Download Mechanical Properties of Pedal Mucus and Their Consequences for

AMER. ZOOL., 24:23-36 (1984)
Mechanical Properties of Pedal Mucus and Their Consequences for
Gastropod Structure and Performance1
MARK W. DENNY
Biological Sciences Department, Stanford University,
Hopkins Marine Station, Pacific Grove, California 93950
SYNOPSIS. This study examines the possiblility that the physical properties of pedal mucus
limit the size and speed of gastropods. At small deformations hydrated pedal mucus is a
viscoelastic solid. At large deformations the mucus yields and becomes a viscous liquid,
the yield strength increasing as the rate of deformation increases. The mucus can 'heal'
if left undeformed, its strength increasing with time. When dehydrated the mucus strength
and stiffness increase substantially. These properties can be used to calculate the maximum
speed of crawling and the maximum size for gastropods. In all the cases examined the
predicted maxima are near those observed in nature, and it seems likely that pedal mucus
indeed places limits on gastropod structure and performance.
INTRODUCTION
MATERIAL PROPERTIES
Many of the basic questions of biology
concern the limits to the structure and performance of organisms. For example it is
often wondered why an organism does not
grow larger than it does, or why it cannot
move faster. For any particular limitation
it is usually possible for a number of factors
to operate simultaneously, and it is generally difficult to decide which has been the
critical limiting mechanism. The biomechanical approach to biology can assist in
this problem by defining the mechanical
limits to structure and performance, and
in many cases an argument can be made
that these mechanical limits have been paramount in the evolution of structure (for
a review see Alexander, 1968, 1971, 1982;
Pedley, 1977). To date, however, only a
few biological materials have been studied
in sufficient detail for these sorts of analyses to be possible. In general these materials (bone, shell, protein rubbers, silks) are
those that most closely match various manmade construction materials (concrete,
ceramics, neoprene, nylon); and a wealth
of engineering data is thus available for
comparison and guidance. The purpose of
this paper is to explore the possible ways
in which mucus, a very unusual sort of
material, places mechanical limits on the
structure and performance of gastropods.
The foot of a gastropod is coated with a
thin layer of mucus which is used for a
variety of functions including adhesion,
lubrication, predator repulsion, and mating (Denny, 1983). This pedal mucus is'
secreted as a gel consisting of approximately 91-98% (weight/weight) water
combined with a small amount (2-9%) of
a high molecular weight glycoprotein
(Grenon and Walker, 1980; Denny, 1983).
As one might expect from this composition
pedal mucus behaves as a slightly elastic
liquid rather than as the slightly viscous
solids such as collagen and bone with which
we are more familiar. Because of the primarily viscous, and therefore time dependent, nature of mucus, its mechanical properties cannot be described simply; strength,
stiffness, and extensibility all depend on
how fast the material is deformed and even
on the nature of previous deformations (see
Wainwright^a/., 1976; AklomsetaL, 1972;
Dorrington, 1981). Further, though
secreted in an initially highly hydrated state,
these mucins as actually employed occur at
a very wide range of hydrations, from the
pedal mucus of the terrestrial slug Ariolimax columbianus (approximately 98% water,
Denny and Gosline [1980]), to the mucus
used by littorine snails to glue their shells
to the substratum (<5% water, personal
observation).
1
From the Symposium on Biomechanics presented
at the Annual Meeting of the American Society of
Zoologists, 27-30 December 1982, at Louisville, Kentucky.
The hydration and deformation regimes
under which pedal mucins operate in nature
fall into three categories: 1) mucus at nearfull hydration and small deformations, 2)
23
24
MARK W. DENNY
mucus at near-full hydration and large
deformations, 3) mucus that is dry or nearly
dry, at small deformations. Denny (1983)
has reviewed the existing knowledge of the
chemistry and mechanical properties of
gastropod pedal mucus. This information
is not extensive, but the properties of A.
columbianus pedal mucus have been examined under each of these conditions and
thus provide a model of the relevant physical properties of mucus.
