Rigid body transformations, model

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] ... 

The rigid-body superposition method RigFit[133] in FlexS is based on some of the above Fourier transform principles. Other examples of the usefulness of Fourier transformation in virtual screening and modeling can be found in [134-138]. 

When we evaluate the Green-Kubo relations for the transport coefficients we solve the equations of motion for the molecules. They are often modelled as rigid bodies. Therefore we review some of basic definition of rigid body dynamics [10]. The centres of mass of the molecules evolve according to the ordinary Newtonian equations of motion. The motion in angular space is more complicated. Three independent coordinates a, =(a,a,-2, ,3), i = 1, 2,. . N where N is the number of molecules, are needed to describe the orientation of a rigid body. (Note that a, is not a vector because it does not transform like a vector when the coordinate system is rotated.) The rate of change of a, is... 

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