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Author Topic:   Population Dynamics - the math behind the evolution of species
RAZD
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Message 1 of 7 (857060)
07-01-2019 2:36 PM


Population Size vs Ecological Carrying Capacity
This is going to be a learning experience for me, as I am weak in this area. Feel free to help.

So let's start with a basic starting point:

quote:
Population dynamics is the branch of life sciences that studies the size and age composition of populations as dynamical systems, and the biological and environmental processes driving them (such as birth and death rates, and by immigration and emigration). Example scenarios are ageing populations, population growth, or population decline.

Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of more than 210 years, although more recently the scope of mathematical biology has greatly expanded. ...

The rate at which a population increases in size if there are no density-dependent forces regulating the population is known as the intrinsic rate of increase. It is

where the derivative dN/dt is the rate of increase of the population, N is the population size, and r is the intrinsic rate of increase. Thus r is the maximum theoretical rate of increase of a population per individual – that is, the maximum population growth rate. The concept is commonly used in insect population biology to determine how environmental factors affect the rate at which pest populations increase. See also exponential population growth and logistic population growth.[2]


so r can be positive (population growth), negative(population decline) or 0 (static population). Making it per individual makes it easier to compare different populations.

quote:
(ibid) ... Population regulation is a density-dependent process, meaning that population growth rates are regulated by the density of a population. Consider an analogy with a thermostat. When the temperature is too hot, the thermostat turns on the air conditioning to decrease the temperature back to homeostasis. When the temperature is too cold, the thermostat turns on the heater to increase the temperature back to homeostasis. Likewise with density dependence, whether the population density is high or low, population dynamics returns the population density to homeostasis. Homeostasis is the set point, or carrying capacity, defined as K.[3]

where ( 1 − N/K ) is the density dependence, N is the number in the population, K is the set point for homeostasis and the carrying capacity. In this logistic model, population growth rate is highest at 1/2 K and the population growth rate is zero around K. The optimum harvesting rate is a close rate to 1/2 K where population will grow the fastest. Above K, the population growth rate is negative. The logistic models also show density dependence, meaning the per capita population growth rates decline as the population density increases. In the wild, you can't get these patterns to emerge without simplification. Negative density dependence allows for a population that overshoots the carrying capacity to decrease back to the carrying capacity, K.[3]

According to r/K selection theory organisms may be specialised for rapid growth, or stability closer to carrying capacity.


If the population exceeds the carrying capacity of the ecology the population size will decrease, if it is below the carrying capacity the population will increase. If it is at K then the population size will be static.

quote:
(ibid) In fisheries and wildlife management, population is affected by three dynamic rate functions.

  • Natality or birth rate, often recruitment, which means reaching a certain size or reproductive stage. Usually refers to the age a fish can be caught and counted in nets.
  • Population growth rate, which measures the growth of individuals in size and length. More important in fisheries, where population is often measured in biomass.
  • Mortality, which includes harvest mortality and natural mortality. Natural mortality includes non-human predation, disease and old age.

If N1 is the number of individuals at time 1 then

where N0 is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the number that immigrated, and E the number that emigrated between time 0 and time 1.

If we measure these rates over many time intervals, we can determine how a population's density changes over time. Immigration and emigration are present, but are usually not measured.


This can be applied to any species, not just fish.

Thus we can define a breeding population of a species and determine if it is growing, declining or static, ... once we know the carrying capacity of the ecological environment it inhabits.

We can see from the numbers/math that each species population will tend to stabilize at the carrying capacity of the ecological environment.

If we see a population that has a static population, we can assume that it is at the carrying capacity of the ecological environment.

If we see a population that is fluctuating, we can assume that it is fluctuating around the carrying capacity of the ecological environment.

Enjoy

Edited by RAZD, : subtitle

Edited by RAZD, : .


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Replies to this message:
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RAZD
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Posts: 19981
From: the other end of the sidewalk
Joined: 03-14-2004
Member Rating: 3.9


Message 2 of 7 (857061)
07-01-2019 2:43 PM
Reply to: Message 1 by RAZD
07-01-2019 2:36 PM


Evolutionary Game Theory
quote:
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinian competition can be modelled. It originated in 1973 with John Maynard Smith and George R. Price's formalisation of contests, analysed as strategies, and the mathematical criteria that can be used to predict the results of competing strategies.[1]

Evolutionary game theory differs from classical game theory in focusing more on the dynamics of strategy change.[2] This is influenced by the frequency of the competing strategies in the population.[3]

