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Mechanism of Evolution
Homoplasy: From Detecting Pattern to Determining Process and
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Homoplasy: From Detecting Pattern
to Determining Process
and Mechanism of Evolution
David B. Wake,1,2* Marvalee H. Wake,1,2Chelsea D. Specht3
Understanding the diversification of phenotypes through time—“descent with modification”—has
been the focus of evolutionary biology for 150 years. If, contrary to expectations, similarity evolves in
unrelated taxa, researchers are guided to uncover the genetic and developmental mechanisms
responsible. Similar phenotypes may be retained from common ancestry (homology), but a phylogenetic
or less likely, that they experienced reversal. Such examples of homoplasy present opportunities to
discover the foundations of morphological traits. A common underlying mechanism may exist, and
components may have been redeployed in a way that produces the “same” phenotype. New, robust
phylogenetic hypotheses and molecular, genomic, and developmental techniques enable integrated
exploration of the mechanisms by which similarity arises.
ically similar, explanation is needed. Homoplasy
is similarity that is the result not of simple an-
a lineage or of independent evolution (conver-
gence, similarity resulting from different devel-
opmental genetic mechanisms; or parallelism,
similarity resulting from the same developmental
elongation in salamanders usually occurs in
parallel in different taxa by addition of vertebrae,
but increased body length may result from elon-
gation of individual vertebrae,an instance of con-
vergence (Fig. 1B). Independent evolution can
result from common adaptive responses to selec-
tion pressures, such as changes in phenotype as-
sociated with a particular life strategy [e.g., a loss
of structural anatomy in aquatic plants; the re-
duction of leaf blade surface in desert plants;
evolution of expanded toe tips (scansors), spe-
more organismal mode of evolution, dependent
on developmental and genetic mechanisms that
are deeply embedded in the evolutionary history
of the lineage and are components of integrated
ary mode—the focus of this essay—hierarchical
perspectives are essential (3). Complex morpho-
logical features of organisms are self-regulating
from developmental genetic and historical per-
by constraints, not all possible morphologies for
as evolution proceeds. Thus, when diver-
gent lineages are found to be morpholog-
a particular organism are realized or expressed.
This inherent limitation on form increases the
likelihood of homoplasy (4).
Phylogenetic analysis is necessary to show
that derived similarity is not the simple result of
common ancestry of taxa being compared.
Usually homoplastic features are consequences
of convergence or parallelism (Fig. 1, B and C).
Structures that appear to have been lost may
reappear, but such instances are uncommon (Fig.
1A). Study of the underlying developmental ge-
netic mechanisms may reveal whether the recur-
or whether the ancestral mechanism has been de-
ployed repeatedly. Thus, the study of homoplasy
requires the integration of genetic, developmen-
tal, and phylogenetic resources and perspec-
tives. However, one does not seek homoplasy—it
“finds” the researcher and compels one to ask
How Is Homoplasy Recognized?
Homology is what is perceived as the same trait
in different taxa and is a true representation of
inheritance and phylogeny at the organismal level
responsible for generating it). Homoplasy is the
diametric opposite of homology (5)—underlying
similarity that does not result from inheritance at
the hierarchical level (e.g., gene, tissue, organ; de-
velopmental pattern) being considered (6, 7).
Homoplasy is recognized by discordance with
other characters in a phylogenetic analysis (Fig.
2). Molecular sequence data have greatly in-
creased our ability to identify homoplastic traits.
The various classes of homoplasy (convergence,
(10). Whereas parallelism and convergence run
along a continuum (11, 12), convergence typical-
ly occurs over relatively greater phylogenetic dis-
the genetics of adaptation (13); convergence gen-
erally results from different genetic mechanisms,
while parallelism typically arises from similar ge-
netic causes, providing a heuristic context. Once
identified, processes that generate the homoplastic
traits become the targets of research.
A New Emphasis on Processes, Mechanisms,
interesting questions for modern developmental
genetics and evolutionary biology. Using devel-
opmental genetic approaches in comparative and
hierarchical contexts is essential for identifying
and defining processes responsible for similar
phenotypes in diverse taxa. Mechanisms respon-
or whole organismal, developmental, epigenetic,
and genetic levels.
