Fluctuations in the Size of Isolated Single-Species Populations and Natural Selection

This paper discusses the basic types of dynamical behavior of populations obtained in discrete models, such as monotonous dynamics, stable limited cycles, and chaotic variations. All these modes are shown to have possibly arisen in the evolution of limited populations under the effect of density-independent selection. This effect together with that of density-dependent non-selective factors has been termed F-selection, which is characterized by independence of relative fitnesses from population density, whereas populations may be ecologically limited; in other words, absolute fitnesses prove to be a function of population size. The characteristic of F-selection is to be not sensitive to changes in population size but to lead to fluctuations, that create conditions for achieving density-dependent selection.     

A Penalized-Likelihood Method to Estimate the Distribution of Selection Coefficients from Phylogenetic Data

We develop a maximum penalized-likelihood (MPL) method to estimate the fitnesses of amino acids and the distribution of selection coefficients (S = 2Ns) in protein-coding genes from phylogenetic data. This improves on a previous maximum-likelihood method. Various penalty functions are used to penalize extreme estimates of the fitnesses, thus correcting overfitting by the previous method. Using a combination of computer simulation and real data analysis, we evaluate the effect of the various penalties on the estimation of the fitnesses and the distribution of S. We show the new method regularizes the estimates of the fitnesses for small, relatively uninformative data sets, but it can still recover the large proportion of deleterious mutations when present in simulated data. Computer simulations indicate that as the number of taxa in the phylogeny or the level of sequence divergence increases, the distribution of S can be more accurately estimated. Furthermore, the strength of the penalty can be varied to study how informative a particular data set is about the distribution of S. We analyze three protein-coding genes (the chloroplast rubisco protein, mammal mitochondrial proteins, and an influenza virus polymerase) and show the new method recovers a large proportion of deleterious mutations in these data, even under strong penalties, confirming the distribution of S is bimodal in these real data. We recommend the use of the new MPL approach for the estimation of the distribution of S in species phylogenies of protein-coding genes.     

The Direction of Linkage Disequilibrium

The previous paper (Langley, Tobari and Kojima 1974) reports that the directional linkage disequilibria, D_ω = P_ABP_ab-P_AbP_aB, tend to be negative for data between allozymes and linked to inversions. A and B stand for the two alleles with the greatest frequency in the population. In this paper we show that linkage disequilibrium in this direction is produced at equilibrium when double homozygotes have fitnesses that are a constant fraction of the product of the two component single homozygote fitnesses, a pattern that is frequently observed in experimental data.     

Natural selection under population regulation

The dynamical behavior of multi-allele, one-locus systems is analyzed under population regulation. Weak selection is assumed. It is shown that for sufficiently large times, t, the nth time derivative of the population number N(t) is of order n}+1 in the selection coefficients. These order relations imply there is an asymptotic “quasi-equilibrium” in which population size and mean fitness change slowly relative to changes in gene frequencies. Consistent with the results of other authors, in quasi-equilibrium the mean fitness is second-order in the selection coefficients. In an effort to understand dynamic behavior beyond the immediate neighborhood of equilibrium, the topology of mean fitness surfaces is explored. In general, population regulation leads to regions of decreasing mean fitness in which there are important changes in gene frequencies. To illustrate this and other related phenomena, I analyze models in which there is logarithmic population control, and in which genotypic fitnesses W_i(x) are linear in the allele frequencies x. Exact solutions for mean fitness W(x) are obtained for two- and three-allele systems with symmetric fertilities and mortalities.     

Genic Variation in Male Haploids under Deterministic Selection

Genic variation in male haploids and male diploids was compared assuming constant fitnesses (derived from computer-generated random numbers) and infinite population size. Several models were studied, differing by the fitness correlation between the sexes (r_s) and genotypes (r_g), and by the intensity of selection as measured by the coefficient of variation (CV) of the fitness distribution. Genic variation was quantified using the proportion of polymorphic loci, P, the gene diversity at polymorphic loci, H_p, and the gene diversity over all loci, H_a. The two genetic systems were compared via variation ratios: variation in male haploidy/variation in male diploidy.—P and H_a were markedly lower for male-haploids than for male diploids, the variation ratios declining with increasing r_s, r_g and CV, but the two genetic systems were similar for H_p. Except for male diploids with r_s = 1, the two sexes had different equilibrium gene frequencies but the sample sizes required to detect such differences reliably were larger than usually possible in surveys of natural populations.—Data from natural populations fit the above trends qualitatively, but the variation ratios are much lower than those from our analyses, except that for H_p, which is higher when Drosophila is excluded. Also, the frequency distribution of most common alleles from electrophoretic data has a deficiency of intermediate frequencies compared to that from the computer-generated sets of fitnesses, possibly reflecting either the influence of stochastic processes shifting frequencies away from equilibrium or the involvement of alleles under selection-mutation balance.——While electrophoretic data suggest that Drosophila has unusually high levels of genic variation, unusually low levels of genic variation in male haploids compared with male diploids are not strongly indicated. However, if further data confirm male haploids as having low levels of genic variation, likely explanations are that the bulk of electrophoretically detected variation involves fixed-fitness balancing selection, selection-mutation balance involving slightly deleterious recessive alleles, new favorable male haploid alleles moving more rapidly to fixation than under male diploidy and thus carrying linked loci to fixation faster, or some combination of these possible factors.     

