# Competitive ligand binding¶

## Scheme¶

Scheme is for competitive binding of $$A$$ and $$B$$ to protein $$P$$:

$A + P \rightleftharpoons PA$
$B + P \rightleftharpoons PB$

To describe this, we use the following equilibrium constants:

$K_{A} = \frac{[PA]}{[P]_{free}[A]_{free}}$
$K_{B} = \frac{[PB]}{[P]_{free}[B]_{free}}$

## Parameters¶

parameter variable parameter name class
association constant for A $$K_{A}$$ K thermodynamic
association constant for B $$K_{B}$$ Kcompetitor thermodynamic
binding enthalpy for A $$\Delta H_{A}$$ dH thermodynamic
binding enthalpy for B $$\Delta H_{B}$$ dHcompetitor thermodynamic
fraction competent fx_competent nuisance
slope of heat of dilution dilution_heat nuisance
intercept of heat of dilution dilution_intercept nuisance

## Species¶

We can only manipulate $$[P]_{total}$$, $$[A]_{total}$$ and $$[B]_{total}$$ experimentally, so our first goal is to determine the concentrations of species such as $$[PA]$$, which we cannot manipulate or directly observe. Start by writing concentrations as mole fractions:

$x_{P} = \frac{[P]_{free}}{[P]_{total}}$
$x_{PA} = \frac{[PA]}{[P]_{total}}$
$x_{PB} = \frac{[PB]}{[P]_{total}}$
$x_{P} + x_{PA} + x_{PB} = 1$

A root of the binding polynomial has been found that describes $$x_{P}$$ only in terms of $$K_{A}$$, $$K_{B}$$, $$[A]_{total}$$, $$[B]_{total}$$ and $$[P]_{total}$$. Start with some convenient definitions:

$c_{A} = K_{A}[P]_{total}$
$c_{B} = K_{B}[P]_{total}$
$r_{A} = \frac{[A]_{total}}{P_{total}}$
$r_{B} = \frac{[B]_{total}}{P_{total}}$

The value of $$x_{P}$$ is given by:

$\alpha = \frac{1}{c_{A}} + \frac{1}{c_{B}} + r_{A} + r_{B} - 1$
$\beta = \frac{r_{A}-1}{c_{B}} + \frac{r_{B} - 1}{c_{A}} + \frac{1}{c_{A}c_{B}}$
$\gamma = -\frac{1}{c_{A}c_{B}}$
$\theta = arccos \Big ( \frac{-2\alpha^{3} + 9\alpha \beta -27\gamma}{2\sqrt{(\alpha^2 - 3 \beta)^3}} \Big)$
$x_{P} = \frac{2\sqrt{\alpha^2 - 3 \beta}\ cos(\theta/3) - \alpha}{3}$

Once this is known $$x_{PA}$$ and $$x_{PB}$$ are uniquely determined by:

$x_{PA} = \frac{r_{A} x_{P}}{1/C_{A} + x_{P}}$
$x_{PB} = \frac{r_{B} x_{P}}{1/C_{B} + x_{P}}$

## Heat¶

The heat for each shot $$i$$ ($$q_{i}$$) is:

$q_{i} = V_{0}P_{total}(\Delta H_{A}(x_{PA,i} - f_{i}x_{PA,i-1}) + \Delta H_{B}(x_{PB,i} - f_{i}x_{PB,i-1})) + q_{dilution},$

where $$V_{0}$$ is the volume of the cell, $$\Delta H_{A}$$ is the enthalpy for binding ligand $$A$$, $$\Delta H_{B}$$ is the enthalpy for binding ligand $$B$$. $$f_{i}$$ is the dilution factor for each injection:

$f_{i} = exp(-V_{i}/V_{0}),$

where $$V_{0}$$ is the volume of the cell and $$V_{i}$$ is the volume of the $$i$$-th injection.

pytc calculates $$x_{PA,i}$$ and friends for the entire titration, correcting for dilution. This means the $$f_{i}$$ term is superfluous. Thus, heats are related by:

$q_{i} = V_{0}P_{total,i}(\Delta H_{A}(x_{PA,i} - x_{PA,i-1}) + \Delta H_{B}(x_{PB,i} - x_{PB,i-1})) + q_{dilution}.$

Note that $$V_{0}$$ is held constant (it is the cell volume) as only that volume is detected, not the neck of the cell.