research papers
Towards the spatial resolution of metalloprotein charge states by detailed modeling of XFEL crystallographic diffraction
^{a}Molecular Biophysics and Integrated Bioimaging Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA, ^{b}SSRL, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA, and ^{c}Department of Biochemistry and Biophysics, University of California, San Francisco, San Francisco, CA 94158, USA
^{*}Correspondence email: nksauter@lbl.gov
Oxidation states of individual metal atoms within a metalloprotein can be assigned by examining Xray absorption edges, which shift to higher energy for progressively more positive valence numbers. Indeed, Xray crystallography is well suited for such a measurement, owing to its ability to spatially resolve the scattering contributions of individual metal atoms that have distinct electronic environments contributing to protein function. However, as the magnitude of the shift is quite small, about +2 eV per valence state for iron, it has only been possible to measure the effect when performed with monochromated Xray sources at synchrotron facilities with energy resolutions in the range 2–3 × 10^{−4} (ΔE/E). This paper tests whether Xray freeelectron laser (XFEL) pulses, which have a broader bandpass (ΔE/E = 3 × 10^{−3}) when used without a monochromator, might also be useful for such studies. The program nanoBragg is used to simulate serial femtosecond crystallography (SFX) diffraction images with sufficient granularity to model the XFEL spectrum, the crystal mosaicity and the wavelengthdependent factors contributed by two differently charged iron centers in the 110aminoacid protein, Bayesian methods are then used to deduce, from the simulated data, the most likely Xray absorption curves for each metal atom in the protein, which agree well with the curves chosen for the simulation. The data analysis relies critically on the ability to measure the incident spectrum for each pulse, and also on the nanoBragg simulator to predict the size, shape and intensity profile of Bragg spots based on an underlying physical model that includes the absorption curves, which are then modified to produce the best agreement with the simulated data. This inference methodology potentially enables the use of SFX diffraction for the study of mechanisms and, in general, offers a more detailed approach to Bragg spot data reduction.
Keywords: metalloproteins; SPREAD; valence; Bayesian methods; XFEL; spectroscopy.
1. Introduction
For proteins containing transition metal sites, a complete understanding of function requires not only the atomic structure, but also the electronic structure and chemical environment of the metal atoms (Kern et al., 2015). Xray absorption spectroscopy has been highly informative, with the extended Xray absorption fine structure (EXAFS) offering a sensitive measurement of metal–metal and metal–ligand distances, whereas the Xray absorption nearedge structure (XANES) classically reveals the and coordination geometry (Yano et al., 2005; Glatzel & Bergmann, 2005). Fundamentally, the Kabsorption edge, corresponding to the removal of a core 1s electron, is shifted to a slightly higher energy when a transition metal is oxidized, as the loss of a valence electron increases the interaction between core electrons and the nucleus (Fig. 1; Sherrell, 2014).
Although these absorptionedge methods have been successful, the usual approach of detecting absorption curves by Kabsorption edge of the metal in question. Absorption is then quantified by refining wavelengthdependent anomalous correction parameters for each metal. Such data have revealed which of two Fe atoms acts as the electron carrier in the [2Fe:2S] cluster of (Einsle et al., 2007), and have been used to characterize the mononuclear Fe binding site and the [Mo:7Fe:9S:C] cofactor of nitrogenase (Zhang et al., 2013; Spatzal et al., 2016). While macromolecular crystallography is commonly thought of as a technique to determine atomic coordinates, these results show that the can readily ascertain the location of a single electron. Spatially resolved (SPREAD) potentially offers an independent check on the assignment of heteroatom valence states based on bond distances, such as those assigned by Suga et al. (2015) for the four manganese ions in the [4Mn:5O:Ca] oxygenevolving complex of photosystem II. It may potentially give a more nuanced view for systems where charge is shared among several metal atoms.
makes it difficult to interpret spectra from metalloprotein systems that have multiple copies of a given metal, due to An alternative that can distinguish distinct metal centers is to detect the through crystallographic diffraction, which inherently provides spatial resolution. In this approach, 3D diffraction datasets are collected from protein crystal(s) using a series of monochromatic energies that span theAlthough the et al., 2014), with crystallography contributing the most relevant information about biological function when the experiment is performed at physiological or room temperature (Keedy et al., 2014, 2015; Russi et al., 2017; Thomaston et al., 2017). However, dispensing with cryopreservation presents a general challenge, as it is the principal method used to protect against radiation damage (Garman & Weik, 2019). Also, with respect to probing the electronic environment, Xray crystallography studies are particularly difficult for metalloproteins, as metal centers are photoreduced at very low Xray doses (Yano et al., 2005; Denisov et al., 2007; Borshchevskiy et al., 2014). Xray (XFEL) sources offer a solution to both problems, as the use of femtosecond pulses enables experiments at ambient temperature by producing diffraction prior to the onset of radiation damage, especially when confined to moderate fluences and pulse durations that are available as standard XFEL configurations^{1} (AlonsoMori et al., 2012, 2016; Kern et al., 2013; see also Lomb, 2011; Barty et al., 2012; Nass et al., 2015). Furthermore, XFEL serial crystallography (wherein the sample is replaced after each shot) has provided highresolution timeresolved structures (in the 100 fs–400 µs range) for metalloproteins including photosystem II (Young et al., 2016; Suga et al., 2017; Kern et al., 2018), cytochrome c oxidase (Shimada et al., 2017) and CO myoglobin (Barends et al., 2015); shottoshot Xray emission spectroscopy can also be used to rule out the presence of unwanted (Fuller et al., 2017; Kern et al., 2018; Fransson et al., 2018). All this provides strong motivation to extend the SPREAD method to the XFEL regime, for it would allow detection of timeresolved redox states of complex reaction mechanisms involving multiple transition metal sites.
and nitrogenase studies were performed on cryopreserved crystals, it is now widely recognized that a macromolecular structure consists of an ensemble of conformations (WoldeyesRealizing this measurement presents profound challenges for both data acquisition and data interpretation. While it is possible to use selfseeding (Amann, 2012) to produce monochromatic pulses [full width at halfmaximum (FWHM) < 1 eV] distributed across the Fe Kedge, the use of monochromatic light reduces the number of Bragg spots observed per shot, making it more difficult to acquire complete data with sufficient multiplicity of coverage. Moreover, as the diffraction from each energy channel is observed independently, it becomes difficult to normalize the observations across Xray wavelengths in order to construct selfconsistent absorption curves such as those illustrated in Fig. 1.
