Evolution under Fertility and Viability Selection

ABSTRACT Evolution at a single multiallelic locus under arbitrary weak selection on both fertility and viability is investigated. Discrete, nonoverlapping generations are posited for autosomal and X-linked loci in dioecious populations, but monoecious populations are studied in both discrete and continuous time. Mating is random. The results hold after several generations have elapsed. With an error of order s [i.e., O(s)], where s represents the selection intensity, the population evolves in Hardy-Weinberg proportions. Provided the change per generation of the fertilities and viabilities due to their explicit time dependence (if any) is O(s 2), the rate of change of the deviation from Hardy-Weinberg proportions is O(s 2). If the change per generation of the viabilities and genotypic fertilities is smaller than second order [i.e., o(s 2)], then to O(s 2) the rate of change of the mean fitness is equal to the genic variance. The mean fitness is the product of the mean fertility and the mean viability; in dioecious populations, the latter is the unweighted geometric mean of the mean viabilities of the two sexes. Hence, as long as there is significant gene frequency change, the mean fitness increases. If it is the fertilities of matings that change slowly [at rate o(s 2)], the above conclusions apply to a modified mean fitness, defined as the product of the mean viability and the square root of the mean fertility.

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The evolution of multilocus systems under weak selection.

Genetics ◽

10.1093/genetics/134.2.627 ◽

1993 ◽

Vol 134 (2) ◽

pp. 627-647 ◽

Cited By ~ 12

Author(s):

T Nagylaki

Keyword(s):

Additive Genetic Variance ◽

Absolute Error ◽

Rate Of Change ◽

Weak Selection ◽

Single Generation ◽

Linkage Disequilibria ◽

Explicit Time Dependence ◽

Change In The Mean ◽

The Mean ◽

Short Time

Abstract The evolution of multilocus systems under weak selection is investigated. Generations are discrete and nonoverlapping; the monoecious population mates at random. The number of multi-allelic loci, the linkage map, dominance, and epistasis are arbitrary. The genotypic fitnesses may depend on the gametic frequencies and time. The results hold for s < cmin, where s and cmin denote the selection intensity and the smallest two-locus recombination frequency, respectively. After an evolutionarily short time of t1 approximately (ln s)/ln(1 - cmin) generations, all the multilocus linkage disequilibria are of the order of s [i.e., O(s) as s-->0], and then the population evolves approximately as if it were in linkage equilibrium, the error in the gametic frequencies being O(s). Suppose the explicit time dependence (if any) of the genotypic fitnesses is O(s2). Then after a time t2 approximately 2t1, the linkage disequilibria are nearly constant, their rate of change being O(s2). Furthermore, with an error of O(s2), each linkage disequilibrium is proportional to the corresponding epistatic deviation for the interaction of additive effects on fitness. If the genotypic fitnesses change no faster than at the rate O(s3), then the single-generation change in the mean fitness is delta W = W-1Vg+O(s3), where Vg designates the genic (or additive genetic) variance in fitness. The mean of a character with genotypic values whose single-generation change does not exceed O(s2) evolves at the rate delta Z = W-1Cg+O(s2), where Cg represents the genic covariance of the character and fitness (i.e., the covariance of the average effect on the character and the average excess for fitness of every allele that affects the character). Thus, after a short time t2, the absolute error in the fundamental and secondary theorems of natural selection is small, though the relative error may be large.

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SIMULATION OF X-LINKED SELECTION IN DROSOPHILA

Genetics ◽

10.1093/genetics/83.3.551 ◽

1976 ◽

Vol 83 (3) ◽

pp. 551-571

Author(s):

Philip W Hedrick

Keyword(s):

Gene Frequency ◽

Frequency Change ◽

Wild Type ◽

Weak Selection ◽

Gene Frequencies ◽

Linked Gene ◽

Males And Females ◽

Genetic Simulation ◽

Linked Selection ◽

Best Fit

ABSTRACT The change in gene frequency for two X-linked mutants, y and w, in a number of experiments was compared to that predicted from a genetic simulation program which utilized estimated differences in relative mating ability, fecundity, and viability. The simulation gave excellent predictions of gene frequency change even when experiments were started with different initial gene frequencies in the males and females or when the two loci were segregating simultaneously. The rate of elimination was slower when there were unequal initial gene frequencies than when males and females had equal initial gene frequencies. Simulation demonstrated that this was a general phenomenon when there is strong selection but that the opposite is true for weak selection. In two other experiments, the mating advantage of wild-type males was balanced by a fecundity advantage in mutant females. In all four replicates of both experiments, the mutant was maintained for several generations at the high initial frequency but then decreased quickly and was eliminated. Results obtained restarting one of these experiments with flies from a generation after the decline in gene frequency indicated that a linked gene and not frequency-dependent selection was responsible for the unpredictable gene-frequency change in the mutant. Using a least squares technique, it was found that a recessive fecundity locus 15 map units from the w locus gave the best fit for bothexperiments.

