research letters
Xray photon correlation spectroscopy of protein dynamics at nearly diffractionlimited storage rings
^{a}European Xray Free Electron Laser Facility, Holzkoppel 4, D22869 Schenefeld Germany, ^{b}Deutsches Elektronen Synchrotron DESY, D22607 Hamburg, Germany, and ^{c}Department Physik, Universität Siegen, D57072 Siegen, Germany
^{*}Correspondence email: christian.gutt@unisiegen.de
This study explores the possibility of measuring the dynamics of proteins in solution using Xray photon correlation spectroscopy (XPCS) at nearly diffractionlimited storage rings (DLSRs). We calculate the signaltonoise ratio (SNR) of XPCS experiments from a concentrated lysozyme solution at the length scale of the hydrodynamic radius of the protein molecule. We take into account limitations given by the critical Xray dose and find expressions for the SNR as a function of beam size, sampletodetector distance and photon energy. Specifically, we show that the combined increase in coherent
and coherence lengths at the DLSR PETRA IV yields an increase in SNR of more than one order of magnitude. The resulting SNR values indicate that XPCS experiments of biological macromolecules on nanometre length scales will become feasible with the advent of a new generation of synchrotron sources. Our findings provide valuable input for the design and construction of future XPCS beamlines at DLSRs.Keywords: materials science; structural biology; nanoscience; radiation damage; SAXS; storage rings; Xray photon correlation spectroscopy; signaltonoise ratio.
1. Introduction
Dynamics in concentrated protein systems are of fundamental interest in fields such as protein crystallization (Durbin & Feher, 1996), (Anderson & Lekkerkerker, 2002), the (Cardinaux et al., 2007) or diffusion in crowded environments (Ellis, 2001), to name just a few. These systems display relatively slow and heterogeneous dynamics ranging from microseconds to seconds on length scales from micrometres down to the singleparticle nanometre scale. Xray photon correlation spectroscopy (XPCS) is well suited to cover this length scale and time window, employing coherent Xray beams and tracing fluctuations in Xray speckle patterns (Sutton, 2002; Grübel et al., 2008; Perakis et al., 2017; Madsen et al., 2018).
However, the highly intense Xray beams of synchrotron storage rings can also cause considerable radiation damage to the samples. Atomic scale XPCS experiments use Xray doses of MGy and beyond, which can lead to beaminduced dynamics even in hardcondensedmatter samples (Ruta et al., 2017; Pintori et al., 2019). Soft and biological matter samples are much more sensitive to radiation damage requiring flowing samples (Fluerasu et al., 2008; Vodnala et al., 2018) or scanning samples with optimized datataking strategies (Verwohlt et al., 2018). Typical critical Xray doses for protein molecules in solution range from 7–10 kGy (BSA) to 0.3 kGy (RNase) after which a degradation of the smallangle Xray scattering (SAXS) patterns becomes visible (Jeffries et al., 2015). These doses are easily reached within milliseconds when using focused beams of modern synchrotron sources. While cryogenic cooling helps to prevent the diffusion of radicals in protein crystallography, such an approach is obviously impossible when studying the dynamics of proteins in solution.
The signaltonoise ratio (SNR) in XPCS experiments ideally scales linearly with the coherent et al., 2000). The fastest accessible times scale with Br^{2}, promising four orders of magnitude faster temporal resolution at the upgraded sources of ESRF and PETRA IV (Einfeld et al., 2014; Weckert, 2015; Schroer et al., 2018), which is one of the key drivers for XPCS at nearly diffractionlimited storage ring (DLSR) sources (Shpyrko, 2014). However, these arguments only hold if radiation damage can be mitigated. Thus, the question arises how much could XPCS experiments of biological/radiationsensitive samples really benefit from the gain in of DLSRs? Here, we show that the combination of (i) larger coherence lengths, (ii) higher photon energy and (iii) the increased coherent yields an increase in SNR of up to one order of magnitude when compared with standard XPCS setups at today's storage rings. We calculate, using the boundary conditions set by the maximum tolerable Xray doses of a lysozyme solution, the XPCS speckle contrast, speckle intensities and the maximum number of images per illuminated spot. We conclude that future DLSRs hold the promise to measure the dynamics of biological samples at length scales of a single protein molecule.
