The theory behind the TLS parameterisation has been presented in
detail by Schomaker and Trueblood (Schomaker and Trueblood, 1968) [3],
with useful summaries in Howlin *et al*. (1989) [2] and Schomaker and
Trueblood (1998) [4].

TLS parameters describe the possible mean square displacements of rigid bodies. In this context, the rigid bodies are groups of atoms in your protein model. How many groups and the make-up of each group must be chosen by the user. TLS parameters describe anisotropic motion - an anisotropic U factor can be derived for each atom in a TLS group. But these U factors are correlated by virtue of belonging to the same rigid body, and only 20 refinement parameters are required for each TLS group. Thus, refinement of TLS parameters is a method of including anisotropic displacements without requiring the large number of parameters of full anisotropic refinement.

TLS refinement can therefore be used at moderate resolution, *e.g.* 2.0Å.
The number of extra parameters depends on the number of TLS groups
defined. A single TLS group for the whole molecule may prove useful,
and only requires 20 extra parameters. Or you may define a
TLS group for every rigid side chain, using a few thousand extra parameters.
TLS refinement may or may not turn out to be useful, but it is unlikely
to do any harm. Individual B factors are refined in addition to the
TLS parameters.

TLS refinement is often useful when there is NCS. It is often the case that different copies of a molecule in the asymmetric unit have different overall displacements. These can be accounted for by refining TLS parameters for each molecule. The residual atomic displacement parameters (B factors) should then be similar between molecules, and NCS restraints can be applied between them.

REFMAC needs the following information to do TLS refinement:

`REFI TLSC 20`- The TLSC subkeyword initiates cycles of TLS refinement, in this example 20 cycles. These cycles are performed after initial estimation of scaling parameters and before refining coordinates and B factors.
`TLSIN`- The
`TLSIN`file specifies the rigid groups to be used. The full specification of this file is given in the Files section. Generally, you only need to give the TLS record (which starts a group definition) and the RANGE record which specifies which atoms are included in the group. `BFAC SET 40`- We have found that TLS refinement works best if individual B factors are first set to a constant value. This is done with the BFAC keyword. B factors are then refined after the TLS parameters have been determined.

*N.B.* There have been some cases where TLS refinement has been
tried in the early stages of refinement, and has not been very
stable. If this happens, then leave it out, and try again later
on when the model is more complete.

Refinement statistics are as in a traditional refinement run. In addition, you get:

- Refined TLS parameters written to the TLSOUT file. This file can be analysed with the auxiliary program TLSANL, see below.
- B factors in the XYZOUT file. These are "residual" B factors that are refined after determining the TLS parameters, and do not contain any contribution from the TLS parameters.

*N.B.* When attempting to interpret the TLS tensors physically, it
is important to bear in mind the following:

- TLS parameters are refined assuming the physical model of a rigid
body,
*i.e.*that all atoms in the TLS groups have amplitudes (in 3 dimensions) appropriate to a rigid body and*that all atoms move in phase*. However, if for example one constructed a model in which all atoms in the TLS group had rigid body amplitudes but with one half moving in anti-phase with the other half, one would get the same derived U values and hence the same fit to the observed structure factors. The same is true of more complicated phase relationships between the atomic displacements. Thus the refinement statistics would be just as good, but the physical interpretation may be quite different. - TLS parameters, like any other form of displacement parameter, will mop up errors as well as a variety of different displacement types. This is particularly true when TLS is the only modelling of anisotropy that is being used - the TLS parameters will attempt to fit anisotropic internal modes and anisotropic errors.

The TLS file and PDB file output from REFMAC can be inputted to the auxiliary program TLSANL for analysis, via:

tlsanl tlsin in.tls xyzin in.pdb xyzout out.pdb <<EOF bresid end EOF

The keyword "bresid" is essential when running TLSANL on the output of REFMAC (it signifies the fact that the B factors in xyzin do not contain any contribution from the TLS parameters in tlsin).

For each group, this gives several representations of the T, L and S
tensors. It also outputs individual anisotropic U factors derived
from the TLS tensors to the file XYZOUT. Full details are can be
found in Howlin *et al.* 1993 [5], but here
are the important bits to look for:

- 1. INPUT TENSOR MATRICES WRT ORTHOGONAL AXES USING ORIGIN OF CALCULATIONS
- This should echo the contents of the tls file, with the values now displayed as matrices.
- 2. FOR TLS TENSOR USING CENTRE OF REACTION:
- (About halfway down.) T and L are real symmetric tensors, and so
can be diagonalised to give principal axes. S is also symmetric for
one particular choice of origin (the Centre of Reaction), and can then
also be diagonalised. This section gives the orientation of the principal
axes of T, L and S in various coordinate frames, and also the magnitudes
along these axes. So for example, if the input TLS tensors are:
INPUT TENSOR MATRICES WRT ORTHOGONAL AXES USING ORIGIN OF CALCULATIONS T TENSOR L TENSOR S TENSOR (A^2) (DEG^2) (A DEG) 0.026 -0.013 0.007 0.973 0.215 -0.130 0.009 0.034 -0.045 -0.013 0.056 -0.016 0.215 5.150 0.082 -0.055 0.010 -0.229 0.007 -0.016 0.005 -0.130 0.082 0.849 0.049 0.016 -0.019

The principal axes of the L tensor are then:AXES OF LIBRATION WRT TO MEAN-SQUARE ANGLE LIBRATION AXES MAKE TO ORTHOGONAL AXES (IN ROWS) DISPLACEMENT ORTHOGONAL AXES (DEG) ABOUT AXES (DEG^2) X Y Z 0.834 -0.033 -0.550 1.050 33.47 91.88 123.40 0.051 0.999 0.017 5.162 87.09 3.07 89.01 0.549 -0.042 0.835 0.759 56.69 92.43 33.42

In this example, there is a dominant libration along the b axis, and we see that the second principal axis is aligned almost exactly along b. The middle column gives the eigenvalues of L, and these can be quoted rather than the entire tensor.

Latest version of TLSANL has keyword AXES for outputting the various axes in a format suitable for molscript.

- [1]
- Winn, M.D., Isupov, M.N. and Murshudov G.N. (2001)
*Acta Cryst.*,**D57**, 122 - 133. - [2]
- Howlin, B., Moss, D.S. and Harris, G.W. (1989)
*Acta Cryst.*,**A45**, 851 - 861. - [3]
- Schomaker, V. and Trueblood, K.N. (1968)
*Acta Cryst.*,**B24**, 63 - 76. - [4]
- Schomaker, V. and Trueblood, K.N. (1998)
*Acta Cryst.*,**B54**, 507 - 514. - [5]
- B.Howlin, S.A.Butler, D.S.Moss, G.W.Harris and H.P.C.Driessen (1993)
*J. Appl. Cryst.*,**26**, 622