Download FFTNS and the shifting balance theory p2

FUNCTIONAL DIVERGENCE TOPIC 1: FFTNS and the shifting balance theory
It is worth revisiting neutral theory before moving on to models of adaptive
evolution. It was clear from the beginning (Kimura 1968; King and Jukes 1968) that
neutral molecular evolution was an implausible mechanism for phenotypic evolution
at the level of the organism. There is virtually no controversy concerning the
inadequacy of neutral theory on this point. This topic introduces two of the more
influential models for adaptive evolution. Neutral theory and models of adaptive
evolution should not be viewed as disjoint entities; there is a role for both in
explaining molecular evolution.
A review of the notion of fitness
Remember that FITNESS is a measure of an individual’s reproductive contribution to
the future, and has two components: (i) an individual’s ability to survive and
reproduce [VIABILITY]; and (ii) an individuals reproductive output [FECUNDITY].
We used the symbols WAA, WAa, and Waa to specify the relative fitness values of the
AA, Aa, and aa genotypes.
Symbolism for selection
AA
Aa
p 02
2p0q0
WAA
WAa
Genotype
Frequency
Phenotype
aa
q 02
Waa
By our definition the fitness ratio is:
WAA : WAa : Waa
and the suvival ratio of AA, Aa, and aa genotypes among the adults will be:
p2WAA : 2pqWAa : q2Waa
The above ratios are correct, but they are NOT frequencies because they no longer
add up to 1! So, to build a model that included selection, we divided the individual
fitness values by the grand total after selection. This normalized the terms so that
the frequencies sum to 1. The grand total is a measure of the AVERAGE FITNESS (
of individuals in the population.
=
)
p2WAA + 2pqWAa + q2Waa
So, we work with a normalized fitness of a genotype; i.e., fitness divided by the
average fitness.
We consider fitness a phenotype because it is an interaction between the genotype
and the environment. It is important to remember that in the following models we
use fitness as phenotype, not phenotypic characters of organisms. It is important
not to confuse these, because in natural populations people often measure
phenotypic traits that they ASSUME correlate with fitness.
Fisher (1930) viewed fitness as the per unit time increase of individuals with
different genotypes, and that is where we will begin.
Fisher’s fundamental theorem of natural selection (FFTNS)
R. A. Fisher (1930) developed a one-locus model of natural selection that implied an
interesting property of population evolution. In words, his theorem states that the
rate of increase in the average fitness of a population (
) is equal to the genetic
component of the variation in fitness. Fisher (1930) gave this model the humble
name: “The fundamental theorem of natural selection”.
Let’s take a look at the basis of Fisher’s theorem. Fisher started with the well known
formula for the RESPONSE OF A POPULATION PHENOTYPE TO SELECTION (R).
R = h2 × S
h2:
The proportion of total phenotypic variance that is
predictably transmitted to next generation (i.e., additive
genetic component of variance)
S:
SELECTION DIFFERENTIAL; the difference between the mean
phenotype of those under selection and the mean
phenotype of the population.
The details of this way of treating selection are interesting, but beyond the scope of
this course. The general equations for R (not shown) will work for any phenotype
that affects fitness, and remember that Fisher was interested in fitness as a
phenotype itself. Fisher wanted to build a model based on R that would answer the
question of what happens to the fitness of a population over time. By substitution
and some clever manipulation (not shown) Fisher obtained the following result:
[This is FFTNS]
The change in population fitness depends on just two parameters.
:
Va(W):
The average fitness of the population
Additive component of the total variation in fitness
Biological implications of FFTNS
FFTNS is a very simple formula, but as a model it has a number of noteworthy
implications for adaptation as a process of microevolution.
1. Populations can not adapt without additive variance in fitness. The parameter
Va(W) can be zero or positive. When it is zero, there can be no change in the
mean fitness of the population; ie., when Va(W)=0,
=0.
2. The rate of change in the mean fitness of the population (
) depends on its
mean fitness ( ). The more fit the population is to start with the slower the
rate of change due to selection.
