Download Forensics and Probability

Forensics and Probability
Origin of Variation?
Charles Darwin from, On the Origin
of Species by Means of Natural
Selection, 1859
"...no-one can say why the same
peculiarity in different individuals....is
sometimes inherited and sometimes not
so: why the child often reverts in certain
characters to its grandfather, or other
much more remote ancestor; why a
peculiarity is often transmitted from one
sex to both sexes, or to one sex alone,
more commonly but not exclusively to the
like sex."
• Genetics is the scientific study of heredity
and variation
• Heredity is the passing on of
characteristics called “traits” from parent to
child
• Variation shows that children differ in
appearance from their parents and their
brothers and sisters
What explains the passing of
traits from parents to offspring?
• The “blending” hypothesis is the idea that
genetic material from the two parents blends
together (like blue and yellow paint blend to
make green)
• The “particulate” hypothesis is the idea that
parents pass on discrete heritable units
(genes)
• Mendel documented a particulate
mechanism through his experiments with
garden peas
http://www.mendelmuseum.org/eng/1online/experiment.htm
Mendel is as important as
Darwin in 19th century science
• Mendel did experiments and analyzed the
results mathematically. His research
required him to identify variables, isolate
their effects, measure these variables
painstakingly and then subject the data to
mathematical analysis.
• He was influenced by his study of physics
and having an interest in meteorology. His
mathematical and statistical approach was
also favored by plant breeders at the time.
Mendel used an Experimental,
Quantitative Approach
• Advantages of pea plants for genetic study:
– There are many varieties with distinct heritable
features, or characters (such as color);
character variations are called traits
– Mating of plants can be controlled
– Each pea plant has sperm-producing organs
(stamens) and egg-producing organs (carpels)
– Cross-pollination (fertilization between different
plants) can be achieved by dusting one plant
with pollen from another
Mendel Planned Experiments
Carefully
• Mendel chose to track only those characters
that varied in an “either-or” manner
• He also used varieties that were “truebreeding” (plants that produce offspring of
the same variety when they self-pollinate)
• He spent 2 years getting “true” breeding
plants to study
• At least three of his traits were available in
seed catalogs of the day
Removed stamens
from purple flower
Transferred spermbearing pollen from
stamens of white
flower to eggbearing carpel of
purple flower
Parental
generation
(P)
Carpel
Stamens
Pollinated carpel
matured into pod
Planted seeds
from pod
First
generation
offspring
(F1)
Examined
offspring:
all purple
flowers
P Generation
(true-breeding
parents)
Purple
flowers
White
flowers
F1 Generation
(hybrids)
F2 Generation
All plants had
purple flowers
Some Terminology
• In a typical experiment, Mendel mated two
contrasting, true-breeding varieties, a
process called hybridization
• The true-breeding parents are the P
generation
• The hybrid offspring of the P generation are
called the F1 generation
• When F1 individuals self-pollinate, the F2
generation is produced
Mendel’s First Law: The Law of
Segregation
• When Mendel crossed contrasting, truebreeding white and purple flowered pea
plants, all of the F1 hybrids were purple
• When Mendel crossed the F1 hybrids, many
of the F2 plants had purple flowers, but
some had white
• Mendel discovered a ratio of about three to
one, purple to white flowers, in the F2
generation
• Mendel reasoned that only the purple flower
factor was affecting flower color in the F1 hybrids
• Mendel called the purple flower color a dominant
trait and white flower color a recessive trait
• Mendel observed the same pattern of
inheritance in six other pea plant characters,
each represented by two traits
• What Mendel called a “heritable factor” is what
we now call a gene
Mendel’s Model
• Mendel developed a hypothesis to explain
