Evolution, dispersal of genetics and Fisher’s equation

Charles Darwin Charles Darwin (12 February 1809 – 19 April 1882) Charles Darwin On the Origin of Species (1859)On the Origin of Species 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 Gregor Mendel ( ) Gregor Mendel 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) (original paper) Mendel compared seven discrete traits: Smoothness of the seeds. Smoothness of the seeds. Color of the seeds. Color of the seeds. Color of the seed coats. Color of the seed coats. Shape of the pods. Shape of the pods. Color of unripe pods. Color of unripe pods. Position of flowers. Position of flowers. Length of the stems. 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. Mendelian inheritancethe modern synthesisMendelian inheritancethe 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 albumen Yellow (I) and green (i) cotyledons Difference in the color of the seed coat Colored (A) and white (a) flowers 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 ) 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) 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) 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) 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) 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 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 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 lawHardy-Weinberg’s law: the frequency remain the same for each genotype

G. H. Hardy ( ) Wilhelm Weinberg Wilhelm Weinberg (1862 — 1937)

Assumptions in Hardy-Weinberg’s Law Expected sex ratio is independent of genotype Expected sex ratio is independent of genotype Mating is random Mating is random Fertility is independent of genotype Fertility is independent of genotype Survivorship is independent of genotype Survivorship is independent of genotype There is no mutation or migration 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 Ronald Fisher ( ) J. B. S. Haldane ( ) Sewall Wright ( ) J. B. S. Haldane Sewall Wright Ronald Fisher J. B. S. Haldane Sewall Wright Darwin-Wallace MedalDarwin-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 It is awarded for "major advances in evolutionary biology"Linnean SocietyCharles DarwinAlfred Russel WallaceOn the Tendency of Species to form Varieties; and on the Perpetuation of Varieties and Species by Natural Means of Selection

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!