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\usepackage{color}
\usepackage{graphicx}
\usepackage{hyperref}

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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\SUCROSE REACTION CHEMISTRY}






\markboth{Sucrose Reaction Chemistry}{Sucrose Reaction Chemistry}






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

\makeStitlePDFLaTex



\begin{abstract}
We offer data on the hydrolysis of sucrose in the presence of catalyst sucrase. We present students with several models and ask which model is best. Additionally, we ask students to estimate parameters for the best model and to validate the model and explain the significance of one of the parameters in the best model.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

At room temperature sucrose is hydrolyzed by the catalytic action of the
enzyme sucrase as follows:

\begin{equation}\label{eq:reaction}
{\rm sucrose} \overset{\rm sucrase}{\longrightarrow} {\rm products} \, .
\end{equation}

Starting with a sucrose concentration of $C_A(0) = 1.0$ millimol/liter and an enzyme
concentration $C_{E0} = 0.01$ millimol/liter, the  kinetic data in Table~\ref{table:data}
are obtained (and plotted in Figure~\ref{fig:1-93-DataPlot}) in a batch reactor (concentrations calculated from optical
rotation measurements):

\begin{table}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|} \hline
$t$ &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hline
  $C_A(t)$ &1 & 0.84 & 0.68 & 0.53 & 0.38 & 0.27 & 0.16 & 0.09 & 0.04 &
   0.018 & 0.006 & 0.0025 \\  \hline
\end{tabular}
\parbox{8 true cm}{\caption{\label{table:data} Data from batch reactor on hydrolysis of sucrose, first time $t$ in hour and then  $C_A(t)$ in millimol/liter.}}
\end{center}
\end{table}


\begin{figure}
\begin{center}
\includegraphics[width=10cm]{1-93-DataPlot.eps}
\vspace*{-.15 true cm}
\parbox{10 true cm}{\caption{\label{fig:1-93-DataPlot} Plot of data (from Table~\ref{table:data}) from batch reactor on hydrolysis of sucrose, first time $t$ in hour and then  $C_A(t)$ in millimol/liter.}}
\end{center}
\end{figure}




Consider each of  the following models and decide if they are appropriate. For one appropriate model find the estimates of the parameters and confirm the model is a reasonable model for this reaction. Furthermore, explain or confirm the significance and units of each of the parameters.

\begin{enumerate}
  \item  Exponential:
  $$ C_A'(t) = - k\, C_A(t) $$
where $k$ is the reaction rate.
\item Effected Limited: 
$$ C_A'(t) = - k_1 e^{-k_2*(C_{E0} - C_A(t))}  $$ 
where $k_1$ and $k_2$ are the reaction rates. 
\item Michaelis-Menten
  $$ C_A'(t) = - k\, C_A(t) \frac{C_{E0}}{C_A(t) + C_M} $$
where $k$ is the reaction rate and $C_M$ is the concentration of sucrose at which the reaction occurs at one half it maximum rate.
\item Logistic:
  $$ C_A'(t) = - k\, C_A(t) \frac{K - C_A(t)}{K} $$
where $k$ is the reaction rate.
\end{enumerate}










\teacher{
\section*{NOTES FOR TEACHERS}

The source of this modeling activity with data is \cite[p.~79]{Levenspiel1999}, a text rich in examples, explanations, and very good exercises based on data.  We have thrown in three bad models and believe students learn to judge models best when finding things wrong. The best model is the Michaelis-Menten model.  Of course, the teacher, can suggest the Michaelis-Menten model is the one to use and proceed without such distractors. 

Upon integration of the Michaelis-Menten model  there is not an explicit solution, rather the solution is implicit.  Nevertheless, students can form a sum of square errors  involving $t$ and $C_A(t)$ (\ref{eq:imp}) which can be minimized in terms of the parameters $k$ and $C_M$, 

\begin{equation}\label{eq:imp}
-C_A(t)-C_M \log (C_A(t))=c+0.01
   k t\, 
\end{equation}
and once the initial condition of $C_A(0) = 1$ is applied we have

\begin{equation}\label{eq:impc}
-C_A(t)-C_M \log (C_A(t))=0.01
   k t-1\, . 
\end{equation}

In an attempt to estimate the parameters $C_M$ and $k$,  one forms the sum of square errors, forcing the sum of expressions of the form $\left( -C_A(t)-C_M \log (C_A(t))- (0.01
   k t-1)\right)^2$ to be as close  to 0 as possible, i.e. minimizing $SSE(k_,C_M)$,
\begin{equation}\label{eq:SSE}
SSE(k_,C_M) = \sum_{i=1}^n \left(-C_A(t_i)-C_M \log (C_A(t_i))- (0.01
   k t_i-1)\right)^2
\end{equation}
where $i= 1, 2, \ldots, n$ with $n = 12$, the number of data points.

Now a nonlinear minimization (in the case of Mathematica, using the FindMinimum command) determines $k = 19.9393$ and $C_M = 0.198$ as the best parameter estimates which minimize (\ref{eq:SSE}) producing a final model:

\begin{equation}\label{eq:imppar}
-C_A(t)-0.198609 \ln (C_A(t))=0.199393 t - 1\, .
\end{equation}


We plot the data and our resulting best fit model in Figure~\ref{fig:1-93-SolPlot}.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=10cm]{1-93-SolPlot.eps}
\vspace*{-.15 true cm}
\parbox{10 true cm}{\caption{\label{fig:1-93-SolPlot} Plot of data (from Table~\ref{table:data}) from batch reactor on hydrolysis of sucrose and best fit model using parameter estimates from the sum of square errors minimization. }}
\end{center}
\end{figure}

Upon completion of this parameter estimation analysis and obtaining the final model (\ref{eq:imppar}) for the Michaelis-Menten reaction:

\begin{equation}\label{eq:MM}
C_A'(t) = - k C_A(t) \frac{C_{E0}}{C_A(t) + C_M} \, ,
\end{equation}
the student should then demonstrate the fact that the estimated parameter, $C_M$, is the concentration of sucrose at which the reaction occurs at one half it maximum rate. This can be seen by noting  that when $C_A(t)$ is infinitely large the ratio $ \frac{C_A(t)}{C_A(t) + C_M}$ approaches 1 (while always being less than 1) and so the maximum reaction rate, $C_A'(t)$, is $-k C_{E0}$. Now when we substitute $C_M$ for $C_A(t)$ in the right side of (\ref{eq:MM}) we obtain
\begin{equation}\label{eq:MMfin}
 - k C_M \frac{C_{E0}}{C_M + C_M}  = -\frac{1}{2} k C_{E0}\, ,
\end{equation}
which is exactly one-half of the maximum reaction rate of $-k C_{E0}$.


In an  Mathematica notebook, 1-93-T-Mma-SucroseReaction-TeacherVersion.nb and an attendant pdf version for those who do not use Mathematica, found in the Supporting Documents for this Modeling Scenario,  we offer  complete analyses, including rationale for rejecting the three other models and a grid approach to parameter estimation for the correct Michaelis-Menten model.
\begin{thebibliography}{99}




\bibitem{Levenspiel1999}  Levenspiel, Octave. 1999. {\it Chemical
Reaction
Engineering
Third Edition\/}. New York: John Wiley \& Sons, Inc.


\end{thebibliography}
}





\end{document}
