AMER. ZOOL., 36:678-694 (1996)
The Importance of Body Stiffness in Undulatory Propulsion1
JOHN H. LONG, JR. AND KAREN S. NIPPER
Department of Biology, Vassar College,
Poughkeepsie, New York 12601
SYNOPSIS. During steady swimming in fish, the dynamic form taken by
the axial undulatory wave may depend on the bending stiffness of the
body. Previous studies have suggested the hypothesis that fish use their
muscles to modulate body stiffness. In order to expand the theoretical
and experimental tools available for testing this hypothesis, we explored
the relationship between body stiffness, muscle activity, and undulatory
waveform in the mechanical context of dynamically bending beams. We
propose that fish minimize the mechanical cost of bending by increasing
their body stiffness, which would allow them to tune their body's natural
frequency to match the tailbeat frequency at a given swimming speed. A
review of the literature reveals that the form of the undulatory wave, as
measured by propulsive wavelength, is highly variable within species, a
result which calls into question the use of propulsive wavelength as a
species-specific indicator of swimming mode. At the same time, the smallest wavelength within a species is inversely proportional to the number
of vertebrae across taxa (r2 = 0.21). In order to determine if intact fish
bodies are capable of increasing bending stiffness, we introduce a method
for stimulating muscle in the body of a dead fish while it is being cyclically bent at physiological frequencies. The bending moment (N m) and
angular displacement (radians) are measured during dynamic bending
with and without muscle stimulation. Initial results from these wholebody work loops demonstrate that largemouth bass possess the capability
to increase body stiffness by using their muscles to generate negative
mechanical work.
Wardle et al, 1995) and stiffen the body
(Johnson et al, 1994; Long et al, 1994).
This
Potential for functional complexity
means that we need a theoretical framework
that
integrates the various forms of muscle
function with the dynamic motion and mechanics of the undulating body. In a visionaT w o r k B l l h t
V
S < 1 9 7 7 ) constructed a qualitative mode1 the
'
"hybrid oscillator," that
integrated muscle activity, the body's mechanical properties, and interactions of the
body with the surrounding water. Quantitative models that have followed emphasized the interaction of either body motion
and hydrodynamics (Hess and Videler,
1984; Cheng and Blickhan, 1994, Liu et al,
1
From the Symposium Aquatic Locomotion: New 1996) or body motion and muscle activity
Approaches to Invertebrate and Vertebrate Biome- ^ ^ Leeuwen et al, 1990). Missing from
cnanics presented at the Annual Meeting of the Society
. .
.
°
for Integrative and Comparative Biology, 27-30 De- Our quantitative toolbox IS a model that forcember 1995, at Washington, D.C.
mulates the interaction of body motion,
678
INTRODUCTION
While myotomal muscles generate the
traveling wave of bending that powers undulatory swimming, how the anatomical
segments interact as functional units is dependent on, among other factors, swimming
speed, behavior, axial position, and species
(Jayne and Lauder, ]995b, 1996). From empirical studies, a picture emerges in which
the axial muscles of fish have a range of
possible functions beyond traditional forceful shortening—for example, by performing
negative work they may transmit force (Altringham et al, 1993; Johnston et al, 1995;
BODY STIFFNESS IN UNDULATING VERTEBRATES
muscle activity and the passive mechanical
properties of the body. Our goal is to lay
the groundwork for a model of undulating
fish as dynamically bending beams of passively and actively variable mechanical
properties.
The mechanical behavior of the fish's
body axis during undulatory swimming is
directly influenced by the body's bending
stiffness (for definitions of stiffness see
"Stiffness and Dynamically Bending Fish"
below). Using freshly-killed pumpkinseed
sunfish, Lepomis gibbosus, undulatory
swimming was generated by electrically
stimulating the axial muscles; when the
bending stiffness of the body and the
amount of active muscle were reduced, the
axial wave kinematics changed and the
swimming speed decreased (Long et al,
1994). McHenry et al, (1995) varied the
flexural stiffness of swimming vinyl models
of pumpkinseed and found that, in accordance with expectations from mechanical
theory, the speed of the undulatory wave
increased with increasing stiffness. As passive bending stiffness decreased from surgical manipulations of the skin in living
longnose gar, Lepisosteus osseus, wave
speed and tailbeat frequency decreased during steady swimming (Long et al, 1996).
The bending stiffness of fish may be, in
some circumstances, actively altered during
swimming by negative work (so-called "active lengthening") generated by the caudal
muscles. In amphibian larvae, Blight (1976,
1977) noted that undulatory swimming
could be accomplished with activity of the
rostral muscle segments alone. This conclusion has been supported in mechanicallydriven tadpoles (Wassersug and Hoff,
1985), electrically-stimulated fish (Long et
al., 1994), and undulating fish models (McHenry et al., 1995). Since only the rostral
muscles are necessary for undulatory propulsion, Blight (1976, 1977) proposed that
the caudal muscles function to adjust the
stiffness of the body to meet the mechanical
demands of different swimming speeds and
behaviors. Tests of this hypothesis in live,
swimming fish remain intractable since
muscle force and work cannot be measured
directly—electromyography measures only
ion flux across the sarcolemma and not
679
cross-bridge interaction or contraction
(Loeb and Gans, 1986). Using a method developed by Josephson (1985), muscle work
during swimming has been inferred by
stimulating excised muscle fibers at physiological strains, strain rates, and stimulus
patterns and measuring the resulting mechanical work (Altringham et al., 1993;
Rome et al., 1993; Johnson et al., 1994;
Johnston et al., 1995). The mechanical
work generated by caudal muscle fibers differs with species, fiber type, axial position,
cycle frequency, and stimulus regime. At
one extreme, the red (slow) fibers of scup,
Stenotomus chrysops, appear to produce
primarily positive mechanical work and virtually no negative work (Rome et al.,
1993). The red fibers of largemouth bass,
Micropterus salmoides, also produce mostly positive work at cycle frequencies associated with steady swimming; at higher frequencies seen during fast starts, negative
work is produced during part of the cycle
(Johnson et al., 1994). Under fast-start conditions, the caudal-most white (fast) fibers
of sculpin, Myoxocephalus scorpius, generate negative work during part of their cycle (Johnston et al., 1995). At the other extreme, the white caudal fibers of saithe, Pollachius virens, generate large amounts of
negative work at cycle frequencies associated with fast steady swimming (Altringham et al, 1993).
