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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due February 26 (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. \\


\noindent \emph{Do}: Create an account on either the Sage Notebook
({\tt http://www.sagemath.org/}) or on the Clemson Sage Server ({\tt
  https://sage.math.clemson.edu:34567}). Email me your username and I
will share with you a Sage worksheet containing the $9$-variable
Boolean model of the \emph{lac} operon that appears below in Problem~3. \\


\begin{enumerate}

\item Consider the first model for the \emph{lac} operon that we saw:
  \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*}
  Convert these equations into polynomials over
  $\F_2:=\{0,1\}$. Recall that $x\wedge y=xy$, $x\vee y=x+y+xy$, and
  $\overline{x}=1+x$. Fully simplify your answers. \\


% Robeva/Hodge, Exercises 1.9 (highly modified)
\item Consider the following system of polynomial equations:
\begin{align*}
  x^2+y^2+xyz&=1 \\
  x^2+y+z^2&=0 \\
  x-z&=0
\end{align*}
To compute a Gr\"obner basis for this system, type the following
commands into Sage, one-by-one, and press Shift$+$Enter after each one:
\begin{verbatim}
                P.<x,y,z> = PolynomialRing(RR, 3, order='lex'); P
                I = ideal(x^2+y^2+xyz-1, x^2+y+z^2, x-z); I
                B = I.groebner_basis(); B
\end{verbatim}

For each system above, use the Gr\"obner basis you computed to write a
simpler systems of polynomial equations that has the same set of
solutions. Sove that system \emph{by hand} (it's not hard) to find all
real solutions to the original system. \\


\item Repeat the steps of the previous problem for this system of
  polynomial equations:
\begin{align*}
  x^2y-z^3&=0 \\ 2xy-4z&=1 \\ z-y^2&=0 \\ x^3-4yz&=0
\end{align*}


\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:
  $(M,P,B,C,R,A,A_\ell,L,L_\ell)=(x_1,x_2,\dots,x_9)$:
  \[
  \begin{array}{lll}
    f_M=\overline{R}\wedge C && \\ f_P=M && f_B=M \\ f_C=\overline{G_e}
    && f_R=\overline{A}\wedge\overline{A_\ell} \\ f_A=L\wedge B &&
    f_{A_\ell}=A\vee L\vee L_\ell \\ f_L=\overline{G_e}\wedge P\wedge L_e && 
    f_{L_\ell}=\overline{G_e}\wedge(L\vee L_e)
  \end{array}
  \]
  Give a well-written one sentence justification for each
  function. For example, $f_M=\overline{R}\wedge C$ could be: ``mRNA
  is produced if the \emph{lac} repressor protein is absent and the
  concentration of the catabolite activator protein CAP is high.'' \\



\item Do \emph{one} of the following problems: 

\begin{enumerate}[(A)]
\item In the $9$-variable Boolean model from the previous problem,
  there is no variable to represent the cAMP receptor protein
  \emph{cmp}.  Could you justify this decision? Propose a new model
  with one additional variable $C_{\small\textsl{AMP}}$ that represents this. 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. Are they biologically feasible? Do you notice any
  qualitative difference between this model and the original?

\item Modify the model from the previous problem 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 Discrete Visualizer of Dynamics
  program ({\tt http://dvd.vbi.vt.edu/}) 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 the first and most basic model of the \emph{lac} operon
  that we saw, which is also in Problem~1:
  \[
  f_M=\overline{G_e}\wedge(L\vee L_e),\qquad f_E=M,\qquad 
  f_L=\overline{G_e}\wedge((E\wedge L_e)\vee(L\wedge\overline{E})).
  \]
  Instead of introducing variables $L_\ell$ and $A_\ell$ that
  represent low concentration levels of \emph{intracellular} lactose
  and allolactose (and thus allowing zero, low, and high levels in
  your model), propose a model that has variables $L_{e,\ell}$ and
  $G_{e,\ell}$ representing low concentration levels of
  \emph{external} lactose and glucose. You can incorporate other
  elements if you want. Use Gr\"obener bases and Sage to compute the
  fixed points of this model. Are you results biologically reasonable?
  If not, explain why.

\end{enumerate}

\end{enumerate}


\end{document}


