Coalescent theory is a
model
A model is an informative representation of an object, person, or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin , .
Models can be divided in ...
of how
allele
An allele is a variant of the sequence of nucleotides at a particular location, or Locus (genetics), locus, on a DNA molecule.
Alleles can differ at a single position through Single-nucleotide polymorphism, single nucleotide polymorphisms (SNP), ...
s sampled from a
population
Population is a set of humans or other organisms in a given region or area. Governments conduct a census to quantify the resident population size within a given jurisdiction. The term is also applied to non-human animals, microorganisms, and pl ...
may have originated from a
common ancestor
Common descent is a concept in evolutionary biology applicable when one species is the ancestor of two or more species later in time. According to modern evolutionary biology, all living beings could be descendants of a unique ancestor commonl ...
. In the simplest case, coalescent
theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
assumes no
recombination, no
natural selection
Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the Heredity, heritable traits characteristic of a population over generation ...
, and no
gene flow
In population genetics, gene flow (also known as migration and allele flow) is the transfer of genetic variation, genetic material from one population to another. If the rate of gene flow is high enough, then two populations will have equivalent ...
or
population structure, meaning that each variant is equally likely to have been passed from one generation to the next. The model looks backward in time, merging alleles into a single ancestral copy according to a random process in coalescence events. Under this model, the expected time between successive coalescence events increases almost
exponentially back in time (with wide
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
). Variance in the model comes from both the random passing of alleles from one generation to the next, and the random occurrence of
mutation
In biology, a mutation is an alteration in the nucleic acid sequence of the genome of an organism, virus, or extrachromosomal DNA. Viral genomes contain either DNA or RNA. Mutations result from errors during DNA or viral replication, ...
s in these alleles.
The mathematical theory of the coalescent was developed independently by several groups in the early 1980s as a natural extension of classical
population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, s ...
theory and models, but can be primarily attributed to
John Kingman. Advances in coalescent theory include recombination, selection, overlapping generations and virtually any arbitrarily complex evolutionary or demographic model in population genetic analysis.
The model can be used to produce many theoretical genealogies, and then compare observed data to these simulations to test assumptions about the demographic history of a population. Coalescent theory can be used to make inferences about population genetic parameters, such as migration, population size and recombination.
Theory
Time to coalescence
Consider a single gene locus sampled from two
haploid
Ploidy () is the number of complete sets of chromosomes in a cell (biology), cell, and hence the number of possible alleles for Autosome, autosomal and Pseudoautosomal region, pseudoautosomal genes. Here ''sets of chromosomes'' refers to the num ...
individuals in a population. The ancestry of this sample is traced backwards in time to the point where these two lineages ''coalesce'' in their
most recent common ancestor
A most recent common ancestor (MRCA), also known as a last common ancestor (LCA), is the most recent individual from which all organisms of a set are inferred to have descended. The most recent common ancestor of a higher taxon is generally assu ...
(MRCA). Coalescent theory seeks to estimate the expectation of this time period and its variance.
The probability that two
lineages coalesce in the immediately preceding generation is the probability that they share a parental
DNA
Deoxyribonucleic acid (; DNA) is a polymer composed of two polynucleotide chains that coil around each other to form a double helix. The polymer carries genetic instructions for the development, functioning, growth and reproduction of al ...
sequence. In a population with a constant
effective population size
The effective population size (''N'e'') is the size of an idealised population that would experience the same rate of genetic drift as the real population. Idealised populations are those following simple one- locus models that comply with ass ...
with 2''N
e'' copies of each locus, there are 2''N
e'' "potential parents" in the previous generation. Under a
random mating
Panmixia (or panmixis) means uniform random fertilization, which means individuals do not select a mate based on physical traits. A panmictic population is one where all potential parents may contribute equally to the gamete pool, and that these ga ...
model, the probability that two
allele
An allele is a variant of the sequence of nucleotides at a particular location, or Locus (genetics), locus, on a DNA molecule.
