Revised rates and times of mitochondrial protein sequences from placental mammals

Mitochondrial protein sequences are used widely in phylogenetic studies and have been particularly popular in studying placental mammals. A desirable feature of these data is relatively long sequences, good taxon sampling and very little missing data. Following alignment, we retained 3660 amino acid sites present in all of 62 placental mammals plus seven outgroup taxa (Table S1 gives the accession numbers of protein sequences). In contrast, large nuclear protein datasets of mammals with diverse taxon sampling [e.g.3], [4] show a large proportion of missing data. Such missing heterogeneous data may lead to complex systematic errors in distance estimation [34]. This is particularly relevant since MVS analyses work best with low stochastic noise (hence relatively long sequences), diverse taxon sampling with a considerable proportion of taxa showing minimal convergent evolution, and minimal bias of the input distances from sources such as missing data.

We adopted a standard two-step procedure to estimate divergence times. The first step is to estimate the phylogenetic tree with unconstrained branch lengths in units of expected numbers of substitutions per site. Given the problems with convergent evolution in mitochondrial data [4], [5], [22], [25], we used the MVS procedure to correct the input distances for convergent evolution. A variant of the core set approach [27] to MVS was used in this instance (Materials and Methods). Fixing the topology to the tree obtained by MVS, which is very similar to trees obtained by previous authors using varied data sets [2]–[5], branch lengths were reestimated using maximum likelihood (ML) in the program PAML [35]. We used the JTT [36]+Gamma model with α = 0.5 for these analyses. The second step uses the tree and its branch lengths, some fossil constrained nodes, and a penalty or cost function to dampen rate changes, to infer divergence times of all nodes and evolutionary rates along all branches.

Three cost functions, F_ADD, F_LOG, and F_IR, were applied to the MVS and ML trees. These functions penalize the fluctuation of rates, and do this on either a linear scale (here called the ADD function), on the log rate (LOG), or on the inverse rates (IR), respectively (see equations (1), (2), and (4) in Materials and Methods). The tree estimated by MVS and F_ADD is denoted MVS-F_ADD and so on (while ML-F_ADD indicates use of ML branch lengths).

To calibrate these trees, we used eight fossil constraints, all taken from previous studies (see Figure 1 caption) [4], [15]. The divergence times were estimated by then minimizing each cost function subject to these constraints (see Materials and Methods). The CI was calculated using a likelihood interpretation (see Materials and Methods) of the cost function residual analogous to the method used in Multidivtime [31]. This captures the error caused by deviations of the tree's branch lengths away from their expected values and includes the unpredictability of rate changes, which might mimic Brownian motion, for example. It will also incorporate variability arising from ancestral polymorphism. Polymorphism will cause fluctuations of the edge lengths of the tree when the analysis has multiple nodes constrained by fossil or other data [c.f. 13]. Generally, the time at the most recent common ancestor of sequences from different species is older than the time at speciation. The extent of this difference varies among internal nodes due to both the stochastic nature of the coalescent and factors governing its expected magnitude, such as population size. The cost function does not take account of this bias, but the CI does include the variance arising from ancestral polymorphism at internal nodes.

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Figure 1. Molecular divergence times and evolutionary rates of placental mammals.

A and B show the MVS-F_IR and ML-F_LOG trees, respectively. The numbers 1–61 denote the ancestral nodes. The red numbers 1–8 indicate the internal nodes with fossil constraints which are as follows: 1, 49–61; 2, 52–58; 3, 45–63; 4, 43–60; 5, <63; 6, >12; 7, 36–55; 8, 54–65 (all in mya) [see 4], [15]. The colors of the branches denote inferred evolutionary rates (in units of ×10⁻⁹/site per year) as follows: black, <0.2; dark blue, 0.2–0.3; light blue, 0.3–0.4; green, 0.4–0.5; brown, 0.5–0.6; yellow, 0.6–0.7; and red, >0.7.

https://doi.org/10.1371/journal.pone.0000384.g001

The MVS-F_IR analysis (Figure 1A) estimated the age of the root at 84.2 mya with a 95% confidence interval (CI) of 80.7–88.4 mya (Table 1). This time is much more recent than the result of the ML-F_L analysis (Figure 1B), i.e., 122.2 (95% CI, 112.2–144.7) mya. Note that, ML-F_L is giving results consistent with that reported in earlier studies [1]–[5], [12]–[15], indicating that the younger root for placentals returned using the function F_IR (see below) is due to the method and not the data. The difference between the F_LOG and the F'_LOG function in table 1 is explained in Materials and Methods. It is the F_LOG method that most closely approximates the Brownian motion assumed by Multidivtime.

