Steady-state Fitness Research Articles

Impact of population size on early adaptation in rugged fitness landscapes.

Owing to stochastic fluctuations arising from finite population size, known as genetic drift, the ability of a population to explore a rugged fitness landscape depends on its size. In the weak mutation regime, while the mean steady-state fitness increases with population size, we find that the height of the first fitness peak encountered when starting from a random genotype displays various behaviours versus population size, even among small and simple rugged landscapes. We show that the accessibility of the different fitness peaks is key to determining whether this height overall increases or decreases with population size. Furthermore, there is often a finite population size that maximizes the height of the first fitness peak encountered when starting from a random genotype. This holds across various classes of model rugged landscapes with sparse peaks, and in some experimental and experimentally inspired ones. Thus, early adaptation in rugged fitness landscapes can be more efficient and predictable for relatively small population sizes than in the large-size limit. This article is part of the theme issue 'Interdisciplinary approaches to predicting evolutionary biology'.

Interpretation of NO<sub>3</sub>–N<sub>2</sub>O<sub>5</sub> observation via steady state in high-aerosol air mass: the impact of equilibrium coefficient in ambient conditions

Abstract. Steady-state approximation for interpreting NO3 and N2O5 has large uncertainty under complicated ambient conditions and could even produce incorrect results unconsciously. To provide an assessment and solution to the dilemma, we formulate datasets based on in situ observations to reassess the applicability of the method. In most of steady-state cases, we find a prominent discrepancy between Keq (equilibrium coefficient for reversible reactions of NO3 and N2O5) and correspondingly simulated [N2O5]/[NO2]×[NO3], especially under high-aerosol conditions in winter. This gap reveals that the accuracy of Keq has a critical impact on the steady-state analysis in polluted regions. In addition, the accuracy of γ (N2O5) derived by steady-state fit depends closely on the reactivity of NO3 (kNO3) and N2O5(kN2O5). Based on a complete set of simulations, air mass of kNO3 less than 0.01 s−1 with high aerosol and temperature higher than 10 ∘C is suggested to be the best suited for steady-state analysis of NO3–N2O5 chemistry. Instead of confirming the validity of steady state by numerical modeling for every case, this work directly provides appropriate concentration ranges for accurate steady-state approximation, with implications for choosing suited methods to interpret nighttime chemistry in high-aerosol air mass.

Groundwater Recharge Assessment Using WetSpass and MODFLOW Coupling: The Case of Hormat-Golina Sub-basin, Northern Ethiopia

Water scarcity in northern Ethiopia, as well as its socio-economic relevance in terms of water demand for agriculture and domestic use, are at the root of the search for new groundwater resources and the development of groundwater models that can be used to control and manage the resource. The groundwater recharge of the Hormat-Golina sub basin was assessed using WetSpass-MODFLOW coupling. The goal of this paper is to assess the groundwater recharge in the Hormat-Golina sub-basin. These findings are then used to simulate the hydraulic head distribution using the MODFLOW groundwater flow simulation model. By comparing measured and simulated hydraulic heads, the steady state groundwater flow calibration was obtained. WetSpass calculated the mean annual evapotranspiration, surface runoff, and groundwater recharge to be 516.6, 204.9, and 35.6 mm, respectively. Groundwater recharge accounted for 4.7% of precipitation, while actual evapotranspiration and surface runoff accounted for 27% and 68% of precipitation, respectively. In such seasonal variations, the groundwater head distribution is 9.37 to 29.86 m in the winter (dry season), 9.53 to 29.89 m in the summer (wet season), and 9.58 to 30.17 m in the annual stress periods (recharges). For all stress periods, the estimated hydraulic heads in steady state fit well with the measured ones, with a correlation coefficient of 0.86 (summer, winter, and annual recharge). To preserve the resource's long-term viability, the balance between groundwater recharge and projected abstraction rates for agriculture and domestic water supply must be considered in future groundwater resource development plans in the valley.

Evolution in alternating environments with tunable interlandscape correlations.

Natural populations are often exposed to temporally varying environments. Evolutionary dynamics in varying environments have been extensively studied, although understanding the effects of varying selection pressures remains challenging. Here, we investigate how cycling between a pair of statistically related fitness landscapes affects the evolved fitness of an asexually reproducing population. We construct pairs of fitness landscapes that share global fitness features but are correlated with one another in a tunable way, resulting in landscape pairs with specific correlations. We find that switching between these landscape pairs, depending on the ruggedness of the landscape and the interlandscape correlation, can either increase or decrease steady-state fitness relative to evolution in single environments. In addition, we show that switching between rugged landscapes often selects for increased fitness in both landscapes, even in situations where the landscapes themselves are anticorrelated. We demonstrate that positively correlated landscapes often possess a shared maximum in both landscapes that allows the population to step through sub-optimal local fitness maxima that often trap single landscape evolution trajectories. Finally, we demonstrate that switching between anticorrelated paired landscapes leads to ergodic-like dynamics where each genotype is populated with nonzero probability, dramatically lowering the steady-state fitness in comparison to single landscapeevolution.

