\chapter{Static Reactors with Molecular Diffusion - Results}
\label{app:mcs_diff}

We describe three series of experiment (where 5 simulations were run in each series) in which the static reactors model with molecular diffusion is employed (Section \ref{sec:McCaskillsModel}). The following diffusion coefficients are employed: $m_1=0.01$,  $m_2=0.05$ and $m_3=0.1$ in respective series.

The following set of fixed parameters is utilised in all simulations:
\begin{itemize}
	\item 30 compartments are utilised and executed in parallel using 30 AMD Opteron 270 (2.0 GHZ) CPUs.
	\item Experiments are run for 3600 seconds.
	\item The maximal compartment carrying capacity is $n_{max}=1000$.
	\item The diffusion probability is set to $p_m=0.05$.
	\item The spontaneous decay probability is set to $d=0.1$.
	\item Similarly to experiences conducted in Chapter \ref{emerSelfMainClosedRN}, the maximal species string length is set to $BD_{Lmax}=500$.	
	\item The per-symbol mutation is set to $p_{sym}=1.0\e{-5}$.
	\item The global spontaneous mutation rate is set to $r_{mut}=0$.
	\item As in the evolutionary experiments presented in Section \ref{riseFall}, each compartment is seeded/initialised with $s_{R_4}=\triangledown0101:\triangledown0101$ molecular species. However the initial amount differs, we seed/fill the compartments with 1000 instances of $s_{R_4}$ species.
\end{itemize}


We present an overview of the dynamics of all simulation runs where the diffusion coefficient is set to $m_1=0.01$, $m_2=0.05$ and $m_3=0.1$. The following chain of events were observed in all runs: 

\begin{enumerate}
	\item We note an initial phase where the system is stable with an average species length of 12 symbols (i.e., the length of $s_{R_4}$ species) and the average population size stagnating at nearly 1000 molecules. Most compartments are thus full during this phase (i.e., the molecular production rate is higher than the decay rate). 
	\item However at some stage, we note that the average length of species starts to increase rapidly. This behaviour suggests that the elongation phenomenon is occurring. 
	\item Shortly after, we observe a rapid decrease in the average population size throughout the 30 compartments. This indicates that the production rate has now become smaller than the decay rate, i.e., the compartments are depleting.
	\item Nevertheless this decrease does not apply to the species length which continues to increase for a period of time, where the species reach a peak in string length. During this period, successive species having an increasing length emerged and displaced each others. As a result we observe a linear increase in the average species length throughout all compartments.
  \item Following the observed peak in the species length, the average species length is then rapidly decreasing (similarly to the average population size) until the system becomes extinct, i.e., all molecules have decayed.
\end{enumerate}

Figures \ref{extinPhase1a} and \ref{extinPhase1b} complement the current analysis by providing more detailed information about the dynamics of each compartment when $m_1=0.01$:

\begin{enumerate}
	\item During the early stable phase, we note that the molecular string length is relatively homogeneous throughout the 30 compartments with little variance occurring. This assertion applies for both the average population size and species string length.
	\item Following this, we observe a divergence in the composition of the compartments (i.e., compartments with different population sizes exist). This variance is maintained until close to the end of the extinction phase. 
	\item Although the composition of compartments start diverging at some point in time, the average species length (which is globally increasing) is more or less homogeneous throughout the compartments for a period of time. Thus during this phase, the mutant species (here classified by their string length) are well diffused throughout the compartments. However as the average molecular population size is already decreasing, it indicates that these mutant species have a production rate lower than the decay rate.
	\item Finally, a significant range of variances is observed in the species string length. This phenomenon suggests that only few reactions leading to the creation of much longer species are succeeding in some compartments.
\end{enumerate}

Figures \ref{extinPhase2a} and \ref{extinPhase2b} detail the extinction phases of the simulation runs with $m_2=0.05$ whereas the extinction phases of the runs with $m_3=0.1$ are depicted in Figures \ref{extinPhase3a} and \ref{extinPhase3b}. In these figures, we distinguish that the range of variances in the average molecular population size actually decreases in constrast to the increase trend reported earlier in the first experiment where $m_1=0.01$. Little variance of average molecular string length is observed over the whole simulation runs excepting during a few seconds where the systems collapse. These results suggest that higher diffusion coefficients allowed for the diffusion equilibrium to be achieved more rapidly. Consequently we identify a homogeneous molecular composition throughout all compartments over time.


\section{5 example simulations with diffusion coefficient \texorpdfstring{$m_1=0.01$}{firstserie}}


\begin{figure}[h]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.01.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.01.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.01.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.01.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.01.eps}}}
\caption{5 example simulations with $m_1=0.01$.}
\end{figure}


\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.01-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.01-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.01-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.01-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.01-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.01-stdev-strlen-zoom.eps}}}
\caption{Extinction phases of simulation runs 1, 2 and 3 - from top to bottom, with diffusion coefficient $m_1=0.01$.}
\label{extinPhase1a}
\end{figure}

\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.01-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.01-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.01-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.01-stdev-strlen-zoom.eps}}}
\caption{Extinction phases of simulation runs 4 and 5 with diffusion coefficient $m_1=0.01$.}
\label{extinPhase1b}
\end{figure}

\clearpage
\section{5 example simulations with diffusion coefficient \texorpdfstring{$m_2=0.05$}{secondserie}}

\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.05.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.05.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.05.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.05.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.05.eps}}}
\caption{5 example simulations with $m_2=0.01$.}
\end{figure}

\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.05-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.05-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.05-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.05-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.05-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.05-stdev-strlen-zoom.eps}}}
\caption{Extinction phases of simulation runs 1,2 and 3 with diffusion coefficient $m_2=0.05$. }
\label{extinPhase2a}
\end{figure}

\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.05-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.05-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.05-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.05-stdev-strlen-zoom.eps}}}
\caption{Extinction phases of simulation runs 4 and 5 with diffusion coefficient $m_2=0.05$.}
\label{extinPhase2b}
\end{figure}

\clearpage

\section{5 example simulations with diffusion coefficient \texorpdfstring{$m_3=0.1$}{thirdserie}}
\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.1.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.1.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.1.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.1.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.1.eps}}}
\caption{5 example simulations with $m_3=0.1$.}
\end{figure}

\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.1-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp0-0.1-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.1-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp1-0.1-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.1-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp2-0.1-stdev-strlen-zoom.eps}}}
\caption{Extinction phases of simulation runs 1,2 and 3 with diffusion coefficient $m_3=0.1$. }
\label{extinPhase3a}
\end{figure}

\begin{figure}[hp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.1-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp3-0.1-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.1-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{appendix/mcs_diff/exp4-0.1-stdev-strlen-zoom.eps}}}
\caption{Extinction phases of simulation runs 4 and 5 with diffusion coefficient $m_3=0.1$.}
\label{extinPhase3b}
\end{figure}