Genetics

Chapter 19

Terms

• Probability:– Number of times an event happens divided by

the total number of possible events• Statistics:

– Measure of certainty of a result• Population genetics:

– Mendel’s laws– How Mendel’s laws affect the alleles present

in a set population

A Match?Three possible outcomes of comparing two

DNA profiles:1. Inconclusive - unknown

– There is not enough data to determine2. Exclusion – non-match

– The profiles are too different to possibly be the same individual

3. Inclusion – if the DNA profiles match– Probability of seeing this match at random is

calculated

Probability of a match• Out of the three possibilities:

– Inconclusive– Exclude– Include

• Only the third requires statistics• Probability of a random match is

calculated based on:– Allele frequencies in a small sampling of the

population– Population genetic principles

Method to use?

• There is often more than one method of calculating the statistics of seeing a random match

• Which statistic test used depends on:– Experience of user– Local legal system– Practicality of the approach– Available data– Question that needs to be answered

Probability

• Number of times an event happens divided by the total number of possible events

• Probability exists on a continuum between zero (0) and one (1)

• Zero probability means event is impossible• Probability of one means event occurred• Most of the time however we are uncertain

– Probability gives a measure of how certain

Probability: The number of times an event occurs divided by the number of trials during which that event could have happened

1 event / 6 possible outcomes =

1/6

Mendel Understood Probability

The probability of rolling a 2 with one roll of one die:

The Multiplication Rule: The probability of two or more independent events occurring simultaneously is the product of their individual probabilities.

The probability of rolling a 2 =

1/6 So rolling two 2’s =

1/6 x 1/6 =1/36

The probability of rolling two 2’s with a pair of dice:

Mendel Understood Probability

The Addition Rule: The probability that an event can occur in two or more alternative ways is the sum of the separate probabilities of the different ways.

(Used to answer “either / or” questions only)

The probability of rolling a 2 or a 5 =

1/6 + 1/6 = 1/3

Mendel Understood Probability

One more thing to remember:

p(a mutually exclusive event) = 1 – p(all the other events)

Probability

Likelihood Ratio

• Comparison of probabilities• Probability that event occurred:

– The defendant committed the murder– Hypothesis of the prosecution (Hp)

• Probability that even did not occur:– Someone else committed the murder– Hypothesis of the defense (Hd)

• Likelihood ratio = Hp/Hd

Likelihood Ratio

• Likelihood ratio = Hp/Hd• More evidence that defendant committed

the murder– Hp becomes larger– LR grows larger – more convincing

• Less evidence– Hd becomes larger– LR shrinks – less convincing

Statistics

• Measure of the certainty of a result• Provides a sense of how reliable a

measurement is• Testing two alternative hypotheses:

– Ho = null hypothesis– Ha or H1 = alternative hypothesis

• Represent the only two possible conditions– This man did it– Or did not do it

Hypothesis Testing

Six steps:1. Formulate two competing hypotheses2. Select the statistical model/test to use3. Determine the level of significance4. Collect and analyze the data5. Does the test statistic meet the level of

significance or not?6. Reject or accept null hypothesis (Ho)

Truth about the populationDecision based on sample examined

Correct decision

Type II error

Type I error Correct decision

H0 True H1 True

Accept H0

Reject H0(Accept H1)

Correct decision

Wrongfully acquitted

Wrongfully accused

Correct decision

(B) Example

Not Guilty

Guilty

Courtroom VerdictNot Guilty Guilty

Defendant

(A) Hypothesis Testing Decisions

Figure 19.2, J.M. Butler (2005) Forensic DNA Typing, 2nd Edition © 2005 Elsevier Academic Press

Chi-Squared Test

• Called the “Goodness of fit” test• How well does the observed data fit with

the expected results?• If the data is what you expect it to be, then

your observed will be very close to your expected – X2 will be low

• If the data is radically different than expected than your Χ2 will be high

Chi-Squared Test

Χ2 = (Observed – Expected)2

Expected• Accept the hypothesis as correct if the

data fits with the expectations– Χ2 is lower than acceptable limit

• Reject the hypothesis as wrong if the data simply does not match to the expected– Because Χ2 is higher than acceptable limit

Confidence Intervals

• How much data fits into your statistical limit?

• Most often people use 95% confidence interval

• This means that you have 95% chance of being correct

• 5% chance of being wrong• p-value = 0.05

Population Genetics

• Determining how frequent alleles and genotypes are within a given population

• Need to know this in order to determine the random chance of seeing one specific DNA profile

• Determined by sampling a portion of the entire population– Because it would take too much time and

money to genotype everyone

Discrete “unit factors” of inheritanceGene -

Physical manifestation of a trait (e.g. Yellow or green seed)

Allelic composition of a trait (e.g. YY, Yy, or yy)

Different forms of a gene (e.g. Y or y)

Genetic Language:

Allele -

Genotype -

Phenotype -

Homozygous – Individuals with two identical copies of a gene

Same allele (yy)

Heterozygous - Individuals with two different copies of a gene

Two different alleles (Yy)

Genetic Language:

Hereditary traits are determined by discrete factors (now called genes) that appear in pairs. During sexual development, these pairs are separated (segregated) into gametes and only one factor from each parent is passed to the offspring.

