\chapter{Evolutionary Capability in Multi-Level Selectional Models}
\label{chap:evoStab}
The preliminary experiments presented in Chapter \ref{emerSelfMainClosedRN} suggested that the \MCS~ cannot support the self-maintenance of closed reaction networks when subjected to perturbations (i.e., mutations). Consequently, this lack of robustness prevented us from evolving closed reaction networks using the \MCS. Although several model variants were proposed to address these robustness and evolvability issues, none of these attempts inhibited the degenerative evolutionary dynamics.

In keeping with this effort to improve the \MCS~framework, we propose further system modifications. However, contrary to previous proposed variants which focused on modifications at the molecular level, we suggest a novel \MCS~implementation which introduces new features at the container level. This \MCS~variant is implemented as a parallel system using distributed computing facilities. The latter were provided by both the ESIGNET project and the Irish Center for High-End Computing (ICHEC). The material presented in this chapter was partially published in several conference articles \citep{alifexi_decraene_158, decraene2008mae, DecraeneBiosignal09}.

\section{Introduction}
We present a novel implementation of the \MCS~which exploits distributed computing facilities. On top of providing benefits in terms of experimental scalability (i.e., we can simulate more molecular species), employing high-computing resources enables us to explore a wider repertoire of AC models. Particularly, we identify two approaches involving compartmentalisation. In these models, molecular species are contained in compartments which result in \textit{multi-level selectional} dynamics \citep{hogeweg2003msm}. Selection applies at both the molecular and compartment levels. These models presented significant results with regards to the self-maintenance of closed reaction networks when subjected to perturbations:

\begin{enumerate}
	\item \textit{Static reactors with molecular diffusion}. \cite{mccaskill2001sec} addressed evolutionary degeneration issues in a spatially resolved stochastic system in which molecular species are contained in distinct compartments. The migration or diffusion of molecules between compartments could occur given a specified diffusion coefficient. 
	
This model considered the effects of mutant species which relied on the host (non-mutant) autocatalytic species to be replicated. This form of parasitism destabilises the system's dynamics and prevents the self-maintenance of the closed reaction network. This degenerative phenomenon is similar to the elongation catastrophe described in Section \ref{riseFall}, which was also caused by the emergence of such parasitic species. McCaskill et al. demonstrated analytically that for such a compartmentalised system, according to the level of mutation rate, a range of diffusion coefficients exists which enables stable cooperation to occur between the molecular species.
	
	\item \textit{Cellular model}. Similarly to McCaskill et al.'s approach, molecular species are contained in distinct compartments. However in cellular models, compartments have the ability to ``grow and divide''. A compartment grows (i.e., molecules are produced within the compartment) until a condition (specific to the cellular model) is satisfied. This condition triggers the division of the compartment. Half of the molecules are selected at random and transferred into another newly created compartment. 
	
This division process is analogous to cellular division, i.e., we regard compartments as biological cells. It has been demonstrated that such cellular models enable the stabilisation and self-maintenance of closed reaction networks when subjected to parasitic phenomena \citep{szathmary1987gse,cronhjort1997ccm,hogeweg2003msm}. Here, parasitised compartments ultimately decay and may be replaced by non-parasitised ones which would result from the division of neighbouring compartments. Cellular models allow the isolation of parasited cells and prevents the invasion of parasite species over the whole cellular population. 
\end{enumerate}


We propose to investigate compartmentalisation and its potential benefits upon evolutionary capability in the \MCS~system. Our distributed implementation of the \MCS~will consequently incorporate compartmentalisation properties. Nevertheless we introduce a supplementary feature which addresses the concurrent nature of biochemical processes. Operations at the compartment level will be executed in a \textit{genuine} parallel manner in contrast to traditional ABMs which rely on a synchronisation mechanism. Implementing the \MCS~ as a distributed/parallel system enables us to explore further selectional models which may resolve the lack of robustness presented in previous chapter. This work will also provide complementary insights on the effects of compartmentalisation and parallelism upon evolutionary capability in agent-based Artificial Chemistries.

In the remainder of this chapter, we present the details of our parallel architecture and its immediate effects on the \MCS. We then evaluate our compartmentalised \MCS~system in which molecular diffusion and cellular division are independently applied. 

\section{A parallel architecture}
Agent-based models typically implement the computational agents on serial computers. Although being conceptually concurrent computational units, agents are not generally executed simultaneously in ABMs. A discrete clock mechanism is usually employed to synchronise the agents' interactions and state updates. The \MCS, presented in Chapter \ref{TheAC} and utilised in Chapter \ref{emerSelfMainClosedRN}, employs a similar clock technique. 

\cite{ray1992eea}, \cite{adami1994ela} attempted to address parallelism by devising models in which agents are executed by multiple virtual CPUs. In Avida, each agent possessed its own memory space and was executed by its own virtual CPU (which speed may vary from other agents' CPUs). Whereas in Tierra, the multiple virtual CPUs executed, in turn, the code (the agents' genotypes) present in the virtual computer memory shared by the different agents. Nevertheless, as in most ABMs, a time-slicing algorithm was employed to simulate the parallel computational processes in both systems.  \cite{lenski2003eoc} also conducted a series of experiments using Avida on a grid computer; but Lenski et al. still relied on a pseudo-parallel system where parallelism was only simulated.

We propose a parallel approach to ABMs, applied here to the \MCS~model. A major difference between this extended \MCS~and Tierra/Avida is that multiple compartments (each of which contains a population of molecular species) are introduced. This compartmentalisation is the key feature by which parallelism is addressed in the extended and distributed version of the \MCS:

\begin{itemize}
	\item Molecular species are contained in distinct compartments each of which possesses a fixed maximum carrying capacity of $n_{max}$ molecules. Within the compartments, molecular interactions are processed in a sequential manner (as in traditional ABMs) through the use of a time-slicing algorithm. Each compartment (conceptually regarded as a meta-agent) is executed on individual distinct CPUs in a parallel fashion. 
	
	\item At the compartment level, we do not introduce any synchronisation mechanisms. Meta-agents are executed in parallel on separate CPUs.	These meta-agents may interact with each other by communicating signals (the effects and nature of these signals are dependent of the specific compartment model utilised and will be presented later). 
\end{itemize}

As introduced earlier, devising these compartment properties enables us to investigate further models which may resolve the \MCS's evolutionary degeneration issues. 

\subsection{Implementation}
Simulations using the distributed/compartmentalised \MCS~are run for a pre-specified length of time (defined in seconds). In a given simulation run, the number of molecular collisions/interactions occurring in each compartment may vary significantly. This variance depends on the level of computations and communications occurring in the compartments during the run.

The integer number $N$ of compartments (i.e., CPUs) used in a simulation is fixed. Compartments are identified by a unique ID number ($1\leq ID\leq N$). $C={c_1, \ldots, c_N}$ is the finite set of all compartments occurring in the universe. Each compartment initialises simultaneously their respective random number generator\footnote{The \MCS~ implementation employs the GNU Scientific Library random number generator. The latter is by default based on the Mersenne twister (MT19937) generator of \cite{matsumoto1998mtd} and has a cycle length of $2^{19937} - 1$.}. These random number generators rely on random seeds which are determined by using the server local time value multiplied by the unique compartment ID. 