STIFFNESS AND DYNAMICALLY
BENDING FISH
Even though it has been proposed that
muscles may, under some conditions, stiffen the body of undulating aquatic vertebrates, it is not immediately clear what the
mechanical consequences of increased stiffness would be. By examining stiffness in
the context of dynamically bending beams,
we present a framework in which body
stiffness determines the bending frequency
at which the minimum force is required to
drive the local body oscillations. If fish alter
their body stiffness so that the force required at that cycle frequency was minimized, large savings in the mechanical
work required to bend the body might be
realized (for review of elastic energy storage, see Pabst, 1996).
680
J. H. LONG, JR. AND K. S. NIPPER
In general terms, stiffness is the mechanical property that characterizes a structure's
deformation when subjected to a given external force (Den Hartog, 1949). In a system whose motion is limited to a single degree of freedom, such as a linear spring
with a mass attached to one end, stiffness,
k (in units of N m), is the proportionality
constant between the externally applied
force (gravity in this case), F (in N), and
the spring's linear deformation, x (in m):
F = kx.
(1)
This equation also works for a cantilevered
beam with a force applied at the free end
and with the displacement, x, measured as
the linear motion of the beam's end (Denny,
1988). This equation is operational if two
conditions are met. First, the structure and
the force must be at equilibrium; that is
they must be motionless, the so-called
"static" testing situation. Second, the stiffness, which can be pictured as the slope of
the force-displacement line, is constant and
hence the line is straight. If, in the case of
a simple hinge, the displacement is angular,
9 (in radians), the external load is a bending
moment, M (in N m), and the proportionality constant, k', is the angular stiffness:
M = k'6.
(2)
If, in the case of beam, the structure is
bending equally along its length under the
influence of the bending moment, the bending displacement can be measured as curvature, K (in irr 1 , the inverse of the radius
of curvature), and the proportionality constant, El, is the flexural stiffness:
In each definition (Eqs. 1-3), the stiffness is constant. However, in most biological systems, like fish axial skeletons (Long,
1992, 1995), and in visco-elastic materials
in general (Ferry, 1961), stiffness is not
constant; it changes as a function of both
displacement and the rate of change of the
displacement, dxldt (Wainwright et al.,
1976; Vincent, 1990):
k = f x,
dx
dt
(4)
If we wish to apply these principles to a
small section of a swimming fish, the situation is further complicated by the fact that
when the system is dynamically bending,
the moment, M, required to cause the dynamic motion, 6 (in rad), is the sum of the
moments due to stiffness, damping, and inertia, as expressed in the equation of motion
for a single-degree-of-freedom system (Den
Hartog, 1956):
M = k'0 + c9' + N8",
(5)
where c is the damping coefficient (in kg
m2 rad~2sec~'), 6' is the first derivative of
the angular displacement (angular velocity,
in rad sec"')> N is the moment of inertia (in
kg m2 rad"3), and 9" is the second derivative
of the angular displacement (angular acceleration, in rad~2sec~2). This second-order,
ordinary differential equation (Eq. 5) can be
readily solved when the angular motion, 9,
is sinusoidal:
(6)
9 = 8max sin(u>t),
where 9max is the maximum amplitude of the
angular displacement, a> is the angular or
M = EIK,
(3) circular frequency (in rad sec"1, where cywhere E is the Young's Modulus, the ma- cle frequency in Hz is the product of w and
terial stiffness (in N irr 2 ), and I is the sec- the inverse of 2ir), and t is the time (in sec).
ond moment of area (m4), a cross-sectional In summary, passive stiffness in a dynamshape factor that takes into account the dis- ically bending biological structure is likely
tribution of the material, which is assumed to be variable with respect to the amplitude
to be homogenous, in relation to the bend- and rate of deformation. Stiffness is but one
ing axis (Wainwright et al., 1976). Because of the mechanical properties that deterof the assumption of material homogeniety, mines the force needed to drive sinusoidal
flexural stiffness must be used with caution oscillations.
when applied to heterogeneous biological
The force needed to drive a system is desystems such as fish bodies; for example, pendent on the driving frequency in a nonone should avoid the temptation to calculate linear manner determined by the system's
a Young's Modulus.
natural frequency of oscillation. In the ab-
BODY STIFFNESS IN UNDULATING VERTEBRATES
sence of a sinusoidal driving force, the frequency at which a system will freely oscillate, following a single displacement, is
called its natural frequency, a>n (in rad
sec"')- In a dynamically bending system,
the natural frequency is determined by the
relative magnitudes of the mechanical properties k', c, and N (modified from Denny,
1988):
c
(7)
N
2N
If the sinusoidal oscillations are being driven by an externally applied force, as in Eq.
5, the maximal bending moment, Mmax, required to drive a sinusoidal motion of constant amplitude, 6max, is a function of the
structure's mechanical properties—k', N,
and, c—and the driving frequency, a>d
(modified from Denny, 1988):
Mmax = 9maxV(k' / ) 2 + (co)d)2.(8)
If amplitude, stiffness, moment of inertia,
and damping coefficient remain constant,
the maximal bending moment varies with
changes in the driving frequency, u)d, and is
lowest at the structure's natural frequency,
con (Fig. 1). The "driver," that is the source
of the bending moment such as the myotomal muscle in fish, benefits from operating
at or near the system's natural frequency—
the minimum bending moment is required
to drive the motion relative to the bending
moment required to operate at other frequencies; in other words, the minimal cost
of driving the motion occurs when the ratio
of the driving and natural frequencies
equals one:
(9)
We predict that fish operate with a natural
frequency at or near their driving frequency
and thus minimize the force required to dynamically bend their body.
