Accuracy and resolution in protein crystallography: a probabilistic approach

The purpose of this paper is to establish an explicit correlation between the values and probable deviations of the observed intensities and the potential accuracy of the structure eventually determined using those intensities. This problem involves several steps: the choice in real space of a reference stochastic model defining the ideal state of ignorance; the determination in reciprocal space of the probability density corresponding to that model and of the probability law corresponding to the experiment; the determination of the information; the transfer of the information from reciprocal to real space; the interpretation of the information in terms of structural resolution. A rigorous treatment of this problem is hindered by the inadequate knowledge of the statistical correlations between different reflections; more realistically, it is convenient to address a simpler problem, corresponding to an ideal experiment in which the intensities relevant to each reflection are measured independently of all the other reflections. In this case the mathematical problems can easily be solved and a parameter introduced - called projection information - whose value can be determined. This information is the sum of two terms, one associated with the modulus, the other with the phase of each reflection. By resorting to a mathematical model it is possible to make use of the projection information to determine the value of a parameter akin to structure resolution. The final result of this work is an operational definition of resolution, based upon the stochastic properties of the experimental observations. This resolution can be used to assess and compare the intrinsic quality of different structure analyses and of different stages of one structure determination, before the structure is solved.