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I am currently working on a project where I study the change of a continuous morphological variable with body size (measured as body weight) across diverse taxa (insects, spiders, lizards, frogs). In order to account for the non-independence of the data, I built an approximate tree based on published phylogenies. I have a reasonable branching pattern down to family or genus level for most taxa, but lack statistically supported branch length.

I find a strong change of the scaling coefficient when using phylogenetic least squares (with a covariance matrix based on my tree) vs a non-corrected least-squares approach, suggesting that most of the change observed in my continuous variable is explained by evolutionary history, rather than by body mass as such. I would now like to investigate how early on in the tree these shifts occur.

Intuitively, I could start collapsing my tree into polytomies down to arbitrary taxonomic levels, eg genus, family, order etc, and re-do my phylogenetic least squares. I would expect a stepwise approximation of the pgls to the uncorrected result (in the extreme case, I would end up with a star phylogeny, which should give an identical result to my uncorrected regression). Such an approach may allow to say something about whether most of the change occurs between genera, families etc, but it requires the use of somewhat arbitrary (and controversial) taxonomic levels. I would be grateful if someone would have an alternative suggestions on how to perform such an analysis, with arbitrary branch length and non-ultrametric trees.

Thanks Thriceguy

## Phylogenetic rate shifts in feeding time during the evolution of Homo

Unique among animals, humans eat a diet rich in cooked and nonthermally processed food. The ancestors of modern humans who invented food processing (including cooking) gained critical advantages in survival and fitness through increased caloric intake. However, the time and manner in which food processing became biologically significant are uncertain. Here, we assess the inferred evolutionary consequences of food processing in the human lineage by applying a Bayesian phylogenetic outlier test to a comparative dataset of feeding time in humans and nonhuman primates. We find that modern humans spend an order of magnitude less time feeding than predicted by phylogeny and body mass (4.7% vs. predicted 48% of daily activity). This result suggests that a substantial evolutionary rate change in feeding time occurred along the human branch after the human–chimpanzee split. Along this same branch, *Homo erectus* shows a marked reduction in molar size that is followed by a gradual, although erratic, decline in *H. sapiens*. We show that reduction in molar size in early *Homo* (*H. habilis* and *H. rudolfensis*) is explicable by phylogeny and body size alone. By contrast, the change in molar size to *H. erectus*, *H. neanderthalensis*, and *H. sapiens* cannot be explained by the rate of craniodental and body size evolution. Together, our results indicate that the behaviorally driven adaptations of food processing (reduced feeding time and molar size) originated after the evolution of *Homo* but before or concurrent with the evolution of *H. erectus*, which was around 1.9 Mya.

Changes in behavior can place an animal under a new suite of selective forces that open new evolutionary pathways. Such adaptations have played a crucial role during the evolution of animal life. Recent evidence suggests that extant humans are biologically adapted for eating cooked and processed food (the cooking hypothesis)—an adaptation that was behaviorally driven by controlled use of fire (1). Food processing would have provided higher caloric intake in the ancestors of modern humans, which likely bestowed significant advantages on reproductive success and survival (2 –4). Malnutrition resulting from a committed raw food diet (5) strongly suggests that eating cooked and processed food is necessary for long-term survival on wild foods in *Homo sapiens* (6, 7). This hypothesis explains the small teeth, jaws, and guts of modern humans and the universal importance that cooking has played in cultures throughout recorded history (1).

Quantitative, phylogenetically based tests of this hypothesis are lacking as well as direct evidence on whether cooking began in the pre-*H. sapiens* lineage in Africa (8). Furthermore, considerable disagreement exists concerning the evolutionary relationships of species within our lineage (9). This phylogenetic uncertainty limits our ability to discern the evolutionary history of many behavioral traits in humans, including obligate food processing. For example, the time that a primate spends feeding as a percentage of its daily activity can be an important constraint on other behaviors (10), and it is expected to be related to metabolic requirements, body size, molar size, and how it socializes with conspecifics and interacts with its environment. However, the relationship between the amounts of time humans spend feeding compared with the time spent by other primates has never been studied.

Feeding time is dependent on the metabolic needs of the organism as well as on ingestion time, chewing time, and bolus formation. The occlusal surface area with which food is chewed also plays an important role in food processing and has long been used to infer shifts in feeding behavior in extinct hominins (11 –14). The reduction of molar size during hominin evolution is thought to be associated with the advent of advanced food processing, because cooking softens food (15) and soft food puts less biomechanical demand on chewing teeth (16). Softer foods also adhere more quickly while being chewed and therefore, are swallowed after fewer chewing cycles (17).

Here, we investigate the amount of time spent feeding by humans compared with other primates, and we use a phylogenetic analysis to distinguish hominin species according to whether changes in molar size are explicable by the overall rate of craniodental evolution. This analysis allows us to test the hypothesis that a major shift in selection pressure involving food processing occurred in the human past. We, thus, use comparative phylogenetic methods to test an explicit phylogenetic prediction of the cooking hypothesis, namely that a significant phylogenetic rate change occurred in molar size and feeding time along the human lineage.

## Materials and methods

We surveyed the literature for data sets on flowering onset and compiled a list of studies spanning at least 20 years, with a minimum of four data points through time per species (after compilation, all usable data sets had at least seven data points through time per species). We searched systematically using Web of Science (using the terms ‘flowering’ and ‘phenolog’) and also followed references to other studies within papers we located. Several data sets were not included because either they did not meet these criteria, or the requisite data for determining shifts in flowering time were not available. We assembled 15 data sets from across the Northern Hemisphere, ranging in duration from 29 to 172 years from 1837 to 2012 (Table 1). The data come from a diversity of habitats and include forbs, grasses, shrubs and trees. From all 15 data sets, we extracted information on whether species had shifted in flowering onset. When that summary information was not available, we used the raw data to test for shifts using simple linear regressions of year against day of year of first bloom. Species shifts in flowering onset were coded as: 1 for significant delay 0 for no shift −1 for significant advance. In a second analysis, we used a continuous measure of shift, the slope of the relationship between first flowering and year, which we could extract or calculate for only eight of the data sets that either reported slopes or provided raw data from which they could be calculated.

Reference | Location | Time span (duration) | Habitat | No. species | Flowering response |
---|---|---|---|---|---|

Abu-Asab et al. ( 2001 ) | Washington DC, USA | 1970–1999 (29) | Metropolitan area | 100 | Shift, Slope |

Bolmgren, Vanhoenacker & Miller-Rushing ( 2013 ) | Sweden | 1934–2006 (72) | Temperate farm | 25 | Shift |

Bradley et al. ( 1999 ) | Wisconsin, USA | 1936–1945, 1977–1998, 1999–2007 added for present study (38) | Tallgrass prairie | 33 | Shift, Slope |

Calinger, Queenborough & Curtis ( 2013 ) | Ohio, USA | 1895–2009 (115) | Temperate woodland and grassland | 141 | Shift |

CaraDonna, Iler & Inouye ( 2014 ) | Colorado, USA | 1974–2012 (38) | Subalpine meadows | 60 | Shift, Slope |

Crimmins, Crimmins & Bertelsen ( 2010 , 2011 ) | Arizona, USA | 1984–2003 (29) 1984–2009 (35) | Semi-arid montane | 428 240 | Shift |

Dunnell & Travers ( 2011 ) | North Dakota and Minnesota, USA | 1910–1961, 2007–2010 (54) | Temperate woodland and grassland | 23 | Shift, Slope |

Ellwood et al. ( 2013 ) | Massachusetts Wisconsin, USA | 1852–1858, 1878, 1888–1902, 2004–2006, 2008–2012 1935–1945, 1977–2012 (66) | Temperate forest, wetland tallgrass prairie | 32 23 | Shift, Slope |

Fitter & Fitter ( 2002 ) | Oxfordshire, England | 1954–2000 (56) | Temperate woodland and grassland | 372 | Shift, Slope |

Menzel, Estrella & Fabian ( 2001 ) | Germany | 1951–1996 (45) | Various | 5 | Shift, Slope |

Miller-Rushing & Primack ( 2008 ) | Massachusetts, USA | 1852–1858, 1878, 1888–1902, 2004–2006 (123) | Temperate forest and wetland | 43 | Shift, Slope |

Molnár et al. ( 2012 ) | Hungary | 1837–2009, 1980–2011 (172, 31) | Various | 39 | Shift, Slope |

Ovaskainen et al. ( 2013 ) | Karelia, Russia | 1960–2010 (50) | Boreal forest | 66 | Shift, Slope |

Panchen et al. ( 2012 ) | Pennsylvania, USA | 1840–2010 (150) | Greater metropolitan area | 28 | Shift, Slope |

Three of the data sets (Bradley *et al*. 1999 Miller-Rushing & Primack 2008 Ellwood *et al*. 2013 ) have species in common from the same locations but for differing time spans. We therefore combined data from these three studies to yield the longest time series possible for species for which data were available in multiple data sets. For example, the data for *Sisyrinchium campestre* as reported in Bradley *et al*. ( 1999 ) indicate there has been no shift in flowering onset for this species however, updating those data to include more recent phenological observations as reported in Ellwood *et al*. ( 2013 ) yielded a significant shift to earlier flowering. We therefore coded *S. campestre* as shifting earlier in our aggregate data set.

For analyses of the pooled data sets, we averaged the data for shift or slope for species for which we had multiple records from different communities because each species is represented only once in the phylogenies and can have only a single value per trait. Mean responses (averaged for 2–7 records per species) were used for 133 (10·7%) of the 1245 species used for phylogenetic analyses with shift as the trait mean responses (averaged for 2–4 records per species) were used for 70 (11·6%) of the 606 species used for phylogenetic analyses with slope as the trait.

To test for phylogenetic signal, we used two phylogenetic trees that were constructed to address similar questions by Davies *et al*. ( 2013 ). The first of the Davies *et al*. ( 2013 ) trees comprises 4494 taxa, was constructed using the Angiosperm Phylogeny Group 3 tree as the backbone in Phylomatic (Webb & Donoghue 2005 ) and is 25% resolved (hereafter the ‘Phylomatic tree’). Following Davies *et al*. ( 2013 ), we also used a molecular phylogeny that differs in topology for comparison this tree was calibrated with penalized likelihood and is fully resolved for 1246 genera (hereafter the ‘molecular tree’). Both trees are available in Davies *et al*. ( 2013 ). Our aggregate data set for shift covered 1245 (27·7%) of the species in the Phylomatic tree and 582 (46·7%) of the genera in the molecular tree. Our aggregate data set for slope covered 610 (13·6%) of the species in the Phylomatic tree and 328 (26·3%) of the genera in the molecular tree. We added species to the molecular tree as polytomies (Davies *et al*. 2013 ), resulting in trees with 1172 and 585 species for shift and slope, respectively. A handful of species in our data set were not included in the Phylomatic tree, and some genera were not in the molecular tree we removed these species/genera (*n* = 14/55) from our compiled data set.

