Download GENERAL GENETICS

GENERAL GENETICS
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The practical outcome of
≅ 1.5 century of genetics
and ≅ 50 years of molecular biology
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Application of genetics -1
(lesson from the field)
A new
species
cobs
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Application of genetics -2
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Application of genetics -3
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Application of genetics -4
rDNA stands for: derived from recombinant DNA
technology
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ALL STARTED WITH …
MENDELISM
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MENDELIAN METHOD
or
MENDELIAN GENETICS
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Johann Gregor Mendel (1822-1884)
Ð 1842 attends the Olmutz Institute of Phylosophy.
Among the other topics (phylosophy, ethics) he selected
also Physics.
This will provide a solid background for his future work.
Scientific Method
Hypothesis
Experimental work
Results
Analysis of the data (by means of mathematics)
Ð 1843 enters the monastery of Saint Thomas – Brünn - Royal Society for the
improvement of agriculture
Ð 1851 enters the University of Wien, where he attended, in addition to 3 courses of
Physics (as an assistant of Christian Doppler), also:
Chemistry, Math, Paleontology, Botanics and Phyisiology
approach)
(multidisciplinary
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Ð 1853 back to Brünn, where he will face the scientific problem:
“What is inherited and how is inherited?”
Fundamental choice
´
model organism
Pisum sativum (fam. Papilionatae)
1. It is easy to cultivate
2. Developing time relatively short (≅ 3 months)
3. The flower is suitable for manipulation
´ sexual elements very well defined:
anther (male )
containing POLLEN
pistill (female)
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Choice of 7 characters that can be easily distinguished by visual
inspection
(Was he lucky, or smart ?!)
Seed color
Pod color
Pod shape
Seed shape
Flower color
Inflated
Stem length
Wrinckled
Flower position
At tip
tall
dwarf
Along stem
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Mendel used some terms derived from German language to
define what we commonly know today as a gene:
Merkmal (characteristic, mark)
Anlage
(plan, design or predisposition)
Elemente
Hypothesis:
Inside each cell there are well defined elements (Elemente) (or factors) that control the
characters (Merkmal).
These Elemente work together as a couple and the two Elemente are derived from both
parents. When the germ cell forms, the two factors are separated (segregation) and either
one or the other is transferred to reproductive cells.
Experimental Approach :
Production of Plant Hybrids using 7 well-defined Merkmal (true breeds)
Instruments:
• Pincers, to remove stamen and pollen,
• Brush, to sprinkle stigma with pollen.
• Cotton hood, to prevent external fertilization.
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Ð 1854 starts teaching at the Realschule of Brünn (secondary school comparable to a
Liceo Scientifico) Physics and Natural Science
Ð 1856 after two years of hard work on 34 varieties of seeds, he managed to obtain 22
true breeds and some preliminary results.
An individual that produces identical offspring when self-fertilized
or hybridized with an individual of like genotype
Crossed Pollination
(in genetic terms: homozygous)
Ð from 1856 until 1863 he carries out with truebreeding plants a set of experiments of seeding,
cross fertilization and analysis of first generation
(F1) and second generation (F2) plants with the
aim of
“… observing how different merkmal are
transmitted to the offspring, and deduce the rules
governing the appearance of these merkmal in the
next generations of individuals”
In other words: “ how hybridization can produce
new species?”
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Results
Conclusions
1) Crossing true breeds produced always the
same trait;
2) F1 progeny appeares as one of the two
parental phenotypes (e.g. smooth seeds);
F1
F2
3) The character that was not present in F1
reappeared in the second generation (F2) with
approx. a 3:1 frequency (75% smooth seeds,
25% rough seeds);
4) The trait which is expressed in F1 progeny is
defined dominant, and the one which is NOT
expressed is called recessive.
5,474 smooth seeds and 1,850 rough seeds
2,96 : 1
F1 plants have a genotype S/r, where S= dominant allele, determining the
smooth character of the seed and r = recessive allele determining the
rough character.
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Mendel used as nomenclature an uppercase letter (S big) for
dominant allele and a lowercase letter (s small) for the recessive
allele
Dominant allele
Recessive allele
Homo-zygous (same zygotic
constitution)?
Hetero-zygous (different zygotic
constitution)?
Number of S alleles in the population?
Number of s alleles in the population?
