Schematic representation of two parabolically replicating systems. (A, B) These systems represent spatially constrained or aggregated, but well-mixed (for example by wet-dry cycles on mineral surfaces) replicator populations. Black arrows indicate the possible chemical reactions, i and j denote master types. (A) Constant population model representing a Moran-process. N denotes the (fixed) total number of replicator molecules. This model allows us to investigate the inherent dynamical properties of parabolic growth, such as its diversity maintaining ability in face of mutations, under fixed and controlled conditions. (B) Chemostat model representing an open system. N(t) denotes the time dependent total number of replicators. This model allows us to investigate the effects of the changes in prebiotic environments on parabolic coexistence. Such alternations in environmental conditions could have involved, for example, variations in resource availability and the consequential shifts in the strength of resource competition. For the explanation of mechanisms and constants, see main text and Table 1.

Model parameters

Replicator coexistence in the constant population model. (A) Time series of the relative frequencies of master types. Panels corresponding to different rows demonstrate distinct replicability distances (δ), panels corresponding to different columns demonstrate distinct population sizes (N). Line colors indicate different replicabilities of the master types (as explained in the legends). The cut-off relative frequency for survival is indicated by horizontal dashed lines. (B) Mean and standard deviation of the number of sustained master types as the function of N and δ calculated from 50 independent runs (the corresponding legend is shown next to panel C). (C) Sum of replicability ranks of the survived master types at different number of coexisting types, corresponding to the simulation results shown in (B). The red curve denotes the maximum of the sum of ranks (ρmax), the dashed red line shows the expected values of the sum of ranks, when the master types are random (ρrand). The default parameter set (see Table 1) was used, unless otherwise indicated.

Sequence effects on replicator abundances in the constant population model. (A) Demonstration of direct sequence effects on replicator abundances. The master types (indicated by differently colored lines) are defined along a GC-content gradient with identical replicabilities (cr = 0,005). (B) Demonstration of removal of sequence effects on replicator abundances. The master types (indicated by differently colored lines) are defined along a replicability gradient (replicability distance: δ = 0.005) with identical GC contents (50%). (C) Means and standard deviation of the relative GC-content in the survived and extinct master types, corresponding to the simulation results shown in Figure 2BC. Columns with the darkest shade represent the average GC content of the randomly generated master sequences constituting the initial populations. Horizontal dashed lines indicate the expected (random) 50% initial GC content. Due to the relatively fast convergence, simulations were terminated after 2 · 106 replication events. Parameters that were used in both simulations: N = 104, pmut = 0.01 (see Table 1).

Effects of the mutation rate on the replicator coexistence in the constant population model. (A) Time series of the relative frequencies of master types at different mutation rates (pmut). Line colors denote master type replicabilities (values are indicated in the legend). The cut-off relative frequency for survival is indicated by horizontal dashed lines. (B) Mean and standard deviation of the coexisting master types as the function of pmut calculated from 20 independent runs. (C) Mean and standard deviation of the sum of relative frequencies of master types in the total population as the function of pmut, corresponding to the simulation results shown in (B). Except for pmut, the default parameter set (see Table 1) was used.

The concentration of master type ( and ) as the function of Q and A in the exponential model (left panel) and in the parabolic replication model (right panel). a = 1 (where a stands for the reproductive value of the mutants) was used.

Effects of resource competition on replicator coexistence in the chemostat system model. (A) Time series of the relative frequencies of master types. Panels corresponding to different columns demonstrate different population sizes (N). Panels corresponding to different rows demonstrate different replicability distances (δ). Line colors indicate master types with distinct replicabilities (as explained in the legends). The cut-off relative frequency for survival is indicated by horizontal dashed lines. (B) Averages for the total population size and the number of coexisting master types (color and numbers on columns) as functions of δ and duplex decay factor (f) at different resource inflow rates (cin). Averages were obtained from 20 individual runs for every parameter combination. (C) Mean and standard deviation of simplex-duplex asymmetry as the function of δ and f at different cin values, corresponding to the simulation results shown in (B). The default parameter set (see Table 1) was used, unless otherwise indicated.

Different outcomes of the dynamics as a function of Q and A. In the darker grey region (low copying fidelity, high reproductive superiority) the master survives only in the parabolic regime. In the lighter grey region (medium copying fidelity range) the master has a higher equilibrium concentration in the parabolic regime than in the exponential regime. The continuous red line is the error threshold for the exponential system (Q*), the dashed black line divides the higher and lower parabolic concentration regions of the parameter space (Q**). a = 1 was used.

Equilibrium concentration of the master in the exponential system with back mutation as a function of Q and B. Parameters are: A = 2, a = 1.

(A–C) The relative differences between the equilibrium concentrations of masters in the two regimes: as the function of the replication accuracy (Q) and the probability of back mutation (B) at A = 1.1, 2, 10, respectively. Blue color indicates higher, red indicates lower parabolic concentrations. (D) The area of the parameter space where the parabolic regime has higher equilibrium concentration. The area shrinks as A increases. If A ≈ a, the parabolic case has a higher concentration in the half of the parameter space, while if A ≫ a, this area is reduced to the quarter of the space (a = 1 was used).