# Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems

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## Abstract

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## 1. Introduction

**Ecological diversity**—maintaining the coexistence of a sufficient number of different species (replicators, sequences, genotypes, etc.) in light of the Gause-principle (see later), which poses a strict limit on the number of coexisting species based on the number of regulating factors.**Ecological stability**—maintaining dynamical stability in a given set of coexistent species against external perturbations.**Evolutionary stability**—maintaining an adequate amount of information (a critical diversity of replicator species) from generation to generation and avoiding information decay (diversity reduction) in spite of frequent mutations and the lack of error correction.

## 2. The Three Pillars of Prebiotics

#### 2.1. Maintaining Diversity

_{1}and r

_{2}, the difference in the densities grows exponentially. This is the core problem of maintaining diversity of exponentially growing populations—and exactly the same dynamics are the indispensable basis of natural selection. It is the requirement of both maintaining diversity and remaining selectable that makes the problem particularly difficult.

#### 2.2. Ecological Stability

#### 2.3. Evolutionary Stability

## 3. Models of Prebiotic Systems

- The Parabolic replicators (PR) model [17]

#### 3.1. Models Assuming no Structure

#### 3.1.1. Hypercycle (HC)

_{w}) and all of its possible mutants lumped together (their total concentration denoted by x

_{m}):

_{W}and A

_{m}are the replication rates (Malthusian growth rates) of the wild-type sequence and its mutants, respectively; and $\Phi $ is the outflow term to keep the total concentration constant. It is evident that in such a system the wild type will go extinct if A

_{m}> A

_{W}. Coexistence, i.e., the survival of the wild type is only possible if QA

_{W}> A

_{m}. Mutational rates are often expressed in units of mutation/nucleotide/replication (μ) instead of replication fidelity. Given a sequence of length L, the fidelity of replication is Q = (1 − µ)

^{L}≅ e

^{−Lµ}. We can then arrive at the inequality of the error threshold setting an upper limit to reliably replicable sequence lengths:

_{W}/A

_{m}) = 1 relationship, we find that a wild type sequence of length 100 but not more, can be stably maintained. Note that since the threshold expression (Equation (6)) is proportional to the logarithmic ratio of the functional and the non-functional replication rates, increasing the replication rate of the wild-type does not increase the length of the maintainable sequence too much. This result yields Eigen’s Paradox [46]: there is no accurate replicase without a large genome and there could be no large genome without an accurate replicase. Thus, the information that can be reliably replicated is less than the information necessary to code for the replicating machinery. This is a key dynamical problem to which the early evolution of life had to find a solution [47,48,49].

_{0}≡ x

_{n}); all other symbols are as above.

_{i}= 0, i.e., the replicators cannot replicate on their own, only with the help of another catalyst, then the system is fully cooperative and all members coexist [51,52,53,54,55,56]. This is the homogeneous hypercycle. Assuming that all catalytic rates are the same, the dynamics leads to a stable fixed point for two-, three- and four-membered hypercycles [51,52]. Furthermore, if there are five or more members in the hypercycle, then the system approaches a stable limit cycle. Theoretically, any number of sequences can coexist but with high numbers of members some replicator concentrations may decrease to very low values during oscillations and with any one of the members lost the whole system collapses. Therefore, for n > 4 the system is unstable.

_{i}> 0) the members are also in competition and if the A

_{i}values are too large compared to the K

_{i}values, then one or more of the sequences can be lost [54]. Again, hypercycles with n ≤ 3 members converge to a stable fixed point [51] and ones with five or more members exhibit oscillatory behaviour (stable limit cycles [57,58]. Hypercycles of six or more members can be unstable [59]. Stability is further affected by differences in the catalytic aid members give to each other [60]. In conclusion, we may say that the hypercycle can show rich dynamics [61,62], although its ability to maintain the coexistence of even a moderate number of different replicators is limited.

_{n}) that can coexist. They find that if all A

_{i}are the same (A) then

#### 3.1.2. Parabolic Replicators (PR)

_{i}takes the following form:

_{max}denotes the largest Malthusian parameter and we used the following inequality: $\sum}_{j=1}^{N}{r}_{j}{x}_{j}^{p}<{\displaystyle \sum}_{j=1}^{N}{r}_{j}<N{r}_{max}.$ Obviously, any replicator has a positive growth rate if its concentration drops below a critical threshold:

#### 3.2 Models Assuming Structure

#### 3.2.1. Spatial Hypercycle (SHC) and Compartmentalized Hypercycle (CHC)

#### 3.2.2. Coexistence in Open Chaotic Flow (OCF)

^{1/(β+1)}[87].

