Mathematical structure-activity

Equation 2). Some of the commonly used variables are indicated in the scheme above. At this very early stage in the development of mathematical structure—activity work a complete set of well understood parameters has not been developed however, the set above has at least allowed us to start serious work. The greatest lack at present is for a suitable numerical way to define the geometry of organic compounds—i.e., one that does not require an inordinate number of terms. 

At about the time of Free and Wilsons work, Kopecky and co-workers introduced a similar mathematical structure-activity model (34, 35). They tested four equations (Equations 8-11) for the expression of the quantitative difference between the log LD50 values of p- (34) and m-(35) disubstituted benzenes and benzene. 

Free SM Jr, Wilson JW. A mathematical contribution to structure-activity studies. J Med Chem 1964 7 395-9. 

With the development of accurate computational methods for generating 3D conformations of chemical structures, QSAR approaches that employ 3D descriptors have been developed to address the problems of 2D QSAR techniques, that is, their inability to distinguish stereoisomers. Examples of 3D QSAR include molecular shape analysis (MSA) [26], distance geometry,and Voronoi techniques [27]. The MSA method utilizes shape descriptors and MLR analysis, whereas the other two approaches apply atomic refractivity as structural descriptor and the solution of mathematical inequalities to obtain the quantitative relationships. These methods have been applied to study structure-activity relationships of many data sets by Hopfinger and Crippen, respectively. Perhaps the most popular example of the 3D QSAR is the com-... 

In 1868 two Scottish scientists, Crum Brown and Fraser [4] recognized that a relation exists between the physiological action of a substance and its chemical composition and constitution. That recognition was in effect the birth of the science that has come to be known as quantitative structure-activity relationship (QSAR) studies a QSAR is a mathematical equation that relates a biological or other property to structural and/or physicochemical properties of a series of (usually) related compounds. Shortly afterwards, Richardson [5] showed that the narcotic effect of primary aliphatic alcohols varied with their molecular weight, and in 1893 Richet [6] observed that the toxicities of a variety of simple polar chemicals such as alcohols, ethers, and ketones were inversely correlated with their aqueous solubilities. Probably the best known of the very early work in the field was that of Overton [7] and Meyer [8], who found that the narcotic effect of simple chemicals increased with their oil-water partition coefficient and postulated that this reflected the partitioning of a chemical between the aqueous exobiophase and a lipophilic receptor. This, as it turned out, was most prescient, for about 70% of published QSARs contain a term relating to partition coefficient [9]. 

Predictions based on structure-activity relationships (SARs), including qualitative and quantitative mathematical models, and the use of read-across data from related chemicals. 

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... 

In many cases of practical interest, no theoretically based mathematical equations exist for the relationships between x and y we sometimes know but often only assume that relationships exist. Examples are for instance modeling of the boiling point or the toxicity of chemical compounds by variables derived from the chemical structure (molecular descriptors). Investigation of quantitative structure-property or structure-activity relationships (QSPR/QSAR) by this approach requires multivariate calibration methods. For such purely empirical models—often with many variables—the... 

Keywords Skin permeability Percutaneous absorption Skin penetration Mathematical model Quantitative structure-activity relationships Permeability coefficient Human skin... 

To overcome this weakness, we are developing a quantitative structure-activity strategy that is conceptually applicable to all chemicals. To be applicable, at least three criteria are necessary. First, we must be able to calculate the descriptors or Independent variables directly from the chemical structure and, presumably, at a reasonable cost. Second, the ability to calculate the variables should be possible for any chemical. Finally, and most importantly, the variables must be related to a parameter of Interest so that the variables can be used to predict or classify the activity or behavior of the chemical (j ) One important area of research is the development of new variables or descriptors that quantitatively describe the structure of a chemical. The development of these indices has progressed into the mathematical areas of graph theory and topology and a large number of potentially valuable molecular descriptors have been described (7-9). Our objective is not concerned with the development of new descriptors, but alternatively to explore the potential applications of a group of descriptors known as molecular connectivity indices (10). 

Odor and taste quality can be mapped by multidimensional scaling (MDS) techniques. Physicochemical parameters can be related to these maps by a variety of mathematical methods including multiple regression, canonical correlation, and partial least squares. These approaches to studying QSAR (quantitative structure-activity relationships) in the chemical senses, along with procedures developed by the pharmaceutical industry, may ultimately be useful in designing flavor compounds by computer. 