We tested for phylogenetic signal in (i) whether and in what direction shifts have occurred (‘shift’) and (ii) the magnitude of shifts (‘slope’) based on the variance of phylogenetically independent contrasts (PIC) for our empirical data set relative to the variance of PIC for randomly reshuffled species identities across the trait data set (iterated 20 000 times). *P*-values assess the fraction of reshuffled data sets that have lower PIC variance scores than our empirical data set, as implemented in the R library ‘picante’ (Kembel *et al*. 2010 R Core Team 2016 ).

We also used Blomberg's K (Blomberg, Garland & Ives 2003 ) and Pagel's λ (Pagel 1999 ) to measure the strength of signal relative to a BM model of trait evolution. K ranges from almost 0 to greater than 1, whereas λ ranges from 0 to 1 for both measures, values of 1 indicate BM evolution. Values of K less than 1 indicate related species resemble each other less than would be expected under BM, implying selection over drift, whereas values of K greater than 1 indicate related species resemble each other more than would be expected under BM (Blomberg, Garland & Ives 2003 ), also implying selection. Because K can depend on tree resolution, and the Phylomatic tree was only 25% resolved, we thinned the tree to eliminate terminal polytomies as recommended by Davies *et al*. ( 2012 ). After randomly removing species to leave only one per node, we then iteratively estimated K on the thinned trees, performing 30 iterations each for the phylogenies used to test for signal in shift and slope. We followed the same procedure to thin the molecular tree (which is fully resolved to genus level but to which species were added as polytomies). For our analyses using individual data sets, we present K values for unthinned trees because no trees were less than 60% resolved (Davies *et al*. 2012 ). However, we note that K values for thinned trees were similar. K and λ were calculated using the R library ‘phytools’ (Revell 2012 ), which provides P-values for the K statistic itself, gained by reshuffling species identities in the trait data set, calculating K for each iteration, and comparing the observed K to this null K distribution. The ‘phytools’ library also provides *P*-values for λ by performing a likelihood ratio test against the null hypothesis that λ = 0. We performed each of these tests for the pooled data sets and each individual data set, using both the Phylomatic and the molecular trees. We excluded two data sets (Menzel, Estrella & Fabian 2001 and Molnár *et al*. 2012 ) from the individual community analyses because after pruning only 3 and 10 species remained respectively.

We also tested the fit of an OU model of trait evolution to determine if shifts in flowering phenology might be constrained by stabilizing selection. We fit a single-optimum OU model for the pooled data and each data set, using both trees. Tests of OU model fit and significance were performed with the R libraries ‘geiger’ (Harmon *et al*. 2008 ), ‘phylolm’ (Ho & Ane 2014 ) and ‘OUwie’ (Beaulieu & O'Meara 2015 ), using likelihood ratio tests to compare the fit of OU vs. BM models.

## Methods

### Terminology

We acknowledge that the history of heterochronic terminology has been tumultuous [1,2,27]. For comparative purposes, here we follow the terminology of Ryan and Bruce [26], which has been the only prior comprehensive treatment of heterochrony in spelerpine plethodontids. We use the terms acceleration and deceleration, respectively, to refer to the relative advancement and delay of the timing of developmental events compared to ancestors (Figure 1 ). These terms are applied to processes effecting somatic and reproductive tissues, which may, or may not, result in a shift between life history categories (direct development, biphasic, paedomorphic). For example, if metamorphosis of a biphasic species occurs significantly earlier than metamorphosis of its biphasic ancestor then this would be an acceleration in the age (timing) of metamorphosis.

Also following Ryan and Bruce [26] and other studies [8], we use the terms neoteny and progenesis primarily to refer to somatic deceleration and reproductive acceleration, respectively, which are processes that can result in larval form paedomorphosis. There are multiple ontogenetic trajectories that can lead to an advancement or delay of a developmental event. Since shifts in maturation and metamorphosis could be considered changes to the ‘onset’ or ‘offset’ of a developmental trajectory, then terms implying a ‘rate’ (neoteny, progenesis, acceleration, deceleration) may not apply. Instead, terms such as predisplacement and postdisplacement have been used to describe such shifts in the timing of reproduction and metamorphosis [2,7].

Data on timing of metamorphosis (age at metamorphosis) and maturation (age of gonadal maturation) in months for 63 plethodontids were primarily derived from the literature and some personal and unpublished observations by colleagues (Additional file 1). This sampling included representatives of most of the major lineages of plethodontids: 26 species of spelerpines from four of the five genera (21 *Eurycea*, 2 *Gyrinophilus*, 2 *Pseudotriton*, and 1 *Stereochilus*), 14 species of desmognathines (*Desmognathus*), and 23 other plethodontids (2 *Aneides*, 1 *Batrachoseps*, 2 *Bolitoglossa*, 1 *Ensatina*, 1 *Hemidactylium*, 1 *Hydromantes*, 14 *Plethodon*, and 1 *Pseudoeurycea*). Our analyses were based on minimum estimates of age at metamorphosis. Direct developing species metamorphose prior to hatching, so we considered their age at metamorphosis to be the time prior to hatching, which for most species was approximately 2 months [21]. Most paedomorphic plethodontids do not metamorphose (obligately paedomorphic), and this is an independently derived state in multiple lineages of spelerpines [25]. Since we were most interested in reconstructing the ancestral timing of metamorphosis, we coded paedomorphic taxa as missing metamorphic data in our analyses of metamorphic timing. We dealt with the evolution of larval form paedomorphosis (compared to direct development or biphasic) in a separate analysis (described below).

We also used minimum age estimates for maturation for all 63 species, and we analyzed male and female maturation times separately. We used minimum age (as opposed to average or maximum age) because it is the most consistent and obtainable metric across species. Most referenced studies are based on evaluating gonadal development across age/size classes. Therefore, we used minimum age at gonadal maturation (which is observed morphologically), as opposed to age at first reproduction (oviposition or spermatophore drop), which are less commonly documented. For example, the minimum age of reproductive maturation for both male and female *Desmognathus ocoee* has been documented at 3 years [36,37]. Even though most female *D. ocoee* may not oviposit until year 4, we used 3 years to be consistent with other studies that are only based on gonadal maturation.

There have been phylogenetic based reconstructions of plethodontid life history: biphasic *vs.* direct development [28] and biphasic (metamorphic) *vs.* paedomophic [25]. However, these three states have not been reconstructed in the same analysis. Therefore, we also reconstructed ancestral life history (direct development, biphasic, paedomorphic) for 100 plethodontids, including all North American and Eurasian genera, as well as a newly described paedomorphic species (*E. subfluvicola*[38]). Life history information for these species is well established and was taken from the literature (Additional file 1). We only included three representative genera from the tropical radiation (bolitoglossines), due to the limited number of lengthy *Rag1* sequences available for this group (see below), but it is clear that this radiation is monophyletic, and all species are thought to be direct developers. In other words, including additional bolitoglossines to our analyses would not significantly change the results presented here. The purpose of this analysis was primarily to reconstruct the origins of paedomorphosis within spelerpines, which was necessary for subsequent tests of progenesis *vs.* neoteny (see below). However, we also performed additional life history reconstructions and included the outgroup families Amphiumidae and Rhyacotritonidae to further test the ancestral life history mode of plethodontids (see Results).

### Phylogeny

We reconstructed two chronograms of plethodontids which included representatives of: (1) all 100 taxa for ancestral life history analysis and (2) the 63 taxa for which we have data on timing of metamorphosis and maturation. The chronograms were each based on complete datasets of 1,033p of the recombination activating gene 1 (*Rag1* Additional file 1). *Rag1* was chosen because it is a conserved nuclear locus that was already available for most taxa included in this study, and provides a close approximation to the topologies and branch lengths of previously reconstructed salamander phylogenies [25,28,34,39-43]. The sequences were primarily derived from previous phylogenetic datasets of plethodontids [28], spelerpines [25], *Plethodon*[44], and additional sequences for nine species from the genus *Desmognathus* that we collected for this study (Additional file 2).

Sequences were aligned using Sequencher v. 4.8 (Gene Codes, Ann Arbor, MI, USA), and the alignment was unambiguous with no missing data. MrModeltest v. 2.2 [45] was used to determine the most appropriate model of nucleotide substitution for each codon position (Additional file 3). The chronograms were estimated using BEAST v. 1.6 [46]. We applied the best-fitting models determined above, and the analysis was based on an uncorrelated lognormal molecular clock and Yule speciation prior across the tree. The fossil record of plethodontids is very limited [47,48], so we used the base of the crown group of extant plethodontids as a calibration point. The estimates of the deepest divergence for this clade are in the range of 41 Mya to 99 Mya, with average estimates at approximately 73 Mya [40,41,49-51]. We applied a normally distributed calibration prior for the crown group of plethodontids, with a mean of 73 Mya and standard deviation of 6 Mya. This combination of parameters yielded a 95% prior distribution between 85 Mya and 65 Mya, representing a reasonable range of potential dates for this clade based on previous studies. Analyses are based on relative branch lengths of the chronograms, and would be the same regardless of the overall time scale. Both analyses were run twice independently for 20 million generations with trees saved every 1,000 generations (total 40,000 trees). Likelihood values across generations were evaluated in Tracer v. 1.5 [52] and the first 25% of generations from both runs (10,000 trees) were conservatively discarded as burnin, which was well beyond stationarity. Both chronograms (100 taxa and 63 taxa) were similar in branch lengths and topology. We used the 30,000 post-burnin trees, from the phylogenetic analysis of each dataset, for their respective reconstructions (see below).

### Ancestral state reconstruction

Ancestral life histories, ages of minimum metamorphosis, and ages of minimum maturation (males and females) of plethodontid salamanders were reconstructed using Bayesian methods. Categorical and continuous ancestral reconstructions were performed in BayesTraits v. 2.0 [53] using �yesMulitState’ [54] and a Markov Chain Monte Carlo (MCMC) model. Reconstructions were based on all 30,000 post-burnin Bayesian chronograms from the phylogenetic analysis in BEAST. Uniform priors from 0 to 100 were applied for each analysis, and acceptance rates were between 20% and 40%. Each analysis was run for 5 million generations with samples taken every 1,000 generations, with the first 1 million generations of each run discarded as burnin (that is, ancestral state results were based on 4 million post-burnin generations =𠂔,000 samples).

Life history was reconstructed as an ordered, categorical trait with three states (direct development, biphasic, paedomorphic). Age-based traits (metamorphosis and maturation) were analyzed using both a 𠆌ontinuous’ (number of months) and categorical coding (number of years). The categorical analyses allowed for testing among alternative states for some key ancestral nodes (for example, the age of metamorphosis and maturation for the clade Spelerpini described below). We divided continuous ages into four metamorphic age categories: 1 =� months or less 2 =� to 23 months 3 =� to 35 months 4 =� months or more. A similar strategy was applied for both minimum male and female maturation, but included two additional age categories: 1 =� months or less 2 =� to 23 months 3 =� to 35 months 4 =� to 47 months 5 =� to 59 months 6 =� months or more.

