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\def\beq{\begin{equation}}
\def\eeq{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\ena{\end{eqnarray}}
 \def\psize{3.5in}
\def\figsize{2.5in}
 
  \long\def\drop#1{}
\mathchardef\rA"0041
\mathchardef\rB"0042
\mathchardef\rC"0043
\mathchardef\rG"0047
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\def\na{{\rm Na}^+}
\def\k{{\rm K}^+}
\def\ca{{\rm Ca}^{++}}
\def\a{\alpha}
\def\b{\beta}
\def\e{\eta}
\def\t{\tau}
\def\g{\gamma}
\def\d{\delta}
\def\eps{\epsilon}
\def\veps{\varepsilon}
\def\k{\kappa}
\def\l{\lambda}
\def\n{\nu}
\def\s{\sigma}
\def\m{\mu}
\def\w{\omega}
\def\z{\zeta}
 
\def\pms{{\rm ms}^{-1}}
 \def\ps{s^{-1}}

\def\ip{{\rm IP}_{3}}
\def\mM{\m{\rm M}}
\def\ea{{\sl et al.}}
\def\kplus{\stackrel{\scriptscriptstyle k_+}{\longrightarrow}}
\def\kminus{\stackrel{\longleftarrow}{\scriptscriptstyle k_-}}

\def\chempl{\raisebox{-8pt}{\mbox{$\stackrel{\kplus}{\kminus}$}}}



\def\chem#1#2{ {
{\scriptscriptstyle#1\atop\longrightarrow}\atop
{\longleftarrow\atop \scriptscriptstyle#2}} }



\begin{document}


\begin{center}
{\bf Homework Exercises for Mathematics 6770 - Fall 2016}
\end{center}
\vskip 5mm
Remark: Solutions may include maple files or matlab files.

\begin{itemize}
\item[~]
Assignment 1: (due Sept. 27, 2016)
\begin{enumerate}

\item \label{ex:trimolecular}In the real world trimolecular reactions are rare, although
trimerizations are not.  Consider the following trimerization reaction in
which three monomers of A combine to form the trimer C,
\bea
\rA + \rA \chem{k_1}{k_{-1}} \rB,\nonumber\\
\rA + \rB \chem{k_2}{k_{-2}}\rC.\nonumber
\ena
\begin{enumerate}
\item
Use the law of mass action to find the rate of production of the trimer C.
\item
Suppose $k_{-1} \gg k_{-2},k_2\rA$.  Use the appropriate quasi-steady state approximation to
find the rates of production of A and C, and show that the rate of production of C is
proportional to [A]$^3$.  Explain in words why this is so.

\end{enumerate}


\item\label{ex:microtubules}
The length of microtubules changes by a process called treadmilling, in which monomer is added to one end of the microtubule and taken off at the other end. 
To model this process, suppose that  monomer $A_1$ is self-polymerizing in that it can form dimer $A_2$ via 
\beq
\rA_1 + \rA_1 {\buildrel k_+\over\longrightarrow}\rA_2.\nonumber
\eeq
Furthermore, suppose $A_1$ can polymerize an $n$-polymer $A_n$ at one end making an $n+1$-polymer $A_{n+1}$ 
\beq
\rA_1 + \rA_n {\buildrel k_+\over\longrightarrow}\rA_{n+1}.\nonumber
\eeq
Finally, degradation can occur one monomer at a time from the opposite end at rate $k_-$.
Find the steady state distribution of polymer lengths after an initial amount of monomer $A_0$ has fully polymerized.
 
\item \label{enzyme:ex_Burk}
An enzyme-substrate system is believed to proceed at a
Michaelis- Menten rate.  Data for the (initial) rate of reaction at
different concentrations is shown in Table \ref{table:ex6}.

\begin{enumerate}
\item
Plot the data $V$ vs. $s$.  Is there evidence that this is a 
Michaelis-Menten type reaction?
\item
Plot $V$ vs. $V/s$.  Is this data well approximated by a straight line?
\item
Use linear regression to estimate $K_m$ and $V_{max}$.
Compare the data to the Michaelis-Menten rate function using these
 parameters.  Does this
provide a reasonable fit to the data?
\end{enumerate}
\begin{table}
\begin{center}
\caption{Data for Problem \ref{enzyme:ex_Burk}.}
\label{table:ex6}
\begin{tabular}{|c|c|}
\hline
Substrate&Reaction\\
\quad Concentration (mM)&\quad Velocity (mM/s)\\
\hline
0.1&0.04\\
0.2&0.08\\
0.5&0.17\\
1.0&0.24\\
2.0&0.32\\
3.5&0.39\\
5.0&0.42\\
\hline
\end{tabular}
\end{center}
\end{table}