The Message Passing Interface (MPI) is employed to handle the communications between the different CPU nodes/compartments. A simplex topology is utilised to condition the interactions between compartments, i.e., the distance between any two-compartments is equal.

Compartments may communicate signals to other compartments. Signals are composed of a vector of molecules. Given a compartment $c_1$, if a condition (specific to the compartment model utilised) is satisfied then a communication occurs. $c_1$ selects the target compartment $c_2$ at random. Following this, $c_1$ sends a \textit{non-blocking} signal to $c_2$ (i.e., $c_1$ continues to operate without having to wait for $c_2$ to receive the signal; this prevents communication deadlocks from happening). The exact nature and effects of the signals will be identified in the models' descriptions below. 

When a compartment checks its ``mailbox'' for incoming signals, several signals may have been received. MPI buffers the incoming signals in an orderly manner (first in first out queue). At each timestep, a single incoming signal is processed by the compartment. Fig.~\ref{fig:paraImp} depicts this algorithm which is run simultaneously on each compartment processes.

\begin{figure}[htbp]
	\centering
		\includegraphics[scale=1.5]{chapter6/paraImp.eps}
	\caption{$N$ compartments/processes running in parallel}
	\label{fig:paraImp}
\end{figure}

\subsection{Introducing chemical kinetics}
\label{subsec:kinetics}
In the non-parallel version of the \MCS, all reactions occurred in a sequential manner. A clock mechanism was utilised to sequence the processing of molecular interactions. At each discrete time step a single molecular collision/interaction occurred. Elastic and catalytic reactions would be equivalent from a temporal point of view. The computational time necessary to process a reaction was not accounted for and did not affect the system's dynamics. Although occurring in a simplified and constrained form, chemical ``kinetics'' were present in the original \MCS~ and dictated all reaction rates to be equal. This constraint is now being relaxed as follows. 

In the distributed implementation of the \MCS, several reactions may be occurring \textit{simultaneously}. Although molecular interactions are processed sequentially within a compartment, $N$ reactions may potentially be processed at the same time throughout the $N$ compartments. Catalytic reactions may result in the modification of the compartments' states. According to the state of a given compartment, a signal can be emitted and addressed to another compartment. These signals may further change the states of both the emitter and receiver compartments. Consequently the speed of chemical reactions occurring in the compartments may indirectly alter the system's dynamics at the compartment level.

Parallelism in this distributed implementation of the \MCS~ altered the system's chemical kinetics and introduced variations in the reaction rates.

\section{Static reactors with molecular diffusion}
\label{sec:McCaskillsModel}
We evaluate a first extended \MCS~model in which both compartmentalisation and molecular diffusion are addressed. 

\subsection{Introduction}
An analytically tractable model was proposed to address the inhibition of degenerative effects due to parasitism in closed reaction networks \citep{mccaskill2001sec}. McCaskill et al. demonstrated that given a compartmentalised model where migrations/diffusions of molecular species occur between the compartments, there exists, according to some parameters (i.e., the replication error rate and decay rate), an optimal diffusion coefficient which allows for the cooperation of molecular species and subsequently stabilises the self-maintaining cycle of the closed reaction network.

The following minimalist model (in which such disruptive parasitic effects are occurring) was employed by McCaskill et al. to examine the effects of compartmentalisation:
\begin{eqnarray*}
2X &\stackrel{Q}{\longrightarrow}&3X\\
2X	&\stackrel{R}{\longrightarrow}& 2X +Y\\
X+Y &\stackrel{2r}{\longrightarrow}& X+2Y\\
X,Y &\stackrel{d}{\longrightarrow}& \emptyset
\end{eqnarray*}

$X$ is an autocatalytic species which is being parasitised by $Y$. The latter is a mutant species of $X$ whose replication can only be catalysed by $X$. $Q$ is the fidelity rate of replication of $X$. $R=1-Q$ is the replication error (mutation) rate leading to the production of mutant species $Y$. $r$ is the recognition coefficient which determines the rate of successful bindings between $X$ and $Y$. The spontaneous decay rate of $X$ and $Y$ is denoted by $d$. This decay rate is fixed and independent of the production rate $Q$. The compartments possess a maximal carrying capacity of $n$ molecules. The diffusion coefficient between the compartments is denoted by $m$. The latter is the key parameter enabling the inhibition of parasitic effects. However $m$ depends on the above various parameters for which minimal and maximal threshold values exist (e.g., the mutation rate has to be \textit{tolerable}). 

Through the use of a stochastic modelling technique (Master equation) which accounted for space, McCaskill et al. showed that a range of diffusion coefficients existed and permitted the survival of the catalyst $X$, despite being parasitised by $Y$. This range was dependent on the mutation and decay rates. As the mutation or decay rate increases, the range of diffusion coefficient decreases until it disappears. 

A number of significant differences exist between McCaskill et al.'s stochastic model and the \MCS: 
\begin{itemize}
	\item The \MCS~is a constructive AC where a rich variety of molecular species exists. If we consider a maximal species string length of molecular species of $BD_{Lmax}=500$, then there are $8^{500}$ possible combinations of molecular species, the nature of whose interactions may vary greatly. In McCaskill et al.'s model, only two species and five reactions are considered. 
	\item The parasites involved in the degenerative evolutionary dynamics (Section \ref{riseFall}) are capable of enzymatic functions. When these particular parasites react with copies of themselves, a further new enzymatic species is generated. We showed that this newly created species is also capable of parasitising both the parent parasite species and ancestor/seed catalysts. Whereas in McCaskill et al.'s model, the replication of mutant species $Y$ can only be catalysed by $X$. 
	\item Mutations are implemented differently in the \MCS. A per-symbol mutation probability is employed. As the length of a species increases, the probability of a mutation to occur increases accordingly. McCaskill et al. employed a fixed mutation rate which is not affected by the nature of the molecular species.
	\item Further differing details exist between the \MCS~ and McCaskill et al.'s model. These points will be addressed in the description of the extended \MCS~ model (Section \ref{subsec:diffmodel}).
\end{itemize}

Although significant differences exist between both models, the parasitic effects and degenerative phenomena studied in the original non-compartmentalised models are essentially similar: The exploitation of catalytic species by mutant parasitic species disrupts the self-maintaining cycle of the closed reaction network. 

Using McCaskill et al.'s work on compartmentalisation and evolutionary capability as an inspiration, we propose an extended version of the \MCS~which accounts for these compartmentalisation and diffusion properties. This extended \MCS~ will then be employed to conduct a set of evolutionary experimentations which will illuminate the effects of compartmentalisation and molecular diffusion upon the system's evolutionary capability.

\subsection{The model}
\label{subsec:diffmodel}
We present an extended \MCS~model which accounts for compartmentalisation and molecular diffusion:
\begin{itemize}
	\item In contrast to the model employed in Chapter \ref{emerSelfMainClosedRN}, successful molecular reactions (generating new molecules) do not lead to the removal of another molecule at random when the reactor is full. Here, a molecular reaction creating a new molecule occurs only if the carrying capacity $n_{max}$ of the reactor has not been reached. Reactors may frequently be filled when the molecular production rate is higher than the decay rate.	
	
