Simulation of ‘hitch-hiking’ genealogies

An ancestral influence graph is derived, an analogue of the coalescent and a composite of Griffiths' (1991) two-locus ancestral graph and Krone and Neuhauser's (1997) ancestral selection graph. This generalizes their use of branching-coalescing random graphs so as to incorporate both selection and recombination into gene genealogies. Qualitative understanding of a ‘hitch-hiking’ effect on genealogies is pursued via diagrammatic representation of the genealogical process in a two-locus, two-allele haploid model. Extending the simulation technique of Griffiths and Tavaré (1996), computational estimation of expected times to the most recent common ancestor of samples of n genes under recombination and selection in two-locus, two-allele haploid and diploid models are presented. Such times are conditional on sample configuration. Monte Carlo simulations show that ‘hitch-hiking’ is a subtle effect that alters the conditional expected depth of the genealogy at the linked neutral locus depending on a mutation-selection-recombination balance. Received: 21 July 2000 / Published online: 5 December 2000     

Phenogenetic drift and the evolution of genotype-phenotype relationships

Maintenance of a stable two-locus polymorphism is analyzed statistically by fitting a logistic regression with a quadratic function of genotypic fitnesses to the probability for a fitness set to maintain a polymorphism. The regression is fitted using a data set containing information on stable equilibria maintained by 32,00 randomly generated fitness sets with three recombination values (0. 005, 0.05, 0.5). Fitted logistic regressions discriminate with 88 to 90% accuracy between fitness sets maintaining and not maintaining a stable internal equilibrium, which implies the existence of a fitness structure (balance of fitnesses) maintaining a two-locus polymorphism. Aspects of the balance of fitnesses revealed by logistic regressions are discussed. It is demonstrated that logistic regression also discriminates between types of a stable polymorphism: globally stable polymorphism, several simultaneously stable polymorphisms, and stable equilibria in addition to a polymorphic one, which implies that different balances of fitnesses are responsible for the maintenance of different types of polymorphism.     

Disequilibrium in Two-Locus Mutation-Selection Balance Models

Equilibrium behavior of two-locus mutation-selection balance models is analyzed using perturbation techniques. The classical result of Haldane for one locus is shown to carry over to two loci, if fitnesses are replaced by marginal fitnesses. If the fitness of the double heterozygote is smaller than would be produced by a multiplicative model, as in additive or quantitative fitness models, the disequilibrium is negative--an excess of gametes with one rare allele. In this case the disequilibrium can be as large as one-half its maximum value possible, if the recombination rate is small, not greater than the strength of selection. If the fitness of the double heterozygote is larger than would be produced by a multiplicative model, the disequilibrium is positive, and is very small relative to its maximum value possible, even if the recombination rate is zero.     

CYCLICAL ENVIRONMENTAL CHANGES AS A FACTOR MAINTAINING GENETIC POLYMORPHISM. 2. DIPLOID SELECTION FOR AN ADDITIVE TRAIT

The subject of this paper is polymorphism maintenance due to stabilizing selection with a moving optimum. It was shown that in case of two-locus additive control of the selected trait, global polymorphism is possible only when the geometric mean fitnesses of double homozygotes averaged over the period are lower than that of the single heterozygotes and of the double heterozygote (with a multiplier [1 – r]^p, which depends on recombination rate r and period length p). But local stability of polymorphism cannot be excluded even if geometric mean fitnesses of all double homozygotes are higher than that of all heterozygotes. We proved, that for logarithmically convex fitness functions, cyclical changes of the optimum cannot help in polymorphism maintenance in case of additive control of the selected trait by two equal loci. However, within the same class of fitness functions, nonequal gene action and/or dominance effect for one or both loci may lead to local polymorphism stability with large enough polymorphism attracting domain. The higher the intensity of selection and closer the linkage between selected loci the larger is this domain. Note that even simple cyclical selection could result in two forms of polymorphic limiting behavior: (a) usually expected forced cycle with a period equal to that of environmental changes; and (b) “supercycles,” nondumping auto-oscillations with a period comprising of hundreds of forced oscillation periods.     

The Alternative Fitness Sets Which Preserve Allele Trajectories: A General Treatment

A general solution is presented of the problem of specifying all alternative, generally frequency-dependent, (absolute) fitness sets which give rise to the same allele frequency changes and population dynamics as a given fitness set. The one- and two-locus cases are analyzed in detail and the method is then extended to the n-locus case. It is shown that if biological constraints can be used to specify the mean fitness of the population and the relative fitnesses of the heterozygotes, then the allele frequency trajectories determine a unique fitness set.     

