USC3002 Picturing the World Through Mathematics

1
USC3002 Picturing the World Through Mathematics

• Wayne Lawton
• Department of Mathematics
• S14-04-04, 65162749 matwml_at_nus.edu.sg

Theme for Semester I, 2007/08 The Logic of
Evolution, Mathematical Models of Adaptation
from Darwin to Dawkins
2
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
3
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 ???
4
History
Who 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 giraffes evolve their long necks
??? Would Lamarcks evolution produce branching
??? How were Lamarcks ideas exploited by
Stalinists ???
5
History
What was the prevailing view of evolution prior
to the publication of Darwins Origins in
1859 ???
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 student Richard
Owen (1804-1892) and Charles Lyell (1797-1875)
whose book Principles of Geology (1830)
criticised Lamarck.
What was the prevailing view on the age of the
Earth ??? Is it relevant to evolution ???
6
History
Charles Robert Darwin (1809-1882)
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
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
October 1838 - reads Malthuss 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.
7
Math Malthus and Exponential Growth
Web search on Malthus
Geometric growth is described by the exponential
function exp R ? (0,infty)
The inverse function, called the natural
logarithm ln (0,infty) ? R is described by the
formula
For
the area of
the shaded region
What is the curved line above the shaded region
???
How might we define ln(x) for x lt 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 ?
Show that for small increments
8
History
1838-1858 Darwin proceeds to work out his theory
Alfred 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
Darwin was then writing an abstract of his full
findings which he published in 1859 under the
title On the Origin of Species
9
History
Darwins theory of evolution, though
controversial in the popular media, was widely
accepted by scientists

• 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

10
Math Graph Theory and Branching
Although many scientist accepted evolution, their
concept of evolution as a progressive process
differed sharply from Darwins 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 ? b
A Tree is a directed graph that contains no
circular loops such as a ?b?c?d?a, a tree is
equivalent to having a partial order. A linear
order is a partial order such that a?b or b?a 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
11
Mendels Experiments
Mendel experimented for 8 years with pea plants
(species Pisum sativum), which exhibit 7 pairs
of phenotypic characteristics for instance seed
color yellow or green
These 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?
12
Mendels 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 G The F2 plants with
yellow seeds were a mixture 1/3 type Y and 2/3
type H
13
Genotype Ratios
14
Random Mating Frequencies
Random mating between pairs of individuals in a
population with genotype frequencies
gives the following frequencies of mating
combinations
Remark
is the frequency of matings where one
parent is Y and the other is H, it can be
computed by
15
Population Dynamics Random Mating
We now combine the genotype frequency table with
the random mating frequencies to compute the
genotype frequencies in the next generation after
random mating
16
Hardy-Weinberg Equilibrium
Define
but amazingly
Clearly
17
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
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
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
18
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
Stigma the receptive apex, on which pollen is
deposited
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
19
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
20
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
21
Homework 1. Due Tuesday 21.08.2007
Question 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
Compute random mating frequencies if this
assumption is not valid using the 3 genotype
frequencies for males and the 3 genotype
frequencies for females.
Question 3. Research and summarize the history of
the discovery of the biological mechanism for
Mendels 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.