Neutron distance measurements

Engelman, D. M.; Moore, P. B.

doi:10.1107/97809553602060000701

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 19.4, p. 442 | 1 | 2 |

Section 19.4.4.2. Neutron distance measurements

D. M. Engelman^a ^* and P. B. Moore^b

^aDepartment of Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06520, USA, and ^bDepartments of Chemistry and Molecular Biophysics and Biochemistry, Yale University, New Haven, CT 06520, USA
Correspondence e-mail: don@paradigm.csb.yale.edu

19.4.4.2. Neutron distance measurements

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In the neutron case, the usual application involves deuterium labelling of positions in the two regions, and the cross term becomes $[I_{x}(Q) = 2(b_{D} - b_{H})^{2} \textstyle\sum\limits_{i}\displaystyle \textstyle\sum\limits_{j}\displaystyle (\sin d_{ij}Q)/d_{ij}Q, \eqno (19.4.4.3)]$ where the first sum runs over all labelled positions in region 1 and the second over all labelled positions in region 2.

The interference experiment as a technique for studying macromolecules in solution was first proposed in 1947 as an X-ray scattering measurement in which the labels are heavy atoms (Kratky & Worthmann, 1947). It has been discussed several times since and tested (Hoppe, 1972; Vainshtein et al., 1970); however, no biological application has ever been published, since the signals are small (Hoppe, 1972). In 1972, the interference idea was discovered again, but in the context of neutron scattering (Engelman & Moore, 1972). Initially, it was thought that the distance measurement would follow from inspection of the damped sinusoidal term, but it soon became evident that the size and shape of the labelled regions would have an important influence. Using a power series expansion like that originally employed by Guinier, the previous expression becomes (for small Q) $[I_{x}(Q) \propto [1 - (1/6)(R_{i}^{2} + R_{j}^{2} + d_{ij}^{2})Q^{2}],]$ where $[R_{i}]$ and $[R_{j}]$ are the radii of gyration of the subunits in situ, and $[d_{ij}]$ is the separation of their centres of mass. It follows that (Moore et al., 1978; Stöckel et al., 1979) $[R_{g}^{2} = (1/2) (R_{i}^{2} + R_{j}^{2} + d_{ij}^{2}). \eqno(19.4.4.4)]$ The second moment of a length distribution, $[M_{ij}]$ , can be related to a radius of gyration. If $[p_{ij}(r)]$ is the length distribution of the cross term $[I_{x}(Q)]$ , then $[M_{ij} = \textstyle\int \displaystyle r^{2}p_{ij}(r)\hbox{ d}r = 2R_{g}^{2}.]$ Hence (Moore et al., 1978) $[M_{ij} = R_{i}^{2} + R_{j}^{2} + d_{ij}^{2}. \eqno(19.4.4.5)]$ Thus, $[M_{ij}]$ is the parameter that contains the information in the difference experiment. If the radii of gyration of the labelled regions are small, then the distance is well measured in a single experiment. This would be the case, for example, if the labels were single heavy atoms in an X-ray experiment. However, in most complexes of macromolecules there will be many pairwise protein–protein relationships where the radii and the separation of centres have comparable magnitudes. One approach to extracting distance information is to know the radii of the subunits in situ, by estimation from their molecular weights, by measurement of the isolated subunits or from triple isotopic substitution (see above). The first approach is the least desirable but the easiest, and the last approach is correct but laborious.

If a complex has eight or more distinct subunits, the number of possible measurements of $[M_{ij}]$ , [n(n - 1)/2] , is sufficient to solve for the distances and radii, enriching the information obtained from the experiment. The minimum number of measurements required is [4n - 6] , and for large complexes, such as the ribosome, a great excess of possible measurements exists, permitting refinement of the information through redundancy. Moore & Weinstein (1979) have described analytical methods for solving the problem with correct error propagation.

References

Engelman, D. M. & Moore, P. B. (1972). A new method for the determination of biological quarternary structure by neutron scattering. Proc. Natl Acad. Sci. USA, 69, 1997–1999.Google Scholar

Hoppe, W. (1972). New X-ray method for the determination of the quaternary structure of protein complexes. Isr. J. Chem. 10, 321–333.Google Scholar

Kratky, O. & Worthmann, W. (1947). Determination of the configuration of dissolved organic molecules by interferometric measurements with X-rays. Monatsch. Chem. 76, 263–281.Google Scholar

Moore, P. B., Langer, J. A. & Engelman, D. M. (1978). The measurement of the locations and radii of gyration of proteins in the 30S ribosomal unit of E. coli by neutron scattering. J. Appl. Cryst. 11, 479–482.Google Scholar

Moore, P. B. & Weinstein, E. (1979). On the estimation of the locations of subunits within macromolecular aggregates from neutron interference data. J. Appl. Cryst. 12, 321–326.Google Scholar

Stöckel, P., May, R., Strell, I., Cejka, Z., Hoppe, W., Heumann, H., Zillig, W., Crespi, H. L., Katz, J. J. & Ibel, K. (1979). Determination of intersubunit distances and subunit shape parameters in DNA-dependent RNA polymerase by neutron small-angle scattering. J. Appl. Cryst. 12, 176–185.Google Scholar

Vainshtein, B. K., Sosfenov, N. I. & Feigin, L. A. (1970). X-ray method for determining the gaps between heavy atoms in macromolecules in solution and its use for studying gramicidin derivatives. Dokl. Akad. Nauk SSSR, 190, 574–577.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 19.4, p. 442