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First published online October 17, 2008
Journal of Experimental Biology 211, 3433-3441 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.020495
To break a coralline: mechanical constraints on the size and survival of a wave-swept seaweed
Hopkins Marine Station of Stanford University, Pacific Grove, CA 93950, USA
* Author for correspondence at present address: Department of Botany, University of British Columbia, Vancouver, BC, Canada, V6T 1Z4 (e-mail: pmartone{at}interchange.ubc.ca)
Accepted 6 September 2008
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Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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Key words: adaptation, biomechanics, breaking stress, Calliarthron, decalcification, drag, flexibility, geniculum, intertidal, macroalgae, material properties
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INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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) and impose great forces on intertidal inhabitants
(Koehl, 1984;
Koehl, 1986;
Carrington, 1990;
Gaylord et al., 1994;
Denny, 1995;
Gaylord et al., 2001;
Helmuth and Denny, 2003). The
severity of wave-induced forces has been hypothesized to limit the maximum
size to which intertidal organisms can grow (e.g.
Denny et al., 1985;
Gaylord et al., 1994;
Denny, 1999). For example,
unlike whales and giant kelps that live in deeper water, intertidal flora and
fauna rarely exceed 0.5 m in any dimension
(Denny et al., 1985).
Blanchette found that intertidal algae transplanted from sheltered to
wave-exposed locations `tattered' back to a smaller size
(Blanchette, 1997). Such damage
is likely to be the result of drag, the primary wave-induced force applied to
intertidal macroalgae (Denny and Gaylord,
2002).
Several studies have measured drag on seaweeds in an attempt to predict the
size to which various species can grow in the intertidal zone
(Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994;
Wolcott, 2007) but have had
mixed success. This may be due in part to the characterization of drag at slow
speeds (<3 m s–1) in re-circulating water flumes and the
need to extrapolate from these data to environmentally relevant water
velocities (20–30 m s–1). Such long-range
extrapolations can be misleading (Vogel,
1994; Bell, 1999).
In particular, drag coefficient (Cd) decreases as flexible
macroalgae bend and reconfigure with increasing water velocity
(Bell, 1999;
Boller and Carrington, 2006a),
but the extent of this reconfiguration and its effect on
Cd have never been characterized at the high velocities
found on wave-swept shores. Here, we introduce a gravity-accelerated water
flume, capable of generating jets of water (meant to mimic crashing waves) up
to 10 m s–1. Thus, for the first time, it is possible to
measure drag and re-configuration of seaweeds at high velocities, reducing the
need for extrapolation.
The articulated coralline alga Calliarthron cheilosporioides Manza
thrives in wave-swept intertidal habitats along the California coast
(Abbott and Hollenberg, 1976).
Unlike fleshy algae, which are flexible along their entire length,
Calliarthron thalli are firmly calcified but have flexible joints
(genicula) that allow fronds to bend when struck by breaking waves. Flexible
genicula also define breakage points along calcified thalli
(Martone, 2006) and are
hypothesized to be especially susceptible to bending stresses, as segmented
bending may locally amplify stress within genicula [see accompanying paper
(Martone and Denny, 2008)].
Nevertheless, Calliarthron fronds can grow to a length of 25 cm,
including more than 100 genicula, and can dominate the most wave-exposed
habitats.
When struck by incoming waves, erect articulated fronds bend in the
direction of flow parallel to the substratum. Most genicula are stretched in
tension by each passing wave, but basal genicula, which are farthest from the
free end of any frond, experience the greatest bending moments (see
Martone and Denny, 2008) and
are hypothesized to be the most prone to breakage
(Martone, 2006). Morphological
characteristics of bending genicula are significantly different from those of
tensile genicula [see table 3
of the accompanying paper (Martone and
Denny, 2008)], helping them increase flexibility and decrease
stress amplification [see figure
7 of accompanying paper
(Martone and Denny, 2008)],
and bending angles may be constrained by the close proximity of neighboring
fronds, further mitigating the amplification of stress within bending
genicula. The mechanical advantages of such traits may be limited, however, if
tensile genicula ultimately break first.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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). Tensile moduli (Et) were calculated as
the slopes of linear stress–strain regressions forced through the
origin; breaking strains (
break) were assumed to be the
strain measurements immediately preceding frond breakage during mechanical
tests. Et and break were correlated, and
a linear regression was fitted to the Et versus
break data. Residuals were calculated for each datapoint
(N=15), and the standard deviation of residuals was calculated.