Alleles can differ at a single position through Single-nucleotide polymorphism, single nucleotide polymorphisms (SNP), ...
s originate from the same parental copy is thus 1/(2''N
e'') and, correspondingly, the probability that they do ''not'' coalesce is 1 − 1/(2''N
e'').
At each successive preceding generation, the probability of coalescence is
geometrically distributed—that is, it is the probability of ''non''coalescence at the ''t'' − 1 preceding generations multiplied by the probability of coalescence at the generation of interest:
:
For sufficiently large values of ''N
e'', this distribution is well approximated by the continuously defined
exponential distribution
In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuousl ...
:
This is mathematically convenient, as the standard exponential distribution has both the
expected value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
and the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
equal to 2''N
e''. Therefore, although the ''expected'' time to coalescence is 2''N
e'', actual coalescence times have a wide range of variation. Note that coalescent time is the number of preceding generations where the coalescence took place and not calendar time, though an estimation of the latter can be made multiplying 2''N
e'' with the average time between generations. The above calculations apply equally to a
diploid
Ploidy () is the number of complete sets of chromosomes in a cell, and hence the number of possible alleles for autosomal and pseudoautosomal genes. Here ''sets of chromosomes'' refers to the number of maternal and paternal chromosome copies, ...
population of effective size ''N
e'' (in other words, for a non-recombining segment of DNA, each
chromosome
A chromosome is a package of DNA containing part or all of the genetic material of an organism. In most chromosomes, the very long thin DNA fibers are coated with nucleosome-forming packaging proteins; in eukaryotic cells, the most import ...
can be treated as equivalent to an independent
haploid
Ploidy () is the number of complete sets of chromosomes in a cell (biology), cell, and hence the number of possible alleles for Autosome, autosomal and Pseudoautosomal region, pseudoautosomal genes. Here ''sets of chromosomes'' refers to the num ...
individual; in the absence of inbreeding, sister chromosomes in a single individual are no more closely related than two chromosomes randomly sampled from the population). Some effectively haploid DNA elements, such as
mitochondrial DNA
Mitochondrial DNA (mtDNA and mDNA) is the DNA located in the mitochondrion, mitochondria organelles in a eukaryotic cell that converts chemical energy from food into adenosine triphosphate (ATP). Mitochondrial DNA is a small portion of the D ...
, however, are only passed on by one sex, and therefore have one quarter the effective size of the equivalent diploid population (''N
e''/2)
The mathematical object one formally obtains by letting ''N
e'' go to infinity is known as the
Kingman coalescent.
Neutral variation
Coalescent theory can also be used to model the amount of variation in
DNA
Deoxyribonucleic acid (; DNA) is a polymer composed of two polynucleotide chains that coil around each other to form a double helix. The polymer carries genetic instructions for the development, functioning, growth and reproduction of al ...
sequences expected from genetic drift and mutation. This value is termed the mean
heterozygosity
Zygosity (the noun, zygote, is from the Greek "yoked," from "yoke") () is the degree to which both copies of a chromosome or gene have the same genetic sequence. In other words, it is the degree of similarity of the alleles in an organism.
Mos ...
, represented as
. Mean heterozygosity is calculated as the probability of a mutation occurring at a given generation divided by the probability of any "event" at that generation (either a mutation or a coalescence). The probability that the event is a mutation is the probability of a mutation in either of the two lineages:
. Thus the mean heterozygosity is equal to
:
For
, the vast majority of allele pairs have at least one difference in
nucleotide
Nucleotides are Organic compound, organic molecules composed of a nitrogenous base, a pentose sugar and a phosphate. They serve as monomeric units of the nucleic acid polymers – deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), both o ...
sequence.
Extensions
There are numerous extensions to the coalescent model, such as the Λ-coalescent which allows for the possibility of multifurcations.
Graphical representation
Coalescents can be visualised using
dendrogram
A dendrogram is a diagram representing a Tree (graph theory), tree graph. This diagrammatic representation is frequently used in different contexts:
* in hierarchical clustering, it illustrates the arrangement of the clusters produced by ...
s which show the relationship of branches of the population to each other. The point where two branches meet indicates a coalescent event.
