Scaling and merging time-resolved pink-beam diffraction with variational inference

Abstract

Time-resolved x-ray crystallography (TR-X) at synchrotrons and free electron lasers is a promising technique for recording dynamics of molecules at atomic resolution. While experimental methods for TR-X have proliferated and matured, data analysis is often difficult. Extracting small, time-dependent changes in signal is frequently a bottleneck for practitioners. Recent work demonstrated this challenge can be addressed when merging redundant observations by a statistical technique known as variational inference (VI). However, the variational approach to time-resolved data analysis requires identification of successful hyperparameters in order to optimally extract signal. In this case study, we present a successful application of VI to time-resolved changes in an enzyme, DJ-1, upon mixing with a substrate molecule, methylglyoxal. We present a strategy to extract high signal-to-noise changes in electron density from these data. Furthermore, we conduct an ablation study, in which we systematically remove one hyperparameter at a time to demonstrate the impact of each hyperparameter choice on the success of our model. We expect this case study will serve as a practical example for how others may deploy VI in order to analyze their time-resolved diffraction data.

FIG. 1.

careless commands used in this work.

FIG. 2.

The origin of harmonic overlap in pink-beam crystallography depicted by the reciprocal lattice for a 2-dimensional crystal. The diffracting condition, shaded gray, is bounded by the three limiting spheres determined by the minimum and maximum wavelength of the spectrum and the resolution limit of the sample. Three reflections from the same central ray are on the diffracting condition. Their scattered beam wavevectors, depicted as arrows, have different magnitudes, inverse wavelengths, but are all parallel. The wavelengths and resolutions of these reflections are recorded in the table to the right. Because the scattered beams are parallel, they will arrive at the same location on the detector.

FIG. 3.

Results of a one-dimensional sweep of the multivariate prior correlation parameter. (a) The active site difference peak heights for various multivariate prior correlation values. There is a clear maximum at 0.990. (b) and (c) RSCCs for ligand alone and for the full model. (d) R_free is provided as an additional global measure of map to model agreement. (e) and (f) CC_Pred was calculated to assess overfitting using either the maximum-likelihood weighted Pearson or Spearman correlation coefficients. Careless produces a posterior distribution for each intensity observation. The mean and standard deviation of this distribution are recorded in the *_predictions_#.mtz files saved after model training. The means of these distributions is typically used to compute CC_Pred or the correlation between observed and predicted intensities. Here, we quantify the uncertainty in CC_Pred using the bootstrap method whereby we resample the predicted intensities recorded in the careless output with replacement 1000 times yielding 1000 estimates of CC_Pred per hyperparameter setting. These bootstrapped estimates are visualized as violin plots. The optimal hyperparameter setting has the highest value while having the smallest gap between test and train. We observe this clearly at 0.990 for $C{C}_{\mathit{\text{Pred}}}(\mathit{\text{Pearson}})$ and at 0.999 for $C{C}_{\mathit{\text{Pred}}}(\mathit{\text{Spearman}})$, indicating that the exact optimum likely lies between the two.

FIG. 4.

The results of the one-dimensional sweep of Student's t degrees of freedom, ν, on active site difference peak heights, RSCCs, and CC_Pred. (a)–(c) The active site difference peak heights and RSCCs have a slight maximum at 32. (d) R_free is provided as an additional global measure of map to model agreement. (e) $C{C}_{\mathit{\text{pred}}}(\mathit{\text{Pearson}})$ has the smallest gap between test and train at 32, which are the optimal results. (f) $C{C}_{\mathit{\text{pred}}}(\mathit{\text{Spearman}})$ is an alternative metric to assess overfitting. It has the best results at 32 with the highest overall value, albeit by a small margin, and the smallest gap between test and train. The distributions visualized in the violin plots were generated by the bootstrap method described in the caption of Fig. 3.

FIG. 5.

Careless infers the spectrum from wavelength metadata. (a) The spectrum of the x-ray beam at BioCARS measured by a channel-cut monochromator. (b) An example diffraction pattern from our DJ-1 dataset with the indicated spots circled and colored by the predicted peak wavelength. (c) A two-dimensional histogram of the scale value (systematic error) predicted by careless and the peak wavelength predicted by Precognition (Renz Research). The skewed distribution is similar to the BioCARS spectrum, indicating that careless can infer spectral information. (d) The equivalent two-dimensional histogram produced from the wavelength-normalization ablation study wherein careless did not have access to the peak wavelength of each reflection observation.