bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439; this version posted January 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license.
1 A Multivariate View of Parallel Evolution
2
3 Stephen P. De Lisle*
4 Daniel I. Bolnick
5
6 Department of Ecology & Evolutionary Biology
7 University of Connecticut
8 Storrs, CT 06269
9
10 * email: [email protected]
11
12 Running title: Parallelism revealed
13
14 Keywords: Convergent evolution, Gasterosteus aculeatus, microevolution, phenotypic vector
15 analysis, random matrix theory
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17
18
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24 Abstract
25 A growing number of empirical studies have quantified the degree to which evolution is
26 geometrically parallel, by estimating and interpreting pairwise angles between evolutionary
27 change vectors in multiple replicate lineages. Similar comparisons, of distance in trait space, are
28 used to assess the degree of convergence. These approaches amount to element-by-element
29 interpretation of distance matrices, and can fail to capture the true extent of multivariate
30 parallelism when evolution involves multiple traits sampled across multiple lineages. We
31 suggest an alternative set of approaches, co-opted from evolutionary quantitative genetics,
32 involving eigen analysis and comparison of among-lineage covariance matrices. Such
33 approaches not only allow the full extent of multivariate parallelism to be revealed and
34 interpreted, but also allow for the definition of biologically tenable null hypotheses against which
35 empirical patterns can be tested. Reanalysis of a dataset of multivariate evolution across a
36 replicated lake/stream gradient in threespine stickleback reveals that most of the variation in the
37 direction of evolutionary change can be captured in just a few dimensions, indicating a greater
38 extent of parallelism than previously appreciated. We suggest that applying such multivariate
39 approaches may often be necessary to fully understand the extent and form of parallel and
40 convergent evolution.
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47 Introduction
48 How repeatable is the evolutionary process? Highpoints in the history of life include taxa that
49 have appeared to independently evolve similar adaptations in response to similar environmental
50 conditions. Such examples represent some of the most striking evidence for adaptive evolution,
51 suggesting not only that evolution can sometimes repeat itself, but that we can also identify the
52 broad environmental factors governing natural selection (Nosil et al. 2002, Langerhans and
53 DeWitt 2004, Losos 2011, Bolnick et al. 2018, Stuart 2019). Thus, this classical question cuts to
54 the core of ongoing debates over the role of chance and determinism in evolution at all
55 timescales.
56 The question of repeatability in evolution has been reframed in light of contemporary
57 approaches to studying evolution and natural selection in the wild. This new work has
58 distinguished the outcome of evolution, convergent (divergent), from the path of evolutionary
59 change, parallel (nonparallel) (Bolnick et al. 2018). Purpose-built statistical tests have been
60 invented to test hypotheses of parallelism and convergence at both the micro (Collyer and Adams
61 2007, Adams and Collyer 2009, Collyer et al. 2015) and macro scale (Mahler et al. 2013). This
62 work has led to some important advances in our empirical understanding of how repeatable
63 evolution can be (Oke et al. 2017, Stuart et al. 2017), yet has also highlighted some fundamental
64 challenges (Bolnick et al. 2018) of linking pattern and process in empirical tests of the
65 repeatability of evolution.
66 Here we emphasize an explicitly-multivariate set of approaches to the study of parallel
67 and convergent evolution. These multivariate approaches surmount three key difficulties: 1)
68 univariate analyses, often employed in studies of parallel and convergent evolution, can be
69 difficult to interpret when the data are multivariate, and can lead to 2) underestimation of the
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70 importance of shared dimensions of evolutionary change, which have been difficult to identify
71 due to the challenge of 3) construction of a statistical null expectation that is also biologically
72 appropriate, against which empirical patterns can be pitted. We show how these challenges can
73 be resolved with some basics of linear algebra that permit analysis of entire matrices of similarity
74 of evolutionary change. We illustrate our points by revisiting a published dataset of parallel
75 evolution in a fish species, where a truly-multivariate approach reveals a far greater extent of
76 parallelism than can be concluded from univariate analysis. Our arguments largely ‘parallel’
77 similar issues raised in a closely aligned subfield, evolutionary quantitative genetics, where it has
78 long been recognized (Lande and Arnold 1983, Phillips and Arnold 1989, Blows and Brooks
79 2003, Blows 2007a, b, Kirkpatrick 2009, Wyman et al. 2013) that understanding selection on and
80 expression of complex traits requires approaches that are explicitly multivariate.
81 Below, we first define parallelism and convergence as separate but related patterns. Next,
82 we outline a set of multivariate geometric techniques to explore the degree and form of
83 parallelism, applying these techniques to a reanalysis of a published dataset of parallelism in a
84 fish species. We then review approaches to assessing multivariate convergent/divergent
85 evolution, independent of parallelism, and highlight a specific published case study leveraging
86 such an approach. We discuss advantages and limitations of these multivariate techniques.
87
88 Defining parallelism and convergence as unique phenomenon
89 Parallel (nonparallel) and convergent (divergent) evolution can be seen as separated but related
90 patterns (Figure 1; see also Bolnick et al. 2018). This separation is potentially important because
91 it is the case that unique evolutionary processes could lead to patterns of evolutionary parallelism
92 without convergence, or vice-versa. But, parallelism and convergence also need not be viewed
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93 as mutually exclusive phenomena. For example, conserved directional selection across lineages
94 could result in parallelism without convergence (Figure 1A). Alternatively, adaptation towards a
95 shared optimum by lineages with unique evolutionary histories, and thus unique ancestral
96 positions in trait space, could result in convergence without parallelism (Figure 1B). Yet,
97 parallel evolutionary processes can lead to divergence if, for example, one lineage evolves faster
98 along a shared trajectory. Thus, separating parallelism and convergence may often be necessary
99 to link evolutionary pattern and process.
100
101 Geometry of parallel evolution
102 An intuitive way to define parallelism is via analysis of evolutionary change vectors (Collyer and
103 Adams 2007, Adams and Collyer 2009), where the vector of multivariate evolutionary change
104 across an environmental gradient or time points a and b (e.g. and ancestral versus descendant
105 population), for a given lineage is
∆ (1)
106 (Lande 1979) perhaps with traits standardized to make units comparable across traits. Note that,
107 beyond morphology, such a vector can be defined using breeding values, sequence variation
108 (Stuart et al. 2017), or gene expression profiles, and that environments can be defined using
109 external abiotic factors or any other feature of interest that defines subpopulations (e.g., sex; De
110 Lisle and Rowe 2017). ∆ is a vector of evolutionary change, or evolutionary response to
111 selection, and a growing number of studies have employed an approach (Collyer and Adams
112 2007, Oke et al. 2017, Stuart et al. 2017) where the angle between ∆ vectors is estimated for
113 each pairwise combination of lineages studied, as
∆ ∆ (2)
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114 where each vector has been normalized to unit length. Statistical significance of the angle is
115 often assessed via permutation of the data, comparing the observed value to a null of perfect
116 parallelism across each pairwise combination of lineage replicates (Collyer and Adams 2007,
117 Adams and Collyer 2009). While appropriate for assessing significance of a specific pair of
118 vectors, for example when only two lineages are sampled or are of particular a priori interest,
119 this approach is problematic for two reasons when multiple lineages are sampled.
120 First, how do we interpret values when more than two lineages are sampled? Lineages
121 would be expected to differ, potentially substantially, in the direction of evolutionary change
122 even when most are evolving in a similar direction. The problem is illustrated in Figure 2A,
123 where three lineages are diverging in multivariate trait space at angles of = 45.5 degrees from
124 each other. Each pairwise angle interpreted alone suggest evolution is largely nonparallel. Yet
125 this element-by-element interpretation misses the fact that there is a shared common axis of
126 divergence in multivariate trait space that all three lineages load strongly onto and that accounts
127 for a large amount (80%) of the among-lineage variation in the direction of evolution.