The terms used here to describe mucus
properties are standard engineering terms
designed to specify mechanical properties
in a manner that is independent of the
dimensions of the sample, and is therefore
a characteristic of the material from which
the sample is made. The mucus was tested
in shear by being deformed between parallel plates. The force of deformation is
expressed as the stress, a, the force divided
by the area over which it is applied; the
strength of a sample is the stress that causes
it to yield (flow as a viscous liquid). Deformation is expressed as the strain, y, the
distance one plate moves relative to the
other divided by the thickness of the sample. Strain rate, y, is strain per second.The
shear modulus, a measure of stiffness, is the
stress divided by the strain. A full explanation of these terms can be found in any
text on materials, e.g., Wainwright et al.
(1976), Gordon (1978), Aklonis et al.
(1972).
HYDRATED
DRY
2
3
4
5
LN TIME (S)
6
7
8
FIG. 1. The creep response of A. columbianus pedal
mucus. A. Hydrated mucus as utilized by a crawling
slug (replotted from data reported by Denny and Gosline [1980]). B. Mucus dried to 11.3% water; means
and standard errors shown. Regression line: J, =
6.51 x 10"9 In t + 3.32 x lO"8, r = 0.987. Note the
different scales for A and B.
and Walker (1980) for the pedal mucus of
the limpet, Patella vulgata. As the frequency of deformation increases (period
decreases) the elastic nature of the material
becomes increasingly more apparent, until
Fully hydrated mucus, small strains
at frequencies on the order of 10 Hz the
The dynamic properties of A. columbi- mucus behaves primarily as a rubbery solid.
anus pedal mucus at small strains (y < 0.1)
and at the hydration normally found under Fully hydrated mucus, large strains
a moving slug (95-97% water) have been
The elastic character of pedal mucus is,
reported by Denny (1980a), Denny and
however,
present only at small strains. At
Gosline (1980), and Denny (1983). If the
a
strain
of
approximately 5-6 the material
mucus is strained sinusoidally with respect
abruptly
yields
and at further extensions
to time, at very low frequencies (<0.1 Hz,
i.e., over periods >10 sec) the mechanical behaves as a viscous liquid (Denny, 1980a,
behavior of the mucus is dominated by its 1983; Denny and Gosline, 1980). The yield
viscous properties. Indeed, over a long strength of the mucus is dependent on
enough period of time the material flows strain rate; the faster the deformation, the
like a liquid, and if subjected to a constant stronger the material. The dependence of
load, continues to increase its strain (creep) yield stress, o\, on strain rate, y, may be
without giving evidence of reaching an described by an exponential function; for
equilibrium strain (Fig. 1). These findings the typical sample shown in Figure 2
are similar to those reported by Grenon
ex, = 165-y0-492
(r = 0.977)
(1)
25
GASTROPOD STRUCTURE AND PERFORMANCE
CO
100
150
200
250
STRAIN RATE
FIG. 2. Yield and flow stress as a function of strain rate for a typical sample of A. columbianus pedal mucus.
Regression lines drawn according to Eqs. 1 and 2 in the text. Yield stress measured after 1 sec of heal time.
where stress is measured in Newtons/m 2 .
Similarly the flow stress, oy, the stress
resulting from the viscosity of the mucus
as it is strained at a given rate, increases as
a function of strain rate
(r= 0.977) (2)
af= 94.97°440
The yield strain does not vary measurably
with strain rate (Denny and Gosline, 1980).
The change in state of the mucus from
solid to liquid is a reversible process (Denny,
1980a, 1983; Denny and Gosline, 1980).
If the mucus is strained past its yield point
(thus becoming a fluid) and the strain rate
is then brought to zero, within a very short
time the gel network of the material has
"healed." Upon again being deformed the
material behaves as an elastic solid. The
yield stress of the material in this subsequent extension increases as the time that
the material is unstrained, the heal time,
increases. The results of a typical experiment showing this effect are presented in
Figure 3; again the relationship is an exponential one, and for this particular sample
(r = 0.936) (3)
where t is the heal time in sec and stress is
again in N/m 2 .
