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\lhead{{\sf Math 9850, Algebraic Systems Biology}: Homework 2}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, September 20 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

$\;$ \vspace{-4mm}

\noindent \textbf{Do}.
\begin{itemize}
\item Download and install Rstudio from \url{https://posit.co/downloads/}
\item Download and install Cyclone from
  \url{https://github.com/discretedynamics/}
\item Familiarize yourself with Macaulay2 at
  \begin{center}
    \url{https://www.unimelb-macaulay2.cloud.edu.au/}
  \end{center}
\item Browse the Gene Intersection Network simulator (GINsim) website
  at
  \begin{center}
    \url{http://ginsim.org/home}
  \end{center}
\end{itemize}

\medskip


\noindent \textbf{Exercise}. 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}[(a)]

\item What other assumptions are made in this model? (E.g., presence
  or absense of extracellular sugars, time-steps, degredation, etc.)
    
  \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 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 Macaulay2 to convert these Boolean functions into
    polynomials over $\F_2=\{0,1\}$. Then find the fixed points using a
    Gr\"obner basis of a particular polynomial ideal.

  \item Draw the wiring diagram of the old and new Boolean model.

  \item Use the BoolNet package in RStudio to find the attractors and plot the state space. Include a print-out or screenshot.
    
  \item Repeat the previous step, but with either Cyclone or GINsim. 

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

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