IUCr99 COMPUTING SCHOOL: Modelling thermal motion and disorder

4.3. Displacement parameter constraints - group displacement parameters

At its simplest, group refinement of displacement parameters can be done by grouping either ADPs or IDPs, i.e. all atoms in the group share the same displacement parameter(s), implying that the group has only translational motion. This is unlikely to be a good model for macromolecules, where rotational oscillation (libration) is likely to play a major rôle. Strictly, the harmonic model is applicable only if the motion is purely translational, but provided the libration amplitudes are not too large it is a good approximation.

4.3.1. Displacement parameter constraints - group TLS parameters

The following analysis is due to Johnson [15]. According to Chasles' theorem [16] the instantaneous displacement u at a point r in a rigid body, resulting from some arbitrary motion of the body, can be specified as a screw motion, that is a rotation of magnitude |l| (in radians) about a suitably positioned and oriented axis l, correlated with a translation t parallel to that axis, all vectors being specified relative to some arbitrarily positioned and oriented fixed Cartesian axes.

In vector notation:

This equation is more conveniently written in matrix notation:

where:

As before, the dispersion matrix for an atom at the point r in a rigid group is the time- and lattice-averaged outer product of the instantaneous displacement vector:

(I is a 3x3 identity matrix).

Group refinement of ADPs reduces the number of variables by defining three new 3x3 matrices (: translational vibration, : libration, : screw-rotational vibration) for each group:

In scalar notation:

Just as the vibration tensor for an atom defined by 3 positional co-ordinates is represented by a 3x3 symmetric matrix, so for a rigid body defined by 6 co-ordinates (3 positional + 3 rotational) it is a 6x6 symmetric matrix:

4.3.2. TLS refinement - some practical considerations

Note that the

and

matrices are symmetric, whereas

is in general not so (because <l_jt_i>

<l_it_j>), though under certain circumstances (see next section) it may become so. In addition the value of the trace of

cannot be determined from diffraction data alone, and it normally remains unknown. It is therefore always constrained to the arbitrary value of zero during refinement; this reduces the number of independent elements by 1 to 8; there are therefore normally 20 (6 + 6 + 8) independent variables per rigid group.

For planar groups of 5 atoms or less (or in general any planar group in which all the atoms lie on a curve that is a conic section), further dependencies arise; however numerical problems can be avoided if in such cases the local origin is defined at an atomic position in the group, and if weak restraints to the starting values are applied to the elements of and . In practice such small groups can only be reliably modelled in terms of TLS matrices if high resolution data (1.5Å or better) data is available. Larger groups such as helices or domains probably only need 3-2.5Å data.

These 20 parameters per rigid group can be treated as variables in the structure refinement; derivatives of the structure factor with respect to them are obtained via the chain rule from the equation relating the atomic ADPs to the elements of the group , and matrices. Note that the relationship between the atomic U and A matrices introduces dependences of the structure factor on the atomic co-ordinates, in addition to that coming from the geometry factor; this will have consequences for the design of the refinement program, as it implies that co-ordinate and displacement parameters cannot be treated as independent.

4.3.3. TLS refinement - simultaneous IDP refinement

An alternative option for large groups which are likely to exhibit significant internal motion, and therefore for which the strict rigid-body model is less appropriate, is to include the atomic IDPs as variables, in addition to the elements of the group

and

matrices. For protein secondary structure elements and whole domains it may also be necessary to exclude non-rigid side-chains from the rigid group. The motions of the atoms in the group are thus simulated as an isotropic "riding" motion superimposed on the anisotropic group motion, so the U used in the structure factor calculation becomes:

If the atomic IDPs in the group are treated as variables in this way, the trace of the

matrix becomes indeterminate (because any change in the trace cancels out in the equation for U' above), which reduces the number of TLS variables per group by 1 to 19.

Updated by Ian Tickle.