An alternative approach is to take full advantage of the natural bandwidth of the XFEL beam. In principle, since the protein specimen is a crystal, differentenergy photons will be split into slightly different diffracted directions, obeying Bragg's law.^{2} In a similar spirit, protein diffraction data could be collected all at once over a range of Xray wavelengths, with the results sorted out computationally, using all the data simultaneously to obtain the scattering factors by a global fit. This concept leads us to focus on two experimental features. Firstly, the use of the full selfamplified (SASE) spectrum of the XFEL source, which has a natural bandwidth on the order of 30 eV when tuned to the metal Kedge. This would avoid the loss of fluence that is a consequence of selfseeding, at the cost of mixing the signal from different energies together. However, it is possible to record a detailed image of the stochastically shaped incident spectrum for each pulse (Zhu et al., 2012), thus providing normalization across energies that can be used to infer statistically the energydependent scattering contribution. Secondly, in order to help resolve the energy dependence, the pixel array detector used for imaging the diffraction pattern can be pulled back to a far distance so that Bragg spots are resolved as radial streaks (Bragg's law dictates that higher energy photons are diffracted to a smaller angle). Even routine XFEL experiments reveal radial streaking that is the result of a combination of energy dispersion and mosaic disorder (Hattne et al., 2014). In this case, we envision the simultaneous use of two imaging detectors for each diffraction pattern: the `right' side would be imaged by a forward detector to cover as many Bragg spots as possible, and thus determine the orientation of the crystal and the by conventional methods, while the `left' side would be imaged further back to resolve the energydependent Bragg streaks, with the limitation that the detector would only subtend a few Bragg spots at midresolution diffraction angles.^{3} The purpose of this paper is to establish the feasibility of the approach by thoroughly modeling such an experiment. We show through simulation that, given current instrumentation, it is feasible to extract SPREAD spectra.
In the following we attempt to advance the computational methods beyond what has recently been done with XFEL protein crystallography data processing, in several regards. Firstly, due to the need to deconvolute the et al., 2014) and still shots (Kirian et al., 2010) to produce simulated diffraction images that emulate the granular details expected from our proposed data collection strategy. We then used the same tools within a Bayesian framework to analyze the simulated data to produce an accurate estimate of the energydependent absorption from each metal atom.
factors at different energies, we explicitly model the diffraction as a linear sum over energychannel contributions. Secondly, since these energy contributions are spread out within each Bragg spot over several pixels, we never explicitly sum the integrated intensity arising from single Bragg spots. Instead, the parameters of interest are refined directly against individual pixel intensities. Finally, since there is no data reduction step (where pixels within a Bragg spot are summed to a single number), and since the parameter required several tens of thousands of images to converge, we had to implement a parallel computational architecture, where the agreement between model and image data was evaluated over many distributed computer nodes. To provide a context for these developments, we adapted previous software for simulating rotation (Holton2. Methods
2.1. Parameters of the data simulation
All data simulations were performed with a CCTBX script archived at github (https://github.com/nksauter/LS49/blob/master/sim/step6_batch.py). In the spirit of previous work (Holton et al., 2014; Holton, 2019), we attempt to use basic physical principles to derive the diffraction pattern expressed in absolute units (photons pixel^{−1} shot^{−1}) so that photoncounting errors may be treated correctly and the experimental feasibility assessed. In addition, the models presented are an attempt to represent the standard configuration of a typical protein crystallography experiment at an XFEL source. This includes the use of a well calibrated, latest generation integrating detector; the delivery of randomly oriented, strongly diffracting crystals by an openair device such as the dropontape conveyor belt (Fuller et al., 2017); and the availability of singleshot Xray spectra that reflect the stochastic nature of the XFEL pulse, as measured in real experimental data. To this end, we treat the experimental parameters as follows.
2.1.1. The imaging detector
We assume an idealized pixel array detector with a gain of 1.0 (one count per photon), consisting of 3000 × 3000 square pixels of size 0.11 mm, situated 141.7 mm from the crystal, which ensures that the inscribed circle captures the diffraction pattern to an outer resolution of 2.1 Å at the Fe Kedge of 7122 eV. It is intended that the analysis of Bragg data in the 2.1–2.5 Å range will allow us to distinguish between the two Fe atoms in the [2Fe:2S] cluster of that are 2.73 Å apart. We assume there is no parallax effect for the detector (Winter et al., 2018) nor any charge sharing as observed for real pixel arrays (Philipp et al., 2011). We assume the detector has 1% calibration noise (systematic pixeltopixel variation that is constant for a given pixel across repeat simulations due to factors such as impurities in the silicon or differing amplifier settings), but no readout noise (random noise due to pixel electronics).
2.1.2. Simulation of the structure factors
Structure factors were derived from PDB entry 1m2a (Yeh et al., 2002), from Aquifex aeolicus, in C2 with unitcell parameters a = 67.2, b = 59.8, c = 47.2 Å, β = 113.2°. This paper deals with two types of structure factors: the ground truth values, F_{true}, which are fed into the program nanoBragg to produce simulated diffraction images, and the fitted values, F_{sim}, deduced from the simulated images by computational processing. We use the generic term F_{model} to describe either quantity. Operationally, we use the CCTBX toolbox to calculate the complex F_{model}(λ) for h_{0} at wavelength λ, as the sum of contributions from the explicit atoms listed in the coordinate file plus the bulk solvent (Afonine et al., 2013; Jiang & Brünger, 1994):
This is the exact procedure used in the program PHENIX (Afonine et al., 2012), with the exception that k_{sol} and B_{sol}, the bulk solvent scale and B factors, are set to 0.435 and 46, respectively, in order to minimize the sum over all low resolution (∞–7 Å) amplitudes F_{model}(λ) in accordance with Babinet's principle. For the present purpose, it is convenient to think of F_{model} (λ) as being arranged into two terms:
and
The F_{fixed} term includes the scattering from all nonFe atoms, calculated by the fast Fourier transform method (Ten Eyck, 1977; GrosseKunstleve et al., 2004). Although the structure does contain other anomalous scatterers such as S and Zn, these anomalous contributions vary only weakly near the central energy of this experiment (7122 eV), so they are evaluated once at that energy and held constant throughout the remainder of the data simulation and analysis. For the of Section 2.4, F_{fixed} is taken to be a known quantity.