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Table 1. The age of the root of placental mammals

https://doi.org/10.1371/journal.pone.0000384.t001

Figure 1 shows the estimated ancestral rates across the tree, while Figure 2 represents the inferred evolutionary rates going from the root to a terminal. The single instance of root to fin whale is shown in figure 2, while figure S1 traces the evolutionary rates inferred by different methods along seven representative lineages. The MVS-F_IR tree identified an abrupt acceleration of evolutionary rate near the common ancestor of both Supraprimates and Laurasiatheria, then a very strong acceleration in just the ancestral lineage of Laurasiatheria (Figures 1A and 2A). This acceleration was followed closely by a strong deceleration along nearly all lineages of Laurasiatheria. In contrast, almost all lineages of Afrotheria and Xenarthra have retained rates close to that of the root. There are also sporadic later accelerations, for example that among hedgehogs and Moon rat (Figure 1A).

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Figure 2. Evolutionary rate changes from the root to the fin whale.

Figures A and B, respectively, trace the estimated rates along edges on analyses using MVS and ML branch lengths. The root time of of the F_ADD analysis in Figure A and B was set at a large value (400 mya) because the numerical calculation continues towards an infinite root time.

https://doi.org/10.1371/journal.pone.0000384.g002

The cost function F_IR detected an acceleration–deceleration pattern near the base of Laurasiatheria using both the MVS and ML branch lengths (the red lines of Figure 2), whereas the function F_LOG showed a far more flat prediction of generally lower evolutionary rates that led to older root times (the green lines of Figure 2). Even more extreme, the cost function F_ADD inferred gradually decreasing rates in all deep branches of the MVS and ML trees (the blue lines of Figure 2), and the root time tended to infinity. Table 1 summarizes the root time values estimated by various models (all the node divergence times from these models are listed in Table S2).

Table 1 also compares the inferred times with those obtained from r8s, a popular program for estimating absolute rates [17], [18]. This program has two options of for its cost functions, F_ADD and F′_LOG (equation (3) in Materials and Methods). It also takes account of stochastic noise caused by the finite length of the sequences used to derive the branch lengths using a penalized-likelihood approach. The fitting to the estimated branch lengths is expressed by the log likelihood, while the weight for the cost function is estimated using cross validation. In our mitochondrial dataset, with its long sequences, the root time values changed little due to the cross validation effect (the reason is explained in Materials and Methods). Further, root divergence time estimates returned by r8s do not show confidence intervals in table 1, since r8s only assesses the sampling variance of node times due to stochastic errors in branch lengths due to finite sequence length (it does this via the bootstrap) and does not include the effect of stochastic variation of evolutionary rate. This source of error remains as sequence lengths go to infinity, and will be dominant with long sequences.

Agreement with fossil data

Figure 3 represents the chronological distribution of the internal node density for the MVS-F_IR tree (Figure 1A) compared with the rate of new species appearing in the well-studied North American fossil record. To avoid potential bias of divergence times in the molecular tree due to species sampling, the scaling constant was determined by a least-squares fit to the fossil data in the period 85–50 mya.

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Figure 3. Comparing molecular and fossil records of placental diversification.

The blue squares show the rate of the appearance of new species based on the fossil record [adapted from 3]. At present, such quantitative data are limited to the well-studied North American record. The red circles represent the chronological distribution of node density (or splits) on the MVS-F_IR tree (Figure 1A). The green triangles represent split frequency on the ML-F_LOG tree (Figure 1B). The node density is given by the number of nodes in an 8 myr sliding window. A scaling constant for the molecular frequencies was calculated via a least-squares fit to the fossil data in the period 85–50 mya.

https://doi.org/10.1371/journal.pone.0000384.g003

The MVS-F_IR analysis reconciles molecular with fossil data in two ways. First, the chronological distribution of the internal node density is clearly largely consistent with the rate of appearance of novel fossil species. This is despite the fact that the paleontological data assessed are limited to the well-studied North American record [10]. Both the MVS-F_IR tree and the fossil record suggest accelerating taxonomic diversity near the K–T boundary, rather than a longer slower buildup [2]–[5], [12]–[15]. Use of the less robust F_LOG criterion seems to suggest a prolonged increase in diversity that agrees far less with the quantitative fossil record. This congruence with fossils does not automatically show that the combination of MVS and F_IR is the best way to analyze this data, but it does reframe the discussion of the relationship between the fossil and molecular times into one where the molecular dates are being looked at far more critically.

