Dynamics of genetic variability in two-locus models of stabilizing selection.

Genetics ◽  
1994 ◽  
Vol 138 (2) ◽  
pp. 519-532
Author(s):  
S Gavrilets ◽  
A Hastings
Keyword(s):  
Linkage Disequilibrium ◽  
Initial Conditions ◽  
Fitness Function ◽  
Stabilizing Selection ◽  
Phenotypic Evolution ◽  
Phenotypic Characteristics ◽  
Locus Model ◽  
Straight Line ◽  
The Difference ◽  
Slow Evolution

Abstract We study a two locus model, with additive contributions to the phenotype, to explore the dynamics of different phenotypic characteristics under stabilizing selection and recombination. We demonstrate that the interaction of selection and recombination results in constraints on the mode of phenotypic evolution. Let Vg be the genic variance of the trait and CL be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that, independent of the initial conditions, the dynamics of the system on the plane (Vg, CL) are typically characterized by a quick approach to a straight line with slow evolution along this line afterward. We analyze how the mode and the rate of phenotypic evolution depend on the strength of selection relative to recombination, on the form of fitness function, and the difference in allelic effect. We argue that if selection is not extremely weak relative to recombination, linkage disequilibrium generated by stabilizing selection influences the dynamics significantly. We demonstrate that under these conditions, which are plausible in nature and certainly the case in artificial stabilizing selection experiments, the model can have a polymorphic equilibrium with positive linkage disequilibrium that is stable simultaneously with monomorphic equilibria.

scholarly journals Maintenance of genetic variability under strong stabilizing selection: a two-locus model.

Genetics ◽  
1993 ◽  
Vol 134 (1) ◽  
pp. 377-386 ◽  
Author(s):  
S Gavrilets ◽  
A Hastings
Keyword(s):  
Genetic Variability ◽  
Fitness Function ◽  
Additive Genetic Variance ◽  
Mean Value ◽  
Stabilizing Selection ◽  
Locus Model ◽  
Large Level ◽  
The Difference ◽  
Different Characteristics ◽  
The Relationship

Abstract We study a two locus model with additive contributions to the phenotype to explore the relationship between stabilizing selection and recombination. We show that if the double heterozygote has the optimum phenotype and the contributions of the loci to the trait are different, then any symmetric stabilizing selection fitness function can maintain genetic variability provided selection is sufficiently strong relative to linkage. We present results of a detailed analysis of the quadratic fitness function which show that selection need not be extremely strong relative to recombination for the polymorphic equilibria to be stable. At these polymorphic equilibria the mean value of the trait, in general, is not equal to the optimum phenotype, there exists a large level of negative linkage disequilibrium which "hides" additive genetic variance, and different equilibria can be stable simultaneously. We analyze dependence of different characteristics of these equilibria on the location of optimum phenotype, on the difference in allelic effect, and on the strength of selection relative to recombination. Our overall result that stabilizing selection does not necessarily eliminate genetic variability is compatible with some experimental results where the lines subject to strong stabilizing selection did not have significant reductions in genetic variability.

scholarly journals Dynamics of polygenic variability under stabilizing selection, recombination, and drift

1995 ◽  
Vol 65 (1) ◽  
pp. 63-74 ◽  
Author(s):  
Sergey Gavrilets ◽  
Alan Hasting
Keyword(s):  
Linkage Disequilibrium ◽  
Genetic Variability ◽  
Transient Dynamics ◽  
Stabilizing Selection ◽  
Genotypic Variance ◽  
Random Drift ◽  
Rates Of Change ◽  
Straight Line ◽  
Selection Experiments ◽  
Slow Evolution

