II. UNDERSTANDING MICROEVOLUTION

1
II. UNDERSTANDING MICROEVOLUTION Topic 10.
Lectures 15-16. Populations and Tools for
Studying Them 1) What is a population? Every
individual belongs to a population of many
similar individuals - this is so familiar that we
take it for granted.
No lonely monsters!
Moreover, we already know why this is the case.
Indeed, there are two reasons 1)
individuals are smaller than their
environments. 2) a lone lineage would soon
go extinct. Now, it is time to consider the key
question - "What exactly is a population?".
2
Simplistic definition a population is a set of
similar individuals living together. This is
mostly correct, but we need to understand the
concept deeper. Cynical definition a
population is a set of individuals that we need
to consider together. This "definition" leads
us to the correct question - Why do we need
populations? Why cannot we consider individuals
one at a time? Let us first assume that
individuals are apomicts, like bdelloid rotifers.
A bdelloid rotifer Philodina roseola (eating
algae). Scale bar, 100 µm.
Even apomictic individuals cannot be considered
separately, if our goal is to study evolution.
Indeed, evolution is a long-term process, so that
we need a durable object. This is the key.
3
In the next generation the lineage of any current
individual may be dead. And after 10 generations,
an individual lineage will be dead with high
probability.
Life is fundamentally unfair a situation on the
right picture is unrealistic.
All future individuals will be descendants of
just ONE current individual after only N
generations, when N is the number of ecologically
equivalent individuals. Conversely, if we trace
lineages of the current individuals back, we will
see them merging into a common ancestor N
generations ago.
4
Strange truth eventually, lineages of all (blue)
current individuals will go extinct, except
lineage of one (red) individual, which will
expand are replace all others.
Why is life so unfair? Due to two reasons -
systematic (selection) and random (drift)
differences in the efficiency of reproduction
between lineages. The red individual is probably
not a bad one, but is not necessarily the best
one, either.
5
Thus, an asexual population is the minimal
durable set of individuals such that, even after
a long time, the descendants of at least one of
them will still be around. Of course,
individuals that belong to the same population
must be ecologically equivalent, which usually
implies genetic similarity. There is no need to
include substantially different individuals into
the same population the lineage of Philodina
roseola (left) will not replace the lineage of
Adineta vaga (right).
Conversely, ecologically equivalent individuals
from the same population must be genetically
similar to each other, because they shared a
recent common ancestor.
6
With amphimixis, two factors force us to consider
populations, instead of individuals. First,
amphimixis usually involves outcrossing, which
abolishes separate lineages of individuals.
Thus we can define sexual population as the
minimal reproductively closed set of individuals,
such that even after a long time a descendant of
each of them will contain genes only from members
of this set, and not from outsiders.
7
Second, differences in rates of propagation of
individuals are still present with amphimixis (of
course). This does not cause coalescence of
lineages of individuals (they do not exists),
but, instead, leads to coalescence of lineages of
short genome segments (loci). All present alleles
at a locus coalesce to a single common ancestor
N generations ago, but amphimixis makes these
common ancestors different for different loci.
Modern human population
Mitochondrial Eve was not married to Y-chromosome
Adam!
Y-chromosome "Adam", 100 Kya
The common ancestor for an autosomal locus,
100-300 Kya
Mitochondrial "Eve", 200 Kya
Thus, we can also define a sexual population as
the minimal durable set of individuals, such
that, even after a long period of time, the
alleles descending from the allele of at least
one of them will still be around at each locus.
8
Theses two definitions of an amphimictic
population, based on genetical and ecological
compatibility, are usually equivalent
individuals that interbreed are also ecologically
equivalent and vice versa. With amphimixis, two
members of the same population typically share a
more recent common ancestor than with apomixis.
Indeed, without inbreeding each amphimictic
individual has 2 parents, 4 grandparents,
1,048,576 (1M) of ancestors 20 generations ago,
and over 1012 ancestors 40 generations
ago. Still, overall levels of variation between
individuals within apomictic and amphimictic
populations do not need to be different.
With amphimixis, an offspring inherits, on
average, only (1/2k)th fraction of its genotype
from each of its 2k ancestors that lived k
generations ago. Thus, sharing a common ancestor
is no big deal under amphimixis.
