Perfect similarity

These equations lead to fomis for the thermal rate constants that are perfectly similar to transition state theory, although the computations of the partition functions are different in detail. As described in figrne A3.4.7 various levels of the theory can be derived by successive approximations in this general state-selected fomr of the transition state theory in the framework of the statistical adiabatic chaimel model. We refer to the literature cited in the diagram for details. 

In case of perfect similarity between two molecules A and B, one would find <7AB = 0, and the more the two density functions differ, the larger will be the value... 

According to this index, a perfect similarity is indicated by CAB=1, and lower values indicate decreasing similarity. 

A different class of indices is the D indices, where similarity is expressed as a distance. With these indices, perfect similarity is characterized by a zero distance. The best known is the Euclidean distance, introduced in Equation 16.5. Again, different connections have been found to exist between C- and D-class indices [59-62]. 

When scaling a process, similarity between the various sizes and processes is sought. As a rule, a perfectly scalable prototype is one that is perfectly similar to its scaled system. A perfectly similar set of systems is one where all the dimensionless numbers or n-groups... 

It can be seen that neither of these leads to a perfectly similar scaled system. 

The two statements are in fact related. In our example, a perfect similarity of theoretical and experimental plots could only be expected if there were a linear relationship between receptor saturation with norepinephrine and muscle contraction. Considering that muscle contraction is triggered quite a bit downstream of receptor activation, there are numerous possible factors that will distort this linearity, and in reality no linear relationship will ever be observed if drug target and drug effect are separated by intervening biochemical cascades. It thus turns out that the shape of a dose-effect relationship will depend very much on the functional proximity of the drug receptor molecule and the observed parameter. 

It is necessary to pay careful attention to these last two expressions where H is considered in conventional length units, which corresponds to a Vj,= 1. For example if v = 1 cm/s, then the conventional unit (cu) is cm, therefore, in the relations, H would be expressed in cm. Another example shows that v = 0.02 m/s so a value of the conventional length unit of 1 cu = 0.02 m is requested to make v = 1 cu/s. If, in this case, the trajectory is 0.2 m, for example, then, for H, H = 0.2/0.02 = 10 is used which corresponds to a dimensionless value. For very large H values, relation (4.55) can be simplified as follows Tjn = 2a(H + 1) 2aH. This simplification can guide us towards various speculative conclusions with respect to the covered linear distance. Categorically, the result obtained can be explained by the perfect similarity of the final relationships with the well-known formulas used in mechanics. 

This body as we perceive, offers but little interest in its properties but it is the more remarkable in its relation to hydrobenzdyl, from the fact, as analysk proves, that it has a perfectly similar composition, and is therefore an isomeric modification of the same, as is apparently shewn by its unintelligible origin from the oil, through the equally inexplicable action of potassa when air is excluded. 

Both these causes probably contribute to produce the excess of hydride of ethyl but the very small amount of gaseous products, compared with the solid ones, convinced me that the production of the former is only an accidental circumstance, which, however it may be interpreted, does not at all affect the principal reaction, viz. the formation of iodide of stanethylium. The gases, evolved by the action of light upon iodide of ethyl and tin, are perfectly similar to those obtained by the action of heat,... 

If we wish to obtain standards of length, time and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wave-length, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules ... 

Still another variation of the hand-glove interaction comes into play, when during the approach, a new type of synthon appears, and the synthons match afterward. For example, in the Hodges superhelical structure (Fig. 13.17), only after formation of the a-helices does it turn out that the leucine and valine side chains of one helix match perfectly similar synthons of the second hehx (the leucine-zipper). 

This generalized cosine index is often called the Carbo index. Naturally the Carbo index is limited to the range (0,1), where Cab = 1 means perfect similarity. Still more (dis)similarity measures have been introduced and will be discussed further in this chapter. The range of the Carbo index naturally agrees with the Schwartz integral inequality " ... 

These C-class indices vary from 0 to 1, 1 denoting perfect similarity, which does not infer equality between the molecules involved. Maggiora, Petke... 

A different class of indices are those that behave like a distance. With these indices, a perfect similarity corresponds to a value 0, and no strict upper limit exists. The best-known of these D-class indices is the Euclidean distance introduced previously in Eqs. [7]-[13]. This Euclidean distance index has been extended to give an index for any operator used by chemists in evaluating the MQSMs by using... 

Respiration is thus a combustion, very slow it is true, but perfectly similar to that of carbon it occurs in the interior of the lungs, without disengagement of visible light since the matter of fire which becomes free is at once absorbed by the humidity of these organs. The heat developed in this combustion communicates itself to the blood which traverses the lungs and from there it is spread over all the animal system. ... 

The relative potency, p, functions here much like a currency exchange rate functions. It converts concentration of one metal into the equivalent concentration of the other. If one were interested in detecting trends away from perfect similar joint action and toward independent action, the absolute difference between estimated slopes from Equations (1.6) and (1.7), i.e., SlopeM - Slope 2 could be used as a metric of deviation from similar mode of action. 

FIGURE 1.8 Deviations from perfect independent (top panel) and perfect similar (bottom panel) action for binary mixtures of metals are predictable from differences in paired metal softnesses. Specific paired metals in mixtures are indicated next to each data point, e.g., MnNi = a binary mixture of Mn and Ni. This figure is Figure 9.8 from Newman, M.C., and W.H. Clements. 2008. Ecotoxicology A Comprehensive Treatment. Boca Raton, FL CRC Press. 

In Chapter 3, a variety of standard plots is presented. It is instructive to plot these as x plots to see how weU they obey the analytical expression. In the following, the x plot fits will be performed only on original data where available. Creation of the standard plot by some fitting routine or simply using a manual spline fit is in itself a distortion of the data. Indeed, the tho-ria and lunar soils standard plots were created using the insights of the x plot, so the standard plot by definition must fit the x plot perfectly. Similar problems are encountered in analyzing heat of adsorption. 

Such behavior is ultimately related to various interactions between the Pluronic molecules in the adsorption layer and between the same molecules, especially their hydrophobic parts and the nonpolar phase. It is reasonable to assume that for surfactant molecules with the same hydrophilic groups the perfect similarity (high affinity) between their hydrophobic groups and the nonpolar phase, that is, perfect adhesion and mutual solubility, result in deep immersion of such groups into this phase, without forming a dense layer. Conversely, lesser physical-chemical similarity (poorer affinity, lesser adhesion) leads to their pushing out, to the aqueous phase,... 

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