128 Moreover, multiple dimensions of parallelism and (anti)parallelism, although a likely feature of
129 multivariate parallel evolution (Figure 2B), cannot be inferred in any intuitive way from
130 interpreting individual angles alone. This challenge in interpreting these pairwise angles
131 individually only becomes greater as the dimensionality of trait space and the number of lineages
132 sampled increases, and is analogous to the within-lineage challenge of using single elements of a
133 genetic covariance matrix to understand the distribution of genetic variation across traits
134 (Kirkpatrick 2009), or the challenge of using single elements of a matrix of nonlinear selection
135 gradients to infer the shape of nonlinear selection (Blows and Brooks 2003, Blows 2007a, b).
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136 Put simply, lineages may share a common axis (or axes) of directional change even when most
137 of the individual angular distances between lineages are significantly non-zero.
138 A second and related difficulty with interpreting angular distances individually is the
139 problem of what the null is for the angle between any given pair of lineages (Bolnick et al.
140 2018). Many investigators may be interested in comparing observed patterns to a null of
141 independent directions of evolution across lineages. This null results in an expected value of 90
142 degrees for any pair of lineages. Yet intuitively, the null distribution should depend on both the
143 number of traits sampled and the total number of lineages (i.e., beyond any given pair in
144 question), as sampling many lineages evolving independently in low-dimensional trait space is
145 expected to often result in some highly parallel pair-wise combinations through chance alone.
146 Moreover, investigators may be interested in testing more sophisticated null hypotheses, for
147 example comparing patterns of parallelism to what may be expected due to biases from genetic
148 correlations when estimate(s) of G are available (Schluter 1996), and it will often be difficult to
149 define such a corresponding null angle that can be tested using the permutation approach
150 typically used in studies of parallelism.
151 These challenges emerge from what is, at its essence, an element-by-element approach to
152 analysis of angular distance matrices, and they can be circumvented by adopting approaches that
153 consider explicitly the entire matrix of similarity among lineage’s evolutionary change vectors.
154 We can define a matrix of ∆ of vectors for n traits from m lineages as
155
∆ , ∆ , (3) ∆ , ∆ ,
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156 with m rows as replicate lineages and n columns as traits. From this data matrix, where each
157 element is the evolutionary change value for a single trait from a single independently evolving
158 lineage, and each row thus represents a ∆ vector from a given lineage, we can calculate two
159 potentially relevant covariance matrices. One is the n×n matrix (where n is the number of
160 measured traits) of among-lineage variances and covariances in the traits’ evolutionary changes
161
(4)
162 This captures the variances and covariances of evolutionary change across replicate lineages for
163 each trait. Spectral decomposition (eigen analysis) of R would be relevant for understanding
164 parallel evolution, since parallel evolution would lead to dimensions of R with very low
165 variance. However, it is a challenge to test hypotheses about null or nearly-null dimensions
166 (Kirkpatrick 2009, Blows and McGuigan 2015). So, for the specific question of, “How parallel is
167 evolutionary change?”, we may be better off focusing on a different product of X,
168
(5)
169 Which is the matrix of vector correlations (assuming the rows of X have been normalized) of
170 evolutionary change vectors for each pair of independent lineages in X. Thus C is a m×m
171 correlation matrix where m is the number of replicate lineages. This matrix contains ones on the
172 diagonal, and each off-diagonal describes the correlation between multivariate evolutionary
173 change vectors across a pair of lineages. By describing the relationship between lineage vectors
174 as a correlation instead of an angle (i.e., a distance), we can use the spectral decomposition
175 (eigen decomposition) of C,
176
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(6)
177 where Q is the matrix of eigenvectors and V is the diagonal matrix of eigenvalues,
178 to assess two major questions on the degree and form of multivariate parallelism that cannot be
179 tested using univariate approaches:
180
181 1) How extensive is parallelism? Parallel evolution (and anti-parallel) will be captured by
182 the leading eigenvector(s) of C. In principle at the extreme, where all evolutionary
183 change is perfectly parallel, C would be of unit rank, with all variance (= m) captured in a
184 single dimension, q1, of shared evolutionary change. At the other extreme, if
185 evolutionary change is completely independent in directional across all lineages sampled,
186 C would be of rank equal to m or n (whichever is lower), with variance distributed
187 uniformly across non-null (if more lineages than traits are sampled) eigenvectors. Thus,
188 the strength of parallelism captured by the leading eigenvector (q1) of C can be expressed
189 as / ∑ , where is the leading eigenvalue of C. True parallelism is reflected when
190 all lineages load positively on the corresponding eigenvector q1, while anti-parallelism
191 would be reflected by a mixture of positive and negative loadings on q1.
192
193 2) How many dimensions of shared evolutionary change exist? (Anti)Parallel evolution in
194 multiple orthogonal directions would be reflected in multiple significant eigenvalues.
195 When multiple dimensions are identified as significant (see below), these dimensions
196 reflect orthogonal axes of parallel and anti-parallel evolution. Multiple dimensions
197 suggest more than one alternative solution to a particular adaptive challenge, or evolution
198 towards more than one phenotypic optimum, and hence reflect parallel evolution among
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199 certain comparisons between replicate descendant populations but non-parallel evolution
200 for other such comparisons. Figure 2B illustrates such a scenario.
201
202 Thus, the eigenvectors and eigenvalues of C provide information on both the extent of
203 multivariate parallelism and contributions of specific lineages to parallel evolution or lack
204 thereof. Eigenvector scores could, in principle, be related to other variables of interest, such as
205 among-lineage environmental variation via canonical correlation or other approaches. We can
206 also relate the eigenvectors of C back to trait space via
(7)
207 Where A is a matrix with m column vectors of length n relating each eigenvector of C back to
208 trait space (Figure 2). We can then see how traits load on these vectors to establish which traits
209 are contributing most to parallelism.
210 Of course, sampling error will create variance in all or most dimensions. This same
211 sampling error will also lead to a skewed distribution of eigenvalues even in the absence of
212 actual parallel evolution (Johnstone 2001). This leads to the problem of constructing a null
213 expectation for evolutionary parallelism against which empirical patterns can be tested.
214 Specifically, what is the null expectation for the distribution of eigenvalues of C? This is a
215 problem in random matrix theory, where the distribution of eigenvalues of random covariance
216 matrices has been of great interest (Tracy and Widom 1996, Johnstone 2001, Tracy and Widom
217 2009, Blows and McGuigan 2015, Sztepanacz and Blows 2017). Generally, the leading
218 eigenvalue of a sample covariance matrix is expected to be follow a Tracy-Widom distribution
219 (Tracy and Widom 1996, 2009, Blows and McGuigan 2015, Sztepanacz and Blows 2017);
220 however, this distribution is sensitive to centering and scaling constants that may make
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221 hypothesis testing difficult (Blows et al. 2015), and we may also be interested in assessing
222 significance of more than one dimension (as suggested above). A null distribution of all m
223 eigenvalues can be obtained by sampling the m-dimensional Wishart distribution with n df,
224 where the hypothesis being tested depends on the covariance structure of the sampled Wishart
225 distribution (Johnstone 2001). If we are interested in testing a null hypothesis of independent
226 evolution across lineages, this would correspond to zero off-diagonals (an identity matrix) in the
227 Wishart covariance structure. An example of such a null distribution under varying n is
228 illustrated in Figure 3. For the case of more lineages than traits, we can simulate random vectors
229 to establish the null empirically, although the strongly skewed distribution of eigenvalues in such
230 a scenario indicates that exceptionally strong parallelism is required to reject the null (Figure 3).
