1. No category

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Mechanical sensitivity reveals evolutionary
dynamics of mechanical systems
P. S. L. Anderson and S. N. Patek
Department of Biology, Duke University, Box 90338, Durham, NC 27708, USA
Research
Cite this article: Anderson PSL, Patek SN.
2015 Mechanical sensitivity reveals evolutionary dynamics of mechanical systems.
Proc. R. Soc. B 282: 20143088.
http://dx.doi.org/10.1098/rspb.2014.3088
Received: 19 December 2014
Accepted: 27 January 2015
Subject Areas:
biomechanics, evolution
Keywords:
evolution, biomechanics, mechanical
sensitivity, mechanical equivalence
Author for correspondence:
P. S. L. Anderson
e-mail: philip.anderson@duke.edu
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspb.2014.3088 or
via http://rspb.royalsocietypublishing.org.
A classic question in evolutionary biology is how form–function relationships
promote or limit diversification. Mechanical metrics, such as kinematic transmission (KT) in linkage systems, are useful tools for examining the evolution
of form and function in a comparative context. The convergence of disparate
systems on equivalent metric values (mechanical equivalence) has been highlighted as a source of potential morphological diversity under the assumption
that morphology can evolve with minimal impact on function. However, this
assumption does not account for mechanical sensitivity—the sensitivity of the
metric to morphological changes in individual components of a structure. We
examined the diversification of a four-bar linkage system in mantis shrimp
(Stomatopoda), and found evidence for both mechanical equivalence and
differential mechanical sensitivity. KT exhibited variable correlations with individual linkage components, highlighting the components that influence KT
evolution, and the components that are free to evolve independently from
KT and thereby contribute to the observed pattern of mechanical equivalence.
Determining the mechanical sensitivity in a system leads to a deeper understanding of both functional convergence and morphological diversification.
This study illustrates the importance of multi-level analyses in delineating
the factors that limit and promote diversification in form–function systems.
1. Introduction
For nearly 200 years, biologists have recognized the importance of the form–
function relationship for understanding the diversity of life [1–6]. Classically,
the mechanics of biological systems are viewed as limits on biological diversity:
the immutable laws of physics restrict how organisms evolve [7]. However, scientists working on the biomechanics and function of form have also understood that
not all morphological variation is equivalent in terms of its functional consequences [3–5,8,9]. Recent advances in both phylogenetic comparative methods
and biomechanical analyses have allowed for a more rigorous exploration of
how mechanical properties influence evolutionary patterns and processes [6].
A standard comparative biomechanics approach is to define a mechanical
metric that allows different systems to be compared; for example, Reynolds
number offers a single, dimensionless number that represents an organism’s
experience in fluids, and mechanical advantage expresses force transfer through
a lever system. The mechanical similarity expressed through these metrics
allows comparisons among diverse systems. The metrics themselves can also
be used as trait data in comparative evolutionary analyses [10,11].
An inherent property of biomechanical metrics is that multiple morphological configurations yield synonymous mechanical outputs, a phenomenon
which we term mechanical equivalence. Mechanical equivalence, alternatively
termed many-to-one mapping or functional redundancy [12], has been
suggested to enhance morphological diversification by allowing morphological
evolution to occur without functional consequence (functional consequence
here is defined as the magnitude of change in the biomechanical metric being
studied) [13 –19]. Thus, researchers have concluded that mechanical equivalence can lead to the decoupling of morphological and functional variation
(figure 1) [13,20,21] and enhanced evolutionary diversification [22]. A key
assumption of the argument that mechanical equivalence enhances diversification is that morphological components can vary without significant effects on
& 2015 The Author(s) Published by the Royal Society. All rights reserved.