For life history reconstructions, transitions were only allowed between biphasic and direct development or biphasic and paedomorphic, but not between direct development and paedomorphic (transitions set to zero probability). Likewise, transitions between categorical age states were also ordered numerically by setting non-numerically adjacent categories to zero probability. For example, for the four metamorphic age categories, transitions were allowed in both directions between categories 1 and 2, 2 and 3, and 3 and 4, but not between 1 and 3, 1 and 4, or 2 and 4. The same strategy was applied to the six maturation age categories for males and females. Implementing ordered categories enforces ancestors to sequentially evolve through age categories (without skipping), and it also reduces the number of possible transitions for our reconstructions. Our analyses with ordered age categories were always a better fit than analyses with unrestricted transitions between categories.

All ordered transitions between states (within traits) were set to equal rates (that is, one-rate models). For each trait we compared the fit of a one-rate model to a model where transition rates (for ordered states) were allowed to vary (multi-rate models). The lowest AIC score indicates the best fitting model. 𢁪IC values σ were considered to be negligible differences between models, values 𢙓 were considered moderately strong, and values � were considered very strong support for rejecting the alternative model with the higher AIC score [55]. For each trait, a one-rate model was a substantially better fit than the multi-rate model (Life history 㥊IC =�.62 Metamorphic Age 㥊IC =𠂘.59 Male Maturation Age 㥊IC =𠂙.12 Female Maturation Age 㥊IC =�.08).

We used BayesTraits to test for differences among ancestral conditions for key nodes in the phylogeny of spelerpines. These analyses were performed by fixing (𠆏ossilizing’) nodes to alternative states and comparing harmonic means (hm) for each run by calculating differences in Log Bayes Factors (LBf). The lowest LBf indicates the best fitting model [53,54]. LBf values σ were considered negligible differences between models, values 𢙓 show were considered moderately strong, and values � were considered very strong support for rejecting the alternative model with the higher LBf.

Continuous trait analyses of the timing of metamorphosis, and male and female maturation were performed in BayesTraits [53,54] using MCMC under a Brownian Motion (𠆌ontinuous Random Walk’) model. Reconstructions were based on all 30,000 post-burnin Bayesian chronograms from the phylogenetic analysis in BEAST. Run generation parameters were the same as described above for multistate analyses, and results were based on 4,000 post-burnin samples.

We used the Bayesian 95% Highest Prior Density (HPD) credibility interval of the ancestral state of spelerpines to determine which taxa have metamorphic ages that arose from acceleration (less than the 95% HPD interval), deceleration (greater than the 95% HPD interval), or stasis (within the 95% HPD interval in other words, showing a lower probability of being different than our ancestral state estimate). Additionally, to test if larval form paedomorphosis arose from neoteny or progenesis, we examined timing of maturation across the evolutionary shift from metamorphosis to paedomorphosis in a large clade of paedomorphic *Eurycea* from the Edwards Plateau of Central Texas (Figure 2 node D). If paedomorphosis arose via progenesis (early maturation [1,26]), we would expect a significant reduction in maturation time concomitant with the evolution of paedomorphosis. In contrast, if paedomorphosis arose from neoteny (delayed somatic development [1,26]) then we would not expect significant differences in ancestral maturation patterns during the transition from metamorphosis to paedomorphosis. We quantified significant changes in ancestral maturation (for males and females separately using both categorical and continuous analyses). For the categorical analyses we fixed the ancestral states at four nodes spanning the evolution of paedomorphosis in Edwards Plateau *Eurycea* (Figures 2 nodes B to E) to the six alternative maturation categories (years). For a given node we used Log Bayes factors to compare which of the maturation category is the best fit, and which categories were significantly worse (methods described above). Again, for progenesis we would expect that the best fitting maturation ages would shift to younger age categories across these nodes, whereas neoteny should show no change (or an increase) in maturation age categories. We further compared the 95% HPD interval of continuous maturation age reconstructions from BayesTraits (above) for these nodes (B to E) to evaluate potential reductions in maturation time (progenesis). Male and female maturation was analyzed separately for both categorical and continuous methods.

**Bayesian reconstruction of ancestral life history modes of plethodontid salamanders.** Three ordered alternative life history states are considered: direct development (yellow), biphasic (dark grey), and paedomorphic (blue). Bayesian ancestral state reconstructions were performed in BayesTraits (see Methods). Pie diagrams at each node show the proportional probability (prob.) of each state, and the highest probability subtends each node. The phylogeny is based on Bayesian analysis of *Rag1* sequences in BEAST. See also Additional file 4.

## Discussion

Plant–soil feedbacks can influence plant species performance and competitive ability, with implications for community assembly (Bever, 2003 Bonanomi *et al*., 2005 ). Consequently, considerable effort has been invested in understanding how plant species respond to their own soil biota and to soil biota cultivated by other species, including whether feedback responses can be predicted from plant species relatedness. Despite evidence that both the identity of biotic partners and the response of plant species to those partners are linked to phylogenetic relatedness (Barrett *et al*., 2016 Hoeksema *et al*., 2018 Giauque *et al*., 2019 ), attempts to identify a phylogenetic signal in feedback responses have produced mixed results. Using an extensive dataset compiled from plant–soil feedback studies (Crawford *et al*., 2019 ), we show that: there is a strong phylogenetic signal in plant–soil feedbacks the phylogenetic signal arises primarily through nondirectional divergence of feedback responses over time with a slight tendency for responses to become more negative with greater phylogenetic distance (see also Crawford *et al*., 2019 ) and the pattern of divergence is consistent with occasional major co-evolutionary shifts between plants and soil microbes rather than continuous gradual divergence.

Much research has examined whether there is a directional trend in feedback responses linked to phylogenetic relatedness. This is due largely to the putative importance of negative feedbacks in promoting coexistence and invasion, and positive feedbacks in promoting dominance by single species (Mehrabi & Tuck, 2015 Fitzpatrick *et al*., 2016 Kempel *et al*., 2018 Kuťáková *et al*., 2018 ). Our findings reiterate those of Crawford *et al*. ( 2019 ) in showing some evidence for a slight negative trend in feedback response with increasing phylogenetic distance. Such an outcome should favour coexistence among more distantly related species and thus promote communities with greater phylogenetic diversity (Bonanomi *et al*., 2005 ).

Nevertheless, our analysis highlights that any negative trend in feedback outcomes is slight compared with the overall increase in variance due to divergence in both directions over time. An increase in the variance of feedback responses over evolutionary time based on data from multiple studies is consistent with our understanding of plant–soil feedbacks, where the net effect of pathogens, mutualists and other components of the soil biota does not consistently alter plant performance in a particular direction (Jiang *et al*., 2020 ). Strong directional trends should only arise in specific situations where there are compelling reasons to expect a disproportionate influence of either pathogens or mutualists on focal species (e.g. Liu *et al*., 2012 ). The slight negative trend we observe could reflect a higher specificity of soil pathogens relative to soil mutualists, which could result in plants benefiting more through the loss of pathogens in soils of more distantly related species, relative to the cost of losing mutualists. The difference between what theory might predict about phylogenetic signals in specific situations or case studies and what theory predicts when integrating across data from multiple studies may be one reason why has proven difficult to identify a clear phylogenetic signal in the outcome of plant–soil feedbacks.

The increase in the variance of feedback responses due to divergence in both directions over evolutionary time implies that close relatives tend to respond to each other’s soil microbiota in similar ways, but that the magnitude and direction of feedback responses become more variable with greater phylogenetic distance. Consequently, it may only be possible to predict feedback outcomes with any accuracy among closely related species: phylogenetic distance is of less help in predicting the response among distantly related species.

Much of the increase in variability in feedback responses over evolutionary time was due to more extreme values than expected under a model of gradual divergence. This is consistent with major shifts associated with some plant lineages being constrained by co-evolution with specialist microbiota. Such lineages should disproportionately benefit from escaping specialist natural enemies or disproportionately suffer from losing specialist mutualists, an outcome known to occur in some plant families. For example, the Orchidaceae (orchids) and Ericaceae (heaths) form specialized associations with orchid and ericoid mycorrhizal fungi, Fabaceae (legumes) rely on soil bacteria (rhizobia) for nitrogen fixation, and Poaceae (grasses) cultivate distinct microbial communities and are more responsive to those communities than other life-forms (Hoeksema *et al*., 2010 Davison *et al*., 2020 ). In the data we analysed, seven family pairs had more extreme feedback responses than average, which included the families Fabaceae and Poaceae (Fig. 5 there were no Orchidaceae in the data). While it is important not to over-interpret these results, because most between-family comparisons involved relatively few species and feedback responses, modelling the variation associated with family-level mean responses (Fig. 5) explained much of the increase in variation in feedback responses with increasing phylogenetic distance, leaving a weaker residual phylogenetic signal (parameter *k* was much closer to zero in model 7 Fig. 3). Hence, increasing divergence in feedback response with greater phylogenetic distance could be largely explained by the differing response of species in certain families to the microbiota associated with species in other families. Understanding variation in feedback responses within and among families may be one way to increase the predictability of feedback outcomes among more distantly related species.

### Conclusions

While relatedness can help to predict the outcome of some biotic interactions (Parker *et al*., 2015 Bufford *et al*., 2016 ), attempts to predict how plant species will respond to each other’s soil microbiota based on relatedness have produced mixed results. We have clarified how phylogenetic signals in plant–soil feedback outcomes could arise and used a recent compilation of data to quantify the nature of the phylogenetic signal. Our results reiterate other studies that provide evidence for, at best, a weak directional trend and highlight that knowledge of plant species relatedness is most likely a weak predictor of community-level outcomes for plant–soil feedbacks. Our results indicate that it is difficult to predict how species will respond to each other’s soil microbiota from a knowledge of the phylogenetic distance between plant species alone, other than to say that more closely related species tend to have more similar responses. Nevertheless, this apparent loss in predictability could be offset by a divergence pattern that suggests feedbacks become constrained in some lineages by co-evolution with specialist mutualists or enemies. If so, feedback outcomes among distantly related species might be predictable from knowledge of the lineages involved and how species in those lineages respond to each other’s soil biota (e.g. Fig. 5). Identifying families for which feedback responses have been constrained by co-evolution with specialist soil microbiota and examining feedback outcomes for species within and among those families could improve our ability to predict outcomes.

## Methods and Results

### MATHEMATICAL AND COMPUTATIONAL DETAILS

We programmed all analyses presented herein in the flexible scientific computing language, R ( R Development Core Team 2010 ). The code we used for the analyses of this paper is available as an Appendix S1 and updated versions will be distributed as part of the R phylogenetics package “phytools” ( Revell 2011 ). The simulation, MCMC, and MCMC diagnostics code, all provided in Appendix S1, call functions from the phylogenetic packages “ape” and “geiger” ( Paradis et al. 2004 Harmon et al. 2008 ), and from the MCMC diagnostics package “coda” ( Plummer et al. 2010 ).

The model presented herein is for the evolution of a single, continuously valued character on a rooted phylogenetic tree with branch lengths in units proportional to time. Under this model, evolution proceeds by a Brownian motion process on the tree ( Cavalli-Sforza and Edwards 1967 Felsenstein 1985 ). The instantaneous variance of the evolutionary process in this model (the evolutionary rate) changes from low to high, or high to low, once and only once in the tree. Accordingly, the model has four parameters: the two evolutionary rates ( σ 2 _{1} and σ 2 _{2} ) that prevail on either side of the rate shift **θ** , a 2 × 1 vector containing the branch identity and the position along the branch at which the evolutionary rate transitions from σ 2 _{1} to σ 2 _{2} (or vice versa) and finally, the ancestral trait value at the root node of the tree ( α ). The data consist of values for a continuously distributed character for all tip species, and a bifurcating or multifurcating rooted phylogeny with branch lengths. We focus on estimating σ 2 _{1} , σ 2 _{2} , and **θ** from the tree and character data.

