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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Monod's Bacterial Growth Model}




\markboth{Monod's Bacterial Growth Model}{Monod's Bacterial Growth Model}

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


\begin{abstract}
Students model growth of bacteria {\it E. coli\/} in a limiting nutrient environment using  data from a historical study.
\end{abstract}



\section*{SCENARIO DESCRIPTION}



\begin{table}[h!]\label{table:Mdata}

\begin{center}
\begin{tabular}{|r|r|r|}\hline \hline
 Time (hr) & E.~coli & Substrate\\ \hline \hline
0.0 & 15.5 & 151.0\\ \hline
 0.54 & 23.0 & 123.0\\ \hline
 0.90 & 30.0 & 105.0\\ \hline
 1.23 & 38.8 & 75.0\\ \hline
 1.56 & 48.5 & 43.5\\ \hline
 1.95 & 58.3 & 14.5\\ \hline
 2.33 & 61.3 & 3.5\\ \hline
 2.70 & 62.5 & 0.5\\ \hline
\end{tabular}

\parbox{12.5 true cm}{\caption{\label{table:Mdata} Data pairs with units  from Monod's original experiments (time in hours, biomass concentration of {\it E. coli\/} where one unit is approximately 0.75 mg dry weight/L, concentration of the growth limiting nutrient (or substrate) lactose in mg/L).}}

\end{center}

\end{table}


We show  Monod's  data in Table \ref{table:Mdata} and plots of the data of {\it Escherichia coli\/} ({\it E. coli\/}) biomass concentration  in Figure~\ref{fig:1-078-EColiPlot} and  growth limiting nutrient concentration in Figure~\ref{fig:1-078-GrowthLimitNutrientPlot}, both as a function of time, from  the 1949 study \cite{Monod1949}  by J.~ Monod.

\begin{quote}
``Monod's bacterial growth model  offered  in 1949 \cite{Monod1949} defines the relationship between specific growth rate and concentration of the limiting nutrient, and the relation between nutrient utilization rate and bacterial growth where $x(t)$ is the biomass concentration at time $t$, $s(t)$ is the concentration of the growth limiting nutrient (or substrate) at time $t$."\cite{LobryFlandrois1990}
\end{quote}

One way to model the consumption of nutrients  ($s(t)$) by bacteria   ($x(t)$)  is given by  (\ref{eq:PreMonod}), a simple (and unlimited) predation model. 


\begin{eqnarray}\label{eq:PreMonod}
x'(t) &=& \mu_m  x(t)  s(t)\\
s'(t) &=& \frac{x'(t)}{Y} \nonumber
\end{eqnarray}


This model has the disadvantage that one unit of bacterial concentration ($x(t)$)  can consume nutrients at a rate proportional to the nutrient level, meaning per unit time we are modeling our bacteria consumption to grow in an unlimited fashion - highly unrealistic. Below is the unlimited growth model.



We offer up Monod's model \cite{Monod1949}, first published in 1949,  which  introduced a satiation or limiting consumption rate term $\frac{s(t)}{K_m+s(t)}$ to replace the unlimited  consumption rate term found in (\ref{eq:PreMonod}), leading to 

\begin{eqnarray}\label{eq:Monod}
x'(t) &=& x(t) \mu_m \frac{s(t)}{K_m+s(t))}\\
s'(t) &=& \frac{x'(t)}{Y} \nonumber
\end{eqnarray}

\noindent where 

\begin{itemize}
\item $x(t)$  is the biomass concentration (one unit  is approximately 0.75 mg dry weight/L) of {\it E. coli} at time $t$  hours;
\item $s(t)$ is the concentration of the growth limiting nutrient (or substrate) lactose in mg/L;
\item $K_m$ is 	the substrate concentration which supports half-maximum {\it E. coli} specific growth rate;
\item $\mu_m$ is  the maximum specific growth rate; and
\item  $Y$ is the yield conversion coefficient.
\end{itemize}

Indeed, from a plot of Monod's data \cite{Monod1949} on the growth of {\it E. coli\/} shown in Figure~\ref{fig:1-078-EColiPlot} we see that growth is NOT unlimited, but rather levels off and stays that way when the nutrient levels ($s(t)$) become less and less.

\begin{figure}


\centering
\includegraphics[width=9.5cm]{1-078-EColiPlot.png}

%\vspace*{-1.7 true cm}
\parbox{8 true cm}{\caption{\label{fig:1-078-EColiPlot} Plot of {\it E. coli\/} ($x(t)$) biomass concentration where one unit is approximately 0.75 mg dry weight/L.}}
\end{figure} 

\begin{figure}


\centering
\includegraphics[width=9.5cm]{1-078-GrowthLimitNutrientPlot.png}

%\vspace*{-1.7 true cm}
\parbox{8 true cm}{\caption{\label{fig:1-078-GrowthLimitNutrientPlot} Plot of concentration of the growth limiting nutrient (or substrate) lactose ($s(t)$)  in mg/L)}}
\end{figure} 


\section*{Assignment}

\begin{enumerate}
\item From the data and solutions of the differential equations (possibly numerical solutions) estimate the best  parameters $K_m$ and $\mu_m$ in (\ref{eq:Monod}). Hint: Use second equation to reduce problem to one differential equation in one variable, $x(t)$.
\item Show that the nutrient concentration level, $K_m$, is actually the level at which the {\it E. coli\/} feeds at one-half its maximum feeding rate. This gives some context and meaning to this parameter in the model and can be measured in a laboratory as well as estimated in a process such as this activity.
\item Use the parameters you obtain for (\ref{eq:Monod}) and show how well your model predicts the {\it E. coli\/} concentration  when compared to the data for {\it E. coli\/} over time as displayed in Table~\ref{table:Mdata}.  
\item Use the parameters you obtain for (\ref{eq:Monod}) and show how well your model predicts the  concentration of the growth limiting nutrient   when compared to the data for {\it E. coli\/} over time.  
\end{enumerate}



\begin{thebibliography}{99}




\bibitem{LobryFlandrois1990} Lobry, J.~R. and J.~P.~Flandrois. 1990. Comparison of estimates of Monod's growth model 
parameters from the same data set. {\it BINARY\/}.  3: 20-23.

\bibitem{Monod1949} Monod, J. 1949. The growth of bacterial cultures. {\it Annual Review of Miocrobiology\/}. 3: 371-394.



\end{thebibliography}






\end{document}
