An efficient molecular-replacement translation function based on the evaluation of direct-methods phase invariants

Traditional molecular-replacement translation functions are based on direct- or reciprocal-space correlations between the observed diffraction amplitudes and the calculated amplitudes and phases of the symmetry-related molecular transforms of the search fragment as a function of the displacement vector. An alternative method that has been described is based on evaluating a list of phase invariants as a function of the position of the search model in the unit cell and seeking those regions which satisfy the expectation value of these invariants as predicted by probability theory. As originally formulated, this procedure required the iterative computation of the phases and the evaluation of the list of invariants as the search model was stepped over the grid points defining the asymmetric portion of the unit cell. A new computational procedure is described whereby the values of the invariants are expressed solely as a function of the displacement vector r as a Fourier series that can be evaluated by a standard fast Fourier transform (FFT) without having to compute and insert the values of the phases based on the search model at each grid point.