Reconstruction of the electron density of molecules with single-axis alignment

Transcription

1 SLAC-PUB-143 Reconstruction of the electron density of olecules with single-axis alignent Ditri Starodub* a, John C. H. Spence b, Dilano K. Saldin c a Stanford PULSE Institute, SLAC National Accelerator Laboratory, 575 Sand Hill Rd, Menlo Par, CA, USA b Dept. of Physics, Arizona State University, PO Box , Tepe, AZ 8587 c Dept. of Physics, University of Wisconsin-Milwauee, PO Box 413, Milwauee, WI 5301 ABSTRACT Diffraction fro the individual olecules of a olecular bea, aligned parallel to a single axis by a strong electric field or other eans, has been proposed as a eans of structure deterination of individual olecules. As in fiber diffraction, all the inforation extractable is contained in a diffraction pattern fro incidence of the diffracting bea noral to the olecular alignent axis. We present two ethods of structure solution for this case. One is based on the iterative projection algoriths for phase retrieval applied to the coefficients of the cylindrical haronic expansion of the olecular electron density. Another is the holographic approach utilizing presence of the strongly scattering reference ato for a specific olecule. Keywords: X-ray coherent iaging, fiber diffraction, phase retrieval 1. INTRODUCTION Due to very sall X-ray atoic scattering cross-sections, usual diffraction ethods for structure deterination require easureent of scattered intensity fro any identical copies of acroolecules. In conventional crystallography this is achieved by using crystals, where identically oriented acroolecules are located in regularly spaced positions. This gives an enhanceent of the scattered signal by a factor of the squared ratio of the crystal volue to that of a unit cell. However, translational syetry liits the scattered intensity sapling to reciprocal lattice of the crystal, which does not provide enough inforation for solution of the phase proble fro the structure factors alone without saple odification and/or using anoalous scattering. Furtherore, large protein coplexes and ebrane proteins, vital for the function of a living cell, are difficult to crystallize, and therefore constitute a ajor challenge for structural biology. In the opposite liit of sall angle scattering, the incoherent su of solution scattering fro acroolecules in rando orientations is recorded, resulting in the spherically averaged diffraction pattern. Then a low resolution structure can be recovered by the Monte Carlo search of a odel that provides a best fit to experiental data. In this type of siulation the proble of uniqueness of the recovered structure is intrinsically present. It was proposed to overcoe difficulties of crystallography and of solution scattering by replacing interolecular forces, orienting acroolecules in a crystal, by interaction of individual isolated olecules with an external field. 1 In this ethod, scattered intensity is collected fro a strea of aligned olecules, and a wea diffraction pattern of the individual olecule is ultiplied by the total nuber of olecules interacting with the incident X-rays. A liited spatial coherence width of the X-ray bea is assued to prevent interference between X-rays scattered by different olecules. Absence of translational periodicity allows arbitrary sapling of the diffraction pattern and subsequent use of well-developed iterative projection algoriths to solve the phase proble. Three-diensional alignent of sall olecules can be achieved by interaction of their polarizability anisotropy with the electric field of an intense elliptically polarized laser pulse, 3 while interaction of the induced dipole oent with the electric field of a linearly polarized laser aligns the olecule s axis of easy polarization along the electric field, 3 if the induced dipole electric field interaction energy is large copared to the theral energy of the olecule. In this field configuration, rotation of the olecule with respect to the alignent axis is not restricted. In any cases realization of the single axis alignent is easier to achieve, and for soe systes, such as syetric-top *starodub@slac.stanford.edu Wor supported in part by US Departent of Energy contract DE-AC0-76SF00515.

