\chapter{Evolving Closed Reaction Networks - Results}

A series of experiments are reported in which closed reaction networks are evolved to promote the growth of a target species $s_t$. 

\section{Two example simulation runs with self-replication reactions enabled}
\label{app:evolSelfRep}
Two evolutionary simulations are conducted and utilise the following set of parameters:
\begin{itemize}
	\item 30 cells are utilised and executed in parallel using 31 AMD Opteron 270 (2.0 GHZ) CPUs.
	\item self-replications reactions are allowed.
	\item Simulations are run for 3600 seconds.
	\item The maximal compartment carrying capacity is $n_{max}=\infty$.
	\item The target molecular species division threshold is set to $n_{target}=200$.
	\item The maximal species string length is set to $BD_{Lmax}=500$.	
	\item The global spontaneous mutation rate is set to $r_{mut}=0$.
	\item The per-symbol mutation probability is set to $p_{sym}=0.00005$.
	\item Each compartment is seeded/initialised with 10 instances of the species $s_1$. 
	\item The target molecular species is $s_t=s_1$.
\end{itemize}
All reactors are seeded with a common closed reaction network which contains the species $s_1$ only (Fig.\ref{fig:selfRep}).

\begin{figure}[htbp]
	\centering
		\includegraphics[scale=0.7]{appendix/selfRep/selfRep.ps}
	\caption{Bipartite reaction graph of seed closed reaction network containing the self-replicase species $s_1$ only.}
	\label{fig:selfRep}
\end{figure}

\begin{figure}[htbp]
	\centering	
	\subfigure{\rotatebox{-90}{\includegraphics[scale=0.29]{appendix/selfRep/evoSelfRep1.eps}}}
	\subfigure{\rotatebox{-90}{\includegraphics[scale=0.29]{appendix/selfRep/evoSelfRep2.eps}}}	
	\caption{Example simulation runs 1 (left) and 2 (right)}
	\label{fig:evoSelfRep1}
\end{figure}

In Fig. \ref{fig:evoSelfRep1}, it is first observed that no displacement occurred in both simulations. A level of cellular species diversity is observed averaging at $\approx 5$ throughout the simulation runs. The seed closed reaction network remained the dominant cellular species throughout the simulations. 

\clearpage

\section{Constructing a minimalist collectively autocatalytic reaction network}
\label{app:consMiniCOll}

In this section we present the construction of a minimalist collectively autocatalytic reaction network. We aim at realising the simplest collectively reaction network in terms of complexity at both the molecular (i.e., using the shortest molecular species) and network level (i.e., involving the least number of species and interactions between the latter). In this reaction network, no self-replication reaction may occur.

We showed that the simplest autocatalytic species that can be built in the \MCS~is $s_{R_0}=*\triangledown:\triangledown$. Indeed the length of $s_{R_0}$ is equal to $BD_{Lmin}$ and $\triangledown$ is the only single enzymatic operator that can both match and output multiple characters (as required to produce an enzymatic molecule having at least 4 symbols).

As self-replication reactions are now disabled then the minimalist closed reaction network would thus contain at least two distinct molecular species. Both of these molecular species would mutually contribute to the maintenance of each other. We first attempt to build such a closed network composed of two distinct molecular species denoted by $s_1$ and $s_2$. 

In this first attempt, $s_1$=$s_{R_0}$. The second molecular species $s_2$ is required to be as well a replicase species being able to replicate $s_1$. Therefore, in this minimalist approach, $s_2$ is to be equivalent to $s_1$ from a enzymatic function point of view with $s_2=*\triangledown:\triangledown$ as a \textit{starting} point. Nevertheless as we cannot decrease the length of $s_2$, an additional symbol is to be inserted to differ $s_2$ from $s_1$. 

If we insert an \textit{executable} symbol in the binding condition or action statement of $s_2$, excluding the structural symbols * or : and the quote symbol as this would inhibit the $s_2$'s enzymatic function, then we would obtain an elongator or ``reductor'' species which by definition cannot replicate $s_1$.

