Turbulent coherent structures and early life below the Kolmogorov scale

Major evolutionary transitions, including the emergence of life, likely occurred in aqueous environments. While the role of water’s chemistry in early life is well studied, the effects of water’s ability to manipulate population structure are less clear. Population structure is known to be critical, as effective replicators must be insulated from parasites. Here, we propose that turbulent coherent structures, long-lasting flow patterns which trap particles, may serve many of the properties associated with compartments — collocalization, division, and merging — which are commonly thought to play a key role in the origins of life and other evolutionary transitions. We substantiate this idea by simulating multiple proposed metabolisms for early life in a simple model of a turbulent flow, and find that balancing the turnover times of biological particles and coherent structures can indeed enhance the likelihood of these metabolisms overcoming extinction either via parasitism or via a lack of metabolic support. Our results suggest that group selection models may be applicable with fewer physical and chemical constraints than previously thought, and apply much more widely in aqueous environments.


Introduction
At major transitions in the history of life, complexity has emerged via cooperation. [1][2][3][4] Most saliently, cooperation between groups of individuals is considered important for the origins of life, as well as the emergence of multicellularity. [3][4][5][6][7][8] One approach to formalize these events generally has been to define "coming together" as the aggregation of independent individuals, and "staying together" as the event where offspring remain close to their ancestors. 4 However, the emergence of cooperation at each of these transitions includes challenges that can be primarily ascribed to the exploitation of cooperators by parasites 6,7,[9][10][11] -if cooperators interact and provide benefits to individuals in the population with equal probability, i.e. the population is "well-mixed", those that receive benefits but do not reciprocate gain a competitive advantage and drive cooperators to extinction, thereby preventing the emergence of more complex entities. 11 To resolve this conflict, population structure models for early life were introduced in which individuals selectively interact with others. 6,[11][12][13] These include lattice models, where individuals are restricted to interactions with particular neighbors (e.g. on a 2D surface), [13][14][15][16] and group (or multilevel) selection models, 17 where individuals only interact with elements within their groups. 6,[11][12][13][18][19][20] In particular, these models 6, 10 and recent experimental studies 21,22 have been most intensively applied to problems in early biology and the origin of life. This is perhaps a natural domain of application because groups may be defined as concrete physical structures (e.g. droplets or protocells 5 ) and the complexity of the underlying processes is relatively minimal.
In the context of abiogenesis, the study of spatial lattice models showed that cooperation can be maintained in "spiral waves" on simple 2D structures without flows. 13,14 One physical realization of these models can be envisioned as rocky surfaces potentially undergoing wet-dry cycles. 13,15,23,24 Group selection research, studied in abstract and also in the context of protocells, suggested that compartments provide necessary functionalities like collocalization of members for reactions, 5,6,25,26 creation of gradients across boundaries (e.g. in lipid membranes 27 ), exclusion of parasites through division 11, 19,20 and rise of diversity through merging, 28 all provide benefits for cooperators.
Many populations of biological organisms exist in an aqueous environment. As such, additional physical mechanisms governing population structure which are compatible with aqueous environments, such as active droplets, 5 slicks of fatty oils, 29 bubbles containing aerosols particles, 30 or surfactant micelles 31 have been considered as early mechanisms for group selection. Interestingly, while the chemical role and necessity of water itself for emergence and maintenance of life is well-appreciated, the potential role of its transport properties (in the absence of additional structures such as those described above) is relatively unexplored. Recent years have seen the beginning of interest in how flows affect population genetics, [32][33][34][35][36] but none of these works address cooperation, which depends much more fundamentally on population structure than does the spread of (dis)advantageous alleles. 37,38 As has been described recently, 34,35 the well-mixed assumption in ecology (an assumption that a population may be represented by a single nonspatial compartment, i.e., all individuals interact with all other individuals at all times) may never be realized in fluid flows, even those which are strongly mixing or ergodic. It is therefore sensible to ask whether the types of population structure which appear naturally in fluids are conducive to the types of cooperation necessary for early evolutionary milestone.
To address this question, we consider the functionalities that motivated classical group selection and ask if moving fluids can reproduce them. We explore in particular if (I) collocalization, (II) division, and (III) merging, which are physical properties of groups, can be replicated. Because many of the aqueous environments pertinent to life are not quiescent, such as the surface of the ocean, ponds or montane streams, we are particularly interested if turbulent flows can support these fundamental aspects of early group selection.
Our interest in these questions is therefore closely related to the recent surge of interest in the stirring and mixing of passive tracers in turbulence, and especially related to the dynamics of coherent Lagrangian structures, [39][40][41][42] long-lasting material surfaces within a flow which roughly divide up the spatial domain into regions with very different transport properties. Depending on the Lyapunov exponent (which measures the convergence or divergence of nearby fluid particle trajectories), this "skeleton" of the flow can be Here we posit that coherent structures, which are a common spatiotemporal motif in turbulence, could provide long-lived safe havens for cooperators. Here we show an example of a two-dimensional fluid flow on a periodic domain, where coherent vortices have formed from random initial conditions. The size of the whole domain is L, and the size of the smallest coherent spatial structures is the Kolmogorov lengthscale, η. White arrows show the velocity field, and colored shading shows vorticity contours. The inset indicates the local flow field near a large vortex, along with individuals from three different species (X,Y,Z) in a cooperative metabolism. Here we show a simple cooperative hypercycle, in which X catalyzes Y, Y catalyzes Z, and Z catalyzes X. While all particles die at the same rate d, birth rates increase under catalyzation, which occurs when a catalyzing particle is within a distance less than the interaction radius R int of another particle. categorized as surfaces that are highly repelling, attracting, or elliptical (meaning particle trajectories remain parallel for long times). 43,44 Lagrangian coherent structures (LCS) can be thought as isolated vortices which trap fluid and thus acts like compartments necessary for group selection. Furthermore, to the naked eye they seem to replicate the qualitative features (I-III) above which are necessary for evolution via cooperation -in real flows, one can observe the creation and destruction, merging and division of LCSs. They tend to arise spontaneously whenever the fluid is put in motion by some large-scale forcing, like surface winds, and they are eventually destroyed by viscous dissipation. 45,46 In the rest of this paper, we substantiate the idea that LCSs in fluids can provide the features (I-III) above that would have been impactful in early cooperative stages of evolution. We study these structures by performing simulations of replicating cooperators embedded within turbulent flows. Furthermore, once these are established, we examine the role of these group-like properties as well as other transport properties of flows on the spread of genetic diversity.