the 3:1 inheritance pattern he observed in
F2 offspring
• Four related concepts make up this model
• These concepts can be related to what we
now know about genes and chromosomes
The First Concept
• Alternative versions of genes account for
variations in inherited characters
• For example, the gene for flower color in
pea plants exists in two versions, one for
purple flowers and the other for white
flowers
• These alternative versions of a gene are
now called alleles
• Each gene resides at a specific locus on a
specific chromosome
Allele for purple color
Locus for flower color gene
Allele for white color
Homologous
pair of
chromosomes
The Second Concept
• For each character, an organism inherits
two alleles, one from each parent
• Mendel made this deduction without
knowing about the role of chromosomes
• The two alleles at a locus on a chromosome
may be identical, as in the true-breeding
plants of Mendel’s P generation
• Alternatively, the two alleles at a locus may
differ, as in the F1 hybrids
The Third Concept
• If the two alleles at a locus differ, then one
(the dominant allele) determines the
organism’s appearance, and the other (the
recessive allele) has no noticeable effect
on appearance
• In the flower-color example, the F1 plants
had purple flowers because the allele for
that trait is dominant
The Fourth Concept
• Known as “the law of segregation”
• Two alleles for a heritable character separate
(segregate) during gamete formation and end up
in different gametes
• Thus, an egg or a sperm gets only one of the
two alleles that are present in the somatic cells
of an organism
• This segregation of alleles corresponds to the
distribution of homologous chromosomes to
different gametes in meiosis
Mendel’s Laws Explain the Data
• Mendel’s segregation model accounts for the 3:1
ratio he observed in the F2 generation of his
numerous crosses
• The possible combinations of sperm and egg
can be shown using a Punnett square, a
diagram for predicting the results of a genetic
cross between individuals of known genetic
makeup
• A capital letter represents a dominant allele, and
a lowercase letter represents a recessive allele
P Generation
Appearance:
Genetic makeup:
Purple
flowers
PP
White
flowers
pp
P
p
Gametes
F1 Generation
Appearance:
Genetic makeup:
Purple flowers
Pp
1
Gametes:
2
1
P
p
2
F1 sperm
P
p
PP
Pp
Pp
pp
F2 Generation
P
F1 eggs
p
3
:1
Some Vocabulary Terms
• Gene: sequence of DNA coding for genetic
information
• Allele: a variant of a single gene, inherited at a
particular location on a chromosome. The
variants can be written as A and a.
• Genotype: The genetic constitution of an
individual. The genotype consists of one
complete set of genes from mother and a
second complete set of genes from father.
• Phenotype: An observable train in an individual.
It is determined by interaction of genotype and
environment.
• Homozygote: individual having two copies of
the same allele at a genetic location (AA or aa)
Some Vocabulary Terms
• Heterozygote: individual having two different
alleles at a genetic location (Aa)
• Dominant: An allele A is dominant when its
phenotype of the heterozygote Aa is the same as
that of the homozygote AA but differs from the
homozygote aa
• Recessive: An allele a is recessive if the
phenotype of the homozygote is different from
that of the heterozygote Aa and homozygote AA,
which are the same.
• Codominant: An allele is codominant if both A
and a contribute to the phenotype of the
heterozygote Aa equally.
LE 14-6
3
Phenotype
Genotype
Purple
PP
(homozygous
Purple
Pp
(heterozygous
1
2
1
Purple
Pp
(heterozygous
White
pp
(homozygous
Ratio 3:1
Ratio 1:2:1
1
The Testcross
• How can we tell the genotype of an
individual with the dominant phenotype?
• This individual must have one dominant
allele, but could be either homozygous
dominant or heterozygous
• The answer is to carry out a testcross:
breeding the mystery individual with a
homozygous recessive individual
• If any offspring display the recessive
phenotype, the mystery parent must be
heterozygous
LE 14-7
Dominant phenotype,
unknown genotype:
PP or Pp?
