Downloaded from rsbl.royalsocietypublishing.org on July 28, 2011 Modelling rate distributions using character compatibility: implications for morphological evolution among fossil invertebrates Peter J. Wagner Biol. Lett. published online 27 July 2011 doi: 10.1098/rsbl.2011.0523 Supplementary data "Data Supplement" http://rsbl.royalsocietypublishing.org/content/suppl/2011/07/21/rsbl.2011.0523.DC1.ht ml References This article cites 19 articles, 13 of which can be accessed free http://rsbl.royalsocietypublishing.org/content/early/2011/07/21/rsbl.2011.0523.full.html #ref-list-1 P<P Published online 27 July 2011 in advance of the print journal. 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Lett. go to: http://rsbl.royalsocietypublishing.org/subscriptions This journal is © 2011 The Royal Society Downloaded from rsbl.royalsocietypublishing.org on July 28, 2011 Biol. Lett. varying selective forces. Moreover, biomechanics, doi:10.1098/rsbl.2011.0523 development and genetics probably create hierarchi- Published online cal interactions among morphological characters [2]. Palaeontology Rates for any one character thus should partially reflect both the number of associated characters and the probabilities of those characters accommodating a Modelling rate distributions change. If overall rates are products of independent prob- abilistic processes [3]and/or hierarchical linkages [4], using character then morphological rates should fit lognormal distributions. compatibility: implications Palaeontologists have combined simulations with for morphological the morphological disparity of character matrices to assess general rates of change (e.g. [5]). Systematists evolution among fossil have used the related concept of compatibility to con- trast general rates for individual characters (e.g. [6]) invertebrates (see the electronic supplementary material for a con- trast of disparity and compatibility). Here, I unite Peter J. Wagner* both traditions to assess how well three basic models National Museum of Natural History, Smithsonian Institution, of character evolution (single rates, gamma distri- Washington DC 20013, USA butions and lognormal distributions) predict fossil *[email protected] morphologies. Rate distributions are important considerations when testing hypotheses about morphological evol- ution or phylogeny. They also have implications 2. MATERIAL AND METHODS about general processes underlying character evol- (a) Compatibility, steps and rates ution. Molecular systematists often assume that Yang [1] suggests using steps implied by parsimony trees to establish rates are Poisson processes with gamma distri- rate distributions. However, when simulated matrix structure matches that of real data, parsimony steps blur rate classes [7]. butions. However, morphological change is the Instead, I use character compatibility to approximate steps. A com- product of multiple probabilistic processes and patible character pair has no necessary homoplasy: a binary pair should theoretically be affected by hierarchical combining for 00, 01 and 11 might not have homoplasy but a pair integration of characters. Both factors predict log- also including combination 10 must have homoplasy [6,8]. Simu- normal rate distributions. Here, a simple inverse lations corroborate the suggestion that compatibility decreases as modelling approach assesses the best single-rate, frequencies of change (and homoplasy) increase, both for whole matrices [7] and for individual characters [9]. gamma and lognormal models given observed State distributions also affect the probability of compatibility. character compatibility for 115 invertebrate Suppose two binary characters change only once. For character A, groups. Tests reject the single-rate model for only one taxon possesses state 1. State 1 cannot be paired with nearly all cases. Moreover, the lognormal outper- two states in any other character, and thus cannot create incompat- forms the gamma for character change rates and ibility. Conversely, character B changes in the middle of the (especially) state derivation rates. The latter in phylogeny so that half of the taxa have state 0 and other half have state 1. Both states 0 and 1 for character B now have four opportu- particular is consistent with integration affecting nities to be paired with a second state for other binary characters. In morphological character evolution. simulations, compatibility given X steps decreases as states become more ‘evenly’ distributed among taxa (see electronic supplementary Keywords: character evolution; compatibility; material). lognormal distribution; gamma distribution; This study uses simulations to inversely model the probability of information theory observed compatibilities and