In the first part of this talk, I will briefly review the evolutionary
forces acting upon natural populations, and in the second part,
show you the results of simulations that illustrate what would
happen in natural populations affected by straying from hatchery
populations. The results are abstract and idealized, and the
simulations have been done in nice symmetrical ways for mathematical
convenience, but it is important to realize that the same principles
operate in a more complicated way in real-life situations. The
task is to understand what the most important principles are and
how you can apply them to a real situation.
I will mention five basic evolutionary forces. We usually do
not think of random mating as an evolutionary force, but it is.
Another force is natural selection, in particular natural selection
to adapt populations to local conditions. A third force that
can potentially change the genetic makeup of a population is mutation.
Since mutation occurs about equally everywhere, it is not in
and of itself a force making populations different or more similar.
Actually, mutation tends to make populations more similar to
each other, but it is a minor force. Migration is another important
force; for this workshop, we are concerned with hatchery straying,
which is one kind of migration. Population geneticists equate
migration with gene flow, the actual incorporation of migrant
genes into a receiving population, and not just with the physical
movement of an individual. The final mechanism that can lead
to genetic changes in a population is random change from random
births and deaths of individuals. This process is called random
genetic drift. I will concentrate on gene flow, random genetic
drift, and natural selection.
To illustrate the simple mathematics of gene flow between natural
populations, let us imagine that we have a genetic locus (a place
on a chromosome) which is variable in a set of populations. The
information encoded by the locus occurs in different forms, called
alleles. Each fish normally has two copies of a gene, one inherited
from each parent, and the frequency of a particular allele in
a natural population can be measured as the proportion of all
the copies of that gene that are of one allele or another. This
proportion is called a gene or allele frequency. Imagine we have
a population represented in Figure 1
as a big square, and we find--using
one biochemical or molecular technique or another--two alleles,
one of which is at a frequency of 0.80, or 80% of the total.
In the next generation, imagine that 70% of the individuals in
the population stayed in that population, but 20% came from population
2, and 10% came from population 3. In populations 2 and 3, the
frequency of the same allele was 0.1 and 0.2, respectively. The
allele frequency in population 1 is simply the weighted average
of the frequencies in the residents and the migrants.
When the mix of individuals in population 1 begins to mate, let
us assume that they mate at random, without regard to where they
came from. If so, the basic units are not whole genotypes, but
individual alleles that re-assort themselves each generation,
and the best way to think about gene flow is to consider the flow
of individual alleles rather than the flow of genotypes. Because
of the peculiarities of sexual reproduction and random mating,
geneticists talk about the frequencies of alleles or genes, which
partially determine the frequencies of genotypes in a population.
In calculating allelic frequencies in the recipient population,
we have to consider migration rates. Migration rates are calculated
as the fraction of new migrants in the recipient population, and
population size can be important when the donating and receiving
populations are very different, as is often the case for salmonid
populations. For example, a migration rate of 20% in a small
recipient population may represent a much smaller fraction of
a large donor population. If you think about the number of individuals
leaving a population, you may get the wrong impression about the
effects of migration.
Suppose that we assume that natural selection is not occurring;
that is, we are not favoring one allele or another and that the
alleles are passively reproducing themselves at random. Following
the frequencies of two alleles from one generation to the next
in a population is much like tossing a coin in which each side
of the coin represents an allele of the gene. If there are 100
individuals in a population, 200 copies of the gene are present--two
copies for each individual. We can simulate random drift by tossing
a coin 200 times to get the frequency in the next generation.
But instead of having a probability of 0.5 for a particular side
of the coin, the probability of getting a particular side in the
toss would be the frequency of the allele in the population before
reproduction, but after migration. As you know, if you toss a
coin several times, you usually do not get the exact proportion
that you expect. In a small number of tosses, which simulates
a small population, the frequencies vary a lot from the expected.
With a large number of tosses (a large population), the frequencies
are closer to the expected proportion. Figure 2 illustrates the
random changes in allele frequencies that might occur in a population.