For example:
\begin{eqnarray}
	*'\triangledown0:\triangledown\ +\ *\triangledown:\triangledown & \rightarrow & \emptyset\\
	*\triangledown0:'\triangledown\ +\ *\triangledown:\triangledown & \rightarrow & \emptyset\\
	*\diamond\triangledown0:\triangledown\ +\ *\triangledown:\triangledown & \rightarrow & \triangledown:\triangledown\\
	*\triangledown0:\diamond\triangledown\ +\ *\triangledown:\triangledown & \rightarrow & *\triangledown:\diamond\triangledown
\end{eqnarray}

As a result we are left with the option to insert a \textit{non-executable} symbol in $s_2$. In this case we may obtain a closed network containing two distinct species. However this also implies that both molecular species are equivalent from a phenotypic point of view. Let $s_2=**\triangledown:\triangledown$. We present an example simulation run in which this reaction network is employed and where the cellular division criterion is to promote the growth of $s_t=s_1$. 

\begin{figure}[htbp]
	\centering
		\subfigure{\rotatebox{-90}{	\includegraphics[scale=0.29]{appendix/noSelfRep/crossMolMutEvoRun1}}}
		\subfigure{\rotatebox{-90}{\includegraphics[scale=0.29]{appendix/noSelfRep/crossMolMutEvoRun2}}}
		\caption{Two example simulation runs where the seed reaction contained 10 instances of both $s_1=*\triangledown:\triangledown$ and $s2=**\triangledown:\triangledown$ with $p_{sym}=0.00005$. and $r_{mut}=0$ }
	\label{fig:noSelfRepUniRepOnly}
\end{figure}

Fig.\ref{fig:noSelfRepUniRepOnly} depicts two simulation runs where the two replicase species $s_1$ and $s_2$ are employed. Although a level of cellular diversity was maintained throughout the simulation runs, no cellular species with a significant difference in fitness emerged. As a result no displacements between  cellular species occurred. These results suggest that the employed seed reaction network is already the optimal catalytic network to produce $s_t$.

In contrast with the above proposal, we now attempt to build a minimalist closed network where the molecular species are phenotypically different from each other and no universal replicases are employed. 

If we were to employ 5 executable symbols to specifiy the molecular species, we would, again, be confronted with that asymmetric relationship between the binding and action statements leading to the elongation/trimming enzymatic functions. We thus propose to relax the minimal molecular string length to 6. The following two molecular species are proposed:

\begin{eqnarray*}
s_1& =& \triangledown0:\triangledown1\\
s_2& =& \triangledown1:\triangledown0
\end{eqnarray*}

However reactions between the molecular species $s_1$ and $s_2$ lead to the production of further molecular species as follows:

\begin{eqnarray*}
s_1\ +\ s_2\ &\rightarrow\ &s_3 \\
s_2\ +\ s_1\ &\rightarrow\ &s_4
\end{eqnarray*}

with $s_3=\triangledown1:\triangledown$ and $s_4=\triangledown0:\triangledown0$.

We enumarate all possible reactions (with $\alpha=4^2$) that may occur between $s_1$, $s_2$, $s_3$ and $s_4$:

\begin{eqnarray*}
s_1\ +\ s_1\ &\rightarrow\ & \emptyset \\
s_1\ +\ s_2\ &\rightarrow\ & s_3 \\
s_1\ +\ s_3\ &\rightarrow\ & \emptyset \\
s_1\ +\ s_4\ &\rightarrow\ & s_1 \\
s_2\ +\ s_1\ &\rightarrow\ & s_4 \\
s_2\ +\ s_2\ &\rightarrow\ & \emptyset \\
s_2\ +\ s_3\ &\rightarrow\ & s_2 \\
s_2\ +\ s_4\ &\rightarrow\ & \emptyset \\
s_3\ +\ s_1\ &\rightarrow\ & s_1 \\
s_3\ +\ s_2\ &\rightarrow\ & \emptyset \\
s_3\ +\ s_3\ &\rightarrow\ & \emptyset \\
s_3\ +\ s_4\ &\rightarrow\ & \emptyset \\
s_4\ +\ s_1\ &\rightarrow\ & \emptyset \\
s_4\ +\ s_2\ &\rightarrow\ & s_4 \\
s_4\ +\ s_3\ &\rightarrow\ & \emptyset \\
s_4\ +\ s_4\ &\rightarrow\ & \emptyset 
\end{eqnarray*}

No further molecular species have been produced in the above reactions. 

