Section 13.3.1. Introduction

A structure determination by the molecular-replacement method traditionally proceeds in two steps (Rossmann, 1972, 1990). The first step involves the determination of the orientation of the search model in the unknown crystal unit cell by the rotation functions (see Chapter 13.2 ). Once the orientation of the search model is known, translation functions are employed in the second step to determine the location of the model in the crystal unit cell. This essentially reduces a six-dimensional problem (three rotational and three translational degrees of freedom) to two three-dimensional problems, which are computationally more manageable. With the speed of modern computers, a strict division between the rotational and the translational components of a molecular-replacement structure solution may no longer be necessary (see Section 13.3.7).

Translation functions are normally formulated to achieve minimum or maximum values when the search molecule is at its correct position in the crystal unit cell. As with the rotation problem, the translation problem is solved as a search. The positional parameters of the model are varied in the unit cell, generally on a grid. Translation functions are evaluated at these search grid points in order to identify those that minimize or maximize the functions.

Most translation functions involve a comparison between the observed structure-factor amplitudes (or squared amplitudes) and those calculated based on the search model. The R factor and the correlation coefficient can be used as indicators for translation searches (Section 13.3.2). The correlation between the observed Patterson map and that which is calculated based on the search model is the foundation of another translation function (Section 13.3.3). If phase information is available from other sources, the correlation between the electron-density maps is the basis for the phased translation function (Section 13.3.4). The power of the translation functions can be enhanced in the presence of noncrystallographic symmetry (Section 13.3.8). Proper packing of the search model in the crystal unit cell is an essential component of a solution to the translation problem (Section 13.3.5).