The stationary distribution of an asymmetrical model of selection in a random environment

The stationary distribution for the asymmetrical form of the SAS-CFF model of selection in a random environment is presented. Also presented are the conditions for the stable coexistence of K alleles. These conditions are the same as the conditions obtained from the classical constant-fitness model with the formal substitution of geometric mean fitnesses for the constant fitnesses of the classical model. Two examples are explored. In the “equally spaced” example, increases in the degree of asymmetry raise the homozygosity, which is accompanied by loss of alleles from the population. In the “best allele” example, increases in the degree of asymmetry raise the homozygosity without the loss of alleles. In both cases the frequency spectra are altered by the changes in the degree of asymmetry.     

A Monte Carlo simulation model for studying evolution in age-structured populations

This model provides for any number of genotypes defined by age-specific survival and fecundity rates in a population with completely overlapping generations and growing under the control of density-governing functions affecting survival or fecundity. It is tested in situations involving two alleles at one locus. Nonselection populations at Hardy–Weinberg equilibrium obey the ecogenetic law; i.e., each genotype follows Lotka's law regarding rate of increase and stable age distribution as if it were an independent true-breeding population. Populations experiencing age- and density-independent selection approximate this situation, and the changes in gene frequency are predicted by relative fitnesses bases on λ, the finite rate of increase of the genotypes. Polymorphic gene equilibria occurring at steady-state population sizes are determined by fitnesses based on R, the net reproductive rate. In examples involving differences in generation time produced by age-dependent differences in fecundity, the allele associated with longer generation time may be favored or opposed by selection, depending on whether the density-governing factor controlling population size affects survival or fecundity. If such genotypes have similar R's, a genetic equilibrium may be established if the population is governed by a density function acting upon fecundity. Received: August 23, 1999 / Accepted: July 13, 2000     

Continuous models for frequency-density-dependent selection

The growth of a panmictic monoecious diploid population with two alleles at one locus is modeled by making fitnesses depend on the genotypes' abundance. This implies an implicit dependence of fitnesses on both density and gene frequency. Equations are derived for the gene frequency and for the population size in the overlapping generation case. A diffusion model for the gene frequency is finally obtained, and the gene frequency transition probability density function is determined in the case of no dominance.     

Biology of a duplicate gene system with glucose 6-phosphate dehydrogenase activity in Drosophila melanogaster: Genetic analysis and differences in fitness components and reaction to environmental parameters among Zw genotypes

There are two structural forms of glucose 6-phosphate dehydrogenase activity in Drosophila melanogaster. Whether one or the other or both show in vitro (and probably in vivo) activity depends on the genotype of a sex-linked locus (Zw). In this article, the relative fitnesses of heterozygotes (with both electromorphs active) and homozygotes (with activity demonstrable for only one or the other electromorph) for the Zw locus are described. It is shown that the relative fitness of heterozygotes increases with increase in population density, or degree of crowding and trophic stress, and that the mean development times of Zw heterozygotes are lower than those of the Zw homozygotes. In addition, and perhaps accounting for the fitness and viability excess of the heterozygotes, one set of evidence strongly suggests that they are better buffered against trophic stress than the homozygotes.     

The evolution of alternative mating strategies in variable environments

Summary We assessed the influence of phenotypic plasticity in age at maturity on the maintenance of alternative mating strategies in male Atlantic salmon,Salmo salar. We calculated the fitness,r, associated with the parr and the anadromous strategies, using age-specific survival data from the field and strategy-specific fertilization data from the laboratory. The fitness of each strategy depended largely on mate competition (numbers of parr per female, i.e. parr frequency) and on age at maturity. Fitness declined with increasing numbers of parr per female with equilibrium frequencies (at which the fitnesses of each strategy are equal) being within the range observed in the wild. Equilibrium parr frequencies declined with decreasing growth rate and increasing age at maturity. Within populations, the existence of multiple age-specific sets of fitness functions suggests that the fitnesses of alternative strategies are best represented as multidimensional surfaces. The points of intersection of these surfaces, whose boundaries encompass natural variation in age at maturity and mate competition, define an evolutionarily stable continuum (ESC) of strategy frequencies along which the fitnesses associated with each strategy are equal. We propose a simple model that incorporates polygenic thresholds of a largely environmentally-controlled trait (age at maturity) to provide a mechanism by which an ESC can be maintained within a population. An indirect test provides support for the prediction that growth-rate thresholds for parr maturation exist and are maintained by stabilizing selection. Evolutionarily stable continua, maintained by negative frequency-dependent selection on threshold traits, provide a theoretical basis for understanding how alternative life histories can evolve in variable environments.     

