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Two protein domain structures can be aligned by superposing their 3D coordinates, with the aim of minimizing the RMSD of the superposition.
Many algorithms have been developed to optimize this complex task.
The task is complex because of the large number of degrees of freedom between two datasets of points in 3D (each molecule has 6 degrees of freedom).
Algorithms
Some approaches use quaternerions to reduce the dimensionality of the space without removing any information (isomorphic translation).
A major class of approaches is based on reducing the representation of the complex protein molecules into its secondary structure.
Secondary structucture elements can be approximatly represented as vectors and aligned in pairs.
Dynamic programming can be used to find local regions of similarity which are then expanded.
The Dali algorithm (named after Salvador Dali) uses netwrok isomorphism between the contact networks of two proteins to perform alignment.