231 By comparing the eigenvalues of C to the expected distribution of eigenvalues under the null
232 expectation, we can establish whether evolution is significantly more parallel in multivariate trait
233 space, than null expectations. And, we can test whether this is true in more than one dimension,
234 which would be reflected in multiple eigenvalues greater than the null expectation.
235 Although a null hypothesis of independent directions of evolution among lineages is
236 likely to be the most appropriate null for many tests of the presence and importance of
237 parallelism, other null hypotheses may be of interest. For example, one may wish to test
238 empirical patterns against a null of perfect parallelism, when researchers seek to identify
239 population-specific (e.g., non-parallel) aspects of evolution. In such cases a null distribution of
240 unit rank matrices would be appropriate. Alternatively, we may also be interested in the degree
241 to which genetic covariances and selection shape parallelism. We can relate C to G, for the case
242 of m < n, by postmultiplying the right hand side of equation 5 by the m dimensional identity
243 matrix in the form of and realizing that under directional selection and constant G,
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244 (Zeng 1988), where B is the covariance matrix of directional selection gradients across
245 lineages. However, because both G and R share the same dimensionality (number of traits, as
246 opposed to lineages as with C) it may be far more straightforward to test hypotheses on the
247 influence of G on evolutionary change via analysis of R instead of C (e.g., see Chenoweth et al.
248 2010). It is likely that constructing more sophisticated null expectations for contrasts in
249 orientation may require development of theoretical models that focus specifically on the
250 geometry of divergence in complex traits (e.g. see Thompson et al. 2019, whose model is
251 constructed specifically to bear relevance to angular contrasts).
252
253 Example: Parallel evolution across a lake-stream gradient in threespine stickleback
254 Threespine stickleback in postglacial inland waters of the Northern hemisphere represent one of
255 the most compelling models for testing the repeatability of evolution. Repeated colonization of
256 freshwater from marine environments has resulted in replicated divergence from a marine
257 ancestral phenotype into multiple freshwater forms (McPhail 1993, Bell and Foster 1994,
258 Colosimo et al. 2005). In a comprehensive study of parallel evolution in this system, Stuart et al.
259 (2017) sampled morphology, genetics, and environmental variables from lake and stream
260 subpopulations from 16 lineages (watersheds) on Vancouver Island, Canada. They found only
261 limited evidence of parallelism in morphological divergence in 84-dimensional trait space
262 between lake and stream subpopulations across the 16 replicate lineages, as inferred in part from
263 a relatively even distribution of angular distances across all 120 pairwise lineage combinations
264 (their figure 3b) with no clear signal of a skew towards values near zero. By correlating these
265 pairwise angular distances in morphological change vectors, with estimates of angular distance
266 in genetic and environmental change vectors, Stuart et al. showed that deviations from
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267 parallelism among lineages is due to non-parallelism in environmental factors, and levels of gene
268 flow across subpopulations. This study is unique in leveraging additional data to explain
269 variation in morphological parallelism, yet the high dimensional nature of the Stuart et al. data
270 exemplifies the challenges of inferring the overall degree of parallelism from angular distance
271 alone.
272 Although element-by-element interpretation of angular distances provides little evidence
273 of strong morphological parallelism in the Stuart et al. data, there are a few traits that were
274 strongly parallel. In a reanalysis of their morphological dataset, we find that spectral
275 decomposition of the matrix of vector correlations in morphological change vectors, C, reveals a
276 strongly skewed distribution of eigenvalues (Figure 4). Comparison of this distribution to that
277 expected under a null hypothesis of random (in direction) evolutionary change across lineages
278 reveals strong statistical support for three dimensions of parallel evolution (Figure 4). The
279 leading eigenvalue is greater than 7, capturing nearly 50% of the variance among lineages in the
280 direction of evolution, and together the three significant dimensions of parallelism explain 77%
281 of the among-lineage variation. Examination of the lineage loadings on the corresponding
282 eigenvectors reveals that the sign of these loadings is shared by most lineages, indicating these
283 shared axes of divergence are largely (although not completely) axes of parallelism, as opposed
284 to antiparallelism (Figure 5). Projecting these vectors back into trait space (via equation 7)
285 indicates that traits classified as those related to defense, swimming, and trophic interactions load
286 most strongly on these three axes, compared to traits that are generally unclassified (Figure 6).
287 This analysis (SAS/IML script as well as estimates of C, Q, V, and A, provided in the
288 supplemental material) suggests an important role for morphological parallelism across lineages,
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289 and suggests traits known to play a role in performance, defense, and resource acquisition to be
290 important in parallel adaptation across a lake/stream gradient.
291 By revealing statistical support for three dimensions of (imperfect) parallelism, our revisit
292 of the Stuart et al. dataset in some ways recapitulates their findings: we do not find evidence of a
293 single dimension of complete parallelism, which would be reflected in only a single significant
294 eigenvalue with an eigenvector that all lineages load positively onto. Thus, patterns of
295 evolutionary change in this system are apparently complex, and influenced by more than the
296 simple environmental classification of ‘lake’ versus ‘stream’, and the analyses in Stuart et al.
297 (2017) show how such deviations from parallelism can be associated with potential causal
298 sources of variation. However, the multivariate approach we take here suggests a far more
299 important role for morphological parallelism than can be inferred from interpretation of
300 individual angular distances: most of the variation in evolutionary change in 84 dimensional trait
301 space across 16 independent lineages can be captured in just three dimensions, indicating
302 overwhelming statistical support (c.f. lead eigenvalue in Figure 5 to null) to reject a null
303 hypothesis that evolution has proceeded in random directions in trait space across these lineages.
304 Moreover, although trait-by-trait univariate linear models suggest that the traits with clearly-
305 defined ecological functions rarely diverge in parallel in univariate space (Stuart et al.’s Figure
306 1A), our multivariate analysis reveals that these traits are in fact the most important contributors
307 to shared axes of parallel divergence in multivariate trait space (Figure 6). This is consistent
308 with biological intuition that traits related to swimming performance, defense from predators,
309 and feeding may play an important role in adaptation to lake versus stream environments. Thus,
310 our reanalysis of the Stuart et al. dataset illustrates the utility of taking a multivariate approach to
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311 assessing the overall degree of parallelism, an approach which complements their original
312 analyses relating deviations from perfect parallelism to gene flow and environmental variation.
313
314 Convergent and divergent evolution in multivariate trait space
315 We can think of convergent versus divergent evolution as a distinct and non-exclusive
316 phenomenon from parallelism (Figure 1; Bolnick et al. 2018). Past workers have proposed
317 separate (from analysis of parallelism) tests of convergence via estimation of the distribution of
318 pair-wise differences in the lengths of evolutionary change vectors
∆ || ∆ || || ∆ || (8)
319 calculated for each pairwise combination of lineages, where non-zero values imply some degree
320 of convergence or divergence (Collyer and Adams 2007, Stuart et al. 2017, Bolnick et al. 2018).