Modeling emerging populations in flows
We consider a population of biological organisms in a moving fluid. We are particularly interested in the problem of cooperation at very small scales, from tens of nanometers (small replicating RNA strands) to microns (e.g. single-celled organisms). Even at these small distances the fluid can be described as a continuum, because the mean free path of water molecules (the average distance travelled without a collision with another molecule) is roughly an angstrom. Thus the Knudsen number (the ratio of the mean free path to the size of the particle) for the smallest object of interest would be no larger than 10 −2 . Hence even the smallest biological particles under consideration undergo many collisions with the surrounding fluid molecules before they can travel an appreciable distance.
Dispersion and clustering of particles in a turbulent flow results from the stirring by the vortices that are the telltale signature of turbulence. While the details of dispersion depend on the statistics of the vortices-set through the choice of forcing and dissipation-some aspects of turbulent dispersion are generic.
In particular trajectories are characterized by random-like walks punctuated by looping motions when the particles are trapped in LCSs. It is during trapping in LCSs that particle clusters can form and stick together for some extended period of time.
We use a minimal two-dimensional point-vortex model, 47 that has been previously employed to study dispersion induced by homogeneous turbulence in two spatial dimensions. [47][48][49] We consider a set of N 2D vortices of strength (circulation) Γ j and position x j = (x j (t), y j (t)). The motion of each vortex, induced by the other vortices is described by the equations, For siimplicity, we assume uniform strength among vortices Γ, but consider clockwise (Γ < 0) and counter-clockwise (Γ > 0) vortices. The vortices are confined to meander within a square box of size L.
To avoid the issues of edge effects, we assume that our domain is doubly-periodic (see Methods).
The point-vortex model is appealing because it qualitatively captures all of the desired features, having spatiotemporal regions of both fast and slow flow, with material surfaces that are attracting, repelling, and neutral. 47,50 In actual applications, the strength Γ and the density of vortices (set by N , keeping the domain size fixed) would be tuned to account for the specific properties of the flow under consideration, but the qualitative results described below are independent of these specific choices.
Note that the point vortices which generate the flow are permanent; only the coherent structures which arise around and between them are impermanent (for an example, see Fig. 4). The permanence of the point vortices allows the flow to propagate forever without a change in its statistical properties, which is desirable since the time-independence of the model statistics is necessary to compare the impact of flow versus biology timescales in different simulations using different biological parameters.
The biology of the particles is modeled by a stochastic birth-death process. Besides advection by the flow, particles can undergo one of two reactions: death or replication. Replication is defined as the appearance of a new particle, of the same species as the parent particle (described below), in the vicinity of the parent. Both birth and death are modeled by Poisson processes: a given particle dies between time t and time t + dt with probability d, so that in a very large population, the total size is decreasing with a rate of roughly d (we set the death rate as a constant). The replication rate, however, is implicitly timeand space-dependent, because of the possibility of cooperation.
We consider several models of early replication metabolisms as also adopted in previous studies. 10,51 These models are summarized in Table 1. To incorporate spatialization, we consider an effective "radius of interaction", R int , within which a individual can provide metabolic benefits to another. Our metabolisms under consideration capture most of the critical dynamics in early cooperation that might arise, no matter the exact chemical pathway. We follow the naming convention as adopted in: 10 Replicase R1 cooperates with any other particle, but also itself, so its replication rate is effectively space-independent (we assume the benefits of cooperation are not additive, so that a particle at any time is either in a state of being cooperated-with or not). All other species are effectively parasites. Replicase R2 differs from Replicase R1 only insofar as a particle cannot cooperate with (replicate) itself (though distinct particles of the replicase type can still cooperate with one another). Finally, we study a hypercycle, representing a system in which an individual of type A can only cooperate with B, B with C, C with D, and so on, with the last type able to cooperate with A, which is a well-studied mechanism for an early metabolism. 51,52 We have used a finite-population, agent-based approach where an emerging population would have faced very basic challenges to its survival; should these challenges be surmounted and the population grow to a much larger size, then issues we omit from our modeling (such as resource limitation, or the possibility of modeling the population as a continuous field) would become relevant. Some of these situations have already received attention in the literature. 32,33,36 Our work therefore bridges the gap between these works, which focus on constant-fitness populations at high numbers in turbulent fluids and the case where Table 1. Reactions for the various metabolisms used in this work. While all particles die at the same rate d, birth rates increase under catalyzation, which occurs when a catalyzing particle is within a distance less than the interaction radius R int of another particle it has the ability to catalyze. In the replicase models, particle A can catalyze both A and B, but B cannot catalyze A -the difference between R1 and R2 is whether a single A particle can (R1) or cannot (R2) catalyze itself. there are no organisms, by considering both small-population effects and frequency-dependent fitness.

