\chapter{Evolving Closed Reaction Networks}
\label{chap:evolClosReacNet}
In Chapter \ref{chap:evoStab}, we evaluated a cellular model applied to the parallel implementation of the \MCS. Our results indicated that this multi-level selectional model provided the \MCS~system with the necessary robustness to control the degenerative effects due to parasitism. This control of evolutionary degenerations permitted the closed reaction networks to self-maintain when subjected to perturbations. This novel \MCS~system enables us to conduct further experiments in which we aim at evolving closed reaction networks to carry out pre-specified information processing tasks. These evolutionary experiments are presented in this chapter which combines some of the material published at several international conferences \citep{DecraeneBiosignal09, DBLP:conf/cec/Decraene09}.

\section{Introduction}
We examine the evolution of closed reaction networks to carry out pre-specified information processing tasks. This chapter is composed of two main sections:
\begin{enumerate}
	\item We first describe an experiment in which we evolve closed reaction networks to perform a signal-amplification function. To drive the evolution of the closed reaction networks to achieve a specified task, we introduce and discuss a novel cellular division criterion.
	\item We then examine the evolution of crosstalking closed reaction networks capable of multitasking. We aim at demonstrating the potential constructive role of crosstalk in enabling the evolution and realisation of closed reaction networks of higher complexity.
\end{enumerate}
Both investigations are assisted with the parallel and cellular model of the \MCS~ which was presented and evaluated in Section \ref{sec:celmod}.


\section{Evolving a signal amplification closed reaction network}
\label{sec:EvolvingSignalProcessing}
We introduce a novel cellular division criterion which aims at driving the evolution of closed reaction networks towards the achievement of a pre-specified task. We then summarise preliminary experiments in which self-replication reactions occur. Following this, we describe an experiment in which collectively autocatalytic reaction networks (where no self-replication reactions may occur) are evolved to carry out a simple target information processing task.

\subsection{Introduction}
\label{subsec:preliExpEvo}
In the experiments conducted with the cellular model of the \MCS~in Section \ref{sec:celmod}, a cell would divide only when $n_{max}$ molecules have been produced (i.e., when the cell is full). These cellular divisions occurred regardless of the nature of the molecular species.

In order to evolve closed reaction networks to carry out pre-specified information processing tasks, we propose to modify the conditions triggering the cellular divisions. The latter determine the implicit fitness of the cellular species. In contrast to fitness functions \textit{explicitly} devised in top-down evolutionary approaches, implicit fitness functions do not directly specify the genotype/phenotype mapping of the candidate species. Here, the cellular agents determine, by themselves, their own actions and ultimately their fitness with regards to the realisation of the target task. Defining new cellular division criteria allows one to indicate the desired target tasks, but not the \textit{actual computations}, that the reaction networks have to perform.

In the remainder of this section, we present a series of experiments in which we utilise a new cellular division criterion devised as follows. A cell divides when the amount of a specific target molecular species, denoted by $s_T$, reaches $n_{target}$. The cellular reproduction rate now depends on the molecular growth rate of $s_T$. The ability of the closed reaction network to both promote the growth of $s_T$ and maintain closure defines the cell's fitness. The pre-specified task assigned to these closed reaction networks is to amplify the ``signal'' $s_T$. Similar \textit{in-vivo} experiments were conducted to maximise the production of target molecular species which had for effect to increase the growth rate of the \textit{E.coli} bacterium \citep{palsson2006systems}. We may also interpret $s_T$ as a necessary molecular species (e.g., a membrane species) to allow the cellular division to occur.

Using the above cellular division criterion, we conducted a series of preliminary experiments where self-replication reactions (i.e., reactions in which the product is syntactically identical to the enzyme and substrate molecules) occur. These self-replication reactions were allowed in the experiments conducted in Chapters \ref{emerSelfMainClosedRN} and \ref{chap:evoStab}. In the current evolutionary simulations (described in Appendix \ref{app:evolSelfRep}), the cells were seeded with the self-replicase $s_{R_4}$ species, the latter was also designated as the target species $s_T$. Our results indicated that the networks would converge toward molecular organisations which are $n$-element hypercycles where $n$ would typically be lower than 5. As these networks of limited complexity were hypercycles, they were fragile and subsequently could not self-maintain under perturbations. These closed reaction networks would typically collapse to the single autocatalytic and target species $s_{R_4}$. 

The above phenomenon was also encouraged by the objective task which was devised to optimise the production of $s_{R_4}$, thus reaction networks containing only $s_{R_4}$ were already the optimal catalytic networks to produce the target species. As these experiments presented a limited interest with regards to the evolutionary growth of complexity, we propose to conduct a similar experiment where self-replication reactions may not occur.

This second series of experiment involving \textit{collectively autocatalytic} reaction networks employs a hand-designed seed reaction network which is presented in the next section.

\subsection{The seed reaction network}
\label{subsec:expEvol}
As self-replication reactions are now disabled, we cannot seed the cells with an ancestor autocatalytic molecular species as in previous experiments. We also demonstrated that the spontaneous emergence of closed reaction networks was unlikely to occur in the \MCS~ given a randomly generated population of molecular species. We thus propose to hand-design a minimalist \textit{collectively} autocatalytic reaction network which will be used as the seed network in the experiment. 

The construction and exploitation of this seed reaction network are clearly characteristic of ``top-down'' approaches (Section \ref{sec:topDown}). Although we advocate for minimising the use of top-down/engineered elements, this remains necessary as no alternative is currently available to explore the evolution of closed reaction networks using the \MCS.

We attempted to construct this seed collectively autocatalytic reaction network in a \textit{minimalist} manner with regards to the complexity at both the molecular (i.e., using simplest/shortest molecular species) and network level (i.e., involving the least number of both molecular species and reactions). Moreover, we include additional constraints to the realisation of this reaction network:
\begin{itemize}
	\item The reaction network is not, by design, the optimal catalytic network to realise the target task.
	\item Molecular species which can perpetually generate new species (such as elongator species) are not allowed. This filter was necessary as such molecular species trigger the spontaneous growth of the reaction network's topology/complexity by default. Although a natural limitation in the growth of the molecular length would be observed (Section \ref{subsec:cellModelExps}), these molecular species are filtered out as they do not facilitate and bias the current investigation on the evolutionary growth of complexity.
\end{itemize}
The construction of this reaction network is documented in Appendix \ref{app:consMiniCOll}. This informal investigation suggested that the reaction network (depicted in Fig.\ref{fig:seedNetworks}) is potentially the minimalist collectively autocatalytic  reaction network, satisfying the above requirements, that can be realised in the \MCS. The molecular species composing this reaction network are listed in Table \ref{tab:seedEvolNet}. 
 
\begin{figure}
	\centering
	\includegraphics[scale=0.5]{chapter7/seedNet.ps}
	\caption{Bipartite reaction network of the seed closed network.}
	\label{fig:seedNetworks}
\end{figure}
 
