Regular environmental cycles.

Environmental cycles would have occurred regularly on the prebiotic Earth (e.g. day-night, tidal, seasonal, hot-cold, freeze-thaw, hydration-dehydration, etc.). In the model presented here we appeal to hydration-dehydration cycles, such as those driven by tidal fluxes or day-night cycling, to provide an energetic source for the assembly of monomers into polymers, where physicochemical properties vary with the environmental phase. Polymerization via spontaneous assembly and USR via template-directed synthesis occur during the hot-dry conditions of the dehydrated phase. Polymer degradation and diffusion of monomers and polymers occur in the cool-wet conditions of the hydrated phase. Additionally, functional polymers (when present in the extant population) only exhibit catalytic activity during the hydrated phase, when cool-wet conditions promote the folding of a polymer into its active state.

Reversible polymerization.

The model is based upon informational polymers with reversible backbone linkages [38]–[43]. During the hydrated phase, condensation polymers are subject to spontaneous degradation (hydrolysis), governed by the first-order rate constant . Monomers liberated via polymer hydrolysis are added back to the local monomer population, creating localized feedback between polymer and monomer concentrations.

Finite monomer concentration.

The total number of monomeric units (e.g. nucleotides) in monomer and as polymeric residues is constant in our model system, with equal numbers of the two monomer species labeled and . The choice to focus on closed mass systems was motivated by our goal of connecting to chemically realistic scenarios, where monomer concentration would have been naturally limited by available resources. For systems with a finite supply of resources, sustainable polymerization requires recycling through turnover of resources via polymer hydrolysis [16], [48]–[50]. An exception is the case where new monomers are generated by a functional polymer that acts as a monomer synthetase, where monomer production is still limited by the availability of precursor molecules, which are also a finite resource (see below).

Surface confinement and limited diffusion.

Incomplete mixing, limited diffusion, and surface attachment have been shown to promote template-directed synthesis [51], [52] and to limit the deleterious effects of “parasites” (non-functional polymers) on selection of functional sequences [31], [35], [53]. In the present model, we therefore confine polymer and monomer movement to a surface, as might occur on a mineral with a thin film of water (or a more viscous solution) covering the surface, or other surface-binding phenomenon that permits the reversible association of monomers and polymers and limited movement in two dimensions. The reaction-surface is modeled as a square lattice (see below) where diffusion of monomers and polymers between neighboring sites is governed by the hopping rates and , respectively, subject to the physical constraint condition . Diffusion only occurs during the hydrated phase of the cycle, when water activity is high. During the dehydrated phase, when the surface is dry, or the viscosity of the thin film is high, diffusion between lattice sites is considered to be negligible, with .

Universal sequence replication (USR).

Recent studies have shown that altering the chemistry of nucleic acid backbone linkages (e.g. with a reversible linkage) [38], [39], [41], [43], or binding of template strands to a surface (e.g. reduced mobility compared to monomers) [51], allows for accurate template-directed synthesis for a wide range of sequences (i.e. approaching a form of USR). The effects of idealized USR are explored in our model, i.e. all polymers have the same intrinsic rate constant for replication, parameterized by the third-order rate constant . Despite the fact that all polymers share the same inherent replicative fitness, replication propensity is not the same for each polymer in each replication cycle, as the replication probability for a given polymer depends on both and the local monomer concentrations. Stochastic variation in the spatial distribution of free monomers results in spatiotemporally varying rates of polymer replication, creating a dynamic fitness landscape. As shown below, in the absence of functionality, this selective advantage is random, acting on local populations rather than individuals, with populations sequestering the most resources having an increased chance of survival, independent of their sequence distribution.

Spontaneous polymer assembly and exploration of sequence space.

