Survival of RNA Replicators is much Easier in Protocells than in Surface-Based, Spatial Systems

Abstract

In RNA-World scenarios for the origin of life, replication is catalyzed by polymerase ribozymes. Replicating RNA systems are subject to invasion by non-functional parasitic strands. It is well-known that there are two ways to avoid the destruction of the system by parasites: spatial clustering in models with limited diffusion, or group selection in protocells. Here, we compare computational models of replication in spatial models and protocells as closely as possible in order to determine the relative importance of these mechanisms in the RNA World. For the survival of the polymerases, the replication rate must be greater than a minimum threshold value, k_min, and the mutation rate in replication must be less than a maximum value, M_max, which is known as the error threshold. For the protocell models, we find that k_min is substantially lower and M_max is substantially higher than for the equivalent spatial models; thus, the survival of polymerases is much easier in protocells than on surfaces. The results depend on the maximum number of strands permitted in one protocell or one lattice site in the spatial model, and on whether replication is limited by the supply of monomers or the population size of protocells. The substantial advantages that are seen in the protocell models relative to the spatial models are robust to changing these details. Thus, cooperative polymerases with limited accuracy would have found it much easier to operate inside lipid compartments, and this suggests that protocells may have been a very early step in the development of life. We consider cases where parasites have an equal replication rate to polymerases, and cases where parasites multiply twice as fast as polymerases. The advantage of protocell models over spatial models is increased when the parasites multiply faster.

Keywords: RNA World; error threshold; evolution of cooperation; membranes; parasites; polymerase; protocell; spatial lattice model.

Figure 1

A cartoon representation of the spatial model with local diffusion dynamics (left) and the protocell models (right). The red strands are polymerases (P), orange strands are complements to polymerases (C), and black strands are parasites (X). The blue arrows indicate the possibility of diffusion to and from the eight neighboring sites.

Figure 2

Average numbers of strands per cell in the PCP model. (a) S₀ = 10, (b) S₀ = 20. k = 25 in both cases. Points are from finite population simulations. Smooth lines are from deterministic theory.

Figure 3

Average numbers of strands per site in the SMF model. (a) N = 100, k = 25, h = 0.4 and (b) N = 400, k = 20, h = 0.4. Points are from finite population simulations. Smooth lines are from deterministic theory.

Figure 4

Comparison of the error threshold of the various models studied as a function of the polymerization rate k. S₀ = 10 in all models except the one per site model, and h = 0.4 in the lattice models. All results are from stochastic simulations except for SMF, which results are from the deterministic method. One strand per site (OSPS) is the one strand per site model from [18]. Other models are defined in Table 1.

Figure 5

Comparison of the error threshold of the various models studied as a function of S₀. k = 25 in all models, and h = 0.4 in the lattice models. Results for SMF are obtained from the deterministic method, except for the points with S₀ > 150, where the deterministic method becomes much slower than the stochastic simulation. The results for the other models are obtained from stochastic simulations. OSPS is the one strand per site model from [18]. Other models are defined in Table 1.

Figure 6

Comparison of the error thresholds of the spatial models with long distance diffusion and local diffusion as a function of the diffusion rate h. Made using S₀ = 10, k = 25.

Figure 7

Error thresholds versus k for PCP and SLD models in which parasites and polymerases have equal replication rates (same as Figure 4) compared with equivalent models where parasites have double the replication rate of polymerases (denoted PCP2X and SLD2X). Both error thresholds are reduced when the parasites multiply faster, but the SLD model is reduced more, meaning that the relative advantage of the protocells over the spatial model is increased. S₀ = 10 and h = 0.4.

Figure 8

Error thresholds versus S₀ for PCP and SLD models in which parasites and polymerases have equal replication rates (same as Figure 5) compared with equivalent models where parasites have double the replication rate of polymerases (denoted PCP2X and SLD2X). Both error thresholds are reduced when the parasites multiply faster, but the SLD model is reduced more, meaning that the relative advantage of the protocells over the spatial model is increased. Note that M_max = 0 for S₀ > 20 for SLD2X, because faster parasites kill the polymerases in the spatial model. k = 25 and h = 0.4.