Molar nonlinear relationships

Isooptoacoustic point A wavelength, wavenumber, or frequency at which the total energy emitted by a sample as heat does not change upon a chemical reaction or physical change of the sample. Its position depends on the experimental conditions. The spectral differences between the isosbestic points and the isooptoacoustic points are the result of the nonlinear relationship between the molar absorption coefficient and the photoacoustic signal. 

In addition to nonlinear lipophilicity relationships for the transport and distribution of drugs, nonlinear relationships on molar refractivity are frequently observed in QSAR studies of enzyme inhibition data (provided that MR values are scaled by a factor of 0.1, as usual) [60,63,64,66-68]. Two such examples are given in Eq. (63) (Escherichia coli DHFR) and Eq. (64) (Lactobacillus casei DHFR) [101]. The differences between both models could be explained after the 3D structure of the enzyme became known. Whereas all substituents of a benzyl ring contribute to biological activities in E. coli DHFR, only the 3- and 4-substituents show up in the QSAR model for L. casei DHFR but not the 5-substituents. This results from a narrower binding pocket in L. casei DHFR a (3-branched leucine hinders the accommodation of 5-substituents, whereas a more flexible methionine in the same position of E. coli DHFR opens a wider binding pocket [101] ... 

Other nonlinear relationships are known in addition to nonlinear lipophilicity-activity relationships. Most common are nonlinear dependences on molar refractivity e.g. resulting from a limited binding site at the receptor for examples see chapter 7.1), but also other types of nonlinear relationships, e.g. with steric parameters, are frequently obtained. Even electronic parameters (eq. 48 chapter 3.5) or molecular weight terms (eq. 56 chapter 3.7) have been used in nonlinear equations. 

The linear polarizability, a, describes the first-order response of the dipole moment with respect to external electric fields. The polarizability of a solute can be related to the dielectric constant of the solution through Debye s equation and molar refractivity through the Clausius-Mosotti equation [1], Together with the dipole moment, a dominates the intermolecular forces such as the van der Waals interactions, while its variations upon vibration determine the Raman activities. Although a corresponds to the linear response of the dipole moment, it is the first quantity of interest in nonlinear optics (NLO) and particularly for the deduction of stracture-property relationships and for the design of new... 

From the viewpoint of developing quantitative correlations it is desirable to seek a linear relationship between descriptor and property, but a nonlinear or curvilinear relationship is adequate for illustrating relationships and interpolating purposes. In this handbook we have elected to use the simple descriptor of molar volume at the normal boiling point as estimated by the Le Bas method (Reid et al. 1987). This parameter is very easily calculated and proves to be adequate for the present purposes of plotting property versus relationship without seeking linearity. 

Model-based nonlinear least-squares fitting is not the only method for the analysis of multiwavelength kinetics. Such data sets can be analyzed by so-called model-free or soft-modeling methods. These methods do not rely on a chemical model, but only on simple physical restrictions such as positiveness for concentrations and molar absorptivities. Soft-modeling methods are discussed in detail in Chapter 11 of this book. They can be a powerful alternative to hard-modeling methods described in this chapter. In particular, this is the case where there is no functional relationship that can describe the data quantitatively. These methods can also be invaluable aids in the development of the correct kinetic model that should be used to analyze the data by hard-modeling techniques. 

The effect of the enantiometric purity of (—)DAIB has also been well documented. The enantioselectivities obtained using optically pure (—)DAIB are higher than those obtained from (—)DAIB having lower ee s. However, the relationship between the enantiometric purity of (—)DAIB and that of the resulting product is nonlinear, and varies as a function of reactants, reactant stoichiometry, and reaction conditions." For example, when benzaldehyde and diethylzinc were reacted in a 1 1 molar ratio in the presence of 8 mol % (—)DAIB, the enantioselectivity of the resulting product, (.S )- -phenyl-1-propanol, was 98% ee when pure (—)DAIB was used. The use of 15% ee (—)DAIB resulted in a product enantioselectivity of 95% ee (92% yield). The mechanism for this dramatic effect has been thoroughly studied. ", ... 

The emission intensity of fluorescence depends on the product of the molar extinction coefficient, the optical path length, the solute concentration, the fluorescence quantum yield of the dye, and the excitation source intensity. In dilute solutions, the intensity is linearly proportional to these parameters. When sample absorbance exceeds about 0.05 in a 1-cm path length, the relationship becomes nonlinear and measurements may be distorted by artifacts such as self-absorption and the inner-filter effect. 

Equilibrium as well as rate constants are related to free energy values AG by relationships of the type of eq. 11 (cliapter 1.2). Thus, only equilibrium constants (e.g. K values or at least IC50 values, not % inhibition at a certain concentration) and rate constants (e.g. log k values, not % absorption or % concentration in a certain compartment) are suited for QSAR studies, which means that all biological data have to be transformed in an appropriate manner before being used in quantitative analyses. In the case of complex biological data resulting from a sequence of several independent processes (in the worst case whole animal data), sometimes one effect predominates e.g. the bioavailability, the penetration of the blood-brain barrier, or the affinity to the receptor site. In other cases several effects overlap, which makes the QSAR analysis much more difficult. Due to the nonlinear characteristics of dose-response relationships, % effect values at a certain dose must not be used in QSAR equations. In each case they have to be transformed to equieffective molar doses (i.e. dose levels which produce or prevent a certain pharmacodynamic effect dose levels that increase the life span of animals to a certain extent dose levels which kill a certain percentage of the animals). 

Molar volume has been studied widely [2, 128, 148, 149] but no simple relationship has been shown between molar volume and the amount of solubilizate dissolved. Stearns et al. [148], studying hexane, heptane, and octane, and benzene, toluene, ethylbenzene, propylbenzene, and butylbcnzene concluded that there was inverse proportionality between the volume of hydrocarbon solubilized and molar volume. The slope of the plots of ml hydrocarbon dissolved per 100 g solution against molar volume of hydrocarbon are different for the aliphatic and aromatic series. Klevens [16], with polycyclic compounds in sodium laurate, found linear relationships between the log volume solubilized and molar volume, the slope of plots for linear polycyclics varying from that for the nonlinear polycyclics. Schwuger [149] reported that the amount of naphthalene, anthracene, pyrene, perylene and dibenzanthracene solubilized by micelles of dodecylpentaglycol ether was inversely related to the molecular size of these solubilizates. 

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