General introduction and Jukes and Cantor model

Distance methods fit a tree to a matrix of N · (N-1)/2 pairwise evolutionary distances, N being the number of sequences considered.  For every two sequences, the distance is a single value estimated from the dissimilarity, i.e. the fraction of positions in which both sequences differ.  Due to the fact that some of these sequence differences are the result of multiple events, this dissimilarity is actually an underestimation of the true evolutionary distance.  Therefore, one usually tries to estimate the number of substitutions that have actually taken place by applying a specific evolutionary model that makes assumptions about the nature of evolutionary changes.  However, since one does not have an exact historical record of events that took place in the evolution of sequences, correct estimation of the evolutionary distance is not self-evident.
One of the first substitution models used in the estimation of evolutionary distances is the one of Jukes and Cantor (1969).  This model starts from the assumptions that all substitutions are independent, that all sequence positions are equally subject to change, that substitutions occur randomly among the four types of nucleotides, and that no insertions or deletions have occurred.  Based on these pre-assumptions, the authors derived an equation for estimating evolutionary distances from observed dissimilarity:

 ,

where f_AB is the dissimilarity (fraction of observed differences) between sequences A and B, and d_AB is the estimated evolutionary distance (fraction of expected substitutions) between sequences A and B. The relationship between dissimilarity and evolutionary distance according to the Jukes and Cantor substitution model is shown graphically in Fig. 1.

Several other equations for the estimation of evolutionary distances have been proposed.  For example, Kimura (1980) has provided a method for inferring  evolutionary distances based on a model of evolution in which transitions and transversions may occur at different rates.  Other equations are based on substitution models in which the four different nucleotides are not used in equal proportions (e.g. Tajima and Nei, 1984), or where a bias in the direction of change is accounted for (e.g. Tamura and Nei, 1993; Zharkikh, 1994).

An important drawback of most of these models is that they do not consider differences in substitution rate among the sites of a molecule (see further).

The substitution model of Jukes and Cantor, also called the one-parameter model, is the simplest one available for estimating the number of nucleotide substitutions per site and is probably still the most used one.

When the fraction of differences between two sequences exceeds 0.75, the distance cannot be computed.  If f > 0.75, TREECON crashes and gives a log-domain error.  When this happens, check your alignment and/or use the more conserved parts of the alignment to construct a tree with.

Kimura - two parameter model

Kimura (1980) provided a method for inferring evolutionary distance in which transitions and transversions are treated separately:

where P is the fraction of sequence positions differing by a transition and Q is the fraction of sequence positions differing by a transversion.

Tajima & Nei

In the general correction of Tajima and Nei (1984), the evolutionary distance is estimated by:

where

 

and f_i is the frequency of the i-th type of nucleotide belonging to the set of possible nucleotide types N (= A, G, C, U or T) in the sequences being compared.  This equation holds for the model of nucleotide substitutions with equal substitution rates between different nucleotides and does NOT take into account unequal rates of substitution among different nucleotide pairs (Tajima and Nei, 1984).  In TREECON, the computed base composition is the average for all the sequences analyzed (as suggested in Swofford et al., 1996).  If the frequencies are 0.25 for all four nucleotides, this equation equals the one of Jukes and Cantor.

Gamma distances

All the previous distance measures start from the assumption that the rate of nucleotide substitution is the same for all nucleotide sites.  However, in real sequences, this assumption rarely holds (see further). Different studies suggest that the rate of nucleotide substitution varies approximately according to the gamma distribution (see Uzzell and Corbin, 1971; Jin and Nei, 1990; Nei, 1991).  This gamma distribution is specified by a parameter a which is the square of the inverse of the coefficient of variation of substitution rate (Nei, 1991).

For the Jukes and Cantor's one parameter model, the distance is computed as follows (Jin and Nei, 1990):

for a=1:

for a=2:

for a=1/2:

A more general equation has been given by Rzhetsky and Nei (1994):

For the Kimura's two parameter evolution model, the distance is computed as follows (Jin and Nei, 1990), where P is the fraction of sequence positions differing by a transition and Q is the fraction of sequence positions differing by a transversion:

for a=1:

for a=2:

for a=1/2:

 

A more general equation has been given by Nei (1991):

Galtier & Gouy

The algorithm of Galtier and Gouy (1995) was developed for estimating evolutionary distances without assuming homogeneity or stationarity of the evolutionary process.  This distance estimate should be useful for phylogenetic analyses when compositional biases are observed in the data.  Two factors are taken into account: the transition/transversion ratio, and the G+C content.

