update version 16-07-25

texmaker/Chapters/Chapter2.tex

@@ -733,7 +733,7 @@ Where $a$ and $b$ correspond to the time points that are associated with the $a_
 the function values at point $a$ and $b$ correspond the approximated metabolite concentrations.
 To approximate the integral over the whole curve, the trapezoids between all collocation points are summed. Since $\tau$ is defined as the sum of metabolites $S,X$ and $Y$ (\ref{problem_simul}), we also have to add up the different metabolites. This can be done easily because of the sum rule for integration:
 \begin{align}
-\int_{0}^{t_{f}}S(t)+X(t)+Y(t)dt = \int_{0}^{t_{f}}S(t)+\int_{0}^{t_{f}}X(t)+\int_{0}^{t_{f}}Y(t)
+\int_{0}^{t_{f}}S(t)+X(t)+Y(t)dt = \int_{0}^{t_{f}}S(t)+\int_{0}^{t_{f}}X(t)+\int_{0}^{t_{f}}Y(t)\,.
 \end{align} 
 Using approximation definitions described above, we can transform the dynamic optimization problem from \ref{problem_simul} to a NLP:
 \begin{flalign}

texmaker/Chapters/Chapter3.aux

@@ -9,10 +9,10 @@
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texmaker/Chapters/Chapter3.tex