Natural fluid timescales may impose selective pressures on early replicators
To summarize the interplay between turbulent transport and biology we introduce the Damköhler number Da, which is the ratio between the timescale over which the velocity experienced by a particle changes and the characteristic timescale of the biological process under consideration. The Damköhler number has proven useful in other studies of biological processes in turbulence, 53 where global extinction can be guaranteed beyond a certain threshold value.
The typical free path of a particle in a flow field generated by point vortices is given by the mean inter-vortex distance, because particles will change direction every time they bounce into a new vortex.
The inter-vortex distance in our setup is given by ∝ L/ √ N . The characteristic velocity experienced by particles in this flow is given by Γ/2π 1 A characteristic timescale for the trajectories is therefore given by distance over velocity τ F ∝ L 2 /N |Γ|. This is essentially the time for which a particle travels in the fluid before bumping into a new vortex. A natural timescale for biology is the expected lifetime of a 1 A single vortex traps particles in regular circular orbits and generates no dispersion. Dispersion of particles is induced by vortex pairs of opposite sign which generate a jet of velocity Γ/2πδ at the midpoint between them; δ is half the inter-vortex distance. In our flow d ∝ . Figure 2. The relative value of the fluid velocity and metabolic timescales is critical for cooperation. The importance of the Damköhler number, and of flows in general, can be understood via the pair covariance G(|x 1 − x 2 |), which gives the probability of finding a pair with interparticle separation |x 1 − x 2 |. Starting from an initial condition of many particles in close proximity (so G(|x 1 − x 2 |) can be approximated by a delta function at G(0)), the evolution of pair covariance is governed by competition between flow and biology. Histograms show an average over 1000 simulations. Times are measured in the expected lifetime of a single biological particle. When particles reproduce, we use the interaction radius R int = 0.03, and the fraction of interacting particles (left of black bar) is given at the top-left. In the absence of flow (green histograms), the initial condition will slowly spread into a large colony, with most particles within the interaction radius, leaving the susceptible to parasites. On the other hand, passive particles in a turbulent flow (grey histograms) obey the Richardson law, in which the interparticle distance increases on average, here quickly reaching a limiting distribution due to the doubly-periodic nature of our spatial domain. In this situation, so few pairs are within the interaction radius that most lineages should be expected to die out and the population on average goes extinct. A replicating population in a flow at Da = O(1) (blue histogram), however, can combine the advantages of both situations, creating rich structure (G(x 1 , x 2 ) having broad support) while also having a higher number of interacting pairs than in pure Richardson diffusion. particle, which is simply τ B = 1/d. The Damköhler number is therefore Da = τ F /τ B = dL 2 /N |Γ|. In our simulations we set d = 1 and keep L fixed, so that effectively Da = (N |Γ|) −1 .
Examining the interplay of physics and biology at extreme Damköhler numbers reveals how biological and physical timescales must be in relative harmony for cooperation to flourish. We expect both extremely low and high Damköhler number to characterize flows where the physics does not help population survival.
At very high Damköhler numbers, particles undergo millions of generations before the particles have a chance to disperse. Particles initialized in a coherent vortex will remain there, along with any parasites initialized nearby; should they survive, their lineage will struggle to spread even by migratory events, since the transit time between coherent structures becomes so long that a large number of comigrating cooperators is necessary to ensure survival. As seen in Supplementary Video 1, an initially randomlydistributed population (here, a two-species hypercycle) simply coarsens into a few dense clusters, with others dying out before they can be brought within distance of other cooperators. High Damköhler number environments therefore satisfy the requirements of collocalization, as "staying together" is almost mandatory. However, without the ability to divide and merge over biologically significant timescales, avoiding parasites (which we may imagine emerge via mutations and therefore arrive at a fairly regular rate in any population) would be quite challenging.
Extremely low Damköhler numbers (Supplementary Video 2) are also detrimental to survival. In this limit, particles disperse quickly throughout the flow on biological timescales, undermining cooperation. We present statistics for this limit in Fig. S3. Most particles spend the majority of their expected lifetime (in the absence of catalytic aid) alone, and do not benefit from cooperation. Indeed, in simulations with small inocula and low Damköhler numbers we always observe extinction.
To interpret results on a whole range of Damköhler numbers, it is useful to introduce the two-particle covariance G(x 1 , x 2 ), defined as the probability that a pair of partciles is found at positions x 1 and x 2 . We plot a relevant example of this quantity in Fig. 2 for cases where the transport and biological timescales are comparable (Da = O(1)), where there is no fluid motion and Γ = 0 (Da >> 1), and where there is no biological effect and d = 0 (Da = 0). Averaged over many realizations, our process is isotropic and homogeneous and the pair covariance is a function of the interparticle distance only, so that the probability of a pair separated by a distance ||x 1 − x 2 || is G(||x 1 − x 2 ||). The relative contributions from biology and flow have competing effects, each with clear advantages and disadvantages for cooperation. In the absence of motion, reproducing populations will always have the largest values of G(||x 1 − x 2 ||) at the smallest values of ||x 1 − x 2 ||, since birth places offspring nearby, whereas death can occur anywhere, leading to cluster formations. 54,55 This means that a large number of pairs are within the interaction radius R inta great benefit in the absence of parasites, but a serious liability in the presence of even a few parasites.
On the other hand, the two point covariance for particles stirred by a point-vortex flow and experiencing no biological effects is known to increase as a power law from zero at zero interparticle separation. 56,57 Therefore, the smaller the value of R int -which are more realistic-the fewer the cooperative interactions and extinction is expected as confirmed in our low Damköhler number simulations.
Intermediate Damköhler number offer a favorable trade-off, because motions prevent particles from aggregating all at small separations (which protects against parasites, as we show below) without completely eliminating particles at small separations (as in the no biology limit). Interestingly, the importance of intermediate Damköhler number for population fitness has already been observed in experiments involving bacterial mutualism where the bacteria are also motile. 58 For the rest of the paper, then, we will focus on the regime Da ∼ O(1).
The covariance function analysis confirms that the likelihood of success of a particular biological metabolism is contingent on the properties of the flow in which it is immersed. Cooperation, in particular, thrives best when the dispersion of particles by the flow acts on timescales comparable to the biological ones at Da ∼ O(1). Given a particular flow, metabolisms that replicate too fast or too slow compared to the physical timescale are quickly extinguished, while those whose natural timescale matches that of dispersion thrive.