2 olecules, the rotational dynaics about their syetry axis cannot be constrained with the elliptically polarized light. The alternatives to the laser alignent include strong electric or agnetic fields and a shear flow. It has been deonstrated that the oderate electric field of V/c is sufficient for the perfect alignent of single wall nanotube (SWNT ethanol suspensions. 4 The dipole oent of a peptide bond is ~3.6 D, and their alignent in α-helices results in buildup of the acroscopic dipole oent roughly in proportion to the nuber of residues. 5 Thus, large proteins can have dipole oents of D, which is sufficient for alignent in a strong electric field. Apparently, inforation is inevitably lost upon rotational averaging of scattering data fro the olecules aligned in one direction. The ai of this wor is to deonstrate that under soe circustances, in particular for high degree or nown olecular cylindrical syetry, the structural inforation on an individual olecule can be deduced fro a single diffraction pattern produced by a large nuber of the single-axis aligned olecules randoly rotated about the alignent axis. Such a diffraction pattern is siilar in character to a fiber diffraction pattern. However, fibrous olecules studied by fiber diffraction display a helical syetry, and therefore they are characterized by soe repeat distance along their long axis. That restricts scattering to the layer planes perpendicular to the direction of the olecular alignent. Unlie fibrous olecules, olecules aligned by an external field do not have periodicity in any direction, and corresponding diffraction patterns are continuous. This allows an oversapling of the scattered intensity with respect to the Shannon sapling interval as deterined by the olecular size. Then a two-diensional iterative projection algorith, which alternatively applies constraints in real and reciprocal space, can be applied to reconstruct at least aziuthally averaged electron density of the saple. 6 In the special cases, when a olecule contains a strongly scattering reference ato, the diffraction pattern can be treated as a hologra, and the aziuthal projection of the olecular electron density about the alignent axis ay be directly obtained by the corresponding transfor of the diffraction pattern. 7. STRUCTURE DETERMINATION FROM CYLINDRICALLY AVERADED DIFFRACTION PATTERN.1 Fourier transfor in cylindrical coordinates for non-periodic object In this section we establish relationship between the electron density of an individual particle and the diffraction pattern produced by a large nuber of aligned identical particles, different only in angle of rotation about alignent axis. The derivation closely follows that for fiber diffraction, 8 with an iportant difference arising fro the fact that we do not have periodicity in the alignent direction. That results in a continuous diffraction pattern as opposed to layer lines observed in fiber diffraction, and allows direct application of iterative projection algoriths for phase retrieval. Since there is no interference between olecules, scattered intensity fro a large enseble of aligned olecules at rando aziuthal angles is equivalent to the diffraction pattern of an individual olecule averaged over all possible rotations. Due to cylindrical averaging it is convenient to introduce cylindrical coordinates ( r, ϕ, z in real space, and corresponding coordinates ( R,ψ, ς in reciprocal space, where z and ζ are parallel to the alignent axis. Any single valued and continuous function f ( r,ϕ, z can be expanded in a Fourier series in ters of the orthogonal set of basis functions exp ( iϕ defined on the unit circle in the plane perpendicular to rotation axis. Then electron density of the olecule can be written as the cylindrical haronic expansion f ( r, ϕ, z = g ( r, z exp( ϕ, (1 where the two-diensional function ( r z i g, is deterined as 1 g ( r, z = f ( r, ϕ, z exp( iϕ dϕ. ( π In the sae fashion, the scattering aplitude of the olecule can be represented in reciprocal space as The scattered intensity is A ( R ψ, ς = G ς exp( ψ,. (3 i