The F_{fit} term is the sum, over all Fe atoms in the of the energydependent contribution evaluated by the usual directsummation formula,
where q_{m} is the occupancy of metal m, S is the magnitude of the scattering vector (= 1/resolution), r_{m} is the position vector of the atom expressed in fractional coordinates, h_{0} is the and B_{m} is the isotropic B factor of the atom. In this expression, f^{0} represents the normal (nonanomalous) scattering factor of the atom, dependent on the scattering vector but not on energy (and we assume negligible dependence on the oxidation state). The Δf′ and Δf′′ terms represent the real and imaginary components of the that are dependent only on energy and valence state. Ground truth (F_{true}) for the present data simulation is for the Fe1 atom to be oxidized and the Fe2 atom to be reduced (Einsle et al., 2007) with corresponding Δf′ and Δf′′ values taken from Sherrell (2014), see Fig. 1. In contrast, for the analysis of simulated images (F_{sim}), the F_{fit}(λ) subterm embodies the (initially unknown) wavelengthdependent anomalous structure factors Δf′ and Δf′′ that we endeavor to recover.
2.1.3. Simulation of the mosaic crystal
To simulate a diffraction image typical of shots taken at beamlines like the Macromolecular Femtosecond Crystallography (MFX) instrument at LCLS (Boutet et al., 2016), we assume that a perfectly collimated beam with a 1 µm^{2} focus intersects a 4 µm path through the crystal. Consistent with the practice of Busing & Levy (1967), we express the crystal orientation in (Sauter et al., 2006) as the matrix
where the B represents the reciprocal basis vectors (a*b*c*) arranged in a conventional reference orientation, and U is a unitary rotation matrix chosen at random for each shot. However, we also wish to model the mosaic disorder of the crystal (Nave, 1998). Therefore, we break up the diffracting crystal volume into 25 congruent but separately rotated domains (blocks), with indices D = 1, …, 25, each of which contributes independently to the diffraction, thus the intensities (not the amplitudes) are summed. We derive an effective orientation matrix for each domain,
orthogonalization matrixwhere the 25 U_{D} are rotation matrices with axes randomly chosen from the unit hemisphere and rotational magnitudes drawn from a Gaussian with a standard deviation of η = 0.05° (Fig. 2). This set of 25 perturbation matrices is generated by the source code at https://github.com/nksauter/LS49/blob/master/tests/tst_mosaic_orientations.py.
2.1.4. Incident Xray pulses
To simulate SASE pulses with properties similar to those at LCLS, we began with actual spectra (Zhu et al., 2012) measured in the frontend enclosure during a 14 min period (run 209) of LCLS user proposal LG36, centered at 7088 eV (Fig. 3). Starting with a separate spectrum for each simulation, we applied a baseline correction above and below the energy region of interest, and an FFTbased lowpass filter to smooth out any features narrower than about 1 eV. Furthermore, we translated the energy scale to center the average spectral maximum at the Fe Kedge (7120 eV), and defined the intensity scale to give an average integrated number of photons over the entire run of 10^{12} photons shot^{–1}. The 7070–7170 eV range was then downsampled into exactly 100 energy channels, thus providing a distribution of stochastic spectral shapes, total fluences and mean energies. We assume that the beam is polarized with the Evector horizontal.
2.2. Simulated diffraction
The kinematic theory (singlescatter from crystals small enough to ignore attenuation) is presented in the classic literature (James, 1962) and has recently been applied to the simulation of both synchrotronbased rotation images (Diederichs, 2009; Holton et al., 2014) and XFELbased still shots (Kirian et al., 2010; KroonBatenburg et al., 2015). Given the wavevectors of the scattered (s_{1}, defined by the pixel position) and incident (s_{0}) Xrays, both vectors of length 1/λ, and defining the scattering vector as S = s_{1} − s_{0}, we compute the crystal diffraction intensity for a single pixel on a femtosecond still shot (photons pixel^{−1}) as follows:
where r_{e} is the classical radius of an electron (2.62 × 10^{−15} m), P is the polarization factor (Kahn et al., 1982) in the direction of the pixel and ΔΩ is the solid angle subtended by the pixel in steradians. Within the detailed summation, J_{0}(λ) is the incident fluence (photons channel^{−1} m^{−2}), and F_{true} the groundtruth energydependent of the [equation (1)] taken at the nearest h_{0} to the position of the pixel in The F_{latt} is the Fourier transform of the finite array of points that make up the crystal or mosaic domain. Like any F_{latt} is the ratio of the scattered wave from the object of interest to that of a single electron at the origin (Hartree, 1925). In the case of F_{latt}, the object is the points themselves and for F_{true} or F_{sim} it is the contents of one F_{latt} is multiplied by F_{true} because the cell is convoluted with the and convolution in real space is a product in At the exact center of each reciprocal point (RLP), where the Laue conditions are met, F_{latt} is equal to the number of unit cells in the mosaic domain, while in the surrounding neighborhood F_{latt} takes on a shape essentially identical to the Fourier transform of the average mosaic domain shape. Smaller mosaic domain size therefore leads to larger spots. In the special case where the crystal is a of dimensions N_{a} × N_{b} × N_{c} (unitcell counts along the a, b and c axes), an exact expression for F_{latt} is a 3D version of the grating function, as employed by Kirian et al. (2010). However, in this study we assume a much larger crystal (Section 2.1.3) consisting of many mosaic domains with a distribution of shapes and sizes (Nederlof et al., 2013), and thus we model the average coherently diffracting volume as a 3D Gaussian. The Fourier transform of this is a Gaussian RLP, which we approximate with the following peak profile:
where Δx is the distance to the center of the RLP expressed in units of the reciprocal domain size:
Here, h is the realvalued corresponding to the pixel (or diffracted ray) of interest,
Miller indices are generally expressed as integers, but because every pixel has a location in h. The nearest integervalued h_{0} is the same used in equations (1) to (3) to select an appropriate F_{model} for each pixel. The factor 0.63 in equation (7) was chosen to force the RLP volume and FWHM to be similar to that from a rectangularvolume domain. The simulations presented here (expressed in the choice of parameters N_{a}, N_{b}, N_{c}) were equivalent to modeling mosaic domains with an average full width at halfmaximum diameter of D_{eff} = 400 nm.
