Binding constant statistical factor

What is the ratio of the concentration of T-state proteins with one ligand bound to the concentration of R-state proteins with one ligand bound The dissociation constant tor a single site in the R state is K[<. For a protein with n sites, there are n possible sites for the first ligand to bind. This statistical factor favors ligand binding compared with a single-site protein. Thus, [R J =... 

Ae Dielectric constant Ellipticity in circular dichroism n.i Statistical factor for ligand-i binding at n sites on a macromolecule... 

Again, the statistical factors 4, 5, etc., arise because Xr and K-y are intrinsic association binding constants, yet the overall expressions for the complexes requires extrinsic parameters e.g., Kr = [RX]/[R][X] = [R4X2]/[R4X][X] because there are three unfilled sites in the R4X molecule and two in each R4X2 molecule (see also Example 9.12). The fractional saturation Y is again defined by Eq. (9.49) ... 

To reduce the likelihood of competitive inhibitors, one should run the assays with high concentrations of substrates relative to Km levels. To favor competitive inhibitors, one should run the assay below the Km values for the substrates. Thus, for making suitable conclusions for assay design, knowledge of the kinetic and binding constants of receptors and enzymes, such as Kd, kCM, Km, Bmax, is useful. Stoichiometric information, such as the number of enzyme molecules per assay, may also be useful because it can serve as a guideline to ensure that the assays are maximally sensitive to the mechanism of action one wants to discover. Problems in assay development often occur when the conditions required for sensitivity to the desired mechanism of action do not yield the best conditions for statistical reproducibility therefore, compromises and balances of these two opposing factors must be often made. 

The constant Mi j.i+i is composed of microscopic constants, as each O2 binding step is composed of multiple microscopic reactions, which is illustrated by the reaction arrows in Fig. 1. Thus, 4 ways exist to bind the first O2, 12 ways to bind the second O2, 12 ways to bind the third O2, and 4 ways to bind the fourth O2. Each microscopic constant is designated by the notation ij of the species formed in the binding process (Fig. 1, Table 1). For each binding step i = 1,2,3, and 4, the macroscopic constant Mi j.i4i represents the average of the microstate constants A ij (i+i)j, with accompanying statistical factors that account for the different isomeric forms of the microstate tetramers, as shown in Table 1. 

Whenever an organic acid contains two or more chemically identical (i.e., stereochemically equivalent) functional groups, statistical factors that originate in the entropy of formation of the acid and/or its conjugate base contribute to the variation of thermodynamic dissociation constants with the degree of dissociation of the acid. Such statistical effects are implicitly included in equations that are often used to describe acid-base equilibria in synthetic and natural polymers. Because those equations have frequently been applied to proton binding by humic substances, a brief discussion of statistical ef-... 

If K is the microscopic equilibrium constant for the association of the monotopic ligand A with B, the stepwise constants are K = 3K Ki =K Ks= K/3. The statistical factors are easily understood in the first equilibrium, three equivalent binding sites A are available for binding in the second, two equivalent binding sites A are available in the forward reaction but two equivalent bonds A—B can dissociate in the reverse reaction in the third, there are three equivalent bonds A—B which can dissociate in the reverse reaction. In general, for the interaction of a symmetrical m-topic ligand with a monotopic metalloporphyrin, the stepwise constants K( are given by Eq. 2, from which Eq. 3 is easily derived [20,29-31]. 

A cychc assembly can be considered as a cychc ohgomer formed by monomers either of the type A—B or of the type A—A + B—B as schematically shown in Fig. 10. The formation of a cyclic -mer requires n - 1 intermolecular bonds and one intramolecular bond. If all the n - 1 intermole-cular processes occur with the same constant JCinter> the stabihty constant of the supramolecular assembly is given by Eq. 13, where at and crp are statistical factors the former accounting for the number of equivalent binding sites of the reactants [cTr = 1 for the case (a), and = 2 for the case (b)],... 