Recessive phenotype,
known genotype:
pp
If Pp,
then 2 offspring purple
and 1 2 offspring white:
If PP,
then all offspring
purple:
p
1
p
P
p
p
Pp
Pp
pp
pp
P
Pp
Pp
P
P
Pp
Pp
Mendel’s Second Law: The Law
of Independent Assortment
• Mendel derived the law of segregation by
following a single character
• The F1 offspring produced in this cross were
all heterozygous for that one character
• A cross between such heterozygotes is
called a monohybrid cross
• Mendel identified his second law of
inheritance by following two characters at
the same time
• Crossing two, true-breeding parents differing
in two characters produces dihybrids in the
F1 generation, heterozygous for both
characters
• A dihybrid cross, a cross between F1
dihybrids, can determine whether two
characters are transmitted to offspring as a
package or independently
LE 14-8
P Generation
YYRR
yyrr
Gametes YR
yr
YyRr
F1 Generation
Hypothesis of
dependent
assortment
Hypothesis of
independent
assortment
Sperm
1
Sperm
1
2
YR
1
2
yr
1
1
2
2
1
4
Yr
1
4
yR
1
4
yr
YR
4
YYRR
YYRr
YyRR
YyRr
YYRr
YYrr
YyRr
Yyrr
YyRR
YyRr
yyRR
yyRr
YyRr
Yyrr
yyRr
yyrr
YR
YYRR
1
YR
Eggs
Eggs
F2 Generation
(predicted
offspring)
4
YyRr
1
Yr
4
yr
YyRr
3
4
yyrr
1
1
yR
4
4
1
Phenotypic ratio 3:1
yr
4
9
16
3
16
3
16
3
16
Phenotypic ratio 9:3:3:1
• The law of independent assortment states
that each pair of alleles segregates
independently of other pairs of alleles
during gamete formation
• Strictly speaking, this law applies only to
genes on different, nonhomologous
chromosomes
• Genes located near each other on the
same chromosome tend to be inherited
together
Probability
• Ranges from 0 to 1
• Probabilities of all possible events must add up
to 1
• Rule o multiplication: The probability that
independent events will occur simultaneously is
the product of their individual probabilities.
• Rule of addition: The probability of an event that
can occur in two or more independent ways is
the sum of the different ways.
Multiplication and Addition Rules
Applied to Monohybrid Crosses
• The multiplication rule states that the
probability that two or more independent
events will occur together is the product of
their individual probabilities
• Probability in an F1 monohybrid cross can
be determined using the multiplication rule
• Segregation in a heterozygous plant is like
flipping a coin: Each gamete has a 1/2
chance of carrying the dominant allele and a
1/2 chance of carrying the recessive allele
½ chance of P and ½
chance of p allele
results in ¼ chance of
each homozygous
genotype.
There are two ways to
get the heterozygous
genotype so it is
¼+¼=½
Three genotypes give
the same phenotype.
Solving Complex Genetics Problems
with the Rules of Probability
• We can apply the rules of multiplication and
addition to predict the outcome of crosses
involving multiple characters
• A dihybrid or other multicharacter cross is
equivalent to two or more independent
monohybrid crosses occurring
simultaneously
• In calculating the chances for various
genotypes, each character is considered
separately, and then the individual
probabilities are multiplied together
YYRR
yyrr
Female Gametes
YyRr
¼
YR
YyRr
¼
Yr
¼
yR
¼
yr
¼ YR
YyRr
Male
gametes
3/16
1/16
YYRr
YyRR
YyRr
YYrR
YYrr
YyRr
Yyrr
YyRR
YyRr
yyRR
yyRr
YyRr
Yyrr
¼ Yr
¼ yR
9/16
3/16
YYRR
¼ yr
yyRr
yyrr
For a dihybrid cross –
the chance that 2 independent events occur together
is the product of their chances of occurring
separately.
