research papers
Beyond integration: modeling every pixel to obtain better structure factors from stills
^{a}Molecular Biophysics and Integrated Bioimaging Division (MBIB), Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA, ^{b}Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA, and ^{c}Department of Biochemistry and Biophysics, UC San Francisco, San Francisco, CA 94158, USA
^{*}Correspondence email: dermen@lbl.gov, nksauter@lbl.gov
Most crystallographic data processing methods use pixel integration. In serial femtosecond crystallography (SFX), the intricate interaction between the ^{4}−10^{6}) of exposures. Although sufficient for generating biological insights, this approach converges slowly, and using it to accurately measure anomalous differences has proved difficult. This report presents a novel approach for increasing the accuracy of structure factors obtained from SFX data. A physical model describing all observed pixels is defined to a degree of complexity such that it can decouple the various contributions to the pixel intensities. Model dependencies include lattice orientation, unitcell dimensions, mosaic structure, incident photon spectra and amplitudes. estimation is used to optimize all model parameters. The application of prior knowledge that amplitudes are positive quantities is included in the form of a reparameterization. The method is tested using a synthesized SFX dataset of ytterbium(III) lysozyme, where each Xray laser pulse energy is centered at 9034 eV. This energy is 100 eV above the Yb^{3+} LIII so the anomalous difference signal is stable at 10 electrons despite the inherent energy jitter of each femtosecond Xray laser pulse. This work demonstrates that this approach allows the determination of anomalous structure factors with very high accuracy while requiring an orderofmagnitude fewer shots than conventional integrationbased methods would require to achieve similar results.
point and the is integrated out by averaging symmetrically equivalent observations recorded across a large number (10Keywords: serial crystallography; freeelectron lasers; SFX; stills; data processing.
1. Introduction
The accuracy of et al., 2011). Ultrafast Xray pulses from such facilities provide unique opportunities to investigate functional, roomtemperature protein states, while probing enzyme dynamics on time scales from femtoseconds to milliseconds (AlonsoMori et al., 2016; Pande et al., 2016; Thomaston et al., 2017; Stagno et al., 2017; Tosha et al., 2017; Nogly et al., 2018; Kern et al., 2018; Nango et al., 2019; Dasgupta et al., 2019; Ibrahim et al., 2020), yet largely avoiding radiation damage (Chapman et al., 2014; Spence, 2017; Fransson et al., 2018). New protein science discoveries commonly arise at the extreme limit of what the signaltonoise of the diffraction data can support, as illustrated by our recent experiences with photosystem II (PSII, Kern et al., 2018; Ibrahim et al., 2020). There, the electron density revealed small timedependent changes, including the appearance of a single substrate oxygen atom at the catalytic site against the backdrop of a 23polypeptide protein complex. Efforts to assign rigorous uncertainties in atomic positions (Ibrahim et al., 2020) showed significant structural changes, yet a clear desire remained to utilize the weaker data at the limiting resolution in order to gain further atomic insights. Ultimately, our interest lies in singleelectron transfers at the four Mn positions of the PSII catalytic cofactor. Spatially resolving the K individually for each Mn center has the potential to elucidate the electronic environment of each Mn atom (Sauter et al., 2020). Such challenging measurements would require quantifying intensities between Friedel pairs ) and among different states at the 1% level of uncertainty.
estimation remains a central experimental focus as we approach the tenth anniversary of serial femtosecond Xray crystallography (SFX) for biological at Xray freeelectron lasers (XFELs, ChapmanDespite this experimental need for accurate interpretation of weak signals, a close look at data analysis pipelines has revealed stubborn and inherent difficulties specific to XFEL diffraction. In general terms, the h = (h, k, l),
amplitude is proportional to the square root of the Bragg spot intensity recorded forwith the proportionality modulated by factors such as incident ; Holton & Frankel, 2010). In particular, the interplay of incident beam divergence, Xray spectrum and crystal mosaicity produces a distribution of diffracted intensities around regions in which satisfy Bragg's condition. The success of the rotation method of data acquisition (Arndt & Wonacott, 1977), long practiced at synchrotron sources, rests on the ability to fully rotate the crystal through the angular range of this distribution, termed the `rocking curve', while summing the diffracted intensity. In this way, the spectral shape and the crystal's mosaic disorder do not contribute to the measurement error, as they are integrated out. Other factors, such as the intensity profile of the incident beam and the size and shape of the illuminated volume vary smoothly with the crystal rotation, hence scaling virtually eliminates errors due to these effects (Evans & Murshudov, 2013). In contrast, the lack of finite rotation during femtosecond exposure gives rise to partial observations that sample each Bragg spot's rocking curve at a single position, meaning the sources of error described above all contribute to the uncertainty in SFX data. As expressed in the work by Kirian et al. (2010), for SFX, averaging repeated measurements of the same across different diffraction patterns can minimize these uncertainties, assuming each measurement samples a rocking curve at random. However, when considering an SFX dataset in its entirety, duplicate measurements of a given form a skewed distribution peaking near zero, resulting from the many partial observations of the rocking curve tails that are lowphoton count and contribute mostly noise, making it difficult to derive an `average' intensity value that accurately represents F_{h}^{2}; see especially Figs. 1 and 5 in the work by Sharma et al. (2017) and Fig. 8 in the work by KroonBatenburg et al. (2015). Further contributing to noise, the rocking curves sampled in each shot are slight variations of one another, owing to the stochastic nature of XFEL pulses and the morphological variations amongst measured crystals.
crystal size, polarization and absorbance (Darwin, 1922Indeed, Kirian et al. (2010) acknowledges that the baseline technique of simply averaging duplicate measurements, the socalled `Monte Carlo' approach, requires 10^{4}–10^{5} individual diffraction patterns for the crystallographic R factor to converge; the 2010 paper thus ushered in ten years of technology development to identify improved algorithms. Most of these algorithms derive a physical model of the data, scored by one of three metrics: the ability to predict Bragg spots in the correct position, the selfconsistency of equivalent Bragg spot intensities and, ultimately, the focus in our paper, the ability to predict not only the position of spots, but their size, shape and intensity profile.
