Anomalous SAXS at P12 beamline EMBL Hamburg: instrumentation and applications

Hardware and software for anomalous small-angle X-ray scattering on biological macromolecules in solution implemented at the P12 beamline of the EMBL (PETRA III storage ring, DESY, Hamburg) are described.

To calibrate the energy setting procedure of the double-crystal monochromator (DCM offset), chromium and copper foils are moved into the beam path and their absorption is measured. The measurements of the absorption edges allow to correct for a possible mechanical offset of the Bragg axis of the DCM. Transmission X-ray measurements of solutions with salts containing relevant ions such as iron or bromine, can also be conducted directly in the SAXS cell for fine energy calibration.
In addition to the Bragg angle, the undulator gap, the distance between the DCM crystals (perp) and the second crystal alignment (pitch) are adjusted such that the trajectory of the outcoming beam is kept constant for all energies and no further adjustment of the downstream optical elements is required.

S3.2. Undulator gap
Monochromatic X-rays of the P12 BioSAXS beamline are sourced from the controllable-gap low-divergence U29 undulator of PETRA III (Schöps et al., 2016;Tischer et al., 2007). Its known dependence between energy of the main harmonics of undulator and its magnetic properties and storage ring properties is given as: where K -magnetic parameter, λ 0 -magnetic period of the undulator, E -storage ring energy, n -undulator harmonics (n=1 or n=3 in for P12). The magnetic parameter reads as: where λ 0 -undulator magnetic period, B 0 -undulator magnetic field, e -electron charge, melectron mass, c -speed of light.
For PETRA III U29 undulator, it is λ 0 =2.9cm and the storage ring energy E=6 GeV.
The magnetic field B 0 can be empirically estimated as follows (Walker, 1996): where G -undulator gap, λ 0 -magnetic period of the undulator, a, b, c -empirical parameters for the particular undulator, depending on the material of the magnets and arrangement.
From (S1) we can express the magnetic parameter K as follows: After substitution (S3) to (S2) and using equation (S4) we obtain following quadratic equation: Solving this equation, one obtains the dependence of the undulator gap G [ ] at the given energy ϵ [keV]: The second root of the equation (S5) does not fit experimental data. Fitting this equation to the experimental dependence of the undulator gap at different energy settings yield empirical parameters a, b, c of the magnetic structure of the undulator and known harmonic .

Figure S5
Geometrical scheme of double-crystal monochromator to illustrate the computation of perp.
Vertical distance between incoming and outgoing beams h ( Figure S5) can be determined using following relation: where θ B -Bragg angle corresponding to the given energy via Bragg's law, z -distance between parallel crystal planes. It can be compensated by vertical translation of the crystals in perpendicular direction to each other. Energy selection can be achieved by changing the Bragg angle θ B which is corresponding to the energy with the well-known Bragg's law. In practical units it can be rewritten as: We need to account for the change of the distance between crystals in response to change of the Bragg angle in order to have the beam at the same vertical position. Assume that we slightly changed the energy (Bragg angle) than this corresponds to the following change in vertical distance h between the beams: Integration of both parts gives: Therefore, overall compensation of the vertical change between beam positions can be written as: h = h 0 (1 + ln (cosθ B )) (S13)

Figure S6
Measured DCM relative perp motor position (circles) at different photon energies. Dashed line represents fit of equation (S13) therefore allowing to calculate optimal perp position of the DCM at an arbitrary energy value.
Equation (S13) was used as a basis for calculation and finding parameters for preliminary distance setting for the DCM. Resulting fit is shown on Figure S6, which emphasizes that correct setting of the perp motor position is more important for the lower energy range than for the higher energy range.

S3.4. Pitch of the second crystal.
Pitch motor of the second crystal is used to ensure the parallelism of two crystal substrates with respect to each other. Slight error in parallelism can cause significant change in the intensity after monochromator. The angle variation should be within the Darwin width of the given set of crystals which is in turn energy-dependent. For preliminary setting of the pitch a cubic polynomial function is used. More precise setting is done via additional scan using piezo-drives with high precision.   [6000,6500,7000,7500,8000,8500,9000,9500,10000,10500,11000,11500,12000,1250 0,13000,13500,14000,14500,15000,15500,16000,16500,17000,17500,18000

S5. Estimation of constant fluorescence background from ASAXS data
The fluorescent signal is estimated by monitoring the change in the background in the SAXS data. Automatic estimation of the fluorescence background was built on method of over-subtraction determination for macromolecular SAXS data. Over-subtracted data usually produced by a mismatch between buffer used for sample preparation and the pure buffer measurement that is used later for the automated subtraction procedure. Detection of over-subtracted data is implemented as a component of the automated SAXS data analysis pipeline SASFLOW (Franke et al., 2012) and in the curated repository for small angle scattering data and models SASBDB (Kikhney et al., 2020).
Let us define "over-subtraction" as the presence of one or more systematically negative data ranges in the background-subtracted scattering curve. For any given over-subtracted scattering curve there is a minimum constant that can be added to the data to make it not over-subtracted. This constant is found using a binary search. of the Longest Consecutive Negative Sequence (LCNS) of intensity values. As a criterion for detection of such systematically negative intensity subsets is likely to occur by chance we have adopted the Correlation Map (Franke et al., 2015). The search for LCNS is repeated multiple times on the same data after averaging every two, three, four etc. subsequent points until either the data is identified as over-subtracted or a predefined minimal number of averaged data points in the scattering curve is reached. In the latter case the curve is considered not over-subtracted.
The problem of detection of "fluorescence" constant from the one-dimensional data curve can be treated as problem of "under-subtraction" when the buffer-subtracted curves still show a significant constant offset. "Under-subtraction" is treated as an inverse oversubtraction problem. The oversubtraction detection approach was adapted to find a maximum constant that can be subtracted from the experimental data without making it systematically negative.   (10 and 12.5; and 9.5 and 17 nm respectively). As it can be seen the differences in p(r) between the 0 2 and 0 2 are indeed small but statistically significant and can be used as an estimate for the sizes of the core and shells of the particles.