9
Occasionally, genetically incompatible
individuals turn out to be ecologically
equivalent, and compete for the same resources.
Such natural selection acting at the level of
species can leads to extinction of one of them
(competitive exclusion).
American grey squirrel Sciurus carolinensis is
now replacing European Red squirrel Sciurus
vulgaris in Britain. In such situations,
populations defined through ecological
equivalence and coalescence are more inclusive
that populations defined through interbreeding.
Indeed, red and grey squirrels in Britain are one
population ecologically, but different
populations genetically. Such situations must be
rare in each case diversity is reduced
drastically (due to extinction of a species),
and slow evolution can restore diversity only
slowly.
10
Every individual is a member of a population due
to two reasons 1) Earth is much larger than
Little Prince's Planet

2) A lineage represented by too few (say, less
than 1000) individuals a long
time will go extinct, due to inefficient
selection and accumulation of
deleterious mutations.
Endangered Florida panther Puma concolor coryi
was represented, for more than a century, by only
100 individuals and experienced progressively
declining fitness.
11
All definitions of a population refer to a "long
period of time". How long? Depending on the
answer, we may delimit populations differently.
Do these two forests harbor two - or only one -
populations of bears? If we are interested in a
short-term process, each forest can be considered
independently. If, however, we consider a
long-term process, all these bears are one
spatially-structured population. Because we are
interested in slow evolution, usually we will
consider long-term, inclusive populations. A
useful way of thinking about this issue is to
ask how far an advantageous mutation will spread?
12
Range of a population P element in Drosophila
melanogaster
P element was acquired by a single D.
melanogaster only 100 years ago in the New
World. In early XX century, most of wild-caught
D. melanogaster were P-negative. Now, however,
every D. melanogaster in the wild is P-positive.
Thus, a single "advantageous" allele took over
the whole species of D. melanogaster in just 100
years. From this perspective, there is just one
world-wide population of D. melanogaster.
13
At any particular moment, diversity of life
mostly consists of groups of similar individuals,
disconnected from and incompatible to other such
groups ("species"). How is a population different
from a species?
Short-term populations are less inclusive that
species, but boundaries of long-term populations
and of species often coincide, as illustrated by
the P-element example.
Still, a species may consist of more than one
long-term population members of the crew
travelling to another galaxy will become a
separate population right after the lift-off.
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For us, the most important property of a
population is that natural selection operates
there. Ecologically-based definitions of a
population can be reformulated as follows
population is a set of individuals that represent
competing lineages (of whole genotypes, with
apomixis, or of individual loci, with
amphimixis), such that expansion of one lineage
must be accompanied by decline of all others.
If a lineage that carries an advantageous
mutation does not displace all other lineages
living in the same area, we are dealing with more
than one population.
15
Key fact of population biology populations
consist of individuals, not organisms.
Population-level analysis CANNOT tell us why
networks of interacting genes are
modular. However, it CAN tell us how rapidly a
derived allele with selective advantage of 1
will displace the ancestral allele.
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2) Populations on fitness landscapes Natural
selection acting within a population is defined
by the fitness landscape and by how a population
sits on it. The range of within-population
variation is not wide, and here we care only
about microscopic properties of fitness
landscapes.
Under a strong enough magnification, every
fitness landscape is close to linear.
17
A linear fitness landscape has two key
properties. First, the fitness of a genotype can
be represented by the sum of constant
contributions from all its constituent
allele. Second, there is just one direction,
known as gradient, in which fitness changes.
Thus, the fitness of a genotype is determined by
its fitness potential, the position alone the
axis that points in the direction of gradient.
Fitness potential axis is in red, and the only
perpendicular axis is in green. Fitness is an
approximately linear function of the fitness
potential of a genotype within the population and
does not depend on its position along the
perpendicular axis.
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Even on the scale of within-population variation,
fitness landscape as a linear function of fitness
potential is not always a good approximation. Any
deviation of fitness landscape from linearity on
the logarithmic scale is called
epistasis. Example of epistasis dominance and
recessivity.
If we consider just one locus A, with alleles A
and a, the log fitness of heterozygote Aa may be
closer to the log fitness of AA (if A is
dominant) or to the log fitness of aa (if A is
recessive), or be the arithmetic mean of the log
fitnesses of AA and aa (intermediate dominance).
