# Binding Polynomial¶

- Binding polynomial for binding at \(N\) sites. Adair constants.
- Freire et al. (2009).
*Methods in Enzymology*455:127-155 (link). - indiv_models.BindingPolynomial

## Scheme¶

The scheme is:

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:

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:

## 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:

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

and

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

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:

we can calculate the change in heat for shot \(j\) as: