scholarly journals THE EVOLUTION OF ONE- AND TWO-LOCUS SYSTEMS

Genetics ◽  
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.

scholarly journals An Exactly Solved Model for Mutation, Recombination and Selection

2003 ◽  
Vol 55 (1) ◽  
pp. 3-41 ◽  
Author(s):  
Michael Baake ◽  
Ellen Baake
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Sufficient Criterion ◽  
Linkage Disequilibria ◽  
Additive Type ◽  
Mean Fitness ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Independent Mutation ◽  
Selection Of

AbstractIt is well known that rather generalmutation-recombination models can be solved algorithmically (though not in closed form) by means of Haldane linearization. The price to be paid is that one has to work with a multiple tensor product of the state space one started from.Here, we present a relevant subclass of such models, in continuous time, with independent mutation events at the sites, and crossover events between them. It admits a closed solution of the corresponding differential equation on the basis of the original state space, and also closed expressions for the linkage disequilibria, derived by means of Möbius inversion. As an extra benefit, the approach can be extended to a model with selection of additive type across sites. We also derive a necessary and sufficient criterion for the mean fitness to be a Lyapunov function and determine the asymptotic behaviour of the solutions.

scholarly journals THE EVOLUTION OF ONE- AND TWO-LOCUS SYSTEMS. II

Genetics ◽  
1977 ◽  
Vol 85 (2) ◽  
pp. 347-354
Author(s):  
Thomas Nagylaki
Keyword(s):  
Continuous Time ◽  
Initial Period ◽  
Natural Populations ◽  
Weak Selection ◽  
Total Change ◽  
Linkage Disequilibria ◽  
Mean Fitness ◽  
Monoecious Population ◽  
Change In Mean ◽  
Age Independent

ABSTRACT Weak selection in a monoecious population is studied in two multiallelic panmictic models. In the first, a single locus is considered with continuous time and age-independent fertilities and mortalities. If the fertilities of the various matings and the genotypic mortalities may be expressed with an error at most of the second order in s (i.e., O(s  2)), where s is the intensity of selection, as sums of terms corresponding to the different genotypes and alleles, respectively, then after several generations the deviations from Hardy-Weinberg proportions are of O(s  2). In the second model, two loci are treated with discrete nonoverlapping generations. It is shown that if the epistatic parameters are of O(s2), then after several generations the linkage disequilibria are reduced to O(s  2). Assuming only weak selection, it is proved that in both models, after several generations, the total change in mean fitness is generally positive. It is likely that the exclusion of the initial period is usually unnecessary in natural populations. Exceptions are discussed.

scholarly journals The transport of vorticity and heat through fluids in turbulent motion

1932 ◽  
Vol 135 (828) ◽  
pp. 685-702 ◽  
Keyword(s):  
Horizontal Plane ◽  
Vertical Axis ◽  
Rate Of Change ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Values ◽  
Plane Element ◽  
The Mean ◽  
Coefficient Of Viscosity ◽  
Change In Mean

In Reynolds’ well-known theory of turbulent flow the effect of turbulence on the mean flow of a fluid is conceived as the same as that of a system of stresses which, like those due to viscosity, may have tangential as well as normal components across any plane element. Taking the case of laminar mean flow, that is when the mean flow is, say, horizontal and constant in direction and magnitude at any given height, the components of stress over a horizontal plane at height z are F x and F y where F x = — ρ uw — , F y = — ρ vw — , and u , v , w are the components of turbulent velocity parallel to two horizontal axes x and y and the vertical axis z . The bar denotes that mean values have been taken over a large horizontal area and ρ is the density of the fluid. The stress F x , is therefore due to the existence of a correlation between u and w. In the extension of Reynolds’ theory due to Prandtl this correlation depends on the rate of change in mean velocity. In its most simplified form the theory may be expressed as follows. A portion of fluid possessing the mean velocity of a level z 0 may be conceived to move upwards to a layer of height z 0 + l preserving the mean velocity U 0 of the layer from which it originated. At this height it is conceived to mix with its surroundings. If l is small the mean velocity of this layer is U 0 + l d U/ dz , U being the mean velocity at height z , so that u = — l d U/ dz , and hence F x = ρ wl — d U/ dz . The quantity ρ wl — is therefore of the same dimensions as viscosity and in Prandtl’s theory it is treated as though it were in fact a coefficient of viscosity, though not necessarily as one which has the same value at all points in the field.

scholarly journals Evolution under Fertility and Viability Selection

Genetics ◽  
1987 ◽  
Vol 115 (2) ◽  
pp. 367-375
Author(s):  
Thomas Nagylaki
Keyword(s):  
Gene Frequency ◽  
Geometric Mean ◽  
Rate Of Change ◽  
Frequency Change ◽  
Weak Selection ◽  
Significant Gene ◽  
Explicit Time Dependence ◽  
Mean Fitness ◽  
The Mean ◽  
Dioecious Populations

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.

Rates of Change in Peak Expiratory Flow and in Diurnal Variation in Peak Flow in Patients Recovering from Acute Severe Asthma

1994 ◽  
Vol 86 (1) ◽  
pp. 59-65 ◽  
Author(s):  
G. E. Packe ◽  
W. Freeman ◽  
Ruth M. Cayton
Keyword(s):  
Diurnal Variation ◽  
Severe Asthma ◽  
Peak Expiratory Flow ◽  
Peak Flow ◽  
Rate Of Change ◽  
Acute Severe Asthma ◽  
Rates Of Change ◽  
The Mean ◽  
Expiratory Flow ◽  
Change In Mean

1. The rates of change in mean peak expiratory flow and in diurnal variation in peak flow were compared in 14 patients recovering from acute severe asthma. 2. Peak expiratory flow was measured on hospital admission, and at 6-hourly intervals for the next 3 weeks. 3. Diurnal variation in peak flow was assessed by measuring the following: amplitude (the highest minus the lowest peak expiratory flow during any given 24 h period), amplitude % mean (the highest minus the lowest peak expiratory flow during any given 24 h period divided by the mean peak expiratory flow over that period) and residual amplitude (the maximum variation about the mean peak expiratory flow during any given 24 h period). 4. Plots of diurnal variation in peak flow and peak expiratory flow against time were constructed for each patient. To enable comparison of changes in peak expiratory flow and diurnal variation in peak flow the data were transformed. 5. The rate of change for mean peak expiratory flow and for the three measures of diurnal variation in peak flow was assessed by fitting an exponential function to each set of data, and calculating the slope of the exponential curve halfway through the period of observation (10.5 days). 6. Median (range) slope for peak expiratory flow was 0.055 (0-2.57). The comparable value for amplitude was −3.15 (−1.27 to −4.22) (absolute median values compared, P = 0.0029), for amplitude % mean was −1.87 (−0.18 to −5.95) (P = 0.012) and for residual amplitude was −1.43 (−0.62 to −3.09) (P = 0.033). 7. Diurnal variation in peak flow therefore takes longer to reach a stable value than does mean peak expiratory flow. We conclude that the magnitude of diurnal variation in peak flow during recovery from an acute attack of asthma is not governed exclusively by mean airway calibre.

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