To describe and analyze this case of parallel evolution of ammonoids, their shell geometry is here quantified by means of eight classical linear measurements, which characterize the major morphological features of the ammonoid shell (Figure ; see also [47
Studied morphological parameters of the ammonoid shell. Scheme showing the linear measurements used to characterize and quantify the morphological evolution of the ammonoid shell in this study (see text for further details).
• The maximal diameter (Dmx) is the maximal shell diameter known for each species and is used to approximate the adult body size of the ammonoid species under consideration.
• The whorl expansion rate (WER) is a measure of the proportional increase of shell diameter through growth (initially defined by [97
], but we used the equation of [47
], which is much easier to apply on actual specimens). It is considered one of the most important and biologically meaningful parameters because it roughly reflects the growth rates of the coiled shell tube and strongly correlates with body chamber length, soft part volume and the syn vivo
-orientation of the shell [58
• The whorl shape compression (WSC) is a measure of the ellipsoid of the whorl section of the ammonoid shell aperture, which is a very important ammonoid taxonomic character due to the accretionary growth of the shell.
• The umbilical width index (UWI) is the ratio between the umbilicus and the shell diameter and thus approximates the amount of shell coiling (degree of involution).
• The imprint zone rate (IZR) describes the relative overlap of two succeeding whorls in terms of height.
• The flank convergence index (FCI; modified after [100
]) approximates the relative compression of the ventral part of the shell compared to its dorsal part (i.e. acute vs. low-arched rounded venter).
• The number of lobes (NLb) approximates the indentation of the suture line, which is the junction line of the septa (chamber walls) with the internal side of the shell. Here, we counted only the lobes on one flank, including those in the plane of symmetry (i.e. internal and external lobes).
• Last, the relative depth of the lateral lobe (= "O-lobe" of [101
]; OLb), which is the ratio between width and height of the lateral lobe, measured from the apertural apices of the neighbouring saddles (see Figure ).
All available specimens of the two families have been measured to quantify these characters. Most data are based on own measurements and some were taken from the literature [66
]. The material referred to in this paper is housed in the following institutional collections: Palaeontological Institute, Moscow (PIN); Palaeontological Institute and Museum, University of Zürich (PIMUZ); National Museum, Prague (L 11705); Institute for Geosciences, University of Tübingen (GPIT).
The eight quantitative parameters are composed of one size measure (Dmx), six ratios (WSC, WER, UWI, IZR, FCI, OLb) and one ordinal count (NLb). For Dmx, FCI, NLb and OLb, only the adult value of each species is reported and not juvenile values, because these parameters always lie near structural boundaries at hatching (e.g., the suture is simple and will necessarily increase its complexity through growth; see discussion).
We here consider parallel evolution in a broad sense by assuming that not all characters are involved in the parallel evolution and by not assuming that evolutionary changes are accomplished by similar alterations in the developmental program (contra [2
]). From these definitions, parallel evolution of some characters can be identified when the evolutionary trajectories of the studied lineages in the morphological space defined by this subset of characters (1) start with the same morphotypes, (2) evolve in parallel and are overlapping, and (3) end with the same forms. In other words, the evolutionary trajectories are identical in origin, magnitude and direction. This pattern of parallel evolution must be distinguished from parallelism in phenotypic space. This different phenomenon concerns lineages having parallel trajectories (same direction), but not necessarily the same origin and/or magnitude (for an example of parallelism but not parallel evolution, see [103
], p. 828, figure 3C). It has to be taken into account that the likelihood of (1) finding statistical support of parallel evolution as well as of (2) parallel evolution itself to occur is dramatically reduced when the evolutionary transformations change more often in several aspects (direction, quality, quantity, proportion) in both lineages. Simple cases of parallel evolution are thus easier to test but less meaningful with respect to selective forces and vice versa.
Before evaluating the parallel evolution of auguritids and pinacitids, we describe the patterns of morphological variation and evolution of these two lineages (Figures , , , ). Patterns of morphological evolution are examined globally by means of a multivariate analysis based on the eight studied quantitative characters of the ammonoid shell (Figure ). We perform a principal component analysis (PCA; [104
]) to examine the variation of the variables within the sample and identify the characters that contribute to observed evolutionary changes by creating high variation. Since the studied characters are of different types (size, ratio, ordinal), the PCA has been performed on the correlation matrix (data standardized to mean zero and unit standard deviation) for all characters. Then, we examine the evolution of each quantitative shell character separately by means of bivariate plots depicting their distribution through the phylogenetic sequence for the two ammonoid lineages separately (Figures and ). These plots enable an empirical evaluation of the presence or absence of directed evolutionary changes (trends) for each character. Bivariate and multivariate exploratory analyses are performed by means of the versatile palaeontological data analysis freeware PAST ([105
), as well as by scripts programmed by C.M. in MATLAB®
Figure 5 Evolution of Dmx, FCI, NLb and OLb in auguritids and pinacitids. Bivariate plots of the evolution of adult size, flank convergence index, number of lobes and relative depth of the lateral lobe through the phylogenetic sequence of the two studied ammonoid (more ...)
