## 1. The Origins of Population Genetics



To understand how population genetics came into being, and to
appreciate its intellectual significance, a brief excursion into the
history of biology is necessary. Darwin's *Origin of
Species*, published in 1859, propounded two main theses: firstly,
that modern species were descended from common ancestors, and secondly
that the process of natural selection was the major mechanism of
evolutionary change. The first thesis quickly won acceptance in the
scientific community, but the second did not. Many people found it
difficult to accept that natural selection could play the explanatory
role required of it by Darwin's theory. This
situation--accepting that evolution had happened but doubting
Darwin's account of what had caused it to happen--persisted
well into the twentieth century (Bowler 1988).



Opposition to natural selection was understandable, for Darwin's
theory, though compelling, contained a major lacuna: an account of the
mechanism of inheritance. For evolution by natural selection to occur,
it is necessary that parents should tend to resemble their offspring;
otherwise, fitness-enhancing traits will have no tendency to spread
through a population. In the *Origin*, Darwin rested his
argument on the observed fact that offspring *do* tend to
resemble their parents--'the strong principle of
inheritance'--while admitting that he did not know why this
was. Darwin did later attempt an explicit theory of inheritance, based
on hypothetical entities called 'gemmules', but it turned
out to have no basis in fact.



Darwin was troubled by not having a proper understanding of the
inheritance mechanism, for it left him unable to rebut a powerful
objection to his theory. For a population to evolve by natural
selection, the members of the population must vary--if all
organisms are identical, no selection can occur. So for selection to
gradually modify a population over a long period of time, in the
manner suggested by Darwin, a continual supply of variation is needed.
Fleeming Jenkin argued that the available variation would be used up
too fast (Jenkin 1867). His reasoning assumed a 'blending'
theory of inheritance, i.e., that an offspring's phenotypic
traits are a 'blend' of those of its parents. (So for
example, if a short and a tall organism mate, the height of the
offspring will be intermediate between the two.) Jenkin argued that
given blending inheritance, a sexually reproducing population will
become phenotypically homogenous in just a few generations, far
shorter than the number of generations needed for natural selection to
produce complex adaptations.



Fortunately for Darwin's theory, inheritance does not actually
work the way Jenkin thought. The type of inheritance that we call
'Mendelian', after Gregor Mendel, is
'particulate' rather than
'blending'--offspring inherit discrete hereditary
particles (genes) from their parents, which means that sexual
reproduction does not diminish the heritable variation present in the
population. (See section 2, 'The Hardy-Weinberg
Principle'.) However this realisation took a long time to come,
for two reasons. Firstly, Mendel's work was overlooked by the
scientific community for forty years. Secondly, even after the
rediscovery of Mendel's work at the turn of the twentieth
century, it was widely believed that Darwinian evolution and Mendelian
inheritance were incompatible. The early Mendelians did not accept
that natural selection played an important role in evolution, so were
not well placed to see that Mendel had given Darwin's theory the
lifeline it needed. The synthesis of Darwinism and Mendelism, which
marked the birth of population genetics, was achieved by a long and
tortuous route (Provine 1971).



The key ideas behind Mendel's theory of inheritance are
straightforward. In his experimental work on pea plants, Mendel
observed an unusual phenomenon. He began with two 'pure
breeding' lines, one producing plants with round seeds, the
other wrinkled seeds. He then crossed these to produce the first
daughter generation (the F1 generation). The F1 plants all had round
seeds--the wrinkled trait had disappeared from the population.
Mendel then crossed the F1 plants with each other to produce the F2
generation. Strikingly, approximately one quarter of the F2 plants had
wrinkled seeds. So the wrinkled trait had made a comeback, skipping a
generation.



These and similar observations were explained by Mendel as follows. He
hypothesised that each plant contains a pair of 'factors'
that together determine some aspect of its phenotype--in this
case, seed shape. A plant inherits one factor from each of its
parents. Suppose that there is one factor for round seeds
\((R)\), another for wrinkled seeds \((W)\). There are then
three possible types of plant: \(RR,\) \(RW\) and
\(WW\). An \(RR\) plant will have round seeds, a \(WW\)
plant wrinkled seeds. What about an \(RW\) plant? Mendel
suggested that it would have round seeds--the \(R\) factor
is 'dominant' over the \(W\) factor. The observations
could then be easily explained. The initial pure-breeding lines were
\(RR\) and \(WW\). The F1 plants were formed by \(RR \times WW\) crosses, so were all of the \(RW\) type and
thus had round seeds. The F2 plants were formed by \(RW \times RW\) crosses, so contained a mixture of the \(RR,
RW\) and \(WW\) types. If we assume that each \(RW\)
parent transmits the \(R\) and \(W\) factors to its
offspring with equal probability, then the F2 plants would contain
\(RR, RW\) and \(WW\) in approximately the ratio
1:2:1. (This assumption is known as *Mendel's First Law*
or *The Law of Segregation*.) Since \(RR\) and \(RW\)
both have round seeds, this explains why three quarters of the F2
plants had round seeds, one quarter wrinkled seeds.



Our understanding of heredity today is vastly more sophisticated than
Mendel's, but the key elements of Mendel's
theory--discrete hereditary particles that come in different
types, dominance and recessiveness, and the law of
segregation--have turned out to be essentially correct.
Mendel's 'factors' are the genes of population
genetics, and the alternative forms that a factor can take (e.g.,
\(R\) versus \(W\) in the pea plant example) are known as
the *alleles* of a gene. The law of segregation is explained by
the fact that during gametogenesis, each gamete (sex cell) receives
only one of each chromosome pair from its parent organism. Other
aspects of Mendel's theory have been modified in the light of
later discoveries. Mendel thought that most phenotypic traits were
controlled by a single pair of factors, like seed shape in his pea
plants, but it is now known that most traits are affected by many
pairs of genes, not just one. Mendel believed that the pairs of
factors responsible for different traits (e.g., seed shape and flower
colour) segregated independently of each other, but we now know that
this need not be so (see section 3.5, 'Two-Locus Models and
Linkage', below). Despite these points, Mendel's theory
marks a turning point in our understanding of inheritance.



The rediscovery of Mendel's work in 1900 did not lead the
scientific community to be converted to Mendelism overnight. The
dominant approach to the study of heredity at the time was biometry,
spearheaded by Karl Pearson in London, which involved statistical
analysis of the phenotypic variation found in natural populations.
Biometricians were mainly interested in continuously varying traits
such as height, rather than the 'discrete' traits such as
seed shape that Mendel studied, and were generally believers in
Darwinian gradualism. Opposed to the biometricians were the
Mendelians, spearheaded by William Bateson, who emphasized
discontinuous variation, and believed that major adaptive change could
be produced by single mutational steps, rather than by cumulative
natural selection a la Darwin. A heated controversy between the
biometricians and the Mendelians ensued. As a result, Mendelian
inheritance came to be associated with an anti-Darwinian view of
evolution.



Population genetics arose in part from the need to reconcile Mendel
with Darwin, a need which became increasingly urgent as the empirical
evidence for Mendelian inheritance began to pile up. A significant
milestone was R.A. Fisher's 1918 paper, 'The Correlation
between Relatives on the Supposition of Mendelian Inheritance',
which showed how the biometrical and Mendelian research traditions
could be unified. Fisher demonstrated that if a given continuous
trait, e.g., height, was affected by a large number of Mendelian
factors, each of which made a small difference to the trait, then the
trait would show an approximately normal distribution in a population.
Since the Darwinian process was widely believed to work best on
continuously varying traits, showing that the distribution of such
traits was compatible with Mendelism was an important step towards
reconciling Darwin with Mendel.



The full reconciliation was achieved in the 1920s and early 30s,
thanks to the mathematical work of Fisher, Haldane and Wright (Fisher
1930, Haldane 1930-32, Wright 1931). These theorists developed
formal models to explore how natural selection, and other evolutionary
factors such as mutation and random drift, would modify the genetic
composition of a Mendelian population over time. This work marked a
major step forward in evolutionary biology, for it enabled the
consequences of various evolutionary hypotheses to be explored
quantitatively rather than just qualitatively. Verbal arguments about
what natural selection could or could not achieve, or about the
patterns of genetic variation to which it would give rise, were
replaced with explicit mathematical arguments. The strategy of
devising formal models to shed light on the evolutionary process
remains the dominant methodology of population genetics today, though
unlike in the 1930s, today's modellers have a wealth of
empirical data against which to test their predictions (Hartl
2020).



Fisher and Haldane were both strong Darwinians--they believed
that natural selection was the most important factor affecting a
population's genetic composition. Wright did not downplay the
role of natural selection, but he believed that chance also played a
crucial role, as did migration between the constituent sub-populations
of a species (See sections 4, 'Random Drift', and 3.3,
'Migration'.) The respective roles of natural selection
and chance (or random drift) in shaping genetic variation, both within
and between species, became a major theme in population genetics, and
is still a live issue today. The issue lay at the heart of the
"neutralist versus selectionist" controversy of the 1960s
and 1970. The neutralists, led by Motoo Kimura, argued that much of
the molecular genetic variation found in natural populations was
likely to be neutral, i.e., the different genetic variants at a given
locus were mostly selectively equivalent (Kimura 1977, 1994). If true,
this suggests a significant role for random drift. Though initially
controversial, the idea of abundant neutral variation in DNA sequence
is today quite standard (Jensen et al. 2019); though this is fully
compatible with natural selection playing the major role in
*adaptive* evolution (Kern and Hahn 2018).