it may conveniently be given a noninteger valueIn addition to the crystal diffraction, our simulation added the diffraction from the liquiddroplet carrier used for sample delivery (Fuller et al., 2017), and from the atmospheric path between the crystal and beamstop. Liquid was represented by a 100 µm path through water, and air by a 10 mm path through N_{2}, as described in the supplementary materials of Holton et al. (2014). No attempt was made to model the effect of diffuse scattering (Wall et al., 2018), and the absorption of the Xray beam in the sample and air was neglected. Once the contributions of crystal, liquid and air were summed, shot noise was added by replacing the expected average photon count Λ_{i} of pixel i with the value k_{i} sampled from a Poissonian distribution, which has the probability density function
Diffraction simulations were performed with randomly chosen crystal orientations (Fig. 4). The original standalone nanoBragg was refactored into a C++ class and provided with Python bindings within the simtbx (simulation toolbox) directory of the CCTBX project (GrosseKunstleve et al., 2002). The Python/C++ interface was configured so as to reuse code objects within the dxtbx (diffraction experiment toolbox) that provide a physical description of the experiment, including the beam, crystal and detector (Parkhurst et al., 2014). Parallel execution was achieved at the Python level by delegating diffraction patterns from independent crystals to separate worker ranks with the message passing interface (MPI), while the C++ loop over image pixels was accelerated using several parallel threads to simulate independent pixels with OpenMP. Overall wallclock calculation times for different highperformance computing systems are shown in Table 1; pixel values on the three systems were numerically identical provided that the same random number seeds were given. A GPUaccelerated version of nanoBragg has also been prototyped (James Holton and Giles Mullen, unpublished work).

2.3. Preliminary analysis of the simulated data
We now switch the point of view, treating the simulated images from Section 2.2 as a real serial crystallography dataset, and ask what data analysis protocols are required to deduce the Fe anomalous corrections Δf′ and Δf′′.
2.3.1. Conventional data reduction
We began with routine data processing with the program dials.stills_process (Brewster, 2016; Brewster et al., 2018), yielding estimates of the unitcell parameters and crystal orientation (Fig. 5), encapsulated in the 3 × 3 orientation matrix A*. Several attempts were needed before it was ultimately possible to deduce the Fe corrections (as judged by r.m.s.d. comparison to the ground truth, see Section 3 below). In Method 1, the matrix A* was refined to fit the data without further restraint. The average unitcell parameters agreed exactly with the ground truth as defined by the PDB file. However the variance levels, with standard deviations of about 0.06% for each parameter [Fig. 6(a), blue traces], were prohibitively large for modeling the scattering factors. We therefore introduced Method 2: the application of isomorphism restraints with the tie_to_target command option of DIALS. Here the initially determined unitcell parameters were used as tightly restrained targets, resulting in very small standard deviations on the order of 0.01% [Fig. 6(a), orange traces]. However, even with these improved unitcell parameters, the crystal orientations were still misaligned from the ground truth with a median missetting angle of 0.046° [Fig. 6(b)], which proved prohibitively large. The cause turned out to arise from mutually inconsistent definitions of the detector origin between the simulation script (following the MOSFLM convention) and the DIALS analysis program, amounting to a 1/2 pixel offset in the horizontal and vertical directions (see Holton, 2019, Section 2.3, paragraph 2).^{4} The corrected detector position was provided back to dials.stills_process as a reference for a third round of indexing and crystal orientational (Method 3). This reduced the median missetting angle to 0.011° [Fig. 6(c), magenta], but it was still insufficient for further progress.
2.3.2. Orientational based on spot profiles (Method 4)
When decomposed into rotational missettings about the horizontal, vertical and Xray beam axes, the only significant contributions were along the horizontal and vertical axes (data not shown). The path forward became clear by using the Method 3 orientation matrices to create nanoBragg image simulations, and noting that the Bragg spot positions in the region of interest (Fig. 4) were up to one pixel out of position, compared with the corresponding original simulations of Section 2.2. We therefore set up a parameter optimization problem to apply horizontal and vertical rotational perturbations to the model, such that the resulting nanoBragg spot simulation would be most consistent with the shape and position of spots on the reference image. As this depends on Bayesian concepts presented below (Section 2.4), the full description of Method 4 is saved for Appendix B. Fig. 6(c) shows the consequent improvement in spot position, as well as the reduction in the median missetting angle to 0.005°.
2.4. A Bayesian approach to modeling the anomalous signal
We now further examine whether the
curves can be extracted separately for each Fe atom.Here we make the following assumptions about what is already known. From conventional data reduction (Section 2.3.1), we have complete knowledge of the nonanomalous Bragg spot intensities. Therefore, we can solve and refine the thus producing a coordinate model that permits us to derive F_{fixed} from equation (2). For the two metal atoms described by F_{m} in equation (3), we know coordinates r_{m} and B factors B_{m}; but are still missing Δf′_{m}(λ) and Δf′′_{m}(λ). We assume that there is accurate knowledge of the unitcell parameters, which have a very narrow distribution [e.g. Fig. 6(a), Method 2], that the mosaic rotation parameter η and effective mosaic domain size D_{eff} are known (Sauter et al., 2014), that the detector geometry and position are known to high precision (Brewster et al., 2018), and that the singleshot spectrometer gives an accurate knowledge of the Xray spectrum J_{0}(λ) that is incident on the crystal. What is still left to model [in addition to Δf′_{m}(λ) and Δf′′_{m}(λ)] is a better estimate of the orientation matrix A* as mentioned above (Section 2.3.1, Method 4), the overall scale factor G_{L} for each image L, and the background photon level g_{x} behind each Bragg spot x.