SummaryWe study the transient dynamics of the genotypic variance of an additive trait under stabilizing selection, recombination and random drift. We show how interaction of these factors determines the form and the rates of change of different components of the genotypic variance. Let Vg be the genie variance of the trait and CL be the contribution of linkage disequilibrium to the genotypic variance. We demonstrate that the dynamics of the system on the plane (Vg, CL) are typically characterized by a quick approach to a straight line with slow evolution along this line afterwards. We show that the number of loci, n, and the population size, N, affect the expected dynamics of Vg mainly through the ratio N/n. We use our analytical and numerical results in interpreting the published results of artificial stabilizing selection experiments. The analysis suggests that it is drift and not selection that most likely led to the reduction of genetic variability in most of these experiments. Even very strong stabilizing selection only slowly removes polygenic variability from populations.

scholarly journals Quantitative genetic models for the balance between migration and stabilizing selection

2000 ◽  
Vol 76 (3) ◽  
pp. 285-293 ◽  
Author(s):  
JARLE TUFTO
Keyword(s):  
Linkage Disequilibrium ◽  
Genetic Basis ◽  
Genetic Variance ◽  
Approximate Model ◽  
Allele Frequencies ◽  
Stabilizing Selection ◽  
Local Optimum ◽  
Symmetric Model ◽  
The Difference ◽  
And Migration

The evolution of a quantitative trait subject to stabilizing selection and immigration, with the immigrants deviating from the local optimum, is considered under a number of different models of the underlying genetic basis of the trait. By comparing exact predictions under the infinitesimal model obtained using numerical methods with predictions of a simplified approximate model based on ignoring linkage disequilibrium, the increase in the expressed genetic variance as a result of linkage disequilibrium generated by migration is shown to be relatively small and negligible, provided that the genetic variance relative to the squared deviation of immigrants from the local optimum is sufficiently large or selection and migration is sufficiently weak. Deviation from normality is shown to be less important by comparing predictions of the infinitesimal model with a model presupposing normality. For a more realistic symmetric model, involving a finite number of loci only, no linkage and equal effects and frequencies across loci, additional changes in the genetic variance arise as a result of changes in underlying allele frequencies. Again, provided that the genetic variance relative to the squared deviation of the immigrants from the local optimum is small, the difference between the predictions of infinitesimal and the symmetric model are small unless the number of loci is very small. However, if the genetic variance relative to the squared deviation of the immigrants from the local optimum is large, or if selection and migration are strong, both linkage disequilibrium and changes in the genetic variance as a result of changes in underlying allele frequencies become important.

Automatic Tread Gaging Machine

1979 ◽  
Vol 7 (1) ◽  
pp. 31-39
Author(s):  
G. S. Ludwig ◽  
F. C. Brenner
Keyword(s):  
Experimental Tests ◽  
Automatic Machine ◽  
Wear Rates ◽  
Handling Device ◽  
Straight Line ◽  
Component Systems ◽  
Before And After ◽  
Good Repeatability ◽  
The Difference ◽  
Straight Lines

Abstract An automatic tread gaging machine has been developed. It consists of three component systems: (1) a laser gaging head, (2) a tire handling device, and (3) a computer that controls the movement of the tire handling machine, processes the data, and computes the least-squares straight line from which a wear rate may be estimated. Experimental tests show that the machine has good repeatability. In comparisons with measurements obtained by a hand gage, the automatic machine gives smaller average groove depths. The difference before and after a period of wear for both methods of measurement are the same. Wear rates estimated from the slopes of straight lines fitted to both sets of data are not significantly different.

scholarly journals Pleiotropy and multilocus polymorphisms.