Contemporary population genetics takes place in a very different
scientific landscape to the one inhabited by Fisher, Haldane and
Wright. They were working in the pre-molecular biology era, when the
"gene" was a purely theoretical entity, posited to explain
observed patterns of inheritance, but whose structure and molecular
composition were unknown. Genetic variation could thus only be
observed indirectly, though the phenotypic variation to which it
(sometimes) gave rise. This meant that there was very little empirical
data against which population genetic models could be tested; so the
enterprise remained an essentially theoretical one. In the intervening
century, the gene has gone from being a theoretical posit to being an
entity whose molecular structure and functioning is understood in
great detail. Since the 1980s, the technology for gene sequencing,
i.e., determining the sequence of nucleotide bases in a length of DNA,
has become increasingly fast and cheap. This has allowed population
geneticists to directly study the genetic variation found in natural
populations, by sampling a number of individuals and sequencing a gene
of interest (or in some cases the whole genome). As a result,
population genetics is now a "data rich" science, the
polar opposite of the situation when the field was founded. Despite
this, many of the models, techniques and conclusions provided by the
earlier theoretical work remain directly relevant today (Charlesworth
and Charlesworth 2017).


## 2. The Hardy-Weinberg Principle



The Hardy-Weinberg principle, discovered independently by G.H. Hardy
and W. Weinberg in 1908, is one of the simplest and most important
principles in population genetics (Hardy 1908, Weinberg 1908). To
illustrate the principle, consider a large population of sexually
reproducing organisms. The organisms are *diploids*, meaning
that they contain two copies of each chromosome, one received from
each parent. The gametes they produce are *haploid*, meaning
that they contain only one of each chromosome pair. During sexual
fusion, two haploid gametes fuse to form a diploid zygote, which then
grows and develops into an adult organism. Most multi-celled animals
and many plants have a lifecycle of this sort.



Suppose that at a given locus, or chromosomal 'slot',
there are two possible alleles, \(A\_1\) and \(A\_2\); the locus is
assumed to be on an autosome, not a sex chromosome. With respect to
the locus in question, there are three possible genotypes in the
population, \(A\_1 A\_1, A\_1 A\_2\) and \(A\_2 A\_2\) (just as in
Mendel's pea plant example above). Organisms with the \(A\_1
A\_1\) and \(A\_2 A\_2\) genotypes are called *homozygotes*;
those with the \(A\_1 A\_2\) genotype are *heterozygotes*. The
proportions, or relative frequencies, of the three genotypes in the
overall population may be denoted \(f(A\_1 A\_1), f(A\_1 A\_2)\) and
\(f(A\_2 A\_2)\) respectively, where \(f(A\_1 A\_1) + f(A\_1 A\_2) + f(A\_2
A\_2) = 1\). It is assumed that these genotypic frequencies are the
same for both males and females. The relative frequencies of the \(A\)
and \(B\) alleles in the population are denoted \(p\) and \(q\), where
\(p = f(A\_1 A\_1) + \frac{1}{2}f(A\_1 A\_2)\) and \(q = f(A\_2 A\_2) +
\frac{1}{2}f(A\_1 A\_2).\) Note that \(p + q = 1\).



The Hardy-Weinberg principle is about the relation between the allelic
and the genotypic frequencies. It states that if mating is random in
the population, and if natural selection, mutation, migration and
drift are absent, then in the offspring generation the genotypic and
allelic frequencies will be related by the following simple
equations:



\[\begin{align}
f(A\_1 A\_1) &= p^2, \\
f(A\_1 A\_2) &= 2pq, \\
f(A\_2 A\_2) &= q^2
\end{align}\]


Random mating means the absence of a genotypic correlation between
mating partners, i.e., the probability that a given organism mates
with an \(A\_1 A\_1\) partner, for example, does not depend on the
organism's own genotype, and similarly for the other two
genotypes.



That random mating will lead the genotypes to be in the above
proportions ("Hardy-Weinberg proportions") is a
consequence of Mendel's law of segregation. To see this, note
that random mating is in effect equivalent to offspring being formed
by randomly picking pairs of gametes from a large 'gamete
pool' and fusing them into a zygote. The gamete pool contains
all the successful gametes of the parent organisms. Since we are
assuming the absence of selection, all parents contribute equal
numbers of gametes to the pool. By the law of segregation, an \(A\_1
A\_2\) heterozygote produces gametes bearing the \(A\_1\) and \(A\_2\)
alleles in equal proportion (on average). Therefore, the relative
frequencies of the \(A\) and \(B\) alleles in the gamete pool will be
the same as in the parental population, namely \(p\) and \(q\)
respectively. Given that the gamete pool is very large, when we pick
pairs of gametes from the pool at random, we will get the ordered
genotypic pairs \(\{A\_1 A\_1\},\) \(\{A\_1 A\_2\},\) \(\{A\_2 A\_1\},\)
\(\{A\_2 A\_2\}\) in the proportions \(p^2 :pq:qp:q^2\). But order does
not matter, so we can regard the \(\{A\_1 A\_2\}\) and \(\{A\_2 A\_1\}\)
pairs as equivalent, giving the Hardy-Weinberg proportions for the
unordered offspring genotypes.



This simple derivation of the Hardy-Weinberg principle deals with two
alleles at a single locus, but can easily be extended to multiple
alleles. (Extension to more than one locus is trickier; see section
3.6 below.) For the multi-allelic case, suppose there are \(n\)
alleles at the locus, \(A\_1 \ldots A\_n\), with relative frequencies of
\(p\_1 \ldots p\_n\) respectively, where \(p\_1 + p\_2 + \ldots + p\_n =
1\). Assuming again that the population is large, mating is random,
evolutionary forces are absent, and Mendel's law of segregation
holds, then in the offspring generation the frequency of the
\(A\_i A\_i\) genotype will be \(p\_i^2\), and the frequency of
the (unordered) \(A\_i A\_j\) genotype \((i \ne j)\) will be
\(2p\_i p\_j\). Note that the two allele case is a special case
of this generalized principle.



Importantly, whatever the initial genotypic proportions, random mating
will automatically produce offspring in Hardy-Weinberg proportions
(for one-locus genotypes). So if generations are non-overlapping,
i.e., parents die as soon as they have reproduced, just one round of
random mating is needed to bring about Hardy-Weinberg proportions in
the whole population; if generations overlap, more than one round of
random mating is needed. Once Hardy-Weinberg proportions have been
achieved, they will be maintained in subsequent generations so long as
the population continues to mate at random and is unaffected by
evolutionary forces. The population is then said to be in
*Hardy-Weinberg equilibrium*--meaning that the genotypic
frequencies are constant from generation to generation.



The importance of the Hardy-Weinberg principle lies in the fact that
it contains the solution to the problem of blending inheritance that
troubled Darwin. Jenkin's argument that sexual reproduction will
rapidly diminish the variation in a population is disproved by the
Hardy-Weinberg principle. Sexual reproduction has no inherent tendency
to destroy genotypic variation, for the genotypic proportions remain
constant over generations, given the assumptions noted above. It is
true that *natural selection* often tends to destroy variation,
and is thus a homogenizing force; but this is a quite different
matter. The 'blending' objection was that sexual mixing
*itself* would produce homogeneity, even in the absence of
selection, and the Hardy-Weinberg principle shows that this is
untrue.



Another benefit of the Hardy-Weinberg principle is that it greatly
simplifies the task of modelling evolutionary change. When a
population is in Hardy-Weinberg equilibrium, it is possible to track
the genotypic composition of the population by directly tracking the
allelic frequencies (or gametic frequencies). That this is so is
clear--for if we know the relative frequencies of all the alleles
(at a single locus), and know that the population is in Hardy-Weinberg
equilibrium, the entire genotype frequency distribution can be easily
computed. Were the population not in Hardy-Weinberg equilibrium, we
would need to explicitly track the genotype frequencies themselves,
which is more complicated.



Primarily for this reason, many population-genetic models assume that
Hardy-Weinberg equilibrium obtains; this is tantamount to assuming
that mating is random with respect to genotype. But is this assumption
empirically plausible? The answer is sometimes but not always. In the
human population, for example, mating is close to random with respect
to ABO blood group, so the genotype that determines blood group is
found in approximately Hardy-Weinberg proportions in many populations
(Hartl and Clark 2006). But mating is not random with respect to
height; on the contrary, people tend to choose mates similar in height
to themselves. So if we consider a genotype that influences height,
mating will not be random with respect to this genotype (see section
3.4 'Non-Random Mating').



The population geneticist W.J. Ewens has written of the Hardy-Weinberg
principle, "it does not often happen that the most important
theorem in any subject is the easiest and most readily derived theorem
for that subject" (1969, p. 1). The main importance of the
principle, as Ewens stresses, is not the gain in mathematical
simplicity that it permits, which is simply a beneficial side effect,
but rather what it teaches us about the preservation of genetic
variation in a sexually reproducing population.


## 3. Population-Genetic Models of Evolution



Evolutionary biologists often define 'evolution' as any
change in a population's genetic composition over time. The
rationale for this definition is the idea that all other aspects of
evolution, e.g., the spread of novel phenotypic traits and the
formation of new species, stem ultimately from changes in gene
frequencies within populations. The four factors that can bring about
such a change are: natural selection, mutation, random genetic drift,
and migration into or out of the population. (A fifth
factor--changes to the mating pattern--can change the
*genotype* but not the *gene* frequencies; many
theorists would not count this as an evolutionary change.) A brief
introduction to the standard population-genetic treatment of each of
these factors is given below.