The integration of genetics, signaling patterns
and regulation, developmental pathways, and
phylogenetics is in its infancy, but promises to
comparing genetic regulatory networks (GRNs),
and conducting experiments to alter them, the
causal basis of development and evolution is
illuminated, and evolutionary pathways that lead
to fundamental changes in morphology can
potentially be reproduced [synthetic experimen-
tal evolution (14)]. Although experiments may
reproduce ancestral phenotypes, alternative de-
velopmental pathways may exist. Exploring the
potential range of phenotypes [evolvable states
(15)] to reveal genetic mechanisms involved
withmacroevolutionaryprocesses islikely tobe
Adaptively Driven Homoplasy
Adaptively driven homoplasy may result from
similar selective pressure, as in the evolution of
reduced body armor and pelvic appendage struc-
that occurred repeatedly in populations that in-
vaded freshwater lakes, which are characterized
by reduced numbers of predators (16, 17). Pelvic
loss results when regulatory mutations occur that
cause deletion of a tissue-specific enhancer asso-
ciated with the Pituitary homeobox transcription
factor 1 (Pitx1) gene (18). Selection for a reduc-
tion in lateral body armor plates involves muta-
findings show that major phenotypic changes can
be associated with regulatory changes in devel-
opmental genetic programs (20).
Homoplasy of individual genes is exempli-
fied by convergent adaptive pigmentation in di-
receptor gene (Mc1r) was found in light-colored
beach mice, as well as a 43,000-year-old mam-
1Museum of Vertebrate Zoology, University of California, Berke-
ley, CA 94720, USA.2Department of Integrative Biology, Uni-
versity of California, Berkeley, CA 94720, USA.3Department of
Plant and Microbial Biology, University of California, Berkeley,
CA 94720, USA.
*To whom correspondence should be addressed. E-mail:
25 FEBRUARY 2011 VOL 331
on March 7, 2011
in the same gene sometimes explain convergent
phenotypes. For example, different mutations in
Mc1r are responsible for blanched phenotypes
of two species of lizards (only distantly related
to each other) from the White Sands of New
Mexico (22), and different mutations in Agouti
are responsible for independently evolved light
coloration in Nebraska Sand Hill and Florida
Coast populations of Peromyscus (maniculatus
and polionotus, respectively). Finally different
genes entirely can be responsible for convergent
phenotypes, as is likely the case for independent-
ly evolved light coloration of Gulf and Atlantic
Coast populations of P. polionotus. These exam-
ples document that phenotypic convergence in-
volves fine- to coarse-grained genetic changes.
plify acomplex hierarchicalevolutionary history.
The perianth (i.e., sterile structures surrounding
reproductive parts of the flower), a defining de-
rived feature for flowering plants, usually com-
prisesboth outer(sepals) andinner(petals)organs.
Petals may have evolved independently at least
six times (23), arising as modified stamens or
bracts through changes in expression patterns of
specific homeotic genes (24). Petals themselves
vary greatly in size, color, shape, orientation, and
function, and have been lost repeatedly. Howev-
er, all petals appear to follow a similar genetic
program that involves the expression of a set of
of the floralmeristem ata specific place (external
to the stamens) and time (following stamen ini-
tiation) (25). Intriguing questions arise when we
genetic regulatory pathway in a similar spatial
context to generate second whorl petals, and in a
different spatial context to create novel structures
(e.g., petaloid stamens in Zingiberales, petaloid
bracts in dogwoods). Although their multiple or-
igins make petals homoplasious, the similarity of
is an example of deep homology (see below).
Hierarchically Determined Homoplasy
Hierarchically determined homoplasy is derived
from the conserved internal organization of orga-
nisms. Homoplasy at the genome level occurs as
an indirect effect, through upward causation.