321 Alternatively, estimation of the change in Euclidean distance between lineage pairs across
322 environments/timepoints a and b
, , , , , , (9)
323 can be calculated, where negative values would indicate convergence between a given lineage
324 pair and positive values divergence (Bolnick et al. 2018), and this approach could perhaps be
325 modified by instead calculating Mahalanobis distance standardized by a pooled covariance
326 matrix. The former approach (equation 8) is problematic because it is difficult to interpret how
327 values of ∆ correspond to convergence vs divergence (Bolnick et al. 2018). The latter
328 approach is essentially an indirect comparison of the variance among lineages in one
329 environment versus the other, which captures multivariate convergence versus divergence in an
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330 intuitive, albeit somewhat indirect (when more than two lineages are sampled) way. Both
331 approaches are problematic in that they do not allow assessment of which traits contribute most
332 to convergence versus divergence, nor do they account for more complex scenarios where the
333 form of convergent/divergent evolution differs across traits or trait combinations (Figure 7).
334 A more direct approach is to compare the among-lineage covariance matrices of trait
335 mean values, D (Lande 1979, Lande 1980), estimated separately for the two environments or
336 timepoints. Given we have vectors of trait means and corresponding to
337 timepoint/environment a and b for each lineage, we can calculate their separate among lineage
338 covariance matrices and . Multivariate convergent evolution would result in a reduction in
339 among-lineage variance, and a reduction in the size of the covariance matrix D. Thus a
340 comparison of the traces of and provides a test of whether evolution has been net-
341 convergent or net-divergent, similar to the analysis of pairwise Euclidean distances as in
342 equation 10.
343 The comparison of the among-lineage covariance matrices provides potential for
344 additional insight when patterns of selection or genetics produce convergence in some
345 combinations of traits, and divergence in others (Figure 7). Although the formal comparison of
346 covariance matrices is a complex statistical problem, the challenge is reduced when only a single
347 pair of matrices is to be compared (as is the case when only two environments or time points are
348 studied) and when one is interested in retaining and contrasting all principle components (Blows
349 et al. 2004). In this case, a Common Principle Component Analysis (CPCA) approach allows for
350 a straightforward test of how and differ that is also easily interpretable (Flury 1988,
351 Phillips and Arnold 1999). In the simplest case, a nested multivariate mixed effects model can
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352 be used to test the null of = , rejection of which indicates convergence (or divergence) in
353 some combination of traits.
354 Finally, we note that trait scaling represents a particular challenge in analysis of
355 convergence regardless of analytic approach. What do we do when traits are expressed in
356 different units, or differ substantially in value, or evolvability? In some cases, analysis of the
357 raw trait values (population means) may appropriate. Alternatively, one could center the
358 population means and scale them by a pooled estimate of evolvability, such as the pooled within-
359 population phenotypic or genetic covariance matrix P or G, if available. Regardless, we
360 emphasize that the issue of trait scale is a complex and critical topic (see Houle et al. 2011), that
361 should be carefully considered in studies of multivariate evolutionary change.
362
363 Example: Multivariate divergent evolution during adaptation to a novel environment
364 Here we provide no novel analyses, but rather highlight a published study employing the
365 approaches outlined above to test hypotheses related to multivariate convergent evolution.
366 Although a number of studies have analyzed eigenstructure of D matrices to test hypotheses on
367 the form of divergence across lineages (Schluter 1996, Blows and Higgie 2003, McGuigan et al.
368 2005, Hohenlohe and Arnold 2008, Kolbe et al. 2011, Punzalan and Rowe 2016, De Lisle and
369 Rowe 2017), the study of Schoustra et al. (2012) provides an example of how formal comparison
370 of D matrices across environments provides a straightforward approach to testing hypotheses
371 related to multivariate convergence. In this study the authors used a laboratory experimental
372 evolution approach to understand how population size influences convergent/divergent
373 evolution. To do so they adapted replicate lineages of filamentous fungus (Aspergillus nidulans),
374 originating from a common ancestor, to a novel laboratory environment under two population
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375 size treatments, low and high. They then measured four traits related to colony formation at the
376 conclusion of 800 generations of adaptation. They used a nested multivariate mixed effects
377 model and found no evidence to reject the null of = , indicating no evidence of
378 multivariate convergence/divergence across population size treatments. They confirmed this
379 result using CPCA, finding no evidence to reject a null of matrix equality in favor of
380 proportionality. Finally, Schoustra et al. (2012) used a factor analytic mixed modelling approach
381 to estimate the rank of the pooled matrix, finding statistical support for a full rank divergence
382 matrix. Together, these analyses indicate divergent evolution in all trait dimensions following
383 adaptation from a common ancestor, and that this divergence was not influenced by population
384 size environments. Importantly, this study illustrates how analysis of D matrices in a study of
385 convergent evolution allows for biological hypotheses related to multivariate convergence to be
386 defined in terms of specific statistical contrasts of D across environments of interest.
387
388 Discussion and Conclusions
389 Here we advocate for an explicit multivariate approach in studies of parallel and convergent
390 evolution. Spectral decomposition and comparisons of two categories of covariance matrices,
391 the matrix of vector correlations between lineage evolutionary change vectors (C) and the
392 among-lineage covariance matrices in trait mean values (D), allow biologists to test hypotheses
393 concerning the extent and form of multivariate parallel or convergent evolution (respectively),
394 and to identify the crucial actual lineages and traits underlying these patterns. Whenever multiple
395 traits from multiple lineages are sampled, such an approach provides a potentially more complete
396 picture of the degree and extent of evolutionary change than does element-by-element analysis
397 (e.g., pairwise angles), as is often employed in such studies.
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398 The general approaches we advocate are certainly not new to evolutionary biology,
399 having been developed, described, and applied extensively in the evolutionary quantitative
400 genetics literature. Moreover, such approaches are likely already conceptually grasped by
401 workers familiar with the mechanics of a principle component analysis, and are readily
402 implemented in existing software capable of performing simple matrix manipulations (e.g., R,
403 SAS/IML, Matlab) and perhaps fitting of mixed effects models. Relevant purpose-built
404 statistical packages are also available (e.g. Melo et al. 2015). The point of our paper, then, is
405 simply to highlight one growing (Bolnick et al. 2018, Stuart 2019) subfield of evolutionary
406 biology – quantitative studies of evolutionary parallelism – where these multivariate approaches
407 may be particularly useful but are often not yet employed. Indeed, in many cases ascertaining the
408 true degree and form of parallel evolution, in particular, may be otherwise impossible.
409 We have suggested that spectral decomposition of among-lineage covariance matrices
410 may often indicate a greater extent of parallelism than is implied through element-by-element
411 interpretation of angular distances. Our reanalysis of a high dimensional dataset of
412 morphological evolution in stickleback lineages support this suggestion, and indicate a greater
413 role for parallelism than can be concluded from individual interpretation of angular distances
414 (Stuart et al. 2017). Of course, whether this suggestion is broadly true is an open empirical
415 question requiring additional studies from a variety of taxa. Moreover, our work suggests that in
416 some cases, especially when trait dimensionality is low, comparison of the eigenvalues of C to
417 an appropriate null may indicate little support for parallelism where an element-by-element
418 approach would suggest otherwise. This is because large correlations between evolutionary
419 change vectors from independent, randomly-diverging lineages are expected when many
420 lineages are sampled relative to the number of traits measured. Although high trait
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421 dimensionality often presents a challenge in analysis of multivariate datasets, analysis of vector
422 correlations transposes the dataframe (cf. equations 4 and 5) in such a way that flips the burden
423 of statistical power to favor sampling of traits. However, an important caveat is that any test of
424 convergent versus divergent evolution via comparison of D matrices, as also suggested here and
425 illustrated in past studies (e.g. Schoustra et al. 2012), will favor sampling of lineages over traits.