### 3.1 Selection at One Locus



Natural selection occurs when some variants in a population enjoy a
survival or reproductive advantage over others. The simplest
population-genetic model of natural selection focuses on a single
autosomal locus with two alleles, \(A\_1\) and \(A\_2\), in a large
population. Random mating is assumed. The three diploid genotypes
\(A\_1 A\_1,\) \(A\_1 A\_2\) and \(A\_2 A\_2\) have different fitnesses,
denoted by \(w\_{11},\) \(w\_{12}\) and \(w\_{22}\) respectively. These
fitnesses are assumed to be constant across generations. A
genotype's fitness may be defined, in this context, as the
average number of successful gametes that an organism of that genotype
contributes to the next generation--which depends on how well the
organism survives, how many matings it achieves, and how fertile it
is. Unless \(w\_{11},\) \(w\_{12}\) and \(w\_{22}\) are all equal, then
natural selection will occur, which may lead the genetic composition
of the population to change.



Suppose that initially, i.e., before selection has operated, the
zygote genotypes are in Hardy-Weinberg proportions and the frequencies
of the \(A\_1\) and \(A\_2\) alleles are \(p\) and \(q\) respectively,
where \(p + q = 1\). The zygotes then grow to adulthood and reproduce,
giving rise to a new generation of offspring zygotes. Our task is to
compute the frequencies of \(A\_1\) and \(A\_2\) in the second
generation; let us denote these by \(p'\) and \(q'\) respectively,
where \(p' + q' = 1\). (Note that in both generations, we consider
gene frequencies at the zygotic stage; these may differ from the adult
gene frequencies if there is differential survivorship.)



In the first generation, the genotypic frequencies at the zygotic
stage are \(p^2 , 2pq\) and \(q^2\) for \(A\_1 A\_1,\) \(A\_1 A\_2,\)
\(A\_2 A\_2\) respectively, by the Hardy-Weinberg principle. The three
genotypes produce successful gametes in proportion to their fitnesses,
i.e., in the ratio \(w\_{11}:w\_{12}:w\_{22}\). The average fitness in
the population is \(w = p^2 w\_{11} + 2pq w\_{12} + q^2
w\_{22}\). Assuming there is no mutation, and that Mendel's law
of segregation holds, then an \(A\_1 A\_1\) organism will produce only
\(A\_1\) gametes, an \(A\_2 A\_2\) organism will produce only \(A\_2\)
gametes, and an \(A\_1 A\_2\) organism will produce \(A\_1\) and \(A\_2\)
gametes in equal proportion (on average). Therefore, the proportion of
\(A\_1\) gametes, and thus the frequency of the \(A\_1\) allele in the
second generation at the zygotic stage, is:



\[\tag{1}
\begin{align}
p' &= \frac{p^2 w\_{11} + \frac{1}{2} (2pq w\_{12})}{w} \\
&= \frac{p^2 w\_{11} + pq w\_{12}}{w}
\end{align}\]


Equation (1) is known as a 'recurrence' equation--it
expresses the frequency of the \(A\_1\) allele in the
second generation in terms of its frequency in the first generation.
The change in frequency between generations can then be written
as:



\[\tag{2}
\begin{align}
\Delta p &= p' - p \\
&= \frac{p^2 w\_{11} + pq w\_{12}}{w} - p \\
&= \frac{pq[p(w\_{11} - w\_{12}) + q(w\_{12} - w\_{22})]}{w}
\end{align}\]


If \(\Delta p \gt 0\), then natural selection has led the
\(A\_1\) allele to increase in frequency; if
\(\Delta p \lt 0\) then selection has led the
\(A\_2\) allele to increase in frequency. If
\(\Delta p = 0\) then no gene frequency change has occurred,
i.e., the system is in allelic equilibrium. (Note, however, that the
condition \(\Delta p = 0\) does *not* imply that no
natural selection has occurred; the condition for that is
\(w\_{11} = w\_{12} = w\_{22}\). It is possible for natural selection to occur
but to have no effect on gene frequencies.)



Equations (1) and (2) show, in precise terms, how fitness differences
between genotypes will lead to evolutionary change. This enables us to
explore the consequences of various different selective regimes.



Suppose firstly that \(w\_{11} \gt w\_{12} \gt w\_{22}\), i.e., the \(A\_1
A\_1\) homozygote is fitter than the \(A\_1 A\_2\) heterozygote, which in
turn is fitter than the \(A\_2 A\_2\) homozygote. By inspection of
equation (2), we can see that \(\Delta p\) must be positive (so long
as neither \(p\) nor \(q\) is zero). So in each generation, the
frequency of the \(A\_1\) allele will be greater than in the previous
generation, until it eventually reaches fixation. Once the \(A\_1\)
allele reaches fixation, i.e., \(p = 1\) and \(q = 0\), no further
evolutionary change will occur, for if \(p = 1\) then \(\Delta p =
0\). This makes good sense intuitively: since the \(A\_1\) allele
confers a fitness advantage on organisms that carry it, its relative
frequency in the population will increase from generation to
generation until it is fixed.



It is obvious that analogous reasoning applies in the case where
\(w\_{22} \gt w\_{12} \gt w\_{11}\). Equation (2) tells us that \(\Delta p\)
must then be negative, so long as neither \(p\) nor \(q\) is
zero, so the \(A\_2\) allele will sweep to fixation.



A more interesting situation arises when the heterozygote is superior
in fitness to both of the homozygotes, i.e., \(w\_{12} \gt w\_{11}\) and
\(w\_{12} \gt w\_{22}\)--a phenomenon known as *heteroygote
superiority*. Intuitively it is clear what should happen in this
situation: an equilibrium situation should be reached in which both
alleles are present in the population. Equation (2) confirms this
intuition. It is easy to see that \(\Delta p = 0\) if either allele
has gone to fixation (i.e., if \(p = 0\) or \(q = 0)\), or, thirdly,
if the following condition obtains:



\[
p(w\_{11} - w\_{12}) + q(w\_{12} - w\_{22}) = 0
\]



which reduces to



\[
p = p^\* = \frac{(w\_{12} - w\_{22})}{(w\_{12} - w\_{22}) + (w\_{12} - w\_{11})}
\]


(The asterisk indicates that this is an equilibrium condition.) Since
\(p\) must be non-negative, this condition can only be satisfied
if there is heterozygote superiority or inferiority; it represents an
equilibrium state of the population in which both alleles are present.
This equilibrium is known as *polymorphic*, by contrast with
the *monomorphic* equilibria that arise when either of the
alleles has gone to fixation. The possibility of polymorphic
equilibrium is quite significant. It teaches us that natural selection
will not always lead to homogeneity; in some cases, selection
preserves the genetic variation found in a population.



Numerous evolutionary questions can be addressed using simple
population-genetic models of this sort. For example, by incorporating
a parameter which measures the fitness differences between genotypes,
we can study the *rate* of evolutionary change, permitting us
to ask questions such as: how long will it take for selection to
increase the frequency of the \(A\_1\) allele from 0.1
to 0.9? If a given deleterious allele is recessive, how much longer
will it take to eliminate it from the population than if it were
dominant? In this way, population genetics converted the theory of
evolution into a quantitatively precise one.



The one-locus model outlined above is unlikely to apply to many
real-life populations, due to the simplifying assumptions it makes. In
reality, selection is rarely the only evolutionary force in operation,
genotypic fitnesses are unlikely to be constant across generations,
Mendelian segregation does not always hold exactly, and not all
evolving populations are large. Much effort in population genetics has
been put into making more realistic models which relax these
assumptions and are thus more complicated. But the one-locus model
illustrates the essence of the population-genetic analysis of
evolutionary change.


### 3.2 Selection-Mutation Balance



Mutation is the ultimate source of genetic variation, preventing
populations from becoming genetically homogeneous. Once mutation is
taken into account, the conclusions drawn in the previous section need
to be modified. Even if one allele is selectively superior to all
others at a given locus, it will not become fixed in the population;
recurrent mutation will ensure that other alleles are present at low
frequency, thus maintaining a degree of polymorphism. Population
geneticists have long been interested in exploring what happens when
selection and mutation act simultaneously.



Continuing with our one locus, two allele model, suppose that the
\(A\_1\) allele is selectively superior to
\(A\_2\), but recurrent mutation from
\(A\_1\) to \(A\_2\) prevents
\(A\_1\) from spreading to fixation. The rate of
mutation from \(A\_1\) to \(A\_2\) per
generation, i.e., the proportion of \(A\_1\) alleles
that mutate every generation, is denoted \(u\). (Empirical
estimates of mutation rates are typically in the region of
\(10^{-6}\).) Back mutation from \(A\_2\) to
\(A\_1\) can be ignored, because we are assuming that
the \(A\_2\) allele is at a very low frequency in the
population, thanks to natural selection. What happens to the gene
frequency dynamics under these assumptions? Recall equation (1) above,
which expresses the frequency of the \(A\_1\) allele in
terms of its frequency in the previous generation. Since a certain
fraction \((u)\) of the \(A\_1\) alleles will have
mutated to \(A\_2\), this recurrence equation must be
modified to:



\[
p' = \frac{(p^2 w\_{11} + pq w\_{12}) (1 - u)}{w}
\]


to take account of mutation. As before, equilibrium is reached when
\(p' = p\), i.e., \(\Delta p = 0\). The
condition for equilibrium is therefore:



\[\tag{3}
p = p^\* = \frac{(p^2 w\_{11} + pq w\_{12}) (1 - u)}{w}
\]