_{i}is the speed of the reaction front for replicator i (= 1,2), p

_{i}is the probability that replicator i is at the boundary of the habitat stripe and the resource and ν = p

_{1}ν

_{1}+ p

_{2}ν

_{2}is the average reaction speed [88]. Due to dimensionality and symmetry reasons

_{i}really follows Equation (15) and, whenever coexistence is observed, the inequalities 0 < α < 1 hold, just as the analysis forecasts. Competition rules can be defined in different ways in the IB model. Interestingly, competitors could not coexist when some rules were applied but these rules always imply α > 1, as expected. (For α = 1 either species 1 or 2 wins the competition depending on other parameters of the model.) Thus, IB models reveal that without knowing the detailed mechanism of competition we cannot determine the dynamical behaviour of the replicators in open chaotic flows [88]. Moreover, although the analysis has been completed only for the two-species model so far, it is straightforward to extend it to many species with ${p}_{i}={\omega}_{i}{x}_{i}^{\alpha}/{\displaystyle \sum}_{i=1}^{m}{\omega}_{i}{x}_{i}^{\alpha}$, where m replicators compete along the fractal set. Using the method presented in [17] it can be shown that any number of replicators coexist if 0 < α < 1, see Section 3.1.2. That is, the dynamics are formally equivalent to those of parabolic replication, although the subexponential term in the replication dynamics follows from imperfect mixing along the fractal set and not from the self-inhibition of replicators [89].

#### 3.2.3. Trait Group Model and Kin Selection (TGM)

#### 3.2.4. Stochastic Corrector Model (SCM)

- Template replicators replicate independently within vesicles (they are not hypercyclically coupled), competing for shared resources (nucleotides, enzymes, space).
- Replicators contribute to a common good (e.g., metabolism) such that they affect the selection of the whole group, thus compartment fitness (group replication rate) depends on composition.
- Replicators are essential: a group can only replicate if both replicator types are present.
- The redistribution of molecules during fission is not biased by any replicator type but is random for each molecule, hence they will follow a hypergeometric distribution in the offspring.
- Compartment size is relatively small and fission happens before equilibrium is reached in cells.
- Replicator migration (or other transposon-effects) from one compartment to another is negligible.

#### 3.2.5. Metabolically Coupled Replicator System (MCRS)

#### The Concept of a Metabolically Coupled RNA World

**x**= (x

_{1}, x

_{2}, …, x

_{s}) is the population density vector for the s different, metabolically essential replicator species with replication rate vector

**r**; M(

**x**) is the monomer supply provided by metabolism at ribozyme densities

**x**; and $\varphi $ is an outflow vector ensuring that the total density $\sum}_{i=1}^{s}{x}_{i$ of all the essential ribozymes remains constant. In accordance with the assumption regarding the metabolic role of essential ribozymes, the metabolic function M(

**x**) must take the value 0 if any one of ${x}_{i}$ is zero. A simple realization of this constraint is using a metabolic function proportional to the geometric mean of replicator densities:

_{i}growth parameters alone, i.e., the metabolic function has no effect on the order of the growth rates r

_{i}M(

**x**) at any time. Therefore, the replicator with the highest growth parameter excludes all the others, in agreement with the Gause-principle.

- the replicase function is given: any RNA sequence is capable of producing a copy of itself by template replication if it has a sufficient concentration of monomers at its disposal.
- all the members of the metabolic replicator set are indispensable in running a simple metabolic reaction network (metabolism) producing the monomers; if any one replicator type is missing from the set, monomers are not produced at all and the system goes extinct.
- the replicators are attached to a 2D mineral surface on which their horizontal mobility is limited; replicators leaving the surface are lost to the system (replicator “death”).
- nutrient compounds (external initial substrates of the metabolic reaction network) are supplied from the third spatial dimension in excess.
- the metabolites (substrates and products of the reactions that the replicators catalyse) are also attached to the surface, on which they may diffuse to a certain distance d before either being detached from the surface and lost, or used in a metabolic reaction or in replication (Figure 8).
- the metabolic contribution to the probability of a certain replicator being copied is dependent on the local ribozyme composition of its metabolic neighbourhood (i.e., within the distance d defining the metabolic neighbourhood of the focal replicator); only metabolically complete neighbourhoods (which have at least one copy of each essential ribozyme) allow for replication.
- the metabolically active set of ribozyme replicators may have enzymatically inactive parasites, i.e., replicators which do not contribute to monomer production but use the monomers produced by the cooperating replicators for their own replication.