The results shown here are typical of the
= 347.
samples tested, but in these tests no effort
was made to ensure a precisely uniform
hydration among samples; consequently the
magnitudes of the scaling coefficients in
the above equations vary among samples.
The values presented here will be used primarily as a guide to the appropriate magnitudes, but it should be kept in mind that
they represent only approximations of the
actual values that might occur in any particular sample drawn from nature.
Dried mucus, small strains
The mechanical properties of pedal
mucus change drastically as the material
dries. Figure 4 is a plot of the shear modulus of A. columbianus pedal mucus as a
function of the hydration. Here mucus was
collected from the slug and air dried at
room temperature (19-21°C) before being
tested in an apparatus similar to that of
Denny and Gosline (1980). The stiffness
varies little at hydrations between 97 and
70%, values ranging from approximately
50-150 N/m 2 . As the water content
decreases below 70% the modulus rises
rapidly until at 10.5% (the lowest hydration obtained at room temperature) the
shear modulus is approximately 5 x 107.
The modulus of Littorina aspera mucus
26
MARK W. DENNY
I
100
50
150
200
HEAL TIME (S)
FIG. 3. Yield stress as a function of heal time for A. columbianus pedal mucus. Regression line drawn according
to Eq. 3 in the text. The strain rate used was approximately 26/sec.
dried to near 0% hydration by being sun
baked at 60°C (the condition in which this
mucus is used in nature) is approximately
9.8 x 108 N/m 2 . For this mucus the yield
stress was measured in tension, and has
been converted to the appropriate modulus by assuming a breaking strain of 0.035
and a Poisson ratio of 0.5 as explained by
Wainwright et al. (1976). The reason for
the rapid rise in stiffness accompanying
dehydration has not been specifically
examined, but it may be speculated that as
the water content decreases various sites
on the glycoprotein molecules that had
previously interacted with water begin to
bond to each other (for a discussion of this
concept see Vincent, 1982). Such bonding
initially serves to increase the number of
crosslinks between glycoprotein chains and
thereby increase the modulus (Ferry, 1980),
but at some hydration the degree of bonding is sufficient to limit the kinetic freedom
of the glycoprotein chains. At this critical
hydration (approximately 60-70%) the
mucus ceases to behave as a rubbery solid
and undergoes a process similar to the glass
transition seen in various man-made materials (see Ferry, 1980). Similar processes
have been shown to occur in elastin (Gosline, 1980) and resilin (Weis-Fogh, 1960).
Regardless of the precise mechanism,
dried mucus is a very different material
from hydrated mucus. Its yield strength is
approximately 108 N/m 2 as opposed to 103
N/m 2 for fully hydrated mucus, and its
breaking strain is approximately 0.035,
much lower than the 5-6 typical of
hydrated pedal mucus. Accompanying this
change in failure parameters is a change
in the long term solidity of the mucus. A
creep test performed on the mucus at
11.3% hydration (Fig. 1) shows that the
compliance of the material is 105 times
lower than that of hydrated mucus, and
consequently its creep rate at equal stress
would be 105 times slower. Even at this low
hydration the material does not reach an
equilibrium stress, though it is still possible
that it would do so at 0% hydration.
The mechanical properties described
here, though a summary of the information available, are far from being a complete description of pedal mucus. A full
description of the effects of hydration on
dynamic and failure properties awaits further research, and the effects of tempera-
GASTROPOD STRUCTURE AND PERFORMANCE
27
o
LLJ
I
05
3
I
99
98
95
80
50
90
% WATER
Fie. 4. The shear stiffness of A. columbianus pedal mucus increases as the water content decreases. Points
plotted represent individual tests (7 = 1 to 3/sec); line fitted by eye. The point at approximately 0% hydration
is for L aspera pedal mucus, calculated as described in the text.
ture on all of the above properties are
totally unknown. However, the information presented here is sufficient for present
purposes.