To simplify our model, we invoke spontaneous polymer assembly as the only means to generate novel sequences (i.e. we do not consider the effects of mutations during replication or of genetic recombination, which have been extensively studied elsewhere, e.g. see [6], [8], [31], [35], [37], [54], [55]). In this framework, polymer degradation allows continual exploration of sequence space as novel sequences are introduced via spontaneous assembly of free monomers. The rate of spontaneous assembly (to be contrasted with template-directed assembly) is governed by the second-order rate constant , and local monomer concentrations. Throughout this work we set , corresponding to a maximum global production rate of approximately new sequences per dehydrated phase when all monomers are free (with lower rates when some monomers are sequestered in polymers). This value is intended to reflect a less efficient process of spontaneous polymerization as compared to template-directed synthesis.

Emergence of a functional polymer.

A critical stage in the origin of life was the emergence of the first functional informational polymers [56]. We therefore explore with our model the case where a functional sequence is discovered by the random appearance of the sequence. Ma and coworkers have previously argued that in an RNA world, where polymer replication is accomplished without the need for enzymatic polymers (i.e. by some mechanism of USR [51], [57], [58]), that a nucleotide synthetase was the first enzyme to emerge [34]. A synthetase would have a clear advantage in an environment where local monomer concentration is the limiting factor for replication. We therefore explore a similar test case. In our simulations, the first polymer to emerge with a beneficial function is referred to as an zyme, a polymer capable of synthesizing monomers from an additional but also finite resource, proto- (). The catalytic activity of the zyme is governed by the second-order rate constant and local concentration. The zyme thereby directly couples the functional dynamics to the local environment (e.g. local monomer abundance). Additionally, the emergence of a synthetase captures some properties of metabolic replicator models [34], [37], demonstrating a possible pathway between the pre-functional stage of replicating nonfunctional sequences and the subsequent stage of functional sequence optimization.

Initial conditions and the reaction surface.

The reaction surface is modeled on a square lattice. Each lattice site represents a locally homogenous reaction domain, characterized by a freely interacting community of monomers and polymers (i.e. a locally well-mixed environment). The lattice size of sites was chosen to be small enough for numerical tractability, yet large enough to allow spatial correlations to be observed for the range of kinetic and diffusive parameters under study (such that any spatial organization observed for the parameter ranges explored here is smaller than the lattice dimensions). We impose periodic boundaries to avoid edge effects. The total number of monomeric units (e.g. nucleotides) in monomer and as polymeric residues is constant, with equal numbers of the two monomer varieties labeled and . The initial conditions are homogeneous, with all mass in monomer: each of the lattice sites is initialized with and monomers, such that monomers each of species and are evenly distributed over the reaction domain at the start of a simulation run. For functional runs in which the zyme appears, each site is also initialized with monomers, and with monomers for simulations where the zyme also appears. No polymers are present at the start of a given simulation run (an exception is for functional runs, which start from an already established pool of random sequence polymers at quasi steady-state, see below).

Defining the polymer species pool.

The number of possible polymer species, , increases exponentially with polymer length: for a polymer of length and two residue types there are possible sequences. We chose to focus our study on polymer populations with a fixed length , meant to approximate the dynamics of a population with a mean length of residues. For the simulations presented here, . Our studies of the dynamics with other fixed polymer lengths, including and , yielded qualitatively similar results. Although using shorter length sequences would have enhanced numerical efficiency, we used oligomers of length 20, as nucleic acids of this length are sufficiently long to adopt folded structures and, arguably, large enough to exhibit initial levels of catalytic activity [61]. Additionally, for the sequence space is sufficiently vast that our simulations never explore all possible sequences within the timescales of our simulations. In other words, the sequence space is larger than what our system can dynamically explore in the space and timescales under study; satisfying the minimal requirement of a system with the potential for unlimited heredity and thus open-ended evolvability [1].

Assigning polymer composition.