3 * ψ, ς = A ψ, ς A * ψ, ς = G ς G ς exp[ i( ψ ] I ψ ψ, (4 where angle bracets stand for angular average over aziuthal angle. The cross ters in the angle bracets < > in Eq. (4 vanish, leaving only diagonal ters: ( R, = G ( R, ς = I ς, (5 where there is no dependence on the aziuthal angle ψ. Thus, in the general case the scattered intensity fro the ultiple olecules aligned parallel to a single axis is a su of independent contributions fro different cylindrical haronics. Eq. (5 also elucidates how the inforation content of the full three-diensional diffraction pattern is reduced by cylindrical averaging. If the cylindrical haronic expansion is terinated at the ter ax, then the right hand side of Eq. (5 can be regarded as the squared nor of the 4 ax vector, with each coefficient ( R,ς G contributing two diensions corresponding to its real and coplex parts. If scattered intensity fro a fixed olecule is easured in a given point of reciprocal space, then odulus constraint for this point is deterined by a two-diensional sphere of radius equal to square root of easured intensity. For the cylindrically averaged scattered intensity at a given pixel, odulus constraint is set by a sphere in 4 ax diensions. Since the X-ray scattering cross section is sall, scattering can be described in the Born approxiation, and in the far field the scattering aplitude is given by the Fourier transfor of the olecular electron density: 1 3 A( q = f ( r exp( iqr d r. (6 3 π ( Substituting here scattering vector q and radius-vector r expressed in cylindrical coordinates, and using Jacobi-Anger expansion 9 ( iz cosθ = i J ( z exp( iθ = exp, (7 where J ( z is the th order Bessel function of the first ind, and using Eq. ( we arrive to the expression A ( R ψ, ς = i rg ( r, z J ( rr exp i( zς + ψ 1, [ ]drdz. (8 π Fro coparison of Eq. (8 and Eq. (3 the relationship between the coefficients of the cylindrical haronic expansions of the olecular electron density and its scattering aplitude can be identified as G i, = drdz. (9 π ( R ς rg ( r, z J ( rr exp( izς The relation inverse to Eq. (9 can be derived in the sae way fro the inverse Fourier transfor of the olecular scattering aplitude as follows: g ( r z ( i ς J ( rr exp( izς, = RG drdz. (10 π Eqs. (9 and (10 represent a cobination of a Hanel transfor in radial direction and a Fourier transfor in the direction of alignent, which we ter a Fourier-Hanel transfor. These coupled two-diensional equations relate the cylindrical haronic expansion coefficients of the scattering aplitude with those of the electron density of the olecule. These relationships are analogous to the Fourier transfor and its inverse, which relate scattering aplitude and corresponding electron density in a standard scattering proble. In a case where the agnitudes of G ς are nown, but not their phases, and the spatial extent of ( r z g, is finite, it would be expected that a two-diensional

4 iterative phasing algorith with appropriate constraints in reciprocal and real space will allow the siultaneous g r, z. deterination of the unnown phases and of (. Analogy with fiber diffraction Due to periodicity of the helical structure in fiber diffraction with repeat distance c along axis z, function ( r z be expanded in Fourier series with respect to arguent z, and Eq. (1 becoes f ( r ϕ, z = g ( r exp[ i( ϕ πlz ] l l c Coponent g l ( r and its counterpart in reciprocal space ( R g ( r = RG ( R J ( rr dr g, can,. (11 G l are related by the Fourier-Bessel transfor l l. (1 The quantity easured in fiber diffraction is scattered intensity of the layer line I l ( R = I = πl c = G ( R l ς. (13 Eqs. (11-(13 of fiber diffraction 8 correspond to Eqs. (1, (10, and (5, respectively, for continuously scattering aligned non-periodic objects. As soon as coponents G l ( R and their phases are deterined, electron density of fibrous olecule can be found fro Eqs. (11 and (1..3 Algorith for phase retrieval As in fiber diffraction, electron density reconstruction in our case is equivalent to the proble of deconvolution the individual contributions in Eq. (5 and deterination of their phases. The nuber of -coponents which should be ept