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\pagestyle{fancy} \lhead{{\sf Homework 9 $|$ Due November 7 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Fall 2017 $|$ M.~Macauley}}


\begin{document}

%$\;$

\noindent \emph{Read}: 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. \medskip

\noindent \emph{Do}:

\begin{enumerate}

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

\item Draw the wiring diagram of the Boolean network
  \[
  f=(f_1,f_2,f_3,f_4,f_5,f_6)=(x_6,\;x_1,\;x_2,\;\overline{x_3},\;x_4,\;\overline{x_5})
  \]
  and then reduce it by eliminating the variables in the order $x_6,x_5,\dots$. At each step along the way, write out the functions and draw the wiring diagram. \bigskip
  
%%---------------------------------------------------------------------

\item Suppose we want to reduce a Boolean network by removing a vertex
  $x_i$ with no self-loop (equivalently, $f_{x_i}$ doesn't depend on
  $x_i$). First, for every vertex $y$ such that $\tikz[baseline=-0.5ex,node distance=1.5cm]{
      \node (xi){$x_i$}; \node (y) [right of=xi] {$y$}; 
      \draw [->] (xi) -- (y);}$,
  substitute $f_{x_i}$ into $x_i$:
  \[
  f_y(x_1\dots,\underbrace{\cdots x_i\cdots}_{\text{pos. $y$}},\dots,x_n)\qquad\text{becomes}\qquad
  f_y((x_1\dots,\underbrace{\cdots f_{x_i}\cdots}_{\text{pos. $y$}},\dots,x_n)\,. 
  \]
  Next, each pair of edges $\tikz[baseline=-0.5ex,node distance=1.5cm]{
      \node (v){$v$}; \node (xi) [right of=v] {$x_i$}; 
      \node (y) [right of=xi] {$y$};
      \draw [->] (v) -- (xi); \draw [->] (xi) -- (y);}$ in the wiring diagram must be replaced with the single edge $\tikz[baseline=-0.5ex,node distance=1.5cm]{
      \node (v){$v$}; \node (xi) [right of=v] {$y$}; 
      \draw [->] (v) -- (xi);}$. The situation is similar for the other three possibilities:
    \begin{enumerate}
    \item $\tikz[baseline=-0.5ex,node distance=1.5cm]{
      \node (v){$v$}; \node (xi) [right of=v] {$x_i$}; 
      \node (y) [right of=xi] {$y$};
      \draw [->] (v) -- (xi); \draw [-|] (xi) -- (y);}$
    \item $\tikz[baseline=-0.5ex,node distance=1.5cm]{
      \node (v){$v$}; \node (xi) [right of=v] {$x_i$}; 
      \node (y) [right of=xi] {$y$};
      \draw [-|] (v) -- (xi); \draw [->] (xi) -- (y);}$
    \item $\tikz[baseline=-0.5ex,node distance=1.5cm]{
      \node (v){$v$}; \node (xi) [right of=v] {$x_i$}; 
      \node (y) [right of=xi] {$y$};
      \draw [-|] (v) -- (xi); \draw [-|] (xi) -- (y);}$
    \end{enumerate}
    For each of these pairs of edges, determine what to replace them with in the wiring diagram of the new network. This would allow one to characterize algorthmically how to construct the wiring diagram of a reduced Boolean network. \bigskip

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

% Robeva, Exercise 6.28 (modified)
\item Consider the following Boolean network model of the \emph{lac}
  operon: 
  \[
  \begin{array}{lll}
    x_1 = \text{\emph{lac} mRNA ($M$)} & f_1 = x_4\And\overline{x_5} \\
    x_2 = \text{\emph{lac} permease ($P$)} & f_2=x_1 \\
    x_3 = \text{$\beta$-galactosidase ($B$)} & f_3=x_1 \\
    x_4 = \text{catabolite activator protein ($C$)} \hspace{8mm} 
    & f_4=\overline{x_{11}} \\
    x_5 = \text{repressor protein ($R$)} 
    & f_5=\overline{x_6}\And\overline{x_7} \\
    x_6 = \text{high allolactose ($A$)} & f_6=x_3\And x_8 \\
    x_7 = \text{medium allolactose ($A_m$)} & f_7=x_6\Or x_8\Or x_9\\
    x_8 = \text{high intracellular lactose $(L)$} 
    & f_8=x_2\And x_{10}\And\overline{x_{11}} \\
    x_9 = \text{medium intracellular lactose $(L_m)$} 
    & f_9=(x_8\Or x_{10})\And\overline{x_{11}} \\
    x_{10} = \text{extracellular lactose $(L_e)$} & f_{10}=x_{10} \\
    x_{11} = \text{extracellular glucose $(G_e)$} & f_{11}=x_{11} 
  \end{array}
  \]
  \begin{enumerate}
  \item Reduce this Boolean network starting from the last
    variable. It is strongly recommended that you use Macaulay2 in
    Sage or at {\tt web.macaulay2.com}.
  \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}

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

  
\end{enumerate}


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