•
•
•
•
The chance of yellow (YY or Yy) seeds= 3/4 (the dominant trait)
The chance of round (RR or Rr) seeds = 3/4 (the dominant trait)
The chance of green (yy) seeds= 1/4 (the recessive trait)
The chance of wrinkled (rr) seeds= 1/4 (the recessive trait)
Therefore:
The chance of yellow and round= 3/4 x 3/4 = 9/16
The chance of yellow and wrinkled= 3/4 x 1/4 = 3/16
The chance of green and round= 1/4 x 3/4 = 3/16
The chance of green and wrinkled= 1/4 x 1/4 = 1/16
Inheritance patterns are often more
complex than predicted by Mendel
• The relationship between genotype and
phenotype is rarely as simple as in the pea
plant characters Mendel studied
• Many heritable characters are not
determined by only one gene with two
alleles
• However, the basic principles of segregation
and independent assortment apply even to
more complex patterns of inheritance
Extending Mendelian Genetics for
a Single Gene
• Inheritance of characters by a single gene
may deviate from simple Mendelian patterns
in the following situations:
– When alleles are not completely dominant or
recessive
– When a gene produces multiple phenotypes
– When a gene has more than two alleles
– The forensic characteristics usually have more
than two alleles
The Spectrum of Dominance
• Complete dominance occurs when
phenotypes of the heterozygote and
dominant homozygote are identical
• In incomplete dominance, the phenotype of
F1 hybrids is somewhere between the
phenotypes of the two parental varieties
• In codominance, two dominant alleles affect
the phenotype in separate, distinguishable
ways
• Forensic Traits are codominant
P Generation
Red
CRCR
White
CWCW
CR
Gametes
CW
Pink
CRCW
F1 Generation
Gametes
1
1
F2 Generation
2
CR
2
CR
1
2
1
CW
Sperm
2
CW
Eggs
1
1
2
2
CR
CRCR
CRCW
CRCW
CWCW
CW
The Relation Between Dominance
and Phenotype
• A dominant allele does not subdue a
recessive allele; alleles don’t interact
• Alleles are simply variations in a gene’s
nucleotide sequence
• For any gene, dominance/recessiveness
relationships of alleles depend on the level
at which we examine the phenotype
• If you look directly at DNA, you can always
detect codominance.
Frequency of Dominant Alleles
• Dominant alleles are not always more
common in populations than recessive
alleles
• For example, one baby out of 400 in the
USA is born with extra fingers or toes
• The allele for this trait is dominant to the
allele for the more common trait of five digits
per appendage
• In this example, the recessive allele is far
more prevalent than the dominant allele in
the population
Multiple Alleles
• Most genes exist in populations in more
than two allelic forms
• For example, the four phenotypes of the
ABO blood group in humans are determined
by three alleles for the enzyme (I) that
attaches A or B carbohydrates to red blood
cells: IA, IB, and i.
Polygenic Inheritance
• Quantitative characters are those that vary
in the population along a continuum
• Quantitative variation usually indicates
polygenic inheritance, an additive effect of
two or more genes on a single phenotype
• Skin color in humans is an example of
polygenic inheritance
LE 14-12
AaBbCc
aabbcc
20/64
Fraction of progeny
15/64
6/64
1/64
Aabbcc
AaBbCc
AaBbcc AaBbCc AABbCc AABBCc AABBCC
Nature and Nurture:
The Environmental Impact on Phenotype
Relating Mendel’s Laws to Cells
• Law of Segregation
• Pairs of
characteristics
(alleles) separate
during gamete
formation
• Each cell has two
sets of chromosomes
that are divided to
one set per gamete.
• Law of Independent
Assortment
• The inheritance of an
allele of one gene
does not influence the
allele inherited at a
second gene.
• Genes on different
chromosomes
segregate their alleles
independently.
Offspring acquire genes from
parents by inheriting chromosomes
• In a literal sense, children do not inherit
particular physical traits from their parents
• It is genes that are actually inherited
• Genes are carried on chromosomes.
• Mendel identified 7 sets of charactersOne per each of the 7 chromosomes in
peas, so his law worked out perfectly.
• Two characters on the same chromosome
are linked together and would have
messed up his law.
