Diffraction ratios for estimating the MAD phasing signal

Smith, J. L.; Hendrickson, W. A.

doi:10.1107/97809553602060000686

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. 14.2, p. 301 | 1 | 2 |

Section 14.2.1.3. Diffraction ratios for estimating the MAD phasing signal

J. L. Smith^a ^* and W. A. Hendrickson^b

14.2.1.3. Diffraction ratios for estimating the MAD phasing signal

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The first consideration in design of a MAD experiment is the choice of anomalous scatterer(s) with consideration of the magnitude of the phasing signal. Estimation of the total scattering by the macromolecule and the potential phasing signal generated by the anomalous scatterer(s) under consideration is informative.

The magnitude of the MAD phasing signal is estimated as the ratio of the expected dispersive or Bijvoet difference to the expected total scattering of the macromolecule. This is based on calculation of the expected root-mean-square structure amplitude $[({\rm rms}|F|)]$ (Wilson, 1942). $[{\rm rms}|F| = \langle |F|^2 \rangle^{1/2} = \left(\textstyle\sum\displaystyle f_{i}^{2}\right)^{1/2} = N^{1/2}f \eqno(14.2.1.12)]$ for N identical atoms. The expected total scattering of the macromolecule is estimated at [s = 0] using an average non-hydrogen atom. Based on atomic frequencies in biological macromolecules, the average values of $[f^{0}]$ are 6.70 e for proteins, 7.20 e for DNA and 7.26 e for RNA. The average number of non-hydrogen atoms and molecular mass per residue are 7.7 atoms and 110 Da for proteins, 21.8 atoms and 292 Da for DNA, and 22.4 atoms and 304 Da for RNA. These averages result in the following expressions for estimated total scattering of biological macromolecules: $[\eqalignno{{\rm rms} |^{0}F_{T}| _{\rm protein} &\approx 6.70 (\hbox{No. of atoms})^{1/2} \approx (346 \times \hbox{No. of amino acids})^{1/2}\cr&\approx (3.14 \times \hbox{molecular mass})^{1/2} &\cr{\rm rms}|^{0}F_{T}| _{\rm DNA} &\approx 7.20 (\hbox{No. of atoms})^{1/2} \approx (1128 \times \hbox{No. of nucleotides})^{1/2}\cr&\approx (3.87 \times \hbox{molecular mass})^{1/2} &\cr{\rm rms}|^{0}F_{T}| _{\rm RNA} &\approx 7.26 (\hbox{No. of atoms})^{1/2} \approx (1183 \times \hbox{No. of nucleotides})^{1/2}\cr&\approx (3.89 \times \hbox{molecular mass})^{1/2}.& (14.2.1.13) \cr}]$ Note: the estimated total scattering of a protein is coincidentally $[\approx (\pi \times \hbox{molecular mass})^{1/2}]$ .

The diffraction ratios relevant to a MAD experiment with N anomalous-scatterer sites are $[{{\rm rms}\|^{\lambda 1}F_{\rm obs} | - | ^{\lambda 2}F_{\rm obs}\| \over {\rm rms}|^{0}F_{T}| } \approx (N/2)^{1/2} {|f'_{\lambda 1} - f'_{\lambda 2}| \over {\rm rms}|^{0}F_{T}| } \eqno(14.2.1.14)]$ for the dispersive signal and $[{{\rm rms}\|^{\lambda}F_{\rm obs}^{+} | - | ^{\lambda}F_{\rm obs}^{-}\| \over {\rm rms}|^{0}F_{T}| } \approx (N/2)^{1/2} {2f''_{\lambda} \over {\rm rms}|^{0}F_{T}| } \eqno(14.2.1.15)]$ for the Bijvoet signal. The diffraction ratios, analogous to similar relations for isomorphous replacement (Crick & Magdoff, 1956), are equivalent to the expected fractional changes in intensity due to anomalous scattering, and, as such, can be compared directly to the $[R_{\rm sym}]$ estimate of error in the experimental data for evaluation of the phasing signal. Of course, the phasing signal may be diminished by partial occupancy or thermal motion, as for normal scattering.

References

Crick, F. H. C. & Magdoff, B. S. (1956). The theory of the method of isomorphous replacement for protein crystals. I. Acta Cryst. 9, 901–908.Google Scholar

Wilson, A. J. C. (1942). Determination of absolute from relative X-ray intensity data. Nature (London), 150, 151–152.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 14.2, p. 301