\chapter{Emergence and Self-Maintenance of Closed Reaction Networks}
\label{emerSelfMainClosedRN}
In the previous chapter we presented an agent-based Artificial Chemistry: the \MCS. We employ this system to investigate organisational closure and the evolution of Cellular Information Processing Networks. Using this formalism we now present our first series of experiments. These experiments examine both the spontaneous emergence and the self-maintenance of closed reaction networks in the \MCS. The content presented in this chapter was published \citep{alifexi_decraene_158} at the Eleventh International Conference on the Simulation and Synthesis of Living Systems (Alife'08) .

\section{Introduction}
\label{sec:introEmerSelfMain}
In keeping with our bottom-up approach to investigating closure in artificial CIPNs, we first examine the conditions required for the spontaneous emergence and self-maintenance of minimal closed reaction networks in the \MCS\footnote{The \MCS~ implementation is object-oriented and was conducted using the C++ language. The \MCS~software package and sources can be downloaded at \url{http://esignet.net/dokumente/upload/WP13}}. This key initial step is necessary to examine and understand the evolutionary dynamics of simple closed reaction networks in our AC. This fundamental knowledge will allow us to study later the emergence and evolution of artificial CIPN functions in these closed reaction networks.

According to our definition of closure, the \textit{simplest} form of closed reaction network that can be identified in the \MCS~are autocatalytic organisations of the following form:
\[
r_j=\ 2\ s_i\ \rightarrow 3\ s_i
\]
$s_1$ is an autocatalytic molecular species also called a \textit{self-replicase}. If we consider a reaction network $C$ containing only the molecular species $s_1$ then $C$ is organisationally closed. As reactions are bimolecular in the \MCS, this reaction involves a trans-acting replicase (requiring the interaction of two distinct molecules to achieve replication). This self-replication scheme differs from the more traditional $s_i\ \rightarrow 2\ s_i$ with $s_i$ growing exponentially. Here, a ``survival of the common'' dynamics applies in which the growth of $s_i$ is hyperbolic \citep{szathmary1997rrf}. The domination of a given species is dependent on both its intrinsic fitness \textit{and} its relative concentration in the molecular population. For another species to displace the dominant one, a significant difference in intrinsic fitness and/or a high/higher initial concentration is necessary.

Both the spontaneous emergence and self-maintenance of such individually autocatalytic molecular species were reported as easily obtained in Alchemy. In the original Tierra system, where autocatalytic reactions are first order (exponential) and not hyperbolic, the spontaneous emergence of autocatalytic molecular species was not expected or reported; however it did arise in the related Amoeba system, specifically devised for this purpose \citep{DBLP:journals/alife/Pargellis01}. 

In this chapter, we first present a series of evolutionary experiments focusing on the spontaneous emergence of autocatalytic molecular species. Following this, we identify the minimal self-replicase $s_{R0}$ that can be specified in the \MCS. We examine the system's dynamics when $s_{R0}$ is manually introduced in a population of randomly generated molecular species. We then investigate the effects of the self-replicases' binding specificity over the system's dynamics. Finally, we describe the system's evolutionary dynamics when a hand-designed self-replicase having a high binding specificity is employed.

\section{Spontaneous emergence of self-replicases}
\label{sec:SponEmergSelfRep}
To examine the spontaneous emergence of autocatalytic species, we perform a first experiment in which the molecular species are generated from the random assembly of monomers. An artefact of the BL's syntax is that it is moderately difficult to observe the spontaneous emergence of an individually autocatalytic molecule. Specifically, there are $4^{8}$ ($65,536$) distinct molecules of length 4 symbols (the minimal length to construct a functional/enzymatic molecule), of which only a single one ($s_{R_0} = *\triangledown:\triangledown$) is autocatalytic. 

Although the probability of spontaneously obtaining such autocatalytic molecules from random assembly of monomers is therefore quite low in \MCS, the intuition is that, \emph{once} such a molecular species does appear, it should be able to rapidly fill the reaction space. This phenomenon was indeed observed in Alchemy and is expected to occur in \MCS. 