Vortices collocalize replicating particles
One of the helpful properties of a protocellular membrane is to collocalize particles, which is essential in allowing cooperators to reach one another and locally increases the concentrations of necessary components. So far we have focused on a rough estimate of the rate at which two particles drift apart from each other, τ F ∝ L 2 /N |Γ|. However in a turbulent flow, like the one generated by the ensemble of point vortices, particles pairs are occasionally trapped into LCSs and remain close to each other for much longer than τ F until the LCS breaks apart and dispersion resumes. The finite-time Lyapunov exponent λ is often used to estimate the rate at which two particles starting a distance δr 0 drift apart, where δr(t) is the particle pair separation at some time t after t = 0. Regions of zero and negative finite-time Lyapunov exponent are regions where passive particles are approaching or staying near one another for a time interval t. A simple illustration of this fact is given in Fig. 3, which shows the average particle separation of an initially-adjacent pair per vortex period. Since particles reside much longer in strong vortices, they are more likely to find new neighbors from within the vortex than without ("coming together") and will remain near their offspring for longer ("staying together").
To confirm that the collocalization we see in a vortex can sustain populations where cooperation is necessary for survival, we simulated two different such metabolisms in a two-vortex flow with a timedependent oscillatory perturbation often used in the chaotic mixing and coherent structure literature. [59][60][61][62] These metabolisms were: a replicase, in which one species increases the replication rate of any other species within the interaction radius, and a hypercycle, as pictured in Fig. 1, where each species requires cooperators from the preceding species in its metabolic cycle in order to survive. We also investigate the two two types of replicases, R1, R2. (as described in Table 1).
Note that we could have simulated a vortical-like motion on a two-dimensional lattice such as those As increases, particles become more likely to cross onto the unstable manifold leading out of the vortex. We define "fixation" as a hundredfold increase in the population size starting from a very small initial condition, where continuum limits previously described 32,33,36 begin to be applicable. (B): The average particle separation (per vortex period) of an initially adjacent pair of particles, versus the strength of the perturbation, in logarithmic scale. (C): Results of numerical simulations of various metabolisms (colors) for varying distances R int at which particles can cooperate with one another. Square markers indicate the likelihood that the population increases 1000-fold starting from an inoculum of 5 particles of each species injected into a random field (no vortex), whereas circular markers indicate the same likelihood for an identical inoculum injected into a finite-size vortex. Note that for Replicase R1 (blue), the establishment probability is equal for both = 1 and = 0.01, so that only the solid line is shown.
that have been employed in previous work. [13][14][15][16] However, we find that automorphisms resembling flows disrupt cooperation in these cases, resulting in extinction (see Supporting Information). This is likely because of the the constant density of particles on a lattice, which does not allow for dense clusters of cooperators to be formed as a result of the flows, whereas the flows still easily break cooperative neighborhoods. We therefore always use a continuous flow, and we furthermore posit that many of the important effects seen here and throughout the paper are contingent on the population being able to organize in clusters with high density of particles.
The outcome of simulations in which a small "inoculum" of these metabolisms is placed in the center of a vortex of finite size is shown in Fig. 3. In addition to varying the nature of cooperation, we varied the range of cooperation (the distance R int at which cooperators are allowed to help one another) and the strength of the unsteady perturbation, . Increasing the value of R int effectively decreases the role of spatial structure, as R int = 1 represents a well-mixed population, in which a population made only of cooperators will always thrive. For lower values of R int , the effect of the vortex collocalization is important, with a stronger vortex being much more useful than a weaker vortex. For instance, with R int = 0.01, an inoculum of Replicase 2 would likely perish in a weak vortex ( = 1), but in a strong vortex ( = 0.01) it multiplied from ten particles to a thousand in roughly 40% of simulations.