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2
mechanical equivalency
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mechanical metric (KT)
(a)
KT morphology
morphology
mechanical metric (KT)
mechanical sensitivity
KT
L1
L2
L3
morphology
Figure 1. Examining the mechanical sensitivity of a mechanical metric to variation in its morphological components is key to understanding the evolutionary
dynamics of mechanically equivalent systems, such as a four-bar linkage. (a) Mechanical equivalency occurs when multiple morphological configurations yield
convergent values of a mechanical metric, such as kinematic transmission (KT). (b) Mechanical sensitivity is indicated by variable effects of components on
the mechanical metric, such that some are tightly correlated to the metric (L1, red) and others are not correlated (L2, orange; L3, blue). Thus, in this example,
the evolutionary diversification of KT is influenced by the component to which it is most sensitive (L1), whereas components to which KT is less sensitive (L2 and L3)
are free to diversify without affecting KT. Combining analyses of mechanical equivalence with analyses of mechanical sensitivity can provide a more accurate portrait
of the evolutionary dynamics of form – function relationships than is offered by analyses at only the mechanical equivalence level. (Online version in colour.)
the focal mechanical metric. Note that ‘equivalence’, as used
here, does not refer to overall mechanical equivalence
between linkages, but only to equivalence in the specific
measurement of KT.
However, in spite of this assumption, morphological shifts in
different components of a biomechanical system may have
differential mechanical consequences [4,5], which we term
mechanical sensitivity. For example, small morphological
changes in one part of a mechanism can cause a large shift in
the magnitude of the biomechanical metric, whereas another
component might be able to vary substantially with minimal
impact on the metric [5]. Evidence of differential mechanical sensitivity has been hinted at in published studies of mechanical
equivalence [23,24] (see [4] for further discussion).
Mechanical sensitivity potentially offers improved insights
into the evolutionary dynamics of mechanical systems over the
current prevailing focus on mechanical equivalence. Indeed,
mechanical equivalence suggests a general lack of correlation
between the morphology and function of a system, whereas
mechanical sensitivity can pinpoint the precise evolutionary
relationships between morphological components and mechanical metrics (figure 1). In order to test a framework that
connects and delineates mechanical equivalence and mechanical sensitivity, we examined these two phenomena via the
evolution of linkage systems.
Linkage systems offer a central and historic focal point
for analyses of mechanical and morphological change
[10,11, 25 –27]. The non-dimensional metric for motion
transferred across a linkage system is called kinematic transmission (KT). KT is an exemplar metric for exploring the
relationship between morphological and mechanical change
as its value is determined by the morphology of a series of
component links. In this study, we analyse KT while considering both overall linkage morphology and variation among
these individual linkage components. Ultimately, this
approach allows analysis of both mechanical equivalence
and differential mechanical sensitivity.
We examined the four-bar linkage system found in the
raptorial appendages of mantis shrimp (Stomatopoda). The
mantis shrimp’s raptorial appendage is a spring-driven,
power-amplified mouthpart used to strike prey items at
ultra-fast speeds (peak speed up to 30.6 m s21, acceleration
up to 154 km s22) [28,29]. Raptorial appendages are morphologically diverse, and are used for a variety of functions
ranging from spearing to smashing (figure 2b,c) [30,31].
Energy released by the spring mechanism is delivered to
the strike via a series of linked mechanisms, including an
exoskeletal, four-bar linkage [28– 32].
In order to assess the dynamics of mechanical equivalence
and mechanical sensitivity, we tested two hypotheses using the
morphological variation offered by mantis shrimp linkage
components (figure 2). First, we tested whether the mantis
shrimp linkage system exhibits a pattern of mechanical equivalence across clades, specifically asking whether clades with
similar mechanical outputs show divergent morphology
of the linkage system and whether the morphology of the
Proc. R. Soc. B 282: 20143088
(b)
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dactyl
(b)
(a)
dactyl
Proc. R. Soc. B 282: 20143088
(d)
(e)
meral-V
(f)
4
1
3
2
Figure 2. The raptorial appendage of the mantis shrimp (Stomatopoda) is a spring-driven, power-amplified appendage used to strike prey items at ultra-fast speeds
(peak speeds up to 30.6 m s21, accelerations up to 154 km s22) [24,25]. (a) A smasher mantis shrimp, Neogonodactylus bredini, is holding the raptorial appendage in a
folded position (black outline). (b) Smashers use the hammer-shaped base of the dactyl to break open hard-shelled prey. (c) Spearers snag softer, evasive prey with an
elongate, spiny dactyl. (d) Force and motion generated by the spring mechanism are transferred to the swinging appendage via a four-bar linkage. The meral-V rotates
distally (to left of page) when the spring is released. This action pushes on the swinging arm, which then rotates distally to perform the strike. (e) Mechanical equivalence was measured using discrete landmarks (red dots) that represent the morphology of the whole four-bar linkage system. (f ) Mechanical sensitivity of individual
components was measured via individual link lengths (coloured and numbered lines). Scale bars, 15 mm. Distal, left; dorsal, top of the page. (Online version in colour.)
