research papers
From deep TLS validation to ensembles of atomic models built from elemental motions. II. Analysis of TLS
results by explicit interpretation^{a}Molecular Biophysics and Integrated Bioimaging Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA, ^{b}Department of Physics and International Centre for Quantum and Molecular Structures, Shanghai University, Shanghai 200444, People's Republic of China, ^{c}Department of Bioengineering, University of California Berkeley, Berkeley, California, USA, ^{d}Centre for Integrative Biology, Institut de Génétique et de Biologie Moléculaire et Cellulaire, CNRS–INSERM–UdS, 1 Rue Laurent Fries, BP 10142, 67404 Illkirch, France, and ^{e}Faculté des Sciences et Technologies, Université de Lorraine, BP 239, 54506 VandoeuvrelesNancy, France
^{*}Correspondence email: pafonine@lbl.gov, sacha@igbmc.fr
TLS modelling was developed by Schomaker and Trueblood to describe atomic displacement parameters through concerted (rigidbody) harmonic motions of an atomic group [Schomaker & Trueblood (1968), Acta Cryst. B24, 63–76]. The results of a TLS are T, L and S matrices that provide individual anisotropic atomic displacement parameters (ADPs) for all atoms belonging to the group. These ADPs can be calculated analytically using a formula that relates the elements of the TLS matrices to atomic parameters. Alternatively, ADPs can be obtained numerically from the parameters of concerted atomic motions corresponding to the TLS matrices. Both procedures are expected to produce the same ADP values and therefore can be used to assess the results of TLS Here, the implementation of this approach in PHENIX is described and several illustrations, including the use of all models from the PDB that have been subjected to TLS are provided.
Keywords: TLS model; TLS refinement; atomic displacement parameters; rigidbody motion; ensemble of atomic models; atomic model validation; PDB.
1. Introduction
1.1. Atomic positions in crystal structures
Describing atomic positions in crystal structures by Cartesian coordinates is a mathematical abstraction. Atomic positions are averages over the diffraction datacollection time and over all of the unit cells in the crystal. The variation of positions may range from large, representing discrete conformations, to small, reflecting atomic motion around a central position.
If a motion is harmonic (in particular, this means that the motion amplitude is small), the probability of a shift of an atom n by a vector r_{n} = Δx_{n}i + Δy_{n}j + Δz_{n}k is defined by individual isotropic (B_{n}) or anisotropic (U_{n}) atomic displacement parameters (ADPs):
These characteristics of atomic mobility are part of the structural information that is associated with models of crystal structures. As discussed in the literature (see, for example, Dunitz & White, 1973; Murshudov et al., 1999; Winn et al., 2001), the atomic displacement is a superposition of various motions that arise from different sources. These include displacement of atoms as part of a group and individual vibrations. A group motion itself can have several sources such as motion of the whole molecule, motion of its domains, sidechain libration etc. Typically, modern programs treat these motions using three separate components: motion of the whole crystal (modelled as an overall anisotropic scale factor), motion of nonoverlapping groups that are considered to be rigid, and individual atomic motions.
1.2. Rigidgroup motion
Atomic displacements arising from rigidbody motions can be accounted for using the TLS model (Schomaker & Trueblood, 1968). Such a model is based on simple geometric considerations allowing the description of elemental harmonic motions of atomic groups in terms of three matrices T, L and S (for a review, see Urzhumtsev et al., 2013). This provides a convenient mathematical way to present these motions in terms of an individual anisotropic ADP,
where
is an antisymmetric matrix expressed using the Cartesian coordinates (x_{n}, y_{n}, z_{n}) of atom n with respect to the origin of the TLS group. The symbol τ denotes the matrix transpose. The TLS approach may be seen as a statistical model for the analytical averaging of atomic positions that vary according to the given elemental motion parameters. The simplest example of a common motion is an isotropic vibration of a group that is equivalent to the assignment of the same B value to all atoms of the group. In the TLS model, the symmetric matrix L corresponds to the libration of a group, the symmetric matrix T corresponds to its common vibrations^{1} (also including a correction for the position of the libration axes) and the matrix S reflects correlations between the motions as well as the position of the axes.
