Refinement of the trial Structure

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Experimental Data ◽  
Least Squares ◽  
Likelihood Methods ◽  
Fourier Methods ◽  
Maximum Likelihood Methods ◽  
Density Map ◽  
Trial Structure ◽  
Phase Angles ◽  
Electron Density Map ◽  
True Structure

When approximate positions have been determined for most, if not all, of the atoms, it is time to begin the refinement of the structure. In this process the atomic parameters are varied systematically so as to give the best possible agreement of the observed structure factor amplitudes (the experimental data) with those calculated for the proposed trial structure. Common refinement techniques involve Fourier syntheses and processes involving least-squares or maximum likelihood methods. Although they have been shown formally to be nearly equivalent—differing chiefly in the weighting attached to the experimental observations—they differ considerably in manipulative details; we shall discuss them separately here. Many successive refinement cycles are usually needed before a structure converges to the stage at which the shifts from cycle to cycle in the parameters being refined are negligible with respect to their estimated errors. When least-squares refinement is used, the equations are, as pointed out below, nonlinear in the parameters being refined, which means that the shifts calculated for these parameters are only approximate, as long as the structure is significantly different from the “correct” one. With Fourier refinement methods, the adjustments in the parameters are at best only approximate anyway; final parameter adjustments are now almost always made by least squares, at least for structures not involving macromolecules. As indicated earlier (Chapters 8 and 9, especially Figure 9.8 and the accompanying discussion), Fourier methods are commonly used to locate a portion of the structure after some of the atoms have been found—that is, after at least a partial trial structure has been identified. Initially, only one or a few atoms may have been found, or maybe an appreciable fraction of the structure is now known. Once approximate positions for at least some of the atoms in the structure are known, the phase angles can be calculated. Then an approximate electron-density map calculated with observed structure amplitudes and computed phase angles will contain a blend of the true structure (from the structure amplitudes) with the trial structure (from the calculated phases).

Genetic and environmental effects on milk yield of Pitangueiras cattle

1984 ◽  
Vol 39 (2) ◽  
pp. 157-163 ◽  
Author(s):  
R. B. Lôbo ◽  
F. A. M. Duarte ◽  
A. A. M. Gonçalves ◽  
J. A. Oliveira ◽  
C. J. Wilcox
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Milk Yield ◽  
Environmental Effects ◽  
Intraclass Correlation ◽  
Statistical Analyses ◽  
Likelihood Methods ◽  
Maximum Production ◽  
Lactation Length ◽  
Maximum Likelihood Methods

ABSTRACTData from 5270 lactation records of 1380 cows sired by 132 bulls and recorded from 1962 to 1977 were analysed. Statistical analyses, using least squares and maximum likelihood methods, showed significant effects for genetic group, age of cow, month and year of calving, and lactation length. Overall mean milk yield was 2780 kg (CV = 0·31) with mean lactation length of 281 days. Maximum production occurred in the fifth lactation (104 or 105 months of age at calving) with a yield of 1·3 times that of the first lactation. Repeatability estimated by intraclass correlation was 0·40 (s.e. 0·03). Heritability estimated from paternal half-sib correlation was 0·16 (s.e. 0·06). Overall results were very similar to those obtained from research with European breeds in temperate areas.

scholarly journals Model morphing and sequence assignment after molecular replacement

2013 ◽  
Vol 69 (11) ◽  
pp. 2244-2250 ◽  
Author(s):  
Thomas C. Terwilliger ◽  
Randy J. Read ◽  
Paul D. Adams ◽  
Axel T. Brunger ◽  
Pavel V. Afonine ◽  
...  
Keyword(s):  
Electron Density ◽  
Main Chain ◽  
Molecular Replacement ◽  
Complete Model ◽  
Working Model ◽  
Density Map ◽  
The Mean ◽  
Electron Density Map ◽  
True Structure ◽  
Sequence Assignment

A procedure termed `morphing' for improving a model after it has been placed in the crystallographic cell by molecular replacement has recently been developed. Morphing consists of applying a smooth deformation to a model to make it match an electron-density map more closely. Morphing does not change the identities of the residues in the chain, only their coordinates. Consequently, if the true structure differs from the working model by containing different residues, these differences cannot be corrected by morphing. Here, a procedure that helps to address this limitation is described. The goal of the procedure is to obtain a relatively complete model that has accurate main-chain atomic positions and residues that are correctly assigned to the sequence. Residues in a morphed model that do not match the electron-density map are removed. Each segment of the resulting trimmed morphed model is then assigned to the sequence of the molecule using information about the connectivity of the chains from the working model and from connections that can be identified from the electron-density map. The procedure was tested by application to a recently determined structure at a resolution of 3.2 Å and was found to increase the number of correctly identified residues in this structure from the 88 obtained usingphenix.resolvesequence assignment alone (Terwilliger, 2003) to 247 of a possible 359. Additionally, the procedure was tested by application to a series of templates with sequence identities to a target structure ranging between 7 and 36%. The mean fraction of correctly identified residues in these cases was increased from 33% usingphenix.resolvesequence assignment to 47% using the current procedure. The procedure is simple to apply and is available in thePhenixsoftware package.

