\documentclass[twoside]{article}
\usepackage{graphicx}

\pagestyle{myheadings}
\markboth{\hfil Modelling interactions of mixtures \hfil EJDE/Conf/10}
{EJDE/Conf/10 \hfil
 Carr, Chambers, Chambers, Oppenheimer, \& Richardson \hfil}

\begin{document}

\title{\vspace{-1in}
\setcounter{page}{89}
\parbox{\linewidth}{\footnotesize\noindent
Fifth Mississippi State Conference on Differential Equations and
Computational Simulations, \newline
Electronic Journal of Differential Equations,
Conference 10, 2003, pp 89-99. \newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu (login: ftp)} 
\vspace{\bigskipamount} 
\\
Modelling the interactions of mixtures of organophosphorus insecticides with
cholinesterase%
%
\thanks{\emph{Mathematics Subject Classifications:} 92C45. 
\hfil\break \indent 
\emph{Key words:} Reaction kinetics, reaction modelling 
\hfil\break \indent 
\copyright 2003 Southwest Texas State University. \hfil
\break \indent 
Published February 28, 2003.
 \hfil\break \indent This work is supported by
a grant from the American Chemistry Council.} }
\date{}

\author{Russell. L. Carr, Howard W. Chambers, Janice E. Chambers, \\
Seth F. Oppenheimer, \& Jason R. Richardson}
\maketitle

\begin{abstract}
The organophosphorus (OP) insecticides are one of the most widely used and
important insecticide classes. These insecticides exert toxicity through
inhibition of the critical nervous system enzyme cholinesterase (ChE) which
functions to rapidly destroy the ubiquitous neurotransmitter acetylcholine. 
When ChE is inhibited, the acetylcholine accumulates, causing hyperactivity
within the cholinergic pathways. Considerable effort has gone into assessing
the risks of various OP insecticides. Unfortunately, people are often
exposed to different OP insecticides in different dosages at different or
overlapping times. The usual statistical methods seem inadequate to the task
of assessing the effect of OP mixtures. This paper will discuss a simple
model using systems of ordinary differential equations. Using this model, we
have had success in predicting the effect of cumulative in vitro OP compound
exposure in terms of ChE inhibition using data from experiments measuring
ChE inhibition by a single OP compound. We will describe our model and
compare our simulations to in vitro experiments where binary mixtures have
been used.
\end{abstract}

\section{Introduction}

The organophosphorus (OP) insecticides are one of the most widely used and
important insecticide classes. These insecticides exert toxicity through
inhibition of the critical nervous system enzyme cholinesterase (ChE) which
functions to rapidly destroy the ubiquitous neurotransmitter acetylcholine. 
Inhibition of ChE by OP compounds is through covalent bond formation, and
the inhibited ChE is quite persistent (half life of reactivation to
uninhibited ChE is hours to days). An additional covalent reaction can
occur, termed aging, which renders the ChE molecule permanently inhibited
and incapable of recovering enzymatic activity. When ChE is inhibited, the
acetylcholine accumulates, causing hyperactivity within the cholinergic
pathways. Considerable effort has gone into assessing the risks of various
OP insecticides. Unfortunately, people are often exposed to different OP
insecticides in different dosages at different or overlapping times. The
usual statistical methods seem inadequate to the task of assessing the
effect of OP mixtures \cite{m1,p1,p2,p3,p4,p5,p6,v1}. This paper will
discuss a simple model using systems of ordinary differential equations. Our
approach will be somewhat different than those used previously in the
toxicology literature \cite{k1}. Using this model, we have had success in
predicting the effect of cumulative in vitro OP compound exposure in terms
of ChE inhibition using data from experiments measuring ChE inhibition by a
single OP compound.

In the second section of this work, we will describe our model. Following
the model description, the next section will give the calibration results.
In the fourth section, we will compare our simulations to in vitro
experiments of simultaneous exposures to two OP inhibitors. The agreement
seen in Section Four is excellent. The fifth section briefly discusses time
asymptotic results. Full information on the experimental methods and results
are forthcoming \cite{c1}.

\section{The model}

We will start by discussing the usual modelling approach.

