I. Darwin's deduction of natural selection
B. Darwin's Crucial assumption
1. Most of the assumptions used by Darwin in his logical deduction
of evolution by natural selection are
self-evident or well established.
2. He had, however, two novel insights, associated with assumption of step 7:
a. Some individuals possess certain traits that make them are better
equipped for competition to obtain
resources; they possess these traits because of their genetic makeup;
and their increased competitiveness
causes them to produce more offspring. (PRINCIPLE OF NATURAL SELECTION)
b. Because these traits are hereditary and individuals possessing
them produce more offspring, the traits
will be more common in the next generation. (PRINCIPLE OF EVOLUTIONARY
CHANGE)
3. Both of these assumptions are clearly open to experimental verification or refutation (see below).
C. Darwin's logic is not circular.
D. Darwinian Fitness
1. Although Darwin's deduction of natural selection is phrased
in terms of differential success among genotypes
in the competition to obtain resources, which suggests that the driving
force behind natural selection is
differential survival, Darwin clearly recongized that natural selection
could also arise due to differential
reproductive success among individuals.
2. His novel insight should thus be rephrased in a more general
fashion: Some individuals have higher FITNESS
because of their genetic makeup; or , more formally, individuals of some
genotypes have a higher fitness, on
average, than individuals of other genotypes.
a. Individual fitness may be defined, at this point, as the number of offspring produced by an individual.
b. Average fitness of individuals of a given genotype is influenced
both by the probability that such an
individual will survive to reproduce and by the number of offspring that
it produces if it survives.
c. In particular, for an annual organism with one generation per year, for a particular genotype,
fitness = W = (probability of survival) x (average number of seeds produced)
E. Natural Selection
1. Definition: Natural selection occurs when genotypes differ in average fitness.
2. Natural selection IS NOT the same thing as Evolution.
a. Natural selection is one cause of Evolutionary change,
as we shall soon see.
b. Processes other than natural selection can cause Evolutionary
change, as we shall also soon see.
II. Testing Darwin's Crucial Assumption about Natural Selection
A. Numerous examples exist of genotypes differing in fitness,
indicating that natural selection is ubiquitous in
nature (e.g. see Endler, J. A. 1986. Natural Selection in
the Wild. Princeton University Press, Princeton, NJ).
B. Example: Leaf-shape polymorphism in Ipomoea hederaceae.
1. This example not only demonstrates that genotypes differ in
fitness, but also illustrates how fitnesses are
calculated experimentally.
2. This example also shows that which allele is favored, or which
genotype has the highest fitness, is not absolute,
but depends on the environment in which a population exists.
III. Formalizing Darwin's argument about natural selection causing evolutionary change.
A. Applicability of Darwin's logic to Mendelian inheritance
1. Although Darwin's argument seems rather impeccable, he derived
it long before anything was known about
the true nature of heredity.
2. Consequently, it behooves us to enquire whether the laws of
Mendel are compatible with that argument and,
if so, to determine whether we can derive a theoretical framework that
will allow us to predict changes in
gene frequency.
3. We will concentrate on developing such a theoretical framework
for predicting evolution at a single locus with
two alleles, since this is relatively straightforward. Incorporating
multiple loci into the framework is
conceptually straightforward but analytically too complicated for this
course.
B. Goal: predict change in gene frequency from one generation to next
1. Information needed for prediction
a. initial gene frequency
b. fitnesses of the genotypes at the locus of interest
2. Possible methods used for prediction
a. Simulation
b. Derivation of a dynamical equation
IV. Predicting gene frequency change through simulation
A. Features of simulation
1. Initial state is a population of zygotes, each of whose genotype is specified and kept track of by the computer
2. The values of fitness for each genotype are used to determine whether an individual survives or not.
a. For each individual, determine from its genotype what its probability
of survival is.
b. The computer determines, based on that probability and selection
of a random number, whether that
individual survives or not. The computer keeps track of individuals
that survive.
