# Survival of RNA Replicators Is Much Easier in Protocells Than in Surface-Based, Spatial Systems

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## Abstract

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_{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.

## 1. Introduction

_{min}, which is required for survival. The polymerases must also replicate accurately enough to pass on their own sequence and avoid invasion by parasitic mutant sequences; hence, there is a maximum value of the mutation probability, M

_{max}, required for survival. RNA replicating systems are most likely to survive in systems with low k

_{min}, and high M

_{max}. Here, we study several alternative versions of protocell and spatial models for RNA replication in order to compare the values of k

_{min}and M

_{max}.

_{0}. One example of a model that is defined in this manner is featured in the study of Branciamore et al. [13]. In their paper, several types of autocatalytic replicators, each catalyzing one reaction in a metabolic network, could be present in a pore. Each pore was assigned a fitness that corresponded to the diversity of replicators it contained, with the requirement that at least one strand of each replicator type be present. Parasitic replicators were introduced through invasion and were also autocatalytic, competing with members of the network for resources without catalyzing any of the reactions [13]. In contrast, instead, we focus on a trans-acting polymerase. The parasitic sequences that we consider are fundamentally different: they cannot replicate without a polymerase present in the site. The polymerases may erroneously produce parasites from an improper replication. Hence, in our study, the central question is the maximum mutation rate the system can sustain rather than the number of different replicator species that can be sustained.

_{0}in the protocell models as the number of strands at which cells divide. Thus, S

_{0}controls the maximum number of strands per cell/site in both types of model. The rules for replication of strands are identical in the two types of models when we define the models in this way. This enables us to focus on the differences between spatial models and protocells: in spatial models, replication occurs locally on one lattice site and the diffusion of strands occurs between neighbouring sites, whereas in protocell models, replication occurs locally in one protocell, new cells arise when cells divide, there is no diffusion of strands between cells, and there is no spatial structure of the cells.

## 2. Materials and Methods

#### 2.1. Overview of Models

_{0}. This produces two daughter cells with the strands that were randomly divided between them. The number of protocells in the population, N, is a fixed parameter in the PCP model (Protocells with Constant Population), and is variable in the PML model (Protocells-Monomer Limited). In the PCP model, whenever a cell divides, another random cell is removed from the population to keep N fixed. This represents a situation where resources, such as lipids or available space, limit population growth. It is analogous to the standard Moran model that was used in population genetics [36]. In each model, there is a limiting factor F in the replication rates, which is required for preventing indefinite increase of either the population or the number of strands (details below in Section 2.2). In the PCP model, the population is already limited by fixing N, therefore no additional limiting factor is needed (F = 1). In the PML model, there is no limit to the number of cells, but the number of strands is limited by the availability of monomers (i.e., nucleotides). The limiting factor is $F=1-{S}_{tot}/{S}_{max}$, where ${S}_{tot}$ is the total number of strands in the whole population and ${S}_{max}$ is the maximum allowed number of strands. We call this limit global, because it applies equally to all cells in the population. In the PML case, when a cell divides, it is not coupled to the removal of another cell. Instead, all empty cells with S = 0 are immediately removed in order to prevent the accumulation of empty cells. In the PCP model, we do not need to immediately remove empty cells because they are eventually removed at random due to the birth and death process of cells.

_{0}controls the number of strands on any one site. The limiting factor is $F=1-S/{S}_{0}$, which means that no further replication is possible on a site when S ≥ S

_{0}. The local motion of strands leads to a build up of correlations between the contents of one site and its neighbouring sites. This correlation causes the clustering of polymerases, which is part of the reason that the spatial model allows for the survival of polymerases and avoids destruction by parasites. Therefore, it is useful to consider the Spatial Model with Mean Field dynamics (SMF) model as a comparison to this. In mean field dynamics, whenever a strand hops to a different site, it is placed on any other site with equal probability, rather than on a neighbouring site. We have previously studied mean field models with small numbers of strands allowed per site [16,27]. If only one strand is allowed per site, then the mean field model is the same as the well-mixed case, which is not useful, because polymerases are always destroyed by parasites. When up to three strands are permitted per site, the mean field model shows the correct qualitative behaviour, but it is still quantitatively very different from the model with local dynamics. We will show here that when many strands are possible per site (S

_{0}= 10 or larger in the examples in this paper), there is very little difference between mean field and local dynamics; hence, the mean field approximation is useful. An advantage of the SMF model is that it is possible to give a deterministic solution; whereas, the SLD model requires stochastic simulations.