Principle of segregation

Alleles are randomly separated into gametes duringmeiosis. One allele, at random, goes into the gamete

and then is passed to baby.

Mendel’s 1st Law:

Independent Assortment

Mendel’s 2nd Law:

Inheritance of a pair of factors for one trait is independent of the simultaneous inheritance of factors for another trait

Two genes will assort independently and randomlyand be inherited completely separately.

2. Independent Assortment

Mendel’s Laws

Two genes will assort independently and randomly from each other

1. Principle of Segregation

Two alleles, of one gene, segregate randomly during formation of gametes

Bi-allelic Gene

• In bi-allelic gene there are only two alleles possible– T or t – for tall or short pea plants– R or r – for wrinkled or round seeds

• p = frequency of the more common of the two alleles

• q = frequency of the less common of the two alleles

Hardy-Weinberg Equilibrium

Where the allele frequencies stay constant from one generation to the next

• Often calculated with a bi-allelic gene(p and q)

Therefore…• p and q remaining constant

Hardy-Weinberg Equilibrium

1. If there are only two alleles then the following must be true:

p + q = 1

The frequency of the two alleles added together must equal the entire population (a frequency of 1)

Hardy-Weinberg Equilibrium2. The genotype frequencies can also be

calculated:p2 + 2pq + q2 = 1

The frequency of each homozygote equals the frequency of the allele squared

The frequency of heterozygote is 2 times p times q

These three genotypes must add to one

Product Rule

Product and Addition Rules

Hardy-Weinberg Equilibrium1. Allele frequencies add to one:

p + q = 1

2. The genotype frequencies can be calculated from the allele frequencies:

p2 + 2pq + q2 = 1

Hardy-Weinberg Equilibrium1. Allele frequencies add to one:

p + q + r = 1

2. The genotype frequencies can be calculated from the allele frequencies:

p2 + 2pq + 2pr + 2rq + q2 + r2 = 1

How it was derived:

A (p) a (q)

A (p)

a (q)

AA Aa(pp) (pq)

Aa aa(pq) (qq)

Frequencies:Allele A = pAllele a = q

Genotype AA = p2

Genotype Aa = 2pqGenotype aa = q2

HWE and Product Rule:

Het = 2pq = 2(.6)(.3) = .36

Genotype Five Bi-allelic Polymorphisms

(ex. SNPs)

Het = 2pq = 2(.5)(.3) = .30Het = 2pq = 2(.15)(.8) = .24

Het = 2pq = 2(.80)(.18)= .29Homo = q2 = (.2)(.2) = .04

HWE and Product Rule:

= .36(.36)(.3)(.24)(.04)(.29) = 0.00031

Or 1/3,226

Therefore, the chance of this matchingthe wrong person is 1/3,226

= .30= .24

= .29= .04

Alleles vs. Genotypes

Allele frequency = number of copies of one allele within a certain population

• Divided by the total number of all alleles within this same population

Genotype frequency = number of individuals with a certain genotype within a certain population

• Divided by the total number of individuals within this sample population

Alleles vs. Genotypes

• Number of alleles will determine how many different genotypes are possible

• n = number of alleles• n(n + 1)/2 = number of genotypes• Theoretically this is true• However some alleles and genotypes are

so rare as to not really exist• Want to use real numbers, rather than

theoretical numbers

Alleles vs. Genotypes

• Table 19.1Part a shows number of alleles ever seen• Theoretical number of genotypes possiblePart b shows number of common alleles• Number of genotypes ever seen• This second table represents actual

number of possibilities– This is what Forensic Labs use for

calculations

Any Questions?

Read Chapter 20

Guest Lecturer –Forensic Anthropology

A Match!Three possible explanations for a match:1. Suspect left DNA at crime scene

– Trial needs to determine whether that information proves suspect committed crime

2. Suspect’s profile matches by chance– This is why statistics of seeing this DNA

profile at random are calculated3. Match is a false positive result

– This is avoided at all times by validating technology, running controls and duplicates

Randomization Trials• Also known as permutation tests• Calculate a statistic to determine how

unusual is the sample– In this case, how rare is the DNA profile?

• Calculate this by randomizing the genotypes 1,000 to 100,000 times

• How many times did you see the same DNA profile at random?

• Get a statistic without any population data

Hardy-Weinberg Equilibrium

1908• Hardy – an English mathematician • Weinberg – a German physician

• Both derived, independently, an algebra calculation for what happens to allele frequencies within a population

• Assuming all those false conditions