The above informal investigation suggests that the above set of molecular species $c_0=\{s_1,s_2,s_3,s_4\}$ is potentially a minimalist collectively autocatalytic reaction network that can be constructed in the \MCS~ given the following condition:
\begin{itemize}
	\item The reaction network is not, by design, the optimal catalytic network to realise the target task. Evolutionary experiments using this reaction network as the seed networks are presented in the next section.
	\item Molecular species which can perpetually generate new species (such as elongator species) are not allowed.
\end{itemize}

\section{10 simulation runs with self-replication reactions disabled}
\label{app:EvolNoSelfRep}
10 evolutionary simuations are executed using $c_0$ and the following set of parameters:
\begin{itemize}
	\item 31 cells are utilised and executed in parallel using 31 AMD Opteron 270 (2.0 GHZ) CPUs.
	\item Simulations are run for 3600 seconds.
	\item The maximal compartment carrying capacity is $n_{max}=\infty$.
	\item The target molecular species division threshold is set to $n_{target}=200$.
	\item The maximal species string length is set to $BD_{Lmax}=500$.	
	\item The global spontaneous mutation rate is set to $r_{mut}=0$.
	\item The per-symbol mutation probability is set to $p_{sym}=1.0\e{-5}$.
	\item Each compartment is seeded/initialised with 10 instances of each species $s_1$, $s_2$, $s_3$ and $s_4$ (as presented in the previous section). 
	\item The target molecular species is $s_t=s_1$.
\end{itemize}

In four of these runs, we observed the emergence and domination of reaction networks which are phenotipically equavalent to $c_1=\{s_1,s_2, s_5,s_6\}$ with $s_5=*\triangledown\triangle:\triangledown0$ and $s_6=*\triangledown\triangle:\triangledown1$. 

In four other runs, the emergence of reaction networks containing the molecular species of either $c_1$ or $c_3$ in addition to some molecular species were noted. However these additional molecular species did not improve the fitness of the cellular species. It is thus conjectured that, given enough time, these reaction networks would displace these extra molecular species and collapse to $c_1$ or $c_3$. In the remaining two runs, the emergence of $c_0$ mutants with no phenotypic differences was observed.

\begin{figure}[htbp]
	\centering	
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp0.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp1.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp2.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp3.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp4.eps}}}
	\caption{Simulation runs 1 to 5 with self-replication reactions disabled}
	\label{fig:evoNoSelfRep}
\end{figure}

\begin{figure}[htbp]
	\centering	
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp5.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp6.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp7.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp8.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.25]{appendix/noSelfRep/exp9.eps}}}
	\caption{Simulation runs 6 to 10 with self-replication reactions disabled}
	\label{fig:evoNoSelfRep2}
\end{figure}


\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|}\hline
	$c_1$ & $c_2$ & $c_3$ \\ \hline
$*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$  \\

$*\triangledown1:\triangledown1$ & $*\triangledown1:\triangledown1$ &	$*\triangledown1:\triangledown1$ \\

$*\triangledown0:\triangledown0$ & $*\triangledown0:\triangledown0$ & $*\triangledown0:\triangledown0$ \\

$*\triangledown1:\triangledown0$	& $*\triangledown1:\triangledown0$ & $*\triangledown1:\triangledown0$\\ 

$1*\triangledown0:\triangledown1$  & $**\triangledown0:\triangledown0$ & $*0:\triangledown0$ \\ 
& &  $*0:\triangledown1$\\
\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 1.}
	\label{tab:succnetsSim1}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|}\hline
	$c_1$ & $c_2$ & $c_3$ \\ \hline
$*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$ \\

$*\triangledown0:\triangledown0$  & $*\triangledown0:\triangledown0$  &	 $*\triangledown0:\triangledown0$\\

$*\triangledown1:\triangledown0$ & $*\triangledown1:\triangledown0$ & $*\triangledown\triangle:\triangledown1$\\

$*\triangledown1:\triangledown1$ & $*\triangledown1:\triangledown1$ & $*\triangledown\triangle:\triangledown0$\\ 

$*\triangledown1\triangledown1$ & $*\triangledown0:\triangledown00$ & $*\triangledown\triangledown0:\triangledown0$ \\
& & $*\triangledown\triangledown0:\triangledown1$\\