3. Zygotes of the next generation are formed by random mating among survivors.
a. For first zygote, chose a male parent randomly from the survivors
and a female parent randomly from
the survivors.
b. Determine the genotype of that zygote according to Mendel's
laws by chosing randomly an allele from
the male parent and an allele from the female parent.
c. Repeat this process until a new generation of zygotes is produced.
4. Repeat steps 2 and 3 for each generation.
B. Results of simulation
1. Set WAA = 0.5, WAa = Waa = 0.67, initial p = 0.001 and population size = 500
2. A simulation of 100 generations shows that p increases to close to 1.0, i.e. the a allele replaces the A allele.
3. A set of 10 simulations shows a similar pattern, though the
exact trajectory of gene frequency change
differs for all 10 simulations.
4. The mean trajectory of the 10 simulations exhibits a steady, monotonic increase in gene frequency of the a allele.
C. Conclusion: simulation indicates that, at least in the case
examined, if fitnesses differ for the three genotypes
(i.e. natural selection occurs), this will be accompanied by evolutionary
change (change in gene frequency).
V. Predicting gene frequency change with a deterministic equation.
A. The equation for gene frequency change at a single locus with two alleles subject to natural selection is
p' = p[pWAA+ qWAa] / [p2WAA+ 2pqWAa + q2Waa] (I) ,
where p is the frequency of the A allele, q = 1 - p = frequency of the a allele, and Wij= fitness of genotype ij.
B. This equation may also be written as
Dp = p' - p = [pq/W..] [p(WAA - WAa ) + q (WAa - Waa ) (II),
where W.. = p2WAA+ 2pqWAa + q2Waa .
C. Using the equation.
1. Plug in initial value of p and values for the fitnesses, Wij , and calculate new gene frequency, p'.
2. Iterate by plugging in the new value of p to calculate subsequent value of p.
D. Comparison of predictions using simulations and predictions using deterministic equation.
1. Using the same values of the Wij , the
deterministic equation yields a predicted trajectory of gene frequency
change very similar to the mean trajectory obtained in the simulations.
2. This result suggests that deterministic equation is an accurate predictor of evolutionary change for one locus.
VI. Derivation of Deterministic Equation for Gene Frequency Change
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VII. Properties of Evolution Revealed by Deterministic Equation(see accompanying Handout)
A. When fitnesses are all equal, no evolutionary change is expected.
1. To see this, substitute the constant W for the Wij
in equation (I) and simplify to get p' = p , i.e. no
evolutionary change.
2. Consistent with Hardy-Weinberg results: if there is no natural selection, expect no change in gene frequencies.
B. We need only know relative ranking the Wij to predict long-term evolutionary change.
1. To see this, first recognize that evolutionary equilibrium is achieved when Dp = 0.
2. All possible relationships among the may be grouped into three categories:
a. WAA <,= WAa <,= Waa , with one inequality strict. (Note that this case pertains to
WAA > WAa > Waaalso, since this is obtained by just relabeling the alleles).
b. WAA < WAa > Waa
c. WAA > WAa < Waa
3. For category a, p will always decrease until the a allele is fixed. This can be seen from Eq. (II), as follows:
WAA <,= WAa <,= Waa implies WAA - WAa < 0 and WAa - Waa < 0 , so that Dp < 0, i.e. the
frequency of the A allele always decreases. (Handout)
4. For category b, a polymorphism will be maintained. To
see this, note that when WAA < WAa >
Waa is true,
a solution to is given by
p* = (WAa - Waa ) / [ (WAa - WAA) + (WAa - Waa ) ]
(plug this value of p into Eq. (II) to verify that it yields Dp = 0).
a. That this equilibrium is STABLE (i.e. if the population is perturbed
away from this equilibrium, it will return to
p*) , can be shown mathematically.
b. Because the proof is too complicated for this course, however,
we rely instead on the results of numerically
iterating Eq. (I) for various starting values of p to demonstrate
the stability of the equilibrium (Handout).