Figure 6 Evolution of WSC, WER, UWI and IZR in auguritids and pinacitids. Box plots of the evolution of whorl shape compression, whorl expansion rate, umbilical index and imprint zone rate through the phylogenetic sequence of the two studied ammonoid lineages, (more ...)
Figure 7 Outlines of shell whorl section and suture line for Auguritidae and Pinacitidae. Cladograms of the lineages leading to, and including, the Eifelian Pinacitidae and the Emsian Auguritidae. Where available, a sketch of the cross section and a suture line (more ...)
Figure 8 Multivariate analysis of the eight measured shell parameters. Results of the principal component analysis (PCA) based on the eight measured shell parameters. The PCA was done on a correlation matrix. The first three principal components axes account for (more ...)
Since apparent trends in evolutionary series can be produced randomly [107
], the previously and empirically identified evolutionary trends are tested statistically. Several methods exist, which are based on random walk models, to test and characterize observed trends and to distinguish the three modes of evolutionary change commonly considered in palaeontological studies: directional change (GRW, general random walk), random walk (URW, unbiased random walk), and stasis [113
]. The evolutionary changes of each character are here evaluated by means of the maximum likelihood method of [116
]. The method is recognized to perform well even when evolutionary sequences are incompletely sampled, which is likely for empirical palaeontological sequences as documented here [116
]. It has been implemented as a package (paleoTS, [116
]) in the freely available statistical environment R (http://www.r-project.org/
). The method evaluates the maximum likelihood of producing the observed trends for three evolutionary modes (GRW, URW, and stasis). The relative support of each of these three models is assessed using well-established statistical means such as Akaike weights ([120
]; for details, see [116
]), which indicate the relative likelihood for each of the three evolutionary models (Figure ). Since auguritids and pinacitids branched off from the same origin, the characters displaying directed trends shared by both lineages and supported by the statistical analysis can potentially
participate to a case of parallel evolution (Figure ).
Figure 9 Evolutionary trajectories for Auguritidae and Pinacitidae and their Akaike weights for three evolutionary models. Standardized evolutionary trajectories of WSC, WER, UWI, and IZR and raw evolutionary trajectories of Dmx, FCI, NLb, and OLb for auguritids (more ...)
In order to assess the parallel evolution of the two studied lineages, we use two different approaches, both based on a multivariate analysis using the subset of characters previously identified to be potentially involved in this case of parallel evolution. Note that univariate approaches can suffer from a "dimensionality bias" and similarities of trajectories in a morphospace should preferably be tested multivariately [121
]. First, the parallel evolution of this subset of characters is evaluated by means of a method developed for comparing evolutionary trajectories of phenotypic change [123
]. According to this method, the phenotypic evolution of a lineage is defined as a trajectory across a set of evolutionary levels in a multivariate morphological space. Attributes of these trajectories (magnitude, direction and shape) are quantified and statistically compared across pairs of taxa by means of a residual randomization permutation method [123
], and a summary statistic is used to determine the extent to which patterns of phenotypic evolution are concordant. Note that the method currently requires that the compared trajectories have the same number of evolutionary levels (i.e. in our case the same number of species). Since more species of pinacitids have been described, the analysis is performed by first merging the data of the phylogenetically closest species of the pinacitid lineage in order to obtain the same number of studied evolutionary levels or steps for both families. The two species of Pinacites
have thus been merged, as well as Foordites succedens
with F. platypleurus
. This constraint of the method reduces the power of the test since the species of different lineages cannot be considered as equivalent.
The second method to test the parallel evolution of these Devonian ammonoids follows the approach proposed by [121
] for comparing ontogenetic trajectories. This method is a permutation test based on within-lineage multivariate regression of the characters hypothesized to be involved in the parallel evolution. If the two lineages evolved in parallel, then their phylogenetic trajectories are identical in the morphological space defined by the subset of characters involved. To test this hypothesis, we first compute for each lineage separately a linear total least square regression, then we sum the squared orthogonal distance for each specimen from its nearest point on the regression curve. This sum provides the original test statistic for subsequent comparison. Then, we randomly resample without replacement a large number of times the taxonomic assignment of studied specimens to the two lineages and recompute the summed squared distances of these permuted families (this provides the permutation distribution). If the two studied lineages evolved in parallel, the original test statistic should not be an outlier in the permutation distribution of summed squared distances (see [121
]). In other words, permuting specimens' affiliation does not increase the residuals of the multivariate regressions and this is possible only if specimens of both families are close together in the studied morphological space.