Classical game theory

Classical non-cooperative game theory was conceived by John von Neumann to determine optimal strategies in competitions between adversaries. A contest involves players, all of whom have a choice of moves. Games can be a single round or repetitive. The approach a player takes in making his moves constitutes his strategy. Rules govern the outcome for the moves taken by the players, and outcomes produce payoffs for the players; rules and resulting payoffs can be expressed as decision trees or in a payoff matrix. Classical theory requires the players to make rational choices. Each player must consider the strategic analysis that his opponents are making to make his own choice of moves.[4][5]

The problem of ritualized behaviour

Evolutionary game theory started with the problem of how to explain ritualized animal behaviour in a conflict situation; "why are animals so 'gentlemanly or ladylike' in contests for resources?" The leading ethologists Niko Tinbergen and Konrad Lorenz proposed that such behaviour exists for the benefit of the species. John Maynard Smith considered that incompatible with Darwinian thought,[6] where selection occurs at an individual level, so self-interest is rewarded while seeking the common good is not. Maynard Smith, a mathematical biologist, turned to game theory as suggested by George Price, though Richard Lewontin's attempts to use the theory had failed.[7]

Adapting game theory to evolutionary games

Maynard Smith realised that an evolutionary version of game theory does not require players to act rationally —– only that they have a strategy. The results of a game shows how good that strategy was, just as evolution tests alternative strategies for the ability to survive and reproduce. In biology, strategies are genetically inherited traits that control an individual's action, analogous with computer programs. The success of a strategy is determined by how good the strategy is in the presence of competing strategies (including itself), and of the frequency with which those strategies are used.[8] Maynard Smith described his work in his book Evolution and the Theory of Games.[9]

Participants aim to produce as many replicas of themselves as they can, and the payoff is in units of fitness (relative worth in being able to reproduce). It is always a multi-player game with many competitors. Rules include replicator dynamics, in other words how the fitter players will spawn more replicas of themselves into the population and how the less fit will be culled, in a replicator equation. The replicator dynamics models heredity but not mutation, and assumes asexual reproduction for the sake of simplicity. Games are run repetitively with no terminating conditions. Results include the dynamics of changes in the population, the success of strategies, and any equilibrium states reached. Unlike in classical game theory, players do not choose their strategy and cannot change it: they are born with a strategy and their offspring inherit that same strategy.[10]


There are quite a number of game models being used, and I recommend reading through the article. For now I'll start with the Hawk Dove game:

quote:
The first game that Maynard Smith analysed is the classic Hawk Dove[a] game. It was conceived to analyse Lorenz and Tinbergen's problem, a contest over a shareable resource. The contestants can be either Hawk or Dove. These are two subtypes or morphs of one species with different strategies. The Hawk first displays aggression, then escalates into a fight until it either wins or is injured (loses). The Dove first displays aggression, but if faced with major escalation runs for safety. If not faced with such escalation, the Dove attempts to share the resource.[1]

Payoff Matrix for Hawk Dove Game

meets Hawk meets Dove
if HawkV/2 − C/2V
if Dove0V/2

Given that the resource is given the value V, the damage from losing a fight is given cost C:[1]

  • If a Hawk meets a Dove he gets the full resource V to himself
  • If a Hawk meets a Hawk – half the time he wins, half the time he loses...so his average outcome is then V/2 minus C/2
  • If a Dove meets a Hawk he will back off and get nothing – 0
  • If a Dove meets a Dove both share the resource and get V/2

The actual payoff however depends on the probability of meeting a Hawk or Dove, which in turn is a representation of the percentage of Hawks and Doves in the population when a particular contest takes place. That in turn is determined by the results of all of the previous contests. If the cost of losing C is greater than the value of winning V (the normal situation in the natural world) the mathematics ends in an ESS, a mix of the two strategies where the population of Hawks is V/C. The population regresses to this equilibrium point if any new Hawks or Doves make a temporary perturbation in the population. The solution of the Hawk Dove Game explains why most animal contests involve only ritual fighting behaviours in contests rather than outright battles. The result does not at all depend on good of the species behaviours as suggested by Lorenz, but solely on the implication of actions of so-called selfish genes.[1]

"ESS" refers to an evolutionarily stable strategy:

quote:
... a strategy (or set of strategies) which, if adopted by a population in a given environment, is impenetrable, meaning that it cannot be invaded by any alternative strategy (or strategies) that are initially rare. It is relevant in game theory, behavioural ecology, and evolutionary psychology. An ESS is an equilibrium refinement of the Nash equilibrium. It is a Nash equilibrium that is "evolutionarily" stable: once it is fixed in a population, natural selection alone is sufficient to prevent alternative (mutant) strategies from invading successfully. The theory is not intended to deal with the possibility of gross external changes to the environment that bring new selective forces to bear.

Another term commonly used is the "fight or flight" reaction to a confrontation, and this applies whether the opponent is a different species or a competing member of the same species.

