TELS: least-squares solution of the structure-invariant equations

Han, Fusen; DeTitta, George T.; Hauptman, H.

doi:10.1107/s0108767391002969

Cited by 8 publications

(7 citation statements)

References 6 publications

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“…The ®rst method suggested by Han et al (1991) was based on a trial-and-error procedure to determine integer values (À1, 0, 1), which when multiplied by 2% and added to 3(h, k) would allow the resultant linear equations to be solved in a straightforward least-squares manner (Woolfson, 1977). The trialand-error aspect of obtaining these integers for M triples equations could in principle be avoided, but required the inversion of an enormous M Â M matrix (Langs & Han, 1995).…”

Section: Ab Initio Solution Methodsmentioning

confidence: 99%

Progress on the direct-methods solution of macromolecular structures using single-wavelength anomalous-dispersion (SAS) data

1999

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In the past few years, a number of strategies have been outlined to resolve the SAS phase ambiguity given that unique estimates 3(h, k) of the triple invariants are available. A new least-squares method is described that can in principle resolve the phase ambiguity to determine macromolecular phases provided that 3(h, k) estimates are unbiased. Limitations of the method in practical applications are discussed. An example is given where the correct solution can be identi®ed by use of the SAS tangent formula in the instance that traditional SAS phasing methods have lead to an incorrect heavy-atom substructure.

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Section: Ab Initio Solution Methodsmentioning

confidence: 99%

Progress on the direct-methods solution of macromolecular structures using single-wavelength anomalous-dispersion (SAS) data

1999

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“…Several methods have been proposed to further take advantage of these SAS triple estimates (Langs, 1986;Han et al, 1991;Hauptman & Han, 1993;Langs & Han, 1995) to ultimately obtain the native crystal phases. Multisolution tangent-formula procedures are easily modi®ed to use SAS invariant estimates and suitable solutions are often identi®ed in a reasonable number of trials by low values of their SAS phase-re®nement residual [Hauptman & Han, 1993, equation (14)].…”

Section: Introductionmentioning

confidence: 99%

Improvement of SAS triple invariant estimates for macromolecular direct-methods phasing

2001

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Single-wavelength anomalous dispersion (SAS) data can in principle be phased by direct methods since a priori estimates of the three-phase structure invariants can be computed from these data. The mean phase error of the most reliable triple estimates for a small protein, however, is typically no better than 60, and does not bode well for applications to larger structures. A procedure is described that can substantially lower the error in these estimates and introduce a larger number of useful triple invariants into the phasing process. The mean phase error of the most reliable triples for a 2.5 A Ê resolution data set from a Pt derivative of a 115-residue protein was reduced from 55 to 25 by this method. It was also possible to identify a signi®cant number of the poorest triple estimates, those with mean phase errors approaching 90, such that they could be reliably down-weighted or excluded from the phasing process.

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“…If ~0n is restricted to lie in the interval 0 to 2Tr, and the known values of the initial p + 3 phases ~o and the known estimates on the right-hand side of (5) are employed, then the number of unknown integer multiples of 2zr to be added to the right-hand side of each equation (5) is greatly restricted. In fact, a simple argument shows that for the additional ql equations (5) involving the single unknown phase 60H there are precisely q~ + 1 different sets of additive integer multiples of 2zr (Han et al, 1991). The least-squares solution of each of these q~ + 1 systems of equations (5), each consisting of q + ql equations in p + 1 unknowns, yields q~ + 1 sets of values for the p +l unknown phases.…”

Section: The 2~-ambiguitymentioning

confidence: 99%

“…The average A value of these seven invariants is 1.03, which is somewhat larger than the average of the total. The origin-fixing reflections are treated as discussed in a previous publication (Han et al, 1991). The second step resolves the integer problem in a different way.…”

Section: Least-squares Solution Of the Structure-invariant Equationsmentioning

confidence: 99%

Phasing macromolecular structures via structure-invariant algebra

Hauptman

Han²

1993

Acta Cryst D

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Owing to the breakdown of Friedel's law when anomalous scatterers are present, unique values of the three-phase structure invariants in the whole range from 0 to 2rr are determined by measured values of diffraction intensities alone. Two methods are described for going from presumed known values of these invariants to the values of the individual phases. The first, dependent on a scheme for resolving the 2rr ambiguity in the estimate toHr of the triplet ~H + tpK + ~p-n-~:, solves by least squares the resulting redundant system of linear equations ~n + ~pK + ~p-a-K = tonK. The second attempts to minimize the weighted sum of squares of differences between the true values of the cosine and sine invariants and their estimates. The latter method is closely related to one based on the 'minimal principle' which determines the values of a large set of phases as the constrained global minimum of a function of all the phases in the set. Both methods work in the sense that they yield values of the individual phases substantially better than the values of the initial estimates of the triplets. However, the second method proves to be superior to the first but requires, in addition to estimates of the triplets, initial estimates of the values of the individual phases.

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TELS: least-squares solution of the structure-invariant equations

Cited by 8 publications

References 6 publications

Progress on the direct-methods solution of macromolecular structures using single-wavelength anomalous-dispersion (SAS) data

Progress on the direct-methods solution of macromolecular structures using single-wavelength anomalous-dispersion (SAS) data

Improvement of SAS triple invariant estimates for macromolecular direct-methods phasing

Phasing macromolecular structures via structure-invariant algebra

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