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THE EVOLUTION OF ONE- AND TWO-LOCUS SYSTEMS

Genetics ◽

10.1093/genetics/83.3.583 ◽

1976 ◽

Vol 83 (3) ◽

pp. 583-600

Author(s):

Thomas Nagylaki

Keyword(s):

Continuous Time ◽

Time Derivative ◽

Second Order ◽

Rate Of Change ◽

Evolutionary Significance ◽

Multiple Alleles ◽

Mean Fitness ◽

The Mean ◽

Change In Mean ◽

Continuous Time Models

ABSTRACT Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O(s 2)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s 2), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s 2). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s 2)), then to O(s 2) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O(s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.

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DMSA-scintigraphy in paediatrics: Is the evaluation of the geometric mean necessary for the calculation of the differential renal function?

Nuklearmedizin ◽

10.1055/s-0038-1625921 ◽

2001 ◽

Vol 40 (04) ◽

pp. 107-110 ◽

Cited By ~ 1

Author(s):

B. Roßmüller ◽

S. Alalp ◽

S. Fischer ◽

S. Dresel ◽

K. Hahn ◽

...

Keyword(s):

Renal Function ◽

Geometric Mean ◽

Region Of Interest ◽

Posterior View ◽

Renal Abnormality ◽

Anterior View ◽

Differential Renal Function ◽

The Mean ◽

The Right ◽

To Receive

SummaryFor assessment of differential renal function (PF) by means of static renal scintigraphy with Tc-99m-dimer-captosuccinic acid (DMSA) the calculation of the geometric mean of counts from the anterior and posterior view is recommended. Aim of this retrospective study was to find out, if the anterior view is necessary to receive an accurate differential renal function by calculating the geometric mean compared to calculating PF using the counts of the posterior view only. Methods: 164 DMSA-scans of 151 children (86 f, 65 m) aged 16 d to 16 a (4.7 ± 3.9 a) were reviewed. The scans were performed using a dual head gamma camera (Picker Prism 2000 XP, low energy ultra high resolution collimator, matrix 256 x 256,300 kcts/view, Zoom: 1.6-2.0). Background corrected values from both kidneys anterior and posterior were obtained. Using region of interest technique PF was calculated using the counts of the dorsal view and compared with the calculated geometric mean [SQR(Ctsdors x Ctsventr]. Results: The differential function of the right kidney was significantly less when compared to the calculation of the geometric mean (p<0.01). The mean difference between the PFgeom and the PFdors was 1.5 ± 1.4%. A difference > 5% (5.0-9.5%) was obtained in only 6/164 scans (3.7%). Three of 6 patients presented with an underestimated PFdors due to dystopic kidneys on the left side in 2 patients and on the right side in one patient. The other 3 patients with a difference >5% did not show any renal abnormality. Conclusion: The calculation of the PF from the posterior view only will give an underestimated value of the right kidney compared to the calculation of the geometric mean. This effect is not relevant for the calculation of the differntial renal function in orthotopic kidneys, so that in these cases the anterior view is not necesssary. However, geometric mean calculation to obtain reliable values for differential renal function should be applied in cases with an obvious anatomical abnormality.

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A MARKOV PROCESS OF GENE FREQUENCY CHANGE IN A GEOGRAPHICALLY STRUCTURED POPULATION

Genetics ◽

10.1093/genetics/76.2.367 ◽

1974 ◽

Vol 76 (2) ◽

pp. 367-377

Author(s):

Takeo Maruyama

Keyword(s):

Genetic Variation ◽

Markov Process ◽

Diffusion Process ◽

Gene Frequency ◽

Transition Probabilities ◽

Large Population ◽

Sample Path ◽

Time Parameter ◽

Frequency Change ◽

Structured Population

ABSTRACT A Markov process (chain) of gene frequency change is derived for a geographically-structured model of a population. The population consists of colonies which are connected by migration. Selection operates in each colony independently. It is shown that there exists a stochastic clock that transforms the originally complicated process of gene frequency change to a random walk which is independent of the geographical structure of the population. The time parameter is a local random time that is dependent on the sample path. In fact, if the alleles are selectively neutral, the time parameter is exactly equal to the sum of the average local genetic variation appearing in the population, and otherwise they are approximately equal. The Kolmogorov forward and backward equations of the process are obtained. As a limit of large population size, a diffusion process is derived. The transition probabilities of the Markov chain and of the diffusion process are obtained explicitly. Certain quantities of biological interest are shown to be independent of the population structure. The quantities are the fixation probability of a mutant, the sum of the average local genetic variation and the variation summed over the generations in which the gene frequency in the whole population assumes a specified value.

