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\lhead{{\sf Algebraic Systems Biology}: Homework 1}
\chead{}\rhead{\thepage}
\lfoot{Due Saturday, 26 April 2025}
\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

$\;$ \vspace{-4mm}

\noindent \textbf{Read}. \emph{Mathematical Concepts in Modern
  Biology}, by Robeva/Hodge. Ch. 1: Boolean models of the \emph{lac}
operon in \emph{E. coli}, and Ch. 2: Bistability in the \emph{lac}
operon: a comparision of ODE and Boolean models, pp. 1--74. \\

%\noindent \textbf{Do}. Familiarize yourself with either Macaulay2,
%Singular, or Sage. \\

\noindent \textbf{Exercises}. \\ 

\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*}
  Assume that each reaction follows the Michaelis-Menten
  kinetics. Also, assume that that the initial enzyme concentration is
  $E_0=[E]+[ES]+[ET]$.
  \begin{enumerate}
  \item Derive rate equations for $P$ and $Q$ in this system in terms
    of $[ES]$ and $[ET]$. That is, determine $d[P]/dt$ and $d[Q]/dt$.
    
  \item Derive rate equations for $ES$ and $ET$. 
    
  \item Assume that the enzyme-substrate complexes reach equilibrium
    quickly: $d[ES]/dt\approx 0$ and $d[ET]/dt\approx 0$. Solve
    for $[E]$ in each of these equations.
    
  \item Equate the two expressions for $[E]$ from Part~(c) and solve for $[ET]$.
    
  \item Solve for $[ES]$ by plugging your answers to Parts (c) and (d)
    into $E_0=[E]+[ES]+[ET]$. You should not have $[E]$ or $[ET]$ in
    your final answer.
    
  \item Plug this into the original ODE for $d[P]/dt$. 
    
  \item Derive an analogous ODE for $d[Q]/dt$. 
    
  \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:
  \[
  E+S\;\stackrel[k_2]{k_1}{\rightleftharpoons}\; ES,\qquad\qquad ES+S\;
  \stackrel[k_4]{k_3}{\rightleftharpoons}\; ES_2\;
  \stackrel{k_5}{\longrightarrow}\; P+E.
  \]
  %\[
  %E+S\;\stackrel[k_2]{k_1}{\rightleftharpoons}\; ES+S\;
  %\stackrel[k_4]{k_3}{\rightleftharpoons}\; ES_2\;
  %\stackrel{k_5}{\longrightarrow}\; P+E.
  %\]
  Assume that the reaction follows the Michaelis-Menten kinetics and that the initial enzyme concentration is $E_0=[E]+[ES]+[ES_2]$. 
  \begin{enumerate}
  \item Derive rate equations for $P$, $ES$, and $ES_2$.

    %Be \emph{very} careful with the $ES\stackrel{k_2}{\longrightarrow}
    %E+S$ reaction when deriving the equation for $[ES]$.

  \item Assume that $d[ES_2]/dt\approx 0$ and solve for $[ES]$.

  \item Assume that $d[ES]/dt\approx 0$. Plug your answer to Part~(b) into 
    this and solve for $[E]$.

  \item Plug your expressions for $[E]$ and $[ES]$ back into
    $E_0=[E]+[ES]+[ES_2]$ and solve for $[ES_2]$. 

  \item Derive an ODE for $[P]$ of the form $d[P]/dt=f([S])$.

  \item Compare your answer to Part~(d) to the Hill equation with Hill
    coefficient $n=2$:
    \[
    \frac{d[P]}{dt}=\frac{V_{\max}[S]^2}{K_m+[S]^2}\,.
    \]
    When do these two equations become roughly the same, and why? \\
  \end{enumerate}


\item Consider a Boolean model of the
  \emph{lac} operon, based on five variables: mRNA ($M$),
  $\beta$-galactosidase ($B$), \emph{lac} permease ($P$), allolactose
  ($A$), and intracellular lactose ($L$), and the following transition
  functions:
  \begin{align*}
    f_M&=A \\
    f_B&=M \\
    f_P&=M \\
    f_A&=A\vee(L\wedge B) \\
    f_L&=P\vee(L\wedge\overline{B})
  \end{align*}
  \begin{enumerate}
  \item As we saw in class, the dynamics do not accurately reflect the
    behavior of the biological system it is meant to model. Therefore,
    something is wrong. Four of these functions are reasonable;
    justify each one in a single well-written sentence.
    
  \item Given the knowledge that $\beta$-galactosidase can also cleave
    allolactose into glucose and galactose, explain why one of these
    functions does not accurately reflect the underlying biology
    and/or the model assumptions. Propose a modification, aimed at
    eliminating the biologically infeasible fixed point,
    $(0,0,0,1,0)$. Give a rationale for your modification and specify
    the biological mechanism or model assumptions that justify the
    change.

  \item Use Cyclone to find the fixed point and
    plot the state space. Include a screenshot.

  \item Justify why the long-term behavior of the system (fixed
    points) agrees with what we should expect biologically.
    
  \end{enumerate}
  
\end{enumerate}
\end{document}

%%---------------------------------------------------------------------  
  

\end{enumerate}


\end{document}