In natural populations, randomness arises from three sources:
randomness of deaths (some individuals may die early), randomness
of births (some pairs may have a lot of offspring and others very
few), and randomness of Mendelian segregation of genes during
gamete formation (only one of two parental genes occurs in each
gamete). These three sources of randomness lead to small changes
in allele frequencies in a population from one generation to the
next. An important characteristic of drift is that these small
changes are cumulative; that is, the starting point for
the next generation is the allele frequency of the present generation,
and not the frequencies of previous generations. If frequencies
change from 0.30 to 0.32 in one generation, the next generation
starts from 0.32 and has no memory that the frequency was ever
at 0.30. Another characteristic of random drift is that the direction
of change is not predetermined. The frequency of each generation
can change up or down, so the frequency can randomly 'walk' away
from the original frequency, then cross back over it again. If
you repeat the same simulation with the same starting allele frequency,
you will not get the same path each time.
Natural selection results in the unequal representation of different alleles in the next generation, owing to differences in survival or reproduction between different genotypes. A particular case that is of importance to this symposium is the adaptation of genotypes to the local environment. We might have an allele, A, that is favored in the local environment but not elsewhere. Thus genotype AA might have fitness 1.05, genotype Aa fitness 1.03, and genotype aa 1.00. For population genetic purposes, it does not matter what units we measure fitness in: all that matters is the ratio of the fitnesses of different genotypes. In this case, we have arbitrarily taken genotype aa to have fitness 1.00. Genotype AA has a 5% higher fitness than aa, and Aa has a 3% higher fitness. The quantities 0.05 and 0.03 here are called selection coefficients (s): they give us a quick idea of how strong natural selection is.
If natural selection occurs in a randomly mating population, with
no migration or genetic drift, we can easily calculate what happens
to the allele frequencies. It will surprise no one that in this
case, allele A will continue to increase in frequency until
it approaches 1. The speed with which this happens is a function
of the selection coefficient. If the selection coefficient is
0.01, it will take hundreds of generations for allele frequencies
to change substantially. For example, with the fitnesses I just
gave (1.05:1.03:1), it will take about 200 generations for the
allele frequency to rise from 0.10 to 0.90. If the selection
coefficient is smaller, it will take proportionally longer. For
selection coefficients one-tenth as great (1.005:1.003:1), it
will take about 2,000 generations instead of 200.
There are many interesting and complex results for more complex
patterns of fitness (overdominance, in which the heterozygote
has the highest fitness, underdominance, in which it has the lowest
fitness, frequency-dependent fitnesses, temporally varying fitnesses,
fitnesses dependent on multiple loci, and so on). But we will
primarily deal with the simple pattern of local adaptation here.
Let us first combine the effects of migration and random drift.
Migration between populations tends to average out allele frequencies
so populations become more and more similar, whereas random drift
tends to make populations different. Figure 3
shows three populations
that are exchanging genes at a particular rate and in some kind
of pattern. The allelic frequencies in each population will wander
over time as they undergo genetic drift, but the amount and direction
of divergence between the populations is constrained by migration
between them. If one population reaches a high allele frequency,
a high proportion of the migrants into the other two populations
will have the high-frequency gene, and migration will tend to
pull the frequencies in the other two populations in the same
direction. At the same time, random drift--thermal noise like
Brownian motion--will tend to pull the frequencies of the three
populations apart. The result is that the whole set of populations,
or the species as a whole, will change at a slower rate than individual
When migration and genetic drift are operating in the absence
of natural selection, the important quantity is four times the
effective population size, Ne, times the migration rate
m, 4Nem. The effective population size is the population
size corrected for other factors that affect the amount of genetic
drift expected in the population. These factors include unequal
contributions of offspring from different individuals in the
population, unequal numbers of males and females, overlapping
generations, and several other factors. These factors usually
reduce the effective population size and cause more genetic drift.
Population genetics theory shows that if 4Nem is much
less than one, the populations act more or less independently
of one another and allelic frequencies in a set of populations
become quite dispersed. If this number is much greater than one,
allele frequencies in the populations tend to be similar to one
another. Note that Nem, the effective population size
times the proportion of migrants coming into a population, is
simply the number of migrants. If the number of migrants for
a set of populations exchanging migrants is less than one per
generation, the populations will tend to drift apart, and this
is true whether the sizes of the populations are 100 or 1 million.
The importance of genetic drift depends not on the proportion
of migrants, but on the number of migrants, and the size
of the population is unimportant. This is strange but true.