A useful simplification of equation (3) can be achieved by making some
assumptions about the genotype fitnesses, and adopting a new notation.
Let us suppose that the \(A\_2\) allele is completely recessive (as is
often the case for deleterious mutants). This means that the \(A\_1
A\_1\) and \(A\_1 A\_2\) genotypes have identical fitness. Therefore,
genotypic fitnesses can be written \(w\_{11} = 1,\) \(w\_{12} = 1,\)
\(w\_{22} = 1 - s,\) where \(s\) denotes the difference in fitness of
the \(A\_2 A\_2\) homozygote from that of the other two
genotypes. \((s\) is known as the *selection coefficient*
against \(A\_2 A\_2)\). Since we are assuming that the \(A\_2\) allele is
deleterious, it follows that \(s \gt 0\). Substituting these genotype
fitnesses in equation (3) yields:




\[
p^\* = \frac{p (1 - u)}{p^2 + 2pq + q^2 (1 - s)}
\]


which reduces to:



\[
p^\* = 1 - \left(\frac{u}{s}\right)^{\frac{1}{2}}
\]


or equivalently (since \(p + q = 1)\):



\[\tag{4}
q^\* = \left(\frac{u}{s}\right)^{\frac{1}{2}}
\]


Equation (4) gives the equilibrium frequency of the \(A\_2\) allele,
under the assumption that it is completely recessive. Note that as
\(u\) increases, \(q\)\* increases too. This is highly intuitive: the
greater the mutation rate from \(A\_1\) to \(A\_2\), the greater the
frequency of \(A\_2\) that can be maintained at equilibrium, for a
given value of \(s\). Conversely, as \(s\) increases, \(q\)\*
decreases. This is also intuitive: the stronger the selection against
the \(A\_2 A\_2\) homozygote, the lower the equilibrium frequency of
\(A\_2\), for a given value of \(u\).



It is easy to see why equation (4) is said to describe
*selection-mutation* balance--natural selection is
continually removing \(A\_2\) alleles from the population, while
mutation is continually re-creating them. Equation (4) tells us the
equilibrium frequency of \(A\_2\) that will be maintained, as a
function of the rate of mutation from \(A\_1\) to \(A\_2\) and the
magnitude of the selective disadvantage suffered by the \(A\_2 A\_2\)
homozygote. Though equation (4) was derived under the assumption that
the \(A\_2\) allele is completely recessive, it is straightforward to
derive similar equations for the cases where the \(A\_2\) allele is
dominant, or partially dominant. In those cases, \(A\_2\)'s
equilibrium frequency will be lower than if it is completely
recessive; for selection is more efficient at removing it from the
population. A deleterious allele that is recessive can
'hide' in heterozygotes, and thus escape the purging power
of selection, but a dominant allele cannot.



Our discussion in this section has focused on deleterious mutations,
i.e., ones which reduce the fitness of their host organism. This may
seem odd, given that beneficial mutations play a key role in the
evolutionary process. The reason is that in population genetics, a
major concern is to understand the causes of the genetic variability
found in biological populations. If a gene is beneficial, natural
selection is likely to be the major determinant of its equilibrium
frequency; the rate of sporadic mutation to that gene will play at
most a minor role. It is only where a gene is deleterious that
mutation plays a major role in maintaining it in a population.


### 3.3 Migration



Migration into or out of a population is a third factor that can
affect its genetic composition. Obviously, if immigrants are
genetically different from the population they are entering, this will
cause the population's genetic composition to change. The
evolutionary importance of migration stems from the fact that many
species are composed of a number of distinct subpopulations, largely
isolated from each other but connected by occasional migration.
Migration between subpopulations gives rise to gene flow, which acts
as a sort of 'glue', limiting the extent to which
subpopulations can diverge from each other genetically.



The simplest model for analysing migration assumes that a given
population receives a number of migrants each generation, but sends
out no emigrants. Suppose the frequency of the \(A\_1\)
allele in the resident population is \(p\), and the frequency of
the \(A\_1\) allele among the migrants arriving in the
population is \(p\_m\). The proportion of
migrants coming into the population each generation is \(m\)
(i.e., as a proportion of the resident population.) So post-migration,
the frequency of the \(A\_1\) allele in the population
is:



\[
p' = (1 - m) p + mp\_m
\]


The change in gene frequency across generations is therefore:



\[\begin{align}
\Delta p &= p' - p \\
 &= -m(p - p\_m)
\end{align}\]


Therefore, migration will increase the frequency of the \(A\_1\) allele
if \(p\_m \gt p\), decrease its frequency if \(p \gt p\_m\), and leave
its frequency unchanged if \(p = p\_m\). We can then derive an equation
giving the gene frequency in generation \(t\) as a function of its
initial frequency and the rate of migration::



\[
p\_t = p\_m + (p\_0 - p\_m)(1 - m)^t
\]


where \(p\_0\) is the initial frequency of the \(A\_1\) allele in the
population, i.e., before any migration has taken place. Since the
expression \((1 - m)^t\) tends towards zero as \(t\) grows large, it
follows that equilibrium will eventually be reached when \(p\_t =
p\_m\), i.e., when the gene frequency of the migrants equals the gene
frequency of the resident population.



This simple model assumes that migration is the only factor affecting
gene frequency at the locus, but this is unlikely to be the case. So
it is necessary to consider how migration will interact with
selection, drift and mutation (cf. Rice 2004, ch.5). A balance between
migration and selection can lead to the maintenance of a deleterious
allele in a population, in a manner analogous to mutation-selection
balance. The interaction between migration and drift is especially
interesting. Genetic drift will often lead the separate subpopulations
of a species to diverge genetically. Migration opposes this
trend--it is a homogenising force that tends to make
subpopulations more alike. Mathematical models suggest that that even
a fairly small rate of migration will be sufficient to prevent the
subpopulations of a species from diverging genetically (Hartl and
Clark 2006). Some theorists have used this to argue against the
evolutionary importance of group selection, on the grounds that
genetic differences between groups, which are essential for group
selection to operate, are unlikely to persist in the face of
migration.


### 3.4 Non-Random Mating



Recall that the Hardy-Weinberg principle was derived under the
assumption of random mating. But departures from random mating are
actually quite common. Organisms may tend to choose mates who are
similar to them phenotypically or genotypically--a mating system
known as 'positive assortment'. Alternatively, organisms
may choose mates dissimilar to them--'negative
assortment'. Another type of departure from random mating is
inbreeding, or preferentially mating with relatives.



Analysing the consequences of non-random mating gets quite
complicated, but some conclusions are fairly easily seen. Firstly,
non-random mating does not in itself affect gene frequencies (so
arguably is not an evolutionary 'force' on a par with
selection, mutation and migration); rather, it affects genotype
frequencies. To appreciate this point, note that the gene frequency of
a population, at the zygotic stage, is equal to the gene frequency in
the pool of successful gametes from which the zygotes are formed. The
pattern of mating simply determines the way in which haploid gametes
are 'packaged' into diploid zygotes. Thus if a random
mating population suddenly starts to mate non-randomly, this will have
no effect on gene frequencies.



Secondly, positive assortative mating will tend to decrease the
proportion of heterozygotes in the population,. To see this, consider
again a single locus with two alleles, \(A\_1\) and \(A\_2\), with
frequencies \(p\) and \(q\) in a given population. Suppose that
initially the population is at Hardy-Weinberg equilibrium, so the
proportion of \(A\_1 A\_2\) heterozygotes is \(2pq\). If the population
then starts to mate completely assortatively, i.e., mating only occurs
between organisms of identical genotype, it is obvious that the
proportion of heterozygotes must decline. For \(A\_1 A\_1 \times A\_1
A\_1\) and \(A\_2 A\_2 \times A\_2 A\_2\) matings will produce no
heterozygotes; and only half the progeny of \(A\_1 A\_2 \times A\_1 A\_2\)
matings will be heterozygotic (on average). So the proportion of
heterozygotes in the second generation must be less than
\(2pq\). Conversely, negative assortment will tend to increase the
proportion of heterozygotes from what it would be under Hardy-Weinberg
equilibrium.



What about inbreeding? In general, inbreeding will tend to increase
the homozygosity of a population, like positive assortment (Hartl
2020, ch. 3). This is because relatives tend to be genotypically
similar. Inbreeding often has negative effects on organismic
fitness--a phenomenon known as 'inbreeding
depression'. The explanation for this is that deleterious
alleles often tend to be recessive, so have no phenotypic effect when
found in heterozygotes. Inbreeding reduces the proportion of
heterozygotes, making recessive alleles more likely to be found in
homozygotes where their negative phenotypic effects become apparent.
The converse phenomenon--'hybrid vigour' resulting
from outbreeding--is widely utilised by animal and plant
breeders.


### 3.5 Two-Locus Models and Linkage



The one-locus model outlined above is unrealistic, since in practice
evolution may occur at multiple loci simultaneously. The simplest
two-locus model assumes two autosomal loci, \(A\) and \(B\), each with
two alleles, \(A\_1\) and \(A\_2, B\_1\) and \(B\_2\). Thus there are four
types of haploid gamete in the population--\(A\_1 B\_1, A\_1 B\_2,
A\_2 B\_1\) and \(A\_2 B\_2\)--whose frequencies we will denote by
\(x\_1, x\_2, x\_3\) and \(x\_4\) respectively. (Note that the \(x\_i\)
are *not* allele frequencies; in the two-locus case, we cannot
equate 'gamete frequency' with 'allelic
frequency', as is possible for a single locus.) Diploid
organisms are formed by the fusion of two gametes, as before. Thus
there are ten possible diploid genotypes in the population--found
by taking every gamete type in combination with every other.