and thus with the source Br (Lumma2. XPCS on protein solutions
XPCS experiments track fluctuations in Xray speckle patterns yielding access to the intermediate scattering function f(q, τ) = S(q, τ)/S(q) by correlating intensities per detector pixel (Sutton, 2002). The measured signal in such experiments is the normalized intensity autocorrelation function (Sutton, 2002; Grübel et al., 2008; Madsen et al., 2018),
with β denoting the speckle contrast and being the scattering vector, depending on the wavelength λ and the scattering angle 2Θ. The time delay between two consecutive time frames is denoted as τ, and is the ensemble average over all equivalent delay times τ and pixels within a certain range of the absolute value q.
The scattering intensity per pixel from a protein solution is given by (Narayanan, 2008)
with F_{c} denoting the incident coherent (photons s^{−1}), t_{fr} denoting the exposure time for one frame, T_{sample} denoting the sample's transmission and ΔΩ_{pix} = (P/L)^{2} denoting the solid angle covered by a single pixel, with P being the pixel size and L being the sampletodetector distance. In the following, we set the sample thickness d(E) to be equal to the absorption length of water d(E) = 1/μ(E) at each respective photon energy E, resulting in a transmission . The differential scattering per unit volume or absolute scattering intensity in m^{−1} of a protein solution is defined as (Glatter & Kratky, 1982; Feigin & Svergun, 1987; Narayanan, 2008)
with P(q) the form factor, S_{eff}(q) the effective and C the protein concentration. We will calculate the SNR for lysozyme as the model protein with a molar mass of M = 14.3 kDa and a specific volume of = 0.74 cm^{2} g^{−1}. The scattering contrast Δρ follows from the chemical composition of lysozyme and shows almost no dependence on energy in the energy range of interest here. With these parameters, the absolute scattering intensity can be expressed as
in good agreement with the measured values of (Mylonas & Svergun, 2007).
P(q) and S_{eff}(q) are modeled following Möller et al. (2012), and displayed in Fig. 1 for a diluted (C = 10 mg ml^{−1}) and a concentrated (C = 250 mg ml^{−1}) lysozyme solution. The q values of interest are within q = 0.5–1.5 nm^{−1}, which corresponds to length scales of 4–12 nm.
The dynamics of the lowconcentrated protein solution can be described as Brownian diffusion with a single exponential autocorrelation function
and relaxation rate
which is proportional to the Stokes–Einstein diffusion constant
where T, η, R_{H} and k_{B} are the temperature, the viscosity of the suspending medium, the hydrodynamic radius of the protein and the respectively. The qdependent relaxation rate is plotted in the upperright inset of Fig. 1 for diluted and concentrated lysozyme solutions. For the diluted case, we assume the viscosity of water and a hydrodynamic radius of R_{H} = 1.9 nm. We use an increased effective solution viscosity by a factor of 8.5 (Garting & Stradner, 2018) in order to illustrate the expected timescales for XPCS experiments on concentrated protein solutions. The time scales of interest range from 100 µs to s.
In practice, XPCS correlation functions are averaged over many pixels in a narrow range of q values. Typical regions of interest are sketched as colored areas in Fig. 1. The same set of regions is additionally depicted in the lowerleft inset, showing the location of the corresponding pixels on an EIGER 4M detector for E = 8 keV and a sampletodetector distance of L = 2 m. In the following, we will always calculate the SNR at the maximum of the structurefactor peak at q = 0.9 nm^{−1}.
3. Signaltonoise ratio
The SNR for the autocorrelation function g_{2}(q, τ) depends on the average intensity per pixel I_{pix}, the contrast β, the number of pixels N_{pix}, the number of frames N_{fr} and the number of repetitions N_{rep} via (Falus et al., 2006)
with N = N_{pix} × N_{fr} × N_{rep}.
Considering N_{fr} = T/t_{fr}, with t_{fr} being the singleframe exposure time and T being the total accumulated time for N_{fr} frames, yields in combination with equation (2), SNR ∝ F_{c}(t_{fr} × T)^{1/2}. This scaling implies that an increase in coherent F_{c} by one order of magnitude gives access to two orders of magnitude faster dynamics for the same SNR. However, this argument only holds when the available detectors are able to measure at the faster frame rates and the sample is capable of handling the increased If a critical dose D_{c} exists, beyond which radiationinduced damage starts to degrade the sample, the longest overall exposure time T depends on F_{c} and the increase of coherent might be less, or not at all beneficial for studying radiationsensitive samples.
