Direct Phasing in Crystallography Fundamentals and Applications

by Giacovazzo, Carmelo

• ISBN13: 9780198500728
• ISBN10: 0198500726
• Format: Hardcover
• Copyright: 1999-02-18
• Publisher: International Union of Crystallography

Summary

Direct methods are, at present, applied to a large variety of cases: X-ray, neutron or electron data; single crystal and powder data; small molecules and macromolecules. While direct methods solved in practice the phase problem for small molecules, their application to macromolecules is recentand still undergoing strong development. The fundamentals of the methods are described: in particular it is shown how the methods can be optimized for powder, neutron or electron data, and how they can be integrated with isomorphous replacement, molecular replacement and anomalous dispersiontechniques. Maximum Entropy methods are also described and discussed. Sets of test structures are used to verify, throughout the various chapters, the mathematical techniques there described and to provide practical examples of applications. This book will appeal to a wide variety of readers -offering both a comprehensive description of direct methods in crystallography and an invaluable reference tool. The first three chapters can be considered as an introduction to the field, with sufficient material to constitute a university course and for allowing the expert use of most directmethods programs. Subsequent chapters are aimed at graduate students and working crystallographers. Basic results are described and discussed in the main body of the text, while the appendices compliment these with in depth mathematical details. The quoted literature is extremely wide and theinterested reader can find suggestions for future work and further reading throughout the book.

Table of Contents

Abbreviations	xxiv
1. Wilson statistics	1	(74)
1.1 Introduction	1	(1)
1.2 The structure factor	1	(1)
1.3 The algebraic form of the structure factor	2	(3)
1.4 Elements of structure factor algebra	5	(6)
Friedel law	6	(1)
Effects of symmetry operators in reciprocal space	6	(1)
Determination of reflections with restricted phase values	7	(2)
Systematic absences	9	(2)
1.5 Structure factor statistics: fixed indices, random positions	11	(1)
1.6 Structure factor statistics in P1	12	(2)
1.7 The influence of symmetry elements on structure factor statistics	14	(3)
Structure factor statistics in P1	14	(1)
Structure factor statistics for space groups with a primitive unit cell	15	(2)
Structure factor statistics for space groups with a centred unit cell	17	(1)
1.8 The distributions P(I)	17	(1)
1.9 The normalized structure factor and the unitary structure factor	18	(4)
1.10 The distributions P(E) and P(z)	22	(2)
1.11 Statistics of structure factors: fixed structure, random indices	24	(3)
1.12 Cumulative distributions	27	(1)
1.13 Statistical criteria for discriminating between centrosymmetric and non-centrosymmetric structures	28	(3)
1.14 Non-ideal distributions of the structure factor	31	(2)
1.15 Absolute scaling of the intensities. The Wilson method	33	(5)
1.16 A list of test structures and of direct methods programs	38	(1)
1.17 About the shape of the Wilson plot	39	(5)
1.18 About the reliability of the Wilson plot scaling method	44	(5)
1.19 The normalization process in the presence of prior information	49	(3)
1.20 The standard reflections and the unique list	52	(1)
1.21 Intensity statistics of some special crystals	53	(3)
Twinning	55	(1)
Incommensurately modulated structures	55	(1)
Appendix 1.A About pseudosymmetries	56	(13)
1.A.1 Pseudosymmetry	56	(2)
1.A.2 Hypersymmetry	58	(11)
Appendix 1.B Fourier series representation of P(E)	69	(4)
1.B.1 The centrosymmetric case	70	(1)
1.B.2 The non-centrosymmetric case	71	(2)
Appendix 1.C A modified Wilson plot procedure	73	(2)
2. Structure invariants, seminvariants and the origin problem	75	(38)