Applications
Disease gene mapping
The utility of coalescent theory in the mapping of disease is slowly gaining more appreciation; although the application of the theory is still in its infancy, there are a number of researchers who are actively developing algorithms for the analysis of human genetic data that utilise coalescent theory.
A considerable number of human diseases can be attributed to genetics, from simple
Mendelian diseases like
sickle-cell anemia and
cystic fibrosis
Cystic fibrosis (CF) is a genetic disorder inherited in an autosomal recessive manner that impairs the normal clearance of Sputum, mucus from the lungs, which facilitates the colonization and infection of the lungs by bacteria, notably ''Staphy ...
, to more complicated maladies like cancers and mental illnesses. The latter are polygenic diseases, controlled by multiple genes that may occur on different chromosomes, but diseases that are precipitated by a single abnormality are relatively simple to pinpoint and trace – although not so simple that this has been achieved for all diseases. It is immensely useful in understanding these diseases and their processes to know where they are located on
chromosome
A chromosome is a package of DNA containing part or all of the genetic material of an organism. In most chromosomes, the very long thin DNA fibers are coated with nucleosome-forming packaging proteins; in eukaryotic cells, the most import ...
s, and how they have been inherited through generations of a family, as can be accomplished through coalescent analysis.
[Morris, A., Whittaker, J., & Balding, D. (2002). Fine-Scale Mapping of Disease Loci via Shattered Coalescent Modeling of Genealogies. ''The American Journal of Human Genetics,'' ''70''(3), 686–707. ]
Genetic diseases are passed from one generation to another just like other genes. While any gene may be shuffled from one chromosome to another during
homologous recombination
Homologous recombination is a type of genetic recombination in which genetic information is exchanged between two similar or identical molecules of double-stranded or single-stranded nucleic acids (usually DNA as in Cell (biology), cellular organi ...
, it is unlikely that one gene alone will be shifted. Thus, other genes that are close enough to the disease gene to be
linked to it can be used to trace it.
Polygenic diseases have a genetic basis even though they don't follow Mendelian inheritance models, and these may have relatively high occurrence in populations, and have severe health effects. Such diseases may have incomplete
penetrance
Penetrance in genetics is the proportion of individuals carrying a particular variant (or allele) of a gene (genotype) that also expresses an associated trait (phenotype). In medical genetics, the penetrance of a disease-causing mutation is the pr ...
, and tend to be
polygenic
A polygene is a member of a group of non- epistatic genes that interact additively to influence a phenotypic trait, thus contributing to multiple-gene inheritance (polygenic inheritance, multigenic inheritance, quantitative inheritance), a type ...
'','' complicating their study. These traits may arise due to many small mutations, which together have a severe and deleterious effect on the health of the individual.
[Rannala, B. (2001). Finding genes influencing susceptibility to complex diseases in the post-genome era. ''American journal of pharmacogenomics'', ''1''(3), 203–221.]
Linkage mapping methods, including Coalescent theory can be put to work on these diseases, since they use family pedigrees to figure out which markers accompany a disease, and how it is inherited. At the very least, this method helps narrow down the portion, or portions, of the genome on which the deleterious mutations may occur. Complications in these approaches include
epistatic
Epistasis is a phenomenon in genetics in which the effect of a gene mutation is dependent on the presence or absence of mutations in one or more other genes, respectively termed modifier genes. In other words, the effect of the mutation is depe ...
effects, the polygenic nature of the mutations, and environmental factors. That said, genes whose effects are additive carry a fixed risk of developing the disease, and when they exist in a disease genotype, they can be used to predict risk and map the gene.
Both the regular coalescent and the shattered coalescent (which allows that multiple mutations may have occurred in the founding event, and that the disease may occasionally be triggered by environmental factors) have been put to work in understanding disease genes.
Studies have been carried out correlating disease occurrence in fraternal and identical twins, and the results of these studies can be used to inform coalescent modeling. Since identical twins share all of their genome, but fraternal twins only share half their genome, the difference in correlation between the identical and fraternal twins can be used to work out if a disease is heritable, and if so how strongly.
