Cooperativity Factor

The difference in the ligands of each tertiary atom is due mainly to the different length of chain segments to which it is bonded. If we consider that, in the absence of cooperative factors, optical activity is a short-range phenomenon, the contribution to the rotatory power of most parts of the chain is zero or near to zero. One can thus foresee that the optical activity of a single enantiomer is very small and falls into the domain of cryptochirality. Only oligomers of low molecular weight can present measurable optical activity. 

In Eqs. (3) and (4), a is defined as a cooperativity factor that represents the efficiency with which the ligand elicits a cellular response. For positive cooperativity, as displayed by agonists, a > 1 for inverse agonists, which exhibit negative cooperativity by hindering receptor activation of G protein, a < 1 (Ghristopoulos, 2002 Christopoulos and Kenakin, 2002 Weiss et al., 1996a,b). 

The ternary complex model allows for cooperative interactions among ligand, receptor, and G protein as defined by the cooperativity factor a. With a lower limit of Kg established from the levels of RG precoupling, the comparisons of K,.A with K l indicate for both / 2AR and FPR that precoupled receptors (RG) would exhibit higher affinity for an agonist... 

Fig. 5. Conceptual schematic of the receptor conformational states elicited by binding to partial (L, ) or full (Ly) agonists, and a depiction of the correlation between the various conformational states and their ability to bind with G proteins. Solid lines show the conformational distributions hypothesized from soluble ternary complex data analyzed by the simple ternary complex model. When a partial agonist binds with a receptor (L R) in this model, the receptor forms a conformational state which has an intermediate affinity for G protein, consequendy leading to formation of intermediate amounts of L RG. On the other hand, the dotted line represents the potential receptor conformations induced by a partial agonist consistent with the extended ternary complex model, which includes the isomerization of receptor between R and R, the only receptor conformation allowed to bind with G protein. For this model, the interactions of a partial agonist with a receptor would result in two populations of ligand-bound receptors with only one (LR ) able to bind with G protein. The x-axis is analogous to the cooperativity factor a.

In general, the overall effect of an allosteric ligand resnlts from the balance between the modulation of affinity and efficacy and it is usually necessary to also measure cooperativity factors and dissociation rates. A description of this rather complicated field is beyond fhe scope of fhis chapter and the interested reader is referred to some recent reviews. " ... 

In the specific case m = 2, where in the absence of cooperativity K2/K1 = 1 /4, an interaction parameter (or cooperativity factor) a = 4K2/K1 can be defined, which provides a quantitative measure of cooperativity in the three cases above, a being = 1, > 1, < 1, respectively [29]. There are other equivalent tests to assess cooperativity, mainly graphical ones, based on the calculation of the occupancy r, that is to say the average number of occupied sites of the m-topic ligand [20,29-31]. A plot of r/[B] as a function of r is known as the Scatchard plot non-cooperative behavior is characterized by a straight line (Eq. 4), whereas a concave downward curve or a concave upward curve are diagnostic for positive or negative cooperativity, respectively. 

Cooperativity can be quantified by an interaction parameter, or cooperativity factor, a given by the ratio of the experimental overall binding constant to the hypothetical overall noncooperative constant calculated by Eq. [41] with i = n ... 

The cooperativity factor a is a dimensionless constant greater than 1 in the case of positive cooperativity, equal to 1 in the case of noncooperativity and smaller than 1 in the case of negative cooperativity. It can be viewed as the equihbrium constant for the conversion of the hypothetical complex with noninteracting sites (noncooperative) into the complex with interacting sites (cooperative), as illustrated in Fig. 13 for the simplest case with = 2. The cooperative interaction may be of any sort (electrostatic, steric, conformational, and so on). 

Figure 13 The allosteric cooperativity factor a is the equilibrium constant for the conversion of the hypothetical noncooperative complex into the cooperative complex. The figure depicts the case of a saturated divalent receptor.

The chelate cooperativity factor jd, defined in Fig. 15 for the case of a cyclic assembly, can be generalized to the case of a multicyclic assembly. For example, consider the equilibrium shown in Fig. 16 in which a trivalent receptor B3 saturated by three molecules of a trivalent Hgand A3 reacts to form a 1 1 bicycHc complex r-A3-B3. 

Figure 16 The chelate cooperativity factor for the case of a trivalent receptor B3 reacting with a trivalent ligand A3.

The bicychc complex is foraied by two consecutive cyclizations, each characterized by its own microscopic effective molarity. Owing to the symmetry of the bicychc complex, it is reasonable to assume that the two EM values are equal. Then, the chelate cooperative factor jS is given by Eq. [49], where 1/9 is the statistical factor of the equilibrium shown in Fig. 16 ... 

With reference to Fig. 17, interannular cooperativity can be quantified by a cooperativity factor, 7, that is given by the ratio of the overall experimental constant to the hypothetical overall noncooperative constant, 7 = EM EM /EM 2. 

---