3. Fitness always increases. The parameter
must be ≥ 0. Hence, the
population can only increase in fitness, and once it goes up in fitness it can
never go back down.
a. This seems trivial, but it means that a population will always go to the
top of the “nearest peak in fitness”. If there exists multiple fitness
peaks, and to get to the global peak a population must go through a
“fitness valley”, FFTNS predicts that the population will be unable to
reach global fitness; it can only go to the top of the local fitness peak.
b. Once at the top of a peak, selection will act to keep it at this point.
Adaptive topography
FFTNS led to the notion of an ADAPTIVE TOPOGRAPHY; a surface of mean fitness for a
population where peaks represent the highest values of mean fitness, and valleys the
lowest values of mean fitness. An adaptive topography is sometimes called an
ADAPTIVE LANDSCAPE. Under FFTNS populations can only go “uphill”.
The simplest case of an adaptive topography can be illustrated with a plot of
population mean fitness for all possible frequencies of two alleles at a single locus
(see plot below).
Population fitness surface for two alleles at one locus under directional selection and overdominant selection
WAA = 0.5
WAa = 1
Waa = 0.1
WAA = 0.1
WAa = 0.75
Waa = 1
Directional selection
Over-dominant selection
In both cases shown in the above figure, there was only one peak in fitness on the
topography, so the population always goes to the peak. In a slightly more complex
case with one locus and three alleles, the adaptive topography has multiple fitness
peaks. In such a case the fate of the population depends on where its starts on the
fitness landscape.
To view the more complex fitness landscape we need a different method of
graphically representing the population allele frequencies. The De Finetti diagram
turns out to be very useful for this but it takes a little time to become familiar with
its layout. To help familiarize you with it, I have provided a generalized version of a
De Finetti diagram below. We assume three alleles whose frequencies sum to 1.
The three vertices of the triangle represent the three points where a population is
fixed for one of the three alleles. Any point in space within the triangle represents
three allele frequencies; the frequency of the allele of interest is the perpendicular
distances from the site opposite the involved vertex to the point in space.
Example of a De Finetti diagram of three allele frequencies at a locus
Allele 1
Indicates:
Alleles 2 and 3 have
freq =0 at this vertex
Allele 1: freq = 0.45
Allele 2: freq = 0.25
Allele 3: freq = 0.60
0.6
0.25
Alleles 1 and 2 have
freq =0 at this vertex
0.45
Allele 2
Allele 3
Alleles 1 and 3 have
freq =0 at this vertex
We are interested in the fitness topography, so we plot the contours of the three
allele frequencies that yield the same population fitness.
An example is shown below that is based on estimates of the fitness of certain
human populations based on three alleles at the beta-globin locus.
Adaptive topography [mean fitness surface] for all possible frequencies of three alleles at a single locus.
Allele 1
Lowest mean fitness
Valley on the fitness landscape
Local peak in mean fitness
Global peak in mean fitness
Allele 2
Allele 3
In the above fitness topography there are two peaks; one is a local peak reflecting a
population polymorphism for alleles 1 and 2, and the other is the globally maximum
peak where allele 3 is fixed in the population. Note the yellow line. This is a FITNESS
VALLEY, a region of locally minimum fitness that separates two peaks on the fitness
topography.
Natural selection cannot move a population across a valley in the adaptive topography.
start here
end here
end here
start here
From FFTNS we see that a population cannot cross a fitness valley by natural
selection, as it cannot go “down-hill”. This leads to an important notion: selection
can actually prevent globally adaptive evolution! It also means that the results of
natural selection are subject to historical effects; i.e., the end-point depends on the
initial allele frequencies.
Difficult assumptions of FFTNS
1. Constant fitness throughout time. Fitness of an allele can vary in different
environments, and consequently as environments vary over time (season of
year, year to year, etc.) so do the fitness levels. The fitness of an allele could be
density dependent, which can vary in time as well.
2. Linkage equilibrium. Alleles at different loci can be in non-random association.
Thus an allele can obey the theorem as it applies to another allele to which it is
closely linked. The most dramatic effect is when a deleterious allele is tightly
linked to a highly beneficial allele; the deleterious allele can increase in
frequency due to the “hitchhiking effect”. If multiple loci affect fitness, then
linkage and rate of recombination must be considered in a model that predicts
their microevolutionary fate.
3. Fitness must be the phenotype. People measure traits in natural populations that
they think correlate with fitness. In order to make prediction under FFTNS, one
must be able to separate genetic from environmental components of variance;
this is not easy to do.