Second, and more indicatively, the cost function F_IR resolves incongruence among the fossil constraints/inferred divergence times in different parts of the molecular tree. The best fossil constraints in Laurasiatheria suggest much older times than constraints in other parts of the tree [4], [37]. The MVS-F_IR, MVS-F_LOG, and MVS-F_ADD trees with all constraints give the root as 84.2, 105.0, and ∞ mya, respectively (Table 1). Removing the whale–hippo constraint gives 82.4, ∞, and 91.2 mya; removing the horse–rhino constraint gives, 82.8, 105.0, and ∞ mya; and removing both sets of constraints gave 82.9, 87.2, and ∞ mya. Thus, in this case only the F_IR tree is insensitive to “constraint sampling”.

Finally, there is a good fossil calibration for tarsier [4] that was withheld because it is not used in both references 4 and 15. It suggests that human and tarsier split around 50–60 mya and the molecular trees suggest the human-loris split was not much earlier (probably <5 myr earlier). Despite our use of whale–hippo and horse–rhino calibrations, the MVS-F_IR tree (Figure 1) gives a human-loris split close to this age, in contrast to earlier studies [4], [15], [33].

Sensitivity and robustness of F_IR assessed by simulations

We highlight two distinct properties of the new function F_IR with the help of evolutionary simulations and worked examples. The first property is its ability to detect a transient acceleration of evolutionary rate. Such an effect might be caused by a burst of positive selection and/or a bottleneck in population size. The second is to assess the effects of both stochastic fluctuations and systematic bias on the robustness of estimated times. Here, we model bias in the form of either a general slowdown or a general acceleration of evolutionary rate across the whole tree.

We first modeled a strong instantaneous acceleration as an analogue to what is inferred by F_IR to have occurred ancestral branch leading to Laurasiatheria. We simulated a 32-taxon symmetric tree in which a molecular clock holds except for an abrupt elevation (by a factor of 10) of evolutionary rate along a short internal branch (the red line in Figure 4A). This example is useful for demonstrating the efficacy of the F_IR function and something similar appears on both the MVS-F_IR and ML-F_IR mitochondrial trees. The times at the internal nodes were set to 48, 56, 64, and 72 mya, and the root time was set to 80 mya. On this weighted tree, the cost functions were minimized with two constrained node times, which corresponded to two fossil calibration points. Only the F_IR function accurately estimated the true divergence times and appeared robust against the abrupt change (Figure 4B). In contrast, the functions F_LOG and F_ADD inferred gradual rate changes (Figure 4E), which are erroneous and lead to overestimation of the root time along with that of many other nodes (Figures 4C and 4D). Table 2 summarizes the root time values inferred by various models. Table 2 includes the result of r8s with the cross-validation method.

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Figure 4. A worked example of the effect of a transient elevation of evolutionary rate upon estimated divergence times.

In this worked example using a symmetric 32-taxon tree (Figure A), a global molecular clock holds, except for a short-term increase in evolutionary rate along one branch (the red line in Figure A). The true root time was set to 80 mya, and the times at the internal nodes are 48, 56, 64, and 72 mya. The deep internal branch (the red line in Figure A) is given an evolutionary rate ten times that of the remaining edges. The various cost functions were minimized subject to two calibrated nodes (the red numbers 1 and 2 in Figure A), using the exact branch lengths of this example as input data. The cost functions F_IR, F_LOG, and F_ADD inferred the weighted trees of Figures B, C and D, respectively. Figures E–G show the trace of evolutionary rates along the lineages from the root to taxa numbers 5, 9, and 25 of Figure A, respectively. The inferred age of the root for each cost function is shown with an arrow. The function F_IR recovered the original pattern of rate change, whereas the other two functions inferred far more gradual changes, which resulted in a substantial overestimation of the root time.