\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|}\hline
	Molecular species\\ \hline
	$s_1=*\triangledown0:\triangledown1$\\ 
	$s_2=*\triangledown0:\triangledown0$\\ 
	$s_3=*\triangledown1:\triangledown0$ \\
	$s_4=*\triangledown1:\triangledown1$ \\ \hline
	\end{tabular}
	\caption{Molecular species present in the seed reaction networks.}
	\label{tab:seedEvolNet}
\end{table}

Figure \ref{fig:seedReactionNetwork} presents the deterministic dynamics (neglecting mutation) of the seed closed reaction network when the initial amount of each molecular species is set to 10. This graph was obtained by solving the ODE system generated from the SBML specification of the seed reaction network using the SBML ODE solver \citep{1182079}. We also conducted a series of 100 simulation runs to measure in real time (i.e., in seconds) the growth of the different molecular species. In each of these independent simulation runs, a single reactor containing the seed reaction network was executed, with no mutations occurring, until 200 $s_1$ molecules were produced. Moreover as the nature (i.e., genotype and phenotype) of the molecular species $s_1$, $s_2$, $s_3$ and $s_4$ is quite similar (i.e., only the symbols 1 and 0 are permuted with each other in the different molecular species) the variations in reaction rates are negligible in these simulation runs. In the latter, the growth dynamics of the molecular species approximately match the deterministic dynamics depicted in Figure \ref{fig:seedReactionNetwork}. The end simulation time was averaged over the 100 runs. This averaged measurement was then employed to rescale the deterministic time course of the different molecular species' growth. This scaling operation is also conducted in the next related Figures \ref{fig:evolvedReactionNetwork} and \ref{fig:comparisonDynamics}.

\begin{figure}
	\centering
	\includegraphics[scale=0.9]{chapter7/graph1.eps}
	\caption{Deterministic dynamics of seed closed reaction network, the species $S1$ and $S4$ are overlapping (top line). $S2$ and $S3$ are similarly overlapping. Time was rescaled using experimental data.}
	\label{fig:seedReactionNetwork}
\end{figure}

In the remainder of this chapter, cellular species are classified by the specific reaction network contained in the cells. The above collectively autocatalytic reaction network or \textit{cellular species} is denoted by $c_0$ and employed as the seed cellular species in the evolutionary experiment presented in the next section. 

\subsection{Experiment}
An evolutionary experiment is presented in which $c_0$ is employed as the seed reaction network and evolved to promote the growth of $s_T=s_1$. A prediction regarding this experiment is proposed as follows. We may first consider the existence of non-closed reaction networks that are more efficient than $c_0$ at producing $s_T$. Nevertheless, in such networks, as the production of the molecules species required to generate $s_T$ molecules cannot be maintained through closure, these molecular species would thus deplete over time. These networks, being not organisationally closed, would therefore not be able to self-maintain over successive cellular reproductions. 

As a result, such non-closed reaction networks would possess a clear selective disadvantage against $c_0$. We may then predict that $c_0$ cannot be displaced by cellular species which contain non-closed reaction networks. In other words, if selective displacements occur, they would necessarily involve novel and organisationally closed reaction networks.

To test this prediction, an evolutionary experiment is conducted 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 $n_{max}=1.0\e{6}$.
	\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 $s_4$. 
	\item The target molecular species is $s_T=s_1$.
\end{itemize}

\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/exp1crossMolMutEvoRun.eps}
	}	\caption{Dynamics of dominant cellular species. A spline function was employed to approximate the different curves.}
	\label{fig:exp1crossMolMutEvoRun}
\end{figure}

During this experiment the average number of interactions per cell per hour was over $4.0\e{7}$. $1235$ different and unique reaction networks were generated due to molecular and cellular mutations. The latter refer to mutations occurring at the cell-level which may result from the stochastic nature of cellular divisions (i.e., some molecular species may not be transferred into the offspring cells, resulting in mutant cellular species). In Fig.\ref{fig:exp1crossMolMutEvoRun}, we can distinguish that three successive displacements occurred (Figure\ref{fig:displacementszoomed}). The reaction networks contained in these three consecutive cellular species are listed in Table \ref{tab:succnets}. 

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|c|}\hline
	$c_0$ & $c_1$ & $c_2$ & $c_3$ \\ \hline
	$s_1=*\triangledown0:\triangledown1$ & $s_1$ & $s_1$	& $s_1$ \\ 
	$s_2=*\triangledown0:\triangledown0$ & $s_2$ & $s_2$	& $s_2$ \\ 
	$s_3=*\triangledown1:\triangledown0$ & $s_5=*\triangledown \diamond : \triangledown0$ & $s_7=*\triangledown \diamond : \triangledown \triangle0$	& $s_9=*\triangledown \triangle : \triangledown0$ \\ 
	$s_4=*\triangledown1:\triangledown1$ &
$s_6=*\triangledown \diamond : \triangledown1$& $s_8=*\triangledown \diamond : \triangledown \triangle1$	& $s_{10}=*\triangledown \triangle : \triangledown1$ \\ \hline
	\end{tabular}
	\caption{Molecular species contained in successive dominant closed reaction networks denoted by $c_1$, $c_2$ and $c_3$.}
	\label{tab:succnets}
\end{table}

\begin{figure}
	\centering
	\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter7/exp0a.eps}}}
	\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter7/exp0b.eps}}}
	\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter7/exp0c.eps}}}
	\caption{Successive displacements that occurred in simulation run 1.}
	\label{fig:displacementszoomed}
\end{figure}

We first observe that, as predicted earlier, the successive dominant closed reaction networks successfully maintained closure.  We first investigate the displacement that occurred between $c_0$ and $c_1$. We examine the reaction network contained in $c_1$ which is depicted in Figure \ref{fig:networks}. The dynamics of the different molecular species' growth are shown in Fig.\ref{fig:evolvedReactionNetwork}.

\begin{figure}
	\centering
	\includegraphics[scale=0.5]{chapter7/evolvedNet.ps}
	\caption{Evolved closed reaction network $c_1$ promoting growth of molecular species $s_1$.}
	\label{fig:networks}
\end{figure}

In $c_1$, we note that both species $s_3$ and $s_4$ have been replaced by the new molecular species $s_5$ and $s_6$. These new molecular species increased the total number of possible reactions from 6 to 9 (see Fig.\ref{fig:networks}), suggesting a higher overall catalytic activity. We compare the dynamics of this evolved closed reaction network against the seed closed reaction network, see Figure \ref{fig:evolvedReactionNetwork}.

\begin{figure}
	\centering
	\includegraphics[scale=0.9]{chapter7/graph2.eps}
	\caption{Dynamics of the evolved closed reaction network, the species $s_5$ and $s_6$ are overlapping (middle line), $s_1$ is the top line and $s_2$ is the bottom line. Each molecular species amount is initialized to 10.}
	\label{fig:evolvedReactionNetwork}
\end{figure}