Our initial model implementation focused on polymers with fixed length and any possible sequence diversity (i.e. any possible arrangement of and monomer residues could appear). However, sequences with a ratio (e.g. composed of monomers and monomers for ) were an attractor for the dynamics under study, having the greatest access to available resources. Any deviations from the mean distribution, yielding a sequence population dominated by sequences with , quickly returned to populations dominated by (as an example consider a pure -residue sequence which only has access to half the resources in the reactor pool and would quickly be outcompeted by other sequences containing some residues which had greater resource availability). Therefore, the simulations were set such that every sequence nucleated contained both monomers and monomers. This greatly simplified polymer identification since each unique species need only be identified by a single ID or lineage number, (a dramatic simplification over tracking the specific sequence structure of each unique lineage). This approximation reduced the size of the relevant sequence space from to possibilities, but still retained our requirement that the relevant sequence space be much greater than what our dynamics could spatiotemporally explore within a given simulation run (see previous section), and allowed larger and longer statistical samplings.

Coupling of diffusive and kinetic events to environmental cycles.

Our numerical implementation of the processes outlined in the previous section explicitly takes into account environmental cycling in order to accurately capture the dynamics of populations of condensation polymers with reversible linkages. The kinetic Monte Carlo implementation is therefore partitioned into two phases: a dehydrated phase, where all lattice sites are diffusively isolated (i.e. diffusion is turned off); and a hydrated phase, where lattice sites interact diffusively through polymer hopping events and monomer diffusion. Polymer assembly and replication occur only in the dehydrated phase, and polymer degradation occurs only in the hydrated phase. For simulations exploring the emergence of a functional sequence in the extant population, an additional kinetic process describing enzymatic catalysis is included in the hydrated phase.

The dehydrated phase.

During the dehydrated phase, each lattice site on the two-dimensional lattice is diffusively isolated and treated individually with the standard Gillespie algorithm [62]. The dehydrated phase supports the kinetic processes of spontaneous assembly and replication. The probabilities of reaction events are weighted by their relative reaction propensities. Spontaneous polymer assembly occurs with (second-order) reaction propensity(1)and USR via template-directed assembly occurs with (third-order) reaction propensity(2)Here , , and are the number of individual monomers, monomers, and polymers of lineage (the species ID number) at a specific site . The total rate for polymer assembly and USR via template-directed assembly are governed by their respective rate constants, and , and the amount of available monomer resource. This dependence is essential to study how local resource availability affects the dynamics of polymer populations (for example, this feature leads to nontrivial spatial patterning, see Results). Since system dynamics are environmentally driven, a polymer may copy itself at most once per hydration/dehydration cycle. Therefore, after each template-directed replication event the total number of polymer templates available for further synthesis is decreased by one unit. In other words, we explicitly take into account that the template will not dissociate from the substrate until the next hydrated phase when dilution can drive dissociation. We note that cycling is often invoked as a mechanism for driving strand dissociation [63], however, it is not always explicitly included in model dynamics. We explicitly include cycling, along with its impacts on generational turnover, to explore the potential evolutionary impact on our model prebiotic system.

We note that USR via template-directed assembly is treated as a third-order process with a total rate dependent on resource availability through the nucleation term (in addition to the rate constant and availability of the template - see eq. 2). This relationship is meant to treat dimer formation as the rate limiting step for formation of full-length polymers, i.e. for the chemistries under investigation here, dimer association with a template is much stronger than monomer association. USR is treated as a one-step process for formation of full-length sequences, since the majority of polymers will form quickly once the first step of dimerization occurs. We therefore consider these approximations to be consistent with the physicochemical processes modeled, with the benefit that they lead to a simplified implementation of the numerical algorithm while still permitting resource dependence in the rates. The choice of kinetics for spontaneous polymer assembly has similar motivation. Dimers are expected to have stronger association to the reaction surface - e.g. mineral or clay - than monomers, thereby promoting polymerization once dimerization has occurred. The choice to implement resource dependence via a nucleation term (rather than or ) is consistent with our implementation of a reduced sequence space.

The hydrated phase.

