Transcript USC3002_2008.Lect5 - Department of Mathematics

USC3002 Picturing the World
Through Mathematics
Wayne Lawton
Department of Mathematics
S14-04-04, 65162749
[email protected]
Theme for Semester I, 2008/09 : The Logic of
Evolution, Mathematical Models of Adaptation
from Darwin to Dawkins
Mendel’s Three Laws
gene = inheritance factor for a trait
alleles = alternative forms of the same gene
1. Law of Dominance : if traits occur in pairs and
crossing parents pure for contrasting traits results in
offspring having ONLY one [dominant] trait. Phenotype
monohybrid-cross Rr x RrRR,Rr,rR,rr ratios are 3:1
2. Law of Segregation : during the formation of gametes
(meiosis) the 2 alleles separate and later at fertilization
combine to determine the genotype of the offspring
3. Law of Independent Assortment : alleles for different
traits are distributed independently, if each trait occurs in
a pair with one member of the pair dominant. Phenotype
dyhybrid-cross RrGg x RrGg  RRGG, RRGg, RrGg,…
ratios are 9:3:3:1
http://www.hobart.k12.in.us/jkousen/Biology/mendel.htm
Punnett Square
http://www.google.com/search?sourceid=navclient&ie=UTF8&rlz=1T4GGLJ_enSG275SG275&q=P%2dSquare+Practice+Page
http://www.hobart.k12.in.us/jkousen/Biology/psquare.htm
Summary
LAW
PARENT CROSS
OFFSPRING
DOMINANCE
TT x tt
100% Tt
tall x short
tall
SEGREGATION
Tt x Tt
INDEPENDENT
ASSORTMENT
tall x tall
RrGg x RrGg
round & green x round & green
75% tall
25% short
9/16 round seeds &
green pods
3/16 round seeds &
yellow pods
3/16 wrinkled seeds
& green pods
1/16 wrinkled seeds
& yellow pods
There you have them, Mendel's huge contributions to the world of science.
A very smart cookie. His work has stood the test of time, even as the
discovery & understanding of chromosomes & genes has developed in the
140 years after he published his findings. New discoveries have found
"exceptions" to Mendel's basic laws, but none of Mendel's things have
been proven to be flat-out wrong. Hail to the "Father of Genetics" !
Combinatorics
Cyanogenesis in white clover is determined by two pairs
of alleles. If the dominant alleles are present (AB) leaves
removed from the tree readily release detectable
amounts of HCN
Gene A
Gene B
Precursor  Substrate  HCN
If only one gene A is present cyanogenesis is very slow.
If A is absent the plant is acyanogenic.
Problem: For a cross between plants heterozygous for
both alleles, what is the probability that
1. one offspring will be cyanogenic?
2. one of the offspring will be acyanogenic?
3. one of the offspring will show slow cyanogenesis?
Combinatoric Answers
1. 9/16
2. ¼
3. 3/16
Chi-Squared Statistic
Experimental results almost always differ from their
expected values. Let us quantify this observation.
 

Is the chi-squared statistic
d2
with n-1 degrees of freedom
e
n classes
Example: From a cross between Drosophila known to be
heterozygous for vestigial wing 100 offspring were
obtained; 30 we vestigial winged and 70 normal. Is this a
significant deviation from the expected 3:1 ratio?
2
phenotypes observed exp ected d d 2 / e
vestigial
30
25
5
1
normal
70
75
 5 0.33
NO, a value at least this large with
2
probability about 0.25, at least this
small about 0.75, each > 0.05
   1.33
Pedigree Diagrams
A baby died from a fatal recessive disease controlled by
a single gene whose frequency in the general
population was 1 in 8 million. What was the genotype of
his parents? What is the probability that his fathers
mother carried the recessive allele?
Molecular Genetics
1. Encoding : How is information arranged ?
2. Processing : How is information processed ?
3. Replication : How is the blueprint replicated
whenever a cell divides so that each new cell
may have a copy ?
4. Mutation : How is the information modified
so as to provide new message material ?
Definitions
Gene: a length of DNA that carries information for
the synthesis of a single polypeptide chain (some
proteins, such a haemoglobin, can consist of a
number of p.c.’s)
Genetic Diversity: "The number of genes range
from about 1,000 in bacteria and 10,000 in some
fungi to 700,000 or more in many flowering plants
and a few animals.”
http://www.google.com/search?sourceid=navclient&ie=UTF-8&rlz=1T4GGLJ_enSG275SG275&q=genetic+diversity
Genetic Code
http://img.sparknotes.com/figures/1/132e5720f3b37836571a1aeb9d163ac3/genetic_code.gif
Definitions
Species: “a group of organisms capable of
interbreeding and producing fertile offspring.”
“can share a common gene pool”
Example: horses and donkeys are not of same species – can
breed but produce infertile mules (female horse with male donkey)
or hinnys; Great Danes and Chihuahua’s belong to the same
species (Canis lupus familiaris) despite the fact that they can not
breed but can share same gene pool through Terriers, etc.
http://en.wikipedia.org/wiki/Species
Sex: “invented some 4 billion years ago by
evolving bacteria as a means for shaking up and
rearranging a cells instructions for making more
cells.” “mutations add new cards to the deck, sex
shuffles the deck”
Chromosome Linkage
http://www.animalgenome.org/edu/genetics/map.html