The dose per second delivered to the sample depends on the E = ℏc/λ, the diameter a of the Xray beam spot size on the sample, and the distance L between sample and detector. In the following, we will establish the dependencies of the different contributions on the SNR, and determine the optimal set of a, E and L values for an XPCS experiment using radiationsensitive samples.
as well as the photon energy, both of which also influence the achievable SNR. Here, we take all those parameters into account and calculate the benefit to the SNR from the increased coherent of DLSRs. We identify three parameters, which we will assume to be nearly free of choice over a wide range of values. These are the photon energyFig. 2 shows the expected increase of coherent as a function of photon energy for a U29 undulator (5 m length) at PETRA III and IV. Additionally, the case of a U18 with 5 and 10 m length will be investigated. The is taken from Schroer et al. (2018). From this, the coherent can be calculated as
with bw denoting bandwidth. Using the referenced ^{11} photons s^{−1}. This is in good agreement with measured values of 2.3 × 10^{11} photons s^{−1}, taking into account transmission effects of beamline components and optics. In the following, the actual coherent on the sample will be calculated by taking into account the same beamline for all undulators.
we calculate the coherent for 8 keV at PETRA III as 3.8 × 103.1. Limitations due to radiation damage
We assume that a critical dose D_{c}, beyond which radiationinduced damage starts to degrade the sample, can be expressed as (Meisburger et al., 2013)
with F_{c} the on the sample, d(E) the energydependent sample thickness, a^{2} the beam area, (1 − T_{sample}) the sample absorption, E the photon energy and T the exposure time. From this, we derive the maximum number of frames which can be measured before radiation damage occurs to be
ignoring the latency time of the detector and absorption within the samplecontainer walls. The sample thickness d(E) is always adapted to the energydependent absorption length of water. One important conclusion from equation (11) is that the SNR scales via SNR ∝ F_{c}(N_{fr})^{1/2} ∝ (F_{c})^{1/2} for radiationsensitive samples. Moreover, with the scalings d(E) ∝ E^{3} and F_{c} ∝ Br(E)/E^{2}, we also find the peculiar relation of N_{fr} ∝ E^{4} favoring higher photon energies if a large number of frames is required.
We illustrate this with the example of a typical spot size for XPCS experiments of a = 4 µm, an exposure time of a single frame of t_{fr} = 1 ms and a critical dose limit for a concentrated lysozyme solution of D_{c} = 1 kGy.
Fig. 3 displays the possible number of consecutive frames as a function of photon energy. A prerequisite for correlation spectroscopy is obviously that the number of consecutive frames is at least two (i.e. N_{fr} ≥ 2), indicated by filled symbols. The coherent of PETRA III already exceeds the critical dose after or during the first image, and beam damage is occurring between two images, for photon energies below 10 keV. Below this energy, an increase in coherent would therefore not be usable for XPCS experiments on protein samples. However, N_{fr} increases with photon energy because of the increasing absorption length of the Xrays. Effectively, the radiation dose is spread over a larger sample volume with increasing photon energy. However, many properties such as the speckle size, the coherent and the longitudinal and transverse coherence lengths decrease with increasing photon energy. Therefore, the disadvantageous influence of these properties on the speckle contrast β and consequently on the SNR of XPCS experiments needs to be taken into account as well.
3.2. Speckle contrast β
The speckle contrast depends on nearly all experimental parameters such as pixel size P, speckle size S ≃ λL/a, beam size a, sample thickness d, wave vector transfer q, and the transverse and longitudinal coherence lengths. It can be written as a product,
in which the first factor β_{cl} corresponds to the reduction of the contrast from unity caused by the finite coherence lengths in transverse and longitudinal direction. The second factor β_{res} corresponds to the finite angular resolution of the experimental setup. This results in a reduction of contrast if the pixel size of the detector P exceeds the size of the speckle S:
with w = 2πPa/Lλ = 2πP/S. Fig. 4 displays the speckle contrast β_{res} as a function of beam size a for sampletodetector distances of L = 5 m and L = 100 m, pixel size P = 75 µm, and photon energies of 8, 15 and 25 keV. The maximum β_{res} is obtained in a highresolution configuration with S ≥ P and scales as β ≃ λ^{2}L^{2}/a^{2}P^{2} in the lowresolution configuration, when . Therefore, XPCS experiments with large beam sizes require long sampletodetector distances in order to resolve the smaller speckles.
