Existence of steady-state probability distributions in multilocus models for genotype evolution

It is shown that a representative Fisher-Wright model withn(≥3) diallelic loci admits a necessary condition for existence of a time-independent steady-state probability distribution. This necessary condition states that a global integral depending on the phenotype fitness functions of natural selection must be larger than a certain quantity depending on the parameters associated with genetic drift.     

Long-term evolution of multilocus traits

 We analyze monomorphic equilibria of long-term evolution for one or two continuous traits, controlled by an arbitrary number of autosomal loci and subject to constant viability selection. It turns out that fitness maximization always obtains at long term equilibria, but in the case of two traits, linkage determines the precise nature of the fitness measure that is maximized. We then consider local convergence to long term equilibria, for two multilocus traits subject to either constant or frequency dependent selection. From a model of long-term dynamics near an equilibrium we derive a criterion of local long-term stability for 2-dimensional equilibria. It turns out that mutation can be a decisive factor for stability. Received 26 January 1994; received in revised form 26 September 1994     

Selective sweeps in multilocus models of quantitative traits

We study the trajectory of an allele that affects a polygenic trait selected toward a phenotypic optimum. Furthermore, conditioning on this trajectory we analyze the effect of the selected mutation on linked neutral variation. We examine the well-characterized two-locus two-allele model but we also provide results for diallelic models with up to eight loci. First, when the optimum phenotype is that of the double heterozygote in a two-locus model, and there is no dominance or epistasis of effects on the trait, the trajectories of selected mutations rarely reach fixation; instead, a polymorphic equilibrium at both loci is approached. Whether a polymorphic equilibrium is reached (rather than fixation at both loci) depends on the intensity of selection and the relative distances to the optimum of the homozygotes at each locus. Furthermore, if both loci have similar effects on the trait, fixation of an allele at a given locus is less likely when it starts at low frequency and the other locus is polymorphic (with alleles at intermediate frequencies). Weaker selection increases the probability of fixation of the studied allele, as the polymorphic equilibrium is less stable in this case. When we do not require the double heterozygote to be at the optimum we find that the polymorphic equilibrium is more difficult to reach, and fixation becomes more likely. Second, increasing the number of loci decreases the probability of fixation, because adaptation to the optimum is possible by various combinations of alleles. Summaries of the genealogy (height, total length, and imbalance) and of sequence polymorphism (number of polymorphisms, frequency spectrum, and haplotype structure) next to a selected locus depend on the frequency that the selected mutation approaches at equilibrium. We conclude that multilocus response to selection may in some cases prevent selective sweeps from being completed, as described in previous studies, but that conditions causing this to happen strongly depend on the genetic architecture of the trait, and that fixation of selected mutations is likely in many instances.     

The long-term evolution of multilocus traits under frequency-dependent disruptive selection

Frequency-dependent disruptive selection is widely recognized as an important source of genetic variation. Its evolutionary consequences have been extensively studied using phenotypic evolutionary models, based on quantitative genetics, game theory, or adaptive dynamics. However, the genetic assumptions underlying these approaches are highly idealized and, even worse, predict different consequences of frequency-dependent disruptive selection. Population genetic models, by contrast, enable genotypic evolutionary models, but traditionally assume constant fitness values. Only a minority of these models thus addresses frequency-dependent selection, and only a few of these do so in a multilocus context. An inherent limitation of these remaining studies is that they only investigate the short-term maintenance of genetic variation. Consequently, the long-term evolution of multilocus characters under frequency-dependent disruptive selection remains poorly understood. We aim to bridge this gap between phenotypic and genotypic models by studying a multilocus version of Levene's soft-selection model. Individual-based simulations and deterministic approximations based on adaptive dynamics theory provide insights into the underlying evolutionary dynamics. Our analysis uncovers a general pattern of polymorphism formation and collapse, likely to apply to a wide variety of genetic systems: after convergence to a fitness minimum and the subsequent establishment of genetic polymorphism at multiple loci, genetic variation becomes increasingly concentrated on a few loci, until eventually only a single polymorphic locus remains. This evolutionary process combines features observed in quantitative genetics and adaptive dynamics models, and it can be explained as a consequence of changes in the selection regime that are inherent to frequency-dependent disruptive selection. Our findings demonstrate that the potential of frequency-dependent disruptive selection to maintain polygenic variation is considerably smaller than previously expected.     