(A) Stochastic five taxon tree. Branch lengths (*v*) are shown above each edge, denoted by the node that they precede. The number below each edge is the fraction of total branch length in the tree represented by the overlying edge. (B) Calculation of **C**_{1} and **C**_{2} for branches painted with blue or red, respectively. Note the hypothesized shift from blue to red occurs fraction *k* along the branch leading to descendant species *B*, *C*, and *D*, and is indicated by **θ**.

We designed our MCMC run as follows. We first initialized the chain with starting values for the parameters in the model. These parameter values can optionally be supplied by the user but to improve computational performance, by default our implementation is programmed to choose reasonable starting values for the parameters (described below). For each generation of the chain, we proceeded to cyclically update each parameter in the model (i.e., to say, updating all four parameters took four generations) with a random step from a proposal distribution for that parameter. We used Gaussian proposal distributions, centered on zero, for changes to the rates ( σ 2 _{1} and σ 2 _{2} ) and the ancestral value ( α ), and we used a symmetric exponential distribution (i.e., an exponential distribution whose density has been halved and reflected across the ordinate) for changes to the shift point ( **θ** ). Negative values, in this case, meant a change toward the root of the tree. We also tried a Gaussian proposal distribution for **θ** , although this made little difference on the test datasets that we analyzed. We have left the symmetric exponential as the default because we believe that it will allow for a more thorough exploration of the tree by the MCMC chain. In our implementation of this MCMC algorithm, the variances of each proposal distribution can be specified by the user.

Changes to **θ** that are larger than the remaining length of the current edge (i.e., branch) also require one or multiple decisions, according to the following algorithm: (1) if going rootward down the tree, and not at the root, we proceeded to the parent edge or the other daughters (allowing for multifurcation) all with equal probability (2) if going tipward up the tree and not at a tip, we proceeded to either daughter edge with equal probability (3) if at the root, we proceeded to any other daughter edge with equal probability and, finally, (4) if at a tip, we reflected the change back along the tip edge exactly the distance it would have otherwise exceeded the terminal node.

If allowed to proceed as an unhindered random walk on the tree, this algorithm will eventually sample all edges on the tree with a probability directly proportional to their lengths (not shown). To ensure proper mixing, we also allowed a small fraction of steps (say, 5%, but this can also be modified by the user) to result in a move to a randomly selected branch with probability proportional to its length. Half of the time that such a move was performed we also switched the values of σ 2 _{1} and σ 2 _{2} .

Our proposal distributions for σ 2 _{1} , σ 2 _{2} , and α are symmetric. Proof of symmetry of the proposal distribution for **θ** is given in Appendix S2. Symmetry of the proposal distributions is an important property because it allows us to set the Hastings ratio to 1.0 ( Hastings 1970 Yang 2006 see below).

In the present study, we used a log-normal prior probability distribution centered on 0.0 for the ratio , and a uniform prior on the log-scale for the geometric mean of σ 2 _{1} and σ 2 _{2} . We also used an unbounded uniform prior for α , and a uniform prior for **θ** . For **θ** , this means that the prior probability of the shift point being on any edge of the tree is exactly proportional to the length of that edge. We explored various distributions for the prior on σ 2 _{1} and σ 2 _{2} and our analyses did not seem to be especially sensitive to the prior, except for under one specific set of conditions that we will discuss at greater length below (see Discussion).

In our implementation of this method, the user can supply randomly or nonrandomly chosen starting parameter values for σ 2 _{1} , σ 2 _{2} , and α to initialize the MCMC run. However, by default, we used a previously derived analytic solution for the maximum likelihood estimate (MLE) of the evolutionary rate under a single rate regime (i.e., σ 2 _{1}=σ 2 _{2}= MLE (σ 2 ) for example, O’Meara et al. 2006 ) and we used the MLE of the root node ancestral value under a single rate for α ( Rohlf 2001 O’Meara et al. 2006 ). This was done mainly to improve computational performance and decrease burn-in by starting with values for σ 2 _{1} , σ 2 _{2} , and α that we might expect to be roughly of the correct magnitude. To initialize **θ** , we randomly selected a location for the shift point between rate regimes on the tree. The probability of choosing a shift point along any branch on the tree was set to be proportional to the branch length meaning, for instance, that one would be exactly twice as likely to start with a random shift point on a branch with length 2*v* than on one with length *v*. This is essentially equivalent to choosing a random value of **θ** from our prior probability distribution for **θ** .

Each posterior sample for this analysis consists of values for σ 2 _{1} , σ 2 _{2} , α , and **θ** a value for the log-likelihood and a list of the tip labels for the set of tips in state σ 2 _{2} . Because σ 2 _{1} and σ 2 _{2} depend both on **θ** and the set of tips in state σ 2 _{2} (i.e., whether σ 2 _{1} or σ 2 _{2} is the derived rate), we propose the following algorithm for preprocessing the posterior sample from our MCMC run. First, we found the median shift point in the posterior sample. This was done by identifying the sampled point with the minimum summed distance to all the other points in the sample (although other options for this are certainly possible, see Discussion). Next, we went through each sample in the posterior, splitting the tree at the shift point for that particular sample, and then assigning the derived and ancestral rates to edges or fractions of edges in the ancestral and derived subtrees, respectively (this might be σ 2 _{1} and σ 2 _{2} , or σ 2 _{2} and σ 2 _{1} , depending on the membership to our list of labels for that sample). Finally, we reattached the two subtrees and computed the average rates rootward and tipward from the median shift point. Note that the collection of edges and fractions of edges rootward or tipward of the median shift point can include one or both rate categories depending on how the estimated shift point for that sample differs from the median shift point. For consistency across samples, we now assigned σ 2 _{2} always as the derived rate and σ 2 _{1} as the ancestral rate.

### SIMULATED EXAMPLE

We generated and analyzed a simulated phylogenetic tree and phenotypic dataset to illustrate the application and results of our method. This simulation also forms the basis for our performance analysis of the method, below (see section Performance analysis).

We first simulated a stochastic pure-birth phylogeny with 100 terminal species. We then randomly selected a position on the tree as the location of the rate shift for our quantitative character. This evolutionary scenario is illustrated by the colored branches of the phylogenetic tree of Figure 2. The rate shift is located in a random position on the labeled branch “147,” where branches are identified by the number of the descendant node, and nodes are numbered according to the conventions of “phylo” objects in the “ape” phylogenetics package for R ( Paradis et al. 2004 Paradis 2006 ). We next evolved a continuous character on the phylogeny under Brownian motion with the starting value α= 0.0 at the root of the tree. The simulated evolutionary process had instantaneous rates σ 2 _{1}= 1.0 on edges rootward of the shift point (blue branches in Fig. 2), and σ 2 _{2}= 10.0 on edges tipward of this point (red branches).

Stochastic, 100 taxon tree used for the simulated example. Phenotypic data were generated on this tree with a 10-fold higher evolutionary rate along the branches painted in red. The node tipward of the rate shift (numbered “147,” by the “phylo” convention in “ape”) is also indicated. Numbers presented in parentheses below or adjacent to branches are the posterior probabilities that the rate shift occurred on each labeled edge from the illustrative example. Only posterior probabilities ≥ 0.001 are reported.

For the MCMC run, we set the following control parameters. We set the standard deviations of the Gaussian proposal distributions for σ 2 _{1} and α to 0.5, and the standard deviation of the proposal distribution for σ 2 _{2} to 1.0. We set the rate parameter ( λ ) for the exponential proposal distribution for shift point moves to λ= 5.0 . Because random deviates from the exponential proposal distribution for tree moves were also assigned random sign with equal probability, the realized proposal distribution for tree moves has the following density: for *x* ≥ 0 and otherwise. We also set the probability of proposing a move to a random point in the tree to 0.05. We set the variance of the log-normal prior for the rate ratio to 2.0 finally, we used a uniform prior for α and **θ** .

We ran the Metropolis–Hastings MCMC algorithm for 100,000 generations, sampling every 10 generations. Figure 3A shows the trace of the log-likelihood sampled every 100 generations (i.e., every 10 samples) from the entire MCMC run. In this example, we can see that the chain converges rapidly. We then preprocessed the posterior sample, as discussed above. Figures 3B and 3C show the frequency histograms of the posterior samples obtained after preprocessing the posterior samples for σ 2 _{1} and σ 2 _{2} , with the first 10,000 generations excluded as burn-in. We computed effective sample sizes (ESSs) and 95% credible intervals (CIs) for the mean of the posterior distribution for σ 2 _{1} , σ 2 _{2} , and α (Table 1). This can be done quite easily using the R package “coda” ( Plummer et al. 2010 ) or, alternatively, in the Java program “Tracer” ( Rambaut and Drummond 2009 ) that has the benefit of a very user friendly graphical interface. We recommend ESSs for the evolutionary rates of at least 100. If an ESS less than 100 is obtained, then the MCMC can be rerun and the post burn-in samples combined (e.g., Ho et al. 2007 ). We computed the estimated values of σ 2 _{1} , σ 2 _{2} , and α as the mean of the preprocessed posterior sample (excluding the burn-in). All were very close to the generating conditions here (Table 1). The choice of the posterior arithmetic mean as an estimator is arbitrary. We might instead compute the posterior median or geometric mean (although in this example, the arithmetic means of the posterior sample are quite close to the generating parameter values). The ESSs for σ 2 _{1} and α were quite high, indicating relatively low autocorrelation in the posterior samples for these parameters however, the ESS was considerably lower for σ 2 _{2} , suggesting that we might do better by adjusting the variance of the proposal distribution for this parameter.

Results from the simulated example. (A) Trace of log(*L*) by generation number for the 100,000 generation MCMC analysis. The log(*L*) was sampled every 100 generations. (B) Frequency histogram of the post burn-in posterior sample for σ 2 _{1} . The generating value of σ 2 _{1}= 1.0 is indicated by the vertical dashed line. The mean from the posterior sample is given by the vertical solid line, whereas the 95% credible interval (CI) is given by the shaded area. (C) Same as (B), but for σ 2 _{2} . The generating condition, in this case, was σ 2 _{2}= 10.0 .

Parameter | Generating value | Estimate (mean from posterior sample) | Effective sample size | 95% credible interval |
---|---|---|---|---|

σ 2 _{1} | 1.0 | 1.074 | 2172 | (0.7373,1.4440) |

σ 2 _{2} | 10.0 | 11.43 | 247.0 | (5.0812,19.8603) |

α | 0.0 | 0.042 | 3520 | (−0.7087,0.7484) |

[147,0.0084] | [147,0.0536] |

We also computed an approximate median rate shift point by computing all pairwise distances between shift points in the post burn-in posterior sample, and then selecting the shift point with the minimum summed distance to all the other points in the sample, as described above. This value, which corresponds very closely to the generating shift point in this example, is also reported in Table 1. This procedure will not be computationally feasible for very long MCMC runs however, in that case one could instead use a sparser sample of shift points from the posterior (taken, say, every 100 or 1000 generations, instead of every 10 generations as in this example). Finally, we computed the posterior probability of the shift point being on each edge of the tree. For all edges with posterior probability ≥ 0.001 , we have plotted these probabilities below or adjacent to the corresponding branches in Figure 2. Nearly, all (96%) of the posterior density for the location of the rate shift is on the generating edge in this case (Fig. 2).