https://doi.org/10.1371/journal.pone.0000384.g004

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Table 2. Estimated age of the root in the presence of an abrupt acceleration then deceleration of ancestral rates

https://doi.org/10.1371/journal.pone.0000384.t002

We next simulated stochastic rate fluctuations by themselves, plus either a prevailing slowing down or acceleration of rates through time. Such simulations are distinct from a Brownian-type process, and are used to gauge the general robustness of the functions. To impose rate fluctuations on the same basic tree as Figure 4A, but without the abrupt rate change, the age of the root was set to 100 mya. We assumed infinite length of sequences and that the branch lengths are known without uncertainty. Next a branch was randomly selected proportional to its duration in time and all its descendant branch lengths were multiplied by a factor (chosen randomly) from a uniform distribution of range 0.5–1.5. A total of 25 such rate changes were placed on the tree to give the final branch lengths for that tree. These branch lengths were used as the input data and the various cost functions were minimized with on calibrated internal node, and then determined all node times. We repeated this procedure 600 times to obtain the mean and standard error of the root time. We also simulated the two other cases. The only difference was the range of the uniform distribution used. Using a range of 1.0–1.75 the model is biased toward rate acceleration through time, while using and 0.25–1 induces a rate slow down through time (a probable situation in the placentals generally). As table 3 shows, the function F_IR gave reasonable estimates with a bias towards deceleration, whereas F_ADD gave an infinite root time in 139 of 600 samples. These undefined root-time values were set arbitrarily to 200 mya before calculating the mean and standard error.

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Table 3. The age of the root inferred by functions F_IR, F_LOG, and F_ADD on simulated data with random auto-correlated rate changes

https://doi.org/10.1371/journal.pone.0000384.t003

Reduced bias in branch length estimation by the MVS model

The MVS model was shown previously to recover the correct tree in a simulation with two strongly convergent lineages [26] that were grouped erroneously by standard methods (including the neighbor joining (NJ) [38] and ML [35] methods). Here, we examined a different question; this is how well the MVS method recovers the true branch lengths, which are of primary importance when estimating divergence times. We simulated the evolution of a sequence of 10⁴ amino acid sites, following the tree depicted in Figure 5. The model of amino acid substitutions used was the JTT [36]. Convergent evolution was then imposed on this data. This was done by sharing parts of the sequences among three ingroup clades plus the outgroup. The red lines in Figure 5 indicate which lineages shared sequence and were therefore subject to a form of convergent evolution.

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Figure 5. A simulation of convergent evolution.

Amino acid substitutions were evolved on the shown weighted tree using the JTT model [36] of amino acid substitutions. After this, 30% of sites were swapped between the four lineages indicated by the red lines.

https://doi.org/10.1371/journal.pone.0000384.g005

Pairwise distances were then estimated from the terminal sequences obtained above using the same JTT model. A modified MVS core-set procedure [27] was able to recognize that the tree consisted of three groups of eight sequences which had additive distances within each group and the outgroup. The MVS procedure then converted the distances within the three ingroups into three sets of perfectly additive distances and sequentially combined them to form a single core set (Materials and Methods). The final MVS tree was obtained by modifying the distances between this single ingroup core set and the outgroup following the rules described in Materials and Methods.

The branch lengths recovered by the MVS model reproduced the true values accurately (Figure 6). The distribution around the diagonal line in Figure 6 represents the stochastic fluctuation of the estimated distances; that is, the magnitude of this fluctuation did not change after the simulation was rerun without convergent evolution. In contrast, the NJ and ML trees returned branch lengths that were affected clearly by convergent evolution. The worst affected branch lengths were ancestral to the groups showing convergent evolution (underestimated) or else were leading to the sister groups of the groups affected by convergent evolution (overestimated).

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Figure 6. Branch lengths estimated by MVS, NJ, and ML for the example of convergent evolution in figure 5.