Population geneticists use abstract models to understand the effects
of random drift and migration on sets of populations with specific
geographic structures. One such model is called the island
model of migration, in which local populations receive immigrants
from a pool of migrants drawn from each population. There is
really no geographic structure in the model. No two populations
are closer to each other than any other two. Another abstract
representation of population structure is called the stepping
stone model, in which migration is limited to neighboring
populations. Stepping stone models can be one-, two-, or even
three-dimensional, depending on the biology of the species being
considered. More realistic models can also be constructed in
which populations can be situated anywhere with specific sizes
and specific migration rates. These kinds of models, however,
are complicated mathematically and are usually studied with numerical
Some work has been done on models similar to the one I will develop
here (Haldane 1930, Hanson 1966), which I call patch swamping.
Let us imagine five populations with stepping stone migration
between them; that is, each population exchanges migrants only
with its two neighbors at a rate m1/2 so that the total
fraction of immigrants is m1 (Fig. 4). An end population
receives migrants from a hatchery population, also with a migration
rate of m1/2. Whatever comes into the population most
distant from the hatchery can get there only through the other
populations by working its way down the chain of populations.
Long-range straying is also possible; I am not sure what kind
of gene flow is most important for salmon. In this long-range
model of straying, migrants can go into any of the populations.
Let us label the exchange rate between neighboring populations
as m1, and the long-distance migration rate as m2.
First of all, let us consider an allele at a gene that has an
adaptive advantage over other alleles in the local populations.
In the absence of migration from the hatchery, this allele will
increase in the natural populations to a frequency of 100%, except
for the small effects of mutation. Next, let us add the effects
of migration from a hatchery population that does not have the
favored allele, so that the frequency of this allele in the hatchery
is 0%. The pattern of allele frequencies among the populations
depends on the relative amounts of local and long-range straying
that we expect to see. The 'simulations' reported here are exact
calculations, by computer, of the allele frequencies that we would
see in the absence of genetic drift.
In the first simulation, we set m2 to 0.10, so that 10%
of the fish in the end population are strays from the hatchery.
We also set selection to 0.10, so that fish carrying the favored
allele have a 10% increase in fitness for each copy of the allele
they carry. If a fish is heterozygous with one non-native allele
and one favored allele, it is 10% better off than a hatchery
fish with two non-native alleles; however, if it is homozygous
with two copies of the favored allele, it is 21% better off.
In the first generation of the simulation, some of the non-adapted
alleles from the hatchery get into the end population, so the
frequency of the adapted allele is only 90% in that population
(Fig. 5A; I have drawn the hatchery population twice so that its
allele frequency is more visible). The frequency of the favored
allele is still 100% in the remaining four populations. The simulations
then continue for 5,000 generations (1, 10, and 5,000 generations
are shown to give you a feel for the rate of change). As the
simulation proceeds, the frequency of the non-native allele begins
to increase down the chain of natural populations, but it is lower
in the more distant populations. The frequency of the favored
allele in the most distant population is still close to 100%,
so this population is resisting the immigration of the non-adaptive
allele from the hatchery. When we set the migration rate to 20%,
we get a similar pattern, except that more hatchery alleles appear
in the natural populations, and the frequency of the favored allele
in the most distant population drops to 98%. At 50% migration,
a smooth geographic pattern appears--a cline--and the frequency
of the favored allele in the end population is 90% when the system
stabilizes. One conclusion from these results is that for a linear
string of populations with stepping stone migration, the populations
have a tremendous ability to resist migration from hatcheries.
But note that the selection coefficient used here was rather
What happens with a favored allele with only a 1% selective advantage?
Such a selective value is not small in evolutionary terms, and
is sufficient to make large changes in allele frequencies over
long periods. In real life, however, it is difficult to measure
a fitness value of only 1%, because humans can measure far fewer
fish than nature can. Researchers are limited to the number of
fish they can measure with the sizes of grants usually available
from funding agencies, whereas nature measures millions of fish.
It is also difficult to get a grant that would last 5,000 generations.
We will still use 10% immigration from neighboring populations.
These results show that after 5,000 generations, more of the
hatchery allele is getting through to the end population, which
has a frequency of the favored allele of 66% (Fig. 5B). This
shows that immigration of non-adaptive alleles is more effective
when selection favoring local adaptation is not strong.