Genetics ◽  
1992 ◽  
Vol 130 (1) ◽  
pp. 223-227
Author(s):  
A Gimelfarb
Keyword(s):  
Genetic Variation ◽  
Linkage Disequilibrium ◽  
Stabilizing Selection ◽  
Additive Genetic Variation ◽  
Very High

Abstract It is demonstrated that systems of two pleiotropically related characters controlled by additive diallelic loci can maintain under Gaussian stabilizing selection a stable polymorphism in more than two loci. It is also shown that such systems may have multiple stable polymorphic equilibria. Stabilizing selection generates negative linkage disequilibrium, as a result of which the equilibrium phenotypic variances are quite low, even though the level of allelic polymorphisms can be very high. Consequently, large amounts of additive genetic variation can be hidden in populations at equilibrium under stabilizing selection on pleiotropically related characters.

scholarly journals XXIII.—On the Disruptive Discharge of Electricity: An Experimental Thesis for the Degree of Doctor of Science, Department A

1878 ◽  
Vol 28 (2) ◽  
pp. 633-671 ◽  
Author(s):  
Alexander Macfarlane
Keyword(s):  
Copper Wire ◽  
Late Summer ◽  
The Other ◽  
Straight Line ◽  
Summer Session ◽  
The Difference ◽  
A Chain ◽  
The One ◽  
Metallic Parts ◽  
The University

The experiments to which I shall refer were carried out in the physical laboratory of the University during the late summer session. I was ably assisted in conducting the experiments by three students of the laboratory,—Messrs H. A. Salvesen, G. M. Connor, and D. E. Stewart. The method which was used of measuring the difference of potential required to produce a disruptive discharge of electricity under given conditions, is that described in a paper communicated to the Royal Society of Edinburgh in 1876 in the names of Mr J. A. Paton, M. A., and myself, and was suggested to me by Professor Tait as a means of attacking the experimental problems mentioned below.The above sketch which I took of the apparatus in situ may facilitate tha description of the method. The receiver of an air-pump, having a rod capable of being moved air-tight up and down through the neck, was attached to one of the conductors of a Holtz machine in such a manner that the conductor of the machine and the rod formed one conducting system. Projecting from the bottom of the receiver was a short metallic rod, forming one conductor with the metallic parts of the air-pump, and by means of a chain with the uninsulated conductor of the Holtz machine. Brass balls and discs of various sizes were made to order, capable of being screwed on to the ends of the rods. On the table, and at a distance of about six feet from the receiver, was a stand supporting two insulated brass balls, the one fixed, the other having one degree of freedom, viz., of moving in a straight line in the plane of the table. The fixed insulated ball A was made one conductor with the insulated conductor of the Holtz and the rod of the receiver, by means of a copper wire insulated with gutta percha, having one end stuck firmly into a hole in the collar of the receiver, and having the other fitted in between the glass stem and the hollow in the ball, by which it fitted on to the stem tightly. A thin wire similarly fitted in between the ball B and its insulating stem connected the ball with the insulated half ring of a divided ring reflecting electrometer.

scholarly journals Initial-Condition Sensitivities and the Predictability of Downslope Winds

2009 ◽  
Vol 66 (11) ◽  
pp. 3401-3418 ◽  
Author(s):  
Patrick A. Reinecke ◽  
Dale R. Durran
Keyword(s):  
Wave Breaking ◽  
Initial Conditions ◽  
Static Stability ◽  
Synoptic Scale ◽  
Wind Speeds ◽  
Downslope Winds ◽  
First Case ◽  
Downslope Wind ◽  
The Difference ◽  
Strong Winds

Abstract The sensitivity of downslope wind forecasts to small changes in initial conditions is explored by using 70-member ensemble simulations of two prototypical windstorms observed during the Terrain-Induced Rotor Experiment (T-REX). The 10 weakest and 10 strongest ensemble members are composited and compared for each event. In the first case, the 6-h ensemble-mean forecast shows a large-amplitude breaking mountain wave and severe downslope winds. Nevertheless, the forecasts are very sensitive to the initial conditions because the difference in the downslope wind speeds predicted by the strong- and weak-member composites grows to larger than 28 m s−1 over the 6-h forecast. The structure of the synoptic-scale flow one hour prior to the windstorm and during the windstorm is very similar in both the weak- and strong-member composites. Wave breaking is not a significant factor in the second case, in which the strong winds are generated by a layer of high static stability flowing beneath a layer of weaker mid- and upper-tropospheric stability. In this case, the sensitivity to initial conditions is weaker but still significant. The difference in downslope wind speeds between the weak- and strong-member composites grows to 22 m s−1 over 12 h. During and one hour before the windstorm, the synoptic-scale flow exhibits appreciable differences between the strong- and weak-member composites. Although this case appears to be more predictable than the wave-breaking event, neither case suggests that much confidence should be placed in the intensity of downslope winds forecast 12 or more hours in advance.