CONSEQUENCES OF MUCUS
MECHANICAL PROPERTIES
Two general questions will be addressed
in light of the properties just described: 1)
how fast can gastropods crawl using pedal
mucus, and 2) how big can gastropods grow
before they become limited by the properties of their pedal slime?
mucus. During locomotion certain portions of the foot (the waves and the rim) are
moved forward, while the remaining parts
of the foot (the interwaves) are stationary
relative to the ground. The waves and
interwaves on the central portion of the
foot divide the foot into a series of lateral
bands (Fig. 5), the waves of movement traveling forward from the posterior end of
the slug. Thus each point on the foot is
alternately moving and stationary. The
time for which a point on the foot is stationary between periods of movement is
t = USU,
How fast?
The general mechanism by which slugs
crawl has been described in detail by Denny
(1981). The animal is attached to the substratum by a thin (10-20 /im) layer of pedal
(4)
where Ls is the length of an interwave and
Su. is the speed at which the waves move
forward. For A. columbianus, Su, = 3.3 S
where S is the overall speed of the slug.
&
MM 4 W
MERNER-PFEIFFER LIBRARY
TENNESSEE \SESL£YAN COLLEGE
28
MARK W. DENNY
FIG. 5. A schematic representation of the ventral
surface of the foot of a slug crawling at speed 5.
mucus depends on two factors: how fast
the mucus is sheared and how long a period
it has had in which to heal. Both of these
factors may be related to the speed of the
animal.
The maximum heal time for this system
is the time spent stationary under an interwave. Thus, by reference to Eq. 3, Fs, the
maximum force that can be resisted by the
interwaves, is
(10)
Thus
= 0.3L,/S
(5)
For an individual adult A. columbianus the
interwave length is fixed, a typical value
being 3 x lfj-'m (Denny, 1981). As a point
on the foot begins to move forward (either
in the waves or the rim) the pedal mucus
beneath it is sheared. After a small initial
deformation (50-100 nm) this mucus yields
and with further movement is in its fluid
form. Consequently the force required to
move these portions of the foot, Fm, can be
described by an equation of the form
(6)
Auaf + Araf
(7)
AJCt(yw)° +
2
= AsKi(0.?,LsyS
(12)
where As is the area of the stationary portions of the foot. The maximum average
shear rate that mucus beneath an interwave can experience and not yield is that
where the material is strained just to its
yield strain in the time it is beneath an
interwave. Thus the maximum strain rate,
/ max* l^
yma, = y,nJt
(13)
where t is again the time spent in the interwave. By reference to Eq. 1 an alternate to
Eq. 12 for the force that can be resisted by
the interwaves is:
where Au. is the wave area (in m ), Ar is the
rim area, yu, is the strain rate caused by the
(14)
movement of the waves, and yr is that
(15)
= i4 J K s [7 M «/(0.3L I /5)]'
caused by the rim. Above and in what follows a, b, c and K with a subscript are empir(16)
ically determined constants. The coeffiThe overall yield stress is proportional
cients K have dimensions suitable to express
force in N {i.e., SI units). The shear rate in to the product of the two conflicting proeach situation is equal to the local speed of cesses described in Eqs. 12 and 16:
the foot divided by the thickness of the
F, oc As(y,naxY(O3Lsy-'S'-b
(17)
mucus layer, T. Thus:
b
= K4As(ymax)'(O.SLsy-'S'(18)
r m — /i u A j^Zo/ 1)
i /l^A. i\o / l )
yo)
The constant of proportionality, K4, can
be determined from empirical measurenoting that the average speed of the foot ments where the heal time, strain rate, yield
in a wave is twice that of the slug overall strain and yield stress are all measured.
Estimates of K4 from examples shown in
(Denny, 1981).