system as a whole (landmark coordinates) correlates with KT
across phylogeny. Second, we tested for variable mechanical
sensitivity among the individual links of the system by comparing the evolution of the mechanical metric (KT) with
evolutionary changes in the individual links (link lengths).
The outcome of these analyses offers new, quantitative insights
into the dynamics between apparent equivalence of mechanical systems and the sensitivity of their mechanics to
morphological change in individual components.
2. Methods
We performed morphological analyses on 195 mantis shrimp
specimens representing 36 species from six superfamilies (see
the electronic supplementary material). This dataset included
smashers (18 taxa), spearers (16 taxa) and two undifferentiated
species (both from the genus Hemisquilla). For comparison
between appendage types, the undifferentiated taxa were
grouped with the spearers owing to their phylogenetic placement outside the monophyletic smasher clade [33,34], as well
as their similarity in functional parameters [31] and muscle
physiology [35] to spearers. The raptorial appendages were
photographed in lateral view using a Nikon D300 digital
camera (12 megapixel; AF micro-Nikkor 60 mm F/2.8D or
105 mm F/2.8D macro lenses).
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(c)
3
The spring system is built of exoskeleton, thus all linkage
measurements were externally visible. The appendage was
measured with the segments folded against the merus in order to
ensure that the landmarks on the merus and carpus segments
were in the same relative orientation to each other on every specimen.
(a) Overall linkage morphology
In order to test for mechanical equivalence, the linkage system
was quantified as a series of landmarks, which are defined by
the system’s four points of rotation, and the proximal and
distal excursion points of the meral-V (figure 2e). Meral-V excursion is externally visible (figure 2d ). When at rest, the meral-V
lies in a distally extended position, representing maximal distal
rotation during striking. When loaded, the meral-V is compressed
against a defined region of the merus exoskeleton. The difference
between the open and closed positions of the meral-V denotes the
total input rotation of the linkage system, and thus provides key
information about the changing mechanics and shape of linkages
during a strike. Landmarks were digitized using TPSDIG v. 2.0 [36].
Landmark morphometric data were aligned using Procrustes
superimposition procedures that remove scale, rotation and
translation from the data [37]. The new, aligned coordinates
were then used to construct a principal component (PC) analysis
to summarize the variation in shape along a series of orthogonal
axes. This resulted in a landmark-based linkage system morphospace. We also calculated average landmark configurations for
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(b) Individual component morphology
(c) Kinematic transmission
In order to compare the morphological data with KT, we used
the minimum KT. KT is the ratio of the rotation of the output
link (link 3: carpus) to the input link (link 2: meral-V). However,
KT is a dynamic metric that changes over the course of rotation
as the orientation of the linkage shifts [32]. Minimum KT refers to
the lowest value of KT for a given linkage over the course of its
full rotation as defined by the meral-V excursion. These data
were collected and published previously from the same
specimens analysed in this study [31].
(d) Analyses
Morphological differences in the stomatopod linkage system
among subclades (superfamilies) were determined using nonparametric multivariate analysis of variance (npMANOVA) as
well as discriminant function analysis (DFA). While npMANOVA was used to determine whether the a priori groups can
be defined by discrete differences in continuous variables (in
this case, the axes of our morphospace), the DFA determines
whether the morphological data actually support the original
groupings. npMANOVA was performed in PAST v. 1.94b [40],
and the DFA was performed in MORPHOJ v. 1.04a [39].