19
Generic epistasis simply means an arbitrary
fitness landscape, with multiple peaks, minima,
etc. Such landscapes are necessary for
consideration of Macroevolution. However, within
a simpler context of Microevolution, it makes
sense to consider two intermediate kinds of
epistasis, which generalize simple linearity but
are still restrictive one-dimensional and
monotonic epistasis. One-dimensional epistasis
inherits, from the simplest linear case, the
assumption that there is just one
fitness-determining variable, fitness potential.
However, now fitness can be an arbitrary function
of this variable.
Examples of one-dimensional epistasis. (left) Log
fitness is plotted no epistasis (green), convex
(blue), and concave (red) fitness functions.
(right) Fitness is plotted no epistasis (green),
unimodal (blue), and bimodal (red) fitness
functions.
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The second restrictive kind of epistasis is
monotonic, meaning that a particular genetic
change never impacts fitness in the opposite
directions. The two restrictive modes of
epistasis are not equivalent one-dimensional
epistasis can be sign epistasis and monotonic
epistasis can be multidimensional.
(left) Monotonic, multidimensional epistasis
high values of both traits are deleterious, and
these deleterious effect reinforce each
other. (right) Sign, one-dimensional epistasis
intermediate values of the trait confer the
highest fitness.
generic fitness landscape generic epistasis
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This analysis of fitness landscapes prepares us
for considering selection. Of course, fitness
landscape alone does not define selection - the
position of the population is also essential.
Selection favors high, intermediate, low, and
extreme values of the trait in populations 1, 2,
3, and 4, respectively.
22
Let us start from the simplest case of unordered
genotypes. Then, just two key modes of selection
are possible, which do not depend on subtle
features of the fitness landscape i) Negative
selection - the most fit of the available
genotypes is common in the population, and less
fit genotypes are rare, ii) Positive selection
- the most fit of the available genotypes is
rare, and the most common genotype is less fit.
The same fitness landscape induces negative
selection in a population with two genotypes if
the common genotype is superior (left) and
positive selection if it is inferior (right).
23
Let us now consider genotypes arranged by their
values of a quantitative trait, which can be a
genotype-level trait such as fitness potential or
a phenotypic trait such as body size. First, we
can classify selection on such genotypes into i)
directional fitness increases or decreases
monotonously, favoring genotypes with one of rare
extreme values of the trait, ii) stabilizing
fitness has one maximum, favoring genotypes
possessing intermediate, common values of the
trait, and iii) disruptive fitness has two
maxima, favoring genotypes with either of the two
extreme values of the trait.
Directional (1 and 3), stabilizing (4), and
disruptive (2) selection.
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Second, we can classify selection on such
genotypes into 1) narrowing, which reduces the
variance of a trait with Gaussian
distribution and 2) widening, which increases
this variance. If log fitness is concave (its
second derivative is negative everywhere)
selection is narrowing, and if log fitness is
always convex (positive second derivative)
selection is widening. When the log fitness is
linear, so that fitness is exponential, the
variance of the Gaussian trait does not change.
Narrowing (blue), widening (red) and exponential
(log-linear, red) selection. Stabilizing
selection is narrowing, disruptive selection is
widening, and directional selection and be both
narrowing and widening.
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Two opposite kinds of selection are possible with
monotonous but multidimensional fitness
landscapes incompatibility (left) and
complementation (right) selection.
Incompatibility selection can lead to speciation.
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Fitness landscape may well be affected by the
population sitting on it. If features of fitness
landscape are aligned to position of the
population, selection is called soft, as opposed
to hard.
Soft selection can naturally result from
competition.
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If direction of selection depends on the position
of the population, selection is called
frequency-dependent.
Frequency-dependent selection in the case of two
genotypes (left) and a quantitative trait
(right). Frequency-dependent selection can
naturally result from different genotypes using
different resources. With apomixis such genotypes
would form different populations, but with
amphimixis they may still interbreed.
28
Finally, selection acting on a trait can be
either real - phenotypes which we watch really
affect fitness - or only apparent - phenotypes
which we watch do not affect fitness but are
connected to some variation which does. This
connection can be due to non-independent
distribution or pleiotropy.
This issue will be addressed later.
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Quiz Draw a fitness landscape and a population
on each such that selection within this
population is 1) directional and
positive, 2) directional and negative, 3)
directional and narrowing, 4) directional and
widening.
