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Application of Singular Value Decomposition to the Analysis of Time-Resolved Macromolecular X-Ray Data

Marius Schmidt^*,, Sudarshan Rajagopal, Zhong Ren^,, and Keith Moffat^,

^* Physik-Department E17, Technische Universitaet Muenchen, 85747 Garching, Germany; Department of Biochemistry and Molecular Biology, University of Chicago, Chicago, Illinois 60637 USA; BioCARS, Argonne National Laboratory, Argonne, Illinois 60439 USA; and Renz Research, Des Plaines, Illinois 60018 USA

Correspondence: Address reprint requests to Marius Schmidt, E-mail: marius{at}hexa.e17.physik.tu-muenchen.de.

Singular value decomposition (SVD) is a technique commonly used in the analysis of spectroscopic data that both acts as a noise filter and reduces the dimensionality of subsequent least-squares fits. To establish the applicability of SVD to crystallographic data, we applied SVD to calculated difference Fourier maps simulating those to be obtained in a time-resolved crystallographic study of photoactive yellow protein. The atomic structures of one dark state and three intermediates were used in qualitatively different kinetic mechanisms to generate time-dependent difference maps at specific time points. Random noise of varying levels in the difference structure factor amplitudes, different extents of reaction initiation, and different numbers of time points were all employed to simulate a range of realistic experimental conditions. Our results show that SVD allows for an unbiased differentiation between signal and noise; a small subset of singular values and vectors represents the signal well, reducing the random noise in the data. Due to this, phase information of the difference structure factors can be obtained. After identifying and fitting a kinetic mechanism, the time-independent structures of the intermediates could be recovered. This demonstrates that SVD will be a powerful tool in the analysis of experimental time-resolved crystallographic data.

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