Regarding the first metric, the ability to predict spots close to their observed positions requires knowledge of the unitcell parameters and the crystal orientation in order to select those et al., 2014). Parameter can optimize the positional match of data and model, which results in more accurate structure factors (Sauter et al., 2014; Yefanov et al., 2015; Waterman et al., 2016; White et al., 2016; Brewster et al., 2018).
points in the diffracting condition (`on the Ewald sphere'), and also to know the mosaic parameters (mosaic domain size and mosaic rotational spread) to determine which lattice points offset from the can still generate reflections (SauterTo examine the selfconsistency of symmetrically equivalent Bragg spot intensities, efforts have been made to estimate the `partiality' of each spot, i.e. a perreflection scale factor that properly accounts for Ewald offset, where lattice points farther from ideal diffraction (and thus weaker) get multiplied by a larger factor to put duplicate observations on the `full spot' scale. Several programs have emerged to treat the problem of partiality, with cxi.merge (Sauter, 2015), prime (Uervirojnangkoorn et al., 2015) and nXDS (Kabsch, 2014) using the cell and mosaicity parameters already mentioned, paired with a monochromatic beam; whereas ccpxfel (Ginn et al., 2016) and partialator (White, 2014) offer a polychromatic model intended to model the spectral width produced by the selfamplified (SASE) obtained at XFEL sources (Emma et al., 2004; Margaritondo & Ribic, 2011). All of these programs employ postrefinement, iteratively optimizing the parameters such that multiple partialitycorrected measurements of the same Bragg reflection yield the most consistent integrated intensity values after scaling.
Finally, the program EVAL exemplifies profile fitting (Duisenberg et al., 2003; Schreurs et al., 2010; KroonBatenburg et al., 2015), utilizing a detailed physical description (source divergence, source dispersion, crystal size, mosaic block size and mosaic rotational spread) to faithfully model the size, shape and intensity profile of each Bragg spot. Along with other diffraction modeling programs such as SIM_MX (Diederichs, 2009), pattern_sim (White et al., 2012) and nanoBragg (Holton et al., 2014; Lyubimov et al., 2016), EVAL allows us to explore how each of these physical parameters influences the appearance of Bragg spots (see also Nave, 1998, 2014), whereas the EVAL paper goes further and uses the profilederived corrections in a simplex minimization to simultaneously optimize unit cells, crystal orientations and pershot scale factors, achieving an optimal set of corrections for alreadyintegrated spots.
Conventional data processing involves a twostep approach, first measuring Bragg spots on individual images, and then scaling and merging them together. For the work presented in this paper, determination of optimal structure factors F_{h}, given the data, occurred in a single step, guided by a likelihood target function dependent on crystal orientations, unit cells, scale factors, mosaic parameters, photon spectra and, importantly, a starting list of structure factors provided by conventional data processing. The likelihood target received contributions from pixels across all images,^{1} with computation of perpixel probabilities facilitated by the forward modeling program nanoBragg. A new program, diffBragg, provided the first derivatives of the forward model with respect to the various parameters, allowing effective navigation of the likelihood gradient in order to deduce the multiparameter In using a likelihood formalism at the pixel level, we utilized an explicit error model for each pixel derived from first principles, hence optimization of the errors, which depend on the model, occurred at each step. Also, the joint of all parameters correctly accounted for covariance. For example, amplitudes and pershot scale factors both directly increase the intensities of spots; however, in the global treatment, other shots measuring the same reflections constrain the structure factors. The integration methods described above all reduce the complicated spot profiles to single numbers; however, in the diffBragg approach, each pixel in the spot profile is tied directly to the model, significantly increasing the number of observations used during model optimization, and leading to higher parameter accuracy from fewer overall shots.
Pixellevel et al. (2020). Having potential access to this level of detail from Bragg scattering opens up new and exciting experimental avenues to consider as the next decade of SFX science begins. Here, using the newly developed tool diffBragg on synthetic data, we further advanced the pixel approach to extract more information from fewer recorded shots: we optimized a set of 13 704 amplitudes using 2023 XFEL diffraction patterns to a comparable accuracy to that achieved from conventional integration of 19 953 diffraction patterns, and we assumed no prior knowledge other than a positivity restraint applied to the structure factors. Furthermore, with the new protocol we combined positional and postrefinement in a single framework. Sections 2.1 and 2.2 below describe the creation of synthetic data with nanoBragg; Section 2.3 introduces the new framework, diffBragg, for iterative parameter fitting. Section 3 details the improvement afforded by the new method, beyond the initial inputs from the program dials.stills_process [Brewster et al. (2016), see also Brewster, Young et al. (2019) for a recent description of the graphical user interface], which represent conventional data analysis. The results presented here required significant computational resources to achieve; however, GPUacceleration, a nearterm goal, will help in this regard (see Section 3.2.3). In addition to setting the ground work for using firstderivatives to perform more complicated refinements, this work reveals the potential to extract sensitive information from fewer recorded shots.
can resolve extremely sensitive details, such as the dispersion corrections arising from two differently oxidized metal atoms, as shown with synthetic data in the work by Sauter2. Methods
2.1. Components of synthetic data
To test the approach, we synthesized realistic lysozyme Yb^{3+} XFEL diffraction images with a mean XFEL pulse energy of 9034 eV, just above the Yb^{3+} LIII (Fig. 1). We chose an anomalous dataset as a stringent test, as anomalous differences are highly sensitive to errors in estimation.
2.1.1. Protein
We used the program CCTBX [Computational Crystallography Toolbox, GrosseKunstleve et al. (2002)] to generate structure factors [see equation 1 of Sauter et al. (2013) for details] from PDB entry 4bs7, a room temperature lysozyme derivative structure with two Yb^{3+} sites (Pinker et al., 2013). Wavelengthindependent scattering factors for all atoms were calculated using Cromer–Mann coefficients [as tabulated in the work by Brown et al. (2006)], and factor corrections f′ and f′′ for all protein atoms were computed using the Henke tables (Henke et al., 1993). However, for ytterbium we specifically used f′ and f′′ from measured data (Shapiro et al., 1995; Hendrickson & Ogata, 1997). Fig. 1 shows the experimentally determined Yb profile of f′′, with a magnitude at the highenergy remote (9034 eV) of 10 electrons (Fig. 1).
2.1.2. Crystal
We synthesized data from tetragonal lysozyme, with the a = b = 79.1, c = 38.4 Å. Each synthetic crystal had mosaic domains consisting of 1000 unit cells (10 along each axis). The crystal mosaic spread was computed by averaging scattering contributions from 100 equivalent mosaic domains. The misorientation of each mosaic domain with respect to the nominal orientation was taken from a normal distribution with a mean of 0° and standard deviation of 0.01° degrees, forming a mosaic texture. The mosaic texture was the same for all synthetic crystals, and the value 0.01° was assumed to be a typically observed value in real crystals (Bellamy et al., 2000). For each shot, the total scattering from this mosaic average was multiplied by a random scale factor Z drawn from a normal distribution with mean μ_{Z} = 1150 and standard deviation σ_{Z} = 115 about the mean. This scale factor Z is the total number of mosaic domains in the crystal. The reason for using only 100 mosaic orientations for spread (instead of Z) was computational expediency. Random variation in the scale factor for each crystal was used to mimic variation in exposed crystal volume expected during each shot at the XFEL. The value 1150 was chosen by hand such that the contrast between low and highresolution spots was typical. We represented the crystal using a standard matrix convention (Busing & Levy, 1967), where each crystal orientation is represented by two matrices: an uppertriangular matrix B, whose columns specify the lattice basis vectors in an aligned orientation defined according to the convention of Arndt & Wonacott (1977), and a threeparameter rotation matrix U_{s} which moves the crystal from its aligned position into its observed position in shot s. For the tetragonal system
where a and c are the realspace unitcell edge lengths.