	\item Moreover a decay probability $d$ is defined, this parameter addresses the continuous decay rate defined in McCaskill et al.'s approach. At each discrete timestep, a molecule is selected at random and may be removed, with the probability $d$, from the compartment. This spontaneous decay of molecules enables further reactions to occur in filled (i.e., saturated) reactors.

	\item	In McCaskill et al.'s model, molecular diffusions occurred continuously in time according to the diffusion coefficient $m$. However time is discretised in the \MCS. Consequently exchanges of molecules occur in a sequential fashion according to the time-slicing algorithm. A probability of molecular diffusion $p_m$ is introduced. At each timestep, an exchange of molecules between two compartments may occur with the probability $p_m$. Devising a probability instead of a fixed time interval was decided to avoid all compartments diffusing at the same time (in which cases the traceability of the system would not have been facilitated). Although the compartments are executed in parallel, all compartments could simultaneously diffuse in some cases, e.g., during the early phase of a simulation run where seed molecules and the nature of computations are equivalent in all compartments. 

	\item When a compartment $c_i \in C$ diffuses, some molecules contained in $c_i$ are selected at random. These selected molecules are removed from $c_i$ and constitute the \textit{diffusion initiation signal}. The size of this signal/vector of molecules is determined by the integer division of the container's current size $c_{i_n}$ by the diffusion coefficient $m$, denoted by $\left\lfloor \frac{c_{i_n}}{m}\right\rfloor$. This signal is then sent from $c_i$ to another compartment $c_j \in C$ which is selected at random. $c_i$ continues to operate following this signal emission.	When $c_j$ receives the diffusion initiation signal from $c_i$, $c_j$ similarly selects and removes a number of molecules according to $\left\lfloor \frac{c_{i_n}}{m}\right\rfloor$ (with  $c_{i_n}$ and $c_{j_n}$ being potentially different). The molecules received from $c_i$ are then inserted in the $c_j$ compartment.	The  molecules removed from $c_j$ constitutes the \textit{response signal} which is sent back to $c_i$. Upon receiving this response signal from $c_j$, the  molecules contained in this signal are similarly inserted into $c_i$. This concludes the exchange of molecules between $c_i$ and $c_j$. 
	
	\item The number of molecules exchanged between compartments is not fixed. Moreover this exchange of molecules may be asymmetric in some cases. Molecular exchanges are symmetric only when the two ``communicating'' compartments contain the same number of molecules. If we consider a molecular exchange between $c_i$ and $c_j$ with $c_{i_n}>c_{j_n}$ then $c_i$ would emit a higher number of molecules than $c_j$ with $\left\lfloor \frac{c_{i_n}}{m}\right\rfloor>\left\lfloor \frac{c_{j_n}}{m}\right\rfloor$. If $c_{i_n}>0$ and $c_{j_n}=0$ then molecules would be diffused only from $c_i$ to $c_j$. Diffusion equilibrium may be reached during subsequent molecular exchanges when $c_{i_n}=c_{j_n}$. 
\end{itemize}

In the following section, we conduct a series of evolutionary experiments using the above extended \MCS~model.
\subsection{The experiments}
We present a series of experiments using the extended parallel version of the \MCS. These experiments aim at demonstrating the effects of compartmentalisation and molecular diffusion upon the system's evolutionary capability.

The differences outlined earlier prevent us from mapping directly the various parameters into the \MCS. As a result we cannot determine analytically the range of suitable parameters, if any, which may allow for the stabilisation of evolution in the \MCS. Nevertheless we attempted to select parameter values that would not \textit{clearly} facilitate the disruptive parasitic effects to occur, e.g., $p_{sym}$ was deliberately set to a relatively low value to diminish the frequency of emergence of potentially parasitic mutant species. The following set of fixed parameters is proposed and utilised in all experiments: 
\begin{itemize}
	\item 30 compartments are utilised and executed in parallel using 30 AMD Opteron 270 (2.0 GHZ) CPUs.
	\item Experiments are run for 3600 seconds (1 hour of ``wall clock'' time).
	\item The maximal compartment carrying capacity is $n_{max}=1000$.
	\item The diffusion probability is set to $p_m=0.05$.
	\item The spontaneous decay probability is set to $d=0.1$.
	\item Similarly to experiences conducted in Chapter \ref{emerSelfMainClosedRN}, the maximal species string length is set to $BD_{Lmax}=500$.	
	\item The per-symbol mutation is set to $p_{sym}=1.0\e{-5}$.
	\item The global spontaneous mutation rate is set to $r_{mut}=0$.
	\item As in the evolutionary experiments presented in Section \ref{riseFall}, each compartment is seeded/initialised with the $s_{R_4}=\triangledown0101:\triangledown0101$ molecular species. However we completely fill each compartment with 1000 instances of $s_{R_4}$ species.
\end{itemize}

Considering the \MCS~ experiments conducted in Chapter \ref{emerSelfMainClosedRN}, McCaskill et al.'s investigation and the specification of the extended \MCS, we propose the following ideal scenario/prediction where degenerative evolutionary outcomes may be inhibited due to compartmentalisation and molecular diffusion in the extended \MCS:

\begin{enumerate}
	\item In this model, compartments which are infected by parasites would present a lower rate of successful catalytic reactions (as shown in Section \ref{riseFall}).
	\item  As a result the molecular production rate would decrease accordingly.
	\item If the production rate becomes lower than the decay rate then the compartment would start depleting. 
	\item Molecular diffusions occurring between non parasitised compartments (in which no depletion occurs) and infected/depleting ones would lead to asymmetric molecular exchanges, i.e., infected compartments would import more non-parasitic species and export less or no parasitic species (limiting the spread of parasites throughout the compartments). 
	\item Consecutive molecular exchanges would allow for the persistence and spreading of the non-parasitic catalysts throughout the compartment population.
\end{enumerate}

In this idealised scenario, compartmentalisation would allow for the isolation of infected compartments. Consequently the closed reaction networks would be able to self-maintain when subjected to disruptive parasitic effects.