2.1 Introduction	75	(1)
2.2 The concept of structure invariants	75	(3)
2.3 Origin shift and symmetry operators	78	(2)
2.4 Allowed or permissible origins and translations in primitive space groups	80	(6)
2.5 The seminvariance concept	86	(6)
2.6 Structure seminvariants	92	(1)
2.7 Origin definition in primitive unit cells	93	(10)
2.8 Origin definition in centred cells	103	(4)
2.9 Enantiomorph definition	107	(2)
2.10 Further considerations on origin and enantiomorph definition	109	(4)
3. Triplet invariant estimation and classical direct phasing procedures	113	(60)
3.1 Introduction	113	(1)
3.2 The positivity of a triplet invariant: algebraic considerations	113	(2)
3.3 The probability density function P((Phi)(h)/ E(h)/, E(k)/, E(h-k)/, Phi(k), Phi(h-k))	115	(5)
3.4 Triplet phase estimation	120	(3)
3.5 The tangent formula	123	(4)
3.6 The distribution of Alpha(h)	127	(2)
3.7 Triplet estimates for centrosymmetric space groups	129	(2)
3.8 About triplet invariant reliability and basic postulates	131	(3)
3.9 Psi-zero triplets	134	(1)
3.10 Scheme of procedure for phase determination	135	(1)
3.11 Set-up of phase relationships	135	(4)
3.12 Fixing the reflections of the `starting set'	139	(3)
3.13 Assignment of the starting phases	142	(4)
3.14 Phase determination	146	(5)
3.15 Finding out the correct solution	151	(5)
3.16 Map interpretation	156	(3)
3.17 The efficiency of the weighted tangent formula based on Cochran estimates	159	(3)
3.18 Structure completion and refinement	162	(5)
Appendix 3.A Other multisolution methods	167	(6)
3.A.1 The symbolic addition procedure	167	(1)
3.A.2 The primary-secondary method	168	(2)
3.A.3 Linear equations method	170	(1)
3.A.4 Cosine least squares method	171	(1)
3.A.5 The X-Y method	172	(1)
4. Direct methods and real space properties	173	(41)
4.1 Introduction	173	(1)
4.2 Wilson statistics and real space properties	173	(2)
4.3 The Sayre equation and its modified form	175	(3)
4.4 The generalization of the Sayre equation and the polynomial method	178	(2)
4.5 The integral criteria	180	(3)
4.6 Direct methods and Patterson map properties	183	(3)
4.7 Positiveness of electron density and Karle-Hauptman determinants	186	(7)
4.8 The algebra of Karle-Hauptman determinants	193	(3)
4.9 Inequalities and probability theory	196	(2)
4.10 Electron density modification procedures	198	(3)
Appendix 4.A Direct methods and Patterson and properties: additional approaches	201	(6)
Appendix 4.B Estimating structure seminvariants from the Harker sections	207	(1)
Appendix 4.C Patterson deconvolution methods	208	(6)
4.C.1 Superposition techniques	208	(4)
4.C.2 The multiple implication function	212	(2)
5. The method of joint probability distributions	214	(29)
5.1 Introduction	214	(2)
5.2 Multivariate distributions in centrosymmetric structures: the case of independent random variables	216	(3)
5.3 Multivariate distributions in non-centrosymmetric structures: the case of independent random variables	219	(3)
5.4 Simplified joint probability density functions in the absence of prior information	222	(3)
5.5 The joint probability density function when some prior information is available	225	(2)
5.6 The calculation of P(E) in P1 in the absence of prior information	227	(3)
5.7 The calculation of P(R, Phi) in P1 in the absence of prior information	230	(3)
5.8 The calculation of P(R, Phi) in any space group in the absence of prior information	233	(1)
Appendix 5.A P(E) and the saddle point method	234	(9)
5.A.1 About the saddle point method	234	(2)
5.A.2 P(E) in P1 via the saddle point method	236	(3)
5.A.3 P(E(1), E(2), ..., E(n)) in P1 via the saddle point method	239	(1)
5.A.4 P(E(1), E(2), ..., E(n)) in any space group via the saddle point method	240	(3)
6. Representations of structure invariants and seminvariants	243	(32)
6.1 Introduction	243	(1)
6.2 The first representation of a structure invariant	243	(4)
6.3 The upper representations of a structure invariant	247	(1)
