VLD algorithm and hybrid Fourier syntheses

2012 ◽  
Vol 45 (6) ◽  
pp. 1287-1294 ◽  
Author(s):  
Maria Cristina Burla ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Electron Density ◽  
Standard Procedure ◽  
Joint Probability ◽  
Distribution Functions ◽  
Medium Size ◽  
Joint Probability Distribution ◽  
Fourier Synthesis ◽  
Acta Cryst ◽  
The Difference ◽  
Difference Fourier

The VLD (vive la difference) phasing algorithm combines the model electron density with the difference electron densityviareciprocal space relationships to obtain new phase values and drive them to the correct values. The process is iterative and has been applied to small and medium-size structures and to proteins. Hybrid Fourier syntheses show properties that are intermediate between those of the observed synthesis (whose peaks should correspond to the most probable atomic positions) and those of the difference synthesis (whose positive and negative peaks should correspond to missed atomic positions and to false atoms of the model, respectively). Thanks to these properties some hybrid syntheses can be used in the phase extension and refinement step, to reduce the model bias and more rapidly move to the target structure. They have been recently revisitedviathe method of joint probability distribution functions [Burla, Carrozzini, Cascarano, Giacovazzo & Polidori (2011).Acta. Cryst. A67, 447–455]. The results suggested that VLD could be usefully combined, forab initiophasing, with the hybrid rather than with the difference Fourier synthesis. This paper explores the feasibility of such a combination and shows that the original VLD algorithm is only one of several variants, all with relevant phasing capacity. The study explores the role of several parameters in order to design a standard procedure with optimized phasing power.

Phasing medium-size structures and proteins by the VLD algorithm

2011 ◽  
Vol 44 (1) ◽  
pp. 193-199 ◽  
Author(s):  
Maria Cristina Burla ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Electron Density ◽  
Random Model ◽  
Atomic Resolution ◽  
Medium Size ◽  
Molecular Models ◽  
Fourier Synthesis ◽  
Large Structures ◽  
Density Maps ◽  
Difference Fourier ◽  
Correct Structure

The VLD algorithm is based on the properties of a new difference Fourier synthesis and allows the recovery of the correct structure starting from a random model. It has previously been applied to a set of small structures. The first aim of this paper is to extend the complexity range to medium-size molecules and to proteins, provided the data have atomic resolution. The algorithm always works in the correct space group and uses electron density maps rather than molecular models. It has been modified in order to (i) provide, at the end of the procedure, molecular models that are automatically interpreted (in a chemical sense), rather than electron density maps, and (ii) show the variety of ways in which VLD may be implemented. The applications show that VLD is able to solve large structures, in favorable cases by using a small number of attempts, and that this property also extends to some of its variants.

scholarly journals Chlorido(pyridine-κN)(5,10,15,20-tetraphenylporphyrinato-κ4N)cobalt(III) chloroform hemisolvate

2012 ◽  
Vol 68 (8) ◽  
pp. m1104-m1105 ◽  
Author(s):  
Yassin Belghith ◽  
Jean-Claude Daran ◽  
Habib Nasri
Keyword(s):  
Bond Length ◽  
Electron Density ◽  
Title Complex ◽  
Acta Cryst ◽  
Π Interactions ◽  
The Difference ◽  
Difference Fourier ◽  
Atom Bond

In the title complex, [CoCl(C44H28N4)(C5H5N)]·0.5CHCl3or [CoIII(TPP)Cl(py)]·0.5CHCl3(where TPP is the dianion of tetraphenylporphyrin and py is pyridine), the average equatorial cobalt–pyrrole N atom bond length (Co—Np) is 1.958 (7) Å and the axial Co—Cl and Co—Npydistances are 2.2339 (6) and 1.9898 (17) Å, respectively. The tetraphenylporphyrinate dianion exhibits an important nonplanar conformation with major ruffling and saddling distortions. In the crystal, molecules are linkedviaweak C—H...π interactions. In the difference Fourier map, a region of highly disordered electron density was estimated using the SQUEEZE routine [PLATON; Spek (2009),Acta Cryst.D65, 148–155] to be equivalent to one half-molecule of CHCl3per molecule of the complex.