426 This suggests that careful consideration of the question of interest (tests of parallelism or tests of
427 convergence) may be critical in the early stages of study design if the approaches advocated here
428 are to be employed. Alternatively, in some cases it could be desirable to identify the traits
429 contributing most to parallelism via eigen analysis of C, and then focus on contrasts of lower
430 dimensional D matrices estimated using only a subset of key traits. Ultimately, the decision of
431 number and type of traits included in any analysis is a biological, rather than statistical or
432 geometric, problem; although high trait dimensionality results in null expectations favorable to
433 uncovering parallelism, including traits arbitrarily may weaken the effects observed through
434 parallel divergence in the traits that matter. Conversely, choosing traits non-randomly can
435 amount to cherry-picking that can overemphasize parallelism.
436 Analysis of the orientation of evolutionary change, either through angular distance or
437 vector correlation, has intuitive appeal for the quantification of parallelism and repeatability in
438 evolution. When evolutionary change involves multiple traits sampled across multiple lineages,
439 the question of how repeatable is the evolutionary process? is one that will often transcend both
440 individual traits and individual lineages, and so will be most clearly addressed with the
441 multivariate approaches presented here.
442
443
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444
445 Acknowledgements
446 We are grateful to Yoel Stuart and David Punzalan for discussion and their helpful comments on
447 the manuscript. Funding was provided by the University of Connecticut to DIB.
448
449
450
451
452
453
454
455
456
457
458
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AB Trait 2 Trait 2
Trait 1 Trait 1
459
460 Figure 1. Parallel and convergent evolution as separate but related patterns. Parallel
461 evolution can be defined as a shared orientation in the direction of evolutionary change across
462 two or more lineages, regardless of their starting (ancestral) position, illustrated for four lineages
463 in panel A showing perfect parallelism. Convergent evolution, however, describes a change
464 (reduction for convergence, increase for divergence) of the dispersion of lineages in trait space,
465 illustrated in panel B. Note that evolution can be any combination of parallel or convergent,
466 depending on the ancestral state of the lineages in question and the evolutionary processes at
467 play.
468
469
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470 471 Figure 2. Parallel evolution in multivariate trait space. In Panel A, Vectors of evolutionary
472 change for three traits sampled from 3 lineages are plotted as blue arrows. Each pairwise
473 combination of lineage vectors differ in orientation by 45.5 degrees in three dimensional trait
474 space. Analysis of these angles alone would indicate evolution is only weakly, if at all, parallel.
475 Yet all three lineages load strongly on a single major axis of evolutionary change that accounts
476 for 80% of the among-lineage variation in the orientation of evolution. The red arrow plots this
477 vector of shared multivariate parallel evolution back into three dimensional trait space (see eqn
478 3-5). Panel B illustrates a similar scenario but sampling four lineages diverging in two-
479 dimensional trait space. Red arrows plot the two orthogonal vectors of shared multivariate
480 parallel evolution, indicating one major axis of parallel evolution upon which all four lineages
481 load positively, in addition to a second axis upon which some lineage evolutionary change
482 vectors load positively, and some negatively, indicating antiparallel evolution in this direction
483 across the sample of lineages.
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484 485 Figure 3. Null distributions of eigenvalues for a 16-dimensional (16 lineage) C matrix.
486 Increasing trait dimensionality relative to lineage replication leads to null distributions favorable
487 to uncovering parallelism. Under the null hypothesis of no relationship between vectors of
488 evolutionary change, which would be reflected in a perfectly uniform distribution of eigenvalues,es,
489 correlations between lineages (manifest as eigenvalues exceeding unity) are still expected to be
490 observed through sampling error. The magnitude of these expected leading eigenvalues increaseses
491 with decreasing trait dimensionality simply due to random sampling. For example, when the
492 number of traits is less than the number of lineages (vectors) sampled, strong correlations are
493 inevitable even when the null hypothesis of random evolutionary divergence across lineages is
494 true. Thus, sampling many traits relative to lineages flattens the null distribution of eigenvalues,
495 increasing power to distinguish true parallel evolution from what is expected purely from
496 sampling error under the null hypothesis. It is important to note, however, that high trait
24 bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439; this version posted January 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license.
497 dimensionality simply creates null distributions favorable to uncovering parallelism; choice of
498 traits to include in any analysis is primarily a biological problem. Distributions obtained by
499 sampling the corresponding Wishart distribution, with the exception of 8 traits which was
500 constructed empirically by placing random vectors in trait space. Dimensionality chosen here for
501 continuity with the empirical example from Stuart et al. 2017.
502
503 504 505 506 507 508 509 510 511 512
513
514
515
516
517
518
519
520
25 bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439; this version posted January 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license.
8 Null distribution (empirical) Null distribution (Wishart) Observed distribution
6
4 Eigenvalue
2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Eigenvalue rank
521 522 Figure 4. Distribution of eigenvalues from matrix C from Stuart et al. 2017 data, showing
523 statistical support for three orthogonal dimensions of parallel change. Yellow and green
524 boxplots indicate the expected distribution of eigenvalues under a null hypothesis of a random
525 direction of evolutionary change cross lineages (zero off-diagonals in the matrix of vector
526 correlations), calculated from sampling the corresponding Wishart distribution or empirically by
527 placing random vectors in trait space (each 1,000 X). Blue indicates values observed from
528 diagonalization of the estimated matrix of vector correlations, C. C was calculated from
529 Supplementary data Table 1in Stuart et al. (2017), with each row of X first normalized to unit
530 length.
531
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532
533
0.2
0.0 Loading on PC1 of C
-0.2
Pachen Freder Pye M Bea Comida Robert Swan V Mo Joe Bo M Nort Kenned Th i uc llag o isty ieme hal v or t h er e y
Population 534 535
0.4
0.2
0.0 Loading on PC2 of C
-0.2 Misty Bo V J Mo C Kenned ThiemeFreder N R Beaver Pye Sw Pachen Much il o o o o o la e mid r ber t g o t a re hy n a a t l Population 536
27 bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439; this version posted January 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license.
0.2
0.0
-0.2 Loading on PC3 of C of PC3 on Loading
-0.4
No Fre Pa Be Vi P Th Muc M M Bo C Ke Robert J Sw ye o r ch a lla ie o isty ot omid n e th der ore ne a e ve g m h n y n r e al a d Population 537 538
539 Fig 5. Loadings for each lineage on the first three eigenvectors (principle components) of C.
540 The observation that most lineages are loading positively on PC1 and PC2 indicate that those are
541 dimensions of (mostly) true parallel evolution, as opposed to anti parallel evolution, which
542 would be represented as a mixture of positive and negative loadings across lineages, for example
543 PC3.
544
545
546
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547
548 549 Fig 6. Each of the three significant dimensions of parallelism, estimated from reanalysis of
550 Stuart et al. 2017, related back to trait space. Trait classifications are taken from Stuart et al.
551 2017.
552
553
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AB
Trait 2 C D
554 Trait 1 555 556 Figure 7. Convergent and divergent evolution in multivariate trait space. Comparison of
557 among lineage variance-covaraince matrices D of mean trait values estimated at timepoint (or
558 environment) one (dashed elipses) and two (solid elispses) indicate the form of convergent or
559 divergent evolution. Arrows in panel A illustriate hypothetical evolutionary change vectors,
560 which are left off of subsequent panels for concision. In panel A, evolution is net-convergent and
561 all traits contribute to this convergence; matrices are proportional. In panel B, evolution is
562 convergent in only a single trait dimension. In panel C, evolution is both divergent and
563 convergent, in different combinations of traits, leading to changes in the orientation of the
564 among-lineage covariance matrix. In panel D, evolution is neither convergent nor divergent in
565 any combination of traits, despite strong parallel change across all lineages, leading to no net
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566 change in the among-lineage covariance. All cases illustrate largely parallel evolutionary change
567 in overall size.