A set of TLS matrices is defined by 21 parameters (six for T, six for L and nine for S). There is a linear constraint on the diagonal elements of the S matrix (Schomaker & Trueblood, 1968), resulting in 20 independent parameters. If individual atomic displacements can be ignored and the assumption that atomic motions are purely rigid can be accepted (at low resolution, for example), then modelling atomic displacements using TLS can significantly reduce the overall number of fitting parameters. In the following, we refer to the parameterization of an atomic group motion using parameters of elemental rigidbody motions as direct parameterization and that using elements of the T, L and S matrices, such as in (2), as indirect parameterization.
Indirect parameterization is mathematically and computationally more straightforward compared with direct parameterization. This is because of the simple relationship between the refinable elements of the T, L and S matrices and the atomic displacement parameters U using (2). In contrast, direct parameterization requires a nontrivial number of mathematical steps that link the parameters of atomic motions (such as the amplitudes of vibration and libration etc.) to the elements of TLS matrices (see, for example, Urzhumtsev et al., 2015). It is thus unsurprising that modelrefinement programs such as phenix.refine (Afonine et al., 2012) and REFMAC (Murshudov et al., 1997) use indirect parameterization for TLS owing to its simplicity; that is, they refine the elements of TLS matrices and not the actual parameters of atomic motions. This approach is inherently problematic because unconstrained or unrestrained of TLS matrices does not guarantee that the derived parameters of atomic motions are physically realistic or comply with TLS theory (see, for example, Zucker et al., 2010; Merritt, 2012; Urzhumtsev et al., 2015). This is very similar to unrestrained of atomic coordinates at typical `macromolecular resolutions' (e.g. 2–3 Å): factually, such would almost certainly result in distorted stereochemistry.
1.3. Two possible interpretations of TLS models
One may think of at least two possible ways to interpret the results of TLS R factors and if the atomic displacement parameters U_{TLS,n} derived from the refined TLS matrices using (2) are realistic (for example, they vary smoothly between neighbouring atoms). A more conservative approach considers TLS modelling to be successful if, in addition to meaningful ADPs and improved modeltodata fit, the TLS parameters comply with the basic assumptions of the corresponding theory set out by Schomaker & Trueblood (1968). This additional requirement is important when atomic motions modelled using TLS parameters are used to describe molecular motions (Trueblood, 1978; Trueblood & Dunitz, 1983; and references therein) or diffuse Xray scattering data (Van Benschoten et al., 2015), or are analysed for biological significance (Kuriyan & Weis, 1991; Harris et al., 1992; Šali et al., 1992; Wilson & Brunger, 2000; Raaijmakers et al., 2001; Yousef et al., 2002; Papiz et al., 2003; Chaudhry et al., 2004); see also discussion in Merritt (1999).
One interpretation considers TLS modelling to be successful if it leads to an improvement in the1.4. Analytical and numerical calculations of ADPs from TLS models
An interpretation of TLS et al., 1993; Urzhumtsev et al., 2015, 2016) provides an opportunity to verify whether the corresponding motion parameters agree with TLS theory. This can be performed in two steps as follows. Firstly, the parameters of elemental group motions extracted from refined TLS matrices can be used to obtain an ensemble of models that samples these motions. In turn, (1) can be used to convert the ensemble back to a single model with the uncertainties in atomic positions described using the corresponding ADP values, U_{ensemble,n}. Secondly, (2) can be used to calculate the uncertainties U_{TLS,n} in atomic positions directly from the TLS matrices. It is intuitive to expect that U_{TLS,n} and U_{ensemble,n} will match within some tolerance. The tolerance is needed to account for rounding errors and the finite number of models in the ensemble. A difference between U_{TLS,n} and U_{ensemble,n} beyond this tolerance may be indicative of various problems with the corresponding TLS set.
results in terms of elemental motions (see, for example, HowlinSince currently used et al., 2015, 2016). The simplest example is the T or L matrices being nonpositive definite. A more subtle example is when the parameters of elemental motions can be extracted from the TLS matrices but may not satisfy the basic assumptions about the TLS model (for example, libration amplitudes being too large, resulting in atomic motions that are anharmonic; see Fig. 1 and §2).