If we increase the amount of migration with a 1% selection coefficient,
we see the patch swamping phenomenon. At 20% migration, allele
frequencies appear to form a cline after a few generations, but
the cline stablizes at very low frequencies. The locally favored
allele is still present, but only at a maximum of 20%, and the
hatchery allele is getting through to the most distant population
(Fig. 5C). At a higher migration rate of 30%, the cline collapses,
and at 5,000 generations only a very small frequency of the favorable
allele is present in the natural populations (Fig. 5D). The patches
of local adaptation have been completely erased by migration from
the hatchery into a single end population. This model does not
take into consideration that the hatchery straying rate may be
much higher than the natural migration rate among wild populations.
It also does not account for long-distance migration beyond neighboring
Let us now incorporate long-distance migration, by using a 1%
long-range straying rate from the hatchery superimposed on a natural
migration rate of 10% between the wild populations. An allele-frequency
cline appears, but many more hatchery alleles move into the most
distant population than would be the case for no long-distance
straying (Fig. 6A). Compare this with the same values for selection
and natural migration, but without long-distance straying (Fig.
5A). Long-distance straying dramatically increases the migration
of hatchery alleles. An increase in long-distance hatchery straying
of 2%, 5%, and 8% progressively depresses the allele-frequency
cline among the natural populations, so that the cline has virtually
collapsed at 8% long-distance straying, and only a very few adaptive
alleles are present in the natural populations (Fig. 6B). The
point is that long-distance straying greatly erodes the populations'
ability to resist the immigration of non-adaptive alleles, because
the non-adaptive alleles can get to the end of chain of populations
in one jump without having to travel through the string of populations.
All of these results show that the collapse of the patch of adaptation
occurs at a critical ratio of the strength of selection to the
migration rate, and depends on which model is used. If the rates
of immigration are larger than the difference in the fitness of
the adaptive and non-adaptive hatchery gene (m > s),
locally adaptive alleles will predictably be swamped by hatchery
alleles. Since this occurs locus by locus, allele by allele,
a situation could arise in which a local population has several
locally adaptive alleles, some strongly favored and others only
weakly favored, in the face of some mix of local and long-distance
migration. Weakly favored alleles may be replaced by hatchery
alleles, but strongly favored alleles may persist in a clinal
pattern. Because of this locus-by-locus complexity, fish in the
populations along the cline will be made up of a mix of adapted
and non-adapted genotypes to varying degrees. If the alleles
in the natural populations are neutral to selection and have differentiated
among populations because of random drift, then hatchery alleles
will push out local alleles and homogenize the frequencies of
alleles among the natural populations. So, fortuitous adaptations
due to genetic drift will not resist invasion from hatchery alleles.
On the other hand, adaptations due to natural selection will
resist the invasion of hatchery alleles to the extent that the
strength of natural selection is greater than the amount of gene
In the simulations presented here, we have assumed that the effects
for one locus are independent of those for other loci. This is
not quite true, because loci are often physically linked together
on the same chromosome. Slatkin (1975) showed that if two genes
are close to each other on a chromosome, and there is little recombination
between them, alleles at the two loci will tend to be associated
with one another in geographically structured populations. For
example, suppose we have two populations: population 1 has all
capital A alleles at locus A and B alleles
at locus B, and population 2 has all a and b
alleles at the two corresponding loci. If individuals from the
two populations are mixed, you would find only A-B and
a-b chromosomes. After random mating, but with very low rates
of recombination because of linkage, you will find not only A-B
and a-b chromosomes, but also double heterozygotes with
the genotype A-B/a-b and very few recombinant chromosomes,
A-b, a-B, which also produce double heterozygotes, A-b/a-B,
but with different states of linkage.
Let us assume that the a-b chromosome is from hatchery
fish and the A-B chromosome is from adapted wild fish.
A correlation appears in the population in which the adapted
alleles at one locus are associated with the adapted alleles at
the other locus. This association has the effect of helping favored
alleles resist migration from non-favored hatchery alleles, because
they travel together and natural selection favors chromosomes
with both adapted alleles over those with just one adapted allele.
Because they are physically linked, selection for one allele
is also selection for the other. The strength of selection is
as though the two individual selection coefficients are added
together. For a chromosome with two linked loci each with alleles
having a selection coefficient of 10%, the total strength of selection
for that chromosome is 20%. Selection for linked loci provides
more resistance to invasion by hatchery alleles than does selection
on two similar, but unlinked loci. So to be able to predict the
effects of hatchery straying in real life, we would have to know
how many genes confer local adaptations, the kind of natural selection
favoring them, and the strength of the linkage on the chromosome.