To test the above prediction, we conduct three series of experiment (where 5 simulations were run in each series) in which the following diffusion coefficients are employed: $m_1=0.01$,  $m_2=0.05$ and $m_3=0.1$.

\begin{figure}[hb]
	\centering
	\rotatebox{-90}{\includegraphics[scale=0.45]{chapter6/exp0-0.01.eps}}
	\caption{Dynamics of the compartments' average population size and molecular species string length (averaged over the species length averages of each compartment) with diffusion coefficient $m_1=0.01$. We depict the dynamics of the molecular population size as it provides an approximate indication about the molecular production rate against the decay rate. For example if the production rate is higher than the decay rate then the occupancy compartments would be maximised and would only fluctuate due to molecular diffusion. On the contrary if the production rate is lower than the decay rate then we would perceive a decrease in the average population size.}
	\label{fig:exp0-0.01}
\end{figure}

Fig. \ref{fig:exp0-0.01} present an overview of the dynamics of an example simulation run where the diffusion coefficient is set to $m_1=0.01$. We identify the following chain of events: 

\begin{enumerate}
	\item We first note an initial phase where the system is stable with an average species length of 12 symbols (i.e., the length of $s_{R_4}$ species) and the average population size stagnating at nearly 1000 molecules. Most compartments are thus full during this phase (i.e., the molecular production rate is higher than the decay rate). 
	\item However from $t\approx 950$ we note that the average length of species starts to increase rapidly. This behaviour suggests that the elongation phenomenon is occurring. 
	\item Then 150 seconds later, at $t \approx 1100$, we observe a rapid decrease in the average population size throughout the 30 compartments. This indicates that the production rate has now become smaller than the decay rate, i.e., the compartments are depleting.
	\item Nevertheless this decrease does not apply to the species length which continues to increase until $t \approx 1270$, where the species length reaches its peak with an average length of 278 symbols. During the phase $950<t<1270$, successive species having an increasing length emerged and displaced each other. As a result we observe a linear increase in the average species length throughout all compartments.
  \item Following the peak of the species length observed at $t \approx 1270$, the average species length is then rapidly decreasing (similarly to the average population size) until $t\approx 1310$ where the system becomes extinct, i.e., all molecules have decayed.
\end{enumerate}

In Fig. \ref{figure:twographics1}, we focus on the extinction phase and present the standard deviations of the compartments' population size and species string length.

\begin{figure}[htbp]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.01-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.01-stdev-strlen-zoom.eps}}}
\caption{Extinction phase of example simulation 1 with diffusion coefficient $m_1=0.01$. For clarity purposes, both curves were plotted using a point interval of 3.}
\label{figure:twographics1}
\end{figure}

Fig.~\ref{figure:twographics1} complements the current analysis by providing more detailed information about the dynamics of each compartment:

\begin{enumerate}
	\item During the early stable phase, we note that the molecular string length is relatively homogeneous throughout the 30 compartments with little variance occurring. This assertion applies for both the average population size and species string length.
	\item However when $t\approx 1100$, we observe a divergence in the composition of the compartments (i.e., compartments with different population sizes exist). This variance is maintained until close to the end of the extinction phase. 
	\item Although the composition of compartments starts diverging at $t\approx 1100$, the average species length (which is globally increasing) is more or less homogeneous throughout the compartments until $t\approx 1230$.
Thus during the phase $1000 < t < 1230$, the mutant species (here classified by their string length) are well diffused throughout the compartments. However as the average molecular population size is already decreasing (since $t>1100$), it indicates that these mutant species have a production rate lower than the decay rate.
	\item When $t>1230$, a significant range of variances is observed in the species string length. This phenomenon suggests that only a few reactions leading to the creation of much longer species are succeeding in some compartments.

\end{enumerate}

The above degenerative scenario is characteristic of the elongation catastrophe phenomenon. We first observed the successive emergence of mutant species having an increasing length. We distinguished successive displacements which led the system to a state where successful catalytic reactions occurred less often. However a decay rate is applied in the current system, and consequently the compartments started to deplete as the production rate became lower than the decay rate. As a result, we ultimately obtained the extinction of the system where all species have decayed.

The behaviour reported above was exhibited in all 5 simulation runs in which $m_1$ was employed. Moreover this degenerative dynamic was also observed in the supplementary experiments in which we set the diffusion coefficient to $m_2$ and $m_3$. The distinctive extinction phases of example simulation runs are shown in Fig.~\ref{figure:twographics3}. All remaining simulation runs are presented in Appendix \ref{app:mcs_diff}.

\begin{figure}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.05-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.05-stdev-strlen-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.1-stdev-popSize-zoom.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.1-stdev-strlen-zoom.eps}}}
\caption{Example simulation runs 2 and 3 using the diffusion coefficients $m_2=0.05$ (top graphics) and $m_3=0.1$ (bottom graphics) respectively.}
\label{figure:twographics3}
\end{figure}

In Fig.~\ref{figure:twographics3}, we distinguish that the range of variances in the average molecular population size actually decreases in contrast to the increase trend reported earlier. Little variance of average molecular string length is observed over the whole simulation runs excepting during a few seconds where the systems collapse. These results suggest that higher diffusion coefficients allowed for the diffusion equilibrium to be achieved more rapidly. Consequently we identify a homogeneous molecular composition throughout all compartments over time.

Moreover we note a second trend related to the speed of the extinction phases which seems to be correlated with the coefficient diffusions. Table \ref{tab:extinctionSpeed} presents the duration of the extinction phase for all experiments. 

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c||c|c|c|c|c||c|}\hline
Series	 & 1  &  2  &  3  &  4  & 5 & Avg. \\ \hline\hline
$m_1$ &	279 & 243 & 223 & 266 & 202 & 242.6 \\ \hline
$m_2$ &	145 & 189 & 149 & 176 & 129 & 122.4 \\ \hline
$m_3$ &	95  & 120 & 103 & 112 & 78  & 101.6 \\ \hline	
	\end{tabular}
	\caption{Duration in seconds of extinction phases for the 3 series of experiments conducted with parameters $m_1$, $m_2$ and $m_3$. We devise an arbitrary threshold (i.e., when the average molecular string length is higher than 20) for determining the start of the extinction phases. This string length criterion is employed as it is a clear symptomatic evidence of the elongation catastrophe phenomenon.}
	\label{tab:extinctionSpeed}
\end{table}

In Table \ref{tab:extinctionSpeed}, we observe that as the diffusion coefficient increases, the duration of the extinction phase decreases. A higher diffusion coefficient accelerates the necessary duration for reaching diffusion equilibrium, which in this case has facilitated the spreading of mutant parasites throughout the compartments. Consequently the elongation catastrophe phenomena occurred at a faster pace. These results also suggest that if we were to decrease the key parameter (i.e., diffusion coefficient), this may only extend the duration of the extinction phase which would still occur quasi-deterministically upon the emergence of mutant-parasitic species. 

The above results obtained using the static reactors \MCS~model with molecular diffusion contradict the prediction that we proposed earlier. The selected range of parameters did not allow for the isolation of infected compartments. 

Nevertheless according to our prediction, for an effective isolation and inhibition of the degenerative phenomenon to occur, we require the depletion of the infected compartments. This depletion is here happening \textit{globally} at a late stage where the parasites have already invaded all compartments (through the exploitation of molecular diffusion). At that stage no healthy compartments remain which prevents any attempts to regulate and stabilise the system via diffusion.

For a rapid depletion to occur \textit{locally} in infected compartments, we suggest that additional experiments should focus on modifying the diffusion probability which affects indirectly the diffusion coefficient and speed of diffusion equilibrium. A significant decrease of this diffusion probability may result in the partial  depletion of infected compartments. This depletion would occur before the mutant species have the opportunity, through molecular diffusion, to spread over all compartments.