An evolutionary experiment is conducted and uses the following parameters:
\begin{itemize}
  \item Each simulation run is initialised with 100 randomly generated, 10-symbol long, molecules.
  \item $n_{max}=1000$ (i.e., the population initially grows without any displacement; but once the total number of molecules reached 1000 it is limited to this value, by displacing one random molecule for each new molecule generated, as previously described).
  \item 30 simulation runs are performed, each for 5000000 molecular interactions (i.e., collisions).
  \item Molecular mutations, as presented in Section \ref{sec:evoOpDesc}, occur with $p_{sym}=0.001$ and $r_{mut}=0.001$. 
\end{itemize}

To identify spontaneously emerging self-replicases, every molecule was tested at each timestep for self-replication functionality. The spontaneously emerging self-replicases identified in these 30 simulation runs are listed in Table~\ref{tab:SRFound}. 

\begin{table}[htpn]
\centering
  \[\begin{array}{|rcl|}\hline
	\multicolumn{3}{|l|}{\mathrm{Self-replicases}}\\ \hline
	00'\triangle & *& \triangledown:\triangle \triangledown \triangledown *0 \\
	1\triangledown 0& *&\triangledown :\triangledown \\
	00'\triangle & *& \triangledown :\triangle \triangledown \lozenge \triangledown *0 \\
	1\triangle \triangledown 0& *&\triangledown :\triangledown  \\
	:1& *&\triangledown :\triangledown \lozenge :1*\triangledown :\triangledown \lozenge  \\
	:0\triangledown \triangledown & *&\triangledown :\triangle \triangle \triangledown  \\
	:& *&\triangledown :\triangledown \triangledown *01 \\ 
	*\triangledown & *&\triangledown :\triangledown \triangle \triangledown \triangle \triangle  \\						
	1\lozenge \triangledown :& *&\triangledown \triangledown :\triangledown : \\ 
	*\triangledown :& *&\triangledown :\triangledown \triangle \triangledown \triangle \triangle  \\
	& *&\triangledown :\triangledown  \\ 					
	& *&\triangledown \triangledown :\triangledown   \\ 
	& *&\triangledown 0\triangledown \triangledown :\triangledown 0 \\ 
	\lozenge \triangledown *\triangledown & *&\triangledown :\lozenge \triangledown \triangledown  \\
	\triangle 1& *&\triangledown :\triangledown \lozenge\\ \hline\end{array}\]
	\caption{Spontaneously emergent self-replicases in \MCS}
	\label{tab:SRFound}
\end{table}

Table~\ref{tab:SRFound} shows that 15 syntactically distinct self-replicases appeared. In the 30 experimental runs, the highest molecular count achieved by any of these spontaneously occurring self-replicases was just a \textit{single} isolated molecule. 

Although these self-replicases are syntactically different, note that it is a property of the BL syntax that some symbols are ignored when functionally interpreted (they are, in a certain sense, ``junk'' symbols). Thus, although 15 distinct self-replicases were identified, it turns out that the core broadcast units (the ``active sites'', after discarding ``junk'' symbols) are, in fact, identical for 14 of these; and are all equivalent to the self-replicase, $s_{R_0} = $ \mbox{$*\triangledown:\triangledown$}.  Only the broadcast device $*\triangledown 0\triangledown \triangledown :\triangledown 0 $ possesses a core broadcast unit of a different form, namely $*\triangledown 0:\triangledown 0$. This is an alternate form of $s_{R_1}$, having just the minimal binding specificity of one symbol, i.e., $s_{R_1}$ may only replicate molecules whose sequence finishes with the symbol 0.

The spontaneous appearance of self-replicases was expected. Results indicated that (self-)replicases do emerge, however they never manage to grow in concentration and would quasi-deterministically be displaced by other molecules. As mentioned earlier, the highest molecular count achieved by any of these spontaneously emerging self-replicases was just a single instance. Nevertheless for a self-replication reaction to occur in the \MCS, the collision of two distinct instances of a self-replicase species is required. Therefore self-replication dynamics could not be observed.

We also propose the following potential underlying phenomena which may have discouraged the emergence and self-maintenance of autocatalytic molecular species in the \MCS:

\begin{itemize}
	\item As already noted, the BL syntax does not strongly facilitate the spontaneous emergence of self-replicases. This syntactical constraint may discourage the spontaneous emergence of such species. The BL syntax may also have an impact on the robustness of these self-replicases against mutation effects. For example, a mutation may lead to the removal or replacement of a structural symbol such as * or : in a given active broadcast device. As a result, this molecule would lose its enzymatic function and become a null broadcast device.
	\item Secondly, if we consider that multiple concurrent instances of a self-replicase species successfully emerge, such molecular species are likely to possess a low molecular concentration when occurring. This low concentration diminishes the capacity of these molecular species to persist against side reactions and mutation events.  
	\item Finally although molecular species with the ability to self-replicate do emerge, these species may also function as replicases being able to catalyse the replication of other species. The latter may be viewed as parasites if these species do not contribute, in return, to the replication of the replicases. In this case the replicases' binding specificity (i.e., the range of species that can be replicated by the replicases) may affect the ability of the replicases to self-maintain in the population. 
\end{itemize}

These three factors, when combined, may significantly have lowered the probability of having a self-replicase spontaneously emerge and self-sustain in the \MCS.