#### Replicator Diversity and Ecological Stability in MCRS Models

#### Evolutionary Stability and Evolvability in MCRS Models

**Phenetic mutations.**Enzymatic control of RNA strand separation during the replication process guarantees that the metabolic replicator community does not follow parabolic growth kinetics (cf. Section 3.1.2) in MCRS, i.e., the populations of all replicator species have the capacity to increase exponentially and this is prerequisite for their evolvability (or, more precisely, selectability). Recent versions of MCRS allow for mutational changes in the structures of all metabolically active replicators so that evolutionary shifts in replicator traits can be simulated and their effects on coexistence and on the stability of the system as a whole can be analysed. In earlier models, only phenotypic changes in the most important replicator functions—replicability, rate of decay and metabolic (ribozyme) activity—had been considered. The phenetic models [35] defined reasonable yet arbitrary trade-off relationships among the three critical traits, assuming, for example, that a mutant replicator that is easier to copy (i.e., features a higher replicability) than its template is less likely to be as good a catalyst (i.e., it has weaker metabolic enzyme activity) and it is probably more exposed to environmental effects, leading to faster decay or loss from the surface (i.e., its decay rate is higher)—all for the very same and, in these phenetic models still implicit, structural reason: a less compact, looser secondary structure. It can be shown in simulations that—even with rather strict phenotypic trade-off constraints enforced between different aspects of replicator performance—it is possible to evolve metabolic replicator sets of nearly optimal values in all these three traits if a small “wobbling” is permitted in the trade-off relationships [35].

**Genetic mutations.**The necessary level of wobbling may indeed be provided by the thermodynamics of RNA folding, as it has been demonstrated in the latest versions of MCRS [38], in which the purely phenotypic, sequence-implicit approach has been relaxed with the assignment of actual nucleotide sequences and the corresponding secondary structures (the latter calculated by the ViennaRNA algorithm [119] on the basis of free energy minimization) to all the replicators present or appearing by mutation in the system (Figure 11). The three critical traits of each sequence on the lattice are direct explicit functions of its primary and secondary structural (i.e., sequence and folding) features. The MCRS mechanism imposes selection on the variations of the resulting phenotypes. The most surprising feature of the dynamics of the extended system is its extreme propensity for robust replicator coexistence through evolving different metabolic functionalities (i.e., distinct ribozyme activity patterns) embodied in replicators of different sequences but highly similar population dynamic and enzyme kinetic properties. For example, simulating the sequence-explicit MCRS with three necessary metabolic ribozyme activities (blue, red and green in Figure 12) and a potentially infinite pool of different parasitic sequences (grey colour in Figure 12), starting from a random sequence distribution with different initial replicator lengths, converges to a stationary distribution with highly similar densities, lengths and enzymatic activities in the evolved set of distinct metabolic replicator species (Figure 12).

**x**) is proportional to the geometric mean of the copy numbers. Obviously, mutations produce metabolically useless parasites in large numbers but they are quickly eliminated by the local regulation mechanism explained in the previous section.

## 4. Discussion

- the stochastic corrector model (SCM), as long as the origin and the maintenance of compartments coupled to replicator population growth are explained;
- the parabolic replicator (PR) model and its dynamical homologue, the open chaotic flow (OCF) model, with the future addition of a scenario for the appearance of a selection regime; and
- the metabolically coupled replicator system (MCRS) model, which meets all the criteria for maintaining diversity and being robust both in the ecological and the evolutionary sense but only for a limited number of replicators as yet.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Genealogy of prebiotic replicator models. The simplest possible model for replicator dynamics is exponential growth, which does not allow coexistence as the fittest always wins. Since it is an idealistic case, all sorts of extensions and deviations from the basic model are intended to make prebiotic systems more realistic and more permissive in terms of coexistence, ultimately crossing the barrier beyond which a sufficient amount of information can be stably maintained on the evolutionary timescale for cellular life to emerge.

**Figure 2.**A 3-membered hypercycle. Each member (R

_{i}) of the hypercycle can catalyse its own replication (A

_{i}) and the replication of the next member in the cycle (K

_{i}

_{+1}).

