Toward the Darwinian transition: Switching between distributed and speciated states in a simple model of early life

It has been hypothesized that in the era just before the last universal common ancestor emerged, life on earth was fundamentally collective. Ancient life forms shared their genetic material freely through massive horizontal gene transfer (HGT). At a certain point, however, life made a transition to the modern era of individuality and vertical descent. Here we present a minimal model for stochastic processes potentially contributing to this hypothesized “Darwinian transition.” The model suggests that HGT-dominated dynamics may have been intermittently interrupted by selection-driven processes during which genotypes became fitter and decreased their inclination toward HGT. Stochastic switching in the population dynamics with three-point (hypernetwork) interactions may have destabilized the HGT-dominated collective state and essentially contributed to the emergence of vertical descent and the first well-defined species in early evolution. A systematic nonlinear analysis of the stochastic model dynamics covering key features of evolutionary processes (such as selection, mutation, drift and HGT) supports this view. Our findings thus suggest a viable direction out of early collective evolution, potentially enabling the start of individuality and vertical Darwinian evolution.

Figure 1

HGT hyperlinks introduce three-genotype interactions to the evolutionary dynamics, creating hypernetwork dynamics. This schematic example illustrates the insertion of an HGT hyperlink (red solid and dashed) to the sequence space of genomes of length $l=4$. Individuals of genotype $A=0100$ take up the first two bases $\underline{11}$ from genotype $B=\underline{11}01$ which are inserted at position three into $A=01\underline{11}00$. After the last two bits are cut from the sequence, a progenote of genotype $A$ becomes of new genotype $C=0111$.

Figure 2

The population dynamics is dominated by a speciated state for low competence and a distributed state for high competence. Shown are example dynamics of the population entropy (2) for competence $c=5$ (a), $c=3$ (b), and $c=1$ (c) in an example population of $N=1000$ progenotes with genome length $l=7$. (a) For high competence $c$ the population entropy almost always fluctuates around a high value for all initial conditions. (b) The dynamics switch stochastically to a low-entropy state and stay there longer for lower values of $c$. (c) The low-entropy state is rendered globally stable for low $c$ so that the population entropy fluctuates slightly above zero for all initial conditions. In the low-entropy state the population dynamics are driven by selection, in the high-entropy state by HGT. Panels (i) show the entropy dynamics, (ii) the average fitness $〈f〉$ of the population corresponding to the entropy dynamics, and (iii) the corresponding HGT rates $r_{\text{HGT}}$ that the population exhibits at time $t$. For low population entropies the fitness is high and HGT rate small, and vice versa for high population entropies. The mutation probability was set to ${\mu }_{ij}=0.0001$. Into the resulting Fujiyama fitness landscape [23] with fitness values between $f_{\text{min}}=0.9$ and $f_{\text{max}}=1.1$ we inserted $m=2000$ HGT links.

Figure 3

The high-entropy state is dynamically stable also for vanishing mutation probabilities. Shown is the measured percentage of time a population stayed in the distributed state for a system with mutation probabilities ${\mu }_{ij}={10}^{-3}$ (blue), ${\mu }_{ij}={10}^{-4},{\mu }_{ij}={10}^{-5}$ (orange), ${\mu }_{ij}={10}^{-6}$ (green), and ${\mu }_{ij}=0$ (gray). Qualitatively, the results are similar, only for higher mutation probabilities the critical transition occurs at a lower value $c_{\text{cr}}$. System parameters were $l=7,N=1000$, and $m=3000$. HGT links were introduced into a Fujiyama fitness landscape with fitness values between $f_{\text{min}}=0.9$ and $f_{\text{max}}=1.1$. Each data point was obtained in a simulation of length $T={10}^6$ with the initial condition $S\left(0\right)=S_{max}$.

Figure 4

The distributed state disappears for low HGT competence. At a critical competence the dynamical fixed point at high population entropy is destroyed in a saddle-node bifurcation. Here we show the analysis of the system yielding the dynamics illustrated in Fig. 2. Panels (a) and (b) show the rate of change of the population entropy due to reproduction and HGT obtained with the methods described in the Supplemental Material [58]. The colors indicate the rate of change for competence values $c=1$ (blue), $c=3$ (red), and $c=5$ (orange). Adding the results from (a) and (b) according to Eq. (5) yields the overall rate of change $\dot{S}$ for the dynamics shown in (c). The arrow indicates the fixed point at high population entropies emerging through a saddle-node bifurcation for increasing the parameter $c$. With Eq. (6) we define a potential $V\left(S\right)$ for the dynamics which is shown in (d) for the competences $c=1$ (blue), $c=3$ (red), and $c=5$ (orange) and additionally for $c=0.5$ (gray), $c=2$ (green), and $c=4$ (black). The potential valley at high population entropies emerges between $c=1$ and $c=2$ so that the critical competence must lie between these two values. Each data set was obtained in simulations measuring the dynamics for a time $T={10}^7$.

Figure 5

A possible scenario for the evolution of distinct species from a pre-Darwinian distributed state. The three time series sketched here are not simulation data but encapsulate the speculations in the text, showing how the average competence, the average fitness, and the population entropy may evolve in the transition from a distributed state to the first distinct species. In the initial state (marked in blue) the competence is high, so that HGT drives the dynamics; the population exhibits a high population entropy and low average fitness. Through a stochastic switching the dynamics reaches a state of low population entropy where the fitness is higher as the population adapts to the fitness landscape. Here the population could evolve slowly toward lower competence. Thus the dynamics switch back and forth between the low- and the high-entropy state, remaining longer and longer in the low-entropy state as the competence decreases (marked in red). When the competence goes below a critical value (marked by the dashed line in the top panel) the high-entropy state disappears (marked in green), the dynamics remains in the low-entropy state, the population's average fitness increases, and the first species may robustly evolve.