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KEY WORDS
Genotype: the genetic make-up of an individual (e.g. RR, Rr or rr)
used with reference to a particular characteristic
Phenotype: the physical expression of an organism’s genes (e.g.
green or yellow seeds)
Homozygous: possessing a pair of identical genes (e.g. RR and rr
plants are homozygous)
Heterozygous: possessing a pair of different genes (e.g. Rr)
Allele: alternative forms of a gene ( or a DNA sequence) found at
a given locus on a chromosome
Female: the circle with the cross at the bottom represents the
mirror of Venus, the goddess of love
Male: the circle with the arrow point sticking outward
represents the shield and spear of Mars, the god of war.
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RESULTS OBTAINED WITH HYBRIDIZATION EXPERIMENTS
(7 CHARACTERS)
p
g
F2 (numeri)
Carattere
F1
(3)
Semi: lisci /rugosi
Tutti lisci
(2)
Semi: giallo/verdi
Tutti gialli
Involucri del seme:grigi/bianchi
Tutti grigi
(1) Tutti porpora
Fiori: porpora/bianchi
Fiori: assiali/terminali
Tutti assiali
(7)
Baccelli: pieni/irregolari (5)
Tutti pieni
Baccelli: verdi/gialli
Tutti verdi
(4)
Stelo: lungo/corto (6)
Tutti lunghi
Totale
F2 (rapporto)
Dominanti
5.474
6.022
705
Recessivi
1.850
2.001
224
Totale
7.324
8.023
929
Domin./Recess.
2.96 : 1
3.01 : 1
3.15 : 1
651
882
428
787
14.949
207
299
152
277
5.010
858
1.181
580
1.064
19.959
3.14 : 1
2.94 : 1
2.82 : 1
2.84 : 1
2.98 : 1
Ð 1866 he published the results and interpretations of his experiments
in a scientific article titled:
“Versuche über Pflanzen-Hybriden” in the Proceedings of the Brünn
Natural History Society.
He also sent 40 reprints of this work to distinguished scientists (e.g.
Charles Darwin): he got a reply from only one, Karl Wilhelm von Nägeli,
professor of Botany at the University of Munchen.
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WHY 3:1 RATIO ?
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R. Punnet – Prof. of Genetics at
Cambridge
(co-discovered genetic linkage)
The genotypes of the
offspring are obtained by
combining the gamete
from each column with the
gamete from each row.
ga
The use of Punnet squares
to predict the outcomes of
genetic experiments when
the parental genotypes
are known.
Number of A alleles ?
Number of a alleles ?
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GENETIC BASIS OF
A PHENOTYPE
Key terms:
GENOTYPE = the genetic
makeup of an individual organism
PHENOTYPE = the observable
outward appearance of an
organism, which is controlled by
the genotype and its interaction
with the environment
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Notes concerning the experimental design
of Mendel’s experiments
Selection of “true breeds” & initial use of
monohybrids (i.e. crosses between plants
differing for a single character)
Prevention of self- and spurious pollination
by removal of anthers (no pollen) and by placing
“hoods” on flowers
The stigmata are “brushed” with pollen derived
from the desired plant
Apply interdisciplinary approach to evaluate
results
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Parental generation → crosses
→ First filial generation (F1) of monohybrids
→ all offsprings display the same (dominant) character
→ self-fertilize monohybrids
→ second filial generation (F2)
→ both phenotypes appear, always in the same ratio ≅ 3
Mendel’s interpretation based on “cell theory”
→ both characters are present in the zygote
→ segregation occurs in the gamets
Reciprocal cross experiments indicate that segregation of
alleles is the same in both male and female parents
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DI-HYBRID EXPERIMENTS
Punnett square illustrating
the genoptypes and
phenotypes of the F2
generation from Mendel’s
dihybrid experiment
I = seed color (yellow or green)
R = seed shape (smooth or rough)
Results: 9 (IR): 3 (Ir): 3 (iR) : 1 (ir)
An expansion of the 3 : 1 ratio
Each gene behaves exactly as it does in a monohybrid cross
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The product law can be used to predict the frequency with
which two independent events will occur simultaneously
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INDEPENDENT ASSORTMENT
Remember cell theory ?