programs utilize an indirect TLS parameterization that does not use restraints or constraints, it may be the case that extracting motion parameters from refined TLS matrices is mathematically impossible (UrzhumtsevWhen motion parameters can be extracted from TLS matrices, comparison of U_{ensemble,n} and U_{TLS,n} requires a measure of and a threshold for the tolerance mentioned above (discussed in §2.1). Since U_{ensemble,n} depends on the number of models in the ensemble, we use a simple test system to estimate how many models are required to sample the group motion accurately and also to estimate a possible threshold value for the similarity of respective matrices (§2.2). The results are then validated using a more realistic protein model (§2.3). These tests highlighted reasons for differences between U_{TLS,n} and U_{ensemble,n} matrices and prompted further improvements for TLS analysis (§2.4). In §3 we discuss the results of the application of our procedures to all models in the PDB (Bernstein et al., 1977; Berman et al., 2000) that contain TLS information. Respective tools have been added to the PHENIX suite (Adams et al., 2010).
2. Anisotropic displacement matrices and the corresponding model ensembles
2.1. Metrics for matrix comparison
To evaluate the similarity of the two sets of anisotropic displacement matrices, U_{TLS,n} and U_{ensemble,n}, for a group composed of N atoms, n = 1, 2, …, N, we arbitrarily choose to use a simple Rfactortype metric,
where U_{1,n} = U_{TLS,n} and U_{2,n} = U_{ensemble,n}. Here, the sums are calculated over all elements of the matrices and over all atoms of the group. Other metrics can also be used (Dunitz & White, 1973; Zucker et al., 2010). Specifically, Kullback–Liebler (KL) divergence (Kullback & Leibler, 1951; Murshudov et al., 2011; Merritt, 2011, 2012) and the (CC_{UV}; Merritt, 1999) seem to be most prominent, with the caveat that they require matrix inversion, which is not always possible in numeric tests where only one single motion can be considered.
The calculation of (4) depends on the randomly generated ensemble models that are used to obtain U_{ensemble,n}. This is a stochastic procedure that depends on random seed values and on the number of models in the ensemble. Below, we analyze how these parameters affect the estimate of U_{ensemble}. Also, we check whether using KL_{UV} or CC_{UV} leads to conclusions that differ from those obtained using R_{U}.
2.2. Illustrations using a oneatom model
For simplicity, in this section we drop the subscript n from U_{ensemble} and U_{TLS} because only a single atom is considered.
2.2.1. Effect of vibration
In this test, we consider a model composed of a single atom vibrating along the Ox axis. For each trial rootmeansquare deviation (r.m.s.d., which we call the vibration amplitude t), we generated M random copies of this atom and then took all of these copies to calculate U_{ensemble} using (1). We then used (4) to compare U_{ensemble} with the corresponding U_{TLS} = T calculated analytically using (2) with L = S = 0. For each trial t we repeated these calculations 100 times, each time with a different random seed. Obviously, for different trials U_{TLS} remains the same while U_{ensemble} varies. Fig. 2(a) shows the average (over 100 trials) R_{U} for different trial values of t and M. The results are essentially independent of t. This is expected since 〈Δx^{2}〉 in (1) is proportional to T_{xx} = t_{x}^{2} in (2) for a sufficiently large number of models. We observe that R_{U} becomes close to 0.01 once the size of the ensemble reaches about 10 000 models.
2.2.2. Effect of libration
Here, we used the same singleatom model as above and the same calculation workflow, except that we varied the libration r.m.s.d. value d. Similarly to the previous example, R_{U} as a function of ensemble size reaches a plateau for about 5000–10 000 models in the ensemble (Fig. 2b); however, the plateau level depends on the value of d. Then we sampled a broad range of d values keeping the ensemble size fixed at 5000 models (Figs. 2c and 2d). We observe that R_{U} remains approximately constant (∼0.02) up to a d_{0} of ∼0.15 rad and then starts increasing monotonically. The d_{0} value obtained in this numerical experiment corresponds to the limit of a linear approximation to the small rotations discussed in Urzhumtsev et al. (2013) and other works cited therein, for example Cruickshank (1956). Owing to rounding errors and the differences between a rotation motion and a linear motion, the R_{U} values never reached zero even for very small d and large ensemble sizes (Fig. 2c). Also, while the average values over several trials are stable, they may vary between individual trials (Fig. 2d). These results allowed us to draw two conclusions. Firstly, generating about 5000–10 000 models is sufficient to estimate U_{ensemble} reliably (in §2.3 we show that this is still the case for realistic macromolecular models). Secondly, we may consider that U_{ensemble} agrees with U_{TLS} for a particular TLS set if R_{U} is approximately 0.05 or less.
