Adaptive dynamics via Hamilton-Jacobi approach and entropy methods for a juvenile-adult model

We consider a nonlinear system describing a juvenile-adult population undergoing small mutations. We analyze two aspects: from a mathematical point of view, we use an entropy method to prove that the population neither goes extinct nor blows-up; from an adaptive evolution point of view, we consider small mutations on a long time scale and study how a monomorphic or a dimorphic initial population evolves towards an Evolutionarily Stable State. Our method relies on an asymptotic analysis based on a constrained Hamilton-Jacobi equation. It allows to recover earlier predictions in Calsina and Cuadrado [A. Calsina, S. Cuadrado, Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics, J. Math. Biol. 48 (2004) 135; A. Calsina, S. Cuadrado, Stationary solutions of a selection mutation model: the pure mutation case, Math. Mod. Meth. Appl. Sci. 15(7) (2005) 1091.] that we also assert by direct numerical simulation. One of the interests here is to show that the Hamilton-Jacobi approach initiated in Diekmann et al. [O. Diekmann, P.-E. Jabin, S. Mischler, B. Perthame, The dynamics of adaptation: an illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol. 67(4) (2005) 257.] extends to populations described by systems.     

From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics

We study the large population limit of the Moran process, under the assumption of weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.     

The adaptive dynamics of function-valued traits

This study extends the framework of adaptive dynamics to function-valued traits. Such adaptive traits naturally arise in a great variety of settings: variable or heterogeneous environments, age-structured populations, phenotypic plasticity, patterns of growth and form, resource gradients, and in many other areas of evolutionary ecology. Adaptive dynamics theory allows analysing the long-term evolution of such traits under the density-dependent and frequency-dependent selection pressures resulting from feedback between evolving populations and their ecological environment. Starting from individual-based considerations, we derive equations describing the expected dynamics of a function-valued trait in asexually reproducing populations under mutation-limited evolution, thus generalizing the canonical equation of adaptive dynamics to function-valued traits. We explain in detail how to account for various kinds of evolutionary constraints on the adaptive dynamics of function-valued traits. To illustrate the utility of our approach, we present applications to two specific examples that address, respectively, the evolution of metabolic investment strategies along resource gradients, and the evolution of seasonal flowering schedules in temporally varying environments.     

A stochastic model of evolutionary dynamics with deterministic large-population asymptotics

An evolutionary birth-death process is proposed as a model of evolutionary dynamics. Agents residing in a continuous spatial environment X, play a game G, with a continuous strategy set S, against other agents in the environment. The agents’ positions and strategies continuously change in response to other agents and to random effects. Agents spawn asexually at rates that depend on their current fitness, and agents die at rates that depend on their local population density. Agents’ individual evolutionary trajectories in X and S are governed by a system of stochastic ODEs. When the number of agents is large and distributed in a smooth density on (X,S), the collective dynamics of the entire population is governed by a certain (deterministic) PDE, which we call a fitness-diffusion equation.     

Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics

The measure dynamics approach to modelling single-species coevolution with a one-dimensional trait space is developed and compared to more traditional methods of adaptive dynamics and the Maximum Principle. It is assumed that individual fitness results from pairwise interactions together with a background fitness that depends only on total population size. When fitness functions are quadratic in the real variables parameterizing the one-dimensional traits of interacting individuals, the following results are derived. It is shown that among monomorphisms (i.e. measures supported on a single trait value), the continuously stable strategy (CSS) characterize those that are Lyapunov stable and attract all initial measures supported in an interval containing this trait value. In the cases where adaptive dynamics predicts evolutionary branching, convergence to a dimorphism is established. Extensions of these results to general fitness functions and/or multi-dimensional trait space are discussed.     

The adaptive dynamics of Lotka-Volterra systems with trade-offs

We analyse the adaptive dynamics of a generalised type of Lotka-Volterra model subject to an explicit trade-off between two parameters. A simple expression for the fitness of a mutant strategy in an environment determined by the established, resident strategy is obtained leading to general results for the position of the evolutionary singular strategy and the associated second-order partial derivatives of the mutant fitness with respect to the mutant and resident strategies. Combinations of these results can be used to determine the evolutionary behaviour of the system. The theory is motivated by an example of prey evolution in a predator-prey system in which results show that only (non-EUS) evolutionary repellor dynamics, where evolution is directed away from a singular strategy, or dynamics where the singular strategy is an evolutionary attractor, are possible. Moreover, the general theory can be used to show that these results are the only possibility for all Lotka-Volterra systems in which aside from the trade-offs all parameters are independent and in which the interaction terms are of quadratic order or less. The applicability of the theory is highlighted by examining the evolution of an intermediate predator in a tri-trophic model.     