In order to assess the relationship between KT and either
whole linkage morphology (landmark data; figure 2e) or individual component morphology (links; figure 2f ), we used linear
regression analysis on the data from individuals (primarily for
visualization purposes) and phylogenetic least-squares regression
(PGLS) on species averages. For the phylogenetic analyses, we
used a previously published pruned and time-calibrated molecular phylogenetic tree based on nucleotide sequence data [34]. Time
calibrations were based on hard-bound calibration points from
fossil data [41]. PGLS analyses were performed using the R package CAPER v. 0.2 [42] with delta and kappa fixed at 1 and lambda
estimated using maximum-likelihood methods. Estimating
lambda allowed the model to deviate from a strict Brownian
motion model of evolution. The PGLS models were used to
measure strength of correlation between morphology and KT
values. The Akaike information criterion was used to assess
which independent variables (PC axes and link lengths) had the
strongest influence on the evolution of KT.
Finally, we tested whether individual morphological components of the linkage (individual links) evolve in a similar
pattern to KT by applying maximum-likelihood techniques
to test alternative adaptive models for morphological traits.
3. Results
(a) Mechanical equivalence
Mantis shrimp four-bar linkages exhibit mechanical equivalence across species. A PC analysis based on linkage system
landmarks illustrates that a wide range of linkage systems
can result in similar values of KT (figures 3 and 4). The morphology of the linkage system (represented by PC scores) is
weakly correlated with KT (all r 2-values less than 0.5). When
species averages are used, the evolutionary correlations
between KT and PC scores are equally weak (table 1). Clades
previously demonstrated to have similar function (KT; Squilloidea, Lysiosquilloidea, Pseudosquilloidea and Parasquilloidea
[31]) have a wide range of divergent morphological forms,
some of which overlap with the mechanically distinct smasher
clade (Gonodactyloidea; figure 5). Permutation MANOVA tests
(F-stat ¼ 39.29, p , 0.0001, based on 10 000 permutations) and
discriminant function analyses (between each pair of subclades) support these observations (electronic supplementary
material, table S1).
(b) Mechanical sensitivity
A combination of phylogenetic regression analyses and
evolutionary modelling indicates differential mechanical sensitivity of KT to the three mobile links of the stomatopod
linkage system. Link 3 (defined as the dorsal –ventral length
of the carpus) has a much stronger correlation with KT
than either links 2 or 4 (figure 4; link 3 r2 ¼ 0.73). PGLS analyses show that link 3 is a better single predictor for the
evolution of KT across stomatopods than the other links
and any of the PC axes from the landmark analyses (table 1).
The evolution of link 3 across stomatopods matches the
pattern previously reported for KT [31]. For both link 3 and
KT, an OU multi-peak model (in which smashers and
spearers evolve towards different optima) is better supported
than either Brownian motion or an OU single-peak model
(table 2). The multi-peak pattern is also the best supported
model of evolution for link 2, although the degree of support
is lower. The evolution of link 4 is best described by a
Brownian motion (random walk) model (table 2).
4. Discussion
The mantis shrimp four-bar linkage system demonstrates
both mechanical equivalence of the whole system and
4
Proc. R. Soc. B 282: 20143088
In order to analyse the mechanical sensitivity of KT to variation of
the individual links of the four-bar system, we measured the lengths
of the three mobile links relative to the immobile link (figure 2f ). The
specifics of the linkage system in mantis shrimp have been covered
in previous work [31,32], and we briefly review them here: link 1 is
the fixed (immobile) link represented by the proximal merus; link 2
is the meral-V, which acts as the input link; link 3 is the carpus,
which acts as the output link; and link 4 is the contracted extensor
muscle, which runs from the carpus to the merus. KT is scale independent; the relative link lengths determine the value of KT. In
order to account for size differences among taxa, we measured
the lengths of links 2, 3 and 4 and divided each by the length of
the fixed link (link 1) following procedures from work on other
four-bar systems [13,20].