The tree topology inferred by these methods was identical to the tree that generated the data, so the estimated branch lengths are plotted against their true values. The blue numbers show the branch index, as used on Figure 5, of outliers. Note, all the outliers are internal branches ancestral to the four lineages undergoing convergent evolution or are ancestral to the sister group to these lineages. Branch lengths ancestral to the groups undergoing convergent evolution are underestimated by the ML and NJ methods, whereas those ancestral to their sister taxa are overestimated.

https://doi.org/10.1371/journal.pone.0000384.g006

Autocorrelated variation of inverse rate

Because the branch lengths of a phylogenetic tree are estimated from the data as the product of the evolutionary rate of a branch and its time duration, these two factors cannot be directly estimated separately. It has become standard to use loose constraints to accommodate uncertainty and it is often wise to exclude constraints if there is no firm basis for them. Information on the age of some internal nodes may be fairly directly available from the fossil record. An example of this is the horse-rhino split [13]. If the molecular clock [16] governs the process of molecular evolution well, we can accurately estimate the times at other nodes given just a single reliable calibration point. However, the assumption of the molecular clock is often rejected by the fit of a clock-like tree to the data.

The most widely used assumption in order to model the evolutionary rate changes away form being constant (or away from a clock) is to introduce a stochastic process. For example, Sanderson [17] suggested the following cost function be minimized:1with rates R_n and R_a(n) being from the nth branch and its ancestral branch a(n), respectively. The exponent of the negative cost function Π_n {(2π)^1/2σ}⁻¹ exp{−(R_n−R_a(n))²/2σ²} implies that this function can be interpreted assumes a priori that rate changes follow a random walk with independent increments randomly drawn from a normal distribution. Likewise, Thorne et al. [31] considered a cost function of the type2in the framework of the hierarchical Bayes procedure. Here T_n is the time interval of the nth branch (the current version of Multidivtime [31] uses rates at the nodes, instead of average rates along branches, as the free parameters). This can be interpreted as an assumption that the log rate undergoes Brownian motion. That is, independent increments of normal random variables, whose variances are proportional to the average time duration between the pair of branches (that is, the average length in time of the two branches). Recently, Sanderson implemented the unweighted cost function3into his program r8s [18].

However, the estimated rates based on any of the above cost functions may overly smooth the change of evolutionary rates when a pronounced transient change of rate has occurred. In turn, biased estimates of evolutionary rates will lead to biased estimates of divergence times. Here we propose a new cost function, which penalizes the local rate deviation from the harmonic mean. For simplicity, we ignore the stochastic variance of branch length [17], [18], [31], which goes to zero with increasingly long sequences. Denote the branch length and inverse rate of the nth branch by B_n and Q_n ( = T_n/B_n), respectively. The local rate variability (F_IR,n) of two successive branches can be expressed as the variance of Q_n and Q_a(n) around their average value, A_n = (Q_nB_n+Q_a(n)B_a(n))/(B_n+B_a(n)), which is equal to the inverse of the harmonic mean of R_n ( = B_n/T_n) and R_a(n), with the weights B_n and B_a(n). That is, F_IR,n = {(Q_n−A_n)²B_n+(Q_a(n)−A_n)²B_a(n)}/(B_n+B_a(n)). Because F_IR,n is rewritten as F_IR,n = (Q_n−Q_a(n))²W_n with the weight W_n = B_nB_a(n)/(B_n+B_a(n))², the inverse-rate cost function, F_IR, is defined as the weighted average of (Q_n−Q_a(n))² over all ancestor–descendant branch pairs in the tree:4which contrasts with previous expressions in terms of R_n.

Because F_IR places a smaller penalty on abrupt rate changes than do previous models, it can confine an abrupt change close to where it occurred on the tree. In contrast, F_ADD and F_LOG a priori put stress upon the smoothness of evolutionary rate change from one branch to the next, and this may propagate the effect of an abrupt rate change over successive branches. In this article, the variable (T_n) (the divergence times) were estimated by minimizing the cost functions (equations 1–4) subject to the (fossil) constraints without introducing other modifiers such as cross validation.