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3) Describing and assaying within-population
variation Three kinds of traits, structureless,
characterized by their entropy or virtual
heterozygosity quantitative, characterized by
their mean, variance, and higher moments and
complex, can be used to describe variation. A
structureless trait A with I states A1, ... AI,
is fully described by Ai (i 1, ..., I), the
frequency of the i-th state of the trait (any I-1
of them are sufficient, because ).
Mathematically, the best characteristic of
variation of a structureless trait is Shannon's
entropy
However, biologists prefer to use virtual
heterozygosity
Virtual heterozygosity is the probability that
two randomly drawn alleles are different. Both E
and H equal to zero only if the population is
monomorphic.
31
Often, we need to consider simultaneously several
variable, structureless traits. Joint
distribution of two structureless traits A and B
is described simply by the frequencies of
individuals carrying each possible combination of
their states, AiBj. Two traits are distributed
independently within the population if AiBj
AiBj. Otherwise, statistical association
between two traits (loci), can be characterized,
in the simplest case when each trait has just two
states (alleles), A1 and A2, and B1 and B2 by
coefficient of association ("linkage
disequilibrium") DA,B DA,B A1B1 A2B2 -
A1B2 A2B1 If traits A and B are distributed
independently of each other, DA,B
A1B1A2B2 - A1B2A2B1 0. Under
given allele frequencies, D deviates from 0
maximally when no more than three genotypes are
present in the population. In other words,
association between loci is maximal, where their
joint distribution is hierarchical. In a special
case when A1 B1 (or A1 B2), DA,B
deviates from 0 maximally, being DA,B A12 (or
DA,B -A12) if this hierarchy is poor, i. e.,
if A1 always occurs together with B1 (or with B2).
32
A quantitative trait, as well as a structureless
trait, can describe both genotypes and phenotypes
of individuals, for example, the number of
deleterious alleles or a body mass. Expressions
for a continuous quantitative trait x are
presented here, but the corresponding expressions
for a discrete trait are analogous. Variation of
a quantitative trait x within the population is
described by its probability density p(x), such
that p(x)dx is the fraction of individuals within
the population which possess the trait values
between x and xdx.
33
A distribution can be characterized by its
moments. The most important of them are the first
moment (mean) and the second central moment
(variance)
Mp
Vp
Often, instead of the variance, it is convenient
to deal with the standard deviation of the trait,
, which has the same
dimensionality as the mean.
34
A joint distribution of two quantitative traits x
and y is described by the corresponding
probability density r(x,y). The two traits are
distributed independently if r(x,y) p(x)q(y).
Otherwise, non-independence of their
distributions can be characterized by the
coefficient of covariance
Cx,y
In a variety of cases, we need to consider
complex traits that can accept values that are
not equally dissimilar from each other but also
cannot be naturally ordered 1. A segment of
sequence consisting, say, of 10 nucleotide sites.
Clearly, some states of such a trait (e. g.,
ATGCATGCAT and ATGCATGCAA) are closer to each
other than others (e. g., ATGCATGCAT and
CGAAGCGTCC), but there is no natural order within
this space of sequences. 2. Many phenotypic
traits (e. g., the shape of a wing) are, in
effect, infinite-dimensional. Again, two shapes
can be similar to different extents, but all
possible shapes cannot be ordered in a useful
way. 3. A trait may be an algorithm used by
individuals in a particular situation. For
example, individuals can form pairs and interact
repeatedly within a pair. If you partner, before
the current moment cooperated, defected,
defected, and cooperated again, what will you do?
35
The salient property of within-population genetic
variation is that, if we ignore extremely rare
variants, it can be resolved into distinct
variable traits (loci). This property follows
from high levels of similarity between genotypes
within a population.
Genotype 1 CTCAAACaAATC---GGGCAAAAgGTGG-TAT
TGAcAGG Genotype 2 CTCAAACaAATCggtGGGCAAAAt
GTGG-TATTGAaAGG Genotype 3
CTCAAACaAATCggtGGGCAAAAgGTGGaTATTGAcAGG Genotype
4 CTCAAACgAATCggtGGGCAAAAgGTGG-TATTGAaAGG A
ncestral State CTCAAACaAATCggtGGGCAAAAgGTGG-TATT
GAaAGG Genotype 1 GCTCCctAACGAAA ...
