Rate = 1				Rate = 0
	A1	G1	C1	T1	A0	G0	C0	T0
A1	-	α	β	γ	δ	0	0	0
G1	α	-	γ	β	0	δ	0	0
C1	β	γ	-	α	0	0	δ	0
T1	γ	β	α	-	0	0	0	δ
A0	κδ	0	0	0	-	0	0	0
G0	0	κδ	0	0	0	-	0	0
C0	0	0	κδ	0	0	0	-	0
T0	0	0	0	κδ	0	0	0	-

The method of covarions, or concomitantly variable codons, is a technique in computational phylogenetics

Computational phylogenetics

Computational phylogenetics is the application of computational algorithms, methods and programs to phylogenetic analyses. The goal is to assemble a phylogenetic tree representing a hypothesis about the evolutionary ancestry of a set of genes, species, or other taxa...

 that allows the hypothesized rate of molecular evolution

Evolution

Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...

 at individual codons in a set of nucleotide sequences to vary in an autocorrelated

Autocorrelation

Autocorrelation is the cross-correlation of a signal with itself. Informally, it is the similarity between observations as a function of the time separation between them...

 manner. Under the covarion model, the rates of evolution on different branches of a hypothesized phylogenetic tree

Phylogenetic tree

A phylogenetic tree or evolutionary tree is a branching diagram or "tree" showing the inferred evolutionary relationships among various biological species or other entities based upon similarities and differences in their physical and/or genetic characteristics...

 vary in an autocorrelated way, and the rates of evolution at different codon sites in an aligned

Sequence alignment

In bioinformatics, a sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. Aligned sequences of nucleotide or amino acid residues are...

 set of DNA

DNA

Deoxyribonucleic acid is a nucleic acid that contains the genetic instructions used in the development and functioning of all known living organisms . The DNA segments that carry this genetic information are called genes, but other DNA sequences have structural purposes, or are involved in...

 or RNA

RNA

Ribonucleic acid , or RNA, is one of the three major macromolecules that are essential for all known forms of life....

 sequences vary in a separate but autocorrelated manner. This provides additional and more realistic constraints on evolutionary rates versus the simpler technique of allowing the rate of evolution on each branch to be selected randomly from a suitable probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 such as the gamma distribution. Covarions is a concrete form of the more general concept of heterotachy

Heterotachy

Heterotachy refers to shifts in site-specific evolutionary rates over time. In the field of molecular evolution, the principle of heterotachy states that the substitution rate of sites in a gene can change through time. It has been proposed that the positions that show switches in substitution rate...

.

Developing a computational algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

 suitable for identifying sites with high evolutionary rates from a static dataset is a challenge due to the constraints of autocorrelation. The original statement of the method used a rough stochastic

Stochastic

Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...

 model of the evolutionary process designed to identify transiently high-variability codon sites. Abandoning the requirement that rates be autocorrelated on a given DNA or RNA molecule allows extension of substitution matrix

Substitution matrix

In bioinformatics and evolutionary biology, a substitution matrix describes the rate at which one character in a sequence changes to other character states over time...

 methods to the covarion model.

The matrix at right represents a covarion-based modification to the three-state Kimura substitution model

Substitution model

In biology, a substitution model describes the process from which a sequence of characters changes into another set of traits. For example, in cladistics, each position in the sequence might correspond to a property of a species which can either be present or absent. The alphabet could then consist...

, where the vertical axis represents the original state and the horizontal axis the destination state. The two rates, 0 and 1, define a pair of mutation states; transitions can occur between state 0 and state 1 at any time, but nucleotides can only mutate in state 1. That is, the rate of mutation in state 0 is 0. Here α and β are the standard Kimura parameters for transition

Transition (genetics)

In genetics, a transition is a point mutation that changes a purine nucleotide to another purine or a pyrimidine nucleotide to another pyrimidine . Approximately two out of three single nucleotide polymorphisms are transitions....

 and transversion mutations, κδ is the rate of transition between a site being invariant (state 0) and variable (state 1), and δ is the rate of transition between a site being variable (state 1) and invariant (state 0). Because nucleotide sequences do not themselves reflect the difference between a 0 or 1 state, an observation of a given nucleotide is treated as ambiguous; that is, if a given site contains a C nucleotide, it is ambiguous between C0 and C1 states.