scholarly journals Chaos and the (un)predictability of evolution in a changing environment

2017 ◽  
Author(s):  
Artur Rego-Costa ◽  
Florence Débarre ◽  
Luis-Miguel Chevin
Keyword(s):  
Evolutionary Dynamics ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Environmental Stochasticity ◽  
Phenotypic Evolution ◽  
Evolutionary System ◽  
Chaotic Oscillations ◽  
Changing Environment ◽  
Environmental Forcing ◽  
Evolutionary Trajectories

Among the factors that may reduce the predictability of evolution, chaos, characterized by a strong dependence on initial conditions, has received much less attention than randomness due to genetic drift or environmental stochasticity. It was recently shown that chaos in phenotypic evolution arises commonly under frequency-dependent selection caused by competitive interactions mediated by many traits. This result has been used to argue that chaos should often make evolutionary dynamics unpredictable. However, populations also evolve largely in response to external changing environments, and such environmental forcing is likely to influence the outcome of evolution in systems prone to chaos. We investigate how a changing environment causing oscillations of an optimal phenotype interacts with the internal dynamics of an eco-evolutionary system that would be chaotic in a constant environment. We show that strong environmental forcing can improve the predictability of evolution, by reducing the probability of chaos arising, and by dampening the magnitude of chaotic oscillations. In contrast, weak forcing can increase the probability of chaos, but it also causes evolutionary trajectories to track the environment more closely. Overall, our results indicate that, although chaos may occur in evolution, it does not necessarily undermine its predictability.

scholarly journals How development affects evolution

2021 ◽  
Author(s):  
Mauricio González-Forero ◽  
Andy Gardner
Keyword(s):  
Niche Construction ◽  
Fitness Landscape ◽  
Stabilizing Selection ◽  
Mathematical Framework ◽  
Phenotypic Evolution ◽  
Landscape Development ◽  
Developmental Constraints ◽  
Evolutionary Pathway ◽  
Punctuated Equilibria ◽  
Adaptive Radiations

How development affects evolution. A mathematical framework that explicitly integrates development into evolution has recently been derived. Here we use this framework to analyse how development affects evolution. We show that, whilst selection pushes genetic and phenotypic evolution uphill on the fitness landscape, development determines the admissible evolutionary pathway, such that evolutionary outcomes occur at path peaks, which need not be peaks of the fitness landscape. Development can generate path peaks, triggering adaptive radiations, even on constant, single-peak landscapes. Phenotypic plasticity, niche construction, extra-genetic inheritance, and developmental bias variously alter the evolutionary path and hence the outcome. Selective development, whereby phenotype construction may point in the adaptive direction, may induce evolution either towards or away landscape peaks depending on the developmental constraints. Additionally, developmental propagation of phenotypic effects over age allows for the evolution of negative senescence. These results help explain empirical observations including punctuated equilibria, the paradox of stasis, the rarity of stabilizing selection, and negative senescence, and show that development has a major role in evolution.

scholarly journals Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

2019 ◽  
pp. 1-12
Author(s):  
Abdualrazaq Sanbo ◽  
Elsayed M. Elsayed ◽  
Faris Alzahrani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Analytical Study ◽  
Initial Conditions ◽  
Specific Form ◽  
Positive Real ◽  
Rational Difference Equations ◽  
Special Cases ◽  
The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

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