@@ -5,12 +5,12 @@ This chapter depicts the results of the solution to the optimization problem fro
 \section{Sequential Approach: Control Vector Parameterization}
 The tool DOTcvpSB was used for the sequential approach, which basically connects the DAE solver and NLP solver in series and loops until convergence. DOTcvpSB offers a MATLAB interface, which enables quick and intuitive model description. Since the sequential approach, in contrast to the simultaneous approach, repeatedly integrates the dynamic system and feeds back information to the NLP solver, the differential equations can directly be entered as constraints in the model description. DOTcvpSB uses integrators from SUNDIALS, a package called CVODES \citep{cvodes}, which is able to solve IVPs. DOTcvpSB offers multiple possibilities for the NLP solver. The results presented here are produced with SRES as the NLP solver. In contrast to all other results which follow in this chapter, this solution is approximated with a heuristic method rather than a deterministic local solver.\\ The model associated with these results has one path constraint, which is the limiting enzyme capacity: $E_{1}+E_{2}+E_{3} \leq 1$, and one end point constraint: $P(t_{f})=0.9$. Figure \ref{plot_dotcvp_metab} displays the concentration curve of each metabolite, which are associated with the best obtained solution. Interestingly, the intervals, which are determined by the optimized switching points, are not equal in size, but vary. This is due to differences in the model description in comparison to the other models. Already in the first interval, metabolite $S$ fully produces metabolite $X$, which has its maximum at the first switching point. The second interval starts with the degradation of metabolite $X$ and therefore, with the production of metabolite $Y$. At the end of the second and the start of the last interval, the concentration of substrate $S$ starts to converge to 0. The production of the final metabolite $P$ is initiated with the degradation of metabolite $Y$ during the last interval. This interval is also significantly larger than the first two intervals and ends with the convergence of the concentration of product $P$. All other metabolites are mostly degraded by the end of interval three. Note that the best solution obtained by the Control Vector Parameterization is 6.637, which is represented by the violet vertical line in figure \ref{plot_dotcvp_metab}.
 The total enzyme concentration is used up by $E_{1}$ in the first interval, indicating the sequential behavior of the enzymes corresponding to the topology of the linear reaction chain. During the second interval, enzymes one and two share the available resources, since, next to the degradation of metabolite $X$, the degradation of metabolite $S$ is not yet fully complete. $E_{3}$, on the other hand, is only active during the last interval, being responsible for the conversion of metabolite $Y$ to the desired product $P$.
-\begin{figure}[!htb]
+\begin{figure}[H]
 \includegraphics[width=0.9\textwidth]{plot_dotcvp_metab}
 \captionsetup{width=\linewidth}
 \caption[Metabolite concentrations sequential $n=3$]{Plotted are the concentration profiles of all metabolites in the 3-step pathway. Those are the concentrations which optimize the transition time, represented in the dotted vertical line. The solution was found by applying the Control Vector Parameterization approach\label{plot_dotcvp_metab}.}
 \end{figure}\noindent
-\begin{figure}[htb!]
+\begin{figure}[H]
 \includegraphics[width=0.9\textwidth]{dotcvp_enzymes}
 \captionsetup{width=\linewidth}
 \caption[Enzyme concentrations sequential $n=3$]{Plotted are the concentration profiles of all enzymes in the 3 step pathway. Those are the concentrations which optimize the transition time. The enzyme concentrations are piecewise constant, where the pieces correspond to the intervals whose lengths are variable and the associated switching points are decision variables that are optimized.\label{Enzyme-concentrations-sequential-$n=3$}}
@@ -27,14 +27,14 @@ This section discusses the results from the Direct Transcription method with AMP
 \captionsetup{width=\linewidth}
 \caption[Metabolite concentrations simultaneous-$n=3$]{Plotted are the concentration profiles of all metabolites in the 3-step pathway. Those are the concentrations which optimize the transition time, represented in the vertical line. The solution was found by applying the Direct Transcription approach.}
 \end{figure}\noindent
-While substrate $S$ is degraded, metabolite $X$ is produced initiating the production of metabolite $Y$, which in turn degrades $X$. Product $P$ is only produced when metabolite $Y$ is at its maximum. As soon as the production of $P$ is initiated, $Y$ is degraded. The corresponding enzyme profile are presented in figure \ref{Enzyme-concentrations-simultaneous-$n=3$}.\begin{figure}[H]
+While substrate $S$ is degraded, metabolite $X$ is produced initiating the production of metabolite $Y$, which in turn degrades $X$. Product $P$ is only produced when metabolite $Y$ is at its maximum. As soon as the production of $P$ is initiated, $Y$ is degraded. The corresponding enzyme profile are presented in figure \ref{Enzyme-concentrations-simultaneous-$n=3$}.
+Each enzyme has three switching points, at which their concentrations can change. $E_{1}$ is only "active" during the first interval as it catalyzes the first reaction, thus degrading substrate $S$ and producing metabolite $X$.  $E_{1}$ does not make use of the total available enzyme concentration, since $E_{2}$ is also active in the first interval. As soon as the concentration of intermediate metabolite $X$ is greater than zero, $E_{2}$ starts catalyzing reaction two by producing $X$, and shares the allowed total enzyme concentration with $E_{1}$ during the first interval.\\
+Note that the best solution, optimized by the Direct Transcription method, is 7.114149, which is slightly worse than the solution that is obtained with CVP.
+\begin{figure}[H]
 \includegraphics[width=0.9\textwidth]{plot1_e}
 \captionsetup{width=\linewidth}
 \caption[Enzyme concentrations simultaneous $n=3$]{Plotted are the concentration profiles of all enzymes in the 3-step pathway. Those are the concentrations which optimize the transition time. The enzyme concentrations are piecewise constant, where the pieces correspond to the intervals whose lengths are determined by $h\cdot t_{f}$ (also optimized).\label{Enzyme-concentrations-simultaneous-$n=3$}}
 \end{figure}\noindent
-Each enzyme has three switching points, at which their concentrations can change. $E_{1}$ is only "active" during the first interval as it catalyzes the first reaction, thus degrading substrate $S$ and producing metabolite $X$.  $E_{1}$ does not make use of the total available enzyme concentration, since $E_{2}$ is also active in the first interval. As soon as the concentration of intermediate metabolite $X$ is greater than zero, $E_{2}$ starts catalyzing reaction two by producing $X$, and shares the allowed total enzyme concentration with $E_{1}$ during the first interval.\\
-Note that the best solution, optimized by the Direct Transcription method, is 7.114149, which is slightly worse than the solution that is obtained with CVP.
-
 
 
 \pagebreak

texmaker/main.aux

@@ -216,8 +216,8 @@
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 \citation{ipopt}
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-\abx@aux@page{101}{50}
+\abx@aux@page{100}{51}
+\abx@aux@page{101}{51}
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 \citation{klipp2}
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texmaker/main.lof

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texmaker/main.toc

@@ -29,7 +29,7 @@
 \contentsline {subsubsection}{Stochastic Ranking Evolution Strategy (SRES)}{47}{section*.167}
 \contentsline {chapter}{\numberline {3}Results}{48}{chapter.168}
 \contentsline {section}{\numberline {3.1}Sequential Approach: Control Vector Parameterization}{48}{section.169}
-\contentsline {section}{\numberline {3.2}Simultaneous Approach: Direct Transcription}{50}{section.172}
+\contentsline {section}{\numberline {3.2}Simultaneous Approach: Direct Transcription}{51}{section.172}
 \contentsline {section}{\numberline {3.3}Multiple Shooting}{53}{section.175}
 \contentsline {chapter}{\numberline {4}Conclusion}{58}{chapter.181}
 \contentsline {chapter}{Bibliography}{61}{chapter*.182}