# Momenta compensation

In the original implementation of torsion angle dynamics in the program DYANA (Güntert et al., 1997), the N-terminus of the protein was fixed in space. This was not a prerequisite of the torsion angle dynamics algorithm of (Jain et al., 1993), but it appeared conceptually more appealing since the system then had only one type of degrees of freedom, the torsion angles, whereas for the description of the global position and orientation of a molecule that can freely reorient in space, one has to introduce six additional degrees of freedom of a different type. As a consequence of fixing the N-terminus of the polypeptide chain, the total angular and linear momenta of the system cannot be conserved, since the conservation laws for these quantities result from the fact that a mechanical system is invariant under global rotation and translation (Arnold, 1978). Therefore, if a part of the molecule is fixed in space, it is in general not possible for the total angular momentum to be zero throughout the calculation. Global rotations associated with a non-vanishing total angular momentum may lead to centrifugal forces that can influence the sampling of conformation space for long, flexible molecules (Güntert & Wüthrich, 2001). Therefore, CYANA allows the molecule to reorient freely in space, and the total angular and linear momenta of the system are conserved and periodically reset to zero.

Fig. 4. Structures and Ramachandran plots of unconstrained Ala60 polypeptide chains. (a) Generated by randomizing the torsion angles. (b) Calculated using the standard CYANA torsion angle dynamics with momentum compensation. (c) Calculated using torsion angle dynamics in CYANA without momentum compensation. Each structure bundle consists of 50 conformers that were superimposed for minimal RMSD (Güntert & Wüthrich, 2001).

## Global orientation of the molecule

The position and orientation of the molecule in the inertial frame are specified, respectively, by the three Cartesian coordinates of the reference point of the base cluster, r0, and by 4 quaternion parameters q = (q0, q1, q2, q3), which are subject to the normalization condition . The rotation matrix that describes the orientation of the base cluster with respect to the inertial frame is given in terms of the quaternion parameters by

.

The following two relations hold between the angular velocity of the base cluster, 0, and the time-derivative of the quaternion parameters, (Kneller & Hinsen, 1994):

and

, [7]

where the superscript T denotes the transpose, and A(q) is given by

.

## Global reorierntation in torsion angle dynamics

The additional degrees of freedom that enable free global reorientation of the molecule are incorporated in the CYANA torsion angle dynamics algorithm by extending the loops over the clusters so as to include the base cluster (cluster 0). Instead of using 00 and v00, the angular and linear velocities of the base cluster are now given by and , respectively, and the value of 0 in the recursive calculation of the torsional accelerations is the solution of the six-dimensional linear system of equations P00z0, in which P0 and z0 are calculated as described above. The second time derivatives of the quaternion parameters are obtained by differentiation of the equation [7] with respect to time. This implementation of the algorithm preserves the total linear momentum and the total angular momentum of the system.

## Compensation of angular and linear momenta

The total angular momentum of the molecule with respect to the origin of the inertial frame is

.

Similarly, the total linear momentum of the molecule is

.

A change in the angular and linear velocity of the base cluster, 00 + 0 and v0v0 + v0, leads to concomitant changes in the velocities of the other clusters to the extent of k0 and vkv00 (rk – r0). Hence, the changes induced in the total angular and linear momenta of the molecule are

, [8]

(with for all three-dimensional vectors x) and

, [9]

where is the total mass of the molecule, and

the position of its center of mass. The angular and linear momentum changes are linear functions of the base cluster velocity changes:

, [10]

where B is a 66 matrix with elements given by equations [8–9]. Solving equation [10] for the case LL and PP yields the values 0 and v0 by which the velocities of the base cluster have to be changed in order to set the angular and linear momenta of the molecule from given values of L and P to zero. Using this procedure, the angular and linear momenta are reset to zero after each sequence of 10 torsion angle dynamics steps. The additional computational effort for the momentum compensation is minimal and scales linearly with the size of the molecule.