Rigid-body transformation

Diamond, R., On the multiple simultaneous superposition of molecular structures by rigid body transformations. Protein Sci, 1992. 1(10) p. 1279-87. 

As a structure-generating tool, nab provides three methods for building models. They are rigid-body transformations, metric matrix distance geometry, and molecular mechanics. The first two methods are good initial methods, but almost always create structures with some distortion that must be removed. On the other hand, molecular mechanics is a poor initial method but very good at refinement. Thus the three methods work well together. 

The problem of comparing two different models of a protein can be formalized as given two sets of points A = ui, az, , an) and B = (bi,bx,. .. b ) in three dimensional space and assuming that they have the same cardinality, i.e., n = m, and that the element a, corresponds to the element b, find the optimal rigid body transformation Gopt between the two sets that minimizes a given distance metric D over all possible rigid body transformation G, as in Eq. [1] ... 

A rigid body transformation is a transformation that does not produce changes in the size, shape, or topology of an object. Mathematically, it can be defined as a mapping G 3 that satisfies the properties ... 

Equation [3] states that distances are conserved, whereas Eq. [4] says that internal reflection is not allowed. Rotations and translations are two examples of rigid body transformation, and in fact, a general rigid body transformation can be expressed as a combination of a rotation R and a translation T. The transformation problem can then be restated as finding the optimal rotation R and optimal translation Tsuch that A — RB — T is a minimum. 

The two subsets A(P) and B(Q) define a correspondence, and p = IA(P)I = IB(Q)I is called the correspondence length. Once the optimal correspondence is defined, it is easy to find the optimal rotation and translation using the rigid-body transformation algorithm described earlier. The concept of optimal correspondence, however, requires more explanation. It is clear that p = l defines a trivial solution to the protein superposition problem Any point of A can be aligned with any point of B, with a cRMS of 0. In practice, we are interested in finding the largest possible value for p under the condition that A(P) and B Q) remain similar. ... 

Various measures of similarity between two sets of points exist. In the section on rigid-body transformation, the cRMS value, which measures... 

STRUCT AL starts with an arbitrary equivalence of atoms between the two proteins A and B. This equivalence defines a list of corresponding residues (represented by their Cd atoms) that are superimposed with the optimal rigid-body transformation. Once the two proteins are superimposed, the program computes a structure alignment matrix SA. SA(i,j) measures the similarity between residue i of protein A and residue j of protein B, based on a function... 

D Rigid Body Transformations A Comparison of Four Major Algorithms. 

Equation (2) is defined in terms of the inertial, local and deformed systems of coordinates, G, E, and D respectively. Transformation matrices are needed to express this equation strictly in terms of the global coordinate system. The rigid-body transformation matrix is. 

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