The usual cautions apply when considering ) the addition of one valence electron to a single Fe atom can change the intensities as much as 5% or more, therefore we expected the desired signal to be embedded in our data. For data analysis, we avoided the routine strategy of integrating the Bragg spots and merging the signal for repeat observations of the same Instead, we took full advantage of positioning the detector far back from the crystal, thus allowing the to act as a spectral analyzer, spreading out the diffracted Xrays over the spectrum of incident energies (Fig. 8). In an idealized case we would simply read the intensity profile along the energy scale illustrated in Fig. 8(a), but in the present case it is more complicated for several reasons. Firstly, we aim to resolve the factors Δf′ and Δf′′ with a spacing of 1 eV on the energy axis, while our simulation was intentionally modeled with a challenging 3.8 eV separation per pixel. Therefore, an appreciable amount of deconvolution will be needed. Secondly, each energy channel contributes a different J_{0}(λ) to the diffraction pattern (Fig. 3); lastly, the effect of crystal mosaicity is to further spread out the diffraction contributed by each wavelength so it is smeared out over several pixels [Fig. 8(e)], as determined by the spot profile factor in equation (6), ∑_{D}F^{2}_{latt}[S(λ)]. All these phenomena lead to the necessity of combining all the available information simultaneously, including the structure factors for the protein (except for the unknown anomalous contribution of the Fe atoms), the recorded spectra and the best orientation and mosaicity of all crystals determined from indexing, all in order to estimate the Δf′ and Δf′′ scattering parameters statistically. We take the normal Bayesian approach, which is nicely introduced in its application to crystallography by McCoy (2004). Bayes's theorem states that the posterior probability of the model (consisting of our parameter estimates), given the data, is proportional to the likelihood of the data given the model and to the prior probability of the model:
from a protein, as the signal is weak. However, for a subset of (Fig. 7The probability of the data P_{data}, which normally appears in the denominator of Bayes's theorem, is constant in our situation and is thus omitted here. P(datamodel) is assumed to be independent for each pixel, therefore the collective likelihood is the product of individual pixel likelihoods, taken over all pixels i, including all Bragg spots observed over all images,
where N_{p} is the number of pixels. We will find the most likely model parameters {z} by minimizing a loss function that takes the negative log of the posterior probability,
where
and
We will use Poissonian statistics [equation (10)] to compute the probability of observing the pixel value k_{i} given the model value Λ_{i}. Combining equations (10) and (13) gives
The Poissonian probability is valid as long as the model and data are expressed in units of photons rather than detector pixel units, so there is an implicit assumption that there is a good understanding of the detector gain. The last term involving k_{i}! is independent of the model parameters, and therefore constant and of no consequence for the parameter fitting, so it is dropped:
2.4.1. Modeling the pixel's photon count Λ_{i}
We intend for the target function to be summed over all pixels i of each rectangular shoebox x containing a strong Bragg spot (as identified by dials.stills_process and illustrated in Fig. 8). Therefore the model must cover the contributions of Bragg diffraction as well as the background g_{x,i} due to liquid and air scatter,
where G_{L} is a scale factor applied to all shoeboxes on a single image L that converts the model value to photons pixel^{−1}, compensating for the arbitrary scale of the model.^{5} Note that F_{sim}(λ) = F_{fixed} + F_{fit}(λ) is the total recovered from the image data, but the subterm F_{fixed} is extracted from F_{true} and not allowed to vary. We treat the background g_{x,i} separately for each Bragg spot observation x using a bestfit plane as employed previously (Rossmann, 1979; Leslie, 1999),
where p_{i} and q_{i} are the slow and fast pixel coordinates of the shoebox, respectively. The background scatter is only weakly dependent on wavelength, so we make it independent of J_{0}(λ) in equation (16). Altogether, the unknown parameters to be determined by fitting are the perimage scale factors G_{L}, the perspot background parameters {a_{x},b_{x},c_{x}} and the {Δf′_{m}(λ), f′′_{m}(λ)} scattering factors for metals Fe1 and Fe2 over 100 energy channels [that determine F_{fit}(λ)].
As for the geometric spot profile ΔΩ_{L,i}∑_{D†}F^{2}_{latt}[S(λ)], it has dependence primarily on the crystal orientation A* determined by DIALS or by profilebased orientational therefore it can be precalculated in a separate step [one value for each energy channel; Fig. 5(d)]. For this purpose we used a large (200 member) ensemble D^{†} to adequately sample the rotational mosaicity [Fig. 2(b)] rather than the small sample [Fig. 2(a)] used for the simulation. We used the ground truth mosaic rotation (0.05°) and domain size (400 nm) for the present calculation, but assume that in real cases these values can be experimentally determined as in the work by Sauter et al. (2014).
2.4.2. Restraints
While the parameters {Δf′_{m}(λ), f′′_{m}(λ)} should be overdetermined by the data, there is still considerable noise, as well as poor energy coverage far away from the 7120 eV set point [Fig. 3(c)]. Thus, there is a danger that the parameter estimates may diverge during the process. As a strategy to avoid this, we take the opportunity to use the P(model) factor in equation (11) to express the prior belief that the scattering curves are smooth as a function of energy, thus imposing restraints on Δf′ and Δf′′ for each metal atom and at each energy step. To cast these model parameters in terms of prior probability, we took the scattering curves for Fe^{2+} and Fe^{3+} in Fig. 1 as a reference distribution. In Fig. 1, the change in scattering factor with respect to energy has an approximately normal (Gaussian) distribution, with mean μ = 0.0 and standard deviation σ_{1} = 0.1 eV^{−1}, while ΔΔf′′/ΔE gives σ_{2} = 0.2 eV^{−1}. Therefore, we express the overall prior probability of the model as a product of probabilities over both metal sites m, over n = 100 independent energy steps in the 7070–7170 eV range, and over both the dispersive and absorbtive corrections,
A corresponding term is incorporated into the loss function of equation (13),
Scattering factors at n = 1 and n = 100 were not refined, thus constraining the values at 7071 and 7170 eV to their starting estimates.