This prediction assumes that the damping
coefficient, c, of the body, which may be
augmented by interactions with the external
water during swimming, is low relative to
the critical damping coefficient, cc. The critical damping coefficient is twice the product of the moment of inertia, N, and the
681
0.8
2 I
I 3
0.6
0.4
0.2
8
°o
bending frequency (Hz)
FIG. 1. Changes in stiffness alter natural frequency.
In a dynamically bending structure, here the intervertebral joint of a blue marlin, driven by an externally
applied bending moment, the maximum moment required to drive the structure at a given angular frequency of constant amplitude (Eq. 10) depends on the
natural frequency of the system (Eq. 9). Each curve is
the maximum moment, given a constant stiffness, required to bend the joint over the range of bending frequencies (see Eq. 10). Note that for each stiffness there
is a single frequency, the natural frequency, u>n, at
which the minimum bending moment is required.
When angular stiffness, k', is increased the natural frequency increases; thus if a fish increased its angular
stiffness as tailbeat frequency increased, it could tune
its body by matching its natural frequency to the driving frequency, thus minimizing the cost to bend the
body. See text for further discussion. The values used
in Eq. 10 to produce these curves are realistic values
chosen from the dynamic mechanical properties of the
intervertebral joints of blue marlin (Long, 1992): k' =
2 to 8 Nm rad"1, damping coefficient, c = 0.1 kg m2
rad~2sec~', angular displacement, 6max = 0.05 rad
(2.9°), and bending frequency = 0 to 3 Hz (angular
frequency, oo, of 0 to 6TT rad sec"'). The moment of
inertia, N = 0.05 kg m2 rad"3, is the equivalent of 5
kg mass concentrated at a radius of 10 cm from the
bending pivot; it is a rough estimate of the body mass
which might be accelerated in a small section of a 70
kg, 1.8 m long marlin (70 kg divided by 18 10-cm
body segments equals 3.9 kg).
natural frequency, a>n (Den Hartog, 1956).
As the ratio of c to cc approaches one, the
driving frequency that incurs the minimal
cost to bend the body will be at values below that of the natural frequency (wyw,, <
1). While the damping coefficient has yet
to be measured for whole fish bodies, values for the intervertebral joints of blue marlin (Long, 1992) yield ratios of c to cc from
0.08 to 0.16, well within the range where
the value for wd at which the cost to bend
the system is minimized is very close to
that of wn (see Fig. 1; also Den Hartog,
1956).
682
J. H. LONG, JR. AND K. S. NIPPER
Changes in driving frequency are important to swimming fish because the thrust
power, T, generated with the axial undulatory wave is proportional to the square of
driving frequency, which is usually measured as F, tailbeat frequency (Hz) (Wu,
1977):
Swimming scallops, Pecten maximus, also
swim at or near the resonance frequency of
their deforming propulsive structures
(DeMont, 1990).
It is important to keep in mind that the
model developed here deals only with a single degree of freedom—simple sinusoidal
motion. While this working assumption
may be appropriate for small sections of the
T oc F2A2D2| 1
(10) undulating fish, it is unlikely that the entire
F\
fish, with its traveling wave of bending, can
where A is the tailbeat amplitude (m), D is be modeled so simply. At the same time,
the depth in the sagittal plane of the trailing there is growing evidence from muscle
edge of the caudal fin (m), v is the swim- function studies that a simple standing
ming velocity (m sec~')> and X. is the wave- wave of force production, which could be
length of the axial propulsive wave (m). modeled with a single degree of freedom,
Kinematic studies reveal that tailbeat fre- may underlie the generation of the traveling
quency increases linearly with increasing wave (Hess and Videler, 1984; Cheng and
steady swimming speed in fish using axial Blickhan, 1994; Long et al., 1994; McHenundulations (for review see Videler, 1993). ry et al., 1995; Jayne and Lauder, 1996).
If, as driving frequency increases to gen- Those wishing to extend the model's deerate more thrust power for higher speeds grees of freedom may seek guidance from
(Eq. 10), the effective stiffness of the body engineering texts {e.g., Den Hartog, 1956;
remains constant («d/wn > 1), the cost of Timoshenko et al., 1974).
movement would increase because of both
The motion of a structure, such as a secthe additional fluid resistance and the ad- tion of a fish body, oscillating with a single
ditional force required to bend the body degree of freedom is that of a standing
(Eq. 8, Fig. 1). Note that this assumes that wave. When degrees of freedom, or secthe passive body stiffness sets an initially tions of the fish, are added, a wave that is
low natural frequency. With increasing generated from a source will propagate
swimming speed, a smart fish would use its through the structure at a speed proportionmuscle to increase body stiffness, thus tun- al to the structure's natural frequency (Tiing its natural frequency to the driving fre- moshenko et al., 1974). Since the natural
quency (wd/con = 1).
frequency, in turn, is proportional to the
One might argue that using muscle to square root of the structure's angular stifftune the body is a costly proposition—why ness (Eq. 7), we predict that the stiffness of
not simply use the muscle to generate more a fish's body will be reflected in the speed
positive work? Because muscle operates at which its undulatory wave propagates or
more efficiently when generating negative travels (McHenry et al., 1995; Long et al.,
work (for review see McMahon, 1984), it 1996). The speed of the undulatory wave is
is likely that tuning saves metabolic energy the product of the driving frequency, F, and
for the swimming fish. Humans use their the propulsive wavelength, \, and it determuscles to increase stiffness during hop- mines, in part, the fish's hydrodynamic
ping, which allows the body to rebound power output (Eq. 10). Thus, for a given
from the ground in a simple, spring-like tailbeat frequency, it is likely that two fish
manner above a preferred frequency (Farley swimming with different propulsive waveet al., 1991). While not modulating stiffness lengths are operating with different body
actively with muscles, the jellyfish, Polyor- stiffnesses.
