Fluctuation X-ray scattering

Fluctuation X-ray scattering (FXS) 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.

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

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

$C_{2}(q,q',\Delta \phi _{q})={\frac {1}{2\pi N}}\sum _{Nimages}\int _{0}^{2\pi }J_{k}(q,\phi _{q})J_{k}(q',\phi _{q}+\Delta \phi _{q})d\phi _{q}$

It can be shown that (under the assumption of a flat Ewald sphere):

$C_{2}(q,q',\Delta \phi _{q})=\sum _{l}B_{l}(q,q')P_{l}(cos[\Delta \phi _{q}])$

where

$B_{l}(q,q')=\sum _{m=-l}^{l}I_{lm}(q)I_{lm}^{*}(q')$

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.

Structure determination from FXS data

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

Algebraic symmetry-constraint imposed 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 and helical models.

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.

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.