2.1.3. Background scattering
We synthesized the diffraction akin to that measured under vacuum at the Coherent Xray Imaging (CXI, Liang et al., 2015) instrument at the Linac Coherent Light Source (LCLS), employing a gas dynamic virtual nozzle (GDVN) for sample delivery (DePonte et al., 2008). The GDVN uses water pressure on a piston to force sample through a capillary and out of a nozzle to produce a liquid jet of sample in the interaction region. We assumed the jet was focused to a 5 µm diameter by a helium gas sheath. For background scattering we modeled the irradiated water volume as that of a cylinder: 5 µm in length multiplied by the beam spot size, a circle with a diameter of 1 µm. We neglected scattering from the helium sheath, and otherwise the path between the interaction region and detector was assumed to be a vacuum.
2.1.4. Beam and detector
We synthesized measurements from a 32panel Cornell–Stanford Pixel Array Detector (CSPAD) as set up at CXI (Hart et al., 2012), where each panel consists of 185 × 388 pixels, each 109.92 µm × 109.92 µm in size. We used a 2 × 2 pixeloversampling rate to minimize aliasing errors from Bragg peaks whose signal might change significantly across the physical pixel dimension. We used a realistic threedimensional CSPAD geometry obtained from CXI, such that the individual panels had relative rotations [see Brewster et al. (2018) for a detailed description]. We defined a `pixel measurement' by both its position on the Xray detector and the diffraction event it represents. Therefore, we let each pixel measurement be X_{i,s} where s refers to an Xray event, or shot, and i is an index specifying the pixel position in the detector pixel array. The index i is equivalent to a triple index (panel, fast, slow) where panel is the CSPAD panel ID (0–31), fast is the panel fastscan pixel coordinate (0–184) and slow is the panel slowscan pixel coordinate (0–387). The CSPAD was placed 124 mm from the interaction region, giving a corner resolution of 1.7 Å; however, we only analyzed scattering out to 2.1 Å. For each synthesized diffraction event we used a unique SASE input spectrum that was a scaled and shifted version of real spectra recorded during an LCLS experiment (run 16, proposal number LS49) using an upstream spectrometer (Zhu et al., 2012). Fig. 1 shows a representative XFEL pulse spectrum used to generate synthetic data, as well as the average spectrum. We assumed a uniform of photons with an average of 8 × 10^{10} photons per pulse, though the total fluence was different for each synthesized shot. We ignored beam divergence effects, as these are typically small at XFELs.
2.2. nanoBragg: computing synthetic data
We used the program nanoBragg to compute, for each pixel during each shot, the expected Bragg scattering I_{i,s,data} and background scattering T_{i,s,data}, and also to apply a realistic noise model.
2.2.1. nanoBragg: generating Bragg scattering
To compute the Bragg scattering measured by each pixel, nanoBragg applies the kinematic theory where the expected number of Braggscattered photons in each pixel is the product of the incident Xray fluence with the scattering of the crystal and the solid angle ΔΩ_{i} of the pixel. In what follows, r_{e} is the classical electron radius, κ_{i} is the Kahn polarization factor for scattered light from a prepolarized incident source (Azároff, 1955; Kahn et al., 1982), J_{s}(λ) is the fluence in photons per area at wavelength λ, Z_{s} is the crystal scale factor randomly sampled for each shot from a normal distribution , F_{h} is the amplitude of the protein in each and I_{0}(Δh_{i,j,s}, m) is the interference factor arising from the periodicity in the The term Δh_{i,j,s} is the Ewald offset at the pixel determined from the orientation of mosaic block j, and m^{3} is the total number of unit cells in the mosaic domain. N_{j} is the number of orientations used to form a mosaic texture, the scattering power of which is then normalized by N_{j} and scaled up by Z_{s} for total scattering equivalent to a crystal that is made up of Z_{s} mosaic blocks. For the synthetic data used here, the expected Bragg scattering was computed according to
where the expression inside the square brackets is the scattering λ. The interference term I_{0}(Δh_{i,j,s}, m) should be maximal when the pixel is exactly probing the diffraction condition (Δh_{i,j,s} = 0), and it should fall off rapidly as Δh_{i,j,s} increases. We let I_{0}(Δh_{i,j,s}, m) take the Gaussian form
of the whole crystal for photons of wavelengthwhere the constant C was chosen such that the full width at halfmaximum of the principal peaks in I_{0} would be equal to that arising from a parallelepiped mosaic block [see Appendix A for a derivation of equation (4); in the work here, we used C = 3.175]. The dimensionless Ewald offset Δh_{i,j,s} is simply the residual between the fractional at each pixel with the nearest integer Miller index
Here q_{i}(λ) is the momentum transfer from incident beam unit vector to the scattered beam unit vector ( points from the interaction region to the pixel):
The transpose of UB in equation (5) is necessary as the columns of B are what define the lattice translation vectors. Note the dependence of I_{0}(Δh_{i,j,s}, m) on m; a larger mosaic domain parameter yields a brighter maximum with a more rapid falloff. For the synthetic data, we let m = 10 and N_{j} = 100. For real data, we expect the true shape of the Bragg peak profiles to be well approximated by Gaussians, or a sum of Gaussians. The parameters describing the synthetic Bragg scattering data are summarized in Table 1.

2.2.2. nanoBragg: generating background scattering
Background scattering was synthesized by estimating the total number of background molecules in the bulk backgroundscattering volume. Here, nanoBragg computed the expected scattering for liquid water:
where V_{H}_{2}_{O}, ρ_{H}_{2}_{O} and u_{H}_{2}_{O} are the volume, density and molecular weight of water, respectively, N_{A} is Avogadro's constant, and F_{H}_{2}_{O}(q_{i}) is the isotropic amplitude per molecule of liquid water which depends on the scattering angle of the pixel, 2θ_{i} = sin^{−1}(q_{i}λ/2). F_{H}_{2}_{O}(q_{i}) was measured independently at the Advanced Light Source beamline 8.3.1 (MacDowell et al., 2004) and adjusted to match the absolute calibration in the work by Clark et al. (2010).