4.
No genetic drift. FFTNS predicts an allele with any advantage will be fixed with
a probability of 1. As we have seen with a beneficial allele in a finite population,
if the product of Ne and s is less than 1, the probability of fixation is just 1/2
Ne .
There are many natural conditions under which we expect that the assumptions of
FFTNS will be violated [e.g., two-sex viability selection, fecundity selection,
ecological complications, etc,]. Nevertheless FFTNS has been influential in providing
a useful way of thinking about population evolution under natural selection
The marginal fitness of an allele
Before we can move on to a real dataset under FFTNS, we need to develop one more
measure of fitness. In contrast to measuring the average fitness of the population,
we also want to measure the average fitness of all the individuals in a population
who bear a certain allele; this latter measure of fitness is called the MARGINAL FITNESS
or AVERAGE AFFECT OF AN ALLELE.
Comparison of the marginal fitness of an allele with the average fitness of the
population tells us where an allele will increase, stay the same or decrease in a
population.
Consider a locus with two alleles:
Allele A (freq = p)
Allele a (freq = q)
The marginal fitness of the
a allele (Wa) is:
Wa = q(Waa) + p(WAa)
Now consider a locus with three alleles:
Allele A (freq = p)
Allele a (freq = q)
Allele c (freq = r)
The marginal fitness of the
a allele (Wa) is:
Wa = q(Waa) + p(WAa) + r(Wac)
The value of this system is that we can extend it to any number of alleles at a locus
and do a quick calculation to determine the influence of selection on an allele
Wa Wa Wa -
= 0; no change in frequency of
a
a allele increases in frequency
< 0; the a allele decreases in frequency
> 0; the
The details of this relationship come from FFTNS; you are referred to population
genetic texts such as Hartl and Clark for a more complete description.
Adaptation in human populations: sickle cell haemoglobin
One of the classic examples of natural selection is overdominant selection for the A
and S alleles of the beta-chain of Haemoglobin in humans of West Africa. The reason
for such selection appears to be a degree of resistance to the malaria parasite (P.
falciparum), whose merozoite stage infects red blood cells.
The three possible genotypes, their phenotypic effects, and their fitness values are
shown below.
Genotypes
AA
AS
SS
Blood Phenotype
Normal
40% sickling of RBCs
Sickle cell anaemia
Mortality1
moderate
Low
very high
Fitness1
0.89
1
0.2
1: Fitness and mortality are estimated as an average over 72 west African populations of
humans. Data from Cavalli-Sforza and Bodmer (1971).
Sickle-cell anaemia in SS heterozygote is so detrimental that fitness is very low in all
environments. Heterozygotes, AS, experience about 40% of RBCs sickling during an
episode. Sickling of a significant proportion of cells will disrupt the development of
the parasite. Although this is not sufficient to kill all the parasites, it slows
reproduction enough for immune system to mount an effective attack on the
remaining parasites.
Based on the data in the above table, the fitness topography is easily determined.
Perhaps surprisingly, fitness of the population is maximized by a frequency of the S
allele of just 0.12.
Adaptive topography for A and S alleles in malarial environment
As before, the fitness of the population can only
go “uphill”. Marginal fitness calculations verify
this result.
Initial freq of S = 0.01:
WS = 0.99 and
= 0.89
WS -
= 0.10; the S allele will increase
Initial freq of S = 0.25:
WS = 0.8 and
= 0.88
WS -
= -0.088; the S allele will decrease
Peak in average fitness of population
The equilibrium frequency of heterozygotes is 0.212, and the average fitness of the
population (
) is 0.91. This means that at equilibrium only 20% of the population
is protected! This is because many people with the normal RBCs survive to
reproduce (the average chance of survival to reproduction is 0.91).
Some populations in West Africa have a third allele, C, that is relevant to malarial
resistance. Individuals homozygous for the C allele have resistance to malaria but no
severe anaemic affects. Let’s look at what happens to a population where the C
allele occurs, and the S allele is not present.
Genotypes
AA
Blood Phenotype
Normal
Fitness
0.89
Data from Cavalli-Sforza and Bodmer (1971).
AC
Normal
0.89
CC
Resistant
1
Adaptive topography for A and C alleles in malarial environment
Marginal fitness calculations verify the plot.