If the slug is to move, this force must be Figures 2 and 3 range from 144 to 345
resisted by the interwaves. The interwaves (with K4 having units appropriate to express
rest on mucus in its solid form, and the Fs in N); the larger value will be used here
maximum force they can resist and remain as giving the largest estimate of speed, and
stationary is one just less than that which therefore is a 'best case' estimate of how
would cause the mucus to yield. As shown fast slugs can crawl.
above (Eqs. 1, 3), the yield strength of pedal
The maximum speed is reached when
= S T - t f ,(2MB. + Ar)
(9)
GASTROPOD STRUCTURE AND PERFORMANCE
29
rate; but, as shown in Eq. 9, this force is
itself a function of the rate of movement.
a = 0.440
Thus the power associated with locomob = 0.214
tion increases in proportion to S1-44 and at
c = 0.492
some speed this power will equal the maxA, = 6 x 10-* m !
imum power output of the pedal muscuAw = 3.5 x 10-> m2
A, = 5.5 x 10-" m2
lature. Without appropriate measure2
K, = 94.9 kg m-' s°ments
of the characteristics of gastropod
1
J
K< = 345 kg iti" s'-»pedal
musculature
it is at present imposL, = 0.003 m
sible to evaluate these alternatives.
T = 2 x 10"5 m
7™» = 6
An examination of Eq. 20 provides some
insight into mechanisms by which a gastropod could control its maximum speed
the force exerted by the moving portion without altering the properties of its pedal
of the foot is just slightly less than that mucus. For example, a decrease in the
capable of being resisted by the stationary interwave length while maintaining the
interwave area results in an increase in
portion; i.e., Fm = Fs. Thus
maximum speed. This suggestion is tenKXS"T-{2"A». + A) =
A^iy^y
tatively born out in nature; Miller (1974)
b
t b
(19) reports that Tegula funebralis decreases the
•(0.3Li) -'S interwave length as it increases its speed,
and solving for speed
and slugs, with 10-17 short interwaves
/a+b c
/a+b c
S = [AJ{2"AW +
Ar)X - {KJKxy instead of 1-3 large ones, are among the
•{ymaxy/a+b-e(0.?>Lsy-<i°+b-<T'1"1+b-<
(20) faster gastropods. It should also be possible
Inserting the appropriate values (shown in to increase speed by increasing the thickTable 1) the maximum speed predicted for ness of the mucus layer, but any increase
A. columbianus is 0.6 mm/sec. The maxi- in thickness directly increases the amount
mum speeds actually observed ranged from of mucus expended in moving a given dis0.8 to 2.3 mm/sec, and speeds for snails tance. Denny (19806) has shown that this
in general range from 0.16 to 12.5 mm/ form of adhesive locomotion is very costly,
sec (Miller, 1974). Given the fact that the primarily due to the cost of mucus procoefficients used here are only approxi- duction; and the mechanism for increasing
mations, it is gratifying to see this close a speed may not be an efficient one. Animals
correspondence between prediction and may also increase their maximum speed by
measurement. A 24% increase in K4 or somehow decreasing the force of movedecrease in /C, would yield a predicted speed ment. A good example of this method is
of 2.3 mm/sec. The mechanical properties the abalone Haliotis rufescens which when
of pedal mucus are thus a likely candidate "running" from predatory starfish clearly
to be the factor which limits the speed at lifts the moving portions of the foot clear
which slugs, and perhaps gastropods in of the substratum, thus moving over a thick
general, can crawl; and this calculation cer- layer of water (viscosity 0.01 poise) rather
tainly provides an argument as to why snails than a thin layer of pedal mucus (viscosdo not rival gazelles and cheetahs for the ity > 50 poise). The consequence, however, is a decrease in adhesive tenacity and
terrestrial speed records.
the animal may thus be exposing itself to
There are, of course, alternative and/or being dislodged from the rock by hydroadditional explanations for the slow rate at dynamic forces.
which gastropods move. It is possible that
the pedal muscles have a slow intrinsic rate
of contraction. It is also possible that the How big?—Wave-swept organisms
animals are limited by the power necessary
Animals living in wave-swept environto move over mucus in its fluid form. This ments such as rocky intertidal shores are
power is equal to the rate of movement generally quite small. The largest animals
times the force required to move at that found intertidally on the Pacific Northwest
TABLE 1. Values used m calculating speed.