Inheritance of Genes
• Genes are the units of heredity
• Genes are segments of DNA
• Each gene has a specific locus on a
certain chromosome
• One set of chromosomes is inherited from
each parent
• Reproductive cells called gametes (sperm
and eggs) unite, passing genes to the next
generation
Sexual Reproduction
• Two parents give rise to offspring that have
unique combinations of genes inherited from the
two parents.
• All humans arise from the joining of 1 egg and 1
sperm cell
• 100% of a person’s DNA is the same within and
throughout a human being’s body.
• Whether you look at the cells of a person’s
blood, skin, semen, saliva or hair, the DNA and
genes will be the same.
Chromosomes Come in Sets
• Each human cell (except gametes) has 46
chromosomes arranged in pairs in its nucleus
• The two chromosomes in each pair are called
homologous chromosomes
• One of each pair came from your mother and the
other came from your father.
• Both chromosomes in a pair carry genes
controlling the same inherited characteristics
• The sex chromosomes are called X and Y
• Human females have a homologous pair
of X chromosomes (XX)
• Human males have one X and one Y
chromosome
• The 22 pairs of chromosomes that do not
determine sex are called autosomes
• Each pair of homologous chromosomes includes
one chromosome from each parent
• The 46 chromosomes in a human somatic cell
are two sets of 23: one from the mother and one
from the father
• The number of chromosomes in a single set is
represented by n
• A cell with two sets is called diploid (2n)
• For humans, the diploid number is 46 (2n = 46)
Meiosis reduces the number of
chromosome sets from diploid to
haploid
The behavior of chromosomes during meiosis
and fertilization is responsible for most of the
variation that arises in each generation
Meiosis is preceded by the
replication of chromosomes
Meiosis takes place in two
sets of cell divisions, called
meiosis I and meiosis II
The two cell divisions result
in four daughter cells
Each daughter cell has only
half as many chromosomes
as the parent cell
Key
Maternal set of
chromosomes
Possibility 2
Possibility 1
Paternal set of
chromosomes
Two equally probable
arrangements of
chromosomes at
metaphase I
Metaphase II
Daughter
cells
Combination 1
Combination 2
Combination 3
Combination 4
8 Gamete Combinations
vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
Maternal set of
chromosomes (n = 3)
2n = 6
Paternal set of
chromosomes (n = 3)
Two sister chromatids
of one replicated
chromosomes
Centromere
Two nonsister
chromatids in
a homologous pair
Pair of homologous
chromosomes
(one from each set)
LE 13-5
Key
Haploid gametes (n = 23)
Haploid (n)
Ovum (n)
Diploid (2n)
Sperm
cell (n)
MEIOSIS
Ovary
FERTILIZATION
Testis
Diploid
zygote
(2n = 46)
Mitosis and
development
Multicellular diploid
adults (2n = 46)
• Homologous pairs of chromosomes orient
randomly at metaphase I of meiosis
• In independent assortment, each pair of
chromosomes sorts maternal and paternal
homologues into daughter cells independently of
the other pairs
• The number of combinations possible when
chromosomes assort independently into
gametes is 2n, where n is the haploid number
• For humans (n = 23), there are more than 8
million (223) possible combinations of
chromosomes
Random Fertilization
• Random fertilization adds to genetic
variation because any sperm can fuse with
any ovum (unfertilized egg)
• The fusion of gametes produces a zygote
with any of about 64 trillion diploid
combinations
• Crossing over adds even more variation
• Each zygote has a unique genetic identity
LE 13-11
Nonsister
chromatids
Prophase I
of meiosis
Tetrad
Chiasma,
site of
crossing
over
Metaphase I
Metaphase II
Daughter
cells
Recombinant
chromosomes
Human Genome
23 Pairs of Chromosomes + mtDNA
Located in cell nucleus
http://www.ncbi.nlm.nih.gov/genome/guide/
Autosomes
2 copies
per cell
Located in
mitochondria
(multiple copies
in cell cytoplasm)
mtDNA
1
2
3
4
5
6
7
8
9
10 11 12
13 14 15 16 17 18 19 20 21 22 X
Nuclear DNA
3.2 billion bp
Y
Sexchromosomes
16,569 bp
Mitochondrial
DNA
100s of copies
per cell
Butler, J.M. (2005) Forensic DNA Typing, 2nd Edition, Figure 2.3, ©Elsevier Science/Academic Press
Gene Pools and Allele
Frequencies
• A population is a localized group of
individuals capable of interbreeding and
producing fertile offspring
• The gene pool is the total aggregate of
genes in a population at any one time
• The alleles at any particular locus can be
• The gene pool consists of all gene loci in all
individuals of the population
Genetic Variation in Populations
• Many genes are monomorphic
– They have only one common allele, i.e. with a
frequency >0.01 (or 1%).