_{0}in the spatial models, and it grows in proportion to the number of strands in the PCP and PML models, V = S. The volume determines the strand concentrations, and hence the reaction rates, as described in Section 2.2. Although it seems natural to keep V constant in the spatial models and to allow it to grow in the protocell models, it is useful for comparison to consider an additional model, PCPCV, in which the volume is kept constant. We will show below that there is a relatively small difference between the PCPCV and PCP models, so the question of whether the protocell volume grows or is fixed is a relatively minor one.

#### 2.2. Model Details

_{p}is the rate of increase in number of strands per cell, not the concentration, so there is an extra factor of V. Hence, K

_{p}depends on $pc/V$, not $pc/{V}^{2}$. Equivalently, we may say that the rate of increase in the number of product strands is proportional to the concentration of polymerases, p/V, times the number of templates, c.

_{0}undergo random division. Cell division is assumed to be rapid once the split size is reached, i.e., all cells with $\mathrm{S}\ge {\mathrm{S}}_{0}$. divide with probability 1 in one time step. The strands from the parent cell are assigned independently with equal probability to one of the two daughter cells. Even though cell division immediately occurs on reaching S

_{0}strands per cell, it is possible for a small number of cells with $\mathrm{S}\ge {\mathrm{S}}_{0}$. to remain in the population after cell division. Firstly, it is occasionally possible to create cells with more than S

_{0}strands, because replications of P, C, and X strands are independently considered; hence, more than one replication can occur in the same cell in one time step. Secondly, it is possible for the random split to occasionally yield S

_{0}strands in one daughter and zero in the other; hence, there will sometimes still be S

_{0}strands after division.

_{0}. We set S

_{max}in PML to NS

_{0}, where N is the fixed population size of the PCP model in order to compare PML with PCP. In the PML model, we begin with ${S}_{max}/2$ cells, each having one P, one C, and one X.

## 3. Results and Discussion

#### 3.1. Error Threshold Behaviour

_{0}= 10 and 20. The smooth lines are obtained from the deterministic theory in the Appendix A, which applies for infinite populations. The points are measured by simulations with N = 1024. These show typical error-threshold behavior. The numbers of P and C strands per cell decrease steadily as the mutation rate is increased, while the number of X strands passes through a maximum. There are always slightly more P than C strands, because of the (p − 1) factor in ${K}_{c}\left(p,c,x\right)$, (i.e., a P cannot replicate itself, whereas a P can replicate all C’s). All three strands die out at the error threshold, M = M

_{max}. The deterministic theory predicts that the strand numbers smoothly decrease to zero as M approaches M

_{max}. Close to this point, the expected number of viable cells in a finite population is very small; hence, the finite population simulations are vulnerable to stochastic fluctuations causing the death of the system. The average number of strands in the simulations in the long-time limit is then zero. This causes the simulated systems to die out at slightly smaller values of M than is predicted by deterministic models.

_{0}values, there is a non-zero parasite population present, even at zero mutation rates. In other words there is a coexistence of non-functional parasites with polymerases, even when the parasites are not replenished by mutations from the polymerases. This is a significant difference from the protocell models that were considered in Figure 2, where the parasites are always purged from the systems at zero mutation rates.

#### 3.2. Comparison of Error Thresholds in Different Models

_{max}(i.e., the maximum sustainable error probability per replication of the whole sequence) and the minimum catalytic rate, k

_{min}

_{,}required for survival of the polymerases. Figure 4 shows M

_{max}measured from simulations as a function of k. The estimates of M

_{max}were obtained by running a series of simulations at each value of k and then gradually adjusting the mutation rate to zero in on the error threshold. A similar method was used to produce Figure 5, where S

_{0}was held fixed.