30
MARK W. DENNY
coast are starfish (Pisaster ochraceous) and
abalone {Haliotis spp.) which in exposed
areas do not exceed 20-30 cm in maximum
dimension (Ricketts et al, 1968; personal
observation). It has generally been assumed
that this small size is a result of some limit
imposed on the organisms by the force of
breaking waves, but the mechanism of this
limitation has not, to my knowledge, been
examined. The problem here is in finding
a force which increases appropriately rapidly with size. As a wave breaks on the shore,
water moves rapidly past the organisms
attached to the rocks. The force imposed
on an organism by the water's velocity is
in part the drag force, F/.
(21)
where p is the density of the water (approximately 1,025 kg/m 3 for seawater), Ap is
the projected area exposed to the flow
(measured in a plane perpendicular to the
direction of flow), U is the water velocity,
and Cd, the drag coefficient, is a function
primarily of the organism's shape. For a
full derivation of this equation see Vogel
(1981). An intertidal gastropod adheres to
the rock with the pedal mucus that underlies its basal surface. The drag force may
be expressed in terms of the stress, <jn,
applied to this pedal mucus by dividing both
sides of Eq. 21 by the area of the foot, Ab
Fd = 0.5pApCdlP
aa = Fd/Ab
= 0.5(Ap/Ab)CdlP
(22)
If the organism grows isometrically, however, the ratio of Ap/Ab is a constant, Nu
and
2
•jU
(23)
aa =
Thus the drag force on an organism of a
given shape is a function of water velocity,
but is not a function of size. If the drag
force were the only force operating we
would be at a loss to predict maximum size
on a mechanical basis. However, the drag
force is not the only hydrodynamic force
present in wave-swept environments. Water
in breaking waves moves not only with rapid
velocities, but also with rapid accelerations
and decelerations. On a gross level water
is accelerated as the wave surges up the
shore and then back down; on a finer scale
the turbulent nature of breaking waves
causes very rapid accelerations. It can be
shown (Batchelor, 1968; Daniel, 1982) that
in accelerational flow a force in addition
to that dependent on the water velocity is
placed on an object, this force is known as
the acceleration reaction, Fa. The total
force placed on a organism in accelerating
flow, Flol, is more accurately expressed as
p =f
1
tot
L
+ p
d
'
A
(24)
c
= 0.5pApCdlP
Cm)V dU/dt
(25)
where the first half of the right hand side
of the equation is the drag force as
described above, and the second half is the
acceleration reaction. The added mass coefficient, Cm, is related to the organism's
shape, dU/dt is the water's acceleration,
and V is the volume of water displaced by
the organism. This force can be expressed
in terms of the stress placed on the basal
adhesive by dividing both sides of Eq. 25
by Ab and noting that a volume divided by
an area is a length
<ra = F,JAb
= O.bpN&JU*
+ p(l + Cm)N2L dU/dt
(26)
where the term V/Ab is expressed as N2L
and L is the maximum length of the organism. Thus, because the acceleration reaction is proportional to volume rather than
area, the overall hydrodynamic stress
placed on an organism's basal adhesive by
an accelerating flow increases with size. If
the various constants and coefficients,
velocities and accelerations can be measured, the maximum applied stress can be
calculated and compared to the adhesive
strength and the maximum length can then
be predicted.
Very few measurements of these values
have been made, but I will take what values
are available and use them to arrive at a
rough prediction for the maximum size of
gastropods in a wave-swept environment.
In a series of experiments on the coast of
Washington, I was able to measure the
forces exerted on intertidal organisms
(Denny, 1982), and to use these values to
estimate maximum water velocity and
acceleration; these are approximately 20
m/sec and 500 m/sec 2 respectively. In
collaboration with Drs. Mimi Koehl and A.
