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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due March 3 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2016 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent \emph{Read}:
\begin{itemize}
\item Chapter 6 of Robeva: \emph{Steady state analysis of Boolean
  models: a dimension reduction approach}. By D.~Murrugarra and
  A.~Veliz-Cuba, pages 121--139.
\item Chapter 3.1--3.4 of Robeva/Hodge: \emph{Inferring the topology
  of gene regulatory networks: an algebraic approach to reverse
  engineering}. By B.~Stigler and E.~Dimitrova, pages 75--88. \medskip
\end{itemize}


\begin{enumerate}

% Robeva, Exercise 6.28 (modified)

\item Consider the following Boolean network model of the \emph{lac}
  operon: 
  \[
  \begin{array}{lll}
    x_1 = \text{\emph{lac} mRNA ($M$)} & f_1 = x_4\And\overline{x_5} \\
    x_2 = \text{\emph{lac} permease ($P$)} & f_2=x_1 \\
    x_3 = \text{$\beta$-galactosidase ($B$)} & f_3=x_1 \\
    x_4 = \text{catabolite activator protein ($C$)} \hspace{8mm} 
    & f_4=\overline{x_{11}} \\
    x_5 = \text{repressor protein ($R$)} 
    & f_5=\overline{x_6}\And\overline{x_7} \\
    x_6 = \text{high allolactose ($A$)} & f_6=x_3\And x_8 \\
    x_7 = \text{allolactose ($A_\ell$)} & f_7=x_6\Or x_8\Or x_9\\
    x_8 = \text{high intracellular lactose $(L)$} 
    & f_8=x_2\And x_{10}\And\overline{x_{11}} \\
    x_9 = \text{intracellular lactose $(L_\ell)$} 
    & f_9=(x_8\Or x_{10})\And\overline{x_{11}} \\
    x_{10} = \text{extracellular lactose $(L_e)$} & f_{10}=x_{10} \\
    x_{11} = \text{extracellular glucose $(G_e)$} & f_{11}=x_{11} 
  \end{array}
  \]
  \begin{enumerate}
  \item Reduce this Boolean network starting from the last
    variable. It is strongly recommended that you use Macaulay2 in
    Sage.
  \item Draw the wiring diagram of the reduced network. Find its fixed
    points and use these to determine the fixed points of the original
    network.
  \item Repeat Part~(a) but using a completely different variable order. \medskip
  \end{enumerate}

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

% Robeva/Hodge, Exercise 3.5 (modified) 
\item Find two finite dynamical systems $f=(f_1,f_2,f_3)$ over $\Z_2$
  that fit the following data:
  \[
  (1,1,0)\stackrel{f}{\longto}(0,0,1)\stackrel{f}{\longto}(0,0,1)\,.
  \]
  You may express your answer using Boolean algebra or as polynomials
  in $\Z_2[x_1,x_2,x_3]$. \medskip

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

% Robeva/Hodge, Exercise 3.7 & 3.8 (modified).
\item Consider the following \emph{time series} in a $3$-node
  polynomial dynamical system over $\Z_3$:
\[
\begin{tikzpicture}
  \node (111) at (0,0) {\small $\s_1=(1,1,1){\color{white}=\t_0}$};
  \node (201) at (0,-1.5) {\small $\s_2=(2,0,1)=\t_1$};
  \node (200) at (0,-3) {\small $\s_3=(2,0,0)=\t_2$};
  \node (022) at (0,-4.5) {\small $\s_4=(0,2,2)=\t_3$};
  \node (same) at (0,-6) {\small ${\color{white}\s_5=}(0,2,2)=\t_4$};
  \draw[-stealth] (111) to (201);
  \draw[-stealth] (201) to (200);
  \draw[-stealth] (200) to (022);
  \draw[-stealth] (022) to (same);
\end{tikzpicture}
\]
For reference, here are the input vectors $\s_i$ and output vectors $\t_i$:
\begin{align*}
  \s_1&=(s_{11},s_{12},s_{13})=(1,1,1)\,,\qquad\t_1=(t_{11},t_{12},t_{13})=(2,0,1)\,,
  \\ 
  \s_2&=(s_{21},s_{22},s_{23})=(2,0,1)\,,\qquad\t_2=(t_{21},t_{22},t_{23})=(2,0,0)\,,
  \\
  \s_3&=(s_{31},s_{32},s_{33})=(2,0,0)\,,\qquad\t_3=(t_{31},t_{32},t_{33})=(0,2,2)\,, 
  \\
  \s_4&=(s_{41},s_{42},s_{43})=(0,2,2)\,,\qquad\t_4=(t_{41},t_{42},t_{43})=(0,2,2)\,.
\end{align*}
\begin{enumerate}
\item For each $j=1,2,3,4$, write down the ideal $I(\s_j)$ of
  polynomials that vanishes on the data point $\s_j$.
\item Use Macaulay2 in Sage to compute the ideal $I$ that vanishes on all of
  the data points. The following commands will be useful:
\begin{verbatim}
   %default_mode macaulay2
   R = ZZ/3[x1,x2,x3,MonomialOrder=>Lex];
   I = intersect{I1, I2, I3, I4};  
\end{verbatim}
Turn in a print-out of your Sage worksheet.
\item Find polynomials $f_1,f_2,f_3$ in $\Z_3[x_1,x_2,x_3]$ that
  fit the data. That is, $f_j(\s_i)=\t_i$ for each $i$ and $j$. Use this to write down the \emph{model space} of the time series.
\end{enumerate}



%In this exercise, we will find a polynomial dynamical systems
%  $f=(f_1,f_2,f_3)$ over $\Z_2$ that fits the following data:
%  \[
%  (1,1,0)\stackrel{f}{\longto}(0,0,1)\stackrel{f}{\longto}(0,0,1)\,.
%  \]


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


  % Robeva/Hodge, Exercise 2.8

%\newpage
%
%$\;$\vspace{.1in}





  
\end{enumerate}


\end{document}


