A combined isotropic and multiple s-shell anisotropic scaling method for multiple data sets

A method to improve the scaling of multiple intensity data sets is presented. A general scaling function K(x, s), which uses the direction cosine of the diffraction vector (x) and (sin θ)/λ (s) as scaling parameters, is developed by combining an isotropic scaling function, K(s) = A exp (Bs²), and a multiple s-shell anisotropic scaling function, K(x)s = (Σ Σ c_ijx_ij)_s. This combined scaling function can greatly reduce the systematic differences in intensities among multiple data sets measured independently. This scaling method for the multiple data sets consists of three steps. In the first step, the individual isotropic scaling functions, K(s), are determined by an indirect least-squares method. Then the weighted mean intensity, 〈I〉, is calculated by applying the K(s) to the individual data sets. In the second step, the data in each data set are divided equally into 20 thin shells of (sin θ)/λ (s). The anisotropic scaling functions, K(x)_s, of each s shell are determined by using the weighted mean intensity, 〈I〉, obtained in the first step as the target quantity in a least-squares minimization, i.e. Σ w_i{〈I〉_i − K(x)_s[K(s)I_i]}². In the final step, the new weighted mean intensity, 〈I〉, is calculated by applying the combined scaling function, K(x, s) = K(s)K(x)_s, to the individual data sets. The new multiple s-shell anisotropic scaling functions are determined using the new weighted mean intensity, 〈I〉, as the target quantity in another least-squares minimization. By repeating this procedure three to five times, the least-squares minimization will converge. The method was successfully used to scale and merge 27 sets of S-adenosylmethionine synthetase data into a single data set. It was also used to scale the isomorphous replacement data sets of the enzyme.