In Figure \ref{fig:evolvedReactionNetwork}, it can be seen that $s_1$ reaches the division threshold at $t\approx0.155$ whereas in the seed closed reaction network $s_1$ attains this threshold at $t\approx0.280$. By producing the target species $s_1$ at a faster rate, the evolved network gained a selective advantage over the seed network. The emergence of the molecular species $s_5$ and $s_6$ had the effect of promoting the growth of $s_1$ whilst maintaining closure. The network closure properties evolved and permitted the network to promote the growth of species $s_1$.

\begin{figure}
	\centering
	\includegraphics[scale=0.9]{chapter7/graph4.eps}
	\caption{Comparison of $s_1$ species' growth in seed and evolved reaction networks. $s_1$ and $s_1'$ depicts the dynamics of $s_1$ with initial molecular amount 10, using the seed and evolved reaction networks respectively.}
	\label{fig:comparisonDynamics}
\end{figure}

Finally Figure \ref{fig:comparisonDynamics} compares the different growth dynamics of $s_1$ using the different networks and a common initial number of molecules. This comparison highlights the improvement in the networks' fitness (i.e., hastened the production of molecular species $s_T$).


We now examine the displacements that occurred between $c_1$, $c_2$ and $c_3$. Although the molecular species contained in $c_1$, $c_2$ and $c_3$ are different from a genotypic point of view, their phenotypes are similar. The symbols $\diamond$ and $\triangle$ act in the same manner when occurring in the condition statement. The symbols $\triangle$ occurring in the action statement of $c_2$ are ignored. Moreover, the genotypic differences yield negligible effects upon the reaction rates ($\triangle$ and $\diamond$ are computationally equivalent and the \textit{ignored} symbol $\triangle$ occurring in $c_2$ has little computational impact). As a result the species contained in $c_1$, $c_2$ and $c_3$ are phenotypically equivalent. These cellular species are thus likely to possess an equivalent fitness.

This common level of fitness suggests that the displacements that occurred between the cellular species $c_1$, $c_2$ and $c_3$ are due to drifts. However, the \textit{abrupt} transitions between these cellular lineages are symptomatic of selective displacements. Whereas displacements due to drift dynamics would take place at a slower speed. All displacements in this simulation run took between 5 and 10 seconds to occur. According to Figure \ref{fig:evolvedReactionNetwork}, a cell may reproduce at least 6 times per second. At the population level ($\times$ 30 cells), this would increase to 180 cellular reproductions per second\footnote{In real experimental conditions, the cellular reproduction rate would usually be higher than 180 reproductions per second as the initial molecular amounts would differ from those employed in Fig.\ref{fig:evolvedReactionNetwork}. Through successive cellular reproductions, the molecular amount distribution may vary mechanically and lead to a faster reproduction speed, being typically twice faster with an initial molecular amount of 100 $s_1$ and fewer molecules $s_2$, $s_5$ and $s_6$.}. If we consider the approximate number of cellular reproductions that may occur in 5 to 10 seconds (between 800 and 1800) and the relatively small cellular population size employed (30 cells), then it may be envisaged that drift dynamics, as proposed by the neutral theory of molecular evolution formulated by \cite{kimura1983neutral}, only could result in those displacements (which would then appear to be abruptly occurring due to the employed timescale) between $c_1$, $c_2$ and $c_3$. 

Also note in Figure \ref{fig:displacementszoomed} that these displacement are associated with a relative increase in the cellular species diversity. This suggests that other cellular species, which are potentially mutants of the pair of dominant cellular species, may have contributed to these displacements. 

Another possible explanation of those displacements, is that $c_2$ and $c_3$ may  have incrementally increased their capacity to resist against potentially disruptive mutation effects (mutations occurring at both the cellular and molecular level). This hypothesis, where the networks have improved their robustness \citep{wagner2005robustness}, could be tested in future work where each cellular lineage would be isolated and examined in details when subjected to mutational perturbations.

Finally with regards to the evolutionary of growth complexity, we note that in this evolutionary experiment, the number of molecular reactions was increased from 6 to 9 when comparing the seed and evolved reaction networks. Nevertheless the complexity of the molecular species remains equivalent with an average string length of 6. Moreover both the seed and evolved reaction networks contain the same number of molecular species. Although the seed reaction network was successfully evolved and optimised to achieve the pre-specified task, we did not observe a significant growth of complexity in this experiment. 

10 additional repetitions of the above experiment were conducted and are described in Appendix \ref{app:EvolNoSelfRep}. In four of these runs, we observed the emergence and domination of either $c_1$ or $c_3$. In four other runs, the emergence of reaction networks containing the molecular species of either $c_1$ or $c_3$ in addition to some other 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.

\subsection{Conclusion}
We introduced a novel cellular division criterion as a means to drive the evolution of closed reaction networks to accomplish a pre-specified task. The latter was devised to promote the growth of a target molecular species (i.e., signal amplification). Preliminary experiments, where self-replication reactions were enabled, showed evolutionary dynamics of limited interest with regards to the evolutionary growth of complexity. We then proposed to conduct further experiments in which we disabled self-replication reactions. A hand-designed minimalist collectively autocatalytic reaction network was presented and used as the seed/ancestor network. A series of evolutionary experiments were conducted in which we identified the common emergence of a ``fitter'' reaction network. We examined this evolved reaction network and demonstrated its ability to produce the target species at a faster rate whilst maintaining closure. Although the seed closed reaction reaction was successfully evolved and optimised to achieve the pre-specified task, a significant evolutionary growth of complexity was not observed in this experiment. In the next section, we extend this work and explore further avenues for evolving closed reaction network of higher complexity.

\section{Crosstalk and the evolution of complexity}
In Section \ref{sec:EvolvingSignalProcessing} , we successfully evolved a simple signal processing ability in a closed reaction network using the cellular and parallel implementation of the \MCS. This resulted in the optimisation of a \textit{minimalist} closed Cellular Information Processing Network capable of a distinct signal-processing function. 

Nevertheless, the previous experiment failed at exhibiting a clear evolutionary growth of complexity. In this section, we extend this preliminary work on the evolution of closed reaction networks and intend to evolve networks of higher complexity. 

To assist this research on the evolutionary growth of complexity, we examine a phenomenon occurring in real biochemical networks: crosstalk. Crosstalk phenomena arise very naturally in such networks due to the fact that molecules from different signalling pathways may share the same physical reaction space (the cell). Depending on the relative specificities of the reactions there is then an automatic potential for any given molecular species to contribute to signal levels in multiple pathways.

In Section \ref{sec:benefits}, we proposed a potential benefit of crosstalk in chemical networks:  Crosstalk is a key mechanism in enabling incremental evolutionary search for more complex closed reaction networks.