**Figure 3.**Evolutionary instability in the hypercycle. (

**a**) A parasite (R

_{M}) that enjoys catalysis from a member of the hypercycle (R

_{2}) but does not take part in the hypercycle organization. (

**b**) A shortcut mutation (red dotted arrow) which changes the specificity of the catalysis offered by a member of the hypercycle (R

_{2}) so that it catalyses the replication of a member it should not catalyse. R

_{1}, R

_{2}and R

_{4}now form a 3-membered hypercycle, which can replicate faster than the 4-membered hypercycle.

**Figure 4.**The motion of particles in an open chaotic flow. The blinking vortex-sink system is used for demonstration. It models the outflow from a large bath tub with two sinks that are opened in an alternating manner. Crosses denote the sinks. (

**a**) Diverging trajectories of two particles that initially are close to each other. In this example, they even leave the bath in different sinks. (

**b**) A snapshot on particles distributed along a fractal set (chaotic saddle) in open chaotic flow generated by the blinking vortex-sink system. (Based on [32].)

**Figure 5.**The distribution of two replicators (red) B and (blue) C competing for the same resource material (white) in the wake of a cylinder. The flow is from left to right. The inset in (

**a**) shows the time-dependence of the population numbers n

_{B}, n

_{C}and clearly indicates the approach to a steady state of coexistence. A blow-up of the region indicated by a rectangle in (

**a**) is seen in (

**b**). B-s replication rate is 4/3 of the C-s, while decay rates are the same for the two species. Coexistence of 35 species is experienced in other simulations. (Based on [31].)

**Figure 6.**Wilson’s trait group (or structured deme) model. (

**A**) Individuals with different traits, black and white dots, form localized trait groups. (

**B**) After ecological interactions and selection, (

**C**) survivors are released to form a pool, where they can reproduce. (

**D**) New groups form (After [39]).

**Figure 7.**The stochastic corrector model. The two replicator types are indicated with filled and empty small circles. Arrows indicate transitions, as individual compartments grow and divide. Compartments with green highlight (after division) contain the optimal composition of replicators (after [29]).

**Figure 8.**Basics of the MCRS algorithm. (1) Metabolic support of the four replicators in the von Neumann neighbourhood of an empty site (black

**X**). Red, green and blue

**S**s denote different, metabolically active replicator species, the yellow

**P**stands for a parasitic replicator. (2) The replicator taking the empty site by the next generation is determined by a random draw, with the empty site to remain empty having a constant claim and the claim of each adjacent replicator depending on its own replicability (R) and the metabolic support it receives from within its own 3 × 3 metabolic neighbourhood. (3) Each replication step is followed by replicator diffusion, implemented as D elementary steps of the Toffoli-Margolus algorithm [114] at random positions of the lattice.

**Figure 9.**Persistence of the MCRS at different sizes of the metabolic and the replication neighbourhood. Neighbourhood sizes are given as side lengths of a square-shaped section of the lattice that is centred on the focal site; N stands for von Neumann neighbourhood. The i/j values inside the table cells specify the numbers of persistent/extinct systems out of five replicate simulations; grayscale values are system densities in percentages of sites occupied by replicators within the lattice after 10,000 generations. Panel (

**a**) D = 0, Panel (

**b**) D = 1, Panel (

**c**) D = 4 and Panel (

**d**) D = 100 (Based on [36]).

**Figure 10.**Initial distribution of the replicators a snapshot and time dynamics of the metabolic network on chaotic advection by an open flow. (

**a**) Replicators are placed into separate stripes initially. Different species are denoted by different colours. (

**b**) The snapshot of spatial distribution of replicators at t = 10 in units of flow’s period. (

**c**) The population size is shown as a function of time measured in units of the flow’s period. The size of the metabolic neighbourhood was σ = 10 for each competitors and their spontaneous decay was δ = 0.02. The replication constants were different for each species, these were k

_{1}= 3 (red), k

_{2}= 4 (green) and k

_{3}= 5 (orange). The potential that an empty site remains empty was C

_{e}= 2 (Based on [32]).

**Figure 11.**The 2D secondary RNA structure is determined from the primary structure (nucleotide sequence) using the thermodynamic condition that the folded molecule should have the smallest possible free conformation energy. The conformation calculations are executed by the ViennaRNA algorithm.

