0.1 Representing proteins in silico and protein forward kinematics  (Page 5/7)

Forward kinematics

As stated earlier, a common operation when manipulating proteins in silico is to retrieve the Cartesian coordinates of each atom in the protein from our knowledge of its dihedral angles and rotations applied to them. For simplicity, assume we have an anchor atom and we are modeling the protein backbone only, that is, the protein consists of a serial linkage composed of consecutive backbone atoms, as shown in Figure 5.

A simple approach

The simplest way to represent a protein chain is to store the Cartesian (x,y,z) coordinates of each atom at all times. These coordinates are relative to some global coordinate frame which is unimportant, for example that in which the atomic positions were obtained by X-Ray crystallography and which are typically read from the PDB files. These coordinates can be changed if so desired. Common changes are to remove the center of mass (thus centering the protein at the global origin), subtract the position of the anchor atom (to center the protein at this atom), etc.

But it was discussed earlier that the "natural" degrees of freedom for kinematic manipulations are usually the dihedral angles alone. This means that algorithms that operate on dihedral angles to achieve their goals will normally require a way to modify the Cartesian coordinates when dihedral rotations are performed, to reflect the new atomic positions. This can be easily done with rotation matrices as follows.

When a rotation of θ degrees around bond i is performed, one can think of all atom positions starting at i+2 rotating around the axis defined by bond i, and all other atoms (from anchor to atom i+1 inclusive) remaining stationary. Thus, upon such a rotation, the Cartesian coordinates of the atoms after the bond need to be updated, and their new values are given by:

Where [x,y,z,1] is the position of a generic atom in homogeneous form, [x',y',z',1]is its position after the rotation (T is the transpose operator), and R(i,θ) is a 4x4 matrix that encodes a rotation of θ degrees around an axis coinciding with bond i that passes through atom $a_i$ , and is given in homogeneous form as:

In the above formula, T(x) is a translation by the vector x and $R_0$ (axis,θ) is a rotation around an axis that goes through the origin of the specified coordinate system. As can be seen, this rotation around an arbitrary point is realized by translating the point to the origin, rotating the target atom around the axis through the origin, and then translating it back (the composition of these 3 transformations yields a unique 4x4 homogeneous matrix that achieves the same effect). The axis of rotation can easily be computed from the positions of atoms i and i+1 and must have unit norm. To perform successive rotations about different bonds, this procedure can be repeated, updating the Cartesian coordinates for each rotation. Note that the convention used for matrix-vector multiplication is to multiply column vectors by matrices on the left , so the rightmost transformation gets applied first, and so on. This is the convention used in most of the literature, but the alternate convention is possible (multiplying row vectors with matrices on the right ; these matrices are the transpose of the column-vector convention).