Now I would think that there is a cost to winning a hawk meets hawk fight, as some injuries are sustained (and this may make the winning hawk less fit for the next confrontation), so his payoff is V-Cwin while the payoff for the loser is 0-Close, where Close is greater than Cwin. And I would think the confrontation costs could make the individual behave as a dove in the next confrontation, and this leads us to:

quote:
Contests of selfish genes

At first glance it may appear that the contestants of evolutionary games are the individuals present in each generation who directly participate in the game. But individuals live only through one game cycle, and instead it is the strategies that really contest with one another over the duration of these many-generation games. So it is ultimately genes that play out a full contest – selfish genes of strategy. The contesting genes are present in an individual and to a degree in all of the individual's kin. This can sometimes profoundly affect which strategies survive, especially with issues of cooperation and defection. William Hamilton,[21] known for his theory of kin selection, explored many of these cases using game theoretic models. Kin related treatment of game contests[22] helps to explain many aspects of the behaviour of social insects, the altruistic behaviour in parent/offspring interactions, mutual protection behaviours, and co-operative care of offspring. For such games Hamilton defined an extended form of fitness – inclusive fitness, which includes an individual's offspring as well as any offspring equivalents found in kin.

Hamilton went beyond kin relatedness to work with Robert Axelrod, analysing games of co-operation under conditions not involving kin where reciprocal altruism comes into play.[24]


These mathematical games help explain such issues as why most animal contests involve only ritual fighting behaviours in contests rather than outright battles, and altruistic behavior seen in different species.

Enough for now

Enjoy

Edited by RAZD, : .

Edited by RAZD, : .

Edited by RAZD, : ..


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 Message 1 by RAZD, posted 07-01-2019 2:36 PM RAZD has responded

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RAZD
Member
Posts: 19981
From: the other end of the sidewalk
Joined: 03-14-2004
Member Rating: 3.9


Message 3 of 7 (857062)
07-01-2019 2:53 PM
Reply to: Message 2 by RAZD
07-01-2019 2:43 PM


Fisherian Runaway Selection
We can also consider sexual/mate selection as a form of confrontation, with male aggression and female choice as parameters for game theory. This takes two forms -- selection for average traits and selection for extreme traits, called Fisherian Runaway Selection.

quote:
Fisherian runaway or runaway selection is a sexual selection mechanism proposed by the mathematical biologist Ronald Fisher in the early 20th century, to account for the evolution of exaggerated male ornamentation by persistent, directional female choice.[1][2][3] An example is the colourful and elaborate peacock plumage compared to the relatively subdued peahen plumage; the costly ornaments, notably the bird's extremely long tail, appear to be incompatible with natural selection. Fisherian runaway can be postulated to include sexually dimorphic phenotypic traits such as behaviour expressed by either sex.

The peacock tail in flight, the classic example of an ornament
assumed to be a Fisherian runaway

Peacock flying on rice field, at Karaikudi, Tamilnadu. Image
taken on 12th November 2012, by Haribabu Pasupathy.

Fisher, in the foundational 1930 book, The Genetical Theory of Natural Selection,[6] first outlined a model by which runaway inter-sexual selection could lead to sexually dimorphic male ornamentation based upon female choice and a preference for "attractive" but otherwise non-adaptive traits in male mates. He suggested that selection for traits that increase fitness may be quite common:

[O]ccasions may not be infrequent when a sexual preference of a particular kind may confer a selective advantage, and therefore become established in the species. Whenever appreciable differences exist in a species, which are in fact correlated with selective advantage, there will be a tendency to select also those individuals of the opposite sex which most clearly discriminate the difference to be observed, and which most decidedly prefer the more advantageous type. Sexual preference originated in this way may or may not confer any direct advantage upon the individuals selected, and so hasten the effect of the Natural Selection in progress. It may therefore be far more widespread than the occurrence of striking secondary sexual characters.

Fisher, R.A. (1930) The Genetical Theory of Natural Selection. ISBN 0-19-850440-3[6]

A strong female choice for the expression alone, as opposed to the function, of a male ornament can oppose and undermine the forces of natural selection and result in the runaway sexual selection that leads to the further exaggeration of the ornament (as well as the preference) until the costs (incurred by natural selection) of the expression become greater than the benefit (bestowed by sexual selection).[5][6]

Fisher outlined two fundamental conditions that must be fulfilled in order for the Fisherian runaway mechanism to lead to the evolution of extreme ornamentation:

  1. Sexual preference in at least one of the sexes
  2. A corresponding reproductive advantage to the preference.[6]

Fisher argued in his 1915 paper, "The evolution of sexual preference" that the type of female preference necessary for Fisherian runaway could be initiated without any understanding or appreciation for beauty.[5] Fisher suggested that any visible features that indicate fitness, that are not themselves adaptive, that draw attention, and that vary in their appearance amongst the population of males so that the females can easily compare them, would be enough to initiate Fisherian runaway. This suggestion is compatible with his theory, and indicates that the choice of feature is essentially arbitrary, and could be different in different populations. Such arbitrariness is borne out by mathematical modelling, and by observation of isolated populations of sandgrouse, where the males can differ markedly from those in other populations.[5][1][2][8][9][10]

Fisher argued that the selection for exaggerated male ornamentation is driven by the coupled exaggeration of female sexual preference for the ornament.

