Evolution Equation of Phenotype Distribution: General Formulation and Application to Error Catastrophe

An equation describing the evolution of phenotypic distribution is derived using methods developed in statistical physics. The equation is solved by using the singular perturbation method, and assuming that the number of bases in the genetic sequence is large. Applying the equation to the mutation-selection model by Eigen provides the critical mutation rate for the error catastrophe. Phenotypic fluctuation of clones (individuals sharing the same gene) is introduced into this evolution equation. With this formalism, it is found that the critical mutation rate is sometimes increased by the phenotypic fluctuations, i.e., noise can enhance robustness of a fitted state to mutation. Our formalism is systematic and general, while approximations to derive more tractable evolution equations are also discussed.Comment: 22 pages, 2 figure

Does the Red Queen reign in the kingdom of digital organisms?

In competition experiments between two RNA viruses of equal or almost equal fitness, often both strains gain in fitness before one eventually excludes the other. This observation has been linked to the Red Queen effect, which describes a situation in which organisms have to constantly adapt just to keep their status quo. I carried out experiments with digital organisms (self-replicating computer programs) in order to clarify how the competing strains' location in fitness space influences the Red-Queen effect. I found that gains in fitness during competition were prevalent for organisms that were taken from the base of a fitness peak, but absent or rare for organisms that were taken from the top of a peak or from a considerable distance away from the nearest peak. In the latter two cases, either neutral drift and loss of the fittest mutants or the waiting time to the first beneficial mutation were more important factors. Moreover, I found that the Red-Queen dynamic in general led to faster exclusion than the other two mechanisms.Comment: 10 pages, 5 eps figure

Mutator Dynamics on a Smooth Evolutionary Landscape

We investigate a model of evolutionary dynamics on a smooth landscape which features a ``mutator'' allele whose effect is to increase the mutation rate. We show that the expected proportion of mutators far from equilibrium, when the fitness is steadily increasing in time, is governed solely by the transition rates into and out of the mutator state. This results is a much faster rate of fitness increase than would be the case without the mutator allele. Near the fitness equilibrium, however, the mutators are severely suppressed, due to the detrimental effects of a large mutation rate near the fitness maximum. We discuss the results of a recent experiment on natural selection of E. coli in the light of our model.Comment: 4 pages, 3 figure

Cryptic Population Dynamics: Rapid Evolution Masks Trophic Interactions

Trophic relationships, such as those between predator and prey or between pathogen and host, are key interactions linking species in ecological food webs. The structure of these links and their strengths have major consequences for the dynamics and stability of food webs. The existence and strength of particular trophic links has often been assessed using observational data on changes in species abundance through time. Here we show that very strong links can be completely missed by these kinds of analyses when changes in population abundance are accompanied by contemporaneous rapid evolution in the prey or host species. Experimental observations, in rotifer-alga and phage-bacteria chemostats, show that the predator or pathogen can exhibit large-amplitude cycles while the abundance of the prey or host remains essentially constant. We know that the species are tightly linked in these experimental microcosms, but without this knowledge, we would infer from observed patterns in abundance that the species are weakly or not at all linked. Mathematical modeling shows that this kind of cryptic dynamics occurs when there is rapid prey or host evolution for traits conferring defense against attack, and the cost of defense (in terms of tradeoffs with other fitness components) is low. Several predictions of the theory that we developed to explain the rotifer-alga experiments are confirmed in the phage-bacteria experiments, where bacterial evolution could be tracked. Modeling suggests that rapid evolution may also confound experimental approaches to measuring interaction strength, but it identifies certain experimental designs as being more robust against potential confounding by rapid evolution

Cryptic Population Dynamics: Rapid Evolution Masks Trophic Interactions

Trophic relationships, such as those between predator and prey or between pathogen and host, are key interactions linking species in ecological food webs. The structure of these links and their strengths have major consequences for the dynamics and stability of food webs. The existence and strength of particular trophic links has often been assessed using observational data on changes in species abundance through time. Here we show that very strong links can be completely missed by these kinds of analyses when changes in population abundance are accompanied by contemporaneous rapid evolution in the prey or host species. Experimental observations, in rotifer-alga and phage-bacteria chemostats, show that the predator or pathogen can exhibit large-amplitude cycles while the abundance of the prey or host remains essentially constant. We know that the species are tightly linked in these experimental microcosms, but without this knowledge, we would infer from observed patterns in abundance that the species are weakly or not at all linked. Mathematical modeling shows that this kind of cryptic dynamics occurs when there is rapid prey or host evolution for traits conferring defense against attack, and the cost of defense (in terms of tradeoffs with other fitness components) is low. Several predictions of the theory that we developed to explain the rotifer-alga experiments are confirmed in the phage-bacteria experiments, where bacterial evolution could be tracked. Modeling suggests that rapid evolution may also confound experimental approaches to measuring interaction strength, but it identifies certain experimental designs as being more robust against potential confounding by rapid evolution