Stable Equilibria at Two Loci in Populations with Large Selfing Rates

The equilibrium structure of two-locus, two-allele models with very large selfing rates is found using perturbation techniques. For free recombination, r = 1/2, the following results hold. If the heterozygotes do not have at least an approximate 30% advantage in fitness relative to homozygotes, a stable equilibrium with all alleles present is possible only if all of the homozygote fitnesses differ at most by approximately the outcrossing rate, t, and all stable polymorphic equilibria have disequilibrium values, D, that are at most on the order of the outcrossing rate. Once the heterozygote fitnesses are above the threshold, there are stable equilibria possible with D near its maximum possible value. The results show that the observed disequilibria in highly selfed plant populations are not likely to result from selection leading to an equilibrium.     

Establishment of allelic two-locus polymorphisms

An exact analysis of necessary and sufficient conditions for the establishment and protectedness of biallelic two-locus polymorphisms is developed for the classical model with constant, sexually symmetric fitnesses and random association of the successful gametes. To demonstrate application of the results to common model types, the model of symmetric viabilities depending on the degree of heterozygosity only is chosen as a paradigm. It is pointed out that a unique locally stable internal equilibrium may exist even though all marginal equilibria (including the fixation states) are locally attractive. This example is quoted as an indication of the priority that analyses of protectedness deserve over analyses of local stability or instability of internal equilibria. Further applications of broader appeal concern the role that recombination plays in protecting polymorphisms. Probably the most interesting finding is that with increasing recombination frequency the chances for protectedness of a polymorphism generally decline. Yet, if a certain hierarchic ordering of the fitnesses with respect to the degree of heterozygosity is realized, the polymorphism is protected for arbitrary amounts of recombination. If recombination is rare, heterozygote advantage is not a universal precondition for persistence of polymorphisms. This phenomenon is utilized to derive conditions under which deleterious recessive mutants can be maintained in a population.     

The significance of over- and underdominance for the maintenance of genetic polymorphisms. I. Underdominance and stability

It is pointed out that the standard selection models in population genetics all require some form of heterozygote advantage in fitness in order to guarantee the maintenance or stability of genetic polymorphisms. Even more recent results demonstrating the existence of stable two-locus polymorphisms with marginal underdominance at both loci are based on certain epistatically acting heterosis assumptions. This raises the question as to whether heterozygote advantage in fitness is indeed a generally valid principle of maintaining polymorphisms. To avoid ambiguity in definition of heterozygote advantage (overdominance) as it appears in multiallele or multilocus systems, a one-locus-two-allele model is considered. This model allows for sexually asymmetric selection and random mating. It is shown that the model produces globally stable polymorphisms exhibiting underdominance in fitness for a considerable and biologically reasonable range of selection values. Having thus properly refuted the general validity of the common overdominance principle, a modified version is suggested which covers the classical viability selection model and its extension to arbitrary, sexually asymmetric viability and fertility selection. This modified overdominance principle is based on the notion of fractional fitnesses and relates protectedness of biallelic polymorphisms to the extent to which each genotype reproduces its own type. The fact that the model treated displays frequency dependent fitnesses which may change in ranking while approaching equilibrium is discussed in relation to problems of the evolution of overdominance and underdominance.     

An alternative stochastic model of generation of oligodendrocytes in cell culture

According to our previous model, oligodendrocyte – type 2 (O-2A) astrocyte progenitor cells become competent for differentiation in vitro after they complete a certain number of critical mitotic cycles. After attaining the competency to differentiate, progenitor cells divide with fixed probability p in subsequent cycles. The number of critical cycles is random; analysis of data suggests that it varies from zero to two. The present paper presents an alternative model in which there are no critical cycles, and the probability that a progenitor cell will divide again decreases gradually to a plateau value as the number of completed mitotic cycles increases. In particular all progenitor cells have the ability to differentiate from the time of plating. The Kiefer-Wolfowitz procedure is used to fit the new model to experimental data on the clonal growth of purified O-2A progenitor cells obtained from the optic nerves of 7 day old rats. The new model is shown to fit the experimental data well, indicating that it is not possible to determine whether critical cycles exist on the basis of these experimental data. In contrast to the fit of the previous model, which suggested that the addition of thyroid hormone increased the limiting probability of differentiation as the number of mitotic cycles increases, the fit of the new model suggests that the addition of thyroid hormone has almost no effect on the limiting probability of differentiation. Received: 6 March 2000 / Revised version: 18 September 2000 / Published online: 30 April 2001     

