Innovation dynamics of the bimodal recombination model.

<p>In (a) we show the different dynamics of <i>N</i>(<i>t</i>) for different aging decays affecting the hyperbolic (<i>k</i> = 2) term. The corresponding <i>γ</i>_<i>H</i> values are indicated in the legend. The rest of the parameter values are: <i>μ</i>_<i>M</i> = 8.5 × 10⁻², <i>γ</i>_<i>M</i> = 1.5 × 10⁻⁴ and <i>μ</i>_<i>H</i> = 5 × 10⁻⁷. In (b) we show again the case <i>γ</i>_<i>H</i> = 2 × 10⁻⁵ (triangles), and the accumulated contribution of the Malthusian (<i>k</i> = 1) recombination to the total patent number <i>N</i>. At any time, the accumulated number of patents that have been generated by hyperbolic (<i>k</i> = 2) recombination corresponds to the difference between the two curves. The red line corresponds to an exponential fit to the first part (up to <i>t</i> = 50) of the time series.</p

Two phases predicted by the generalised model of technological evolution with power law ageing.

<p>The white area includes all parameter combinations allowing a singularity to emerge through a hyperbolic growth process.</p

Evolution of technological diversity.

<p>The main plot (a) shows the accelerated increase of the total number patents <i>N</i>(<i>t</i>) as provided by the USPTO dataset. In (b) we show the same data in linear-log form. As an alternative illustration, we also display the spindle diagrams for <i>N</i>(<i>t</i>) associated to (c) the overall pattern and (d) the early development linked to the Industrial Revolution [corresponding to the period indicated by the open square in (a)].</p

Improved environments can slow down the front of synthetic mutualists.

<p><i>a)</i> Invasion speed of the mutualistic strains according to a minimal reaction-diffusion model. The gray area indicates the domain where the mutualistic interaction favours hyperbolic growth over Malthusian competition. The maximum Malthusian growth rates <i>μ</i>_<i>CI</i> = 9.13 × 10⁻² and <i>μ</i>_<i>CL</i> = 2.18 × 10⁻¹ hr⁻¹ (for <i>I ^-</i> and <i>L^-</i>, respectively) correspond to monoculture growth rates observed in the competition scenario (see <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005689#pcbi.1005689.s001" target="_blank">S1 Fig</a>). <i>b)</i> Observed front speeds exhibit a slowing-down in facultative mutualism scenarios that is not captured by the RD model (average and standard deviation values over 5 replicates are shown). <i>c)</i> According to agent based simulations, the slow down in facultative mutualism scenarios is correlated with a decay in the fraction of active cells. <i>d)</i> Snapshots of simulated fronts (darker colours depict stagnant cells). The red arrow indicates a patch of <i>I ^-</i> cells formed by local consumption of environmental amino acids. Once amino acids are locally depleted, a high number of cells in the patch become stagnant.</p

Cell dynamics in the agent-based model is governed by binary decisions (Heaviside behaviour) that depend on extracellular concentration thresholds of nutrients, amino acids and antibiotics.

<p>The scheme shows the logical steps that determine cell behavior according to our model.</p

Environmental conditions determine the fate of parasites during range expansions.

<p><i>a)</i> Obligate mutualism scenario (absence of supplemented amino acid) leads the strain <i>P</i> to act as a parasite in well-mixed conditions, while competition is observed at 10⁻⁴M supplementation of <i>iso</i> and <i>leu</i>. Average and standard deviation values over 9 replicates are shown. <i>b)</i> Spatial structure leads the mutualists to conquer the edge of the population front, defeating the parasite <i>P</i>. Yellow arrows indicate regions where the parasite has been excluded from the population front (red arrow indicates one of the few regions in which the parasite still surfs at the edge of the front). Note that the front curvature is enhanced at regions governed by the mutualists, a hallmark of an enhancement of the front speed at these regions. The grey rectangle indicates the magnified area on the right. <i>c)</i> Frequency of the <i>P</i> strain at the edge of the front for two different scenarios (0 and 100<i>μ</i>M extracellular ampicillin). <i>d)</i> The <i>P</i> strain offers cross-protection to the mutualists when threatened by antibiotics, leading to the survival of the <i>P</i> strain at the edge of the front. <i>e)</i> Scheme of the complex mutualistic interaction (which involves cross-feeding and cross-protection) between the three species in the presence of antibiotics. Each species lacks a different ability needed to survive in the system, but the ensemble may be able to survive if able to develop the corresponding division of labour. <i>f)</i> Three-species spatial structure in a simulated heterogeneous environment with non-isotropic antibiotic concentration at <i>t</i> = 0. While the <i>P</i> strain is conserved in the areas where cross-protection is essential for the mutualistic ensemble, <i>P</i> cells are excluded from the front in areas where the antibiotic concentration does not reach the growth inhibition threshold.</p

Natural and synthetic cooperative loops and their parasites.

<p>Cooperative feedback loops are widespread in ecological systems, and three examples are shown in (a-c). Here we indicate in (a) the mutual support between vegetation (grasses) and earthworms and in (b) a more complex cycle composed by vegetation, cattle and earth worms (and other invertebrates). In (c) the image shows a small area within a semiarid ecosystem including a plant surrounded by biological soil crust. Formal models of these types of interactions are described closed feedback interactions. In (d) we display the basic logical scheme of interactions for a two-component cooperative loop (a two-member hypercycle in the molecular replicators literature). In (e) we show an extended model where a parasitic species (colour circle) takes advantage of one of the species but gives no mutual feedback. In models of molecular replicators, it has been shown that parasites can easily damage cooperation, but this effect is reduced or suppressed under the presence of oscillations and spatial diffusion when spiral waves get formed (f). Here different colours indicate different molecular species in a <i>n</i> = 8 member hypercycle. In this paper we examine the role played by space and parasites in synthetic ecosystems.</p

Resource availability alters interactions between synthetic mutualists and modulate genetic diversity during range expansions.

<p><i>a)</i> We use a pair of engineered bacterial strains (yellow depicts <i>I ^-</i> cells and blue stands for <i>L^-</i>) that engage in mutualistic interactions by cross-feeding amino acids. <i>b)</i> Both strains are able to grow in liquid cocultures lacking both amino acids, but monocultures exhibit no growth in this conditions (Obl. Mutualism). When amino acids are supplemented at 10⁻⁴M (Competition), monocultures grow to comparable levels while the <i>L^-</i> strain overcomes its partner in cocultures. Error bars show the standard deviation across 9 replicates. <i>c)</i> Bacterial mutualists develop single-strain patches during range expansions, whose spatial structure is influenced by environmental conditions (concentration of supplemented amino acids), see also <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005689#pcbi.1005689.s003" target="_blank">S3 Fig</a>. <i>d)</i> Width of single-strain sectors as the range expansion takes place. Obligate mutualism and facultative mutualism scenarios correspond to environments supplemented with 0 and 10⁻⁵ <i>μ</i>M of <i>iso</i> and <i>leu</i>, respectively, both leaving an approximately constant patch width. In contrast, the competition scenario (10⁻⁴ <i>μ</i>M of <i>iso</i> and <i>leu</i>) leads to an increasing patch width as the range expansion progresses. Curves show the patch width for single colonies, see replicates in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005689#pcbi.1005689.s003" target="_blank">S3 Fig</a>.</p