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J. theor. Biol. (1999) 200, 183}191
Article No. jtbi.1999.0986, available online at http://www.idealibrary.com on
The Pattern of Variation in Centipede Segment Number as an Example
of Developmental Constraint in Evolution
WALLACE ARTHUR*- AND MALCOLM FARROW?
*Ecology Centre and ?School of Computing, Engineering & ¹echnology, ;niversity of Sunderland,
Sunderland SR1 3SD, ;.K.
(Received on 11 March 1999, Accepted in revised form on 9 June 1999)
The range of animal morphologies observed in nature is partly determined by natural
selection.However, there is no agreement yet regarding whether it is also partly determined by
developmental constraint. Testing for the e!ects of constraint has been di$cult due to the lack
of both an appropriate null model and a su$ciently simple system capable of yielding
unambiguous results regarding the model's plausibility. Here we examine the case of variation
in segment number in geophilomorph centipedes. Curiously, while this ranges between 29 and
191, there are no species in which an evennumber of segments is observed, in contrast to about
1000specieswith odd numbersof segments.It seems unlikelythatthis distributionofcharacter
values is determined by selection alone. Using an approach based on Bayesian inference, we
attempt to quantify the probability of obtaining the observed distribution of values given
a null model in which developmentalconstraintis absent. Since this probability is in the region
of10???, we concludethat constraintmustbe involved. We discussvarious implicationsof this
conclusion, and comment on the unexpected absence of neoteny and progenesis in centipede
evolution.
? 1999 Academic Press
1. Introduction
Why do animals take the forms they do, and not
others?Why,as Ra!(1996) hasasked,are all land
vertebrates &&tetrapods''*except for cases of sec-
ondary loss, for example snakes*while none
have six, eight, or many legs? Why is the situation
precisely reversed for land arthropods? In
general, why are certain areas of multicellular
morphospace densely populated with many rep-
resentative species, while other areas, apparently
characterizing viable designs, are unoccupied by
any extant or extinct animals?
There are two very di!erent answers to these
questions, representing two opposing schools of
-Author to whom correspondence should be addressed.
E-mail: wallace.arthur@sunderland.ac.uk.
thought on the relative importance of natural
selection and developmental constraint in deter-
mining the actual distribution of morphologies
that we observe*either now, or at any point in
evolutionary history, or cumulatively. One is the
&&pan-selectionist'' view that variation is poten-
tially available in all directions from any given
phyletic starting-point, and that selection deter-
mines which subset of variants prevails. The
alternative is the &&developmental constraint''
view that many of the gaps we observe between
di!erentmorphologies do not arise from the non-
adaptiveness of the absent forms but rather from
the di$culty of making them through an onto-
genetic process.
The pan-selectionistviewcan be tracedback to
Wallace (1870), who considered variation to be
0022}5193/99/018183#09 $30.00/0
? 1999 Academic Press
Page 2
omnipresent and available in all phenotypic di-
rections imaginable, apparently without even
a quantitative bias in any direction. He refers
(p. 290) to &&Universal variability*small in
amount but in every direction'', and Mayo (1983,
p. 104) boldly states that &&The major constraint
on natural selection as an agent of change is
naturalselection as a stabilizing force'', apparent-
ly relegating any kind of developmental constraint
to a minor role at best.
In contrast, Gould & Lewontin (1979) argue
that organisms are &&so constrained by phyletic
heritage, pathways of development and general
architecture that the constraints themselves be-
come more interesting and more important in
delimiting the pathways of change than the
selective force that may mediate change when
it occurs''. Goodwin (1994) makes a similar point
but in a speci"c context, namely the arrangement
of leaves (phyllotaxis) in angiosperms. He points
out that more than 80% of the quarter-millionor
so extant species have a spiral arrangement, and
suggests that &&the frequency of the di!erent phyl-
lotactic patterns in nature may simply re#ect the
relative probabilities of the morphogenetic tra-
jectories of the various forms and have little to do
with natural selection''.