Salamanders have the largest genomes among
growth among chromosomes from transposons
and retrotransposons. Genome size is positively
correlated with cell size. Because of constraints
on organismal size (terrestrial salamanders rarely
are corresponding constraints on cell number per
organ,affecting oganismalform.Cellsizeis neg-
cell the slower it divides. Small animals have
fewer and more slowly dividing cells. Slowing
cell division, reducing the numbers of limb blas-
tema cells (cells that differentiate into the various
tissuesthatcompose thelimb),can decrease digit
number in both frogs and salamanders (27) in
phylogenetically determined patterns (innermost
Thus, increased genome size may retard ontoge-
netic trajectories and result in a simplification of
in brain morphology has been documented in
large-genomed salamanders, frogs, and caecil-
Deep homology. Homoplastic traits that are
found to share a “deeper” developmental genetic
Fig. 1. Homoplasy in a phylogenetic context. (A) Example of a reversal in
plethodontid salamanders where larvae (L) were lost and replaced by direct
development (DD). Larvae re-evolved within the deeply nested genus
Desmognathus (47). (B) Convergence and parallelism in two clades of
plethodontid salamanders that show similar body elongation due to different
pathways. The ancestral mode (I) has 14 trunk vertebrae (highlighted in
yellow; every fifth vertebra shown in orange). One mode (which has evolved
in two independent lineages) adds vertebrae (II); the alternative mode
elongates individual vertebrae (III) (48). (C) The evolution of viviparity (live-
bearing reproduction) in caecilians (Amphibia: Gymnophiona) shows
parallelism in five different lineages (49, 50). Red branches indicate lineages
with viviparity; red squares indicate genera in which one or more species
have evolved the trait; ? indicates unknown reproductive mode, assumed to
be viviparous as are other members of the clade; blue branches and circles
indicate oviparous (egg-laying) clades. Phylogram after Wilkinson and
VOL 331 25 FEBRUARY 2011
on March 7, 2011
mon developmental genetic mechanisms have
been shown to underlie features that long were
considered classic examples of convergent evo-
lution (29–31). The paired appendages of tetra-
pods (e.g., salamanders, lizards, mammals) and
arthropods (e.g., flies, lobsters, spiders) evolved
independently, but integration of phylogenetics,
development, and genetics in a hierarchical con-
text shows that homologous gene clusters sharing
ancient common ancestry are responsible for the
initialoutgrowthsfrom thebodythat become pat-
etc.) (29, 30). Patterning in tetrapod appendages,
despite considerable variation among taxa, is
largely governed by relatively late expression of
long-conserved homologous Hox genes during
development. This also happens in fish fins; the
same fundamental process might control even
fins and limbs (30). Thus, while the morpholog-
ical structures expressed in adults (e.g., legs of
and fin rays of zebrafish) are not homologous
(because they were not present in a shared an-
cestor), homology may lie within the organiza-
tion of Hox genes and their regulatory networks,
although specific genes might have different ex-
pressions. This deep homology (29) breaks the
ideological constraints associated with homo-
plasy (5) and reveals a continuum rather than a
dichotomy (11, 12) of convergence and parallel-
ism at different levels within an organism and
among diverse taxa within a clade.
The image-forming eyes of invertebrate and
vertebrate taxa are convergent organs that share
some core developmental genetic mechanisms
that exemplify deep homology (32). All eyes, in-
vertebrate and vertebrate, develop through a cas-
cade (32) of similar transcription factors despite
vast phylogenetic distances. These networks in-
clude genes (e.g., Pax6) that have been deployed
in different ways at different times, and specific
pathways that have re-evolved in different line-
ages by mutation, gene duplication, and interca-
which contain homologous genes and members
of the same gene families, are not genetically
identical. Thus, the end phenotypes might be gen-
eral homologs at a deep hierarchical level but
convergent with respect to end phenotype and
phylogeny. Indeed, what has historically been
termed “convergence” and attributed to indepen-
dent evolution in unrelated taxa has a common
genetic system associated with trait development
insufficient if they are not integrated appropriate-
Metameric growth of plants (production of
repeating units) requires the identification of ho-
mology in positional, developmental, and func-
tional levels to detect homoplasy (33). The outer
whorl (sepals) in some monocots (e.g., bananas,
Zingiberales: Musaceae) appears identical to the
second whorl(petals) (Fig.3),yetinothermono-
cots (e.g., gingers, Zingiberales: Zingiberaceae