The relation between multilocus population genetics and social evolution theory

Evolution at multiple gene positions is complicated. Direct selection on one gene disturbs the evolutionary dynamics of associated genes. Recent years have seen the development of a multilocus methodology for modeling evolution at arbitrary numbers of gene positions with arbitrary dominance and epistatic relations, mode of inheritance, genetic linkage, and recombination. We show that the approach is conceptually analogous to social evolutionary methodology, which focuses on selection acting on associated individuals. In doing so, we (1) make explicit the links between the multilocus methodology and the foundations of social evolution theory, namely, Price's theorem and Hamilton's rule; (2) relate the multilocus approach to levels-of-selection and neighbor-modulated-fitness approaches in social evolution; (3) highlight the equivalence between genetical hitchhiking and kin selection; (4) demonstrate that the multilocus methodology allows for social evolutionary analyses involving coevolution of multiple traits and genetical associations between nonrelatives, including individuals of different species; (5) show that this methodology helps solve problems of dynamic sufficiency in social evolution theory; (6) form links between invasion criteria in multilocus systems and Hamilton's rule of kin selection; (7) illustrate the generality and exactness of Hamilton's rule, which has previously been described as an approximate, heuristic result.     

Molecular evolution and multilocus sequence typing of 145 strains of SARS-CoV

In this study, we have identified 876 polymorphism sites in 145 complete or partial genomes of SARS-CoV available in the NCBI GenBank. One hundred and seventy-four of these sites existed in two or more SARS-CoV genome sequences. According to the sequence polymorphism, all SARS-CoVs can be divided into three groups: (I) group 1, animal-origin viruses (such as SARS-CoV SZ1, SZ3, SZ13 and SZ16); (II) group 2, all viruses with clinical origin during first epidemic; and (III) group 3, SARS-CoV GD03T0013. According to 10 special loci, group 2 again can be divided into genotypes C and T, which can be further divided into sub-genotypes C1-C4 and T1-T4. Positive Darwinian selections were identified between any pair of these three groups. Genotype C gives neutral selection. Genotype T, however, shows negative selection. By comparing the death rates of SARS patients in the different regions, it was found that the death rate caused by the viruses of the genotype C was lower than that of the genotype T. SARS-CoVs might originate from an unknown ancestor.     

Inbreeding depression and the evolution of dispersal rates: a multilocus model

Inbreeding depression is one of the possible reasons organisms disperse. In this article, we present a two-locus model for the evolution of dispersal in the presence of inbreeding depression. The first locus codes for a modifier of the migration rate, while the second locus is a selected locus generating inbreeding depression. We express the change in frequency of the migration modifier as a function of allele frequencies and genetic associations and then use a quasi-equilibrium assumption to express genetic associations as functions of allele frequencies. Our model disentangles two effects of inbreeding depression: it gives an advantage to migrant individuals because their offspring are on average less homozygous, but it also decreases the degree of population structure, thus decreasing the strength of kin selection for dispersal. We then extend our model to include an infinite number of selected loci. When the cost of dispersal is not too high, the model predictions are confirmed by multilocus simulation results and show that inbreeding depression can have a substantial effect on the dispersal rate. For high costs of dispersal, we observe discrepancies between the model and the simulations, probably caused by associations among selected loci, which are neglected in the analysis.     