Bend-to-break tests
In the field, Calliarthron fronds often break near the base
(Martone, 2006), where bending
moments and bending stresses are greatest
(Martone and Denny, 2008). To
explore breakage in bending, Calliarthron fronds (N=7) were
collected from the field site described above. Branches were removed from each
frond by cutting below the first dichotomy, and the remaining straight chains
of segments were tested as follows. Individual fronds were gripped in clamps
by the first few genicula and held horizontal
(Fig. 1A). To quantify the
force to bend genicula to failure, a second clamp was secured near the tenth
genicula (numbered from the clamp) and masses were hung, in 20 g and 50 g
increments, from the clamp until fronds broke
(Fig. 1B). As bending genicula
experienced increasing force, they broke gradually
(Fig. 1C). An image analysis
revealed that unbroken genicular cells approached, but had not yet exceeded,
the average breaking strain (break) of genicula loaded in
tension (P.T.M. and M.W.D., unpublished data).
Dimensions of broken genicula were quantified as described in
figure 1 of the accompanying
paper (Martone and Denny,
2008). Genicular radii (r1,
r2) and intergenicular radii (y) were measured with
an ocular dial-micrometer. Genicular lengths (
) and gap lengths
(–2x) were measured in wet, long-sectioned
genicula adjacent to broken genicula and briefly decalcified in HCl. Average
length measurements were assumed for broken genicula. Intergenicular lip
length (x) of broken genicula was estimated as half the difference
between mean and mean gap length.
Modeling breakage in bending
The mathematical model that we present in our other study
(Martone and Denny, 2008) was
augmented to allow genicula to break gradually in order to estimate the force
to bend experimental genicula to failure. The distribution of breaking strains
was assumed to be normal with a mean of
,break and standard
deviation of s.d. ,break, and when tensile moduli
(Et) were plotted against breaking strains
(break), residuals were assumed to be normally distributed
around the linear regression with a mean of 0 and standard deviation of
s.d.residual. To generate unique bivariate normal pairs of
Et and break for each iteration of the
model, a random value was chosen from the break distribution
and a corresponding Et was calculated from the normal
distribution of values around the regression. As virtual fronds deflected, the
model eliminated any portion of the first geniculum whose strain exceeded
break (see Appendix). This reduction of tissue was factored
into the internal moments and neutral axis positions calculated by the model.
In some cases, reduction of genicular cross-section created a positive
feedback, increasing bending angles and further increasing strain. Breakage
occurred when >break across the entire geniculum.
All other components of the original bending model were unchanged (see
Martone and Denny, 2008).
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Breaking force predictions
To estimate breakage of fronds in the field, morphological dimensions of 10
fronds were used from our other analysis [see
table 3 in Martone and Denny
(Martone and Denny, 2008)].
Forces to break first genicula (nearest the base) of these experimental fronds
in bending were calculated from 1000 model iterations (means ± s.d.;
N=10).
Calliarthron grow in dense clusters in the field, and fronds
emerging from individual bases are tightly packed together
(Fig. 2A). The spatial density
of fronds probably limits bending angles of basal genicula. To evaluate this
`neighbor effect', the spatial density of fronds was measured in 15
Calliarthron individuals growing in the low intertidal zone at HMS.
Average distance between fronds (D) was calculated from the average
diameter of basal intergenicula (2y=1.334 mm) [see
table 3 in Martone and Denny
(Martone and Denny, 2008)] and
the number of fronds growing within 1 cm of one another (N=15;
Fig. 2B). Given average
intergenicular diameter and spacing, the maximum bending angle
(
i) of each frond depends upon the bending angle
(i–1) of the neighboring frond:
 | (1) |
When fronds bend over, most genicula experience drag force in tension [see
right panels of figure 4 in
Martone and Denny (Martone and Denny,
2008)]. Tensile forces required to break tenth genicula in the 10
experimental fronds were estimated using genicular cross-sectional area and
breaking stresses sampled from a normalized distribution with
,break=25.9 MN
m–2 and s.d.,break=2.3 MN
m–2 [determined by Martone
(Martone, 2006)]. Mean and
standard deviation of 1000 breaking force estimates were calculated for each
geniculum. Forces to break tenth genicula in tension and to break first
genicula in bending were compared, assuming fronds would break at the lesser
force. Mean and standard deviation of forces to break fronds were calculated,
and these values were used to predict breakage of articulated fronds in the
field.
