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:

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