From a random to the correct structure: the VLD algorithm

2010 ◽  
Vol 43 (4) ◽  
pp. 825-836 ◽  
Author(s):  
Maria Cristina Burla ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Electron Density ◽  
Random Model ◽  
Good Representation ◽  
Large Set ◽  
Target Structure ◽  
Difference Electron Density ◽  
Fourier Synthesis ◽  
Acta Cryst ◽  
The Difference ◽  
Difference Term

A recent probabilistic reformulation of the difference electron-density Fourier synthesis [Burla, Caliandro, Giacovazzo & Polidori (2010).Acta Cryst.A66, 347–361] suggested that the most suitable Fourier coefficients are the sum of the classical difference term (mF−DFp) with a flipping term, depending on the model and on its quality. The flipping term is dominant when the model is poor and is negligible when the model is a good representation of the target structure. In the case of a random model the Fourier coefficient does not vanish and therefore could allow the recovery of the target structure from a random model. This paper describes a new phasing algorithm which does not require use of the concept of structure invariants or semi-invariants: it is based only on the properties of the new difference electron density and of the observed Fourier synthesis. The algorithm is cyclic and very easy to implement. It has been applied to a large set of small-molecule structures to verify the suitability of the approach.

Fuzzy Multi-Objective Programming With Joint Probability Distribution

2019 ◽  
pp. 263-295
Keyword(s):  
Probability Distribution ◽  
Goal Programming ◽  
Joint Probability ◽  
Chance Constraints ◽  
Fuzzy Goal Programming ◽  
Probabilistic Constraints ◽  
Joint Probability Distribution ◽  
Multi Objective ◽  
Fuzzy Goal ◽  
The Difference

In this chapter, a fuzzy goal programming (FGP) model is employed for solving multi-objective linear programming (MOLP) problem under fuzzy stochastic uncertain environment in which the probabilistic constraints involves fuzzy random variables (FRVs) following joint probability distribution. In the preceding chapters, the authors explain about linear, fractional, quadratic programming models with multiple conflicting objectives under fuzzy stochastic environment. But the chance constraints in these chapters are considered independently. However, in practical situations, the decision makers (DMs) face various uncertainties where the chance constraints occur jointly. By considering the above fact, the authors presented a solution methodology for fuzzy stochastic MOLP (FSMOLP) with joint probabilistic constraint following some continuous probability distributions. Like the other chapters, chance constrained programming (CCP) methodology is adopted for handling probabilistic constraints. But the difference is that in the earlier chapters chance constraints are considered independently, whereas in this chapter all the chance constraints are taken jointly. Then the transformed problem involving possibilistic uncertainty is converted into a comparable deterministic problem by using the method of defuzzification of the fuzzy numbers (FNs). Objectives are now solved independently under the set of modified system constraints to obtain the best solution of each objective. Then the membership function for each objective is constructed, and finally, a fuzzy goal programming (FGP) model is developed for the achievement of the highest membership goals to the extent possible by minimizing group regrets in the decision-making context.

Joint probability distribution functions when a model is available: Fourier syntheses

2013 ◽  
Author(s):  
Carmelo Giacovazzo
Keyword(s):  
Probability Distribution ◽  
Current Model ◽  
Joint Probability ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Target Structure ◽  
Probability Distribution Functions ◽  
Structure Factors ◽  
Joint Probability Distributions