Certain remarkable consequences do, however, follow ... in a species in which the preferences of … the female, have a great influence on the number of offspring left by individual males. ... development will proceed, so long as the disadvantage is more than counterbalanced by the advantage in sexual selection … there will also be a net advantage in favour of giving to it a more decided preference.

Fisher, R.A. (1930) The Genetical Theory of Natural Selection. ISBN 0-19-850440-3[6]

Over time a positive feedback mechanism will see more exaggerated sons and choosier daughters being produced with each successive generation; resulting in the runaway selection for the further exaggeration of both the ornament and the preference (until the costs for producing the ornament outweigh the reproductive benefit of possessing it).

Several alternative hypotheses use the same genetic runaway (or positive feedback) mechanism but differ in the mechanisms of the initiation. The sexy son hypothesis (also proposed by Fisher) suggests that females that choose desirably ornamented males will have desirably ornamented (or sexy) sons, and that the effect of that behaviour on spreading the female's genes through subsequent generations may outweigh other factors such as the level of parental investment by the father.[11]


The thread Sexual Selection, Stasis, Runaway Selection, Dimorphism, & Human Evolution argues that sexual selection can lead to stasis by selection for average traits:

quote:
From (an abundance of) this kind of evidence one can conclude that sexual selection involves actively (whether consciously or not) choosing mates that best represent {the species icon} based on visual, olfactory and behavioral clues, where {the species icon} represents the "ideal mate" not just for the individual but for the population.

If we assume {the species icon} represents average values of features within the population, then in the absence of survival selection pressure within a species population, this active choice mechanism will lead to choosing the more "average" individuals for mates (and excluding the least normal individuals) whenever possible, a process that will essentially guarantee stasis within the species population.


These average individuals would be the most fit for a stable ecology due to natural selection. Traits new and old would regress to the mean.

It further argues that Run-away Selection would occur

quote:
If we assume that {the species icon} does not represents an average value for any one (or more) {choice feature(s)} within the population, then in the absence of survival selection pressure within a species population, this active choice mechanism will lead to choosing the more "extreme" individuals displaying the desired feature(s) for mates (and excluding the more normal individuals) whenever possible, a process that will essentially guarantee transformation within the species population over time, until a point is reached where it can be taken no further. Mature populations without survival stress and without any other reason to {change\evolve} would trend towards an {extreme individual icon}, even to the point where species survival could be jeopardized: a species "fatal attraction" if you will.

The "run-away" aspect of this mating behavior develops because not only is the selection for individuals with the {choice feature(s)} but it is by individuals that prefer the {choice feature(s)}. Thus, to use a popular example of run-away sexual selection, not only does the male peacock have a large and extremely ornate tail, but the female peacock prefers males with the largest and most extremely ornate tail.

Richard Dawkins in The Blind Watchmaker also discusses an experiment with a long tailed bird where the male's long tail feathers were cut and then glued back with (1) shorter (2) same length and (3) longer tail feathers and then monitored for breeding success compared to (4) unmodified (control) males. There was no difference between group (2) and (4) (ie the effect of the glue process was eliminated as a variable), but the ones with the artificially long tail feathers were selected above the others: the sexual preference was for expression of the feature to an extent not seen within the population. Natural selection prevented the males from developing the longer tails after a certain point had been reached, but female preference was still for even longer tails.


Finally it argues that humans show some traits typical of runaway selection:

quote:
Do humans exhibit any features that show the effects of run-away sexual selection: features that have no survival advantage, that may impose a survival burden, and where selection continues to push selection towards a skewed end of natural variation within the population?

Elsewhere I have discussed this, and have posted these features:
(a) Long head hair (longer than any other primate)
(b) Music\Dance\Artistic Creativity (more than any other primate)
(c) Sexual signal features (larger than any other primate)
(d) Skin hair thinness (more than any other primate)

I am not alone with this. See The mating mind: human sexual selection (click) for a discussion on the role of sexual selection in the evolution of the human brain size, complexity and ability -- with both pro and con arguments.