• Other genes are polymorphic
– They have two or more alleles with
frequencies >0.01. Examples of polymorphic
loci include the ABO and Rh blood groups.
Mice in the Gene Pool:
Calculating Genotype Frequency
vs. Allele Frequency
4 + 4 + 2 = 10
8 +4 =12 B
4 + 4 =8 b
12 + 8 =20
12/20=0.6 B
8/20=0.4 b
4/10=0.4 BB
4/10=0.4 Bb
4 BB
4Bb
2/10=0.2 bb
2bb
Calculating the HW Law
• Chance combinations of alleles of a gene in a
population can be expressed by the binomial
expansion.
• For a two-allele locus, let p = frequency of allele
G1 and q = frequency of allele G2.
• Since there are no other alleles, p + q = 1.0.
• The distribution of genotypes would be
(p + q)2 = p2 + 2pq + q2 = 1,
• where p2 and q2 are the frequencies of the two
homozygotes and 2pq is the frequency of the
heterozygote.
The Hardy-Weinberg Theorem
• The Hardy-Weinberg theorem describes a
population that is not evolving
• It states that frequencies of alleles and
genotypes in a population’s gene pool
remain constant from generation to
generation, provided that only Mendelian
segregation and recombination of alleles are
at work
• Mendelian inheritance preserves genetic
variation in a population
Population Genetics and Human
Health
• We can use the Hardy-Weinberg equation to
estimate the percentage of the human population
carrying the allele for an inherited disease
• Take the square root of the recessive to solve for
its allele frequency or q.
• Subtract that frequency from 1 to get the other
frequency, p.
• The frequency of being a carrier is 2pq.
• The frequency of being a homozygote is p2
LE 23-5
Gametes for each generation are
drawn at random from the gene pool
of the previous generation:
80% CR (p = 0.8)
20% CW (q = 0.2)
Sperm
CR
CW
(20%)
p2
pq
64%
CRCR
16%
CRCW
(20%)
CR
(80%)
CW
Eggs
(80%)
qp
4%
CWCW
16%
CRCW
q2
Conditions for Hardy-Weinberg
Equilibrium
• The Hardy-Weinberg theorem describes a
hypothetical population
• In real populations, allele and genotype
frequencies do change over time
• The five conditions for non-evolving
populations are rarely met in nature:
– Extremely large population size
– No gene flow
– No mutations
– Random mating
– No natural selection
Modeling Genetics with Candy
• Each type of candy is like a gene
• Each Flavor is like an Allele
• We can only have two pieces of each type
of candy- one from Mom, One From Dad
• Doesn’t matter how many flavors are
possible, you just get two, although they
can be the same flavor (homozygous) or
different flavors (heterozygous)
Modeling Genetics with Candy
• Some candy has only one flavor- likewise most
genes DO NOT vary. They have essential
functions.
• Probability of each flavor is 25% (for both Kisses
and Starburst)
• So probability follows HW Equilibrium
predictions:
– The frequency of being a carrier is 2pq.
– The frequency of being a homozygote is p2
• Each type of candy was independently inherited
so probabilities are multiplied times one another
to get joint probability.