In the one-locus case, we saw that in a large randomly-mating
population, there is a simple relationship between the gametic and
zygotic frequencies. In the two-locus case, the same relationship
holds. Thus for example, the frequency of the \(A\_1 B\_1 / A\_1 B\_1\)
genotype will be \((x\_1)^2\); the frequency of the \(A\_1 B\_1 / A\_2
B\_1\) genotype will be \(2x\_1 x\_3\), and so-on. (This can be
established rigorously with an argument based on random sampling of
gametes, just as in the one-locus case.) The first aspect of the
Hardy-Weinberg principle--genotypic frequencies given by the
square of the array of gametic frequencies--therefore transposes
neatly to the two-locus case. However, the second aspect of
Hardy-Weinberg--stable genotypic frequencies after one round of
random mating--does not generally apply in the two-locus
case.



A key concept in two-locus population genetics is that of
*linkage*, or lack of independence between the two loci. To
understand linkage, consider the set of gametes produced by an
organism of the \(A\_1 B\_1 / A\_2 B\_2\) genotype, i.e., a double
heterozygote. If the two loci are *unlinked*, then the
composition of this set (on average) will be \(\{ \frac{1}{4} A\_1 B\_1,
\frac{1}{4} A\_1 B\_2, \frac{1}{4} A\_2 B\_1, \frac{1}{4} A\_2 B\_2\}\),
i.e., all four gamete types are equally represented. (This assumes
that Mendel's first law holds at both loci.) So unlinked loci
are independent--which allele a gamete has at the \(A\) locus
tells us nothing about which allele it has at the \(B\) locus. The
opposite extreme is perfect linkage. If the two loci are perfectly
linked, then the set of gametes produced by the \(A\_1 B\_1 / A\_2 B\_2\)
double heterozygote has the composition \(\{\frac{1}{2} A\_1 B\_1,
\frac{1}{2} A\_2 B\_2\}\); this means that if a gamete receives the
\(A\_1\) allele at the \(A\) locus, it necessarily receives the \(B\_1\)
allele at the \(B\) locus and vice versa.



In physical terms, perfect linkage means that the \(A\) and
\(B\) loci are located close together on the same chromosome; the
alleles at the two loci are thus inherited as a single unit. Unlinked
loci are either on different chromosomes, or on the same chromosome
but separated by a considerable distance, hence likely to be broken up
by recombination. Where the loci are on the same chromosome, perfect
linkage and complete lack of linkage are two ends of a continuum. The
degree of linkage is measured by the *recombination fraction*
\(r\), where \(0 \le r \le \frac{1}{2}\). The composition of
the set of gametes produced by an organism of the
\(A\_1 B\_1 /
A\_2 B\_2\) genotype can be written
in terms of \(r\), as follows:




|  |  |
| --- | --- |
| \(A\_1 B\_1\) | \(\frac{1}{2} (1 - r)\) |
| \(A\_1 B\_2\) | \(\frac{1}{2} r\) |
| \(A\_2 B\_1\) | \(\frac{1}{2} r\) |
| \(A\_2 B\_2\) | \(\frac{1}{2} (1 - r)\) |



It is easy to see that \(r = \frac{1}{2}\) means that the loci are
unlinked, while \(r = 0\) means that they are perfectly
linked.



In a two-locus model, the gametic (and therefore genotypic)
frequencies need not be constant across generations, even in the
absence of selection, mutation, migration and drift, unlike in the
one-locus case. (Though *allelic* frequencies will be constant,
in the absence of these evolutionary forces.) It is possible to derive
recurrence equations for the gamete frequencies, as a function of
their frequencies in the previous generation plus the recombination
fraction:



\[\begin{align}
x\_1' &= x\_1 + r(x\_2 x\_3 - x\_1 x\_4) \\
x\_2' &= x\_2 + r(x\_2 x\_3 - x\_1 x\_4) \\
x\_3' &= x\_3 + r(x\_2 x\_3 - x\_1 x\_4) \\
x\_4' &= x\_4 + r(x\_2 x\_3 - x\_1 x\_4)
\end{align}\]


(See Ewens 1969 or Edwards 2000 for an explicit derivation of these
equations.)



From the recurrence equations, it follows that gametic (and thus
genotypic) frequencies will be stable across generations, i.e., \(x\_i'
= x\_i\) for each \(i\), under either of two conditions: (i) \(r = 0\),
or (ii) \(x\_2 x\_3 - x\_1 x\_4 = 0\). Condition (i) means that the two
loci are perfectly linked and thus in effect behave as one; condition
(ii) means that the two loci are in 'linkage equilibrium',
so that alleles at the \(A\)-locus are in random association with
alleles at the \(B\)-locus. More precisely, linkage equilibrium means
that the population-wide frequency of the \(A\_i B\_i\) gamete is equal
to the frequency of the \(A\_i\) allele multiplied by the frequency of
the \(B\_i\) allele.



An important result in two-locus theory shows that, given random
mating, the quantity \((x\_2 x\_3 - x\_1 x\_4)\) will decrease
every generation until it reaches zero--at which point the
genotype frequencies will be in equilibrium. So a population initially
in linkage disequilibrium will approach linkage equilibrium over a
number of generations, at a rate that depends on \(r\), the
recombination fraction. Note the contrast with the one-locus case,
where just one round of random mating is sufficient to bring the
genotype frequencies into equilibrium.


## 4. Random Drift



Random genetic drift refers to the chance fluctuations in gene
frequency that arise in finite populations. In many evolutionary
models, including those outlined in section 3 above. the population is
assumed to be very large (technically, infinite) precisely in order to
abstract away from such fluctuations. But though mathematically
convenient, this assumption is often unrealistic. In real life
populations, particularly those of small size, stochasticity is an
important source of evolutionary change. Thus a given allele may
increase or decrease in frequency not because of any effect it has on
organismic survival or reproduction, but simply by chance.
Understanding such stochastic changes in gene frequency, and their
interaction with natural selection, is a major topic in population
genetics, past and present.



The term "random drift" has both a narrow and a broad
sense (Kimura 1964; Rice 2004; Millstein 2016). In the narrow sense,
it refers to gene frequency changes that arise from the random
sampling of gametes to form the offspring generation. (The point here
is that organisms produce many more gametes than will ever make it
into a fertilized zygote, and only half of a diploid organism's
genes are transmitted to each gamete). In the broader sense, drift
refers to gene frequency changes arising from all stochastic factors,
including for example random fluctuations in selection intensities, or
in survival and mating success. The narrower sense of the term is used
here.



Random drift greatly complicates the task of the population
geneticist. For in the presence of drift, it is no longer possible to
deduce the genetic composition of the population in generation
\(t+1\) from its composition in generation \(t\); so no
recurrence relation for an allele's frequency, of the sort
expressed in equation (1) above, can be derived. Instead, the aim must
be to deduce the *probability distribution* over all the
possible genetic compositions of the population in generation
\(t+1\). From this, it is sometimes possible to extract a
prediction about the long-term fate of an allele.



The simplest and most widely-used model for analysing random drift is
known as the *Wright-Fisher model*. This model deals with a
finite population containing \(N\) diploid organisms. \(N\)
is assumed constant over generations (perhaps because of ecological
constraints). Generations are non-overlapping, meaning that parents
die as soon as they have reproduced, and mating is random. Selection,
migration and mutation are assumed absent. The offspring generation is
formed by randomly sampling \(2N\) of the gametes produced by the
parental generation. Time is discrete, with one generation
corresponding to one time period. Consider a particular allele of
interest. Let \(X(t)\) denote the number of copies of
the allele in the population in generation \(t\), where \(0 \leq
X(t) \leq 2N\) (since organisms are diploid). The allele's
frequency \(p(t)\) is then equal to \(\frac{1}{2N}X(t),\)
where \(0 \leq p(t) \leq 1\).



We are interested in \(X(t+1)\) and \(p(t+1)\), the number of copies
and frequency of the allele in generation \(t+1,\) respectively. (They
are related by \(p(t+1) = \frac{1}{2N}X(t+1).\)) Now, \(X(t+1)\) is a
random variable that can take any of \(2N+1\) possible values from the
set \(\{0, 1, 2, \ldots ,2N\}\). Since the offspring generation is
formed by random sampling from the parental gamete pool, the
probability distribution of \(X(t+1)\) is given by the binomial
distribution:



\[
\textit{Prob}(X(t+1) = x) = \binom{2N}{x}p(t)^x(1-p(t))^{2N-x}
\]


This formula tells us, for each possible value of \(X(t+1)\), what its
probability is as a function of the population size \(N\) and the
allele's frequency in the parental generation \(p(t).\) From
this, we can easily compute the expected value of \(X(t+1)\), denoted
\(E(X(t + 1))\), which turns out to simply be equal to \(X(t)\). This
is quite intuitive: since the second generation is formed by random
sampling, the number of copies of the allele is just as likely to
increase as to decrease, so the expected number of copies in
generation \(t+1\) equals the actual number of copies in generation
\(t.\) It follows that the expected change in allele frequency from
generation \(t\) to \(t+1,\) denoted \(E(\Delta p)\), is equal to
zero.



The fact that \(E(\Delta p) = 0\) does not imply that drift will have
no evolutionary effect. For \(\Delta p\) may have a substantial
variance around the mean of zero (depending on the value of \(N\)), so
it may be quite probable that \(\Delta p\) deviates from zero by a
substantial amount. (Similarly, if one flips a fair coin 20 times, the
expected number of heads is 10, but the probability that the actual
number of heads is 8 or less is quite substantial -
approximately 25%). In the Wright-Fisher model, the variance of
\(\Delta p\) turns out to be \(Var(\Delta p) = \frac{1}{2N}p(1-p)\).
Thus as the population size increases, the variance of \(\Delta p\)
gets smaller and smaller, which illustrates the point that random
drift is more important in small than in large populations.