We argue that the above benefit may be achieved through crosstalk by allowing distinct closed reaction networks to cooperate with each other when occurring in the \textit{same} reaction space. We thus propose that crosstalk may enable the merging of distinct crosstalking closed reaction networks to form a new closed reaction network of higher complexity. 

This work is thus naturally related to the symbiogenesis theory which was originally postulated by \cite{mereschkowsky1910tzp}. According to this theory, separate organisms may merge with each other to form new organisms of higher complexity \citep{margulis1981sce,margulis2002agt}. Barricelli was the first to conduct computer-based experiments in which symbiogenetic organisms were artificially evolved \citep{barricelli1957sep,barricelli1963nte}.

More specifically, the work presented in this chapter is inspired by seminal related experiments conducted in Alchemy \citep{fontana1994aft}. Based on this preliminary work, we hypothesise that crosstalk enables the cooperation and subsequently, the evolutionary growth of complexity of biochemical networks. We develop further this hypothesis using the \MCS. 

\subsection{Introduction}
This investigation on crosstalk and complexity was inspired by specific experiments carried out by \citep{fontana1994aft} with the Alchemy system. When mixing two collectively autocatalytic reaction networks (which were obtained from previous independent experiments) in the same reaction space, two outcomes could be observed according to the level of interaction (i.e., crosstalk) between the two reaction networks:
\begin{enumerate}
		\item If no molecular interaction (i.e., no crosstalk) exist between the two networks then one would displace the other network. 
	\item If, on the contrary, some molecular interactions occur between the two crosstalking networks then a ``meta'' hierarchical closed reaction network emerges which contains and maintains both seed closed reaction networks.
\end{enumerate}
These observations suggest that crosstalk may be responsible for the emergence of molecular organisations of higher complexity. 
To develop further this hypothesis we extend this seminal investigation using the \MCS. However a number of key differences exist between Alchemy and the \MCS:

\begin{itemize}
	\item Alchemy is based on the $\lambda-$calculus formalism, whereas the \MCS~employs the broadcast language. Although both agent-based ACs employ term-rewriting systems, the specification of molecular species and reactions (binding rules and enzymatic capabilities) varies greatly. For example, there is only a single level of enzymatic/computational transformation that can be defined in molecular reactions within the \MCS. Whereas in Alchemy, the analogous maximum number of reduction steps was set to 10000.

	\item We define mutation operators at both the molecular and cellular level. No evolutionary operators were specified in Alchemy. In Alchemy the molecular diversity resulted from the initial randomly generated molecular population and subsequent catalytic molecular reactions. The \MCS~introduces a greater space exploration of molecular species by implementing molecular mutations. This molecular and cellular variance allows for \textit{evolutionary} dynamics to occur in the \MCS.

	\item Similarly to Alchemy, molecular species may interact and compete with each others. In addition to this first level of selection we introduced a higher level of selection: Molecules are contained in multiple reactors (i.e., cells) which are capable of competing with each others through cellular divisions. As demonstrated in Chapter \ref{chap:evoStab} this second level of selection was necessary to allow closed reaction networks to self-maintain in the \MCS. As no mutations could occur in Alchemy, no evolutionary degenerations were observed in this system where only a single level selection was implemented.
	
		\item In our approach we evolve closed reaction networks to carry out pre-specified tasks. In Alchemy, the reaction networks were not driven to perform any target functions. Introducing a target task in the \MCS~affects the implicit fitness function devised in this system. In addition to performing self-maintenance (as in Alchemy), closed reaction networks in the \MCS~have to carry out a pre-specified task. Both the ability to perform self-maintenance and the pre-specified task affect the fitness of a given closed reaction network in the \MCS. The fitness landscape in the \MCS~ is thus multi-dimensional where the dimensions are:
		
\begin{enumerate}
	\item The cellular reproduction rate which depends on the ability of the cell to achieve the target task.
	\item The ability to maintain closure which is required to control potentially disruptive mutational effects. Note that closure properties may themselves evolve to better control evolutionary degenerations.
\end{enumerate}
 
\end{itemize}

We now report three series of experiments addressing crosstalk and the evolutionary growth of complexity in closed reaction networks. Although these experiments may be directly related to level 2 organisation experiments conducted in Alchemy, they partially diverge from the Alchemy experiments as molecular and cellular mutations occur and introduce the potential for evolutionary dynamics. The first and second series of experiments involve non-crosstalking and crosstalking closed reaction networks respectively. In the first two experiments, only cell-level mutation applies\footnote{Cell-level mutations occur during cellular reproductions where the stochastic distribution of molecules may result in mutant cells. Cell-level mutations are an inherent feature of the compartmentalised \MCS~and cannot be disabled.}. Therefore the diversity of molecular species in these two experiments is limited. The only molecular species that may appear in the simulation runs, involving non-crosstalking networks, are the initial molecular species contained in the seed networks. Molecular-level mutations are disabled to facilitate the comparison with the Alchemy system in which no mutations were implemented. $\alpha$ being the total number of molecular species that may appear and $\nu$ the number of initial molecular species, we have $\alpha=\nu$. The crosstalking networks based experiment includes further molecular species which may result from novel reactions occurring between the crosstalking molecular species, therefore $\alpha\leq \frac{\nu^2}{2}$. Cell-level mutations may produce mutant cellular species in which only the specific assortment of molecular species (given this limited set of molecular species) may vary. The third experiment examines systems of crosstalking closed reaction networks where both cellular \textit{and} molecular mutations occur. The potential diversity of molecular species in this final experimental series is thus significantly increased to $\alpha=\sum_{L=1}^{BD_{Lmax}}|\lambda|^L$.

\subsection{The seed reaction networks}
In the following experiments, no self-replication reactions (i.e., reactions in which the product is syntactically identical to the enzyme and substrate molecules) are allowed (as was the case in analogous Alchemy experiences). As briefly discussed in Section \ref{subsec:preliExpEvo}, enabling self-replication would prevent any relatively complex molecular organisations from emerging. However we showed that the spontaneous emergence of closed reaction networks was unlikely to occur given a randomly generated population of molecular species. Thus, similarly to Section \ref{sec:EvolvingSignalProcessing}, we employ seed closed reaction networks to initialise the molecular populations in the experiments.

