(A) Random cleavage model. A random break in a chain of type c generates two primer fragments, which are elongated to give rise to two chains of type c. Elongation requires a complementary template of type (B) An example of catalyzed cleavage given by hammerhead ribozyme [32]. Note that the right cleavage fragment is perfectly complementary to the blue sequence, while the left one contains an extra non-complementary base C. (C) Catalyzed cleavage model. A cleavage of the red chain a catalyzed by the blue chain gives rise to two primers aL (red) and aR (purple). Because of an extra non-complementary base (see Panel B) the aL primer can only elongate to a, while the aR primer to either a or b depending on its first hybridization partner. Similar processes involving complementary chains āand (not shown) result in the replication of templates.

Dynamical phase portraits for different catalytic cleavage rates β. (A) The phase portrait for a small catalytic cleavage rate β = 6 has two non-cooperative steady state solutions marked with red and blue stars corresponding to pure a/ ā and pure [ineaq] subpopulations respectively. These solutions are maintained by random rather than catalytic cleavage. (B) The phase portrait for intermediate catalytic cleavage rate β = 10 in addition to two non-cooperative steady states marked with red and blue stars has a cooperative steady state marked with the green star in which all four subpopulations coexist. One can reach this state e.g. starting from the non-cooperative steady state (the blue star) and adding a relatively small subpopulation of a/ ā> 2e 5 crossing the saddle point separating blue and green trajectories. (C) The phase portrait for a large catalytic cleavage rate β = 18 again has only two non-cooperative cleavage steady states marked with red and blue stars. All three panels were obtained by numerically solving dynamical equations (7-11) with random cleavage rate β0 = 0.015, elongation asymmetry factor λ = 2, and dilution factor δ = 1.

(A) The relationship between parameters of the cooperative state. plotted vs β for λ = 2, δ = 1 and increasing values of β0: 0.015 (green), 0.03 (purple) and 0.045 (red). Lines are given by the parametric equation describing the state and derived in the SI Appendix (Eq. S14), while open circles are obtained by direct numerical solution of dynamical equations (7-11). Monotonically increasing branches (solid lines) correspond to the stable cooperative fixed point, while the decreasing branches (dashed lines) to the dynamically unstable saddle points separating different steady state solutions in Fig. 2B. (B) Phase diagram of the cooperative state. The shaded region marks the values of β/δ and β0/δ for which the cooperative solution exists. Green, purple and red lines show the ranges of β for which the cooperative solution exists for the corresponding value of β0 shown in Panel A. Increasing the parameter β0 makes the range of β for which the cooperative solution exists progressively smaller until it altogether disappears above β0/δ ≈ 0.057.

The fitness landscape of our system. (A) the threedimensional plot and (B) the heatmap of the fitness of the cooperative state in our system as a function of the catalytic cleavage rate enhancement β/β0 and elongation asymmetry λ. The co-evolution of λ and β would increase together. A typical evolutionary trajectory in which small changes in these parameters are independent of each other is shown as a dashed line.