Anuran species richness,complementarity and conservation conflicts in Brazilian Cerrado

Broad-scale correlations between species richness and human population suggest that processes driving species richness, mainly related to high ecological productivity, may also drive human populations. However, it is still under debate if this coincidence implies conflicts between biodiversity conservation and human development. In this paper, we analyzed the relationships among human population size, species richness and irreplaceability in Brazilian Cerrado. We analyzed a dataset with 131 species of anurans distributed in 181 cells with 1° of spatial resolution covering the biome. We found a positive correlation between human population size and anuran species richness (r = 0.46; P = 0.033 with 19.5 geographically effective degrees of freedom, v*), but the irreplaceability of each cell was poorly correlated with human population size (r = 0.075; P = 0.323; v* = 173.9). The 17 cells in the 97 optimal reserve networks contained a total human population ranging from 2942,195 to 4319,845 people, representing on average 11.8% of the human population in the entire Cerrado grid. The comparison of these observed values with 10,000 values from randomly generated networks suggests a relatively high flexibility in optimal complementarity sets for reserve selection. Our results indicated that correlation between richness and human population does not necessarily result in conflicts, given the opportunities for conciliating conservation and development. However, the analyses performed here are initial explorations within the framework of conservation biogeography, so more detailed studies are necessary to establish conservation planning at regional and local scales.     

Natural selection and fitness entropy in a density-regulated population

The entropy H(p_o,p*) of a population with the initial allele frequency p_o given the equilibrium polymorphic frequency p* has been proposed as a measure of natural selection. In the present paper, we have extended this concept to include a particular aspect of density-dependent selection. We compared size trajectory of a population initially at genetic equilibrium, N(t), with the size trajectories of populations not initially at p*,N(t), but which do eventually converge to a common equilibrium allele frequency and equilibrium density, N*. The following experimentally testable hyopthesis was established. The total area defined by the difference between the trajectories of N(t) and N(t) as they converge to N* is directly proportional to the fitness entropy when population size is transformed using the density-dependent fitness value. Two properties of this relationship were noted. First, it is independent of the magnitude of natural selection and, secondly, it does not depend upon the initial population density as long as the equilibrium and nonequilibrium populations have the same initial numbers. This hypothesis was evaluated with experimental data on the flour beetle Tribolium castaneum.     

Evolution of dispersal in density regulated populations: A haploid model

The evolution of dispersal is explored in a density-dependent framework. Attention is restricted to haploid populations in which the genotypic fitnesses at a single diallelic locus are decreasing functions of the changing number of individuals in the population. It is shown that migration between two populations in which the genotypic response to density is reversed can maintain both alleles when the intermigration rates are constant or nondecreasing functions of the population densities. There is always a unique symmetric interior equilibrium with equal numbers but opposite gene frequencies in the two populations, provided the system is not degenerate. Numerical examples with exponential and hyperbolic fitnesses suggest that this is the only stable equilibrium state under constant positive migration rates (m) less than . Practically speaking, however, there is only convergence after a reasonable number of generations for relatively small migration rates ( ). A migration-modifying mutant at a second, neutral locus, can successfully enter two populations at a stable migration-selection balance if and only if it reduces the intermigration rates of its carriers at the original equilibrium population size. Moreover, migration modification will always result in a higher equilibrium population size, provided the system approaches another symmetric interior equilibrium. The new equilibrium migration rate will be lower than that at the original equilibrium, even when the modified migration rate is a nondecreasing function of the population sizes. Therefore, as in constant viability models, evolution will lead to reduced dispersal.     

A simple stochastic gene substitution model

If the fitnesses of n haploid alleles in a finite population are assigned at random and if the alleles can mutate to one another, and if the population is initially fixed for the kth most fit allele, then the mean number of substitutions that will occur before the most fit allele is fixed is shown to be

$\text{1}\text{2}\text{+}\text{1}\text{k}\text{+}\text{∑}\text{i=2}\text{k?1}\text{(i+3)}\text{(2i(i+1))}$

when selection is strong and mutation is weak. This result is independent of the parameters that went into the model. The result is used to provide a partial explanation for the large variance observed in the rates of molecular evolution.     

