The evolutionary distance between two biological sequences is generally based on the assumption of a first-order Markovian process of amino acid evolution. This implies two biological assumptions, common to all standard models of evolution: no memory and position-independence. The substitutional processes are described in the form of substitution matrices, defining mutation probabilities from each character to every other character for a given evolutionary distance. These matrices are either parametrical models of sequence evolution or empirically based substitution matrices. Parametrical models are often employed for nucleotide substitution (e.g. Jukes-Cantor 6 or Hasegawa-Kishino-Yano 7), while empirical matrices (based on counted substitutions of large sets of sequence alignments) are widely used for peptide replacements in proteins. Pioneered by Dayhoff in the 1970s 8, these models have been improved with more sequence data becoming available in the 1990s (e.g. the updated Dayhoff matrices by Gonnet-Cohen-Benner 9 or Jones-Taylor-Thornton (JTT) 10). Codon substitutions have been described by parametrical (e.g. 11) as well as empirical (e.g. 12) matrices.

Because of the additivity of distances computed under the Markovian model of sequence evolution. substitution matrices for a wide range of evolutionary distances can be derived from a single substitution matrix M(d₀) through the equation M(d₀)^x = M(xd₀), which is a special form of the Chapman-Kolmogorov equation for Markov chains. It is common and computationally more efficient to formulate this process in terms of a rate matrix Q from which the probability matrices for distance d are derived as M(d) = e^dQ. We normally measure d in PAM units 8, which completely defines Q.

Evolutionary distances are best estimated by maximum likelihood (ML). In case of a pair of sequences, the ML estimation is well known and practical (see Methods part). When more sequences are under consideration, the complexity of distance estimation by ML increases very steeply, mainly because it requires a multiple sequence alignment (MSA) and the inference of the phylogenetic tree topology, two difficult procedures for which the optimal solution can currently only be computed in exponential time with respect to the number of sequences. A common strategy for tackling this problem is to work on the basis of pairs, such as in distance tree methods. In this article, we focus on the specific problem of estimating, in a triplet of homologs X,Y,Z (Fig. 1). the difference Δ between two distances d_XY and d_XZ. In such case, the multidimensional ML approach over the triplet is still practical. We call the estimator of Δ obtained by this method Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet. Alternatively, Δ can be estimated by a simple algebraic relation over pairwise distances over X, Y, Z estimated individually. We call this alternative estimator Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise. Details about the computation of Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet and Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise are provided in the Methods section.

In terms of computational complexity, the two estimators differ significantly. Given m sequences of length n, Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet requires the separate treatment of each O(m³) triplet, and considering that an optimal 3-way alignment by dynamic programming (DP) is O(n³), the time complexity is O(m³n³). In contrast, all Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise can be computed on the basis of O(m²) pairs of sequences aligned by DP in O(n²), yielding a time complexity of O(n²m²). Typically, whenever an analysis involves more than a few thousand proteins, millions of triplets have to be considered and Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise is the only practical approach of the two. In terms of accuracy, both estimators are asymptotically unbiased: in the case of Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet, it is a property of the ML estimator, while in the case of Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise, it is the consequence of the linearity of the expected value (see Methods). We compared the two estimators by simulation over a large number of triplets (length: 300 AA), generated randomly according to the PAM model of evolution with different distances d_OX, d_OY, d_OZ (Fig. 1). In each experiment, both estimators were converging toward the correct value for the difference, which confirms that the asymptotic behavior is a reasonable assumption for protein sequences of typical length. In terms of statistical power; surprisingly, the observed variance of the estimates obtained by Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise was on average less than 1% larger than the observed variance of the ML estimator over the triplet, suggesting that Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise, although much faster to compute, is on average almost as accurate as Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet.

The variance of Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet can be computed exactly (see Methods section). But there is no direct estimator of the variance of Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise, since it results from an algebraic relation over pairwise distances estimated individually, whose covariances are therefore unknown. There are indirect ways of estimating that variance, through the sampling distribution when doing simulation such as the one mentioned above, or bootstrapping when handling real data. However, such procedures are very time consuming. To overcome this problem, we devised a numerical approximation of σ²(Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise) as function of the pairwise distance estimates.