2.2.3. nanoBragg: generating realistic measurement noise
With nanoBragg, we applied three types of measurement noise arising from the inherent randomness of signal amplification and detector readout. These noise operations are computed sequentially, modeling the various stages of photon measurement. In the first measurement stage, shot noise produces counting error; the number of photons arriving in the pixel is a random number sampled from a with mean I_{i,s,data} + T_{i,s,data}. In the second stage, the charge produced by detected photons is amplified by each pixel slightly differently. The amplifiers are fixed in the circuitry, however there is always error in their calibration (gain). Here, the photon gain for each pixel was assigned by randomly selecting numbers from a normal distribution with a mean of 28 ADUs per photon and standard deviation equivalent to 3% of the mean, or 0.84 ADUs. These perpixel gain values are typical for the CSPAD, and once selected they are fixed for all shots. Note that this type of calibration error assumes all pixels are the same physical size; pixeltopixel nonuniformity could be computed by adjusting the expected number of photons before computing the Poisson deviates above. Lastly, there is noise associated with detector readout due to electronic switching during the readout event itself and darkcharge accumulation during the exposure. CSPADs are dominated by noise, but here we make no distinction and lump all errors associated with readout into a single Gaussian process. We included readout noise of the detector by adding a random number to the final computed values, where that random number was drawn from a normal distribution with a mean of 0 and a standard deviation of 3 ADUs. We modeled darksubtracted data, where an average dark signal had already been subtracted from each shot, hence why the readout error fluctuates about 0. Perpixel readout and counting noise terms fluctuate across every pixel and every shot; however, pixel gain calibration errors, though different for each pixel, are constant for all shots. It is typical to divide the observed pixel values by a nominal gain value for the entire camera (28 in this case) so that a unit pixel increment is the same size as the signal from a single photon, hence we have for the pixel value
where the photon count N_{i,s} is randomly sampled from a with the I_{i,s,data} + T_{i,s,data}, the pixel gains g_{i} form a normal distribution , and the readout noise r_{i,s} is randomly sampled from a normal distribution . For a full description of the noise options available in nanoBragg, see the supplemental material from the work by Holton et al. (2014).
2.3. estimation using pixel data
Fig. 2 shows several examples of the Bragg reflections from the synthetic diffraction patterns, illustrating the variability of repeated Bragg reflection measurements performed at XFELs (see Fig. S1 of the supporting information for the same images in the absence of noise). This pershot variation in observations is what makes the analysis of XFEL still shots with conventional protocols subject to large uncertainties. By summing together neighboring Bragg spot pixels, one loses the intricate perpixel intensity variations that encode the incident photon spectra and crystal morphology. This information could otherwise be used to constrain complicated physical models. Rather than integrate Bragg spots we seek to use their pixelated profiles to optimize a model, thereby disentangling the various contributions to the scattering, and obtaining a more accurate measure of the amplitudes F_{h}.
2.3.1. target
The goal was to treat our synthetic images as an experimental dataset, and then use the pixel values X_{i,s} within all the Braggspot shoeboxes (Fig. 2) to optimize a global model to obtain increased accuracy. We accomplished this through iterative parameter estimation, using a target. Let p(X_{i,s}Θ) represent the probability of observing X_{i,s} given the full set of model parameters Θ needed to describe the observed pixel values. We will explicitly define Θ in Section 2.3.7. The likelihood of the entire dataset is given by the product of the individual pixel probabilities
provided we neglect interpixel effects such as crosstalk: a fair approximation for the sake of defining an optimization target. Other interpixel effects such as pointspread are discussed in Section 4, but are not necessarily applicable. The set of parameters Θ_{ML} that maximizes the likelihood of the data
is called the
estimate, or the most probable model, given the data.2.3.2. Probability of observing pixel values
In order to express the likelihood of the data given by equation (9), we must define p(X_{i,s}Θ), the probability of observing X_{i,s} given a set of model parameters Θ describing the scattering. In this case we know the precise model that was used to generate X_{i,s}, but the arguments below are general and applicable to real data. In what follows, we let R_{X} represent a random variable for a pixel measurement, where X_{i,s} is a sample from the distribution governing R_{X}. We assume R_{X} describes an observation after division by the nominal gain, and after subtraction of an average dark pedestal, as shown in equation (8). Using the algebra of random variables, we can define R_{X} as an expression involving three independent random variables:
Here R_{n} represents randomness in R_{g} represents randomness in signal amplification and R_{r} represents randomness in signal readout. The random variable R_{n} is governed by a whose mean μ_{n} represents the for the number of photons captured by the pixel during the diffraction event. We can simplify the statistics by approximating as a normal distribution with equivalent mean and variance: . This approximation breaks down when the number of scattered photons approaches 0; however, in this regime we expect the readout term to dominate R_{X}. The random variable R_{g} arises due to error in detector gain calibration; even though we divide through by the nominal gain value, we do not know the precise gain of each pixel. Therefore, we let be the distribution governing the gain calibration error. One can estimate σ_{g} by recording flat illumination on the detector, e.g. scattering from a copper foil far upstream of the detector, but the result is never perfect. The product of the two normally distributed random variables R_{n}R_{g} is in general not a normal random variable, however in certain limits we can make that approximation. Specifically, if we assume that R_{n} and R_{g} are independent random variables, then for large μ_{n}/σ_{n} and μ_{g}/σ_{g} we can approximate (SeijasMacías & Oliveira, 2012)
Note, we used the fact that μ_{g} = 1 and as shown above. The second approximation in equation (12), stating that is valid for small σ_{g}, yet becomes worse as μ_{n} increases. If , the approximation error is equivalent to μ_{n}. For σ_{g} = 0.03 (0.84/28, which we used for the synthetic data) this occurs when μ_{n} = 1.1 × 10 ^{3} photons, and for the work presented here most pixels received far fewer photons. The random variable R_{r} describes a random offset applied to each pixel measurement which is governed by the underlying electronics of the detector modules, and it follows a normal distribution . Usually during an experiment with a CSPAD, a dark measurement is recorded and subtracted from subsequent measurements. For a given exposure time this dark offset is generally stable, but there is always a random component that fluctuates on a shottoshot basis. This readout noise will result in a positive or negative offset applied to each pixel, and it is these offsets that are represented by R_{r}. The value σ_{r} is easy to estimate by closing the Xray shutter and observing the pixel values fluctuating in the absence of Xrays. We used a value of σ_{r} = 3/28 throughout the analysis in this paper, in line with equation (8). With the above definitions we now define the distribution f_{R}_{X} governing R_{X}. This distribution is the convolution of f_{RnRg} and f_{Rr} (true for the sum of any two random variables). In the case that both f_{Rn}_{Rg} and f_{Rr} are normal, then f_{RX} will also be normal: , where and * is a convolution operator. With this we can express the probability of observing X_{i,s} photons as
where we used the definition to be indexexplicit, and where the modeldependent variance is given by
For the remainder of this report, we use n_{i,s}(Θ) to represent the model for the expected number of photons in pixel i during shot s. Note this is different from N_{i,s} used in equation (8), which is a randomly drawn sample, given an n_{i,s}(Θ). It is noteworthy that the probability model in equation (13) allows the observed data X_{i,s} to be negative, something mathematically forbidden when modeling the observations using Poisson statistics alone. In other words, a photon count by itself can never be negative, only when coupled with additional terms such as the readout noise can a pixel report a negative value. Negative pixel values occur in regions of weak scattering where readout noise dominates. This can easily happen and is indeed expected after subtraction for invacuum lowbackground measurements at facilities like CXI (see Fig. 2).
