1.14.23: Enzyme-Substrate Interaction
- Page ID
- 375473
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We consider the formation in aqueous solution of an enzyme –substrate complex \(\mathrm{ES}\) by an enzyme \(\mathrm{E}\) and substrate \(\mathrm{S}\). The system is prepared using \(\mathrm{n}^{0} (\mathrm{E})\) moles of enzyme and \(\mathrm{n}^{0} (\mathrm{S}\)) moles of substrate; equation (a)
| \(\mathrm{E}(\mathrm{aq}) +\) | \(\mathrm{S}(\mathrm{aq}) \Leftrightarrow\) | \(\mathrm{ES}(\mathrm{aq})\) | |
| At \(t = 0\) | \(\mathrm{n}^{0}(\mathrm{E})\) | \(\mathrm{n}^{0}(\mathrm{S})\) | \(0 \mathrm{~mol}\) |
| At \(t = \infty\) | \(n^{0}(E)-\xi\) | \(\mathrm{n}^{0}(\mathrm{~S})-\xi\) | \(\xi \mathrm{~mol}\) |
The upper limit of the extent of interaction \(\xi\) is controlled by whichever is the smallest amount, either \(\mathrm{n}^{0}(\mathrm{E})\) or \(\mathrm{n}^{0}(\mathrm{S})\). The latter two variables determine the total amount of \(\mathrm{ES}(\mathrm{aq})\) which can be formed in the limit of tight binding.
The approach described above can be extended to more complicated schemes involving multip-step reactions. In the following we consider the case where enzyme \(\mathrm{E}\) converts substrate \(\mathrm{A}\) into product \(\mathrm{D}\). The system is prepared using \(\mathrm{n}^{0}(\mathrm{E})\) moles of enzyme and \(\mathrm{n}^{0}(\mathrm{A})\) moles of substrate \(\mathrm{A}\) such that there are two intermediates \(\mathrm{EB}(\mathrm{aq})\) and \(\mathrm{EC}(\mathrm{aq})\), product \(\mathrm{D}\) being liberated from the bound state in the final step. The equilibrium state can be represented by the following scheme.
\[\begin{aligned}
&\mathrm{E}(\mathrm{aq}) \quad+\mathrm{A}(\mathrm{aq}) \Leftrightarrow \mathrm{EB}(\mathrm{aq}) \Leftrightarrow \mathrm{EC}(\mathrm{aq}) \Leftrightarrow \mathrm{E}(\mathrm{aq})+\mathrm{D}(\mathrm{aq})\\
&\mathrm{n}^{0}(\mathrm{E})-\xi_{1}+\xi_{3} \mathrm{n}^{0}(\mathrm{~A})-\xi_{1} \quad \xi_{1}-\xi_{2} \quad \xi_{2}-\xi_{3} \quad \mathrm{n}^{0}(\mathrm{E})-\xi_{1}+\xi_{3} \xi_{3}
\end{aligned} \nonumber \]
The key point is that at equilibrium the amounts of enzyme \(\mathrm{E}(\mathrm{aq})\) identified at both ends of the reaction must be the same. Further a mass balance shows that the total amount of enzyme present equals \(\mathrm{n}^{0}(\mathrm{E}) \mathrm{~mol}\). Other features are interesting; \(\xi_{3}\) must be zero if \(\xi_{2}\) is zero. The method can be applied to more complicated reaction schemes including those where the path from reactant to product involves parallel reactions.