Segregation of coherent structures by flows can insulate cooperators from parasites
While co-localization is necessary for cooperation to take place, it is prone to parasitism. If some elements become defective over time and cease providing cooperative support (effectively becoming parasites), the entire population of cooperators becomes vulnerable to collapse. 9 The likelihood of this collapse increases still further if we assume that cooperators pay some "cost", a sacrifice in fitness, to provide their metabolic benefit, resulting in prisoner's dilemma dynamics . 1 In compartment models, group selection has been offered as a solution to this problem, as selection works against groups that contain a large number of parasites.
Since turbulent flows naturally generate multiple LCSs, it would seem that group selection can also preserve cooperators by the same mechanism. However, the analogy is not perfect. LCSs are impermanent, even if they are long-lived. Further, since they are not physical membranes and furthermore merge and divide with some regularity, they allow a degree of mixing which is sometimes forbidden in theoretical compartment models. It therefore remains to be shown that LCSs provide the kind of population segregation necessary for cooperators to escape parasites.
To demonstrate the role of turbulent flow in isolating and removing defective elements, we consider the population dynamics of an R2-replicase in a system of N=7 point vortices. All particles within range of a cooperator are catalyzed, even if there is a single cooperator. Furthermore, we do not enforce a resource limitation or maximum population size.
In the well-mixed limit, be it by design or by happenstance in realizations where a cluster is so dense that every particle is within the range of interaction of all others, cooperators and parasites are equally fit. The absorbing state is therefore one of complete extinction, which is achieved when all cooperators die out due to stochastic fluctuations. In this limit the fitness of each individual is constant, so that it is reasonable to consider dynamics where the population size does not change, given e.g. by a Wright-Fisher process, 63 and simply consider the fractions of replicases and parasites fluctuate until one goes extinct.
If the parasites go extinct, the system persists unchanged for all time; if the replicases go extinct, then the population would collapse. This collapse occurs with probability f , where f is the initial fraction of parasites (we ignore mutation between types). 64 This is a useful benchmark for the success of cooperation in our non-well-mixed, non-population-limited model, and so we considered a variant of our model with constant population size and Wright-Fisher updating. The results are shown in Fig. 4. Starting from an initial population, randomly distributed in space, with an initial fraction of half-parasites and halfreplicases, we would expect parasites to fix in half of the simulations in the absence of flow; however, due to the flow, which can isolate parasites and replicases, we found that replicases can do slightly better than parasites for certain values of R int and Da. As seen in previous work, 34 the well-mixed results are seen at much lower values of R int than those which span the system, with the results already visually indistinguishable from the well-mixed prediction for R int ≈ 0.1.
We then lifted the restriction of fixed population size, simply tracking the relative sizes of the parasite and replicase populations in our normal branching Replicase-R2 system. Over 1, 000 simulations conducted until the population had grown from size 50 to size 5000 (or extinction), we never once saw extinction. Rather, the fraction of replicases seems to be roughly normal or possibly log-normal, suggesting that perhaps some fraction of realizations would eventually lead to extinction. This shows that the results from the fixed population size simulations only provide some intuition for how replicases and parasites become segregated, since if parasites were fixed the population would go extinct. Instead, it seems the system can sustain a fraction of both replicases and parasites, similar to what has been seen in other models of cooperation with fluctuating population size. 65 We also tracked the pair correlations between replicase-replicase and replicase-parasite in these simulations. In aggregate (e.g., all possible pairs in all realizations, binned together), the replicase-replicase and replicase-parasite pair correlation functions exactly match that which was shown in Fig. 2 (the maximum difference between any bin and the corresponding bin in Fig. 2 was less than 10 −4 ). However, on a realization-by-realization basis, there seemed to be some correlation between the emerging of structures which segregated replicases from parasites and their eventual fractions.
When cooperators are spatially isolated from parasites, they flourish and grow in numbers, whereas the isolated parasites quickly die out. While the mixing due to the flow ensures this is not a permanent situation, it is one that occurs more often in some realizations than others, leading to a much more robust community of cooperators, which in turn increases the probability that cooperators will survive for longer timescales than we would expect under well-mixed dynamics. Returning to a birth-death process, where the population either goes extinct or grows to infinite size, we tracked the average replicase fraction of the population over 1, 000 simulations from t = 0 to t = 10, with roughly 50 members of each species at t = 0. We never saw either population go extinct when starting from even such a small size; rather, the population tends to sustain both parasites and replicases, with on average slightly more replicases than parasites.
Flows can create rich population structure via division and merging of coherent structures In the previous section, the existence of multiple LCSs facilitated compartmentalization of the population such that the deleterious effect of parasites was avoided. However, in order for LCSs to have been truly useful in early life, they must have further properties -namely, they must be able to permute their biological contents by division and merging.
As discussed in the Introduction, LCSs continuously appear, merge and disappear in a turbulent flow as a result of nolinear dynamics. Thus LCSs seem to replicate the qualitative features of division and merging even in the absence of life.
To investigate this idea, we simulated different metabolisms in many different turbulent flows. Comparing passive particles and reproducing particles allows us to understand what properties of the demographics seen in the simulations are due purely to the flow and what properties are due to the birth-death process, and also allows us to compare the relative importance of the two.
Analyzing passive particle behavior also suggests when the landscape of LCSs is changing without necessitating the costly calculations to actually map the finite-time-Lyapunov exponent (indeed, many algorithms for detecting LCSs rely on analyzing the dynamic-graph properties of passive particles 62, 66, 67 ).
For instance, Fig. 5 shows a relevant demographic measure, the size of the largest particle cluster, as well as pictures of the LCS landscape at times of great demographic flux. Here, it seems obvious that the combination of LCSs during a fast increase in the largest cluster size are related, as are a fast decrease and the splitting of LCSs.
We can use this logic to identify demographic shifts even in reproducing populations that are almost certainly induced by the flow and not by biology. Focusing on demographic shifts above a certain size allows us to bypass the challenges inherent in actually tracking many LCSs over time in many simulations.
The probability that large, monotone demographic shifts arise due to biology can easily be calculated from the negative binomial distribution, assuming the best and worst possible scenario for replicators. Paying attention only to large swings which are highly improbable (p < 0.005) given the biology alone allows us to infer the effects of merging and division of structures without having to constantly calculate the Lyapunov exponent landscape.
Examining these extreme, flow-driven population shifts over many simulations generates the statistics shown in Fig. 5. While passive particles provide a reasonable first approximation, especially for the frequency of division and merging events, the impact of the demographic shifts (as measured by the percentage change in the largest cluster) are much lessened for more complicated metabolisms. This is likely due to the natural behavior of these systems to locally increase concentration gradients. It is clear from previous work 68, 69 that analysis of the gradients in particle concentrations alone is inadequate for understanding systems undergoing a birth-death process, since true birth-death processes with motion (or, "superprocesses" 55, 70-72 ) tend to be much patchier and support larger, less smooth gradients in particle number. Therefore, as an initially spread-out population of cooperators coarsens into dense clusters, its ability to "sense" the boundaries of a coherent structure decreases; as a consequence, it is not always affected if a section of a coherent structure it is not occupying is cleaved off.