\item \label{enzyme:ex_Hill}
Suppose the maximum velocity of a chemical reaction is known
to be 1 mM/s, and the measured velocity $V$ of the reaction at different
concentrations $s$ is shown in Table
\ref{table:enzyme:ex_Hill}.

\begin{table}
\begin{center}
\caption{Data for Problem \ref{enzyme:ex_Hill}.}
\label{table:enzyme:ex_Hill}
\begin{tabular}{|c|c|}
\hline
Substrate&Reaction\\
\quad Concentration (mM)&\quad Velocity (mM/s)\\
\hline
0.2&0.01\\
0.5&0.06\\
1.0&0.27\\
1.5&0.50\\
2.0&0.67\\
2.5&0.78\\
3.5&0.89\\
4.0&0.92\\
4.5&0.94\\
5.0&0.95\\
\hline
\end{tabular}
\end{center}
\end{table}

\begin{enumerate}
\item
Plot the data $V$ vs. $s$.  Is there evidence that this is a Hill type reaction?
\item
Plot $\ln({V\over V_{max}-V})$ vs. $\ln(s)$.  Is this approximately a straight line, and if
so, what is its slope?
\item
Use linear regression to estimate $K_m$ and the Hill exponent $n$.
Compare the data to the Hill rate function with these parameters.  Does this
provide a reasonable fit to the data?
\end{enumerate}


\item \label{enzyme:ex8}
Suppose that a substrate can be broken down by two
different enzymes with different kinetics. (This happens, for
example, in the case of cAMP or cGMP, which can be hydrolyzed by two
different forms of phosphodiesterase).
\begin{enumerate}
\item
Write  the reaction
scheme and differential equations, and nondimensionalize, to get the
system of equations 
\bea
{d\s  \over dt} &=& - \s+\alpha_1(\mu_1+\s)x +
\alpha_2(\mu_2+\s)y,\label{soleq:ex8.1}\\
    \epsilon_1{dx\over dt} &=& {1\over \lambda_1} \sigma (1-x)
-x,\label{soleq:ex8.2}\\
    \epsilon_2{dy\over dt} &=& {1\over \lambda_2}\sigma (1-y)
-y.\label{soleq:ex8.3}
\ena
where $x$ and $y$ are the nondimensional concentrations of the
two complexes.  Identify all parameters. 
\item
Apply the quasi-steady-state approximation to find the equation governing the dynamics
of substrate
$\s$.  Under what conditions is the
quasi-steady state approximation valid?
\item
Solve the differential equation governing $\s$.
\item 
For this system of equations, show that the
solution can never leave the positive octant $\s,x,y \ge 0$. By
showing that $\s+\eps_1\lambda_1\alpha_1 x + \eps_2 \lambda_2\alpha_2y$ is  decreasing everywhere in the positive octant, show
that the solution approaches the origin for large time.
\end{enumerate}



\item ATP is known to inhibit its own
dephosphorylation\index{ATP!dephosphorylation}\index{dephosphorylation!of
ATP}. One possible way for this to occur is if ATP binds with the enzyme,
holding it in an inactive state, via
$${\rm S}_1 + {\rm E} \chem{k_4}{k_{-4}} {\rm S}_1{\rm E}.$$

\begin{enumerate}
\item
Add this reaction to the Sel'kov model for glycolysis and derive the corresponding 
equations governing glycolysis of the form
\begin{eqnarray}
{d\sigma_1 \over d\tau} &=& \nu -f(\sigma_1,\sigma_2),\label{eq:gly1}\\
{d\sigma_2 \over d\tau} &=&
\alpha f(\sigma_1,\sigma_2) - \eta
\sigma_2\label{eq:gly2}.
\end{eqnarray}

Explain from the model why this
additional reaction is inhibitory.
\item
Give an analysis of these equations using xpp.  In particular, modify the file selkov.ode and find phase portraits of periodic solutions as well as the bifurcation diagram, similar to Fig.~1.9 in the text.



