**Reviewer 1: Omer Markovitch, Weizmann Institute of Science (nominated by Doron Lancet).**

The authors argue that a phase diagram, exhibiting living and non-living states (when the templating rate is respectively bigger or equal to the spontaneous polymerization rate), is a generic feature of replicator-first origin of life models. The authors then develop a simple lattice model in which a site can contain up to 3 molecules, templating can occur when exactly 2 monomers exist in a site and diffusion between adjacent sites is allowed. Their spatial model exhibits patches of replicating molecules, some of which can eventually grow to dominate the system akin to a percolation transition, in analogy to a bistable diagram. They find that rapid diffusion in a large lattice inhibits the non-life to life transition because fluctuations are smaller. The manuscript deals with spatial effects in replicating polymers, which did not receive full attention before, and presents the results in a clear manner, thus deserves publication.

The discussion of 3-molecules-limit vs. no-limit is suggested to be extended. The authors should consider performing an intermediate test in order to verify that carrying capacity is qualitatively similar to concentration limit.

Response: *We would not call this a percolation transition. The analogy we are making is with a nucleation phenomenon, such as nucleation of solid crystals in a liquid followed by subsequent rapid growth of the crystals.*

In the RNA polymerization model, resources are explicitly modelled, and therefore no carrying capacity is necessary. In the Two’s Company it is necessary to impose a carrying capacity in order to avoid unbounded growth. It would also be possible to add a resource molecule explicitly to the Two’s Company model, but this would go against our intention of making this model as simple as possible.

1) The abstract will benefit from being toned down, example given from “The key step is…” to “A key step is…”.

Response: *We changed this to A in the abstract.*

2) Providing information on the existence of alternative views to the RNA world is suggested.

Response: *We added citations to Shapiro [*[23]*] and Segre and Lancet [*[24]*] in the introduction.*

3) Section 2 (A Generic Phase Diagram for Replicating Molecules) is too long. It is suggested to provide fewer details of a previously published work and stress the stress the conclusion right at the beginning.

Response: *We have made only very slight reductions in this section. All the equations are necessary for what follows in this paper, and the textual discussion is important. The first two sentences already state the purpose and conclusion of this section.*

4) The reader could benefit from an illustration / diagram of the “Two is a company” model.

Response: *A new Figure*
3
*has been added showing the rules for this model.*

**Reviewer 2: Claus Wilke, University of Texas, Austin.**

Comment: This article uses mathematical modeling and computer simulations to study a simple origin-of-life model. The two main contributions this article makes are (i) a simplified model of the origin of life that maps in a straightforward fashion to a more complex model and (ii) a detailed analysis of the origin-of-life model in a spatial setting. The authors address the conditions under which a living state can emerge from a non-living state, and find that a spatially extended system with moderate diffusion can greatly facilitate the emergence of the living state. Most importantly, in this regime, the probability of emergence scales with the size of the system, so that the emergence of life somewhere is virtually guaranteed for sufficiently large systems. Overall, this is a very nice contribution. The modeling is sound, and the authors use multiple avenues (different models, different modeling approaches) to support their theory.

In the expression *k = k*
_{
0
}
*f*, wouldn’t *f* be vanishingly small for standard RNA sequences, possibly so small that even a large *k*
_{
0
} would not be sufficient to yield a reasonable *k*? The authors give some qualitative discussion of this topic towards the end of their paper, and I’d like to see the arguments fleshed out a bit more and possibly made quantitative. Do we have a sense of what a reasonable *k*
_{
0
} would look like, and what k needs to be to make the origin of life happen within, say, a billion years? What would the corresponding *f* be, and is it reasonable?

Response: *We agree that if only a single sequence is functional and if the sequence is very long, then f would be very small. We also agree that if f is very small then k*
_{
0
}
*must be large. However, it is probable that many sequences folding to the same secondary structure would have had similar functions, which makes f larger. This argument also suggests that it may be easier to begin with rather short ribozymes with a rather low catalytic ability than with longer, but rarer sequences that are better catalysts. These questions are clearly important, but we do not know how to put concrete numbers on these quantities. To give a quantitative answer to the time scale and required reaction rates, we would need to the concentrations of reagents on the primitive Earth, the reaction pathways by which nucleotides and RNAs were synthesized, and the temperature, pressure and pH at which these reactions were occurring. Current knowledge of these things is very incomplete.*

Comment: I think the manuscript would improve if it had a separate Methods section containing some of the more technical details of the simulations and derivations. Section 4 in particular mixes results with highly specific implementation details that don’t move the story forward. Similarly, other sections contain material that doesn’t directly impact the overall story, such as the proof that the Two’s Company model agrees with Eq. (*5*). I’m not asking the authors to delete this material, just reorganize it such that the a reader can get the most important results without also having to read all the supporting evidence and materials.

