SAXS calculation

MD Simulations often complement conventional experiments, such as X-ray crystallography, Nuclear Magnetic Resonance (NMR) spectroscopy and Atomic-Force Microscopy (AFM). X-ray crystallography is a method with which the structure of molecules can be resolved. X-rays of wavelength 0.1 to 100 Å are scattered by the electrons of atoms. The intensities of the scattered rays are amplified by creating crystals containing a multitude of the studied molecule positionally ordered. The molecule is thereby no longer under physiological conditions. The study of structures in a solvent should be done under physiological conditions (in essence a disordered system), wherefore X-ray crystallography does not represent the ideal method. Small-Angle X-ray Scattering (abbreviated to SAXS) allows for measurements to be made on molecules in solutions. With this method the shape and size of the molecule and also distances within it can be obtained. That SAXS provide information on generally larger objects can be realized from the Bragg-Equation

\[n \cdot \lambda = 2 \cdot d \cdot \sin(\theta)\]

with \(n \in \mathbb{N}\), \(\lambda\), the wavelength of the incident wave, \(d\) the size of the diffracting object, and \(\theta\) the scattering angle. \(d\) and \(\theta\) are inversely proportional which means larger objects scatter X-rays at small angles.

Experiments

The measured quantity in SAXS experiments is the number of elastically scattered photons as a function of the scattering angle \(2\theta\), i.e. the intensity of the scattered rays across a range of small angles. The general set-up of a SAXS experiment is shown in figure below.

The experiments are carried out by placing the sample of interest in a highly monochromatic and collimated (parallel) X-ray beam of wavelength \(\lambda\). When the incident rays with wave vector \(\boldsymbol{k}_i\) reach the sample they scatter. The scattered rays, with wave vector \(\boldsymbol{k}_s\), are recorded by a 2D-detector revealing a diffraction pattern.

The scattering agents in the sample are electrons and so diffraction patterns reveal the electron density. Because the scattering is elastic the magnitudes of the incident and scattered waves are the same: \(|\boldsymbol{k}_i| = |\boldsymbol{k}_s| = 2\pi/\lambda\). The scattering vector is \(\boldsymbol{q} = \boldsymbol{k}_s - \boldsymbol{k}_i\) with a magnitude of \(q = |\boldsymbol{q}| = 4\pi \sin(\theta)/\lambda\). From the intensity of the scattered wave, \(I_s(\boldsymbol{q})\), and each particle`s form factor \(f (q)\), the structure factor can be obtained.

Simulations

In simulations the structure factor \(S(\boldsymbol{q})\) can be extracted directly from the positions of the particles. MAICoS’ Saxs module calculates these factors. The calculated scattering intensities can be directly compared to the experimental one without any further processing. We now derive the essential equations. \(S(\boldsymbol{q})\) is defined as

\[S(\boldsymbol{q}) = \frac{1}{\sum_{j=1}^N f_j^2(q)} I_s(\boldsymbol{q}) \,.\]

The form factor as a function of \(q\) is specific to each atom and relates to the amplitude of the scattered waves.

The scattering intensity is expressed as

\[I_s(\boldsymbol{q}) = A_s(\boldsymbol{q}) \cdot A_s^*(\boldsymbol{q}) \,,\]

with the amplitude of the elastically scattered wave

\[A_s(\boldsymbol{q}) = \sum\limits_{j=1}^N f_j(q) \cdot e^{-i\boldsymbol{qr}_j} \,,\]

\(f_j(q)\) is the form factor and \(\boldsymbol{r}_j\) the position of the \(j\) th atom out of \(N\) atoms. The complex conjugate of the amplitude is

\[A_s^*(\boldsymbol{q}) = \sum\limits_{k=1}^N f_k(q) \cdot e^{i\boldsymbol{qr}_j} \,.\]

The intensity therefore can be written as

\[I_s (\boldsymbol{q}) = \sum\limits_{j=1}^N f_j(q) e^{-i\boldsymbol{qr}_j} \cdot \sum\limits_{k=1}^N f_k(q) e^{i\boldsymbol{qr}_k} \,.\]

With Euler’s formula \(e^{i\phi} = \cos(\phi) + i \sin(\phi)\) the intensity is

\[I_s (\boldsymbol{q}) = \sum\limits_{j=1}^N f_j(q) \cos(\boldsymbol{qr}_j) - i \sin(\boldsymbol{qr}_j) \cdot \sum\limits_{k=1}^N f_k(q) \cos(\boldsymbol{qr}_k) - i \sin(\boldsymbol{qr}_k) \,.\]

Multiplication of the terms and simplifying yields the final expression for the intensity of a scattered wave as a function of the wave vector and with respect to the particle’s form factor

\[I_s (\boldsymbol{q}) = \left[ \sum\limits_{j=1}^N f_j(q) \cos(\boldsymbol{qr}_j) \right ]^2 + \left[ \sum\limits_{j=1}^N f_j(q) \sin(\boldsymbol{qr}_j) \right ]^2 \,.\]

For an isotropic systems containing only one kind of atom the structure factor is

\[S(\boldsymbol{q}) = \left\langle \frac{1}{N}\sum\limits_{j=1}^N f_j(q) \cos(\boldsymbol{qr}_j) \right \rangle^2 + \left\langle \frac{1}{N} \sum\limits_{j=1}^N f_j(q) \sin(\boldsymbol{qr}_j) \right \rangle^2 \,.\]

The structure factor of systems with more than one atom type is the sum of partial structure factors normalised by the form factor

\[S(\boldsymbol{q}) = \left\langle \frac{1}{\sum_{j=1}^N f_j^2(q)}\sum\limits_{j=1}^N f_j(q) \cos(\boldsymbol{qr}_j) \right \rangle^2 + \left\langle \frac{1}{\sum_{j=1}^N f_j^2(q)} \sum\limits_{j=1}^N f_j(q) \sin(\boldsymbol{qr}_j) \right \rangle^2 \,.\]

The form factors \(f(q)\) of a specific atom can be approximated with

\[f(\sin\theta/\lambda) = \sum_{i=1}^4 a_i e^{-b_i \sin^2\theta/\lambda^2} + c \,.\]

Expressed in terms of the scattering vector we can write

\[f(q) = \sum_{i=1}^4 a_i e^{-b_i q^2/(4\pi)^2} + c \,.\]

The coefficients \(a_{1,\dots,4}\), \(b_{1,\dots,4}\) and \(c\) are documented in Prince[1].

References

[1]

E. Prince. International Tables for Crystallography,Volume C: Mathematical, Physical and Chemical Tables. Springer, Dordrecht, 3rd ed. edition edition, January 2004. ISBN 978-1-4020-1900-5.