We modelled trait evolution using functions in the R package
OUCH v. 2.8-1 [43], which estimates the Brownian motion
rate parameter (s 2), strength of selection (a), optimal trait
values (u) and support (Akaike information criterion corrected
for finite sample sizes, AICc) for each model. We compared
three distinct models of evolution: a Brownian motion model
(a ¼ 0 and u ¼ 0), a single-peak Ornstein – Uhlenbeck (OU)
model (Brownian motion pulled towards a single adaptive
peak (u) for each parameter), and an OU model that was set to
pull towards two adaptive peaks, one for smashers and one for
spearers þ Hemisquilla [44]. The multi-peak OU model differentiates the evolution between smashers and spearers; therefore,
ancestral state reconstructions are necessary to assign appendage
types to the internal nodes of the tree. We used previously published ancestral state reconstructions [31] based on a rerooting
method [45] as implemented in PHYTOOLS v. 2.9 [38].
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each species in order to create a species-based morphospace, generate coordinates for phylogenetic comparative analyses and
construct a phylomorphospace using PHYTOOLS v. 2.9 [38]. All
morphometric analyses were performed in the software package
MORPHOJ v. 1.04a [39].
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5
0.2
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0.1
KT (log)
PC2
0
Proc. R. Soc. B 282: 20143088
1.1
1.0
0.9
0.8
0.7
0.6
0.5
smasher
specimen
range
spearer
specimen
range
–0.1
undifferentiated
specimen
range
–0.2
–0.1
0.1
0
PC1
Figure 3. The stomatopod linkage system shows the classic pattern of mechanical equivalence in which a mechanical metric (KT; colour map) is overlaid onto a
morphospace based on linkage morphology. The morphospace is composed of PC axes 1 and 2 from a landmark-based PC analysis of 195 stomatopod specimens
(each point represents an individual specimen). A total of 36 species are represented with 1 – 10 specimens per species. KT is overlaid onto the morphospace as a
‘heat map’ showing a pattern of decoupled KT from morphology. Three ecological groups defined by appendage mechanics (smashers, spearers and undifferentiated)
are identified in morphospace and show overlap in morphology and KT. (Online version in colour.)
(a)
logKT
1.0
0.8
0.6
–0.2
–0.1
0
–0.1
0.1
0
PC2
PC1
0.1
0.2
–0.1
0
PC3
0.1
(b)
logKT
1.0
0.8
0.6
0.6
0.7
0.8
link 2
0.9
1.0
0.05
0.10
0.15
link 3
0.20
1.0
1.2
1.4
link 4
Figure 4. By examining the mechanical sensitivity of the stomatopod four-bar system, it is possible to identify the influence of individual morphological components
on the evolution of mechanical outputs. (a) Plots of kinematic transmission (KT) versus the overall morphology of the four-bar system, denoted here by PC axes 1 –
3, show a decoupling of form from function as expected in a mechanically equivalent system. (b) However, plots of KT versus the individual components of the
system (the lengths of links 2, 3 and 4 all divided by link 1, as explained in the Methods) show that one component (link 3) is tightly correlated with KT, which
potentially allows the other components (link 2 and link 4) to vary independently from KT. (Online version in colour.)
differential mechanical sensitivity of KT to individual linkage components. Multiple stomatopod lineages converge
on similar mechanical outputs via very different morphologies. However, analyses of individual link lengths
revealed that certain components are tightly associated
with KT, with strong evolutionary correlations between
link 3 and KT. These results offer a new lens on the relationships among morphology, function and evolution, and,
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0.15
6
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0.10
PC2
0.05
0
–0.20
–0.10
–0.15
–0.05
PC1
0
0.05
Figure 5. This phylomorphospace of the stomatopod linkage system illustrates the mechanically equivalent nature of kinematic transmission (KT). The morphospace is composed of PC axes 1 and 2 from a landmark-based PC analysis of 36 stomatopod species representing overall morphology of the linkage system. Previous work showed that the
four-bar system in spearing clades (Lysiosquilloidea, Parasquilloidea, Pseudosquilloidea and Squilloidea; shown in colour) all evolve towards high KT values. However, this
phylomorphospace indicates that each clade does so with distinctive morphologies (sometimes multiple morphologies within clades). Furthermore, certain taxa from the
Parasquilloidea and Pseudosquilloidea clades inhabit morphospace adjacent to the smasher clade (Gonodactyloidea; shown in black). (Online version in colour.)
Table 1. Phylogenetic generalized least-squares (PGLS) analyses support link 3 as the best single predictor of KT evolution across Stomatopoda. Link 3 has a
higher correlation with the mechanical metric (KT) than the other links, or any of the PC axes from the whole system morphological analysis (PCs 1– 8).
independent variable
l
d.f.