The title of this chapter may seem a little strange; it relates Fourier syntheses, an algebraic method for calculating electron densities, to the joint probability distribution functions of structure factors, which are devoted to the probabilistic estimate of s.i.s and s.s.s. We will see that the two topics are strictly related, and that optimization of the Fourier syntheses requires previous knowledge and the use of joint probability distributions. The distributions used in Chapters 4 to 6 are able to estimate s.i. or s.s. by exploiting the information contained in the experimental diffraction moduli of the target structure (the structure one wants to phase). An important tool for such distributions are the theories of neighbourhoods and of representations, which allow us to arrange, for each invariant or seminvariant Φ, the set of amplitudes in a sequence of shells, each contained within the subsequent shell, with the property that any s.i. or s.s. may be estimated via the magnitudes constituting any shell. The resulting conditional distributions were of the type, . . . P(Φ| {R}), (7.1) . . . where {R} represents the chosen phasing shell for the observed magnitudes. The more information contained within the set of observed moduli {R}, the better will be the Φ estimate. By definition, conditional distributions (7.1) cannot change during the phasing process because prior information (i.e. the observed moduli) does not change; equation (7.1) maintains the same identical algebraic form. However, during any phasing process, various model structures progressively become available, with different degrees of correlation with the target structure. Such models are a source of supplementary information (e.g. the current model phases) which, in principle, can be exploited during the phasing procedure. If this observation is accepted, the method of joint probability distribution, as described so far, should be suitably modified. In a symbolic way, we should look for deriving conditional distributions . . . P (Φ| {R}, {Rp}) , (7.2) . . . rather than (7.1), where {Rp} represents a suitable subset of the amplitudes of the model structure factors. Such an approach modifies the traditional phasing strategy described in the preceding chapters; indeed, the set {Rp} will change during the phasing process in conjunction with the model changes, which will continuously modify the probabilities (7.2).

The approach of the joint probability distribution functions: the SIR-MIR, SAD-MAD and SIRAS-MIRAS, cases

2002 ◽  
Vol 217 (12) ◽  
Author(s):  
C. Giacovazzo ◽  
M. Ladisa ◽  
D. Siliqi
Keyword(s):  
Probability Distribution ◽  
Joint Probability ◽  
Distribution Functions ◽  
Joint Probability Distribution ◽  
Probability Distribution Functions

AbstractThe method of the joint probability distribution functions has been recently applied to SIR-MIR, SAD-MAD and SIRAS-MIRAS cases. The capacity of the method to treat various forms of errors (i.e., errors in measurements, possible lack of isomorphism, errors in a substructure model when a model is

scholarly journals Success run statistics defined on an urn model

2007 ◽  
Vol 39 (04) ◽  
pp. 991-1019 ◽  
Author(s):  
Frosso S. Makri ◽  
Andreas N. Philippou ◽  
Zaharias M. Psillakis
Keyword(s):  
Joint Probability ◽  
Nonparametric Tests ◽  
Distribution Functions ◽  
Binary Sequences ◽  
Binomial Coefficients ◽  
Joint Probability Distribution ◽  
Urn Model ◽  
Success Runs ◽  
Mean Values ◽  
Success Run

Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

About difference electron densities and their properties

2017 ◽  
Vol 73 (6) ◽  
pp. 460-473 ◽  
Author(s):  
Maria Cristina Burla ◽  
Benedetta Carrozzini ◽  
Giovanni Luca Cascarano ◽  
Carmelo Giacovazzo ◽  
Giampiero Polidori
Keyword(s):  
Crystal Structure ◽  
Electron Density ◽  
Probability Distribution Function ◽  
Strong Support ◽  
Joint Probability ◽  
The Other ◽  
Joint Probability Distribution ◽  
Electron Densities ◽  
Structure Solution ◽  
Joint Probability Distribution Function

Difference electron densities do not play a central role in modern phase refinement approaches, essentially because of the explosive success of the EDM (electron-density modification) techniques, mainly based on observed electron-density syntheses. Difference densities however have been recently rediscovered in connection with theVLD(Vive la Difference) approach, because they are a strong support for strengthening EDM approaches and forab initiocrystal structure solution. In this paper the properties of the most documented difference electron densities, here denoted asF−Fp,mF−FpandmF−DFpsyntheses, are studied. In addition, a fourth new difference synthesis, here denoted as {\overline F_q} synthesis, is proposed. It comes from the study of the same joint probability distribution function from which theVLDapproach arose. The properties of the {\overline F_q} syntheses are studied and compared with those of the other three syntheses. The results suggest that the {\overline F_q} difference may be a useful tool for making modern phase refinement procedures more efficient.

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