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Mendel’s Principles
a) Principle of SEGREGATION
In the formation of pollen and ovules, the alleles of each pair
“segregate” from each other into different gametes
b) Principle of PARENTAL EQUIVALENCE
In the formation of both male and female gametes, segregation of
alleles is the same (exceptions: sex-linked genes, mitochondrial genes)
c) Principle of INDEPENDENT SEGREGATION
or INDEPENDENT ASSORTMENT
The inheritance of alleles at one locus does not influence the inheritance
of alleles at another locus
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PRE
GEN REQUI
S
ETI
CS C ITE FO
RG
MIT
OUR
ENE
OS I
SE:
RA L
S–
- CR
M
O
EIO
SSI
DI F
N
G O SIS I
FE R
V
ENC
ES ER – D MEIO
S
E TA
ILS IS II
A ND
Segregation explained using cell theory
Diploid
2n
Haploid
n
F1
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2n
F1
Meiosis ensures that when an individual
is heterozygous at any locus, half of its
gametes will carry one allele (A), and
half will carry the other (a).
n
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Meiosis not only preserves the genome size of sexually
reproducing eukaryotes but also provides 3
mechanisms to diversify the genomes of the offspring.
1. Crossing Over (Prophase I)
Chiasmata represent points where earlier (and unseen) nonsister chromatids had
swapped sections. The process is called crossing over. It is reciprocal; the segments
exchanged by each nonsister chromatid are identical (but may carry different alleles).
Each chromatid contains a single molecule of DNA. So the problem of crossing over
is really a problem of swapping portions of adjacent DNA molecules. It must be done
with great precision so that neither chromatid gains or loses any genes. In fact,
crossing over has to be sufficiently precise that not a single nucleotide is lost or
added at the crossover point if it occurs within a gene.
Otherwise a frameshift would result and the resulting gene would produce a
defective product or, more likely, no product at all.
In this photomicrograph (grasshopper chromosomes),
a tetrad shows 5 chiasmata.
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When the chromosomes of the pair cross over, the chiasmata do not always occur at
the same points along the lengths of the chromosomes; there are different sites at
which chiasmata may occur: this gives endless different chromosome combinations
after meiosis and NO two chromosomes will be identical in the gametes.
Diagram 3 illustrates this point: the same pair of chromosomes, one black and one
white, have formed chiasmata at different sites, producing different chromosomes to
those in Diagram 2.
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2. Random Assortment
In meiosis I, the orientation of paternal and
maternal homologues at the metaphase plate
is random. Therefore, although each cell
produced by meiosis contains only one of
each homologue, the number of possible
combinations of maternal and paternal
homologues is 2n, where n = the haploid
number of chromosomes.
In this diagram, the haploid number is 3, and
8 (23) different combinations are produced.
Ch.1
Ch.2
Ch.2
Ch.3
Ch.1
Random assortment of homologues in humans produces 223 (8,388,608)
different combinations of chromosomes.
Furthermore, because of crossing over, none of these chromosomes is "pure"
maternal or paternal. The distribution of recombinant and non-recombinant
sister chromatids into the daughter cells at anaphase II is also random.
So it is safe to conclude that of all the billions of sperm produced by a man
during his lifetime (and the ≅ 500 eggs that mature over the life of a woman),
no two have exactly the same gene content.
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3. Fertilization
By reducing the number of chromosomes from 2n to n,the
stage is set for the union of two genomes. If the parents
differ genetically, new combinations of genes can occur in
their offspring.
Taking these three mechanisms together, it is safe to
conclude that no two human beings have ever shared an
identical genome unless they had an identical sibling;
that is a sibling produced from the same fertilized egg
(monoovular twins).
The behavior of chromosomes during meiosis (2n → n)
and fertilization (n + n → 2n) provides the structural basis
for Mendel's rules of inheritance.
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Can we PREDICT if a
purple flowered plant is
heterozygous or
homozygous?
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A TESTCROSS can be
used to determine
whether an individual
is heterozygous (Aa)
or homozygous (AA)
The tester organism is
homozygous recessive
(aa white-flowered plant)
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THE MOLECULAR BASIS OF DOMINANCE vs RECESSIVENESS
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(This is related to Incomplete Dominance)
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The effect of dominance
A dominant allele
a recessive allele
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Discovery of Mendel’s
data
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- Hugo De Vries (NL)
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- W. Bateson (ENG)
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- K.H. Correns (D)
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- E. von T.-Seysenegg (A)
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- A. Garrod (ENG)
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“Inborn Errors of
Metabolism” 1908
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- Giuseppe Cuboni (1903)
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GENETICS OF CHARACTERS SHOWING
ALTERNATIVE FORMS
[Discontinuous variation]
GENETICS OF QUANTITATIVE CHARACTERS
(e.g. body size, human weight, height). The
majority of measures fall in the middle of a
bell-shaped frequency distribution with a
range of observation tailing away on either
side → continuous variation
[many gene loci and environmental factors]
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Thomas Hunt Morgan
Deviations from Mendel's
Predicted Ratios
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DISCOVERY OF SEX
LINKAGE
Genes on the Same
Chromosome, DO NOT
Assort Independently always
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Drosophila melanogaster as MODEL ORGANISM:
- Very easy to isolate and cultivate
- Larvae feed not on the vegetable matter itself but on
the yeasts and microorganisms present on the decaying
breeding substrate.
