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\pagestyle{fancy} \lhead{{\sf Homework 8 $|$ Due February 25 (Thursday)
}} \rhead{\thepage} \cfoot{{\sf Math 4500 $|$ Mathematical Modeling
    $|$ Spring 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent \emph{Read}: Chapter 2.3--2.7: Bistability in the lactose
operon on \emph{Escherichia coli}: A comparison of differential
equation and Boolean network models. By R.~Robeva and N.~Yildirim,
pages 47--73. \\


\begin{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 ($B$), allolactose
  ($A$), intracellular lactose ($L$), and \emph{lac} permease ($P$),
  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 Print out the state space for this model using ADAM. 
\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 print 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 $k\geq 2$. \\
\end{enumerate}


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


% Robeva/Hodge, Exercises 2.6 (highly modified) 
\item Recall the $3$-variable ODE model of the \emph{lac} operon
  proposed by Yildirim and Mackey in 2004, where $M(t)=$ mRNA, $B(t)=$
  $\beta$-galactosidase, and $A(t)=$ allolactose (concentrations),
  respectively.
  \begin{align*}
    \frac{dM}{dt}&=
    \alpha_M\frac{1+K_1(e^{-\mu\tau_M}A_{\tau_M})^n}{K+K_1(e^{-\mu\tau_M}A_{\tau_M})^n}
    -\widetilde{\gamma_M}M
    \\ \frac{dB}{dt}&=\alpha_Be^{-\mu\tau_B}M_{\tau_B}-\widetilde{\gamma_B}B
    \\ \frac{dA}{dt}&=
    \alpha_AB\frac{L}{K_L+L}-\beta_AB\frac{A}{K_A+A}-\widetilde{\gamma_A}A
  \end{align*}
  Suppose the exponential decay constants are estimated from the
  literature to be $\widetilde{\gamma_M}=.441$,
  $\widetilde{\gamma_B}=.031$, and
  $\widetilde{\gamma_A}=.55$.
  %%
  \begin{enumerate}
  \item Compute the half life for $M$, $B$, and $A$.
  \item Justify the following Boolean model by explaining the logical
    expression defining each transition function:
    \[
    \begin{array}{ll}
    f_M=A & f_{B_{\old}}=\overline{M}\And B
    \\ f_B=M\Or\left(B\And\overline{B_{\old}}\right)\hspace{10mm} &
    f_A=(B\And L)\Or L_{\high}
  \end{array}
  \]
  What approximate timestep is assumed by this model?

  \item Assuming $(M,B,B_{\old},A)=(x_1,x_2,x_3,x_4)$, write the
    polynomial form of the Boolean model above. The easiest way to do
    this is to go into Sage and type the following commands (hit
    shift-enter after each one):
\begin{verbatim}
    %default_mode macaulay2
    R = ZZ/2[x1,x2,x3,x4] / ideal(x1^2-x1, x2^2-x2, x3^2-x3, x4^2-x4);
    RingElement | RingElement :=(x,y)->x+y+x*y;
    RingElement & RingElement :=(x,y)->x*y;
\end{verbatim}
Now, if you want to store $f=(x_1\Or x_2)\And x_3$, you can enter \verb!f=(x1 | x2) & x3!. Typing \verb!f! will give the polynomial form of $f$. 

\item Open a new tab of Sage (so you aren't in the Macaulay2
  environment), and find the fixed points of the Boolean network by
  solving $\{f_1+x_1=0,\dots,f_4+x_4=0\}$.

\item Does this model exhibit bistability? Justify your answer. \\

%\begin{verbatim}
%   P.<x1,x2,x3,x4,x5,x6> = PolynomialRing(GF(2), 6, order='lex'); P
%\end{verbatim}

\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. Alternately, this can be done implicitly through the
  logical assumptions built into the transition functions. For
  example, if we choose to work with $\gamma_A=1.8\times 10^{-4}$,
  then the degradation time for $A$ will be slower than both $M$ 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 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}

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





  
\end{enumerate}


\end{document}


