Garib N. Murshudov^{1,2*},Gideon J. Davies^{1}, Mikhael Isupov^{3}, Szymon Krzywda^{4}, Eleanor J. Dodson^{1}.

^{1}Chemistry Department, University of York, York, U.K.^{2}CLRC, Daresbury Laboratory, Warrington, Daresbury, U.K.^{3}Chemistry Department, University of Exeter, Exeter, U.K.^{4}Crystallography Department, Faculty of Chemistry, Adam Mickiewicz University, Poznan, Poland,^{*}e-mail garib@yorvic.york.ac.uk

It was noted by S.Gamblin (1996): ``in the absence of an overwhelming argument(such as cubic space group), it is always safest to assume that diffraction is anisotropic''. This fact should be taken into account in refinement as well as in data collection strategy and data processing. Figure 1 shows that sometimes only at high resolution does the anisotropicity of data becomes apparent on the diffraction images. Anisotropicity of data might also cause problems in data collection. If one accidentally collects the first image in the direction of high thermal motion, one might not make the optimal decision for data collection. An image perpendicular to the first should also be collected to observe the true behaviour of the crystal.

The anisotropicity of the data should be taken into account during the data processing stage. This is possible with the CCP4 SCALA program, written by Phil Evans. In the absence of prior information about the contents of the crystal, refinement of the overall anisotropicity at the data processing stage will remain ill determined. In that case, any residual anisotropy should be taken into account at the refinement stage or alternatively refinement and data processing could be alternated. An even better approach would be simultaneous refinement and data processing, but this would only be possible if the structure had already been solved.

In the first ever paper on least-squares refinement of crystal structures, Hughes (1941) noted the existence of anisotropicity and described improved refinement behaviour by the introduction of anisotropic scale factors. It is surprising that in highly mobile and large structures, such as proteins, this fact has not been taken into account until recently.

A second source of anisotropicity is the movement of whole molecules as a rigid bodies within the crystal lattice. This can be described by TLS parameters (20 more per molecule) which are independent of the crystal form (Schomaker and Trueblood 1968). The RESTRAIN program (Moss et al., 1996) is able to evaluate these, and the correction has been shown to be valuable in some situations.

A third source of anisotropicity is vibration along torsion angles. In principle this might be described by refining the torsion angles themselves, and estimating their displacement parameters. However there are problems, since these parameters are highly correlated and such refinement may be sensitive to small perturbations of one or several of these. It may be better to deduce the displacement parameters of the torsion angles from the individual anisotropic atomic U values.

To summarise, the observed atomic anisotropic **U** values can be written as:

| (1) |

Care should be taken in the refinement of these different contributions as they are highly correlated. To
overcome this difficulty they could be refined at the different levels. I.e. first
**U**_{crystal}, second **U**_{TLS}, third **U**_{torsion} and finally
**U**_{atom}, or alternatively refine **U**_{crystal}, **U**_{TLS} and along internal degrees of freedom as described by Diamond (1990).

**U**_{crystal} is in principle sum of two
factors: 1) those remaining after data processing and 2) common mode from U_{TLS}.

Here effect of the anisotropic scaling only will be discussed. For refinement of individual atomic anisotropic thermal parameters see Murshudov et al. (1998)

(2) |

where the scale factor k = k_{0} e^{-hT U* h}. U^{*} is symmetric reciprocal
space anisotropic tensor. The space group puts constraints on the anisotropic
**U** tensor. For example, cubic space groups do not have an overall anisotropic **U**.
The space group P4_{2} 2_{1} 2 has 2 parameters and so on. In the implementation in the program REFMAC (Murshudov et al. 1997) this fact has been taken into account. As in this treatment anisotropic **U** is the difference between the observed and
calculated structure factors, there is no need to use positive definite
constraints. At each cycle of refinement, the program refines anisotropic
scale factors and applies them to calculated ones. There is also an option to apply
anisotropic scale to the observed structure factors. At this stage,
application of this option is not recommended since it changes the observed structure
factors. Thus, the calculated R-values would not be comparable with each other. If
anisotropic **U** values would be applied to observed structure factors then:

(3) |

It is clear from this equation that at each cycle calculated R-value is in fact
weighted R-value with weights k_{1}^{2}(s), where k_{1}(s) = [1/ k(s)]. If k(s) is refined at each cycle then behaviour of R and R_{k} could be different.