chis penicillatus, appears to swim at a cycle
frequency at or near the resonance frequenBODY STIFFNESS AND SWIMMING MODE
cy of its bell, thus maximizing the bell's
In the search for a relationship between
deformation and thrust with minimal input axial structure and swimming mode in fishof energy (Demont and Gosline, 1988). es, a single hypothesis has dominated the
BODY STIFFNESS IN UNDULATING VERTEBRATES
field: stiffness of the body is proportional
to the relative length of the body's undulatory or propulsive wave, \ (in body
lengths, L). First codified by Breder (1926),
the hypothesis stems from observations (see
also the mechanical argument in the previous paragraph) that long, slender and presumably flexible (low stiffness) fish appear
to swim with undulatory waves of small
length relative to that of other species (always \ < 1, Lindsey, 1978). The spectrum
of axial swimming can then be divided into
modes, with "anguilliform" (eel-like)
swimmers possessing apparently flexible
bodies and "carangiform" (jack-like)
swimmers possessing stiffer bodies (see Table VII, Breder, 1926). What structures
might determine the range of flexibility
across species? Nursall (1958) grouped
fishes into three categories based on the observation that anguilliform swimmers have
many more vertebrae than their carangiform
counterparts. As a result of these observations, and the importance of propulsive
wavelength in hydrodynamic models (Eq.
10; see also Lighthill, 1975), the number of
vertebrae is sometimes used as either the
primary structural basis for swimming
mode or as a proxy for body shape, flexibility, or body segment number, features
which in turn determine swimming mode.
As evidence of the pervasiveness of this
concept, recent evolutionary studies of fish
populations have focused on vertebral number as a target for natural selection of individual differences in swimming performance (Reimchen and Nelson, 1987;
Swain, 1992a, b). Also, the number of vertebrae are correlated with water temperature
(Lindsey and Arnason, 1981), suggesting a
developmental and/or evolutionary response in the propulsive system to temperature-associated changes in the kinematic
viscosity and hydrodynamic forces (for review of fluid properties, see Denny, 1993).
Given the potential importance of the relationship between the number of vertebrae
and the relative propulsive wavelength of
the body, it is striking that, to our knowledge, this relationship has never been examined statistically. In order to test the robustness of the relationship, we sought out
studies which measured variation in pro-
683
pulsive wavelength during steady swimming. This criterion was met in twenty species of fish and eight species of amphibians
and reptiles (Fig. 2, Table 1).
In each case, the reported intraspecific
variation represents the greatest range of
values measured in a particular study (Fig.
2). The source of that variation depends on
the nature of the study undertaken; propulsive wavelength varies with individual fish
and with changes in axial position, swimming speed, body length, and the method
of calculation (Table 1). Furthermore, variation between studies is introduced by the
method of measurement (see Webb et al,
1984). There are three common methods
used to measure propulsive wavelength: (1)
from the axial speed, c, of the propulsive
wave (where c is the product of tailbeat frequency and propulsive wavelength), (2)
from the internode distance of super-imposed midlines with the tail tip at the lateral-most positions (abbreviated "ind-lat"
in Table 1), (3) from the internode distance
of super-imposed midlines with the tail tip
at its lateral mid-point ("ind-mid" in Table
1). Another source of variation is introduced by the measurement of either full or
half waves. To provide the greatest intraspecific range of propulsive wavelengths,
we developed the following protocol: (1)
When given the mean value (±SE) of propulsive wavelength (L), we added and subtracted two times the standard error from
the mean (96% confidence interval), (2)
when given half-wavelength values, we
doubled the smallest and the largest reported extremes to yield full wavelengths, (3)
when given values for wavelength in absolute units without association to specific
individuals, we divided the smallest wavelength (m) by the largest individual (m) and
vice versa, and (4) when individual values
were given, we took the extreme values reported.
To test the hypothesis that propulsive
wavelength during steady swimming is inversely proportional to number of vertebrae
in fish, we regressed the largest and smallest propulsive wavelength (L) for each species onto the smallest and largest number
of vertebrae, in turn, reported for that species. For the purpose of the regression analGANSERUBRAKf
MILLERSVILLE. PA 17681
684
J. H. LONG, JR. AND K. S. NIPPER
"carangiform"
"anguilliform"
I
vertebral columns
number of vertebrae
J!
Jill.
FIG. 2. Range of propulsive wavelength (L) in swimming fishes, amphibians, and reptiles as function of vertebral number. The propulsive wavelengths measured during steady swimming in fish (black bars) and in amphibians and reptiles (white bars) varies with axial position on the body, swimming speed, body size, and
measurement technique (see Table 1). To test the hypothesize that propulsive wavelength was inversely proportional to number of vertebrae in swimming fish (for numbers see Table 1), a regression analysis was performed.
A significant relationship was detected between the smallest propulsive wavelength and vertebral number (see
text for details). The horizontal line at 1.0 wavelengths represents the traditional demarcation between anguilliform and carangiform swimming mode. Species are arranged by increasing number of vertebrae, with overlapping ranges indicated by horizontal arrows and accompanying vertebral numbers.
ysis, the three fish species (A. fulvescens, L.
fluviatilis, and P. marinus) with persistent
adult notochords and no mineralized vertebral centra (placed under the heading
"notochords" in Fig. 2) were given "vertebral numbers" of 300, which placed them
just beyond the species with the largest
number of vertebrae in this study (297 in
the corn snake, Elaphe guttata, Table 1).
The value of 300 was chosen with the assumption that structurally continuous notochords are more flexible than any vertebral
column, even one with 297 vertebrae. Values for reptiles and amphibians were ex-
cluded from the regression analysis to minimize bias caused by features correlated
with shared phylogeny (Felsenstein, 1985);
relative to the more evenly distributed fish
(Fig. 2), the reptiles and amphibians possessed either high vertebral numbers (5. intermedia, N. fasciata, and E. guttata) or notochords in the caudal region, as in the tadpoles of the frogs (B. americanus, R. catesbeiana, R. clamitans, R. septentrionalis,
and X. laevis; placed under the heading
"notochords" in Fig. 2). All species possessing mineralized vertebral centra along
the entire length of their axial skeleton were
TABLE 1. Sources of variation* in propulsive wavelength (see also Fig. 2).