To investigate the above propositions, we conduct a second series of experiments in which we manually design and introduce a minimalist self-replicase. Following this, we will explore the role of the replicases' binding specificity upon the system's dynamics.

\section{No selective advantages for universal replicases}
\label{specificity}
We present a second experiment in which a minimal self-replicase is devised and manually introduced in a reactor in addition to randomly generated molecules. 

The behaviour of the minimal self-replicase $s_{R_0}$, which recurrently emerged in the previous experiment, is as follows. The matching condition is defined by a single symbol, $\triangledown$, which designates a multiple character wildcard. This indicates that $s_{R_0}$ may bind to any molecular species. In addition when reactions occur between $s_{R_0}$ and substrate species $s_i$, $\triangledown$ is assigned a value, being the matched substring of $s_i$. In this case, this will be the complete string $s_i$. A unique symbol $\triangledown$ also constitutes the action part of $s_{R_0}$. This specifies that the output string of $s_{R_0}$ is exactly the string bound by the $\triangledown$ in the condition part, i.e., a copy of $s_i$'s string. Therefore the broadcast device $s_{R_0}$ is actually a ``universal'' replicase; which, by definition, means that it is also a \emph{self-}replicase (in the special case that it binds to another instance of itself, i.e., $s_i=s_{R_0}$). The ``specificity'' of $s_{R_0}$ is said to be \emph{null}.

Figure \ref{fig:popSR0} presents a first experiment examining the behaviour of $s_{R_0}$ averaged over 30 simulation runs. In this experiment, the following parameters are employed:

\begin{itemize}
	\item The reactor is seeded with 900 randomly generated molecules, each of length 10 symbols. These initial molecules are independently and randomly generated for each simulation run.
  \item In addition, 100 instances of $s_{R_0}$ are inserted.
  \item $n_{max}$ designates the fixed maximum number of molecules that may be contained in the universe, $n_{max}=1000$.
  \item Molecular interactions occur as follows: two molecules $m_e$ and $m_s$ are picked at random. $m_e$ is considered as an enzyme and $m_s$ as a substrate. If $m_e$ can bind and react with $m_s$ then a molecule $m_p$ is produced and added to the population. A molecule $m_x$ (other than the $m_e$, $m_s$ and $m_p$) is picked at random and removed from the population. 
  \item No mutation may occur in these experiments in order to facilitate our investigation on replicases. 
\end{itemize}
A high initial molecular amount (100 instances) of $s_{R_0}$ was chosen to satisfy the trans-replication constraint (i.e., at least two distinct molecules are required to achieve self-replication) and minimise early extinction due simply to stochastic fluctuation. 

\begin{figure}[ht]
	\centering
		\includegraphics[scale=0.9]{chapter5/SR-STDEV}
	\caption{Relative population growth of replicators $s_{R_0}$ averaged over 30 simulation runs. Solid line is
average concentration; error bars denote standard deviation.}
	\label{fig:popSR0}	
\end{figure}

From Figure~\ref{fig:popSR0} it is clear that the species $s_{R_0}$ never grows to take over the population; rather, it consistently diminishes, contrary to the original, informal, prediction. A formal explanation of this outcome is given by modelling the system with the (approximate, continuous) catalytic
network equation \citep{156971}. The state of the system is described by the concentration vector $x=(x_1,\ldots,x_n$) with $x_1+\ldots+x_n=1$ and $x_i>0$, where $x_i$ refers to the concentration of a molecular species (or collection of ``chemically equivalent'' species) $s_i$. The general dynamic behaviour is then given by:

\begin{eqnarray}
\dot{x_k}=\sum_{i=1}^n\sum_{j=1}^n 
\alpha_{ij}^kx_ix_j-x_k\sum_{i,j,l=1}^n \alpha_{ij}^lx_ix_j 
\label{caNetEq}
\\
\textrm{with } k=1,\ldots,n\nonumber
\end{eqnarray}

In Eq. \ref{caNetEq}, the second term represents the dilution flow: A molecule may be removed (at random)  when a successful reaction occurs in the single reactor model (Section \ref{subsec:singleReacMod}). $\alpha_{ij}^k$ are the rate constants for each reaction $s_i+s_j\rightarrow s_i+s_j+s_k$. 

