USC3002 Picturing the World Through Mathematics

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Transcript USC3002 Picturing the World Through Mathematics

USC3002 Picturing the World
Through Mathematics
Wayne Lawton
Department of Mathematics
S14-04-04, 65162749
[email protected]
Theme for Semester I, 2007/08 : The Logic of
Evolution, Mathematical Models of Adaptation
from Darwin to Dawkins
= 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
= 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 ???
Who were Maupertius, Diderot, Erasmus Darwin ???
Jean-Baptiste Lamarck (1744-1829), in his book
Philosophie 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 ???
What was the prevailing view of evolution prior to
the publication of Darwin’s ‘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 (17971875) whose book Principles of Geology (1830)
criticised Lamarck.
What was the prevailing view on the age of the
Earth ??? Is it relevant to evolution ???
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 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.”
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:
ln ( x)  
1 s
For x  1 the area of
the shaded region
What 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’ ?
Show that for small increments x  0
exp( x  x)  exp( x)  x exp( x)
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
Darwin’s 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
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  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
Mendel’s 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?
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 G
The F2 plants with yellow seeds were a mixture:
1/3 type Y and 2/3 type H
Genotype Ratios
Male Y
1.00 Y
0.50 Y
0.50 H
0.50 Y
0.50 H
0.25 Y
0.50 H
0.25 G
0.50 H
0.50 G
0.50 H
0.50 G
1.00 G
Random Mating Frequencies
Random mating between pairs of individuals in a
population with genotype frequencies PY , PH , PG
gives the following frequencies of mating combinations
P  P ,P
 P ,P
PYH  2PY PH , PYG  2PY PG , PHG  2PH PG
Remark: P
is the frequency of matings where one
parent is Y and the other is H, it can be computed by
PYH  prob([father Y and mother H ] or [father H and mother Y ])
 prob(father Y and mother H )  prob( father H and mother Y )
 prob(father Y ) prob(mothe rH )  prob( father H )prob(mother Y )
 PY PH  PH PY  2PY PH
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
P  PYY prob(Y | YY )  PYH prob(Y | YH )  PHH prob(Y | HH )
 PYY  PYH  PHH  P  PY PH  P
P  PYH prob( H | YH )  PYG prob( H | YG )
 PHH prob( H | HH )  PHG prob( H | HG )
 12 PYH  PYG  12 PHH  12 PHG  PY PH  2 PY PG  12 PH2  PG PH
PG'  PGG prob(G | GG)  PGH prob(G | GH )  PHH prob(G | HH )
 PGG  PGH  PHH  P  PG PH  P
Hardy-Weinberg Equilibrium
y  y( PY , PH , PG )  PY  12 PH
g  g ( PY , PH , PG )  PG  12 PH
Clearly y  g  PY  PH  PG  1 but amazingly
P  P  PY PH  P  y
P  PY PH  2 PY PG  P  PH PG  2 yg
P  PG  PG PH  4 P  g
y  y ( P , P , P )  P  P  y  yg  y
g  g ( P , P , P )  P  P  g  yg  g
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
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
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
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 Tuesday 21.08.2007
Question 1. Complete the derivation of the random
mating frequencies PYG  2 PY PG , PHG  2 PH PG
Question 2. In the derivation we assumed that genotype
was uncorrelated with the sex of parents, this means
prob(father U )  prob(mothe rU )  PU for U  Y , H , G
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 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.