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.mendel-museum.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 chose to track only those characters that varied in an “either-or” manner

• He also used varieties that were “true-breeding” (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

Mendel Planned Experiments Carefully

Removed stamensfrom purple flower

Transferred sperm-bearing pollen fromstamens of whiteflower to egg-bearing carpel ofpurple flower

Carpel Stamens

Parentalgeneration(P)

Pollinated carpelmatured into pod

Planted seedsfrom pod

Examinedoffspring:all purpleflowers

Firstgenerationoffspring(F1)

P Generation

(true-breedingparents)

F1 Generation

(hybrids)

F2 Generation

Purpleflowers

Whiteflowers

All plants hadpurple flowers

• 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

Some Terminology

Mendel’s First Law: The Law of Segregation

• When Mendel crossed contrasting, true-breeding 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

• 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

The First Concept

Allele for purple color

Homologouspair ofchromosomes

Allele for white color

Locus for flower color gene

• 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 Second Concept

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

Appearance:

P Generation

Genetic makeup:

Gametes

F1 Generation

Appearance:Genetic makeup:

Gametes:

F2 Generation

Purple flowersPp

P p1 21 2

P p

F1 sperm

F1 eggsPP Pp

Pp pp

P

p

3 : 1

Purpleflowers

PP

Whiteflowers

pp

P p

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 14-6Phenotype

Purple

Purple3

Purple

Genotype

PP(homozygous

Pp(heterozygous

Pp(heterozygous

pp(homozygous

1

2

1

Ratio 1:2:1

White

Ratio 3:1

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?

If PP,then all offspring

purple:

p p

P

P

Pp Pp

Pp Pp

If Pp,then 1

2 offspring purpleand 1

2 offspring white:

p p

P

Ppp pp

Pp Pp

Recessive phenotype,known genotype:

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

F1 Generation

YYRR

Gametes YR yr

yyrr

YyRr

Hypothesis ofdependentassortment

Hypothesis of independent assortment

SpermEggs

YR

Yr

yrYR

YR

yr

Eggs

YYRR YyRr

YyRr yyrr yR

yrPhenotypic ratio 3:1

F2 Generation(predictedoffspring)

YYRR YYRr YyRR YyRr

YYRr YYrr YyRr Yyrr

YyRR YyRr yyRR yyRr

YyRr Yyrr yyRr yyrr

Phenotypic ratio 9:3:3:1

YR Yr yR yr

Sperm

12

14

14

14

14

1 43

4

12

12

12

14

916

316

316

316

14

14

14

• 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

YyRr

Male gametes

Female Gametes

YyRr

YyRr

¼

¼

¼

¼

¼ ¼ ¼ ¼ YR Yr yR yr

YR

Yr

yR

yr

YYRR YYRr

YYrR YYrr

YyRR YyRr

YyRr Yyrr

YyRR YyRr

YyRr Yyrr yyRr

yyRr

yyrr

yyRR9/16

3/16

3/16

1/16

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 = 9/16The chance of yellow and wrinkled= 3/4 x 1/4 = 3/16The chance of green and round= 1/4 x 3/4 = 3/16The 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

RedCRCR

Gametes

P Generation

CR CW

WhiteCWCW

PinkCRCW

CRGametes CW

F1 Generation

F2 Generation Eggs

CR CW

CR

CRCR CRCW

CRCW CWCW

CW

Sperm

12

12

12

12

12

12

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 AaBbcc AaBbCc AABbCc AABBCc AABBCC

AaBbCcAaBbCc

20/64

15/64

6/64

1/64

Fra

ctio

n o

f p

rog

eny

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 characters-

One 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 ofchromosomes

Paternal set ofchromosomes

Possibility 1 Possibility 2

Combination 2Combination 1 Combination 3 Combination 4

Daughtercells

Metaphase II

Two equally probablearrangements ofchromosomes at

metaphase I

vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

Maternal set ofchromosomes (n = 3)

2n = 6

Paternal set ofchromosomes (n = 3)

Two sister chromatidsof one replicatedchromosomes

Two nonsister chromatids in a homologous pair

Pair of homologouschromosomes(one from each set)

Centromere

8 Gamete Combinations

LE 13-5

Key

Haploid (n)

Diploid (2n)

Haploid gametes (n = 23)

Ovum (n)

Spermcell (n)

TestisOvary

Mitosis anddevelopment

Multicellular diploidadults (2n = 46)

FERTILIZATIONMEIOSIS

Diploidzygote(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 13-11Prophase Iof meiosis

Tetrad

Nonsisterchromatids

Chiasma,site of crossingover

Recombinantchromosomes

Metaphase I

Metaphase II

Daughtercells

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p:/

/ww

w.n

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/gu

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1 2 3 4 5 6 7 8 9 10 11 12

13 14 15 16 17 18 19 20 21 22 X Y

Human Genome 23 Pairs of Chromosomes + mtDNA

Sex-chromosomes

mtDNA

16,569 bp

Autosomes

Mitochondrial DNA

Nuclear DNA

3.2 billion bp

Located in cell nucleus

Located in mitochondria

(multiple copies in cell cytoplasm)

2 copies per cell

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

8 +4 =12 B

4 + 4 =8 b

12 + 8 =20

12/20=0.6 B

8/20=0.4 b

4 BB 4Bb 2bb

4 + 4 + 2 = 10

4/10=0.4 BB 4/10=0.4 Bb 2/10=0.2 bb

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 aredrawn at random from the gene pool

of the previous generation:

80% CR (p = 0.8) 20% CW (q = 0.2)

SpermCR

(80%)CW

(20%)

pqp2

16%CRCW

64%CRCR

Eg

gs

CW

(20%

)C

R

(80%

)

16%CRCW

qp4%

CWCW

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.