# Competitive ligand binding¶

- Model binding where two molecules compete for binding to a single other molecule.
- Sigurskjold BW (2000)
*Analytical Biochemistry*277(2):260-266 (link). - indiv_models.SingleSiteCompetitor

## Scheme¶

Scheme is for competitive binding of \(A\) and \(B\) to protein \(P\):

To describe this, we use the following equilibrium constants:

## 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:

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:

The value of \(x_{P}\) is given by:

Once this is known \(x_{PA}\) and \(x_{PB}\) are uniquely determined by:

## Heat¶

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

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:

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:

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