2.2.3. Checking other metrics
To illustrate that the results obtained in previous tests are independent of matrixcomparison metrics, we repeated the test described in §2.2.2 using other metrics such as KL_{UV} and CC_{UV}. Since these metrics require a matrix inversion, we had to use a minor modification consisting of adding a small value to all of the diagonal elements of the respective matrices U_{ensemble} and the corresponding U_{TLS}, which is a convolution with an isotropic vibration. For example, the modified KL_{UV} metric is KL_{UV∊} = ∊tr(U_{∊}V_{∊}^{−1} + V_{∊}U_{∊}^{−1} − 2I) with U_{∊} = U + ∊I and V_{∊} = V + ∊I. The scale factor ∊ before `tr' is used to put the results on a similar scale (to facilitate comparisons). By trial and error, we found that a value of ∊ in the range 10^{−8}–10^{−6} allows the calculation of KL_{UV∊} and CC_{UV∊} but does not significantly affect the results. Overall, KL_{UV∊}, CC_{UV∊} and R_{U} do not contradict each other (Fig. 2c), with CC_{UV∊} showing a much stronger dependence on ∊ (Fig. 2e) and both KL_{UV∊} and CC_{UV∊} showing a less prominent drop (Figs. 2e and 2f) at a d_{0} of ∼0.10–0.15 rad (see Cruickshank, 1956) compared with R_{U}. In the following we use R_{U} because the original matrices can be used without modification and it has a predictable range of values, unlike KL_{UV∊}.
2.3. Illustrations using a protein model
As a more realistic example, we selected the model of IgGbinding domain III (PDB entry 2igd; S. Butterworth, V. L. Lamzin, D. B. Wigley, J. P. Derrick & K. S. Wilson, unpublished work) refined at 1.1 Å resolution using individual anisotropic ADP values. We chose the core of this model as a single TLS group containing residues 6–61 (leaving out the flexible Nterminus).
We considered two models derived from these data. One model contained C^{α} atoms only (56 in total) and the other model included all mainchain atoms (C^{α}, O, C and N). Each of the two models was treated as a single TLS group. For each model we fitted TLS matrices to individual anisotropic U_{n} values (ANISOU records from the PDB file) using the phenix.tls tool; we refer to these matrices as TLS_{CA} and TLS_{MC}, respectively. Then, using each of the two TLS sets (TLS_{CA} and TLS_{MC}) we calculated U_{TLS,n} using (2) and generated U_{ensemble,n} as described in Urzhumtsev et al. (2015). Similarly to as described in §2.2, we sampled a range of different numbers of models per ensemble.
The blue dashed curve in Fig. 3(a) shows that for the C^{α}only model (TLS_{CA}) R_{U} becomes smaller than 0.05 when the ensemble contains about 5000 models; using a larger ensemble does not change R_{U} significantly. This agrees with the conclusions derived in §2.2. For the mainchain model R_{U} reaches a plateau for ensembles containing the same number of models, but the value of R_{U} does not decrease below 0.09 (red dashed curve in Fig. 3a). To investigate the source of such a significant difference in R_{U} we performed the following tests.
Firstly, we note that the only difference between the two models is their composition and TLS matrices (Fig. 4). To determine which of the two, the composition or TLS matrices, contributes to the large R_{U} value, we repeated the calculations above using the C^{α}only model with TLS_{MC} matrices and the mainchain model with TLS_{CA} matrices. In the first case R_{U} was 0.09 and in the second case it was 0.05. This shows that the difference in R_{U} is owing to the TLS matrices and is not owing to the model composition. To find out which features of TLS_{MC} are responsible for the increased R_{U}, we performed a further analysis.