Initial freq of C = 0.01:
WC = 0.891 and
= 0.890
WC -
= 0.001; the C allele will increase [slowly]
Initial freq of C = 0.25:
WC = 0.917 and
= 0.90
WC -
Frequency of “C” allele
= 0.02; the C allele will increase
Peak in average fitness of population
In the case of the C allele above, the population will go to fixation, with 100% of the
population protected (as compared with only 20% protection under AS over
dominance).
An interesting phenomenon is observed when we consider a population where both
the S and C alleles are present. The fitness values for West Africa are as follows.
Genotypes
AA
AS
SS
AC
SC
CC
Frequency
p2
2pq
q2
2pr
2qr
r2
Fitness
0.89
1
.2
.89
.71
1.31
Mortality1
moderate
low
Very high
moderate
moderate
low
Anaemia
none
some
severe
none
some
none
1: Fitness and mortality are estimated as an average over 72 west African populations of humans. Data
from Cavalli-Sforza and Bodmer (1971).
Consider the case where the population is at equilibrium for AS, and the C allele
occurs by mutation. To investigate evolution in this case we can compare the
marginal fitness of the S and A alleles to the average fitness of a population with
both alleles present.
Frequency of A = 0.879 = p
Frequency of S = 0.120 = q
Frequency of C = 0.001 = r
The marginal fitness of the S allele is:
WS = q(WSS) + p(WAS) + r(WSC)
WS = 0.12(0.2) + 0.879(1) + 0.001(0.71)
WS = 0.903
The marginal fitness of the C allele is:
WC = r(WCC) + p(WAC) + q(WSC)
WC = 0.001(1.31) + 0.879(0.89) + 0.12(0.71)
WC = 0.869
The average fitness of the population is computed in the usual way.
= 0.903
The fate of the S allele:
WS -
= 0; no change in the population
The fate of the C allele:
WC -
= -0.03; the C allele cannot invade!
What if both the S and C allele are rare?
Frequency of A = 0.879 = p
Frequency of S = 0.05 = q
Frequency of C = 0.01 = r
WS = 0.957
WC = 0.885
= 0.898
WS WC -
= 0.06; The S allele invades!
= -0.01; the C allele cannot invade!
The result is that natural selection will push this population to a strategy where only
20% are protected from malaria, despite an optimal solution of 100% protection
(i.e., fixing the C allele) being available.
We can understand this result if we take a look at the adaptive topography for these
three alleles:
Our example from above comes from the three allele haemoglobin case.
S allele
In this region the C
allele only needs to
be a littler more than
10%
The C allele has to
start with a high
frequency in order for
it to invade
C allele
A allele
The fitness valley prevents the population from achieving the globally optimum state
when the initial frequency of the C allele is low. Natural selection, on its own, simply
cannot cause the population to cross this region. Once it begins going up the local
peak, in the direction of a stable AS polymorphism, it cannot go back (FFTNS).
Let’s check it out by testing the marginal fitness of the C allele when it is in a higher
frequency in the population.
Frequency of A = 0.64 = p
Frequency of S = 0.25 = q
Frequency of C = 0.11 = r
WS = 0.768
WC = 0.891
= 0.877
WS -
= -0.10; The S allele is a gonner!
WC -
= 0.01; the C invades!
The reason for the difference between the C and S alleles is due to the effectiveness
of selection on heterozygotes. When an allele is rare it is most often found in the
heterozygote. In the case of the AS heterozygote (fitness = 1), the phenotype is
beneficial; in the case of the AC (fitness = 0.89) and SC (fitness = 0.71)
heterozygote the phenotype is deleterious. Because most of the C alleles are found
in the heterozygous genotypes when it is rare, it cannot invade a population that
also has S alleles.
We have seen that the C allele can invade when its allele frequency exceeds 10%.
This represents the far side of a fitness valley. Once the C allele gets across this
valley, FFTNS take over and the population moves to the global peak, where 100%
have CC genotype. We noted earlier that natural selection cannot get the population
across the valley on its own. Two other population genetic forces can provide a
means of getting the population across:
1. Strong drift effect. A large change in frequencies due to a bottleneck or a
founder event could get the frequency of the C allele high enough in one
generation that it comes under the influence of the global peak.
