Statistical methods, structure

B. Widom, Structure and Thermodynamics of Interfaces, in Statistical Mechanics and Statistical Methods in Theory and Application, Plenum, New York, 1977, pp. 33-71. 

To understand the recommendations for structure descriptors in order to be able to apply them in QSAR or drug design in conjunction with statistical methods or machine learning techniques. 

A structure descriptor is a mathematical representation of a molecule resulting from a procedure transforming the structural information encoded within a symbolic representation of a molecule. This mathematical representation has to be invariant to the molecule s size and number of atoms, to allow model building with statistical methods and artificial neural networks. 

The abbreviation QSAR stands for quantitative structure-activity relationships. QSPR means quantitative structure-property relationships. As the properties of an organic compound usually cannot be predicted directly from its molecular structure, an indirect approach Is used to overcome this problem. In the first step numerical descriptors encoding information about the molecular structure are calculated for a set of compounds. Secondly, statistical methods and artificial neural network models are used to predict the property or activity of interest, based on these descriptors or a suitable subset. A typical QSAR/QSPR study comprises the following steps structure entry or start from an existing structure database), descriptor calculation, descriptor selection, model building, model validation. 

The data analysis module of ELECTRAS is twofold. One part was designed for general statistical data analysis of numerical data. The second part offers a module For analyzing chemical data. The difference between the two modules is that the module for mere statistics applies the stati.stical methods or rieural networks directly to the input data while the module for chemical data analysis also contains methods for the calculation ol descriptors for chemical structures (cl. Chapter 8) Descriptors, and thus structure codes, are calculated for the input structures and then the statistical methods and neural networks can be applied to the codes. 

Completely ah initio predictions can be more accurate than any experimental result currently available. This is only true of properties that depend on the behavior of isolated molecules. Colligative properties, which are due to the interaction between molecules, can be computed more reliably with methods based on thermodynamics, statistical mechanics, structure-activity relationships, or completely empirical group additivity methods. 

To gain the most predictive utility as well as conceptual understanding from the sequence and structure data available, careful statistical analysis will be required. The statistical methods needed must be robust to the variation in amounts and quality of data in different protein families and for structural features. They must be updatable as new data become available. And they should help us generate as much understanding of the determinants of protein sequence, structure, dynamics, and functional relationships as possible. 

A common use of statistics in structural biology is as a tool for deriving predictive distributions of strucmral parameters based on sequence. The simplest of these are predictions of secondary structure and side-chain surface accessibility. Various algorithms that can learn from data and then make predictions have been used to predict secondary structure and surface accessibility, including ordinary statistics [79], infonnation theory [80], neural networks [81-86], and Bayesian methods [87-89]. A disadvantage of some neural network methods is that the parameters of the network sometimes have no physical meaning and are difficult to interpret. 

P Stolorz, A Lapedes, Y Xia. Predicting protein secondary structure using neural net and statistical methods. J Mol Biol 225 363-377, 1992. 

The Number of Resonance Structures. In calculating the number of resonance structures per atom, vhypel for hyperelec-tronic metals with v = z+ 1/2, we use the same statistical method as for hypoelectronic metals except that the factor 2m is introduced to correct for the fact that there are two kinds of atoms forming z + I bonds, M+ and M, which differ in that M has an unshared electron pair and M+ does not have one. The equation for vhyper is... 

The hrst step in theoretical predictions of pathway branching are electronic structure ab initio) calculations to define at least the lowest Born-Oppenheimer electronic potential energy surface for a system. For a system of N atoms, the PES has (iN — 6) dimensions, and is denoted V Ri,R2, - , RiN-6)- At a minimum, the energy, geometry, and vibrational frequencies of stationary points (i.e., asymptotes, wells, and saddle points where dV/dRi = 0) of the potential surface must be calculated. For the statistical methods described in Section IV.B, information on other areas of the potential are generally not needed. However, it must be stressed that failure to locate relevant stationary points may lead to omission of valid pathways. For this reason, as wide a search as practicable must be made through configuration space to ensure that the PES is sufficiently complete. Furthermore, a search only of stationary points will not treat pathways that avoid transition states. 

The statistical distribution of r values for long polymer chains and the influence of chain structure and hindrance to rotation about chain bonds on its root-mean-square value will be the topics of primary concern in the present chapter. We thus enter upon the second major application of statistical methods to polymer problems, the first of these having been discussed in the two chapters preceding. Quite apart from whatever intrinsic interest may be attached to the polymer chain configuration problem, its analysis is essential for the interpretation of rubberlike elasticity and of dilute solution properties, both hydrodynamic and thermodynamic, of polymers. These problems will be dealt with in following chapters. The content of the present... 

Once the particular branching process that specifies the probability measure on the set of macromolecules of a polymer specimen has been identified, the statistical method provides the possibility to determine any statistical characteristic of the chemical structure of this specimen. In particular, the dependence of the weight fraction of a sol on conversion can be calculated by formulas [extending those (55)] which are obtainable from (61) provided the value of dummy variable s is put unity ... 

The goal of EDA is to reveal structures, peculiarities and relationships in data. So, EDA can be seen as a kind of detective work of the data analyst. As a result, methods of data preprocessing, outlier selection and statistical data analysis can be chosen. EDA is especially suitable for interactive proceeding with computers (Buja et al. [1996]). Although graphical methods cannot substitute statistical methods, they can play an essential role in the recognition of relationships. An informative example has been shown by Anscombe [1973] (see also Danzer et al. [2001], p 99) regarding bivariate relationships. 

Rotaxanes are the compounds consisting of noncovalent entities called rotor and axle [77], Figure 21 illustrates them schematically. Initially, attempts were made to prepare them by statistical methods, so that the yields were necessarily very low [78-80], Recently, methods have been proposed for their more efficient synthesis, with renewed interest in their unique structure and properties. Section 4.1 summarizes some of the typical results obtained. 

Empirical statistical methods, which are based upon data generated from studying proteins of known three-dimensional structure and correlation of such proteins primary amino acid sequences with structural features. 

The early structure-activity studies [69, 70] were limited in the number of compounds studied but showed that reasonable correlations could be drawn between the structure of compounds and their biological activity without a complete understanding of the underlying mechanisms involved. Chou and Jurs [71] expanded the approach to structure-activity relationships by applying computer-assisted mathematical and statistical methods to a large set of N-nitroso compounds. These methods... 

Basak, S. C., Mills, D., Hawkins, D. M., Kraker, J. J. Quantitative structure-activity relationship (QSAR) modeling of human blood air partitioning with proper statistical methods and validation. Chem. Biodivers., accepted. 

Statistical methods based on generation of branched and crosslinked structures from units in different reaction states. 

Despite its simplicity, the statistical method has been quite successful in predicting the effect of various chemical variables on network formation (cf. e.g. [29, 30, 34-37]). Since the internal structure of the gel can be characterized to a certain degree by the statistical method (e.g. average size of dangling chains and weight fraction of material in them), these methods offer a basis for correlations between structure and viscoelastic properties. 

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