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
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589 References
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690
691 Supplemental Material doi:
692 Tables https://doi.org/10.1101/2020.01.26.920439
Table S1. Matrix C of vector correlations between lineage evolutionary change vectors from Stuart et al 2017 data Village Beaver Boot Comida Frederick Joe Kennedy Misty Moore Muchalat Northy Pachena Pye Roberts Swan Thiemer Bay
Beaver 1 ------available undera
Boot -0.215 1 ------Comida 0.611 0.1764 1 ------Frederick 0.806 0.6859 1 ------0.0927 CC-BY 4.0Internationallicense
Joe 0.092 0.4975 0.5374 0.2702 1 ------; this versionpostedJanuary27,2020. Kennedy -0.607 0.5037 -0.4296 -0.6763 0.0232 1 ------
Misty -0.115 0.6279 0.1237 -0.0905 0.2473 0.5395 1 ------
Moore 0.298 0.0589 0.4003 0.3499 0.1781 -0.0741 0.4100 1 ------Muchalat 0.544 0.5165 0.6601 0.0204 -0.7952 0.1722 1 ------0.4092 0.5312 - - Northy -0.335 0.2551 -0.4651 -0.3115 0.5977 0.4289 -0.597 1 ------0.2904 0.1105 - - Pachena 0.753 0.5767 0.8734 0.1251 -0.7509 0.3215 0.844 -0.465 1 - - - - -
0.3524 0.3357 . - - Pye 0.690 0.5544 0.8042 0.2033 -0.6206 0.3474 0.713 -0.450 0.8711 1 - - - - 0.2233 0.2439 - -
Roberts 0.460 0.6201 0.4360 0.4897 -0.3416 0.3868 0.423 -0.622 0.5117 0.5155 1 - - - The copyrightholderforthispreprint 0.2462 0.1805 - - Swan 0.247 0.5223 0.2682 0.5166 -0.2108 0.1145 0.297 -0.580 0.2811 0.2563 0.8601 1 - - 0.1401 0.2532 - Thiemer -0.595 0.4242 -0.5560 -0.5848 0.8036 0.5697 0.0630 -0.680 0.730 -0.6423 -0.5329 -0.6453 -0.648 1 - 0.2125 Village Bay 0.450 0.3726 0.5420 0.5722 0.4027 -0.1337 0.2773 0.3688 0.247 -0.102 0.4878 0.5747 0.2645 0.051 -0.077 1
693
694
35 (which wasnotcertifiedbypeerreview)istheauthor/funder,whohasgrantedbioRxivalicensetodisplaypreprintinperpetuity.Itmade bioRxiv preprint
695
Table S2. Spectral decomposition of C from Stuart et al. 2017 doi:
Eigenvalues Eigenvectors, Q https://doi.org/10.1101/2020.01.26.920439
V lineage q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16
7.405 Beaver 0.29 0.06 0.22 0.03 0.53 -0.08 -0.36 -0.12 0.41 -0.14 0.42 0.11 0.09 0.19 0.03 0.09
3.010 Boot -0.12 0.45 -0.06 -0.42 -0.14 -0.19 -0.02 0.16 0.45 -0.28 -0.27 -0.03 0.30 -0.24 0.13 0.10
1.983 Comida 0.27 0.27 -0.09 -0.07 0.13 -0.34 0.02 0.49 -0.18 0.19 0.23 -0.45 -0.34 -0.13 -0.03 0.07
0.998 Frederick 0.31 0.13 0.24 -0.11 0.24 -0.01 0.29 -0.11 0.05 -0.09 -0.40 0.19 -0.44 0.03 0.29 -0.44 available undera 0.598 Joe 0.11 0.36 -0.37 -0.27 -0.11 0.08 0.41 -0.48 -0.04 0.02 0.43 0.04 -0.04 0.13 -0.12 -0.03
0.543 Kennedy -0.29 0.22 -0.14 0.12 0.05 0.37 -0.10 0.31 0.37 0.18 0.11 0.16 -0.23 -0.08 -0.38 -0.42
0.347 Misty -0.14 0.46 0.09 0.19 0.16 -0.32 -0.07 -0.11 -0.20 0.56 -0.20 0.31 0.26 0.11 -0.03 0.04
0.292 Moore 0.12 0.30 0.13 0.70 -0.29 -0.20 0.13 -0.08 0.08 -0.43 -0.02 -0.12 -0.01 0.04 -0.22 -0.04 CC-BY 4.0Internationallicense ; 0.199 Muchalat 0.31 -0.15 0.13 -0.06 -0.34 -0.10 0.27 0.46 0.09 0.11 0.23 0.30 0.35 0.32 0.08 -0.24 this versionpostedJanuary27,2020.