We define the different reaction networks $X,Y$ and $Z$ which are utilised throughout these series of experiments, see Table \ref{tab:nets}. 
\begin{table}[htbp]
	\centering
	\begin{tabular}{|c|c|c|}\hline
	$X$ & $Y$ & $Z$ \\ \hline
	$s_1=*\triangledown00:\triangledown01$ & $s_5=*\triangledown10:\triangledown11$	& $s_9=*\triangledown10:\triangledown00\ \ \ \ $ \\ 
	$s_2=*\triangledown00:\triangledown00$ & $s_6=*\triangledown10:\triangledown10$ & 	$s_{10}=\triangledown1*\triangledown00:\triangledown10$\\
	$s_3=*\triangledown0\diamond:\triangledown00$ & $s_7=*\triangledown1\diamond:\triangledown10$ & $s_{11}=*\triangledown10:\triangledown10\ \ \ \ \ $ \\
	$s_4=*\triangledown0\diamond:\triangledown01$ & $s_8=*\triangledown1\diamond:\triangledown11$ & $s_{12}=\triangledown1*\triangledown00:\triangledown00$\\ \hline
	\end{tabular}
	\caption{Molecular species contained in seed closed reaction networks $X$, $Y$ and $Z$}
	\label{tab:nets}
\end{table}

No molecular species from $X$ interact with any molecular species from $Y$ and \textit{vice versa}. $X$ and $Y$ are declared as \textit{non-crosstalking} reaction networks. The species $s_1, s_2, s_3$ and $s_4$ from $X$ may interact with species $s_9$ and $s_{12}$ from $Z$, whereas species $s_{10}$ and $s_{12}$ may interact with $s_2$ and $s_3$ from $X$. $X$ and $Z$ are declared as \textit{crosstalking} reaction networks. 

\begin{figure}[htbp]
	\centering
	\includegraphics[scale=0.45]{chapter7/RNX.ps}
	\includegraphics[scale=0.45]{chapter7/RNZ.ps}
	\caption{Bipartite reaction network graphs of networks $X/Y$ and $Z$. The topology of molecular interactions of $X$ and $Y$ are equivalent, e.g., the reaction $R4$ would involve the molecular species $s_2$ and $s_3$ in $X$, whereas $R4$ would involve the molecular species $s_6$ and $s_7$ in $Y$.}
	\label{fig:RNX}
\end{figure}

$X,Y$ and $Z$ were obtained from previous experiments in which they were evolved to maximise the production of molecular species $s_1, s_5$ and $s_9$ respectively. Fig. \ref{fig:RNX} depicts the bipartite reaction network graphs of networks $X,Y$ and $Z$. Note that $X$ and $Y$ possess the same network topology. The number of instances of a given molecular species $s_j \in S$ contained in a cell $c_i \in C$ is denoted as $n^i_j$. All simulations are run for a pre-defined amount of time $t_{max}=3600$ (seconds in real time) using thirty AMD Opteron 270 (2.0 GHZ) CPUs. 

\subsection{Non-crosstalking networks}
In this first series of experiment, we investigate the dynamics of a system in which the non-crosstalking closed reaction networks $X$ and $Y$ are used. 30 concurrent cells are employed and initialised with 10 molecules from each species from both $X$ and $Y$. 

As previously mentioned, $X$ and $Y$ were evolved to optimise the production of species $s_1$ and $s_5$ respectively. We devise a new cellular division criterion which accounts for both molecular species. The motivation to this criterion is to encourage the maintenance of both networks $X$ and $Y$.  Ultimately we aim at evolving/obtaining a more complex network capable of ``multitasking'', i.e., a network which is able to carry out the pre-specified tasks of both $X$ and $Y$. Therefore, a cell $c_i$ divides if $n^i_1\geq200 \wedge n^i_5\geq200$. In this experiment, only mutations at the cellular level occur (i.e., mutations at the molecular level are excluded at present and $\alpha=8$).

\begin{figure}[htbp]
	\centering
		\subfigure{\rotatebox{-90}{	\includegraphics[scale=0.267]{chapter7/noCrossReactors}}}
		\subfigure{\rotatebox{-90}{\includegraphics[scale=0.278]{chapter7/noCrossReactorsCellRep}}}
		\caption{Growth of molecular species $s_1$ and $s_5$ (left) and number of cellular reproductions (right). Both graphics depict dynamics at the cellular population level in an example simulation run. }
	\label{fig:popNoCrosstalk0}
\end{figure}

Fig. \ref{fig:popNoCrosstalk0} depicts the growth of $s_1$ and $s_5$ and number of cellular reproductions at the cell population level in a single simulation run. Five additional simulation runs were conducted to explore any significantly differing dynamics. The dynamics described here were found to be exhibited in all of these runs. 

The number of cellular reproductions provides an approximate indication of the cells' fitness, i.e., an absence of cellular reproduction would suggest that the cells are not producing both $s_1$ and $s_5$ sufficiently to trigger their division. In such cases the cells would thus possess a relatively low fitness.

In Fig.\ref{fig:popNoCrosstalk0}, we first observe an early phase where both the number of $s_1$ and $s_5$ molecules vary between $10000$ and $11000$. Moreover this phase is associated with recurrent cellular reproduction events.

At $t\approx32$ we note that the number of $s_5$ is now rapidly increasing, reaching up to $1.0\e{5}$ when $t\approx 80$ whereas $s_1$ increases up to $5.0\e{4}$. From $t>400$, no further cellular reproductions  occur.

Throughout this simulation run, $12$ different and unique reaction networks were generated due to cell-level mutations. The growth dynamics of these cellular species are depicted in Fig.\ref{fig:popNoCrosstalk}. 
\begin{figure}[htbp]
	\centering
\subfigure{\rotatebox{-90}{\includegraphics[scale=0.29]{chapter7/noCrossSpeciesDom}}}
\subfigure{\rotatebox{-90}{\includegraphics[scale=0.29]{chapter7/noCrossSat}}}
		\caption{Growth dynamics of the 12 cellular species (left) and saturated cells (right). The two dominant cellular species, the red and blue curves, are denoted by $c_1$ and $c_2$ respectively. $c_1$ includes the molecular species from both $X$ and $Y$, $c_1=X+Y$. Whereas $c_2$ is a mutant cellular species of $c_1$ which does not include the molecular species $s_3$, $c_2=\{s_1,s_2,s_4\}+Y$. All reaction networks contained in remaining cellular species are subsets of $X+Y$. A cell $c_i$ is considered as ``saturated'' if $n_1^i>2000\vee n_5^i>2000$. }
	\label{fig:popNoCrosstalk}
\end{figure}

In this run, we distinguish the following chain of events at the cell population level (Fig.\ref{fig:popNoCrosstalk}):

\begin{itemize}
	\item We note that the early phase $0 \leq t \leq 32$ mentioned earlier, where repeated cellular reproductions are observed, is driven only by the cellular species $c_1$ (which contains both closed reaction networks $X$ and $Y$). From $t\geq 32$, we distinguish the emergence of various mutant cellular species which contain reaction networks being subsets of $X+Y$. This emergence of mutant species is associated with the sudden decline in the number of cellular reproductions.

	\item Moreover we observe that the number of ``saturated'' cells increases rapidly when $t\approx32$ which correlates with previous observations reported in Fig. \ref{fig:popNoCrosstalk0} where the number of $s_1$ and $s_5$ molecules starts to increase rapidly.

	\item This cellular saturation suggests that although some $c_1$ cells are still present (which are  capable of producing both molecular species $s_1$ and $s_5$), these cells are overpopulated/saturated with either species $s_1$ or $s_5$. As a result, this cellular saturation and the ``survival of the common'' dynamics (Section \ref{sec:introEmerSelfMain}) occurring here, cause the production rate of $s_1$ and $s_5$ to become highly \textit{asymmetric} (as depicted in Fig.\ref{fig:noCrossReactor1}).