_{min}is the value of k at which M

_{max}becomes zero. For the PCP and PML models, k

_{min}is approximately 3, whereas it is approximately 18 for SLD and SMF. Thus, there is a substantial range 3 ≤ k ≤ 18 where replication is possible in protocells and not in spatial models. All of these rates should be thought of as relative to the breakdown rate, because we have set v = 1.

_{min}

_{,}we find M

_{max}is around 0.36 for PCP, but only approx 0.075 for SLD and 0.09 for SMF. Thus, the protocell models are four- to five-fold more tolerant of error. These Figures are per-sequence. If they are converted to per-base error rates, this implies that there is a four- to five-fold greater limit in the maximum length of replicating sequences that can be maintained in protocells relative to spatial models.

_{0}in the same way as it is in the spatial models, and it therefore eliminates a minor difference in the definitions of protocell and spatial models. The error threshold of PCPCV is reduced slightly relative to PCP, but it is still much higher than the spatial models. Therefore, the issue of whether the protocell volume is fixed or grows with the number of strands is only a minor effect. PCP seems more realistic, because in reality a cell cannot keep constant volume when it divides.

_{max}intermediate between the protocell models and the other spatial models, and has k

_{min}that is almost equal to the protocell models. This comparison is interesting from a theoretical point of view, as it highlights the fact that the local limitation on growth that applies in the SLD and SMF models leads to much lower error thresholds than the global limitation in the SML model. However, there are problems with the SML model that mean that is not a biologically realistic model. Replication is fastest on sites with the largest number of polymerases. Strands tend to pile up with very large numbers of strands on a very small number of sites, and with many other sites being empty, as there is no local limit on the number of strands per site in the SML model. This cannot be realistic, because sooner or later, local limits must take effect. Either the monomer limit becomes local, because the concentration of available monomers becomes depleted on sites when there is a lot of replication, or the local limit of space takes effect. Thus, we consider the SLD to be the most realistic of the spatial models, and the comparison between the SLD and the two protocell models as the most valid comparison of the differences between spatial models and protocells.

_{0}, which controls the number of strands per site/cell, has important effects on the error threshold, as shown in Figure 5. For a site/cell to be viable, there must be a minimum of either two P’s or one P and one C. When S

_{0}is small, there are many sites that are not viable, and the whole system dies out. Once S

_{0}is above this minimum size for viability, M

_{max}rapidly increases with S

_{o}and then decreases slowly as S

_{0}becomes large. For very large S

_{0}, each site is a well-mixed model, and there is no more clustering or group selection. Therefore, M

_{max}must tend to zero for very large S

_{0}. The SML model is an outlier here, in that it is not affected by high S

_{0}values. In the SML model, the only effect of S

_{0}is to determine the total number of strands, because S

_{max}= S

_{o}N, and it does not limit the number of strands on one site, as it does in the other models. The one strand per site model from Tupper et al. [18] is also shown as a comparison, but there is no equivalent of S

_{0}in this case. It should be remembered that polymerases replicate templates on the neighbouring sites in ref. [18], but on the same site in this paper. Hence, it is not possible to have S

_{0}= 1 in the spatial models in this paper. Once again, in Figure 5, we see that the PCP and PML models are very similar, and that the PCPCV is only slightly lower than the PCP model. The most useful comparison is between the PCP/PML models and the SLD model, and this shows a substantially larger error threshold for the protocell models, by a factor of 4 to 10.

#### 3.3. Effect of Diffusion Rate in the Spatial Models

_{max}is approximately 0.32 for the protocell models with S

_{0}= 10 and k = 25, as shown in Figure 2a). Furthermore, there is no reason in nature why diffusion should be tuned to the optimum value. For most of the range of h, the error threshold for the spatial models would be even lower than those that are shown in Figure 4 and Figure 5, and the system cannot survive at all (M

_{max}= 0) if h is too high or too low. The problem of tuning diffusion does not arise in the protocell models, which is another advantage of protocells.