At first glance, it seems most intuitive to use a penalty such as (rate₁−rate₂)²; this has been the implicit assumption until now [31] and is the basis of published programs such as r8s and Multidivtime [17], [18], [31]. It rests on the implicit expectation that any reasonable type of smoothing of changes in rate will give a similar answer. However, a simple linear form for the difference in rates is misleading, and this is illustrated by considering the general problem of estimating velocity (rate) given distance (in our case branch length). Going from point a to b at velocity v₁, then back at velocity v₂, the average velocity (v_av) is equal not to the standard arithmetic mean, but to the harmonic mean, 1/v_av = (1/v₁+1/v₂)/2. Further, going from point (node) a to b at velocity (rate) v₁, and from point b to c at velocity v₂, then the distances traveled (d_ab and d_bc) need no longer be equal so we must use weights. Specifically, v_av = (d_ab+d_bc)/(t₁+t₂), where t₁ = d_ab/v₁ and t₂ = d_bc/v₂. As a result, 1/v_av = (d_ab/v₁+d_bc/v₂)/(d_ab+d_bc). This is the same form as the average quantity A_n used to obtain F_IR (equation 4). The use of the harmonic mean becomes important when v₁ differs greatly from v₂.

The penalty in hierarchical Bayes and penalized likelihood approaches

The Bayesian approach estimates the divergence times and evolutionary rates in the form of a posterior distribution, which is summarized approximately as P(B|R,T) exp(−λF). Here B, R, and T are the vectors of the estimated branch lengths, evolutionary rates, and divergence times, respectively, while F is the cost function. The value of λ expresses the weight for the penalty, and is called a hyperparameter. Introducing the distribution for the hyperparameter is done via a so-called hyper-prior; the hyperparameter is then estimated concurrently with the posterior distribution. The penalized likelihood approach maximizes log P(B|R,T)−λF. The weight for the penalty λ is estimated using cross-validation. When the sequences are long enough, for example concatenated mitochondrial protein sequences, it is often safe to assume that the branch lengths are estimated accurately, that is with minimal stochastic error (but not necessarily without systematic error, something we address in this paper using MVS). Accordingly, it is assumed that the products of rates and times are known exactly. Thus in a situation of long, or very long sequences (e.g. genomic alignments), the results returned by using just the penalty function will converge to those returned by either a Bayesian or a penalized likelihood approach with the same type of cost function.

MVS core-set correction of biased evolutionary distances

In the MVS method, the additivity of evolutionary distances is converted using the expected orthogonality among branch vectors in a multidimensional Euclidean space [25]. If the estimated pairwise distances do not satisfy additivity, MVS provides an index to measure deviations from orthogonality without specifying a tree structure. This index enables one to diagnose whether a distance matrix is compatible with a tree and to modify the biased distances until they satisfy orthogonality, that is, with a zero value for the index. Here, a revised core-set approach, that is, an extension of [27], is applied to correct the observed pairwise distances for convergent evolution.

The first step of this approach is to divide the set of taxa into subgroups and to correct the distances between pairs in each subgroup. We decompose the taxa based on partitions with both strong biological support and high bootstrap support. The deviation from additivity of pairwise distances is much smaller within each subgroup than across the whole distance matrix. If an anomalously large deviation is observed associated with a single taxon within a subgroup, that taxon is removed temporarily from the analysis. When the only deviations between distances within a subset of taxa are judged to be due to stochastic noise, the pairwise distances are modified by solving the equation of motion (a method using a many body kinetic equation in physics), which uses the index of the deviation from additivity as the potential energy [25], [26]. The modified distance matrix within a subgroup is interpreted as the true additive distance matrix (core set) achieved by minimal modification of the original distances. Generally, the modified distances are very close to the distances estimated by the NJ method for just these taxa. Then, the excluded taxa are deposited into this core set tree by solving the equation of motion with the same potential function. Assuming that convergent evolution and long-branch attraction are the main source of deviation from additivity, we allow for only positive corrections of the biased distances between the core set and excluded taxa (that is, distances can only get bigger).

The second step is a clustering of subgroups. The pairwise distances between two core-set groups are corrected by assuming attractions between them (that is, some distances are underestimated) and by minimal enlargement of the distances between the two groups to satisfy additivity. In this way, the core-set approach used in this article is in the direction opposite to that used in an earlier procedure that proceeds from the whole tree down to subtrees [26]. This revised method seems more appropriate here because the branching pattern within subgroups becomes more apparent and there are smaller deviations from additivity.