	\item When $200<t<400$ we note that the number of saturated cells decreases, resulting from the  
cellular reproduction events occurring sporadically during this period (Fig.\ref{fig:popNoCrosstalk0}). Nevertheless, from $t>600$ the number of saturated cells starts to increase again, indicating that either the species $s_1$ or $s_5$ are being produced despite no further cellular reproductions occur.
\end{itemize}

\begin{figure}[htbp]
	\centering
	\rotatebox{-90}{\includegraphics[scale=0.40]{chapter7/noCrossReactor1}}
		\caption{Growth of molecular species $s_1$ and $s_5$ in an example cell containing the network $c_1$ when $24\leq t \leq 80$. }
	\label{fig:noCrossReactor1}
\end{figure}

A complementary investigation revealed that some of the non-saturated cells contained reaction networks in which no successful reaction could occur. These reaction networks were essentially composed of target species $s_1$ and $s_5$ only. Therefore no chemical interaction could occur within these reaction networks which resulted from cell-level mutations (as defined in Section \ref{subsec:cellModel}). 

During the successive cellular reproduction events, the numbers of molecules $s_1$ and $s_5$ contained in a given cell increasingly deviated from each other until one of these two species started to take over the cell. As a result the production rate of $s_1$ and $s_5$ molecules in the reaction networks (including closed ones) was highly asymmetric. In such cells saturated by $s_1$ or $s_5$, further divisions (if any) resulted in offspring cells which were likely to contain a majority of, or only, $s_1$ or $s_5$ molecules. 

Such resulting mutant cellular species did not contain all molecular species necessary to maintain closure for both $X$ and $Y$. As the closure of $X$ and $Y$ was not maintained in these cells, this consequently penalised the production of further molecules $s_1$ or $s_5$. As a result, the cellular division threshold became increasingly more difficult and ultimately impossible to reach in these cellular lineages.

Although based on a significantly different Artificial Chemistry, these experiments essentially exhibited the same dynamics as in related experiments conducted in Alchemy: the different reactors were rapidly and quasi-deterministically dominated by one of the seed closed reaction networks. In other words, when two non-crosstalking closed reaction networks are mixed together, one displaces the other one.

This above experiment also relates to results obtained with the stochastic corrector model proposed by \cite{szathmary1987gse}. In Szathmary and Demeter's cellular model, the survival of the cells depended on the concentration of two distinct self-replicases having differing growth rates. The difference in growth rate was due to the ability of one the self-replicases to parasite the other one. Cells in which the concentration of the replicases deviated too importantly from each other were selected against. 

Although major differences exist between our model and Szathmary and Demeter's one (e.g., we do not employ self-replicase species but collectively autocatalytic reaction networks each of which possesses an initial common growth rate), these models share similarities where the survival of the cells depends on the concentration of two distinct molecular species. 

Using the stochastic corrector model, Szathmary and Demeter demonstrated that multi-level selection in such a cellular model was capable of controlling parasitism. However the regulation of the degenerative outcomes due to stochastic variations, occurring during the transmission of molecular species into offspring cells, was possible only when the number of self-replicases was small. This limitation may be involved in the current experiment where: 1) 8 distinct molecular species were necessary to maintain closure. 2) The cellular division threshold required 200 molecular instances of both $s_1$ and $s_5$. This may subsequently have affected the survival of the cells over successive cellular divisions.

Therefore, we may argue that given two distinct non-crosstalking collectively autocatalytic reaction networks containing fewer molecular species and a cellular division threshold requiring less molecules, the current cellular model would theoretically be able to select against cells in which significant deviations occur between both target molecular species. As a result, this model may potentially regulate the degenerative effects due to stochastic variations occurring during successive cellular divisions. However, this hypothesis could not be applicable in Alchemy since the latter lacks the multi-level selectional regime specific to the \MCS~and stochastic corrector model.

\subsection{Crosstalking networks}
\label{secCrossCellLevelMut}
We investigate the effects of crosstalking closed reaction networks upon the system's dynamics. In this experiment, the cells are initialised with molecular species from the crosstalking reaction networks $X$ and $Z$. A cell $c_i$ divides if $n^i_1\geq200 \wedge n^i_9\geq200$. The number of molecular species that may appear in the simulation runs is $\alpha\leq \frac{8^2}{2}=32$. Any other experimental conditions are identical to those described in the previous section.

Our results showed that the interactions between molecular species from $X$ and $Z$ led to the production of new molecular species $s_{13}, s_{14}, s_{15}$ and $s_{16}$ (which may engage in novel reactions with existing molecular species). This new cellular species, denoted as $c_1$, contains both networks $X$ and $Z$, and presents an increased level of complexity (the reaction network now contains 12 molecular species and 55 reactions, see Fig. \ref{fig:RNR1}). Moreover these $c_1$ cells were able to self-maintain for a sustained period of time ($\approx$ 400 seconds). This first observation also applied in analogous experiments conducted in Alchemy, in which a meta-reaction network emerged and had the ability to self-sustain and maintain both seed closed reaction networks.

However, an additional phenomenon occurred which was not observed in the Alchemy system. We distinguish a selective displacement event between $c_1$ and a new cellular species. In this simulation run, a level of diversity (see Fig.~\ref{fig:disC1_c2}) was maintained due to cell-level mutations (27 unique reaction networks appeared during this run), a feature specific to the \MCS. At $t\approx380$ we note the emergence of a new cellular species, denoted as $c_2$ and shown in Fig. \ref{fig:RNR2}, which later displaced $c_1$ at $t\approx 400$. During this displacement phase, we note that the cell diversity also increased suggesting that other cellular species may also have contributed to the displacement of $c_1$ cells.

\begin{figure}[htpb]
	\centering
		\includegraphics[scale=0.1]{chapter7/crosstalkNoMutBegin.ps}
		\caption{Reaction network of cellular species $c_1$ which contains all molecular species from networks $X$ and $Z$ in addition to new molecular species $s_{13}, s_{14}, s_{15}$ and $s_{16}$.}
	\label{fig:RNR1}
\end{figure}

\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/limitedCrossEvoRun.eps}
	}	\caption{Cellular species displacement between $c_1$ and $c_2$. The cellular species diversity refers to the number of different (from a qualitative - topological point of view) reaction networks present at a given timestep.}
	\label{fig:disC1_c2}
\end{figure}

In Fig. \ref{fig:cmpPerfXvsY} we compare the fitness of reaction networks $c_1$ and $c_2$. The fitness of a given cell $c_i$ is reciprocal of time, where both the condition $n^i_1\geq200 \wedge n^i_9\geq200$ and the necessary time $t_{c_i}$ to satisfy this condition are accounted for. With the present parallel system, as the speed of production of species $s_1$ and $s_9$ increases (augmenting the reproduction rate of the cell), the fitness of the cell increases accordingly.