GTAAAattgATCCC Genotype 2 GCTCCgaAACGAAA
... GTAAA----ATCCC Genotype 3
GCTCCgaAACGAAA ... GTAAAatcgATCCC Genotype
4 GCTCCgaAACGAAA ...
GTAAAattgATCCC Ancestral State GCTCCgaAACGAAA
... GTAAAattgATCCC
36
Every opportunity to simply things further, by
lumping some elementary sequence-level traits
together, should be used. For example, if we
study the balance between deleterious mutations
and negative selection against them within a
protein-coding gene, all drastic sequence
variants can be lumped into one allele, as they
all have the same impact on fitness. HB407
17761,C?T 116,Arginine?Stop Calgary 24
17756,-G Frameshift UK 246 17782,T?C
123,Serine?Proline Sao Paulo 4 20360,T?G
Splicing acceptor splice A sample of
descriptions of loss-of function alleles of an
X-linked gene that encodes the protein known as
factor IX of blood coagulation in humans. All
these alleles lead, in males, to severe
hemophilia B.
37
Complete description of genetic variation
consists of frequencies of all possible
genotypes, i. e., of combinations of alleles at
all the polymorphic loci. The number of such
combinations, 2n, for n diallelic loci, can be
huge. However, a simpler description may be
enough. Frequencies of individual alleles at all
loci are sufficient to describe the population,
as long as alleles of different loci are
distributed independently. In a diploid
population with one locus A and two alleles A1
and A2 an individual is characterized by its
maternal and paternal alleles, so that four
genotypes A1A1, A1A2, A2A1, and A2A2 are
possible. Then, a population is fully described
by frequencies of any 3 genotypes, because A1A1
A1A2 A2A1 A2A2 1. We often can
treat two reciprocal heterozygotes together,
using the same notation A1A2 for both. Then,
there are only three genotypes A1A1, A1A2, and
A2A2, and frequencies of any two of them are
enough. Moreover, amphimixis with random mating
leads to independent assortment of alleles and,
if maternal and paternal allele frequencies are
the same,   A1A1 A1A1 A1A2
2A1A2 A2A2 A2A2   These
relationships are known as Hardy-Weinberg law. If
so, only one number, for example the frequency of
allele A1, (because A1 A2 1), is
sufficient to describe the population.
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Similarly, if we consider two diallelic loci A
and B, two variables, frequencies of alleles A1
and B1, may sufficient to describe a haploid
population, instead of any 3 of the 4 genotype
frequencies (A1B1, A1B2, A2B1, and A2B2),
as long as alleles at the two loci are
distributed independently and DA,B 0. When
many variable loci are considered simultaneously
and their alleles are not distributed
independently, it may still be possible to take
into account only pairwise associations between
them, and ignore higher-order associations. In
contrast, distributions of alleles at tightly
linked loci may be so strongly correlated that
only some combinations of their alleles, for
example A1B1 and A2B2 are present at most
moments. Such tightly linked loci may be
approximately treated as one locus. Even knowing
frequencies of all genotypes may be not enough -
but this is crazy.
39
We need to consider the smallest portion of the
genome which evolves more or less independently
of the rest. The main factor which forces us to
consider different loci together is epistatic
selection.
Fitness landscape over two loci, corresponding to
two sites that harbor nucleotides interacting in
an RNA secondary structure. High fitness is
conferred by Watson-Crick pairs GC (the best)
and AU, as well as by a relatively stable GU
pair, and other pairs lead to lower fitness.
Clearly, neither of these loci can be studied
separately. Bottom line divide et impere.
40
Variable population is a constellation of points
within the space of genotype, and each point has
its own brightness (frequency). Microevolution is
a slow movement of this constellation.
41
Alternatively, we can think of the space of all
such constellations, or of possible compositions
of the population. Simple example if we
consider haploids with one locus A and two
alleles, A1 and A2, the space of genotypes
consists of just two points. The corresponding
space of population compositions is the segment
of a line from 0 to 1, as we describe the
population by A1 (which is sufficient, because
A1 A2 1).
The same space of compositions can be used to
describe a diploid population, where 3 different
genotypes can be present, if it obeys
Hardy-Weinberg law.