5. For category c, there is again an equilibrium at
p* = (WAa - Waa ) / [ (WAa - WAA) + (WAa - Waa ) ]
but this equilibrium is unstable.
a. Again, we rely on numerical iteration of the Eq. (I) from various
starting p to show this. (Handout)
b. These iterations show that the system will evolve to either
fixation or elimination of the allele, depending
on whether the initial gene frequency is greater or lower than the unstable
equilibrium frequency.
VIII. Conclusions
A. Evolution is a predictive science: Eq. (I) predicts the long-term
course of evolutionary change at a single locus
with two alleles.
B. If we know precise values of fitnesses for the different genotypes
and are willing to assume that these values do
not change over time, then Eq. (I) predicts the precise trajectory of
evolutionary change.
C. In most cases, because fitness is difficult to measure precisely,
the best we will be able to do is estimate the
relative rankings of the fitnesses of the different genotypes.
D. However, even though we may not be able to predict the exact
trajectory of evolutionary change in these
cases, the above results indicate that knowing just the relative rankings
of fitness does allow us to predict
the evolutionary equilibrium that will eventually be reached. Such
a prediction of course necessitates
assuming that the relative rankings
do not change, but in many cases the biology of the situation will make
this assumption reasonable.
IX. Testing the Predictive Equation
A. We have seen that numerous experiments have demonstrated that genotypes
in natural populations frequently differ
in fitness, i.e. that Natural Selection is ubiquitous in nature.
B. We have also developed a theoretical framework for understanding
how natural selection causes changes in gene
frequencies (i.e. causes evolution) for a single locus with two alleles.
C. Several questions arise from these two observations:
1. Is the equation for gene frequency change we have derived quantitatively
correct, i.e. given that we can
measure fitnesses accurately, does it actually predict the correct evolutionary
trajectory a population will follow?
2. Are the qualitative predictionsof the equation acurate, i.e., given
that we can rank fitness of different genotypes,
even if we can't measure them with great accuracy, do the rankings predict
the evolutionary outcome in
natural populations?
3. Given that we have confidence, from answers to questions 1 and
2, that we understand how natural selection
causes evolutionary change, what other types of evidence can be used to
infer that populations have undergone
divergent evolution due to natural selection?
D. In this lecture, we address each of these questions.
X. Laboratory Experiments: Demonstrating Quantitative agreement
with equation for gene
frequency change.
A. Laboratory experiments as microcosms
1. Ideally, to assess whether the predictive equation for gene frequency
change accurately and quantitatively
describes evolution by natural selection, we would like to measure fitnesses
in a genetically variable natural
population, follow gene frequency change over time, and determine whether
the actual trajectory of gene
frequency change matched that predicted by the equation.
2. Several things make this extremely difficult:
a. Fitness is very difficult to measure accurately in the field.
b. Fitnesses do not remain constant, as assumed by the predictive
equation.
--Environmental fluctuations cause generation to generation changes in the
value of a genotype's fitness,
even if such fluctuations are not drastic enough to change the fitness ranking
of different genotypes.
--Therefore, it is generally not possible to measure fitnesses once in the
field and use those values to predict
gene frequency change over long periods of time.
3. One way around these difficulties is to assess the quantitative
accuracy of the predictive equation in laboratory
populations, in which the environment can be held constant and in which
accurate measures of fitness can be obtained.
4. This approach is legitimate because we the predictive equation
is completely general: it should apply in all
environments, including artificial laboratory environments.
B. Example: Drosophila melanogaster and the lt locus (B. Wallace. 1963. American Naturalist 97: 65-66).
1. Characteristics of the lt locus
a. lt locus carried on CyL (CurlyLobed) chromosome.