Nevertheless if this probability of diffusion is set too low, we would obtain a model where infected compartments would first decay entirely before being subjected to molecular diffusion. In such cases, there would be \textit{no exchange} of molecules, the migration would only be one way. Such a model would then less account for molecular diffusion and the nature of the reactor model would not be as \textit{continuous} (i.e., we would observe the successive depletions and growths of molecular species in the compartments). This resulting model would become more similar to the \textit{cellular} model approach (which will be described later) than McCaskill et al.'s original proposed model.

Further experiments are necessary to identify a \textit{potentially} suitable range of parameters where the degenerative effects of parasitism can be controlled, whilst \textit{exchanges} of molecules still occur between compartments.

\subsection{Conclusion}
\label{sec:McCaskillConc}
We introduced the static reactors with molecular diffusion model which was examined by McCaskill et al. The latter demonstrated that for such models, according to the mutation and decay rates, there exists a range of diffusion coefficients which allow for the stabilisation of the self-maintaining cycle of closed reaction networks in which parasitic effects are occurring.

We presented the extended model of the \MCS~which accounted for compartmentalisation and molecular diffusion. Using this system we conducted a series of evolutionary experiments to examine evolutionary capability. 

Our experiments suggested that according to the parameters used, compartmentalisation and molecular diffusion do not allow for the inhibition of degenerative evolutionary phenomena. 

However as we did not conduct a systematic evaluation of parameters due to time constraints and the  multi-dimensional nature of the search space (example dimensions are the mutation and decay rate, diffusion probability and coefficient). We only explored one area of this vast search space where it may still be possible to find a suitable set of parameters which would stabilise evolution in the \MCS.

More specifically we discussed the potential role of the diffusion probability, a feature introduced in the extended \MCS~ but not present in McCaskill et al.'s continuous model. We argued that decreasing this parameter may allow for the local depletion of infected compartments to occur. This isolation of infected compartments would prevent the parasitic species from invading the remaining non-parasitised compartments through molecular diffusion. Therefore this alternate solution may lead to the inhibition of degenerative dynamics due to parasitism.

Nevertheless, although this model with a low diffusion probability may control the elongation catastrophe phenomenon, this resistance against parasistism would not be directly due to molecular diffusions. In McCaskill et al.'s model, molecular diffusion was the key regulator where the number of molecules, contained in the reactors, remained more or less static. In this proposed model, degenerative outcomes would be prevented by allowing the parasitised compartments to deplete/decay before integrating species incoming from neighbouring and non-parasitised compartments. Parasitised compartments would first decay and molecular diffusion would occur in one direction, from non-decayed compartments to decayed ones. Following this unidirectional molecular exchange, the received species would increase in number in the formerly parasitised compartment. In such cases, the resulting model would thus be more adequately captured as a dynamic reactors model with unidirectional molecular diffusions. Such a model is, to some extent, quite analogous to the cellular model presented in the next section.

\section{A cellular model}
\label{sec:celmod}
We present a second multi-level selectional model applied to the \MCS~in which, similarly to biological cells, compartments may grow and divide. This version of the \MCS~is independent from the previously extended model and does not include the properties (complementary to compartmentalisation) introduced in Section \ref{sec:McCaskillsModel}.
\label{sec:multi}

\subsection{Introduction}
\cite{cronhjort1997ccm} proposed a deterministic model (using PDEs) where \textit{dynamic} clusters (regarded here as compartments) of molecules would spontaneously arise, grow, divide and die in a two-dimensional space. 

In Cronhjort and Blomberg's model, molecules would aggregate spontaneously and form compartments without possessing a complex membrane structure. As mentioned earlier, compartments have also the ability to divide spontaneously. It was shown that, when parasites are introduced in a compartment, they rapidly destabilise the self-maintaining cycle of closed reaction networks contained in the compartment. This destabilisation ultimately leads to the decay of all molecules. Moreover when a compartment is infected with parasites, the latter spread to other compartments through parasitising the molecular species which are present \textit{between} the compartments. By exploiting this propagation/diffusion technique, the parasites can invade and ``kill'' all remaining compartments. 

However Cronhjort and Blomberg introduced a cut-off rule which restricted the spread of parasites throughout the compartments. 
This rule sets to zero (with a defined cut-off probability) the concentration of the molecular species which occur between the compartments (without affecting the molecular concentration in the compartments). Using a suitable cut-off value, infected compartments may become isolated in space. As a result, parasitic species decay locally without invading the rest of the compartments. Following this, an empty space emerges which may be occupied by a new compartment resulting from the division of a neighbouring ``healthy'' non-parasitised  compartment.

In this model, compartmentalisation isolated the infected compartments which successfully prevented the invasion of parasitic species over the rest of the compartments. Cronhjort and Blomberg demonstrated that this cellular model may provide resistance against disruptive parasitic effects. We utilise this work as an inspiration and implement compartmentalisation and cellular division features in the \MCS. The goal is to realise a sufficiently robust \MCS~system which would prevent the degenerative evolutionary dynamics from occurring.

Finally McCaskill et al. and  Cronhjort and Blomberg's models share common properties (e.g., definition of continuous production and decay rates, nature and interactions of catalytic/parasitic, etc) which differ from the \MCS. These differences are presented and addressed in the following section.

\subsection{The model}
\label{subsec:cellModel}
In the remainder of this section, compartments which can grow and divide are referred to as ``cells''. We present an extended \MCS~model which accounts for compartmentalisation and cellular division:
\begin{itemize}
	\item In contrast to Cronhjort and Blomberg's model, we define clear compartmental boundaries. Molecular species are contained in distinct compartments/cells which are simulated on individual CPUs. The cells possess a maximal carrying capacity of $n_{max}$ molecules.
	\item We employ a simplex topology to condition the interactions between cells as opposed to the two-dimensional space utilised by Cronhjort and Blomberg.
	\item Similarly to McCaskill et al.'s approach, catalytic reactions creating new molecules do not lead to the removal of another one in the cell. However, these reactions may occur only if the cell is not full (i.e., iff $c_i \in C, c_{i_n}<n_{max}$).

	\item When a cell $c_i$ becomes full (i.e., $c_{i_n}=n_{max}$), a cellular division occurs as follows.
	 $c_i$ selects $\frac{n_{max}}{2}$ molecules at random. These molecules are removed from the cell and constitute the signal to be sent to a target cell $c_j \in C$. The latter is selected at random. $c_i$ continues to operate with no further interactions (directly associated with the current division event) with $c_j$. When $c_j$ receives the signal sent by $c_i$, all molecular species contained in $c_j$ are removed. Following this, $c_j$ inserts the molecules included in the signal into its own reaction space. This step concludes the cellular division process of $c_i$ which offspring effectively displaced $c_j$.	