\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 2.}
	\label{tab:succnetsSim2}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|}\hline
	$c_1$   \\ \hline
$*\triangledown0:\triangledown1$  \\
$*\triangledown0:\triangledown0$ \\
$*\triangledown\triangle:\triangledown1$  \\
$*\triangledown\triangle:\triangledown0$ \\ 
$*0:\triangledown1$ \\
$*0:\triangledown0$ \\ \hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 3.}
	\label{tab:succnetsSim3}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|}\hline
	$c_1$   \\ \hline
$*\triangledown0:\triangledown1$  \\
$*\triangledown0:\triangledown0$ \\
$*\triangledown\triangle:\triangledown1$  \\
$*\triangledown\triangle:\triangledown0$ \\ \hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 4.}
	\label{tab:succnetsSim4}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|}\hline
	$c_1$   &$c_2$  \\ \hline
$*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$ \\
$*\triangledown0:\triangledown0$ & $*\triangledown0:\triangledown0$\\
$*\triangledown1:\triangledown1$ & 1 \\
$*\triangledown1:\triangledown0$ & 0 \\
$**\triangledown0:\triangledown1$ & $*\triangledown\triangle:\triangledown0$\\
&  $*\triangledown\triangle:\triangledown1$\\
&  $*\triangledown\triangle:1$\\
\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 5.}
	\label{tab:succnetsSim5}
\end{table}


\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|}\hline
	$c_1$    \\ \hline
$*\triangledown0:\triangledown1$ \\
$*\triangledown0:\triangledown0$ \\
$*\triangledown\triangle:\triangledown1$  \\
$*\triangledown\triangle:\triangledown0$ \\
\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 6.}
	\label{tab:succnetsSim6}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|}\hline
	$c_1$ & $c_2$ & $c_3$   \\ \hline
$*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$& $*\triangledown0:\triangledown1$\\
$*\triangledown0:\triangledown0$ & $*\triangledown0:\triangledown0$& $*\triangledown0:\triangledown0$\\
$*\triangledown1:\triangledown1$ & $*\triangledown\triangle:\triangledown0$& $*10:\triangledown1$\\
$*\triangledown1:\triangledown0$ & $*\triangledown\triangle:\triangledown1$& $*10:\triangledown0$ \\
$*\triangledown01:\triangledown1$ & $*\triangledown0:\triangledown\triangle0$& $*\triangledown\triangle:\triangledown1$\\
$*\triangledown01:\triangledown0$ & $*\triangledown0:\triangledown\triangle1$& $*\triangledown\triangle:\triangledown1$\\
\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 7.}
	\label{tab:succnetsSim7}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|c|}\hline
	$c_1$ & $c_2$ & $c_3$  & $c_4$   \\ \hline
$*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$& $*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$\\

$*\triangledown0:\triangledown0$ & $*\triangledown0:\triangledown0$& $*\triangledown0:\triangledown0$& $*\triangledown0:\triangledown0$\\

$*\triangledown1:\triangledown\triangle1$ & $*\triangledown\triangle:\triangledown1$ & $0*\triangledown\triangle:\triangledown1$ & $0*\triangledown\triangle\triangledown:\triangledown1$\\

$*\triangledown1:\triangledown\triangle0$ & $*\triangledown\triangle:\triangledown0$& $0*\triangledown\triangle:\triangledown0$ & $0*\triangledown\triangle\triangledown:\triangledown0$\\
\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 8.}
	\label{tab:succnetsSim8}
\end{table}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|}\hline
	$c_1$ & $c_2$ & $c_3$ \\ \hline
$*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$ & $*\triangledown0:\triangledown1$ \\

$*\triangledown0:\triangledown0$  & $*\triangledown0:\triangledown0$  &	 $*\triangledown0:\triangledown0$\\

$:*\triangledown1:\triangledown0$ & $*\triangledown\triangle:\triangledown0$ & $*\triangledown\triangle\triangle:\triangledown1$\\

$:*\triangledown1:\triangledown1$ & $:*\triangledown\triangle:\triangledown1$ & $*\triangledown\triangle\triangle:\triangledown0$\\ 


\hline
		\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks in simulation run 10.}
	\label{tab:succnetsSim9}
\end{table}
\clearpage