2.3.3. Selecting pixels for estimation
In principle one can evaluate the likelihood shown in equation (9) for every pixel in the camera, but for the work shown here we only included a selection of pixels that were expected to be in the vicinity of Bragg scattering, referred to throughout this text as shoeboxes. Fig. 2 shows several such shoeboxes. Each synthetic CSPAD diffraction pattern is made up of 32 × 185 × 388 = 2 296 960 pixels, but by limiting the analysis to shoeboxes, we only used an average of 1.35 × 10^{5} pixels per image. This made estimation approximately 17 times faster than it would be if including all pixels. Interestingly, we found that overpredicting shoeboxes (in order to guarantee inclusion of all observed spots) did not hurt the This is a key distinction from integration methods, where overprediction can be problematic.
2.3.4. Modeling expected photons in each pixel
We modeled n_{i,s}(Θ) as a sum of Bragg scattering I_{i,s} and background scattering T_{i,s}:
We will proceed to define the background and Bragg scattering models, after which we will define the full list of ). Note the subscript `model' is used to distinguish these expressions from those in equations (3) and (7) describing the synthetic data.
parameters (summarized in Table 2

2.3.5. Bragg scattering model
We modeled the Bragg scattering similarly to that shown in equation (3):
Here is the total fluence across all photon energies in the XFEL pulse, and = is computed using the central wavelength of each XFEL pulse and a single mosaic domain at orientation U_{s}. The scale factor G_{s} relates primarily to the crystal size variation, but other factors can also affect the scale during real measurement. One can use equation (4) directly to model the full energy spectra; however, here we use equation (16) purely for computational efficiency and accept that it is slightly inaccurate (see Section S1 and Fig. S2 of the supporting information). Also, given the relatively small mosaic domain size used for the synthetic data, we assumed mosaicity would dominate the spot profile shapes, as opposed to Even though all crystals have the same mosaic parameter m and unitcell matrix B, we modeled each crystal as having a unique mosaic parameter m_{s} and unitcell matrix B_{s}.
2.3.6. Background scattering model: tilt planes
The measured background scattering arose from the solvent, and we did not model it using first principles. Instead, we fit a linear expression, or tilt plane, to the pixel measurements at the periphery of each Bragg spot shoebox (Rossmann, 1979), and the resulting tilt plane was used to evaluate the background intensity under the Bragg peaks. For fitting, we used weighted linear least squares. To obtain each tilt plane we solved the linear system
for the tilt plane coefficients t = [t_{1}, t_{2}, t_{3}], where x_{i}, y_{i} are the fastscan, slowscan coordinates of the N_{T} shoebox pixels selected for the tilt plane fit, and where X_{i} are the observed values of those pixels. Rewriting equation (17) as At = b, we can then write the solution as where W is a diagonal matrix whose diagonal entries are the reciprocals of crude variance estimates for each pixel value in b. Given a pixel value X_{i}, we approximated its variance (for tilt plane fitting purposes only) as , where σ_{r} is the readout noise standard deviation (3/28 = 0.11 in the synthetic data), and we used this information to compute a signaltonoise estimate for each Bragg reflection (see Leslie, 1999). Fig. 2 shows these signaltonoise estimates for several simulated reflections. Recalling equation (14), it becomes obvious that the estimate v_{i,crude} uses the approximation n_{i} ≃ X_{i}. In other words, v_{i,crude} approximates an with a single measurement, providing a suitable guess of the spot variance in the absence of a model. We emphasize that a unique background vector t was computed for each shoebox, that is, for each Bragg spot prediction on each shot. Ideally the pixels used in equation (17) do not include contributions from Bragg scattering. After solving for t, we modeled the background for pixels in the corresponding shoebox as
This linear fit exhibits no curvature and is therefore best applicable to local regions on the detector where the background signal is slowly varying. During the t_{1}, t_{2}, t_{3}] were initialized as the solutions to equation (17) and were then fixed. Though the least squares solution is analytical, it is dependent on the proper distinction between background pixels and Bragg pixels. For weakly scattering data, tilt planes can be evaluated at levels close to zero intensity, sometimes giving rise to negative n_{i,s}(Θ), especially for pixels at the shoebox periphery. This is a result of using a nonphysical background model, and it poses a risk to violate equation (13) which requires that v_{i,s}(Θ) remain positive. To guard against this occurrence, we filtered all shoeboxes whose tilt planes dipped below 0 for any pixel in the shoebox. This resulted in the removal of an average of 2.3 out of 580 shoeboxes per shot.
parameter optimization, the tilt plane coefficients [2.3.7. Unknown model parameters
We now explicitly define Θ, the list of all unknown model parameters that we determined via A parameter can be placed into one of two categories: local or global. A local parameter belongs to a particular XFEL diffraction shot. We let Γ_{s} represent the local parameters of shot s, namely the crystal orientation matrix U_{s}, the matrix B_{s}, the scale factor G_{s} and the mosaic domain parameter m_{s}:
In total each Γ_{s} represents seven parameters: three Euler angles describing crystal orientation, two unknown unitcell constants, a single scale factor and a single mosaic parameter. On the other hand, a global parameter is shared across all shots in the experiment. Global parameters in the model were the set of all structure factors, which we refer to here as {F_{h}}. The full list of parameters for N_{s} diffraction patterns was
For the
we imposed tetragonal symmetry, leading to 13 704 unique amplitudes out to 2.1 Å.2.3.8. Optimization using diffBragg
Solving for Θ_{ML} given by equation (10) was carried out using the quasiNewton optimization algorithm LBFGS (Liu & Nocedal, 1989) as implemented in CCTBX. LBFGS requires the first derivative of the likelihood expression f(Θ) in equation (9) with respect to each parameter of interest defined in equation (20). In practice it is typical to minimize the negative logarithm of the likelihood, both to maintain numerical accuracy when multiplying the large number of probabilities, and to make use of standard minimization algorithms. Therefore, we solved the equation
where
The first derivative of the loglikelihood for a parameter (needed for LBFGS) is then given by [recalling equation (14)]
We developed a new program alongside nanoBragg (dubbed diffBragg) for computing the derivatives of the expected scattering n_{i,s} with respect to each parameter. For example, when computing the derivative of the likelihood expression with respect to the structure factors, we used diffBragg to evaluate
which follows from equation (16). The results were then substituted into equation (23) to compute the gradients of the likelihood expression needed for optimization.