2.4.3. Model optimization
Iterative parameter estimation was performed with the limitedmemory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm (Liu & Nocedal, 1989) as implemented in CCTBX. Initial estimates for the background parameters {a_{x},b_{x},c_{x}} for each Bragg spot were obtained by masking out the Bragg signal (with a pixel mask determined by DIALS) and modeling the peripheral shoebox pixels only; however, subsequent iterations considered the entire shoebox when refining the background model. Requisite first derivatives are listed in the Appendices.
2.4.4. Implementation
Table 2 lists the computational resources used for data analysis. Parallel execution with Pythonmediated MPI was critical for keeping run times to within 30 h. However, work parcels were distributed in distinct patterns for various steps. The geometrical profiles ΔΩ_{L,i}∑_{D†}F^{2}_{latt}[S(λ)] depend primarily on the crystal orientation A* and the parameters {a_{x}, b_{x}, c_{x}, G_{L}}, but only weakly on the factors Δf′ and Δf′′. The profiles are therefore prerefined as step (d), which also happens to be the most computationintensive step, while also refining {A*, a_{x}, b_{x}, c_{x}, G_{L}}, after which the geometric profiles are fixed. We then perform repeated macrocycles of step (e), refining {a_{x}, b_{x}, c_{x}, G_{L}}, and step (f), refining {Δf′_{m}(λ), Δf′′_{m}(λ)}. Although all the refineable parameters of steps (e)–(f) are, in principle, interdependent, and thus subject to simultaneous optimization, as a practical matter it is easier to refine the two parameter sets alternately until convergence is achieved. The anomalous correction step (f) in particular has a complex implementation with respect to parallel execution. At each iteration within LBFGS, the structure factors F_{sim}(λ)^{2} are initially calculated in MPI rank 0 and broadcast to all ranks. Individual ranks then calculate the separate contributions to from various diffraction images, which are finally summed up by MPI.reduce() and are thereby available to rank 0 for the line search. In this programming pattern, the contributions of the restraints are handled by rank 0.

3. Results
A total of 100 000 simulated diffraction patterns were processed with dials.stills_process (Method 3, Fig. 6). The 67 936 patterns with ≥3 DIALSidentified Bragg spots in the region of interest (Fig. 4) yielded 305 777 `shoeboxes' (rectangular boxes each containing a Bragg spot plus background, Fig. 8), representing 100% of the 8241 unique in the C2 that span the 2.1–2.5 Å resolution range, implying an average 37fold multiplicity of observation. These contained a total of 106 628 830 pixels (both background and Bragg spot) to be used for estimation of the energydependent parameters at the two iron centers in ferredoxin.
LBFGS parameter optimizations are summarized in Table 3, highlighting various starting models and conditions. For ease of comparison among many trials, Table 3 reports the rootmeansquared deviation of model scattering factors versus ground truth. Progress is best visualized (Fig. 9) by plotting the energydependence of the factors {Δf′_{m}(λ), Δf′′_{m}(λ)}.

Of key interest is whether the inferred scattering curves can be used to distinguish valence states. Scattering from reduced (Fe^{2+}) and oxidized (Fe^{3+}) states is expected to differ in several regards (Einsle et al., 2007; Sherrell, 2014; Fig. 1): the absorption Kedge [as shown by Δf′′_{m}(λ)] shifts 1–2 eV to a higher energy for the oxidized state, over roughly the 7115–7125 eV window, and corresponding changes are also seen in the dispersion spectrum [Δf′_{m}(λ)] at the preedge (7117 eV) and peak (7122–7132 eV) windows.
Table 3 indicates that the correct valence configuration is indeed readily determined by the analysis of our simulated data. We performed five parameter estimations, four of which started with guesses that incorrectly assign the valence state. One differed by switching the electron to the wrong Fe site, two by either the overall loss or the gain of one electron, and one involved the gain of five electrons (modeling the iron centers as metallic Fe^{0}, which gives a very poor r.m.s.d. comparison of the ground truth). In all cases, including the use of the ground truth as the starting guess, the model refined to a state with a high degree of similarity to the ground truth.
These results suggest that our approach to parameter estimation is well behaved. Various starting guesses for the scattering factors yield essentially the same result, showing that we are comfortably within the radius of convergence (restraints described in Section 2.4.2 are necessary; data not shown). Convergence was achieved using a cohort of 50 000 input diffraction images. Utilizing a different cohort of 50 000 produces very similar agreement to ground truth. However, taking progressively smaller subsets degrades the performance, such that results from fewer than 25 000 images would be suspect. Shortcuts that involve fewer than three macrocycles (Figs. 5 and 9) would also be inadvisable as the interdependent treatments of {a_{x}, b_{x}, c_{x}, G_{L}} and {Δf′_{m}(λ), Δf′′_{m}(λ)} would have insufficient opportunity to crossrefine. Finally, we performed an important negative control: analysis with h = [H, K, L] replaced by [H, K, L + 1] in equation (1), leading to completely wrong scattering factors as expected.
To summarize, our simulation of XFEL diffraction patterns from a homogeneous and isomorphous population of 4 µm Δf′(λ) and Δf′′(λ) for each of the two Fe atoms with sufficient precision to distinguish between the ferrous and ferric oxidation states. The calculation assumes an air path of 10 mm, a water path of 100 µm, and neglects diffuse scattering and absorption. It is assumed that the upstream singleshot spectrometer provides a good estimate of the incident spectra at the sample. It is also assumed that the unitcell parameters are identical over the crystal population.^{6}
crystals with well characterized mosaicity shows that the pixelprofile analysis of Bragg spots in a small region of interest centered at 2.3 Å (from 50 000 patterns) determines the anomalous corrections4. Discussion
The et al., 2007), but such a measurement has yet to be achieved under the timeresolved, physiologically relevant conditions that are possible with XFELs. Severalatom cofactors such as the [4Mn:5O:Ca] oxygenevolving complex of photosystem II have been investigated using Xray emission spectroscopy at the Kedge, but this does not distinguish among the multiple Mn sites (Kern et al., 2018). There are certainly many practical challenges: the contribution is small compared with the overall diffraction (Fig. 7), the XFEL pulse's broad bandpass smears out the energydependence of the signal (Figs. 1 and 8), and it has been notoriously difficult to scale XFELmeasured Bragg spots into selfconsistent amplitudes. However, the consideration of simulated data (Table 3; Fig. 9) suggests that the technique is possible with present XFEL instrumentation, provided that the incident Xray spectra are measured to normalize the energy dependence (Zhu et al., 2012; Fig. 3), the highresolution Bragg diffraction is imaged by a pixel array positioned far enough back to spread out the energies (Fig. 8) and detailed physical modeling (such as nanoBragg) is applied to the signals from each pixel using sufficiently large datasets that are best analyzed by current petascale supercomputers.