So apparently sexual selection for creativity made our brains larger ...

How can we measure this? Probably by looking at the zygosity of the species to see where the average traits are distributed in the breeding population and whether there are peak values near one of the extreme ends of the available traits -- eg size traits distributed around the middle or clustered to one end of the spectrum.

Enjoy

Edited by RAZD, : ..

Edited by RAZD, : ...


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RAZD
Member
Posts: 19981
From: the other end of the sidewalk
Joined: 03-14-2004
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Message 4 of 7 (857063)
07-04-2019 3:00 PM
Reply to: Message 3 by RAZD
07-01-2019 2:53 PM


Zygosity
quote:
In population genetics, the concept of heterozygosity is commonly extended to refer to the population as a whole, i.e., the fraction of individuals in a population that are heterozygous for a particular locus. It can also refer to the fraction of loci within an individual that are heterozygous.

Typically, the observed (Ho) and expected (He) heterozygosities are compared, defined as follows for diploid individuals in a population:

Observed

where is the number of individuals in the population, and are the alleles of individual at the target locus.

Expected

where m is the number of alleles at the target locus, and is the allele frequency of the allele at the target locus.

Heterozygosity values of 51 worldwide human populations.[7] Sub-Saharan Africans
have the highest values in the world.
By David López Herráez , Marc Bauchet , Kun Tang , Christoph Theunert,
Irina Pugach, Jing Li, Madhusudan R. Nandineni, Arnd Gross, Markus Scholz,
Mark Stoneking - López Herráez D, Bauchet M, Tang K, Theunert C, Pugach I,
Li J, et al. (2009) Genetic Variation and Recent Positive Selection in Worldwide
Human Populations: Evidence from Nearly 1 Million SNPs. PLoS ONE 4(11): e7888.
doi:10.1371/journal.pone.0007888 http://journals.plos.org/plosone/article?i
d=10.1371/journal.pone.0007888, CC BY 2.5, https://commons.wikimedia.org/
w/index.php?curid=54659673

Just to be clear, the value for the observed heterozygosity function is 0 when the individual in question is homozygous, and ai1 = ai2. The function just counts up the number of individuals that are heterozygous at locus i.

quote:
A heterozygote advantage describes the case in which the heterozygous genotype has a higher relative fitness than either the homozygous dominant or homozygous recessive genotype. ...

When two populations of any sexual organism are separated and kept isolated from each other, the frequencies of deleterious mutations in the two populations will differ over time, by genetic drift. It is highly unlikely, however, that the same deleterious mutations will be common in both populations after a long period of separation. Since loss-of-function mutations tend to be recessive (given that dominant mutations of this type generally prevent the organism from reproducing and thereby passing the gene on to the next generation), the result of any cross between the two populations will be fitter than the parent.


Interesting that (non-lethal) deleterious mutations are involved here. This explains how "hybrid vigour" can occur.

quote:
Heterosis, hybrid vigor, or outbreeding enhancement, is the improved or increased function of any biological quality in a hybrid offspring. An offspring is heterotic if its traits are enhanced as a result of mixing the genetic contributions of its parents. These effects can be due to Mendelian or non-Mendelian inheritance.

The physiological vigor of an organism as manifested in its rapidity of growth, its height and general robustness, is positively correlated with the degree of dissimilarity in the gametes by whose union the organism was formed … The more numerous the differences between the uniting gametes — at least within certain limits — the greater on the whole is the amount of stimulation … These differences need not be Mendelian in their inheritance … To avoid the implication that all the genotypic differences which stimulate cell-division, growth and other physiological activities of an organism are Mendelian in their inheritance and also to gain brevity of expression I suggest … that the word 'heterosis' be adopted.

When a population is small or inbred, it tends to lose genetic diversity. Inbreeding depression is the loss of fitness due to loss of genetic diversity. Inbred strains tend to be homozygous for recessive alleles that are mildly harmful (or produce a trait that is undesirable from the standpoint of the breeder). Heterosis or hybrid vigor, on the other hand, is the tendency of outbred strains to exceed both inbred parents in fitness. ...


BUT the range of genetic difference between the two populations where interbreeding occurs doesn't always produce hybrid vigor (heterosis):

quote:
(ibid) Not all outcrosses result in heterosis. For example, when a hybrid inherits traits from its parents that are not fully compatible, fitness can be reduced. This is a form of outbreeding depression.

So once the differences become too great, too incompatible, then you get outbreeding depression, increasing until a point is reached where outbreeding fails to produce viable offspring ... and you have speciaton instead, the populations are reproductively isolated.

Enjoy

Edited by RAZD, : .