The usual models for chemical kinetics involve elementary reactions of the
following form \cite{g1,k2}: 
\[
aA+bB\rightarrow cC+dD,
\]%
where the uppercase letters represent concentrations in moles per liter or a
similar set of units and the lower case letters are natural numbers. In this
case of an elementary reaction, the standard model is given by the following
differential equations:%
\begin{eqnarray*}
\frac{dA}{dt} &=&-akA^{a}B^{b} \\
\frac{dB}{dt} &=&-bkA^{a}B^{b} \\
\frac{dC}{dt} &=&ckA^{a}B^{b} \\
\frac{dD}{dt} &=&dkA^{a}B^{b}.
\end{eqnarray*}%
That is, the rate of reaction is always proportional to products of the
concentration raised to the number of molecules that participate in each
reaction. The overall order of the reaction, $a+b$, is also the molecularity
of the reaction, where molecularity is number of molecules taking part in
the reaction. It has been observed by some investigators that not all such
reactions are quite so simple. It may be because of intermediate reactions
or multiple reaction paths, but the above model can fail. An alternative
model is proposed in \cite{s1} of the form 
\begin{eqnarray*}
\frac{dA}{dt} &=&-akA^{\alpha }B^{\beta } \\
\frac{dB}{dt} &=&-bkA^{\alpha }B^{\beta } \\
\frac{dC}{dt} &=&ckA^{\alpha }B^{\beta } \\
\frac{dD}{dt} &=&dkA^{\alpha }B^{\beta }
\end{eqnarray*}%
where the exponents, called the partial orders, do not necessarily have a
relationship with the molecularity and are in fact obtained empirically.
This is the modeling approach we will follow. We will consider single
reactions first. Let us consider our case of ChE and a variety of
inhibitors. Let $c$ be the molarity in solution of ChE at time $t$ and $x_{i}
$ be the molarity of the $i^{th}$ inhibitor at time $t$. We will model the
reaction of ChE with the inhibitor by the following system of differential
equations%
\begin{eqnarray*}
\frac{dc}{dt} &=&-k_{i}c^{\alpha _{i}}x_{i}^{\beta _{i}} \\
\frac{dx_{i}}{dt} &=&-k_{i}c^{\alpha _{i}}x_{i}^{\beta _{i}} \\
c(0) &=&c^{0} \\
x_{i}(0) &=&x_{i}^{0}
\end{eqnarray*}%
where $k_{i}$ is called the rate coefficient and $\alpha _{i}$ and $\beta
_{i}$ are the partial orders. These constants are all empirically derived
from the single inhibitor experiments. The reader will observe that we are
not modelling the concentration of the result of the reaction. This is
because the rate of the back reaction (reactivation) is so slow compared to
the time scale of the experiments, fifteen to thirty minutes, that the back
reaction will have negligible effect on the experimental results. We will
denote the solution of equation (1) as $c\left( \alpha _{i},\beta
_{i},k_{i},x_{i}^{0};t\right) .$

The experimental data used in this work are in the form of inhibition
curves. That is, the experimentalists will start with several samples
containing a fixed concentration of ChE. Different amounts of an inhibitor
are added and the fraction of the ChE which is inhibited is measured at
fifteen minutes. That is, the data sets consist of several different initial
concentrations of inhibitor $i$, $\left\{
x_{i1}^{0},\dots,x_{iN}^{0}\right\} $ and the percentage of ChE inhibited
after fifteen minutes $I_{ij}.$ Here, $x_{ij}^{0}$ is the initial
concentration of inhibitor $i$ in the $j^{th}$ experiment. For each
inhibitor, fifteen experiments were carried out and $N=15$. (We observe that
this is quite a simplification of the actual experimental procedure and the
reader is referred to \cite{c1,r1}.) The method of determining the unknown
constants $\alpha _{i},\beta _{i},$ and $k_{i}$ was as follows. We used the
built--in numerical program ODE45 in the Mathlab programming language \cite%
{t1} to approximate the solution of equation (1) with initial inhibitor
concentration $x_{ij}^{0}$ at time 15 minutes, $c( \alpha _{i},\beta
_{i},k_{i},x_{ij}^{0};15)$, and compute the percent inhibition at 15 minutes
for this set of parameters 
\[
p\left( \alpha _{i},\beta _{i},k_{i},x_{ij}^{0};15\right)  =\Big( 1-\frac{c(
\alpha _{i},\beta _{i},k_{i},x_{ij}^{0};15) }{c^{0}}\Big) 100. 
\]
For inhibitor $i$ we define the objective function%
\[
O\left( \alpha _{i},\beta _{i},k_{i}\right) =\sum_{j=1}^{15}\left(
I_{ij}-p\left( \alpha _{i},\beta _{i},k_{i},x_{ij}^{0};15\right) \right)
^{2}. 
\]
We find the parameters $\alpha _{i},\beta _{i},$ and $k_{i}$ by minimizing
the objective function using the Matlab function fmins.