\end{enumerate}
\end{enumerate}

% 
\end{itemize}
\end{document}
\item[~]
Assignment 2: (due Oct. 21, 2014)

\begin{enumerate}
 \item
Suppose a semi-infinite tube with cross-sectional area $A$ initially has only water, and that the concentration of a chemical species (with diffusion coefficient $D$), in a large bath at the end of the tube has fixed concentration $C_0$.  Find the total number of molecules in the tube at time $t$.




\item
The following data were used  by Segel, Chet and Henis (1977)  to estimate
the  diffusion coefficient for bacteria.  With the
external concentration $C_0$ at  $7\times 10^7 {\rm ml}^{-1}$, at times $t$ = 2, 5, 10,
12.5, 15, and 20 minutes, they counted $N$ of 1,800, 3,700, 4,800, 5,500, 6,700, and 8,000
bacteria, respectively, in a capillary of length 32 mm with 1 $\mu$l total capacity.  In
addition, with  external concentrations $C_0$ of 2.5, 4.6, 5.0, and
12.0 $\times 10^7$ bacteria per milliliter, counts of 1,350, 2,300, 3,400, and 6,200
were found at $t = 10$ minutes.  Estimate  $D$.

\item 
Find the maximal enhancement for diffusive transport  of carbon
dioxide  via binding with
myoglobin  using $D_s=1.92\times 10^{-5}\,{\rm cm}^2/{\rm
s}$, $k_+=2\times10^8\,{\rm cm}^3/{\rm M}\cdot{\rm s}$,
$k_{-}=1.7\times10^{-2}/s$. Compare the amount of
facilitation\index{facilitated diffusion} of carbon dioxide
transport\index{carbon dioxide transport} with that of oxygen at similar
concentration levels.

\item
A fluorescent dye with a diffusion coefficient of
$D = 10^{-7}\ {\rm cm}^2$/s and binding equilibrium of $K_{\rm eq} = 30$ mM is
used to track the spread of hydrogen ($D_h = 4.4\times 10^{-5}{\rm
cm}^2$/s).  Under these conditions the measured diffusion coefficient is
$8\times 10^{-6}{\rm cm}^2$/s.  How much dye is present? (Assume the dye
is a fast buffer of hydrogen.)


\drop{
\item\label{ex:gibbs_donnan} Suppose the  intracellular macromolecule X can bind $b$ molecules of the ion S via $X + bS \stackrel{\longrightarrow}{\longleftarrow} XS$.  What is the effect of this buffering on the Gibbs-Donnan equilibrium potential?
}

\item \label{cell:ex7} 
Almost immediately upon entering a cell, glucose is
phosphorylated\index{phosphorylation!of
glucose}\index{glucose!phosphorylation} in the first reaction step of
glycolysis\index{glycolysis}. How does this rapid and nearly unidirectional
reaction affect the transmembrane flux of glucose (Find an expression for glucose flux that incorporates this reaction.) 
 How is this
reaction affected by the concentration of ATP?


\item \label{cell:ex:chips}
A 1.5 oz bag of potato chips\index{potato chips} (a typical single serving)
contains about 200 mg of Na$^+$. When eaten and absorbed into the body,
how many osmoles does this bag of potato chips represent?  


\item   
\begin{enumerate}
\item  Consider a vertical tube with a cross-sectional area of 1 cm$^2$. The bottom of
the tube is closed with a semi-permeable membrane and 1 gram of sugar is placed in the
tube. The membrane-closed end of the tube is then put into an inexhaustible supply of pure
water at $T=300$K.  What will be the height of the water in the tube at equilibrium?  (The weight of
a sugar molecule is $3\times 10^{-22}$ gm, and the density of water is 1 gm/cm$^3$).
\item Two columns with cross-sectional area 1 cm$^2$ are initially
filled to a height of one meter with water at $T=300^\circ\,$K. Suppose 0.001 gm of sugar is
dissolved in one of the two columns. How high will the sugary column be when equilibrium
is reached?.
\item Suppose in the previous question 1 gm of sugar is dissolved in one of the two
columns. What is the equilibrium  height of the two columns? 
\end{enumerate}

