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\lhead{{\sf Algebraic Biology}: Homework 1}
\chead{}\rhead{\thepage}
  \lfoot{Due Monday, March 26 2018}\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

$\;$ \vspace{-4mm}

\noindent \textbf{Read}. \emph{Mathematical Concepts in Modern
  Biology}, 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 \textbf{Do}. Familiarize yourself with either Macaulay2,
Singular, or Sage. \\

\noindent \textbf{Exercises}.

\begin{enumerate}

\item Given a Boolean network $(f_1,f_2,f_3,f_4,f_5)$, suppose that
  $\{x_1+x_5,\;x_3+1,\;x_2+x_5+1,\;x_4\}$ is a Gr\"obner basis of the
  ideal $I=\<f_i+x_i\mid i=1,\dots,5\>$. What can you
  deduce from this? Your answer should consist of several clear and complete sentences, and \emph{you should be as specific as possible}. \medskip



  % 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*}

\begin{enumerate}

\item Use a software package to compute a Gr\"obner basis for this
  system over $\Q$. Include a print-out of the code that you used.

\item Use the Gr\"obner basis you just computed to write a simpler
  system of polynomial equations that has the same set of
  solutions. Solve that system \emph{by hand} (it's not hard) to find
  all (complex-valued) solutions to the original system.

\item Repeat the previous two parts, but over the binary field,
  $\F_2=\{0,1\}$.

\item Now, do Parts~(a) and (b) but over the ternary field,
  $\F_3=\{0,1,2\}$. \medskip

\end{enumerate}

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

\item Consider the following simple model of the \emph{lac} operon:
  \[
    \begin{array}{lll}
    f_M=\overline{R}\hspace{15mm} && f_R=\overline{A} \\ f_P=M && f_A=L\And B
    \\ f_B=M && f_L=P
  \end{array}
  \]
  For this problem, make the convention that
  $(x_1,x_2,x_3,x_4,x_5,x_6)=(M,P,B,R,A,L)$.
  \begin{enumerate}
    \item Sketch the wiring diagram of this Boolean network.
    \item Justify each function in a single sentence. What other
      assumptions are made in this model? (E.g., presence or absense
      of external lactose and glucose?)
    \item Write each function as a polynomial over
      $\F_2=\{0,1\}$. Then, write out a system of equations whose
      solutions are the fixed points of the Boolean network.
    \item Find the fixed points by computing a Gr\"obner bases with a
      software package.
    \item Compute and print out the entire phase space of your model
      with the help of the Cyclone algorithm on AlgoRun, at
      \url{http://algorun.org/browse}. \\
  \end{enumerate}

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

\item Consider the following Boolean network:
  \[
  (f_1,f_2,f_3)=\big(\Not{x_1}\And
  x_2\And\Not{x_3},\;(x_1\And\Not{x_3})\Or(\Not{x_1}\And
  x_3),\;\Not{x_1}\And x_2\And\Not{x_3}\big).
  \]
  \begin{enumerate}
  \item Sketch the wiring diagram of this Boolean network.
  \item Write these functions as polynomials in
    $\F_2[x_1,x_2,x_3]/\<x_1^2-x_1,x_2^2-x_2,x_3^2-x_3\>$. Feel free
    to use Macaulay2, Sage, or Singular.
  \item Sketch the (synchronous) phase space of this Boolean
    network. Feel free to use Algorun.
  \item Sketch the asynchronous phase space of this Boolean network,
    and find all strongly connected components.
    \item Classify the nodes of the asynchronous phase space as
      transient points, cyclic attractors, or complex attractors. \\
  \end{enumerate}

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

\item 

\end{enumerate}


\end{document}