It is important during
that the target function be equally sensitive to all parameters. To this end we applied reparameterizations of the formwhere θ_{o} is the initial value of the parameter and σ_{θ} represents the expected variation of the parameter during (Hammersley, 2009). With this change of variables, all parameters started with an initial value of 1. If the target equation appeared exceptionally sensitive to certain parameters, the corresponding σ_{θ} values were incremented by factors of 10 until we observed the first several LBFGS iterations updating parameters by sensible amounts. In this specific problem we also applied bound restraints to certain parameters. This was accomplished by reparameterizations of the form
such that the parameter θ will always be greater than θ_{min} ≥ 0. For example, to ensure the amplitudes remained positive during we made the substitution . The parameters refined without restraints; however, the resulting remained positive quantities. A similar restraint was used on the scales G_{s} and mosaic parameters m_{s}. With these reparameterizations the updated derivatives of the target equation can be written as
where ∂θ/∂x is computed for each parameter according to the corresponding reparameterization scheme. See Appendix B for derivative expressions of the remaining model parameters. Optimization was carried out using the National Energy Research Scientific Computing Center (NERSC).
3. Results
3.1. Initialization of model parameters
Initial estimates of the orientation matrices U_{s} and the crystal unitcell matrices B_{s} were provided by the program dials.stills_process after successfully indexing each diffraction pattern (see Fig. 3) using the algorithm of Steller et al. (1997). The output from dials.stills_process also provided an estimate of the mosaic domain size for the measured crystals (Sauter et al., 2014). We used these estimates to construct an initial guess of 13.7 for the mosaic parameters m_{s}. The quantity 13.7 is the median mosaic domain size from dials.stills_process divided by the cubed root of the unitcell volume, (a^{2}c)^{1/3}. The initial estimates for F_{h} came from running the standard integrationbased XFEL merging application in CCTBX (see Table 3), without postrefinement (see Table S1 of the supporting information for a comparison with postrefinement). The scale factors G_{s} were each initially set to a very high number, in this case 10^{6}. This number was chosen to ensure that the model initially predicted finite Bragg scattering in most of the shoeboxes. The background tiltplanes for all shoeboxes were initialized by solving equation (17) for each reflection shoebox.

3.2. Parameter estimation carried out in stages
Once the parameters were initialized,
parameter optimization was carried out in two main stages.3.2.1. Stage 1 refinement
Here we refined shots one at a time in a series of two steps: first, for each shot, we refined the scale G_{s} and the mosaic parameter m_{s} while keeping all other parameters fixed. We did this using only the lowresolution shoeboxes (less than 5 Å) with signaltonoise ratio greater than 3. Second, using the optimized values for G_{s} and m_{s}, the matrix B_{s} and the crystal rotation matrix U_{s} were refined while keeping all other parameters fixed. This was done for all spots with signaltonoise ratio greater than 0.2 and resolution less than 2.1 Å. After this stage we identified images which refined poorly and removed them prior to the global stage 2 For the 2023 shot example, we removed 25 out of 2048 shots after stage 1 by examining the resulting distributions for all a_{s}, c_{s}, m_{s} and G_{s} (a_{s} and c_{s} are the unitcell constants that fall out of the optimization of each B_{s}). The results of stage 1 are shown in Fig. 3 for 2023 exposures. For reparameterization in this stage of we let σ_{Us} = 0.001°, σ_{Bs} = 0.1 Å, σ_{Gs} = 1 and σ_{ms} = 0.1 for all s, where σ_{Us} corresponds to the expected variation in the three angles defining the crystal rotation matrices U_{s} and σ_{Bs} corresponds to the expected variation in the unitcell edge parameters. Timing statistics for this stage of are shown in Table 4.

3.2.2. Stage 2 refinement
Here we refined the F_{h} and the scale factors G_{s} as part of a global over all images. All other parameters were kept fixed. At the start of this stage, we set all m_{s} equal to the median of the values obtained during stage 1 [see Fig. 3(c)]. This was done because the pershot scales G_{s} and the pershot mosaic domain parameters m_{s} are correlated (both directly increase the number of expected photons in shot s). Also, at the start of this stage, all scale factors G_{s} were set equal to the median of the results obtained for the scale factors during stage 1 [see Fig. 3(d)], but then they optimized to different values for each shot. Fig. 4 illustrates stage 2 results for the 2023 image set. This stage of utilized all shoeboxes with a signaltonoise ratio greater than 0.2 and resolution less than 2.1 Å. For reparameterization during this stage of we let σ_{Fh} = σ_{Gs} = 1 for all h and shots s.
amplitudes 3.2.3. Processing
Standard message passing interface (MPI) was used to accelerate the analysis, parallelizing over images; however, beyond that no attempt was made to optimize the runtime. Timing tests for stage 1 and stage 2 ^{6} Bragg spot shoeboxes for a total of 2.7 × 10^{8} pixels. For 2023 shots, stage 1 (including input/output overhead) was completed in 12 min, utilizing all 40 hardware threads. Stage 2 ran at a rate of 23.5 s per LBFGS iteration using all 40 hardware threads (40 MPI ranks), though this number can be decreased by utilizing multiple compute nodes simultaneously. Future work will also utilize GPUacceleration. The nanoBragg program used to compute the pixel values in equation (3) includes a GPU kernel which for our specific usen case currently offers a 528fold speedup over the CPU code, so we expect similar speedups for the minimization. Thus far, however, the methods used by diffBragg to compute the loglikelihood and its derivatives have only been written for CPUs.
were conducted on a single compute node at NERSC comprising two 2.4 GHz Intel Xeon Gold processors for a total of 40 hardware threads. The complete set of data for all 2023 shots comprises 1.17 × 103.3. Comparison of optimized parameters with ground truth values
We judged the success of the diffBragg estimation by comparing refined parameters with the ground truth parameters used to synthesize the data. In this section we discuss three metrics important for judging the success of the namely the pershot lattice misorientation with respect to the ground truth, the ground truth R factor and the anomalous difference correlation with the ground truth.
3.3.1. Misorientation ΔU_{s}
A useful metric to observe during optimization is the misorientation ΔU_{s} of the optimized crystal rotation matrices U_{s} from the known crystal rotation matrices used to synthesize each shot. Fig. 3(e) shows ΔU_{s} before and after stage 1 For 2023 exposures, starting with a median ΔU_{s} of 0.038° (as given by the DIALS indexing results), we were able to refine the crystal orientations such that the median ΔU_{s} was 0.0028° [Fig. 3(e)].