and Costaceae), sepals and petals are distinct
(24), indicating that positioning in the outer whorl
alone does not necessarily imply homology of
development or function. Furthermore, petaloid
staminodes (organs that resemble petals) replace
the outer stamens in four of the eight families of
the Zingiberales (Fig. 3) (34). Similar heterotopic
(displacement from normal position) modifications
of stamens and petals are found in other flower-
ing plants such as members of the Ranunculales
distant from the monocots. Deducing the lineage-
specific genetic program underlying sepal, petal,
stamen, and staminode identity among closely
related taxa such as the Zingiberales, and more
Determine whether each trait is
Map traits of interest on the
Select a phylogram of relationships
of taxa under study
An ancestor and its
Two (or more) lineages
that lack a recent
Convergence or parallelism
An ancestor, but not in its
but present in a
Fig. 2. Flow diagram of the process of detecting
C class (AG)
B class (GLO, DEF)
Fig. 3. A model for the developmental evolution of Zingiberales floral organs. The canonical ABC model
(52) forfloralorganidentityintaxaoutside Arabidopsisindicatesthatthe SEPALLATA(SEP)genesaloneare
expressed in the sepal whorl (30), B-class genes [GLOBOSA (GLO) and DEFICIENS (DEF) homologs] code for
petals, B- plus C-class genes [AGAMOUS (AG) homologs] code for stamens, and C-class alone (AG) codes for
carpels. Expression of members of the SEP gene family is likely present in all whorls (53). The phylogeny of
whorl are indicated: green, sepal; orange, petal; yellow, fertile stamen; blue, carpel. The petal-like sepals of
Musa and petal-like stamens of the four ginger families are indicative of the potential of different genetic
programs underlying the positional homoplasy (functional homology). For each family, the hypothesized
pattern of B-class (yellow bar) and C-class (blue bar) gene expression for each whorl is indicated. (a and b)
Musa (Musaceae) flower with petal-like organs in sepal and petal whorls and filamentous fertile stamens.
(cand d)Monocostus (Costaceae) flower withdistinctsepaland petalwhorlorgans,fusedouterand inner
petaloids forming the labelum, and (d) a single fertile petaloid stamen from the inner stamen whorl.
25 FEBRUARY 2011 VOL 331
on March 7, 2011
distant taxa comprising all angiosperms, requires Download full-text
both a phylogenetic framework to fully explain
the mechanisms of organ homology and evo-
lution (Fig. 3) and an understanding of the na-
ture of gene regulation to determine at what level
“the same thing” operates in disparate organisms
displaying homoplastic parts. Thus, the distinc-
tion between convergent and parallel homoplasy
(5, 10, 30, 31, 36) fades, to be replaced with new
When Phylogenies Do Not Resolve
they are not always sufficient to resolve major
questions involving parallelism and reversibility.
Mesoevolution (36) connotes the problem of
parallelism, a transitional condition between true
homology (recent common ancestry) and true
convergence (independent evolution of similarity),
as well as between microevolution and macro-
networks connected with them, and selection,
established, bias and constraint are established
environments. Co-option of this genetic system
transitions microevolutionary processes govern-
ing parallelism (with its reliance on common
mechanisms) to macroevolutionary convergence
(independent evolution of the same trait). Con-
vergent morphologies may arise from gene and
genome duplications, followed by co-option of
pathways or parts of pathways and shifts in
timing and position of expression. The CYC/TB1
across angiosperms; monosymmetric (zygomor-
phic) flowers have evolved homoplastically from
radially symmetric ancestors several times, using
the same toolkit of CYC-like genes (37). Copy
number and expression patterns of the homolog-
irreversible—has been challenged (38, 39); how-
ever, examples of homoplastic reversion of or-
gans to an ancestral state are not convincing
(40, 41). Although atavism (the sporadic appear-
ance of ancestral traits) long has been invoked as
evidence of evolution, such traits do not become
fixed. Serially repeated structures (teeth, verte-
brae, segments, numbers of phalanges, wings)
have re-evolved in different positions within ani-
mal bodies, but the developmental genetic and
morphogenetic underpinnings are likely to have
digits, the embryonic condensations of lost digits
might have remained and been redeployed, in
which case an understanding of the recurrence
way rather than the expressed trait (39, 43),
assuming objections to the phylogenetic hypoth-
esis can be overcome (41). Reversals, however,
also include regaining the ancestral traits of con-
ditions or states, not specifically organs (Fig. 1A),
but the mechanistic bases for such trait reversals
usually are not known.
