NMR spectroscopy may be used to determine the equilibrium constant for dissociation of a

Chapter 15, Problem 15.23

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NMR spectroscopy may be used to determine the equilibrium constant for dissociation of a complex between a small molecule, such as an enzyme inhibitor I, and a protein, such as an enzyme E: EI5E + I KI = [E][I]/[EI] In the limit of slow chemical exchange, the NMR spectrum of a proton in I would consist of two resonances: one at I for free I and another at EI for bound I. When chemical exchange is fast, the NMR spectrum of the same proton in I consists of a single peak with a resonance frequency given by = fII + fEIEI, where fI = [I]/([I] + [EI]) and fEI = [EI]/([I] + [EI]) are, respectively, the fractions of free I and bound I. For the purposes of analysing the data, it is also useful to define the frequency differences = I and = EI I. Show that, when the initial concentration of I, [I]0, is much greater than the initial concentration of E, [E]0, a plot of [I]0 against 1 is a straight line with slope [E]0 and y-intercept KI.