This chromosome carries the Curly (Cy) and Lobed (L)
genes that give visible phenotypic effects.
b. Two alleles: lt+ and lt
c. lt+/ lt+ individuals: viable, bright
red eyes
d. lt+/ lt individuals:
viable, orange-yellow eyes
e. lt / lt individuals:
inviable
f. These characteristics indicate that the lt allele is a RECESSIVE
LETHAL, i.e. when homozygous it is lethal,
but it has no apparent effects on viability when it is heterozygous (i.e.
its lethal effects are recessive; note,
however, its effects on eye color are not recessive).
2. Genotype of an individual may be ascertained by mating to
a CyL/Pm individual, where Pm is an alternate
version of the CyL chromosome that lacks the Cy and L
markers, but contains the Pm marker:
a. Cross CyL lt /Pm with individual to be scored and discard
offspring that show the Pm phenotype because
these will not be informative about eye color. Examine eyecolor of
remaining flies.
b. If individual to be scored is CyL lt+ /CyL lt+,
then all surviving offspring will be lt+/ lt+
and hence will have
bright red eyes.
c. By contrast, if individual to be scored is CyL lt+
/CyL lt, then half of surviving offspring will be lt+/
lt+
and have bright red eyes, while half of the offspring will be lt+/
lt and will have orange-yellow eyes.
d. Thus, if get half red, half orange-yellow eyed flies, the tested
individual was lt+/ lt, while if get all red eyes, the
tested individual was lt+/ lt+ .
3. Experimental procedure
a. Start a laboratory population consisting entirely of heterozygous
lt+/ lt individuals.
b. Each generation, remove a sample of flies and test cross each to
CyL lt /Pm and determine the genotype
of each fly of the sample by scoring eye color of its offspring.
c. From these test crosses, determine the gene frequency of lt
in the population.
d. Transfer the remaining flies to a new population cage to found
the next generation.
e. Repeat steps b-d over a number of generations.
4. Theoretical expectations from a recessive lethal
a. Wlt+/lt+ = Wlt+/lt = 1,
Wlt/lt = 0 .
b. Plugging these values into our predictive equation yields
p' = [p(p x Wlt/lt + q x Wlt+/lt )/(p2 x Wlt/lt + 2pq x Wlt+/lt + q2 x Wlt+/lt+ )]
= [p(p x 0 + q x 1)/(p2 x 0 + 2pq x 1 + q2 x 1)]
= p/(1-p)
5. Comparing theoretical expectations with observed gene frequencies.
a. Match is good (see Graph)
b. Conclude: Predictive equation accurately and quantitatively
describes the trajectory of gene frequency
change in this case.
C. Example: Drosophila pseudoobscura and inversion polymorphisms
(T. Dobzhansky. 1954. Proc. 9th Int. Cong.
Genetics, in Caryologia, 435-449).
1. Natural history
a. D. pseudoobscura, a native of western North America, exhibits
polymorphisms for chromosomal inversions.
b. There are two "alleles", ST and CH
`
c. The frequencies of these alleles undergo regular seasonal oscillations
(see Graph), which can only be
accounted for by fluctuating natural selection.
d. This result implies that the three genotypes ST/ST, ST/CH and CH/CH
differ in fitness, which should be
detectable and measurable under laboratory conditions.
e. Given these differences in fitness, it should be possible to use
this system to test the quantitative accuracy
of the predictive equation for gene frequency change.
2. Measuring Fitness (Handout).
a. Dobzhansky estimate the relative fitnesses of ST/ST, ST/CH and
CH/CH to be 0.90, 1, and 0.41 respectively.
b. These fitnesses imply that a polymorphism should be actively maintained
in the laboratory environment.
3. Experiment
a. Established four replicate population cages with initial ST "allele"
frequency of 0.2.
b. Allowed population to reproduce naturally for approximately 15
generations.
c. Approximately every two generations, removed a random sample of
flies and determined their genotype,
from which the "allele" frequency could be calculated.
4. Results (see Graph)
a. Excellent match between trajectory from predictive equation and
observed "allele" frequencies.
b. Evolution led to establishment of a stable polymorphism, as expected
with heterozygote superiority.