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First published online January 18, 2008
Journal of Experimental Biology 211, 289-291 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.008912
JEB Classics |
FOUNDATIONS OF ANIMAL HYDRAULICS: GEODESIC FIBRES CONTROL THE SHAPE OF SOFT BODIED ANIMALS
University of British Columbia
shadwick{at}zoology.ubc.ca
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). While their results may have been of interest to only a few
biologists at that time, the essential idea of this paper, that a structure
composed of inextensible fibres could accommodate large extensibility, has
endured and its application has become widespread, first in numerous
biomechanical case studies and, more recently, in modern biomimetics and
mechanical engineering. The basic model that was developed is now entrenched
as a design principle in biomechanics. The paper continues to be cited, and
its summary concept (as seen in Fig.
1) has been reproduced in numerous texts or reviews over the past
several decades (e.g. Alexander,
1968; Alexander,
1979; Alexander,
1988; Chapman,
1975; Clark, 1964;
Gray, 1968;
Vogel, 1988;
Vogel, 2003;
Wainwright et al., 1976;
Wainwright, 1988).
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;
Chapman, 1950;
Chapman and Newell, 1947;
Chapman and Newell, 1956;
Harris and Crofton, 1957;
Newell, 1950;
Wells, 1944); phenomenological
explanations were established and the concept of a `hydrostatic skeleton' came
into use. This term defines a system in which muscles shorten to act against a
contained volume of fluid, rather than rigid skeletal elements, to maintain
shape and effect movement. These earlier papers described the basic workings
of hydrostatic skeletons of cnidarians, annelids and nematodes.
The initial focus of the Clark and Cowey research was to find a functional
explanation for the high extensibility of some nemerteans (such as the
extremely long bootlace worm, Lineus longisimus). To a certain extent
the role of circular and longitudinal muscles in producing dimensional changes
in some invertebrates was understood, but the mechanism responsible for
limiting shape changes was not. For example, it was not clear exactly how
length changes were related to diameter changes produced by circular muscle
contractions, or how a worm-like animal could move if it had only one set of
muscles (as in the case of nematodes). So Cowey and Clark set out to match
morphology with mechanics, an approach that had been well established at the
Journal of Experimental Biology under the influence of Sir James
Gray. The idea that the geometry of the reinforcing fibres in the body wall
was the key to the solution was first explored in a study of structure and
extensibility of a common British nemertean, Amphiporus lactifloreus,
and published in a 1952 paper in which Cowey affirms the partnership by
acknowledging the assistance of R. B. Clark `in mathematical matters'
(Cowey, 1952). They showed
that the epidermal basement membrane of Amphiporus was invested with
silver-staining `reticulin' (primarily collagen) fibres that were nearly
inextensible and laid down in crossed arrays in a latticework that could
change shape, `just as in the extension and retraction of lazy-tongs'. They
then suggested that the orientation and inextensibility of the fibres set the
physical limits on length changes that can occur, and showed how these limits
can be extreme, with changes of more than 5 fold being observed in
Amphiporus. This subject was explored further in the 1958 paper,
which presented additional experimental data and a more detailed exposition of
their geometric model.
The essence of this model is presented in
Fig. 1, which shows the
familiar representation of a worm as a fluid-filled tube stiffened by helical
wrappings of inextensible fibres. The length of a segment bounded by one full
turn of the fibre is controlled by the fibre angle 
, defined as the
inclination of the fibres to the longitudinal axis. With extension, the
segment's diameter and fibre angle both decrease; conversely, with segment
shortening the fibre angle and diameter increase. If the segment maintains a
circular cross-section, its volume, V, will vary, according to the
curve in Fig. 1C. V
decreases towards zero as goes to 0° (a long, thin thread) or
90° (a flat disc), and it peaks at an intermediate angle of 54.74°.
But an extensible worm, in most cases, does not change volume, so it cannot
follow the curve. However, according to Clark and Cowey, `The system can
always contain less than this volume if the cross-section is elliptical
instead of circular,' allowing a worm to adopt a flattened, elliptical
cross-section as it changes length along a horizontal line of constant volume,
as shown in Fig. 1C. The
extremes of shortening and lengthening occur where this line intersects the
V vs curve and only here will the worm be circular. The
greatest degree of flattening occurs when =54.74°, also the angle
where circumferential and longitudinal stresses in a pressurized cylinder
balance (Wainwright et al.,
1976). This is probably the reason why Clark and Cowey observed
that a worm fully relaxed by anaesthesia adopts a length where
55°. The vertical position of the extensibility line is
determined by the degree of ellipticity, or flattening, that the segment
adopts when relaxed (i.e. the ratio of major to minor axes, n). In
theory, a flatter worm should have a higher range of extensibility, because of
its lower position on the plot in Fig.
