The method of Blow & Crick

Matthews, B. W.

doi:10.1107/97809553602060000685

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. 14.1, pp. 294-295 | 1 | 2 |

Section 14.1.4. The method of Blow & Crick

B. W. Matthews^a ^*

^aInstitute of Molecular Biology, Howard Hughes Medical Institute and Department of Physics, University of Oregon, Eugene, OR 97403, USA
Correspondence e-mail: brian@uoxray.uoregon.edu

14.1.4. The method of Blow & Crick

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Blow & Crick pointed out that in practice the phase angle φ can never be determined with complete certainty. Rather, there is a finite probability that any arbitrary phase angle may be the correct one. Consider the vector diagram shown in Fig. 14.1.4.1, in which $[{\bf F}_{H}]$ is known and we wish to determine the probability $[P(\varphi)]$ that the arbitrary phase angle φ is the correct phase of $[{\bf F}_{P}]$ . Strictly, one should allow for the possibility of errors in $[{\bf F}_{H}, F_{P}]$ and $[F_{PH}]$ , and should consider the probability that the vector $[{\bf F}_{P}]$ occupies all possible positions in the Argand diagram. However, Blow & Crick suggested that the analysis might be considerably simplified by assuming that $[F_{P}]$ and $[{\bf F}_{H}]$ are known accurately and that all the error lies in the observation of $[F_{PH}]$ . In other words, it was assumed that the vector $[{\bf F}_{P}]$ must lie on the circle of radius $[F_{P}]$ , and the probability distribution of $[F_{P}]$ could be evaluated as a function of φ only.

Figure 14.1.4.1| top | pdf |

Vector diagram illustrating the lack of closure, ɛ, of an isomorphous-replacement phase triangle.

For an arbitrary phase angle φ, the phase triangle (Fig. 14.1.4.1) will not close exactly. If we define $[F_{C}]$ to be the vector sum of $[{\bf F}_{H}]$ and $[F_{P} \exp (i\varphi)]$ , then the lack of closure of the phase triangle is given by $[\varepsilon = F_{C} - F_{PH}. \eqno(14.1.4.1)]$ Following Blow & Crick, if E is the r.m.s. error associated with the measurements, and the distribution of error is assumed to be Gaussian, then the probability P(φ) of the phase φ being the true phase is $[P(\varphi) = N \exp (-\varepsilon^{2}/2E^{2}), \eqno(14.1.4.2)]$ where N is a normalizing factor such that the sum of all probabilities is unity. The un-normalized probability distribution corresponding to Fig. 14.1.4.1 (and Fig. 14.1.2.1a) is shown in Fig. 14.1.2.1(b). The two most probable phase angles ( $[\varphi = \varphi_{1}]$ and $[\varphi = \varphi_{2}]$ ) are the alternative phases of $[F_{P}]$ for which the phase triangle is closed.

Individual probability distributions for the additional heavy-atom derivatives are derived in an analogous manner and may be multiplied together to give an overall probability distribution. The joint probability distribution corresponding to Fig. 14.1.3.1(a) is shown in Fig. 14.1.3.1(b), and in this case the most probable phase is that which simultaneously fits best the observed data for the two isomorphous derivatives.

The main objection which may be made to the Blow & Crick treatment is that it assumes that there is no error in $[F_{P}]$ . In practice, however, this is not a serious limitation.

References

International Tables for Crystallography (2006). Vol. F. ch. 14.1, pp. 294-295