Global and local motions in ribonuclease A: a molecular dynamics study

The understanding of protein dynamics is one of the major goals of structural biology. A direct link between protein dynamics and function has been provided by x-ray studies performed on ribonuclease A (RNase A) (B. F. Rasmussen et al., Nature, 1992, Vol. 357, pp. 423-424; L. Vitagliano et al., Proteins: Structure, Function, and Genetics, 2002, Vol. 46, pp. 97-104). Here we report a 3 ns molecular dynamics simulation of RNase A in water aimed at characterizing the dynamical behavior of the enzyme. The analysis of local and global motions provides interesting insight on the dynamics/function relationship of RNase A. In agreement with previous crystallographic reports, the present study confirms that the RNase A active site is constituted by rigid (His12, Asn44, Thr45) and flexible (Lys41, Asp83, His119, Asp121) residues. The analysis of the global motions, performed using essential dynamics, shows that the two beta-sheet regions of RNase A move coherently in opposite directions, thus modifying solvent accessibility of the active site, and that the mixed alpha/3(10)-helix (residues 50-60) behaves as a mechanical hinge during the breathing motion of the protein. These data demonstrate that this motion, essential for RNase A substrate binding and release, is an intrinsic dynamical property of the ligand-free enzyme.     

Comparison of monovalent and divalent ion distributions around a DNA duplex with molecular dynamics simulation and a Poisson‐Boltzmann approach

The ion atmosphere created by monovalent (Na⁺) or divalent (Mg²⁺) cations surrounding a B‐form DNA duplex were examined using atomistic molecular dynamics (MD) simulations and the nonlinear Poisson‐Boltzmann (PB) equation. The ion distributions predicted by the two methods were compared using plots of radial and two‐dimensional cation concentrations and by calculating the total number of cations and net solution charge surrounding the DNA. Na⁺ ion distributions near the DNA were more diffuse in PB calculations than in corresponding MD simulations, with PB calculations predicting lower concentrations near DNA groove sites and phosphate groups and a higher concentration in the region between these locations. Other than this difference, the Na⁺ distributions generated by the two methods largely agreed, as both predicted similar locations of high Na⁺ concentration and nearly identical values of the number of cations and the net solution charge at all distances from the DNA. In contrast, there was greater disagreement between the two methods for Mg²⁺ cation concentration profiles, as both the locations and magnitudes of peaks in Mg²⁺ concentration were different. Despite experimental and simulation observations that Mg²⁺ typically maintains its first solvation shell when interacting with nucleic acids, modeling Mg²⁺ as an unsolvated ion during PB calculations improved the agreement of the Mg²⁺ ion atmosphere predicted by the two methods and allowed for values of the number of bound ions and net solution charge surrounding the DNA from PB calculations that approached the values observed in MD simulations. © 2014 Wiley Periodicals, Inc. Biopolymers 101: 834–848, 2014.     

Resident-invader dynamics and the coexistence of similar strategies

We study the resident-invader dynamics for a given class of models of unstructured populations of finite-dimensional strategies. We prove various results on the existence and uniqueness of -limit sets in the interior of the resident-invader population state space, and we classify the generically possible types of dynamics in terms of the invasion conditions when the resident and invader strategies are similar to one another.This work was supported by the Academy of Finland     

The role of trade-off shapes in the evolution of parasites in spatial host populations: an approximate analytical approach

Given the substantial changes in mixing in many populations, there is considerable interest in the role that spatial structure can play in the evolution of disease. Here we examine the role of different trade-off shapes in the evolution of parasites in a spatially structured host population where infection can occur locally or globally. We develop an approximate adaptive dynamic analytical approach, to examine how the evolutionarily stable (ES) virulence depends not only on the fraction of global infection/transmission but also on the shape of the trade-off between transmission and virulence. Our analysis can successfully predict the ES virulence found previously by simulation of the full system. The analysis confirms that when there is a linear trade-off between transmission and virulence spatial structure may lead to an ES virulence that increases as the proportion of global transmission increases. However, we also show that the ESS disappears above a threshold level of global infection, leading to maximization. In addition just below this threshold, there is the possibility of evolutionary bi-stabilities. When we assume the realistic trade-off between transmission and virulence that results in an ESS in the classical mixed model, we find that spatial structure can increase or decrease the ES virulence. A relatively high proportion of local infection reduces virulence but intermediate levels can select for higher virulence. Our work not only emphasizes the importance of spatial structure to the evolution of parasites, but also makes it clear that situations between the local and the global need to be considered. We also emphasize the key role that the shape of trade-offs plays in evolutionary outcomes.     

The application of a model of glucose and insulin dynamics to explain an observed effect of leptin administration in reversal of developmental programming

A dynamical model describing the glucose-insulin physiological system was applied to an experiment on the administration of the adipokine leptin between neonatal days 3 and 13 to rats whose dams were subject to different levels of nutrition during gestation. The effect of leptin treatment on the glucose-insulin equilibrium point and on the levels of other associated metabolites showed a significant change in direction that depended on the level of prenatal nutrition. Leptin has been shown to affect two factors that affect the equilibrium levels of glucose and insulin, gluconeogenesis and insulin sensitivity. Each effect is described by a parameter in the dynamical model. Mathematical analysis shows that changes in these parameters in the manner promoted by leptin would indeed increase or decrease the glucose-insulin equilibria depending on their initial equilibrium levels which might be induced by the level of prenatal nutrition. This analysis explains the results of the leptin experiment in the context of the dynamics of the glucocorticoid system. It also proposes a physiological mechanism for the expression of plasticity in the organism based on the status of the glucose and insulin equilibria.     