Chaotic flows induce small-scale migration events that begin new colonies
In addition to the division and merging of well-defined LCSs explored above, particles which are trapped with one CLS can be captured by another, a process which we call migration to maintain the distinction with division, and to observe that this is an additional mechanism which is not commonly found in theoretical compartment models. Migration events occur due to the imperfect trapping by permanent vortices and the very slight stochasticity arising from a particle's offspring being placed very close to, but not completely atop them; at times, regions of positive Lyapunov exponent can move into an otherwise perfectly-trapping vortex without splitting it, but causing some particles to "bleed out" onto the unstable manifold of the flow. These particles then move very quickly along the unstable manifold until they find a new vortex or temporary coherent structure. Based purely on the birth-death aspect of our process, large changes in the size of the largest particle cluster are improbable; for instance, the probability of monotonic changes greater than 10 or so particles is always less than 0.001%. In (A) this is highlighted by removing birth and death, leaving only passive particles; large changes in cluster demographics can therefore only arise from the flow. Such changes are therefore likely due to dramatic events in the fluid Lyapunov exponent landscape, such as division and merging of LCSs. Restricting our attention to only these large demographic swings over one thousand realizations yields histograms on the number of (B) splits, (C) merges, as well as (D) the percentage decrease in size of the largest cluster after a split and the (E) percentage increase in size of the largest cluster after a merge. Dots show the center of histogram bins, while the value on the y-axis shows the height of the bin; full bars have not been drawn to aid visualization. While all statistics for different metabolisms can be roughly approximated by the behavior of passive particles, the inherent patchiness arising from increasingly baroque metabolisms leads to larger merge and split frequencies with smaller average effects on demographics.
Put another way, migration refers to the changes in population structure that occur due to the chaotic, rather than structured, part of the flow. For instance, one might imagine the rather unnatural scenario in which perfectly coherent vortices are "frozen in" to a fluid, generating a steady flow field which perfectly traps passive particles within the vortices. Even in this limit, however, the lineage of a particle initially placed in one vortex can reach another via migration events. Because its offspring appear at an extremely small (but finite) distance apart, the lineage will execute a random walk, and eventually some individuals The distinction between migration and division is important, because whereas in the previous section, where division of one non-permanent coherent structure into two or more leads to a division of roughly equal fractions of particles, migration events do not involve the creation or annihilation of LCSs and involve only small changes in the demographics of the parent vortex, since typically only a small fraction of particles are cleaved from the parent vortex. However, these small fractions were often seen to successfully "seed" an empty coherent structure or permanent vortex, growing to much larger fractions and thereby promoting their lineage disproportionately, somewhat akin to "gene surfing" seen in other studies. 73 For instance, Fig. S2 shows a realization in which a seven-vortex system initially seeded with a large inoculum of a three-species hypercycle "dyed" by lineage spreads via splitting, merging, and migration. By focusing only on the population very close to the seven point vortices, we ensure that diversity spreading via splitting is not tracked. Migration events eventually lead to all seven vortices being populated, and also contribute to the spread of different lineages, contributing to genetic diversity. Over many realizations, migration processes consistently contributed not only to genetic diversity, but also insulation by creating smaller colonies within larger LCSs, as was suggested in the previous section due to the lessened impact of divisions and merges on more sparse metabolisms. Figure 6 illustrates the average number of clusters (interacting components, equivalent to components of the geometric graph 74 induced on the population by R int ) in different metabolisms over time, as well as the gradual mixing of different lineages, as measured by the mean cluster heterozygosity (H = 1 − 3 l=1 f 2 l , where f l represents the frequency of lineage l in a cluster).
While the inherent patchiness of replicating metabolisms (as discussed in previous sections) leads to a higher number of particle clusters in a rather trivial way, the evolution of heterzygosity in living populations when compared to passive particles is quite surprising. Passive particles exhibited by far the greatest heterozygosity, and in some simulations exhibited a nearly perfect mixing of lineages after a short time. Living populations, on the other hand, had much less diversity on average. However, this can also be understood in terms of the patchiness that arises from the "death-anywhere, birth-locally" property of our process. This introduces number fluctuations at two levels not present in passive particles.
Firstly, the number of lineages is not guaranteed to be preserved, as stochastic effects can even eliminate a whole lineage, drastically decreasing the maximum possible heterzygosity. Additionally, the "gene surfing" effect mentioned above guarantees that newly-formed clusters have zero heterozygosity, and demographic fluctuations even in a large cluster would tend to diminish heterozygosity. The balance between the role of the fluid and of the biology in determining genetic variation is therefore a rich and interesting one.