2. Inbreeding: Inbreeding changes genotype frequencies, not allele frequencies.
The adaptive topography changes because dominance effects are reduced
(homozygosity is increased); this favours the C allele and disfavours the S
allele.
3. Large Migration event of individual carrying the C allele in their genome.
Wright’s shifting balance model of adaptive evolution
The notion of a complex adaptive topography, with both adaptive peaks and valleys,
is actually due to Sewall Wright, who proposed the shifting balance theory (SBT) of
evolution. Wright (1932) was motivated by a particular problem: “The problem as I
see it is that of a mechanism by which species may continually find its way from
lower to higher peaks…” Wright (1932, 1977) proposed the idea of an adaptive
topography as a conceptual model of the relationship between the mean fitness of a
population and the frequencies of the involved genotypes.
Wright’s solution was a model of adaptation he called the SHIFTING BALANCE THEORY; a
model that proposes how a population can move from one locally stable peak in
mean fitness to another, and in the process cross an adaptive valley, by means of
genetic drift and population selection. In SBT genetic drift provides the mechanism
for a population to “drift” about the local peak it finds itself on. You can think of
genetic drift as allowing a population to explore the local surface of the adaptive
topography; this is not possible under FFTNS, as that model is always kept on the
peak by selection. On rare occasions, with a strong enough drift effect; a population
can get across a valley.
Before we get to the details of SBT, we must review its underlying assumptions:
Assumption 1: A large amount of multi-locus polymorphism is
maintained in an equilibrium state. This assumption is conditional on
two premises: (i) such variation is relevant to fitness (i.e., generally not
neutral; after all, this is a model of adaptation); and (ii) fitness variation
is relevant to “minor factors”. Minor factors refer to small fitness
consequences [small s], mutation selection equilibrium points [low
frequencies], recurrent mutations, etc.
Assumption 2: A single genes has effects on multiple phenotypic
characters. This is called PLEIOTROPY.
Assumption 3: A complex adaptive landscape with multiple fitness peaks
and valleys.
Assumption 4: Multiple, partially isolated populations. This increases the
chance that an allele will get across a fitness saddle and get fixed in one
of the local populations.
Let’s take a look at a single population under assumptions 1, 2, and 3, and with a
very large effective size (Ne). In this case drift does not move them around much.
The figures below illustrate how such a population will appear on an adaptive
topography.
A small portion of a hypothetical adaptive
topography. The lines are fitness contours This
portion has two peaks (+) and two valleys (-).
The landscape is a multidimensional field of
allele frequencies, represented in 2 dimensions
for easy visualization.
Mass selection under uniform conditions. The
population (shaded) is held tightly to the peak
This case:
1. effective selection (Nes very large)
2. low mutation (Neµ lower)
3. drift is NOT moving the popn (Ne very
large)
Mutation pressure pushes the population out in
all directions relative to the case above
This case:
1. less effective selection (Nes still large)
2. more mutation (Neµ higher)
3. drift is NOT moving the popn (Ne very
large)
Now let’s add in the effects of genetic drift by substantially reducing the effective
size of the population.
This population has a very small effective size.
It has very high levels of population inbreeding
and moves more erratically over the landscape
as movement is purely by drift. Such a
population will be small with little variation. Over
time, this population will fix too many deleterious
mutations, leading to degeneration and
extinction.
This population has an intermediate effective
size. It also movers over the landscape but
stays in the vicinity of the peak. This population
is in a perpetual state of “SHIFTING BALANCE” of
altered gene frequencies and selection
coefficients. On very rare occasions such a
population could cross a “saddle” on the adaptive
topography.
Now let’s put a structured species on the landscape.
This represents a very large population that is
subdivided into numerous local populations. The
local populations are connected to each other
via individuals that move among populations and
reproduce (i.e., gene flow). Each local
population is part of a continually shifting
balance.
The complete model of SBT is specified by three phases of evolution:
1. Phase of genetic drift: Random drift moves the population, and recurrent
mutation spreads it out. Selection acts to push the population back up towards the
peak. Migration among local populations prevents them from evolving independently
of each other. So alleles lost via drift in some populations can be regained by
mutation and migration between populations. This results in a continually shifting
balance that allows “exploration” of the local topography.
