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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\DNA Degradation in Rat Plasma}




\markboth{Kinetic Modeling of Plasmid DNA Degradation in Rat Plasma}{Kinetic Modeling of Plasmid DNA Degradation in Rat Plasma}

\author{Brian Winkel,
Director SIMIODE\\ 
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
We ask students to use the  system of first order linear differential equations given in a source paper \cite{HoukEtAl1999} and estimates of the data from laboratory procedures from a plot to estimate the parameters and complete the modeling process. Then we seek to compare the results of the final model with estimated parameters (as best we can) with our ``estimated" or ``rough" data to that in the paper.

\end{abstract}


\section*{SCENARIO DESCRIPTION}

We quote the abstract from a source paper \cite{HoukEtAl1999}:
\begin{quote}
A major obstacle in gene delivery is the transport of intact plasmid DNA (pDNA) to target sites. We sought to investigate the kinetic processes underlying the degradation of pDNA in a rat plasma model, as this is one of the main components responsible for the clearance of pDNA after intravenous administration. We further sought to construct a complete kinetic model to describe the degradation of all three topoforms (supercoiled, open circular, and linear) of pDNA in a rat plasma model. Supercoiled pDNA was incubated in isolated rat plasma at 37$^{\rm o}$C in vitro. At various time points, the plasma was assayed by electrophoresis for the amounts of supercoiled, open circular, and full-length linear pDNA remaining. The calculated amounts remaining were fit to linear differential equations describing this process. In this model, pDNA degradation is considered to be a unidirectional process, with supercoiled degrading to open circular and then to the linear topoform. The calculated kinetic parameters suggested that supercoiled pDNA degrades in rat plasma with a half-life of 1.2 minutes, open circular pDNA degrades with a half-life of 21 minutes, and linear pDNA degrades with a half-life of 11 minutes. Complexation of pDNA with liposomes resulted in a portion of the supercoiled plasmid remaining detectable through 5.5 hours.
\end{quote}

The complete paper, \cite{HoukEtAl1999},  is availabe in Supporting Docs for this Modeling Scenario. 

We present the system of linear differential equations model (\ref{eq:model}) and the data in Table~\ref{table:data} from the article \cite{HoukEtAl1999} and ask you to explain the reasonableness of the model, to solve the model, to estimate the parameters in the model, and then to compare their model results with the data.

 The model concerns three kinds of pDNA, denoted $SC$ (supercoiled), $OC$ (open circular), and $L$ (linear) present in rats at time $t$  in minutes and attempts to model the degradation of the pDNA, starting at the supercolied state, through the open circular form, until a final linear form is reached for the pDNA.

We obtained the data presented for each of the three types of pDNA on concentrations of the respective pDNA in ng/$\mu$l by enlarging Figure 3 from the source \cite{HoukEtAl1999}. We show the plots in Figure~\ref{fig:pDNA}.

\begin{figure}[t]
\centering

\includegraphics[width=12cm]{pDNA.jpg}



\parbox{12 true cm}{\caption{\label{fig:pDNA} Plot of data and model from\cite{HoukEtAl1999} with caption, ``Experimental and fitted data based on the kinetic model described in the text. Data points represent actual experimental data. Lines represent values predicted by the model. The values are expressed as $\pm$ SD of 6 rats. Key: + supercoiled, x open circular, * linear.''}}

\end{figure}

\section*{The Model}

In \cite{HoukEtAl1999} the following model (\ref{eq:model}) is posed for the transition between the three types of pDNA.

\begin{eqnarray}
\label{eq:model} \nonumber
SC'(t) &=& - k_s\cdot SC(t)\\ 
OC'(t) &=& k_s\cdot SC(t) - k_o\cdot OC(t)\\ \nonumber
L'(t) &=& k_o\cdot OC(t) - k_l L(t)\, . 
\end{eqnarray}



We extracted the  data (by visual estimation) in Table~\ref{table:data}  from Figure~\ref{fig:pDNA} and encourage you to try your hand at such extraction to corroborate our data and/or obtain more data points. 