6.4 Representations and multipoles	248	(4)
6.5 Representation theory extended to isomorphous structures	252	(4)
6.6 The first representation of a structure seminvariant	256	(4)
6.7 Algebraic properties related to the rank of the s.s.s	260	(3)
6.8 Finding the first representation of an s.s. Applications	263	(7)
6.9 Upper representations of structures seminvariants	270	(1)
6.10 A branch: the method of complementary seminvariants	271	(1)
Appendix 6.A Two algebraic propositions	272	(3)
7. The probabilistic estimation of triplet invariants	275	(36)
7.1 Introduction	275	(1)
7.2 The estimation of Phi in P1 in the absence of prior information	275	(4)
7.3 The estimation of Phi in P1 in the absence of prior information	279	(1)
7.4 The estimation of Phi in any space group via the first representation: no prior information available	280	(4)
7.5 The estimation of Phi from its first representation when prior information is available	284	(3)
7.6 Recovering of the complete from a partial structure (first representation formula)	287	(5)
7.7 The estimate of a triplet phase when pseudotranslational symmetry is present (first representation formula)	292	(2)
7.8 The estimate of the triplet phase via its second representation	294	(8)
7.9 The estimation of Phi from its second representation in the presence of pseudotranslational symmetry	302	(4)
7.10 The probabilistic estimation of non-measured diffraction magnitudes	306	(3)
Appendix 7.A Cosine invariant computation techniques and the P(6) formula	309	(2)
8. The probabilistic estimation of quartet invariants	311	(41)
8.1 Introduction	311	(1)
8.2 The estimation of Phi in P1 from its first representation: the Hauptman formula	311	(4)
8.3 The estimation of Phi in P1 from its first representation: the Hauptman formula	315	(1)
8.4 The estimation of Phi in P1 and P1 from its first representation: the Giacovazzo formula	316	(2)
8.5 Finding quartets	318	(1)
8.6 About quartet reliability	319	(6)
8.7 The estimation of Phi from its first representation in any space group	325	(2)
8.8 The role of the quartets in SAYTAN and SHELX	327	(5)
8.9 The integration of triplet and quartet relationships in SIR92	332	(5)
8.10 The integration of strong triplets, negative quartets, and psi-zero triplets in the SIR92 phasing process	337	(3)
8.11 Improving quartet estimates via their second representation	340	(1)
8.12 Quartets as a figure of merit	340	(2)
Appendix 8.A Mathematical derivation of the phase relationship (8.2)	342	(4)
Appendix 8.B Mathematical derivation of the sign relationship (8.9)	346	(2)
Appendix 8.C Special quartets	348	(4)
9. The probabilistic estimation of quintet invariants	352	(8)
9.1 Introduction	352	(1)
9.2 The estimation of Phi = Phi(h) + Phi(k) + Phi(l) + Phi(m) -- Phi(h+k+l+m) in P1 and P1 via its first representation	353	(2)
9.3 Estimation of Phi in any space group from its first representation: some general criteria	355	(1)
9.4 The role of the quintets in direct procedures	356	(4)
10. The probabilistic estimation of one- and two-phase structure seminvariants	360	(12)
10.1 Introduction	360	(1)
10.2 Estimation of one-phase structure seminvariants of first rank via their first representation	360	(2)
10.3 Estimation of one-phase structure seminvariants of first rank via their second representation	362	(3)
10.4 The estimation of Phi(h) via the method of quartet complementary invariants	365	(1)
10.5 The estimation of two-phase structure seminvariants of first rank via their first representation	366	(2)
10.6 About the role of seminvariants in direct procedures	368	(2)
Appendix 10.A Some mathematical details about Sigma(1) estimates	370	(2)
11. Probabilistic determinantal approaches	372	(9)
11.1 Introduction	372	(1)
11.2 The maximum determinant rule	372	(3)
11.3 The regression equation	375	(2)
11.4 About determinantal approaches	377	(1)