in Eq. (5 depends on the saple and required resolution. It has been noted 10 that a Bessel function of order has negligible values if its arguent x is saller than. Therefore, it would be expected that for a given x = rr contribution of all cylindrical haronics with > x to the diffraction pattern intensity vanishes. The axiu value of x is defined by the axiu radial coordinate of any ato of the olecule relative to the olecular rotation axis, which can be approxiated by the olecule radius a, and required resolution d = π/r ax. Then all ters with > πa/d in Eq. (5 can be neglected. If the olecule is characterized by M-fold rotational syetry, then cylindrical haronic expansion of its electron density contains only ters of odulo M, i.e. = 0, M, M, Provided M > πa/d, the ter with = 0 is the overwhelingly doinant contributor to scattered intensity, and G 0 ς can be approxiated by the square root of the easured diffraction pattern. The phases of G 0 ς can be deterined by iteratively solving Eqs. (9 and (10 for = 0 to find the solution that siultaneously belongs to the set of objects, whose transfor (9 of aziuthal projection results in the easured diffraction pattern, and the set of objects, satisfying available constraints in the real space, such as support constraint, which sets the electron density outside the nown object boundaries to zero. In a ore general case, it ight be necessary to find a few ters in Eq. (5 beyond = 0. Note that for a real object only ters with 0 need to be considered. Expansion of the real electron density in cylindrical haronics (1 can be written as * Using identity G ς = ( 1 G ( R ς f ( r,, z = g ( r, z + Re[ g ( r, z ( iϕ ] 1 ϕ 0 exp. (14, for expansion coefficients of the scattering aplitude for a real valued object, we obtain in place of Eq. (5 I = G ς + G ς + G ( R ς ( ς. (15 0, 1

5 Figure 1. Bloc diagra of the phasing algorith. The expansion coefficients of scattering aplitude are independent, however they can be related by the positivity or other constraints in real space after their transfor (10. The standard odulus projection which eeps the phases of current iterate G' ς of scattering aplitude and replaces agnitudes by the easured intensities in diffraction pattern, is odified to the for G = G' ς I ς ς + G' ς + G' ( R ς ( ς, (16 G ' 0, where suation is perfored over all non-vanishing positive. This is the sae projection that was used in phasing powder diffraction pattern to separate overlaps of non-equivalent reflections having the sae agnitudes of scattering vector, 11 originating fro spherical averaging of powder diffraction patterns. The ethod has been generalized by Elser and Millane 1 for the reconstruction proble in the case of diffraction pattern incoherently averaged over a discrete syetry group. It was shown that with positivity constraint available, reconstructions are successful for up to syetry group order of 4. Applied to our proble, that iplies that just one coponent corresponding to non-zero can be included in Eq. (14. However, it can be expected that this condition will be relieved if additional constraints are available in real space. Bloc diagra of the phasing algorith is presented in Fig. 1. As a starting point, the scattered intensity is equally distributed aong N expansion coefficients included in the diffraction pattern decoposition, with each of those coefficients assigned a value of G ς = I ς N, and a set of rando phases χ ( R,ς is generated for the. After transfor (10 for each -coponent, they have been cobined via Eq. (14 to obtain a first estiation of electron density. This estiate is a real function which generally contains both positive and negative values. A suitable object-doain operation utilizing projection on the set restricted by object support and positivity is applied, and subsequent inverse Fourier-Hanel transfor (9 gives a new estiate of scattering aplitude expansion coefficients. Their phases and relative values for different -coponents are retained, but aplitudes are renoralized according to projection (16. The algorith iterates between two constrained sets until convergence occurs.