### PERFORMANCE ANALYSIS

To assess the performance of the method more generally, we conducted two sets of simulation tests of the method. First, we conducted the following simulation 80 times in total (20 times for each of the four sets of generating rates, described below): (1) We simulated a stochastic, pure-birth, *N* = 100 species phylogenetic tree with branch lengths. (2) We picked a shift point at random on the tree. Although in theory our method should be appropriate to detect evolutionary rate shifts in subclades of any size, we anticipate that the method will suffer from low power when the number of species rootward versus tipward of the rate shift is extremely unbalanced. Thus, to avoid this issue in our early performance analysis of the method, we decided to exclude randomly chosen shift points with fewer than 20 or more than 80 descendant species (in other words, we excluded splits for simulation in which more than 80% of the taxa in the phylogeny were on one side of the split). (3) We then simulated data on the tree with generating conditions as follows: the branches ancestral to the shift point evolved with rate σ 2 _{1}= 1.0 , whereas the derived branches evolved with rates σ 2 _{2}= 0.1 , 1.0, 5.0, or 10.0 (20 simulations each). (4) We initiated the MCMC chain as in the illustrative example, above, and ran the chain for 100,000 generations. Normally users would probably run multiple MCMC chains, adjusting the control parameters to ensure proper mixing and convergence. Here, we merely adjusted the control parameters (mainly the variances of the proposal distributions or the number of generations in the chain) and reran any MCMC for which the ESSs for σ 2 _{1} or σ 2 _{2} was less than 100. We did this until we obtained ESSs greater than 100 for all runs. (5) We computed summary statistics on the posterior sample, excluding the first 20,000 generations of the sample. In addition to the summary measures reported in Table 1, we also computed the patristic distance between the inferred shift point () and the generating value of **θ** for each replicate (i.e., the minimum edge distance connecting the two points in the tree).

A summary of the results from these analyses is given in Table 2. Results for all of the 80 simulations are in Appendix S3. For each set of simulation conditions, Table 2 gives the arithmetic means of σ 2 _{1} , σ 2 _{2} , and α (the geometric means and medians for σ 2 _{1} and σ 2 _{2} are reported in Appendix S3), the proportion of simulations in which the correct node was inferred, and the fraction of simulations in which the 95% CIs for σ 2 _{1} did not overlap (our estimate of σ 2 _{2} ) and vice versa. This latter frequency is analogous to the statistical “power” of the method (or its type I error, for the generating conditions σ 2 _{1}=σ 2 _{2}= 1.0 alternatively, 1.0 minus this fraction is the type II error rate of the method if σ 2 _{1}≠σ 2 _{2} ). This procedure is somewhat ad hoc as the model itself explicitly assumes that σ 2 _{1}≠σ 2 _{2} however, the results of Table 2 and Appendix S1 suggest that we will only be infrequently mislead to believe that σ 2 _{1}≠σ 2 _{2} if they are in fact equal. We also report the mean distance between and **θ** for each simulation condition. In general, parameter estimates are pretty good, and 95% CIs nearly always included the generating parameter value. The method also has excellent success in identifying the position of the rate shift to a specific edge in the tree, particularly when the proportional difference between σ 2 _{1} and σ 2 _{2} was high (Table 2).

Simulation 1: σ 2 _{1}= 1.0, σ 2 _{2}= 0.1 | Simulation 2: σ 2 _{1}= 1.0, σ 2 _{2}= 1.0 | Simulation 3: σ 2 _{1}= 1.0, σ 2 _{2}= 5.0 | Simulation 4: σ 2 _{1}= 1.0, σ 2 _{2}= 10.0 | |
---|---|---|---|---|

(SD) | 1.05 (0.175) | 1.06 (0.172) | 1.241 (0.389) | 1.16 (0.158) |

(SD) | 0.129 (0.0618) | 1.08 (0.173) | 4.326 (1.463) | 10.66 (3.522) |

(SD) | −0.137 (0.132) | 0.013 (0.409) | 0.0400 (0.0331) | −0.083 (0.057) |

ESS(σ 2 _{1}) | 679.6 | 691.0 | 324.0 | 746.4 |

ESS( σ 2 _{2} ) | 618.1 | 633.8 | 368.8 | 419.4 |

ESS(α) | 773.8 | 780.2 | 806.5 | 780.8 |

On CI( σ 2 _{1} ) | 1.00 | 0.95 | 0.80 | 1.0 |

On CI(σ 2 _{2}) | 0.85 | 0.95 | 0.80 | 0.9 |

No CI overlap | 1.00 | 0.00 | 0.75 | 1.0 |

Correct edge | 0.95 | 0.15* | 0.70 | 0.85 |

Distance (SD) | 0.0531 (0.0473) | 0.195* (0.146) | 0.0821 (0.0533) | 0.0533 (0.0587) |

Second, we also explored the performance of the method on smaller and larger phylogenies than the stochastic *N* = 100 trees described above. To do this, we used the following procedure: (1) We simulated 20 pure-birth phylogenies with each of the following sizes *N* = 30 , 50, 70, and 200. (2) On each tree, we chose a random rate shift location such that no less than 20% and no more than 80% of the species in the tree were found tipward from that point. (3) We then simulated the evolution of a continuously valued character with σ 2 _{1}= 1.0 and σ 2 _{2}= 10.0 as the ancestral and derived rates, respectively. (4) We ran our MCMC chain on each simulated dataset and tree, using the conditions described previously, and then computed summary measures from the posterior sample. Again, we reran MCMCs for which ESSs of either σ 2 _{1} or σ 2 _{2} was less than 100.

The results from these analyses are summarized in Table 3 and specific results from all analyses are given in Appendix S3. In general, we found that the method had remarkable success in identifying the location of the rate shift in the tree. Evolutionary rates were biased on small trees (in particular, such that and were more similar) but this bias nearly vanishes for the larger simulated phylogenies in the study.

N=30 | N=50 | N=70 | N=200 | |
---|---|---|---|---|

(SD) | 1.93 (0.909) | 1.33 (0.414) | 1.17 (0.353) | 1.066 (0.146) |

(SD) | 8.80 (5.20) | 10.17 (5.09) | 10.02 (3.20) | 9.54 (1.34) |

(SD) | 0.0955 (0.401) | −0.156 (0.404) | −0.029 (0.466) | 0.020 (0.493) |

ESS(σ 2 _{1}) | 483.6 | 526.7 | 683.8 | 763.0 |

ESS( σ 2 _{2} ) | 393.1 | 277.4 | 314.6 | 558.2 |

ESS(α) | 815.0 | 858.1 | 859.1 | 828.3 |

On CI( σ 2 _{1} ) | 0.90 | 1.00 | 0.95 | 0.95 |

On CI(σ 2 _{2}) | 0.85 | 0.85 | 0.90 | 1.00 |

No CI overlap | 0.55 | 0.85 | 0.95 | 1.00 |

Correct edge | 0.80 | 0.85 | 0.95 | 0.90 |

Distance (SD) | 0.173 (0.232) | 0.133 (0.144) | 0.062 (0.060) | 0.057 (0.048) |

### EMPIRICAL TEST

Finally, we also examined the performance of the method using an empirical dataset and tree. We analyzed the evolution of body size (measured as log-SVL: “snout-to-vent length”) in a 32 species subtree extracted from the 100 taxon *Anolis* phylogeny of Mahler et al. (2010) . We chose to analyze this subtree rather than the whole Caribbean *Anolis* phylogeny because the results of prior studies (e.g., Butler and King 2004 ) suggest that more than two different evolutionary processes might govern the evolution of this lizard group in the Caribbean. We focused on the subtree given in Figure 4, which contains the *Anolis sagrei* group on Cuba the *A. distichus* group on Hipaniola *A. cristatellus* and related Puerto Rican lizards and, finally, the endemic radiation of six *Anolis* species on Jamaica.

Phylogeny of a subtree from the radiation of Caribbean *Anolis*. Rates and posterior densities are from an analysis of body size evolution on this tree. Posterior probabilities of the rate shift being on each edge of the tree (if > 0.01 ) are represented by the filled fraction of each pie graph. Most of the posterior density for a rate shift suggests an increase in the evolutionary rate at the base of or within the Jamaican diversification of *Anolis*. This finding is consistent with prior studies showing that the rate of evolution is increased on newly colonized islands, as Jamaica is the only island that was colonized de novo in this subtree (all other inferred colonizations are secondary or back-colonizations see Mahler et al. 2010 ). The total tree length is scaled to 1.0 in this example.

We optimized the MCMC as follows. We used Gaussian proposal distributions for σ 2 _{1} , σ 2 _{2} , and α , with variances 0.015, 0.050, and 0.90. We used a prior on the ratio of the log-transformed rates with variance of 4.0. A summary of the results from this analysis is given in Figure 4. We found that the vast majority of the posterior density for a rate shift in our model was found either at the base of or within the Jamaican radiation. The estimated rate tipward of the shift was about 8.5 times higher than the estimated rate rootward of the shift. Note that Jamaica is the only island in this phylogeny that is hypothesized to have been colonized de novo in this subtree (see Mahler et al. 2010 ). Consequently, this result is consistent with our impression based on prior studies that the evolutionary rate is higher on newly colonized islands when ecological opportunity is high ( Mahler et al. 2010 ).

## Results and Discussion

### Simulation

#### Varying the magnitude of the selection shift

I simulate data sets assuming a single divergence point in a topology relating eight hypothetical taxa (Figure (Figure1). 1 ). I use a discrete gamma distribution with four rate categories and the same shape parameter *α* to approximate the variation in both spatial and temporal rates [25]. To explore the sensitivity of the method to the strength of the temporal selection shift, I vary the degree of rate variation (smaller *α* implies greater rate variation) and the fraction of sites *θ* experiencing a selection shift across the divergence point. When a site is selected to shift rate, it randomly selects a new rate class from the discrete gamma distribution. All other parameters do not vary in this first set of simulations. In particular, the simulated divergence point is located at relative position *l* = 0.9 on branch *b* = 8 in the topology *τ* of Figure Figure1 1 with *t*_{j}= 0.1 expected mutations (transition/transversion ratio *κ* = 2) per site along each branch *j* = 1. 13. C code implementing Markov chain Monte Carlo sampling of the posterior distribution analyzes each simulated alignment. Posterior statistics of model parameters are computed along with the Bayes factor *B*_{DP}in favor of a divergence point somewhere in the tree. When log_{10}*B*_{DP}> 1, there is strong support for a divergence point, which then allows conditional estimation of *θ*, *l*, and the Bayes factor *B*_{j}favoring a divergence point located specifically on branch *j*. All these latter statistics are based on the subset of MCMC samples that have a divergence point.

