Identical sites

The rate constants reported in Table 4 are double the specific rate constant for a given site when two identical sites are present, so that the actual position reactivity is half of the level indicated. Since the rates are reported as seeond order, the... 

The three most common types of inhibitors in enzymatic reactions are competitive, non-competitive, and uncompetitive. Competitive inliibition occurs when tlie substrate and inhibitor have similar molecules that compete for the identical site on the enzyme. Non-competitive inhibition results in enzymes containing at least two different types of sites. The inhibitor attaches to only one type of site and the substrate only to the other. Uncompetitive inhibition occurs when the inhibitor deactivates the enzyme substrate complex. The effect of an inhibitor is determined by measuring the enzyme velocity at various... 

Figure 4-8. NMR absorption by a hypothetical two-identical site system with chemical exchange (/I) Slow exchange limit. (B) Moderately slow exchange. (D) Coalescence. (F) Fast exchange limit.

From the results of this kinetic study and from the values of the adsorption coefficients listed in Table IX, it can be judged that both reactions of crotonaldehyde as well as the reaction of butyraldehyde proceed on identical sites of the catalytic surface. The hydrogenation of crotyl alcohol and its isomerization, which follow different kinetics, most likely proceed on other sites of the surface. From the form of the integral experimental dependences in Fig. 9 it may be assumed, for similar reasons as in the hy-drodemethylation of xylenes (p. 31) or in the hydrogenation of phenol, that the adsorption or desorption of the reaction components are most likely faster processes than surface reactions. 

Cellular automata are simple mathematical idealizations of natural systems. They consist of a lattice of discrete identical sites, each site taking on a finite set of say integer values. The values of the sites evolve in discrete time steps according to deterministic rules that specijy the value of each site in terms of the values of neighboring sites. Cellular automata may thus be considered as... 

Invertebrate prey species contain analogous, but not identical, sites to those considered above. In many phylla, calcium channels play the role normally ascribed to sodium channels in vertebrates. In addition, the peripheral locomotor neurotransmitter is not acetylcholine but amino acids such as gamma amino butyric acid (GABA). In other phylla, the channels which underly locomotion remain poorly understood. 

We now consider what would happen if the binding of the first molecule of agonist altered the affinity of the second identical site. The dissociation equilibrium constants for the first and second bindings will be denoted by KA(U and KM2), respectively, and E is defined as before. 

MHI possess at least two different polymerization initiating sites. The identical sites are selective for a particular class of monomers, and thus the resulting ji-star consists of chemically different arms. In order to obtain well-defined //-stars, these identical active sites should have equal reactivity and furthermore, initiation should be faster than propagation. It is not always possible to achieve these requirements since differentiation in the topology of... 

Fig. 7 shows the progressive transformation of montmorillonite to iliite/smectite interlayers by the gradual development of both the characteristic Cs and Rb high selectivity profiles observed for pure illite and the high Cs-Rb selectivity at+ race fadings. The data can be simulated (see table VI for the Ca - Cs case) using a consistent set of intrinsic selectivity coefficients and identical site group capacities for the Ca-Cs and... 

In some cases where the canonical PF of all the ( ) specific configurations are equal, we say that the system has m identical sites in the strict sense. Only in this case does the last equality on the rhs of Eq. (1.4.3) hold. Thus, from Eq. (1.4.1) to (1.4.3) we have proceeded from Jik) to Qjik), the latter being the canonical PF of an adsorbent molecule with k (unspecified) sites occupied by ligands. 

We now turn to the case of identical sites. Actually, we require that the sites be identical in a strict sense, as we explain below. We use again the three-site case, but instead of three different sites a, b, and c we assume that the sites are identical. Since we are dealing with localized molecules, the sites are still distinguishable. The canonical PFs listed in Eqs. (2.2.2)-(2.2.5) now reduce to... 

We proceed with the case of three identical sites in the strict sense and define the corresponding intrinsic constants... 

Sometimes, the term equivalent is used instead of strict sense. This can be confusing. For instance, in an equilateral triangle the three sites are equivalent, but in a linear case they are not equivalent. However, the term equivalent might not be suitable to distinguish between square and tetrahedral models. In both cases, identical sites are also equivalent because of symmetry. Yet, one has strict identical sites and the other weak identical sites in the sense defined here. For more details, see Chapter 6. 

These equations relate the sequential thermodynamic constants (first, second, and third) to the sequential intrinsic constants. The difference between the two sets arises from the requirement to specify the sites in the latter but not in the former. The generalization to m identical sites (in the strict sense) is quite straightforward. 

Note that in the case of two identical sites, occupied and empty sites are denoted by 1 and 0, respectively. When the sites are different we replace the 1 by the symbol used to name the site, say a or b and insert zero when the site is empty. 

We start with two identical sites. Since the sites are distinguishable (the system is localized), there are altogether eight states for the system, shown in Fig. 4.8. The GPF of a single system is... 

Figure 4.8. The eight possible configurations of a two-state system with two identical sites. The L form is represented by a rectangle (white) and the H form by an ellipse (black).

The sign of the direct correlation depends on the direct interaction between the ligands. The sign of the indirect correlation is always positive (for two identical sites see below for two different sites), and is independent of (7(1,1), but dependent on the difference of binding energies t/ - f/, and the difference of energies of the two conformers... 

THREE STRICTLY IDENTICAL SITES NONADDITIVITY OF THE TRIPLET CORRELATION... 

In two-site systems, there is only one correlation function which characterizes the cooperativity of the system. In systems with more than two identical sites, for which additivity of the higher-order correlations is valid, it is also true that the pair correlation does characterize the cooperativity of the system. This is no longer valid when we have different sites or nonadditivity effects. In these cases there exists no single correlation that can be used to characterize the system, hence the need for a quantity that measures the average correlation between ligands in a general binding system. There have been several attempts to define such a quantity in the past. Unfortunately, these are valid only for additive systems, as will be shown below. 

If the system is known to have m identical sites, either in a strict or in a weak sense, then K.J. = mk, where k is the intrinsic binding constant. On the other hand, if we know that the system has m different binding sites, each having a different intrinsic binding constant k, then we must determine each of these from the limiting slope of the corresponding individual BI, i.e.. 

On the other hand, for a system of four identical sites in the strict sense, i.e., when all = k and all the correlations of any given order are identical, such as the arrangement of four identical subunits at the vertices of a perfect tetrahedron, we... 

For the general case of m identical sites, in the strict sense we have... 

We shall see in the next subsection that g(C) can change dramatically as a function of concentration. However, if one insists on having a single measure of the average cooperativity, a convenient choice for the general case could be C = 1 [measured in the same units of ( ) ]. A more convenient choice for the case of identical sites (in either respect) is C = k , in which case Eq. (5.8.24) reduces to... 

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