Information content and retrieval in solution scattering studies. I. Degrees of freedom and data reduction

As a consequence of several technological developments, the speed and sensitivity of solution X-ray (and neutron) scattering studies have recently risen by factors which may be as large as 10⁵; formal problems - like assessing the information content of the data and retrieving that information in structural terms - have thus become of immediate practical interest. The general problem of the information content of scattering experiments is discussed; its mathematical expression is derived, which depends on both the experimental data (observed values and estimated accuracy) and the a priori stochastic assumptions on the structure of the sample. The practical application of these notions to solution scattering studies involves several steps, three of which - choice of the degrees of freedom, data reduction and error analysis - are dealt with in this work. The first step is to specify the minimal number of independent parameters necessary and sufficient to describe the whole of the scattering properties of the system. Whenever the solute particles are of finite dimensions the entire scattering curve is defined by its values at a one-dimensional lattice; if, moreover, the asymptotic trend of the scattering curves is known, then the degrees of freedom are the ideal intensities at a finite number of points plus a small number of parameters describing the asymptotic trend. It is also possible to include among the degrees of freedom a few subsidiary parameters like the normalization factors. The next step is, starting from a composite set of data, to determine the most probable numerical value of each degree of freedom and to evaluate its range of uncertainty. This is discussed within the framework of variable-contrast studies, assuming that the invariant-volume hypothesis is fulfilled. An algorithm is formulated which treats all the experimental observations and determines simultaneously all the degrees of freedom and the error matrix. The algorithm also allows one to introduce additional linear constraints on the degrees of freedom. As an example, the algorithm is applied to solution X-ray scattering data recorded with a low-density serum lipoprotein. The determination of the maximal chord of the particle - an important parameter in the informational analysis - turns out to be rather tricky.