Continual evolution through coupled fast and slow feedbacks.

Continual evolution describes the unceasing evolution of at least one trait involving at least one organism. The Red Queen Hypothesis is a specific case in which continual evolution results from coevolution of at least two species. While microevolutionary studies have described examples in which evolution does not cease, understanding which general conditions lead to continual evolution or to stasis remains a major challenge. In many cases, it is unclear which experimental features or model assumptions are necessary for the observed continual evolution to emerge, and whether the described behavior is robust to variations in the given setup. Here, we aim to find the minimal set of conditions under which continual evolution occurs. To this end, we present a theoretical framework that does not assume any specific functional form and, therefore, can be applied to a wide variety of systems. Our framework is also general enough to make predictions about both monomorphic and polymorphic populations. We show that the combination of a fast positive and a slow negative feedback between environment, population, and evolving traits causes continual evolution to emerge even from the evolution of a single evolving trait, provided that the ecological timescale is sufficiently faster than the timescales of mutation and the negative feedback. Our approach and results thus contribute to a deeper understanding of the evolutionary dynamics resulting from biotic interactions.

Fatou Components of Attracting Skew-Products.

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004, the non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov (Fatou theory in two dimensions, PhD thesis, University of Michigan, 2004). In 2014, it was shown in Astorg et al. (Ann Math, arXiv:1411.1188 [math.DS], 2014) that wandering domains can exist near a parabolic invariant fiber. In Peters and Vivas (Math Z, arXiv:1408.0498, 2014), the geometrically attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew-products; this class contains the maps studied in Peters and Vivas (Math Z, arXiv:1408.0498, 2014). Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.

Metabolic states with maximal specific rate carry flux through an elementary flux mode.

Specific product formation rates and cellular growth rates are important maximization targets in biotechnology and microbial evolution. Maximization of a specific rate (i.e. a rate expressed per unit biomass amount) requires the expression of particular metabolic pathways at optimal enzyme concentrations. In contrast to the prediction of maximal product yields, any prediction of optimal specific rates at the genome scale is currently computationally intractable, even if the kinetic properties of all enzymes are available. In the present study, we characterize maximal-specific-rate states of metabolic networks of arbitrary size and complexity, including genome-scale kinetic models. We report that optimal states are elementary flux modes, which are minimal metabolic networks operating at a thermodynamically-feasible steady state with one independent flux. Remarkably, elementary flux modes rely only on reaction stoichiometry, yet they function as the optimal states of mathematical models incorporating enzyme kinetics. Our results pave the way for the optimization of genome-scale kinetic models because they offer huge simplifications to overcome the concomitant computational problems.