A comparison of least squares and maximum likelihood methods using sine fitting in ADC testing

Measurement ◽  
2013 ◽  
Vol 46 (10) ◽  
pp. 4362-4368 ◽  
Author(s):  
Ján Šaliga ◽  
István Kollár ◽  
Linus Michaeli ◽  
Ján Buša ◽  
Jozef Lipták ◽  
...  
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Likelihood Methods ◽  
Maximum Likelihood Methods ◽  
Fitting In ◽  
Adc Testing

The derivation of trial structures. I. Analytical methods for direct phase determination

2010 ◽  
Author(s):  
Jenny Pickworth Glusker ◽  
Kenneth N. Trueblood
Keyword(s):  
Crystal Structure ◽  
Structure Determination ◽  
Direct Methods ◽  
Analytical Techniques ◽  
Patterson Function ◽  
Structure Factors ◽  
Term Trial ◽  
Trial Structure ◽  
Electron Density Map ◽  
True Structure

As indicated at the start of Chapter 4, after the diffraction pattern has been recorded and measured, the next stage in a crystal structure determination is solving the structure—that is, finding a suitable “trial structure” that contains approximate positions for most of the atoms in the unit cell of known dimensions and space group. The term “trial structure” implies that the structure that has been found is only an approximation to the correct or “true” structure, while “suitable” implies that the trial structure is close enough to the true structure that it can be smoothly refined to give a good fit to the experimental data. Methods for finding suitable trial structures form the subject of this chapter and the next. In the early days of structure determination, trial and error methods were, of necessity, almost the only available way of solving structures. Structure factors for the suggested “trial structure” were calculated and compared with those that had been observed. When more productive methods for obtaining trial structures—the “Patterson function” and “direct methods”—were introduced, the manner of solving a crystal structure changed dramatically for the better. We begin with a discussion of so-called “direct methods.” These are analytical techniques for deriving an approximate set of phases from which a first approximation to the electron-density map can be calculated. Interpretation of this map may then give a suitable trial structure. Previous to direct methods, all phases were calculated (as described in Chapter 5) from a proposed trial structure. The search for other methods that did not require a trial structure led to these phaseprobability methods, that is, direct methods. A direct solution to the phase problem by algebraic methods began in the 1920s (Ott, 1927; Banerjee, 1933; Avrami, 1938) and progressed with work on inequalities by David Harker and John Kasper (Harker and Kasper, 1948). The latter authors used inequality relationships put forward by Augustin Louis Cauchy and Karl Hermann Amandus Schwarz that led to relations between the magnitudes of some structure factors.

Phasingviapure crystallographic least squares: an unexpected feature

2018 ◽  
Vol 74 (2) ◽  
pp. 123-130
Author(s):  
Maria Cristina Burla ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Least Squares ◽  
Structure Refinement ◽  
Current Model ◽  
Main Tool ◽  
Solution Procedure ◽  
Medium Size ◽  
Density Map ◽  
Electron Density Map ◽  
First Time ◽  
Least Squares Techniques

Crystallographic least-squares techniques, the main tool for crystal structure refinement of small and medium-size molecules, are for the first time used forab initiophasing. It is shown that the chief obstacle to such use, the least-squares severe convergence limits, may be overcome by a multi-solution procedure able to progressively recognize and discard model atoms in false positions and to include in the current model new atoms sufficiently close to correct positions. The applications show that the least-squares procedure is able to solve many small structures without the use of important ancillary tools:e.g.no electron-density map is calculated as a support for the least-squares procedure.

scholarly journals Statistical Inference for Periodic Self-Exciting Threshold Integer-Valued Autoregressive Processes

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 765
Author(s):  
Congmin Liu ◽  
Jianhua Cheng ◽  
Dehui Wang
Keyword(s):  
Least Squares ◽  
Likelihood Inference ◽  
Simulation Studies ◽  
Autoregressive Processes ◽  
Disability Benefits ◽  
Practical Application ◽  
Likelihood Methods ◽  
Short Term ◽  
Time Series Dataset ◽  
Maximum Likelihood Methods

This paper considers the periodic self-exciting threshold integer-valued autoregressive processes under a weaker condition in which the second moment is finite instead of the innovation distribution being given. The basic statistical properties of the model are discussed, the quasi-likelihood inference of the parameters is investigated, and the asymptotic behaviors of the estimators are obtained. Threshold estimates based on quasi-likelihood and least squares methods are given. Simulation studies evidence that the quasi-likelihood methods perform well with realistic sample sizes and may be superior to least squares and maximum likelihood methods. The practical application of the processes is illustrated by a time series dataset concerning the monthly counts of claimants collecting short-term disability benefits from the Workers’ Compensation Board (WCB). In addition, the forecasting problem of this dataset is addressed.

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