analysis presented above offers a path for using XFEL diffraction as a spatially resolved spectroscopic method. has the potential for distinguishing the electronic environment at metalloprotein metal sites (EinsleThe main result, the possibility of distinguishing valence states, rests on the relevance of the conditions we chose for the data simulation. We made every attempt to pick conservative parameters describing the crystal size and mosaicity, the liquid and air paths, the Xray spectrum and intensity, and the solid angle subtended by the pixel array. Although early generation XFEL imaging detectors may have lacked the large et al., 2016), Jungfrau (Leonarski et al., 2018) and AGIPD (Allahgholi et al., 2015) offer the level of measurement stability that is incorporated into our assumptions. As for data analysis, it was important to model the contributions to each pixel from distributions of mosaic rotations and beam energies, the socalled `raytracing' approach, and to properly weight the shotnoise statistical probability of each pixel value with equation (15). We note that an alternate calculation using only singleenergy assignments for each pixel, and using an equally weighted nonlinear least squares pixel treatment, failed to stably refine the anomalous corrections (data not shown). Translating our simulation into a real experiment will inevitably present additional systematic corrections such as proper calibration of the incident Xray spectrometer, parallax effects in the imaging detector (Holton et al., 2014; Winter et al., 2018) and treatment of unexpectedly complex mosaic texture. Uncertainty in quantities such as the structurefactor phase angle from nonmetals (in F_{fixed}) may have to be integrated out (McCoy, 2004).
linear response and well characterized gain needed to achieve our goals, we estimate that currentgeneration devices such as the ePix (SikorskiThere may be additional scientific potential beyond what is anticipated in our simulation. Although most protein crystallography literature treats the anomalous corrections Δf′ and Δf′′ as scalar quantities, anisotropy has been reported in some cases, such that the scattering factor is represented by a tensor quantity, reflecting the complex chemical environment of the absorbing atom (Hendrickson et al., 1988; Schiltz & Bricogne, 2008, 2010). In XFEL diffraction the crystals are examined in random orientations with respect to the polarized Xrays, which may thus offer the unique opportunity to sample the full rotational variation of the scattering, yielding additional details of the chemical environment.
Apart from its usefulness in modeling the et al., 2012, 2016; Kabsch, 2014; Brewster et al., 2018) rely on pixel summation to obtain the signal intensity for each Bragg spot. In contrast, for synchrotronbased experiments that involve goniometer rotation, an alternate and more accurate method has long been available based on profile fitting. This has been achieved because there are standard theoretical frameworks for profile prediction (Otwinowski & Minor; 1997; Kabsch, 2010). Profile prediction has been discussed for XFEL work (KroonBatenburg et al., 2015; White, 2014; Ginn et al., 2015) but has not been widely applied. Fig. 8 illustrates the potential for quantitative description, showing that (i) each pixel of the Bragg spot represents a different average photon energy, (ii) each energy channel contributes to a narrow band of pixels, with adjacentchannel bands overlapping, and (iii) different energies contribute unequally to different spots, depending on how far (Δx) the point is from the energyspecific Incorporating nanoBragg profile predictions into a data processing workflow such as dials.stills_process would provide a means for normalizing the Bragg spot intensities against the stochastically shaped incident spectra that can be measured for each pulse.
detailed physical modeling as described above might play a future role in general XFEL data processing. Current XFEL data integration programs (WhiteIn a related matter it is interesting to speculate on what role the nanoBragg approach might play in optimizing the model parameters describing the crystal (the unitcell parameters, orientation and mosaic texture). Two types of objective function have recently played a role in XFEL data modeling: the agreement of observed and predicted spot positions, and the agreement of observed and predicted spot intensities (postrefinement). Our results with positional (using dials.stills_process, Fig. 6) illustrate that centroid spot positions do not give parameters such as crystal orientation to high accuracy. In postrefinement, the parameters are further refined to achieve the best intensity agreement among duplicate measurements after scaling for spot `partiality', essentially the falloff of spot intensity as Δx increases. Many postrefinement approaches have been explored for XFEL data (White, 2014; Kabsch, 2014; Sauter, 2015; Uervirojnangkoorn et al., 2015; Ginn et al., 2015; KroonBatenburg et al., 2015). However, none of these were considered for use in this paper since the pixel summation step fundamentally erases the energydependent information that we sought to extract. Our approach represents a third type of objective function [equation (15)], which takes account of the nuanced spot sizes, shapes, and intensity profiles that are accessible when the analysis is done on a pixeltopixel basis. Indeed, spots that overlap for other reasons, such as multiple lattices or nonmerohedral twins could be deconvoluted in this way. The material presented here provides a basic framework, and our initial results indicate that it is possible to refine crystal orientation to high accuracy (Fig. 6), at least with simulated data. The details of how to transfer these ideas to real experimental data remain to be worked out.
5. Conclusions
The availability of XFEL beamlines has facilitated the study of proteins under physiological conditions free from radiation damage. For metalloenzymes in particular, time resolution has also been key for the study of catalytic mechanisms. In order to fully exploit the potential of timeresolved measurements, we have previously developed multimessenger techniques, simultaneously combining the results from Xray diffraction for reporting the atomic structure, and Xray emission spectroscopy for reporting the electronic state of active site transition metals (Kern et al., 2013, 2018; Young et al., 2016; Fuller et al., 2017; Fransson et al., 2018). Now, based on the current results, there is the potential of adding a third reporter to follow the timedependence of the spatially resolved factors and the underlying metal chemistry over the course of the reaction cycle. This information can be obtained without additional experiments, provided that the Xray diffraction is collected at the metal hence avoiding the problems of normalization and comparability between different separate measurements. We hope that this approach will be a driver for future experimental design, and with respect to detectors and beam spectrometers, for XFEL endstation development.