In this experiment, these simplify to:

\begin{equation}
	\alpha_{ij}^k=\left\{
	\begin{tabular}{l}
		1\ if $s_i+s_j\rightarrow s_i+s_j+s_k $\\
		0\ \textrm{otherwise}
	\end{tabular}
	\right.
\end{equation}

To explain the results presented in Fig.\ref{fig:popSR0}, we propose a \textit{simplified} analysis which focuses on the dynamics of universal replicases. Consider the case where only universal replicases ($s_{R_0}$) and the set of all non-enzymatic molecular species ($S_{NE}$) (that may only act as substrates) are present. This is clearly the \emph{most} favourable case for the growth of $s_{R_0}$. Denote the molecular concentrations of $s_{R_0}$ and $S_{NE}$ by $x_1$ and $x_2$
respectively. Then $\alpha_{ij}^1=1$ if $i=1, j=1$; otherwise $\alpha_{ij}^1=0$. Similarly, $\alpha_{ij}^2=1$ if $i=1, j=2$; otherwise $\alpha_{ij}^2=0$. Inserting into Eq.~\ref{caNetEq}, we obtain:

\begin{equation}
\dot{x_1}=x_1^2-x_1(x_1^2+x_1x_2)\\
\end{equation}
But given that $x_2=1-x_1$:
\begin{eqnarray}
\dot{x_1}&=&x_1^2-x_1^3-x_1^2+x_1^3 \nonumber \\
\dot{x_1}&=&0 
\end{eqnarray}
whereas the growth rate of molecules $S_{NE}$ is:
\begin{eqnarray}
\dot{x_2}&=&x_1(1-x_1)-(1-x_1)[x_1^2+x_1(1-x_1)]\\
\dot{x_2}&=&x_1-x_1^2-(1-x_1)(x_1^2+x_1-x_1^2) \nonumber \\
\dot{x_2}&=&x_1-x_1^2-x_1+x_1^2 \nonumber \\
\dot{x_2}&=&0
\end{eqnarray}

Thus, both molecular species $s_{R_0}$ and $S_{NE}$ share a common zero ``expected'' growth.  Under the stochastic conditions of the reactor this would yield a random drift in relative concentrations---as opposed to a quasi-deterministic growth of the $s_{R_0}$ species. 

In such systems driven by random drift dynamics, only two possible outcomes may be observed where the system reaches steady state. If both species $s_{R_0}$ and $S_{NE}$ are initialised with a common concentration, then both outcomes would deterministically occur with equal chances: 

\begin{enumerate}
	\item The universal replicase species eventually displaces all $S_{NE}$ molecules in the reactor.
	\item All replicases $s_{R_0}$ \textit{but one} are displaced by $S_{NE}$ molecules. This single replicase molecule remains and cannot be displaced since: 1) Reactants are not consumed during reactions in the \MCS. 2) No side reactions, involving $S_{NE}$ molecules only, may occur. 
\end{enumerate}

Qualitatively the above phenomenon is due to the fact that any \mbox{(self-)}replicase having low or zero specificity, such as $s_{R_0}$, will not only replicate itself but also replicate any other molecules; and therefore cannot selectively displace these molecules. But recall that this was the \emph{best case} situation for growth of $s_{R_0}$, where none of the other molecules had any enzymatic activity. In the practical case of Figure~\ref{fig:popSR0} the collection of such additional side reactions will give a nett negative growth rate for $s_{R_0}$, which therefore, quasi-deterministically, decays until $x_1=0$.

With regards to the spontaneous emergence and domination of self-replicases given a set of randomly generated molecular species, this analysis is consistent with the results described in Section \ref{sec:SponEmergSelfRep}. Self-replicases of very low specificity (which do spontaneously occur) cannot grow to significant concentrations and would therefore be commonly displaced by other molecules.

In Section \ref{sec:SponEmergSelfRep}, we also mentioned that the replicases' binding specificity may potentially affect the system's dynamics. We examine the role of binding specificity in the following section.

\section{Specificity and domination of the replicases}
To investigate the role of binding specificity, we proceed to a series of experiments in which we incrementally increase the specificity of the \mbox{(self-)}replicases.  Table~\ref{tab:SR} shows the different replicases employed in these experiments. $s_{R_1}$ designates a molecular species that would only react with molecules whose strings end with the symbol ``1''. As the latter occurs at the rightmost position of $s_{R_1}$, it may react with itself, producing another instance of $s_{R_1}$. Similarly, $s_{R_2}$ only binds to molecular strings containing the suffix $01$. This tag forms a constraint on the replicases, allowing them to react only with a progressively more restricted set of substrate molecular species. This impacts directly on these molecules' binding specificity.