It is necessary, at this point, to clarify the
nature of the debate. There are probably no
evolutionary biologists left who deny a role of
some sort for natural selection. So there is no
&&pan-constraintist''school of thought.Also, while
the pan-selectionists criticized by Gould &
Lewontin (1979) do indeed deny any role for
developmental constraint, most neo-Darwinians
are not pan-selectionists. The relationship be-
tween neo-Darwinism and developmental con-
straint is best expressed by Wagner (1988) as
follows. &&It is true that the concept of develop-
mental constraints is implicitly contained in
neo-Darwinian theory. Nevertheless, it is also
true that this concept has almost never had an
in#uence on the main stream of research that was
done by neo-Darwinists.'' There are now some
signs that the situation may be changing: see the
recent work of Guerra et al. (1997) and Pezzoli
et al. (1997).
In the past, much debate about the importance
of developmental constraint in evolution has
been di!use and inconclusive. This is partly due
to the lack of an agreed de"nition, partly to the
lack of a clear question regarding the role of
constraint for which we can seek an unambigu-
ous answer, and partly to the lack of a &&model
system'' which encapsulates a form of constraint
thatis su$ciently simple
sional that it can be dealt with productively. We
attempt to rectify all three of these problems
below.
and unidimen-
2. What is 99Constraint::?
There is a great danger that discussions of any
form of &&constraint'' in evolution degenerate into
a &&hopeless exercise in semantics'' (Antonovics
& van Tienderen, 1991). Various authors have
attempted de"nitions and categorizations in at-
tempt to avoid this danger, notably Maynard
Smith et al. (1985) and Resnik (1995). We will
make use of these de"nitions, and a criticism of
the former authors by Williams (1992), to make
clear exactly what we mean by developmental
constraint in this article.
We adopt a de"nite strategy here: we deliber-
ately choose a narrow, sensu stricto, de"nition.
We do this not because it corresponds best to the
predominant usage in the literature (which it
does not) but because (a) it increases the clarity of
the argument and (b) it is &&conservative'': if we
can show that narrowly de"ned developmental
constraint is important, then it follows that any
more broadly de"ned counterpart must be at
least equally important, if not more so.
Our starting point is the rather broad de"ni-
tion of Maynard Smith et al. (1985) which is as
follows: &&A developmental constraint is a bias on
the production of variant phenotypes or a limita-
tion on phenotypic variability caused by the
structure, character, composition or dynamics of
the developmental system.'' Narrowing down
from this, we will deal only with what both
Maynard Smith et al. (1985) and Resnik (1995)
call local (as opposed to universal) constraints.
We are focusing herein on centipedes, and on
questions speci"c to a segmented body plan. We
are thus not interested, in the present context, in
the fact that all animals "nd it hard to produce
wheels (Gould, 1983, chapter 12), interesting
though this is, because it is an example of a uni-
versal constraint. Also, we will include only the
184
W. ARTHUR AND M. FARROW
Page 3
TABLE 1
¹he six orders of centipedes (class Chilopoda), together with their segment
numbers, species numbers and mode of development. (¸BS"leg-bearing
segments; ana- and epimorphic are explained in the Discussion; Devonobio-
morpha is an extinct group2see Shear & Bonamo, 1988)
No. of extant
species
Development
Order No. of LBS
Scutigeromorpha
Lithobiomorpha
Craterostigmomorpha
Devonobiomorpha
Scolopendromorpha
Geophilomorpha
15
15
15
?
130
1100
Anamorphic
Anamorphic
Anamorphic
?
Epimorphic
Epimorphic
2
0
21 or 23
29}191*
550
1000
*Odd numbers only throughout this range.
complete non-production of certain variant
phenotypes rather than a restriction in (or &&bias''
against) their production, even though the
broader &&bias'' de"nition is useful in a more gen-
eral way as a counter-view to Wallace's (1870)
notion of e!ectively unbiased variation in all
directions.
We will try, as far as possible, to divorce devel-
opmental from selective constraints, though this
is in factvery di$cult.We are followingWilliams'
(1992) advice here that &&including "tness costs as
developmental constraints'' is not sensible, be-
cause &&If selection is listed among the constraints,
then what is it that is being constrained?'' How-
ever, the problem is that while the complete ab-
sence of (for example) centipedes with 20 pairs of
legs (see Table 1) appears to be a case of non-
production rather than selective elimination,
sinceno such centipedeshave everbeen observed,
the possibility of selective elimination of embryos
with 20 leg-pairs before egg-hatching cannot be
excluded on the basis of the limited number of
observations on centipede embryology conduc-
ted to date (Heymons, 1901; Johannsen & Butt,
1941; Whitington et al., 1991). If selection is
involved here, it is probably &&internal selec-
tion''*see Discussion.