What will happen in the long run? Under the assumptions of the
Wright-Fisher model, the sequence of allele frequencies in successive
generations \(\{p(0), p(1), p(2), \ldots \}\)constitutes what is known
as a *Markov chain*, that is, a sequence of random variables
(stochastic process) where the probability distribution of any
variable depends only on the value of the immediately preceding
variable. That is, the probability that the allele has a frequency of
(say) 0.8 in generation \(t+1\), denoted \(\textit{Prob}(p(t+1) = 0.8)\),
depends on the value of \(p(t),\) the allele's frequency in
generation \(t\), but not on its frequency in earlier
generations. Importantly, this Markov chain has a special feature,
namely that the the two extremal values of \(p(t)\), i.e., 0 and 1,
are *absorbing boundaries,* meaning that if the system reaches
one of these boundaries it stays there. That is, if the allele goes
extinct in generation \(t\), so has a frequency of zero, then in
all subsequent generations its frequency will also be zero (since we
are ignoring mutation). Similarly, if the allele goes to fixation in
generation \(t\), it will remain fixed in subsequent generations.
Formally, we can express these facts as: \(\textit{Prob}(p(t+1) = 0 \mid p(t)
= 0) = 1\) and \(\textit{Prob}(p(t+1) = 1 \mid p(t) = 1) = 1\). Since there is
no upper bound on the number of generations, eventually random drift
must lead the allele to go extinct or to become fixed in the
population (and similarly for other alleles). This is because of the
absorbing boundary assumption, which implies that every stochastic
trajectory must eventually end up at \(p(t) = 0\) or \(p(t) = 1\), for
some value of \(t.\)



This leads naturally to the following question. What is the
probability that the allele will become fixed in the population,
rather than going extinct? The Wright-Fisher model yields a very
simple answer to this question. If a given (neutral) allele has a
frequency \(p(t)\) in generation \(t\) , then the
probability that it eventually fixes is simply \(p(t).\)
This is a fairly intuitive result. For if the allele is rare, it is
quite likely that it will be lost from the population by chance.
Conversely, if the allele is common, it is most unlikely to be lost
from the population, as this would require an improbable series of
chance events to occur together. An immediate consequence of this
result is that the probability of fixation of a novel (neutral)
genetic variant, that has arisen in the population by sporadic
mutation, is \(\frac{1}{2N}\) - since initially there is one copy of
the novel variant in the population. Thus for appreciable \(N\),
it is overwhelmingly likely that any given novel variant will be lost
to genetic drift. This illustrates the general fact that genetic drift
has a homogenizing tendency over many generations, reducing the
genetic variation in a population.



Importantly, the equality between an allele's current frequency
and its probability of becoming fixed assumes that the allele in
question, and other alleles at the same locus, are selectively
neutral--meaning that random drift is the sole determinant of the
changes in frequency. If this assumption is relaxed, matters become
more complicated. The fate of an allele then depends on both drift and
on its selective advantage or disadvantage. This takes us beyond the
confines of the simple Wright-Fisher model since now there are two
evolutionary factors at work--random drift and natural selection.
In a finite population, an allele that is selectively advantageous, so
has a positive selection coefficient, will have a higher probability
of fixation that a neutral allele; and conversely for one that is
selectively disadvantageous. This is fairly obvious. What is less
obvious, but still true, is that even if a novel variant arises that
confers a significant selective advantage, it is still more likely to
be eliminated by drift than to become fixed. To quantitatively study
the combined effects of selection and drift, population geneticists
use an advanced probabilistic technique known as diffusion analysis,
which lies beyond the scope of this article (see Rice 2004 ch.5, Hartl
2020, ch.6. or Otto and Day 2007, ch.15). But one key result deserves
mention, which is that the eventual fate of an allele depends on the
relative magnitude of two quantities, namely \(4N\_e\) and \(s\).
Here \(N\_e\) denotes the "effective population size"
(which corrects the actual population size to take account of
deviations from the idealized assumptions of the Wright-Fisher model),
and \(s\) is the selection coefficient, which is a measure of the
relative fitness of organisms with the allele compared to organisms
without, where \(0 \leq s \leq 1\). It turns out that if \(4N\_e s
\gg 1\) then natural selection will determine the fate of an
allele, while if \(4N\_e s \ll 1\) then drift will determine its
fate.



The respective roles of drift and natural selection in molecular
evolution was the subject of the selectionist versus neutralist
controversy in the 1960s and 1970s, as noted in the Introduction (cf.
Dietrich 1994). The neutralist camp, headed by Kimura, argued that
most molecular variants had no effect on phenotype, so were not
subject to natural selection; random drift was instead the main
determinant of their fate. Kimura argued that the apparently constant
rate at which the amino acid sequences of proteins evolved, and the
extent of genetic polymorphism observed in natural populations, could
best be explained by the neutralist hypothesis (Kimura 1977, 1994).
Selectionists countered that natural selection was also capable of
explaining the observed polymorphism. The controversy ended without a
clear victory for either side, in part due to paucity of data. However
the opposition between selection and drift remains a central topic in
molecular population genetics today, where there is an abundance of
data on DNA sequence variation in natural populations. Sophisticated
methods have been developed to allow researchers to hunt for
signatures of past selection in the genomes of modern organisms. It
has become clear that there is indeed much neutral molecular variation
in DNA sequence (in part due to "synonymous" mutations
that leave unchanged the amino acid sequence of the protein that the
gene codes for). However, there is also much evidence showing that the
genomes of contemporary species have been substantially influenced by
natural selection (Casillas and Barbadilla 2017, Kern and Hahn 2018).
Moreover, the idea that drift is the sole determinant of a neutral
variant's frequency, as the original neutralists held, is not
necessarily true. Another possibility, championed by J. Gillespie
(2004), is that a neutral variant's spread in a population may
be heavily influenced by selection at linked loci, a process known as
"hitchhiking" or "genetic draft"; see Skipper
(2004) for discussion. A recent evaluation of the selection versus
drift issue concludes that "the extent to which DNA sequence
evolution is caused by selection versus drift remains an important
unanswered general question" (Charlesworth and Charlesworth
2017, p.6).



Though random drift is well-understood mathematically, and is the
subject of much empirical research in biology, a number of
philosophers have suggested that it is conceptually less clearcut than
one might think. Thus for example Millstein (2002) has argued that the
term "random drift" as used by biologists is often
ambiguous as between a process (such as random sampling) and an
outcome (such as change in gene frequency). Millstein's point
has given rise to a considerable philosophical literature on how
exactly the terms "drift" and "selection"
should be defined; see the entry on
 genetic drift
 for discussion and references.


### 4.1 Coalescence



Traditional population genetics models of the sort sketched above are
"forward-looking", in that their goal is to predict the
future genetic composition of a population, or the fate of an allele,
based on various assumptions about the evolutionary processes at work.
Starting in the 1980s, a different approach to population genetics was
developed known as "coalescent theory", originally as a
result of work in applied probability (Kingman 1982). Coalescent
theory has a "backwards-looking" orientation: it aims to
make inferences about a population's history based on a sample
of genes drawn from the current population (Wakeley 2008). Compared to
the traditional forwards-looking models, coalescent theory allows a
different set of evolutionary questions to be asked, and also yields
simpler ways of calculating certain quantities of interest in the
traditional models, such as fixation probabilities (Rice 2004, ch. 5).
Also, coalescent theory yields predictions, for example about the
amount of DNA sequence variation we should expect to find in a sample
of genes from a natural population, that can be directly tested
against data.



Coalescent theory is all about tracing lines of ancestry between genes
in a (diploid) population. Ordinarily we think of ancestor-descendant
lineages of organisms, but we can equally (indeed more easily) think
in terms of lineages of genes at a locus, while simply ignoring the
organisms that the genes are housed within. The starting point of
coalescent theory is the observation that all the genes at a locus in
a current population must ultimately stem from a single ancestral gene
copy in the past (the "most recent common ancestor" or
MRCA). This is in effect the flip-side of genetic drift. If we go back
far enough in a population's ancestry, we must arrive at a point
at which all of the genes bar one have left no descendants in the
current population. This is because, at every round of reproduction, a
given gene copy has a certain chance of not leaving any descendants in
the next generation, i.e., being eliminated by drift. This implies
that, as we trace back in time, the gene lineages will join up, or
"coalesce".



The simplest approach to coalescence uses the Wright-Fisher model,
expounded above. Recall that this model involves a diploid population
of fixed size \(N\), in which selection is absent, mating is random,
and generations are non-overlapping. Each new generation is formed by
randomly sampling \(2N\) of the gametes produced by the previous
generation. To illustrate coalescence, suppose that we pick two gene
copies at random from the current population. There are then two
possibilities: either both derive from a single copy in the preceding
generation, or they do not. These two events occur with probabilities
\(\frac{1}{2N}\) and \((1 - \frac{1}{2N})\) respectively. To see this, note that
the first gene we pick must have some parent or other in the previous
generation; so the probability that the two genes we pick derive from
a single copy in the previous generation is simply the probability
that the second gene has the same parent as the first; since there are
\(2N\) possible parents, this equals \(\frac{1}{2N}\). So the alternative
possibility, that the two genes do not coalesce in the previous
generation, has the complementary probability of \((1 - \frac{1}{2N})\).



