Electronic Supplementary Material, Appendix S1.R script to assess the effects of species selection, immigration, and anagenetic change on mean mammalian body size change across the latest Paleocene and earliest Eocene in the Bighorn and Clarks Fork basins of Wyoming using the extended Price equation.Each of the provided datasets (see electronic supplementary material, dataset S1-S4) centerson a single boundarybetween two time faunal intervals (e.g., latest Clarkforkian/earliest Wasatchian), and must be analyzed with the provided script separately.
########################################################
###Extended Price equation for a single boundary
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#Read in csv file
dat<-read.csv(file.choose())
ancestors<-c(dat[,5])
#Identify taxa withancestors within the dataset
has_ancestor<-function(num) {if (num>0){return(1)} else if (num==0){return(0)}}
#Zero the counting variable
count<-0
#Count and store the number of ancestor/descendant relationships within the dataset
for (a in ancestors){count<-count+has_ancestor(a)}
links<-count
#Count the number of taxon occurrences (i.e., total number of occurrences in both the #older or younger time intervals)
countn<-function(num){length(num[!is.na(num)])}
#Count the number of descendants in the older time interval and the number of ancestors #in the younger time interval
desc_i<-countn(dat[,4])
anc_j<-countn(dat[,3])
#Calculate the mean body size of ancestors and the mean body size of descendants, #omitting missing data
mean_anc_size_i<-mean(dat[,3], na.rm=TRUE)
mean_desc_size_j<-mean(dat[,4], na.rm=TRUE)
#Calculate the difference between the mean ancestor and mean descendant sizes
xbar_anc_desc<-mean_anc_size_i-mean_desc_size_j
#Set the index variable to the first column of the dataset
index<-c(dat[,1])
#Calculate the number of ancestors for each taxon
for (x in index){dat[x, "no_anc"]<-has_ancestor(c(dat[x,5]))}
#Separate immigrants in dataset
dat$no_anc<-ifelse((!is.na(dat$size_i==TRUE)&(dat$ancestor==0)),NA,(dat$no_anc))
#Calculate the number of descendants for each taxon
for (x in index) {if ((!is.na(dat[x,"size_i"])==FALSE)&((!is.na(dat[x,"size_j"])==TRUE))) {dat[x,"no_desc"]<-NA} else {dat[x, "no_desc"]<-nrow(dat[dat$ancestor==x,])}}
#Restrict the dataset to taxa with ancestors
non_missing_ancs<-na.omit(as.numeric(dat[,"no_anc"]))
#Calculate average number of ancestors, omitting missing data
average_ancestors<-mean(non_missing_ancs)
#Remove taxa with no descendants
non_missing_desc<-na.omit(as.numeric(dat[,"no_desc"]))
#Calculate the average number of ancestors, omitting missing data
average_descendants<-mean(non_missing_desc)
#Calculate relative fitness for each taxon
for (x in index) {dat[x, "rel_fitness"]<-dat[x,"no_desc"]/average_descendants}
#Create separate column for ancestor size
for (x in index) {if (dat[x,"ancestor"]>0) {dat[x,"anc_size"]<-dat[(dat[x,"ancestor"]),"size_i"]}}
#Calculate the difference in mean body size between descendants and ancestors
dat$size_diff<-dat$size_j-dat$anc_size
#Calculate the contribution of anagenetic changes (i.e., term three of the extended Price #equation)
transmission_bias<-mean(dat$size_diff, na.rm=TRUE)
#Calculate the difference between the number of descendants in the older time interval#and mean number of descendants per ancestor in the older time interval
dat$diff_desc_i_mean<-as.numeric(dat$no_desc) - average_descendants
#Calculate the difference between the number of ancestors of the younger time interval#and mean number of ancestors per descendant in the younger time interval
dat$diff_anc_j_mean<-as.numeric(dat$no_anc) - as.numeric(average_ancestors)
dat$diff_anc_size_i_mean <-(dat$size_i)-mean_anc_size_i
dat$anc_i_covar_contr<-(dat$diff_desc_i_mean)*(dat$diff_anc_size_i_mean)
dat$diff_desc_size_j_mean<-(dat$size_j)-mean_desc_size_j
dat$desc_j_covar_contr<-(dat$diff_anc_j_mean)*(dat$diff_desc_size_j_mean)
#Remove missing data to calculate the covariance of the ancestor contribution
nonmissing_covar_anc<-na.omit(as.numeric(dat[,"anc_i_covar_contr"]))
#Remove missing datato calculate the covariance of the descendant contribution
nonmissing_covar_desc<-na.omit(as.numeric(dat[,"desc_j_covar_contr"]))
#Calculate the contribution of species selection (i.e., term one of the extended Price #equation)
covar_term_1<-(sum(nonmissing_covar_anc)/anc_j)/(links/anc_j)
#Calculate the contribution of immigration (i.e., term two of the extended Price #equation)
covar_term_2<-(sum(nonmissing_covar_desc)/desc_i)/(links/desc_i)*-1
Electronic Supplementary Material, Appendix S2.