0.179 Northy -0.26 0.11 0.29 0.03 0.47 0.30 0.52 0.19 -0.18 -0.16 0.06 -0.21 0.33 0.02 -0.05 0.06
0.138 Pachena 0.33 -0.02 0.23 0.01 -0.06 0.17 0.19 0.03 0.04 0.11 0.03 0.40 -0.09 -0.57 -0.27 0.44
0.122 Pye 0.31 0.06 0.22 -0.01 -0.18 0.35 -0.01 -0.17 0.31 0.43 -0.24 -0.53 0.12 0.18 -0.08 0.07
0.068 Roberts 0.27 0.09 -0.36 0.33 0.09 0.27 -0.09 0.00 -0.07 0.06 0.09 -0.04 0.32 -0.42 0.49 -0.23
0.053 Swan 0.21 0.03 -0.53 0.13 0.20 0.20 0.08 0.25 0.05 -0.11 -0.35 0.18 -0.07 0.41 -0.04 0.42
0.039 Thiemer -0.31 0.18 0.20 0.15 -0.23 0.18 0.07 0.08 0.15 0.11 0.24 0.08 -0.33 0.13 0.60 0.35 .
0.027 Village Bay 0.16 0.38 0.22 -0.21 -0.18 0.40 -0.44 0.10 -0.50 -0.25 0.01 0.10 0.01 0.15 -0.05 -0.03 The copyrightholderforthispreprint 696
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36 (which wasnotcertifiedbypeerreview)istheauthor/funder,whohasgrantedbioRxivalicensetodisplaypreprintinperpetuity.Itmade bioRxiv preprint
700 doi: https://doi.org/10.1101/2020.01.26.920439 available undera CC-BY 4.0Internationallicense ; this versionpostedJanuary27,2020. . The copyrightholderforthispreprint
37 bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439Table S3. Matrix A relating; this eigenvectorsversion posted of January C to trait 27, space 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made a1 a2 a3 availablea4 a5under aCC-BYa6 4.0a7 International a8 licensea9 . a10 a11 a12 a13 a14 a15 ------Standard length 0.384 0.218 0.044 0.045 0.018 0.021 0.006 0.032 0.002 0.016 0.029 0.015 0.021 0.019 0.014 ------First dorsal spine length 0.366 0.169 0.159 0.044 0.062 0.159 0.075 0.098 0.015 0.018 0.025 0.012 0.030 0.013 0.000 ------Second dorsal spine length 0.297 0.206 0.191 0.118 0.062 0.215 0.078 0.185 0.003 0.001 0.032 0.019 0.010 0.022 0.001 ------Dorsal fin length 0.232 0.070 0.000 0.029 0.062 0.008 0.040 0.026 0.037 0.027 0.024 0.016 0.014 0.000 0.007 ------Caudal peduncle depth 0.494 0.011 0.102 0.111 0.002 0.064 0.007 0.040 0.038 0.017 0.012 0.018 0.003 0.008 0.008 ------Anal. fin length 0.214 0.182 0.028 0.116 0.101 0.012 0.006 0.017 0.014 0.015 0.007 0.003 0.024 0.015 0.010 ------Pectoral fin insertion length 0.269 0.153 0.026 0.055 0.054 0.087 0.022 0.065 0.015 0.028 0.004 0.056 0.036 0.009 0.005 ------Body depth 0.598 0.225 0.151 0.018 0.090 0.018 0.049 0.019 0.049 0.042 0.070 0.020 0.029 0.041 0.009 - - - - Mouth Length 0.218 0.152 0.025 0.033 0.008 0.054 0.013 0.003 0.024 0.006 0.010 0.006 0.045 0.038 0.046 ------Snout length 0.386 0.242 0.047 0.025 0.134 0.060 0.079 0.004 0.033 0.012 0.016 0.037 0.028 0.012 0.011 - - - Eye length 0.236 0.294 0.079 0.037 0.036 0.144 0.016 0.050 0.016 0.049 0.008 0.067 0.010 0.004 0.012 ------Head length 0.364 0.142 0.009 0.052 0.115 0.065 0.020 0.054 0.002 0.031 0.012 0.008 0.013 0.003 0.012 ------Pectoral fin width 0.287 0.050 0.051 0.087 0.046 0.034 0.056 0.107 0.022 0.132 0.003 0.014 0.019 0.002 0.021 - - - - Pectoral fin length 0.267 0.160 0.043 0.067 0.051 0.028 0.126 0.024 0.121 0.041 0.010 0.008 0.025 0.003 0.004 ------Pectoral fin perimeter 0.260 0.036 0.093 0.083 0.026 0.095 0.015 0.098 0.094 0.123 0.016 0.005 0.025 0.007 0.044 - - - - - Pectoral fin area 0.273 0.272 0.079 0.041 0.089 0.085 0.019 0.100 0.098 0.084 0.007 0.027 0.008 0.037 0.030 - - - - Buccal cavity length 0.380 0.090 0.014 0.122 0.139 0.022 0.037 0.035 0.043 0.021 0.071 0.049 0.021 0.022 0.026 - - - - - Gape width 0.441 0.003 0.026 0.098 0.016 0.020 0.023 0.074 0.016 0.038 0.004 0.042 0.019 0.047 0.004 ------Body width point 1 0.368 0.016 0.065 0.153 0.012 0.003 0.061 0.021 0.005 0.038 0.013 0.059 0.027 0.039 0.037 ------Body width point 2 0.439 0.052 0.076 0.020 0.120 0.029 0.120 0.067 0.015 0.028 0.011 0.028 0.006 0.014 0.015 ------Pelvic girdle width 0.574 0.106 0.104 0.049 0.163 0.036 0.070 0.004 0.014 0.014 0.062 0.043 0.016 0.005 0.017 ------Pelvic girdle diamond width 0.563 0.103 0.060 0.045 0.091 0.027 0.043 0.011 0.061 0.035 0.062 0.057 0.054 0.016 0.024 ------Pelvic girdle length 0.344 0.270 0.088 0.131 0.105 0.018 0.030 0.064 0.068 0.009 0.064 0.045 0.029 0.006 0.004 Pelvic girdle diamond ------length 0.363 0.217 0.015 0.079 0.056 0.012 0.022 0.039 0.018 0.026 0.014 0.008 0.047 0.001 0.007 ------Body width point 3 0.370 0.066 0.058 0.005 0.055 0.015 0.131 0.060 0.019 0.067 0.013 0.011 0.080 0.019 0.016 ------Body width point 4 0.373 0.080 0.011 0.047 0.048 0.020 0.066 0.046 0.028 0.067 0.013 0.033 0.052 0.054 0.010 ------Left side plate number 0.064 0.046 0.106 0.293 0.168 0.137 0.023 0.040 0.026 0.064 0.073 0.054 0.022 0.034 0.012 ------Right side plate number 0.055 0.092 0.083 0.321 0.131 0.167 0.020 0.067 0.020 0.088 0.069 0.040 0.002 0.020 0.006 - - - - Right side gill raker number 0.227 0.479 0.166 0.022 0.093 0.029 0.005 0.032 0.149 0.109 0.056 0.059 0.078 0.010 0.007 ------First longest raker length 0.256 0.253 0.020 0.004 0.001 0.152 0.067 0.090 0.040 0.008 0.033 0.059 0.048 0.026 0.016 Second longest raker ------length 0.262 0.228 0.029 0.001 0.009 0.138 0.102 0.070 0.053 0.005 0.017 0.081 0.034 0.020 0.006 ------Third longest raker length 0.235 0.227 0.022 0.022 0.004 0.142 0.053 0.080 0.041 0.022 0.023 0.055 0.039 0.029 0.005 ------Raker density 0.392 0.065 0.007 0.136 0.016 0.101 0.026 0.050 0.017 0.036 0.057 0.001 0.023 0.001 0.007
38 bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439; this version posted January 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license.
