# Assembly Auto Inhibition¶

- Ligand binding that promotes protein oligomerization, which is auto-inhibited by saturation of ligand (related to the prozone effect).
- Model contributed by: Martin Rennie, PhD
- Rennie & Crowley (2019).
*ChemPhysChem*(link). - indiv_models.AssemblyAutoInhibition

## Parameters¶

parameter | variable | parameter name | class |
---|---|---|---|

macroscopic association constant
for binding of the first ligand
to the protein monomer
(M^{-1}) |
\(K_{1}\) | `Klig1` |
thermodynamic |

average* association constant
for binding of the remaining
ligands to the protein monomer
(M^{-1}) |
\(K_{2}\) | `Klig2` |
thermodynamic |

average* association constant
for formation of the protein
oligomer (M^{-1}) |
\(K_{3}\) | `Kolig` |
thermodynamic |

enthalpy change for binding of the first ligand to the protein monomer | \(\Delta H_{1}\) | `dHlig1` |
thermodynamic |

enthalpy change for binding of the remaining ligands to the protein monomer | \(\Delta H_{2}\) | `dHlig2` |
thermodynamic |

enthalpy change for formation of the protein oligomer | \(\Delta H_{3}\) | `dHolig` |
thermodynamic |

stoichiometry of ligands in the saturated protein monomer, must be ≥2 | \(m\) | `m` |
thermodynamic |

stoichiometry of ligands in the protein oligomer | \(n_{L}\) | `n_lig` |
thermodynamic |

stoichiometry of proteins in the protein oligomer | \(n_{P}\) | `n_prot` |
thermodynamic |

fraction competent protein | — | `fx_prot_competent` |
nuisance |

fraction competent ligand | — | `fx_lig_competent` |
nuisance |

slope of heat of dilution | — | `dilution_heat` |
nuisance |

intercept of heat of dilution | — | `dilution_intercept` |
nuisance |

*equilibrium constants for the higher order equilibria are “averaged” using \(\sqrt[N]{K}\),
where \(N\) is the order of the equilibrium, such that the units are M^{-1}
(see Rennie & Crowley (2019). *ChemPhysChem* (link))

## Species¶

\[[P_{T}]_{i} = [P]_{i} + [PL]_{i} + [PL_{m}]_{i} + n_{P}[P_{olig}]_{i}\]

\[[L_{T}]_{i} = [L]_{i} + [PL]_{i} + m[PL_{m}]_{i} + n_{L}[P_{olig}]_{i}\]

\[[PL]_{i} = K_{1}[P]_{i}[L]_{i}\]

\[[PL_{2}]_{i} = K_{1}K_{2}^{m-1}[P]_{i}[L]_{i}^{m}\]

\[[P_{olig}]_{i} = K_{3}^{n_{L}+n_{P}-1}[P]_{i}^{n_{P}}[L]_{i}^{n_{L}}\]

## Heat¶

\[\begin{split}q_{i} = V_{cell}\Big ( \Delta H_{1}^{\circ}([PL]_{i} - [PL]_{i-1}(1-v_{i}/V_{cell})) \\
+ (\Delta H_{1}^{\circ} + \Delta H_{2}^{\circ})([PL_{2}]_{i} - [PL_{2}]_{i-1}(1 - v_{i}/V_{cell})) \\
+ \Delta H_{3}^{\circ}([P_{olig}]_{i} - [P_{olig}]_{i-1}(1 - v_{i}/V_{cell})) \Big ) + q_{dil}\end{split}\]

where: \([P_{T}]_{i}\) is the total cell concentration of protein at the \(i^\text{th}\) injection (independent variable); \([L_{T}]_{i}\) is the total cell concentration of ligand at the \(i^\text{th}\) injection (independent variable); \(V_{cell}\) is the volume of the cell; \(v_{i}\) is the volume of the \(i^\text{th}\) injection; \(q_{i}\) is the heat generated from the \(i^\text{th}\) injection; \(q_{dil}\) is the heat of dilution.