The dependence of β_{cl} on beam size a, sample thickness d, transverse coherence length ξ_{h}, bandwidth and q value is taken into account via (Sutton, 2002)
with q{ 1[(1/4) q^{2}/k^{2}]}^{1/2}, and k = 2π/λ. In the vertical direction, we assume a completely coherent beam. In the horizontal direction, a coherence length is estimated as
with R being the distance between the source and the beamdefining aperture, and σ being the RMS source size. With σ_{h} = 36 µm (P10, lowβ source, 10 keV, R = 90 m), this results in a horizontal coherence length at E = 10 keV of ξ_{h} = 50 µm. A reduced horizontal source size at PETRA IV of σ_{h} = 12 µm would result in an increased horizontal coherence length of ξ_{h} = 148 µm at the same energy. These values reduce to 20 µm and 74 µm at an energy of E = 25 keV, respectively. The full energy dependence of ξ_{h} is shown in Fig. 5 (top).
Using a partially coherent source like a undulator for coherentscattering experiments, a slice of the incident Xray beam is required in order to obtain a nearly fully transversely coherent beam. Therefore, a beamdefining aperture is set to an opening size equal to the transverse coherence length. Smaller beam sizes can be achieved with additional focusing elements. For our calculations, we will consider the resulting focused beam as fully coherent with a ξ_{h} equal to the beam size. For larger beam sizes, ξ_{h} is calculated following equation (15).
The temporal or longitudinal coherence length can be calculated as
depending on the bandwidth of the monochromator used [Δλ/λ ≃ 1.4 × 10^{−4} for an Si(111) monochromator and Δλ/λ ≃ 3 × 10^{−5} for Si(311)].
The results for β_{cl} as a function of beam size and Xray energy are shown in Fig. 5 (bottom) for a q value of q = 0.9 nm^{−1}, corresponding to the peak of the shown in Fig. 1.
We observe a reduction in speckle contrast with increasing beam size and a reduced contrast for smaller beam sizes as a function of photon energy. Both reductions can be explained by the scattering volume, defined by spot size a and sample thickness d, exceeding the coherence volume defined by the longitudinal and transverse coherence lengths.
3.3. Number of pixels
Changing the photon energy and sampletodetector distance has direct implications on the number of pixels which can be covered within an area of a certain q range. The scattering signal may be in a circular region of interest on the detector of width Δq and radius q. In the SAXS regime, q = (4π/λ)Θ and Δq = (4π/λ)ΔΘ, and the diffraction ring has a width on the detector of Δθ × L and a circumference of 2π(2Θ)L. The number of illuminated pixels is thus
However, if the size of the speckles exceeds the size of the detector pixel S > P, the number of independent detecting pixels decreases, following in this case
4. XPCS of protein solutions
Having established the dependence of the SNR on the experimental parameters, we can use the expression
to characterize the influence of the improved
of the new generation of Xray sources on XPCS experiments with radiationsensitive samples.In Fig. 6, we display the SNR for a standard XPCS setup. We assume that an EIGER 4M detector (Johnson et al., 2014) is used, with a sampletodetector distance of L = 2 m, which corresponds at a photon energy of E = 8 keV to the inset of Fig. 1. In order to match the speckle size to the pixel size, an Xray spot size of a = 4 µm is required, corresponding to the calculations shown in Fig. 3. Further parameters are shown in Table 1.

The red data points correspond to the photonbeam properties of PETRA III, and the green, blue and cyan points correspond to the improved coherent F_{c} offered by PETRA IV with different undulators. As can be seen, the increasing coherent offers theoretically improved SNR values of more than one order of magnitude. However, as marked with open symbols, the highest theoretically possible SNR of each configuration corresponds to experimental conditions where the critical dose limit of the sample is reached within two sequential acquisitions (i.e. N_{fr} ≤ 2), and hence the sample would suffer from radiationdamage effects during the measurement. Therefore, the maximum increase in SNR can not be reached in practice for this setup, and the upgrade to PETRA IV would not lead to a significant increase in SNR.