Response: *A methods section has been added at the end of the paper and the technical sections have been moved to the methods. The separate sections on the Two’s Company have now been merged into one main section.*

**Reviewer 3: Nobuto Takeuchi, National Center for Biotechnology Information, National Library of Medicine, National Institutes of Health, USA. (Nominated by Eugene Koonin).**

Comment: In their manuscript, Wu and Higgs present a minimal replicator model that exhibits a stochastic transition from a “non-living” to a “living” state. Such a transition was originally proposed and investigated by Dyson in a different model [32, 49]. Using this minimal model, the authors investigate the factors determining the waiting time for such a transition. In particular, the authors calculate the waiting time as a function of the diffusion rate of replicators. The calculation reveals that the waiting time is smallest when the diffusion rate assumes an intermediate value. This result is intuitively explained by the authors as follows: Too strong diffusion diminishes the stochasticity of dynamics and so impedes the transition, whereas too weak diffusion prevents replicators from encountering each other and, thus, from catalyzing each other’s replication. Moreover, the authors examine the waiting time as a function of the system size. The calculation shows that the waiting time decreases as the system size increases, provided that the diffusion is finite. This result sharply contrasts with that obtained for infinite diffusion. The authors again give an intuitive explanation: If diffusion is finite, a non-living-to-living transition first occurs in a local region and then propagates through the entire system; thus, the larger the system, the greater the number of local regions (assuming finite diffusion), and so is the probability of a transition per unit time. Also, the authors check this result in a more complex model that considers RNA polymerization.

The manuscript describes its content in a clear manner. I obtained only one specific question regarding the results (more general comments to follow). The transition time is shortest when the diffusion rate takes an intermediate value in the minimal replicator model. Does this result also hold in the RNA polymerization model in general?

Response: *Yes, although we have not fully investigated the regime of very low diffusion rate in the RNA polymerization model. Very low diffusion rate should be unfavourable because the catalysts can only be effective if monomers and growing chains encounter the catalysts and if newly created catalysts can spread beyond the site in which they were created.*

Comment: I have three general comments (and a question). First, the non-living-to-living transition exhibited by the models rests on the combination of the two processes: the stochastic occurrence of rare events and the subsequent deterministic amplification thereof. Interestingly, such a combination also plays an important role for the stability of a spatially extended RNA-like replicator system [40]. Namely, the stable coexistence between catalysts and parasites is enabled by the formation of traveling wave patterns, whose emergence rests on the combination of the two processes: rare stochastic events in which a few catalysts are spatially segregated from parasites and the subsequent expansion of these catalysts through replication (followed by infection with parasites). This kind of dynamics, in which rare stochastic events are deterministically amplified, might be a generic feature of (life-like) systems that have a finite system size and positive feedback (see also the last paragraph).

Response: *We agree that it is interesting to note that stochastic effects and limited spatial diffusion favour both the origin of life and the stable coexistence of replicators and parasites.*

Comment: My second comment is concerned with the implication of the current study in relation to Dyson’s study [32, 49]. Dyson’s model is conceived under the metabolism-first scenario for the origin of life: it is created as a simplest model of metabolic systems and assumes no replicators. By contrast, the minimal replicator model investigated in the current study, of course, assumes replicators. The RNA polymerization model of Wu and Higgs is conceived under the RNA world hypothesis, but does not actually assume replicators (i.e., it does not assume template-directed polymerization). Despite these differences, all these models exhibit essentially the same general result: two stable states that are regarded as non-living or living and stochastic transitions between them. Therefore, the general result seems to be independent of whether the model assumes metabolism, replicators, or the RNA world. This independence is also implied by the reason why these models produce the same result. Wu and Higgs identify two features shared by their models that are essential for the general result: resource limitation and nonlinear (positive) feedback—the latter does not necessarily imply replication. These features are clearly shared by Dyson’s model as well. Taken together, the results of the current study seems to imply that Dyson’s proposal of identifying the origin of life as a stochastic transition between two stable states is orthogonal to whether one considers the metabolism-first scenario, the replicator-first scenario, or the RNA world hypothesis.

Response: *Although we like the idea of the stochastic transition to life that was introduced first by Dyson, we feel that Dyson was not clear on what was meant by the ‘active’ and ‘inactive’ states and why the active state corresponds to life. In both our RNA polymerization model and the Two’s Company model, the high concentration of catalysts in the living state is maintained by the reactions carried out by the catalysts. The definition of the living state in our models is that it has autocatalytic biopolymers. Although we could presumably write down a metabolism-only model without biopolymers and without replication that would have two stable states and a possibility of a stochastic transition, this would be a purely chemical system and it would not constitute a transition to life, in our view, because the high-concentration state would not have replication, heredity and evolutionary potential.*

Comment: My last comment is concerned with the authors’ treatment of the linear growth term in the minimal replicator model. In investigating this model, the authors regard this term as less important than the other terms. However, I find three arguments to challenge this view. First, the linear growth term can play an important, though negative, role for the reported results of the model. Namely, if the linear growth rate r is greater than the decay rate u, the model has only one stable state (Figure 2 shows results for r < u). Second, replicators whose growth is described by a linear term (exponentially growing replicators, for short) are theoretically the simplest possible replicators [50]. Moreover, they have been experimentally synthesized [12].