\begin{table}[htbp]
	\centering
		\begin{tabular}{|c|l|} \hline
			Replicase & Broadcast device\\ \hline
			$s_{R_0}$ & $*\triangledown:\triangledown$ \\
			$s_{R_1}$ & $*\triangledown1:\triangledown1$ \\
			$s_{R_2}$ & $*\triangledown01:\triangledown01$ \\
			$s_{R_3}$ & $*\triangledown101:\triangledown101$\\
 			$s_{R_4}$ & $*\triangledown0101:\triangledown0101$ \\ \hline
		\end{tabular}
		\caption{\mbox{(self-)}replicases with 
increasing binding specificity}
		\label{tab:SR}
\end{table}


\begin{figure}
	\centering
		\includegraphics[scale=0.9]{chapter5/nPop}
	\caption{Population growth of replicators $s_{R_0}, s_{R_1}, s_{R_2}, s_{R_3}$ and $s_{R_4}$. Each line represents the average concentration of corresponding replicase over 30 simulation runs.}
	\label{fig:nPopSR}	
\end{figure}

The results depicted in Figure~\ref{fig:nPopSR} suggest the potential role of binding specificity in encouraging the domination of replicase species. The ability of a \mbox{(self-)}replicase to dominate the reaction space, in which a random initial population of molecules is generated, increases progressively with its binding specificity. Fig.\ref{fig:nPopSRDetail} depicts the growth of the replicases in each of the 30 simulation runs conducted independently in the 5 experimental series.

\begin{figure}
\subfigure{	\includegraphics[scale=0.55]{chapter5/nPop1.eps}}
\subfigure{	\includegraphics[scale=0.55]{chapter5/nPop2.eps}}
\subfigure{	\includegraphics[scale=0.55]{chapter5/nPop3.eps}}
\subfigure{	\includegraphics[scale=0.55]{chapter5/nPop4.eps}}
\subfigure{	\includegraphics[scale=0.55]{chapter5/nPop5.eps}}
\caption{5 series of experiment in which the binding specificity of the seed replicase molecules was incrementally increased using $s_{R_0}$, $s_{R_1}$, $s_{R_2}$, $s_{R_3}$ and $s_{R_4}$. For each experiment, 30 simulation runs were conducted.}
\label{fig:nPopSRDetail}
\end{figure}

Fig.\ref{fig:nPopSRDetail} shows that as the replicases' binding specificity increases, the number of simulation runs in which $x_1$ reaches 1 increases accordingly. When non-universal replicases are employed, an initial increase of $x_1$ is commonly observed. However this increase in $x_1$ appears, in particular simulation runs, to be followed by a random drift dynamics (observed at different $x_1$ level), preventing the replicases to fully dominate the reaction space, echoing the results presented in Section \ref{specificity}. 

As in the previous section, we examine and explain this phenomenon through the use of an ODE model. To facilitate our investigation, this analysis employs a simplified model illustrating a \textit{best case} scenario. Although being less intricate, this model contains the core elements of the system employed in one of the above experiments (where $s_{R_1}$ species are employed). In this model, we consider a reactor containing only the following molecular species:

\begin{itemize}
\item Replicases $s_{R_1}$ which only replicate molecules terminating with the symbol ``1'' (which includes $s_{R_1}$ molecules themselves).
\item A variety of non-enzymatic molecules $S_{NE}$ which are randomly generated. $S_{NE_1}\subseteq S_{NE}$ is the subset of molecules whose strings terminate with the designated symbol. These molecules contained in $S_{NE_1}$ can be replicated by molecules $s_{R_1}$.
\end{itemize}

The concentration vector is given by $x=(x_1, x_2, \ldots, x_n)$ with $x_1+x_2+\ldots+x_n=1$ where $x_1$ is the concentration of $s_{R_1}$ and $x_2$ is the sum of concentrations of molecules in $S_{NE_1}$. The growth rate of the different molecular species in this reactor are as follows:
\begin{eqnarray}
\dot{x_1}&=&x_1^2-x_1(x_1^2+x_1x_2)  \\
\dot{x_1}&=&x_1^2-x_1^3-x_1^2x_2 \nonumber \\
\dot{x_1}&=&x_1^2(1-x_1-x_2)
\end{eqnarray}
The growth rate of molecules $S_{NE_1}$ is:
\begin{eqnarray}
\dot{x_2}&=&x_1x_2-x_2(x_1^2+x_1x_2)  \\
\dot{x_2}&=&x_1x_2-x_1^2x_2-x_1x_2^2   \nonumber \\
\dot{x_2}&=&x_1x_2(1-x_1-x_2)
\end{eqnarray}
Since $x_1+x_2+\ldots+x_n=1$, we have $x_1+x_2<1$ and therefore $\dot{x_1}>0$ and $\dot{x_2}>0$. 