We have now arrived at a narrow de"nition
which encapsulates only a subset of the phe-
nomena that would be included in Maynard
Smith et al.1s (1985) broader de"nition. Ours
reads as follows: Developmental constraint is the
non-production of variant phenotypes caused by
the nature of the developmental system. Let us now
put this de"nition to work.
3. Questions and Models
The question we pose in this article is as
follows: is the distribution of &&segment number''
in centipedes such that it can only be explained
by invoking developmental constraint? The way
we proceed towards an answer is (a) to construct
a model or hypothesis of what the distribution
should look like under the assumption that all
variants are possible and the range actually
found is determined entirely by selection; and
(b) to examine the plausibility of this model. This
is the approach advocated by Antonovics & van
Tienderen (1991) who put it as follows. &&A con-
straint seems to have little relevance without
a speci"c reference to a null model.2We there-
fore strongly urge that authors should state the
null model explicitly; usually this null model de-
scribes what phenotypicvariation to expect given
a set of assumptions''.
The &&null'' description derives from the fact
that developmental constraint is presumed ab-
sent. What we need is some means of measuring
the strength of evidence (see below) with respect
to the null model. If the null model is su$ciently
implausible, an alternative model incorporating
developmental constraint is favoured, though of
course there are many possible such models and
there is no way, as yet, of distinguishing between
them. (For examples of the null model approach
DEVELOPMENTAL CONSTRAINT IN EVOLUTION
185
Page 4
in community ecology, where it has been more
widely utilized, see Strong et al., 1984.)
4. A Null Model for Centipede Segment Number
4.1. BAYESIAN INFERENCE
The approach we take here uses Bayesian in-
ference (see, for example, O'Hagan, 1994) where
the starting point is the identi"cation of a &&prior''
probabilitydistribution for the variable &&segment
number'', given a limited knowledge of centipede
morphology on the part of an observer. The
argument in favour of a Bayesian approach
seems particularly strong in this case where the
evidence consists of the observed results of the
evolutionary process rather than any repeat-
able experiment. However, since many readers
may be more familiar with the frequentist ap-
proach to statistical inference, we give a brief
outline of the relevant ideas as follows.
Probability is used to represent &°ree of
belief''. Hence, if there are J competing hypothe-
ses H?,2,H?, each is assigned a probability to
measure the certainty attached to it. The usual
rules of probability apply so, for example, if
H?,2,H?, are mutually exclusive and exhaust-
ive and the probability of H?is P?, then ?P?"1.
Before observing the data D, we have prior prob-
abilities P??,2,P??. When the data are observ-
ed these are converted to posterior probabilities
P??,2,P??using Bayes' theorem to calculate
the conditional probabilities of the hypotheses
given the data. That is
P??"
P??Pr(D?H?)
?????P??Pr(D?H?),
(1)
where Pr(D?H?) is the conditional probability of
the data given the hypothesis H?. In the case of
simplehypotheses whichdo not involveunknown
parameters, Pr(D?H?) is called the likelihood and
depends only on the hypothesis and the data. In
the case of composite hypotheses, its calculation
may involve prior probabilities speci"ed for un-
known parameters.
The prior odds in favour of H?over H?are
P??/P??and similarly the posterior odds are
P??
P??
"P??
P??
Pr(D?H?)
Pr(D?H?).
(2)
The ratio Pr(D?H?)/Pr(D?H?) is called a Bayes
factor.
4.2. A SIMPLE ANALYSIS
Now, we will attempt to apply these ideas for
measuring the evidence from the numbers of leg-
bearing segments of known species of centipedes.
For simplicity, we will discuss only the Geo-
philomorpha (see Table 1). One hypothesis is H?:
there is no developmental constraint. If there is
developmental constraint, then, before we ob-
serve the data, it could take any one of many
di!erent forms and these are represented by
H?,H?,2. We need to be able to identify what it
might have been reasonably imagined that these
would be, before the data had been seen. We
might imagine that we had not counted the leg-
bearing segments of centipedes before but had, at
least, some idea of what the animals would look
like and roughly how many segments they would
have. We can then ask whether the evidence,
when we see it, should convince us that only
odd numbers of leg-bearing segments are pos-
sible, which is the curious pattern actually ob-
served*see Fig. 1. (Odd numbers of leg-bearing
segments may correspond to even total numbers
of segments (Minelli & Bortoletto, 1988), though
there is some di$culty in interpreting the seg-
mental basis of the head and genital regions.)