11.5 Practical aspects and applications to small molecules	378	(3)
12. Phasing via neutron and electron data	381	(29)
12.1 Introduction	381	(1)
12.2 Electron scattering	381	(3)
12.3 Collecting electron diffraction data	384	(1)
12.4 Perturbations to electron diffraction amplitudes	385	(3)
12.5 A typical experimental procedure for electron diffraction studies	388	(4)
12.6 Electron microscopy, image processing, and direct methods	392	(2)
12.7 Neutron scattering	394	(3)
12.8 About the violation of the positivity postulate	397	(6)
Appendix 12.A About the elastic scattering of electrons: the kinematic approximation	403	(4)
Appendix 12.B About HREM image formation	407	(3)
13. Direct methods versus powder data	410	(35)
13.1 Introduction	410	(2)
13.2 About the experimental powder patterns	412	(5)
13.3 About powder-pattern indexing	417	(2)
13.4 The recovery of the (F)^2	419	(4)
13.5 Pitfalls in phasing powder data	423	(10)
13.6 Prior information versus the full pattern decomposition programs	433	(7)
13.7 A practical case	440	(1)
13.8 Powder data and anomalous dispersion effects	441	(4)
14. Macromolecular crystallography techniques and traditional direct methods	445	(64)
14.1 Introduction	445	(1)
14.2 The isomorphous replacement method	446	(11)
The isomorphous heavy atom derivative	449	(1)
The determination of heavy atom positions	450	(3)
The SIR method	453	(2)
The MIR method	455	(2)
Refinement of heavy atom parameters	457	(1)
14.3 Density modification techniques in macromolecular crystallography	457	(7)
Positivity and atomicity constraints to electron density modifications	458	(1)
Solvent flattening	459	(1)
The electron density histogram method	459	(3)
Skeletonization techniques	462	(1)
Molecular averaging	463	(1)
14.4 Finding the heavy atom positions by direct methods	464	(1)
14.5 Early direct methods applications: phase extension and refinement	465	(3)
Tangent techniques	465	(1)
The Sayre equation	466	(1)
Determinantal method	467	(1)
14.6 About the limits of the tangent formula and of the Sayre equation	468	(7)
14.7 Traditional direct methods and the use of prior information	475	(2)
14.8 `Shake and bake' and `half-bake': two direct methods programs for macromolecules	477	(6)
The shake and bake method	478	(3)
The half-bake approach	481	(2)
Appendix 14.A The Crick and Magdoff relation	483	(2)
Appendix 14.B Protein phase estimate incorporating the treatment of errors	485	(5)
Appendix 14.C Solvent flattening	490	(5)
14.C.1 Wang, Leslie, and other procedures	490	(5)
Appendix 14.D About Fourier syntheses of particular use in macromolecular crystallography	495	(5)
Appendix 14.E Solvent content according to Matthews	500	(1)
Appendix 14.F The ab initio calculation of low-resolution envelopes	501	(8)
15. The integration of direct methods with isomorphous replacement techniques	509	(42)
15.1 Introduction	509	(1)
15.2 The joint probability distribution of the two normalized structure factors E(ph), E(dh) and related distributions	510	(2)
15.3 The joint probability distribution P(E(ph), E(pk), E(pl), E(dh), E(dk), E(dl)) with h + k + l = 0	512	(2)
15.4 Native protein-heavy atom derivative. The distributions for two-phase invariants	514	(3)
15.5 Native protein-heavy atom derivative. The distributions for three-phase invariants	517	(4)
15.6 A procedure for the direct phasing of proteins	521	(9)
15.7 About the quality of the electron density maps	530	(5)
15.8 Estimate of the quartet invariants	535	(5)
15.9 The triplet invariant estimate via its second representation	540	(2)
15.10 The relation between traditional SIR and direct methods techniques	542	(4)
15.11 Improving direct methods phases with heavy atom information	546	(3)
Appendix 15.A Karle's algebraic rule R(iso)	549	(1)
Appendix 15.B About the local scaling	550	(1)
16. The integration of direct methods with anomalous dispersion techniques	551	(51)
16.1 Introduction	551	(1)
16.2 Violation of the Friedel law	552	(4)