	\item In contrast to both McCaskill et al. and Cronhjort and Blomberg's models, we do not define a decay rate or decay probability as we did in the previous extended \MCS~model. In the static reactors model with molecular diffusion, reactions could occur in a given reactor only if the latter was not saturated. In such a saturated compartment, the spontaneous decay of molecules enabled further reactions to occur. In Cronhjort and Blomberg's model, the decay of a parasitised compartment led to the creation of a vacant space which then allowed for the formation of a new compartment (resulting from the division of a neighbouring non-parasitised compartment). Both the saturation of compartments and the displacement of decayed ones are dealt with by our specification of our cellular division mechanism. The latter prevents compartments from becoming saturated (i.e., half all of the molecules are ``diffused'' when $n_{max}$ is reached) and also triggers the displacement of potentially decayed compartments. Consequently the specification of a spontaneous decay rate is not required in the current extended \MCS~model.
 
 	\item Finally, in contrast to Cronhjort and Blomberg's model, \textit{any} cells (including parasitised ones) may still divide upon producing $n_{max}$ molecules. Therefore even though parasites have invaded a given cell, the latter may still spread the parasitic species through cellular division.
\end{itemize}

In the following section we present and examine a series of experiments using the above extended \MCS~ model. Following this, we analyse the effects of chemical kinetics in this particular extension of the \MCS.

\subsection{Cell-level mutations}
During cellular divisions, molecules are randomly selected and transferred into the offspring cells. Due to the stochastic nature of these processes, some molecular species may not be transmitted to the offspring cells. Moreover, the concentration of the transmitted molecular species may also significantly vary.

Such ``mutant'' cells would contain novel reaction networks exhibiting differing dynamics. We refer to these error-prone transmissions as \textit{cell-level mutations}.

Note that the term ``mutation'' is here relaxed and does not specifically refer to the commonly known phenomenon where variance may be introduced over \textit{strings of monomers}. We nevertheless use this notion to capture the stochastic recombination involved in this process which may affect the dynamics/phenotype of the cellular species.

\subsection{The experiments}
\label{subsec:cellModelExps}
We present a series of experiments using the extended parallel version of the \MCS. These experiments aim at investigating the effects of compartmentalisation and cellular division upon the system's evolutionary capability.

We propose the following set of fixed parameters which is utilised in all experiments:
\begin{itemize}
	\item 30 compartments are utilised and executed in parallel using 30 AMD Opteron 270 (2.0 GHZ) CPUs.
	\item The maximal compartment carrying capacity is $n_{max}=1000$.
	\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 Each compartment is seeded/initialised with 250 instances of the $s_{R_4}=\triangledown0101:\triangledown0101$ molecular species. 
\end{itemize}

As mentioned in the conclusion of Section \ref{sec:McCaskillsModel}, we predict that in such compartmentalised models, infected compartments should ultimately decay. However in the current model no spontaneous decay is implemented. Infected cells would here present a low or null growth rate which would prevent the cell from dividing, as opposed to non-parasitised cells which may still grow, divide and displace any cells. As shown by Cronhjort and Blomberg, this isolation of parasitised cells prevents parasitic species from spreading over the rest of the cells. Compartmentalisation and cellular division may then inhibit the degenerative phenomena from occurring in the \MCS. However a mutation error rate threshold may still exist. Indeed, if the mutation rate is too high, then parasitic species may emerge too rapidly in the cells and potentially kill all cells.

We conduct three series of 5 simulation runs each. In each of these series, we decrease the per-symbol mutation probability using the following parameters: $p_{sym1}=1.0\e{-4}$, $p_{sym2}=5.0\e{-5}$ and $p_{sym3}=1.0\e{-5}$.  
As the mutation rate is decreased in the successive experiments, the execution time of the simulation runs is increased as follows: 3600, 7200 and 36000 seconds in series 1, 2 and 3 respectively.
	
\begin{figure}[ht]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.0001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp1-0.0001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp2-0.0001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp3-0.0001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp4-0.0001.eps}}}
\caption{5 example simulation runs with $p_{sym1}=1.0\e{-4}$. The average reaction success rate represents the average number of catalytic reactions divided by the number of molecular interaction per second, e.g,  if the reaction network is complete then the ratio is 1. In the contrary if only elastic reactions occur then the ratio is 0. As the cells are seeded with species $s_{R_4}$, the initial reaction networks are complete.}
\label{figure:seriesA}
\end{figure}

In the first series of experiment (where $p_{sym1}$ is employed, see Fig.~\ref{figure:seriesA}) we note that the degenerative dynamics occur in the simulation runs 1, 3 and 4. The ``extinction phases'' observed in these simulation runs are all associated with a sudden increase in the molecular species' length, suggesting the elongation catastrophe phenomenon. However in runs 2 and 5, no extinction phenomena occur. In both simulation runs, we observe little variance in the reaction networks' level of catalytic activity which is described here by the average reaction success rate. 

\begin{figure}[ht]
	\centering
	\includegraphics[scale=1.4]{chapter6/exp0-0.0001-3d.eps}
	\caption{Classification and evolution of cellular lineages from simulation run 1. The cells are classified by the average length of molecular species at a given timestep. We employ a crude classification criterion to discriminate the parasitised cells from the non-parasitised ones.}
	\label{chapter6/exp0-0.0001-3d.jpg}
\end{figure}

Nevertheless, in all simulation runs, we identify a comparable level of variance in the average species string length. This indicates that, as expected, parasitic species emerged and invaded a number of cells, as shown in Fig.~\ref{chapter6/exp0-0.0001-3d.jpg} with the sporadic presence of cells containing species with a higher average string length (which causes the high variances/peaks in the average species length). Moreover we distinguish a clear initial lineage of cells where the average species length vary around 12 (the length of the seed species $s_{R_4}$). 

\begin{figure}[ht]
	\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.00005.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp1-0.00005.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp2-0.00005.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp3-0.00005.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp4-0.00005.eps}}}
\caption{5 example simulation runs with $p_{sym2}=5.0\e{-5}$.}
\label{figure:seriesB}
\end{figure}

In the second series of experiments, the mutation rate is decreased to $p_{sym2}=5.0\e{-5}$, see Fig.~\ref{figure:seriesB}. Here, we distinguish the survival of the cells in all simulation runs. Moreover, we observe a phenomenon which was not present in the first series:
\begin{itemize}
	\item In all runs we identify the emergence of cellular lineages in which the molecular species have a longer average string length (two example lineages are depicted in Fig.~\ref{figure:seriesBa}). However we do not observe an exponential growth of the species' string length as commonly observed.
	\item  These species remain stable and self-maintain for a period of time after which they may eventually be displaced by other species having, similarly, a longer average string length.
	\item  These cellular lineages are due to mutant species which were able to gain catalytic support from the ancestor species. However in contrast to the parasites involved in the elongation catastrophe phenomena, those species do not catalyse the production of further parasite species. Those species can in fact catalyse the production of species which are equivalent to the ancestors species from a phenotypic point of view. 
	\item However an additional function of those species is also to elongate the substrate species. This elongation occurs in a linear fashion as opposed to the exponential string growth observed in the elongation catastrophe phenomenon. 
	\item This slow/linear growth of species length had no immediate \textit{negative} effects for the cells (in contrast to an exponential string growth which would rapidly cause the cell to degenerate) and therefore permitted the diffusion of these slightly longer species throughout the cellular population.
	\item Nevertheless as the species length increased, it soon became a selective disadvantage for cells to contain such ``elongator'' species. As a result, these cells were selected against and the increase in the average string length ceased. This fitness penalty did not apply to the cells from the start,           when these elongators emerged (and shortly after),	as the difference in the average molecular species length was then not significant from a computational point of view. 
\end{itemize}


The difference in average string length in both lineages is due to the nature of the patterns that were concatenated to substrate molecules. This pattern is longer in simulation runs 2 and 5 and explains the higher average molecular string length. 