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     

Mutation and recombination with tight linkage

An exact solution of the mutation-recombination equation in continuous time is presented, with linear ordering of the sites and at most one mutation or crossover event taking place at every instant of time. The differential equation may be obtained from a mutation-recombination model with discrete generations, in the limit of short generations, or weak mutation and recombination. The solution relies on the multilinear structure of the dynamical system, and on the commuting properties of the mutation and recombination operators. It is obtained through diagonalization of the mutation term, followed by a transformation to certain measures of linkage disequilibrium that simultaneously linearize and diagonalize the recombination dynamics. The collection of linkage disequilibria, as well as their decay rates, are given in closed form. Received: 26 January 1999 / Revised version: 20 October 2000 / Published online: 10 April 2001     

A deterministic size-structured population model for the worm Naidis elinguis

In this paper we model the population dynamics of the worm Nais elinguis, which reproduces by division into two unequal parts. By using renewal theory we derive the asymptotic behaviour of a Naidis elinguis population. In particular we prove a certain relation between the fraction of the population that was born small (respectively the fraction that was born large) and the inter-division times. Received 20 January 1999 / Revised version: 1 August 1999?Published online: 10 April 2001     

Archaeobotanical analysis of some early Neolithic settlements in the Kujawy region, central Poland, with potential plant gathering activities emphasised

Neolithic settlements in the Kujawy region of central Poland are represented by seven archaeological sites which have botanical material archaeologically dated to the Linear Pottery culture (LBK) (ca. 5400-5000 cal. B.C.) and the Lengyel culture (ca. 4400-4000 cal. B.C.). The composition of plant remains suggests that Stipa pennata s.l. played a certain role in the economy of the Neolithic settlers. The presence of this xerothermic grass is best explained by local gathering rather than distant transport or coming into the sediment by chance. The finding of Hierochlo? cf. australis grains represents the first identification of this plant in archaeobotanical material from Poland. This, now rare, plant contains coumarin and for this reason could have been useful in prehistory. Other plants such as Bromus spp., Chenopodium album type, Fallopia convolvulus and Galium spp. were found in large quantities, and although common weeds now, they could also have been collected by the Neolithic settlers. Plants such as Corylus avellana and Vaccinium vitis-idaea that are typically considered to be collected as foods are present in the studied material but in very small quantities. Received September 17, 2001 / Accepted March 13, 2002     

Convergence of multilocus systems under weak epistasis or weak selection

 The convergence of multilocus systems under viability selection with constant fitnesses is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multiallelic loci, the linkage map, dominance, and epistasis are arbitrary. It is proved that if epistasis or selection is sufficiently weak (and satisfies a certain nondegeneracy assumption whose genericity we establish), then there is always convergence to some equilibrium point. In particular, cycling cannot occur. The behavior of the mean fitness and some other aspects of the dynamics are also analyzed. Received: 15 November 1997 / Revised version: 25 May 1998     

Characterisation of the thiol–disulphide chemistry of desmopressin by LC, μ-LC, LC-ESI-MS and Maldi-Tof

Summary. To date, the majority of therapeutic peptides and proteins have to be administered via parenteral routes, which are painful and inconvenient. In order to gain sufficient high blood concentrations after oral application, various barriers in the gastrointestinal tract have to be overcome. Apart from a poor membrane uptake and intense enzymatic degradation, this study will demonstrate that thiol–disulphide reactions are an underestimated essential part of the presystemic metabolism. Glutathione, integrative part of the antioxidant defence system in the gastrointestinal tract, may play an important role in the inactivation of orally given peptides and proteins. In order to verify this hypothesis, desmopressin which bears a single disulphide bond was used as model peptide drug. Desmopressin was incubated with GSH in various concentrations, and the extent of thiol/disulphide exchange reactions between the peptide drug and GSH was investigated in dependence on pH and ratio of reactants determined as a function of time via HPLC, LC-MS and Maldi-Tof-MS analyses. Results showed that desmopressin is degraded by 1% reduced glutathione at pH 4 and pH 5.5. In presence of 0.01%, 0.1% and 1% of reduced glutathione 6.1%, 19.4% and 52.1% of desmopressin, respectively, were degraded. The masses of the conjugates after deconvolution measured by liquid chromatography and electrospray ionisation mass spectrometric detection were m/z 1069.67, m/z 1376.50, m/z 1683.40 and m/z 2138. These molecular masses, confirmed by Maldi-Tof-MS analysis, correspond with the masses of conjugates expected in theory. Under defined conditions, these results reveal that thiol–disulphide exchange reactions have a considerable impact on the alteration of peptide drugs and proteins.     

