Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 2.3, pp. 244-245   | 1 | 2 |

Section Interpretation of isomorphous difference Pattersons

M. G. Rossmanna* and E. Arnoldb

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: 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[link]). 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(] 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(] 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(] 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 ([link] will now be compared with the `feedback' method (Dickerson et al., 1967[link], 1968[link]) of verifying heavy-atom sites using SIR phasing. Inspection of Fig.[link] 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[link]; Dickerson et al., 1961[link]). 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}).]


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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(] This expression is similar to ([link] derived by consideration of a Patterson search. It differs from ([link] in two respects: the Fourier coefficients are different and expression ([link] 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 ([link] and ([link] are indeed rather similar. The second term in ([link] relates to the use of the search atoms in phasing and could be included in ([link], 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.


First citation Argos, P. & Rossmann, M. G. (1976). A method to determine heavy-atom positions for virus structures. Acta Cryst. B32, 2975–2979.Google Scholar
First citation Blow, D. M. & Crick, F. H. C. (1959). The treatment of errors in the isomorphous replacement method. Acta Cryst. 12, 794–802.Google Scholar
First citation 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
First citation 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
First citation 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

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