16.3 OAS: finding the positions of the anomalous scatterers	556	(3)
16.4 OAS: a basic algebraic formula	559	(2)
16.5 OAS: the resolution of phase ambiguity via algebraic methods	561	(3)
16.6 OAS: the resolution of phase ambiguity via traditional direct methods	564	(2)
16.7 MAD phasing methods	566	(3)
16.8 SIRAS and MIRAS: the algebraic methods	569	(8)
16.9 Wilson statistics for the OAS case	577	(4)
16.10 The distribution P(E(h), E(-h)) and related distributions in n.cs. crystals	581	(4)
16.11 The estimation of Phi = Phi(h) + Phi(-h) in cs. crystals or for symmetry restricted phases in cs. space groups	585	(1)
16.12 Direct methods for the OAS case: a first step	585	(2)
16.13 Algebraic estimates of triplet invariants: Karle's rule	587	(4)
16.14 The joint probability distribution P(E(h)(1), E(h)(2), E(h)(3), E(-h)(1), E(-h)(2), E(-h)(3))	591	(3)
16.15 The first applications of the triplet formula	594	(1)
16.16 Perspectives for the integration of direct methods with anomalous dispersion techniques	595	(1)
Appendix 16.A Probabilistic treatment of the errors in the SIRAS case	595	(1)
Appendix 16.B Coefficients in the distribution (16.76)	596	(3)
Appendix 16.C The SAS maximal principle	599	(1)
Appendix 16.D Anomalous scattering terms and MAD performances	600	(2)
17. Molecular replacement techniques and direct methods	602	(53)
17.1 Introduction	602	(2)
17.2 Rotation functions	604	(4)
17.3 Translation functions	608	(7)
17.4 Direct methods and the rotation problem	615	(4)
17.5 The translation problem: the role of the triplet invariants	619	(9)
17.6 Quartet invariants and the translation problem	628	(6)
17.7 The translation problem in polar space groups	634	(1)
Appendix 17.A Calculation of the rotation function in orthogonalized crystal axes	635	(10)
Appendix 17.B About the fast rotation function	645	(6)
Appendix 17.C Rotation functions using phases	651	(2)
Appendix 17.D The estimation of ((F^2)(F(mod))^2))	653	(2)
18. Maximum entropy techiques for crystal structure solution	655	(44)
18.1 Introduction	655	(1)
18.2 Entropy as a measure of the missing information	656	(2)
18.3 Jaynes' maximum entropy formalism	658	(6)
18.4 The maximum entropy criterion when a set of structure factors are know a priori	664	(7)
18.5 Maximum entropy and saddle point methods	671	(2)
18.6 Phase extension by maximum entropy	673	(3)
18.7 Entropy and likelihood: a phasing procedure	676	(2)
18.8 About various implementations of the ME formalism	678	(6)
18.9 Some applications of maximum entropy	684	(3)
Appendix 18.A Bayesian statistics and maximum entropy	687	(4)
Appendix 18.B A short recap of Newton's method	691	(4)
18.B.1 The Newton-Raphson method	691	(2)
18.B.2 Minimizing a function of several variables: gradient methods	693	(1)
18.B.3 Constrained optimization	694	(1)
Appendix 18.C About entropical properties of Gaussian and von Mises distributions	695	(4)
Appendix M.A Some basic results in probability theory	699	(16)
M.A.1 Probability distribution functions	699	(1)
M.A.2 Moments of a distribution	700	(1)
M.A.3 The characteristic function	700	(2)
M.A.4 Cumulants of a distribution	702	(1)
M.A.5 The normal or Gaussian distribution	703	(2)
M.A.6 The central limit theorem	705	(1)
M.A.7 Multivariate distribution	706	(4)
M.A.8 Evaluation of the moments in the structure factor distributions	710	(2)
M.A.9 Joint probability distributions of the signs of the structure factors	712	(1)
M.A.10 Some measures of location and dispersion in the statistics of directional data	713	(2)
Appendix M.B Congruences	715	(3)
Appendix M.C The gamma function	718	(2)
Appendix M.D The Hermite and Laguerre polynomials	720	(3)
Appendix M.E Some results in the theory of Bessel functions	723	(6)
M.E.1 Bessel functions	723	(5)
M.E.2 Generalized hypergeometric functions	728	(1)
Appendix M.F Some definite integrals and formulae of frequent application	729	(4)
References	733	(32)
Index	765