More rigorously, we are really considering how
convinced we should be that any new species
discovered would also have an odd number of
leg-bearing segments.
For simplicity, let us, at least initially, only
consider constraints which take the form of peri-
odicities in the possible numbers of leg-bearing
segments. That is, only every jth number is pos-
sible. E.g., if j"2, then either only odd numbers
are possible or only even numbers are possible. If
j"3 then only 1,4,7,2or only 2,5,8,2or only
3,6,9,2 are possible. We might reasonably
restrict our attention to fairly small values of j as
con"rmationof large values would require obser-
vation of species with implausibly large numbers
of segments.
If we label the hypotheses as H?: there is peri-
odicity with period j, then it is immediately ap-
parent from the data that Pr(D?H?)"0 and
hence P??"0 for j'2. Determining Pr(D?H?)
186
W. ARTHUR AND M. FARROW
Page 5
FIG. 1. The distribution of the number of leg-bearing segments among all known species of geophilomorphs. Note that
variable species contribute to several adjacent bars, while "xed species (family Mecistocephalidae) contribute to a single bar.
From Minelli & Bortoletto (1988), with permission.
and Pr(D?H?) is not so straightforward.However,
we present a very simple argument and argue
that our conclusions are likely to be conser-
vative compared to those from more detailed
analyses.
The data consist of the numbers X of leg-
bearing segments in the known species. We make
a number of simplifying assumptions here, all of
which should cause our approach to be &&conser-
vative''. First, we ignore the intraspeci"c com-
ponent of the variation, even though this is itself
strongly suggestive of developmental constraint.
In the following, X values can be taken to be
the precise number of trunk segments for
invariant species and the modal number for
variable ones. Second, it does not seem reason-
able to consider the species to be independent. It
could be, for example, that many of the species
with X"69 have evolved from a common ances-
tor with X"69. This means that the weight of
evidence from the number of species with X"69
is less than it would be if they were independent.
The limit of how small it can be is given when we
treat all species with a given X value as a single
observation. We will do this to obtain a conser-
vative value for the Bayes factor.
Third, it is also likely that neighbouring seg-
ment numbers are not independent. We might
expect the observed numbers to be clustered to-
gether,as in factthey are.Thatis, if we discovered
a species with X"69 this might lead us to expect
to "nd other species with similar X values. This
point is reinforced by a recent cladistic analysis
of the Geophilomorpha by Foddai (1998). We
might also expect some of the numbers to be
spread by the e!ects of competition, through
the co-evolutionary process of character dis-
placement (Brown & Wilson, 1956), though in
fact, the evidence for such a process is rather
weak (Arthur, 1982). Similarly, there may be
some values of X, for example very large ones,
which are less likely a priori than others. Such
considerations would have a similar e!ect to
reducing the number of available segment
numbers. If we ignore this, then again we will
obtain a conservative value for the Bayes factor.
Taking this extremely simplistic, conservative,
approach we argue as follows. Suppose that there
DEVELOPMENTAL CONSTRAINT IN EVOLUTION
187
Page 6
are 2n possible values of X from which to choose.
Under H?the number of the possible selections
of m values is
?
2n
m?"
(2n)!
m!(2n!m)!.
(3)
Under H?selections containing only odd num-
bersand selections containingonly evennumbers
are possible, but not selections containing both
odd and even numbers. Thus, the number of
possibilities is
2?
n
m?"
2n!
m!(n!m)!.
(4)
Assuming, conservatively, that all selections
are equally likely, the probability of any given
selection is the reciprocal of the number of pos-
sible selections. To obtain Pr(D?H?) this pro
bability must be multiplied by the probability of
the value of m, the number of di!erent X values,
given H?. If only odd numbers are possible then
we might reasonably expect only half as many
di!erent values of X, therefore giving each value
of m approximately twice the probability. How-
ever, it could be argued that this is not so and,
even if both odd and even numbers were possible,
we might not expect to observe all of the possible
X values in a range. Therefore, we again take the
conservative choice and assume that the prob-
ability for a given value of m does not di!er
between H?and H?. In this case it cancels out in
the Bayes factor.
