Interpretation of isomorphous difference Pattersons

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. 244-245 | 1 | 2 |

Section 2.3.3.5. Interpretation of isomorphous difference Pattersons

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

2.3.3.5. Interpretation of isomorphous difference Pattersons

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Difference Pattersons have usually been manually interpreted in terms of point atoms. In more complex situations, such as crystalline viruses, a systematic approach may be necessary to analyse the Patterson. That is especially true when the structure contains noncrystallographic symmetry (Argos & Rossmann, 1976). Such methods are in principle dependent on the comparison of the observed Patterson, $[P_{1}({\bf x})]$ , with a calculated Patterson, $[P_{2}({\bf x})]$ . A criterion, $[C_{P}]$ , based on the sum of the Patterson densities at all test vectors within the unit-cell volume V, would be $[C_{P} = {\textstyle\int\limits_{V}} P_{1}({\bf x}) \cdot P_{2}({\bf x})\;\hbox{d}{\bf x}.]$ $[C_{P}]$ can be evaluated for all reasonable heavy-atom distributions. Each different set of trial sites corresponds to a different $[P_{2}]$ Patterson. It is then easily shown that $[C_{P} = {\textstyle\sum\limits_{\bf h}} \Delta_{\bf h}^{2} E_{\bf h}^{2},]$ where the sum is taken over all h reflections in reciprocal space, $[\Delta_{\bf h}^{2}]$ are the observed differences and $[E_{\bf h}]$ are the structure factors of the trial point Patterson. (The symbol E is used here because of its close relation to normalized structure factors.)

Let there be n noncrystallographic asymmetric units within the crystallographic asymmetric unit and m crystallographic asymmetric units within the crystal unit cell. Then there are L symmetry-related heavy-atom sites where [L = nm] . Let the scattering contribution of the ith site have $[a_{i}]$ and $[b_{i}]$ real and imaginary structure-factor components with respect to an arbitrary origin. Hence, for reflection h $[E_{\bf h}^{2} \left({\textstyle\sum\limits_{L}} a_{{\bf h}i}\right)^{2} + \left({\textstyle\sum\limits_{L}} b_{{\bf h}i}\right)^{2} = L + {\!\!\!\!\!}{\textstyle\sum\limits_{\!\hskip1.0pc i \neq j}^{N}}\!\!\! {\textstyle\sum\limits^{N}} (a_{{\bf h}i} a_{{\bf h}j} + b_{{\bf h}i} b_{{\bf h}j}).]$ Therefore, $[C_{P} = {\textstyle\sum\limits_{\bf h}} \Delta_{\bf h}^{2} \left[L + 2\!\! {\textstyle\sum\limits_{\hskip1.0pc i \neq j}}\!\!\! {\textstyle\sum} (a_{{\bf h}i} a_{{\bf h}j} + b_{{\bf h}i} b_{{\bf h}j})\right].]$ But $[{\textstyle\sum_{\bf h}} \Delta_{\bf h}^{2}]$ must be independent of the number, L, of heavy-atom sites per cell. Thus the criterion can be re-written as $[C'_{P} = \textstyle\sum\limits_{\bf h} \Delta_{\bf h}^{2} \left[\!\!\!{\textstyle\sum\limits_{\hskip1.0pc i \neq j}}\!\!\! {\textstyle\sum} (a_{{\bf h}i} a_{{\bf h}j} + b_{{\bf h}i} b_{{\bf h}j})\right]. \eqno(2.3.3.3)]$ More generally, if some sites have already been tentatively determined, and if these sites give rise to the structure-factor components $[A_{\bf h}]$ and $[B_{\bf h}]$ , then $[E_{\bf h}^{2} = \left(A_{\bf h} + {\textstyle\sum\limits_{N}} a_{{\bf h}i}\right)^{2} + \left(B_{\bf h} + {\textstyle\sum\limits_{N}} b_{{\bf h}i}\right)^{2}. \eqno(2.3.3.4)]$ Following the same procedure as above, it follows that $[C'_{P} = {\textstyle\sum\limits_{\bf h}} \Delta_{\bf h}^{2} \left[(A_{\bf h} a_{\bf h} + B_{\bf h} b_{\bf h}) + {\!\!\!}{\textstyle\sum\limits_{\hskip1.0pc i \neq j}}\!\!\! {\textstyle\sum} (a_{{\bf h}i} a_{{\bf h}j} + b_{{\bf h}i} b_{{\bf h}j})\right], \eqno(2.3.3.5)]$ where $[a_{\bf h} = {\textstyle\sum_{i=1}^{L}} a_{{\bf h}i}]$ and $[b_{\bf h} = {\textstyle\sum_{i=1}^{L}} b_{{\bf h}i}]$ .