Once the constants have been found, we can give the model for any
combination of two inhibitors:%
\begin{eqnarray}
\frac{dc}{dt} &=&-k_{i}c^{\alpha _{i}}x_{i}^{\beta _{i}}-k_{n}c^{\alpha
_{n}}x_{n}^{\beta _{n}}  \label{2} \\
\frac{dx_{i}}{dt} &=&-k_{i}c^{\alpha _{i}}x_{i}^{\beta _{i}}  \nonumber \\
\frac{dx_{n}}{dt} &=&-k_{n}c^{\alpha _{n}}x_{n}^{\beta _{n}}  \nonumber \\
c(0) &=&c^{0}  \nonumber \\
x_{i}(0) &=&x_{i}^{0}  \nonumber \\
x_{n}(0) &=&x_{n}^{0}  \nonumber
\end{eqnarray}
This is the model we will use below for the simultaneous exposures. We note
that this model is much more general than we indicate here. Using it in
slightly different form, we can handle any number of inhibitors, any
exposure regime, reactivation, and aging. If the various inhibitors react
with each other or other more complex interactions occur, these basic models
still form the building blocks of the model.

We observe that in the work below the initial ChE concentration was not
measured directly, but was obtained using data in \cite{r1} and a published
abstract \cite{k3}. The value used is $c_{0}=4.045\times 10^{-13}$ moles per
liter.

\section{Model Calibration}

We will report on the calibration of the model for two OP inhibitors in this
section, chlorpyrifos-oxon and paraoxon. As noted in the previous section,
the unknown parameters are found by minimizing the objective function $%
O\left( \alpha _{i},\beta _{i},k_{i}\right) $. We will subscript the
parameters associated with chlorpyrifos-oxon with $C$ and we will subscript
the parameters associated with paraoxon with $P$.

Below, we will give the identified parameters, a chart comparing the
inhibition predicted by the calibrated model, and a graph of the model
inhibition curve with the experimental data points.

\subsection*{Chlorpyrifos-oxon}

The identified parameters for chlorpyrifos-oxon are 
\begin{eqnarray*}
k_{C} &=&.0498 \\
\alpha _{C} &=&1.1616 \\
\beta _{C} &=&.9670
\end{eqnarray*}

We now give a chart comparing the model's predictions with the experimental
data. Observe that the experimental error can be on the order of $\pm 3$\%
inhibition:

\begin{center}
\begin{tabular}{|p{22mm}|p{22mm}|p{22mm}|p{22mm}|}
\hline
Nanomoles of chlorpyrifos\-oxon per liter at time 0 & Percent inhibition of
ChE observed at 15 minutes & Percent inhibition of ChE predicted at 15
minutes & Predicted $-$ observed \\ \hline
{.5} & {8.7} & {10.2} & {-1.5} \\ \hline
{.5} & {9.0} & {10.2} & {-1.2} \\ \hline
{.5} & {9.6} & {10.2} & {-.6} \\ \hline
{.5} & {9.97} & {10.2} & {-.23} \\ \hline
{.5} & {10.3} & {10.2} & {.1} \\ \hline
{1} & {17.7} & {18.8} & {-1.1} \\ \hline
{1} & {19.6} & {18.8} & {.8} \\ \hline
{1} & {19.7} & {18.8} & {.9} \\ \hline
{1} & {20.2} & {18.8} & {1.4} \\ \hline
{1} & {20.9} & {18.8} & {2.1} \\ \hline
{1.7} & {27.0} & {29.1} & {-2.1} \\ \hline
{1.7} & {28.0} & {29.1} & {-1.1} \\ \hline
{1.7} & {28.5} & {29.1} & {-.6} \\ \hline
{1.7} & {28.8} & {29.1} & {-.3} \\ \hline
{1.7} & {31.3} & {29.1} & {2.2} \\ \hline
\end{tabular}
\end{center}

The information may be summarized statistically as follows. The maximum of
the absolute error is 2.2, the mean error is $-.082$, the median error is $%
-.3$, and the standard deviation in the error is 1.1308. All of the units
are percent inhibition.