- Fast development time = 10 - 12 days
- Sexual dimorphism
- Only 4 pairs of chromosomes
- Larvae salivary glands provide polytenic chromosomes.
The term "Drosophila", means "dew-loving"
“melanogaster” means “black belly”
1. narcotize (ether)
2. collect
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3. observe with stereomicroscope
SEX LINKAGE
First experimental evidence of characters linked to sex was produced by
T.H. Morgan in 1910 who found a white eyed male Drosophila and made the
following crosses:
Parental
F1
F2
wild female x white male
all wild offspring – red eyes (mated together)
all females were wild :
1/2 wild males and 1/2 white males
White eye character is recessive, but is found ONLY in males
“the fly room”
> 13.000.000 flies
examined
female
male
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X = “straight
chromosome”
Y = “bent
chromosome”
white recessive in
only one type of cross !
3470 R
783 W
all ♂
50% RED
50% WHITE
Based on these results, Morgan postulated that the gene for white eyes
was on the X chromosome. Morgan deduced that the X-chromosome
carried a number of discrete hereditary units, or factors.
Other characters (variations) were found to be X-linked.
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The abstract ideas of gene become tangible;
genes are discrete physical units
Morgan adopted the term gene, which was introduced by the Danish
botanist Wilhelm Johannsen in 1909.
This was the first experimental proof that: i) genes are located on
chromosomes; ii) genes are possibly arranged in a linear fashion on
chromosomes.
Calvin Bridges
Alfred Sturtevant
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Using crossover data, A. H. Sturtevant and his coworkers
mapped other Drosophila genes in linear arrays at particular
genetic loci.
This figure depicts an abbreviated GENETIC MAP (or LINKAGE MAP) of a
chromosome in Drosophila
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Morgan found that some alleles do not separate (segregate);
instead they tend to “travel together”
B = wild type body (grey)
b = mutant body (black)
Vg= wild type (normal wings)
vg= mutant (vestigial wings)
Experiment:
tester
Test-cross with F1 individuals
Expected ratio: 1 : 1 : 1 : 1
2300 Drosophila examined
Expected results
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Obtained results
The tester organism is
homozygous recessive
(bb vgvg)
Expected results
Obtained results
nonparental combinations t recombinant
An exchange event between homologous chromosomes, crossingover,
results in the recombination of genes in the homologous chromosomes.
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Frequency of recombination = frequency of nonparental combinations
(recombinant phenotypes) of the traits over the total number of phenotypes
The FR values allow to determine:
1. The linear order of loci along the chromosome
2. The relative distance between them
1 + 2 = LINKAGE MAP
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B = wild type body (grey)
b = mutant body (black)
= 17 cM
Vg= wild type (normal wings)
vg= mutant (vestigial wings)
2300 Drosophila examined
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- Maps of relative positions, (the orders ) of linked genes on a chromosome can be
constructed by noting the frequencies of crossing-over between genes. The closer
two genes are together, the less likely they will show crossing-over.
- Conversely, the greater the distance between two genes on a chromosome, the
greater the chance that a cross-over will happen between them.
-The probability of crossing over between any two genes can be expressed as a
distance or value (the % of crossing-overs that occurs between 2 points on the
chromosome).
-One map unit, (m.u.) is the distance between linked genes in the space where
1% of crossing-over occurs, or is the distance between genes for which one result
of meiosis out of 100 is recombinant.
1 map unit = 1% recombinant = 1cM (centi-Morgan)
In humans 1 cM ≅ 1.000.000 bp
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The position on the map (LINKAGE MAP) where a gene is located is called the
gene locus. On Drosophila chromosome 1, for instance, the locus of the crossveinless wings (cv) is 13.7.