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Message 5 of 7 (857065)
07-05-2019 8:46 AM


Thread Copied from Proposed New Topics Forum
Thread copied here from the Population Dynamics - the math behind the evolution of species thread in the Proposed New Topics forum.
    
RAZD
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Message 6 of 7 (857217)
07-06-2019 4:43 PM
Reply to: Message 4 by RAZD
07-04-2019 3:00 PM


Sexual Incompatibility
So once the differences become too great, too incompatible, then you get outbreeding depression, increasing until a point is reached where outbreeding fails to produce viable offspring ... and you have speciaton instead, the populations are reproductively isolated.

So how can we tell when this point has been reached?

quote:
The population genetics of speciation: the evolution of hybrid incompatibilities.

Speciation often results from the accumulation of "complementary genes," i.e., from genes that, while having no deleterious effect within species, cause inviability or sterility when brought together with genes from another species. Here I model speciation as the accumulation of genic incompatibilities between diverging populations. Several results are obtained. First, and most important, the number of genic incompatibilities between taxa increases much faster than linearly with time. In particular, the probability of speciation increases at least as fast as the square of the time since separation between two taxa. Second, as Muller realized, all hybrid incompatibilities must initially be asymmetric. Third, at loci that have diverged between taxa, evolutionarily derived alleles cause hybrid problems far more often than ancestral alleles. Last, it is "easier" to evolve complex hybrid incompatibilities requiring the simultaneous action of three or more loci than to evolve simple incompatibilities between pairs of genes. These results have several important implications for genetic analyses of speciation.


The link goes to the abstract (above) for this paper, but there is a link for a free download of the full paper in PDF format.

quote:
... DOEZHANSKY (1936) and MULLER (1939, 1940) early in the modern synthesis. Each produced genetic models showing that two populations could come to produce completely sterile or inviable hybrids even when no substitution caused any sterility or inviability within either population. Their models were very simple: two allopatric populations begin with identical genotypes at two loci (aa, bb). In one population, an A allele appears and is fixed; the Aabb and AAbb genotypes are perfectly viable and fertile. In the other population, a B mutation appears and is fixed; aaBb and aaBB are also viable and fertile. The critical point is that, although the B allele is compatible with a, it has not been “tested” on an A genetic background. It is thus possible that B has a deleterious effect that appears only when A is present. If the two populations meet and hybridize, the resulting AaBb hybrid may be inviable or sterile.

As MULLER (1942) pointed out, it makes no difference whether the substitutions occur in both populations, as above, or in one only. If one population retains the ancestral aabb genotype and the other becomes AAbb and then AABB, the B allele may well be incompatible with the a allele among the AaBb hybrids. In either case, reproductive isolation results from “complementary” or “reinforcing” epistasis between loci A and B (CROW and KIMURA 1970, p. 81): the lethal or sterile effect of an allele at one locus depends on the background genotypes at other loci. Of course, complementary genes need not have a complete effect-a pair of complementary genes might cause only partial hybrid sterility or inviability.

DOBZHANSKY and MULLER’s model of speciation is important for two reasons. First, it shows that the evolution of hybrid sterility or inviability need not involve any intermediate, maladaptive step. Perhaps more important, it shows that, while the problem of the origin of species can be reduced to the origin of reproductive isolation, this in turn -- at least for postzygotic isolation -- can be reduced to the building up of complementary genes.

THE BASIC MODEL

The central assumption of the DOBZHANSKY-MULLER model of speciation is that alleles cause no sterility or inviability on their normal “pure species” genetic background. Instead, an allele can lower fitness only when brought together with genes from another species. Any particular hybrid incompatibility might cause partial or complete hybrid sterility or inviability. For most of this paper, I assume that hybrid incompatibilties involve interactions between pairs of genes, as in DOBZHANSKY and MULLER’S verbal models. Later, I consider three-locus and higher interactions. I also assume that multiple substitutions do not occur at the same locus, an assumption that is reasonable during the early divergence of taxa. I assume nothing about the evolutionary causes of substitutions. The DOBZHANSKY-MULLER model of speciation requires only that substitutions occur and assumes nothing about whether they are brought about by natural selection or genetic drift.

Because I consider the cumulative effects of interactions between many loci -- which quickly gets complicated -- it is useful to picture this process diagramatically. Figure 1 offers a simple way to picture the accumulation of complementary genes between two haploid populations. Each of the two heavy lines represents a lineage descended from a common ancestor. The two allopatric populations begin with identical “ancestral” lowercase genotypes at all loci (a b c . . .). Time runs upward, with the first substitution occurring at the a locus, the second at the b locus and so on.