6 .4 Holographic approach If a olecule contains a strong scatterer, an alternative approach to reconstructing an aziuthal projection of the olecule can be applied. In this case diffraction pattern can be treated as a hologra, consisting of the square odulus of a superposition of a relatively large reference wave of nown for and an unnown object wave. Then, the scattered intensities becoe linear in the unnown object wave and its conjugate, thus allowing the reconstruction of the object giving rise to object wave. This ethod was eployed to reconstruct a siulated laser aligned syetric top olecule CF 3 Br. 7 The for factor of the Br atos located on the rotation axis of the olecule doinates over those of F and C. It was deonstrated that application of the transfor (10 with = 0 directly to the diffraction pattern gives the aziuthal projection of the CF 3 Br olecule, along with a twin iage, irror syetric relative to the plane perpendicular to the rotation axis, as shown in Fig. 1. Figure. (a X-ray diffraction pattern fro CF 3 Br olecules perfectly aligned with respect to horizontal axis perpendicular to the X-ray bea. (b The sae for iperfect alignent due to the theral oscillations at 1 K. Green area corresponds to the non easurable issing edge due to the Ewald sphere curvature; (c-(d The aziuthally projected electron density holographically reconstructed fro the siulated diffraction patterns (a and (b, respectively. 3. NUMERICAL EXAMPLE As the test object for our siulations we choose the E. coli chaperonin GroEL 14 GroES 7 (ADP AlF x 7 protein coplex, constituted of 59,76 non-hydrogen atos. The length of the coplex is 0 n, and diaeter 14.5 n. GroEL contains 14 identical subunits of olecular ass 58 Da, and GroES contains 7 subunits of olecular ass 10 Da. They for a structure consisting of three distinctive rings. The atoic coordinates have been obtained fro the Protein Data Ban (entry 1SVT and converted to cylindrical coordinates (r, φ, z, where the subscript specifies ato. Contributions of different cylindrical haronics into diffraction pattern can be coputed fro the discrete counterpart of Eqs. ( and (9: G R, ς = f exp i ςz ϕ π J Rr, (17 ( {[ ( ]} ( where f is the for factor of ato. Aziuthal projection of the protein coplex coputed by inverse Fourier-Hanel transfor (10 of G 0ς is shown in Fig. 3(a. Since the structure of GroEL GroES coplex is characterized by a 7- fold rotational syetry about a long olecular axis, only expansion coefficients for = 0, ±7, ±14 will have nonvanishing values. A diffraction pattern was calculated on a regular grid to the axiu scattering vector in both R and ζ directions of 5 n -1. Incident X-ray bea polarization and grid distortion due to the Ewald sphere curvature are not considered, since these effects can be accounted for during data post-processing. The presence of the wedge with issing data is also ignored, because it is sall for low resolution diffraction pattern (3 pixels at the detector edge for incident X-ray energy of 10 ev, and the values of the data in the issing wedge can be estiated by allowing the to float in the iterative phasing algorith. With olecular radius of 7.5 n, cylindrical haronics of the order up to = 35 should be included in the su (13, but for the test purposes we truncate the expansion at = 14. The diffraction

7 Figure 3. (a Aziuthal projection g ( r, z 0 of the GroEL-GroES protein coplex. (b Cylindrically averaged diffraction pattern for the GroEL-GroES acroolecule siulated fro the atoic coordinates. Figure 4. Aziuthal projection of the GroEL-GroES electron density reconstructed fro the diffraction pattern in Fig. 3(b using (a = 0 cylindrical haronics only, and (b = 0, ±7 haronics of scattering aplitude to represent scattered intensity. (c Real space root ean square error for the phasing algorith for different nuber of cylindrical haronics included in the diffraction pattern decoposition. (d Contributions of the basis functions with = 0 and = 7 (solid blac and gray lines, respectively into the total olecular scattering aplitude in the plane ζ = 0, siulated fro atoic coordinates. Sybols of corresponding colors show reconstructed contributions. pattern coputed fro Eqs. (15 and (17 using tabulated atoic for-factors 13 is shown in Fig. 3(b. For the iterative schee, we eployed cobination of the hybrid input-output (HIO algorith (0 cycles, based on the negative