The genomic distribution of heterozygosity
The human
single-nucleotide polymorphism
In genetics and bioinformatics, a single-nucleotide polymorphism (SNP ; plural SNPs ) is a germline substitution of a single nucleotide at a specific position in the genome. Although certain definitions require the substitution to be present in a ...
(SNP) map has revealed large regional variations in heterozygosity, more so than can be explained on the basis of (
Poisson-distributed) random chance. In part, these variations could be explained on the basis of assessment methods, the availability of genomic sequences, and possibly the standard coalescent population genetic model. Population genetic influences could have a major influence on this variation: some loci presumably would have comparatively recent common ancestors, others might have much older genealogies, and so the regional accumulation of SNPs over time could be quite different. The local density of SNPs along chromosomes appears to cluster in accordance with a
variance to mean power law and to obey the
Tweedie compound Poisson distribution. In this model the regional variations in the SNP map would be explained by the accumulation of multiple small genomic segments through recombination, where the mean number of SNPs per segment would be
gamma distributed in proportion to a gamma distributed time to the most recent common ancestor for each segment.
History
Coalescent theory is a natural extension of the more classical
population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, s ...
concept of
neutral evolution and is an approximation to the
Fisher–Wright (or Wright–Fisher) model for large populations. It was discovered independently by several researchers in the 1980s.
Software
A large body of software exists for both simulating data sets under the coalescent process as well as inferring parameters such as population size and migration rates from genetic data.
BEASTand
BEAST 2
BEAST 2 is a cross-platform program for Bayesian analysis of molecular sequences. Using MCMC, it estimates rooted, timed phylogenies using a range of substitution and clock models, and a variety of tree priors. There is an associated tool, cal ...
–
Bayesian inference package via
MCMC with a wide range of coalescent models including the use of temporally sampled sequences.
BPP– software package for inferring phylogeny and divergence times among populations under a multispecies coalescent process.
– software for simulating genetic data under the coalescent model.
DIYABC– a user-friendly approach to
ABC for inference on population history using molecular markers.
DendroPy– a Python library for phylogenetic computing, with classes and methods for simulating pure (unconstrained) coalescent trees as well as constrained coalescent trees under the multispecies coalescent model (i.e., "gene trees in species trees").
GeneRecon– software for the fine-scale mapping of
linkage disequilibrium Linkage disequilibrium, often abbreviated to LD, is a term in population genetics referring to the association of genes, usually linked genes, in a population. It has become an important tool in medical genetics and other fields
In defining LD, it ...
mapping of disease genes using coalescent theory based on a
Bayesian MCMC framework.
genetree software for estimation of
population genetics
Population genetics is a subfield of genetics that deals with genetic differences within and among populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as Adaptation (biology), adaptation, s ...
parameters using coalescent theory and simulation (the
R package "popgen"). See als
Oxford Mathematical Genetics and Bioinformatics GroupGENOME– rapid coalescent-based whole-genome simulation
IBDSim– a computer package for the simulation of genotypic data under general isolation by distance models.
– IMa implements the same Isolation with Migration model, but does so using a new method that provides estimates of the joint posterior probability density of the model parameters. IMa also allows log likelihood ratio tests of nested demographic models. IMa is based on a method described in Hey and Nielsen (2007 PNAS 104:2785–2790). IMa is faster and better than IM (i.e. by virtue of providing access to the joint posterior density function), and it can be used for most (but not all) of the situations and options that IM can be used for.
Lamarc– software for estimation of rates of population growth, migration, and recombination.
– a program which implements coalescent algorithms for a maximum likelihood analysis (using
Importance Sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally at ...
algorithms) of genetic data with a focus on spatially structured populations.
Migrate–
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stati ...
and
Bayesian inference
Bayesian inference ( or ) is a method of statistical inference in which Bayes' theorem is used to calculate a probability of a hypothesis, given prior evidence, and update it as more information becomes available. Fundamentally, Bayesian infer ...
of migration rates under the ''n''-coalescent. The inference is implemented using
MCMC
MaCS– Markovian Coalescent Simulator – simulates genealogies spatially across chromosomes as a Markovian process. Similar to the SMC algorithm of McVean and Cardin, and supports all demographic scenarios found in Hudson's ms.