Data points which correspond to beam conditions where at least two sequential acquisitions are possible are displayed as filled symbols. It is evident that higher beam energies with thicker samples would ease the effect of a higher L = 2 m, a = 4 µm). However, as displayed in Fig. 6 (bottom part), this also results in much reduced speckle contrasts and, therefore, the beneficial effect of an increased coherent on the SNR is largely lost.
and make XPCS experiments possible also at a standard configuration (4.1. Optimizing the experimental setup
In order to use the increased coherent a and sampletodetector distances L. At each point in the a–L plane, the SNR is calculated as a function of photon energy and the maximum is plotted. However, only values are considered which correspond to N_{fr} ≥ 2 at 1 ms exposure. The maximum SNR for each pair of a and L values is displayed in Fig. 7.
for XPCS experiments, one has to adapt the experimental setup in terms of focusing, photon energy and sampletodetector distance. Therefore, we repeat the previously presented calculations for a set of different beam sizesIt can be seen that the previously discussed setup with a small beam and large speckle (marked by a red dot) does not give the best SNR, already evident in the case of PETRA III. With a sampletodetector distance of L = 5.5 m and an Xray spot size of a = 9 µm, the expected SNR increases by 25%.
However, in the case of PETRA IV (U1810m), an overall increase in SNR by about one order of magnitude can be achieved, without exceeding the critical radiation dose of the sample. This setup would feature a sampletodetector distance of L = 26 m and a spot size of a = 24 µm at E = 14.7 keV.
The resulting parameters for the optimized experimental setups are summarized in Table 2 for each of the considered undulators. We note that for higher coherentflux setups, the optimized setups feature an increase in beam size a, sampletodetector distance L and photon energy E.

As a general trend, it is evident that the sample volume, spanned by the sample thickness d and spot size a, needs to be increased when the coherent increases. In order to compensate for the consequently decreasing angular speckle size, the sampletodetector distance needs to increase so that the speckle size can maintain its value of S ≃ 75 µm, matching the pixel size. However, it can be seen that one can still observe a decrease in speckle contrast β, even though the speckles have the same size on the detector for all four setups presented. This effect is due to the second contribution to the speckle contrast β_{cl}, see equation (14), originating from the limited longitudinal coherence length of the Xray beam.
4.2. Si(311) monochromator
Here, we investigate how an additional increase of the longitudinal coherence length by using a Si(311) monochromator benefits the achievable SNR. We repeat the previous calculations, however, with a reduced bandwidth of 3 × 10^{−5} and a reduced compared with the Si(111) calculations by 74%.
The resulting SNRs are displayed in Fig. 8 and Table 3. We find that the use of a Si(311) monochromator improves the SNR by an additional 30% compared with the Si(111), thus leading to an overall SNR gain of a factor of 13 when comparing PETRA III with PETRA IV. Interestingly, the achievable SNR at PETRA III decreases when a Si(311) monochromator is used instead of a Si(111), showing that XPCS measurements at existing synchrotron sources can be considered as limited, whereas an increase in SNR can be noted for all three undulator types studied at PETRA IV.

4.3. Multipleframe XPCS and twotime correlation functions
It becomes evident that with SNR values ∝ 3–5, XPCS measurements from protein solutions are possible at DLSRs with adapted experimental setups. As a direct consequence of the presented results, the optimized dataacquisition scheme differs from conventional XPCS measurements. Instead of taking many hundreds to thousands of images at one spot, the scheme with maximum SNR for protein XPCS consists of `doubleshot' exposures. Therefore, a fullcorrelation function from one spot on the sample cannot be measured, but rather only one data point of g_{2} for each illuminated sample spot. Consequently, the correlation function would be constructed from many such doubleshot exposures, each on a new sample spot and with a different delay time τ between the two frames (see e.g. Verwohlt et al., 2018). The required sample volume therefore scales with the desired number of data points of g_{2}.