The graph of the predicted fifteen minute inhibition curve along with the
experimental data may be seen in Figure 1.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1.eps}
\end{center}
\end{figure}

\subsection*{Paraoxon}

The identified parameters for paraoxon are 
\begin{eqnarray*}
k_{P} &=&.0050 \\
\alpha _{P} &=&1.1160 \\
\beta _{P} &=&1.0133
\end{eqnarray*}

We now give a chart comparing the model's predictions with the experimental
data. Observe that the experimental error can be on the order of $\pm 3$\%
inhibition:

\begin{center}
\begin{tabular}{|p{22mm}|p{22mm}|p{22mm}|p{22mm}|}
\hline
Nanomoles of paraoxon per liter at time 0 & Percent inhibition of ChE
observed at 15 minutes & Percent inhibition of ChE predicted at 15 minutes & 
Predicted $-$ observed \\ \hline
3.5 & 9.7 & 10.16 & 0.46 \\ \hline
3.5 & 9.7 & 10.16 & 0.46 \\ \hline
3.5 & 9.97 & 10.16 & 0.19 \\ \hline
3.5 & 10.0 & 10.16 & 0.16 \\ \hline
3.5 & 11.7 & 10.16 & -1.54 \\ \hline
7.5 & 19.6 & 20.57 & 0.97 \\ \hline
7.5 & 20.0 & 20.57 & 0.57 \\ \hline
7.5 & 20.8 & 20.57 & -0.23 \\ \hline
7.5 & 21.0 & 20.57 & -0.43 \\ \hline
7.5 & 22.0 & 20.57 & -1.43 \\ \hline
12 & 28.8 & 30.78 & 1.98 \\ \hline
12 & 30.5 & 30.78 & 0.28 \\ \hline
12 & 31.0 & 30.78 & -0.22 \\ \hline
12 & 31.3 & 30.78 & -0.52 \\ \hline
12 & 33 & 30.78 & -2.22 \\ \hline
\end{tabular}
\end{center}

The information may be summarized statistically as follows. The maximum of
the absolute error is 2.22, the mean error is $0.1013$, the median error is $%
-0.16$, and the standard deviation in the error is 1.0521. All of the units
are percent inhibition.

The graph of the predicted fifteen minute inhibition curve along with the
experimental data may be seen in Figure 2.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig2.eps}
\end{center}
\end{figure}

\section{Simultaneous exposure to a binary mixture}

Using the parameters obtained above, we will use the model 
\begin{eqnarray}
\frac{dc}{dt} &=&-k_{P}c^{\alpha _{P}}x_{P}^{\beta _{P}}-k_{C}c^{\alpha
_{C}}x_{n}^{\beta _{C}}  \label{3} \\
\frac{dx_{P}}{dt} &=&-k_{P}c^{\alpha _{P}}x_{P}^{\beta _{P}}  \nonumber \\
\frac{dx_{C}}{dt} &=&-k_{C}c^{\alpha _{C}}x_{n}^{\beta _{C}}  \nonumber \\
c(0) &=&c^{0}  \nonumber \\
x_{P}(0) &=&x_{P}^{0}  \nonumber \\
x_{C}(0) &=&x_{C}^{0}  \nonumber
\end{eqnarray}
to predict the percent ChE inhibition when two inhibitors, chlorpyrifos-oxon
and paraoxon, are present.

Using this model we got excellent agreement with experiment. By excellent,
we mean that the inhibition predicted by the model was within experimental
error of the observed inhibition. This is particularly significant as the
model was calibrated independently of any of the binary mixture data. We
present a table summarizing our results.