The Bayes factor in favour of H?over H?is thus
B"?
2n
m?
2?
n
m?
"?
1
2??
2n
m
2n!1
m!122n!m#1
1 ?
??
n
m
n!1
m!12n!m#1
1 ?
??
"?
1
2??
2n
n
2n!1
n!122n!m#1
n!m#1?. (5)
So, as nPR, BP2???. For n(R, B'2???.
For the Geophilomorpha m"76 so, bearing in
mind our discussion above, we conclude that
B should be at least 2??+3.8?10??.
In order to see how this Bayes factor turns into
a posterior probability, we need to introduce
some prior probabilities. At "rst glance, this
appears to be di$cult as there are many possible
patterns which we could observe. However, we
can argue as follows. Suppose that the probabil-
ity that there is constraint of some sort is ?. Let
? be the conditional probability that, given that
there is constraint, this takes the form of a simple
periodicity as described above. Let ? be the con-
ditional probability that, given that there is con-
straint and that it is a simple periodicity, the
period is 2. Then the prior probability of H?is
1!? and the prior probability of H?is ???. The
prior odds in favour of H?over H?are therefore
???/(1!?). Readers may supplytheir own values
for ?, ? and ? but, if we are willing to state, for
example, that ?*0.1, ?*0.1 and ?*0.1, this
leads to the conclusion that the prior odds are at
least 10??/9. This in turn leads to posterior odds
of greater than 4?10?? and a posterior probabil-
ity of no constraint of around 10???.
This analysis has been very simple and un-
sophisticated. However, we put it forward to
show the kind of approach that might be taken
and to stimulate discussion. We acknowledge
that many re"nements could be made to the
argument. For example, we have not discussed
the possibility of a system where even numbers
are possible but, for some reason, very unlikely to
be observed. We hope to have the opportunity to
presenta moredetailedanalysisin a futurepaper.
5. Discussion
Our conclusion is clear: some mixture of selec-
tion and constraint determines the observed pat-
tern of variation in centipede segment number.
At this stage, attempts to evaluate the relative
importance of the two processes are probably
futile; it would be more productive to focus on
the nature of these and other developmental con-
straints*thatis, to ask whatcellularmechanisms
underly the di$culty of producing certain forms.
This question can be asked in both molecular
and &&systems'' terms, and much of the ongoing
188
W. ARTHUR AND M. FARROW
Page 7
work on the molecular basis of development will
eventually help to elucidate the nature of con-
straints. One model of how mechanisms of seg-
ment production might lead to constraint in
this particular context is given by Minelli &
Bortoletto (1988). These authors propose a
multiplicative system of segment production in
which pairs of segments are derived from single
&&eosegments'' in early embryogenesis.
Centipedes are sometimes divided into two
sub-classes based on the way segments are pro-
duced in ontogeny (Lewis, 1981): Anamorpha
(a paraphyletic group that includes the litho-
biomorphs*see below) in which segment num-
ber increases from egg-hatching to adulthood by
posterior addition of segments through progress-
ive juvenile stages; and Epimorpha (a derived
monophyletic group including the geophilo-
morphs) in which the adult segment number is
established during embryogenesis, and does
not increase after the egg hatches. Ironically,
the longest centipedes are epimorphic while
the shortest are anamorphic, which seems
counter-intuitive and is itself an intriguing
problem.
