778 Acta Cryst. (1976). A32, 778 A Packing Function for Delimiting the Allowable Locations of Crystallized Macromolecules BY WAYNE A. HENDRICKSON AND KEITH B. WARD Laboratory for the Structure of Matter, Naval Research Laboratory, Washington, D.C. 20375, U.S.A. (Received 8 October 1975; accepted 24 January 1976) A function is proposed for evaluating the likelihood of packing arrangements of maeromolecules in crystals. It is based on the simple principle that the constituent molecules of a crystal should not inter- penetrate. This packing function has been successfully applied in the structure solution of a hemerythrin by using the molecular shape previously determined for myohemerythrin. An increasingly important method in protein crys- tallography is the use of molecular search techniques to solve crystal structures composed of molecules that are nearly isostructural with a known molecular structure. A typical procedure uses the rotation func- tion of Rossmann & Blow (1962) to determine the orientation of the known molecule in the unknown crystal and then finds the position of the properly oriented molecule by the translation function of Tollin (1966) or Crowther & Blow (1967). One criterion frequently invoked to establish the validity of a resulting proposed structure is the reasonableness of its packing arrangement. Indeed the modes of packing available to a given macromolecule of globular shape in the lattice of a particular crystal are usually quite limited. Thus an analysis of packing could also be used a priori to delimit the allowable locations of crys- tallized macromolecules. It is the purpose of this note to describe a simple, general packing function which proved to be crucial to the solution of the structure of Phascolopsis gouldii hemerythrin B (Ward, Hen- drickson & Klippenstein, 1975) when attempts with a conventional translation function had failed to yield the structure. In order to deal quantitatively with the crystal packing of macromolecules, it is first necessary to describe the shape of the molecules from which the crystal is supposed to be composed. This can be done by defining a molecular shape function by 1 if x is intramolecular M(x)= 0 if x is elsewhere. (1) Such a shape function can be determined by any of several different means. Among the possibilities are definition through an analytic functional form such as an ellipsoid, evaluation of the van der Waals envelope of an atomic model, or row-by-row delimita- tion of a single contiguous molecule from an electron- density map. A second prerequisite to an analysis of packing is a definition of the transformations required to place the known molecule into the unknown crystal structure. Each independent molecule of the crystal structure must be transformed from points given by x in the coordinate frame of the known molecular structure to points given by x' in the coordinate frame of the unknown crystal. This transformation can be accom- plished by the relation x'= Iqx + t. (2) Here t is a translation vector along the three axes of the unknown crystal and FI is the rotational trans- formation matrix which reorients the known molecule. The matrix Iq is in general a function of three orienta- tion angles and the parameters of the two coordinate frames. Rossmann & Blow (1962) give the elements of R in terms of typical variables and parameters. Once a molecule has been transformed by (2), it remains to generate the other molecules related to it by the crys- tallographic symmetry operations of the unknown crystal. This can be done by applying the transforma- tions, x'~= Aix' + di (3) where At is the rotation matrix and d, is the transla- tion vector which relate points in the ith molecule of the crystal to equivalent ones in the unique molecule. An evaluation of the molecular packing in a proposed crystal structure can be based on the simple principle that the constituent molecules of ~ crystal should not interpenetrate. Packing arrangements that minimize the intersection of the molecular spaces from the several molecules of the crystal which impinge on a given unit cell present the most probable crystal struc- tures. Maximization of tke union of molecular spaces is equivalent to minimization of the intersection. Thus the volume represented by the union of all intramo- lecular space in a proposed crystal structure can be taken as a measure of likelihood for that packing arrangement. With definitions of the molecular shape function and coordinate transformations already in hand, this idea can readily be expressed in functional form. For the case of a crystal with a single molecule of known structure in the asymmetric unit, a suitably normalized measure of packing likelihood is the packing function given by