Yasuhiro Kitazoe; Hirohisa Kishino; Peter J. Waddell; Noriaki Nakajima; Takahisa Okabayashi; Teruaki Watabe; Yoshiyasu Okuhara

“ 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. ”

“ Just as the effective size of fluctuating populations is represented by the harmonic mean over time, the mean evolutionary rate among lineages is expressed better by the harmonic mean. This approach is especially useful when branch lengths measured in the expected number of substitutions per site (the products of rates and times) are estimated accurately, and there are either rapid transient changes of rate (hence large rate heterogeneity) or a general bias towards a speed up or slow down in rates through time. ”

“ The root of living placental mammals has been reported to be in the range of 100–140 mya [5] , [8] – [9] even with application of rate-adjustment techniques [5] , [9] . The molecular consensus of an old root is becoming strong enough that it may become dogma without a close examination of the assumptions being made in such analyses. The true age of the orders and superorders has important implications for determining the overall paleoecology and biogeography of placental mammals during a period that included the breakup of continents and the extinction of terrestrial dinosaurs. ”

“ While some methods will detect potential convergent evolution, for example Hadamard conjugations and SplitsTree [17] , Kitazoe et al. have developed multidimensional vector space (MVS) representation methods to both detect convergent evolution among estimated distances and to also correct for this bias [18] – [19] .
A further problem facing molecular dating is the evolutionary rate constancy, or lack of constancy, over long periods. Since its proposal in 1965 [20] , the molecular clock hypothesis has been one of the most hotly debated subjects in evolutionary biology. ”

“ Using this new procedure, an analysis of 69 mitochondrial protein sequences (3660 amino acid sites in total) from placental mammals identified a rapid acceleration of evolutionary rate for the lineage directly leading to the common ancestor of Supraprimates and an even more marked one for the lineage leading to Laurasiatheria. This acceleration was followed closely by a strong deceleration, which persisted in nearly all lineages of Laurasiatheria. In contrast, almost all lineages of Afrotheria and Xenarthra seem to have retained rates similar to that of the root. ”

“ Application to complete mitochondrial genomes identified marked rate accelerations and decelerations, which are not obtained with current methods. The root of placental mammals is estimated to be ∼18 million years more recent than when assuming a log Brownian motion model. Correcting the pairwise distances for convergent evolution using MVS lowers the age of the root about another 20 million years compared to using standard maximum likelihood tree branch lengths. ”

“ The robustness of the new procedure is assessed using simulations that show the types of change that most concern biologists; speedups or slowdowns through time, transient rate changes, and rate changes that are do not follow a normal or transformed normal distribution, as well as stochastic variation. A revised estimate of the origin of placental mammals is as young as 84 mya, which is much more recent than current estimates using molecular data. ”

“ 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.Log-likelihood profiles of the FIR, FLOG, and FADD cost functions about the estimated ages of the root.This figure shows the profile of the cost function with respect to the age of the root estimated by three different cost functions. ”

“ 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) . ”

“ 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. ”

“ 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 FIR tree is insensitive to “constraint sampling”.
Finally, there is a good fossil calibration for tarsier [5] 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) . ”

“ Ba (n) ) / (Bn+Ba (n) ) , which is equal to the inverse of the harmonic mean of Rn ( = Bn/Tn) and Ra (n) , with the weights Bn and Ba (n) . That is, FIR,n = { (Qn−An) 2Bn+ (Qa (n) −An) 2Ba (n) }/ (Bn+Ba (n) ) . Because FIR,n is rewritten as FIR,n = (Qn−Qa (n) ) 2Wn with the weight Wn = BnBa (n) / (Bn+Ba (n) ) 2, the inverse-rate cost function, FIR, is defined as the weighted average of (Qn−Qa (n) ) 2 over all ancestor–descendant branch pairs in the tree ”

“ The root time of of the FADD analysis in Figure A and B was set at a large value (400 mya) because the numerical calculation continues towards an infinite root time.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 [25] . 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-FIR tree (Figure 1A) . ”

“ Finally a strong acceleration effect is achieved by the use of the range 1.0–1.75. The function FADD gave an age of the root tending to infinity on 139 of the weighted trees simulated under the deceleration model. In such cases the root time is set to 200 mya in order to allow the mean and standard deviation of this cost function to be calculated.Competing Interests: The authors have declared that no competing interests exist.Funding: This work was supported by the Technology Agency (JST) and the Japan Society for the Promotion of Science (JSPS grant SR B-16300086 to YK and HK) . ”