In essence, the numerical approximation described here was obtained through regression over a large number of samples. We settled for this approach after discovering that the analytical solution to this problem, even when using a simpler model of evolution (all amino-acid mutations with equal probability). requires solving a polynomial of degree 23. The details of this investigation are reported in the Appendix. In view of this inherent complexity, the regression cannot be exact, but it turns out to be a surprisingly precise numerical approximation for comparisons that involve proteins that have an evolutionary distance smaller than 250 PAM units, which corresponds to percentage sequence identity greater or equal to 19.68%. We generated random triplets in the following way: a random-length (uniform 100..500) sequence was chosen as the origin O. Three random PAM distances (uniform 1..125) were selected for d_OX, d_OY and d_OZ. The sequence O was mutated according to these distances to obtain X,Y and Z, our triplet. We generated about 30,000 triplets for three types of scoring matrix: updated Dayhoff matrices 9, DNA for coding genes and JTT 10. The DNA scoring matrices were computed from a very large set of entire coding gene alignments from mammals. It is used in the OMA project 4 to align entire coding genes and is based on a 4-symbol alphabet. For each triplet, we computed pairwise distance estimates and their variances as input for the approximation. Given that Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise is almost as powerful as Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet, we computed and used σ² (Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet) as reference value for σ²(Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise).

We examined a large number of regressions and one approximation stood out of the rest due to its efficiency, low average error and other minor indications. Table 1 shows the coefficients of the approximation for the three types of scoring matrices.

For example, the approximation for DNA variances is

σ ˜ 2 ( Δ ^ p a i r w i s e ) = d ^ Y Z 1.3182 σ 2 ( d ^ Y Z ) 0.3026 . ( σ 2 ( d ^ X Y ) + σ 2 ( d ^ X Z ) ) 1.0933 ( σ 2 ( d ^ X Y ) σ 2 ( d ^ X Z ) ) 0.1181 ( d ^ X Y + d ^ X Z ) 1.2449 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqabeGabaaabaacciGaf83WdmNbaGaadaahaaWcbeqaaiabikdaYaaakiabcIcaOiqbfs5aezaajaWaaSbaaSqaaiabdchaWjabdggaHjabdMgaPjabdkhaYjabdEha3jabdMgaPjabdohaZjabdwgaLbqabaGccqGGPaqkcqGH9aqpdaWcaaqaaiqbdsgaKzaajaWaa0baaSqaaiabdMfazjabdQfaAbqaaiabigdaXiabc6caUiabiodaZiabigdaXiabiIda4iabikdaYaaaaOqaaiab=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@9B87@

Readers familiar with numerical analysis will find an analogy between the approximation presented here and standard approximations for transcendental functions. For example, it is customary to approximate exp(x) through a quotient of polynomials p(x)/q(x), for some limited range of x.

The relative error is in all the three cases less than 10%. Furthermore, since we normally use the square root of the variance, the relative error is in such cases half of the indicated. The last column indicates the dimension of the approximation which should be 2 in perfect conditions, and is indeed quite close.

The fact that very different matrices have very similar coefficients, the low error and the almost correct dimensionality reassures us of the quality of the approximation.

To test the accuracy/applicability of the approximation, as well as the other two methods to obtain the variance, we compared the 95 and 99% confidence level obtained using the appropriate number of standard deviations to the actual percentage of correct decisions obtained in a simulation over 400, 000 protein triplets generated as described above. The results are shown in Table 2.

As expected, the ML estimator over the entire triplet (first row) yields a precise variance estimate. On the other hand, we see that assuming independence for the estimation of the variance (last row) leads to very inaccurate confidence intervals. Estimating the variance of Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise by bootstrapping (10,000 re-samples) gives good confidence intervals, but the procedure is even more computationally intensive than Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet, and therefore of little practical use in the present context. Using σ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFdpWCgaacaaaa@2E85@²(Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise) in conjunction with the variance of the ML estimator works remarkably well (third and fourth row). And surprisingly, applying the numerical approximation (fourth row) happened to give slightly more accurate results than the exact triplet variance (third row).