#### 3.4. Rapidly-Replicating Parasites

_{0}. The error threshold for PCP2X is substantially reduced relative to PCP for larger S

_{0}. Nevertheless, there is a non-zero error threshold in PCP2X up to at least S

_{0}= 275. On the other hand, there is only a very narrow range of S

_{0}(approximately 7–20) where the error threshold is non-zero for SLD2X, and even within this range, the error threshold is extremely low. For S

_{0}> 20 in the SLD2X model, fast replicating parasites multiply and lead to destruction of the polymerases (and themselves), even in the limit of zero mutation rate. We allowed the system to reach a steady state with only P and C strands present to test the limit of zero mutation rate. A very small number of parasites were then added, and replication continued with zero mutation rate. For S

_{0}> 20, the initial few parasites multiply and destroy the system, even though there is no further production of parasites by mutation. In contrast, there is a finite error threshold for the PCP2X model at high S

_{0}, as we just noted. Thus, once again, the advantage of the protocell over the spatial model is increased when we consider faster replicating parasites.

## 4. Conclusions

_{0}but not in protocell models. In the case where parasites multiply faster than polymerases, the advantage of the protocells over the spatial models is increased (as in Figure 7 and Figure 8). In Figure 8, there is a qualitative difference between the protocells (PCP2X) and spatial model (SLD2X). For the protocells, there is a finite error threshold, even for the largest S

_{0}considered, whereas parasites destroy the spatial system, even in the limit of zero mutation rate.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**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**) S

_{0}= 10, (

**b**) S

_{0}= 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. S

_{0}= 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 S

_{0}. 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 S

_{0}> 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 S

_{0}= 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. S

_{0}= 10 and h = 0.4.

**Figure 8.**Error thresholds versus S

_{0}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 S

_{0}> 20 for SLD2X, because faster parasites kill the polymerases in the spatial model. k = 25 and h = 0.4.

Model | Dynamics | Limiting Factor | Volume |
---|---|---|---|

PCP—Protocells with Constant Population | Division when $S\ge {S}_{0}$ N fixed | No limit, $\mathrm{F}=1$ | Grows with cell $\mathrm{V}=\mathrm{S}$ |

PCPCV—Protocells with Constant Population and Constant Volume | Division when $S\ge {S}_{0}$ N fixed | No limit, $\mathrm{F}=1$ | Constant $\mathrm{V}={\mathrm{S}}_{0}$ |

PML—Protocells Monomer Limited | Division when $S\ge {S}_{0}$ N variable | Global limit, $F=1-{S}_{tot}/{S}_{max}$ | Grows with cell $\mathrm{V}=\mathrm{S}$ |

SLD—Spatial Model with Local Diffusion | Local diffusion rate h | Local limit, $F=1-S/{S}_{0}$ | Constant $\mathrm{V}={\mathrm{S}}_{0}$ |

SMF—Spatial Model with Mean Field dynamics | Mean field diffusion rate h | Local limit, $F=1-S/{S}_{0}$ | Constant $\mathrm{V}={\mathrm{S}}_{0}$ |

SML—Spatial Model Monomer Limited | Local diffusion rate h | Global limit, $F=1-{S}_{tot}/{S}_{max}$ | Constant $\mathrm{V}={\mathrm{S}}_{0}$ |

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**MDPI and ACS Style**

Shah, V.; de Bouter, J.; Pauli, Q.; Tupper, A.S.; Higgs, P.G.
Survival of RNA Replicators Is Much Easier in Protocells Than in Surface-Based, Spatial Systems. *Life* **2019**, *9*, 65.
https://doi.org/10.3390/life9030065

**AMA Style**

Shah V, de Bouter J, Pauli Q, Tupper AS, Higgs PG.
Survival of RNA Replicators Is Much Easier in Protocells Than in Surface-Based, Spatial Systems. *Life*. 2019; 9(3):65.
https://doi.org/10.3390/life9030065

**Chicago/Turabian Style**

Shah, Vismay, Jonathan de Bouter, Quinn Pauli, Andrew S. Tupper, and Paul G. Higgs.
2019. "Survival of RNA Replicators Is Much Easier in Protocells Than in Surface-Based, Spatial Systems" *Life* 9, no. 3: 65.
https://doi.org/10.3390/life9030065