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GENETIC LOAD IN NATURAL POPULATIONS: IS IT COMPATIBLE WITH THE HYPOTHESIS THAT MANY POLYMORPHISMS ARE MAINTAINED BY NATURAL SELECTION?

Genetics ◽

10.1093/genetics/77.3.569 ◽

1974 ◽

Vol 77 (3) ◽

pp. 569-589

Author(s):

Martin L Tracey ◽

Francisco J Ayala

Keyword(s):

Drosophila Melanogaster ◽

Natural Selection ◽

Natural Population ◽

Natural Populations ◽

Genetic Load ◽

Structural Genes ◽

Invertebrate Species ◽

Average Proportion ◽

Mean Fitness ◽

The Mean

ABSTRACT Recent studies of genetically controlled enzyme variation lead to an estimation that at least 30 to 60% of the structural genes are polymorphic in natural populations of many vertebrate and invertebrate species. Some authors have argued that a substantial proportion of these polymorphisms cannot be maintained by natural selection because this would result in an unbearable genetic load. If many polymorphisms are maintained by heterotic natural selection, individuals with much greater than average proportion of homozygous loci should have very low fitness. We have measured in Drosophila melanogaster the fitness of flies homozygous for a complete chromosome relative to normal wild flies. A total of 37 chromosomes from a natural population have been tested using 92 experimental populations. The mean fitness of homozygous flies is 0.12 for second chromosomes, and 0.13 for third chromosomes. These estimates are compatible with the hypothesis that many (more than one thousand) loci are maintained by heterotic selection in natural populations of D. melanogaster.

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Accumulation of mutations in sexual and asexual populations

Genetics Research ◽

10.1017/s0016672300026938 ◽

1987 ◽

Vol 49 (2) ◽

pp. 135-146 ◽

Cited By ~ 98

Author(s):

Pekka Pamilo ◽

Masatoshi Nei ◽

Wen-Hsiung Li

Keyword(s):

Weak Selection ◽

Population Variance ◽

Asexual Population ◽

Term Evolution ◽

A Genome ◽

Analytical Formulae ◽

The Mean ◽

Asexual Populations ◽

Mutant Genes

SummaryThe accumulation of beneficial and harmful mutations in a genome is studied by using analytical methods as well as computer simulation for different modes of reproduction. The modes of reproduction examined are biparental (bisexual, hermaphroditic), uniparental (selfing, automictic, asexual) and mixed (partial selfing, mixture of hermaphroditism and parthenogenesis). It is shown that the rates of accumulation of both beneficial and harmful mutations with weak selection depend on the within-population variance of the number of mutant genes per genome. Analytical formulae for this variance are derived for neutral mutant genes for hermaphroditic, selfing and asexual populations; the neutral variance is largest in a selfing population and smallest in an asexual population. Directional selection reduces the population variance in most cases, whereas recombination partially restores the reduced variance. Therefore, biparental organisms accumulate beneficial mutations at the highest rate and harmful mutations at the lowest rate. Selfing organisms are intermediate between biparental and asexual organisms. Even a limited amount of outcrossing in largely selfing and parthenogenetic organisms markedly affects the accumulation rates. The accumulation of mutations is likely to affect the mean population fitness only in long-term evolution.

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The volume coefficients of expansion of several gases at pressures below one metre

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1934.0016 ◽

1934 ◽

Vol 143 (850) ◽

pp. 487-505 ◽

Cited By ~ 2

Keyword(s):