The locus of cut wings (ct) is 20.0, so the distance is 6.3 m.u. The relationship
could be shown like this:
cv
ct
__|___________________________|__
6.3
If the recombination frequency between cv and ct is 6.3, and ct and vermillion
eyes (v) is 13, the order on the chromosome could either be cv-ct-v, or ct-cv-v.
We can determine which of these is correct by measuring the recombination
frequency between cv and v. If cv and v are found to recombine with a frequency
of 19.3 %, then we deduce that ct is located between them.
The genetic mapping of linked genes is an important research tool in genetics
because it enables a new gene to be assigned to a chromosome and often to a
precise position relative to other genes within the same chromosome. Genetic
mapping in the pre-genomic era was the first step in the identification and
isolation of a new gene and the determination of its DNA sequence.
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[1 cM = the distance between two loci determined by 1 % frequency of recombination]
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MATHS & GENETICS
A theoretical framework to study
population genetics (1908)
p+q=1
(p + q)2 = 1
(p + q)2 = p2 + 2pq + q2
Godfrey Harold
Hardy
Wilhelm Weinberg
W.E. Castle (1903) and S. Chetverikov
1. How can “O” be the most common of the blood types if it is a
recessive trait?
2. If Huntington's disease is a dominant trait, shouldn't threefourths of the population have Huntington's while one-fourth
have the normal phenotype?
3. Shouldn't recessive traits be gradually “swamped out' so they
disappear from the population?
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F1
4/4 = 100% purple
F2
3/4 purple
1/4white
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Vocabulary
Evolution – changes in population allele frequencies over time. The population is
the smallest unit which can evolve.
Population – any group of organisms coexisting at the same time and place that
are capable of interbreeding with one another.
Allele frequency – proportion of a particular allele among all
other alleles in a population. They are represented by the letters
p and q (range 0 – 1).
Gene Pool – all of the alleles at all loci in the population.
Natural Selection – differential survival and reproduction of individuals in a
population due to trait differences.
Genetic Drift – changes in the gene pool of a small population due to chance.
Random changes due to sampling errors in propagation of alleles.
Bottleneck Effect – population undergoes a drastic reduction in size as a result of
chance events. A cause of genetic drift.
Founder Effect – a small group of individuals becomes separated from the larger
population. A cause of genetic drift.
Gene flow – movement of genes between populations. Gain or loss of alleles
from a population due to migration of fertile individuals, or from the transfer of
gametes.
Allele fixation – when a gene has only one allele. When one allele of a gene
becomes the only allele, while the alternatives are eliminated from the
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population.
The Hardy-Weinberg Theorem states that:
the allele frequencies of a gene in a population will remain constant, as
long as evolutionary forces are not acting. H-W therefore provides a
baseline (a null expectation) for a population that is not evolving.
For a population to be in H-W equilibrium, the following conditions or
assumptions must be met:
1. The population is very large; there is no genetic drift
2. Matings are random
3. There is no mutation
4. There is no migration
5. There is no selection
If one of these conditions is broken, an evolutionary force is acting to
change allele frequencies, and the population may not be in H-W
equilibrium.
Natural populations probably seldom meet all of these conditions; H-W
provides a nice model to study EVOLUTION via deviations from H-W
equilibrium.
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Frequency of Allele O
Hardy-Weinberg Equation
Basic Relations:
p+q=1
where
A = dominant allele
a = recessive allele
applies to all populations with only two alleles at one locus
p = frequency of A allele
and q = frequency of a allele
p2 + 2pq + q2 = 1 this equation (binomial square) provides the
genotype frequencies.
Where
p2 =
2pq =
q2 =
frequency of homozygous AA genotype
frequency of heterozygous Aa genotype
frequency of homozygous aa genotype
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If p equals the frequency of allele A and q is the frequency of
allele a, union of gametes would occur as follows:
p
q
p
p2
pq
q
pq
q2
In the above table the genotypic frequency for AA is p2, the genotypic
frequency for Aa is 2pq and the genotypic frequency for aa will be q2
These are the values that are predicted by the law.
The prediction is that the frequencies of the two alleles will remain the same from
generation to generation. The following is a mathematical proof of the prediction.
To determine the allelic frequency, they can be derived from the genotypic
frequencies as shown above.
p = f(AA) + ½f(Aa) (substitute from the above table)
p = p2 + ½(2pq)
p = p (p + q)
(p + q =1; therefore q =1 - p)
p = p [p + (1 - p)] (subtract and multiply) p = p
Therefore, gene frequencies do not change in one generation !