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 | (2) |
is seawater density (approximately 1025 kg m–3),
U is water velocity, A is algal planform area, and
Cd is the drag coefficient, a dimensionless index of shape
change and reconfiguration of flexible fronds
(Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994;
Bell, 1999) (see also
Boller and Carrington,
2006a). To quantify the effect of frond size and growth on drag force, Calliarthron fronds (N=24) were collected from the low intertidal zone at HMS and were tested in re-circulating and gravity-accelerated water flumes. In both flume types, fronds were attached with cyano-acrylate glue to custom-made force transducers. In the re-circulating flume, drag force was measured on fronds (N=8) at 0.23, 0.46, 0.69, 0.92, 2.0 and 3.6 m s–1. In the gravity-accelerated water flume, drag force was measured as fronds were struck with jets of water (Fig. 3). Flow was fully turbulent as it fell through the 10 cm diameter pipe, and velocity was adjusted by varying the distance through which the water fell. Fronds were tested at 6.8 m s–1 (N=6) and 10.0 m s–1 (N=10).
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) and genicula
are approximately the same size in young and old fronds
(Martone, 2007). Drag on whole
fronds was measured, apical branches of fronds were removed, and drag force
was re-measured. Then sub-apical branches of fronds were removed, and drag
force was re-measured. This process was repeated until all branches had been
removed. Severed branches were digitally photographed and planform areas were
measured using an image analysis program (ImageJ, NIH Image,
http://rsb.info.nih.gov/ij/).
The correlation between frond planform area and drag force was plotted for
fronds at all eight water velocities.
To compare the performance of Calliarthron fronds with that of
streamlined bodies and fleshy algae in flow, Vogel's E
(Vogel, 1984) was calculated
as the slope of a linear regression fitted to a log–log plot of
speed-specific drag (Fdrag/U2) versus
velocity (U).
Calculating drag coefficients, Cd
Given drag force measurements, drag coefficients were calculated for every
combination of frond planform area (A) and water velocity
(U) by re-arranging Eqn
2:
 | (3) |
 | (4) |
is the kinematic viscosity of water
(1x10–6 m2 s–1 at
20°C). Drag coefficients decreased with increasing Ref
according to the power curve:
 | (5) |
).
Drag force extrapolation
Re-arranging Eqn 4 and
squaring both sides yields:
 | (6) |
 | (7) |
Predicting breakage in the field
Fronds are expected to break in the field when drag force experienced by
genicula exceeds breaking force. This expectation can be represented as
follows:
 | (8) |
 | (9) |
Field measurements
From November 2003 to November 2006, Calliarthron fronds were
collected every few months, totaling eight collections, from the intertidal
field site described above. During each collection we searched for the largest
available fronds. Collections typically consisted of 10–20 fronds.
Fronds were digitally photographed and frond planform areas were measured
using image analysis (ImageJ). Maximum frond area was noted on each date over
the 3 year span.
From November 2005 to August 2006, maximum water velocities were measured.
On 2 November 2005, three dynamometers
(Bell and Denny, 1994;
Helmuth and Denny, 2003) were
installed at mean lower low water approximately 0.75 m apart, spanning the
field site. Dynamometers were first checked and reset on 4 November 2005 and
were checked and reset during sufficiently low tides (13 times) until 10
August 2006. The maximum water velocity recorded by any dynamometer was noted
on each date over the 9 month span.
Field measurements were compared with breakage predictions to determine whether water velocities in the field were sufficient to generate drag forces that would equal forces experimentally determined to break Calliarthron fronds.
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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break of genicula was 1.18±0.44 (mean ±
s.d.), and mean Et of genicula was 27.7±12.4 MN
m–2 (mean ± s.d.). break and
Et were significantly negatively correlated
(Fig. 4;
R2=0.62, P<0.001), such that:
 | (10) |
Bend-to-break tests and model validation
Predicted and observed breaking forces were similar
(Table 1) and, on average, were
not significantly different (P=0.47, Student's paired
t-test).
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Breaking force predictions
Without neighbors, bending genicula were predicted to break before tensile
genicula because, on average, tensile genicula resisted significantly more
force (P<0.01, paired t-test;
Fig. 5). However, fronds were
predicted to resist greater forces in bending when supported by neighboring
fronds (Fig. 5). Neighboring
fronds were spaced 0.4 mm apart, on average, limiting bending angles to
approximately 54 deg. With neighbors, bending genicula and tensile genicula
were predicted to resist similar forces
(Fig. 5), which, on average,
were not significantly different (P=0.30, paired t-test).
Mean force to break first genicula in bending with neighbors was 26.3 N, and
the mean force to break tenth genicula in tension was 22.7 N
(Table 2). Assuming fronds
would break at the lesser of the two breaking forces for each frond, mean
force to break Calliarthron fronds was 20.0±3.8 N (mean
± s.d.; Table 2).