Evolutionary trajectories in rugged fitness landscapes

We consider the evolutionary trajectories traced out by an infinite population undergoing mutation-selection dynamics in static, uncorrelated random fitness landscapes. Starting from the population that consists of a single genotype, the most populated genotype \textit{jumps} from a local fitness maximum to another and eventually reaches the global maximum. We use a strong selection limit, which reduces the dynamics beyond the first time step to the competition between independent mutant subpopulations, to study the dynamics of this model and of a simpler one-dimensional model which ignores the geometry of the sequence space. We find that the fit genotypes that appear along a trajectory are a subset of suitably defined fitness \textit{records}, and exploit several results from the record theory for non-identically distributed random variables. The genotypes that contribute to the trajectory are those records that are not \textit{bypassed} by superior records arising further away from the initial population. Several conjectures concerning the statistics of bypassing are extracted from numerical simulations. In particular, for the one-dimensional model, we propose a simple relation between the bypassing probability and the dynamic exponent which describes the scaling of the typical evolution time with genome size. The latter can be determined exactly in terms of the extremal properties of the fitness distribution.Comment: Figures in color; minor revisions in tex

Evolutionary instability of Zero Determinant strategies demonstrates that winning isn't everything

Zero Determinant (ZD) strategies are a new class of probabilistic and conditional strategies that are able to unilaterally set the expected payoff of an opponent in iterated plays of the Prisoner's Dilemma irrespective of the opponent's strategy, or else to set the ratio between a ZD player's and their opponent's expected payoff. Here we show that while ZD strategies are weakly dominant, they are not evolutionarily stable and will instead evolve into less coercive strategies. We show that ZD strategies with an informational advantage over other players that allows them to recognize other ZD strategies can be evolutionarily stable (and able to exploit other players). However, such an advantage is bound to be short-lived as opposing strategies evolve to counteract the recognition.Comment: 14 pages, 4 figures. Change in title (again!) to comply with Nature Communications requirements. To appear in Nature Communication

Live to cheat another day: bacterial dormancy facilitates the social exploitation of beta-lactamases

The breakdown of antibiotics by β-lactamases may be cooperative, since resistant cells can detoxify their environment and facilitate the growth of susceptible neighbours. However, previous studies of this phenomenon have used artificial bacterial vectors or engineered bacteria to increase the secretion of β-lactamases from cells. Here, we investigated whether a broad-spectrum β-lactamase gene carried by a naturally occurring plasmid (pCT) is cooperative under a range of conditions. In ordinary batch culture on solid media, there was little or no evidence that resistant bacteria could protect susceptible cells from ampicillin, although resistant colonies could locally detoxify this growth medium. However, when susceptible cells were inoculated at high densities, late-appearing phenotypically susceptible bacteria grew in the vicinity of resistant colonies. We infer that persisters, cells that have survived antibiotics by undergoing a period of dormancy, founded these satellite colonies. The number of persister colonies was positively correlated with the density of resistant colonies and increased as antibiotic concentrations decreased. We argue that detoxification can be cooperative under a limited range of conditions: if the toxins are bacteriostatic rather than bacteridical; or if susceptible cells invade communities after resistant bacteria; or if dormancy allows susceptible cells to avoid bactericides. Resistance and tolerance were previously thought to be independent solutions for surviving antibiotics. Here, we show that these are interacting strategies: the presence of bacteria adopting one solution can have substantial effects on the fitness of their neighbours

Signatures of arithmetic simplicity in metabolic network architecture

Metabolic networks perform some of the most fundamental functions in living cells, including energy transduction and building block biosynthesis. While these are the best characterized networks in living systems, understanding their evolutionary history and complex wiring constitutes one of the most fascinating open questions in biology, intimately related to the enigma of life's origin itself. Is the evolution of metabolism subject to general principles, beyond the unpredictable accumulation of multiple historical accidents? Here we search for such principles by applying to an artificial chemical universe some of the methodologies developed for the study of genome scale models of cellular metabolism. In particular, we use metabolic flux constraint-based models to exhaustively search for artificial chemistry pathways that can optimally perform an array of elementary metabolic functions. Despite the simplicity of the model employed, we find that the ensuing pathways display a surprisingly rich set of properties, including the existence of autocatalytic cycles and hierarchical modules, the appearance of universally preferable metabolites and reactions, and a logarithmic trend of pathway length as a function of input/output molecule size. Some of these properties can be derived analytically, borrowing methods previously used in cryptography. In addition, by mapping biochemical networks onto a simplified carbon atom reaction backbone, we find that several of the properties predicted by the artificial chemistry model hold for real metabolic networks. These findings suggest that optimality principles and arithmetic simplicity might lie beneath some aspects of biochemical complexity

Integrated information increases with fitness in the evolution of animats

One of the hallmarks of biological organisms is their ability to integrate disparate information sources to optimize their behavior in complex environments. How this capability can be quantified and related to the functional complexity of an organism remains a challenging problem, in particular since organismal functional complexity is not well-defined. We present here several candidate measures that quantify information and integration, and study their dependence on fitness as an artificial agent ("animat") evolves over thousands of generations to solve a navigation task in a simple, simulated environment. We compare the ability of these measures to predict high fitness with more conventional information-theoretic processing measures. As the animat adapts by increasing its "fit" to the world, information integration and processing increase commensurately along the evolutionary line of descent. We suggest that the correlation of fitness with information integration and with processing measures implies that high fitness requires both information processing as well as integration, but that information integration may be a better measure when the task requires memory. A correlation of measures of information integration (but also information processing) and fitness strongly suggests that these measures reflect the functional complexity of the animat, and that such measures can be used to quantify functional complexity even in the absence of fitness data.Comment: 27 pages, 8 figures, one supplementary figure. Three supplementary video files available on request. Version commensurate with published text in PLoS Comput. Bio