$S_{NE_2}=S_{NE}-S_{NE_1}$ is the set of non-enzymatic molecules species that cannot be replicated by $s_{R_1}$.
Let us set $x_3=\sum_{i=3}^n x_i$, $\dot{x_3}$ is the growth rate of species $S_{NE_2}$ and is given by:
\begin{eqnarray}
\dot{x_3}& = & 0 -x_3(x_1^2+x_1x_2)
\label{eqXn}
\end{eqnarray}
In Eq.~\ref{eqXn}, we note that any given molecules $s_i \in S_{NE_2}$ possess a negative growth rate which indicate that these molecules would deterministically be displaced by molecules $s_{R_1}$ and $S_{NE_1}$. 

The displacement of $S_{NE_2}$ molecules allows both $s_{R_1}$ and $S_{NE_1}$ to increase in concentration. However when $S_{NE_2}$ molecules are fully displaced, we then obtain $x_1+x_2=1$ (in contrast to the initial $x_1+x_2<1$ condition). The system's dynamics are now equivalent to those described in Section \ref{specificity} where both species have a common null growth rate. In this case where a random drift dynamics applies, the species having the highest relative concentration is more likely to displace the other one. 

The replicases' binding specificity limited the initial concentration of $S_{NE_1}$ species (resulting from the random initialisation of the molecular population). As the replicases's binding specificity increased, the initial concentration of $S_{NE_1}$ decreased.  Consequently the replicases having a higher \textit{relative} specificity possessed a higher chance to displace the parasitic species (once $S_{NE_2}$ species were first fully displaced). 

The replicases' binding specificity conditioned the initial concentration of parasitic species which explains the behaviour observed in Figure~\ref{fig:nPopSR}, in which replicases with higher specificity are more likely to take over the reactor space. Therefore in this system, for replicase molecules to successfully dominate a randomly generated molecular population, a significant binding specificity is required. 

Regarding the related spontaneous emergence and domination of autocatalytic species in the \MCS, given a set of randomly generated molecular species, it is progressively more difficult for self-replicases of higher specificity to \textit{spontaneously} arise by chance (due to their greater length, and relatively rare frequency as defined by the BL syntax). The spontaneous emergence of a ``sustainable'' self-replicase (i.e., of sufficient specificity to establish itself) remains theoretically possible in \MCS.  However, both the experimental results and the analysis presented here suggest that the expected emergence time would be extremely (perhaps infeasibly) long. While we have not formally quantified this, it appears that \MCS~therefore shares this property with the Tierra system. 

It is conjectured that this binding specificity property may have been implicated in the dynamics of a variety of previously reported artificial chemistries. For example, Fontana and Buss reported (in level 0 Alchemy experiments) the recurrent emergence and domination of the universal replicase $\lambda x.x$ \citep{fontana1994aft}. Nevertheless Fontana and Buss also mentioned that if non-enzymatic species are not filtered out then an accumulation of such inert species would occur. This observation suggests that the lack of binding specificity may have, as well, affected the system's dynamics where non-enzymatic species prevented the domination of emerging universal replicases. As suggested, this underlying phenomenon has potentially been involved in previous AC-based research. Nevertheless it has never been examined and explicitly isolated in the manner presented here. 

Finally, we may also consider the potential effects of molecular mutations. Even though replicase species having a significantly high binding specificity are employed, mutations may lead to the emergence of non-enzymatic mutant species possessing the replicases' tag (enabling these mutants to be replicated by the replicases). The replicases' binding specificity, which was originally high, would thus become again \textit{relatively} low or null. As a result, the replicases' binding specificity  may not prevent \textit{potentially disruptive} effects from occurring. To test this hypothesis, we carry out a final evolutionary experiment ``a la Tierra'' in which a hand-designed self-replicase (i.e., an ancestor molecule) having a ``high'' binding specificity capable of self-sustaining is employed. This experiment is presented in the next section.	

\section{Rise and fall of the fittest}
\label{riseFall}
In the Tierra system, a hand-designed molecule called the ``ancestor'' is manually introduced into the space.  This initially grows to saturate the available core memory. The population subsequently evolves into a variety of collectively autocatalytic reaction networks (where Tierra ``creatures'' or programs are here considered analogous to ``molecules''). Accordingly, our next step is to mirror this methodology, and introduce a hand-designed self-replicase of relatively high specificity into the \MCS~system.

