UltraScan Version

Manual

SOMO HPLC-SAXS Module Gaussian analysis theory

Last updated: December 2017

NOTICE: the Gaussian decomposition method is being developed by E. Brookes, J. Perez, P. Vachette, and M. Rocco.
Portions of this help file are taken from the Supplementary Materials of Brookes et al., "Fibrinogen species as resolved by HPLC-SAXS data processing within the UltraScan SOlution MOdeler (US-SOMO) enhanced SAS module", J. Appl. Cryst. 46:1823-1833 (2013), and from Brookes et al. "US-SOMO HPLC-SAXS Module: Dealing with Capillary Fouling, and Extraction of Pure Component Patterns from Poorly Resolved SEC-SAXS Data", J. Appl. Cryst. 49:1827-1841 (2016).

Given the matrix I containing columns i of I(q) and rows j of I(t), the principles of Gaussian analysis can be schematized as follows.

Single curve fitting:
Pick a row i of I and define a set of p Gaussians Gaussian definition, with amplitudes Gaussian amplitudes, centers Gaussian centers, and widths Gaussian widths. Then:

Gaussian eq1

In the US-SOMO HPLC-SAXS module, we let the user visually place the centers Gaussian centers, and subsequently provide several methods for fitting (see below) by minimizing over, in general, 3p variables, Gaussian amplitudes, Gaussian centers, and Gaussian widths:

Gaussian eq2

or in the case that Gaussian widths (i.e., the ith row of the matrix S containing the data-associated SDs has no zero elements):

Gaussian eq3

In the program, there are options to fix a combination of individual Gaussian curves k, amplitudes a, centres b, and widths c, which would result in fewer than 3p variables during the minimization. Constraints, in percentage from previous value or from the initial value, are also available for a, b, and c.

Global Gaussians:
In the US-SOMO program, entering the Global Gaussian mode does a fit of the preset single curve Gaussian against every curve i = 1, ..., m, keeping the centers b and widths c fixed. This provides an initialization of the amplitudes a for all curves as a starting point for global fitting or for refinement/extension to other datasets a previous global fitting on a subset of data.

Global fitting:
Given a Gaussian for a specific row i = l from the result of a single curve fitting, one can globally fit over the amplitudes Gaussian amplitudes by utilizing common centers, Gaussian common centers for = {1, ...,m; i ≠ l}, and common widths, Gaussian common widths for = {1, ...,m; i ≠ l}, and then doing a global minimization over the pm + 2p variables Gaussian amplitudes, Gaussian centers, Gaussian widths, as above. Global fitting is currently only available with a Levenberg-Marquardt minimization routine. As in the single Gaussian fitting, there are options to fix a combination of individual Gaussian curves k, amplitudes a, centres b, and widths c, which would result in fewer variables during the minimization. Constraints, in percentage from previous value or from the initial value, are also available for a, b, and c.

www contact: Emre Brookes

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Last modified on December 13, 2017.