In general, because relaxed selection leads to
erosion of unused developmental genetic path-
ways involved in trait production, lost structures
are unlikely to be re-evolved and evolutionary
reversals, especially at the level of organs and
complex features, are rare at best. The transition
either from relaxed selection on the blue pathway
(which leads to its degradation) or from stabilizing
selection on the red pathway, is sufficiently com-
plex that reappearance of the original condition
does not occur (44). Similar irreversible losses
have been observed for self-incompatibility (a
postpollination mechanism that prevents self-
fertilization) among angiosperms (45).
What Does the Future Hold for Understanding
Homoplasy, and Thereby Evolution?
Similar environmental pressures are expected to
elicit similar adaptive morphologies, suggesting
that phenotypic homoplasy is often a conse-
similarity may resultfrom homoplasy atdifferent
hierarchical levels [different mutations of the same
gene, different genes, or different gene functions
available variation upon which natural selection
can act, thus influencing the course of evolution-
ary change (5, 6). Convergent evolution may
provide insight into both ultimate and proximate
mechanisms generating diversity and can inform
regarding the extent to which the evolutionary
process is both repeatable and predictable (5).
Sets of developmental genetic mechanisms are
deployed repeatedly, under the control of genetic
regulatory and epigenetic factors, and the effects
can belarge (30).Morphologicallydisparate taxa
that are only remote relatives share toolkits of
body-building and body-patterning genes (31).
Bounded variation on such general morphoge-
study can illuminate the underlying processes.
Although some think that such processes have
ing phenotypic evolution. It is in this context that
study of homoplasy has its greatest promise. Ex-
ploration of homoplasy will illuminate the limits
on phenotypic evolution, the nature and reasons
for biases in its direction, and why “descent with
modification” may follow predictable pathways.
References and Notes
1. E. Jablonka, G. Raz, Q. Rev. Biol. 84, 131 (2009).
2. S. F. Gilbert, D. Epel, Ecological Developmental Biology:
Integrating Epigenetics, Medicine, and Evolution (Sinauer
Sunderland, MA, 2009).
3. M. H. Wake, Biol. Theory 3, 213 (2008).
4. D. B. Wake, A. Larson, Science 238, 42 (1987).
5. S. J. Gould, The Structure of Evolutionary Theory
(Harvard Univ. Press, Cambridge, MA, 2002).
6. D. B. Wake, Am. Nat. 138, 543 (1991).
7. D. B. Wake, in Key Words and Concepts in Evolutionary
Developmental Biology, B.K. Hall, W. M. Olson,
Eds. (Harvard Univ. Press, Cambridge, MA, 2003),
8. B. K. Hall, Biol. Rev. Camb. Philos. Soc. 78, 409 (2003).
9. B. K. Hall, J. Hum. Evol. 52, 473 (2007).
10. R. Diogo, Biol. Philos. 20, 735 (2005).
11. J. Arendt, D. N. Reznick, Trends Ecol. Evol. 23, 26 (2008).
12. J. Arendt, D. N. Reznick, Trends Ecol. Evol. 23, 483 (2008).
13. B. S. Leander, Trends Ecol. Evol. 23, 481, author reply
14. E. H. Davidson, Nature 468, 911 (2010).
15. M. J. Donoghue, R. H. Ree, Am. Zool. 40, 759 (2000).
16. M. D. Shapiro et al., Nature 428, 717 (2004).
17. A. Y. Albert et al., Evolution 62, 76 (2008).
18. Y. F. Chan et al., Science 327, 302 (2010).
19. D. Schluter, K. B. Marchinko, R. D. H. Barrett,
S. M. Rogers, Philos. Trans. R. Soc. B 365, 2479 (2010).
20. M.-C. King, A. C. Wilson, Science 188, 107 (1975).
21. M. W. Nachman, H. E. Hoekstra, S. L. D’Agostino,
Proc. Natl. Acad. Sci. U.S.A. 100, 5268 (2003).
22. M. Manceau, V. S. Domingues, D. R. Linnen,
E. B. Rosenblum, H. E. Hoekstra, Philos. Trans. R. Soc. B
365, 2439 (2010).