\item  
Ouabain is known to compete with K$^+$ for external
potassium binding sites of the Na,K-ATPase\index{sodium--potassium
pump!and ouabain}.  Many animal cells swell and burst when treated with the drug
ouabain\index{ouabain}.  Why? (Explain this using a model of cell volume
control.)
\end{enumerate}


\item[~] Assignment 3 (due Nov. 13, 2014)

\begin{enumerate}
 
   
\item Suppose the $\na$ Nernst potential of a cell is 56 mV, its
resting potential is $-70$ mV, and the
extracellular $\ca$ concentration is 1 mM. At what intracellular
$\ca$ concentration is the flux of a three-for-one $\na$--$\ca$
exchanger 
zero? (Use that $RT/F = 25.8$ mV at $27^\circ C$.)

\item Intestinal epithelial 
cells have a glucose--$\na$ symport  that
transports one $\na$ ion and one glucose molecule from the intestine into
the cell. Model this transport process. Is the transport of
glucose  aided
or hindered by the cell's negative membrane potential?

\drop{\item
Write a computer program to simulate the behavior of the stochastic
three-state $\na$ channel\index{stochastic models} shown in Fig.~\ref{chgatfig2}, assuming it starts in the closed state.  Use  $\a=1$/ms, $\b=0.4$/ms, $\g=1.6$/ms and
$\d=1$/ms.  Take the ensemble average of many runs to reproduce its macroscopic
behavior. Using the data from simulations,
reconstruct the open-time distribution, the latency distribution,
and the distribution of $N$, the number of times the channel opens.
From these distributions estimate the rate constants of the
simulation and compare with the known values.

\begin{figure}[h]
\begin{center}
\includegraphics[width=4cm]{./figs/nachan.eps}
\end{center}
\caption{A: Schematic diagram of the states of the $\na$ channel. C, O, and I
denote the closed, open, and inactivated states, respectively.}
\label{chgatfig2}
\end{figure}
}

\item Explore the behavior of the reduced Hodgkin-Huxley model (you may use the  code hhred.ode).  In particular, for what values of applied current are there oscillatory solutions?  Produce phase portraits for several different parameter values showing the different types of possible behaviors, and use xppaut to produce a bifurcation diagram.
 
\item \label{ex:morrislecar}
Morris and Lecar (1981) proposed the following two-variable model of membrane potential for a barnacle muscle fiber:
\begin{eqnarray}
C_m{dV\over dT} + I_{\rm ion}(V,W) = I_{\rm app}, \label{morrislec1}\\
{dW\over dT} = \phi \Lambda (V)[W_{\infty}(V) - W], \label{morrislec2}
\end{eqnarray}
where $V={}$membrane potential, $W={}$fraction of open $\k+$
channels\index{potassium channels!in barnacle muscle fibers},
$T={}$time, $C_m={}$membrane capacitance, $I_{\rm
app}={}$externally applied current, $\phi={}$maximum rate for closing
$\k+$ channels, and
\bea
I_{\rm ion}(V,W) &=& g_{\rm Ca}M_{\infty}(V)(V - V_{\rm Ca}^0) +
g_{\rm K}W(V-V_{\rm K}^0) + g_{\rm L}(V - V_{\rm L}^0),\\
M_{\infty}(V) &=& {1\over 2}\left(1 + \tanh \left({V -V_1\over
V_2}\right) \right),\\ W_{\infty}(V) &=& {1\over 2}\left(1 +
\tanh\left({V-V_3\over V_4}\right) \right),\\
\Lambda(V) &=& \cosh \left({V-V_3\over 2V_4}\right).
\ena
Typical parameter values for these equations are shown in Table \ref{table:1}.