F-statistic
p-value
r2
AICc
log likelihood
link 2
link 3
0.857
0.911
2.34
2.34
0.439
94.53
0.648
,0.0001
20.016
0.73
230.84
278.16
17.6
41.26
link 4
0.878
2.34
0.0182
0.982
20.029
230.44
17.4
PC1
PC2
0.874
0.883
2.34
2.34
2.231
0.031
0.123
0.97
0.034
20.028
232.71
230.45
18.54
17.41
PC3
PC4
0.878
0.875
2.34
2.34
0.087
0.04
0.916
0.961
20.027
20.028
230.52
230.46
17.44
17.41
PC5
PC6
1.000
0.859
2.34
2.34
16.29
2.765
,0.0001
0.077
0.304
0.048
240.37
233.21
22.37
18.78
PC7
0.832
2.34
22.6
,0.0001
0.38
248.59
26.48
PC8
0.899
2.34
0.043
0.065
233.85
19.11
3.452
more broadly, the interplay between physical laws and
evolutionary processes.
(a) Mechanical equivalence
Mantis shrimp four-bar linkage systems exhibit a pattern of
mechanical equivalence that is similar to the findings in
other biological linkage systems. While our data confirmed
the presence of mechanical equivalence in mantis shrimp, it
is key to note that it would have been surprising not to find
mechanical equivalence given that mantis shrimp have convergent KT. Similar KTs are, by definition, mechanically
equivalent. Thus, our findings confirm the expected mechanical equivalence of KT and offer another excellent example
of biological diversification essentially exploring a space of
general physical rules.
(b) Mechanical sensitivity
Analyses of mechanical sensitivity pinpoint the components
of a system to which the mechanical output is most sensitive.
The evolution of link 3 is tightly correlated with the evolution
of KT based on both PGLS and evolutionary trait modelling
(table 2). Link 3 is the output link and transfers motion from
the linkage system to the weaponry of the raptorial appendage (the distally rotating propodus and dactyl segments
that hammer or spear prey; figure 2). Therefore, evolutionary
shifts in link 3 simultaneously affect linkage KT and the lever
system formed by the rotating weaponry [31]. The mechanics
of link 3 thereby tie both the mechanics and evolution of the
linkage and lever system together, reflecting the coupled
evolutionary pattern previously identified in this system [31].
Mechanical sensitivity analyses can also be used to identify the components of a mechanical system to which the
output metrics are insensitive. In the stomatopod linkage
system, mobile links 2 and 4 show little correlation with KT
(figure 4). Link 2 represents the meral-V (the input link),
whereas link 4 is a muscle –tendon element that connects
the carpus to the stationary link of the merus and incorporates the saddle (figure 2f ). As figure 4 illustrates, the
relative insensitivity of KT to these links probably forms the
Proc. R. Soc. B 282: 20143088
–0.10
Gonodactyloidea
Hemisquilloidea
Lysiosquilloidea
Parasquilloidea
Pseudosquilloidea
Squilloidea
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Table 2. Evolutionary model comparisons for the mechanical metric (KT) and individual morphological components (link lengths) across stomatopod phylogeny.
Italicized AICc values indicate best-fit model. s 2, Brownian motion rate parameter; a, strength of selection; u, optimal trait value.
s2
a
u
Brownian
OU single peak
226.219
223.13
0.0564
0.0789
n.a.
0.749
n.a.
1.928
OU multi peak
link 2
229.414
0.103
2.027
1.598, 1.99
Brownian
291.85
0.0091
n.a.
n.a.
OU single peak
OU multi peak
289.74
294.6
0.0136
0.020
0.952
2.597
0.793
0.675, 0.806
link 3
Brownian
2178.14
0.00083
n.a.
n.a.
OU single peak
OU multi peak
2183.90
2187.74
0.00189
0.01694
2.739
34.45
0.106
0.115, 0.0987
Brownian
OU single peak
282.022
281.81
0.012
0.029
n.a.
2.377
n.a.