23. M. J. Zanis, P. S. Soltis, Y. L. Qiu, E. Zimmer, D. E. Soltis,
Ann. Mo. Bot. Gard. 90, 129 (2003).
24. L. P. Ronse De Craene, Ann. Bot. (London) 100, 621 (2007).
25. V. F. Irish, J. Exp. Bot. 60, 2517 (2009).
26. T. R. Gregory, Biol. J. Linn. Soc. Lond. 79, 329 (2003).
27. P. Alberch, E. Gale, Evolution 39, 8 (1985).
28. G. Roth, K. C. Nishikawa, D. B. Wake, Brain Behav. Evol.
50, 50 (1997).
29. N. Shubin, C. Tabin, S. Carroll, Nature 388, 639 (1997).
30. N. Shubin, C. Tabin, S. Carroll, Nature 457, 818 (2009).
31. S. B. Carroll, Cell 134, 25 (2008).
32. J. Piatigorsky, Evo. Edu. Outreach 1, 403 (2008).
33. R. M. Bateman, in Homoplasy: The Recurrence of
Similarity in Evolution. M. J. Sanderson, L. Hufford,
Eds. (Academic Press, San Diego, CA, 1996), pp. 91–130.
34. B. K. Kirchoff, L. P. Lagomarsino, W. H. Newman,
M. E. Bartlett, C. P. Specht, Am. J. Bot. 96, 580 (2009).
35. E. M. Kramer, Annu. Rev. Plant Biol. 60, 261 (2009).
36. E. Abouheif, Evol. Dev. 10, 3 (2008).
37. J. C. Preston, L. C. Hileman, Trends Plant Sci. 14, 147
38. R. Collin, M. P. Miglietta, Trends Ecol. Evol. 23, 602 (2008).
39. T. Kohlsdorf, G. P. Wagner, Evolution 60, 1896 (2006).
40. E. E. Goldberg, B. Igić, Evolution 62, 2727 (2008).
41. F. Galis, J. W. Arntzen, R. Lande, Evolution 64, 2466,
discussion 2477 (2010).
42. C. R. Marshall, E. C. Raff, R. A. Raff, Proc. Natl. Acad. Sci.
U.S.A. 91, 12283 (1994).
43. T. Kohlsdorf, V. J. Lynch, M. T. Rodrigues, M. C. Brandley,
G. P. Wagner, Evolution 64, 2466 (2010).
44. R. A. Zufall, M. D. Rausher, Nature 428, 847 (2004).
45. B. Igic, R. Lande, J. R. Kohn, Int. J. Plant Sci. 169,
46. H. E. Hoekstra, J. A. Coyne, Evolution 61, 995 (2007).
47. R. L. Mueller, J. R. Macey, M. Jaekel, D. B. Wake,
J. L. Boore, Proc. Natl. Acad. Sci. U.S.A. 101, 13820 (2004).
48. G. Parra-Olea, D. B. Wake, Proc. Natl. Acad. Sci. U.S.A.
98, 7888 (2001).
49. M. H. Wake, J. Exp. Zool. 266, 394 (1993).
50. D. J. Gower, V. Giri, M. S. Dharne, Y. S. Shouche, J. Evol.
Biol. 21, 1220 (2008).
51. M. Wilkinson, R. A. Nussbaum, in Reproductive Biology
and Phylogeny of Gymnophiona, J.-M. Exbrayat, B. Jamieson,
Eds. (Science Publishers, Enfield, NJ, 2006), pp. 39–78.
52. E. S. Coen, E. M. Meyerowitz, Nature 353, 31 (1991).
53. G. Ditta, A. Pinyopich, P. Robles, S. Pelaz, M. F. Yanofsky,
Curr. Biol. 14, 1935 (2004).
54. Supported by the U.S. National Science Foundation
(EF 034939 to D.B.W. and M.H.W., IOS 0845641
and DEB 0816661 to C.D.S.) and the Hellman Faculty
Fund (C.D.S.). We thank R. Mueller, P. O’Grady, and
E. Rosenblum for comments; Y. Zeng, M. Koo,
and T. Renner for assistance with Figs. 1 (A and B),
2, and 3 (A to D), respectively; and many colleagues
and students for discussions of principles of
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