Evidently eaeh next stability constant is smaller than the preceding one due to only statistical factors. Any deviation from the statistical ratio K,+ /K, implies a nonequivalence of binding sites or some sort of interaction between sites, in particular due to the allosteric effect. In the latter case, deviations from statistical binding are commonly referred to as cooperativity, which can be positive or negative, depending on whether the ratio of successive stability constants is higher or lower than the statistically expected value. 

Each stepwise binding constant Kj can be factored into the product of a statistical factor and the corresponding microscopic equilibrium constant K. The statistical factor is easily obtained considering that in the forward reaction there are n—j+1 empty sites available for the ligand, whereas in the reverse reaction there are j Hgands that may dissociate from the receptor. Accordingly, Eq. [39] holds ... 

Under the given conditions, there are only four possible states for the receptor free BB, the partially bound 1 1 open complex o-AA-BB, the fuUy bound 1 1 cyclic complex c-AA-BB and the 1 2 complex BB-(AA)2. The macroscopic equffibrium constants in Scheme 24 have been factored as the product of statistical factors and microscopic equffibrium constants specifically, K is the microscopic intermolecular constant that expresses the strength of the binding interaction between A and B, and EM is the microscopic effective molarity, defined as the microscopic equffibrium constant of the reaction in Eq. [6] (see Section 5.1). It is useful to recall that positive allosteric cooperativity is characterized by a low concentration of partially bound species. In the most extreme cases only the unbound and... 

Considering the binding of a bipyridine bgand to a Cu ion as the reference intermolecular process, we can extract the microscopic intermolec-ular constant, log K = 3.4, by applying Eq. [51] to the self-assembly of 17, that is, Ksa(i7) = K. The statistical factor for the formation of... 

Here, n is the valency number, F is statistical factor defined by the system, and s = 30/(interreceptor distance (A)). An application is illustrated by a monovalent P trisaccharide ligand, which binds to pentavalent Shiga toxin (AB5) with a Ka of 10 M . If this monovalent ligand is converted to a polyvalent ligand through conjugation to polyacrylamide, a binding constant can be estimated for the multivalent interaction with the toxin pentamer as follows in (8). 

The microscopic binding constants, Ki, are related to the intrinsic hgand binding to a site, and therefore reflect the intrinsic binding affinities to each site. The relationship between macroscopic and microscopic binding constants is a statistical factor given by... 

Under the above conditions with Ca constant, this set of equations can be solved for each intermediate, as shown in the discussion of consecutive first order reactions in section 3.1. The statistical factors attached to the rate constants are dependent on the number of free sites (for ) or occupied sites (for k ) available on a molecule. For instance, there are four ways of occupying the first site, while there is only one way to dissociate. However, the fourth site can be occupied in one way only, while there are four ways in which dissociation can occur. This is discussed in more detail for binding equilibria by Edsall Gutfreund (1983). The general rate equation for liganding the / s site on a molecule with a total of n sites is... 

The use of statistical factors requires extra care when the reaction introduces or removes an asymmetric centre. For example, the addition of a hydrogen atom to an asymmetric methyl radical leads to two enantiomers, because the H atom may bind to either side of the symmetry plane (Figure 6.3). In the forward direction we have Cf = 2, but in the reverse direction, cr,=l and the equilibrium constant is multiplied by a factor of two... 

Figure 3.3 Kinetics of four site ligand binding curves (1) to (5) represent the zero-, mono-, di-, tri- and fully liganded tetramers. Curve (6) represents the total concentration of liganded sites, (a) is based on the reaction of identical and independent sites with statistical multiplication factors (see p. 68) given to the intrinsic rate constant it = Is , (b) is based on a cooperative system (identical sites) with intrinsic rate constants of 0.04 for the first step and 1.0 for the subsequent three steps of ligand binding. In this graph lines 2,3 and 4 are omitted because they all straddle the baseline.

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