deletions | additions       diff --git a/subsection_Shape_Indexing_and_Similarity__.tex b/subsection_Shape_Indexing_and_Similarity__.tex  index 2e03a6b..4f6084f 100644  --- a/subsection_Shape_Indexing_and_Similarity__.tex  +++ b/subsection_Shape_Indexing_and_Similarity__.tex 
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Alternatively, instead of serving as hard constraints, they can also be used to rank molecular shapes by similarity to the shape constraint query. We use the shape Tanimoto\cite{RushIII2005} to compute the similarity of two shapes:  $$\delta(A,B) = \frac{A \cap B}{A \cup B}$$  where a larger score indicates a greater degree of similarity.  We consider both similarities with a single query ligand and similarities with shape constraints. The similarity of a shape, $A$, with the minimum, $MIN$, and maximum, $MAX$, shape constraints is computed by adding averaging  their shape Tanimotos: $$\delta(A,MIN,MAX) = \delta(A,MIN) \frac{\delta(A,MIN)  + \delta(A,MAX) \delta(A,MAX)}{2}  = \frac{A \frac{1}{2}(\frac{A  \cap MIN}{A \cup MIN} + \frac{A \cap MAX}{A \cup MAX }$$ })$$