\begin{figure}[htpb]
	\centering
		\includegraphics[scale=0.1]{chapter7/RNRendNoMut.ps}
		\caption{Reaction network present in cellular species $c_2$ in which molecular species $s_2, s_{10}$ and $s_{12}$ from $c_1$ are absent.}
	\label{fig:RNR2}
\end{figure}

\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/limitedCrossXvsY.eps}
	}	\caption{Comparison of molecular growth of species $s_1$ and $s_9$ in $c_1$ and $c_2$. Similarly to Fig.\ref{fig:seedReactionNetwork}, the time courses of the species' growth were rescaled using experimental data.}
	\label{fig:cmpPerfXvsY}
\end{figure}

We note in Fig.~\ref{fig:cmpPerfXvsY} that $c_2$ cells produce molecular species $s_1$ and $s_9$ at a faster rate than $c_1$ cells (i.e., $t_{c_2}<t_{c_1}$). According to our definition of fitness, $c_2$ cells are fitter than $c_1$ cells. Both the evolved qualitative properties of $c_2$ and the exploitation of crosstalk led to the maximisation of the production of molecular species $s_1$ and $s_9$. We also identify this increase in fitness in Fig. \ref{fig:crossCellMutDisp}, in which we distinguish a net increase in the overall cellular reproduction rate following the displacement event. The multitasking $c_2$ cells were able to self-maintain throughout the entire simulation while cell-level mutations continued to occur. This clear selective displacement event occurred as a direct consequence of both:
\begin{itemize}
	\item  The cell-level mutations which increased cellular diversity.
	\item  The more elaborate implicit fitness function which affected the fitness landscape. This multi-dimensional fitness landscape allowed for an incremental evolutionary improvement to occur.
\end{itemize}



The above properties are specific to the \MCS~and were not present in Alchemy. As a result, comparable evolutionary dynamics described in this section have not previously been reported using Fontana and Buss's system.

\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/crossCellMutDisp.eps}
	}	\caption{Crosstalking networks with no molecular mutation - Dynamics of cellular reproductions and diversity. A spline function was employed to approximate the number of cellular reproductions and cellular species diversity curves.}
	\label{fig:crossCellMutDisp}
\end{figure}

We also note that the complexity (i.e., the number of molecular species and reactions) actually decreased in $c_2$ (9 species and 32 reactions) when compared with $c_1$ (12 species and 55 reactions). This observation suggests that the increased level of complexity of $c_1$ did not provide any beneficial features, but on the contrary, reduced the speed of reproduction (i.e., fitness) of $c_1$. The lower level of complexity of $c_2$ led to a lower computational cost (and consequently a faster reproduction speed) whilst maintaining closure. This ultimately provided $c_2$ with a selective advantage over $c_1$.

Finally, as some molecular species from $X$ and $Z$ have been removed in $c_2$, the latter was thus no longer maintaining the seed original reaction networks $X$ and $Z$. As we cannot identify $X$ and $Z$ in $c_2$, a natural open-question follows: Does $c_2$ still contain crosstalking closed reaction networks? Such a question could be addressed if we employ an adequate formalism and identify distinct closed reaction networks as subsystems in $c_2$. This issue is nevertheless beyond the scope of this investigation but the reader may find further details in \cite{wolkenhauer2007acm}, where an abstract cell model is proposed to investigate such issues.
		


\subsection{Crosstalking networks with molecular mutations}
\label{subsecCrossMolMut}
We finally examine the effects of molecular mutation in systems where the crosstalking reaction networks $X$ and $Z$ are used. Molecular mutations introduce a higher level of both molecular and cellular diversity, which may potentially lead to more complex molecular organisations and richer evolutionary dynamics. Molecular mutations occur with the following probability: $p_{sym}=5.0\e{-5}$. Complementary experimental parameters are identical to those presented in Section \ref{secCrossCellLevelMut}. Using these conditions, we conduct an experiment in which we identify the following distinctive behaviour. 

We first note in Fig. \ref{fig:crossMolMutDisp} that the dynamics of the cellular reproduction rate shares some similarities with analogous dynamics shown in the previous experiment (Fig. \ref{fig:crossCellMutDisp}). Indeed we observe a common early phase where the cellular activity is  approximately equal to 16 cellular reproductions per second, then at $t\approx250$ the cellular reproduction rate starts to increase. This common early dynamic is driven by the same cellular species $c_1$ (i.e., the meta-reaction network containing both seed reaction networks $X$ and $Z$) which was also able to self-maintain for a period of time. However due to molecular mutations occurring, a significant difference exists in the cellular species diversity. Here a higher average level of cellular diversity per second is observed, being roughly 20 times higher than in the previous experiment, and is maintained throughout the evolutionary simulation. During this run, 37863 unique reaction networks appeared.
\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/crossMolMutDisp.eps}
	}	\caption{Crosstalking networks with molecular mutations - Dynamics of cellular reproduction rate and diversity when molecular mutations occur. A spline function was employed to approximate the cellular reproductions and cellular species diversity curves}
	\label{fig:crossMolMutDisp}
\end{figure}

In Fig. \ref{fig:crossMolMutEvoRun}, we note that two cellular species displacements occurred at $t\approx475$ and $t\approx2500$. The cellular species $c_1$ is similarly displaced by a mutant cellular species, denoted by $c_2$ which contains a reaction network that is phenotypically equivalent to $c_2$ described in previous experiment (Section \ref{secCrossCellLevelMut}). The third emergent dominant cellular species is denoted by $c_3$. The cellular species $c_3$ shares an equivalent level of complexity (containing 13 molecular species and 66 reactions) with $c_1$ cells.

In addition, it can be observed that the cellular species' subpopulation rarely exceeded half of the total population. The dominating cellular species have not once succeeded at \textit{fully} displacing the other species for a sustained period of time.

\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/colorCrossMolMutEvoRun.eps}
	}	\caption{Dynamics of the major cellular species. Only the cellular species which invaded, at least once, one third of the cellular population during the simulation run are shown (14 cellular species are plotted). }
	\label{fig:crossMolMutEvoRun}
\end{figure}


\begin{figure}[htpb]
	\centering
	\rotatebox{-90}{
		\includegraphics[scale=0.45]{chapter7/limitedCrossXYvsZ.eps}
	}	\caption{Comparison of molecular growth of species $s_1$ and $s_9$ in $c_1, c_2$ and $c_3$.}
	\label{fig:cmpPerfXYvsZ}
\end{figure}

In typical evolutionary simulations it is usually expected to observe incremental improvements in the species' fitness. However when comparing the fitness of the different successive dominant cellular species (Fig. \ref{fig:cmpPerfXYvsZ}), we note that this incremental evolutionary improvement did not occur according to our definition of fitness. As $t_{c_2}<t_{c_3}<t_{c_1}$, it may be argued that the cellular species $c_3$ is fitter than $c_1$ and less fit than $c_2$. We would thus expect the cellular species $c_2$ to be the dominant species and not $c_3$). Therefore our definition of fitness is unsatisfactory here.

