动物学报 50(4) :645 - 656 , 2004 Acta Zoologica Sinica Received April 04 , 2004 ; accepted May 24 ,20043This research was funded by a grant from Biotechnology and Biological Sciences Research Council to Z. Y. 33E2mail : z. yang ucl.ac. uk2004动物学报Acta Zoologica SinicaA heuristic rate smoothing procedure for maximum likelihood esti2 mation of species divergence times3Ziheng YANG33Department of Biology , University College London , Darwin Building , Gower Street , London WC1E 6BT , EnglandAbstract Estimation of species divergence times is well2known to be sensitive to violation of the molecular clock assum p2 tion (rate constancy over time) . However , the molecular clock is almost always violated in comparisonsof distantly related species , such as different orders of mammals. Thus it is im portant to take into account different rates among lineages when divergence times are estimated.The maximum likelihood method provides a framework for accommodating rate variation and can naturally accommodate heterogeneous datasetsfrom multiple loci and fossil calibrations at multiple nodes. Previous implementations of the likelihood method require the researcher to assign branches to different rate classes. In this paper , I implement a heuristic rate2smoothing algorithm (the AHRS algorithm) to automate the assignment of branches to rate groups. The method combinesfeaturesof previous likelihood , Bayesian and rate2smoothing methods. The likelihood algorithm is also improved to accommodate missing sequences at some loci in the combined analysis. The new al2 gorithms are applied to estimate the divergence times of Malagasy mouse lemurs using a dataset of mammalian mitochon2 drial genes and compared with previous likelihood and Bayesian Markov chain Monte Carlo analyses Acta Zoologica Sini2 ca50 (4) : 645 - 656 , 2004. Key words Rate smoothing , Molecular clock , Divergence times , Maximum likelihood , Combined analysis一个适于最大似然法估计物种分化年代的进化速率平滑近似 方法3Ziheng YANG33Department of Biology , University College London , Darwin Building , Gower Street , London WC1E 6BT , England摘 要 众所周知,物种分化年代的估计对分子钟(进化速率恒定)假定很敏感。另一方面,在远缘物种(例如哺乳纲不同目的动物)的比较中,分子钟几乎总是不成立的。这样在估计分化时间时考虑不同进化区系的速率差异至为重要。最大似然法可以很自然地考虑这种速率差异,并且可以同时分析多个基因位点的资料以及同时利用多重化石校正数据。以前提出的似然法需要研究者将进化树的树枝按速率分组,本文提出一个近似方法以使这个过程自动化。本方法综合了以前的似然法、贝斯法及近似速率平滑法的一些特征。此外,还对算法加以改进,以适应综合数据分析时某些基因在某些物种中缺乏资料的情形。应用新提出的方法来分析马达加斯加的倭狐猴的分化年代,并与以前的似然法及贝斯法的分析进行了比较动物学报50 (4) : 645 - 656 , 2004。关键词 速率平滑 分子钟 分化年代 最大似然法 综合分析The assumption of molecular clock , that is , con2 stancy of the evolutionary rate among lineages (Zuck2 erkandl and Pauling , 1965) , provides a simple and powerful way of dating species divergences. This as2 sumption predicts that the expected genetic distance between species is proportional to the time of their di2 vergence.Thus the estimated branch lengths or se2 quence distances can be converted into absolute diver2gence times through fossil calibration.While the clock assumption appears to hold in closely related species , for example , within the hominoids , it is most often violated in distant comparisons , for exam2 ple, among different orders of mammals (Hasegawa et al. , 2003 ; Springer et al. , 2003 ;Yoder and Yang , 2000) . The effects of the clock assumption on divergence time estimation is well2characterized (e.g. , Aris2Brosou andYang , 2002 ;Rambaut and Bromham , 1998) . In the past few years , much effort has been taken to account for such rate variation when divergence times are estimated. Likelihood methods account for the rate variation by assigning indepen2 dent rates to branches on the phylogeny ( Kishino and Hasegawa , 1990 ; Rambaut and Bromham , 1998 ; Yoder and Yang , 2000) . This approach has recently been extended to deal with multiple fossil calibration points and multiple genes ( Yang and Yoder , 2003) . In the Bayesian framework , Thorne et al. (1998) . Kishino et al. (2001) uses a stochastic model of evolu2 tionary rate change to specify the prior distribution of rates, and , together with a prior for divergence times, calculates the posterior distributions of times and rates. Markov chain Monte Carlo (MCMC) is used to make the computation feasible. The algorithm isrecentlyextendedtoanalyzemultiplegenes (Thorne and Kishino , 2002) .The Bayesian algo2 rithm followed the seminal work of Sanderson (1997 , 2002) , who developed heuristic rate smoothing meth2 ods for joint estimation of times and rates.Fig11 Example trees to explain the theory (a) Master tree for eight species. Two calibration points are used so that node agest3andt6are fixed , while the ages of other nodes are parameters to be estimated. (b) Six species are sequenced at locus 1 , for which the gene tree isconstructed from the master tree. Different branches may have different evolutionary rates , represented by the thicknessof the branches , which are accommodated in the likelihood analysis. (c) Five species are sequenced at locus 2.A drawback of the likelihood method (Yan