\begin{center}
\begin{tabular}{|p{18mm}|p{18mm}|p{18mm}|p{18mm}|p{18mm}|}
\hline
Nanomoles per liter of paraoxon at time 0 & Nanomoles per liter of
chlorpyrifos-oxon at time 0 & Observed inhibition & Predicted inhibition & 
Observed - predicted \\ \hline
{3.5} & {.5} & {18.1} & {19.17} & -1.07 \\ \hline
{3.5} & {1} & {27.3} & {26.8} & .5 \\ \hline
{3.5} & {1.7} & {36.33} & {35.95} & .38 \\ \hline
{7.5} & {.5} & {27.1} & {28.41} & -1.31 \\ \hline
{7.5} & {1} & {35} & {35.06} & -.06 \\ \hline
{7.5} & {1.7} & {42} & {43.06} & -1.06 \\ \hline
{12} & {.5} & {35} & {37.48} & -2.48 \\ \hline
{12} & {1} & {41.7} & {43.19} & -1.49 \\ \hline
{12} & {1.7} & {49} & {50.07} & -1.07 \\ \hline
\end{tabular}
\end{center}

The information may be summarized statistically as follows. The maximum of
the absolute error is 2.48, the mean error is $-0.8511$, the median error is 
$-1.07$, and the standard deviation in the error is 0.9604. All of the units
are percent inhibition.

The graph of the predicted fifteen minute inhibition surface along with the
experimental data may be seen in Figure 3.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig3.eps}
\end{center}
\end{figure}

\section{Results for larger times}

Our goal in this work has been to develop models for Cholinesterase
inhibition by mixtures of OP insecticides that are more accurate and
flexible than those currently available. We did so in the context of
standard experimental protocols for obtaining fifteen minute inhibition
curves. However, we shall consider some results, both theoretical and
experimental for longer times. We note that as time increases, we expect
other mechanisms, such as reactivation, to come into play, thus examining
this model in isolation over longer times has limited utility.

We observe for the model with only one OP%
\begin{eqnarray*}
\frac{dc}{dt} &=&-kc^{\alpha }x^{\beta } \\
\frac{dx_{i}}{dt} &=&-kc^{\alpha }x^{\beta } \\
c(0) &=&c^{0} \\
x(0) &=&x^{0}
\end{eqnarray*}
that for each $t>0$, $x(t)-x^{0}=c(t)-c^{0}$. We may therefore write 
\[
\frac{dc}{dt}=-kc^{\alpha }\left( c(t)+x^{0}-c^{0}\right) ^{\beta }. 
\]%
If we assume that $x^{0}>c^{0}$ as we had in all of our experiments we obtain%
\[
\frac{dc}{dt}\leq -kc^{\alpha }\left( x^{0}-c^{0}\right) ^{\beta }. 
\]
From this it is easy to see that if $a<1$ then $c$ reaches zero in finite
time and if $\alpha \geq 0$ then $\lim_{t\rightarrow \infty }c\left(
t\right) =0$. Of course, a residual of the OP will be left of the amount $%
x^{0}-c^{0}$. For the case of multiple OP's we can construct a similar upper
bound and obtain analogous results. We note that the OP for which $\alpha $
is smaller will dominate the reaction. If the $\alpha $'s are equal, then
the term with the largest $k$ will dominate.

For longer time periods, experimental measurements become more difficult.
However, we do have a comparison of the thirty minute inhibition curve for
paraoxon as predicted by the model and experimental data. We will plot the
model inhibition curve along with forty five data points.

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig4.eps}
\end{center}
\end{figure}

The information may be summarized statistically as follows. The maximum of
the absolute error is 7.6, the mean error is 2.27, the median error is 1.98,
and the standard deviation in the error is 2.69. All of the units are
percent inhibition. Considering the range of experimental values, this is
not too bad.

\paragraph{Conclusion}

We developed a new model for enzyme inhibition by an OP insecticide. We
showed that this model accurately modelled single inhibitor experiments.
More significantly, using parameters obtained independently of any data from
experiments using binary mixtures, the model successfully predicted the
results of in vitro experiments using two inhibitors.

\begin{thebibliography}{00}
\frenchspacing

\bibitem{c1} J. E. Chambers, R. L. Carr, J. R. Richardson, H. W. Chambers,
and S. F. Oppenheimer. The In Vitro Inhibition of Rat Brain Cholinesterase
Following Simultaneous or Sequential Exposure to Binary Mixtures of
Organophosphate Compounds. In preparation.

\bibitem{g1} I. Glassman. {Combustion, Second Edition}. Academic Press,
Orlando, 1987.

\bibitem{k1} Stacey A. Kardos and Lester G. Sultatos. Interactions of the
Organophosphates Paraoxon and Methyl Paraoxon with Mouse Brain
Acetylcholinesterase, Toxicological Sciences Vol. 55, 118--126 (2000).