The three terms of the Price equation are not best thought of as statistical estimates of unknown population parameters. The Price equation does not treat the observed data as sample data, and does not depend on any assumptions about the sampling process (e.g., regarding randomness and independence of observations, the distribution from which the observations were sampled, etc.). Rather, the Price equation takes the observed data as given and reveals what those data imply about the species selection, immigration, and anagenetic change. Rice [1,2] developed a prospective extension of the Price equation in which phenotypes and fitnesses are unknown and so are treated as random variables described by probability distributions; however, this approach does not apply to fossil data describing evolution that has already occurred.
Nevertheless, our results are subject to various sources of uncertainty. Quantifying these various sources of uncertainty is desirable in principle but difficult in practice for our dataset. In order to quantify uncertainty in the terms of the Price equation, it would be necessary to quantify sources of error in the observed data, and then propagate those errors through the Price equation. For instance, we estimated the body size of each species as the mean of the natural logarithm of lower first or second molar area. These means are subject to sampling error, with the measured molars of each species being a sample from the population of molars. In principle, this sampling error could be quantified by bootstrapping. Resampling with replacement from the molar measurements of each species would generate bootstrap replicates, each of which could be analyzed with the Price equation, yielding a bootstrapped distribution for each Price equation term. In practice, however, this cannot be accomplished since measurements of individual fossils are unavailable for some species, and because some species are known from only a single specimen or only a few specimens at our study site (see electronic supplementary material, table S2). Other sources of error are more difficult to quantify. New fossil discoveries that added new species to our dataset, or altered the timing of species’ first and last appearances would alter our results. So would revisions to species’ ancestor-descendant relationships and classification.
Assessing whether the observed values of species selection, anagenesis, and non-random immigration across the PETM are different than would be expected under some “null” model with a randomization test is another possibility. However, a biologically-relevant “null” hypothesis is impossible to define. For instance, randomly shuffling body sizes among species and then applying the extended Price equation to the randomized data sets would provide the distribution of results expected under a “null” model in which all species are equally likely to speciate, go extinct, or change in size by any amount. It would not be scientifically informative to either reject this model, or fail to reject it, because this model obviously is biologically implausible. Similarly, randomly shuffling which descendant species descended from which ancestral species would provide the distribution of results expected under an implausible null model in which all ancestor-descendant links are equally probable. In principle, it would be informative to simulate data expected under alternative plausible evolutionary scenarios. The available data, however, are insufficient to guide and constrain development of models of different scenarios.
Electronic Supplementary Data, Table S1.Latest Palaeocene and earliest Eocene mammals from the Bighorn and Clarks Fork Basins, Wyoming, USA, studied to examine the macroevolution of mean body size using the extended Price equation. Lineages, with respective sources, are: Palaeanodon [3], Esthonyx [3, unpublished data], Azygonyx [3,4], Macrocranion [4,5, unpublished data], Phenacolemur [6], Niptomomys [4,7], Carpolestes [8,9], Teilhardina [4,10,11], Tetonius [12], Cantius [3,4, unpublished data], Arfia [3], Galecyon [13], Didymictis [14], Viverravus [4,14], Chriacus [3,4], Hyopsodus [15], Haplomylus [16], Phenacodus [4,17], Ectocion [3,4,17], Copecion [17], Diacodexis [3,4, unpublished data], Sifrhippus sandrae/Arenahippusgrangeri [3,4,18, unpublished data], and Paramys [19].
Order / Suborder / Family / Genus / Species?Pholidota / Palaeanodonta / Metacheiromyidae / Palaeanodon / Palaeanodonparvulus
Palaeanodonnievelti
Cimolesta / Tillodonta / Esthonychidae / Esthonyx / Esthonyxspatularius
Azygonyx / Azygonyxxenicus
Azygonyxanyclion
Azygonyxlatidens
Azygonyxgrangeri
Azygonyxgunnelli
Azygonyx sp.