%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
\label{table:morrislecar}
$$
\begin{array}{|rcl|rcl|}
\hline
C_m &\!\!\!=\!\!\!& 20\ \mu{\rm F/cm}^2 & I_{\rm app} &\!\!\!=\!\!\!& 0.06\ {\rm mA /cm}^{2}\\
[6bp] g_{\rm Ca} &\!\!\!=\!\!\!& 4.4\ {\rm mS/cm}^{2} & g_{\rm K} &\!\!\!=\!\!\!& 8\ {\rm
mS/cm}^{2}\\ 
[6bp] g_{\rm L} &\!\!\!=\!\!\!& 2\ {\rm mS/cm}^{2} & \phi &\!\!\!=\!\!\!& 0.04\
\pms\\ 
[6bp] V_1 &\!\!\!=\!\!\!& -1.2\ {\rm mV} & V_2 &\!\!\!=\!\!\!& 18\ {\rm mV}\\ [6bp]
V_3 &\!\!\!=\!\!\!& 2 & V_4 &\!\!\!=\!\!\!& 30\ {\rm mV}\\ [6bp]
V_{\rm Ca}^0 &\!\!\!=\!\!\!& 120\ {\rm mV} & V_{\rm K}^0 &\!\!\!=\!\!\!& -84\ {\rm mV}\\ [6bp]
V_{\rm L} &\!\!\!=\!\!\!& -60\ {\rm mV} & &\!\!\! \!\!\!& \\ [6bp]
\hline
\end{array}
$$
\caption{Typical parameter values for the Morris--Lecar model.}
\label{table:1}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%

\begin{enumerate}

\item 
Make a phase portrait  
 for the Morris--Lecar equations. Plot the nullclines and show some typical trajectories, demonstrating that the model is excitable.

\item For what range of applied current is this system oscillatory?  Use XPP to find the bifurcation diagram  and determine the types of bifurcations.  (You can use the file ML.ode to get started.)

 \end{enumerate}
 
 
 \item The Hodgkin-Huxley equations are
\bea
C_m{dv\over dt} &=& -\bar g_{\rm K} n^4(v-v_{\rm K}) - \bar g_{\rm Na} m^3h(v-v_{\rm Na}) -\bar g_{\rm L}(v-v_{\rm L}) + I_{\rm app},\  \label{excit11}\\
{dm \over dt} &=& \alpha_m(1-m) - \beta_m m,\label{excit12}\\
{dn \over dt} &=& \alpha_n(1-n) - \beta_n n,\label{excit13}\\
{dh \over dt} &=& \alpha_h(1-h) - \beta_h h,\label{excit14}
\ena
with the functions $\alpha_j$ and $\beta_j$, in units of (ms)$^{-1}$,
\bea
\alpha_m &=& 0.1\, {25 - v \over \exp\left({25 -v \over
10}\right)-1}, \label{eq:am}\\
\beta_m &=& 4\, \exp\left({-v \over 18}\right), \label{eq:bm}\\
\alpha_h &=& 0.07\, \exp\left({-v\over 20}\right), \label{eq:ah}
\\
\beta_h &=& {1\over\exp({30-v\over 10})+1}, \label{eq:bh} \\
\alpha_n &=& 0.01\, {10-v\over \exp({10-v\over
10})-1}, \label{eq:an}\\
\beta_n &=& 0.125\, \exp\left({-v\over 80}\right). \label{eq:bn}
\ena
For these expressions, the potential $v$ is the deviation from rest, measured in units of mV, current density is in
units of $\mu {\rm A/cm}^2$, conductances are in units of ${\rm m
S/cm}^2$, and capacitance is in units of $\mu {\rm F/cm}^2$. The
remaining parameters are
\beq
	\bar g_{\rm Na} = 120, \qquad \bar g_{\rm K} = 36, \qquad \bar g_{\rm L} = 0.3,\qquad C_m=1,
\eeq
and
with shifted  equilibrium potentials
\beq
 v_{\rm Na} = 115,\qquad  v_{\rm K} = - 12, \qquad v_{\rm L} = 10.6.
 \eeq
 \begin{enumerate}
 \item
 Simulate these equations with current input $ I_{\rm app}=0$.  What are the steady state values for all variables?
 \item
 Starting with all variables at steady state, apply a piecewise constant current that is nonzero for 0.5 ms.    Plot the voltage response for several different values of $I_{\rm  app}$.  Identify the threshold value of $I_{\rm  app}$.
 \item
 For what range of constant applied current is this system oscillatory?  Use XPP to find the bifurcation diagram  and determine the types of bifurcations. 
 \end{enumerate}
 \end{enumerate}
%\end{itemize}
%\end{document}
%\item
%Using typical values for the internal and external sodium concentration, and that the cell membrane is 8 nm thick, evaluate the dimensionless parameter $\lambda^2$ in equation (3.28).  For this model, which approximation (long channel or short channel) is more appropriate? 