When comparing the overall cellular reproduction rate depicted in Fig. \ref{fig:crossCellMutDisp} and Fig. \ref{fig:cmpPerfXYvsZ}, we identify a roughly equivalent level of cellular reproduction rate ($\approx$22 cellular reproductions per second). This would thus indicate that although $c_2$ are fitter (producing molecular species $s_1$ and $s_9$ more rapidly) than $c_3$, the latter (or potentially the cell population as a whole) may have developed other features which maintained a similar cellular reproduction rate.

The details of these particular evolutionary dynamics remain unclear and have not been examined within the timeframe of this thesis. Nevertheless we formulate a number of potential explanations that might merit future investigation:
\begin{itemize}
	\item Our simplistic view of fitness may not be appropriate in the current experiment. As molecular mutation is now occurring, the cellular species \textit{or} the cellular population as a whole may have developed new features to cope with negative mutation effects. These features may have enabled the cellular population to maintain a competitive overall cellular reproduction rate while mutations occur. Such features improving the cellular reproduction rate and robustness \citep{wagner2005robustness} should then be accounted for in the cellular species' fitness. This reflects the complex multi-level nature of fitness in chemical reaction networks \citep{1004466}. 
	\item Our classification of cellular species may not expose the dominant cellular species adequately. A different classification scheme may be defined which would be based on some key properties of the cell's reaction network (and not only on the molecular species being present in the cell).
	\item The chaotic nature of the dominant cellular species dynamics (Fig. \ref{fig:crossMolMutEvoRun}) may also suggest that the observed displacements might not only be due to selection. This chaotic behaviour may have resulted from the relatively small cellular population size employed here. This parameter may have increased the sensitivity of the cellular population to statistical fluctuations. Note that the choice of population size was essentially driven the number of CPUs available in the experimental cluster.
	\item Finally we note that in the different closed networks employed and evolved in Appendix \ref{app:consMiniCOll}, Section \ref{sec:EvolvingSignalProcessing} and in the current section, the number of molecular species necessary to maintain closure was successively increased. When comparing the dynamics observed in these experiments, we remark that, as this diversity of molecular species increased, the maintenance and domination of cellular species became more chaotic. For example, drift dynamics only appeared in the experiments, conducted in Section  \ref{sec:EvolvingSignalProcessing}, in which the closed reaction networks were composed of four distinct molecular species.

We propose that this diversity of molecular species may have been implicated in the different dynamics described in this chapter. Indeed, \cite{szathmary1987gse} demonstrated, using the stochastic corrector model, that variations due to the stochastic transmission of molecular species during cellular divisions may result in degenerative outcomes (i.e., the cellular species cannot self-maintain over time) when the number of both the molecular species and molecules required for the survival of the cells is too important.

If this constraint is in effect implicated in the experiments presented in this chapter, then dealing with more molecular species and instances (i.e., more complex information) would increasingly become more difficult using the \MCS, averting any significant evolutionary growth of complexity. This would \textit{ultimately} suggest the limitations of the \MCS~to encode and process more complex information using autocatalytic networks only. To overcome this limit, it may be conjectured that a complementary mechanism enabling the stable storing and subsequently processing of more complex information might be required in the current model. A genetic subsystem could for example address this requirement and potentially lead to the evolution of higher forms of digital organisms. This would thus naturally relate to a major evolutionary transition as proposed by \cite{smith1997mte}.
	
\end{itemize}
This experiment presented a range of interesting and unexpected issues which resulted directly from the key differences existing between Alchemy and the \MCS~system. Further analytical work using adequate tools such as Organisation Theory \citep{dittich2007} may also illuminate these complex evolutionary dynamics.

\subsection{Conclusion}
Inspired by specific experiments related to crosstalk conducted with Alchemy by Fontana, we investigated a potential constructive role of crosstalk: To allow distinct closed reaction networks to cooperate with each other when occurring in the same reaction space. This cooperation would then lead to the emergence of molecular organisations of higher structural and functional complexity. We indicated the similarities and key differences between the Alchemy system and the \MCS.
Three series of experiments were then detailed: 
\begin{enumerate}
	\item Two non-crosstalking closed reaction networks were employed. Although significant differences exist between the \MCS~and Alchemy, we essentially identified a similar behaviour: one reaction network would displace the other.
	\item Two crosstalking closed reaction networks were utilised. We first noted a phenomenon (which also occurred in the corresponding Alchemy experiments), in which a meta-reaction network emerged and contained both seed closed-reaction networks. This new cellular species was able to self-maintain for a sustained period of time. However a second phenomenon occurred (which was not observed in Alchemy), in which a selective displacement took place. A mutant cellular species emerged and displaced the meta-reaction network. This mutant cellular species was no longer maintaining the seed reaction networks but was in fact fitter at performing the pre-specified tasks. This resulted in a faster cellular reproduction rate which provided the mutant species with a selective advantage.
	\item Two crosstalking closed reaction networks were used and molecular mutations were applied. We identified a common selective displacement as reported in previous experiment. However an additional cellular species displacement was observed and presented evolutionary dynamics which are not fully understood. We discussed some of the possible pitfalls of our analysis and outlined potential explanations. 
\end{enumerate}
These experiments demonstrated the constructive role of crosstalk in enabling cooperation to occur between closed reaction networks. The evolutionary process was also able to optimise the reaction networks and their crosstalk properties to carry out the pre-defined multitask function.

The resulting evolved networks presented a higher level of functional and structural complexity which supports our initial hypothesis: Crosstalk is a key mechanism enabling the evolutionary growth of complexity in biochemical networks. More precisely, crosstalk enabled the symbiogenesis of separate closed reaction networks to occur, leading to the emergence of novel closed reaction networks of higher complexity. However future work remains necessary as the final series of experiments presented very complicated and difficult to interpret evolutionary dynamics.

\section{Summary}
Using the extended version of the \MCS~ where a cellular model is employed, we conducted a series of experiments focusing on the evolution of closed reaction networks to carry out pre-specified information processing tasks. We first presented a simple experiment in which a closed reaction network was evolved to perform a signal amplification function. Following this, we extended this work and examined networks of higher complexity. 
We hypothesised the constructive role of crosstalk to allow the evolution of more complex closed reaction networks. This work was inspired by the symbiogenesis theory and preliminary experiments conducted by Fontana and Buss. We demonstrated that crosstalk was in fact necessary for the cooperation of distinct closed reaction networks. This cooperation subsequently permitted the evolutionary growth of complexity of crosstalking closed reaction networks.
This chapter demonstrated the possibility of evolving closed reaction network which are capable of performing information processing tasks.