**Figure 12.**Trait convergence in the sequence-explicit version of MCRS. (

**a**) Relative replicator frequencies, (

**b**) (metabolic) ribozyme activities and (

**c**) replicator lengths at the stationary states of the simulations, after two million generations, as functions of the selection pressure against sequence length (“length penalty”). Open triangles in panel (

**a**) are the proportion of surviving systems out of 100 parallel simulations; Red, green and blue dots represent the three different metabolically active replicator types, grey dots represent all the parasitic (metabolically inactive) replicators. Persistent MCRS systems are efficiently selected for convergence in all the fitness-related traits of the replicators (Based on [38]).

**Table 1.**Categorization of dynamical models with respect to their temporal and structural resolution. For details of the models see the main text and references. Note that unstructured replicator models in discrete time are generally lacking as fully (i.e., in both space and time) continuous models are much easier to handle analytically.

Structure/Time | Discrete Time | Continuous Time | |
---|---|---|---|

Without structure (only global interactions) | - | QS, HC, PR | |

With structure (global and local interactions) | Compartmentalized | SCM | CHC, TGM |

Spatial | MCRS | SHC, CM |

**Table 2.**An assessment of each model in the context of the three main criteria and their “evolvability”, the scope for the adoption of mutant replicators with a useful function into the system.

Diversity-Maintaining Ability | Ecological Stability | Evolutionary Stability | Evolvability | |
---|---|---|---|---|

HC | An arbitrary number of sequences can coexist if there is no population stochasticity; otherwise some species can be lost. | The cooperative nature of organization ensures that, given high enough catalytic aid, the system is stable. | Selfish parasites and short-cut mutants can destroy the system. | No |

SHC | Due to the importance of local interaction, the number of potentially coexisting sequences is limited. | The cooperative nature of organization ensures that, given high enough catalytic aid, the system is stable. | The organization is stable against selfish parasites but short-cut mutants could still take over. The system still cannot evolve new hypercycle members. | No |

CHC | The number of sequences is limited due to the random assortment into daughter cells. | The cooperative nature of organization ensures that, given high enough catalytic aid, the system is stable. | Group selection can probably maintain existing diversity but the system still cannot evolve new hypercycle members. | N/A |

PR | An arbitrary number of sequences can coexist at arbitrarily small concentrations. | The continuous advantage of rarity of any replicator ensures coexistence at any external parameter combination. | - -
- Non-Darwinian regime.
- -
- No classical selection.
- -
- Any number of new replicators can invade the community without outcompeting others.
| No |

SCM | N/A | N/A | - -
- Stochastic replication/degradation.
- -
- Random assortment during fission.
- -
- Shared metabolism.
| Yes |

OCF | An arbitrary number of sequences can coexist at arbitrarily small concentrations but locally dense populations (the concentration at the boundary of fractals can be very high). | The continuous advantage of rarity for any replicator ensures the coexistence at any external parameter combination. | - -
- Non-Darwinian regime.
- -
- No classical selection.
- -
- Any number of new replicators can invade the community without outcompeting others.
| No |

MCRS | A limited number of ribozyme replicators coexist in a highly robust system. | Advantage of rarity due to the mandatory metabolic cooperation of all replicator species maintains stability across the parameter space. | - -
- Darwinian selection for fitness homogeneity.
- -
- Dynamical trait convergence with functional diversification.
- -
- Sequence-dependent replicator functionality.
- -
- Parasite resistance.
- -
- Parasite “adoption” for useful functions.
- -
- No need for membrane compartments.
| Yes |

TGM | N/A | N/A | - -
- Small compartment size.
- -
- Low diffusion rate.
- -
- Rapid extinction of inferior compartments.
- -
- Selfish replicators are coupled to cooperative ones.
| Yes |

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Szilágyi, A.; Zachar, I.; Scheuring, I.; Kun, Á.; Könnyű, B.; Czárán, T.
Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems. *Life* **2017**, *7*, 48.
https://doi.org/10.3390/life7040048

**AMA Style**

Szilágyi A, Zachar I, Scheuring I, Kun Á, Könnyű B, Czárán T.
Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems. *Life*. 2017; 7(4):48.
https://doi.org/10.3390/life7040048

**Chicago/Turabian Style**

Szilágyi, András, István Zachar, István Scheuring, Ádám Kun, Balázs Könnyű, and Tamás Czárán.
2017. "Ecology and Evolution in the RNA World Dynamics and Stability of Prebiotic Replicator Systems" *Life* 7, no. 4: 48.
https://doi.org/10.3390/life7040048