If we consider one haploid locus with 4 alleles,
A, T, G, and C, the space of population
compositions is a part of a 3-dimensional cube
(excluding one of the alleles, for example, C),
limited by a plane A T G 1.
42
For "points" in the infinite dimensional space of
populations compositions if we characterize
individuals consider one quantitative trait.
The space becomes just a two-dimensional
Euclidean space, if the trait always has Gaussian
distribution.
Bottom line describe the population as simply as
possible, but not too simple.
43
The composition of a population must be inferred
from a sample of individuals. How to deduce the
value of an unobservable parameter of the
population from the observed data on a sample?
The common approach is maximal likelihood
(remember phylogenetic reconstructions?). Here,
our hypotheses H are different values of the
unobservable parameter (e. g., allele frequency
in the population), and our data D are the value
of this parameter within the sample. As before,
we seek such a hypothesis that produces the data
with the maximal probability, and interpret this
probability P(DH) as the likelihood of the
hypothesis given the data. For example, if 500
individuals from a sample of size 1000 have
genotype A1 (D 0.5), the ML estimate of H is
also 0.5 the chances of finding exactly 50 of
A1 individuals within the sample are the highest
when A1 0.5.
44
A ML, or any other estimate of H is just a
number, which "most likely" corresponds to its
real, unobservable, value. Because the chances of
estimating H exactly are slim, we also need to
know how good our estimate is. Confidence
interval with probability q is an interval that
includes the true, unobservable value of H with
probability q. 95 confidence intervals are
commonly used.
The concept of a confidence interval. 4 out of 5
independently computed intervals include the true
value of the unobservable parameter H.
45
In a sample of K individuals, we will observe k
A1 individuals with the probability   Thus,
under given H and K, k (which is our data D) has
a binomial distribution. The average value of
our data D k/K is H, and the standard deviation
of D is D is an unbiased ML estimate of H.
When K is large enough, binomial distribution
approaches Gaussian distribution. For a Gaussian
distribution, the probability of deviating, in
either direction, by more than 2 standard
deviations from the mean is 2.5. Thus, 95
confidence interval for this estimate of H is

k/H
.
46
4) Studying dynamics of within-population
variation Studies of any changes are based on
investigating dynamical models. A dynamical model
consists of a sufficiently detailed description
of the changing object at a particular moment of
time and of a transformation law that describes
how these changes occur. Time can be treated as
continuous or as discrete. The description of an
object is provided by variables, and all possible
combinations of their values constitute its phase
space. Changes of the object can be represented
by trajectories within the phase space. In
addition to variables, a model usually contains
parameters. Let us build and study a simple
deterministic dynamical model with one variable.
Consider a population of N individuals with
two possible genotypes, A and a. Individuals
breed true. Generations do not overlap, so that
time is discrete. The expected numbers of
offspring of an individual of genotypes A and a
are wA and wa, respectively. Unless wA and wa are
identical, selection operates within the
population.
47
The numbers of offspring with genotypes A and a
will be almost precisely NAwA and Nawa,
respectively, as long as N is large enough. The
frequency of A in the next generation, An1, is
provided by the ratio of the number of offspring
of genotype A over the total number of offspring.
Because a 1 - A, full description of our
population consists of just one number, for
example A, and we obtain the following
transformation law An1
wAA/wAAwa(1-A)
It appears that dynamics
of the population depends on two parameters, wA
and wa. However, if we divide both the numerator
and the denominator of the right-hand side by,
say, wA, we can see that this is not
so   An1 A/A(wa/wA)(1-A)
The following four statements summarize what we
achieved so far 1) if N is very large, the
model is deterministic, 2) the phase space of
our model is one-dimensional, 3) the
transformation law of our model does not depend
of N, 4) only the ratio of fitnesses of the two
genotypes, wa/wA, is important.

48
Let us create a continuous-time version of the
model. Dynamics with discrete and continuous time
can be very different. Still, if selection is
weak, i. e. that wa and wA are close to each
other, there will be no long jumps and time can
be treated in either way. Let us define selective
advantage of A over a as s 1 - wa /wA. s 0 if
fitnesses of A and a are equal, s gt 0 if wa lt
wA, and s lt 0 if wa gt wA. Then   An1
A/A (1-s)(1-A) A/1 - s(1-A)
  Selection is weak if s is small, so that
wa/wA is close to 1. Then, we can use an
approximation 1/(1-e) 1 e O(e2) (e means a
small number)   An1 A
sA(1-A)   Assume that velocity of A,
dA/dt, is equal to its increment between two
successive moments in discrete-time treatment,
An1 - A   dA/dt sA(1-A) This
differential equation describes the most
important process in Microevolution, an allele
replacement driven by natural selection.