Optimal Recombination Rate in Fluctuating Environments

The optimal recombination rate which maximizes the long-term geometric average of the population fitness is studied for a two-locus haploid model, assuming that the fitnesses of genotypes AB, Ab, aB and ab are 1 + s(t), 1 - s(t), 1 - s(t), and 1 + s(t), respectively, where s(t) follows various stationary stochastic processes with the average zero. With positive recombination, the polymorphism is stably maintained at both loci. After an initial transient phase, the dynamics are reduced to one dimension, and are analyzed for weak selection limit, strong selection limit, and selection with two state Markovian jump. Results are: (1) If the environmental fluctuation has a predominant periodic component, ropt is approximately inversely proportional to the period irrespective of selection intensity. (2) If the fluctuation is a superposition of many periodic components, the one with the longest period is the most effective in determining ropt because the genetic dynamics cannot track very quick fluctuations (low pass filter effect). (3) If the power spectrum density is decreasing with the frequency, as in pink, or 1/f noises, ropt is small when selection is weak, and increases with the selection intensity. Numerical calculation of the genetic dynamics of a recombination modifier supports all these predictions for the evolutionarily stable recombination rate.     

The interaction of thin-film flow, bacterial swarming and cell differentiation in colonies of Serratia liquefaciens

 The rate of expansion of bacterial colonies of S. liquefaciens is investigated in terms of a mathematical model that combines biological as well as hydrodynamic processes. The relative importance of cell differentiation and production of an extracellular wetting agent to bacterial swarming is explored using a continuum representation. The model incorporates aspects of thin film flow with variable suspension viscosity, wetting, and cell differentiation. Experimental evidence suggests that the bacterial colony is highly sensitive to its environment and that a variety of mechanisms are exploited in order to proliferate on a variety of surfaces. It is found that a combination of effects are required to reproduce the variation of bacterial colony motility over a large range of nutrient availability and medium hardness. Received: 29 April 1999     

Gene conversion and the evolution of euryarchaeal chaperonins: a maximum likelihood-based method for detecting conflicting phylogenetic signals

Recombination is well known as a complicating factor in the interpretation of molecular phylogenies. Here we describe a maximum likelihood sliding window method based on a likelihood ratio test for scanning DNA sequence alignments for regions of incongruent phylogenetic signals, such as those influenced by recombination. Using this method, we identify several instances of gene conversion between paralogous chaperonin genes in euryarchaeote Archaea, many of which are not detected by two other widely used methods. In the Thermococcus/Pyrococcus lineage, where a gene duplication producing a and b paralogues predates the divergence of Thermococcus strains KS-1 and KS-8, gene conversion has homogenized portions of the a and b genes in KS-8 since the divergence of these two strains. A region near the 3′ end of the a and b paralogues in the methanogen Methanobacterium thermoautotrophicum also appears to have undergone gene conversion. We apply the method to two additional test data sets, the argF gene of Neisseria and a set of actin paralogues in maize, and show that it successfully identifies all the recombinant regions that were previously detected with other methods. Our approach is relatively insensitive to the presence of divergent sequences in the alignment, making it ideal for detecting recombination between both closely and distantly related genes.     

Perturbation analysis of a two-locus model with directional selection and recombination

A population genetic two-locus model with additive, directional selection and recombination is considered. It is assumed that recombination is weaker than selection; i.e., the recombination parameter r is smaller than the selection coefficients. This assumption is appropriate for describing the effects of two-locus selection at the molecular level. The model is formulated in terms of ordinary differential equations (ODES) for the gamete frequencies x = (x ₁, x ₂, x ₃, x ₄), defined on the simplex S ₄. The ODEs are analyzed using first a regular pertubation technique. However, this approach yields satisfactory results only if r is very small relative to the selection coefficients and if the initial values x(0) are in the interior part of S ₄. To cope with this problem, a novel two-scale perturbation method is proposed which rests on the theory of averaging of vectorfields. It is demonstrated that the zeroth-order solution of this two-scale approach approximates the numerical solution of the model well, even if recombination rate is on the order of the selection coefficients.