\bibitem{k2} James Keener and James Sneyd. {Mathematical Physiology}.
Springer, New York, 1998.

\bibitem{k3} A. K. Kousba and L. G. Sultatos. Estimation of
Acetylcholinesterase Concentrations for Pharmacodynamic Models. The
Toxicologist Vol. 60, 241 (2001).

\bibitem{m1} J. P. Michaud, A. J. Gandolfi, and K. Brendel. Toxic Responses
to Defined Chemical Mixtures: Mathematical Models and Experimental Designs.
Life Sciences, Vol. 55 (\textbf{9}), 635--651 (1994).

\bibitem{p1} John L. Plumber and Tim G. Short. Statistical Modeling of the
Effects of Drug Combinations. Journal of Pharmacological Methods Vol. 23,
297--309 (1990).

\bibitem{p2} Gerald P\"{o}ch, Peter Dittrich, and Sigrid Holzmann.
Evaluation of Combined Effects in Dose-Response Studies by Statistical
Comparison with Additive and Independent Reactions. Journal of
Pharmacological Methods Vol. 24, 311--325 (1990).

\bibitem{p3} Gerald P\"{o}ch, Peter Dittrich, R. J. Reiffenstein, W. Lenk,
and A. Schuster. Evaluation of Experimental Combined Toxicity by Use of
Dose-Frequency Curves: Comparison with Theoretical Additivity as Well as
Independence. Can. J. Physiol. Pharmacol. Vol. 68, 1338--1345 (1990).

\bibitem{p4} Gerald P\"{o}ch and Sonja N. Pancheva. Calculating Slope and ED$%
_{50}$ of Additive Dose-Response Curves, and Application for These Tabulated
Parameter Values. Journal of Pharmacological and Toxicological Methods Vol.
33, 137--145 (1995).

\bibitem{p5} Gerald P\"{o}ch, R. J. Reiffenstein, and H.-P. Baer.
Quantitative Estimation of Potentiation and Antagonism by Dose Ratios
Corrected for Slopes of Dose-Response Curves Deviating From One. Journal of
Pharmacological and Toxicological Methods Vol. 33, 197--204 (1995).

\bibitem{p6} Ross L. Prentice. A Generalization of the Probit and Logit
Methods for Dose Response Curves. Biometrics Vol. 32, 761-768 (1976).

\bibitem{r1} Jason R. Richardson, Howard W. Chambers, and Janice E.
Chambers. Analysis of the Additivity of in Vitro Inhibition of
Cholinesterase by Mixtures of Chlorpyrifos-oxon and Azinphos-methyl-oxon.
Toxicology and Applied Pharmacology Vol. 172, 128--139 (2001).

\bibitem{s1} Garrison Sposito. Chemical Equilibria and Kinetics in Soils.
Oxford University Press, Oxford, 1994.

\bibitem{t1} The Math Works Inc., The Student Edition of MATLAB, Prentice
Hall, New Jersey, 1992.

\bibitem{v1} Aage V\O lund. Application of the Four-Parameter Logistic Model
to Bioassay Comparison with Slope Ratio and Parallel Line Models. Biometrics
Vol. 34, 357--365 (1978).

\end{thebibliography}

\noindent \textsc{Russell L. Carr} (e-mail: rlcarr@cvm.msstate.edu)\newline
\textsc{Janice E. Chambers} (e-mail: chambers@cvm.msstate.edu)\newline
Center for Environmental Health Sciences, P.O. Box 6100\newline
Mississippi State University, MS 39762, USA  \smallskip

\noindent\textsc{Howard W. Chambers} \newline
Department of Entomology and Plant Pathology\newline
P.O. Box 9775, Mississippi State University, MS 39762, USA \smallskip

\noindent\textsc{Seth F. Oppenheimer}\newline
Department of Mathematics and Statistics, Drawer MA\newline
Mississippi State University, MS 39762, USA\newline
e-mail: seth@math.msstate.edu \smallskip

\noindent \textsc{Jason R. Richardson} \newline
Center for Neurodegenerative Disease,
Emory University \newline
Whitehead Biomedical Research Building 575\newline
615 Michael Street,
Atlanta, GA 30322, USA \\
e-mail: jricha3@emory.edu

\end{document}