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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due March 14 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 450 $|$ Mathematical Modeling
    $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent \emph{Read}: Chapter 2: Bistability in the lactose operon of
\emph{Escherichia coli}: A comparison of differential equation and
Boolean network models, by R.~Robeva, and N.~Yildirim.  pages
57--73. \\

\begin{enumerate}


% Robeva/Hodge, Exercise 2.8
\item Recall our original Boolean network model for the \emph{lac}
  operon:
  \begin{align*}
    f_M&=\overline{G_e}\wedge(L\vee L_e) \\ f_E&=M
    \\ f_L&=\overline{G_e}\wedge((E\wedge
    L_e)\vee(L\wedge\overline{E})).
  \end{align*}
  Suppose now that $L_e$ instead stands for external concentration of
  lactose that being at least medium. Introduce a new parameter
  $L_{e_{\text{\tiny high}}}$ to denote high levels of external
  lactose.
  \begin{enumerate}
  \item Modify the transition functions above to get a model that
    exhibits bistable behavior for medium lactose concentrations.
  \item For the three possibile concentration levels of lactose; low:
    $(L_e,L_{e_{\text{\tiny high}}})=(0,0)$, medium:
    $(L_e,L_{e_{\text{\tiny high}}})=(1,0)$, and high:
    $(L_e,L_{e_{\text{\tiny high}}})=(1,1)$, sketch or print out the
    phase space of this model. Use the DVD software ({\tt
      dvd.vbi.vt.edu}). Clearly show how this model exhibits
    bistability. \\
  \end{enumerate}


\item So far, we have used designated ``old'' variables to separate
  the time scales of dilution and degradation processes from those of
  synthesis. An alternate to introducing new variables this is to
  properly modify the transition functions. \\

  Consider a model of the \emph{lac} operon using variables $M$, $B$,
  and $A$, and parameters $L$ and $L_{\text{high}}$.  Suppose that the
  degradation time for $A$ is much larger than that for $M$ and
  $B$. In this case, choosing a proper timestep allows us to neglect
  the degradation times for $M$ and $B$. Instead of introducing an
  $A_{\text{old}}$ variable, we propose the following model, which
  builds this feature into the function:
  \begin{align*}
    f_M&=A, \\
    f_B&=M, \\
    f_A&=(B\wedge L)\vee L_{\text{high}}\vee(A\wedge\overline{B}).
  \end{align*}
  \begin{enumerate}
  \item Justify the three equations in this model, and why this
    captures the delay in the degradation of $A$. Your answer should
    be clear and convincing.
  \item Use the DVD software to sketch (or print) the phase space of
    this model for the three levels of lactose concentration: low,
    medium and high.
  \item Does this model exhibit bistability? Why or why not?
  \end{enumerate}

\end{enumerate}


\end{document}



  % Robeva/Hodge, Exercise 2.6
\item In the model of the \emph{lac} operon from class, we assume that
  the timestep was $t=10$ minutes. This meant that  If a larger timestep is considered,
  say $t=20$ minutes, then we can elimiate the variables
  $A_{\text{old}}$ and $B_{\text{old}(2)}$. Consider the following
  model:
  \[
  \begin{array}{lll}
    f_M=A && f_{B_{\tiny \text{old}}}=\overline{M}\wedge B
    \\ f_B=M\vee (B\wedge \overline{B_{\text{old}}}) && f_A=(B\wedge
    L)\vee L_{\text{high}}
  \end{array}
  \]
  \begin{enumerate}
    \item Justify the model by explaining the logical expression
      defining each of the four transition functions.
    \item Explain why the model would be consistent with choosing a
      time step $t=20$ min.
    \item Use DVD ({\tt dvd.vbi.vt.edu}) to analyze the model and
      obtain the state space diagram. ???
  \end{enumerate}