%\item
%Chapter 4, Problems 3, 8, 11.
%\end{itemize}
%\end{enumerate}

%\end{document}



\item[~]
Assignment 4: (due Dec. 18, 2014)
\begin{enumerate}
\item
One of the earliest models of $\ca$ oscillations was the two-pool model of Goldbeter, Dupont and Berridge
(1990). They assumed that $\ip$ causes an influx, $r$, of $\ca$ into the cell and that this influx causes
additional release of $\ca$ from the ER via an $\ip$-independent mechanism. Thus,
\bea
    {dc\over dt} &=& r - kc - f(c,c_e),\\
    {dc_e\over dt} &=&    f(c,c_e),\\
    f(c,c_e) &=& J_{\rm uptake}- J_{\rm release} - k_fc_e, 
\ena
where 
\bea
    J_{\rm uptake} &=& {V_1c^n \over K_1^n+c^n},\\
    J_{\rm release} &=& \left({V_2c_e^m \over K_2^m+c_e^m}\right)
       \left({c^p \over K_3^p+c^p}\right). \label{Jrelease}
\ena
Here, $k_fc_e$ is a leak from the ER into the cytoplasm. Typical parameter values are given in Table \ref{table:twopoolparams}. (All the concentrations in this model are with respect to the total cell volume, and thus there is no need for the correction factor $\gamma$  to take into account the difference in the ER and cytoplasmic volumes.)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}
$$
\begin{array}{|lcl|lcl|}
\hline
k &=& 10\ \ps       &  K_1 &=& 1\ \mM     \\
K_2 &=& 2\ \mM      &  K_3 &=& 0.9\ \mM   \\
V_1 &=& 65\ \mM\ps  &  V_2 &=& 500\ \mM\ps \\
k_f &=& 1\ \ps      &  m &=& 2             \\
n&=&2               &  p &=& 4             \\
\hline
\end{array}
$$
\caption{Typical parameter values for the two-pool model of $\ca$
oscillations (Goldbeter \ea, 1990).}
\label{table:twopoolparams}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin {enumerate}
\item 
Nondimensionalize these equations.  How many non-dimensional parameters are there?
\item
 Show that in a closed cell (i.e., one without any
interaction with the extracellular environment) the two-pool model cannot
exhibit oscillations.
\item
How does the steady state solution depend on influx?
\item
Use a bifurcation tracking program such as XPPAUT to plot the bifurcation diagram of this model, using $r$
as the bifurcation parameter. Find the  Hopf bifurcation points and locate the 
branch of stable limit cycle solutions. Plot some typical limit cycle solutions for different values of $r$.
\end{enumerate}


\item
Use XPPAUT to provide a fast-slow analysis of the polynomial system
\begin{eqnarray}
{dx\over dT}&=& y - x^3 + 3x^2 + I - z,\label{hr3}\\
{dy\over dT}&=&1 - 5x^2 - y,\label{hr4}\\
{dz\over dT}&=&r[s(x-x_1) - z],\label{hr5}
\end{eqnarray}
where $x_1 =
-{1\over 2}(1+\sqrt{5})$ is the $x$-coordinate of the resting state in the reduced
two-variable model  and where $I$, is the applied current, with  $r=0.001$ and $s=4$.
\begin{enumerate} 
\item
Make a bifurcation diagram of the reduced two variable system, as a function of the "parameter" $z$, with $I=2$.
\item
Superimpose this bifurcation diagram over trajectories of the full system in the $x$-$z$ plane.
\item 
Describe the mechanism of this bursting.  What happens to the bursting pattern if $I$ is increased or decreased?
\end{enumerate}

\item
\begin{enumerate}
\item Numerically simulate the system of differential equations
\bea
{dv\over dt} &=&f(v) -w -gs(v-v_\theta),\\
5{dw\over dt} &=& w_\infty(v)-w,\\
{ds\over dt} &=& f_s(s)+\a_s(x-0.3),\\
{dx\over dt} &=& \b_x\left((1-x)H(v)-x\right),
\ena
where $f(v) = 1.35v(1-{1\over 3}v^2), f_s(s) = -0.2(s-0.05)(s-0.135)(s-0.21), w_\infty(v) = \tanh(5v)$, and $H(v) = {3\over 2} (1 + \tanh(5v-2.5))$, and
$v_\theta = -2, \a_s = 0.002, \b_x = 0.00025, g =0.73$.

