# Binding Polynomial¶

## Scheme¶

The scheme is:

$S + iT \rightleftharpoons ST_{i}$

where $$S$$ is the stationary species and $$T$$ is the titrant. This is an overall binding polynomial, meaning that we account for the total loading of $$i$$ molecules of $$T$$ onto $$S$$. The equilibrium constants (Adair constants) are:

$\beta_{i} = \frac{[ST_{i}]}{[S][T]^{i}}$

This model is entirely general (and therefore phenomenological), but is an appropriate starting point for analyzing a complex binding reaction. The Adair constants can be related to a sequential binding model by:

$S + T \rightleftharpoons ST$
$ST + T \rightleftharpoons ST_{2}$
$...$
$ST_{i-1} + T \rightleftharpoons ST_{i}$
$K_{i} = \frac{[ML_{i}]}{[ML_{i-1}][L]} = \frac{\beta_{i}}{\beta_{i-1}}$

## Parameters¶

parameter variable parameter name class
Adair constant for site 1 $$\beta_{1}$$ beta1 thermodynamic
binding enthalpy for site 1 $$\Delta H_{1}$$ dH1 thermodynamic
This will have as many $$\beta$$ and $$\Delta H$$ terms as sites defined in the model.
fraction competent fx_competent nuisance
slope of heat of dilution dilution_heat nuisance
intercept of heat of dilution dilution_intercept nuisance

## Species¶

The first thing to note is that the binding polynomial $$P$$ is a partition function:

$P = \sum_{i=0}^{n}\frac{[ST_{i}]}{[S]} = \sum_{i=0}^{n} \beta_{i}[T]^{i}$

This allows us to write equations for the average enthalphy and number of ligand molecules bound:

$\langle \Delta H \rangle = \frac{\sum_{i=0}^{n} \Delta H_{i} \beta_{i}[T]^{i}} {\sum_{i=0}^{n} \beta_{i}[T]^{i}}$

and

$\langle n \rangle = \frac{\sum_{i=0}^{n} i \beta_{i}[T]^{i}} {\sum_{i=0}^{n} \beta_{i}[T]^{i}}$

This means that obtaining the relative populations of species in solution is (relatively) simple:

$[T]_{total} = [T]_{bound} + [T]_{free}$
$[T]_{total} = \langle n \rangle[S]_{total} + [T]_{free}$
$0 = \langle n \rangle[S]_{total} + [T]_{free} - [T]_{total}$
$0 = \frac{\sum_{i=0}^{n} i \beta_{i}[T]_{free}^{i}} {\sum_{i=0}^{n} \beta_{i}[T]_{free}^{i}}[S]_{total} + [T]_{free} - [T]_{total}$

This can then be solved numerically for a value of $$[T]_{free}$$.

## Heat¶

We can relate the heat at shot to the average enthalpies calculated using the value of $$T_{free}$$ over the titration. Recalling:

$\langle \Delta H \rangle = \frac{\sum_{i=0}^{n} \Delta H_{i} \beta_{i}[T]_{free}^{i}} {\sum_{i=0}^{n} \beta_{i}[T]_{free}^{i}}$

we can calculate the change in heat for shot $$j$$ as:

$q_{j} = V_{0} S_{total,j} (\langle \Delta H \rangle_{j} - \langle \Delta H \rangle_{j-1}) + q_{dilution,i}.$