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Evolution, dispersal of
genetics and Fisher’s equation
Charles Darwin
(12 February 1809 – 19 April 1882)
•On the Origin of Species (1859)
•Many individuals of s species are destined to
die before reaching reproduction age.
•Advantageous gene tends to be persevered,
thus change the characteristics of the species
•Evolution by natural selection (slight
modifications are passed on through
generations)
•But how does it work?
Gregor Mendel (1822-1884)
•Study of the inheritance of traits in pea plants.
Over a period of 7 years he bred and counted
about 28,000 pea plants.
•Traits are passed through generations unchanged,
a child can inherit the trait from either its mother
or its father
•The trait is determined by two factors, one
inherited from each parent, and it comes with
probability from either parent
•For pairs of contrasting traits, one of the two is
dominant and always overrules the other
(recessive) factor
Trait - a variant for a character
Character - a heritable feature
Factor – Mendel’s factor is gene today
Experiments on Plant Hybridization, Gregor Mendel
(1865, Proceedings of the Natural History Society)
http://www.mendelweb.org/Mendel.html (original paper)
Mendel compared seven discrete traits:
• Smoothness of the seeds.
• Color of the seeds.
• Color of the seed coats.
• Shape of the pods.
• Color of unripe pods.
• Position of flowers.
• Length of the stems.
Through experimentation, Mendel discovered that one inheritable trait would
invariably be dominant to its recessive alternative. This model, later known
as Mendelian inheritance or Mendelian genetics, provided an alternative to
blending inheritance, which was the prevailing theory at the time.
Unfortunately, Mendel's work received little attention from the scientific
community and was largely forgotten. It was not until the early 20th
century that Mendel's work was rediscovered and his ideas used to help
form the modern synthesis.
Plant material Mendel used: sweet pea
Examples of Mendel traits
Difference in the
form of the ripe
seeds
Round (R) and
wrinkled (r)
cotyledons
Difference in the
color of the seed
coat
Colored (A)
and white (a)
flowers
Difference in the
color of the seed
albumen
Yellow (I) and
green (i)
cotyledons
Difference in the
color of the unripe
pods
Green (Gp) or
yellow (gp) pod
wall
Genetic Terms
• Phenotype - the outward, physical appearance of a particular trait
(Pea: round or wrinkled seed phenotype; yellow or green seed phenotype)
• Genotype - genetic make-up of a particular trait, the specific allelic
combination of a certain gene (AA, BB, or AB)
• Allele - one alternative form of a given allelic pair (A or B)
• Homozygote - an individual which contains only one allele at the allelic
•
•
•
pair (AA or BB)
Heterozygote - an individual which contains one of each member of the
gene pair (AB)
Dominant - the allele that expresses itself at the expense of an alternate
allele; the phenotype that is expressed in the F1 generation from the cross
of two pure lines
Recessive - an allele whose expression is suppressed in the presence of a
dominant allele; the phenotype that disappears in the F1 generation from
the cross of two pure lines and reappears in the F2 generation
More general, the probabilities
(frequencies) of allele A or a in a
population can be p and q, where
p+q=1. Then the probability of each
genotype in F2 is
AA: p^2 Aa: 2pq aa: q^2
Hardy-Weinberg’s law: the frequency
remain the same for each genotype
1
pn 1  pn  2 pn qn  p n
2
1
2
qn 1  qn  2 pn qn  qn
2
2
G. H. Hardy
(1877-1947)
Wilhelm Weinberg
(1862 — 1937)
Assumptions in Hardy-Weinberg’s Law
•
•
•
•
•
Expected sex ratio is independent of genotype
Mating is random
Fertility is independent of genotype
Survivorship is independent of genotype
There is no mutation or migration
Different survival rate or fertility rate for the two genotypes could
break Hardy-Weinberg’s law
Evolution of genes under selection:
Fisher-Haldane-Wright equation
Ronald Fisher (1890-1962)
J. B. S. Haldane (1892-1964)
Sewall Wright (1889-1988)
Darwin-Wallace Medal is a medal awarded by the Linnean Society of
London every 50 years, beginning in 1908, 50 years after the joint presentation
by Charles Darwin and Alfred Russel Wallace of two scientific papers - On the
Tendency of Species to form Varieties; and on the Perpetuation of Varieties and
Species by Natural Means of Selection - to the Linnean Society of London on 1
July 1858. It is awarded for "major advances in evolutionary biology"
Some special cases
Evolution to advantageous gene:
numerical solutions by Matlab
(upper): dominant and advantageous, left: s=0.002, right: s=0.2
(lower): dominant and recessive, left: s=0.002, right: s=0.2
Finally, Fisher’s equation with diffusion !
Traveling wave of Fisher’s equation
Finding Traveling wave solution
Phase portrait Analysis
Problem solved!