Fluctuation X-ray scattering

Fluctuation X-ray scattering (FXS)^[1][2] is an X-ray scattering technique similar to SAXS, but is performed using X-ray exposures below sample rotational diffusion times. This technique, ideally performed with an ultra-bright X-ray light source, such as a free electron laser, results in data containing significantly more information as compared to traditional scattering methods.^[3]

FXS can be used for the determination of (large) macromolecular structures,^[4] but has also found applications in the characterization of metallic nanostructures,^[5] magnetic domains^[6] and colloids.^[7]

The most general setup of FXS is a situation in which fast diffraction snapshots of models are taken which over a long time period undergo a full 3D rotation. A particularly interesting subclass of FXS is the 2D case where the sample can be viewed as a 2-dimensional system with particles exhibiting random in-plane rotations. In this case, an analytical solution exists relation the FXS data to the structure.^[8] In absence of symmetry constraints, no analytical data-to-structure relation for the 3D case is available, although various iterative procedures have been developed.

A fluctuation scattering experiment collects a series of X-ray diffraction snapshots of multiple proteins (or other particles) in solution. An ultrabright X-ray laser provides fast snapshots, containing features that are angularly non-isotropic (speckle), ultimately resulting in an detailed understanding of the structure of the sample.

Overview

An FXS experiment consists of collecting a large number of X-ray snapshots of samples in a different random configuration. By computing angular intensity correlations for each image and averaging these over all snapshots, the average 2-point correlation function can be subjected to a finite Legendre transform, resulting in a collection of so-called B_l(q,q') curves, where l is the Legendre polynomial order and q / q' the momentum transfer or inverse resolution of the data.

Mathematical background

A visual representation of mathematical relations in Fluctuation X-ray Scattering illustrates the relation between the electron density, scattering amplitude, diffracted intensities and angular correlation data. Image modified from ^[3]

Given a particle $\rho(\mathbf{r})$, the associated three-dimensional complex structure factor $A(\mathbf{q})$ is obtained via a Fourier transform

$A(\mathbf{q}) = \int_V \rho(\mathbf{r}) \exp[i\mathbf{qr}] d \mathbf{r}$

The intensity function corresponding to the complex structure factor is equal to

$I(\mathbf{q}) = A(\mathbf{q}) A(\mathbf{q}) ^*$

where $^*$ denotes complex conjugation. Expressing $I(\mathbf{q})$ as a spherical harmonics series, one obtains

$I(\mathbf{q}) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} I_{lm}(q) Y_l^m(\theta_q,\phi_q)$

The average angular intensity correlation as obtained from many diffraction images $J_k(q,\phi_q)$ is then

$C_2(q,q',\Delta\phi_q) = \frac{1}{2\pi N}\sum_{N images} \int_0^{2\pi} J_k(q,\phi_q) J_k(q',\phi_q+\Delta\phi_q) d \phi_q$

It can be shown that:

$C_2(q,q',\Delta\phi_q) = \sum_l B_l(q,q') P_l( \cos( \theta_q )\cos( \theta_{q'} ) + \sin( \theta_q )\sin( \theta_{q'} ) \cos[ \Delta \phi_q] )$

where

$\theta_q = \arccos( \frac{q\lambda} {4 \pi} )$

with $\lambda$ equal to the X-ray wavelength used, and

$B_l(q,q') = \sum_{m=-l}^{l} I_{lm}(q) I_{lm}^*(q')$

$P_l(\cdot)$ is a Legendre Polynome. The set of $B_l(q,q')$ curves can be obtained via a finite Legendre transform from the observed autocorrelation $C_2(q,q',\Delta\phi_q)$ and are thus directly related to the structure $\rho(\mathbf{r})$ via the above expressions.

Additional relations can be obtained by obtaining the real space autocorrelation $\gamma(\mathbf{r})$ of the density:

$\gamma(\mathbf{r}) = \int_V \rho(u)\rho(\mathbf{r}-\mathbf{u}) d \mathbf{u}$

A subsequent expansion of $\gamma(\mathbf{r})$ in a spherical harmonics series, results in radial expansion coefficients that are related to the intensity function via a Hankel transform

$I_{lm}(q) = \int_0^{\infty} \gamma_{lm}(r) j_l(qr) r^2 d r$

A concise overview of these relations has been published elsewhere^[1][3]

Basic relations

A generalized Guinier law describing the low resolution behavior of the data can be derived from the above expressions:

$\log B_l(q) - 2 l \log q \approx \log B_l^{*} - \frac{2q^2R_l^2}{2l+3}$

Values of $B_l^*$ and $R_l$ can be obtained from a least squares analyses of the low resolution data.^[3]

The falloff of the data at higher resolution is governed by Porod laws. It can be shown ^[3] that the Porod laws derived for SAXS/WAXS data hold here as well, ultimately resulting in:

$B_l(q) \propto q^{-8}$

for particles with well-defined interfaces.

Structure determination from FXS data

Currently, there are three routes to determine molecular structure from its corresponding FXS data.

