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\lhead{{\sf Algebraic Biology}: Homework 2}
\chead{}\rhead{\thepage}
  \lfoot{Due Tuesday, March 31 2020}\cfoot{}\rfoot{Written by M.~Macauley}
  \pagestyle{fancy}

\begin{document}

$\;$ \vspace{-4mm}

\noindent \textbf{Read}: Chapter 2 of \emph{Mathematical Concepts and
  Methods in Modern Biology} (Robeva \& Hodge, 2013). Bistability in the
lactose operon of \emph{Echerichia coli}: A comparison of differential
equation and Boolean network models. By R.~Robeva and N.~Yildirim. \\

\noindent Chapter 6 of \emph{Algebraic and Discrete Mathematical Methods for
  Modern Biology} (Robeva, 2015). Steady state analysis of Boolean
models: a dimension reduction approach. By D.~Murrugarra and
A.~Veliz-Cuba, pages 121--139. \bigskip


%\noindent \textbf{Do}: Familiarize yourself with either Sage,
%Singular, or Macaulay2. \\

\noindent \textbf{Exercises}.

\begin{enumerate}

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

\item In this exercise, you will construct the finite field of order $9$.
\begin{enumerate}
\item Find an irreducible polynomial of degree 2 in $\F_3[x]$. Note
  that any such $f\in\F_3[x]$ for which $f(c)\neq 0$ for all
  $c\in\F_3=\{0,1,2\}$ will work.
\item Write down all $9$ elements of $\F_9\cong\F_3[x]/I$, where
  $I=\<f\>$ is the ideal generated by the polynomial you found in
  Part~(a). All of the elements should be of the form $g+I$, for some
  $g\in\F_3[x]$.
\item Construct the addition table of $\F_9$ and the multiplication
  table of $\F_9^*:=\F_9\setminus\{0\}$, like what we did for $\F_4$
  in class. You should omit the ``$+I$'' for clarity of notation. \\
\end{enumerate}

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

\item Consider the reactions where two substrates $S$ and $T$ compete
  for binding to an enzyme $E$ to produce two different products $P$
  and $Q$:
  \begin{align*}
    &E+S\;\stackrel[p_2]{p_1}{\rightleftharpoons}\; ES\;
    \stackrel{p_3}{\longrightarrow}\; P+E
    \\ &E+T\;\stackrel[q_2]{q_1}{\rightleftharpoons}\; ET\;
    \stackrel{q_3}{\longrightarrow}\; Q+E
  \end{align*}
  Assume that each reaction follows the Michaelis-Menten
  kinetics. Also, assume that that the initial enzyme concentration is
  $E_0=[E]+[ES]+[ET]$.
  \begin{enumerate}
  \item Derive rate equations for $P$ and $Q$ in this system in terms
    of $[ES]$ and $[ET]$. That is, determine $d[P]/dt$ and $d[Q]/dt$.

  \item Derive rate equations for $ES$ and $ET$.

  \item Assume that the enzyme-substrate complexes reach equilibrium
    quickly: $d[ES]/dt\approx 0$ and $d[ET]/dt\approx 0$. Solve
    for $[E]$ in each of these equations.

  \item Equate the two expressions for $[E]$ from Part~(c) and solve
    for $[ET]$.

  \item Solve for $[ES]$ by plugging your answers to Parts (c) and (d)
    into $E_0=[E]+[ES]+[ET]$. You should not have $[E]$ or $[ET]$ in
    your final answer.

  \item Plug this into the original ODE for $d[P]/dt$.

  \item Repeat the previous three steps but solve for $[ES]$ instead
    of $[ET]$.

  \item Explain the effects of the competition occuring. \\

  \end{enumerate}

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

\item Recall our original $3$-variable Boolean model of 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*}
  For each of the $4$ possible initial conditions, $G_e,L_e\in\F_2^2$,
  the model had one connected component with the biologically correct
  fixed point. Compute the probability that this would have happened
  purely by chance. (Assume a uniform distribution.) \\
  
%%--------------------------------------------------------------------- 

% 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:
    \[
    \begin{array}{ll}
    f_M=A & f_{B_{\old}}=\overline{M}\And B
    \\ f_B=M\Or\left(B\And\overline{B_{\old}}\right)\hspace{10mm} &
    f_A=(B\And L_m)\Or L
  \end{array}
  \]
  What approximate timestep is assumed by this model?
  \item Find the fixed points of this model using computational algebra.
    Use the variable order $(M,B,B_{\old},A)=(x_1,x_2,x_3,x_4)$, and include
    your code from Macaulay2, Singular, or Sage.
  \item Does this model exhibit bistability? Justify your answer. \\
\end{enumerate}
  
%%--------------------------------------------------------------------- 

\end{enumerate}


\end{document}


