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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due April 10 (Friday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2015 $|$ 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}

\item Recall the model of Stigler and Veliz-Cuba for the \emph{lac}
  operon, where a global system state is a $9$-varible Boolean vector
  $(M,P,B,C,R,A,A_\ell,L,L_\ell)=(x_1,x_2,\dots,x_9)\in\F_2^9$:
  \[
  \begin{array}{lll}
    f_M=\overline{R}\wedge C && \\ f_P=M && f_B=M \\ f_C=\overline{G_e}
    && f_R=\overline{A}\wedge\overline{A_\ell} \\ f_A=L\wedge B &&
    f_{A_\ell}=A\vee L\vee L_\ell \\ f_L=\overline{G_e}\wedge P\wedge L_e && 
    f_{L_\ell}=\overline{G_e}\wedge(L\vee L_e)
  \end{array}
  \]
  Modify this model by removing the variables $L_\ell$ and $A_\ell$,
  which represent low concentrations of intracellular lactose and
  glucose, respectively. Use Gr\"obener bases and Sage to compute the
  fixed points of this model for each of the four possibilities for
  $(L_e,G_e)$, external lactose and glucose. Do the fixed points make
  sense biologically? Since the state space has size $2^7=128$, the
  Analysis of Dynamic Algebraic Models (ADAM) toolbox, at
  \url{http://adam.plantsimlab.org/}, should be able to handle it. Are there
  any periodic cycles that are not fixed points? Do you notice any
  shortcomings of this new model? \\
  

  

\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 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 using ADAM. Clearly show how this model exhibits
    bistability. \\
  \end{enumerate}




  % Robeva/Hodge, Exercise 2.8
\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 ADAM 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}
 
 
% Robeva/Hodge, Exercises 1.12--14 (highly modified)
\item 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}[(a)]
\item Sketch the wiring diagram for this model.
\item Sketch the state space for this model. Feel free to use
  ADAM, 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 ADAM 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}



  
\end{enumerate}


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