Comparison of the expansion process for different strengths of the sequence-information parameter . Means of 200 simulations are shown, and for all panels the color code given in panel (c) holds. (a) The network size measured by the number of enzymes attached to the network is shown over time. The expansion velocity increases with in both normalized time and in absolute time (inset) obtained from the Gillespie algorithm. (b) The number of attachable enzymes at every step in the expansion process can be understood as the evolvability of the network. Using sequential information leads to a less evolvable but thus denser network. (c) How quickly do we expand to the border of existing knowledge? At every step in enzyme time, we plot the number of detected metabolites which only participate in one reaction in KEGG. Higher approach the border faster supporting the assumption of a smarter expansion. (d) Mean sequence distances between every new enzyme and its duplication partner. The curve decreases, since by chance for any new enzyme on average a similar sequence can be found if more enzymes are present in the current network. For higher , isolated sequences without any similarity to all others are preferentially found at the end resulting in an increase.
The acquisition of new enzymes happens in bursts of increasing strength for larger sequence sensitivity. Here we show one example run in (a)–(c) and means of 200 runs in (d) and (e). (a) Spike train with 1 bar at every incident of a new enzyme. The panel shows a window of 500 new enzymes for each on its particular normalized time. While for the enzymes appear almost equidistantly, larger leads to enzyme bursts. (b) Distribution of time intervals between any two new enzymes (IEI). For higher the distributions are shifted to smaller distances and exhibit multiple peaks. (c) The coefficient of variation measured in sliding frames of 100 enzymes indicates multiple characteristic time scales. The peaks point to times of evolutionary explosions. (d) The autocorrelation of IEIs supports the bursting behavior further. For large IEIs are strongly correlated on a short time scale whereas small lead to no significant correlation. (e) The fit of the data to the Fano factor of biased Brownian motion enables to estimate the correlation time . [For all color panels the legend of panel (e) holds.]
Time order of appearance of enzymes, amino acids and nucleotides, and entire organisms. (a) Time-ordered ranking of enzyme appearance for . From the graph of all time-ordered pairs of enzymes with , pairs also appearing in the -case are removed and only the paths of length 3 higher are shown (order-precision 100%). Time runs from top to bottom; the seed enzymes as root nodes are omitted for simplicity. (b) Appearance of amino acids (top part) and nucleotides (bottom part) sorted by the appearance and averaged over 200 runs. The order is very similar (rank correlation 0.7) to the order of robustness observed in the E. coli network (Ref. 33). Further, aromatic amino acids (labeled by ) are synthesized late. The -curves look similar indicating that the order strongly originates from stoichiometry rather than from sequence relations. (c) Every enzyme defined by its EC number is mapped to its genes and thus to the corresponding organisms. An organism is assumed to have evolved if 80% of its annotated enzymes are discovered. The x-axis depicts the mean enzyme time of birth of a new organism while the y-axis shows the size of the organisms given by the enzyme repertoire. For higher organisms, the appearance time correlates well with the size of the organisms but this is not the case for bacteria and archaea. See Ref. 39 for a list of all organisms and the appearance time.
Coefficients of variation and parameters of Fano factor fits averaged over 200 runs. The coefficient of variation is measured in the domain of the first 6000 enzymes. The data are fitted to the Fano factor equation (4) via parameter diffusion coefficient and correlation time .
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