Equation Sheets
Equation Sheets

BIOEN 470 System Immunology and Immunoengineering

  
Contents
Definition of Terms
  • Variables
    • [R]\ce{[R]} - Receptor concentration
    • [RL]\ce{[RL]} - Receptor-ligand complex concentration
  • Parameters
    • [L]\ce{[L]} - Ligand concentration
    • [RXT]=[R]+[RL]\ce{[R_T] = [R] + [RL]} - Total receptor concentration
    • k1k_1 - Forward (binding) rate
    • k−1k_{-1} - Reverse (unbinding) rate
  • Derived parameters
    • K1=k1k−1K_1 = \dfrac{k_1}{k_{-1}} - Association constant
    • K−1=k−1k1K_{-1} = \dfrac{k_{-1}}{k_1} - Dissociation constant
  • Relationships
    • K−1=K1−1K_{-1} = K_1^{-1}
Derivation

Suppose reversible ligand binding to receptor is given by

R+L⇌kX−1kX1RL \color{blue} \ce{R + L <=>[\mathit{k}_1][\mathit{k}_{-1}] RL}

The total number of receptors is constant [RXT]\ce{[R_T]}

[RXT]=[R]+[RL]. \ce{[R_T] = [R] + [RL]}.

By steady-state approximation,

d[RL]dt=k1[R][L]−k−1[RL]Principle of detailed balance0=k1[R][L]−k−1[RL]Steady state approximation d[RL]dt=00=k1([RXT]−[RL])[L]−k−1[RL]Total receptor constraint [RXT]=[R]+[RL]k1[L][RXT]=k1[L][RL]+k−1[RL][RL]=k1[L][RXT]k1[L]+k−1[RL]=(k1[L][RXT])/k1(k1[L]+k−1)/k1Divide by k1[RL]=[L][RXT][L]+k−1k1[RL]=[L][RXT][L]+K−1Dissociation coefficient K−1=k−1k1 \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}

Results

The concentration of the bound receptor-ligand complex depends on concentration of ligands, total concentration of receptors, forward rate, and backward rate:

[RL]=[L][RXT][L]+K−1 \boxed{\ce{[RL]} = \dfrac{\ce{[L][R_T]}}{\ce{[L]} + K_{-1}}}

The [RL]\ce{[RL]} vs [L]\ce{[L]} graph has the following features

lim⁡[L] →∞[RL]=[RXT]lim⁡[L] →KX−1[RL]=12[RXT] \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}

Updated on 2022-04-15
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