8 feedbac, and the error reduction algorith (00 cycles, which siply projects bac and forth between two constrained sets. Cylindrical support of the height of 3 n and radius of 13 n has been used, providing required oversapling. Convergence of the iterative procedure was onitored by error etric in real space, equal to the noralized aount of charge-density reaining outside the support. Fig. 4 copares algorith perforance for two cases: (1 the aziuthal projection = 0 only is included in the algorith, and ( the = ±7 ters are additionally used to synthesize the threediensional electron density ap and diffraction pattern. The first procedure converges to a solution with high probability for a hypothetical exercise when only the 0th ter of the scattering aplitude expansion contributes to diffraction pattern. However, after the higher degree ters are added to the diffraction pattern siulation, the perforance of the algorith drastically deteriorates. In particular, the success rate drops to 6%, and even then the final iage significantly deviates fro a true solution, as coparison of Fig. 3(a and Fig. 4(a reveals. Including ters = ±7 in the reconstruction algorith results in increase of success rate to 49% in 00 trials with rando initial phases. The reconstructed iage of the olecular aziuthal projection after one run is shown in Fig. 4(b, and deonstrates uch higher siilarity to the original iage than that in Fig. 4(a. The difference in the behavior of the two variants of the algorith is also anifested in the behavior of the error etrics, illustrated in Fig. 4(c. When haronics = ±7 are taen into account, the final error is twice as sall as copared to the case when = 0 only is included. Additionally, error spies during HIO cycles becoe uch saller. Relative values of the = 0 and = 7 contributions of the scattering aplitude at ζ = 0 as deterined by the algorith are shown in Fig. 4(d by blac circles and grey squares, respectively. They agree well with corresponding curves directly siulated fro atoic structure, and shown by solid lines. Unfortunately, the algorith was unable to separate correctly contributions fro the = 7 and = -7 expansion coefficients for ζ 0, which prevented us fro reconstructing a three-diensional olecular envelope. However, the = 0 coponent was still properly extracted, allowing for the reconstruction of the aziuthal olecular projection. Note that the resulting decoposition of the scattering aplitude into coponents originated fro different is not sensitive to the input initial guesses for their contributions into the easured diffraction pattern. 4. DISCUSSION The theory of fiber diffraction is applied to fibrous olecules with helical structure, characterized by a repeat distance along the helical axis. The periodicity along this direction results in the appearance of the layer lines in the diffraction pattern, separated in reciprocal space by an inverse repeat distance. An enseble of the isolated olecules, aligned parallel to a single olecular axis, lacs a translational periodicity inherent to a fiber structure. Therefore, the coposite diffraction pattern produced by an incoherent su of diffraction patterns fro these olecules, is continuous in all directions. This property allows for oversapling of the easured scattered intensity, required for application of iterative projection phase retrieval algoriths. We have shown that for olecules with high degree of the nown rotational syetry, when a three-diensional scattering aplitude can be odeled by a few ters of expansion in cylindrical haronics, such algoriths are capable of extracting a 0th ter in this expansion, corresponding to the cylindrically averaged scattering aplitude. This is in turn related to the aziuthally averaged electron density by a two-diensional transfor, coprised by a Fourier transfor in the direction of olecular alignent, and a Hanel transfor of order zero in the radial direction. Then iterations of the Fourier-Hanel transfor between two constrained sets allow reconstruction of the aziuthally projected electron density. This schee is directly applicable to the scattering geoetry, where only alignent along a single axis is achieved, and olecular rotation about this axis and translation otion are not restricted. This situation could be realized in the laser alignent, alignent by electric and agnetic field, or flow alignent. ACKNOWLEDGEMENTS DS was supported through the PULSE Institute at the SLAC National Accelerator Laboratory by the U.S. Departent of Energy, Office of Basic Energy Sciences. JCHS and DKS jointly acnowledge support fro DOE grant DE-SC and DKS also acnowledges grant DE-FG0-06ER4677 for financial support. REFERENCES [1] Spence, J. C. H. and Doa, R. B., Single olecule diffraction, Phys. Rev. Lett. 9, (004. [] Marchesini, S., A united evaluation of iterative projection algoriths for phase retrieval, Rev. Sci. Instru. 78, (007.

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