6. Software availability
The program nanoBragg is available as a standalone C program at https://bl831.als.lbl.gov/~jamesh/nanoBragg/. In this work, nanoBragg was ported into the opensource Python/C++ framework of CCTBX and can be downloaded at https://github.com/cctbx/cctbx_project. All scripts for reproducing this work are at https://github.com/nksauter/LS49, and in particular see the README file under paper1.
APPENDIX A
Derivatives for the image scale factor, the spot background and the anomalous corrections.
At each minimization step, LBFGS requires the first derivative of with respect to each parameter z. With respect to the likelihood term
Fig. 5(e) describes the of the overall scale factor for each image G_{L}, and the background plane parameters a_{x},b_{x},c_{x} for each Bragg spot. Considering each parameter in turn,
where the pixel index i is over all Bragg spots of interest on a single image L, and
where the pixel index i spans a single Bragg spot x. Fig. 5(f) mentions the of the anomalous corrections of metal m at wavelength λ,
where Δf_{m}(λ) is either Δf′_{m}(λ) or Δf′′_{m}(λ), and I_{sim}(λ) is the intensity at the h_{0} associated with pixel i [computed as the dot product F_{sim}(λ)·F_{sim}(λ)]. Therefore,
Also, since the complex N_{M} atoms within the class m,
is a sum of individual atom contributions, we only need to consider theand finally for the sth atom,
For the restraints expressed in the prior probability term, derivatives for the nth value of λ are
and
for n = 2, …, 99.
APPENDIX B
Derivatives for the crystal orientation
As mentioned in Section 2.3.2, the crystal orientation A* derived from DIALS (Method 3) is insufficient to model a spot profile close enough to that observed. Instead, it was necessary to produce an orientation that was modified (refined) by slight rotational perturbations,
where R_{H} and R_{V} are matrices that encode rotation about horizontal and vertical axes (perpendicular to the Xray beam). The rotation values φ_{H} and φ_{V} are small, on the order of ±0.01°. To derive optimal values for each crystal [Fig. 5(d)] we use LBFGS parameter fitting, requiring the first derivative of with respect to φ_{H} and φ_{V}; these are refined while also refining scale factor G_{L} for each image and background plane parameters a_{x}, b_{x}, c_{x} for each Bragg spot in the framework of equations (15)–(17). However, in contrast to Appendix A where analytical derivatives are calculated for all other parameters (in order to determine the step gradient), we did not use an analytical form for the derivatives of equation (16) with respect to φ_{H,V}. Although such an exercise is possible given the material presented above, it was expedient simply to use finite difference derivatives,
where φ_{c} is the current value of at any given step of parameter refinement.
Footnotes
^{1}With a 2 µm (FWHM) focus at the Linac Coherent Light Source, an Xray fluence of 10^{12} photons per 50 fs pulse at 9.5 keV produces no in Mn_{2}^{III, IV}O_{2}terpyridine (AlonsoMori et al., 2012).
^{2}One of the earliest demonstrations of [Coster et al., (1930); see also James (1962), p. 33]; involved the 85 eV splitting of the gold Lα doublet to either side of the crystal's Zn edge. See also Arndt et al. (1982).
^{3}Other XFEL experiments have used backposition detectors to magnify the diffraction pattern (Chapman et al., 2011; Brewster et al., 2018; Duyvesteyn et al., 2018).
^{4}Rather than treating the detector offset as a software issue to be recoded, we treated it as a beam position in need of precise as discussed by Brewster et al. (2018). Briefly, the initial models from 2000 processed images were combined into a single file with dials.combine_experiments. The program dials.refine was then used to determine a single detector position consistent with all 2000 experiments.
^{5}In contrast to equation (6), where the simulated units are completely defined, equation (16) is intended to model actual experimental data. In this case, the absolute scale of several factors is unknown, including the number of incident photons J_{0}(λ) measured on the Xray spectrometer and the number of diffracting unit cells that enters into the F^{2}_{latt} factor.
^{6}In a real experiment, the true parameter distributions would need to have standard deviations of ≤0.02% in order to produce Bragg streaks, illustrated in Fig. 8(a), that are aligned along the energy axis to within ∼1 eV.
Acknowledgements
We thank Aaron Brewster, Asmit Bhowmick, Robert Bolotovsky, Franklin Fuller, Derek Mendez and Diling Zhu for helpful discussions. Scattering factor measurements for
were provided by Darren Sherell and Graham George. Vittal Yachandra, Limei Zhang, Allen Orville, Alexander Batyuk, Jason Koglin, Roberto AlonsoMori, Anton Loukianov, Sheraz Gul, Iris Young, Kyle Sutherlin, Joshua Heinemann, Cindy Pham, Ruchira Chatterjee, InSik Kim and Chi Chung Lee participated in an April 2018 LCLS experiment attempting to measure spatially resolved factors (for which the data analysis has not been completed and is not presented here).Funding information
Research was supported by NIH grants GM117126 (NKS), GM124149, GM124169, GM103393 and GM082250 (JH) for dataprocessing methods; and GM110501 (JY) and GM126289 (JK) for experimental methods development and metalloenzyme serial crystallography. JH acknowledges support from NSF DBI 1625906 and the US Department of Energy (DOE) (contract No. DEAC0205CH11231). NKS acknowledges support from the Exascale Computing Project (17SC20SC), a collaborative effort of the DOE Office of Science and the National Nuclear Security Administration. JY and JK acknowledge support by the Director, Office of Science, Office of Basic Energy Sciences (OBES), Division of Chemical Sciences, Geosciences, and Biosciences of the Department of Energy (DOE). Portions of this research were carried out at LCLS at the SLAC National Accelerator Laboratory, supported by the DOE Office of Science, OBES (contract No. DEAC0276SF00515). Data processing was performed in part at the National Energy Research Scientific Computing Center, supported by the DOE Office of Science (contract No. DEAC0205CH11231).
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