Carbon Dioxide ◽

Linear Function ◽

Rate Of Change ◽

Linear Extrapolation ◽

Satisfactory Explanation ◽

Volume Coefficient ◽

Temperature Ranges ◽

The Mean ◽

Permanent Gases ◽

Limiting Value

There have not appeared recently any new determinations of the rate of change of the volume coefficient of expansion of condensable gases at pressures in the neighbourhood of a half to one metre. The work of Henning and Heuse and Heuse and Otto has been confined to a study of the permanent gases, their results leading to the conclusion that up to a pressure of 1 metre the rate of change of either the pressure or volume coefficient is a linear function of the pressure. Our knowledge of the behaviour of the condensable gases in this connection rests almost entirely on the very careful work of Chappuis, who in 1907 made a series of accurate determinations of the volume coefficient of expansion of carbon dioxide at a series of pressures from 1500 mm. to 500 mm. and over several temperature ranges. The investigation led to one unexpected conclusion which Chappuis left largely unexplained. On linear extrapolation to zero pressure of the graph of pressure against the mean coefficient of expansion over temperature intervals 0-20º, 0-40º, 0-100ºC., the limiting value of the coefficient rose from the normal value of 0.003661 for the 0-20º determinations to 0.003671 for those made over the range 0-100ºC. Chappuis concludes "that condensation on the reservoir surface plays a part in the irregularities but it is difficult to obtain a satisfactory explanation." As far back as 1853 Magnus demonstrated that the adsorption of sukphur dioxide on glass was sufficient to affect measurements of the expansion coefficient of gasses, and the importance of this error was recognized by Chappuis who in 1879 applied a correction to Regnault's measurements. Richards and Mark and Baly and Ramsay have pointed out the necessity for a knowledge of the amount of adsorption on the walls of the containing vessels when undertaking such determinations.

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Cadmium concentrations in kidneys of sheep and cattle in Western Australia. 1. Regional distribution

Australian Journal of Agricultural Research ◽

10.1071/ar9940851 ◽

1994 ◽

Vol 45 (4) ◽

pp. 851 ◽

Cited By ~ 11

Author(s):

PW Morcombe ◽

DS Petterson ◽

HG Masters ◽

PJ Ross ◽

JR Edwards

Keyword(s):

Western Australia ◽

Geometric Mean ◽

Regional Distribution ◽

Adult Cattle ◽

Agricultural Region ◽

Wet Weight ◽

The Mean ◽

Winter Annual ◽

Rate Of Accumulation ◽

Cattle And Sheep

A sample of 4973 kidneys from sheep stratified by age and shire of origin within the Agricultural Region of Western Australia, was analysed for cadmium (Cd) content during the period August 1989 to April 1991. The geometric mean Cd concentration in the kidney of hogget ewes was 0.9 mg/kg, in 4-tooth ewes 1.47 mg/kg and in adult ewes 3.34 mg/kg on a wet weight basis. The mean Cd concentrations of either ewe or hogget flocks from different Divisions of the Agricultural Region did not differ from each other. The annual increase in Cd concentration of kidney from hogget sheep was estimated to be 0-65 mg/kg. The rate of accumulation of Cd in kidney from cattle and sheep grazing the same properties was similar. Kidneys from a sample of 354 adult cattle from the Kimberley Region and 483 aged sheep from the Pastoral Region, both areas of unimproved rangelands, had geometric mean Cd concentrations of 0.15 mg/kg and 0-31 mg/kg respectively. A higher Cd concentration in flocks from the divisions adjacent to the Agricultural Region may have resulted from the establishment of some volunteer species of winter annual pastures in the rangeland.

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A comparison of two methods for predicting changes in the distribution of gene frequency when selection is applied repeatedly to a finite population

Genetics Research ◽

10.1017/s0016672300002834 ◽

1969 ◽

Vol 13 (2) ◽

pp. 117-126 ◽

Cited By ~ 3

Author(s):

Derek J. Pike

Keyword(s):

Iterative Procedure ◽

Gene Frequency ◽

Finite Population ◽

Transition Matrices ◽

Single Cycle ◽

Gene Frequencies ◽

Mean And Variance ◽

The Mean ◽

The One ◽

Selection Of

Robertson (1960) used probability transition matrices to estimate changes in gene frequency when sampling and selection are applied to a finite population. Curnow & Baker (1968) used Kojima's (1961) approximate formulae for the mean and variance of the change in gene frequency from a single cycle of selection applied to a finite population to develop an iterative procedure for studying the effects of repeated cycles of selection and regeneration. To do this they assumed a beta distribution for the unfixed gene frequencies at each generation.These two methods are discussed and a result used in Kojima's paper is proved. A number of sets of calculations are carried out using both methods and the results are compared to assess the accuracy of Curnow & Baker's method in relation to Robertson's approach.It is found that the one real fault in the Curnow-Baker method is its tendency to fix too high a proportion of the genes, particularly when the initial gene frequency is near to a fixation point. This fault is largely overcome when more individuals are selected. For selection of eight or more individuals the Curnow-Baker method is very accurate and appreciably faster than the transition matrix method.

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