Finally, we compared the different estimators on real biological sequences, using data obtained from the OMA orthologs project 4, Triplets of orthologous sequences from various eukaryotes were randomly selected and aligned using the multiple sequence alignment package from Darwin 13. All positions containing gaps were excluded, and variances were then estimated on the ungapped triplets using the various estimators (Fig. 2). The variance estimates from the approximation formula deviate very little from the results obtained by the two more expensive methods – for simulated as well as empirical alignments. Additionally, the plots illustrate the high correspondence between the results from the ML estimation and the bootstrapping, and show that the estimator based on an assumption of independence often yields overestimates of the variance. The difference between simulated and empirical data probably arises from the limitations of the Markovian model of evolution. Worth noticing is that the agreement of our estimator with bootstrapping is comparable to the one of the ML variance estimator: this implies that our approximation has a similar robustness when applied to real data.

In the following, we provide two examples of applications that benefit from the increase in statistical power of the estimator Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise enabled by the approximation: detection of asymmetric evolution and identification of the closest relative in a set of homologs. Furthermore, in 14, we show how our result can be used in the context of paralogy detection.

We first define three indicator functions that will be used in these comparisons. They decide whether the pair of proteins X, Y is significantly closer than X, Z at the confidence level expressed by the number of standard deviations k. The first and second ones both use the estimator Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise, but the first definition uses as variance of the estimate the upper bound that is obtained by assuming independence of d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@_XY and d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@_XZ (see Methods), whereas the second use the approximation σ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFdpWCgaacaaaa@2E85@²(Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_pairwise) of the variance. The third indicator function uses the estimator Δ^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuqHuoargaqcaaaa@2E22@_triplet.

c l o s e r i n d ( X , Y , Z , k ) = { true if  Δ ^ p a i r w i s e < − k ⋅ σ i n d ( Δ ^ p a i r w i s e ) false otherwise MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@87DB@

c l o s e r a p p ( X , Y , Z , k ) = { true if  Δ ^ p a i r w i s e < − k ⋅ σ ˜ ( Δ ^ p a i r w i s e ) false otherwise MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@83AF@

c l o s e r t r i p l e t ( X , Y , Z , k ) = { true if  Δ ^ t r i p l e t < − k ⋅ σ ( Δ ^ t r i p l e t ) false otherwise MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@86B8@

The likelihood of an alignment A at an evolutionary distance d is defined 252627 as

L ( A | d ) = ∏ [ x , y ] ∈ A f ( x ) M x , y ( d ) = ∏ [ x , y ] ∈ A f ( x ) [ e d Q ] x , y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@660D@

with x and y being aligned characters (e.g. amino acids, bases, but no deletions), and f(x) the background frequency of the character x. Maximizing L(A | d) in terms of d gives the ML estimator d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@ of the evolutionary distance. This is usually done numerically using the Newton-Raphson method. The variance of the ML estimator d^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaqcaaaa@2E0D@ can be computed from the second derivative of the log-likelihood:

σ 2 ( d ^ ) = − ( ∂ 2 L ( A | d ^ ) ∂ d 2 ) − 1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaahaaWcbeqaaiabikdaYaaakiabcIcaOiqbdsgaKzaajaGaeiykaKIaeyypa0JaeyOeI0YaaeWaaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaeGOmaidaaOGaemitaWKaeiikaGIaemyqaeKaeiiFaWNafmizaqMbaKaacqGGPaqkaeaacqGHciITcqWGKbazdaahaaWcbeqaaiabikdaYaaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiabigdaXaaaaaa@4576@

Notice that the variance is obtained for free as it is already computed in Newton's iteration.