The last argument concerns the very core of the current study. The non-living-to-living transition conceived by the authors entails rare stochastic events and the deterministic amplification thereof. Such a transition might be exhibited by exponentially growing replicators as well if one abandons the presupposition that the non-living state must be dynamically stable, as follows. Let us suppose that RNA molecules are randomly synthesized as assumed in the RNA polymerization model. Moreover, some RNA sequences are capable of self-replication; however, these sequences form but a tiny subset of the sequence space. Now, the non-living state corresponds to the state in which the system hardly contains any self-replicating molecules; the living state, the state in which the system contains self-replicating molecules in abundance. The rare stochastic event corresponds to the emergence of a (few) self-replicating molecule(s) through random synthesis; the deterministic amplification, the expansion of the nascent population of these molecules through self-replication. This argument suggests a topic for further discussion: Must “non-living” states be dynamically stable?

Response: *If r > u in our Two’s Company model, there will be no non-living state and there will be spontaneous multiplication of replicators without any waiting. This does not seem realistic to us. It is important to have non-linearity in some way in order to have two stable states. In the Two’s Company model there is only one way to do this, which is the quadratic term representing a two-molecule replication process. However, in more complex models there are many ways to do it. In the RNA polymerization models, the nonlinearity can arise either by feedback in the polymerization rate or in the monomer synthesis rate [*[27]*]. It is true that the ligase reaction in [*[12]*] corresponds to an exponentially growing replicator, but this system only works if it is supplied with very specific complementary oligomer strands. This example does not seem very close to the r reaction in our model. For the r reaction, we envisage a template directed process (e.g. on the surface of a clay catalyst) in which a single strand acts rather passively as a template. Although the template ability of different RNAs might depend to some extent on the sequence, it seems likely that most sequences could act as a passive template to some degree. If the mineral catalyzed r process was already good enough to support exponential growth (r > u), then there would be immediate multiplication of large numbers of RNAs with no sequence specificity, which seems unreasonable. The reviewer’s suggestion that the non-living state need not be dynamically stable makes sense as a logical possibility, but it seems unlikely to us, because the origin of life would just be too easy in this case.*

The possible interplay between template directed processes and ribozyme-catalyzed processes is interesting and is not fully captured in the models we studied so far. In our view, a non-specific *r* process could be important in generating diversity of RNA strands prior to life, but life itself would only arise when specific ribozymes came along to carry out the *k* process. Note that the ligase replicator in [12] is not acting as a template to specify the sequence of the new strand, because the sequence is already specified in the oligomer strands that are supplied to it. For an RNA strand to act as a catalyst it needs to fold to a specific structure. It is difficult to see how any strand could be a folded catalyst and an unfolded template at the same time. Therefore we are led almost inevitably to consider two-molecule replication processes.

Reply to the authors’ reply: *Given the persistent differences in the definitions of life, it is all the more interesting that these models—conceived under different views of life—produce the same general result based on the same principle. The authors consider that there is unlikely to be a replicator that grows exponentially and requires specific sequence patterns to act both as a catalyst and as a template (I assumed such replicators when I suggested that the non-living state need not be dynamically stable). The ligase replicator of Lincoln and Joyce (2009) is an example of such replicators as elaborated below. Although it does not answer the question I raised (its growth requires substrates prepared by humans), it suggests great possibilities of what RNA molecules can do (it was generated by (only) six rounds of in vitro selection).*

Contrary to the authors’ interpretation, the ligase replicator of Lincoln and Joyce (2009) does act as a template to specify the sequence of the new strand as follows. Their replicator consists of two ligases (denoted by E and E’), each composed of two substrates (denoted by A and B, and A’ and B’, respectively). The substrates A and B’ contain sequences complementary to each other, and so do the substrates A’ and B. Based on this complementarity, each ligase (say E) forms a complex with two substrates (A’ and B’) and catalyzes the ligation between them, synthesizing the other ligase (E’). In this way, a ligase acts both as a template and as a catalyst for the synthesis of the other ligase. More important, Lincoln and Joyce (2009) generated 12 sets of distinct substrates (denoted by An, Bn, A’n, and B’n where n ranges from 1 to 12). During a serial transfer experiment, base pair mismatches generate “recombinants” such as a ligase composed of A5 and B3 (denoted by E5,3). Such a recombinant can replicate; for example, an E5,3 forms a complex with A’3 and B’5 and catalyzes the synthesis of an E’3,5, which, in turn, catalyzes the synthesis of an E5,3. In this way, the replicator can transmit “genetic information”, thanks to the ability to act as a template.