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Analysis of Self-Associating Proteins by Singular Value Decomposition of Solution Scattering Data

Tim E. Williamson ^*, Bruce A. Craig , Elena Kondrashkina , Chris Bailey-Kellogg  and Alan M. Friedman ^*

^* Department of Biological Sciences, Markey Center for Structural Biology, Purdue Cancer Center and Bindley Bioscience Center, Purdue University, West Lafayette, Indiana; Department of Statistics, Purdue University, West Lafayette, Indiana; BioCAT, Argonne National Lab, Argonne, Illinois; and Department of Computer Science, Dartmouth College, Hanover, New Hampshire

Correspondence: Address reprint requests to Bruce A. Craig, Dept. of Statistics, Purdue University, West Lafayette, IN 47907. E-mail: bacraig{at}purdue.edu; Chris Bailey-Kellogg, Dept. of Computer Science, Dartmouth College, Hanover, NH 03755. E-mail: cbk{at}cs.dartmouth.edu; or Alan M. Friedman, Dept. of Biological Sciences, Purdue University, West Lafayette, IN 47907. E-mail: afried{at}purdue.edu.

We describe a method by which a single experiment can reveal both association model (pathway and constants) and low-resolution structures of a self-associating system. Small-angle scattering data are collected from solutions at a range of concentrations. These scattering data curves are mass-weighted linear combinations of the scattering from each oligomer. Singular value decomposition of the data yields a set of basis vectors from which the scattering curve for each oligomer is reconstructed using coefficients that depend on the association model. A search identifies the association pathway and constants that provide the best agreement between reconstructed and observed data. Using simulated data with realistic noise, our method finds the correct pathway and association constants. Depending on the simulation parameters, reconstructed curves for each oligomer differ from the ideal by 0.05–0.99% in median absolute relative deviation. The reconstructed scattering curves are fundamental to further analysis, including interatomic distance distribution calculation and low-resolution ab initio shape reconstruction of each oligomer in solution. This method can be applied to x-ray or neutron scattering data from small angles to moderate (or higher) resolution. Data can be taken under physiological conditions, or particular conditions (e.g., temperature) can be varied to extract fundamental association parameters (H_ass, S_ass).