Two interesting points emerge from this
anamorphic/epimorphic distinction. First, con-
sider the juvenile anamorphic centipede ¸itho-
bius variegatus shown in Fig. 2, which has 12
pairs of legs. Its existence demonstrates (a) that
a centipede with an even number of trunk seg-
ments and leg pairs can be a perfectly viable,
well-adapted entity capable of moving, feeding,
and generally surviving, in a natural envi-
ronment; and (b) that what is &&prohibited'' in
developmental terms is not the production of
an even-segment-number centipede, but rather
an ontogenythat culminatesin an evennumber of
segments. In fact, it is quite remarkable that
throughout the evolution of the Lithobiomorpha
(currently more than 1000 species*see Table1)
there appears not to have been a single hetero-
chronic change (neither neoteny nor progenesis)
that has resulted in a species in which an even-
segment-number juvenile becomes reproductive-
ly mature and the ancestral 15-segment adult
stage is lost. Heterochronic processes are com-
mon in other taxonomic groups (reviewed by
Gould, 1977 and McKinney & McNamara,
1991), suggesting that the developmental}genetic
FIG. 2. A juvenile ¸ithobius variegatus with 12 leg-bear-
ing segments. Note also the three pairs of limb buds (poste-
rior) which will give rise to leg-pairs at the "nal sub-adult
moult. From Eason (1964), with permission.
DEVELOPMENTAL CONSTRAINT IN EVOLUTION
189
Page 8
basis of such changes is widespread and that this
kind of evolution is relatively &&easy'' for natural
selection to produce, along with its spatial equiv-
alent, heterotopy. So why it has not occurred
here is a mystery.
Second, consider the embryonic development
of an epimorphic centipede, Scolopendra morsi-
tans. Although this is the most-studied species of
centipede from an embryological perspective (see
Heymons, 1901; Johannsen & Butt, 1941; Gilbert
& Raunio, 1997), it is uncertain whether embryos
with even segment numbers, destined to be even-
numbered hatchlings and adults, are simply
never produced, or alternatively are produced
occasionally but always die before hatching. If
the latter, then it is a moot point whether this is
more appropriately considered as developmental
constraint or selection. Perhaps, in such a case,
the choice of stance should depend on the cause
of mortality. If embryos with even numbers of
segments are more susceptible to mortality
caused by an external agent such as an egg-pred-
ator, then this is clearly selection. However, such
a scenario seems most unlikely. Alternatively, if
these embryos die due to internal &&scrambling'' of
the developmental process, then this could be
considered to be developmental constraint, or,
perhaps better, &&internal selection''. Indeed, these
two can be hard to separate*see Whyte (1965)
and Arthur (1997). (Note that &&developmental
selection'' or &&clonal selection'' is an entirely dif-
ferent process to internal selection*see Buss,
1987, Frank, 1997).
The pattern of variation in segment number
also has an implication for the size of morpho-
logical step involved in directional evolutionary
change. The ancestral &&stem species'' centipede
probably had 15 trunk segments (see Shear
& Bonamo, 1988; Borucki, 1996; Giribet et al.,
1999, but, for a counter-view, Shultz & Regier,
1997). If so, then to get to the next largest number
observed*21 segments*the smallest step pos-
sible is two segments at a time. Since even well-
preserved fossil centipedes are rarely complete
(see Mundel, 1979 for an exception), it is hard to
get direct evidence concerning whether steps
were 2/2/2, 2/4, 4/2 or 6. Also, if Minelli &
Bortoletto's (1988) &&octonary'' model is correct, it
may be that evolution took a path 15P 23P 21,
in which case the "rst step was # 8 segments.
Even the smallest possible addition of two seg-
ments to 15 is a sizeable proportional change
(13%), while adding eight segments is a change
of 53%. This does not, of course, mean that
we should all become neo-Goldschmidtians be-
lieving in the preponderance of saltations (see
Goldschmidt, 1940). However, it does indicate
that a view of morphological evolution wherein
individual mutations always cause only imper-
ceptible shifts of the value of a character such as
&&body length'' is too extreme in the other direc-
tion. Evolution must also involve what Dawkins
(1986) calls &&stretched DC8'' mutations (as op-
posed to those that turn DC8's into 747's), of
which a mutationally elongated centipede is
a classic example.
Finally, it is important to distinguish between
demonstration of the existence of developmental
constraint and demonstration of its importance.
Wagner (1988) makes a link between highly con-
strained basic body plans and the evolution of
high degrees of phenotypic complexity and
functional integration. (He uses the insects as an
example of this link.) It seems likely that some
forms of constraint do indeed &&accidentally''
facilitatethe evolutionaryproductionof morpho-
logical complexity, while others have no such
e!ect. The apparent prohibition of centipede on-
togenies that culminate in an even number of
trunk segments may well be an example of the
latter.
We thank Alec Panchen for his helpful comments
on the manuscript.
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