During the hydrated phase, monomers and polymers diffuse, and polymers may degrade or, if functionally active, perform catalysis. Individual lattice sites are diffusively coupled in the hydrated phase, and the dynamics are therefore modeled with a spatially-explicit hybrid kinetic Monte Carlo algorithm [59], [60]. All kinetic events are treated locally, occurring within an individual lattice site which is modeled as a locally homogeneous reaction domain, and only diffusive events occur between sites. A natural partition between rare and common events occurs due to the large separation in the population densities of monomer and polymer observed in our simulations. The simulations are therefore hybridized such that monomer site hopping is coarse-grained and treated via mass-action kinetics (see Supporting Information for more details, Methods S1). All other events in the hydrated phase are treated stochastically. We have verified that a full stochastic treatment (via a standard spatially-explicit Gillespie algorithm [64]) reproduces our hybrid results. Against the background of coarse-grained monomer diffusion, the rare polymer events of diffusion and degradation occur. The probabilities of (rare) reaction events are weighted by their relative reaction propensities. Polymer hydrolysis (degradation) occurs with reaction propensity(3)and polymer diffusion (hopping between nearest-neighbor lattice sites) occurs with propensity(4)Here hydrolysis is all or none, and degradation is therefore treated as a first-order one-step process which is stochastically determined. This approximation is made based on the implicit assumption that shorter polymer lengths are less stable, as might occur for cases where mers can maintain stable folded conformations whereas shorter length polymers cannot (i.e. we assume increasingly shorter length polymers become increasingly less stable to hydrolysis).

Data analysis.

Data presented for quasi-steady state distribution averages, for explorations of both kinetic (i.e. replication/hydrolysis) and diffusive parameter space, are the combined result of time-averages over cycles and ensemble averages over a small statistical sampling of runs (averaged over and runs for kinetic and diffusive parameter space exploration, respectively). Small statistical samples are sufficient given the small spread in simulation values and the length of simulations with large time-sampling statistics. The quasi-steady state distribution is defined as the period when the ratio of polymer to monomer achieves an equilibrium value (with fluctuations due to stochastic effects). We used the term “quasi” here to indicate that the sequence population is not static, even at steady-state (see Results). Quasi-steady state distributions are calculated starting at cycles (typically steady-state is achieved at cycles depending on simulation parameters). Time averages were taken from cycles. Data point error bars correspond to sample standard deviation on the mean time-averaged values. In comparing kinetic and diffusive processes for the results presented in this work it is important to note that the simulation dynamics are dependent on the overall rates of processes, which are dependent on both monomer and polymer abundances. The kinetic rate constants (, , and ) and the diffusive hopping rate constants ( and ) are useful in providing measures of the relative strengths of the processes under study, but must not be confused with the actual rates for the different kinetic and diffusive processes, which are dynamically determined by the ratios of monomer to polymer in a given simulation and their spatial distribution (as defined above, eqs. 1, 2, 3, and 4).

Simulating functionality.

In simulations including the zyme, which catalyzes formation of monomers from monomers, two additional processes are added to the hydrated phase, diffusion of monomers and catalytic conversion of . Diffusion of monomers, like diffusion of and monomers, is treated via mass-action kinetics in the hybridized algorithm. Enzymatic catalysis by the zyme is added to the rare events of the hydrated phase, with the probability of catalysis calculated from the reaction propensity(5)where is the number of monomers on a local site , is the catalytic rate constant for conversion of in the presence of the zyme, and is the total number of zymes on the local site . To illustrate the impact of the emergence of a functional sequence, data was saved at cycles for all details of a given simulation run. This data provided the initial starting distribution of monomers and polymer species for the functional runs. The choice of starting at cycles is somewhat arbitrary given that the systemic dynamics in the quasi-steady state evolution are time-independent, but was chosen to be sufficiently late in the system evolution that a quasi-steady state had been established (i.e. the ratio of monomer to polymer was relatively constant). To this initial condition, 60 monomers were added to each site (to model a previously untapped resource in the environment) and a single polymer representing the zyme sequence, or a nonfunctional sequence, was inserted on the lattice as a spontaneous assembly event. Results were averaged over twenty-five runs, each with a randomly chosen insertion point for the inoculated sequence. Selection of the inoculation site was weighted by the propensities for spontaneous assembly (i.e. inoculation was not completely random but determined by the resource distribution in the system as done for any other spontaneous assembly event). The simulations were permitted to run until the inoculated sequence (functional or nonfunctional) died out, or until the sequence had survived for cycles. Lifetimes were averaged over survival times for the inoculated sequence lineage (defined as the duration of time where at least one individual of the inoculated sequence is still on the lattice) taken over twenty-five simulation runs. Population size averages, the average number of extant species, and exploration rate were averaged over the sequence lifetime. For example, a polymer that lives cycles is only averaged over cycles, and therefore yields much higher variance in the data than nonfunctional simulations, which are averaged over entire populations of thousands of sequences, over thousands of cycles. Simulations including a zyme, catalyzing , were inoculated in a similar manner with the initial simulation time taken at  = 4000 cycles.

