1. Autocovariance Structures for Radial Averages in Small Angle X-Ray Scattering Experiments Andreea Erciulescu, Jay Breidt, Mark van der Woerd Departments of Statistics and Biochemistry & Molecular Biology, Colorado State University, Fort Collins Supported in part by Award R01GM096192 from the Joint NSF/NIGMS Initiative to Support Research in the Area of Mathematical Biology. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute Of General Medical Sciences or the National Institutes of Health.

2. Outline Small Angle X-Ray Scattering = SAXS – One useful way to analyze molecular shape Brief introduction to SAXS – From experiments to images – From images to data – From data to analysis The problem of autocorrelation – Empirical analysis – Theoretical model Future directions

3. SAXS experiments and images Different concentrations Different detectors and exposure times

4. Data processing 2D image, centered & 1D Intensity profile Fourier Transform Particle shape reconstruction (biochemical and biological interest)

5. Importance of autocorrelation We need to know confidence intervals on intensity data and derived data interpretation, ultimately for molecular shape reconstructions Current interpretation methods assume Gaussian error model and independent pixels We apply correlation analysis to our two-dimensional images to test if individual pixels in the images are uncorrelated.

6. Remove mean to study autocorrelation ― = Consider the difference in replicate images to subtract the mean Problems: heteroskedasticity and residual mean structure Are local sample autocorrelations significantly different from zero?

7. Nearest neighbors show autocorrelation Locally-estimated autocorrelations appear small, globally homogeneous, and isotropic (not direction-dependent).

8. Now consider a range of experimental conditions Samples 1. Buffer (solvent) 2. Protein only (Glucose Isomerase) 3. DNA only 4. Both protein and DNA Instruments Two different instruments

9. Autocorrelation results are stable across experiments Asymptotic Bartlett Bounds: ± 1.96 : √m m=32 in our case => expect exceedances of about 5% under no correlation. The autocorrelation is about 0.3 at lag 1 and close to 0 after lag 1.

10. Autocorrelation results for diagonal neighbors Asymptotic Bartlett Bounds: ± 1.96 : √m m=32 in our case => expect exceedances of about 5% under no correlation. The autocorrelation is about 0.09 at lag 1 and close to 0 after lag 1.

11. Work in progress: theoretical model for autocorrelation Sample autocorrelations are consistent with “kernel convolution” model: where { } an iid random field with mean 0 and variance 1. This has a physical interpretation due to detector engineering

12. Future directions Investigate model properties Incorporate globally-estimated autocorrelations into statistical tests. Investigate various experimental conditions with new data sets. Ensure best data quality possible for molecular envelope reconstructions. Appropriately track uncertainty throughout the process.

13. Thank you!