20.109(S20):Analyze titration curves (Day5)
Introduction
The antigen, lysozyme (L), and antibody (Ab) form a complex (C), which can be written
$ L + Ab \rightleftharpoons\ ^{k_f}_{k_r} C $
At equilibrium, the rates of the forward reaction (rate constant = $ k_f $) and reverse reaction (rate constant = $ k_r $) must be equivalent. Solving this equivalence yields an equilibrium dissociation constant $ K_d $, which may be defined either as $ k_r/k_f $, or as $ [Ab][L]/[C] $, where brackets indicate the molar concentration of a species. Meanwhile, the fraction of antibody that are bound to antigen at equilibrium, often called y or θ, is $ C/Ab_{TOT} $, where $ Ab_{TOT} $ indicates total (both bound and unbound) receptors. Note that the position of the equilibrium (i.e., y) depends on the starting concentrations of the reactants; however, $ K_d $ is always the same value. The total number of antibody $ Ab_{TOT} $= [C] (L-bound Ab) + [Ab] (unbound Ab). Thus,
$ \qquad y = {[C] \over Ab_{TOT}} \qquad = \qquad {[C] \over [C] + [Ab]} \qquad = \qquad {[L] \over [L] + [K_d]} \qquad $
where the right-hand equation was derived by algebraic substitution. If the antigen concentration is in excess of the concentration of the antibody, [L] may be approximated as a constant, L, for any given equilibrium. Let’s explore the implications of this result:
- What happens when L << $ K_d $?
- →Then y ~ $ L/K_d $, and the binding fraction increases in a first-order fashion, directly proportional to L.
- What happens when L >> $ K_d $?
- →In this case y ~1, so the binding fraction becomes approximately constant, and the antibodies are saturated.
- What happens when L = $ K_d $?
- →Then y = 0.5, and the fraction of antibodies that are bound to ligand is 50%. This is why you can read $ K_d $ directly off of the plots. When y = 0.5, the concentration of lysozyme (our [L]) is equal to $ K_d $. This is a great rule of thumb to know.
The figures below demonstrate how to read $ K_d $ from binding curves. You will find semilog plots (right) particularly useful today, but the linear plot (left) can be a helpful visualization as well. Keep in mind that every L value is associated with a particular equilbrium value of y, while the curve as a whole gives information on the global equilibrium constant $ K_d $.
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