The fast rotation function

Rossmann, M. G.; Arnold, E.

doi:10.1107/97809553602060000556

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 255-258 | 1 | 2 |

Section 2.3.6.5. The fast rotation function

M. G. Rossmann^a ^* and E. Arnold^b

^aDepartment of Biological Sciences, Purdue University, West Lafayette, Indiana 47907, USA, and ^bCABM & Rutgers University, 679 Hoes Lane, Piscataway, New Jersey 08854-5638, USA
Correspondence e-mail: mgr@indiana.bio.purdue.edu

2.3.6.5. The fast rotation function

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Unfortunately, the rotation-function computations can be extremely time-consuming by conventional methods. Sasada (1964) developed a technique for rapidly finding the maximum of a given peak by looking at the slope of the rotation function. A major breakthrough came when Crowther (1972) recast the rotation function in a manner suitable for rapid computation. Only a brief outline of Crowther's fast rotation function is given here. Details are found in the original text (Crowther, 1972) and his computer program description.

Since the rotation function correlates spherical volumes of a given Patterson density with rotated versions of either itself or another Patterson density, it is likely that a more natural form for the rotation function will involve spherical harmonics rather than the Fourier components $[|{\bf F}_{\bf h}|^{2}]$ of the crystal representation. Thus, if the two Patterson densities $[P_{1} (r, \psi, \varphi)]$ and $[P_{2} (r, \psi, \varphi)]$ are expanded within the spherical volume of radius less than a limiting value of a, then $[P_{1} (r, \psi, \varphi) = {\textstyle\sum\limits_{lmn}} a_{lmn}^{*}\; \hat{j}_{l} (k_{ln} r) \hat{Y}_{l}^{m^{*}} (\psi, \varphi)]$ and $[P_{2} (r, \psi, \varphi) = {\textstyle\sum\limits_{l'm'n'}} b_{l'm'n'} \hat{j}_{l'} (k_{l'n'} r) \hat{Y}_{l'}^{m'} (\psi, \varphi),]$ and the rotation function would then be defined as $[ R = {\textstyle\int\limits_{\rm sphere}} P_{1} (r, \psi, \varphi) \hbox{\scr R} P_{2} (r, \psi, \varphi) r^{2} \sin \psi\;\hbox{d}r\;\hbox{d}\psi\;\hbox{d}\varphi.]$ Here $[\hat{Y}_{l}^{m} (\psi, \varphi)]$ is the normalized spherical harmonic of order l; $[\hat{j}_{l} (k_{ln} r)]$ is the normalized spherical Bessel function of order l; $[a_{lmn}]$ , $[b_{lmn}]$ are complex coefficients; and $[ \hbox{\scr R} P_{2} (r, \psi, \varphi)]$ represents the rotated second Patterson. The rotated spherical harmonic can then be expressed in terms of the Eulerian angles $[\theta_{1}, \theta_{2}, \theta_{3}]$ as $[ \hbox{\scr R} (\theta_{1}, \theta_{2}, \theta_{3}) \hat{Y}_{l}^{m} (\psi, \varphi) = {\textstyle\sum\limits_{q = -l}^{l}} D_{qm}^{l} (\theta_{1}, \theta_{2}, \theta_{3}) \hat{Y}_{l}^{q} (\psi, \varphi),]$ where $[D_{qm}^{l} (\theta_{1}, \theta_{2}, \theta_{3}) = \exp (iq\theta_{3}) d_{qm}^{l} (\theta_{2}) \exp (im\theta_{1})]$ and $[d_{qm}^{l} (\theta_{2})]$ are the matrix elements of the three-dimensional rotation group. It can then be shown that $[R(\theta_{1}, \theta_{2}, \theta_{3}) = {\textstyle\sum\limits_{lmm'n}} a_{lmn}^{*} b_{lm'n} D_{m'm}^{l} (\theta_{1}, \theta_{2}, \theta_{3}).]$ Since the radial summation over n is independent of the rotation, $[c_{lmm'} = {\textstyle\sum\limits_{n}} a_{lmn}^{*} b_{lmn},]$ and hence $[R(\theta_{1}, \theta_{2}, \theta_{3}) = {\textstyle\sum\limits_{lmm'}} c_{lmm'} D_{m'm}^{l} (\theta_{1}, \theta_{2}, \theta_{3})]$ or $[R(\theta_{1}, \theta_{2}, \theta_{3}) = {\textstyle\sum\limits_{mm'}} \left[{\textstyle\sum\limits_{l}} c_{lmm'} d_{m'm}^{l} (\theta_{2})\right] \exp [i(m'\theta_{3} + m\theta_{1})].]$ The coefficients $[c_{lmm'}]$ refer to a particular pair of Patterson densities and are independent of the rotation. The coefficients $[D_{m'm}^{l}]$ , containing the whole rotational part, refer to rotations of spherical harmonics and are independent of the particular Patterson densities. Since the summations over m and m′ represent a Fourier synthesis, rapid calculation is possible.

As polar coordinates rather than Eulerian angles provide a more graphic interpretation of the rotation function, Tanaka (1977) has recast the initial definition as $[ \eqalign{ R(\theta_{1}, \theta_{2}, \theta_{3}) &= {\textstyle\int\limits_{\rm sphere}} [\hbox{\scr R} (\theta_{1}, \theta_{2}, \theta_{3} = 0) P_{1} (r, \psi, \varphi)]\cr &\quad \times [\hbox{\scr R} (\theta_{1}, \theta_{2}, \theta_{3}) P_{2} (r, \psi, \varphi)]\;\hbox{d}V\cr &= {\textstyle\int\limits_{\rm sphere}} [P_{1} (r, \psi, \varphi)][\hbox{\scr R}^{-1} (\theta_{1}, \theta_{2}, \theta_{3} = 0)\cr &\quad \times \hbox{\scr R} (\theta_{1}, \theta_{2}, \theta_{3}) P_{2} (r, \psi, \varphi)]\;\hbox{d}V.}]$ He showed that the polar coordinates are now equivalent to $[\kappa = \theta_{3}]$ , $[\psi = \theta_{2}]$ and $[\varphi = \theta_{1} - \pi/2]$ . The rotation function can then be expressed as $[\eqalign{ R(\kappa, \psi, \varphi) &= {\textstyle\sum\limits_{lmm'}} \left({\textstyle\sum\limits_{n}} a_{lmn}^{*} b_{lm'n}\right) {\textstyle\sum\limits_{q}} \{d_{qm}^{l} (\psi) d_{qm'}^{l} (\psi) (-1)^{(m' - m)}\cr &\quad \times \exp [i(\kappa q)] \exp [i(m' - m) \varphi]\},}]$ permitting rapid calculation of the fast rotation function in polar coordinates.

Crowther (1972) uses the Eulerian angles α, β, γ which are related to those defined by Rossmann & Blow (1962) according to $[\theta_{1} = \alpha + \pi/2]$ , $[\theta_{2} = \beta]$ and $[\theta_{3} = \gamma - \pi/2]$ .

References

Crowther, R. A. (1972). The fast rotation function. In The molecular replacement method, edited by M. G. Rossmann, pp. 173–178. New York: Gordon & Breach.Google Scholar

Rossmann, M. G. & Blow, D. M. (1962). The detection of sub-units within the crystallographic asymmetric unit. Acta Cryst. 15, 24–31.Google Scholar

Sasada, Y. (1964). The differential rotation function. Acta Cryst. 17, 611–612.Google Scholar

Tanaka, N. (1977). Representation of the fast-rotation function in a polar coordinate system. Acta Cryst. A33, 191–193.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 255-258