Turbulent coherent structures and early life below the Kolmogorov scale

Abstract

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.

Fig. 1. Compartmentalization appears naturally in fluid flows.

Segregation of particles into different regions can occur in two-dimensional flows for a variety of reasons. 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 (white arrows: velocity vectors, blue shade: local vorticity). 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.

Fig. 2. A summary of the model.

a We consider the situation of cooperative catalyzation in a turbulent fluid. For computational tractability we restrict our attention to two dimensions, since the trapping regions (Lagrangian coherent structures, LCS) which are of qualitative interest arise here from simple models. We consider the full domain of interest to be of size L, with L ≫ η, where η is the Kolmogorov scale, the scale of the smallest advective spatial structures in turbulence. b Our modeling of the flow. Many qualitative as well as quantitative features of turbulence are well-captured by a toy model consisting of multiple point vortices (here, 7 vortices: blue = clockwise, red = counterclockwise). To avoid edge effects, the domain is doubly periodic. In this snapshot, there is a quiescent region circled in green, whose size represents the Kolmogorov scale. If this spatial region remains quiescent for long times, effectively trapping particles (exhibiting a negative finite-time-Lyapunov-exponent), it will represent an LCS. c Within an LCS, particles separate much more slowly from one another than outside (black arrows). Over the lifetime of this LCS, the particles will, in addition to advection, diffuse a distance given by the Batchelor scale λ_B, and so we could consider them “blurred” over this distance. For a nucleotide in water, λ_B ≈ 0.02 × η. Biological particles (colored X,Y,Z) can catalyze each other if they are within a distance R_int, and therefore a single snapshot induces a dynamic graph structure of cooperators and defectors. d ×5 magnification of η as in b as well as the other relevant lengthscales drawn to scale; here we show the value R_int = 0.03*L typically used in our simulations (see “Methods“ section), and λ_B = 0.02η (thickened to a box to aid the eye).

Fig. 3. 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(|x₁ − x₂|), which gives the probability of finding a pair with interparticle separation |x₁ − x₂|. Starting from an initial condition of many particles in close proximity (so G(|x₁ − x₂|) 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 tracers in a turbulent flow (gray histograms) obey a known power 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 $Da=\mathcal{O}\left(1\right)$ (blue histogram), however, can combine the advantages of both situations, creating rich structure (G(x₁, x₂) having broad support) while also having a higher number of interacting pairs than in the case of no biology.

Fig. 4. Coherent structures collocalize replicating particles.

a A sample velocity field consisting of a circular flow perturbed by an unsteady oscillation of amplitude ε = 0.1 atop a plot of the finite-time Lyapunov exponent, a common measure of the local “chaoticity” in a flow. As ε increases, particles (colored dots, schematizing different species) become more likely to cross onto the unstable manifold and separate from nearby neighbors. We define “establishment” as a hundredfold increase in the population size starting from a very small initial condition, where continuum limits previously described^,, begin to be applicable. b The average particle separation (per flow period) of an initially adjacent pair of particles, versus the strength ε 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 five particles of each species injected into a chaotic field (no coherent structure, ε = 1), whereas circular markers indicate the same likelihood for an identical inoculum injected into a nearly pure coherent structure (ε = 0.01). Note that for Replicase R1 (blue), the establishment probability is equal for both ε = 1 and ε = 0.01, so that only the solid line is shown.

Fig. 5. Trapping and segregation by coherent structures gives cooperators a boost.

In a two-species Replicase R2, one species (the replicase) plays the role of a cooperator (at no cost, provides a catalytic benefit of size β to all non-self organisms within a radius R_int), while the other species plays the role of a parasite. a Chaotic flows generate LCSs with finite lifetimes, which act as traps for particles. Here a six-vortex (teal dots) example is shown with the finite-time Lyapunov exponent (FTLE) at left and trapping regions (negative FTLE) at right. b In a well-mixed population obeying a Wright–Fisher update rule on constant population size, each species would be equally likely to fix (reflected by P_coop, the fixation probability of cooperators shown on the y-axis, equaling 1/2) if they began at equal fractions. To test whether flow-segregation helped replicases, we performed Wright–Fisher simulations on a population of size N = 200 at different R_int and Da. Intermediate values of Da show a boost for cooperation at specific values of R_int. c 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 1000 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.

Fig. 6. Merging and splitting of LCSs leads to rich, dynamic population 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 >10 or so particles is always <0.001%. In a this is highlighted by removing birth and death, leaving only passive tracers; 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 tracers, the inherent patchiness arising from increasingly baroque metabolisms leads to larger merge and split frequencies with smaller average effects on demographics.

Fig. 7. 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 Supplementary Fig. 1. 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” section) 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 three vortices.