The geometry of random drift II. The symmetry of random genetic drift

1977 ◽  
Vol 9 (2) ◽  
pp. 250-259 ◽  
Author(s):  
Peter L. Antonelli ◽  
Jared Chapin ◽  
G. Mark Lathrop ◽  
Kenneth Morgan
Keyword(s):  
Velocity Field ◽  
Genetic Drift ◽  
Gene Frequency ◽  
Radial Symmetry ◽  
Divergence Times ◽  
Random Genetic Drift ◽  
Initial State ◽  
Random Drift ◽  
Arithmetic Average ◽  
Frequency Space

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher–Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

The geometry of random drift II. The symmetry of random genetic drift

1977 ◽  
Vol 9 (02) ◽  
pp. 250-259 ◽  
Author(s):  
Peter L. Antonelli ◽  
Jared Chapin ◽  
G. Mark Lathrop ◽  
Kenneth Morgan
Keyword(s):  
Velocity Field ◽  
Genetic Drift ◽  
Gene Frequency ◽  
Radial Symmetry ◽  
Divergence Times ◽  
Random Genetic Drift ◽  
Initial State ◽  
Random Drift ◽  
Arithmetic Average ◽  
Frequency Space

It has been conjectured that a certain transformation of gene frequency space due to Fisher and Bhattacharyya will map the random genetic drift process, or its diffusion approximation, into one with radial symmetry. This paper proves rigorously that the Fisher–Bhattacharyya mapping does not do this. This implies that the initial state of an evolving ensemble can only be unbiasedly estimated from the means of a sample if we weight by the proper divergence times. If the ensemble is known not to have begun at the centroid of frequency space, the estimate of the initial state vector is not simply the arithmetic average, as symmetry analysis of the Christoffel velocity field shows.

The geometry of random drift III. Recombination and diffusion

1977 ◽  
Vol 9 (02) ◽  
pp. 260-267 ◽  
Author(s):  
Peter L. Antonelli ◽  
Kenneth Morgan ◽  
G. Mark Lathrop
Keyword(s):  
Velocity Field ◽  
Markov Chains ◽  
Genetic Drift ◽  
Present Model ◽  
Random Mating ◽  
Recombination Fraction ◽  
Random Genetic Drift ◽  
Random Drift ◽  
Intrinsic Geometry ◽  
And Diffusion

A new diffusion model for random genetic drift of a two-locus di-allelic system is proposed. The Christoffel velocity field and the intrinsic geometry of the diffusion is computed for the equilibrium surface. It is seen to be radically non-spherical and to depend explicitly on the recombination fraction. The model has not been shown to be a limit of discrete Markov chains. For large values of the recombination, the present model is radically different from that of Ohta and Kimura, which is an approximation to the discrete process of random mating in the limit as the value of the recombination fraction goes to zero.

The geometry of random drift III. Recombination and diffusion

1977 ◽  
Vol 9 (2) ◽  
pp. 260-267 ◽  
Author(s):  
Peter L. Antonelli ◽  
Kenneth Morgan ◽  
G. Mark Lathrop
Keyword(s):  
Velocity Field ◽  
Markov Chains ◽  
Genetic Drift ◽  
Present Model ◽  
Random Mating ◽  
Recombination Fraction ◽  
Random Genetic Drift ◽  
Random Drift ◽  
Intrinsic Geometry ◽  
And Diffusion

A new diffusion model for random genetic drift of a two-locus di-allelic system is proposed. The Christoffel velocity field and the intrinsic geometry of the diffusion is computed for the equilibrium surface. It is seen to be radically non-spherical and to depend explicitly on the recombination fraction. The model has not been shown to be a limit of discrete Markov chains. For large values of the recombination, the present model is radically different from that of Ohta and Kimura, which is an approximation to the discrete process of random mating in the limit as the value of the recombination fraction goes to zero.

Mutational order: a major stochastic process in evolution

1990 ◽  
Vol 240 (1297) ◽  
pp. 29-37 ◽  
Keyword(s):  
Stochastic Process ◽  
Computer Simulations ◽  
Genetic Drift ◽  
Molecular Clock ◽  
Quantitative Character ◽  
Evolutionary Divergence ◽  
Random Genetic Drift ◽  
Random Drift ◽  
Large Populations ◽  
Different Populations

Computer simulations in which selection acts on a quantitative character show that the randomness of mutations can contribute significantly to evolutionary divergence between populations. In different populations, different advantageous mutations occur, and are selected to fixation, so that the populations diverge even when they are initially identical, and are subject to identical selection. This stochastic process is distinct from random genetic drift. In some circumstances (large populations or strong selection, or both) mutational order can be greatly more important than random drift in bringing about divergence. It can generate a ‘disconnection’ between evolution at the phenotypic and genotypic levels, and can give rise to a rough ‘molecular clock’, albeit episodic, that is driven by selection. In the absence of selection, mutational order has little or no effect.