Species
Fish
Acipenser fulvescens
Anguilla rostrata
Carassius auratus
Carcharhinus limbatus
Esox Indus X masquinongy
Gadus morhua
Lampetra fluviatilis
Lepisosteus osseus
Lepomis macrochirus
Lepomis gibbosus
Micropterus salmoides
Oncorhynchus mykiss
Oncorhynchus nerka
Petromyzon marinus
Pholidichthys leucotaenia
Pollachius virens
Salmo salar
Scomber scombrus
Sphryna tiburo
Triakis semifasciata
Source (method)
i, v (half; ind-lat)
i, p (half; ind-lat)
p (half, c)
i,v (whole, c)
i (?; ind-lat)
v (whole, c)
i (whole, c)
i (whole; ind-?)
Study
Vertebral number
(whole; c)
/ (whole; c)
i (whole; c)
/ (whole; c)
Webb, 1986
Gillis, unpublished
Bainbridge, 1963
Webb and Keyes, 1982
Webb, 1988
Videler and Wardle, 1978
Williams et al. 1989
Webb et al. 1992
Webb, 1992
Long et al. 1994
Jayne and Lauder, 1995a
Webb et al. 1984
Webb, 1973
Shepherd, unpublished
Shepherd, unpublished
Videler and Hess, 1984
Tang and Wardle, 1992
Videler and Hess, 1984
Webb and Keyes, 1982
Webb and Keyes, 1982
notochord
103-111 (Smith, 1985)
26-27 (Smith, 1985)
163-203 (Compagno, 1979)
57-65 (Etnier and Starnes, 1993)
49-53 (Wheeler, 1969)
notochord
61-62 (Smith, 1985)
28-29 (Etnier and Starnes, 1993)
28-29 (Etnier and Starnes, 1993)
30-32 (Etnier and Starnes, 1993)
62-65 (Smith, 1985)
63-72 (Smith, 1985)
notochord
71-79 (Springer and Friehofer, 1976)
53-56 (Wheeler, 1969)
57-60 (Wheeler, 1969)
30-31 (Wheeler, 1969)
135-212 (Compagno, 1979)
129-150 (Compagno, 1979)
Amphibians
Bufo americanus
Rana catesbeiana
Rana clamitans
Rana septentrionalis
Siren intermedia
Xenopus laevis
i. /, v (?)
/. /, v (?)
i, /, v (?)
/. /, v (?)
p (ind-lat)
i. v (?)
Wassersug and Hoff, 1985
Wassersug and Hoff, 1985
Wassersug and Hoff, 1985
Wassersug and Hoff, 1985
Gillis, unpublished
Hoff and Wassersug, 1986
notochord
notochord
notochord
notochord
97, Gillis, unpublished
notochord
Reptiles
Elaphe guttata
Nerodia fasciata
i, p (half, ind-lat)
i, p (half, ind-lat)
Jayne, 1985
Jayne, 1985
275-297 (Jayne, 1985)
185-203 (Jayne, 1985)
i, p (half; ind-mid)
i, p. v (half; ind-lat)
i, / (half; ind-lat)
i, p (half, ind-lat)
p, I (half, ind-lat)
* Sources of biological variation are abbreviated as follows: " i " is individual; "/" is length; "p" is axial position; "v" is swimming velocity.
Method of wavelength measurement is abbreviated as follows: " c " is from wave speed; "ind-lat" is from internodal distance when tailtip is at the lateral-most
position; "ind-mid" is from internodal distance when tailtip is at the mid-point of its excursion; "half" and "whole" are measures from twice the half-wavelength
either anterior or posterior half-wave) or the whole wavelength (anterior and posterior half-waves together), respectively. The " ? " symbol means that the method
could not be ascertained.
O
0
686
J. H. LONG, JR. AND K. S. NIPPER
grouped under the heading "vertebral columns" in Figure 2.
The most striking result of this analysis
is that propulsive wavelength, in fish, amphibians, and reptiles, varies tremendously
within a species (Fig. 2). As a result, distinctions of functional groups based on anguilliform (A. < 1 L) and carangiform (A. >
1 L) swimming mode seem inappropriate.
While some may view this result with trepidation, it raises the exciting possibility that
the biological sources of variation—axial
position, speed, and size—are related to
changes in the stiffness of the body. For
example, Blight (1977) predicted that
changes in the propulsive waveform with
axial position might be caused, in part, by
changes in the stiffness of the body relative
to the local fluid forces. An axial gradient
has been measured in the passive structures
of fish: relative axial flexibility varies in
different fish bodies (Aleev, 1969), angular
stiffness of the intervertebral joints increases caudally in blue marlin, Makaira nigricans (Long, 1992), and the mechanical
properties of the notochord of white sturgeon, Acipenser transmontanus, vary with
axial position (Long, 1995). J. Webb (1973)
predicted that the body stiffness of swimming amphioxus, Branchiostoma lanceolatum, is actively modulated by paramyosin
contraction in the notochord.
Several interesting results emerge for the
swimming amphibians and reptiles (Fig. 2).
First, the swimming snakes, Nerodiafasciata and Elaphe guttata, have a range of propulsive wavelengths that are similar to that
of swimming fish (Jayne, 1985). Second,
the tadpoles of the frog Xenopus laevis possess the widest range of propulsive wavelengths measured in any undulatory swimmer; as swimming speed increases, the tadpoles progressively recruit more of their tail
(Hoff and Wassersug, 1986). Wassersug
(1989) has noted that some tadpoles have
been observed holding their hind limbs
against the base of their tail during swimming, a behavior which may stiffen the tail
and thus expand the range of propulsive
wavelengths.
