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Molecular replacement (MR) is a well-established computational method for phasing in macromolecular crystallography. In MR searches, spaces of motions are explored for determining the appropriate placement of rigid single-body (or articulated multi-rigid-body) models of macromolecules. By determining a priori which portions of motion space correspond to non-physical packing arrangements with symmetry mates in collision, it becomes feasible to construct more efficient MR techniques which avoid searching in these non-realizable regions of motion space. This paper investigates which portion of the motion space is physically realizable, given that packing of protein molecules in a crystal are subject to the constraint that they cannot interpenetrate, and gives explicit expressions for the volume of the non-realizable regions for crystals in two-dimensions.
We study the conditional distribution of zeros of a Gaussian system of random polynomials (and more generally, holomorphic sections), given that the polynomials or sections vanish at a point p (or a fixed finite set of points). The conditional distribution is analogous to the pair correlation function of zeros but we show that it has quite a different small distance behaviour. In particular, the conditional distribution does not exhibit repulsion of zeros in dimension 1. To prove this, we give universal scaling asymptotics for around p. The key tool is the conditional Szegő kernel and its scaling asymptotics.
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