– Richard Hudson's original program for generating samples under neutral models and an extension which allows
recombination hotspots.
msms– an extended version of ms that includes selective sweeps.
msprime– a fast and scalable ms-compatible simulator, allowing demographic simulations, producing compact output files for thousands or millions of genomes.
PhyloCoalSimulations- a Julia package to simulate gene trees under the coalescent along a phylogenetic network / admixture graph. The model allows for possible correlated inheritance at reticulations, which represent introgression, gene flow or hybridization events.
Recodon and NetRecodon– software to simulate coding sequences with inter/intracodon recombination, migration, growth rate and longitudinal sampling.
CoalEvol and SGWE– software to simulate nucleotide, coding and amino acid sequences under the coalescent with demographics, recombination, population structure with migration and longitudinal sampling.
SARG– structure Ancestral Recombination Graph by Magnus Nordborg
simcoal2– software to simulate genetic data under the coalescent model with complex demography and recombination
TreesimJ– forward simulation software allowing sampling of genealogies and data sets under diverse selective and demographic models.
References
Sources
Articles
* Arenas, M. and Posada, D. (2014) Simulation of Genome-Wide Evolution under Heterogeneous Substitution Models and Complex Multispecies Coalescent Histories
''Molecular Biology and Evolution'' 31(5): 1295–1301* Arenas, M. and Posada, D. (2007) Recodon: Coalescent simulation of coding DNA sequences with recombination, migration and demography
''BMC Bioinformatics'' 8: 458* Arenas, M. and Posada, D. (2010) Coalescent simulation of intracodon recombination
''Genetics'' ''184(2)'': 429–437* Browning, S.R. (2006) Multilocus association mapping using variable-length markov chains
''American Journal of Human Genetics'' 78:903–913* Cornuet J.-M., Pudlo P., Veyssier J., Dehne-Garcia A., Gautier M., Leblois R., Marin J.-M., Estoup A. (2014) DIYABC v2.0: a software to make Approximate Bayesian Computation inferences about population history using Single Nucleotide Polymorphism, DNA sequence and microsatellite data
''Bioinformatics'' '30': 1187–1189* Degnan, JH and LA Salter. 2005. Gene tree distributions under the coalescent process. Evolution 59(1): 24–37
pdf from coaltree.net/* Donnelly, P., Tavaré, S. (1995) Coalescents and genealogical structure under neutrality. ''Annual Review of Genetics'' 29:401–421
*
* Ewing, G. and Hermisson J. (2010), MSMS: a coalescent simulation program including recombination, demographic structure and selection at a single locus
''Bioinformatics'' 26:15* Hellenthal, G., Stephens M. (2006) msHOT: modifying Hudson's ms simulator to incorporate crossover and gene conversion hotspot
''Bioinformatics'' AOP*
* Hudson RR (1983b) Properties of a neutral allele model with intragenic recombination. ''Theoretical Population Biology'' 23:183–201.