Spatial patterning.

Before discussing system metric results, it is instructive to consider the effects of varying and on the spatial distribution of polymers, as these diffusion-dependent distributions are important for understanding results for all other system metrics, as well as being interesting in their own right. As shown in Figure 3, simulations that have completed hydration-dehydration cycles exhibit polymer spatial organization that is highly dependent on the diffusivities and , and their relative magnitude. All systems shown in Figure 3 had identical initial conditions – the system was initialized with a uniform distribution of monomers and no polymers at . For monomer diffusive hopping rates , the total number of polymers in the system is fairly constant, with approximately polymers on the lattice. However, large variations in the distribution of polymers are clearly visible. In simulations where monomers and polymers have low diffusivities (e.g. with ), polymer species tend to stay spatially isolated and localized near their nucleation sites. Localized resource recycling sustains these small communities, with small polymer clusters being homogeneously distributed across the simulation lattice. As is increased from to , with fixed at 0.001, polymer cluster size gradually increases until only one or two dominant clusters are observed after hydration-dehydration cycles. A similar trend of increasing cluster size is observed for simulations with a -fold greater polymer diffusion rate, (e.g. ), with larger, more diffuse clusters observed for greater polymer diffusion rates. Further increase in (i.e. and in Figure 3) leads to a loss of defined clusters, or clustering on a scale that is larger than the simulation lattice. In simulations with no polymer movement, , new sequences can only be introduced at a grid point by spontaneous polymerization, and clustering is not observed regardless of value. Additional spatial maps are provided in Supporting Information (Figures S1, S2, S3, S4, S5).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 3. Spatial maps of polymer density.

Each column of images corresponds to simulations run with a different value for monomer diffusivity (in sites/cycle), and each row has a different value for polymer diffusivity (in sites/cycle). All data shown are for kinetic rate constants , , and . All maps correspond to cycles. The color scale is in units of polymers/site. Simulations were only run for cases in which monomer diffusivity is greater than or equal to the polymer diffusivity.

https://doi.org/10.1371/journal.pone.0034166.g003

The clustering patterns observed in Figure 3 emerge due to the underlying stochasticity of the dynamics. As the first polymers nucleate and replicate, stochastic fluctuations in populations lead to inhomogeneities in polymer population densities. Populations with a slight initial excess of polymer grow by sequestering free monomers that diffuse to their local vicinity. Isolated polymers migrate toward these regions of concentrated resources or go extinct due to resource competition. Thus, clustering emerges as a result of an indirect form of cooperativity between replicating polymers, where early populations with higher polymer densities gain a fitness advantage. Polymers that exist within a cluster enjoy a local recycling dynamic in which the polymers of a cluster act as a reservoir of monomers: fresh monomer resources for replication become available upon polymer hydrolysis, where monomers are sequestered into polymers before these recycled resources have the opportunity to diffuse away from the cluster. This dynamic can occur because the overall rate of polymerization within a cluster is larger than the polymer diffusion rate, thereby resulting in localization of polymer populations (for polymer diffusivities with in Figure 3 strong clustering is not observed). Between the clusters, polymer density fluctuates near zero. These low population regions act as physical barriers to the transport of information between clusters, leading each aggregate to have a unique sequence population. In contrast, these polymer-depleted regions permit monomer transport, which, through stochastic fluctuations, allow growing clusters to acquire resources from shrinking clusters, even though the polymer clusters may not be in direct contact. The results shown in Figure 3 also illustrate that, for the parameters used in these simulations, polymer population growth is greatly inhibited if monomer diffusion is too slow. In particular, simulations carried out with were almost devoid of polymers.