**Simulation tree**. The phylogenetic tree used for simulation. There are eight taxa, labeled 0 to 7, related according to the depicted topology with all branch lengths equal. Each simulation assumes a single divergence point (DP) located at a distance *l* = 0.9 away from the right end of the middle branch, known as branch 8. The vertical line, splits the full phylogeny into two subtrees. Subtree 2 has a stubbed branch where it connected to subtree 1 of length *lt*_{8}. Subtree 1 has a stubbed branch of length (1 – *l*)*t*_{8}.

Figure Figure2 2 plots the method type I error rate and power for the various simulation conditions (blue bars) when the null hypothesis is homotachy. Here, type I errors result when there is no divergence point, but the user concludes one because log_{10} *B*_{DP}> 1. Type I errors do not occur for any of the 500 simulations without a divergence point. Power is the probability that the method strongly supports a divergence point when one is simulated. For frequentist methods, the type II error rate is one minus the power, i.e. the probability of accepting the null hypothesis when the alternative hypothesis of heterotachy is actually true. Bayesian analyses are advantageous when it comes to assessing the strength of the null hypothesis. In this case, one should not commit to the null hypothesis of homotachy unless it receives strong support, e.g. log_{10}*B*_{DP}< -1, which here occurs for only nine of 2500 datasets simulated with a divergence point and only when *θ* = 0.1. A more important concern for the Bayesian method is the decreasing power to detect the divergence point as rate variation and the fraction of sites subject to rate shifts decrease. When *θ* = 0.1, the divergence point becomes effectively undetectable. For all other simulated values of *θ*, the divergence point is detectable given sufficient rate variation. When *α* = 2, the method never works well, and the rates for the four discrete categories are 0.3, 0.7, 1.1 and 2.0, yielding less than seven-fold differences in rate. Susko et al. [11] use a regression technique to estimate the size of rate differences between eukaryotic and archaebacterial amino acid sequences of elongation factor 1*α* and find rate variation roughly between 3 and 15-fold, just straddling the level of rate variation detectable in this simulation.

**Method comparison**. The type I error rate and power of the new method are compared with two other methods, that of Ané et al. [33] and Lopez et al. [13]. The null hypothesis is no temporal rate variation or homotachy. Type I error means the method rejects the null when it is actually true. Power is the probability the method correctly rejects the null when there actually is a divergence point. For the Bayesian method, the null hypothesis is rejected if log_{10} *B*_{DP}> 1. Error bars indicate 95% confidence intervals accounting for simulation error.

When a simulated divergence point is highly supported, the identification of the branch with the divergence point is exceptionally successful via Bayes factor *B*_{j}for branch *j*. Out of 1195 simulations with high support for the divergence point, only 4 failed to also identify the true branch 8 as highly likely (log_{10} *B*_{8} > 1) to carry that divergence point. Only twice, another branch is incorrectly found to strongly favor a divergence point somewhere along its length. These results demonstrate the method can not only detect the presence of a divergence point in a phylogenetic tree, but also pinpoint the affected branch with high confidence.

Table Table1 1 records the posterior mean (indicating accuracy) and width of the 95% Bayesian credible intervals (indicating precision) for parameters *α*, *θ* and *l* averaged across simulated data sets. Figure Figure3 3 plots the distributions of posterior means for these parameters as well as *κ* and two branch lengths: *t*_{8} is the length of the branch carrying the divergence point and *t*_{12} is that of a randomly selected terminal branch. Each entry in Table Table1 1 and boxplot in Figure Figure3 3 is based on 100 estimated values except those for *θ* and *l*, which may be estimated in far fewer simulations. Estimates of *α*, which are logged before plotting in Figure 3(a) , tend to overestimate the true value, especially when true *α* = 0.01 and as the fraction of heterotachous sites increases. Evolutionary rate parameter *κ*, the transition/transversion ratio, is fairly well estimated but with a slight upward bias when site-to-site rate variation is high (*α* < 0.5). In contrast, estimation of *θ* is poor. While there is relatively low posterior uncertainty in *θ* (as compared to *l*), the estimates are dramatically and increasingly downward biased as true *θ* climbs above 0.1. The effect is not just a consequence of the prior, which would tend to pull estimates toward the prior mean of 0.5, because even for true *θ* ≤ 0.5, the bias is downward. The bias is most noticeable for those datasets that detect the divergence point. When simulating *θ* = 0.9 and *α* = 0.01, the divergence point is always detected with high confidence, but the 95% Bayesian credible intervals for *θ* never contain the true value. For estimating the divergence point location *l*, the fact that the estimates pull toward the true value 0.9 as heterotachy increases suggests that there is some information in the data about this parameter, however the information is weak as demonstrated by the very wide Bayesian credible intervals in Table Table1. 1 . Given this result, a model that simply places divergence points at internal nodes of the tree may have just as much power to detect divergence events, while simplifying the MCMC algorithm and convergence. Finally, estimates of all branch lengths tend to be less precise with increasing site-to-site rate variation. In addition, branch 12 is increasingly overestimated as the amount of noticeable heterotachy increases. Since temporal and spatial rate variation can be somewhat or completely confounded [29], it is not surprising to find estimation of *α* and *θ* somewhat entangled. Additionally, failure to adequately account for site-to-site rate variation, in this case because *α* is overestimated, is known to produce biased branch length estimates [46].

**Parameter estimation**. Boxplots of posterior mean estimates of (a) *α*, (b) *κ* (c) *θ*, (d) *l*, (e) branch length of the 8th branch *t*_{8} (the one with the divergence point), and (f) branch length of the 12th branch *t*_{12}. Each boxplot is based on 100 simulations, except (c) and (d), where posterior means of *θ* and *l* are only estimated for those simulations strongly supporting the divergence point. Results are grouped by simulated *θ* value as marked on the x-axis. There are five simulations per group, with simulated *α* *decreasing* 2.0, 1.0, 0.5, 0.1, 0.01. The arrangement is such that temporal rate variation is generally increasing from left to right. The estimates of *α* are logged before plotting to better show the variation in the smaller values. The location of the true value(s) of each parameter are marked by a + just right of its plot.

### Table 1

Estimation of α | Estimation of θ | Estimation of l | |||||||||||||

θα | 2.0 | 1.0 | 0.5 | 0.1 | 0.01 | 2.0 | 1.0 | 0.5 | 0.1 | 0.01 | 2.0 | 1.0 | 0.5 | 0.1 | 0.01 |

0.0 | 2.05 | 0.99 | 0.49 | 0.10 | 0.04 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA |

1.90 | 0.61 | 0.23 | 0.05 | 0.07 | NA | NA | NA | NA | NA | NA | NA | NA | NA | NA | |

0.1 | 2.18 | 1.07 | 0.54 | 0.11 | 0.05 | 0.76 | 0.68 | NA | 0.17 | 0.15 | 0.55 | 0.58 | NA | 0.51 | 0.51 |

2.14 | 0.69 | 0.28 | 0.08 | 0.08 | 0.99 | 0.96 | NA | 0.21 | 0.17 | 0.97 | 0.99 | NA | 0.97 | 0.97 | |

0.3 | 2.77 | 1.23 | 0.59 | 0.11 | 0.05 | NA | 0.49 | 0.38 | 0.26 | 0.26 | NA | 0.53 | 0.57 | 0.58 | 0.57 |

3.42 | 0.91 | 0.36 | 0.08 | 0.08 | NA | 0.55 | 0.48 | 0.22 | 0.19 | NA | 0.97 | 0.97 | 0.96 | 0.97 | |

0.5 | 3.27 | 1.41 | 0.60 | 0.11 | 0.06 | 0.79 | 0.56 | 0.43 | 0.40 | 0.40 | 0.37 | 0.56 | 0.57 | 0.63 | 0.61 |

4.59 | 1.21 | 0.40 | 0.09 | 0.10 | 0.85 | 0.73 | 0.47 | 0.23 | 0.21 | 0.88 | 0.99 | 0.96 | 0.96 | 0.95 | |

0.7 | 4.87 | 1.45 | 0.62 | 0.12 | 0.08 | 0.80 | 0.62 | 0.52 | 0.55 | 0.54 | 0.50 | 0.53 | 0.58 | 0.66 | 0.65 |

7.85 | 1.49 | 0.40 | 0.08 | 0.10 | 0.89 | 0.74 | 0.50 | 0.24 | 0.22 | 0.95 | 0.96 | 0.95 | 0.94 | 0.94 | |

0.9 | 6.74 | 1.62 | 0.65 | 0.13 | 0.10 | 0.86 | 0.74 | 0.66 | 0.69 | 0.69 | 0.47 | 0.56 | 0.61 | 0.67 | 0.66 |

11.44 | 1.83 | 0.40 | 0.08 | 0.10 | 0.86 | 0.72 | 0.48 | 0.24 | 0.23 | 0.92 | 0.97 | 0.94 | 0.94 | 0.93 |

Each pair of entries summarizes the posterior estimation of model parameters *α*, *θ*, and *l* from 100 random data sets simulated assuming various choices of *θ* (rows) and *α* (columns). The first row, where *θ* = 0 is for data simulated *without* a divergence point. First in each pair is the average posterior mean, summarizing accuracy second is the average width of the 95% Bayesian credible interval, summarizing precision. Statistics for *θ* and *l* are based only on those simulations strongly supporting a divergence point, which may be substantially fewer than 100 simulations. NA means a divergence point was never strongly supported for that simulation condition.

Whether the performance in simulation translates to real biological sequences is still questionable. Previous analyses suggest that biological site-to-site rate variation falls in the range *α* ∈ (0.1, 10), with nonsynonymous (and more likely *selected*) rate variation tending to fall below *α* = 1 [24,46,47]. While there is less information about biologically relevant ranges for *θ*, Gu [36] estimates *θ* = 0.46 for a study of the cyclooxygenase gene family at the amino acid level. Comparison of rates between pre-defined monophyletic groups shows very high proportions of sites eventually experience heterotachy during evolution, even in functionally conserved sequences, for example, 66% of rRNA sites [48] or as high as 47% of the cytochrome *b* amino acids [49]. Yang and Nielsen [42] find the proportion of codons undergoing positive selection during episodic evolution along particular lineages to be between 0.03 and 0.2, depending on the gene analyzed. Thus, it appears that the power of this method to detect divergence points may falter near the boundary of biological relevance. In the next section, additional simulations investigate the amount of information, as measured by the sequence divergence, alignment length, and number of taxa, needed to detect divergence points.

#### Varying the amount of data and evolution

To explore the power of the method to detect the divergence point for varying amounts and diversity of input data, I generate simulated data sets under a variety of conditions. I start by again simulating data alignments using the tree of Figure Figure1. 1 . For these simulations, I set *α* = 0.7 and *θ* = 0.5 and vary both the branch length (all branches of the topology are equal) and the length of the alignment. Figure Figure4 4 displays the results, showing that increasing diversity, as measured by the branch length, and data, as measured by the alignment length, both improve the power of the method to detect the divergence point. In particular, the divergence point is detectable for these *α* and *θ* when the branch length is above 0.07 and the alignment length is over 5000. Similar patterns are observed for different *α* and *θ* combinations. As long as *θ* ≥ 0.3, the method achieves good power at least for the simulation with *t*_{j}= 0.9 and 7500 base pairs (data not shown).