Algebraic phasing

By assuming a specific symmetric configuration of the final model, relations between expansion coefficients describing the scattering pattern of the underlying species can be exploited to determine a diffraction pattern consistent with the measure correlation data. This approach has been shown to be feasible for icosahedral^[9] and helical models.^[10]

Reverse Monte Carlo

By representing the to-be-determined structure as an assembly of independent scattering voxels, structure determination from FXS data is transformed into a global optimisation problem and can be solved using simulated annealing.^[3]

Multi-tiered iterative phasing

The multi-tiered iterative phasing algorithm (M-TIP) overcomes convergence issues associated with the reverse Monte Carlo procedure and eliminates the need to use or derive specific symmetry constraints as needed by the Algebraic method. The M-TIP algorithm utilizes non-trivial projections that modifies a set of trial structure factors $A(\mathbf{q})$ such that corresponding $B_l(q,q')$ match observed values. The real-space image $\rho(\mathbf{r})$, as obtained by a Fourier Transform of $A(\mathbf{q})$ is subsequently modified to enforce symmetry, positivity and compactness. The M-TIP procedure can start from a random point and has good convergence properties.^[11]

References

1. Kam, Zvi (1977). "Determination of Macromolecular Structure in Solution by Spatial Correlation of Scattering Fluctuations". Macromolecules 10 (5): 927–934. Bibcode:1977MaMol..10..927K. doi:10.1021/ma60059a009.
2. Kam, Z.; M. H. Koch, and J. Bordas (1981). "Fluctuation x-ray scattering from biological particles in frozen solution by using synchrotron radiation". Proceedings of the National Academy of Sciences of the United States of America 78 (6): 3559–3562. Bibcode:1981PNAS...78.3559K. doi:10.1073/pnas.78.6.3559.
3. Malmerberg, Erik; Cheryl A. Kerfeld and Petrus H. Zwart (2015). "Operational properties of fluctuation X-ray scattering data". IUCrJ 2 (3): 309–316. doi:10.1107/S2052252515002535. PMID 25995839.
4. Liu, Haiguang; Poon, Billy K.; Saldin, Dilano K.; Spence, John C. H.; Zwart, Peter H. (2013). "Three-dimensional single-particle imaging using angular correlations from X-ray laser data". Acta Crystallographica Section A Foundations of Crystallography 69 (4): 365–373. doi:10.1107/S0108767313006016. ISSN 0108-7673.
5. Chen, Gang; Modestino, Miguel A.; Poon, Billy K.; Schirotzek, André; Marchesini, Stefano; Segalman, Rachel A.; Hexemer, Alexander; Zwart, Peter H. (2012). "Structure determination of Pt-coated Au dumbbellsviafluctuation X-ray scattering". Journal of Synchrotron Radiation 19 (5): 695–700. doi:10.1107/S0909049512023801. ISSN 0909-0495.
6. Su, Run; Seu, Keoki A.; Parks, Daniel; Kan, Jimmy J.; Fullerton, Eric E.; Roy, Sujoy; Kevan, Stephen D. (2011). "Emergent Rotational Symmetries in Disordered Magnetic Domain Patterns". Physical Review Letters 107 (25). Bibcode:2011PhRvL.107y7204S. doi:10.1103/PhysRevLett.107.257204. ISSN 0031-9007.
7. Wochner, Peter; Gutt, Christian; Autenrieth, Tina; Demmer, Thomas; Bugaev, Volodymyr; Ortiz, Alejandro Díaz; Duri, Agnès; Zontone, Federico; Grübel, Gerhard; Dosch, Helmut (2009). "X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter". Proceedings of the National Academy of Sciences 106 (28): 11511–11514. Bibcode:2009PNAS..10611511W. doi:10.1073/pnas.0905337106. ISSN 0027-8424.
8. Kurta, R. P.; Altarelli, M.; Weckert, E.; Vartanyants, I. A. (2012). "X-ray cross-correlation analysis applied to disordered two-dimensional systems". Physical Review B 85 (18). arXiv:1202.6253. Bibcode:2012PhRvB..85r4204K. doi:10.1103/PhysRevB.85.184204. ISSN 1098-0121.
9. Saldin, D. K.; H.-C. Poon, P. Schwander, M. Uddin, and M. Schmidt (2011). "Reconstructing an icosahedral virus from single-particle diffraction experiments". Optics Express 19 (18): 17318–17335. arXiv:1107.5212. Bibcode:2011OExpr..1917318S. doi:10.1364/OE.19.017318.
10. Poon, H.-C.; P. Schwander, M. Uddin, & D. K. Saldin (2011). "Fiber Diffraction without Fibers" (PDF). Physical Review Letters 19 (18): 17318–17335. Bibcode:2013PhRvL.110z5505P. doi:10.1103/PhysRevLett.110.265505.
11. Donatelli, Jeffrey J.; Peter H. Zwart, and James A. Sethian (2015). "Iterative phasing for fluctuation X-ray scattering" (PDF). Proceedings of the National Academy of Sciences of the United States of America 112: 10286–10291. doi:10.1073/pnas.1513738112. early edition online ahead of publication