1C, i.e. there is a greater range of lengths possible between the
extremes bounded by the V curve.
The elegantly simple experimental component of the paper tested this hypothesis. In nine species of nemertean and turbellarian worms, Clark and Cowey determined the range of possible lengths the worms could achieve, with the maximum based on passive stretching of an anaesthetized worm, while the fully contracted length was achieved by dropping an unanaesthetized worm in formaldehyde. The degree of flattening, n, was measured at the relaxed length and from this the volume relative to the maximum possible if the worm was circular was calculated as 2n/(n2+1). This volume established the position of the horizontal extensibility lines for each species shown in Fig. 2, and allowed predictions of the maximum and minimum lengths to be compared with the measured values. The worms with moderate or low extensibility showed remarkable agreement with the theory, leading to the conclusion that, `the geodesic fibres are the operative factor in limiting length changes.' Interestingly, the flattest species which had the greatest theoretical extensibilities had the least observed ones. This was explained as being the result of additional connective tissue elements and muscles in these species that impose restrictions to deformability, although the fibre system was still regarded as setting an overall limit to their changes in shape. It is also likely that the method used underestimated minimum length because it involved full contraction of circular and longitudinal muscles together, rather than contraction of just the latter.
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=75° is the only antagonist to the longitudinal muscles; no
circular muscles exist. Because these worms maintain a high internal pressure
they are cylindrical and thus occupy a position on the right-hand side of the
volume curve in Fig. 1, where
circular muscles are unnecessary. It is now recognized that energy-saving
mechanisms based on tensile fibre lattices are potentially at work in many
skeletal systems, such as fish skin, cetacean subdermal tissues or squid
mantle wall (see Wainwright,
1988; Pabst, 1996;
Gosline and Shadwick,
1983).
My first introduction to the helical winding principle of Clark and Cowey
was from Animal Mechanics
(Alexander, 1968), which I used
as a text in an undergraduate biophysics course in 1974. This was one of
several key examples in Alexander's book that made the idea of using mechanics
to understand animal function very appealing. In a subsequent graduate course
using Mechanical Design in Organisms
(Wainwright et al., 1976), the
Clark and Cowey model was espoused as one of the key design principles of
structural systems in biology. Perhaps the greatest advocate of the usefulness
of analysing hydrostatic skeletons according to this model has been Steve
Wainwright, who wrote a book on the subject [Axis and Circumference
(Wainwright, 1988)] and
encouraged many students to do research in this area. A sampling of these
efforts reveals studies on such topics as elephant trunks, lizard tongues,
cephalopod arms, notochord development, fish skin, echinoderm tube feet, and
cetacean dermal and subdermal structures
(Hebrank, 1980;
Kier and Smith, 1985;
Kier and Stella, 2007;
Koehl et al., 2000;
Long et al., 1996;
McCurley and Kier, 1995;
Pabst, 1996;
Wainwright et al., 1978).
In discussing the significance of their work, Clark and Cowey state that,
`helical bounding systems... may be quite widespread, if not general, in
softbodied, worm-like animals,' suggesting that their findings could have
broad application in biology. Indeed this has proven to be the case; time has
shown that helical fibre winding is ubiquitous in nature, including plants,
animals (see Vogel, 1988;
Vogel, 2003;
Wainwright, 1988) and even
bacteria (Wolgemuth et al.,
2003). Engineers, too, have taken inspiration from the worms.
Efforts to create compliant actuators for robotics have adopted the geodesic
fibre-reinforcing model, based on that described by Clark and Cowey. For
example, a crossed helical fibre-reinforced flexible tube will change shape
when pressurized, according to the rules laid out in
Fig. 1. If the resting
is <55° then increasing pressure will tend to increase volume and
, driving the shape up the left-hand side of the volume curve. This
results in shortening and provides the basis for an artificial muscle and its
control (e.g. Liu and Rahn,
2003). Similarly, polymer hydrogels encased in helical fibre
lattices, again mimicking the worm model, are effective high-force actuators
that can be controlled by gel swelling changes in the presence of water (e.g.
Santulli et al., 2005). What
started simply as a search for an explanation of the curiously high
extensibility of `certain nemertean worms' has provided good service to
biomechanicists interested in support and locomotion over the past five
decades, and has now entered the realm of biomimetics and robotics. Certainly,
such endurance and breadth of application of the original work in this 1958
JEB paper is the hallmark of a true `classic'.
Footnotes
Robert Shadwick discusses R. B. Clark and J. B. Cowey's 1958 paper entitled `Factors controlling the change of shape of certain nemertean and turbellarian worms'. A copy of the paper can be obtained from http://jeb.biologists.org/cgi/reprint/35/4/731.
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