1.07
OU multi peak
279.32
0.029
2.359
1.08, 1.07
link 4
basis of the mechanical equivalence pattern found when comparing overall morphology with KT. For example, link 4
shows no evolutionary correlation with KT (figure 4 and
table 1), and evolutionary trait modelling gives strong support for an evolutionary pattern in link 4 that is distinct
from KT and the other links (Brownian motion; table 2). Identifying morphological components to which a metric is
mechanically insensitive allows for a better understanding
of how mechanically equivalent systems diversify at the
evolutionary scale.
Mechanical sensitivity offers an approach for examining the
interactions between the morphology of a system and its function beyond single metrics such as KT. In stomatopods, link 3 is
evolutionarily correlated with KT, whereas links 2 and 4 are not.
However, that does not mean that links 2 and 4 are not mechanically important. Both the meral-V (link 2) and the saddle (link 4)
have previously been identified as part of the elastic mechanism
underlying the extreme power amplification of the appendage
[28,47–49]. We can hypothesize that the evolution of those components might be tied more closely with energy storage, as
opposed to KT. By exploring mechanical sensitivity in this
system, further evolutionary hypotheses can address the
relationship between morphology and multiple functions.
(c) Mechanical metrics and evolutionary analyses
Analysing the evolution of biomechanical systems requires
the identification of traits that allow comparisons across systems. Mechanical metrics, such as KT, enable comparisons of
analogous mechanical systems, such as linkage systems in
both modern [10,11] and fossil fishes [19], and between vertebrates and invertebrates (fish and mantis shrimp) [32].
When mechanical equivalence is invoked, it is generally in
the context of a single mechanical metric (such as KT).
Similarly, mechanical sensitivity refers to the sensitivity of a
particular mechanical metric (such as KT).
However, KT, like most mechanical metrics, can account
only for a portion of the function of a mechanism. While multiple linkage configurations can share the same KT, the
variation in absolute lengths of the components can affect
other aspects of linkage function not represented by KT
[27,46]. For example, KT does not incorporate scaling. If a
linkage is scaled to a larger size, KT will be identical, but
the kinematics of the system (speed, vector of motion, absolute distance travelled) will change. These important aspects
of linkage mechanisms have largely not been analysed in biological systems, given the prevailing focus on KT, and thus
merit consideration in the broader context of animal function,
performance and multi-functionality.
(d) Evolutionary biomechanics
While mechanical equivalence has been suggested to have
strong evolutionary implications, analyses of mechanical sensitivity are critical for assessing the evolutionary dynamics of
mechanical systems. Differential mechanical sensitivity is a
general phenomenon in form – function relationships involving multi-part mechanisms [4,5]. There is evidence in fish
linkage systems, often used as a key example of the evolutionary effects of mechanical equivalence, for differential
mechanical sensitivity. Barel [4] suggested that the lower
jaw link in cichlids is more ‘functionally effective’ than the
other links, allowing morphological variation in the rest of
the system with little functional consequence. Recent work
on cichlids from Lake Malawi found a tight correlation
between the ratio of two links (lower jaw and maxilla) of the
oral four-bar and KT [24]. Differential mechanical sensitivity
has also been identified in the jaw system of shrews [23].
Using evolutionary correlations to identify differential
mechanical sensitivity offers new insights into the evolution
of biomechanical systems and the effects of mechanical equivalence on diversification. Combining the concepts of mechanical
equivalence and mechanical sensitivity into a single framework
Proc. R. Soc. B 282: 20143088
AICc
rspb.royalsocietypublishing.org
KT
7
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offers a comprehensive understanding of biomechanical
evolution and the role of physics in evolutionary processes.
digitally at Dryad Digital Repository (doi:10.5061/dryad.4m7v7).
Acknowledgements. We thank K. Smith, M. Rosario, P. Green and
R. Crane for helpful suggestions on drafts of this manuscript and
to D. Polly for helpful advice on comparative methods. We are
especially grateful to two reviewers for their insightful comments
that have greatly improved the quality of this paper. We also thank
Funding statement. This research was supported by a National Science
Foundation grant (IOS-1149748) awarded to S.N.P.
Competing interests. We have no competing interests.
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