Discussion
In this work, we have focused on qualitative features of turbulent flows and their similarity to theoretical notions of compartments in biological modeling. In such a generic setting, it is impossible to try to achieve exact quantitative results, as the types of biological metabolisms, cooperative mechanisms, and flow-kinetic parameters could vary widely.
While the parameter space for our model is indeed quite large, we suspect that reasonable variations in many parameters (such as the difference between the death rate and birth rate of particles) will not cause qualitative differences in our results. However, there is one critical parameter, the Damköhler number  Figure 6. Dynamic population structure and small-scale chaotic migration effects increase cluster diversity. Results of many simulations beginning with three dense clusters of particles near three distinct vortices of a seven-vortex flow, of which one realization is shown in Fig. S2. The three clusters are considered to be "dyed" so that their lineages can be tracked over time. (Left-hand panel) The diversity of each cluster having at least 10 particles, as measured by the mean cluster heterozygosity (which here has the value 0 for a cluster of all one lineage and value 2/3 for a cluster with all three lineages represented equally). Different colors mark different metabolisms. The solid line represents the average over many simulations (see Methods) and the opaque background represents one standard deviation. (Right-hand panel) The number of clusters consisting of 10 or more particles, averaged over all simulations, starting from the initial 3 vortices.
in other works combining flow/migration and biological reproduction, 53, 58 an O(1) Damköhler number is optimal for catalytic cooperation. We interpret this result to suggest that early metabolisms may have faced selective pressures based on whether or not their own internal timescales were commensurate with those at which they were separated by the fluid flow.
The timescales of eddy turnover and the lifetimes of LCSs differ according to the energy in the flow, but more importantly according to whether the flow is fully three-dimensional or (quasi)-two-dimensional. This is a distinction that we have largely sidestepped in this work, as the framework we have chosen in which LCSs are created and destroyed is appropriate as a general framework in any dimensionality. However, importance of LCSs as seen here suggest that another selective force may have been dimensionality itself. Since LCSs are more common and longer lived in quasi-two-dimensional flows, such as thin films, bubbles, and pycnoclines, our results suggest that these are environments which might have been much more conducive to the origins of life.
For the small replicating molecules that are found in all theories of the origins of life, motion and Krieger et al.
Turbulent coherent structures and early life below the Kolmogorov scale spatialization does not only provide a mechanism to protect cooperators from parasites. 14 It can, additionally, provide many of the benefits associated with membranes, 26 including algorithmic-like procedures such as merging and splitting collocalized colonies of particles, thereby facilitating the spread and combination of genetic material, and also allowing for the removal of parasites by group-like selection. These results relax the requirements for theories of early life, as it suggests that the cell membrane or some primitive imitation may not have been necessary in the earliest stages in order to perform the critical operations of collocalization and group selection. Furthermore, it suggests that single-celled organisms may have benefited from group selection if and when their life-cycles synchronized with the properties of the turbulent structures around them.

Simulations
Simulations were performed in MATLAB by combining flow kinematics, incorporated via the function ode23, and a stochastic birth-death process implemented via a time-dependent Gillespie simulation. 75 In more detail, the simulations update the population according to the following steps: • The system is initialized with a certain number of each particle species localized in the unit torus at positions picked uniformly-at-random (in Fig. 3, with distance from the vortex core not exceeding 1/50). Units of time are measured in a frame for which the death rate d for particles is d = 1. • Between fluid motions, the distance between particles is assessed via the MATLAB function knnsearch.
Depending on the metabolism used, particles which receive a catalytic boost from a neighbor are assigned one accordingly.
• Stochastic birth-death events operate according to a Gillespie process, adjusted for the fact that the rates (namely, the birth rate, due to time-dependent catalyzation) are time-dependent. Between fluid motions, when the state of catalysis for particles is assessed, a local propensity vector p = In all of these cases, we believe the cause of hypercycle collapse is clear -because particles can only be catalyzed by other particles in their Moore neighborhood, which consists of no more than eight particles at any given time, catalyzation is simply a rare event. This, however, is an artifact of the lattice model. As seen in the main text, real flows allow any number of nearby catalysts, and the continuous versions of the flows used here change the local density of passive particles over time, thereby avoiding this collapse.

Collocalization
For this initial section, exploring only the role of a single vortex in facilitating the important properties of "coming together" and "staying together", we used an unsteady double-gyre, a flow commonly used as a benchmark for vortex recognition and FTLE calculation algorithms: ψ(x, y, t) = A sin(πf (x, t)) sin(πy) (S1) f (x, t) = sin(ωt)x 2 + (1 − 2 sin(ωt))x (S2) where A = 0.5 and ω = 2π were used and ranged from 10 −2 to 10 0 . Although this system consists of two vortices (so that the total circulation in the system is zero), we are mainly interested in the confining force of one vortex as the amount of noise is increased, and so we simply measured the residence time as the time in which a passive particles stayed in one half of the system if initialized in the center of one vortex.
For all metabolisms, 10, 000 simulations were performed for differing values of and R int , the radius of possible cooperative interaction. For each metabolism, we initialized an "inoculum" of 5 particles per species type at positions chosen uniformly-at-random near (within a radial distance of 1/20) the vortex core (r = 0). If the population increased by a factor of 1, 000, the simulation was considered a success; if the population died out completely, the simulation was considered a failure. The sum total of successes, divided by 10, 000, yields the numbers plotted in Fig. 3.