In the following, we show that the analytical solution of the maximum-likelihood estimator for the distances of a triplet is very complex, even for a simplified model of mutation. The k-state model 29 is an idealized situation where each position has k possible states and the transition probabilities are all identical and only depend on the time t. For k = 4 this is equivalent to the Jukes-Cantor model 6. Whatever is the initial state, the probability of a mutation after time t is given by

p ( t ) = k − 1 k ( 1 − r t ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGWbaCcqGGOaakcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiabdUgaRjabgkHiTiabigdaXaqaaiabdUgaRbaacqGGOaakcqaIXaqmcqGHsislcqWGYbGCdaahaaWcbeqaaiabdsha0baakiabcMcaPaaa@3D8D@

where r is

r = 1 − k 100 ( k − 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCcqGH9aqpcqaIXaqmcqGHsisldaWcaaqaaiabdUgaRbqaaiabigdaXiabicdaWiabicdaWiabcIcaOiabdUgaRjabgkHiTiabigdaXiabcMcaPaaaaaa@3A25@

so that t is measured in PAM units. (Measuring in PAM units is proportional to any other measure, and it means that at t = 1 one percent of the characters are changed, i.e. p(1) = 1/100.) and that all transitions are equally likely, and only depend on the PAM distance. Under this model, the log-likelihood can be expressed in terms of the counts of matches/mismatches of the triplet (X, Y, Z), i.e. N_xxx is the number of positions where all the characters are identical, N_xxz is the number of positions where X and Y coincide but Z differs, etc.

l ( A | t ) = N x x x log ⁡ ( P x x x ) + N x x z log ⁡ ( P x x z ) + N x y x log ⁡ ( P x y x ) + N x y y log ⁡ ( P x y y ) + N x y z log ⁡ ( P x y z ) P x x x = ( 1 − p x ) ( 1 − p y ) ( 1 − p z ) + p x p y p z ( k − 1 ) 2 P x x z = ( 1 − p x ) ( 1 − p y ) p z + p x p y ( 1 − p z ) k − 1 + ( k − 2 ) p x p y p z ( k − 1 ) 2 P x y x = ( 1 − p x ) p y ( 1 − p z ) + p x ( 1 − p y ) p z k − 1 + ( k − 2 ) p x p y p z ( k − 1 ) 2 P x y y = p x ( 1 − p y ) ( 1 − p z ) + ( 1 − p x ) p y p z k − 1 + ( k − 2 ) p x p y p z ( k − 1 ) 2 P x y z = k − 2 k − 1 ( ( 1 − p x ) p y p z + p x ( 1 − p y ) p z + p x p y ( 1 − p z ) + ( k − 3 ) p x p y p z k − 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@BCE8@

where p_x is the probability of mutating from the origin to X and similarly for p_y and p_z. Taking partial derivatives of the likelihood with respect to p_x, p_y and p_z gives a system of 3 rational polynomial equations (all the logarithms disappear) in 3 unknowns and 6 parameters. Such a system of equations has a solution that will be an algebraic function of the parameters (a root of a polynomial, where the coefficients of the polynomial involve the parameters). Despite its simple appearance, this system of equations is beyond the capabilities of current computer algebra systems to resolve. And this is not a complete surprise, as the algebraic numbers/functions involved are at least of degree 23. The special case where two of the branches have the same length, has been solved exactly in 30, they find that their solution is an algebraic function of degree 11. This unfortunately is not applicable as we are interested in the cases where the branches away from the origin are of different lengths.

We have computed the exact solution for concrete values of the parameters, in particular N_xxx = 10, N_xxz = 5, N_xyx = 4, N_xyy = 3, N_xyz = 2, k = 3 using Maple and the value of p_x is a root of the irreducible polynomial

-6582435840000 + 189590785228800 z - 2438333515038720 z² + ...

... + 10304020514917800 z²¹ - 1635488137841976 z²² + 99990709180560 z²³

This means that the general solution will be an algebraic function of degree 23 or higher, it cannot be lower. If an instantiation of the polynomial with values gives this irreducible polynomial, then the general polynomial must be irreducible of degree 23 or higher (some terms could have simplified in the instantiation). This makes the usefulness of an exact solution inexistent. it is more difficult to solve the polynomial and select the right root than to maximize the likelihood and/or solve the system of equations by numerical methods.