deletions | additions       diff --git a/subsection_Shape_Indexing_and_Similarity__.tex b/subsection_Shape_Indexing_and_Similarity__.tex  index 80112be..218fa36 100644  --- a/subsection_Shape_Indexing_and_Similarity__.tex  +++ b/subsection_Shape_Indexing_and_Similarity__.tex 
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Shape constraints combined with shape indexing provide a rapid way to filter a virtual library. 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.  The similarity of a shape, $A$, with the included, $I$, minimum, $MIN$,  and excluded, $E$, maximum, $MAX$,  shape constraints is computed by combining the shape Tanimoto with the included constraint with the their  shape Tanimoto with the \textit{inverse} of the excluded constraint:  $$\delta(A,I,E) Tanimotos:  $$\delta(A,MIN,MAX)  = \delta(A,I) \delta(A,MIN)  + \delta(A,\overline{E}) \delta(A,MAX)  = \frac{A \cap I}{A \cup I} + \frac{A \cap \overline{E}}{A \cup \overline{E} }$$ The closer a shape is to meeting the included constraint, the larger the value of $\delta(A,I)$, while the more a shape violates the excluded constraint, the smaller the value of $\delta(A,\overline{E})$. The more an shape exceeds the included constraint, the more it is penalized by the $\delta(A,I)$ term, but, to the extent that its volume avoid the conflicting with the excluded shape constraint, it is rewarded by the $\delta(A,\overline{E})$ term.