Lipotyphla / Erinaceomorpha / Amphilemuridae / Macrocranion / Macrocranionjunnei
Macrocranionnitens
Primates / Plesiadapiformes / Paromomyidae / Phenacolemur / Phenacolemurpagei
Phenacolemur praecox
Microsyopsidae / Niptomomys / Niptomomysdoreenae
Carpolestidae / Carpolestes / Carpolestesnigridens
Carpolestessimpsoni
Euprimates / Omomyidae / Teilhardina / Teilhardinabrandti
Teilhardinagingerichi
Teilhardinaamericana
Tetonius / Tetoniusmatthewi
Notharctidae / Cantius / Cantiustorresi
Cantiusralstoni
Cantiusmckennai
Creodonta / Hyaenodontidae / Arfia / Arfiajunnei
Arfiazele
Arfiashoshoniensis
Galecyon / Galecyonperegrinus
Carnivora / Viverravidae / Didymictis / Didymictis proteus
Didymictis leptomylus
Didymictis protenus
Viverravus / Viverravus politus
Condylarthra / Arctocyonidae / Chriacus / Chriacus sp.
Chriacusbadgleyi
Phenacodontidae / Ectocion / Ectocion osbornianus
Ectocion parvus
Copecion / Copecionbrachypternus
Copeciondavisi
Phenacodus / Phenacodusintermedius
Phenacodustrilobatus
Phenacodusvortmani
Hyopsodontidae / Hyopsodus / Hyopsodusloomisi
Haplomylus / Haplomyluspalustris
Haplomylussimpsoni
Haplomyluszalmouti
Haplomylusspeirianus
Artiodactyla / Diacodexeidae / Diacodexis / Diacodexisillicis
Diacodexismetasiacus
Perissodactyla / Equidae / Sifrhippus / Sifrhippus sandrae
Arenahippus / Arenahippusgrangeri
Rodentia / Sciuromorpha / Ischyromyidae / Paramys / Paramystaurus
Electronic Supplementary Data, Table S2. Mean of the natural logarithm of the first or second molar area (mm) of latest Palaeocene and earliest Eocene mammals from the Bighorn and Clarks Fork Basins, Wyoming, USA, studied to examine the macroevolution of mean body size using the extended Price equation. Sample sizes are provided in parentheses.
Species / Cf1 / Cf2 / Cf3 / Wa0 / Wa1/2Palaeanodonparvulus / 3.09 (2) / 3.24 (2)
Palaeanodonnievelti / 2.80 (2) / 2.82 (6)
Esthonyxspatularius / 3.69 (2) / 3.77 (27)
Azygonyxxenicus / 3.88 (2)
Azygonyxanyclion / 4.03 (2) / 4.18 (4)
Azygonyxlatidens / 4.42 (6)
Azygonyxgrangeri / 4.72 (5)
Azygonyxgunnelli / 4.25 (2)
Azygonyx sp. / 4.45 (4) / 4.45 (3)
Macrocranionjunnei / 0.68 (3)
Macrocranionnitens / 0.73 (1)
Phenacolemurpagei / 1.43 (1) / 1.53 (3) / 1.49 (5)
Phenacolemur praecox / 1.66 (4) / 1.78 (1) / 1.76 (15)
Niptomomysdoreenae / 0.06 (3) / 0.40 (6)
Carpolestesnigridens / 1.01 (7) / 1.07 (5)
Carpolestessimpsoni / 0.85 (4) / 0.84 (11)
Teilhardinabrandti / 1.09(9)
Teilhardinagingerichi / 1.29 (1)
Teilhardinaamericana / 1.13 (20)
Tetoniusmatthewi / 1.55 (4)
Cantiustorresi / 2.32 (3)
Cantiusralstoni / 2.46 (163)
Cantiusmckennai / 2.55 (67)
Arfiajunnei / 2.88 (4)
Arfiazele / 3.19 (3)
Arfiashoshoniensis / 3.46 (18)
Galecyonperegrinus / 2.89 (1) / 3.30 (3)
Didymictis proteus / 3.69 (3) / 3.94 (13) / 3.98 (9) / 3.61 (3)
Didymictis leptomylus / 3.79 (13)
Didymictis protenus / 4.32 (5)
Viverravus politus / 2.53 (2) / 2.98 (4) / 2.89 (1) / 3.20 (5)
Chriacus sp. / 3.15 (1) / 3.01 (1) / 3.19 (2) / 3.16 (2)
Chriacusbadgleyi / 2.93 (8)
Ectocion osbornianus / 3.59 (37) / 3.55 (46) / 3.50 (157) / 3.54 (181)
Ectocion parvus / 3.17 (35)
Copecionbrachypternus / 3.56 (1) / 3.61 (2) / 3.56 (11)
Copeciondavisi / 3.20 (4)
Phenacodusintermedius / 4.77 (10) / 4.80 (14) / 4.82 (31) / 4.72 (4) / 4.67 (9)
Phenacodustrilobatus / 4.55 (11)
Phenacodusvortmani / 4.39 (2) / 4.24 (2) / 4.23 (5) / 4.17 (4) / 4.07 (20)
Hyopsodusloomisi / 2.10 (33) / 2.30 (215)
Haplomyluspalustris / 1.58 (1)
Haplomylussimpsoni / 1.82 (6) / 1.98 (11)
Haplomyluszalmouti / 1.17 (7)
Haplomylusspeirianus / 1.65 (162)
Diacodexisillicis / 2.38 (14)
Diacodexismetasiacus / 2.55 (174)
Sifrhippus sandrae / 3.44 (87)
Arenahippusgrangeri / 3.55 (294)
Paramystaurus / 1.85 (1) / 1.93 (2) / 1.95 (4) / 1.89 (1)
Electronic Supplementary Data, Figure S1. The number of descendant species descended from each ancestral species, as a function of mean ancestral body size, approximated from the natural logarithm of the lower first or second molar areafor (a) the early to middle Clarkforkian, (b) middle to late Clarkforkian, (c) late Clarkforkian to earliest Wasatchian, and (d) earliest Wasatchian to early Wasatchian. Each point gives data for one ancestral species. Lines are linear regressions. Insets provide the three subcomponents of species selection (equation 2): the selection gradient (regression slope) β, the variance in body size among ancestral species, σ2, and the mean number of descendant species to which an ancestral species gives rise, .