However, this acquisition scheme is not suitable for samples displaying heterogeneous dynamics or aging effects. In such cases, a moviemode acquisition scheme with more than two frames per spot is needed. Fig. 9 displays the resulting SNR values in the a–L plane if the minimum number of frames is set to N_{fr} = 2, 5, 25 or 100 (for the case of PETRA IV U1810m).
We find that with an increasing number of images, N_{fr}, the value of the maximum SNR decreases only slightly, and its position in the a–L plane shifts towards larger beam sizes a and larger sampletodetector distances L. For realizing the higher number of frames, an increase in the photon energy and in the beam size is required [from equation (11) we find the scaling a ∝ (N_{fr})^{1/2}]. The resulting degradation of speckle contrast is partially counterbalanced by improving the angular resolution via a larger sampletodetector distance. For example, for N_{fr} = 100 frames, the optimum SNR is still 3.2 at a = 75 µm, L = 82 m and E = 15.4 keV. Generally speaking, we find at the maximum of the SNR a scaling of L ∝ a ∝ [N_{fr}Br(E)]^{1/2}.
In practice, the realization of a beamline with up to 100 m sampletodetector distance, which would also require a detector with a very large number of pixels, is challenging. However, it can be seen from Fig. 9 that at shorter sampletodetector distances L the SNR is still significantly larger than 1 for N_{fr} ≤ 100. Therefore, we investigate SNR optimization if the length of the beamline is a fixed value of L and if a certain number of frames N_{fr} is required to track the physics of the protein solution.
We demonstrate this by fixing the sampletodetector distance to L = 30 m, which is already available at specialized ultrasmallangle scattering beamlines (see e.g. Möller et al., 2016; Zinn et al., 2018), and using a Si(311) monochromator. We plot both the SNR and the maximum number of possible frames as a function of beam size a (Fig. 10, top) for photon energies of E = 13.1 keV (solid), E = 17.0 keV (dashed) and E = 23.7 keV (dashed–dotted). Fig. 10 (bottom) displays the same data but plotted as SNR as a function of N_{fr} for the different photon energies. The benefit of using even higher photon energies than 13 keV for multiframe acquisitions is obvious as it allows either an increase of the SNR value at fixed N_{fr} or the ability to record more images at a fixed value of the SNR. We find that with the source parameters of PETRA IV (U1810m) the resulting values of the SNR can be well above 1, even for several hundreds of frames recorded. Specifically, we may take the example of N_{fr} = 100 and find an SNR value of 2.5 at 17 keV. The best combination of photon energy and spot size depends on the required number of frames, i.e. the timescales which are investigated. It can be seen that higher energies are beneficial in order to record multiframe acquisitions below the radiationdamage threshold and with high SNR. Additionally, the higher energies shift the scattering angles to lower values, which consequently reduces the size of the detector required.
5. Conclusion
We determined the signaltonoise ratios for XPCS experiments of a concentrated lysozyme solution at length scales of the hydrodynamic radius of a single protein molecule. We showed that by adapting the experimental setup, XPCS measurements can be performed below the radiationdamage threshold and with strongly increased SNR. The results show that the SNR values can be increased up to one order of magnitude at future upgraded storage rings when compared with existing facilities.
With this, the required measuring time for multiframe acquisitions would reduce by two orders of magnitude making dynamic studies of protein solutions at nanometre length scales feasible. However, in order to take full advantage of the properties of the future sources, XPCS experiments require experimental setups with larger beam sizes and longer sampletodetector distances than usually available at standard XPCS beamlines. We showed that the optimal photon energy for softmatter doubleshot XPCS measurements at the upgraded storage ring PETRA IV will be between 12.5 and 14 keV, with sampletodetector distances in the range of 8 to 15 m. Additionally, we showed that multiframe acquisitions on protein solutions are possible with up to several hundreds of frames by further increasing the photon energy and increasing the length of the beamline (E ≥ 17 keV, L ≥ 30 m).
We hope that this study shows the opportunities offered by nearly diffractionlimited storage rings and may additionally serve as a guide for the design of softmatter XPCS beamlines.
Acknowledgements
The authors would like to thank the PETRA IV project team, and especially C. Schroer, M. Tischer and S. Klumpp, for useful discussions and support.
Funding information
Funding for this research was provided by: Bundesministerium für Bildung und Forschung (grant No. 05K19PS1).
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