Rotations

orientation: orientation from reference coordinates.

The block orientation {...} returns the same optimal rotation used in the rmsd component to superimpose the coordinates $\{\mathbf{x}_{i}(t)\}$ onto a set of reference coordinates $\{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$ . Such component returns a four dimensional vector $\mathsf{q} = (q_0, q_1, q_2, q_3)$ , with $\sum_{i} q_{i}^{2} = 1$ ; this quaternion expresses the optimal rotation $\{\mathbf{x}_{i}(t)\} \rightarrow \{\mathbf{x}_{i}^{\mathrm{(ref)}}\}$ according to the formalism in reference [50]. The quaternion $(q_0, q_1, q_2, q_3)$ can also be written as $\left(\cos(\theta/2), \, \sin(\theta/2)\mathbf{u}\right)$ , where $\theta$ is the angle and $\mathbf{u}$ the normalized axis of rotation; for example, a rotation of 90$^{\circ}$ around the $z$ axis is expressed as ``(0.707, 0.0, 0.0, 0.707)''. The script quaternion2rmatrix.tcl provides Tcl functions for converting to and from a $4\times{}4$ rotation matrix in a format suitable for usage in VMD.

As for the component rmsd, the available options are atoms, refPositionsFile, refPositionsCol and refPositionsColValue, and refPositions.

Note: refPositionsand refPositionsFile define the set of positions from which the optimal rotation is calculated, but this rotation is not applied to the coordinates of the atoms involved: it is used instead to define the variable itself.

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• refPositions: see definition of refPositions (rmsd component)
• refPositionsFile: see definition of refPositionsFile (rmsd component)

• refPositionsCol: see definition of refPositionsCol (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue (rmsd component)

• closestToQuaternion $\langle\,$ Reference rotation$\,\rangle$
  Context: orientation
  Acceptable values: ``(q0, q1, q2, q3)'' quadruplet
  Default value: (1.0, 0.0, 0.0, 0.0) (``null'' rotation)
  Description: Between the two equivalent quaternions $(q_0, q_1, q_2, q_3)$ and $(-q_0, -q_1, -q_2, -q_3)$ , the closer to (1.0, 0.0, 0.0, 0.0) is chosen. This simplifies the visualization of the colvar trajectory when sampled values are a smaller subset of all possible rotations. Note: this only affects the output, never the dynamics.

Tip: stopping the rotation of a protein. To stop the rotation of an elongated macromolecule in solution (and use an anisotropic box to save water molecules), it is possible to define a colvar with an orientation component, and restrain it through the harmonic bias around the identity rotation, (1.0, 0.0, 0.0, 0.0). Only the overall orientation of the macromolecule is affected, and not its internal degrees of freedom. The user should also take care that the macromolecule is composed by a single chain, or disable wrapAll otherwise.

orientationAngle: angle of rotation from reference coordinates.

The block orientationAngle {...} accepts the same base options as the component orientation: atoms, refPositions, refPositionsFile, refPositionsCol and refPositionsColValue. The returned value is the angle of rotation $\theta$ between the current and the reference positions. This angle is expressed in degrees within the range [0$^{\circ}$ :180$^{\circ}$ ].

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• refPositions: see definition of refPositions (rmsd component)
• refPositionsFile: see definition of refPositionsFile (rmsd component)

• refPositionsCol: see definition of refPositionsCol (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue (rmsd component)

orientationProj: cosine of the angle of rotation from reference coordinates.

The block orientationProj {...} accepts the same base options as the component orientation: atoms, refPositions, refPositionsFile, refPositionsCol and refPositionsColValue. The returned value is the cosine of the angle of rotation $\theta$ between the current and the reference positions. The range of values is [-1:1].

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• refPositions: see definition of refPositions (rmsd component)
• refPositionsFile: see definition of refPositionsFile (rmsd component)
• refPositionsCol: see definition of refPositionsCol (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue (rmsd component)

spinAngle: angle of rotation around a given axis.

The complete rotation described by orientation can optionally be decomposed into two sub-rotations: one is a ``spin'' rotation around e, and the other a ``tilt'' rotation around an axis orthogonal to e. The component spinAngle measures the angle of the ``spin'' sub-rotation around e.

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• refPositions: see definition of refPositions (rmsd component)
• refPositionsFile: see definition of refPositionsFile (rmsd component)
• refPositionsCol: see definition of refPositionsCol (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue (rmsd component)
• axis $\langle\,$ Special rotation axis (Å)$\,\rangle$
  Context: tilt
  Acceptable values: (x, y, z) triplet
  Default value: (0.0, 0.0, 1.0)
  Description: The three components of this vector define (when normalized) the special rotation axis used to calculate the tilt and spinAngle components.

The component spinAngle returns an angle (in degrees) within the periodic interval $[-180:180]$ .

Note: the value of spinAngle is a continuous function almost everywhere, with the exception of configurations with the corresponding ``tilt'' angle equal to 180$^\circ$ (i.e. the tilt component is equal to $-1$ ): in those cases, spinAngle is undefined. If such configurations are expected, consider defining a tilt colvar using the same axis e, and restraining it with a lower wall away from $-1$ .

tilt: cosine of the rotation orthogonal to a given axis.

The component tilt measures the cosine of the angle of the ``tilt'' sub-rotation, which combined with the ``spin'' sub-rotation provides the complete rotation of a group of atoms. The cosine of the tilt angle rather than the tilt angle itself is implemented, because the latter is unevenly distributed even for an isotropic system: consider as an analogy the angle $\theta$ in the spherical coordinate system. The component tilt relies on the same options as spinAngle, including the definition of the axis e. The values of tilt are real numbers in the interval $[-1:1]$ : the value $1$ represents an orientation fully parallel to e (tilt angle = 0$^\circ$ ), and the value $-1$ represents an anti-parallel orientation.

List of keywords (see also for additional options):

• atoms: see definition of atoms (rmsd component)
• refPositions: see definition of refPositions (rmsd component)
• refPositionsFile: see definition of refPositionsFile (rmsd component)
• refPositionsCol: see definition of refPositionsCol (rmsd component)
• refPositionsColValue: see definition of refPositionsColValue (rmsd component)
• axis: see definition of axis (spinAngle component)