Stochastic parabolic growth promotes coexistence and a relaxed error threshold in RNA-like replicator populations

Reviewer #1 (Public Review):

Summary: Szathmary and colleagues explore the parabolic growth regime of replicator evolution. Parabolic growth occurs when nucleic acid strain separation is the rate-limiting step of the replication process which would have been the case for non-enzymatic replication of short oligonucleotide that could precede the emergence of ribozyme polymerases and helicases. The key result is that parabolic replication is conducive to the maintenance of genetic diversity, that is, the coexistence of numerous master sequences (the Gause principle does not apply). Another important finding is that there is no error threshold for parabolic replication except for the extreme case of zero fidelity.

Strengths:
I find both the analytic and the numerical results to be quite convincing and well-described. The results of this work are potentially important because they reveal aspects of a realistic evolutionary scenario for the origin of replicators.

Weaknesses:
There are no obvious technical weaknesses. It can be argued that the results represent an incremental advance because many aspects of parabolic replication have been explored previously (the relevant publications are properly cited). Obviously, the work is purely theoretical, experimental study of parabolic replication is due. In the opinion of this reviewer, though, these are understandable limitations that do not actually detract from the value of this work.

Reviewer #2 (Public Review):

Summary:

A dominant hypothesis concerning the origin of life is that, before the appearance of the first enzymes, RNA replicated non-enzymatically by templating. However, this replication was probably not very efficient, due to the propensity of single strands to bind to each other, thus inhibiting template replication. This phenomenon, known as product inhibition, has been shown to lead to parabolic growth instead of exponential growth. Previous works have shown that this situation limits competition between alternative replicators and therefore promotes RNA population diversity. The present work examines this scenario in a model of RNA replication, taking into account finite population size, mutations, and differences in GC content. The main results are (1) confirmation that parabolic growth promotes diversity, but that when the population size is small enough, sequences least efficient at replicating may nevertheless go extinct; (2) the observation that fitness is not only controlled by the replicability of sequences, but also by their GC content ; (3) the observation that parabolic growth attenuates the impact of mutations and, in particular, that the error threshold to which exponentially growing sequences are subject can be exceeded, enabling sequence identity to be maintained at higher mutation rates.

Strengths:

The analyses are sound and the observations are intriguing. Indeed, it has been noted previously that parabolic growth promotes coexistence, its role in mitigating the error threshold catastrophe - which is often presented as a major obstacle to our understanding of the origin of life - had not been examined before.

Weaknesses:

Although all the conclusions are interesting, most are not very surprising for people familiar with the literature. As the authors point out, parabolic growth is well known to promote diversity (Szathmary-Gladkih 89) and it has also been noted previously that a form of Darwinian selection can be found at small population sizes (Davis 2000). Given that under parabolic growth, no sequence is ever excluded for infinite populations, it is also not surprising to find that mutations have a less dramatic exclusionary impact.

A general weakness is the presentation of models and parameters, whose choices often appear arbitrary. Modeling choices that would deserve to be further discussed include the association of the monomers with the strands and the ensuing polymerization, which are combined into a single association/polymerization reaction (see also below), or the choice to restrict to oligomers of length L = 10. Other models, similar to the one employed here, have been proposed that do not make these assumptions, e.g. Rosenberger et al. Self-Assembly of Informational Polymers by Templated Ligation, PRX 2021. To understand how such assumptions affect the results, it would be helpful to present the model from the perspective of existing models.

The values of the (many) parameters, often very specific, also very often lack justifications. For example, why is the "predefined error factor" ε = 0.2 and not lower or higher? How would that affect the results? Similarly, in equation (11), where does the factor 0.8 come from? Why is the kinetic constant for duplex decay reaction 1.15e10−8? Are those values related to experiments, or are they chosen because specific behaviors can happen only then?

The choice of the model and parameters potentially impact the two main results, the attenuation of the error threshold and the role of GC content:

Regarding the error threshold, it is also noted (lines 379-385) that it disappears when back mutations are taken into account. This suggests that overcoming the error threshold might not be as difficult as suggested, and can be achieved in several ways, which calls into question the importance of the particular role of parabolic growth. Besides, when the concentration of replicators is low, product inhibition may be negligible, such that a "parabolic replicator" is effectively growing exponentially and an error catastrophe may occur. Do the authors think that this consideration could affect their conclusion? Can simulations be performed?

Regarding the role of the GC content, GC-rich oligomers are found to perform the worst but no rationale is provided. One may assume that it happens because GC-rich sequences are comparatively longer to release the product. However, it is also conceivable that higher GC content may help in the polymerization of the monomers as the monomers attach longer on the template (as described in Eq.(9)). This is an instance where the choice to pull into a single step the association and polymerization reactions are pulled into a single step independent of GC content may be critical. It would be important to show that the result arises from the actual physics and not from this modeling choice.

Some more specific points that would deserve to be addressed:

- Line 53: it is said that p "reflects how easily the template-reaction product complex dissociates". This statement is not correct. A reaction order p<1 reflects product inhibition, the propensity of templates to bind to each other, not slow product release. Product release can be limiting, yet a reaction order of 1 can be achieved if substrate concentrations are sufficiently high relative to oligomer concentrations (von Kiedrowski et al., 1991).

- Population size is a key parameter, and a comparison is made between small (10^3) and large (10^5) populations, but without explaining what determines the scale (small/large relative to what?).

- In the same vein, we might expect size not to be the only important parameter, but also concentration.

- Lines 543-546: if understanding correctly, the quantitative result is that the error threshold rises from 0.1 in the exponential case to 0.196 in the parabolic. Are the authors suggesting that a factor of 2 is a significant difference?

- Figure 3C: this figure shows no statistically significant effect?

- line 542: "phase transition-like species extension (Figure 4B)": such a clear threshold is not apparent.