When the propulsive wavelength in fish
is regressed onto vertebral number, a significant negative slope is detected only for
the smallest wavelengths. The smallest propulsive wavelength is predicted by the following equations:
y = 0.7979 - 0.001 Ox,
(H)
where y is the smallest propulsive wavelength and x is the fewest number of vertebrae in a given species (r2 = 0.211, P =
0.024, n = 20). When x is the greatest number of vertebrae, the regression is still significant but it explains less of the variation
in propulsive wavelength (r2 = 0.168, P =
0.041, n = 20):
y = 0.7941 - 0.0009x.
(12)
2
While only 21% (r = 0.21) of the variation
in propulsive wavelength is explained by
the variation in number of vertebrae (Eq.
11), the relationship is consistent with the
hypothesis that the two variables are mechanically linked. The mechanical hypothesis that remains to be tested is that body
stiffness varies in proportion to the number
of vertebrae. A portion of the remaining intraspecific variation (79%) may be caused
by uncorrelated structural factors, different
hydrodynamic interactions of the body with
the water, or a systematic bias in the detection of smaller wavelengths in species with
greater number of vertebrae. The relationship between vertebral number and body
waveform may extend beyond steady
swimming; in four species of fast-starting
fish, E. L. Brainerd and S. Petak (in preparation) found that maximal body curvature
increases with increasing number of vertebrae.
The largest propulsive wavelength in
each species does not vary significantly
with changes in vertebral number. It is highly variable (±0.359, 2 SD) about a mean
value of 1.02 L. That the mean value falls
close to the demarcation between anguilliform and carangiform modes again highlights the danger in using these terms to distinguish functional groups based solely on
propulsive wavelength.
Several general conclusions can be
drawn from these data. First, propulsive
wavelength is not constant for a given species; any conclusions or theories based on
the assumption of a species-specific wavelength or a wavelength invariant with axial
BODY STIFFNESS IN UNDULATING VERTEBRATES
position, size, or swimming speed must be
viewed with caution. The functionally
meaningful distinction between undulatory
swimmers may be hydrodynamic in nature—some fish may produce continuous
forward thrust with all of their axial body
elements while others may produce pulsatile thrust with only their caudal fins (Aleev,
1969; Lighthill, 1975; Wardle et al, 1995).
Second, the smallest propulsive wavelength
in a species decreases significantly with increasing number of vertebrae, suggesting
that this structural feature, and/or a correlated feature such as body stiffness, number
of muscle segments, or number of spinal
nerves, places a limit on the shape of the
axial wave form in fish.
687
ulate whole myotomal muscle as it is undergoing cyclic strain within the body; the
body is held in a dynamic bending machine
that applies the strain and measures the resulting bending moment (Nm). This technique combines ideas from Josephson's
(1985) single-fiber work loop technique and
Johnsrude and Webb's (1985; see also
Webb and Johnsrude, 1988) half-myotome
stimulation experiments. The WBWL technique has the advantage over the single-fiber work loop technique of allowing the
working muscle to interact with the serial
and parallel elastic components of the body
as it would during swimming.
METHODS
A largemouth bass of 30 cm total length
(L) was obtained from Candlewood Lake,
Connecticut, transported to Vassar College,
The strongest evidence we have for fish and held in a 110 L tank at 20°C for four
muscles actively stiffening the body during weeks. For the experiment, the individual
swimming comes from studies on isolated was killed with an overdose of tricaine
muscle fibers generating negative work (see (Finquel, Argent Chemical Company) and
Introduction). This function of muscle is immediately mounted in the dynamic bendfeasible in fishes because it is employed by ing machine (Fig. 3), modified from Long
other vertebrate species. The trunks of ele- (1992). The fish was gripped by two pairs
phants can adjust their apparent stiffness to of concave surfaces to which sandpaper had
match the external load they bear (Wilson been glued. Each surface was pressed
et al, 1991). Trunks are muscular hydros- against the left and right sides of the fish
tats, systems in which antagonistic muscles by an array of screws attached to a frame,
operate to provide structural support and which in turn was rigidly attached to the
stiffness (Kier and Smith, 1985). Evidence machine. The position of the edge of the
for dynamic modulation of stiffness comes grips on the fish was at 0.39 L anteriorly
from hopping humans, who behave like and 0.47 L posteriorly, leaving 2.5 cm and
simple mass-spring systems above a pre- approximately 8 intervertebral joints free to
ferred frequency, increasing vertical stiff- bend. This region was chosen because preness with increasing speed and hopping fre- vious experiments had shown that the cauquency (Farley et al., 1991; for other tet- dal region was insufficiently stiff to permit
rapods see McMahon, 1985). Furthermore, accurate measurement of the passive stiffness with our apparatus. Five scales were
muscles acting to stiffen limb joints are cru- then removed epaxially on the left side and
cial to the coordination of standing and a platinum subdermal needle electrode
walking in paraplegic humans whose leg (Grass Instruments, 1 cm active tip) was inmuscles are stimulated by external sources serted. This entire procedure was completed
(Bajd et al, 1995; Fung and Barbeau, 1994; in ten minutes. Previous experiments indiMulder et al, 1992; Stein and Capaday, cated that the muscle maintains 90% of its
1988).
initial capacity to generate work for up to
Because techniques are lacking to direct- one hour after death.
ly measure muscle force in a swimming
With muscle temperature maintained at
fish, we have developed a means to measure force and stiffness in a dynamically 20°C, the bass was sinusoidally bent at a
bending fish, a method we call whole-body frequency of 2.6 Hz and a total angular diswork loops (WBWL). We electrically stim- placement of ±4.6°. This frequency was
CAN FISH USE THEIR MUSCLES TO
INCREASE BODY STIFFNESS?