The first substitution involves the replacement of the a allele by the A allele in population 1 (uppercase letters indicate only that an allele is “derived”; no dominance is implied). The A allele cannot cause any hybrid sterility or inviability: because A is obviously compatible with the genetic background of population 1, it must be compatible with the identical background of population 2. The second substitution, at the B locus (in population 2), could be incompatible with only one locus: A, as the B allele has not been “tested” for compatibility with A. The third substitution, at C, could be incompatible with the B or a alleles. As we continue this process, it is clear that we can identify all possible (i.e., evolutionarily allowed) incompatibilities by drawing an arrow from each derived allele to each “earlier” (lower) allele carried by the other species. Thus D can be incompatible with c, B, and a. This arrow-drawing device will repeatedly prove useful.

Several facts immediately emerge from Figure 1. First, hybrid incompatibilities only occur between two loci that have both experienced a substitution: in Figure 1, arrows never run up toward loci that have not diverged (e.g., locus e). This follows from the fact that the two populations carry identical alleles at all undiverged loci, so that any substitution must be compatible with these loci in both species.

Several other less trivial facts also emerge from Figure 1:

  • All incompatibilities are asymmetric. For example, although B might be incompatible with A, b cannot be incompatible with a.
  • Evolutionarily derived (uppercase) alleles are involved in more potential incompatibilities than ancestral (lowercase) alleles.
  • Later substitutions cause more possible incompatibilities than earlier ones (e.g., although the substitution of B produces one possible incompatibility, the later substitution of D produces three). This suggests that the strength of reproductive isolation might increase faster than linearly with time.

Note that this means that ancestral (lowercase) alleles cannot on their own cause speciation, and that at least two derived/mutated (uppercase) alleles are needed to cause reproductive incompatibilities.

Thus for hybrid vigor to occur it must be early in the evolution of two diverging daughter populations. This also says that the mutations need not be (non-lethal) deleterious mutations as mentioned in Message 4 for the incompatibilities to arise.

Continuing:

quote:
THE ROLE OF DERIVED VS. ANCESTRAL ALLELES IN HYBRID INCOMPATIBILITIES

Figure 1 also shows that derived (uppercase) alleles tend to be involved in hybrid incompatibilities more often than ancestral (lowercase) alleles. Once again, this is a trivial consequence of the types of incompatibilities that are possible: when a new derived allele (say C) is substituted, it might be incompatible with either another derived allele (B) or with an ancestral allele (a). Thus, both derived-derived (DD) and derived-ancestral (DA) incompatibilities occur. The only type of incompatibility that does not arise is ancestral-ancestral (AA). The reason is obvious: all ancestral alleles must be compatible as they represent the initial genotype. Thus, restricting our attention to those loci that have experienced a substitution, the alleles causing postzygotic isolation will be derived more often than ancestral. We would like to know how much more often.

Imagine that a fraction f of all substitutions occur in population 1 and 1-f in population 2. ... By making all such comparisons it is easy to see that the probabilities of the various possible incompatibilities are

where D represents a derived allele, A an ancestral allele, and the subscripts identify the two populations

From this, we can tabulate the expected frequency with which derived vs. ancestral alleles from each population will be involved in hybrid incompatibilities:

Obviously P(D1) + P(A1) = P(D2) + P(A2), i.e., the total frequency with which alleles from population 1 are involved equals the frequency with which alleles from population 2 are involved, as every incompatibility must involve an allele from each species. Finally, the ratio P(D) : P(A) of derived-to-ancestral alleles causing hybrid incompatibilities is

In words, the number of derived vs. ancestral alleles causing reproductive isolation depends on the proportion of substitutions that occur in each population. When there are equal rates of evolution in the two lineages ( f = 1/2), derived alleles are three times more likely to be involved in hybrid incompatibilities than ancestral alleles, as noted by ORR (1993). Indeed, derived alleles are always more likely to cause hybrid problems unless all evolution occurs in one lineage (f = 1). In that case, P(D) : P(A) = 1:1 since the only possible type of incompatibility is derived--ancestral. The ratio P(D) : P(A) plays an important role in one possible explanation of Haldane’s rule (see ORR 1993).


Note that the possibility of two ancestral alleles being involved in incompatibility is

... because they were the same allele in the parent population.

And it should be readily apparent that speciation -- defined as sexual incompatibility -- depends on mutations occurring in the sub-populations.

Enjoy

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RAZD
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Message 7 of 7 (857343)
07-07-2019 7:49 PM
Reply to: Message 6 by RAZD
07-06-2019 4:43 PM


Speciation
quote:
The population genetics of speciation: the evolution of hybrid incompatibilities.