There would be less type O blood in that next generation, but not less O alleles !
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F2
Allele frequency = p (0.5); q (0.5)
Genotype frequency =
25% Homozygous dominant;
50% Heterozygous;
25% Homozygous recessive
Phenotype frequency = 75% purple
25% white
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Table 1: Punnett square for
H–W equilibrium
Females
A (p)
a (q)
A (p)
AA
(p2)
Aa
(pq)
a (q)
Aa
(pq)
aa
(q2)
Males
Probability of being homozygous AA: the probability of receiving A from the mother
parent is p and the probability of receiving A from the father parent is p.
Combined probability is p2.
Probability of being homozygous aa: the probability of receiving “a” from the mother
parent is q and the probability of receiving “a” from the father parent is q.
Combined probability is q2.
Probability of being heterozygous Aa: Combined probability is 2pq.
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If 9% of a population is born with a severe form of “disease r” (rr),
what percentage of the population will be heterozygous (Rr) ?
What percentage of the population will be homozygous (RR) ?
Start from q:
9% =0.09 = rr = q2
(f)r = q = Square root of 0.09 = 0.3
p+q=1
→
p = 1 - 0.3 = 0.7
2pq = 2 (.7 x .3) = .42 = 42% of the population are heterozygotes Rr
(carriers)
In a certain population of 1000 fruit flies, 640 have red eyes while the
remainder have sepia eyes. The sepia eye trait is recessive to red eyes.
How many individuals would you expect to be homozygous for red eye
color?
Calculations:
Start from q:
q2 for this population is 360/1000 = 0.36
q = √0.36 = 0.6
p = 1 - q = 1 - 0.6 = 0.4
The homozygous dominant frequency = p2 = (0.4)(0.4) = 0.16
Therefore, you can expect 16% of 1000, or 160 individuals, to be homozygous
dominant for red eye color.
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Gene pool frequencies are inherently stable.
That is to say, they do not change by themselves.
Despite the fact that evolution is a common occurrence in
natural populations, allele frequencies will remain unaltered
indefinitely unless evolutionary mechanisms such as
mutation and natural selection cause them to change.
Example: f(A/A)
f(A/a)
f(a/a)
I
0.3
0.0
0.7
II
0.2
0.2
0.6
III
0.1
0.4
0.5
Frequency p of allele A in the three populations is:
I
p = f(A/A) + ½ f(A/a) = 0.3 + ½ (0) = 0.3
II
p=
0.2 + ½(0.2) = 0.3
III
p=
0.1 + ½(0.4) = 0.3
Conclusion: these three populations have different genotypic
composition, but share the same allelic frequencies.
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Before Hardy and Weinberg, it was thought that dominant alleles must, over
time, inevitably swamp recessive alleles out of existence. This incorrect
theory was called "genophagy" (literally "gene eating"). According to this
wrong idea, dominant alleles always increase in frequency from generation
to generation.
Hardy and Weinberg were able to demonstrate with their equation that
dominant alleles can just as easily decrease in frequency.
… as G. H. Hardy stated in 1908, 'There is not the slightest foundation
for the idea that a dominant trait should show a tendency to spread over
a whole population, or that a recessive trait should die out.'
Gene frequencies can be high or low no matter how the allele is
expressed, and can change, depending on the conditions that exist.
It is the changes in gene frequencies over time that result in evolution.
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Amish community
Chi-square is used to test whether or not some observed distributional
outcome fits an expected pattern. Since it is unlikely that the observed genotype
frequencies will be exactly as predicted by the Hardy-Weinberg equation, it is
important to look at the nature of the differences between the observed and
expected values and to make a judgment as to the "goodness of fit" between
them. In the chi-square test, the expected value is subtracted from the observed
value in each category, and this value is then squared. Each squared value is
then weighted by dividing it by the expected value for that category. The sum of
these squared and weighted values, called chi square (denoted as χ2), is
represented by the following equation:
χ2 = Σ (observed - expected)2
expected
In the chi-square test, two hypotheses are tested. The null hypothesis (Ho)
states that there is no difference between the two observed and expected
values; they are statistically the same and any difference that may be detected
is due to chance. The alternative hypothesis (Ha) states that the two sets of
data, the observed and expected values, are different; the difference is
statistically significant and must be due to some reason other than chance.
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