GASTROPOD STRUCTURE AND PERFORMANCE
R. Palmer I measured the drag coefficient
for a variety of organisms (barnacles, snails,
limpets) and estimate the values for a limpet shaped organism (at 20 m/sec) as 0.4.
Dr. T. Daniel was kind enough to measure
the added mass coefficient for limpets; Cm
is approximately 0.7. The methods by
which these data were measured are
reported by Denny, Daniel and Koehl in a
paper submitted to Ecological Monographs. The shear tenacity of one species
of limpet {Patella vulgata) has been carefully measured by Grenon and Walker
(1982), an average value being 5.0 x 104
N/m 2 . Inserting these values into Eq. 26,
along with appropriate constants JV, and
iV2 (0.318 and 0.125 respectively, calculated assuming the organism to be a cone
with a height = x/i the diameter) the maximum size predicted for limpet shaped
organisms is 17.3 cm. Large limpets and
abalones range in size to maximally 25 cm,
thus the estimate calculated here for maximum size is quite close to that observed
in nature, and it seems likely that mechanical factors play a role in limiting the size
of wave-swept gastropods.
These calculations depend on the shear
strength of pedal mucus, and at present it
is not possible to explicity account for the
shear tenacity of gastropods on the basis
of the known properties of mucus. Certainly the measured value of 5 x 104 N /
m2 is much higher than the yield values
predicted by Eq. 2 for any reasonable strain
rate; a value of 5 x 104 N/m 2 would
require a strain rate of 1.5 x 106, well
above the rate of approximately 300
actually used in this determination (Grenon
and Walker, 1982). It seems likely, as suggested by Grenon and Walker (1982), that
much of the shear tenacity is due to the
friction between the shell and the substratum. This in turn depends on how forcefully the shell is pulled down onto the rock,
and is ultimately a function of how forcefully the pedal muscles contract and how
strong the pedal mucus is in tension. When
limpets are dislodged by forces directed
away from the rock (thus stressing the pedal
mucus in tension) the mucus fails when a
crack is propagated through it (Grenon and
Walker, 1982; personal observation). A
31
typical yield stress under these conditions
is 1-3 x 105 N/m 2 (Grenon and Walker,
1982; Branch and Marsh, 1978; Denny,
unpublished). If the size of the initial crack
can be estimated; in this case a crack being
any imperfection in the pedal mucus layer
under the foot, the equations of fracture
mechanics can be used to relate the properties of the pedal mucus to the adhesive
tenacity. For a full explanation of fracture
mechanics see Wainwright et al. (1976),
Gordon (1976), or Andrews (1981). An
attempt was made to measure the size of
potential cracks under limpets by allowing
limpets to attach to glass plates to which a
sparse coating of beach sand had been glued
with a transparent glue. The underside of
the foot of the limpets could thus be examined while they adhered to a rugose surface
similar to that in nature. The limpets were
found to be extraordinarily adept at molding their foot to the surface, but even the
best had numerous spots 10-100 /urn long
where the foot could not accurately follow
the surface contour. Using this range as a
rough estimate of crack size, 500 N/m 2 as
an estimate of the mucus's tensile modulus,
E (=3 times the shear modulus, assuming
a Poisson ratio of 0.5), and 105 N/m 2 as
the yield strength the work of fracture, Wo,
can be calculated according to Gordon
(1976)
Wo = ire*Lt/2E
= irl0'°(10- 4 to 10- 5 )/l,000
= 314 to 3,142 J/m 2
(27)
(28)
(29)
These estimated values of the work of
fracture are lower than those obtained for
other biomaterials (i.e., wood = 104 (Gordon, 1976)), but are nonetheless respectable given the fact that mucus is greater
than 90% water. The surface energy of
water (0.07 J/m 2 at 20°C) is quite low, and
the energy required to break the covalent
bonds of the glycoprotein is not much
higher (approximately 2-9 J/m 2 using values cited by Andrews (1981) and assuming
the pedal mucus to be 9% glycoprotein
(Grenon and Walker, 1980)). Together
these energies can account for only a small
fraction of the work of fracture. The
remaining energy must be due to various
32
MARK W. DENNY
dissipative processes in the mucus (rounding of the crack tip, viscous flow) but there
are no data concerning these mechanisms
at the present time.