2. Phase of Mass selection: Occasionally a population will cross a saddle in the
adaptive topography. Once across, selection becomes a strong force to move the
population close to the “new peak”. This transition is expected to occur rapidly, until
a new shifting balance is reached.
conditions of phase1.
Thus this phase ends with a return to the
3. Phase of inter-population selection: Once a transition to a new peak is made, gene
flow among populations is a force for moving the population. If there is sufficient
excess reproduction coming from the population on the new peak, the entire set of
local populations could be shift to an equilibrium surrounding the new peak. This
effect is based on the assumption that a population that comes under the control of a
higher fitness peak produced more offspring than populations under the control of
relatively lower fitness peaks.
Greater success and reproduction of individuals
with equilibrium allele frequencies determined by
the peak in the lower right quadrant means that
there is net change in the allele frequencies of
all the populations in that direction. The
direction of the change to a new equilibrium is
indicated by the arrow.
Difficulties with SBT
SBT sounds great; however, the theory has been controversial since the very
beginning.
R. A. Fisher, one of the founders of the discipline of population genetics, immediately
criticized the theory on two points: (i) species never get stuck on peaks, as there are
always paths that are going up; and (ii) the effective population sizes in nature were
too large for drift to move them around in the required amount. These two points
are not conclusive. Specific cases appear consistent with the notion of valleys
between peaks (e.g., C allele in human haemoglobin).
Moreover population
structures and effective population sizes can vary considerable over macroevolution
time periods; so SBT needs only for the appropriate conditions to be met at some
point in time.
Recently, more focused criticisms have been levelled at SBT:
1. Low migration rates are required for exploration and transition during phases
1 and 2, yet higher migration rates are necessary to complete phase 3.
2. Population structures typical of natural populations seem to be too small for
phase 1
3. Group selection is a weak force for evolution, and hence unlikely to result in a
shift in equilibrium. Because it is weak, an extremely high amount of
migration is required among sub-populations for inter-population selection to
work as suggested in phase 3.
4. Alternatives seem more likely.
Alternatives to genetic drift as the force for movement on the topography
Wright (1932) initially proposed several alternatives:
1. Change in environment changes the landscape. Essentially, the peak moves
out from under you, or a new “ridge” appears in the fitness landscape.
Remember industrial melanism; dramatic changes in the fitness topography
clearly can result from changes in the environment.
2. Mutation. In this case a mutation occurs that allows a population to evolve to
a higher fitness without going through a deleterious intermediate stage.
Perhaps a population has a rare recessive allele that would be beneficial but
the heterozygotes have severely reduced fitness. Such a population might
need to “wait” for a compensatory mutation to occur.
We know that
compensatory mutations can be very strongly selected (e.g., the stem
structures of RNA molecules)
3. Change in the strength of selection. In this case the structure of the adaptive
topography stays the same but the relative height of the peaks, or depth of
valleys, changes. A flattening of the topography would allow a population to
move under FFTNS.
These alternatives require a population to “wait” in time for such events to occur.
The criticisms in the previous section point out the conditions for SBT that might not
be optimal in natural populations.
However, the alternative models require
populations to “wait” for changes in the landscape or for certain mutations; why not
expect that populations must also “wait” to work on those occasions when structure,
or levels of gene flow, are appropriate for SBT?
It is worth comparing SBT with Neutral theory one more time. SBT concerns events
that are extremely rare as compared with events modeled under neutral theory. The
rarity of such events means that the chance of observing data from populations that
reflect evolution under SBT is very low. Hence, SBT does not provide us with many
predictions we can test directly. Alternatively, most natural populations are likely to
exhibit some data relevant to tests of neutral evolution. Some have suggested that
SBT can neither be proven nor disproved (e.g., Whitlock 2002). Lastly, we cannot
directly observe adaptive valleys, peaks, saddles etc, so we can never be completely
sure if a population ever really did cross a valley, as opposed to following FFTNS to a
fitness peak.
Despite these difficulties, SBT has been important in evolution. It has provided us
with a useful conceptual model, and encouraged further development of the theories
of genetic drift, group selection, and complex adaptive topographies.
SBT and the evolution of heterostyly
[I will try to get it into the lecture in the future; no time this year]