Diffusive dependence of system metrics.

In Figure 4 the six system metrics are shown for the set of simulations in which was varied from to , and from to . For nonzero polymer diffusion rates, Average Species Lifetime tends to decrease with monomer diffusion rates of (Figure 4A). This trend is associated with the observed clustering at higher monomer diffusion rates. High diffusion rates allow for a small number of species to spread and sequester the majority of available resources, which causes high extinction rates for later-nucleating sequences. Thus, a species will, on average, have a shorter mean lifetime when a smaller number of species are able to dominate the sequestration of resources. The effect of diffusion on species growth is perhaps more easily appreciated by considering two other system metrics: Average Species Population Size and Number of Extant Species. Average Species Population Size (Figure 4B) shows a positive correlation between population size and polymer (and monomer) diffusion rates, with the average population size increasing with increased diffusivity. This result illustrates that increasing polymer and monomer diffusion rates leads to a decrease in the number of species that are able to dominate the acquisition of resources. Likewise, the Number of Extant Species (Figure 4C) shows that the mean number of extant species, like Average Species Lifetime, decreases with increasing polymer and monomer diffusion rates. In the special case of no polymer diffusion (), Average Species Lifetime and Number of Extant species increase, for the most part, with increasing monomer diffusion rates. This distinct trend is apparently due to increased monomer diffusion rates allowing for the replication of polymers before spontaneous hydrolysis, but without the ability for sequences to spread and “colonize” other regions of the surface, which also limits species population size. As noted above, the total number of polymers in a simulation after cycles was relatively independent of , for (Figure 4D).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 4. Exploring diffusive parameter space.

Plots of quasi steady-state values of Average Species Lifetime (in units of number of cycles), Average Species Population, Extant Species, Total Population Size, Sequence Exploration Rate (in units of number of novel sequences generated per cycle), and Average Local Diversity. Time averages were taken from cycles, and each point is the ensemble average over ten realizations. Error bars denote the sample standard deviation (most are smaller than symbols). The kinetic rate constants are , , and . The blue data set shows the case where the , where polymers are completely immobile.

https://doi.org/10.1371/journal.pone.0034166.g004

In contrast to the results presented above for variations in hydrolysis and replication rates, we observe that the Sequence Exploration Rate (Figure 4E) is essentially independent of polymer diffusion rate and only modestly dependent on the monomer diffusion rate for the parameter ranges explored in Figures 3 and 4. Thus, when considering the ability for a system to evolve through sequence exploration, variations in monomer and polymer diffusivities are more likely to affect the capacity for survival of a newly nucleated sequence than the system's capacity to discover new sequences. For example, in the case of no polymer diffusion (), a sequence is not able to reap the benefits of spatial expansion, which permits population growth. On the other hand, a sequence that emerges in a system with high polymer mobility will experience strong competition, and may not take hold in the system. The ability for a functional sequence to overcome these pressures in explored below. Finally, the metric Average Local Diversity shows a unique response to changes in monomer and polymer diffusion rates. Like Average Species Population Size, Average Local Diversity increases with . This positive correlation with polymer diffusion illustrates how more rapid polymer movement leads to more complete spatial mixing of polymer species. In contrast, unlike the five other system metrics, Average Local Diversity is apparently independent of , an observation that, taken with other system observations, demonstrates how variations in monomer diffusion rates can affect the spatial distribution of polymers without affecting the local spatial distribution of species diversity.

Functional selection.