Open quasispecies models: Stability, optimization, and distributed extension

We suggest a natural approach that leads to a modification of classical quasispecies models and incorporates the possibility of population extinction in addition to growth. The resulting modified models are called open. Their essential properties, regarding in particular equilibrium behavior, are investigated both analytically and numerically. The hallmarks of the quasispecies dynamics, viz. the heterogeneous quasispecies distribution itself and the error threshold phenomenon, can be observed in our models, along with extinction. In order to demonstrate the flexibility of the introduced framework, we study the inverse problem of fitness allocation under the biologically motivated criterion of steady-state fitness maximization. Having in mind the complexity of numerical investigation of high-dimensional quasispecies problems and the fact that the actual number of genotypes or alleles involved in a studied process can be extremely large, we also build continuous-time distributed open quasispecies models. The obtained results may serve as an initial step to developing mathematical models that involve directed therapy against various pathogens.

Entropic constraints on the steady-state fitness of competing self-replicators.

Recent developments in nonequilibrium statistical mechanics suggest that the history of entropy production in a system determines the relative likelihood of competing processes. This presents the possibility of interpreting and predicting the self-organization of complex active systems, but existing theories rely on quantities that are challenging to obtain. Here, we address this issue for a general class of Markovian systems in which two types of self-replicating molecular assemblies (self-replicators) compete for a pool of limiting resource molecules within a nonequilibrium steady state. We derive exact relations that show that the relative fitness of these species depends on a path function, ψ, which is a sum of the entropy production and a relative-entropy term. In the limit of infinite path length, ψ reduces to the entropy production. We demonstrate use of the theory by numerically studying two models inspired by biological systems, including a simplified model of a competition between strains of the yeast prion Sup35 in the presence of driven disaggregation by the ATPase Hsp104.

Decoding the Evolution of Complex Biological Networks- A Theory for the Dynamics of Modularity in Populations

Biological systems have long been recognized to be modular. The modular structure of the network is represented by an intrinsic parameter, modularity. Modularity is driven by the fitness of the evolving network. A number of observations show that modularity increases under increased environmental pressure. We developed a quasispecies theory for the dynamics of modularity in populations of complex networks. We show how the steady-state fitness in a randomly changing environment can be computed. We derive a relationship between rate of environmental changes and rate of growth of modularity. This relationship explains how complex networks cope with challenging environment change in order to survive and thrive.

Critical temperature and accretion rate of outbursts in long-period dwarf novae

Dwarf nova outbursts are nonlinear phenomena, and a time-dependent disk model is necessary to account for observations in detail. However, it is also necessary to elaborate a simpler steady-state fit to interpret observations. To know in what condition the outburst is initiated, understanding of the dwarf nova outburst is important. The parameterized, steady-state fitting formulae are suggested by Smak (Acta Astron. 52, 429 (2002); ibid 60, 83 (2010)) for the critical disk temperature and mass accretion rate above which the disk becomes thermally unstable. The fits give a single-valued temperature and accretion rate and are radius-independent whereas the observations show that the outbursts are radius-dependent phenomena of the ionizaton propagating in the disk. The fits have been tested to account for the observed outbursts only for systems with orbital periods shorter than a half day. Therefore, we examine the fits for orbital period as long as 2 days and compare the fits to the time-dependent model of a long-period dwarf nova GK Per. The fits are not much different from the time-dependent result for the critical temperature. However, the fits for the critical mass accretion rate above which the disk enters the hot state overestimate the time-dependent model for a long-period system like GK Per. The critical mass accretion rate in the intermediate state is consistent with that from the time-dependent disk model. However, the fit value should be treated as a maximum possible value below which the disk maintains the intermediate state, which is consistent with an interpretation for the observations of the Z Cam stars.

A differential equation for the asymptotic fitness distribution in the Bak–Sneppen model with five species

The Bak–Sneppen model is an abstract representation of a biological system that evolves according to the Darwinian principles of random mutation and selection. The species in the system are characterized by a numerical fitness value between zero and one. We show that in the case of five species the steady-state fitness distribution can be obtained as a solution to a linear differential equation of order five with hypergeometric coefficients. Similar representations for the asymptotic fitness distribution in larger systems may help pave the way towards a resolution of the question of whether or not, in the limit of infinitely many species, the fitness is asymptotically uniformly distributed on the interval [fc, 1] with fc ≳ 2/3.