3.3.2. R factor with ground truth
During optimization we monitored the R factor between the refined amplitudes and the ground truth (GT) amplitudes:
Here, k is a scale factor chosen to minimize R_{GT}. Fig. 4 shows the evolution of R_{GT} throughout stage 2 for different resolution bins. At each point in we computed a new k using the Adaptive Nelder–Mead Simplex method (Gao & Han, 2012) as implemented in the SciPy software for Python.
3.3.3. Anomalous difference correlation with ground truth
Because we targeted Yb^{3+} bound to lysozyme, we expected a strong anomalous component to be present. To this end we observed the correlation of anomalous difference amplitudes with those from the ground truth . Specifically, we computed the Pearson CC_{ano}^{*} between and . The correlation CC_{ano}^{*} is discussed in great detail in the work by Terwilliger et al. (2016) where it was shown to be directly proportional to the peak height at sites of the absorptive heavy atoms in an anomalous difference density map, making it a good indicator of one's ability to solve a SAD dataset. Note, it is common practice to report CC_{ano} when discussing real data where one cannot know the ground truth model. This is accomplished using an empirical relationship outlined in the work by Terwilliger et al. (2016). Here, however, we are explicitly computing CC_{ano}^{*} according to a ground truth model, hence the use of an asterisk in the defining symbol. Fig. 5 shows CC_{ano}^{*} versus resolution for both the integration method and the method, where it is obvious that 2023 shots using is comparable to 19 953 shots using integration. Overall values of R_{GT} and CC_{ano}^{*} for various trials are shown in Table 5, for both the integration method and the method. With only 2023 exposures we achieved overall values of R_{GT} and CC_{ano}^{*} equal to 4.9% and 79%, respectively. Using the integration approach on the same images we got values of 11.0% and 48.4%, respectively. It is noteworthy that additional cycles of stage 1 and stage 2 provided little improvement beyond the initial cycle (Fig. S3).

4. Discussion
The ability to accurately determine protein F_{h} in the presence of large experimental uncertainties largely governs the success of an SFX experiment. The program presented here, diffBragg, provides a direct protocol for decoupling the various contributions to the scattering which would otherwise obscure the amplitudes. These noisy contributions include variable pershot incident photon spectra, crystal morphology and Ewald offset (partiality), all of which can be detrimental in situations involving weak signals, such as anomalous difference amplitudes used for the spatial resolution of heavy atoms (Sauter et al., 2020), or for experimental phasing (Schlichting, 2017). With 2023 shots we achieved similar quality anomalous differences (revealed by CC_{ano}^{*}) to those obtained by the conventional processing of 19 953 shots. Our approach eliminates the twostep process of measuring Bragg spot intensities on individual images, followed by merging. Rather, we refine the structure factors themselves against the raw data, in a restrained manner along with all other model parameters, to arrive at a stable solution. Also, the pixelbased approach provides more terms (one for each pixel) with which to restrain the optimization of the amplitudes, with the based optimization implicitly incorporating an errorbased weighting scheme derived from the physical interpretation of signal measurement.
amplitudes On a single compute node, shows how quality improves with each iteration). On the same compute node, it took 157 s to index and integrate 2023 images with conventional methods, which were used to initialize the optimization. Therefore, the strategy going forward is to reduce this 75× wallclock disparity by applying GPU acceleration to the step. We note that several beamline facilities, including LCLS, offer GPUaccelerated servers, with which it may be possible to run conventional integration and diffBragg in parallel with data collection, to better gauge experimental progress. We also plan to incorporate diffBragg as part of a broader effort centered on enabling leadershipclass computing for rapid analysis of XFEL data. New compute facilities such as the preexascale system at NERSC (Perlmutter) and exascale systems at Argonne National Laboratory and Oak Ridge National Laboratory (Aurora and Frontier, respectively) will provide data processing at speeds to match the increased data collection rates expected for nextgeneration XFEL facilities (LCLSII). The need for this level of computing (which includes GPU acceleration) becomes apparent by recalling equation (16), where we replaced the polychromatic spectrum of each shot with a simplified version. By instead including, for example, 100 energy channels for polychromatic model the 2023 shot would take approximately 100 times longer, or 2350 s per iteration on a single 40core CPU compute node.
of 2023 shots took approximately 208 min, including 12 min for pershot (stage 1) and 196 min for 500 iterations of global (stage 2) running at 23.5 s per iteration (Fig. 4We clarify that both stage 1 and stage 2 modeling are necessary to maximize the information extracted from the data. To illustrate this, we used the results from stage 1 CC_{ano}^{*} beyond that obtained with uncorrected integrated intensities; however, stage 2 consistently yielded the best results (see Section S2 and Table S2). For example, using 2023 shots we were able to boost CC_{ano}^{*} to 0.57: an improvement over uncorrected integration, yet still worse than the value of 0.79 obtained with diffBragg.
to compute modelderived partiality corrections for each integrated spot. By correcting each integrated spot intensity using these new terms, we were able to increaseIt is common to model the aggregate effect of energybandwidth and mosaicity in a single Gaussian equation (Parkhurst, 2020, Chapter 5), yet diffBragg is a general framework. Notably, the work shown is directly applicable to twocolor serial diffraction (Hara et al., 2013; Lu et al., 2018; Lutman et al., 2014) and pinkbeam serial diffraction (Dejoie et al., 2013; Milne et al., 2017; MartinGarcia et al., 2019) where indexing, and data reduction protocols are in early development (Gevorkov et al., 2020). The work presented here assumed a perfect detector geometry; however, we demonstrated the robustness of diffBragg to typical levels of detector panel displacement (see Section S3, Figs. S4 and S5, and Table S3). We also assumed a detector with minimal pixel crosstalk and a well calibrated linear response, attributes that are realized in currentgeneration detectors like the ePix (Sikorski et al., 2016), JUNGFRAU (Leonarski et al., 2018) and AGIPD (Allahgholi et al., 2015). On the other hand, significant pointspread occurs in detectors such as the widely used Rayonix (Holton et al., 2012; Ke et al., 2018). Using results derived in the work by Holton et al. (2012), we applied a pointspread function to the synthesized data and observed that optimization could still proceed, provided the pointspread kernel was also applied to the model n_{i,s}(Θ) (see Section S4, and Figs. S6 and S7). Sources of error we have not described include measurement parallax, perimage Debye–Waller factors, intracrystal unitcell variation and multiple lattice scattering, all of which contribute to error in the determined structure factors. Thus far we have neglected any mention of errors, but we know proper error estimation can aid in solving real systems (Brewster, Bhowmick et al., 2019). In order to obtain error estimates for the amplitudes in the current context, one can consider the second derivative of the loglikelihood expression evaluated at the estimate:
Here, is called the Fisher information matrix, is the loglikelihood defined in equation (22), and u and v are row and column indices. The variances of the parameters Θ_{ML}, including the structure factors {F_{h}}, are given by the diagonal elements of (Pawitan, 2001). The matrix is large, and inversion should be performed using sparse matrix algebra. Our future efforts will involve implementing this computation.