Of the four system parameters explored above, the monomer hopping rate, , was selected for variation in a series of simulations in which the zyme sequence “spontaneously” appears. This parameter was chosen because diffusion of monomers into the vicinity of an zyme, as well as the diffusion of newly synthesized monomers away from a zyme, was expected to affect the ability for an zyme to become established in a pre-existing population. In Figure 5, the Species Lifetime and Population Size of zyme lineages are shown for simulations in which was varied from to . The polymer hopping rate, , was fixed at , and all other parameters were the same as those used in the simulations presented in Figures 3 and 4. The selective advantage of the zyme over nonfunctional sequences is clear in these simulations, with the lifetime of an zyme sequence (shown in black) being to times as many cycles as the average lifetime of nonfunctional polymers (shown in green) in simulations with identical parameters, but without the appearance of an zyme. The population size of the zyme, for all values explored, was approximately two times larger than the average population size of nonfunctional polymers in simulations where no zyme appeared.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 5. Selection for functional sequences.

Plots illustrating the propagation of a functional zyme, compared to nonfunctional sequences. A: Average Species Lifetime (in units of number of cycles), and B: Average Population Size. Each data point is the ensemble average over twenty-five runs, with error bars denoting the sample standard deviation. Kinetic rate constants are , , and , with a polymer diffusion rate constant of sites/cycle. The green points represent overall population statistics for realizations with no zyme (plotted in green in Figure 4). The black points represent statistics for a single functional zyme. The blue points represent statistics for a nonfunctional sequence introduced at the same location and cycle as in the zyme simulations, except with no functionality.

https://doi.org/10.1371/journal.pone.0034166.g005

It is important to note that Species Lifetime and Population Size for simulations with nonfunctional polymers are weighted towards longer lifetimes and larger population sizes by sequences that appear early in the simulations. Early-time sequences are able to attain relatively large populations before the free monomer concentration begins to limit replication. Therefore, the average lifetimes and populations of early-time sequences are typically much greater in magnitude than those of nonfunctional sequences that emerge later, e.g. after DKS has been established. Thus, the observed enhanced species lifetime and population size of the late-appearing zyme is even more significant than it first appears relative to the green curve in Figure 5. To illustrate this point, simulations were carried out in which a nonfunctional polymer was introduced at the same cycle time and lattice site as the zyme (without the introduction of the zyme). As shown by the blue curve in Figure 5A, the selective advantage of the zyme sequence is, as expected, more dramatic when compared to the measured lifetimes and population sizes of the late-appearing nonfunctional sequence for all values of .

The lifetime of an zyme sequence appears to be less dependent on than the non-functional sequences, being of similar lifetime from to . One exception is the considerable increase in zyme lifetime observed in simulations with . Under these conditions of very slow monomer diffusion, the lifetime of nonfunctional polymer sequences, even those that appear early in a simulation, are too short for any polymer lineages to become established and for a considerable population of any nonfunctional sequence to subsist. Thus, the results shown in Figure 5 also demonstrate that the zyme is able to survive in an environment that cannot sustain nonfunctional polymer populations. In other words, the zyme, by significantly enhancing its own survival, can become the first sustainable extant sequence in a highly dynamic and previously unsustainable environment.

System-level benefits of a functional sequence.

The population of the zyme is plotted in Figure 6A as a function of time (in units of cycles) for a simulation with . A comparison of this plot with those of simulations with only nonfunctional sequences reveals that the population growth of this sequence is faster and to a similar level of the most successful early-appearance polymers (e.g. Sequence ID 33 in Figure 1A). The effect of zyme appearance on the overall system can be appreciated by comparing the total polymer population after the appearance of the zyme to the steady-state polymer population that is maintained by nonfunctional sequences (Figure 6B). In these simulations, the zyme becomes well established in the population, but its population growth represents only about of the total increase in polymer population. Thus, most of the newly formed monomers are used to generate new sequences and to replicate existing nonfunctional sequences. While this result might, at first, be considered deleterious for the zyme and a waste of resources on “parasitic” nonfunctional sequences, it must be realized that the allocation of resources to other sequences is positive at a system level, as the capacity to search sequence space and to propagate other functional sequences is enhanced.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Figure 6. Spatial distribution maps for no functional species, one functional species, and two functional species.