There are several properties of a chemical that are related to exposure potential or overall reactivity for which structure-based predictive models are available. The relevant properties discussed here are bioaccumulation, oral, dermal, and inhalation bioavailability and reactivity. These prediction methods are based on a combination of in vitro assays and quantitative structure-activity relationships (QSARs) [3]. QSARs are simple, usually linear, mathematical models that use chemical structure descriptors to predict first-order physicochemical properties, such as water solubility. Other, similar models can then be constructed that use the first-order physicochemical properties to predict more complex properties, including those of interest here. Chemical descriptors are properties that can be calculated directly from a chemical structure graph and can include abstract quantities, such as connectivity indices, or more intuitive properties, such as dipole moment or total surface area. QSAR models are parameterized using training data from sets of chemicals for which both structure and chemical properties are known, and are validated against other (independent) sets of chemicals. 

Since one of the main aims of green chemistry is to reduce the use and/or production of toxic chemicals, it is important for practitioners to be able to make informed decisions about the inherent toxicity of a compound. Where sufficient ecotoxicological data have been generated and risk assessments performed, this can allow for the selection of less toxic options, such as in the case of some surfactants and solvents [94, 95]. When toxicological data are limited, for example, in the development of new pharmaceuticals (see Section 15.4.3) or other consumer products, there are several ways in which information available from other chemicals may be helpful to estimate effect measures for a compound where data are lacking. Of these, the most likely to be used are the structure-activity relationships (SARs, or QSARs when they are quantitative). These relationships are also used to predict chemical properties and behavior (see Chapter 16). There often are similarities in toxicity between chemicals that have related structures and/or functional subunits. Such relationships can be seen in the progressive change in toxicity and are described in QSARs. When several chemicals with similar structures have been tested, the measured effects can be mathematically related to chemical structure [96-98] and QSAR models used to predict the toxicity of substances with similar structure. Any new chemicals that have similar structures can then be assumed to elicit similar responses. 

In contrast to the qualitative better and worse approach of traditional SAR, many researchers have sought to develop a method that is able to quantitatively link molecular structure changes to biological activity, a quantitative structure-activity relationship (QSAR). As a method, QSAR strives to develop a mathematical formula to relate biological activity as a function of molecular or substituent properties. 

In addition to using PMs, predictions of toxic hazard can also be made by using structure-activity relationships (SARs). A quantitative structure-activity relationship (QSAR) can be defined as any mathematical model for predicting biological activity from the structure or physicochemical properties of a chemical. In this chapter, the premodifer quantitative is used in accordance with the recommendation of Livingstone (1995) to indicate that a quantitative measure of chemical structure is used. In contrast, a SAR is simply a (qualitative) association between a specific molecular (sub)structure and biological activity. 

Structure-activity relationships (SARs) and quantitative structure-activity relationships (QSARs), referred to collectively as QSARs, can be used for the prediction of physicochemical properties, environmental fate parameters (e.g., accumulation and biodegradation), human health effects, and ecotoxicological effects. A SAR is a (qualitative) association between a chemical substructure and the potential of a chemical containing the substructure to exhibit a certain physical or biological effect. A QS AR is a mathematical model that relates a quantitative measure of chemical structure (e.g., a physicochemical property) to a physical property or to a biological effect (e.g., a toxicological endpoint). 

Quantitative Structure Activity Relationship (QSAR) is a method that makes predictions by the quantitative description of molecular properties with the use of descriptors of the chemical structure (Dearden 2003). This means QSAR models describe the quantitative or calculated relationship between a chemical structure and their biological activity (e.g. toxicity) with the help of chemical descriptors that are generated from the molecular structure (Durham and Pearl 2001). This relationship is described in from of a mathematical equation (e.g. log 1/C = a tt + b a +. .. + const). QSAR models generally show better predictivity if all compounds of a dataset involved in the prediction are derived from a congeneric series of compounds, that means they should all act by the same mechanism of action, since the physico-chemical and structural descriptors used in the QSAR reflect the same mechanism of action. Sometimes it is difficult to determine the mechanism of action, so series of compounds involved in a QSAR model are often restricted to a given chemical class in the hope that this will ensure a single mechanism of action (Dearden 2003). 

---