*Native Catalase at 1.5Å*- Crystals of catalase from the
bacterium
*Mycrococcus lysodeikticus*are almost perfect. Data from these crystals have now been collected at 0.9Å resolution. Even in this case one can see that anisotropic scale factor improves R-value and free R-value (Table 1) *Catalase frozen at 1.96Å*- Data from catalase soaked in peracetic acid
solution were used in order to obtain the reaction intermediate. Data were collected, from a frozen crystal, using
CuK
_{a}radiation and the RAXIS II as detector. In this case it can be seen that effect of anisotropic scaling is much larger than in native room temperature data (Table 1). *Cellulase*- Data for this enzyme were collected from frozen crystals in the home laboratory to 1.6Å resolution and the MIR structure determination was essentially trivial (Davies et al., 1998) The refinement, although straightforward, converged with unusually high values for both R and Rfree, both above 20%. It was only upon collection of atomic (0.9Å) resolution data that the anisotropic nature of the diffraction became easily apparent to the authors from inspection of the diffraction images. At this point, the anisotropic data scaling became available resulting in immediate reductions in R and R free of over 6% (Table 1).
*Myoglobin*- This case was one of the prime reasons for speeding up the implementation of anisotropic scale factor refinement. There were 10 different data sets of mutant and native myoglobins with different complexes. All data sets were collected from frozen crystals. Refinement with overall isotropic B values stuck with R 22%, R-free around 29%. Refinement of overall anisotropic scale factor immediately reduced R-value and free R-value (Table 1).
*Oxoindolyl-L-alanyn complexed tryptophanase*- The complex of tryptophanase from
*P.vulgaris*with competetitve inhibitor Oxindolyl-L-alanine have been crystallised in the space group P2_{1}2_{1}2 with a=152.480 , b= 213.694 , c=63.518 which was different from holotryptophanase crystals. The structure has been solved by molecular replacement using the holotryptophanase coordinates. The conventional REFMAC refinement at 18-3 Å converged with R/R-free = 28.4/31.3%. Refinement with anisotropic scaling reduced R/R-free to 24.88/27.8. Moreover, the refinement went on to R/R-free=18.0/24.7 (Table 1)

Another problem related to anisotropic scaling is the overall molecular motion
(TLS) in the unit cell. Future development of REFMAC will incorporate this information. This can be achieved easily since the fast refinement of individual **U** values by FFT now is available.

In principle, the treatment of anisotropic **U** values should start from the data
processing so that many factors contributing to anisotropic scale factor could be accounted for. Then the refinement protocol could be used to model the residual
overall anisotropic scale factor.

- CCP4. Collaborative Crystallographic Project, Number
4. (1994)
*Acta Cryst.***D50**, 760-763 - Cruickshank, D.W. (1956)
*Acta Cryst.***9**, 747-753 - Davies, G.J., Dauter, M., Brzozowski, A.M., Bjornvad, M.E., Andersen, K.V. &
Schulein, M. (1998)
*Biochemistry***37**, 1926-1932 - Diamond, R. (1990)
*Acta Cryst.***A46**425-435 - Gamblin, S.J. (1996) in
*Macromolecular Refinement. Proceedings of the CCP4 Study Weekend*Ed. Dodson,E., Moore,M., Ralph,A., Bailey, S. 163-170 - Hughes, E.W. (1941)
*J. Am. Chem. Soc.***63**, 1737-1752 - Isupov et al., manuscript in preparation
- Murshudov, G.N., Lebedev, A., Vagin, A.A., Wilson, K.S., Dodson, E.J. (1998) submitted to
*Acta Cryst. D* - Murshudov, G.N, Melik-Adamyan, W.R., Grebenko, A.I., Barynin, V.V., Vagin, A.A., Vainshtein, B.K., Dauter, Z. & Wilson, K.S. (1992)
*FEBS letters***312**, 127-131 - Murshudov, G.N., Vagin, A.A. & Dodson, E.J. (1997)
*Acta Cryst.***D53**, 240-253 - Moss, D.S., Tickle, I.J., Theis, O. & Wostrack, A. in ``
*Macromolecular refinement''*Proceedings of the CCP4 Study Weekend, 105-113, Ed. Dodson, E., Moore, M., Ralph, A. & Bailey, S. CCLRC, Dareesbury Laboratory - Schomaker, V. & Trueblood, K.N. (1968)
*Acta Cryst.***B24**63-76

MLC1 | MLC2 | CELL | MB | OIA | |

d(Å) | 1.5 | 1.96 | 1.8 | 1.8 | 1.5 |

R/R-free(iso %) | 11.7/14.0 | 17.3/22.6 | 20.2/25.1 | 22.1/28.8 | 28.4/31.3 |

R/R-free(aniso %) | 11.6/13.9 | 15.1/20.6 | 14.3/18.0 | 20.6/27.0 | 18.0/24.7 |

B_{11} | -0.3 | -3.4 | 9.1 | 5.6 | -9.8 |

B_{22} | -0.3 | -3.4 | -4.2 | 1.0 | -15.5 |

B_{33} | 0.7 | 7.1 | -4.8 | -6.2 | 33.6 |

B_{12} | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

B_{13} | 0.0 | 0.0 | 0.0 | 2.9 | 0.0 |

B_{23} | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |

MLC1 - MLC native data collected at room temperature

MLC2 - MLC soaked in peracetic acid collected from frozen crystals

MB - Myoglobin collected from frozen crystals

OIA - Oxiindolyl-L-alanine complex of Tryptophanase

CELL - Cellulase

B_{11}, B_{22}, B_{33}, B_{12}, B_{13}, B_{23} are elements of anisotropic **B** tensor. **B** = 8p^{2}**U**

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