Electronic Supplementary Data, Figure S2. The number of ancestors for each descendant species, as a function of mean descendant body size, approximated as the natural logarithm of the lower first or second molar area, for (a) the early to middle Clarkforkian, (b) late Clarkforkian to earliest Wasatchian, and (c) earliest Wasatchian to early Wasatchian.The middle to late Clarkforkian boundary is omitted because there was no immigration across this boundary. Each point gives data for one descendant species. Lines are linear regressions. Insets provide the three subcomponents of the effect of non-random immigration (equation 3): the regression slope β, the variance in body size among descendant species, σ2, and the mean number of ancestral species per descendant species, (equal to the proportion of descendant species that are not immigrants, and so have one ancestor in the dataset).
References
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Captions for Electronic Supplementary Material, Datasets
Electronic Supplementary Material, Dataset S1. Mean body sizes,approximated fromthe natural logarithm of the lower first or second molar area, and the proposed evolutionary relationships between mammalian genera from the early and middle Clarkforkian (Cf1 to Cf2) of the Bighorn and Clarks Fork Basins, Wyoming, USA. Notably, each taxon is assigned an index number (i.e., ‘Index’ column). The mean body size of each taxon is provided for each of the time intervals that the taxon occurs (i.e., ‘size_i’ is the mean body size of the taxon in the older time interval and ‘size_j’ is the mean body size of the taxon in the younger time interval; NA indicates that the taxon does not occur in that time interval). Evolutionary relationships are indicated with specific reference to the index number of taxa, with a number in the ‘ancestor’ column indicating that the taxa is a descendant of the referenced index number; zeros in the ‘ancestor’ columns indicates that there is no ancestor for that taxon in the dataset. For instance,Haplomylus_palustris(i.e., Index taxon 1) does not have an ancestor in the dataset and a zero is recorded within the ‘ancestor’ column for this taxon. However, in the same dataset, Haplomylus_simpsoni(i.e., Index taxon 2) does have an ancestor in the dataset; this ancestor is Haplomylus_palustrisand a 1is recorded in‘ancestor’ column referring to the index number of Haplomylus_palustris.
Electronic Supplementary Material, Dataset S2.Mean body sizes,approximated fromthe natural logarithm of the lower first or second molar area, and the proposed evolutionary relationships between mammalian genera from the middle and late Clarkforkian (Cf2 to Cf3) of the Bighorn and Clarks Fork Basins, Wyoming, USA. For details of the dataset see caption for electronic supplementary material, dataset S1.
Electronic Supplementary Material, Dataset S3.Mean body sizes,approximated fromthe natural logarithm of the lower first or second molar area, and the proposed evolutionary relationships between mammalian genera from the late Clarkforkian and earliest Wasatchian (Cf3 to Wa0) of the Bighorn and Clarks Fork Basins, Wyoming, USA. For details of the dataset see caption for electronic supplementary material, dataset S1.
Electronic Supplementary Material, Dataset S4.Mean body sizes,approximated fromthe natural logarithm of the lower first or second molar area, and the proposed evolutionary relationships between mammalian genera from the earliest to early Wasatchian (Wa0 to Wa1/2) of the Bighorn and Clarks Fork Basins, Wyoming, USA. For details of the dataset see caption for electronic supplementary material, dataset S1.
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