**Sample size and power**. The proportion of simulations that strongly support the simulated divergence point when *α* = 0.7 and *θ* = 0.5. Each group of bars corresponds to a different alignment length ranging through 1000, 2500, 5000, and 7500. Within the group, there are four different branch lengths assigned to every branch in the simulation topology of Figure 1, either 0.03, 0.05, 0.07, or 0.09. Error bars indicate 95% confidence intervals accounting for simulation error.

To test the impact of including more sequences, I simulate data with increasing numbers of taxa in each subtree. This time, *α* = 0.5, *θ* = 0.5, and all branch lengths *t*_{j}= 0.1 are selected to demonstrate a range of outcomes in resulting power. The original simulation tree of Figure Figure1 1 has 8 taxa. I also simulate sequences with 4, 12, or 16 taxa, maintaining the divergence point on the middle branch and adding taxa in a balanced fashion to both subtrees. Because all branch lengths are held constant, any effect of adding more taxa could be a consequence of the additional taxa or the increase in total evolutionary time simulated. The power of the method to detect the divergence point increases substantially with the number of taxa in each subtree (Figure (Figure5). 5 ). Unfortunately, the computational cost also increases substantially. Roughly, based on informal observation only, 8 taxa take ten times as long as 4 taxa, and computational times double for every 4 additional taxa after that. Finally, I also examine the probability of detecting heterotachy when the divergence point is placed on a terminal branch rather than the internal branch of Figure Figure1. 1 . This time *α* = *θ* = *t*_{j}= 0.1. Not surprisingly, power of the method to detect a terminal branch divergence point is substantially compromised (Figure (Figure5), 5 ), indicating that balanced subtrees including many taxa provide the ideal conditions for detecting a divergence point.

**Power as a function of group size**. The power of the method to detect the divergence point with strong support as the branch location of the divergence point or the size of the topology changes. For the first set, the divergence point is located on the middle branch or a terminal branch of the 8-taxa tree of Figure 1 and *α* = 0.5, *θ* = 0.5, *t*_{j}*=* 0.1, and *L =* 1000. For the second set, the divergence point is located on the middle branch of a 4-, 8-, 12-, or 16-taxa tree, and *α* = 0.1, *θ* = 0.1, *t*_{j}= 0.1, and *L* = 1000. Error bars indicate 95% confidence intervals accounting for simulation error.

#### Comparison to existing heterotachy detection methods

Figure Figure2 2 compares the power of the Bayesian divergence point method with two other statistical tests for heterotachy given pre-defined subgroups. Ané et al. [33] recently describe a parametric bootstrap test of the covarion model that tests the degree of independence in the proportion of invariant sites in the two subgroups. When applied to the first set of simulated data, this method demonstrates a low type I error rate in the absence of heterotachy and comparable power to the Bayesian method in the presence of heterotachy except when *θ* = 0.1 and *α* is small. However, the method is not ideally matched to the simulations since it specifically tests the covarion model with invariant sites, but the simulation model allows no truly invariant sites. A more appropriate test is suggested by Lopez et al. [49], who describe a method to compare the number of substitutions in each subgroup at each site. Under the homotachous model, the number of substitutions at a site should be proportional to the amount of evolution, or tree length, of each subtree. Substantial deviations from this expectation, as measured by a chi-square statistic, indicate a change in evolutionary rate between the two subtrees. As expected, this method has more power than the Ané et al. and also beats the Bayesian method. In particular, it is better able to detect heterotachy when *α* > 0.5 and there is low site-to-site rate variation. However, these conditions are also the ones where the method's type I error rate begins to exceed expectation (see Figure Figure2, 2 , No divergence point and *θ* = 0.1). Thus, it may be that the conservative behavior of both the Ané et al. and Bayesian methods in the presence of low rate variation are desirable.

As HIV spread into the human population in the last century, genetically distinct lineages arose [50]. These so-called subtypes have distinct geographic distributions [51]. In particular, subtype B dominates throughout much of the non-African and non-Asian world, while subtype C dominates in southern and eastern Africa, parts of the Middle East, and India [51]. Much of the geographic restriction of subtypes can be explained by the travels of a few infected individuals [52], however there is also evidence of population level selection on the virus, particularly in relation to immune selection [53]. I hypothesize that if the virus encounters substantial population-specific selection pressures when entering a new population, a selection shift signature may be detectable on the branches of phylogenetic trees that separate subtypes.

To test the hypothesis, I align 10 HIV sequences, five from subtype B and five from subtype C. Summaries of the marginal posterior distributions for each continuous parameter of the model are shown in Table Table2. 2 . The reported potential scale reduction factors [54] demonstrate healthy agreement between the six independent MCMC runs and all six runs are combined for statistical estimation. The Bayes factor in favor of a divergence point cannot be computed because the support for a divergence point is unanimous in the posterior sample. The model clearly identifies a highly supported divergence point on the branch separating subtypes B and C, with log_{10} *B*_{BC}= 4.08, where indexing is meant to indicate the branch separating B and C. Figure Figure6 6 shows the location of the estimated divergence point along with its 95% Bayesian credible interval on the phylogeny drawn with branch lengths at their posterior means. The precise location of the divergence point along the branch is poorly estimated, but the selected branch is highly supported. Considering the estimated values of *α* = 0.23 and *θ* = 0.28, the alignment length *L* = 6610, and the average branch length ( = 0.04) of this data set, simulation results (not shown) suggest that the method *just* has enough power to detect the presence of a divergence point. It may not be possible to detect heterotachy for shorter regions of HIV.

**Phylogenetic tree of HIV subtypes B and C**. Phylogenetic tree inferred from HIV data. The topology is not estimated, but branch lengths are shown at their posterior means. The posterior mean location of the divergence point is shown along with a parallel bar demarcating the 95% Bayesian credible interval. The numbers indicate the conditional posterior probability that the indicated branch carries the divergence point given there is a divergence point in the tree. For reference, the length of the middle branch is 0.12.

### Table 2

Parameter | Posterior Mean | LBCI | UBCI | PSRF |

θ | 0.27 | 0.18 | 0.38 | 1.01 |

l | 0.56 | 0.21 | 0.89 | 1.03 |

α | 0.23 | 0.19 | 0.26 | 1.00 |

κ | 5.32 | 4.89 | 5.77 | 1.01 |

t_{3} | 0.01 | 0.00 | 0.01 | 1.02 |

log_{10} B_{DP}≈ inf (decisive support for) | ||||

log_{10} B_{BC}= 4.08 (decisive support for) |

The posterior mean, upper and lower 95% Bayesian credible interval bounds (UBCI and LBCI) and potential scale reduction factor (PSRF) [54] for all continuous parameters of the model. Data for the branch length with largest PSRF is reported. The last two rows report the Bayes factor in support of a divergence point and the Bayes factor in support of a divergence point along the middle branch.

Like HIV, HBV has diverged into genetically distinct lineages with nonuniform geographic distribution around the world [55]. In the case of HBV, these lineages are called genotypes. Although the origins of HBV are unclear, HBV is most likely to have evolved with humans since our emigration from Africa [56]. The genotypes and their geographic distribution can thus be associated with major migration events, but it remains unclear whether the genotypes express distinct disease phenotypes [57-59]. HBV genotypes F and H are restricted to the Americas, probably arriving on these continents with the first human immigrants [57]. Genotype H is found much less frequently than F, and its origin is uncertain [60]. In fact, its classification as a separate genotype is controversial [61]. Given the best estimate of HBV origins, it is not likely that the spread of HBV into new human populations has exerted recent selective pressure on the virus, however co-evolution of the virus along with the human host may create divergence points along branches where humans and viruses co-adapted to new ecological niches.

To look for divergence points related to the emergence of HBV genotypes F and H, I align seven genotype F sequences and three genotype H sequences. Posterior summaries are in Table Table3. 3 . Figure Figure7 7 displays the estimated phylogeny relating these 10 sequences with the branch lengths drawn proportional to their posterior means. The number accompanying each branch is the conditional posterior probability that the divergence point lies somewhere along that branch given one exists somewhere in the tree. In contrast to the HIV results, a divergence point is not supported by the data with log_{10} *B*_{DP}= -0.64. Considering only the posterior sample supporting a divergence point (1123 samples), no branch shows evidence of strong heterotachy, although the posterior distribution across branches is significantly different from the uniform prior (p-value < 0.001). The alternative hypothesis of homotachy is substantially, but not strongly supported, and the method may simply have insufficient power to detect heterotachy in this data set. Notably, because H is a poorly sampled genotype, the three representatives included here are highly similar, thereby forcing any potential genotype-associated divergence point onto what is effectively a terminal branch. Table Table4 4 suggests power is low under this condition, but I performed no simulations with parameters matching the HBV data, so it is unclear whether the method should have sufficient power to estimate the presence of a divergence point. Evidence of site-to-site rate variation is high, with the posterior mean *α* = 0.04, however the low diversity (average branch length 0.015) and short alignment (3,215 base pairs) sharply reduce the power of the method.

**Phylogenetic tree of HBV subtypes F and H**. Phylogenetic tree inferred from HBV data. The topology is not estimated, but branch lengths are shown at their posterior means. The numbers by each branch indicate the conditional posterior probability that the indicated branch carries the divergence point given a divergence point is present in the tree. In fact, there is substantial support against a divergence point in this data set. For reference, the length of the middle branch is 0.07

### Table 3

Parameter | Posterior Mean | LBCI | UBCI | PSRF |

θ | 0.37 | 0.01 | 0.94 | 1.01 |

l | 0.42 | 0.02 | 0.96 | 1.01 |

α | 0.04 | 0.00 | 0.08 | 1.03 |

κ | 4.00 | 3.38 | 4.70 | 1.00 |

t_{11} | 0.01 | 0.00 | 0.01 | 1.02 |

log_{10} B_{DP}= -0.64 (substantial support against) | ||||

log_{10} B_{FH}= -0.71 (substantial support against) |

See the caption of Table 2. The second Bayes factor is the support for a divergence point along the middle branch separating the two genotypes.

### Table 4

For each parameter listed, a new value (starred, e.g. *κ**) is proposed according to the listed distribution. Branch lengths *t*_{i} are updated one at a time for *i* = 1. 2*N* - 3. Tuning parameters are subscripted by *t*. Updates of either *l* or (*b, l*) are mixed with probability *m*_{t}. Update of *d* is a trans-dimensional move. The Metropolis-Hastings acceptance ratios are given in the last column. Dependence on parameters not involved in the update is not shown.

