BIOEN 470 System Immunology and Immunoengineering
Receptor-Ligand Kinetics
- Variables
- $\ce{[R]}$ - Receptor concentration
- $\ce{[RL]}$ - Receptor-ligand complex concentration
- Parameters
- $\ce{[L]}$ - Ligand concentration
- $\ce{[R_T] = [R] + [RL]}$ - Total receptor concentration
- $k_1$ - Forward (binding) rate
- $k_{-1}$ - Reverse (unbinding) rate
- Derived parameters
- $K_1 = \dfrac{k_1}{k_{-1}}$ - Association constant
- $K_{-1} = \dfrac{k_{-1}}{k_1}$ - Dissociation constant
- Relationships
- $K_{-1} = K_1^{-1}$
Suppose reversible ligand binding to receptor is given by
$$ \color{blue} \ce{R + L <=>[\mathit{k}_1][\mathit{k}_{-1}] RL} $$
The total number of receptors is constant $\ce{[R_T]}$
$$ \ce{[R_T] = [R] + [RL]}. $$
By steady-state approximation,
$$ \begin{aligned} \dfrac{d\ce{[RL]}}{dt} &= k_1 \ce{[R][L]} - k_{-1} \ce{[RL]} && \footnotesize\text{Principle of detailed balance} \\ 0 &= k_1 \ce{[R][L]} - k_{-1} \ce{[RL]} && \footnotesize\text{Steady state approximation } \frac{d\ce{[RL]}}{dt} = 0 \\ 0 &= k_1\ce{([R_T] - [RL])[L]} - k_{-1} \ce{[RL]} && \footnotesize\text{Total receptor constraint } \ce{[R_T] = [R] + [RL]} \\ k_1\ce{[L][R_T]} &= k_1\ce{[L][RL]} + k_{-1}\ce{[RL]} \\ \ce{[RL]} &= \dfrac{k_1\ce{[L][R_T]}}{k_1\ce{[L]} + k_{-1}} \\ \ce{[RL]} &= \dfrac{(k_1\ce{[L][R_T]})/k_1}{(k_1\ce{[L]} + k_{-1})/k_1} && \footnotesize\text{Divide by } k_1 \\ \ce{[RL]} &= \dfrac{\ce{[L][R_T]}}{\ce{[L]} + \frac{k_{-1}}{k_1}} \\ \ce{[RL]} &= \dfrac{\ce{[L][R_T]}}{\ce{[L]} + K_{-1}} && \footnotesize\text{Dissociation coefficient } K_{-1} = \frac{k_{-1}}{k_1} \end{aligned} $$
The concentration of the bound receptor-ligand complex depends on concentration of ligands, total concentration of receptors, forward rate, and backward rate:
$$ \boxed{\ce{[RL]} = \dfrac{\ce{[L][R_T]}}{\ce{[L]} + K_{-1}}} $$
The $\ce{[RL]}$ vs $\ce{[L]}$ graph has the following features
$$ \begin{aligned} \lim_{\ce{[L] \to \infty}} \ce{[RL]} &= \ce{[R_T]} \\ \lim_{\ce{[L] \to K_{-1}}} \ce{[RL]} &= \frac{1}{2}\ce{[R_T]} \end{aligned} $$