In addition, we would have to know how much local and how much
long-distance migration is occurring.
The genetic makeup of natural populations is potentially influenced
by an interacting mix of evolutionary forces. In the absence
of natural selection, the quantity 4Nem, four times the
number of migrants, is an important quantity. If Nem
is greater than one, then differentiation among natural populations
from random genetic drift is unimportant. When natural selection
is overlaid on migration and genetic drift, patch swamping will
occur when immigration from hatcheries is greater than the strength
of locally adapted selection. Patch swamping also occurs more
quickly with long-distance hatchery migration than with migration
into a single natural population. Linkage between loci with adapted
alleles, however, increases a wild population's ability to resist
the invasions of non-adapted alleles.
Haldane, J. B. S. 1930. A mathematical theory of natural and
artificial selection. VI. Isolation. Proceedings of the Cambridge
Philosophical Society 26:220-230.
Hanson, W. D. 1966. Effects of partial isolation (distance),
migration, and different fitness requirements among environmental
pockets upon steady state gene frequencies. Biometrics 22:453-468.
Slatkin, M. 1975. Gene flow and selection in a two-locus system.
Question: Mike Lynch: If we know a specific straying rate, what
you showed was the greater the strength of adaptive selection,
the lower the equilibrium frequency of deleterious alleles. So
from a fishery point of view, the question might be that, given
a particular amount of migration and a particular strength of
selection, how does a natural population 'feel'? Is this analogous
to mutational load where the load on the population does not depend
on selection, only on mutation?
Answer: Joe Felsenstein: Yes, the analogy with mutational load
is correct. The fitness of a wild population will be controlled
in much the same way that the fitness of a population receiving
deleterious mutations is reduced. This is the concept of mutational
load. The effects of hatchery straying would be called migrational
load, and you can use the principle that deleterious alleles will
sooner or later be selected out. When deleterious alleles are
selected out, you have one reproductive failure (death) for each
copy of the deleterious allele that comes into the population.
On the other hand, if the individuals being eliminated through
selection carry multiple hatchery alleles, then there will be
less than one death per allele, because each death eliminates
more than one deleterious allele.
If migration is 10% in a set of populations that have formed a
cline because the hatchery alleles are being resisted, the fitness
of a population is reduced by 10%, and you do not need to know
what the actual selection coefficient of the allele is. It is
the migration rate that is most important in reducing fitness.
However, if the adapted patches are swamped by hatchery alleles,
you can 'calculate' the effects by saying that the natural populations
used to have an adaptive allele, but do not have it any more.
In this case, the reduction in fitness depends on the selection
Question: Gary James: You have assumed that hatchery alleles
are not favorable in natural habitats, but what if the hatchery
alleles do show favorable traits for local adaptation?
Answer: Joe Felsenstein: If that is the case, then only genetic
drift is important. With more than 1/4 of a migrant per generation,
the natural populations will all have similar allelic frequencies.
If not, they will genetically diverge by genetic drift. Allele-frequency
clines would not appear, patch swamping would be unimportant,
and fitness in the wild populations would not change.
Question: Audience: It seems from what salmon biologists know
about hatchery straying that the value of Nem is probably
larger than one, so how do we use these principles?
Answer: Joe Felsenstein: If this value is greater than one,
the frequencies of selectively neutral genes in the natural populations
will be pushed toward the frequencies of these genes in the hatchery
population. If wild alleles have higher fitness than hatchery
alleles, the effects depend on the balance between the migration
rate and the selection coefficient.
Question: Audience: Why did you use 5,000 generations in these
simulations? Most salmon populations we work with have not been
in the rivers for that long because of events in the Late Pleistocene,
and selection has changed over that time and will continue to
change in the future.
Answer: Joe Felsenstein: I thought 5,000 generations would be
enough to show whatever might happen, but most of the changes
took place in tens of generations. In most of these simulations,
equilibrium conditions arrived in a short while; I continued the
simulations just to make sure. I hope you do not come away with
the impression that these are only very long-term problems.
Comment: Robin Waples: One of the limitations of theoretical population genetics is that it is difficult to make the transition from dealing with frequencies of alleles at a single locus to what happens in organisms as a whole, which have thousands of gene loci affecting fitness.
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