions, and it is assumed that this will also be the case forisolated protein molecules [26,31,32]. Of course, the objectbeing studied would be randomly oriented when entering theX-ray beam path, and since no biological sample will survivemore than a single radiation pulse, a series of these single,randomly oriented particles would have to be injected into thebeam during the course of the diffraction experiment.Recording the two-dimensional diffraction patterns fromindividual particles and using averaging procedures developedfor electron cryo-microscopy [33,34] would allow thethree-dimensional structures of these particles to bereconstructed. Phase assignment would be achieved byoversampling as discussed above, yielding the correctthree-dimensional images of the objects being studied. A

recent model calculation study by Miao et al. [9] usingribulose 1,5-bisphosphate carboxylase (Rubisco), a relativelylarge protein (Mr ~ 106 kDa) whose structure has alreadybeen determined crystallographically, demonstrated that intheory, this approach worked very well for a single molecule ofthis protein, for which a correct phase set and a clear electrondensity map at 0.25 nm resolution were obtained.

Conclusions

X-ray crystallography continues to produce >80% of the newlydetermined structures of biological macromoleculesdeposited in the Research Collaboratory for StructuralBioinformatics (RCSB) Protein Data Bank, the remainderbeing largely elucidated by NMR spectroscopy. Whereas thelatter method has severe limitations with respect to thesolubility and the size of the molecules that can be studied, itcan also be estimated that 25% or more of all biologicalmacromolecules are difficult or impossible to crystallize,including many of the important membrane proteins. Electroncryo-microscopy (cryo-EM), combined with advanced imagereconstruction methods, is rapidly developing, but currentlythe maximum resolution achievable is limited to about 7 Å[35]. The novel approach reviewed in this article combinescontinuous diffraction from single molecules or non-crystallinemolecular assemblies, with averaging techniques developedfor cryo-EM and the unique properties of X-ray free electronlasers presently under development. It has to be emphasizedthat the expectations currently raised are, to a large extent,based upon theoretical model calculations and that any realproof-of-principle will only be possible when the appropriatehardware becomes available. However, if successful, this newtechnology will provide a means of determining the structuresof biological specimens ranging from single molecules to largemolecular clusters, subcellular organelles and even wholecells. The major challenge will be to overcome the problem ofradiation damage. In this respect, it is important tounderstand that according to the model calculations reviewedhere, the X-ray free electron laser could be the solution to theradiation damage problem rather than its cause, as one tendsto assume a priori. Finally, it is pleasing to note that in spite ofthe apparent dichotomy in these approaches, using crystallinesamples on one hand and single molecules on the other, thenew methods described in this review owe a great deal to theprinciples and techniques that have been developed in thefield of X-ray crystallography, which will undoubtedly continueto make significant future contributions to the study of singlemolecules.

Acknowledgement We gratefully acknowledge support by

BMBF (through DESY-HS), TMWFK and DFG (SFB 604/C2),

as well as by the Fonds der Chemischen Industrie, U.S.

Department of Defense and the Massachusetts Department

of Public Health. RH thanks Massimo Altarelli (Sincrotrone

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Perspectives on Single Molecule DiffractionUsing the X-Ray Free Electron Laser

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Trieste) for discussions. This contribution is dedicated to the

memory of Björn Wiik, the late Director of DESY.

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