Consider the dimerization 2 A 5A2, with forward rate constant ka and backward rate

Chapter 22, Problem 22.14

(choose chapter or problem)

Consider the dimerization \(2 \mathrm{~A} \rightleftharpoons \mathrm{A}_2\), with forward rate constant \(k_{\mathrm{a}}\) and backward rate constant \(k_{\mathrm{b}}\).

(a) Derive the following expression for the relaxation time in terms of the total concentration of protein, \([\mathrm{A}]_{{tot }^{\prime}}=[\mathrm{A}]+2\left[\mathrm{~A}_2\right]\):

\(\frac{1}{\tau^2}=k_{\mathrm{b}}^2+8 k_{\mathrm{a}} k_{\mathrm{b}}[\mathrm{A}]_{tot}\)

(b) Describe the computational procedures that lead to the determination of the rate constants \(k_{\mathrm{a}}\) and \(k_{\mathrm{b}}\) from measurements of \(\tau\) for different values of \([\mathrm{A}]_{tot}\).

(c) Use the data provided below and the procedure you outlined in part (b) to calculate the rate constants \(k_{\mathrm{a}}\) and \(k_{\mathrm{b}}\), and the equilibrium constant K for formation of hydrogen-bonded dimers of 2-pyridone:

\(\begin{array}{llllll}{[\mathrm{A}]_{\mathrm{tot}} /\left(\mathrm{mol} \mathrm{dm}^{-3}\right)} & 0.500 & 0.352 & 0.251 & 0.151 & 0.101 \\ \tau / \mathrm{ns} & 2.3 & 2.7 & 3.3 & 4.0 & 5.3\end{array}\)