Estimation of overall atomic displacement parameters from observed data

Department of Applied Biochemistry, Tokai University
â—‹Ayako T Nakata Ryuta Mizutani

Overall atomic displacement parameters represent average displacement of atoms over the whole crystal entities which contribute diffraction. The Wilson plot is known as the primary method for the estimation of the overall isotropic displacement parameter and absolute scaling from observed data. This plot utilizes the logarithmic fall-off property of observed intensity in the higher resolution. Results of statistical analysis indicate that the Wilson plot gives displacement parameter lower than model average, since high resolution data largely depend on atoms with small displacements.
It has been reported that the overall isotropic displacement parameter can be estimated from the Gaussian profile of the origin peak of the Patterson function. The Patterson origin peak represents a sum of self vectors of electron densities, and is computed from the entire observed data. This method, called Patterson scaling, principally gives displacement parameters deviated from model, since the Patterson function is a Fourier transform of a product of the atomic scattering factor and the Debye-Waller factor, resulting in deterioration of shape of the origin peak giving deviated displacement parameters.
Assuming the static atomic electron density has spherical symmetry in average, overall atomic displacement can be determined from the Patterson origin distortion from that calculated using the static models. The origin peak distortion is computed from a Patterson-like function
PE(r) = Int. {|E(h)T(h)|2 exp(-2pi i hr)} dv*,
where E(h) is the normalized structure factor and T(h) the Debye-Waller factor. Overall isotropic and anisotropic displacement parameters can be estimated from a plot of PE(r), giving the best agreement with structure model.