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Drag force measurements and drag coefficient estimates
For all water velocities, drag force increased with frond planform area,
and fronds of any given area experienced more drag force at greater velocities
(Fig. 6). Vogel's E
was calculated to be –0.68 (R2=0.70;
Fig. 7).
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 | (11) |
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 | (12) |
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Substituting these values for Ref,crit in
Eqn 9, the following three
equations were generated to predict water velocities that would break fronds
of given planform area in the field (Fig.
11):
 | (13) |
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DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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),
Calliarthron genicula are clearly well adapted to resist mechanical
failure in wave-swept habitats. Even without the support of neighbors, fronds
located near the periphery of aggregations are able to bend to 90 deg., and
several experimental fronds supported more than 1 kg of weight (>9.8 N) in
this position (Table 1). When
neighboring fronds are considered, bending angles of central fronds are
constrained, allowing first genicula to resist approximately twice the force
of fronds near the periphery. Ultimately, fronds growing within dense
populations are just as likely to break at tenth genicula in tension as they
are to break at first genicula in bending.
These data suggest that genicula are not `over-designed' in an evolutionary
sense. The tensile strength of calcified intergenicula in
Calliarthron (28.5 MN m–2)
(Martone, 2006) is similar to
that of coral skeleton (25.6 MN m–2)
(Vosburgh, 1982) and to that
of several bivalve and gastropod shells that appear similar to
Calliarthron cell walls (`homogeneous' type, 30 MN
m–2; `foliated' type, 38.3 MN m–2)
(Currey, 1980). This suggests
an upper limit to the tensile strength of biologically deposited calcium
carbonate within Calliarthron cell walls. This mechanical constraint
may be biologically linked to genicula, whose tissue is equally strong (25.9
MN m–2) (Martone,
2006). Indeed, genicular tissue is far stronger than other algal
tissues (up to an order of magnitude)
(Martone, 2006) but may be
biologically constrained from growing any stronger. Given this putative
maximum tensile strength of genicular tissue, there may not be a selective
advantage to adjusting the spatial density of fronds or the dimensions of
bending genicula, if tensile genicula would ultimately break first. In other
words, growing more densely packed clusters of fronds or decreasing the length
of intergenicula (see Martone and Denny,
2008) may indeed increase the breaking force of bending genicula,
but such fronds would probably break at tensile genicula anyway; and breakage
of either bending or tensile genicula near the base is likely to be
disadvantageous, resulting in the loss of an entire frond that may have been
reproductive and several years old
(Johansen and Austin,
1970).
Together, these data are consistent with the engineering theory of optimal
design [Maxwell's Lemma (see Wainwright et
al., 1982)], which states that all components in a mechanically
stressed system should be equally strong to avoid wasting resources in their
construction. The fact that bending and tensile genicula are morphologically
distinct (Martone and Denny,
2008) but resist similar drag forces suggests that these
structures may have been shaped by selective pressures imposed by breaking
waves. Morphological differences among tensile and bending genicula,
therefore, may represent adaptations to hydrodynamic stress.
Environmentally relevant drag coefficient
To our knowledge, this is the first study to report drag coefficients for
an intertidal seaweed at high, environmentally relevant water velocities. Drag
coefficients reported here for Calliarthron are up to an order of
magnitude lower than those reported for several other algae at slow water
velocities (Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994). Our
data suggest that drag coefficient continues to decrease as water velocity
increases, at least up to 6 m s–1
(Fig. 8), contrary to the
assumption that reconfiguration is solely a low-velocity phenomenon (see
Bell, 1999). Vogel's E
(–0.68) suggests that the drag coefficient of reconfiguring
Calliarthron fronds drops faster than that of a typical streamlined
body (–0.50) (Vogel,
1984) and faster than those of several branched red algae,
including a congeneric species (–0.35±0.13, mean ± s.d.,
N=7 species), although slower than those of flat bladed algae
(–1.11±0.10, N=3 species;
Fig. 7) [data compiled from
table 4 in Gaylord et al.
(Gaylord et al., 1994)].
Without high-speed data for other seaweeds, it is unknown at this time whether
such low drag coefficients are a distinct characteristic of
Calliarthron or a shared feature of intertidal algae. Our data
emphasize the importance of measuring drag at high water velocities to avoid,
or at least improve, extrapolation. For example, Gaylord and colleagues
extrapolated fivefold beyond their data to generate drag predictions
(Gaylord et al., 1994); in
contrast, we would need to extrapolate less than twofold beyond our high-speed
flume velocities (i.e. a short distance along the logRef axis of
Fig. 9) to generate the same
predictions. The accuracy of past predictions awaits verification in the
high-speed flume.
