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\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 19 (Tuesday)
}} \rhead{\thepage} \cfoot{{\sf MthSc 450 $|$ Mathematical Modeling
    $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent \emph{Read}: Chapter 1: Mechanisms of gene regulation: Boolean
network models of the lactose operon in \emph{Escherichia coli}, by
R.~Robeva, B.~Kirkwood, and R.~Davis, pages 1--35. \\


% Robeva/Hodge, Exercises 1.12--14 (highly modified)
\noindent Consider a Boolean model of the \emph{lac} operon, based on
five variables: mRNA ($M$), $\beta$-galactosidase ($B$), lac permease
($P$), intracellular lactose ($L$), and allolactose ($A$), and the
following transition functions:
\begin{align*}
  f_M&=A \\
  f_B&=M \\
  f_A&=A\vee(L\wedge B) \\
  f_L&=P\vee(L\wedge\overline{B}) \\
  f_P&=M
\end{align*}
This model does not have any parameters -- it assumes that
extracellular lactose is always available and extracellular glucose
is always unavailable, and thus it is only able to describe the
behavior of the system under the conditions.
\begin{enumerate}
\item Sketch the wiring diagram for this model.
\item Sketch the state space for this model. Feel free to use the
  \emph{Discrete Visualizer of Dynamics} (DVD) software: {\tt
    dvd.vbi.vt.edu}, but let $x_1=M$, $x_2=B$, $x_3=A$, $x_4=L$, and
  $x_5=P$.
\item There are 3 fixed points: $(0,0,0,0,0)$, $(1,1,1,1,1)$, and
  $(0,0,0,1,0)$. Give a biological interpretation of the first two.
\item Explain why the fixed point $(0,0,0,1,0)$ does not make sense
  biologically.
\item Since the dynamics do not accurately reflect the behavior of
  the biological system it is meant to model, something is
  wrong. For each function, decide if it accurately reflects the
  underlying biology and/or the model assumptions. 
\item Propose a modification of the transition functions aimed at
  eliminating the biologically infeasible fixed point. Give the
  rationale for your modification and specify the biological
  mechanism or model assumptions that justify the change.
\item Draw the wiring diagram and state space of your modified
  model. Use the DVD software.
\item Analyze your model. How many fixed points are there? Do they
  all correspond to biologically realistic situations? Note that
  there should be no limit cycles of size $k\geq 2$. (Why?) 
\end{enumerate}

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