A quasispecies approach to viral evolution in the context of an adaptive immune system

A deeper understanding of the mechanisms that determine viral evolution in the context of an adaptive immune system is vital for the development of efficient strategies to defeat viral infections. The problem of describing these mechanisms is discussed using the concept of quasispecies. Conditions for both an optimal immune response and for highest viral viability are derived from theoretical models and are supported by empirical data.     

The rate of environmental change drives adaptation to an antibiotic sink

Recent accelerated trends of human-induced global changes are providing many examples of adaptation to novel environments. Although the rate of environmental change can vary dramatically, its effect on evolving populations is unknown. A crucial feature explaining the adaptation to harsh environments is the supply of beneficial mutations via immigration from a 'source' population. In this study, we tested the effect of immigration on adaptation to increasing concentrations of antibiotics. Using experimental population of Pseudomonas aeruginosa, a pathogenic bacterium, we show that higher immigration rates and a slow increase in antibiotic concentration result in a more rapid evolution of resistance; however, a high immigration rate combined with rapid increases in concentration resulted in higher maximal levels of resistance. These findings, which support recent theoretical work, have important implications for the control of antibiotic resistance because they show that rapid rates of change can produce variants with the ability to resist future treatments.     

Heterogeneous animal group models and their group-level alignment dynamics: an equation-free approach

We study coarse-grained (group-level) alignment dynamics of individual-based animal group models for heterogeneous populations consisting of informed (on preferred directions) and uninformed individuals. The orientation of each individual is characterized by an angle, whose dynamics are nonlinearly coupled with those of all the other individuals, with an explicit dependence on the difference between the individual's orientation and the instantaneous average direction. Choosing convenient coarse-grained variables (suggested by uncertainty quantification methods) that account for rapidly developing correlations during initial transients, we perform efficient computations of coarse-grained steady states and their bifurcation analysis. We circumvent the derivation of coarse-grained governing equations, following an equation-free computational approach.     

The NOS/NO system in an example of extreme adaptation: The African lungfish

African dipnoi (lungfish) are aestivating fish and obligate air breathers that, throughout their complex life cycle, undergo remarkable morpho-functional organ readjustment from biochemical to morphological level. In the present review we summarize the changes of the NOS/NO (Nitric Oxide Synthase/Nitric Oxide) system occurring in lungs, gills, kidney, heart, and myotomal muscle of African lungfish of the genus Protopterus (P. dolloi and P. annectens), in relation to the switch from freshwater to aestivation, and vice-versa. In particular, the expression and localization patterns of NOS, and its protein partners Akt, Hsp-90 and HIF-1α, have been discussed, together with the apoptosis rate, evaluated by TUNEL technique.We hypothesize that all these molecular components are crucial in signalling transduction/integration networks induced by environmental challenges (temperature, dehydration, inactivity)experienced at the beginning, during, and at the end of the dry season.     

Aging mechanism as the "down side" of adaptation: a network approach

Many diverse hypotheses on aging are in play. All from "aging genes" over decreasing telomere length to increased level of gene mutations has been suggested to determine an organism's lifespan, but no unifying theory exists. As part of a growing interest toward more integrative approaches in the field we propose a simplistic model based on the "use-it-or-lose-it" concept: we hypothesize that biological aging is a systemic property and the down side of adaptation in complex biological networks at various levels of organization: from brain over the immune system to specialized tissues or organs. The simple dynamical model undergoes three phases during its lifetime: (1) general plasticity (childhood), (2) optimization/adaptation to given conditions (youth and adolescence) and (3) steady state associated with high rigidity (aging). Furthermore, our model mimics recent data on the dynamics of the immune system during aging and, although simplistic, thus captures essential characteristics of the aging process. Finally, we discuss the abstract model in relation to current knowledge on aging and propose experimental setups for testing some of the theoretical predictions.     

Adaptive dynamics of Lotka-Volterra systems with trade-offs: the role of interspecific parameter dependence in branching

We determine the adaptive dynamics of a general Lotka-Volterra system containing an intraspecific parameter dependency--in the form of an explicit functional trade-off between evolving parameters--and interspecific parameter dependencies--arising from modelling species interactions. We develop expressions for the fitness of a mutant strategy in a multi-species resident environment, the position of the singular strategy in such systems and the non-mixed second-order partial derivatives of the mutant fitness. These expressions can be used to determine the evolutionary behaviour of the system. The type of behaviour expected depends on the curvature of the trade-off function and can be interpreted in a biologically intuitive manner using the rate of acceleration/deceleration of the costs implicit in the trade-off function. We show that for evolutionary branching to occur we require that one (or both) of the traded-off parameters includes an interspecific parameter dependency and that the trade-off function has weakly accelerating costs. This could have important implications for understanding the type of mechanisms that cause speciation. The general theory is motivated by using adaptive dynamics to examine evolution in a predator-prey system. The applicability of the general theory as a tool for examining specific systems is highlighted by calculating the evolutionary behaviour in a three species (prey-predator-predator) system.