Note that the reproduction rate of the cells containing molecular species with a significantly longer string length should be lower (due to a higher computational demand) than the reproduction rate of cells containing shorter molecules. We may thus hypothesise that over time, these cell lineages containing longer molecular species might be displaced if a mutant molecular species emerges and possesses the following intrinsic properties:
\begin{enumerate}
	\item The molecular string length is short enough to exhibit a significant difference in fitness as autocatalysis is here hyperbolic (Section \ref{sec:introEmerSelfMain}).
	\item The ability to counter-act the exploitation of the elongators by developing an immunity to these pseudo-parasites or being similarly able to exploit the elongators to be replicated/or produce smaller molecules. 
\end{enumerate}
The emergence of such molecular species has not been observed in any of the simulation runs.

\begin{figure}
\subfigure{	\includegraphics[scale=0.285]{chapter6/exp0-1-0.00005-3d.eps}}
\caption{Evolution of cellular lineages in simulation run 1 (left) and 2 (right). A level of homogeneity in the molecular composition of most cells is observed at any given timesteps.}
\label{figure:seriesBa}
\end{figure}

Table \ref{tab:elongators} compares example elongators species triggering the elongation catastrophe (with exponential increase of string length) with those present in the above experiment (with linear increase of string length). Both species are mutants (resulting from a single mutation) of the self-replicase $s_{R4}=*\triangledown0101:\triangledown0101$.

\begin{table}[ht]
\centering
\begin{tabular}{|l|} \hline
Elongator species $s_1$ with exponential length growth\\  \hline
\hspace{4.45cm}$s_1^0=*\triangledown0101:\triangledown00101$ \\
$2\ s_1^0\rightarrow\ 2\ s_1^0\ +\ s_1^1$ with $s_1^1=*\triangledown0101:\triangledown000101$\\
$2\ s_1^1\rightarrow\ 2\ s_1^1\ +\ s_1^2$ with $s_1^2=*\triangledown0101:\triangledown00000101$ \\
$2\ s_1^2\rightarrow\ 2\ s_1^2\ +\ s_1^3$ with $s_1^3=*\triangledown0101:\triangledown000000000101$ \\
$\ldots$\\
$length(s_1^i)=length(s_1^0)+i^2$\\ \\
Moreover if we consider the reaction between two differing $s_1$ species \\generations with $i\neq j$ then:
$s_1^i\ +\ s_1^j\rightarrow\ s_1^i\ +\ s_1^j+\ s_1^{i+j}$
\\  \hline
Elongator species $s_2$ with linear length growth\\ \hline
\hspace{4.45cm}$s_2^0=*\triangledown0101:0\triangledown0101$\\
$2\ s_2^0\rightarrow\ 2\ s_2^0\ +\ s_2^1$ with $s_2^1=0*\triangledown0101:0\triangledown0101$ \\
$2\ s_2^1\rightarrow\ 2\ s_2^1\ +\ s_2^2$ with $s_2^2=00*\triangledown0101:0\triangledown0101$ \\
$2\ s_2^2\rightarrow\ 2\ s_2^2\ +\ s_2^3$ with $s_2^3=000*\triangledown0101:0\triangledown0101$\\
$\ldots$\\
$length(s_2^i)=length(s_2^0)+i$\\
$s_2^i\ +\ s_2^j\rightarrow\ s_2^i\ +\ s_2^j+\ s_2^{j+1}$ \\ \hline
\end{tabular}
\caption{Example elongator species $s_1$ and $s_2$ with exponential and linear string length growth respectively.  Both $s_1^0$ and $s_2^0$ represent the first generation of $s_1$ and $s_2$ species, whereas $s_1^i$ and $s_2^i$ are the $(i+1)^{th}$ generation.}
\label{tab:elongators}
\end{table}

Fig.~\ref{figure:seriesC} presents the final series of experiments where the mutation probability $p_{sym3}$ is employed. We note that all cells successfully resisted against the disruptive parasitic effects. We also distinguish a net decrease in the density of parasitised cells (see Table \ref{tab:densitySpike}) which consequently affected the average reaction success rate. The latter remained stable with little variance occurring (when compared with previous experiments) throughout the simulation runs. 

\begin{figure}[ht]
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp0-0.00001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp1-0.00001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp2-0.00001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp3-0.00001.eps}}}
\subfigure{	\rotatebox{-90}{\includegraphics[scale=0.28]{chapter6/exp4-0.00001.eps}}}
\caption{5 example simulation runs with $p_{sym3}=1.0\e{-5}$.}
\label{figure:seriesC}
\end{figure}

\begin{table}[htbp]
	\centering
	\begin{tabular}{|c||c|c|c|c|c||c|}\hline
Series	   &   1  &  2  &  3  &  4  & 5 & Avg. \\ \hline\hline
$p_{sym1}$ &	0.141 & 0.145 & 0.204 & 0.165 & 0.145 & 0.160 \\ \hline
$p_{sym2}$ &	0.060 & 0.029 & 0.056 & 0.071 & 0.020 & 0.047\\ \hline
$p_{sym3}$ &	0.021 & 0.016 & 0.012 & 0.021 & 0.013 & 0.017 \\ \hline	
	\end{tabular}
	\caption{Density of high variances in the average molecular species length, i.e., number of peaks where the species length increased significantly (where the average species length at time $t$ is 1.5 times higher than at $t-1$) per second.}
	\label{tab:densitySpike}
\end{table}


The probability of mutation $p_{sym3}$ employed in this final series of experiment also allowed for the self-maintenance of closed reaction networks in all cells throughout the simulation runs. Similarly to the second series of experiment, the emergence of non-lethal ``elongator'' species increasing the average string length is observed but no degenerative outcomes occur. 

As the mutation rate decreased, the probability of emergence of mutant parasitic molecules decreased. As the latter diminished, the number of cells where the elongation phenomenon may occur \textit{simultaneously} reduced, as shown in Table \ref{tab:densitySpike} with the number of peaks (high variance in average species length). These results suggest that a minimal number of infected cells is necessary to spread (rapidly) enough parasites throughout the cellular population, and consequently to provoke extinction at the system level. Moreover these results indicate that a mutation error threshold may exist in the range $p_{sym}<1.0\e{-4}$ in the \MCS/cellular model. This final series of experiment suggests that the cellular model of the \MCS~ is able to provide the closed reaction networks with resistance against parasites and associated degenerative effects. However this control in evolutionary degeneration is possible given a tolerable rate of mutation. 

In the next section, we analyse how chemical kinetics may also have contributed to this control of parasitic effects by providing a selective advantage to non-parasitised cells over the parasitised ones in specific cases.