Point-vortex flow
Because our simulations already involve some operations which can be quite costly as the population grows (for instance, knnsearch takes O(n log n) time for a population of size n), we wanted to employ a model of turbulence which could generate an infinite number of distinct (yet statistically identical) flows in an efficient manner. We did not concern ourselves with more complicated physics (such as influx of energy by forcing or outflux by viscous dissipation), instead opting for a perfectly conservative fluid with a perfectly conserved number of point vortices, whose motion is given by Eqns. (??)where Γ j is the circulation of the j-th entity, having value of either +1 or −1 for a point vortex and 0 for a particle, and d ij is the L 2 distance between entity i and j. The ramifications of these equations is that both particles Typically, one would have to employ a technique such as Ewald summation to deal with the doubly-infinite sum over m. However, the reader can easily check that the size of the terms drops off incredibly fast, with the size of the |m| = 2 terms already being O(10 −20 ) or smaller. We therefore truncated the sum at |m| = 2.
The number of vortices allows for rich dynamics, as a generic realization produces elliptic structures at (at least) three different scales: near a vortex, between two vortices (of any sign combination), and within short-term bound four-vortices. As seen in the main text, we often observed long-term elliptic structures that fell into none of these three categories as well.

Identifying coherent structures
The identification of LCSs in Fig. 4, Fig. 5 We note that the code used was not designed to be employed on a doubly-periodic surface, and therefore we believe that the values of the FTLE reported at the gridpoints closest to the domain boundary are incorrect. However, the interior LCSs indicated in Figs. 4 and 5 were also identified by a different algorithm, 62 also from the Dabiri group, which "colors" structures based on the kinematic similarity of passive particles trajectories (generated for the same flow whose field was originally used to create Figs. 4 and 5).

Escaping parasites
To test whether LCSs would allow spatially-dependent metabolisms to escape from parasites in a manner similar to proto-membranes, we simulated the Replicase R2 metabolism, which represents a cooperator and a parasite. To check if their relative fitness was affected by the flow, we examined a Wright-Fisher type process instead of our usual branching process. The Wright-Fisher process employed has a fixed population size of N = 200, with the entire population updating at fixed discrete time intervals T upd , at which time a new generation is formed by picking, for each of the 200 members, a parent with probability proportional to the parent's relative fitness in their population. The initial condition is distributed uniformly-at-random.
The Wright-Fisher simulations were continued until either all parasites were extinct, or the whole population (including parasites, which cannot survive once all healthy particles have been extinguished) were extinct. The former case was considered a success, the latter case a failure. The number of instances of the latter, divided by 10, 000, gives the points plotted in Fig. 4.
We then reverted to our branching process, starting from a population of 50 per species localized near the core of one out of seven vortices, of which a fraction f = 1/2 were chosen uniformly-at-random to be parasites. Once the population grew to a size of 5000, we recorded the average fraction of cooperating replicases. Performing this over 1, 000 simulations gives the histogram in Fig. 4.

Division and merging
Calculating the FTLE field is quite computationally costly, and counting elliptic LCSs without fear of error would require an extremely high spatial resolution, and therefore making full evaluation of the flow properties in a simulation is a much more involved process than a full evaluation of the biological properties. Furthermore, it is easy to predict when demographic changes aren't due to the biological process from this evaluation. For instance, we can assign a minimum and maximum probability to the number of birth-death events occurring within a certain timespan given the approximate population size, and can also predict the extremal probabilities of the outcome in which all (or most) such birth-death events were either birth or death. The latter is particularly easy; the likelihood of M death events in a row before a birth must be bounded above by NB (

Migration and diversity
For Figure S2, seven vortices were used and were assigned an integer label which was maintained throughout the simulations. The curves shown were generated by simulating 1, 000 runs per metabolism, initializing 100 of each particle species in the vortices labeled 1-3 (therefore implicitly assuming that the metabolism had experienced some reproductive success in one vortex). Simply tracking the dynamics of particles from the three original lineages over time generated the left-hand side of 6. The right-hand side was given by counting the number of components (with 10 or more particles included) in a geometric graph 74 induced by drawing an edge between two particles iff they were at a distance (in the Euclidean metric) less than R int .  t Figure S2. Migration leads to genetic diversity. In addition to the concept of LCSs splitting and merging, we also observed "migration" events in which a handful particles were cleaved from a parent vortex and traveled quickly along a high-FTLE ridge to a new vortex or transient coherent structure. This mechanism proved especially effective at combining genetic material from different vortices as they passed by one another, for instance in this realization of a seven-vortex system supporting a 3-species hypercycle. We initialized three "flourescently dyed" lineages in three parent vortices at t = 0 and observed the spread of these lineages (shown here, the fraction of the disk in a particular color represents the total fraction of all particles from a particular lineage) over time. Drawn here is one realization demonstrating founder-like effects, in which the first migrating particles to reach a new vortex tend to occupy a large fraction of the structure for long times.  Figure S3. Very low Damköhler flows make cooperation challenging. While decreasing the Damköhler number greatly increases the cost of simulations on replicating populations, we can get an idea of the challenges to cooperation in this limit by considering the statistics for a population of 500 passive particles in a thousand simulations. (Top-left) The average time that a passive particle spends at a distance greater than R int = 0.03 from any other particle divided by the expected particle lifetime (t = 1). (Top-right) The likelihood of retaining a particular neighbor over one timestep in our simulations (∆t = 0.01). (Bottom-left) The mean number of particles in a "cluster" -in a given snapshot of the flow, most particles are isolated from one another. (Bottom-right) The standard deviation in cluster size.