\item
Give a fast--slow analysis of this burster. Hint: The equations for
$v, w$ comprise the fast subsystem, while those for $s, x$ comprise
the slow subsystem.

\item
Describe the bursting mechanism in this model. For what kind of burster
might this be a reasonable model?
\end{enumerate}
 
\item \label{ex:Griffith}
Suppose that the production of an enzyme is turned on by $m$ molecules of the enzyme according to
$$\rG + m\rP \chempl \rX,$$
where G is the inactive state of the gene and X is the active state of the gene. Suppose that mRNA\index{RNA!messenger RNA, mRNA} is produced when the gene is in the active state and the enzyme is produced by mRNA and is degraded at some linear rate. Find a system of differential equations governing the behavior of mRNA and enzyme. Give a phase portrait analysis of this system and show that it has a ``switch-like'' behavior.
\end{enumerate}
\end{itemize}
\end{document}

------------------------------------------------------------
Derive the correct reduction of the Othmer-Tang model (Find the correct
version of equation 5.56)
\item Simulate both the full model and the reduced model and compare
them. 
\end{enumerate}
\item
Chapter 6, Problem 1  (Warning: There is a typo in this problem.
$H(v)= {3\over 2}(1+\tanh(5v-2.5))$.)
\item
Turn in your project/paper.
\end{enumerate}

\end{enumerate}




\item
Formulate and solve one problem that would be appropriate for an
undergraduate course on the material in this chapter.

\item Hint and Extra Credit:


The simplest way to solve problem 2 numerically is with  an iterative
technique that is based on the maximum principle for elliptic equations.
The iteration scheme is to define the
$k^{th}$ function
$\sigma_k(y)$ by
\beq
{1\over y^2}{d\over dy}(y^2{d\sigma_k\over dy})
-\alpha^2{\sigma_{k-1}
\over
\sigma_{k-1} +1}=0
\eeq
with $\sigma_0(y) = \sigma_0$.  It follows from the maximum principle that
the sequence of functions $\{\sigma_k(y)\}_k$ is a decreasing sequence
with
$\sigma_k(y) < \sigma_{k-1}(y)$ for $0<y<1$, and that $\sigma_k(y)>0$.  As
a result, the sequence converges monotonely to the solution of the
nonlinear boundary value problem.

The numerical solution for each iterate is a simple quadrature.  That is,
the solution of 
\beq
{1\over y^2}{d\over dy}(y^2{d\sigma\over dy}) = g(y)
\eeq
is found by integrating the first order system 
\beq {d\sigma\over dy}={u\over y^2},\qquad {du\over dy} = y^2g(y)
\eeq
with initial data $u(0)=0, \sigma(0) = 0$, and then adding the
appropriate constant so that $\sigma(1) = \sigma_0$.

For Extra Credit:  Prove that the iterates are monotonely decreasing, and prove that the solution of the boundary value problem is a monotonely decreasing function of $\alpha^2$.  To do this you may assume without proof that if
\beq
Lu = {1\over y^2}{d\over dy}(y^2{du\over dy}) - f(y)u \ge 0
\eeq
where $f(y) \ge 0$ and $u(1) = u'(0) = 0$, then $u(y) \le 0$.
\end{enumerate}
\item[~]
Assignment 3: (due Nov. 20, 2001)
\begin{enumerate}
\item


\item
The Aliev model is the set of differential equations 
\bea
{du\over dt} &=& 
-g_a(u-\eps_2)(u-a)(u-1)-uv\\
{dv\over dt} &=& \eps(u,v)(-v-g_s(u-\eps_2)(u-a-1))
\ena
where 
\beq
\eps(u,v)=\eps_1+{\mu_1v\over u+\mu_2}
\eeq
and typical parameter values are
$a=0.05$, $g_a=8.0$, $g_s=8.0$, $\mu_1=0.1$, $\mu_2=0.3$, $\eps_1=0.002$,
$\eps_2=0.0001$.
\begin{enumerate}
\item
Simulate the Aliev model to demonstrate that it is an excitable system,
and plot the phase portrait.
\item
Explore what happens when this model is stimulated with a constant current
input.  Are there ranges of input for which the response is oscillatory?
\end{enumerate}
\item
Formulate and solve one problem that would be appropriate for an
undergraduate course on the material in Chapter 3 or 4.