------Mean pelvic spine length 0.386 0.284 0.256 0.079 0.066 0.217 0.119 0.006 0.022 0.003 0.069 0.100 0.041 0.015 0.003 ------Jaw opening inlever length 0.346 0.150 0.073 0.011 0.024 0.013 0.005 0.064 0.101 0.003 0.077 0.046 0.015 0.046 0.024 - - - - - Jaw closing inlever length 0.285 0.188 0.001 0.016 0.001 0.026 0.010 0.007 0.036 0.017 0.055 0.006 0.018 0.010 0.025 ------Jaw outlever length 0.289 0.188 0.016 0.010 0.018 0.046 0.005 0.046 0.043 0.029 0.058 0.022 0.013 0.048 0.001 Epaxial muscle cross------sectional area 0.786 0.283 1.206 0.303 0.030 0.105 0.053 0.014 0.003 0.006 0.002 0.021 0.016 0.008 0.002 ------Neurocranium length 0.298 0.106 0.083 0.032 0.048 0.017 0.020 0.021 0.026 0.049 0.012 0.025 0.016 0.021 0.038 ------Epaxial muscle height 0.302 0.027 0.092 0.011 0.005 0.021 0.081 0.025 0.003 0.038 0.110 0.019 0.000 0.022 0.005 ------Opercular 4-bar fixed length 0.285 0.159 0.051 0.013 0.061 0.012 0.057 0.035 0.055 0.011 0.020 0.042 0.012 0.015 0.015 Opercular 4-bar coupler ------length 0.265 0.124 0.032 0.032 0.088 0.034 0.073 0.046 0.033 0.007 0.009 0.028 0.011 0.026 0.024 Opercular 4-bar input ------length 0.308 0.086 0.060 0.009 0.009 0.022 0.047 0.058 0.053 0.053 0.074 0.062 0.021 0.012 0.022 Opercular 4-bar output ------length 0.244 0.123 0.037 0.002 0.011 0.008 0.035 0.040 0.056 0.008 0.026 0.050 0.013 0.025 0.021 Opercular 4-bar diagonal ------length 0.265 0.129 0.022 0.035 0.096 0.028 0.046 0.015 0.032 0.017 0.030 0.021 0.006 0.020 0.020 ------Centroid size 0.387 0.221 0.029 0.035 0.028 0.032 0.005 0.034 0.010 0.016 0.035 0.007 0.029 0.021 0.005 Landmark 1 GPA ------coordinate X 0.134 0.097 0.076 0.095 0.192 0.112 0.002 0.107 0.004 0.052 0.033 0.018 0.015 0.010 0.009 Landmark 1 GPA - - - - coordinate Y 0.090 0.027 0.038 0.090 0.077 0.072 0.089 0.020 0.022 0.028 0.054 0.007 0.017 0.002 0.039 Landmark 2 GPA ------coordinate X 0.177 0.256 0.060 0.105 0.054 0.153 0.129 0.010 0.010 0.045 0.033 0.095 0.014 0.027 0.011 Landmark 2 GPA ------coordinate Y 0.155 0.072 0.034 0.077 0.140 0.045 0.069 0.016 0.014 0.038 0.030 0.003 0.028 0.017 0.034 Landmark 3 GPA - - - - coordinate X 0.065 0.192 0.059 0.012 0.125 0.071 0.094 0.154 0.101 0.063 0.009 0.038 0.013 0.050 0.048 Landmark 3 GPA ------coordinate Y 0.381 0.319 0.065 0.056 0.070 0.028 0.024 0.124 0.023 0.050 0.010 0.010 0.024 0.030 0.005 Landmark 4 GPA ------coordinate X 0.030 0.144 0.111 0.301 0.031 0.024 0.071 0.029 0.038 0.047 0.018 0.013 0.030 0.009 0.031 Landmark 4 GPA ------coordinate Y 0.453 0.460 0.195 0.112 0.080 0.010 0.005 0.006 0.003 0.001 0.034 0.027 0.024 0.004 0.014 Landmark 5 GPA ------coordinate X 0.135 0.039 0.031 0.142 0.064 0.059 0.130 0.018 0.015 0.044 0.066 0.023 0.033 0.020 0.033 Landmark 5 GPA ------coordinate Y 0.340 0.405 0.165 0.170 0.062 0.010 0.031 0.031 0.004 0.006 0.038 0.019 0.032 0.013 0.006 Landmark 6 GPA ------coordinate X 0.081 0.132 0.046 0.181 0.132 0.073 0.104 0.051 0.008 0.109 0.004 0.001 0.029 0.034 0.002 Landmark 6 GPA ------coordinate Y 0.244 0.322 0.095 0.141 0.037 0.081 0.025 0.095 0.044 0.024 0.020 0.001 0.017 0.013 0.024 Landmark 7 GPA ------coordinate X 0.165 0.128 0.034 0.054 0.103 0.017 0.011 0.038 0.045 0.060 0.062 0.014 0.005 0.030 0.023 Landmark 7 GPA ------coordinate Y 0.005 0.265 0.014 0.020 0.007 0.053 0.062 0.001 0.003 0.030 0.069 0.019 0.040 0.034 0.017 Landmark 8 GPA - - - - - coordinate X 0.425 0.047 0.020 0.129 0.018 0.036 0.042 0.042 0.028 0.041 0.066 0.024 0.026 0.005 0.037 Landmark 8 GPA - - - - - coordinate Y 0.014 0.015 0.007 0.111 0.076 0.057 0.035 0.001 0.037 0.017 0.019 0.002 0.027 0.006 0.004 Landmark 9 GPA ------coordinate X 0.323 0.120 0.014 0.050 0.044 0.061 0.029 0.056 0.048 0.062 0.017 0.024 0.019 0.022 0.010 Landmark 9 GPA ------coordinate Y 0.004 0.056 0.026 0.114 0.097 0.036 0.038 0.008 0.062 0.046 0.009 0.021 0.003 0.006 0.003 Landmark 10 GPA ------coordinate X 0.423 0.115 0.014 0.077 0.042 0.055 0.017 0.021 0.090 0.120 0.030 0.002 0.007 0.025 0.035 Landmark 10 GPA - - - - - coordinate Y 0.013 0.101 0.079 0.126 0.099 0.050 0.012 0.009 0.046 0.025 0.018 0.002 0.008 0.017 0.001 Landmark 11 GPA ------coordinate X 0.070 0.102 0.009 0.052 0.074 0.069 0.031 0.040 0.034 0.020 0.016 0.007 0.067 0.003 0.044 Landmark 11 GPA - - - - - coordinate Y 0.165 0.147 0.124 0.051 0.005 0.009 0.069 0.099 0.057 0.053 0.050 0.019 0.015 0.014 0.026
39 bioRxiv preprint doi: https://doi.org/10.1101/2020.01.26.920439; this version posted January 27, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license.
Landmark 12 GPA ------coordinate X 0.066 0.109 0.082 0.188 0.154 0.052 0.045 0.067 0.019 0.000 0.014 0.026 0.001 0.003 0.000 Landmark 12 GPA - - - - - coordinate Y 0.273 0.235 0.063 0.038 0.029 0.114 0.030 0.100 0.136 0.081 0.019 0.003 0.006 0.019 0.010 Landmark 13 GPA ------coordinate X 0.240 0.049 0.184 0.297 0.112 0.121 0.102 0.011 0.017 0.050 0.000 0.012 0.058 0.007 0.033 Landmark 13 GPA ------coordinate Y 0.104 0.181 0.003 0.088 0.151 0.084 0.077 0.032 0.011 0.014 0.003 0.005 0.023 0.008 0.017 Landmark 14 GPA ------coordinate X 0.069 0.095 0.045 0.058 0.049 0.138 0.070 0.040 0.011 0.023 0.005 0.036 0.025 0.053 0.016 Landmark 14 GPA - - - - coordinate Y 0.242 0.311 0.034 0.036 0.159 0.104 0.030 0.020 0.003 0.011 0.036 0.084 0.005 0.025 0.016 Landmark 15 GPA - - - - - coordinate X 0.041 0.226 0.048 0.117 0.063 0.013 0.004 0.075 0.116 0.002 0.056 0.038 0.018 0.090 0.026 Landmark 15 GPA ------coordinate Y 0.238 0.340 0.052 0.000 0.115 0.117 0.008 0.024 0.007 0.020 0.004 0.088 0.008 0.025 0.004 Landmark 16 GPA ------coordinate X 0.163 0.316 0.059 0.037 0.159 0.046 0.050 0.065 0.028 0.040 0.040 0.062 0.046 0.041 0.013 Landmark 16 GPA ------coordinate Y 0.212 0.211 0.055 0.036 0.093 0.034 0.156 0.017 0.013 0.005 0.079 0.023 0.010 0.021 0.012 Landmark 17 GPA ------coordinate X 0.017 0.167 0.088 0.172 0.009 0.117 0.104 0.062 0.022 0.002 0.049 0.062 0.038 0.008 0.033 Landmark 17 GPA ------coordinate Y 0.165 0.070 0.024 0.034 0.124 0.028 0.070 0.010 0.018 0.036 0.068 0.022 0.022 0.007 0.039 Landmark 18 GPA ------coordinate X 0.213 0.048 0.076 0.054 0.126 0.056 0.138 0.047 0.090 0.014 0.049 0.021 0.013 0.031 0.019 Landmark 18 GPA ------coordinate Y 0.058 0.044 0.073 0.084 0.004 0.080 0.117 0.081 0.058 0.023 0.022 0.049 0.010 0.012 0.018 Landmark 19 GPA ------coordinate X 0.022 0.238 0.060 0.040 0.073 0.074 0.086 0.074 0.057 0.072 0.007 0.042 0.020 0.019 0.019 Landmark 19 GPA ------coordinate Y 0.040 0.060 0.077 0.144 0.079 0.126 0.016 0.051 0.020 0.022 0.022 0.043 0.005 0.016 0.056
701
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