Representation of nonepistatic selection models and analysis of multilocus Hardy-Weinberg equilibrium configurations

Summary The paper develops conditions for the existence and the stability of central equilibria emanating from selection recombination interaction with generalized nonepistatic selection forms operating in multilocus multiallele systems. The selection structure admits a natural representation as simple sums of Kronecker products based on a common set of marginal selection components. A flexible parametrization of the recombination process is introduced leading to a canonical derivation of the transformation equations connecting gamete frequency states over successive generations. Conditions for the existence and stability of multilocus Hardy-Weinberg (H.W.) type equilibria are elaborated for the classical nonepistatic models (multiplicative and additive viability effects across loci) as well as for generalized nonepistatic selection expressions. It is established that the range of recombination distributions maintaining a stable H.W. polymorphic equilibrium is confined to loose linkage in the pure multiplicative case, but is not restricted in the additive model. In the bisexual case we ascertain for the generalized nonepistatic model the stability conditions of a common H.W. polymorphism.This paper was supported in part by NIH Grant GM 10452-14 and NSF Grant MCS 75-23608.     

Mathematical limits of multilocus models: the genetic transmission of bipolar disorder.

We describe a simple, graphical method for determining plausible modes of inheritance for complex traits and apply this to bipolar disorder. The constraints that allele frequencies and penetrances lie in the interval 0-1 impose limits on recurrence risks, KR, in relatives of an affected proband for a given population prevalence, KP. We have investigated these limits for KR in three classes of relatives (MZ co-twin, sibling, and parent/offspring) for the general single-locus model and for two types of multilocus models: heterogeneity and multiplicative. In our models we have assumed Hardy-Weinberg equilibrium, an all-or-none trait, absence of nongenetic resemblance between relatives, and negligible mutation at the disease loci. Although the true values of KP and the KR''s are only approximately known, observed population and family data for bipolar disorder are inconsistent with a single-locus model or with any heterogeneity model. In contrast, multiplicative models involving three or more loci are consistent with observed data and, thus, represent plausible models for the inheritance of bipolar disorders. Studies to determine the genetic basis of most bipolar disorder should use methods capable of detecting interacting oligogenes.     

Considerations on study designs using the extreme sibpairs methods under multilocus oligogenic models

Several issues pertinent to study designs employing extreme sibpairs (ESP) methods to detect complex oligogenic quantitative trait loci (QTL) are investigated in the setting of genome-wide multipoint scans. We demonstrate that when stringent alpha-levels are imposed (e.g., alpha = 0.00022 as recommended by Landers and Kruglyak), the power to detect a susceptibility locus could drop from 83.6% under a one-locus model down to a hopeless 22.8% under a two-locus model of the same heritability h(2) = 0.5 and gene frequency (p = 0.1). We introduce the notion of joint power that is the power to detect linkage to at least one location over a given panel of markers across a genomic region and describe the effect of several design factors on such joint power in a multipoint scan. Moreover, power of analysis conditional on the IBD sharings of ESPs at a known/detected locus is examined and shown to increase substantively (to 93.3% under the previous two-locus model) in detecting novel trait loci. We conclude that with such remedies, the ESP design continues to be a relatively powerful design for mapping oligogenic QTL. However, when the effect of individual contributing loci becomes less tractable, especially when their contributions are "asymmetric," deliberation on balancing two types of statistical errors and a careful examination of possible contributions from multiple genetic factors and/or interaction effects are a must in designing an efficient study.     

Chiasma-based models of multilocus recombination: increased power for exclusion mapping and gene ordering

Cytological evidence indicates that the number of chiasmata which can occur on any given chromosome arm is limited. In this paper a Monte-Carlo simulation study is used to compare the power of a model of multilocus recombination parameterized in terms of chromosome-specific chiasma distributions with the traditional model parameterized by recombination fractions in adjacent intervals. Two specific gene mapping problems are considered: excluding a test locus from a given chromosome map and ordering a test locus with respect to a fixed map of syntenic marker loci. We show that the chiasma-based models require significantly fewer observations to exclude or order a test locus and that they are quite robust to errors in specifying the underlying true chiasma distribution.     

Genetic variation maintained in multilocus models of additive quantitative traits under stabilizing selection.

