\subsection*{ Shape Constraints}

Since we assume all shapes are registered to a common coordinate system defined by the anchor fragment, it is possible to exactly specify regions of space within this coordinate system that a molecule should and should not occupy.  Following the nomenclature of VAMS \cite{VAMS}, we refer to these constraints as minimum and maximum shape constraints. These constraints combine to form an expressive and exacting specification for a desired target shape.
A \textit{minimum shape constraint} sets a strict lower bound on the volumetric shape of a target shape.  Every voxel within the minimum shape constraint must be contained within the target shape.  A minimum shape constraint can be used to require that a target shape has a specific binding mode and minimum bulkiness.
A \textit{maximum shape constraint} sets a strict upper bound on the volumetric shape of a target shape.  Every voxel of the target shape must be contained within the maximum shape constraint.  The maximum shape can be used to constrain the total volume of the target shape and prevent the target shape from overlapping undesirable areas, such as space filled by a receptor. 
Shape constraints are distinct from shape similarity. Unlike shape similarity, which produces a continuous ranking of similarity with respect to a query shape, shape constraints are binary filters: a shape either matches the constraints or does not.

The structure of a receptor-ligand complex provides a natural starting point for the development of shape constraints.  The inverse of the receptor shape provides a starting point for defining a maximum shape constraint.  The constraint can be expanded within the binding site as needed to incorporate tolerance for small steric clashes and receptor flexibility while the shape can be shrunk to avoid excessive exposure of the target ligand to solvent. 
For visualization purposes we display the \textit{inverse} of the maximum shape, which corresponds to the region of space the desired shape is excluded from, to more clearly visualize the minimum shape, which is, by definition, fully subsumed in the maximum shape.

Similarily, the ligand shape can be used to develop a minimum shape constraint.  However, constructing a minimum shape constraint directly from the ligand shape, even if the shape is shrunk or skeletonized, results in a highly specific shape query that may be limited in its ability to retrieve novel chemical scaffolds.
As an alternative to directly using the shape of a bound ligand, we derive \textit{interaction points} from the ligand-receptor complex. Interaction points are points at the center of clusters of ligand atoms that are interacting with the receptor.  Interacting ligand atoms are defined to be those no more than 6{\AA} away from a receptor atom, as measured between atom centers.  These atoms are then clustered into groups of three or more that are no more than 4{\AA} across.  The center of the cluster is an interaction point.  An example of interaction points identified from a ligand-receptor complex is shown with the corresponding minimum and maximum shape constraints in Figure~\ref{iptsshape}.  Interaction points provide a more general specification of the binding mode of a ligand that is less dependent on the ligand chemical scaffold than the full ligand shape and ignores non-interacting components of the ligand.