The elemental motions encoded by TLS matrices are three screw librations (around three mutually orthogonal axes l_{x}, l_{y}, l_{z}) coupled with three vibrations, also about three mutually orthogonal axes v_{x}, v_{y}, v_{z}. In the following, d_{x}, d_{y}, d_{z} stand for libration amplitudes, t_{x}, t_{y}, t_{z} stand for vibration amplitudes, s_{x}, s_{y}, s_{z} stand for the corresponding screw parameters and w_{x}, w_{y}, w_{z} stand for the points that belong to the respective libration axes (for formal definitions, see Urzhumtsev et al., 2013). As discussed in §2.2, the condition U_{ensemble,n} ≃ U_{TLS,n} (for a sufficiently large number of models in the ensemble) may break down owing to inadequate librations and not owing to vibrations. To separate the contribution of vibrations and librations, we first derived the set of parameters of elemental motions
from the corresponding TLS matrices as described in Urzhumtsev et al. (2015). Here, and in the following, to simplify the text we drop the parameters l_{x}, l_{y}, l_{z}; w_{x}, w_{y}, w_{z}; v_{x}, v_{y}, v_{z} from the list in (5) since they are invariant within these tests. Then, using the parameters in (5) (Table 2) and the C^{α}only model, we calculated U_{ensemble,n} and U_{TLS,n} for the following different scenarios.

Firstly, we considered a scenario where all three librations are used together, including their screw components, while vibrations are excluded:
Excluding vibrations led to an increase in R_{U} for both TLS_{MC} and TLS_{CA}; the values in the R_{U}(no V) column in Table 1 are ∼1.5 times larger^{2} than those for R_{U}(all).

Secondly, we calculated R_{U} separately for each individual libration, including its corresponding screw component (Table 1, columns 5–7),
and without it,
(Table 1, columns 8–10). For the screw librations, R_{U} is large for the rotation around l_{x} (Table 1), which is likely to be owing to a large magnitude of the screw component s_{x}. This component is two and a half times greater for TLS_{MC} compared with TLS_{CA} (−5.70 versus −2.07; Table 2), resulting in an about twofold larger R_{U} value. Removing all screw components results in an R_{U} of the order of 0.01 for all librations [R_{U}(d_{x}), R_{U}(d_{y}) and R_{U}(d_{z}) columns in Table 1].
These tests let us draw two conclusions. Firstly, the ensemble size required for reliable calculation of U_{ensemble,n} does not depend on the model size and, similarly to the oneatom case (§2.2), 5000–10 000 models are sufficient. Secondly, large values of the screw components are responsible for the disagreement between U_{ensemble,n} and U_{TLS,n} and the large reesulting R_{U}. This conclusion prompted us to revisit the TLS decomposition algorithm described in Urzhumtsev et al. (2015).
2.4. Improvement of the TLS decomposition
In the decomposition of the TLS matrices into elemental motions, some parameters, including libration amplitudes and axes, are defined unambiguously. However, the screw parameters are derived using the S matrix, which is not unique but is defined with an arbitrary constant σ that can be added to or subtracted from its diagonal elements (Schomaker & Trueblood, 1968). This freedom in the definition of S does not change the ADP and provides the possibility for alternative (and possibly better) decompositions of the TLS matrices. In Urzhumtsev et al. (2013) we discussed a possible argument for the traditional choice of σ from the condition
Here S_{xx}, S_{yy}, S_{zz} are the diagonal elements of the matrix S expressed in the basis [L] of the principal libration directions; these directions are eigenvectors of the matrix L. In Urzhumtsev et al. (2015) we showed that (9) may result in TLS matrices that do not correspond to elemental motions, and to address this issue we suggested a better choice for the t value,
under some additional constraints on σ that are discussed in that paper.
