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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 12 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent \emph{Read}: Chapter 2.3: Modeling biochemical reactions
with differential equations, by R.~Robeva, and N.~Yildirim.  pages
40--46. \\

\begin{enumerate}

\item Consider the reactions where two substrates $S$ and $T$ compete
  for binding to an enzyme $E$ to produce two different products $P$
  and $Q$:
  \begin{align*}
    &E+S\;\stackrel[p_2]{p_1}{\rightleftharpoons}\; ES\;
    \stackrel{p_3}{\longrightarrow}\; P+E
    \\ &E+T\;\stackrel[q_2]{q_1}{\rightleftharpoons}\; ET\;
    \stackrel{q_3}{\longrightarrow}\; Q+E
  \end{align*}
  \begin{enumerate}
  \item Assuming that each reaction follows the Michaelis-Menten
    kinetics, derive rate equations for $P$ and $Q$ in this
    system. That is, determine $d[P]/dt$ and $d[Q]/dt$.
  \item Explain the effects of the competition occuring. \\
  \end{enumerate}



\item The Hill equation is an approximation for multi-molecule binding
  and it assumes simultaneous binding of $n$-moecules of a substrate
  $S$ to the enzyme $E$. Suppose that two molecules of the substrate
  $S$ are undergoing a reaction with an enzyme in an ordered manner as
  follows:
  \[
  E+S\;\stackrel[k_2]{k_1}{\rightleftharpoons}\; ES+S\;
  \stackrel[k_4]{k_3}{\rightleftharpoons}\; ES_2\;
  \stackrel{k_5}{\longrightarrow}\; P+E.
  \]
  \begin{enumerate}
  \item Derive a rate equation under the steady state assumption and
    compare it with the Hill equation with Hill coefficient $n=2$:
    \[
    \frac{d[P]}{dt}=\frac{V_{\max}[S]^2}{K_m+[S]^2}\,.
    \]
  \item When do these two equations become roughly the same? \\
  \end{enumerate}


\end{enumerate}


\end{document}


