Simple model for the Darwinian transition in early evolution

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 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 led to the emergence of vertical descent and the first well-defined species in early evolution. A nonlinear analysis of a 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 route from early collective evolution to the start of individuality and vertical Darwinian evolution, enabling the emergence of the first species.Comment: 9 pages, 5 figures, under review at Physical Review

Frequency-dependent fitness induces multistability in coevolutionary dynamics

Evolution is simultaneously driven by a number of processes such as mutation, competition and random sampling. Understanding which of these processes is dominating the collective evolutionary dynamics in dependence on system properties is a fundamental aim of theoretical research. Recent works quantitatively studied coevolutionary dynamics of competing species with a focus on linearly frequency-dependent interactions, derived from a game-theoretic viewpoint. However, several aspects of evolutionary dynamics, e.g. limited resources, may induce effectively nonlinear frequency dependencies. Here we study the impact of nonlinear frequency dependence on evolutionary dynamics in a model class that covers linear frequency dependence as a special case. We focus on the simplest non-trivial setting of two genotypes and analyze the co-action of nonlinear frequency dependence with asymmetric mutation rates. We find that their co-action may induce novel metastable states as well as stochastic switching dynamics between them. Our results reveal how the different mechanisms of mutation, selection and genetic drift contribute to the dynamics and the emergence of metastable states, suggesting that multistability is a generic feature in systems with frequency-dependent fitness.Comment: 12 pages, 6 figures; J. R. Soc. Interface (2012

Non-monotonic response to regular inputs as observed in MNTB-neurons from whole-cell patch-clamp recordings.

<p>(a,b) Experimentally obtained response curves of two different MNTB-neurons for different pulse currents [(a) green: 375pA, orange, red: 425pA, blue: 450pA (b) orange: 575pA, red: 600pA, blue: 650pA]. The dashed lines indicate the major <i>n:1</i>-locking states predicted —no free fit parameter. Error bars indicate the estimated error made by calculating the mean output frequency from a finite number of output spikes. (c–f) Membrane potential dynamics for different locking types: (c) <i>1:1</i>-locking, (d) <i>2:1</i>-locking, (e) <i>3:1</i>-locking, (f) unlocked, irregular dynamics. The letters in (b) indicate the data points where these dynamics were observed.</p

Non-monotonic response in a Hodgkin-Huxley neuron receiving periodic input via a static excitatory synapse.

<p>(a) In the response curve <i>n:1</i>-locking regions are interrupted by broad transition regions (b), magnified from (a). In the transition regions nonperiodic, irregular dynamics arise. (c) and (d) show example dynamics of the membrane potential (c) in the <i>3:1</i>-locking region (<i>λ</i>_in = 170Hz) and (d) in the irregular regime (<i>λ</i>_in = 140.2Hz). Simulation parameters were <i>C</i> = 2, <i>V</i>_Na = 50, <i>V</i>_K = −77, <i>V</i>_L = −54.4, <i>g</i>_Na = 120, <i>g</i>_K = 36, <i>g</i>_L = 0.3, <i>I</i>₀ = 5 and <i>ε</i> = 9. We used an alpha-function <math><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi><mi>t</mi><msub><mi>τ</mi>ex</msub><mi>exp</mi><mrow><mo stretchy="true">(</mo><mo>−</mo><mi>t</mi><mo>/</mo><msub><mi>τ</mi>ex</msub><mo stretchy="true">)</mo></mrow></mrow></math> with time constant <i>τ</i>_ex = 1 to model the synaptic inputs (<i>e</i> is the Euler constant to normalize <i>K</i>(<i>t</i>)).</p

Non-monotonicity of response curves is robust against irregularity of the input.

<p>Panels (a)–(c) show the input-output response of the LIF model system receiving input spike sequences with Gamma-distributed inter-spike intervals. The system parameters in (a) and (b) are identical to the system parameters in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.g002" target="_blank">Fig. 2(a),(b)</a>. (c) shows the response of the system for parameters <i>τ</i> = 4, <i>μ</i> = 1, <i>u</i> = 0.2, <i>c</i> = 0.5 and <i>V</i>_eq = 0, where the inset demonstrates that for small input rates no output is generated. Inset of (b) shows the distribution for one fixed input rate, normalized to one. The shape parameter of the distribution was set to <i>α</i> = 100, so that the relative standard deviation of the input intervals is <math><mrow><msub><mi>σ</mi><mrow><mi>Δ</mi><mi>t</mi></mrow></msub><mo>/</mo><mrow><mi>Δ</mi><mi>t</mi></mrow><mo stretchy="true">¯</mo><mo>=</mo><mn>0.1</mn></mrow></math>.</p

Non-monotonic response functions of the idealized LIF synapse-neuron system.

<p>Input-output response (a) for resource recovery that is much slower than the time scale of membrane potential leakage (<i>τ</i> = 1, <i>μ</i> = 10), for system with dynamics shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.g001" target="_blank">Fig. 1</a>, (b) for both processes occurring on the same time scales (<i>τ</i> = 1, <i>μ</i> = 1). In (a) only downward jumps, in (b) upward as well as downward jumps occur. Further parameters were <i>u</i> = 0.2, <i>c</i> = 0.5, <i>V</i>_eq = 0.8 for (a) and <i>u</i> = 0.4, <i>c</i> = 0.8, <i>V</i>_eq = 0 for (b).</p

Non-monotonic response to regular input spike sequences: increasing the input spike frequency may increase but also decrease the output spike frequency.

<p>Bottom panel: input spike frequency that slowly increases ten-fold. Top three panels: output spike responses (LIF: leaky integrate-and-fire neuron with depressive synapse, FHN: Fitzhugh-Nagumo and HH: Hodgkin-Huxley neuron, both with static synapses). Time is rescaled so that all three data sets fit in this Figure. For details of models see equations (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.e001" target="_blank">1</a>)–(<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.e002" target="_blank">2</a>) for LIF, equations (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.e012" target="_blank">12</a>)–(<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.e013" target="_blank">13</a>) for FHN and equations (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.e014" target="_blank">14</a>)–(<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004002#pcbi.1004002.e017" target="_blank">17</a>) for HH.</p

Non-monotonic response in a Fitzhugh-Nagumo neuron receiving periodic input via a static synapse.

<p>(a) Input-output response exhibits dominant <i>n:1</i>-locking interrupted by broad transition regions (b), magnified from (a). Several locking ratios <i>n:m</i> are indicated. In the transition regions, periodic, <i>n:m</i>-locked as well as nonperiodic, irregular dynamics arise. (c,d) Membrane potential dynamics (c) in the <i>4:1</i>-locking region and (d) in the irregular regime. The model parameters were <i>a</i> = 0.139, <i>b</i> = 2.54, <i>c</i> = 0.5, <i>μ</i>′ = 125 and <i>K</i>(<i>t</i>) = 2(exp(−t)−exp(−2<i>t</i>)).</p