Polynomials binding

Binding Polynomials Are Used to Compute Binding Curves 

Consider the multiple binding equilibria of a ligand X with / different binding sites on a protein or pol Tner P  

What are the fractions of all the P molecules in solution that are i-liganded To get these quantities, you first need the sum of the concentrations of all of 

534 Chapter 28. Multi-site Cooperative Ligand Binding 

The fraction of P molecules that are in the t-liganded state is [PA,]/(Qrf D-A key quantity for comparing models with experiments is the a erage number = i of ligands bound per P molecule, as a function of the ligand concentration x. To compute this average, use (i) = Xipd), where p(i) = PXi KQ P ) is the probability that a P molecule is in state i, to get 

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BI Binding isothenn BP Binding polynomial BPG D-2,3-bisphosphoglycerate CPF Canonical partition function FG Functional group GPF Grand partition function IHP Inositol hexaposphate Ihs Left-hand side PF Partition function rhs Right-hand side H( )0 Hydrophobic H( )I Hydrophilic... 

Indeed, there are two approaches to the theory of binding phenomena. The first, the older, and the more common approach is the thermodynamic or the phenomenological approach. The central quantity of this approach is the binding polynomial (BP). This polynomial can easily be obtained for any binding system by viewing each step of the binding process as a chemical reaction. The mass action... 

ATP also binds magnesium ions as shown by the three complex ion dissociation constants in Table 1.2. Incorporating these species into equations 1.3-1 and 1.3-2 yields the following binding polynomial for ATP ... 

Now the binding of hydrogen ions is given by the following partial derivatives of the binding polynomial ... 

When magnesium ions or other metal ions are hound reversibly and a wider range of pH is considered, equation 1.4-3 becomes more complicated. Therefore it is convenient to use the nomenclature of binding polynomials introduced in equation 1.3-8. The binding polynomial of ATP is given in equation 1.3-12, and the binding potentials for ADP and P, are as follows ... 

For these five forms of the tetramer, AfG"°(TotT) is the same as the binding potential n defined by Wyman (1948,1964) as 11 = RT n P, except for the difference in sign. The binding polynomial P is defined as (see Section 1.3)... 

This apparent equilibrium constant can be written in terms of the binding polynomials of tetramer and dimer and °K2 ... 

If kx [Pg.133]

The abbreviations for reactants represent sums of species at a specified pH, and the expression for the equilibrium constant is written in terms of concentrations because K is taken to be a function of ionic strength as well as pH. The pH dependence of K can be written in terms of the binding polynomials of the three reactants (see equation 1.4-8) ... 

The corresponding binding polynomial for the protein with the binding site occupied (PLtot) is given by... 

The impact of K+ and Mg2+ binding on the apparent equilibrium constant may be calculated by incorporating binding of these species into the binding polynomials for the reactants in a given reaction. For example, for the ATP hydrolysis reaction, the binding polynomials for the three reactants become ... 

When K+ and Mg2+ binding are considered, the apparent equilibrium free energy and equilibrium constant are computed as functions of the K+- and Mg2+-dependent binding polynomials. 

Carbon dioxide in solution does not significantly bind to hydrogen ions or other metal ions. Thus it is possible to treat CO2 as a reactant composed of a single species in biochemical reactions, without introducing a binding polynomial or multiple ion-bound states for C02. However, C02 can be hydrolyzed to H2C03 via the reaction... 

There are conservation statements for total magnesium and potassium concentrations that are analogous to that for hydrogen ion of Equation (2.20). Write the potassium and magnesium conservation statements in terms of the binding polynomials. 

Note that only states that are expected to be significant in the pH and ionic range studied are included in these calculations. Therefore some binding polynomials do not include terms for all possible cation-bound states. Given these forms of the binding polynomials, the relationships between the reference species concentrations and the reactant concentrations take the usual form ... 

Note that this equation differs from those for the reactions of Complex I, III, and IV in two important ways. First, the binding polynomials, PAtpx, Padpx, and Ppix,... 

The binding polynomials for PI in the matrix and intermembrane space are denoted PPIx and Ppn, respectively. The hydrogen-ion dissociation constant Kpj is listed in Table 6.1. The activity of the transporter is xpht = 2-0 x 107 mol-s 1-M 1 (1 mito)-1 and the Michaelis-Menten constant is Kpht = 1.0 mM. 

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