* Hudson RR (1991
Gene genealogies and the coalescent process.''Oxford Surveys in Evolutionary Biology'' 7: 1–44
* Hudson RR (2002) Generating samples under a Wright–Fisher neutral model
''Bioinformatics'' 18:337–338* Kendal WS (2003) An exponential dispersion model for the distribution of human single nucleotide polymorphisms. ''Mol Biol Evol'' 20: 579–590
* Hein, J., Schierup, M., Wiuf C. (2004) ''Gene Genealogies, Variation and Evolution: A Primer in Coalescent Theory'' Oxford University Press
* Kaplan, N.L., Darden, T., Hudson, R.R. (1988) The coalescent process in models with selection. ''Genetics'' 120:819–829
*
* Kingman, J.F.C. (2000) Origins of the coalescent 1974–1982
''Genetics'' 156:1461–1463* Leblois R., Estoup A. and Rousset F. (2009) IBDSim: a computer program to simulate genotypic data under isolation by distanc
''Molecular Ecology Resources'' 9:107–109* Liang L., Zöllner S., Abecasis G.R. (2007) GENOME: a rapid coalescent-based whole genome simulator. ''Bioinformatics'
23: 1565–1567* Mailund, T., Schierup, M.H., Pedersen, C.N.S., Mechlenborg, P. J. M., Madsen, J.N., Schauser, L. (2005) CoaSim: A Flexible Environment for Simulating Genetic Data under Coalescent Model
''BMC Bioinformatics'' 6:252* Möhle, M., Sagitov, S. (2001) A classification of coalescent processes for haploid exchangeable population models ''The Annals of Probability'' 29:1547–1562
* Morris, A. P., Whittaker, J. C., Balding, D. J. (2002) Fine-scale mapping of disease loci via shattered coalescent modeling of genealogie
''American Journal of Human Genetics'' 70:686–707*
Neuhauser, C., Krone, S.M. (1997) The genealogy of samples in models with selectio
''Genetics'' 145 519–534* Pitman, J. (1999) Coalescents with multiple collisions ''The Annals of Probability'' 27:1870–1902
*Harding, Rosalind, M. 1998. New phylogenies: an introductory look at the coalescent. pp. 15–22, in Harvey, P. H., Brown, A. J. L., Smith, J. M., Nee, S. New uses for new phylogenies. Oxford University Press ()
* Rosenberg, N.A., Nordborg, M. (2002) Genealogical Trees, Coalescent Theory and the Analysis of Genetic Polymorphisms. ''Nature Reviews Genetics'' 3:380–390
* Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestral lines ''Journal of Applied Probability'' 36:1116–1125
* Schweinsberg, J. (2000) Coalescents with simultaneous multiple collisions ''Electronic Journal of Probability'' 5:1–50
* Slatkin, M. (2001) Simulating genealogies of selected alleles in populations of variable size ''Genetic Research'' 145:519–534
* Tajima, F. (1983) Evolutionary Relationship of DNA Sequences in finite populations. ''Genetics'' 105:437–460
* Tavare S, Balding DJ, Griffiths RC & Donnelly P. 1997. Inferring coalescent times from DNA sequence data. ''Genetics'' 145: 505–518.
* The international SNP map working group. 2001. A map of human genome variation containing 1.42 million single nucleotide polymorphisms. ''Nature'' 409: 928–933.
* Zöllner S. and
Pritchard J.K. (2005) Coalescent-Based Association Mapping and Fine Mapping of Complex Trait Loc
''Genetics'' 169:1071–1092* Rousset F. and Leblois R. (2007) Likelihood and Approximate Likelihood Analyses of Genetic Structure in a Linear Habitat: Performance and Robustness to Model Mis-Specificatio
''Molecular Biology and Evolution'' 24:2730–2745
Books
* Hein, J; Schierup, M. H., and Wiuf, C. ''Gene Genealogies, Variation and Evolution – A Primer in Coalescent Theory''.
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books ...
, 2005. .
* Nordborg, M. (2001
Introduction to Coalescent Theory* Chapter 7 in Balding, D., Bishop, M., Cannings, C., editors, ''Handbook of Statistical Genetics''. Wiley
* Wakeley J. (2006) ''An Introduction to Coalescent Theory'' Roberts & Co
Accompanying website with sample chapters* Rice SH. (2004). ''Evolutionary Theory: Mathematical and Conceptual Foundations''. Sinauer Associates: Sunderland, MA. See esp. ch. 3 for detailed derivations.
* Berestycki N. "Recent progress in coalescent theory" 2009 ENSAIOS Matematicos vol.16
* Bertoin J. "Random Fragmentation and Coagulation Processes"., 2006. Cambridge Studies in Advanced Mathematics, 102.
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...
, Cambridge, 2006. ;
* Pitman J. "Combinatorial stochastic processes" Springer (2003)
External links
EvoMath 3: Genetic Drift and Coalescence, Briefly— overview, with probability equations for genetic drift, and simulation graphs
{{Population genetics
Population genetics
Statistical genetics