Essentially the same equation also plays the key
role in population ecology, where it describes
population growth with self limitation (r is per
capita growth rate and K is carrying
capacity)   dN/dt rN(1-N/K)
49
Two ways of presenting the same model with
discrete time graphically by vectors that
describe jumps from a value A to the
corresponding value of An1 (two top figures)
and by a function An1 f(A) (bottom).
50
Two ways of presenting the same model with
continuous time graphically by vectors that
describe velocities of A corresponding to its
current values (two top figures) and by a
function dA/dt f(A) (bottom).
51
Comprehensive solution of a dynamical model is a
family of trajectories which show, for all
possible initial values, how the variables will
changes in the future. Model of selection-driven
allele replacement is simple enough to be solved
explicitly (x A)
Gather different variables at different sides
(useful mneumonics)
Rewrite the differential equation in integral
form
52
The right-hand side integral is simply s(t-t0),
and the left-side integral is
because
. Further, the right-habd side is
Thus, we now need to recover x(t) from
53
This family of x(t) is a comprehensive solution
of our model. Each trajectory corresponds to its
own initial frequency of A, with the value x0 at
time t0. Moreover, the dynamics of the frequency
of A also depend on s, and for each value of s
there exists its own family of trajectories.
Naturally, x(t) increases with time if s gt 0,
decreases if s lt 0, and does not change if s 0.
54
We can also investigate our model only
qualitatively, finding attractors and their
stability. This this only way, when there is not
explicit comprehensive solution. There are two
exceptional initial frequencies of A, x1 0 and
x2 1. Trajectories with such initial
frequencies are flat, i. e. if x is equal to 0 or
to 1 at some moment, it never changes and retains
this value forever. Biologically, this result is
obvious. Values of variables that do not change
are called equilibria. With s gt 0, equilibrium x1
0 is unstable, in the sense that a small
deviation from it will increase and equilibrium
x2 1 is stable, because a small deviation from
it will decrease. It is the other way around with
s lt 0. With s 0, every value of x is an
equilibrium, and all these equilibria are neutral.
Stable, unstable, and neutral equilibria are
blue, red, and green, respectively.
55
To find equilibria, we replace a dynamical
equation with an algebraic equation. With
discrete time, we ask that the next state is
identical to the current state A
A/A(wa/wA)(1-A)   which is a quadratic
equation with two roots, A1 0 and A2 1.
Only if wa/wA 1, every value of A satisfies
this equation. With continuous time, we ask that
the velocity is zero 0 sA(1-A)
which has the same roots. Local stability of
an equilibrium is determined by whether small
deviations from it increase or decrease.
With continuous time, equilibrium is stable if
dx/dt is a decreasing function at it, and
unstbale if it is increasing.
56
To complete qualitative investigation of the
model, we need to understand transitions between
its qualitatively different modes of dynamics.
Here, there are three such modes s lt 0, s 0,
and s gt 0.
When, for example, a negative s starts increasing
very slowly, the rate of decline of allele A
frequency will diminish, until everything freezes
at s 0, after which the frequencies will start
growing slowly.
57
Direct problem of dynamical theory we know what
forces affect our object, how will it
change? Inverse problem of dynamical theory we
know ow our object changes, what forces are
affecting it? For example, one can try to
estimate the strength of selection from the rate
of changes of the genotype frequencies
If the frequency of allele A changed from 50 to
51 in one generation, its selective advantage
must be 0.04. Factors that can affect dynamics
of within-population variation are mutation,
selection, mode of reproduction, population
structure, and drift. Microevolution is due to
their joint action. However, before dealing
with this joint action, we will first need to
consider variation and each of the 5 factors
acting separately.
58
Quiz Solve differential equation describing
a positive selection-driven allele replacement.
Make sure you understand every step this is the
key equation in both Microevolution theory and
population ecology. Please do this not by just
copying from text! Seek help if needed.