This reasoning can be repeated in a natural way. Suppose that the two
genes we pick do not coalesce in the immediately previous generation,
i.e., they have different parents. Then, those two parents genes will
themselves either derive from a single copy in the previous
generation, or they will not. If so, then the two genes we have picked
will coalesce two generations ago, i.e., derive from a single
grandparent gene. The probability of this is \(\frac{1}{2N} \times (1 -
\frac{1}{2N})\). By iterating this reasoning, we can work out the
probability distribution that two randomly chosen gene copies in the
current generation derive from a common ancestor *t*
generations ago. This is given by:



\[
\textit{Prob}(\text{coalescence}~t~\text{generations ago}) = \frac{1}{2N}
\times \left(1 - \frac{1}{2N}\right)^{t-1}.
\]


The next question to ask is what the expected value of this
distribution is, that is, what the mean time to coalescence is? The
answer turns out to be approximately \(2N\). Thus on average, a
pair of randomly picked genes at a given locus will coalesce after
\(2N\) generations, where \(N\) is population size. However,
there is considerable variability about this mean, meaning that there
is a significant chance that coalescence will occur much quicker, or
much slower, than this. By building on this simple analysis,
coalescent theory allows a range of more complicated questions to be
answered, involving, for example, multiple alleles, sub-divided
populations, populations that change in size over time, plus other
deviations from the assumptions of the basic Wright-Fisher model. For
example, coalescent theory yields a straightforward calculation of how
many generations back we must go, on average, to find the MRCA of a
number of different genes at a locus (Otto and Day 2007, ch. 13).


## 5. Population Genetics and its Critics



The status of population genetics in contemporary biology is an
interesting issue. Despite its centrality to evolutionary theory, and
its historical importance, population genetics is not without its
critics. Some argue that population geneticists have devoted too much
energy to developing theoretical models, often with great mathematical
ingenuity, and too little to actually testing the models against
empirical data (Wade 2005). This was probably a fair criticism at one
time, however the recent flourishing of molecular population genetics
has changed the situation, allowing much greater contact between
theory and data (Hahn 2018). Others argue that population-genetic
models are usually too idealized to shed any real light on the
evolutionary process, and are limited in what they can teach us about
phenotypic evolution (Pigliucci 2008). Still others have argued that,
historically, population genetics has had a relatively minor impact on
the actual practice of most evolutionary biologists, despite the
lip-service often paid to it (Lewontin 1980). However, not all
biologists accept these criticisms. Thus the geneticist Michael Lynch
(2007), for example, has written that "nothing in biology makes
sense except in the light of population genetics", in a twist on
Dobzhansky's famous dictum; see Bromham (2009) and Pigliucci
(2008) for discussion of Lynch's arguments. And in a recent
survey of 50 years of population genetics, Charlesworth and
Charlesworth (2017) argue that population-genetic analysis has not
only enabled us to understand the nature and causes of molecular
genetic variation, but has also provided deep insights into a variety
of topics in evolutionary biology.



Population-genetic models of evolution have sometimes been criticised
on the grounds that few phenotypic traits are controlled by genotype
at a single locus, or even two or three loci. (Multi-locus
population-genetic models do exist, but they are inevitably very
complicated.) There is an alternative body of theory, known as
quantitative genetics, which deals with so-called
'polygenic' or 'continuous' traits, such as
height, which are thought to be affected by genes at many different
loci in the genome, rather than just one or two; see Falconer (1995)
or Walsh and Lynch (2018) for good overviews. Quantitative genetics
employs a quite different methodology from population genetics. The
latter, as we have seen, aims to track gene and genotype frequencies
across generations. By contrast, quantitative genetics does not
directly deal with gene frequencies; the aim is to track the phenotype
distribution, or moments of the distribution such as the mean or the
variance, across generations. Though widely used by animal and plant
breeders, quantitative genetics is usually regarded as a less
fundamental body of theory than population genetics, given its
'phenotypic' orientation. Nonetheless, the relationship
between population and quantitative genetics is essentially
harmonious.



A different criticism of the population-genetic approach to evolution
is that it ignores embryological development; this criticism really
applies to the evolutionary theory of the 'modern
synthesis' era more generally, which had population genetics at
its core. As we have seen, population-genetic reasoning assumes that
an organism's genes somehow affect its phenotype, and thus its
fitness, but it is silent about the details of how genes actually
build organisms, i.e., about embryology. The founders of the modern
synthesis treated embryology as a 'black box', the details
of which could be ignored for the purposes of evolutionary theory;
their focus was on the transmission of genes across generations, not
the process by which genes make organisms (see the entry on
 evolution and development).
 This strategy was perfectly reasonable, given how little was
understood about development at the time. But since the 1990s, great
strides have been made in molecular developmental genetics, which has
renewed hopes of integrating the study of embryological development
with evolutionary theory and has led to the emergence of the
discipline of 'evolutionary developmental biology', or
evo-devo (see Arthur 2021 for a recent overview or the entry on
 evolution and development). 
It is sometimes argued that evo-devo is in tension with traditional
neo-Darwinism (e.g., Amundson 2007), but it is more plausible to view
them as complementary ways of studying evolution that have different
emphases.



In a 2005 book, Sean Carroll, a leading evo-devo researcher, argued
that population genetics no longer deserves pride-of-place on the
evolutionary biology curriculum. He writes: "millions of biology
students have been taught the view (from population genetics) that
'evolution is change in gene frequencies' ... This
view forces the explanation toward mathematics and abstract
descriptions of genes, and away from butterflies and zebras, or
Australopithecines and Neanderthals" (2005 p. 294). A similar
argument has been made by Pigliucci (2008). Carroll argues that
instead of defining evolution as 'change in gene
frequencies', we should define it as 'change in
development', in recognition of the fact that most morphological
evolution is brought about through mutations that affect organismic
development. Carroll may be right that evo-devo makes for a more
accessible introduction to evolutionary biology than population
genetics, and that an exclusive focus on gene frequency dynamics is
not the best way to understand all evolutionary phenomena; but
population genetics arguably remains indispensable to a full
understanding of the evolutionary process.



In recent years, a vigourous debate has opened up about whether the
"modern synthesis" (MS) is still adequate to the needs of
biology. The MS, or neo-Darwinian synthesis, is the intellectual
edifice that arose in the 20th century from the integration of
Darwin's theory of evolution with Mendelian genetics, which had
population genetics at its core, as we have seen. Proponents of the
"extended evolutionary synthesis", or EES, argue that the
main principle of the MS--that adaptive evolution arises from
natural selection acting on sporadic genetic mutation--is not
wrong but is not the whole story either, and needs supplementation in
the light of recent discoveries (Pigliucci and Muller eds. 2010;
Laland et al. 2014, 2015). They point to phenomena such as niche
construction, epigenetic inheritance, multi-level selection,
phenotypic plasticity and developmental bias, which, they claim, do
not fit easily with the MS's emphasis on gene-based evolution.
The EES's proponents are typically somewhat suspicious of
population genetics, and seek to downplay its explanatory
significance. However, their views are controversial. Defenders of the
traditional MS argue that the empirical phenomena in question are of
relatively minor evolutionary importance and / or that they can be
accommodated within the MS without any major paradigm shift (Wray et
al. 2014; Walsh and Lynch 2018; Charlesworth, Barton and Charlesworth
2017). This debate looks set to continue.



Despite the criticisms levelled against it, population genetics has
certainly had a major influence on our understanding of evolution. For
example, the well-known 'gene's eye' view of
evolution, developed by G.C. Williams (1966), W. D. Hamilton (1964)
and R. Dawkins (1976), stems directly from population-genetic
reasoning; indeed, important aspects of gene's eye thinking were
already present in Fisher's writings (Okasha 2008, Ewens 2011).
Proponents of the gene's eye view argue that genes are the real
beneficiaries of the evolutionary process; genotypes and organisms are
mere temporary manifestations. Natural selection is at root a matter
of competition between gene lineages for greater representation in the
gene pool; creating organisms with adaptive features is a
'strategy' that genes have devised to secure their
posterity (Dawkins 1976, 1982). Gene's eye thinking has
revolutionised many areas of evolutionary biology in the last fifty
years, particularly in the field of animal behaviour (cf. Agren 2021),
but in many ways it is simply a colourful gloss on the conception of
evolution implicit in the formalisms of population genetics.


## 6. Philosophical and Conceptual Issues in Population Genetics



Population genetics raises a number of interesting conceptual and
philosophical issues. One such issue concerns the concept of the gene
itself. As we have seen, population genetics came into being in the
1920s and 1930s, long before the molecular structure of genes had been
discovered. In these pre-molecular days, the gene was a theoretical
entity, postulated in order to explain observed patterns of
inheritance in breeding experiments; what genes were made of, how they
caused phenotypic changes, and how they were transmitted from parent
to offspring were not known. Today we do know the answers to these
questions, thanks to the spectacular success of molecular genetics and
genomics. The gene has gone from being a theoretical entity to being
something that can actually be manipulated in the laboratory.



The relationship between the gene of classical (pre-molecular)
genetics, and the gene of modern molecular genetics is a subtle and
much discussed topic (Beurton, Falk and Rheinberger (eds.) 2000,
Griffiths and Stotz 2006, Moss 2003, Meunier 2022). In molecular
genetics, 'gene' refers, more or less, to a stretch of DNA
that codes for a particular protein--so a gene is a unit of
function. But in classical population genetics, 'gene'
refers, more or less, to a portion of hereditary material that is
inherited intact across generations--so a gene is a unit of
transmission, not a unit of function. In many cases, the two concepts
of gene will pick out roughly the same entities--which has led
some philosophers to argue that classical genetics can be
'reduced' to molecular genetics (Sarkar 1998). But it is
clear that the two concepts do not have precisely the same extension;
not every molecular gene is a classical gene, nor vice-versa. Some
theorists go further than this, arguing that what molecular biology
really shows is that there are no such things as classical genes.



