# Emergence of a \"Cyclosome\" in a Primitive Network Capable of Building \"Infinite\" Proteins +
## 2 A Primitive Network at the Origin of Life  
In our hypothesis, amino acids were concentrated around the AL, which acted as a \"proto-nucleus\" to allow the first \"organ\" or \"cyclosome\" to synthesize peptides. Insofar as an object corresponds to a discontinuity in a field of connectivity [26], the boundary of this cyclosome corresponded to a discontinuity in the gradient of peptides around the AL. The boundary of the first functional \"machine\" able to build peptides can be defined as a peptide gradient boundary centred on the \"proto-nucleus\" AL, resulting from an amino acid confinement around the AL favoring the occurrence of peptide bonds. This \"organ\" functioned as a \"cyclosome\" in a \"proto-membrane\", thus as a \"proto-cell\" with a circular organization. This proto-cell is a solution to the problem of how to obtain autopoiesis: Peptide synthesis favored by the AL was necessary to repair the proto-cell membrane made of hydrophobic peptides and lipids, which reciprocally protected the AL against denaturation by ensuring the integrity of the proto-nucleus. The autopoiesis network underlying this organization has been studied in [27; 28; 29] and exhibits exponential growth if the peptide proto-membrane allows the entry of nucleic acids for AL replication. We can represent its dynamics by defining the variables of the network and their interactions using a system of differential equations (1) whose Jacobian graph is given in Figure 1: Let us denote by R, A, B, E, M, and P for, respectively, the concentration of AL Ring, Amino acids, nucleotide Bases, hydrophilic Enzyme peptides, hydrophobic Membrane peptides, and the Pool of lipids plus the elements C, N, and H\({}_{2}\)O:  
\[ \begin{array}{c}\mathrm{dR/dt}=\mathrm{d_{R}\Delta R}+\mathrm{k_{B}B}- \mathrm{k_{R}R}\\ \mathrm{dA/dt}=\mathrm{d_{A}\Delta A}+\mathrm{k_{P}P}-\mathrm{k_{A}RA}-\mathrm{ k_{{}^{\prime}E}RA}\\ \mathrm{dB/dt}=\mathrm{d_{B}\Delta B}+\mathrm{k^{\prime}P}-\mathrm{k_{B}B}\\ \mathrm{dE/dt}=\mathrm{d_{E}\Delta E}+\mathrm{k^{\prime}E}\mathrm{RA}-\mathrm{ k_{E}E}\\ \mathrm{dM/dt}=\mathrm{d_{M}\Delta M}+\mathrm{k_{A}RA}-\mathrm{k_{M}M}\\ \mathrm{dP/dt}=\mathrm{d_{P}\Delta P}+\mathrm{k_{R}R}+\mathrm{k_{M}M}-\mathrm{ k^{\prime}P}-\mathrm{k_{P}P}\end{array}  \tag{1}\]  
In the absence of diffusion (diffusion coefficients \(\mathrm{d_{i}}\)'s equal to 0), the differential system (1) has an initial exponential growth behaviour and tends, if P is constant and initial values of variables are not zero (which corresponds to an unstable steady state), towards the unique stable stationary state:  
\[\mathrm{(R^{*}A^{*}B^{*}E^{*}M^{*})}=\mathrm{(K/k_{R},K^{*}k_{R}/K(k_{A}+k^{ \prime}E),K/k_{B},K^{*}k_{E}/k_{E}(k_{A}+k^{\prime}E),K^{*}k_{A}/k_{M}(k_{A}+k ^{\prime}E))} \tag{2}\]  
with the following Jacobian matrix \(\mathrm{J^{*}}\) equal to:  
\[ \begin{array}{cccc}-\mathrm{k_{R}}&0&\mathrm{k_{B}}&0&0\\ -\mathrm{K^{*}k_{R}/K}&-\mathrm{K(k_{A}+k_{E})/k_{E}}&0&0&0\\ 0&0&-\mathrm{k_{B}}&0&0\\ \mathrm{K^{*}k_{R}k_{E}/K(k_{A}+k^{\prime}E)}&\mathrm{KK^{*}k_{R}}&0&-\mathrm{ k_{E}}&\\ \mathrm{K^{*}k_{R}k_{A}/K(k_{A}+k^{\prime}E)}&\mathrm{Kk_{A}/k_{R}}&0&0&- \mathrm{k_{M}}\end{array} \]  
whose characteristic polynomial (\(\mathrm{k_{R}+\lambda}\))(K(\(\mathrm{k_{A}+k^{\prime}E}\))/\(\mathrm{k_{E}+\lambda}\))(\(\mathrm{k_{B}+\lambda}\))(\(\mathrm{k_{E}+\lambda}\))(\(\mathrm{k_{M}+\lambda}\)) = 0 has only negative eigenvalues, ensuring the stability of the stationary state (2) (\(\mathrm{R^{*}A^{*}B^{*}E^{*}M^{*}}\)). If we add a diffusion term for the different metabolites, this dynamics leads to the spatial segregation of R, A, B, M, and P into structures like a protein-synthesizing machine made of the AL or the anti-sense AL (R), proto-cytoplasm (A, B), proto-membrane (M), and building blocks (P). The AL serves as a template for the formation of hydrophilic enzymatic peptides E (Figure 1) able to activate (or inhibit depending on their catalytic properties) the AL or anti-sense AL replication with a reaction constant \(\mathrm{k_{r}}\)[30].