The transition/transversion ratio is assumed to be the same in all lineages.  It is estimated once, from the whole dataset.  In the substitution model of Galtier and Gouy, the ratio between the sums of transition and of transversion rates equals a/2.  This ratio is estimated for each sequence pair (A, B):

where P(A,B) is the observed proportion of sites in sequences A and B showing a transition difference, and Q(A,B) is the observed proportion of sites showing a transversion difference.

The estimate of parameter a is given by the mean of a(A,B) values for all sequence pairs.  This estimate is used for all pairwise distance computations.  In TREECON the transition/transversion ratio is estimated before the actual computation of the evolutionary distances starts.  Since this value is based on all pairwise comparisons, this in fact doubles the time needed for the estimation of distances between sequences.  In the bootstrap analysis, the transition/transversion ratio is only once computed, based on the actual set of sequences.

The evolutionary distance is estimated as follows:

with

,

and

where theta1 Is the G+C content of sequence 1, and theta2 is the G+C content of sequence 2.

Transversion analysis

Sometimes, it can be interesting to estimate the evolutionary distance on the basis of transversions only (see e.g. Woese et al., 1991; Van de Peer et al., 1996b).  The evolutionary distance is then estimated by (Tajima and Nei, 1984; Swofford et al., 1996):

where Q is the fraction of transversions and

and  f_A + f_G being the fraction of purines, and f_C + f_U or f_T being the fraction of pyrimidines, computed over the complete alignment.

It is also possible not to correct for superimposed mutations (the so-called p-distance)

 

DISTANCE ESTIMATION FOR AMINO ACID SEQUENCES

Overall, the computation of evolutionary distances for amino acid sequences is similar to the computation of distances for nucleic acid sequences.

The distance between two amino acid sequences is computed starting from the assumption that the rate of amino acid substitution at each site follows the poisson distribution (e.g. Zuckerkandl and Pauling, 1965; Dickerson, 1971):

where f_AB is the fraction of different amino acids between two sequences (dissimilarity).

Kimura

This formula (Kimura, 1983) should be a good approximation to the Dayhoff model  (Hasegawa and Fujiwara, 1993):

Tajima & Nei

The evolutionary distance is calculated as:

where b=0.95 (i.e. 19/20) for amino acid replacements (Tajima and Nei, 1984).

Gamma distances

When one starts from the assumption that the rate of amino acid substitutions varies from site to site and follows the gamma distribution, the evolutionary distance is computed as (Kumar et al., 1993; Ota and Nei, 1994):

GENETIC DISTANCE ESTIMATION FOR PATTERN ANALYSIS

At the moment, two methods to compute the genetic distance for pattern analyses such as RFLP, RAPD, and AFLP are implemented.

The genetic distance is computed as follows:

 ,

where Nxy is the number of fragments (bands) shared in lines x and y, and Nx is the number of fragments in line x, and Ny is the number of fragments in line y.

Method 2 (Link et al., 1995)

The genetic distance is computed as follows:

,

where Nx is the number of bands in line x and not in line y, Ny is the number of bands in line y and not in line x, and Nxy is the number of bands shared in lines x and y.

INSERTIONS AND DELETIONS

In TREECON, the user can choose whether to take insertions and deletions (indels) into account or not.  If they are, they are counted separately since the correction for superimposed events does not hold for insertions and deletions.  For example, in TREECON, the evolutionary distance according to the Jukes and Cantor model is then computed as (Van de Peer et al., 1990):

where I is the number of identical nucleotides, f_U is the number of positions showing a substitution and G is the number of gaps in one sequence with respect to the other.  T is the sum of I, f_U and G.  The first term of the equation accounts for substitutions and comprises the Jukes and Cantor correction factor for multiple mutations per site (see above).  The second term accounts for deletions and insertions.  A row of adjacent gaps is treated one gap, regardless of its length.  Ambiguities (not A, G, C, T or U) are not taken into account.  Comparison of the imaginary sequences:

A G C U G G - - - G C N U A
- - C U A G A A A G A C U A

would give the values I=6, f_U=2, G=2, and T=10.

HOW TO CHECK DISTANCE ESTIMATES

Distance estimates are saved in the file ‘mat.out’.  The format of saving may seem a bit awkward but this is for compatibility reasons with other (older) software.

If one wants to know the evolutionary distance between two sequences, this is how to interpret the distance matrix (5 sequences were compared):

For example, the evolutionary distance (number of substitutions per nucleotide) between sequences 2 and 4 equals 0.25016 (the values in the distance matrix are multiplied by 100).

In future, I will make it possible to display the distance estimates in a more user-friendly way.