As shown in the previous paragraph, excessively large screw parameters lead to significant discrepancies between U_{ensemble,n} and U_{TLS,n}. This suggests that a better alternative to (9) and (10) might be to choose σ such that it minimizes the norm of the screw vector s. The new condition is then
Here, according to equations (5) and (8) in Urzhumtsev et al. (2015), S_{xx} − σ = s_{x}〈d_{x}^{2}〉 and L_{xx} = 〈d_{x}^{2}〉 (and similar expressions for the four other terms) are the diagonal elements of the matrices S and L given in the basis [L].
In order to test the new approach for adjusting the S matrix, we used the same models and sets of TLS matrices as described in §2.3. For each set of matrices, TLS_{MC} or TLS_{CA}, we extracted elemental motions using (11), generated U_{ensemble,n} using corresponding models and then computed R_{U} values using the previously obtained U_{TLS,n}. Table 1 shows that the updated R_{U} values calculated using (11) to adjust S are acceptably low not only for the total motion but for each of the individual components, both for TLS_{CA} and for TLS_{MC}. Fig. 3(a) shows R_{U} plots as a function of the number of generated models. The curves are nearly identical for both models, showing even lower values for R_{U} than the original curve for the C^{α}only model.
A more striking result is obtained when applying the new correction method to a reallife example: PDB entry 4muy (Span et al., 2014). In all tests TLS groups were used as defined in the PDB file. The 4muy model is composed of 40 TLS groups, and we focus on group No. 6 (residues 65–77 in chain A). The decomposition of the reported TLS matrices into motion parameters using the approach described previously (Urzhumtsev et al., 2015) suggests removing σ = 10^{−5} from the diagonal elements of the S matrix (expressed in Å rad). The R_{U} corresponding to these matrices is very high at 0.61, indicating a large disagreement between U_{ensemble,n} and U_{TLS,n} (Figs. 5a and 5b and the dashed curve in Fig. 3b). We suspected that this disagreement was owing to a very large value of the screw parameter s_{x} of 303.6 for the screw rotation around the axis l_{x} (Table 1). Applying (11) to adjust the S matrix resulted in σ increasing to 42 × 10^{−5}, which in turn reduced s_{x} to 0.1 and also reduced the respective R_{U} from 0.89 to 0.02. Fig. 5(c) shows the thermal ellipsoids obtained using the screw libration parameters extracted with (11): clearly, U_{ensemble,n} and U_{TLS,n} are much more similar (compare with Fig. 5a). Fig. 6 shows the variation of R_{U} and of s as a function of the σ value; indeed, the minimum of R_{U} is observed for σ obtained using (11). The R_{U} for the overall motion decreased to an acceptable value of 0.05, and for the libration alone it decreased from 0.85 to 0.11. The latter value is still high, possibly because by reducing s_{x} the procedure increased the magnitudes of s_{y} and s_{z} (from 2.90 to −3.38 and from −3.11 to −5.14, respectively; Tables 1 and 2). This test shows both the advantage of the new approach (11) compared with (9) and (10) and also its limitations. In this test using other norms, in particular max{s_{x}, s_{y}, s_{z}}, in (11) did not improve the result. In general, there is no guarantee that (11) always results in the best screw parameters and further improvements may be needed, for example by using a local search around the σ value obtained with (11).
Fig. 3(b) shows R_{U} as a function of the ensemble size for the 4muy model generated using parameters obtained with (10) and (11). It shows the significant difference between the results of the two approaches for correcting the S matrix and also confirms the previous observation that 5000–10 000 models are sufficient.
3. PDB analysis and improvement of the TLS decomposition
3.1. Model selection and analysis setup
The PDB (as of 14 November 2016) contains 123 954 entries, of which 32 162 contain TLS records. Since each PDB entry may contain more than one TLS record, a total of 260 353 TLS groups are available in the PDB. For each of these groups we tried to determine the corresponding elemental motions. This was performed using phenix.tls_as_xyz as described in Urzhumtsev et al. (2015).
88 697 groups could be interpreted in terms of elemental motions. In 263 of these cases all three matrices were composed of zeros. Some further 314 groups were excluded because the deposited TLS information was corrupted in a number of different ways (missing TLS group origin, noninterpretable atomic or TLS records etc.).