In addition, strong spatial rate variation may not translate to strong temporal rate variation in the case of HBV. Normally, the magnitude of temporal rate variation is expected to approximately match the magnitude of spatial rate variation, because choosing a new function for a site is roughly equivalent to selecting a new site at random from the same protein [29,62]. The model makes this assumption by using the same rate class distribution for spatial and temporal rate variation. Strong purifying selection combined with an error-prone reverse transcriptase is expected to produce highly heterogeneous rates in HBV, with widespread conservation due to overlapping reading frames interrupted by a limited number of mutation-tolerant sites [63]. But for a dual-coding nucleotide to temporally shift rate class, it must acquire a new function in both reading frames. This dual constraint may eliminate the possibility of divergence points in HBV and certainly reduces both the magnitude of temporal rate shifts and the number of affected sites. In short, the biology of HBV may limit both the presence of and the power to detect divergence points. Increasing the number of sampled sequences per genotype may restore power, but this option is not examined further here.

## Discussion

Three main patterns were observed in the data. First, under the simplest scenario for the evolutionary process, constant-rate genetic drift, there was no relationship between the evolutionary rate and phylogenetic signal ( Fig. 2). This suggests that low phylogenetic signal should not generally be interpreted as evidence of high evolutionary rate. Second, many different evolutionary processes produced similar phylogenetic signals ( Fig. 2, Fig. 3, Fig. 4, Fig. 5). This suggests that our ability to infer evolutionary process from the measurement of phylogenetic signal is probably limited. This is especially true for observations of low phylogenetic signal—as all but one evolutionary process simulated in this study produced depressed phylogenetic signal under some circumstances ( Fig. 3, Fig. 4, Fig. 5). Finally, some processes increased phylogenetic signal relative to the neutral expectation. In particular, the scenario of a high rate of early peak shifts (niche occupancy) increased phylogenetic signal ( Fig. 5b *b*_{ρ} < 0.0). Although fewer of the processes simulated in this study produced high phylogenetic signal, the prospect for evolutionary inference from high phylogenetic signal is limited by the fact that nonadaptive processes, such as heterogeneous rate genetic drift, can also produce similarly elevated signal ( Fig. 5a *b*_{μ} < 0.0).

### Phylogenetic Signal and Evolutionary Rate

When evolution was unconstrained—in other words, under pure genetic drift—there was no relationship between the evolutionary rate and phylogenetic signal ( Fig. 2). This is not a surprising result as phylogenetic signal for continuous characters should be viewed as primarily a consequence of the evolutionary process, not the evolutionary rate ( Blomberg and Garland, 2002 Blomberg et al., 2003). However, evolutionary rate can affect phylogenetic signal when evolution is bounded ( Fig. 3b). It is well appreciated that evolutionary rate affects phylogenetic signal for discrete characters, such as genetic sequence data, when the number of states for the character is limited ( Hillis and Huelsenbeck, 1992). This effect is more severe as the number of possible states for the character decreases or the rate increases ( Donoghue and Ree, 2000 Ackerly and Nyffeler, 2004). Bounds on morphospace have an analogous effect (e.g., Whitehead and Crawford, 2006).

Under fluctuating selection when the position of the optimum moved by Brownian motion, phylogenetic signal was low when the rate of movement of the optimum was low ( Fig. 4a). This is because when the rate of evolution is very low the optimum moves negligibly over the course of the simulation and as a consequence all the species at the tips are effectively experiencing stabilizing selection to the same optimum (as in scenario 2A Fig. 3a). Variation among species means is then a consequence only of an inability to perfectly track the selective optimum (maladaptation), which will not have a phylogenetic component. This finding is consistent with Hansen and Martins' (1996) predictions regarding phylogenetic covariances under stabilizing selection to a static optimum (which is the limiting case for process 3A, as the rate of movement of the optimum approaches 0.0), and as well as with our findings for constant stabilizing selection (scenario 2A ω 2 = 10).

In general, however, evolutionary rate did not affect phylogenetic signal for continuous characters under the assumption of most comparative methods—i.e., when the evolutionary process approximates Brownian motion (as in genetic drift, and some conditions of bounded evolution and fluctuating selection). However, under other circumstances, such as fluctuating natural selection when the rate of fluctuation is low and functional constraint when the bounds relative to the rate are small, evolutionary rate will affect phylogenetic signal.

### Phylogenetic Signal and Evolutionary Process

Although rate can influence signal, a much larger source of variability in phylogenetic signal among our simulations arose from the evolutionary process. Phylogenetic signal was more or less invariant under only two processes. Phylogenetic signal was consistently high and not significantly different from *K* = 1.0 for all conditions of constant-rate genetic drift ( Fig. 2), whereas phylogenetic signal was consistently low for all conditions of divergent selection ( Fig. 4c). Under all other scenarios, phylogenetic signal was low or high depending on the simulation conditions. No time-independent process seemed to increase phylogenetic signal over that expected under neutral conditions (*K* = 1.0). In contrast, heterogeneity of the evolutionary parameters over time both decreased and significantly increased the phylogenetic signal under different conditions ( Fig. 5).

For constant stabilizing selection, phylogenetic signal was low (and, in fact, very near 0.0) for all conditions of strong stabilizing selection ( Fig. 3a, low ω 2 ). This finding is consistent with Hansen and Martins' (1996) finding that the phylogenetic covariances among species will be low for stabilizing selection to a single optimum. Phylogenetic signal increased under progressively weaker stabilizing selection around a constant optimum (larger ω 2 ). As ω 2 tends towards ∞, the simulation tends towards genetic drift (scenario 1) in which phylogenetic signal is invariably high ( Fig. 2, Fig. 3). For evolution with fixed bounds, phylogenetic signal was low or high depending on the rate, as discussed above however, the process of evolution with bounds generally tended to decrease signal, particularly as the evolutionary rate relative to the bounds was increased.

For fluctuating natural selection, in which the position of the optimum moved according to a Brownian motion process, phylogenetic signal was high so long as the rate of fluctuation was sufficiently high ( Fig. 4a). When the rate of fluctuation was low, phylogenetic signal was decreased: these simulations reflect our findings for scenario 2A, discussed above.

We found low phylogenetic signal for simulations of rare stochastic peak shifts, when the size of rare niche shifts was small ( Fig. 4b). This parallels the situation for fluctuating natural selection at a low rate (discussed above) and stasis of the fitness optimum (scenario 2A Fig. 3a). As the size of rare niche shifts increased, so did phylogenetic signal ( Fig. 4b). Punctuated divergent selection (divergent selection at speciation events) resulted in low phylogenetic signal under all conditions ( Fig. 4c). Although we simulated divergent selection at speciation events, phylogenetic signal would also be low for any condition in which peak shifts are sufficiently common and random with respect to the prior state for the peak, as they are in our punctuated divergent selection simulations (scenario 3C).

Phylogenetic signal was generally decreased when the rate of evolution by genetic drift was initially low but increased over time (*b*_{μ} > 0.0 Fig. 5a). This is because an increasing evolutionary rate tends to concentrate evolutionary change along branches towards the tips of the tree. This will tend to cause the variances of a character across the tips to increase without a concordant increase in the covariances among taxa. Conversely, high initial rate decreasing over time (*b*_{μ} < 0.0) will increase the covariances among tips relative to that expected under constant-rate Brownian motion and will consequently increase phylogenetic signal ( Fig. 5a).

Similarly, phylogenetic signal was decreased when the rate of fitness peak shifts, or niche shifts, was initially low but increased over time (*b*_{ρ} > 0.0 Fig. 5b). In this case, niche shifts are also concentrated towards the tips of the tree. The paucity of niche shifts on internal branches will decrease the covariances among tips relative to the neutral expectation and depress phylogenetic signal. Conversely, an initially high rate of niche differentiation (occupancy), which decreases towards the present (*b*_{ρ} < 0.0), will tend to increase phylogenetic signal relative to the neutral expectation. This is because most niche shifts, and thus most evolutionary changes, are concentrated towards the root of the tree. This will tend to increase the covariances among tips relative to the neutral expectation and thus enhance phylogenetic dependence ( Fig. 5b). This latter niche occupancy model corresponds fairly closely to the niche differentiation model described in Price (1997), and perhaps expected during an adaptive radiation ( Schluter, 2000).

### On the Evolutionary Parameters

The specific values of the evolutionary parameters simulated in this study have not been justified. However, most are poorly known empirically (see Jones et al., 2003 Revell, 2007a). Even for situations in which estimates for the evolutionary parameters are available (such as for the mutation rate Kimura, 1968), we could not practically use available rates because other parameters have been specified unrealistically, usually for computational reasons. For example, due to the small simulated effective population sizes in this study, realistically small mutation rates would lead to unrealistically negligible standing genetic variances (also discussed in Jones et al., 2003), and similarly negligible divergence among species. An implicit assumption of increasing one parameter while decreasing another is that so doing will have compensatory effect with regard to the evolutionary process and outcome. This assumption has been made in previous similar simulation studies ( Jones et al., 2003, 2004 Revell, 2007a), and a compensatory effect has been demonstrated explicitly in one study ( Revell, 2007a). However, future studies might consider using biologically realistic parameter values in similar analyses.

### Artifacts of Molecular Phylogeny Estimation and Phylogenetic Signal

Blomberg et al. (2003) and Ives et al. (2007) point out two sources of bias in the empirical estimation of *K*. Error in phylogenetic topology and error in the estimation of species means will both, on average, downwardly bias the calculation of phylogenetic signal ( Blomberg et al., 2003 Ives et al., 2007).

Consideration of the time-dependent simulations in this study, and in particular of the observation that factors affecting the covariances among species tend to affect *K*, suggests two other types of phylogenetic error that will also create bias in *K* due to their tendency to cause misestimation of the time of shared history between taxa. These are model underparameterization and gene coalescence, and they work in the opposite direction of bias caused by topological error and error in the estimation of species means.

Sequence model underparameterization tends to cause early branches in a molecular phylogeny to be disproportionately shortened relative to later branches ( Revell et al., 2005). This will cause estimated *K* to be upwardly biased by decreasing the expected covariances among species relative to their expected values were the true tree known without error. Even if species exhibit only the expected amount of covariance under Brownian motion evolution on the true tree, these data will have covariance in excess of that predicted based on the (underestimated) lengths of internal branches, and will thus exhibit inflated phylogenetic signal.

In addition, gene lineage coalescence tends to precede speciation by, on average, 2 · *N*_{e} generations, in which *N*_{e} is the effective population size ( Pamilo and Nei, 1988 Hein et al., 2005). This will not cause internal branches of a gene genealogy to be elongated compared to the branches of the true phylogeny, because all branches are, on average, lengthened by 2 · *N*_{e} towards the root and shortened by 2 · *N*_{e} towards the tips. However, tip branches are only lengthened by this phenomenon. This will tend to cause species expected variances (estimated from the gene genealogy) to be enlarged over their true values were the phylogeny (rather than the gene genealogy) known without error, and this occurs without an associated increase in species expected covariances. This will also cause an increase in the estimated value of *K*, because species expected covariances are underestimated relative to their true values in terms of species expected variances.

Thus, although topological error and error in the estimation of species means will tend to decrease estimated phylogenetic signal, we have identified two sources of upward bias in *K* that are expected to result as artifacts of molecular phylogeny estimation. It should be established in a future study to what extent these four sources of error—topological error ( Blomberg et al., 2003), error in the esimtation of species means ( Ives et al., 2007), model underparameterization, and gene coalescence—are likely to affect the estimation of phylogenetic signal in empirical studies.

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