In summary, we used the program nanoBragg to generate realistic XFEL diffraction images, which we analyzed using both the standard integration protocol and the new program, diffBragg. With diffBragg, we utilized all measured pixels simultaneously to estimate highaccuracy structure factors while requiring an order of magnitude fewer diffraction events compared with the conventional method. Reducing the number of required shots for a dataset can greatly benefit the general SFX experiment, with scarce beam time routinely plagued by unforeseen interruptions, limiting the amount of data one can realistically collect. Future work will aim to apply this method to real measurements and develop a userfriendly application for the general SFX researcher.
6. Software availability
The tools used for diffBragg are included in CCTBX. Scripts for computing and processing the synthetic data are available at https://github.com/dermen/cxid9114.
APPENDIX A
The interference term I_{0} for a parallelepiped mosaic block consisting of N_{a} × N_{b} × N_{c} unit cells stacked along the crystal axes a, b, c is given in the work by James (1962),
where the subscript f is used to indicate a fractional evaluated at a pixel. I_{0} exhibits principal maxima at integers h, k, l and subsidiary maxima along lines between h, k, l grid points as observed in the work by Chapman et al. (2011). As I_{0} is the modulus squared of the Fourier transform of all unitcell origins in the mosaic block, each principal maximum is equal to the squared number of unit cells (N_{a}N_{b}N_{c})^{2}. One can verify this claim by applying L'Hôpital's rule to find the limit of equation (30) as (h_{f}, k_{f}, l_{f}) tends to (0, 0, 0). We wish to model the principal peaks in I_{0} using Gaussians, but for that we also have to analyze the full width at halfmaximum, W, of the principal peaks. W can be found numerically by solving the transcendental equation
for x = x_{fwhm} along the interval −0.5 < x < 0.5, such that W = 2x_{fwhm}. Numerically solving equation (31) for a range of N (5 ≤ N ≤ 100) reveals an inverse N dependence on W, as expected given the physical interpretation of I_{0} (larger crystals produce smaller peaks):
With this we can write the standard deviation of an appropriate Gaussian form as
hence, we can approximate
where Δh, Δk, Δl represent the distance from each value h_{f}, k_{f}, l_{f} to its nearest integer value h, k, l (e.g. Δh = h_{f} − h). Substituting equation (33) into (34) we arrive at
Letting N_{a} = N_{b} = N_{c} = m brings us to the form shown in equation (4), with C = 1/0.286 (though we actually used C = 2/0.63 during all of this work as that was the default parameter in nanoBragg). The precise value of the constant in equation (35) can change for different lattices, and one can potentially refine it along with the mosaic parameter m for each lattice in order to obtain a better fit to the data. The Gaussian approximation is useful computationally as it lends itself to simpler derivatives, while also retaining the main properties of the analytical expression in equation (30).
APPENDIX B
Scale factor derivative. We define its derivative as
The scale factor G_{s} should always be a positive quantity, therefore to each G_{s} we apply a reparameterization of the form , such that (after applying the chain rule). This ensures the scale never becomes negative during refinement.
U_{s}matrix derivative. In practice, rather than minimizing the absolute misorientation angles in the matrix U_{s}(ϕ_{x}, ϕ_{y}, ϕ_{z}), we instead define three parameters representing angles about the standard laboratoryframe basis vectors, e.g. for the laboratory (lab) x axis we define the rotation operator
We define similar matrices for the laboratory y and z axes. The matrix U_{s} is then redefined as U_{s} = R_{x}R_{y}R_{z}U_{s,o} where U_{s,o} is the initial unitary matrix determined by the indexing protocol and then held fixed during We refine the parameters , after initializing them all to 0. The Ewald offset is now redefined as , where h = (h, k, l) is the integer and the derivative of the expected scattered photons with respect to the angular offset parameter is given by
where
and C is the parameter describing the lattice transform (see Appendix A). Similar expressions follow for and . No restraints were applied to the rotation angles during hence the reparameterizations used were of the form (1/σ_{Δϕ})Δϕ + 1.
Bmatrix derivative. The derivative of the mean scattered photons with respect to the unitcell matrix is similar to that derived for U_{s}. We show here the case for the a edge:
where
A similar result follows for the c edge. Reparameterization could be applied in order to keep a and c within a valid range; however, we never encountered a situation where a or c diverged. We therefore applied reparameterizations of the form x_{a} = (1/σ_{a})(a − a_{o}) + 1.
Derivative of mosaic parameter m. The derivative of the mean scattered photons with respect to m is given by
For this parameter we performed a reparameterization to ensure that m was always greater than 3, hence we use the reparameterization x_{m} = (1/σ_{m})[ln(m − 3) − ln(m_{o} − 3)] + 1.
Supporting information
Supporting figures and tables detailing the effects of mosaic spread and geometry errors on the https://doi.org/10.1107/S2052252520013007/zf5012sup1.pdf
DOI:Footnotes
^{1}Recently, `whole pattern' methods have demonstrated the fitting of entire serial diffraction datasets to a global threedimensional intensity model (Dilanian et al., 2016; Lan et al., 2018), avoiding the integration step completely. Lan et al. (2018) employed the expand–maximize–compress (EMC) algorithm (Loh & Elser, 2009), derived for singleparticle imaging, to process serial millisecond crystallography data. By employing the approach, they extracted intensities from weak signals in the context of a stable synchrotron source.
Acknowledgements
We thank Peter Zwart, Jeff Donatelli, Billy Poon, Dorothee Liebschner and Pavel Afonine for useful discussions. Giles Mullen wrote the GPU version of nanoBragg used for simulating data. The synthetic experiment in this paper was modeled after LCLS proposal LD91 conducted in 2014 in collaboration with Stanford faculty Soichi Wakatsuki, Axel Brunger and William Weis, and future work will apply this technique to that data.
Funding information
Research was supported by the National Institutes of Health (grants GM117126 to NKS; GM124149, GM124169, GM103393 and AI150476 to JH; GM110501 to JY; and GM126289 to JK) and the Exascale Computing Project (grant 17SC20SC to NKS), a collaborative effort of the Department of Energy (DOE) Office of Science and the National Nuclear Security Administration. JK and JY were supported by the Director, Office of Science, Office of Basic Energy Sciences (OBES), Division of Chemical Sciences, Geosciences, and Biosciences (CSGB) of the DOE; JH was supported by the Integrated
Technologies program of the DOE Office of Biological and Environmental Research (OBER); and data processing was performed at the National Energy Research Scientific Computing Center, supported by the DOE Office of Science; all three under DOE contract DEAC0205CH11231. This work made use of the GPU nodes allocated for the NERSC Exascale Science Applications Program (NESAP).References
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