The three scenarios shown are all identical up to cycles, at which time the system has achieved a quasi-steady state distribution. In the first scenario, no functional sequences appear. In the second scenario, a functional zyme appears at  = 2500. In the third scenario, the same functional zyme appears at  = 2500, and a functional zyme also appears at  = 4000. A: Time evolution of the Species Populations of the zyme and zyme. The units of time are in number of cycles. The red curve corresponds to the second scenario, having only the zyme, while the black and blue curves correspond to the third scenario with both enzymes emerging. B: The time evolution of the Total Polymer Population for the three scenarios. C: The spatial distribution of the polymer (total) and monomer concentrations, at  = 5000 cycles. White arrow indicates contour containing % of zyme polymers, cyan arrow indicates contour containing 95% of zyme polymers. Kinetic rate constants are , , and , and diffusive rate constants are and sites/cycle.

https://doi.org/10.1371/journal.pone.0034166.g006

A comparison of spatial maps of polymer and monomer densities for systems with and without the introduction of the zyme sequence further illustrates the effects of a functional zyme on a system of otherwise nonfunctional polymers (Figure 6C). The location of the initial appearance of the zyme sequence is essentially devoid of polymers at cycles in the simulation containing only nonfunctional polymers. In contrast, introduction of the single zyme sequence at cycles results in a substantial change in the local and global polymer distribution. A cluster grows around the site of zyme introduction, which eventually merges with nearby clusters of nonfunctional sequences. Within this larger cluster the zyme coexists with nonfunctional sequences that benefit from the temporal increase in local free -monomer concentration. Dramatic differences in -monomer and -monomer densities are also observed across the simulation surface (Figure 6C). The monomer is no longer a limiting factor in polymer production, and eventually increases to higher than pre-zyme levels at every point on the simulation domain. In contrast, the monomer becomes a more strongly limiting reagent to polymer production, with -monomer levels dropping well below those of the pre-zyme levels across the simulation surface as more resources are consumed by the larger polymer population.

Functional cooperation over time and space.

Having demonstrated that a single functional sequence can become established within a system of nonfunctional polymers, we next investigated the effects of adding a second functional sequence that acts as a catalyst for the conversion of a proto- monomer () to monomer. As shown in Figure 6A, in simulations where 60 monomers were initially present at each lattice site, the appearance of a single zyme sequence at cycles ( cycles after the appearance of the zyme) quickly results in a burst of zyme population growth. As was observed for the zyme, the growth in zyme population represents only a fraction of the total increase in the global polymer population size (Figure 6B). A plot of polymer density cycles after the appearance of the zyme shows the growth of a large cluster with the zyme population at its center. Other polymer clusters on the surface show substantial growth as a result of the appearance of the zyme.

An important feature of the system explored here is that the selective pressure for a functional polymer can be transient in time and space. Each simulation run shows slightly different dynamical evolution after the appearance of a functional polymer. zyme and zyme population plots and 2 D density plots for a second example are provided in the Supporting Information (Figure S6) for a simulation with (as compared to  = 10 for the simulations shown in Figure 6). For this simulation where only the zyme appears at 2500 cycles (the zyme does not appear), the zyme sequence goes extinct within the next 6000 cycles. In contrast, when the zyme is nucleated near the center of zyme activity, survival of both functional sequences is enhanced. We note that extinction of the zyme, in the absence of zyme appearance, does not terminate system evolution. Figure S6B illustrates that the pool of polymers benefited from the transient activity of the zyme. Furthermore, the zyme had nearly exhausted its benefit to the system at the time of extinction, having had converted nearly all to monomers. However, a stable cluster emerged where the zyme nucleated (Figure S6C), leading to localized enhancement of polymer density and number of extant species. This result demonstrates that the zyme (or any other functional sequence) is not required to live indefinitely in order to have a positive impact on a system undergoing continuous rounds of USR. Moreover, once the -monomer is no longer a limiting reagent, it would be a distinct disadvantage for the zyme population to remain high: for continued system-level evolution, it is more advantageous for the monomers in the zyme sequences to be recycled into polymers with functions that are needed at later times.