Genetic and strategic models for the evolution of mating systems

Male and female fitnesses in the Shaw-Mohler equation are partitioned into components which putatively determine mating systems. The resultant genetic models provide criteria for evolutionary stable population states and yield strategic models based on maximization principles and fitness sets.     

Frequency-dependent selection with dominance: a window onto the behavior of the mean fitness

Selection in which fitnesses vary with the changing genetic composition of the population may facilitate the maintenance of genetic diversity in a wide range of organisms. Here, a detailed theoretical investigation is made of a frequency-dependent selection model, in which fitnesses are based on pairwise interactions between the two phenotypes at a diploid, diallelic, autosomal locus with complete dominance. The allele frequency dynamics are fully delimited analytically, along with all possible shapes of the mean fitness function in terms of where it increases or decreases as a function of the current allele frequency in the population. These results in turn allow possibly the first complete characterization of the dynamical behavior by the mean fitness through time under frequency-dependent selection. Here the mean fitness (i) monotonically increases, (ii) monotonically decreases, (iii) initially increases and then decreases, or (iv) initially decreases and then increases as equilibrium is approached. We analytically derive the exact initial and fitness conditions that produce each dynamic and how often each arises. Computer simulations with random initial conditions and fitnesses reveal that the potential decline in mean fitness is not negligible; on average a net decrease occurs 20% of the time and reduces the mean fitness by >17%.     

A macroscopic approach to population genetics

Conventional population genetics uses as primitive variables the frequencies and fitnesses of individual genes. This paper develops a formalism whose primitive variables are the frequencies and fitnesses of genotypes and environmental histories in a population. From the mathematical relation that describes genetic variation and selection of genotypes and environmental histories we derive a sequence of more specialized equations, including those of the conventional theory. Some familiar formulas of the conventional theory (including Fisher's fundamental theorem, the formula relating the rate of change of a metric character to selection pressure, and the definitions of broad and narrow heritability) are shown to be special cases of simpler and more general formulas. It is shown that the “genotypic value” of a trait, together with its heritability, may depend strongly on genotype-environment correlations.A generalization of Fisher's fundamental theorem shows that the rate of evolution of a trait depends on the skewness of its fitness distribution. An equation relating the second derivative of the mean fitness to the skewness is derived.Finally, the formalism is applied in a preliminary way to a recent theory of genetic variation (Layzer,1978a), according to which the genetic variability of a trait is selected along with the trait itself. It is shown that there is positive feedback between the two kinds of selection.     

Multiple adaptive substitutions during evolution in novel environments

We consider an asexual population under strong selection-weak mutation conditions evolving on rugged fitness landscapes with many local fitness peaks. Unlike the previous studies in which the initial fitness of the population is assumed to be high, here we start the adaptation process with a low fitness corresponding to a population in a stressful novel environment. For generic fitness distributions, using an analytic argument we find that the average number of steps to a local optimum varies logarithmically with the genotype sequence length and increases as the correlations among genotypic fitnesses increase. When the fitnesses are exponentially or uniformly distributed, using an evolution equation for the distribution of population fitness, we analytically calculate the fitness distribution of fixed beneficial mutations and the walk length distribution.     

Identifying Signatures of Selection in Genetic Time Series

Both genetic drift and natural selection cause the frequencies of alleles in a population to vary over time. Discriminating between these two evolutionary forces, based on a time series of samples from a population, remains an outstanding problem with increasing relevance to modern data sets. Even in the idealized situation when the sampled locus is independent of all other loci, this problem is difficult to solve, especially when the size of the population from which the samples are drawn is unknown. A standard χ²-based likelihood-ratio test was previously proposed to address this problem. Here we show that the χ²-test of selection substantially underestimates the probability of type I error, leading to more false positives than indicated by its P-value, especially at stringent P-values. We introduce two methods to correct this bias. The empirical likelihood-ratio test (ELRT) rejects neutrality when the likelihood-ratio statistic falls in the tail of the empirical distribution obtained under the most likely neutral population size. The frequency increment test (FIT) rejects neutrality if the distribution of normalized allele-frequency increments exhibits a mean that deviates significantly from zero. We characterize the statistical power of these two tests for selection, and we apply them to three experimental data sets. We demonstrate that both ELRT and FIT have power to detect selection in practical parameter regimes, such as those encountered in microbial evolution experiments. Our analysis applies to a single diallelic locus, assumed independent of all other loci, which is most relevant to full-genome selection scans in sexual organisms, and also to evolution experiments in asexual organisms as long as clonal interference is weak. Different techniques will be required to detect selection in time series of cosegregating linked loci.