Our data demonstrate effects of both planform area and water velocity on
drag coefficient, suggesting yet another source of error in previous studies
that treated drag coefficient strictly as a function of velocity and tested
only a narrow size range of fronds (e.g.
Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994;
Bell, 1999). Flexible algal
fronds have lower drag coefficients in faster water because of the increased
reconfiguration that occurs as fronds bend. Similarly, larger fronds are
likely to have lower drag coefficients because of their capacity to re-arrange
their branches and collapse to be more streamlined, unlike smaller fronds
whose sparse branches are perhaps less capable of reconfiguration
(Fig. 8).
It is important to note that, in this and previous studies of algal
reconfiguration, drag coefficient is calculated using frond planform area
(i.e. flattened and photographed from above). This methodology assumes a
constant area term, allowing the drag coefficient to absorb any change in
shape due to reconfiguration; all else being equal, larger fronds will tend to
have lower drag coefficients (see Eqn
3). This contrasts sharply with a recent study
(Boller and Carrington, 2006a)
that calculated drag coefficients using frond projected area (i.e.
photographed from upstream in flow) and tracked the independent decline of
both projected area and drag coefficient as macroalgae reconfigured with
increasing water velocity – a method that had previously been applied to
reconfiguring gorgonians (Sponaugle and
LaBarbera, 1991). Their data show that as flow increases,
projected areas and drag coefficients both decline; fronds with larger
projected areas have higher drag coefficients. Unlike our method, theirs
generates drag coefficients that are directly comparable to other engineering
shapes. However, because projected area cannot be visualized or measured in
turbulent water at high speeds (e.g. in the gravity-accelerated water flume or
under breaking waves), predictions of projected area, like those for drag
coefficient, will inevitably rely upon long-range extrapolations. Thus, even
though the drag coefficients presented here cannot be compared with those of
standard shapes, our method reduces extrapolation and so seems suited to
exploring the maximum size to which wave-swept fronds can grow.
Limits to frond size in the intertidal zone
Forces estimated to break Calliarthron fronds in the field are
consistent with forces previously measured to break genicula in tension
(Martone, 2006). An average
Calliarthron frond can resist approximately 20 N of force before
breaking – a substantial amount of force. For example, one large
experimental frond (30 cm2) experienced only 5 N of drag force at
10 m s–1 (Fig.
6) – far below the threshold breaking force. This suggests
that Calliarthron may be well adapted to resist drag imposed by
intertidal water velocities. However, velocities as high as 35 m
s–1 have been recorded at HMS (M.W.D., unpublished data), and
articulated fronds may ultimately be size limited when water velocities
approach this extreme.
Indeed, data presented here suggest that the size of Calliarthron
fronds may be limited by drag forces imposed by intertidal water velocities.
According to our breakage model, the maximum water velocity measured at the
field site (22.1 m s–1) closely predicts the mean maximum
size (40.9±7.8 cm2, mean ± s.d.) of
Calliarthron fronds expected to survive there
(Fig. 11). These data suggest
that the broad range of frond sizes observed at the field site may, at least
in part, be a consequence of variation in forces to break genicula. Our
simplified breakage model ignores any possible drag-reducing effects of
neighboring algae (Boller and Carrington,
2006b) and, because intertidal water velocities vary widely in
both space and time (Denny and Wethey,
2001; Helmuth and Denny,
2003; O'Donnell,
2005), characterizing years of wave-forces with only a few
dynamometer measurements is a broad generalization. For example, if breaking
waves during some storm event actually generated 28 m s–1
water velocities at the field site – a distinct possibility – then
the size of the largest frond predicted to survive, including 95% model error,
would closely match the observed size of the largest frond (51.9
cm2; Fig. 11). The
close correlation between maximum velocity and frond size across the field
site suggests that, although Calliarthron is well adapted to
resisting breakage, growth may ultimately be limited by wave-induced drag
forces. Observations of larger fronds growing subtidally (K. A. Miller,
personal communication), where drag is lower, support this conclusion but have
yet to be properly quantified.
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APPENDIX |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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break. After calculating frond deflection,
the model determined the position within the geniculum, 
max,
where =break and subtracted the genicular area where
>max (and therefore
>break; Fig.
A1):
 | (A1) |
|
Acknowledgments |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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