\subsection{Effects of chemical kinetics}
We examine the effects of chemical kinetics upon the system's evolutionary dynamics. A side effect of the parallel implementation of the \MCS~ is the alteration of chemical kinetics, as presented in Section \ref {subsec:kinetics}. We discuss here the potential role of chemical kinetics in improving resistance against parasites in particular cases.

As mentioned earlier, parasitised cells may still divide as they may \textit{even} contain reaction networks that are still complete (i.e., all molecular collisions lead to successful creations of new parasitic molecules). Let us consider the cells $c_i \in C$ and $c_j \in C$ which both contain complete reaction networks. The cell $c_i$ contains non-parasitic species only, as opposed to $c_j$ in which we insert parasites only. 

If all reaction rates are equal, then both $c_i$ and $c_j$ would exhibit an equal growth rate. As any reactions occurring in both cells would occur at the same pace, both cells would thus possess an equivalent growth rate and may ultimately divide simultaneously. Both cellular species would have equal chances to displace each other, this scenario applies \textit{until} the parasitised cell has started to decay (with the reaction success rates decreasing). 

However, if we consider variations in the reaction rates, which is addressed here by the necessary computational time to process a reaction in the parallel version of the \MCS, then we argue that non-parasitised cells possess a selective advantage. In Section \ref{riseFall}, we showed that the elongation catastrophe was due to mutant parasitic species which had the effect of increasing their string length repeatedly. However in the parallel system, as the length of species increases, the computation necessary to process reactions between these species increases accordingly. Reactions involving parasite species as the reactants would therefore be computationally more demanding. As time matters in this parallel system, the growth/production rate of parasitised cells would be lower than the rate of non-parasitised cells.

Therefore we hypothesise that, in the current extended version of the \MCS, parallelism improves resistance against parasitism where the elongation catastrophe phenomena occur.

To test the above hypothesis, we present a simple experiment in which we employ the following set of parameters:

\begin{itemize}
	\item 30 compartments are utilised and executed in parallel using 30 AMD Opteron 270 (2.0 GHZ) CPUs.
	\item Experiments are run for 60 seconds.
	\item The maximal compartment carrying capacity is $n_{max}=1000$.
	\item The maximal string length is set to $BD_{Lmax}=500$.	
	\item A single cell $c_1$ is seeded with 2 instances of $s_{R_0}=\triangledown0:\triangledown0$.
	\item The remaining 29 cells $c_2, \ldots, c_{30}$ are seeded with 2 instances of $s'_{R_0}=\triangledown0:\triangledown00$. $s'_{R_0}$ triggers the elongation catastrophe.
	\item Both the global spontaneous mutation rate $r_{mut}$ and mutation per symbol probability $p_{sym}$ are set to 0.
\end{itemize}

Mutation is turned off in order to restrict the diversity of species to only $s_{R_0}$ and the lineage of parasites species $s'_{R_0}$.

Only the single healthy cell $c_1$ was specified against 29 parasitised cells, which is clearly the most disadvantageous situation for the survival of $c_1$. In a non-parallel system, $c_i$ would have only a $\frac{1}{30}$ chance not to be displaced during the first ``round'' of cellular divisions.

5 distinct simulations are run. The results are shown in Fig.~\ref{fig:concurrency}.

\begin{figure}
	\centering
		\includegraphics[scale=0.8]{chapter6/concurrency.eps}
	\caption{Dynamics of average number of non-parasitised cells (averaged over 5 independent experiments. The error-bars denote the minimal and maximal number of non-parasitised cells throughout all experiments at a given timestep. The reaction success rate, not described here, was maintained to one during all simulation runs (due to the completeness of the reaction networks.}
	\label{fig:concurrency}
\end{figure}

In Fig.~\ref{fig:concurrency}, an early phase is observed where the average species length is increasing, suggesting that the elongation catastrophe is occurring in the cells. However this trend diminishes (down to 6, i.e., the length of $s_{R_0}$) in contrast to the average number of non-parasitised cells which increases until all parasitised cells have been displaced. This displacement occurred due to the fast reproduction rate of non-parasitised cells which were provided here with a selective advantage over the parasitised cells. Fast reactions permitted the non-parasitised cells to divide more rapidly and were subsequently given the opportunity to displace the parasitised cells.

This experiment demonstrates the potential improvement in evolutionary capability due to the new chemical kinetics properties introduced in the parallel \MCS. Nevertheless, this experiment was an ideal scenario where no mutation could occur and all cells where initiated at the same time. Consequently, the non-parasitised cell was always the first to trigger its cellular division and \textit{successive ones} over the rest of the parasitised cells. 

\subsection{Conclusion}
We introduced Cronhjort and Blomberg's cellular model in which compartmentalisation and cellular division prevented degenerative effects due to parasistism from occurring. Using this work as an inspiration, we proposed an extended parallel implementation of the \MCS~ which accounts for these cellular features. 

Using this extended version of the \MCS~, we conducted and examined a series of evolutionary experiments in which we varied the mutation rates. Our results showed that as the mutation rate decreased, the emergence and density of parasitised cells decreased. We showed that a level of parasitised cells is necessary for the parasites to spread rapidly enough throughout the cellular population and to  eventually cause extinction at the system level. If no such rapid diffusions occur, then the parasitised cell(s) would decay locally and will be replaced by non-parasitised cells resulting from the division of other cells.

We also showed how chemical kinetics may improve, in particular cases, resistance against parasites. This feature resulted from the parallel implementation of the \MCS.

This cellular model successfully improved the \MCS's evolutionary capability and, according to the level of mutation rates, prevented the degenerative elongation catastrophe phenomena from occurring.

\section{Summary}
We presented two distinct multi-level selectional models applied to the \MCS. This work aimed at resolving the evolutionary degeneration issues of the \MCS. 

We first examined a parallel implementation of the \MCS~which was inspired by the static reactors model with molecular diffusion investigated by McCaskill et al. Our result suggested that the range of parameters selected in our experiments could not allow for the self-maintenance of closed reaction networks when subjected to perturbations. Nevertheless we mentioned that future work could illuminate further on the application of this model upon the \MCS~to improve evolutionary capability. 

Our second attempt addressed compartmentalisation and cellular division. These features were present in a model previously examined by Cronhjort and Blomberg. We conducted a series of evolutionary experiments using a novel parallel implementation of the \MCS~ where we introduced compartmentalisation and cellular division properties. Our results indicated that this cellular model was able to improve the resistance of closed reaction networks against the disruptive effects due to parasitism, including the elongation catastrophe phenomena. 

The above investigations provide complementary insights on the potential effects of compartmentalisation over evolutionary capability in agent-based Artificial Chemistries (ACs). As no analytically tractable methods are currently available for the study of such complex ACs, empirical investigations are necessary and may provide guidance on the construction and analysis of future evolutionary systems.

The conclusive outcomes obtained with the cellular model provided us with a novel and robust \MCS~system which we may now employ to conduct further evolutionary experiments. The latter will focus on the evolution of closed reaction networks to carry-out pre-specified information processing tasks. This follow-up investigation is presented in the next chapter.


 