USC3002 Picturing the World Through Mathematics

Wayne LawtonDepartment of Mathematics

S14-04-04, 65162749 matwml@nus.edu.sg

Theme for Semester I, 2008/09 : The Logic of Evolution, Mathematical Models of Adaptation from Darwin to Dawkins

Evolution= change in the form and behavior of organisms between generations, Darwin called it “descent with modifications”, p.4

Excludes p.4-5- developmental change, e.g. growth, and change in the composition of ecosystems- common use of the word ‘evolution’ to describe changes in human politics, economics, history, technology, and scientific theories

Darwin proposed that evolution gave rise to a repeated ‘splitting of lineages’ that exhibits a clear branching, tree-like structure of species (biologically defined as ‘interbreeding natural populations, p.351)page numbers from EVOLUTION by Mark Ridley

Adaptation= properties of living things that enable them to survive and reproduce in nature p.6

Examples p.6- woodpeckers’ beaks enables them to ??? - camouflage used by ??? enables them to ???- other examples ???

Natural Selection = some kinds of individuals in a population reproduce more than others p. 6

How might natural selection be used to explain adaptation ???

HistoryWho were Maupertius, Diderot, Erasmus Darwin ???

Jean-Baptiste Lamarck (1744-1829), in his bookPhilosophie Zoologique (1809) argued that species change over time and proposed - as the primary mechanism for that change an “internal force” that caused offspring to differ slightly from their parents- and a secondary mechanism the inheritance of acquired characters ( = characteristics), an idea proposed by the philosopher Plato ~350BCE

How could giraffe’s evolve their long necks ???Would Lamarck’s evolution produce branching ???How were Lamarck’s ideas exploited by Stalinists ???

History

Anatomist Georges Cuvier (1769-1832) studied the design of organisms, proposed that animals were divided into four branches: vertebrates, articulates, mollusks, and radiates, established that some species had gone extinct, and firmly promoted the idea that each species had a separate origin.

His views were supported by his studentRichard Owen (1804-1892) and Charles Lyell (1797-1875) whose book Principles of Geology (1830) criticised Lamarck.

What was the prevailing view of evolution prior to the publication of Darwin’s ‘Origins …’ in 1859 ???

What was the prevailing view on the age of the Earth ??? Is it relevant to evolution ???

History

Thinks that the finches may have evolved from a common ancestor, but struggled to explain why. Rejected existing theories because he thought that they failed to explain adaptation. p.10

Charles Robert Darwin (1809-1882)

October 1838 - reads Malthus’s Essay on Population “and being well prepared to appreciate the struggle for existence which everywhere goes on from long continued observation of the habits of animals and plants, it at once struck me that under these circumstances favourable variations would tend to be preserved and unfavourable ones to be destroyed. The result would be the formation of a new species.”

Ponders, after his voyage on the Beagle (1832-37) that each Galapagos island had its own species of finches, the geographical diversity of S. American rheas, etc

Math: Malthus and Exponential Growth

The inverse function, called the natural logarithm ln : (0,infty) R is described by the formula:

Web search on MalthusGeometric growth is described by the exponential function exp : R (0,infty)

sdxx

s1

1)(ln For 1x the area of

1 xWhat is the curved line above the shaded region ???How might we define ln(x) for x < 1 ???What is that funny symbol that looks like a long ‘S’ ???Show geometrically that ln(xy) = ln(x) + ln(y)What is meant by the ‘inverse function’ ?

the shaded region

Show that for small increments)exp()exp()exp( xxxxx 0x

History

Darwin was then writing an abstract of his full findings which he published in 1859 under the title “On the Origin of Species

1838-1858 Darwin proceeds to work out his theoryAlfred Russel Wallace (1823-1913) travels to Malaysia and writes Darwin about his similar ideas, Charles Lyle and Joseph Hooker arrange for simultaneous announcement of their ideas at a meeting on the Linnean Society in London in 1858. p. 10

History

- Joseph Dalton Hooker (1817-1911) who conducted a botanical expedition to Sikkim in 1849- Thomas Henry Huxley (1825-1829)- Carl Gegenbauer (1826-1903) traced evolutionary relationships between animal groups- Ernst Haeckel (1834-1919) proposed his “ontology recapitulates phylogeny” theory

Darwin’s theory of evolution, though controversial in the popular media, was widely accepted by scientists

Math: Graph Theory and Branching Although many scientist accepted evolution, their concept of evolution as a progressive process differed sharply from Darwin’s concept of a branching process

A Graph is a set V of vertices together with a set E of unordered pairs {a,b}, a,b in V called edges. A Directed Graph is a set V of vertices and a set of ordered pairs (a,b), a, b in V called (directed) edges. In a directed edge we can write (a,b) as a bA Tree is a directed graph that contains no ‘circular loops’ such as a bcda, a tree is equivalent to having a partial order. A linear order is a partial order such that ab or ba for nodes a, b

Show that progressive processes are described by linear orders whereas branching processes are described by more general trees that are not partially ordered but not linearly ordered

Mendel’s Experiments Mendel experimented for 8 years with pea plants (species: Pisum sativum), which exhibit 7 pairs ofphenotypic characteristics – for instance seed color: yellow or greenThese plants can be easily domesticated (selectively crossed) so as to produce types Y and G such that all decendents obtained from crossing type Y (G) with type Y (G) ONLY produce plants with yellow (green) seeds

Mendel crossed type Y and type G pea plants and noticed that all of the resulting hybrid plants had all yellow seeds. Type Y is dominant and type G recessive.