The remaining 88 120 TLS sets were subjected to three independent rounds of decomposition into elemental motions, each applying corrections to the S matrix using (9), (10) and (11), respectively. When using (10) and (11) the constraints on σ described in Urzhumtsev et al. (2015) were applied. For each set of extracted parameter values we analyzed the following motions.
3.2. Analysis of the elemental motions using (9)
Table 3 shows the overall statistics and the number of TLS groups with R_{U} ≤ 0.05. For the overall motion combining all elemental components together (5), about half of the TLS groups for which we could extract the motion parameters pass the test condition R_{U} ≤ 0.05 (this is approximately 17% of the total number of deposited TLS groups). The same condition applied when considering libration components only (6) reduced the number of acceptable groups roughly by half. In the case of considering librations individually (equations 7 and 8) the criterion R_{U} ≤ 0.05 selects only 2.3% of the TLS groups (6107 groups).
There are 45 sets where using pure rotations gives R_{U} > 0.05, all of which correspond to large libration amplitudes. Thus, the main source of the discrepancies between U_{ensemble,n} and U_{TLS,n} are the screw components.
We checked (Fig. 7a) the distribution of the TLS groups as a function of R_{U} calculated for the overall motion (5), for the motion excluding vibrations (6) and separately for the screw components (7). The first distribution (maroon full rectangles) has a peak in the interval 0.02–0.05 which corresponds to the TLS matrices that comply with the underlying study. Nevertheless, for a significant number of sets this value is above 0.05. Major problems come from screw components, for which many TLS groups have an R_{U} above 0.10 or even above 0.20 (blue full rectangles).
The largest value of s observed across all TLS groups is greater than 1000 Å. Such a large value means that for a rotation of 0.01 rad, i.e. approximately 0.6°, the rotated atoms would move by 10 Å in the direction of the rotation axis, which is clearly physically unrealistic. Fig. 7(b) shows that there are many groups with large values of s. The larger the screw parameter s, the larger the R_{U} values (Fig. 7c). However, since a particular screw motion also depends on the libration amplitude and on the positions of the axes, this does not allow anharmonic rotations to be discriminated unambiguously using this value alone (Fig. 7c).
3.3. Analysis of the elemental motions using (10) and (11)
Using the approach in (10) allows motion parameters to be extracted for 6700 more TLS sets compared with (9). Table 3 and Fig. 7(b) show that the distributions of R_{U} values and the screw parameters s are similar to those using (9).
Repeating the same calculations using (11) shows a significantly greater difference compared with using (10) (Table 3). Considering all motions together, the number of groups for which R_{U} ≤ 0.05 increased results in more than 12 000 groups compared with using (9). Considering only screw librations, the number of groups satisfying the condition R_{U} ≤ 0.05 is doubled compared with using (9) (`Individual screw' column in Table 3). Fig. 7(b) shows that the number of rotations with a large value of the screw parameter s is significantly reduced. The largest value of s fell to below 700 Å (which is still overly large).
Fig. 7(a) shows that using the approach in (11) instead of that in (9) significantly shifts all three distributions to the left (compare the open rectangles with the full rectangles in Fig. 7a), i.e. it improves the similarity between U_{ensemble,n} and U_{TLS,n}. In particular, R_{U} ≤ 0.10 for the majority of TLS sets when analyzing only the matrices for the total motion (5). However, considering vibrations alone, R_{U} > 0.10 for more than a third of the models even when using the improved decomposition into elemental motions (11).