Figure~\ref{fig:popMut} presents an example of such an experiment in which ancestor molecules are manually introduced. The results indicate that \MCS~does \emph{not} exhibit an evolutionary dynamic at all comparable to Tierra in this case. This evolutionary dynamic was moreover systematically observed in repeated simulation runs. The ancestor self-replicators do, at first, quickly dominate the reaction space, just as expected. However, this population immediately collapses again. The average molecular length then increases dramatically, while the overall reaction rate (indicating the average rate of binding between random molecules in the population) also collapses. In this particular run, molecules are arbitrarily limited to a maximum length of $BD_{Lmax}=1.0\e{6}$. 

\begin{figure}[ht]
	\centering
		\includegraphics[scale=0.9]{chapter5/popMut}
	\caption{Effects of molecules length growth upon overall system reactions rates. In this experiment, an ancestor ($s_{R_4}=\triangledown0101:\triangledown0101$) is inserted (with initial concentration $[s_{R_4}]=0.1$) in addition to randomly generated molecules. Moreover mutation per molecule and per symbol is turned on with $p_{sym}=0.001$ and $r_{mut}=0.001$.}
	\label{fig:popMut}
\end{figure}

As with the experiments discussed earlier, these results were not expected. In fact, certain mutants of the original autocatalytic molecule developed a distinct advantage over the ancestor. That is, these mutants could be replicated by the ancestor molecules but only to the cost of these ancestors, i.e., an asymmetric relationship.  Moreover, some of these mutants also lose their ability to self-replicate, explaining the rapid decrease in the global number of self-replicases. By exploiting their molecular tag and the ancestors, these non-autocatalytic molecules succeed in displacing the dominant ancestors.

To illustrate this phenomenon, we present a simple example of such a case in which we define two molecules: $s_{R_4}=*\triangledown 0101:*\triangledown0101$ and $s_{R_4}'=*\triangledown 0101:*\triangledown00101$. The latter is a readily accessible mutant of $s_{R_4}$. Once it appears, the mutant $s_{R_4}'$ allows for a runaway degenerative scenario to occur. The possible reactions between species $s_{R_4}$ and $s'_{R_4}$ are as follows:

\begin{eqnarray*}
	s_{R_4}+s_{R_4} & \longrightarrow & 3s_{R_4} \\ 
	s_{R_4}+s_{R_4}' & \longrightarrow & s_{R_4}+2s_{R_4}' \\
	s_{R_4}'+s_{R_4} & \longrightarrow & s_{R_4}+2s_{R_4}' \\
	s_{R_4}'+s_{R_4}' & \longrightarrow & 2s_{R_4}'+s_{R_4}''	
\end{eqnarray*}

The product $s_{R_4}''$ is of the form $*\triangledown 0101:*\triangledown000101$ and similarly has a selective advantage over both $s_{R_4}$ \emph{and} $s_{R_4}'$. The reaction $s_{R_4}''+s_{R_4}''$ would result in the production of a molecule $s_{R_4}'''$ of the form $*\triangledown 0101:*\triangledown00000101$ and clearly shows the potential for unlimited elongation in molecule length. Of course, as molecule length increases, the per-molecule mutation rate also increases, leading to progressively more frequent disruptive changes to molecular structure.  The observed consequences of these mutation effects are twofold:

\begin{itemize}
\item Molecules may become inactive (i.e., lose all enzymatic activity). This is a direct consequence of the BL syntax. A mutation leading to the removal or insertion of structural symbols such as $*$ or $:$ would ``break'' the active site. This degenerative effect may be regarded as a consequence of syntactic ``brittleness'' of BL.

\item The binding specificity may be increased.  This arises when mutations lead to the insertion of informational symbols such as $0s$ and $1s$. As a result, although some molecules may still possess an active site capable of some enzymatic function, their high specificity decreases the variety of target molecules that it can bind to; ultimately meaning there may be few, if any, functional targets for it left in the population.
\end{itemize}

Both of these phenomena result in a continual decrease in the overall reaction rate until reactions effectively cease completely (i.e., system death). Figure~\ref{fig:elongationCatastrophe} summarises this cascade of events.  Note that this system level degeneration (the ``elongation catastrophe'') occurs precisely because of the stepwise emergence of molecules which are progressively ``fitter'' at the molecular level. Nevertheless this notion of fitness differs from its meaning usually implied in Artificial Chemistries. In typical ACs, molecular species or more specifically ``digital organisms'' improve in fitness by evolving their intrinsic properties. Such agents become fitter by incrementally ameliorating their ability to perform a target task, e.g., competing for CPU resources in Tierra. Here the molecules are fitter, not due to their intrinsic functionality, but due to their ability to exploit their interactions with the other molecules. This interpretation of fitness applies specifically in the case of catalytic reaction networks in contrast with the more general class of self-replicating multi-agent systems.