Conceptually, the first step of MVS core set is to decompose the whole tree into putative monophyletic groups that show minimal bias internally. The approach developed as part of this particular study resembles k-means clustering, that is, minimizing the deviation (variance) within groups while maximizing the deviation (variance) between groups. At present, the decomposition is done in a supervised, as opposed to an automatic, manner. In the second step, and when it is difficult to connect two groups definitely, we first exclude taxa with the largest deviations from additivity and then complete a new core set for the remaining sequences. We then locate the excluded sequences on the core set tree using minimal modification of distances (a parsimony-like criterion) under the constraint of only enlarging the distances identified as being biased (thus the only biases are assumed to be due to attraction).

When analyzing the mitochondrial protein sequences, we estimated pairwise distances initially under the JTT+Gamma model with α = 0.5. Given moderately high bootstrap support, the placental tree was decomposed into four major groups: Laurasiatheria, Supraprimate, Xenarthra, and Afrotheria. Laurasiatheria was decomposed into five subgroups, namely Cetartiodactyla, Perissodactyla, Carnivora, Chiroptera, and Euliptotyphyla. Supraprimates was decomposed into two subgroups: Primates and Glires. The core-set analysis began by generating additive distances within these subgroups, after which the clustering procedure described above was applied to connect these subgroups. In this analysis, we analyzed sequences of Xenarthra and Afrotheria together in a single core-set. This is because a relatively short internal branch separates them and Xenarthra contains but a single sequence in this analysis. Finally, we tried to modify the distances between placental mammals and the outgroups, but we could not determine the position of the root with confidence because of very strong attractions between many placental mammals and the outgroups. The consensus [3]–[5], [15] has been that the most probable placement of the root separates Afrotheria from the other placental mammals. The most likely alternative placement for the root is between Xenathra plus Afrotheria (Atlantogenata [2], [20]) and all other placentals (Boreotheria). Recent analyses including LINE sequences and indels [43] support this view. When we place the root in this position, the divergence times, including that of the root, change little (results not shown).

Further details of the current MVS procedure of distance modification are documented in Methods S1. The derived additive distance matrix was converted to a Newick tree file with the help of the NJ method, and this was used as the input data for divergence time estimation by the cost functions.

Log-likelihood profiles of the cost function with respect to the age of the root

We note that least-squares estimators can be interpreted as ML estimators when errors in the data follow a multi-normal distribution. Accordingly, the least-squares residual can be treated as twice the log-likelihood ratio for the three cost functions G_J(T_root) (J = IR, ADD, or LOG); that is, as G_J(T_root) = N log{F_J(T_root)/F_J(T_M)}. Here, F_J(T_root) gives the minimum residual value with all internal node times reestimated, except for the root time T_root, and it becomes truly minimal at T_root = T_M. Here, N is the number of terms in the cost function, which is equal to 2t−4, where t is the number of tips on the tree (here t = 62, so N = 120). The 95% CI can be obtained using the threshold of G_J(T_root) = 3.84. Here, the minimum for criterion F_ADD was arbitrarily set to F_ADD(T_root) at T_root = 200 because the profile decreased monotonically for all greater values of the root time. In all evaluations, we avoid putting arbitrary constraints near the root of the tree (for example, that the rate at the root is the same as that of one of its descendants) because while arbitrary constraints may bound the minimum, they will also create unknown biases. Note that the MVS-F_IR profile is symmetric and parabolic around the estimated root time, whereas those of F_LOG and F_ADD are not. We see the same thing even in worked examples using a perfectly clock-like tree.

Our novel way of estimating CIs jointly takes into account some, but not all, of the sources of error noted in a previous publication [13]. These include errors in edge length estimation (because of a finite sampling of sites), the unpredictability of fluctuating evolutionary rates, and coalescent time. The assumptions of this method, vis-à-vis independence of errors, are the same as those used in programs by Thorne et al. [31]. Note that even if edge lengths have no error due to sampling of sites (e.g., the type of error identified by the bootstrap), they may still show four wide CIs because of the other two factors, which do not go to zero with increasing sequence length. This, in turn, shows that using the sequence bootstrap to infer errors on divergence times [17], [18] is not comprehensive and therefore the CIs are too narrow. We expect that this empirical approach to estimating standard errors will become increasingly accurate with increased numbers of internal nodes and calibration points.

Fortran source code and executable versions of the programs used in this study are downloadable from 〈http://www.kochi-ms.ac.jp/∼ct_cmis/kitazoe/〉. The operation of these programs, using the data in this analysis, is documented in Methods S2.