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\lhead{{\sf Math 9850, Algebraic Systems Biology}: Homework 3}
\chead{}\rhead{\thepage}
\lfoot{Due Friday, September 27 2024}
\cfoot{}\rfoot{Written by M.~Macauley}
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\begin{document}

$\;$ \vspace{-4mm}

\begin{enumerate}

\item \textbf{Fixed points}. Consider the following $10$-variable model
  of the \emph{lac} operon, from class, where the $3$ parameters are
  added with ``frozen'' update functions.
\begin{alignat*}{2}
  f_M&=\Not{R} \wedge \Not{R_m} \wedge C\hspace{20mm} &f_{A_m}&=L\vee L_m \\
  f_B&=M & f_L&=\Not{G_e}\wedge P\wedge L_e \\
  f_P&=M & f_{L_m}&=  \Not{G_e}\wedge((L_{em} \wedge P)\vee L_e) \\
  f_C&=\Not{G_e} & f_{L_e}&=L_e \\
  f_R&=\Not{A}\wedge\Not{A_m} & f_{L_{em}}&=L_{em} \\
  f_{R_m}&=(\Not{A} \wedge \Not{A_m}) \vee R & f_{G_e}&=G_e \\ 
  f_A&=L\wedge B &   
\end{alignat*}
Use Macaulay2 to compute a Gr\"obner basis of the ideal
$I=(f_{x_1}+x_1,\dots,f_{x_{13}}+x_{13})$, in the quotient ring
$\F_2[x_1,\dots,x_{13}]/(x_i^2-x_i)$. Solve the resulting system by hand to find all fixed points of the original Boolean model. Code is provided on the course website. \\


\item \textbf{Bistability}. Consider the following model of the tryptophanase (\emph{tna}) operon in E. coli:
  \[
  \begin{array}{llll}
    f_A=M\And\Not{\gamma} \qquad &&& f_P=\Not{W}\And\Not{W_m} \\
    f_B=M &&& f_W=\omega_e\And B \\
    f_C=\Not{\gamma} &&&  f_{W_m}=(\omega_{em}\And B)\Or \omega_e\Or W \\
    f_M=C\And\Not{P}
  \end{array}
  \]
  There are three parameters: $\gamma$ is glucose, $\omega_e$ is high
  levels of extracellular tryptophan, and $\omega_{em}$ represents (at
  least) medium levels of extracellular tryptophan.
  \begin{enumerate}
  \item Let $\gamma$, $\omega_e$, and $\omega_{em}$ be ``frozen''
    variables, making this a $10$-variable model. Use Cyclone to find
    the attractors. Code is provided on the course website.
  \item Fix $(\gamma,\omega_e,\omega_{em})=(0,0,1)$ as constants to
    get a $7$-variable model, which assumes no glucose and medium
    levels of extracellular tryptophan. Use BoolNet to find the
    attactors under an asynchronous update, and summarize your
    findings. This can be done with the command
\begin{verbatim}
  >  getAttractors(tnaModel,type="asynchronous")
\end{verbatim}
How does this compare to the synchronous state space?
\item Explain how/why this model exhibits bistability.
    

  \end{enumerate}


    \end{enumerate}

\end{document}
  
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