THE RATE OF SPECIATION

Later substitutions cause more potential incompatibilities than earlier ones (Figure 1). As already noted, the first substitution at the A locus cannot cause any hybrid incompatibility, while the second substitution could be incompatible with only one locus: the B allele has not been tested with the A allele. In general, the Kth substitution can be incompatible with K-1 loci from the other population. It is obvious, then, that the total number of incompatibilities separating two taxa increases faster than linearly with the number of substitutions that have occurred between them. This, in turn, implies that the strength of reproductive isolation -- or the probability of speciation -- between two taxa increases faster than linearly with time. This important effect is easily quantified. I consider two cases. First, I assume that complete reproductive isolation results from a single incompatibility between two complementary genes. Second, I assume that reproductive isolation results from the cumulative effects of many small incompatibilities. As we will see, both cases yield similar results.

Single-incompatibility speciation: Reproductive isolation here results from a single incompatibility between two complementary genes and the level of isolation suddenly leaps from zero to unity. Although it is unclear just how common this situation is, there is considerable evidence that reproductive isolation sometimes has a simple genetic basis (HOLLINGSHEAD 1930; GERSTEL 1954; WITTBRODT et al. 1989).

I calculate the probability that speciation has occurred as a function of the number of substitutions separating two diverging populations. For simplicity, assume that a new derived allele has a fixed probability, p, of being incompatible with each of the loci that has previously experienced a substitution. Because substitution K may be incompatible with K-1 loci, the Kth substitution has a probability 1-(1-p) K-l of causing speciation. The cumulative probability of speciation, S, is simply the probability that at least one incompatibility occurs, given by

(4)

When p is small (which it surely is) and K is large,

(5)

It is important to note that Equations 4 and 5 do not depend on the proportion of substitutions that occur in each population nor on the order in which substitutions arise in population 1 vs. 2.

Interestingly, we can also find the expected “time” to speciation. If Ks is the number of substitutions required until the appearance of a hybrid incompatibility, then
P(Ks > K) = (1 - p)K(K-1)/2. Thus the expectation of Ks is

which is approximately

(7)

Thus if the probability that any two genes are incompatible in hybrids is p = 10-5, an average of 400 substitutions is required for speciation. Because the time to speciation is an inverse function of it is not as sensitive to p as one might expect. Doubling the probability that an incompatibility occurs, for example, does not halve the time to speciation but reduces it by a factor of .

Multiple incompatibility speciation: I now consider the case where speciation results from the cumulative effects of several to many smaller incompatibilities. We will see that the above results do not depend on the assumption that speciation is caused by a single incompatibility of complete effect.

The reduction in hybrid fitness, r (0 ≤ r ≤ l), resulting from an interaction between any two genes has some frequency distribution f(r). Obviously, the mean

must be very small or speciation would be nearly instantaneous; indeed, most interactions between genes in hybrids may have no effect on hybrid fitness (r = 0).

I assume that different incompatibilities act independently: if one incompatibility reduces fitness to (1-rl) and another reduces fitness to (1-r2), both together reduce fitness to (1-rl)(1-r2). Thus the effect of substitution K on hybrid fitness is given by Thus the effect of substitution K on hybrid fitness is given by

where wK is fitness, considering only those incompatibilities that involve the Kth substitution. If the ri are all fairly small, this is roughly

Considering the cumulative effects of all K substitutions,

where L is the strength of reproductive isolation (or the fitness “load” among hybrids due to complementary gene interactions). Thus, early in the divergence of two taxa (L << l), the strength of reproductive isolation increases as the square of the number of substitutions:

Thus the chance of speciation increases much faster than linearly with K (or time) whether speciation typically results from a very small number of genes of large effect (as in the first model) or a large number of genes of smaller effect (as in the second). The number of substitutions having a substantial effect on reproductive isolation also increases faster than linearly with time. Thus, if one were to double the time since divergence, one would more than double the number of genes having a large effect on hybrid fitness. ...

The role of early vs. late substitutions: The above discussion might seem to imply that a gene of known large effect on hybrid fitness was more likely a later than earlier substitution. This is incorrect. Although the probability that a substitution causes hybrid sterility or inviability increases with time, any gene afflicting hybrids is just as likely to have been the first gene to diverge as the last. This is because a late diverging gene must be incompatible with something, in particular with some locus that diverged earlier.


The bottom line is that speciation is caused by multiple (at least 2) mutations, and the longer populations are isolated the higher is the probability that incompatible mutations or mutation combinations arise in either of the daughter populations.

Enough for now. Note that I have taken some liberties with formating so that the formulas that were embedded in the text are on separate lines for clarity.

Enjoy



Note:

The math may seem daunting for those not familiar with the symbols, but they are fairly easy to comprehend when you know what the symbols represent. For instance:

means you sum each iteration from n=1 to n=4

and

means you multiply each iteration from n=1 to n=4


Edited by RAZD, : .

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