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\lhead{{\sf Math 9850, Algebraic Systems Biology}: Homework 4}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, October 11 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

\item \textbf{Reduction}. Consider the following Boolean network model of the \emph{lac} operon.
  \[
  \def\arraystretch{1.4}
  \begin{array}{llll}
    f_1=x_4\And\Not{x_5}\And\Not{x_6}\qquad &&& f_8=x_9\Or x_{10} \\
    f_2=x_1 \qquad &&& f_9=x_3\And\Not{G_e}\And L_e \\
    f_3=x_1 \qquad &&& f_{10}=\Not{G_e}\And\big(L_e\Or(x_3\And L_{em})\big) \\
    f_4=\Not{G_e} \qquad &&& f_{G_e}=G_e\\
    f_5=\Not{x_7}\And\overline{x_8} &&& f_{L_e}=L_e \\
    f_6=x_5\Or(\Not{x_7}\And\Not{x_8}) &&& f_{L_{em}}=L_{em} \\
    f_7=x_2\And x_9 &&&
  \end{array}
  \]
  Here, the variables represent
  \[
  (x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10})=(M,B,P,C,R,R_m,A,A_m,L,L_m).
  \]
  \begin{enumerate}
  \item Use Macaulay2 to reduce this Boolean network, as much as
  possible.
\item Draw the wiring diagram of the reduced network. Find its fixed
  point(s) and use these to determine the fixed point(s) of the
  original network by back-substitution.
  \end{enumerate}
  


    \end{enumerate}

\end{document}
  
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