Expression (2.3.3.5) will now be compared with the `feedback' method (Dickerson et al., 1967, 1968) of verifying heavy-atom sites using SIR phasing . Inspection of Fig. 2.3.3.4 shows that the native phase, α, will be determined as $[\alpha = \varphi + \pi]$ (φ is the structure-factor phase corresponding to the presumed heavy-atom positions) when $[|{\bf F}_{N}| \gt |{\bf F}_{H}|]$ and $[\alpha = \varphi]$ when $[|{\bf F}_{N}| \ll |{\bf F}_{H}|]$ . Thus, an SIR difference electron density, $[\Delta \rho({\bf x})]$ , can be synthesized by the Fourier summation $[\eqalign{\Delta \rho ({\bf x}) = &{1 \over V} {\sum} m(|{\bf F}_{NH}| - |{\bf F}_{N}|) \cos (2\pi {\bf h}\cdot {\bf x} - \varphi_{{\bf h}})\cr &\quad \hbox{from terms with } \Delta_{{\bf h}} = |{\bf F}_{NH}| - |{\bf F}_{N}| \gt 0\cr &+ {1 \over V} {\sum} m(|{\bf F}_{NH}| - |{\bf F}_{N}|) \cos (2\pi {\bf h}\cdot {\bf x} - \varphi_{{\bf h}} - \pi)\cr &\quad \hbox{from terms with } \Delta_{{\bf h}} \lt 0\cr = &{1 \over V} {\sum} m|\Delta_{{\bf h}}| \cos (2\pi {\bf h}\cdot {\bf x} - \varphi_{{\bf h}}),}]$ where m is a figure of merit of the phase reliability (Blow & Crick, 1959; Dickerson et al., 1961). Now, $[{\bf F}_{{\bf h}} = A_{{\bf h}} + iB_{{\bf h}} = F_{H} \cos \varphi_{{\bf h}} + iF_{H} \sin \varphi_{{\bf h}},]$ where $[A_{{\bf h}}]$ and $[B_{{\bf h}}]$ are the real and imaginary components of the presumed heavy-atom sites. Therefore, $[\Delta \rho ({\bf x}) = {1 \over V} \sum {m|\Delta_{{\bf h}}| \over |{\bf F}_{H}|} (A_{{\bf h}} \cos 2\pi {\bf h}\cdot {\bf x} + B_{{\bf h}} \sin 2\pi {\bf h}\cdot {\bf x}).]$

Figure 2.3.3.4 | top | pdf |

The phase α of the native compound (structure factor $[{\bf F}_{N}]$ ) is determined either as being equal to, or 180° out of phase with, the presumed heavy-atom contribution when only a single isomorphous compound is available. In (a) is shown the case when $[|{\bf F}_{N}| \gt |{\bf F}_{NH}|]$ and $[\alpha \simeq \varphi + \pi]$ . In (b) is shown the case when $[|{\bf F}_{N}| \lt |{\bf F}_{NH}|]$ and $[\alpha = \varphi]$ , where φ is the phase of the heavy-atom structure factor $[{\bf F}_{H}]$ .

If this SIR difference electron-density map shows significant peaks at sites related by noncrystallographic symmetry, then those sites will be at the position of a further set of heavy atoms. Hence, a suitable criterion for finding heavy-atom sites is $[C_{\rm SIR} = {\textstyle\sum\limits_{j=1}^{n}} \Delta \rho ({\bf x}_{j}),]$ or by substitution $[C_{\rm SIR} = \sum\limits_{j = 1}^{n} {1 \over V} \sum\limits_{\bf h} {m|\Delta_{{\bf h}}| \over |{\bf F}_{H}|} (A_{{\bf h}} \cos 2\pi {\bf h}\cdot {\bf x}_{j} + B_{{\bf h}} \sin 2\pi {\bf h}\cdot {\bf x}_{j}).]$ But $[a_{{\bf h}} = {\textstyle\sum\limits_{j = 1}^{n}} \cos 2\pi {\bf h}\cdot {\bf x}_{j} \quad \hbox{and} \quad b_{{\bf h}} = {\textstyle\sum\limits_{j = 1}^{n}} \sin 2\pi {\bf h}\cdot {\bf x}_{j}.]$ Therefore, $[C_{\rm SIR} = {1 \over V} \sum\limits_{{\bf h}} {m|\Delta_{{\bf h}}| \over |{\bf F}_{H}|} (A_{\bf h}a_{\bf h} + B_{\bf h}b_{\bf h}). \eqno(2.3.3.6)]$ This expression is similar to (2.3.3.5) derived by consideration of a Patterson search. It differs from (2.3.3.5) in two respects: the Fourier coefficients are different and expression (2.3.3.6) is lacking a second term. Now the figure of merit m will be small whenever $[|{\bf F}_{H}|]$ is small as the SIR phase cannot be determined well under those conditions. Hence, effectively, the coefficients are a function of $[|\Delta_{{\bf h}}|]$ , and the coefficients of the functions (2.3.3.5) and (2.3.3.6) are indeed rather similar. The second term in (2.3.3.5) relates to the use of the search atoms in phasing and could be included in (2.3.3.6), provided the actual feedback sites in each of the n electron-density functions tested by $[C_{\rm SIR}]$ are omitted in turn. Thus, a systematic Patterson search and an SIR difference Fourier search are very similar in character and power.

References

Argos, P. & Rossmann, M. G. (1976). A method to determine heavy-atom positions for virus structures. Acta Cryst. B32, 2975–2979.Google Scholar

Blow, D. M. & Crick, F. H. C. (1959). The treatment of errors in the isomorphous replacement method. Acta Cryst. 12, 794–802.Google Scholar

Dickerson, R. E., Kendrew, J. C. & Strandberg, B. E. (1961). The crystal structure of myoglobin: phase determination to a resolution of 2 Å by the method of isomorphous replacement. Acta Cryst. 14, 1188–1195.Google Scholar

Dickerson, R. E., Kopka, M. L., Varnum, J. C. & Weinzierl, J. E. (1967). Bias, feedback and reliability in isomorphous phase analysis. Acta Cryst. 23, 511–522.Google Scholar

Dickerson, R. E., Weinzierl, J. E. & Palmer, R. A. (1968). A least-squares refinement method for isomorphous replacement. Acta Cryst. B24, 997–1003.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 244-245