Quasispecies theory for evolution of modularity.

Biological systems are modular, and this modularity evolves over time and in different environments. A number of observations have been made of increased modularity in biological systems under increased environmental pressure. We here develop a quasispecies theory for the dynamics of modularity in populations of these systems. We show how the steady-state fitness in a randomly changing environment can be computed. We derive a fluctuation dissipation relation for the rate of change of modularity and use it to derive a relationship between rate of environmental changes and rate of growth of modularity. We also find a principle of least action for the evolved modularity at steady state. Finally, we compare our predictions to simulations of protein evolution and find them to be consistent.

Recombination in one- and two-dimensional fitness landscapes

We consider many-site mutation-recombination models of molecular evolution, where fitness is a function of a Hamming distance from one (one-dimensional case) or two (two-dimensional case) sequences. For the one-dimensional case, we calculate the population distribution dynamics for a model with zero fitness and an arbitrary symmetric initial distribution and find an error threshold transition point in the single-peak fitness model for a given initial symmetric distribution. We calculate the recombination period in the case of a single-peak fitness function, when the original population is located at one sequence, at some Hamming distance from the peak configuration. Steady-state fitness is calculated with finite genome length corrections. We derive analytical equations for the two-dimensional mutation-recombination model.

Beneficial mutation selection balance and the effect of linkage on positive selection.

When beneficial mutations are rare, they accumulate by a series of selective sweeps. But when they are common, many beneficial mutations will occur before any can fix, so there will be many different mutant lineages in the population concurrently. In an asexual population, these different mutant lineages interfere and not all can fix simultaneously. In addition, further beneficial mutations can accumulate in mutant lineages while these are still a minority of the population. In this article, we analyze the dynamics of such multiple mutations and the interplay between multiple mutations and interference between clones. These result in substantial variation in fitness accumulating within a single asexual population. The amount of variation is determined by a balance between selection, which destroys variation, and beneficial mutations, which create more. The behavior depends in a subtle way on the population parameters: the population size, the beneficial mutation rate, and the distribution of the fitness increments of the potential beneficial mutations. The mutation-selection balance leads to a continually evolving population with a steady-state fitness variation. This variation increases logarithmically with both population size and mutation rate and sets the rate at which the population accumulates beneficial mutations, which thus also grows only logarithmically with population size and mutation rate. These results imply that mutator phenotypes are less effective in larger asexual populations. They also have consequences for the advantages (or disadvantages) of sex via the Fisher-Muller effect; these are discussed briefly.

The solitary wave of asexual evolution.

Using a previously undescribed approach, we develop an analytic model that predicts whether an asexual population accumulates advantageous or deleterious mutations over time and the rate at which either process occurs. The model considers a large number of linked identical loci, or nucleotide sites; assumes that the selection coefficient per site is much less than the mutation rate per genome; and includes back and compensating mutations. Using analysis and Monte Carlo simulations, we demonstrate the accuracy of our results over almost the entire range of population sizes. Two limiting cases of our results, when either deleterious or advantageous mutations can be neglected, correspond to the Fisher-Muller effect and Muller's ratchet, respectively. By comparing predictions of our model (no recombination) to those of simple single-locus models (strong recombination), we show that the accumulation of advantageous mutations is slowed by linkage over a broad, finite range of population size. This supports the view of Fisher and Muller, who argued in the 1930s that progressive evolution of organisms is slowed because loci at which beneficial mutations can occur are often linked together on the same chromosome. These results follow from our main finding, that distribution of sequences over the mutation number evolves as a traveling wave whose speed and width depend on population size and other parameters. The model explains a logarithmic dependence of steady-state fitness on the population size reported recently for an RNA virus.

Improving Steady-State Fit of Dynamics to Data Using Adjoint Equation for Gradient Preconditioning

Abstract The steady-state fit of Arakawa and Lamb's shallow-water equation model to time-mean sea surface height (SSH) has been examined, seeking better performance of the steady-state fit without requiring longer time integrations. It is shown that the minimization problem with steadiness penalties involving one time step is ill conditioned. The difficulty is due to disparity among components of the cost function gradient, when the cost function surface is characterized by steep slopes in some parameter directions and by flat slopes in others. Preconditioning the gradient of cost function by means of an adjoint model is examined. Results show improvement of condition number of the Hessian of the cost function and quality of the fit while retaining the economy of the one-time-step approach. The method is extended to fitting the time-mean SSH from eddy-active simulations.