4. Discussion
Validation of atomic models is now routine in macromolecular crystallography and is an integral part of structure submission to the Protein Data Bank (Read et al., 2011; Gore et al., 2017). It requires nomenclature compliance and fit to experimental data. Atomic coordinates are subjected to validation that includes analysis of stereochemistry and molecular packing. Atomic displacement parameters (ADPs) are also subjected to validation. For isotropic ADPs the existing validation criteria are rather simple: their values must be positive, not excessively large and not vary too much between neighbouring atoms. For anisotropic ADP values the criteria are somewhat more complex (Hirshfeld, 1976; Schneider, 1996). Similarly to atomic coordinates and displacement parameters, TLS matrices are model parameters and therefore should be subjected to some form of validation. Depending on the accepted paradigm (§1.3) the scope of TLS validation may refer to two questions: (i) how well does the the TLS approximation explain the experimental data and how well does it describe the atomic displacement parameters (see, for example, Merritt, 2011, 2012) and (ii) are the particular descriptors of the TLS model also consistent with the TLS formalism in addition to (i). Addressing the first question does not require analysis of the TLS matrices themselves but only of the derived ADP values. This includes making sure that the ADPs are positive definite and vary smoothly between adjacent atoms and TLS groups (Winn et al., 2001; Zucker et al., 2010; Merritt, 2011, 2012). The current work addresses the second question, which focuses exclusively on the analysis of TLS matrices and the parameters of group motion that they encode. Since modern atomic model packages use an indirect TLS parameterization (§1.3), i.e. they refine the elements of the TLS matrices and not the parameters of group motions, it is unsurprising to find that some TLS matrices do not comply with the assumption of harmonic motion that the TLS modelling theory is built upon. The number of such cases may vary based upon the different measures or thresholds that are used. For example, using the criteria discussed above we find that only 2.3% of the TLS groups reported in the PDB can be interpreted in terms of elemental harmonic motions. We envisage two reasons for this. Firstly, the validation of TLS results, focusing on TLS matrices and corresponding group motions, has never been enforced. Secondly, the implementation of TLS in modern packages does not allow control of the parameters of group motion by means of restraints or constraints (see a discussion in Painter & Merritt, 2006) because these parameters are refined indirectly. Unsurprisingly, such unrestrained provides no guarantee of TLS matrices that are interpretable in terms of harmonic elemental motions.
In this work, we have developed methods and a software implementation in the PHENIX suite to analyze the results of TLS refinements. These methods are based on comparison of individual atomic displacement parameters calculated analytically from the TLS matrices with ADPs derived numerically using parameters of elemental motions extracted from the TLS matrices. We theorise that large differences between these matrices indicate problematic TLS parameters. In particular, this may indicate a suboptimal choice of TLS groups or protocol. We show that a postrefinement correction of the deposited TLS matrices makes it possible to curate some but not all of the problematic TLS groups.
The analyses presented in this work rely on the choice of particular criteria (metrics and thresholds). These criteria may be optimized further, which is a nontrivial project and may help to diagnose the problem while still not addressing it. A possibly better investment of effort would be to improve TLS ). This would make it possible to control the refinable parameters directly during and therefore keep them physically realistic without the need for postrefinement corrections. This is a major undertaking both mathematically and algorithmically, which may be considered as a future improvement to the PHENIX software.
protocols so that they operate in terms of elemental parameters of motions, which has been proposed previously (Tickle & Moss, 1999The methods and tools discussed in this manuscript have been implemented and are available in PHENIX 1.12 and later. Data and scripts that can be used to reproduce the figures and tables are available at https://phenixonline.org/phenix_data/afonine/tls2/.
Footnotes
^{1}Traditionally, the term translation is used to denote the pure shift of the group without rotation. We prefer to use the term vibration, which is more coherent when using the term libration, and not rotation. This allows different components of the T matrix to be distinguished: those owing to the proper vibrations and those owing to apparent shifts.
^{2}U_{TLS,vibr} and U_{ensemble,vibr} from vibration cancel each other in the numerator of (4) since pure vibrations in the TLS model are always harmonic. Therefore, the denominator of (4) is larger for the full U matrices than for the matrices for librations only (this is easier to see in the coordinate system where U_{TLS,vibr} is diagonal; the diagonal elements of all U matrices are always nonnegative). This means that R_{U} for the overall motion can increase after excluding the vibration component.
Acknowledgements
We thank Dr N. W. Moriarty for critical reading of the manuscript and useful discussions.
Funding information
This research was supported by the NIH (grant GM063210) and the Phenix Industrial Consortium. This work was partially supported by the US Department of Energy under Contract DEAC0205CH11231. AU acknowledges the support and the use of the resources of the French Infrastructure for Integrated Structural Biology (FRISBI; ANR10INBS05) and of InstructERIC.
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