Many biological reactions are very sensitive to \(\mathrm{pH}\). This can
readily be incorporated into the rate laws because protolytic reactions can be
assumed to be much faster than other rates in most cases. For example, in
enzyme mechanisms the ionization states of a few key protein side chains are
often critical. Suppose that two ionizable groups on the enzyme are critical
for catalytic activity and that one of them needs to be protonated and the
other deprotonated. The protolytic reactions can be written as
$$
\mathrm{EH}_{2} \rightleftharpoons \mathrm{EH}+\mathrm{H}^{\prime}
\rightleftharpoons \mathrm{E}+2 \mathrm{H}^{\prime}
$$
If only the species \(\mathrm{EH}\) is catalytically active and the protolytic
reactions are much more rapid than the other steps in the reaction, all of the
rate constants that multiply the free enzyme concentration in the rate law
have to be multiplied by the fraction of enzyme present as EH. A. Calculate
the fraction of free enzyme present as EH at a given pH. Your answer should
contain the concentration of \(\mathrm{H}^{+}\)and the ionization constants of
the two side chains, \(\left.K_{\mathrm{E}
1}=[\mathrm{E}]\left[\mathrm{H}^{+}\right] / \mathrm{EH}\right]\) and
\(\left.K_{\mathrm{E} 2}=[\mathrm{EH}]\left[\mathrm{H}^{+}\right] /
\mathrm{EH}_{2}\right]\).
B. Assume that ES in the Michaelis-Menten mechanism (Eq. 5-2) also exists in
three protonation states, \(\mathrm{ESH}_{2}\), ESH, and ES, with only ESH being
catalytically active. Calculate the fraction of the enzyme-substrate complex
present as EHS. Designate the ionization constants as \(K_{\mathrm{ESI}}\) and
\(K_{\mathrm{ES} 2}\).
C. Use the results of parts \(\mathrm{A}\) and \(\mathrm{B}\) to derive equations
for the \(\mathrm{pH}\) dependence of \(V_{\mathrm{m}^{\prime}} K_{\mathrm{M}}\),
and \(V_{\mathrm{m} /} K_{\mathrm{M}}\). Measurement of the \(\mathrm{pH}\)
dependence of the steady. state parameters permits determination of the
ionization constants, and sometimes identification of the amino acid side
chains.