688
J. H. LONG, JR. AND K. S. NIPPER
A.
stationary
grip
stimulus
site
oscillating
- oscillating
platform
RVDT—>
oscillating
platform
left side stimulated
(negative work)
C.
left side stimulated
(positive work)
FIG. 3. The whole-body work loop method to measure dynamic changes in body stiffness. A. Left lateral
diagram of largemouth bass mounted in the dynamic bending machine. As a motor (not figured) drives the
oscillating platform through a sinusoidal motion, the strain gauge and rotary variable differential transducer
(RVDT) measure the resulting bending moment (Nm) and angular displacement (rad), respectively. To determine
the ability of the muscle to alter body stiffness, the muscle on the left side is electrically stimulated at the
stimulus site during bending. B-C. Dorsal view of set up showing lateral placement of bending grips and lateral
bending of a small section of the body. B. Muscle is stimulated while lengthening to produce negative work.
C. Muscle is stimulated while shortening to produce positive work.
chosen because it represented a physiological value for steady swimming of bass
(Johnson et al, 1994; Jayne and Lauder,
1995a); the angular displacement, which is
±0.6° per joint, is also within the physiological range measured in rostral positions
in bass during steady swimming (Jayne and
Lauder, 1995a). Muscle was stimulated at
supramaximal voltage (20 V) for a duration
of 35 msec (Grass Instruments stimulator).
In preliminary experiments, where muscle
was stimulated and emg signals were recorded through an array of six electrodes in
the white muscle, we estimated that all of
one side of the myotomal muscle mass was
stimulated under these conditions. Since we
were stimulating both red and white fibers,
we chose a short stimulus duration of 35
689
BODY STIFFNESS IN UNDULATING VERTEBRATES
msec (9% of cycle) to approximate that
measured during swimming behaviors in
which white muscle was recruited (Johnson
et al., 1994; Jayne and Lauder, 1995c). To
cause the muscle to generate both negative
and positive work, the phase of the stimulus
with respect to body bending was varied
from 50° to 220°, where 0° and 180° are
phases for which the side of the body being
stimulated is bent maximally convex and
concave, respectively.
Voltage signals from the stimulator, angular strain transducer (Schaevitz, model
R30D), and the bending moment transducer
(Omega, model DMD 520 bridge amplifier
and a 120 Q. two gauge bridge) were simultaneously recorded at sample rates of 12
kHz per channel using an analog-to-digital
converter (National Instruments, model
NB-MIO-16L) in a microcomputer (Apple
Corporation, Macintosh Ilfx). Ten experiments of different stimulus phase were run.
During each, 20 bending cycles were recorded: passive baseline (no stimulus, 5 cycles), stimulus (5 cycles with stimulus each
separated by a recovery cycle), and passive
baseline (no stimulus, 5 cycles). We took
the average of the baseline before and after
stimulation as the baseline for that trial.
The mechanical work (J) required to
bend the fish over a complete tailbeat cycle
was measured by finding the area circumscribed by the graph of bending moment
over angular displacement (Fig. 4). The raw
signals for left-to-right and right-to-left
strain were fit with separate third-order
polynomial equations. The difference between the areas under these curves, as determined by integration, yielded the work
to bend the fish. Positive work performed
by the muscles was measured as a reduction
in the work required to bend the fish over
one cycle; negative work was measured as
an increase in the work required to bend the
fish over one cycle. Angular stiffness (k')
of the body was measured as the overall
slope of the work loop, yielding an average
stiffness. Relative measures of work and
stiffness (% change) were calculated for
each trial relative to the average passive
baseline values in the trial. This method
was calibrated using a vinyl model of a
4
z
0
-
/
J
£f
Jr
stimulus on
stimulus off
-4
angular displacement (°)
FIG. 4. Typical work loop from a largemouth bass.
The area circumscribed by the curves is the work (J)
required to bend the body through one complete cycle.
Each curve (solid line) is generated from a third-order
polynomial regression of the bending moment (Nm)
onto the angular displacement (in degrees here; degrees converted to radians for calculations in text). The
inset (upper left corner) shows that time is parameterized in the work loop in a clockwise direction. The
onset and offset of the electrical stimulus to the muscle
is indicated; note that, in this case, the stimulated muscle reduces the work required to bend the body, as
indicated by the reduction in area in that region of the
loop. The muscle is thus performing positive mechanical work.
largemouth bass of known flexural stiffness
(for fabrication see McHenry et al, 1995).
RESULTS
The results presented here are from ten
experiments on a single bass. They are given for the purpose of demonstrating the feasibility of the whole-body work loop method and of providing a qualitative answer to
the question of whether fish bodies are capable of modulating body stiffness dynamically. Quantitative conclusions drawn from
these results should be done so with extreme caution. A complete study with statistically meaningful sample sizes will be
published elsewhere.
The mean passive baseline work to bend
the bass was 0.105 ± 0.007 J (1 SE, n =
10). Of the ten trials, the largest decrease
in the work to bend the fish (positive muscle work) was 0.022 J; the largest increase
in the work to bend the fish (negative mus-
690
J. H. LONG, JR. AND K. S. NIPPER
cle work) was 0.014 J. From the largest
positive muscle work value we calculated
the maximum power per kg of muscle mass
as follows: maximum net power output per
cycle was the product of the maximum net
muscle work and the cycle frequency (Altringham et al., 1993). That calculation
yielded a maximum positive power of
0.057 W (0.022 J times 2.6 Hz). To estimate
the mass of muscle stimulated, we used the
proportion of the body bent (2.5 cm divided
by 30 cm) times one half (the left half) the
total body mass (0.171 kg). Since this assumes an even distribution of mass along
the body, the value of 0.01425 kg is most
likely an underestimate of the mass in the
robust precaudal region; at the same time
0.01425 is probably an overestimate of the
muscle, since it neglects the contribution of
the non-muscle tissues and organs to the
mass. The maximum positive muscle power
per mass of muscle thus calculated is 4.0
W kg"1 (0.057 W divided by 0.01425 kg).
