Mathematical Biology
Mathematical biology models populations, genomes, and biochemical networks, from coalescents and reaction dynamics to the algorithms that read biological sequences.
Population genetics follows allele frequencies under drift, selection and mutation. Over half of it is the diffusion theory of the Wright–Fisher diffusion, from Kimura's fixation probability \((1 - e^{-2\sigma x})/(1 - e^{-2\sigma})\) for an allele at frequency \(x\) with scaled selection coefficient \(\sigma = 2Ns\) to Wright's beta stationary law under two-way mutation, and the Ewens sampling formula gives the law of the allelic partition of a sample under the neutral infinite-alleles model. Coalescent theory traces genealogies backward in time: under Kingman's coalescent Watterson's estimator of \(\theta\) is unbiased, and the multispecies coalescent has an anomaly zone, where a gene tree that disagrees with the species tree is more probable than the one that agrees.
Bioinformatics is the largest field, with about a sixth of the articles, and over half of it is sequence alignment, from Needleman–Wunsch and Smith–Waterman to affine-gap, linear-space and wavefront alignment and profile hidden Markov models. Phylogenetics reconstructs trees from sequences, with the Jukes–Cantor model, neighbor joining and Felsenstein's pruning algorithm. Statistical genetics treats genome-wide association studies, with LD score regression and Mendelian randomization; population structure uses Patterson's f-statistics and admixture graphs; quantitative genetics has Fisher's decomposition of genetic variance; and in systems biology the deficiency-zero theorem gives a weakly reversible mass-action network of deficiency zero exactly one equilibrium in each positive stoichiometric compatibility class.
- Population genetics
- Coalescent theory
- Detecting natural selection
- Population structure and admixture
- Phylogenetics
- Quantitative genetics
- Statistical genetics
- Cancer genomics
- Bioinformatics
- Systems biology
- Population and evolutionary dynamics
- Reaction–diffusion and pattern formation
- Mathematical epidemiology
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