\begin{figure}[ht]
	\centering		
\includegraphics[scale=1]{chapter5/elongationCatastrophe}
	\caption{Elongation catastrophe in \MCS}
	\label{fig:elongationCatastrophe}
\end{figure}

\section{Solving the elongation catastrophe problem}
In this section, we describe different qualitative modifications conducted on the \MCS, which are aimed at preventing the elongation catastrophe from occurring. These various technical modifications are directed at limiting the string length of product molecules. Following this, the different outcomes are briefly presented.

\begin{enumerate}

\item The multiple symbol wildcard $\triangledown$ was altered so that it would not be able to pass an unlimited number of symbols from the input molecule (substrate) to the output molecule (product). An integer parameter $1\leq c \leq c_{max}$ represents the number of symbols that can be matched and passed
  by $\triangledown$, i.e., the capacity of the wildcard. This capacity may be subjected to a form of ``parametric'' mutation, where its value would change randomly in $\left[1 , c_{max} \right]$ over time.
	
\item Similarly to (1), a finite total number of ``free'' symbol objects (monomers) available in the reactor was defined. This reservoir of (untyped) monomers is reduced when new molecules are produced, and increased when molecules are destroyed. If insufficient monomers are available to complete a
  reaction, the reaction fails. This should favour smaller molecules over longer ones suffering from elongation catastrophe.
	
\item Proposal (2) was extended, further constraints were defined to limit the number of particular symbols available in the universe. Different arbitrary symbol distributions were employed (e.g. structural and informational symbols such as $*$,$:$,$1$,$0$ could be made more common than multiple
  symbol wildcards such as $\triangledown$).
	
\item Another extension to (3) was to introduce a probability of successful reaction which would depend on the product's length. Smaller molecules could then be given a selective advantage over the longer ones.

\end{enumerate}

In summary, the above system changes generally produced one of the following outcomes:

\begin{itemize}
\item Did not prevent the elongation catastrophe.
\item The system evolved towards a population of inactive and relatively small ($[1-4]$ symbols long) molecules.  The system's global success reaction rate was also close to zero. Such an accumulation of inactive or inert molecules was also reported to occur in Alchemy when inert molecules were not filtered out.
\item The system converged towards a population where enzymatic molecules were still present but could not react with any other molecules present in the reaction space. The specificity continuously increased until no further reactions occurred.
\end{itemize}

Thus, although a range of modifications were implemented, the different outcomes do not differ substantially from the degenerative cases presented above (Section \ref{riseFall}). 
	
\section{Conclusion}
We conducted a series of experiments using the \MCS~system. These experiments focused on the emergence, self-maintenance and evolution of closed molecular organisations. 

We first reported an experiment focusing on the spontaneous emergence of autocatalytic species in the \MCS. The results showed that, autocatalytic species do emerge but cannot significantly grow in concentration and eventually get displaced by other molecules. To explain this phenomenon, a number of potential explanations were proposed and developed in following experiments. 

The second experiment introduced minimalist self-replicases that were manually inserted in the reaction space in addition to randomly generated molecular species. Using a simplified model, we showed that these species being universal replicases do not possess any selective advantages over non-enzymatic species. Under the stochastic conditions of the reactor this would yield a random drift in relative concentrations of universal replicases and non-enzymatics species. 

We then exposed the effects of the replicases' binding-specificity over the system's dynamics. We showed that this binding specificity conditioned the capability of the replicases to displace the other molecular species present in a randomly generated molecular population. As the replicases' binding specificity increased, the capability of the replicases to displace the other species increased accordingly.

Similarly to the Tierra system, we hand-designed an ancestor molecular species, which possessed a high binding specificity ensuring its ability to self-maintain. We described the results obtained from an evolutionary experiment in which these ancestor molecules were employed. Our results presented unexpected degenerate evolutionary dynamics in which the closed reaction networks were not able to self-maintain. This degenerate scenario was due to an elongation catastrophe phenomenon. To address this robustness issue, we explored several model variants which all failed to prevent the degenerate outcomes to occur. These results indicated counter-intuitive outcomes when compared with a variety of other AC systems in the literature.  

In the next chapter, we explore additional models of the \MCS~which are implemented on a parallel architecture. These models aim at enhancing the robustness and subsequently the evolvability of the \MCS~system. On top of providing computational benefits, a distributed implementation allows us to explore novel system models. These modifications introduce a second level of selection at the reactor level. These extended \MCS~ systems are thus multi-level selectional models where selection occurs at both the molecular and reactor level. 