When Mendel crossed the first generation of hybrid plants with themselves – he was surprised !What do you think he found?

Mendel’s Ratios The first generation F1 of hybrid plants had yellow seeds

The second generation F2 of hybrid plants (F1 plants crossed with themselves or with other F1 plants) gave a mixture: 75% yellow seed plants, 25% green seed plants

Crossing F1 (hybrids) with Y plants gave: 100% yellow seed plants; with G plants gave: 50% yellow seed plants, 50% green seed plants

The third generation F3 of hybrid plants (F2 plants crossed with themselves or with other F2 plants) showed that there were only three types of plants (genotypes): Y, G and H ( same as all of the F1 generation of hybrids)All plants with the green seeds were type GThe F2 plants with yellow seeds were a mixture: 1/3 type Y and 2/3 type H

Genotype Ratios Male

Female

Y H G

Y 1.00 Y 0.50 Y

0.50 H

1.00H

H 0.50 Y

0.50 H

0.25 Y

0.50 H

0.25 G

0.50 H

0.50 G

G 1.00H 0.50 H

0.50 G

1.00 G

Random Mating FrequenciesRandom mating between pairs of individuals in a population with genotype frequencies

gives the following frequencies of mating combinationsGHY PPP ,,

222 ,, GGGHHHYYY PPPPPP GHHGGYYGHYYH PPPPPPPPP 2,2,2

Remark: YHP is the frequency of matings where one

parent is Y and the other is H, it can be computed by

])mother and father[or ]mother and father([prob YHHYPYH

)mother and father prob()mother and father(prob YHHY )mother(prob)father prob()rprob(mothe )father(prob YHHY

HYYHHY PPPPPP 2

Population Dynamics: Random MatingWe now combine the genotype frequency table with therandom mating frequencies to compute the genotype frequencies in the next generation after random mating

)|(prob)|(prob)|(prob' HHYPYHYPYYYPP HHYHYYY 2

412

41

21

HHYYHHYHYY PPPPPPP

)|(prob)|(prob)|(prob' HHGPGHGPGGGPP HHGHGGG 2

412

41

21

HHGGHHGHGG PPPPPPP

)|(prob)|(prob' YGHPYHHPP YGYHH )|(prob)|(prob HGHPHHHP HGHH

HGHGYHYHGHHYGYH PPPPPPPPPPP 221

21

21

21 2

Hardy-Weinberg Equilibrium

Define

22412' yPPPPP HHYYY

22412' gPPPPP HHGGG

ygPPPPPPPP GHHGYHYH 22 221'

HYGHY PPPPPyy 21),,( HGGHY PPPPPgg 2

1),,( Clearly 1 GHY PPPgy but amazingly

yygyPPPPPyy HYGHY 2'21''''' ),,(

gyggPPPPPgg HGGHY 2'21''''' ),,(

Bio-Jargon

Gamete : a reproductive cell (or germ cell) having the haploid number of chromosomes, especially a mature sperm or egg capable of fusing with a gamete of the opposite sex to produce a fertilized egg

Chromosomes: a threadlike linear combination of DNA and associated proteins in the nucleus of animal and plant cells that carries the genes and functions in the transmission of hereditary information

Haploid: having half the number of sets of chromosomes as a somatic cell (or body cell).

Diploid: having two sets of chromosomes : diploid somatic cells

Botanical Jargon

Pistil: the female, ovule-bearing organ of a flower, including the stigma, style, and ovary

Ovule: a minute structure in seed plants, containing the embryo sac and surrounded by the nucellus, that develops into a seed after fertilization

Nucellus: the central portion of an ovule in which the embryo sac develops

Style: the usually slender part of a pistil, situated between the ovary and the stigma

Ovary: the ovule-bearing lower part of a pistil that ripens into a fruit

Stigma: the receptive apex, on which pollen is deposited

Botanical Jargon

Stamen: the pollen-producing reproductive organ of a flower, usually consisting of a filament and an anther

Pollen: the fine, powder like material consisting of pollen grains that is produced by the anthers of seed plants

Filament: the stalk that bears the anther in a stamen

Anther: the pollen-bearing part of the stamen

Evolution Mathematics Dynamics : growth of populations, population genetics, both discrete and continuous

Combinatorics: graph theory, permutations, string matching

Probability and Statistics: population genetics, Markov processes, genetic drift, Binomial, Poisson, and Gaussian (normal) distributions

Game Theory: models for altruism and competition

Physical Modeling: X-ray structure of proteins, radioactive dating of fossils

Artificial Intelligence: evolution of language and reasoning ability

Homework 1. Due Monday 25.08.2008Question 1. Complete the derivation of the random mating frequencies

Question 2. In the derivation we assumed that genotype was uncorrelated with the sex of parents, this means

Question 3. Research and summarize the history of the discovery of the biological mechanism for Mendel’s findings in terms of gametes, chromosomes, genes, etc. (not the molecular level DNA mechanism)

Question 4. Derive the HW equations using probability methods and the mechanism of gametes, genes, etc.

GHHGGYYG PPPPPP 2,2

GHYUUU ,,for P)rprob(mothe )father(prob U Compute random mating frequencies if this assumption is not valid using the 3 genotype frequencies for males and the 3 genotype frequencies for females.