The geometry of random genetic drift VI. A random selection diffusion model

1980 ◽  
Vol 12 (01) ◽  
pp. 50-58
Author(s):  
Peter L. Antonelli ◽  
Jared Chapin ◽  
B. H. Voorhees
Keyword(s):  
Diffusion Process ◽  
Diffusion Model ◽  
Genetic Drift ◽  
Random Selection ◽  
Jacobi Field ◽  
Random Genetic Drift ◽  
Field Equations ◽  
Frequency Space ◽  
Quadratic Dependence ◽  
Selection Parameters

The ray solution of Felsenstein's n-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.

The geometry of random genetic drift VI. A random selection diffusion model

1980 ◽  
Vol 12 (1) ◽  
pp. 50-58 ◽  
Author(s):  
Peter L. Antonelli ◽  
Jared Chapin ◽  
B. H. Voorhees
Keyword(s):  
Diffusion Process ◽  
Diffusion Model ◽  
Genetic Drift ◽  
Random Selection ◽  
Jacobi Field ◽  
Random Genetic Drift ◽  
Field Equations ◽  
Frequency Space ◽  
Quadratic Dependence ◽  
Selection Parameters

The ray solution of Felsenstein'sn-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.

scholarly journals NUMERICAL ANALYSIS OF WEAK RANDOM DRIFT IN A CLINE

Genetics ◽  
1979 ◽  
Vol 92 (1) ◽  
pp. 297-303
Author(s):  
Mitchell Luskin ◽  
Thomas Nagylaki
Keyword(s):  
Gene Frequency ◽  
The Other ◽  
Random Genetic Drift ◽  
Gene Frequencies ◽  
Random Drift ◽  
Linear Habitat ◽  
Asymptotic Formulae ◽  
Genotype A ◽  
Average Position ◽  
Uniform Population

ABSTRACT The equilibrium state of a diffusion model for weak random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric, and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a non-zero constant. The correlation between the gene frequencies at two points decreases monotonically to zero as the separation is increased with the average position fixed; the decrease is asymptotically exponential. The correlation decreases monotonically to a positive constant depending 011 the separation as the average position increasingly deviates from the center of the cline with the separation fixed. The correlation also decreases monotonically to zero if one of the points is fixed and the other is moved outward in the habitat, the ultimate decrease again being exponential. All the results are parameter free. Some asymptotic formulae are derived analytically.

CYTO-NUCLEAR EPISTASIS: TWO-LOCUS RANDOM GENETIC DRIFT IN HERMAPHRODITIC AND DIOECIOUS SPECIES

Evolution ◽  
2006 ◽  
Vol 60 (4) ◽  
pp. 643 ◽  
Author(s):  
Michael J. Wade ◽  
Charles J. Goodnight
Keyword(s):  
Genetic Drift ◽  
Random Genetic Drift ◽  
Dioecious Species

scholarly journals Influence of Random Genetic Drift on Human Immunodeficiency Virus Type 1envEvolution During Chronic Infection

Genetics ◽  
2004 ◽  
Vol 166 (3) ◽  
pp. 1155-1164 ◽  
Author(s):  
Daniel Shriner ◽  
Raj Shankarappa ◽  
Mark A. Jensen ◽  
David C. Nickle ◽  
John E. Mittler ◽  
...  
Keyword(s):  
Human Immunodeficiency Virus ◽  
Human Immunodeficiency Virus Type ◽  
Virus Type ◽  
Genetic Drift ◽  
Chronic Infection ◽  
Random Genetic Drift ◽  
Immunodeficiency Virus

NUMERICAL ANALYSIS OF RANDOM DRIFT IN A CLINE

Genetics ◽  
1980 ◽  
Vol 94 (2) ◽  
pp. 497-517
Author(s):  
Thomas Nagylaki ◽  
Bradley Lucier
Keyword(s):  
Natural Populations ◽  
The Other ◽  
Random Genetic Drift ◽  
Gene Frequencies ◽  
Random Drift ◽  
Linear Habitat ◽  
Asymptotic Formulae ◽  
A Chain ◽  
Average Position ◽  
Uniform Population

ABSTRACT The equilibrium state of a diffusion model for random genetic drift in a cline is analyzed numerically. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneouq symmetric and independent of genotype. A single diallelic locus with a step environment is investigated in the absence of dominance and mutation. The flattening of the expected cline due to random drift is very slight in natural populations. The ratio of the variance of either gene frequency to the product of the expected gene frequencies decreases monotonically to a nonzero constant. The correlation between the gene frequencies at two points decreases monotonically to zero as the separation is increased with the average position fixed; the decrease is asymptotically exponential. The correlation decreases monotonically to a positive constant depending on the separation as the average position increasingly deviates from the center of the cline with the separation fixed. The correlation also decreases monotonically to zero if one of the points is fixed and the other is moved outward in the habitat, the ultimate decrease again being exponential. Some asymptotic formulae are derived analytically.—The loss of an allele favored in an environmental pocket is investigated by simulating a chain of demes exchanging migrants, the other assumptions being the same as above. For most natural populations, provided the allele would be maintained in the population deterministically, this process is too slow to have evolutionary importance.

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