Fluctuation X-ray scattering

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 a detailed understanding of the structure of the sample.

Fluctuation X-ray scattering (FXS)[1][2] is an X-ray scattering technique similar to small-angle X-ray scattering (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.

## Contents

* 1 Overview
* 2 Mathematical background
* 3 Basic relations
* 4 Structure determination from FXS data
* 4.1 Algebraic phasing
* 4.2 Reverse Monte Carlo
* 4.3 Multi-tiered iterative phasing
* 5 References

## 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 Bl(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 with density distribution [math]\displaystyle{ \rho(\mathbf{r}) }[/math], the associated three-dimensional complex structure factor [math]\displaystyle{ A(\mathbf{q}) }[/math] is obtained via a Fourier transform

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

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

[math]\displaystyle{ I(\mathbf{q}) = A(\mathbf{q}) A(\mathbf{q}) ^* }[/math]

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

[math]\displaystyle{ I(\mathbf{q}) = \sum_{l=0}^{\infty} \sum_{m=-l}^{l} I_{lm}(q) Y_l^m(\theta_q,\phi_q) }[/math]

The average angular intensity correlation as obtained from many diffraction images [math]\displaystyle{ J_k(q,\phi_q) }[/math] is then

[math]\displaystyle{ 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 }[/math]

It can be shown that

[math]\displaystyle{ 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] ) }[/math]

where

[math]\displaystyle{ \theta_q = \arccos( \frac{q\lambda} {4 \pi} ) }[/math]

with [math]\displaystyle{ \lambda }[/math] equal to the X-ray wavelength used, and

[math]\displaystyle{ B_l(q,q') = \sum_{m=-l}^{l} I_{lm}(q) I_{lm}^*(q') }[/math]

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

Additional relations can be obtained by obtaining the real space autocorrelation [math]\displaystyle{ \gamma(\mathbf{r}) }[/math] of the density:

[math]\displaystyle{ \gamma(\mathbf{r}) = \int_V \rho(u)\rho(\mathbf{r}-\mathbf{u}) d \mathbf{u} }[/math]

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

[math]\displaystyle{ I_{lm}(q) = \int_0^{\infty} \gamma_{lm}(r) j_l(qr) r^2 d r }[/math]

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:

[math]\displaystyle{ \log B_l(q) - 2 l \log q \approx \log B_l^{*} - \frac{2q^2R_l^2}{2l+3} }[/math]

Values of [math]\displaystyle{ B_l^* }[/math] and [math]\displaystyle{ R_l }[/math] 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:

[math]\displaystyle{ B_l(q) \propto q^{-8} }[/math]

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 [math]\displaystyle{ A(\mathbf{q}) }[/math] such that corresponding [math]\displaystyle{ B_l(q,q') }[/math] match observed values. The real-space image [math]\displaystyle{ \rho(\mathbf{r}) }[/math], as obtained by a Fourier Transform of [math]\displaystyle{ A(\mathbf{q}) }[/math] 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. ↑ 1.0 1.1 Kam, Zvi (1977). "Determination of Macromolecular Structure in Solution by Spatial Correlation of Scattering Fluctuations". Macromolecules 10 (5): 927–934. doi:10.1021/ma60059a009. Bibcode: 1977MaMol..10..927K.
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. doi:10.1073/pnas.78.6.3559. PMID 6943555. Bibcode: 1981PNAS...78.3559K.
3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 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 69 (4): 365–373. doi:10.1107/S0108767313006016. ISSN 0108-7673. PMID 23778093.
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. PMID 22898947.
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): 257204. doi:10.1103/PhysRevLett.107.257204. ISSN 0031-9007. PMID 22243108. Bibcode: 2011PhRvL.107y7204S.
7. ↑ Wochner, Peter; Gutt, Christian; Autenrieth, Tina; Demmer, Thomas; Bugaev, Volodymyr; Ortiz, Alejandro Díaz; Duri, Agnès; Zontone, Federico et al. (2009). "X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter". Proceedings of the National Academy of Sciences 106 (28): 11511–11514. doi:10.1073/pnas.0905337106. ISSN 0027-8424. PMID 20716512. Bibcode: 2009PNAS..10611511W.
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): 184204. doi:10.1103/PhysRevB.85.184204. ISSN 1098-0121. Bibcode: 2012PhRvB..85r4204K.
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. doi:10.1364/OE.19.017318. PMID 21935096. Bibcode: 2011OExpr..1917318S.
10. ↑ Poon, H.-C.; P. Schwander, M. Uddin, & D. K. Saldin (2011). "Fiber Diffraction without Fibers". Physical Review Letters 19 (18): 17318–17335. doi:10.1103/PhysRevLett.110.265505. PMID 23848897. Bibcode: 2013PhRvL.110z5505P. http://hermes.phys.uwm.edu/publications/PhysRevLett.110.265505.pdf.
11. ↑ Donatelli, Jeffrey J.; Peter H. Zwart, and James A. Sethian (2015). "Iterative phasing for fluctuation X-ray scattering". Proceedings of the National Academy of Sciences of the United States of America 112 (33): 10286–10291. doi:10.1073/pnas.1513738112. PMID 26240348. PMC 4547282. Bibcode: 2015PNAS..11210286D. http://www.pnas.org/content/early/2015/07/30/1513738112.full.pdf. early edition online ahead of publication

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