The maximum negative muscle power per
mass of muscle calculated in the same manner is 2.6 W kg"1. For comparison, the
maximum positive power output per mass
of muscle of white muscle in saithe is 63
W kg"1, as determined from single-fiber
work loop experiments (Altringham et al.,
1993). The fact that our maximal power
values are an order of magnitude lower in
bass, relative to those in saithe, suggests
that we did not acheive supramaximal stimulation of the muscle throughout the section
being bent. Thus the use of more stimulating electrodes, positioned throughout the
muscle, may elicit much greater positive
and negative muscle work and hence greater changes in body stiffness than those measured in this experiment.
The mean baseline angular stiffness of
the 2.5 cm section of the bass was 42.7 ±
0.57 N m rad"1 (1 SE, n = 10). Of the ten
trials, the largest increase in stiffness was
to 44.6 N m rad"1 (change of 3.1% relative
to baseline of 41.6 N m rad"1 for that trial);
the largest decrease in stiffness was to 39.4
N m r a d 1 (change of 5.2% relative to baseline of 43.2 N m rad"1 for that trial). When
the values of angular stiffness (% change
from baseline) were regressed against those
of the relative muscle work (% change), a
2 -
-
0
•3
3
-2 h
00
§
-4 -
1
-20
-10
10
20
30
relative muscle work (% baseline)
FIG. 5. Dynamic changes in body stiffness of largemouth bass during sinusoidal bending. The angular
stiffness (N m rad"') of the body changes, relative to
the passive baseline stiffness, as a function of the relative work done by the muscles, as shown by the significant regression line (see Eq. 13). See text for details
of experimental protocol and results.
significant relationship was detected (r2 =
0.62; P = 0.007; n = 10):
y = 0.147 - 0.147x,
(12)
where y is the relative angular stiffness and
x is the relative muscle work (Fig. 5).
CONCLUSIONS
These results demonstrate that the muscles of a dynamically bending bass, left attached to the rest of the body, are capable
of altering the body's angular stiffness. Increased body stiffness is generated by negative muscle work; decreased body stiffness
is generated by positive muscle work (Fig.
5, Eq. 7). This whole-body work loop method provides a means to directly measure
muscle work and dynamic body stiffness in
situ. This method has the advantage over in
vitro single-fiber work loop techniques in
that the muscles are left attached to connective tissues; these structural elements
may alter, because of their compliance,
muscle function in a way that cannot be discerned in vitro (see McMahon, 1985).
It should be noted that these WBWL experiments are not completely realistic for
BODY STIFFNESS IN UNDULATING VERTEBRATES
two reasons. First, the bending moment is
applied externally by the bending machine;
in swimming fish the bending force would
be applied internally by muscles and by
previously loaded elastic tissues (see Wainwright, 1983; Pabst, 1996). Given the spatial and temporal complexity of muscle activity, contraction, and axial motion (for review see Jayne and Lauder, 1996), it is not
clear how the internal loading of the system
would alter the results or how the WBWL
technique, in its current form, could be
modified to mimic that complexity. Second,
the WBWL method is limited in that red
and white muscle, and separate myomeres,
cannot be stimulated independently. It is
possible that this short-coming may be
overcome by direct stimulation of the motor
nerves or descending interneurons (see Fetcho, 1987; Fetcho and Faber, 1988). Once
a technique for separate stimulation is
achieved, a greater range of physiological
conditions can be examined. Timing, duration, intensity (number of motor units), and
region activated are all variables of muscle
function that are likely to affect the magnitude of stiffness modulation (Hasan et al,
1985).
In addition to the muscles generating
negative work, other muscular and nonmuscular factors may contribute to variations in body stiffness seen within and between individuals and species. The contractile properties of fish muscles can be altered
by conditioning, acclimation temperature,
and adaptation temperature (Ball and Johnston, 1996; Guderley and Johnston, 1996).
The bending stiffness of the skeletal structures of fish, such as skin (Wainwright et
al, 1978; Long et al, 1996) and axial skeleton (Long, 1992, 1995), varies with
changes in strain and rate of strain (Eq. 4).
Furthermore, small changes in the crosssectional shape of the body, brought on by
weight gain/loss or body taper, will alter the
flexural stiffness (Eq. 3).
From the context of dynamically bending
beams, this paper proposes that the ability
of fish to actively control their body stiffness as they change swimming speed is
central to their efficient functioning. While
single-fiber and whole-body work loop
studies may demonstrate that muscle is ca-
691
pable of generating negative work and
hence changes in body stiffness, tests of the
variable stiffness hypothesis await our ability to directly measure the timing, magnitude, and orientation of muscle force in
live, swimming fish (Wainwright, 1983).
ACKNOWLEDGMENTS
Many of the ideas in this paper have been
strongly influenced by discussions and collaborations, for which we are grateful, with
Nick Boetticher, Hugh Crenshaw, Olaf Ellers, Ted Goslow, Dave Hall, Matt Healy,
Bruce Jayne, Matt McHenry, Ann Moore,
Ann Pabst, Charles Pell, Bart Shepherd,
Steve Vogel, Steve Wainwright, and Thelma Williams. John Gosline was instrumental in the redesigning of our dynamic bending machine. Without the help of Donald
Caggiano, a professional fisherman, it is
likely that we would still be fishing. Ted
Grand and Leathern Mehaffey assisted in
the development of the whole-body work
loop technique. We are indebted to our colleagues who allowed us to cite their soonto-be-published data: Elizabeth Brainerd
and Sheila Patek at University of Massachusetts (fast start and body curvature in
fish); Gary Gillis at University of California, Irvine (wavelengths of Siren intermedia and Anguilla rostrata), and William
Shepherd III at Vassar College (wavelengths of Pholidichthys leucoteania and
larval and adult Petromyzon marinus). Bob
Suter, Oren Rosenberg, and members of the
Biomechanics Advanced Research Kitchen
at Vassar College helped improve the analysis of vertebral number and propulsive
wavelength. We also thank Wyatt Korff,
Chuck Pell, Bart Shepherd, Steve Wainwright, and two anonymous reviewers for
comments on the manuscript. This work
was made possible by a grant to JHL from
the Office of Naval Research (grant number
N00014-93-1-0594).
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