Whatever one's view of this debate, it is striking that
virtually all of the central concepts of population genetics were
devised in the pre-molecular era, when so little was known about what
genes were; the basic structure of population-genetic theory has
changed little since the days of Fisher, Haldane and Wright
(Charlesworth and Charlesworth 2017). This reflects the fact that the
empirical presuppositions of population-genetic models are really
quite slim; the basic presupposition is simply the existence of
hereditary particles that obey the Mendelian rules of transmission,
and that somehow affect the phenotype. Therefore, even without
knowing what these hereditary particles are made of, or how they exert
their phenotypic effects, the early population geneticists were able
to devise an impressive body of theory. That the theory continues to
be useful today illustrates the power of abstract models in
science.



This leads us to another facet of population genetics that has
attracted philosophers' attention: the way in which abstract
models, that involve simplifying assumptions known to be false, can
illuminate actual empirical phenomena. Idealized models of this sort
play a central role in many sciences, including physics, economics and
biology, and raise interesting methodological issues. In particular,
there is often a trade-off between realism and tractability; the more
realistic a model the more complicated it becomes, which typically
limits its usefulness and its range of applicability. This general
problem and others like it have been extensively discussed in the
philosophical literature on modelling (e.g., Godfrey-Smith 2006,
Weisberg 2006, Frigg and Hartmann 2006), and are related to population
genetics by Plutynski (2006).



It is clear that population genetics models rely on assumptions known
to be false, and are subject to the realism / tractability trade-off.
The simplest population-genetic models assume random mating,
non-overlapping generations, infinite population size, perfect
Mendelian segregation, frequency-independent genotype fitnesses, and
the absence of stochastic effects; it is very unlikely (and in the
case of the infinite population assumption, impossible) that any of
these assumptions hold true of any actual biological population. More
realistic models, that relax one of more of the above assumptions,
have been constructed, but they are invariably much harder to analyze.
It is an interesting historical question whether these
'standard' population-genetic assumptions were originally
made because they simplified the mathematics, or because they were
believed to be a reasonable approximation to reality, or both. This
question is taken up by Morrison (2004, 2014) in relation to
Fisher's early population-genetic work.



Another philosophical issue raised by population genetics is
reductionism. It is often argued that the population-genetic view of
evolution is inherently reductionistic, by both its critics and its
defenders. This is apparent from how population geneticists define
evolution: change in gene frequency. Implicit in this definition is
the idea that evolutionary phenomena such as speciation, adaptive
radiation and diversification, as well as phenotypic evolution, can
ultimately be reduced to gene frequency change. But do we really know
this to be true? Many biologists, particularly 'whole
organism' biologists, are not convinced, and thus reject both
the population-genetic definition of evolution and the primacy
traditionally accorded to population genetics within evolutionary
biology (Pigliucci 2008).



This is a large question, and is related to the issues discussed in
section 5 above. The question can be usefully divided into two: (i)
can microevolutionary processes explain all of evolution?; (ii) can
all of microevolution be reduced to population genetics?
'Microevolution' refers to evolutionary changes that take
place within a given population, over relatively short periods of time
(e.g., a few hundred generations). These changes typically involve the
substitution of a gene for its alleles, of exactly the sort modelled
by population genetics. So over microevolutionary time-scales, we do
not typically expect to see extinction, speciation or major
morphological change -- phenomena which are called
'macroevolutionary'. Many biologists believe that
macroevolution is simply 'microevolution writ large', but
this view is not universal. Authors such as Gould (2002) and Eldredge
(1989), for example, have argued persuasively that macroevolutionary
phenomena are governed by autonomous dynamics, irreducible to a
microevolutionary basis. Philosophical discussions of this issue
include Sterelny (1996), Grantham (1995) and Okasha (2006). A useful
overview is provided by Turner and Havstad (2019).



Setting aside the reducibility of macro to microevolution, there is
still the issue of whether an exclusively population-genetic approach
to the latter is satisfactory. Some reasons for doubting this have
been discussed already; they include the complexity of the
genotype-phenotype relation, the fact that population genetics treats
development as a black-box, and the idealizing assumptions that its
models rest on. Another point, not discussed above, is the fact that
population genetics models are (deliberately) silent about the
*causes* of the fitness differences between genotypes whose
consequences they model (Sober 1984, Glymour 2006). For example, in
the simple one-locus model of section 3.1, nothing is said about why
the three genotypes leave different numbers of successful gametes. To
fully understand evolution, the ecological factors that lead to these
fitness differences must also be understood. While this is a valid
point, the most it shows is that an exclusively population-genetic
approach cannot yield a complete understanding of the evolutionary
process. This does not really threaten the traditional view that
population genetics is fundamental to evolutionary theory.



A final suite of philosophical issues surrounding population genetics
concerns causation. Evolutionary biology is standardly thought of as a
science that yields causal explanations: it tells us the causes of
particular evolutionary phenomena (Okasha 2009, Otsuka 2016a). This
causal dimension to evolutionary explanations is echoed in population
genetics, where selection, mutation, migration and random drift are
often described as causes, or 'forces', that lead to gene
frequency change (Sober 1984). The basis for this way of speaking is
obvious enough. If the frequency of allele \(A\) in a population
increases from one generation to another, and if the population obeys
the rules of Mendelian inheritance, then as a matter of logic one of
three things must have happened: (I) migrants bearing allele
\(A\) entered the population (II); there was mutation to allele
\(A\) from another allele; (III) the average number of
descendants left by each copy of \(A\) in the parental generation
exceeded the average for all genes. It is straightforward to verify
that if none of (I)-(III) occurred, then the frequency of allele
\(A\) would have been unchanged. Note that case (III) covers both
selection and random drift, depending on whether copies of \(A\)
left more descendants than average by chance, or because of some
systematic effect of \(A\) on organismic fitness.



Despite this point, a number of philosophers have objected to the idea
that evolutionary change can usefully be thought of as caused by
different factors, including natural selection (e.g., Matthen and
Ariew 2009, Walsh 2007). A variety of objections to this apparently
innocent way of speaking have been levelled; some of these seem to be
objections to the metaphor of 'forces' in particular,
while others turn on more general considerations to do with causality
and chance. The status of these objections is a controversial matter;
see Reisman and Forber (2005), Brandon and Ramsey (2007), Sarkar
(2011) and in particular Otsuka (2016b) for critical discussion. The
'non-causal' (or 'statisticalist' as it is
sometimes called) view of evolution is certainly a radical one, since
the idea that natural selection, in particular, is a potential cause
of evolutionary change is virtually axiomatic in evolutionary biology,
and routinely taught to students of the subject. As Millstein (2002)
points out, if one abandons this view it becomes hard to make sense of
important episodes in the history of evolutionary biology, such as the
selectionist / neutralist controversy mentioned above.



A full resolution of this issue cannot be attempted here; however, it
is worth making one observation about the idea that mutation,
selection, migration and drift should be regarded as
'causes' of gene frequency change. There is an important
difference between drift on the one hand and the other three factors
on the other. This is because mutation, selection and migration are
directional; they typically lead to a non-zero expected change in gene
frequencies (Rice 2004 p. 132). Random drift on the other hand is
non-directional; the expected change due to drift is by definition
zero. As Rice (2004) points out, this means that mutation, selection
and migration can each be represented by a vector field on the space
of gene frequencies; their combined effects on the overall
evolutionary change is then represented by ordinary vector addition
(which arguably lends substance of the "force" metaphor).
But drift cannot be treated this way, for it has a magnitude but not a
direction. In so far as proponents of the 'non-causal'
view are motivated by the oddity of regarding drift, or chance, as a
force, they have a point. However this line of argument is specific to
random drift; it does not generalize to all the factors that affect
gene frequency change. And it does not alter the fact that, in
attributing the spread of a gene to random drift, we have given a
bona fide scientific explanation that is potentially falsifiable; for
we have said that the gene's spread was *not* due to any
systematic advantage that the gene conferred on organisms that had it.




A final area of philosophical concern to which population genetics is
relevant is the debate over the status of racial categories. This is a
central topic within the philosophy of race, a burgeoning sub-field of
philosophy (see the entry on
 race
 for an overview). One influential line of argument has it that racial
categories are social constructions rather than natural kinds. On this
view, the division of the human species into distinct
"races" (which can be done in multiple ways), should be
understood primarily in terms of the social and political role that such
racial categories play. This view is sometimes called
"anti-realism" about race, since it holds that the usual
ways of dividing humans into races (as employed in national censuses,
for example) are conventional rather than real. Opponents of
anti-realism have sometimes drawn on human population genetics to
bolster the rival position that racial divisions are in fact real
(Andreasen 2007, Sesardic 2010, Spencer 2015). Empirical studies of
how genetic variation in humans is structured, using machine learning
techniques, have found genetic clusters that appear to roughly
correspond to certain broad racial groupings (such as
"African" and "Eurasian") (Rosenberg et al.
2002, Li et al. 2008). However, the precise bearing of these empirical
results on the question of the reality versus conventionality of
racial divisions is a matter of ongoing philosophical dispute (Kopec
2014, Winther 2014).



To conclude, there are a number of interesting philosophical issues
surrounding population genetics discussed in the recent literature. In
addition to these, population genetics is indirectly relevant to a
still broader set of philosophical concerns, given its centrality to
evolutionary biology, a science replete with implications for many
branches of philosophy.