Stabilizing selection for an intermediate optimum is generally considered to deplete genetic variation in quantitative traits. However, conflicting results from various types of models have been obtained. While classical analyses assuming a large number of independent additive loci with individually small effects indicated that no genetic variation is preserved under stabilizing selection, several analyses of two-locus models showed the contrary. We perform a complete analysis of a generalization of Wright's two-locus quadratic-optimum model and investigate numerically the ability of quadratic stabilizing selection to maintain genetic variation in additive quantitative traits controlled by up to five loci. A statistical approach is employed by choosing randomly 4000 parameter sets (allelic effects, recombination rates, and strength of selection) for a given number of loci. For each parameter set we iterate the recursion equations that describe the dynamics of gamete frequencies starting from 20 randomly chosen initial conditions until an equilibrium is reached, record the quantities of interest, and calculate their corresponding mean values. As the number of loci increases from two to five, the fraction of the genome expected to be polymorphic declines surprisingly rapidly, and the loci that are polymorphic increasingly are those with small effects on the trait. As a result, the genetic variance expected to be maintained under stabilizing selection decreases very rapidly with increased number of loci. The equilibrium structure expected under stabilizing selection on an additive trait differs markedly from that expected under selection with no constraints on genotypic fitness values. The expected genetic variance, the expected polymorphic fraction of the genome, as well as other quantities of interest, are only weakly dependent on the selection intensity and the level of recombination.     

Network models for sequence evolution

We introduce a general class of models for sequence evolution that includes network phylogenies. Networks, a generalization of strictly tree-like phylogenies, are proposed to model situations where multiple lineages contribute to the observed sequences. An algorithm to compute the probability distribution of binary character-state configurations is presented and statistical inference for this model is developed in a likelihood framework. A stepwise procedure based on likelihood ratios is used to explore the space of models. Starting with a star phylogeny, new splits (nontrivial bipartitions of the sequence set) are successively added to the model until no significant change in the likelihood is observed. A novel feature of our approach is that the new splits are not necessarily constrained to be consistent with a treelike mode of evolution. The fraction of invariable sites is estimated by maximum likelihood simultaneously with other model parameters and is essential to obtain a good fit to the data. The effect of finite sequence length on the inference methods is discussed. Finally, we provide an illustrative example using aligned VPl genes from the foot and mouth disease viruses (FMDV). The different serotypes of the FMDV exhibit a range of treelike and network evolutionary relationships.Correspondence to: A. von Haeseler     

Strategic and genetic models of evolution

A new model which allows both for the effect of behavioural patterns on productive matings and for parental investment in the survival of offspring to maturity is considered. This combines ideas from genetics and evolutionary game theory, and provides a more realistic formulation to describe mating behaviour than the traditional 'battle of the sexes' model. Allowing individuals to migrate leads to spatial versions of both models. The saddle point method is used to obtain the speed of first spread of new genes/strategies in both spatial systems.     

HIV-specific probabilistic models of protein evolution

Comparative sequence analyses, including such fundamental bioinformatics techniques as similarity searching, sequence alignment and phylogenetic inference, have become a mainstay for researchers studying type 1 Human Immunodeficiency Virus (HIV-1) genome structure and evolution. Implicit in comparative analyses is an underlying model of evolution, and the chosen model can significantly affect the results. In general, evolutionary models describe the probabilities of replacing one amino acid character with another over a period of time. Most widely used evolutionary models for protein sequences have been derived from curated alignments of hundreds of proteins, usually based on mammalian genomes. It is unclear to what extent these empirical models are generalizable to a very different organism, such as HIV-1-the most extensively sequenced organism in existence. We developed a maximum likelihood model fitting procedure to a collection of HIV-1 alignments sampled from different viral genes, and inferred two empirical substitution models, suitable for describing between-and within-host evolution. Our procedure pools the information from multiple sequence alignments, and provided software implementation can be run efficiently in parallel on a computer cluster. We describe how the inferred substitution models can be used to generate scoring matrices suitable for alignment and similarity searches. Our models had a consistently superior fit relative to the best existing models and to parameter-rich data-driven models when benchmarked on independent HIV-1 alignments, demonstrating evolutionary biases in amino-acid substitution that are unique to HIV, and that are not captured by the existing models. The scoring matrices derived from the models showed a marked difference from common amino-acid scoring matrices. The use of an appropriate evolutionary model recovered a known viral transmission history, whereas a poorly chosen model introduced phylogenetic error. We argue that our model derivation procedure is immediately applicable to other organisms with extensive sequence data available, such as Hepatitis C and Influenza A viruses.     