state distributions given the rates of character state change. For a clade of S taxa with Nch characters, each simulation: 1. INTRODUCTION — evolves a tree of S taxa; — evolves Nch characters (retaining the observed state num- Palaeontologists are interested in rates of character bers and missing entries as the data) until the observed state change both for testing macroevolutionary compatibility is reached; hypotheses and for inferring phylogeny. However, — iteratively removes simulated characters matching the com- patibility of observed characters; palaeontologists have paid much more attention to — re-simulates the ‘matching’ character with 1, ..., S steps; and rate variation over time and among clades than to — tallies how often we find observed compatibility and state rate variation among characters. Indeed, most rate distributions given 1, ..., S steps. and systematic studies tacitly assume a single rate for all characters. We have not yet explored whether There are three critical differences between this approach and permutation tests assessing character compatibilities (e.g. [10–12]). there are any general rules for rate distributions, or First, phylogeny underlies the character state distributions even even whether single-rate models are poor ones. under high rates of change (part 1). Second, each character is com- Molecular systematists often model rate variation pared with a ‘remaining’ matrix having the same compatibility as with gamma distributions. This assumes a collection the observed ‘remaining’ matrix (parts 2 and 3). Finally, results are tied to specific amounts of change (parts 4 and 5). of Poisson processes with different ‘waiting times’ Now, we have the likelihoods of 1, ..., S changes (steps) given the between events [1]. However, palaeontologists use observed compatibility for each character. However, model evalu- only fossilizable morphology. Morphological change is ation requires numbers of characters with X steps (see below). This, in turn, requires posterior probabilities rather than likelihoods. probably the product of waiting time and probabilistically Assuming flat priors for step numbers, the probability of character C having X steps is: Electronic supplementary material is available at http://dx.doi.org/ 10.1098/rsbl.2011.0523 or via http://rsbl.royalsocietypublishing.org. p[compatibility of CjX steps P S : One contribution to a Special Feature on ‘Methods in Palaeontology’. Y¼1 p[compatibility of CjY steps Received 20 May 2011 Accepted 7 July 2011 This journal is q 2011 The Royal Society Downloaded from rsbl.royalsocietypublishing.org on July 28, 2011 2 P. J. Wagner Compatibility and rate distributions (a)(b) 30 25 20 15 10 echinoderm clades 5 0 35 30 25 20 15 mollusc clades 10 5 0 45 40 35 30 25 20 trilobite clades 15 10 5 0 0.10 0.20 0.30 0.40 0 0.04 0.08 0.12 0.16 0.20 0.24 Akaike’s weight (w1) Akaike’s weight (w1) Figure 1. Akaike’s weights (w1) for single-rate models of (a) character change and (b) state derivation. These are akin to Bayesian probabilities that the model is the most correct of the models considered. The estimated number of characters with X steps now is: at least one derivation for each ‘other’ state (e.g. the two ‘other’ states for character B above must be derived once each). At four XNch steps, we have two additional derivations, so for each ‘other’ state: p½Xstepsjcompatibility of C: C¼1 p½0 additional derivations¼ p½2 additional derivations¼0:25; Character evolution models, such as the Mk model [13], describe p½1 additional derivation¼0:50: rates of character change and rates of state derivation. This distinc- tion is critical when characters have different numbers of states. As there are two ‘other’ states, we expect 2 Â 0.25 ¼ 0.50 states Consider simple transition matrices for two-state character A and to be derived either once or three times in total, and 2 Â 0.50 ¼ 1.0 a three-state character B, where the off-diagonals give p[state to be derived twice the total. The number of states from each char- derivation] and the diagonal gives p[stasis] (i.e. 1 2p[change d]): acter expected to be derived Y times now is the sum of p[Y derivations j X steps] Â p[X steps j data] for X ¼ 1, ..., S.This states 0 1 2 is summed over all characters. Analyses then use that final summation d d 01À d B B to evaluate different models (see electronic supplementary material). states 0 1 B 2 2 01À dA dA versus dB dB _ (b) Rate distribution models 1 1 À dB 1 dA 1 À dA 2 2 The single-rate model varies one parameter: mean rate, d. Gamma dis- d d tributions use a scale and a shape parameter to vary rates around the 2 B B 1 À d 2 2 B mean rate (d). Following Yang [1], I set the scale equal to shape, which leaves one freely varying parameter.