USC3002 Picturing the World Through Mathematics Wayne Lawton Department of Mathematics S , Theme for Semester I, 2008/09 : The Logic of Evolution, Mathematical Models of Adaptation from Darwin to Dawkins

Mendel’s Three Laws 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 gene = inheritance factor for a trait alleles = alternative forms of the same gene

Punnett Square 8&rlz=1T4GGLJ_enSG275SG275&q=P%2dSquare+Practice+Page

LAWPARENT CROSSOFFSPRING DOMINANCE TT x tt tall x short 100% Tt tall SEGREGATION Tt x Tt tall x tall 75% tall 25% short INDEPENDENT ASSORTMENT RrGg x RrGg round & green x round & green 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" ! Summary

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 Precursor  Substrate  HCN Gene AGene B 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 with n-1 degrees of freedom 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? NO, a value at least this large with probability about 0.25, at least this small about 0.75, each > 0.05

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.”

Genetic Code

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. 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