General aspects of angiosperm evolution and macrosystematics

Taxonomy makes increasing use of significant results from many fields of research including the rapidly developing fields of micro– and macromolecular chemistry, ultrastructure and micromorphology in combination with macromorphology, anatomy, embryology, cytology, paleontology, biological interaction and distribution. Some of these results have contributed to make the current systems of classifications more concordant. Recent studies on Cretaceous fossils are related to present–day angiosperms and their floral types. It is concluded that pleiomerous flowers with helically arranged parts (corresponding to the Magnolia type, though probably less elaborate), on the basis of recent evidence can still be regarded as the probably earliest floral type in angiosperms. But the trimerous flowers must also have appeared very early, at least in the Albian. There is also evidence that the monocotyledons had differentiated as a separate group at that time. Similarities between certain extant monocotyledons and certain dicotyledons, in particular between some Dios–coreales and some Annonales–Aristolochiales, indicate that the monocotyledons had their roots in early Cretaceous pro–Magnoliiflorae. Fossil petaliferous flowers from Cenomanian layers, and later of a variety of flower types, such as the obdiplostemo–nous, petaliferous, epigynous Scandianthus (similar to extant saxifragaceous genera), or flowers with secondarily pleiomerous androecia of the theaceous type are discussed in relation to the distribution of corresponding floral types in extant dicotyledons. The main features of the author's classification of angiosperms are outlined with notes on important, though often neglected, aspects and critical problems. Finally, an updated table of classification down to family rank is presented.     

General models for segregation analysis.

Theoretical details are given of various oligogenic models for segregation analysis that are available as a general segregation analysis ("GENSEG") package, programmed in FORTRAN iv. The models allow for up to two autosomal loci and one X-linked locus, normally distributed or dichotomous phenotypes, variable age of onset, and various ascertainment functions (including one that allows the probability of becoming a proband to be dependent on the age of onset). Current programs are limited to the analysis of 2-generational data, using the joint likelihood of the sibship and parental phenotypes, unless it can be assumed that the pedigrees being analyzed are a random sample from the population; half-sibships and twins, however, are explicitly allowed.     

Assessing site-interdependent phylogenetic models of sequence evolution

In recent works, methods have been proposed for applying phylogenetic models that allow for a general interdependence between the amino acid positions of a protein. As of yet, such models have focused on site interdependencies resulting from sequence-structure compatibility constraints, using simplified structural representations in combination with a set of statistical potentials. This structural compatibility criterion is meant as a proxy for sequence fitness, and the methods developed thus far can incorporate different site-interdependent fitness proxies based on other measurements. However, no methods have been proposed for comparing and evaluating the adequacy of alternative fitness proxies in this context, or for more general comparisons with canonical models of protein evolution. In the present work, we apply Bayesian methods of model selection-based on numerical calculations of marginal likelihoods and posterior predictive checks-to evaluate models encompassing the site-interdependent framework. Our application of these methods indicates that considering site-interdependencies, as done here, leads to an improved model fit for all data sets studied. Yet, we find that the use of pairwise contact potentials alone does not suitably account for across-site rate heterogeneity or amino acid exchange propensities; for such complexities, site-independent treatments are still called for. The most favored models combine the use of statistical potentials with a suitably rich site-independent model. Altogether, the methodology employed here should allow for a more rigorous and systematic exploration of different ways of modeling explicit structural constraints, or any other site-interdependent criterion, while best exploiting the richness of previously proposed models.