\begin{center}
\begin{table}[h]\label{table:data}\begin{minipage}{3 true cm}
\begin{center}
\begin{tabular}{|c|c|}\hline
Time& Supercoiled \\
(min)& pDNA\\ \hline
0&7.5\\
1.0& 6.0\\
3.5& 1.25\\
5.0& 1.0\\
7.5& 0.1\\ \hline
\end{tabular}
\end{center}
\end{minipage}\  \hspace*{2 true cm} \
\begin{minipage}{3 true cm}
\begin{center}
\begin{tabular}{|c|c|}\hline
Time& Open Circular \\
(min)& pDNA \\ \hline
 0.5 & 4.0 \\
 2 & 6.0 \\
 5 & 7.9 \\
 7.5 & 7.8 \\
 10 & 7.9 \\
 15 & 7 \\
 20 & 7.4 \\
 30 & 4.0 \\
 40 & 3.5 \\
 50 & 2.6 \\ \hline
\end{tabular}
\end{center}
\end{minipage}\  \hspace*{2 true cm} \
\begin{minipage}{3 true cm}
\begin{center}
\begin{tabular}{|c|c|}\hline
Time& Linear \\
(min)& pDNA\\ \hline
 1 & 0 \\
 5 & 0.8 \\
 10 & 1.25 \\
 15 & 2.5 \\
 20 & 2.5 \\
 30 & 3.0 \\
 40 & 2.4 \\
 50 & 2.3 \\
 60 & 1.2 \\\hline
\end{tabular}
\end{center}
\end{minipage}

\caption{\label{table:data} Data on three different pDNA's extracted from Figure 3 in \cite{HoukEtAl1999}.}
\end{table}

\end{center}

\section*{Activities}

\begin{enumerate}
\item Explain the reasonableness of the mode (ref{eq:model}). 
\item Solve the model. 
\item Estimate the parameters in the model. 
\item Compare their model results with the data.
\item Confirm the half-life data from the Abstract above.
\end{enumerate}

\begin{thebibliography}{98}

\bibitem{HoukEtAl1999} Houk, B.~E., G.~Hochhaus, and J.~A.~Hughes. 1999. Kinetic Modeling of Plasmid DNA Degradation in Rat Plasma. {\it AAPS PharmSci\/}. 1(3): 15-20.


\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

We offer complete analyses in a Mathematica notebook 5-010-T-Mma-DNADegradation-Teacher Version.nb and an accompanying pdf version for those who do not access Mathematica files. Figure~\ref{fig:pDNAFinalModel} renders the model curves obtained thus plotted over the respective extracted data sets.

We believe this Modeling Scenario is a demonstration of the role of systems of linear differential equations in the study of life science issues. For while the model is presented, indeed the complete analysis (except the details of the parameter estimation method) is offered in the source article \cite{HoukEtAl1999}, the paper shows the thinking of life scientists in trying to model the degradation of the pDNA. 

The main focus of this Modeling Scenario is the defense of the model, the solution of the system, the estimation of the parameter, and the comparison of model and data. Students should be encouraged to ``take'' their own data points from Figure~\ref{fig:pDNA}.




\begin{figure}[h]
\begin{center}

\includegraphics[width=12cm]{pDNAFinalModel.jpg}



\parbox{12 true cm}{\caption{\label{fig:pDNAFinalModel} Plot of extracted data and model obtained from the analysis in our Mathematica notebook found in the Supporting Docs for this Modeling Scenario.}}
\end{center}

\end{figure}

We believe with a short introduction and motivation of the model offered in the Theoretical section of \cite[p.3]{HoukEtAl1999} the students could be directed to perform the activities on their own with proper tools provided, in our  case, Mathematica, to obtain  the closed form solution of (\ref{eq:model}) with all the parameters and initial conditions as constants. This would be in  preparation for  using a minimization of the sum of squared errors over all three forms of pDNA and finally comparing the model to the data as a homework assignment. 

}





\end{document}
