Figure 1.
The three replicator systems made of the four types of polymerases.
The notation is as follows: Rp and Dp denote RNA polymerase and DNA polymerase respectively. The superscripts to Rp and Dp denote whether the polymerase is in RNA-form (catalyst and template) or in DNA-form (template). The prefixes to Rp and Dp denote the type of templates a polymerase depends on: Rd stands for RNA-dependent, and Dd stands for DNA-dependent. Solid arrows represent the template-product relationship. Broken arrows represent the catalyst-reaction relationship. A: Self-replication system consists of an RNA replicase (RdRp). B: Transcription system consists of a transcriptase (DdRp) and a DNA replicase (DdDp). C: Reverse transcription system consists of a transcriptase (DdRp) and a reverse transcriptase (RdDp).
Figure 2.
The evolution of the transcription system in the surface model.
The model was initialized such that the system consisted of a population of RNA polymerase (Rp) and parasites. The simulation was first run with the mutation converting Rp into Dp disabled (). After the system reached evolutionary equilibrium (panel A), the mutation was enabled (
), and the resulting evolutionary dynamics are depicted in panel B to D. The larger panels depict snapshots of simulations taken at different times as indicated above panels. The color coding is indicated at the bottom of the figure. RNA and DNA are not distinguished. The timescale is scaled such that it has the same meaning as that of the ordinary differential equation model that describes the replicator dynamics with the same rate constants as in the CA model (the timescale is scaled in this manner throughout the paper). The smaller panels within the larger panels depict a two-dimensional histogram of
and
. See the main text for the description for each panel. The parameters (rate constants) used in this simulation were as follows:
(replication);
(decay);
(diffusion);
(parasite advantage);
(mutation rate of
and
);
(mutation step);
(mutation rate from Rp to Dp);
(mutation rate to parasites). The size of CA was 1024×1024 squares. The boundary had no flux.
Table 1.
Summary of the results with the surface models.
Figure 3.
Spatial pattern generated by the transcription system in the absence of the self-replication system.
The surface model was initialized such that the system consisted of the transcription system (see below for the parameter values). No mutation processes were enabled except for the mutation converting molecules into parasite (). The color coding is indicated in the figure. The parameters were as follows:
and
for both Rp and Dp;
;
; the size of CA was 512×512 squares; the other parameters were the same as in Figure 2.
Figure 4.
The evolutionary dynamics of the transcription system after the self-replication system was removed.
After the surface model reached evolutionary equilibrium (Figure 2D), the whole population of the RNA replicase (i.e. the self-replication system) was removed from the system (Figure 4A), and the simulation was continued. The resulting evolutionary dynamics are depicted (Figure 4B–F). The figure has the same format as that of Figure 2. See the main text for the explanation of each panel. The parameters where as follows: ; the size of CA was 512×512 squares; the other parameters were the same as in Figure 2.
Figure 5.
The evolutionary dynamics of the surface model without explicitly predefined parasites.
The surface model was initialized with a population of Rp (no parasites were introduced in the system). The simulation was run in the same manner as in Figure 2 with the mutation converting molecules into parasites disabled (). The format of the figure is the same as that of Figure 2. For the explanation of each panel, see the main text. The parameters were as follows:
; the size of CA is 512×512 squares; the other parameters were the same as in Figure 2.
Figure 6.
The schematic depiction of the causal (A) and historical (BCD) relationship among the evolution of each species of replicators present at equilibrium in the surface model.
Dual-Rp denotes a dual specificity Rp. In B, C and D, the evolutionary dynamics progress from top to bottom. For the explanation, see the main text.
Figure 7.
The evolution of the transcription-like system in the compartment model.
The compartment model was initialized, and the simulation was run in the same way as in Figure 2. The model was initialized such that the system consisted of a population of Rp enclosed in a compartment. The simulation was first run with the mutation converting Rp into Dp disabled (). After the system reached evolutionary equilibrium (Figure 7A), the mutation (
) was enabled. The resulting evolutionary dynamics are depicted in panel B to E. The left picture of each panel shows a snapshot of the simulation taken at different times as indicated above panels. The color coding is indicated in the upper left corner of the figure. DNA and RNA are not distinguished. The insets depict two-dimensional histogram of
and
. The right picture of each panel shows a snapshot with a different color coding, which indicates the value of
. Distinction is not made between Dp and Rp and between DNA and RNA. The insets depict a histogram of
with the same color coding as in the larger pictures that contain them. For the explanation of each panel, see the main text. The parameters were as follows:
(the volume threshold for division of compartments);
(the target volume is set to the number of internal replicators multiplied by
);
; the size of the CA is 512×512 squares; the other parameters were the same as in Figure 2.
Table 2.
Summary of the results with the compartment models.
Figure 8.
The invasion of compartments containing only RpRNA and their eventual extinction, which happens repeatedly after the system reached evolutionary equilibrium in the compartment model.
The figure depicts the same simulation and in the same format as in Figure 7. The time is reset to zero at an arbitrary moment after the time in Figure 7E. For the explanation of each panel, see the main text.
Figure 9.
The time course of evolutionary deterioration of catalysts under well-mixed condition with a large population size for various replication systems.
The model was modified such that interactions between molecules happen globally regardless of the location of molecules (the system is effectively well-mixed). The model was initialized with a population of RpRNA in panel A, with a population of RpRNA, RpDNA, DpRNA and DpDNA in equal proportion in panel B and C, and with a population of RpRNA, RpDNA and DpRNA in equal proportion in panel D. The initial value of Rrec and Drec were set as indicated in the figure (at time = 0). The parameters were as follows: (effectively); the size of the CA is 512×512 squares; the other parameters were the same as in Figure 2.
Table 3.
Summary of the results with the ODE models.
Table 4.
Summary of the results with the well-mixed CA model.