This analysis also assumes that snails grow
isometrically such that their adhesive area
increases as [weight]*7'. It would be possible,
at least in theory, for gastropods to change
their shape as they grew such that the foot
area kept pace with volume. Two simple
observations indicate that, in fact, this does
not happen. First consider an organism with
a circular aperture, and allow that organism to grow by increasing both in height
and radius. How must the shape of this
structure change as it grows so that the
aperture area (as for a limpet a measure of
maximum foot area) increases in direct
proportion to its volume. Expressing the
radius as a function, f(h), of shell height,
h, volume/aperture area is a constant if
FIG. 6. A cross-section through a hypothetical shell
with a circular basal area that increases in direct proportion to the volume contained within the shell.
Drawn in accordance with Eq. 31 in the text.
of sizes, an indication that the shells conform to a more or less constant shape.
Though this method is a very crude way
of looking at shape and is certain to miss
subtle variations, it is nonetheless a good
indication that within each broad class of
shell type gastropods have not adjusted shell
shape to increase foot area at larger sizes.
Littorine snails
P(h) dh \/TP(h) = C
(30)
This condition is met if
f(/t) =
(31)
A shell with this shape would be a very
unlikely looking structure (Fig. 6), and in
searching the extensive literature available
on the shell shapes of intertidal gastropods
I have found no organism even remotely
resembling this shape. Another approach
to this question consists of measuring the
foot area and weight (here used as an estimate of volume) for a variety of gastropods
of different maximum shell lengths. If
shape is a constant the foot loading (weight/
foot area) should increase linearly as a
function of length; if shape changes in a
manner so as to keep the foot loading constant, weight/foot area should be independent of length. Figure 7 shows a plot of
foot loading vs. shell length for a variety
of gastropods which inhabit the rocky
shores of Washington, Oregon, and California. The data have been grouped into
two classes based on shell shape: 1) snails
(i.e., gastropods with coiled shells, abalones
excluded), 2) limpets and abalones. Within
each class the foot loading increases
approximately linearly over a wide range
Snails which inhabit high intertidal rocks
are stressed not only by hydrodynamic
forces, but by heat and desiccation as well;
and a variety of methods have been evolved
to cope with these factors. Some snails hide
during the heat of the day and thus avoid
the issue altogether; others cool themselves evaporatively, trading water for a
livable temperature. Littorine snails, common inhabitants of upper intertidal rocks,
employ a method which, in contrast, con-
SHELL LENGTH (CM)
FIG. 7. Foot loading versus shell length for 17 species
of intertidal snails (squares) and 5 species of intertidal
limpets and abalone (circles). The data for snails were
taken from Miller (1972). Points plotted are for representative individuals. Regression lines: snails, foot
loading (g/cm2) = 0.766 shell length (cm) + 1.075,
r = 0.868; limpets and abalone, foot loading (g/cm2) =
0.255 shell length (cm) + 0.266, r = 0.989.
GASTROPOD STRUCTURE AND PERFORMANCE
33
fore assume the thickness to be size-independent. The area of glue holding the snail
in place is equal to the arc length times the
mucus thickness, and thus varies directly
as the shell length. A stress is placed on
the glue by the snail's weight, a maximum
value being
(32)
a = MY/I
where M is the moment due to the snail's
weight, Y is the maximum distance from
the point of contact between the shell and
FIG. 8. The mechanics of a littorine snail attached
the rock as shown in Figure 8, and / is the
to a vertical rock face. A. A side view. The shell con- second moment of area. For the derivation
tacts the rock only at the lowest point of the apertural
of this equation see Timoshenko and Gere
arc and is held in place by a thin arc-shaped layer of
(1972). The moment M that must be
mucus. The force due to the animal's mass acts through
the center of gravity, C.G. B. The length of the shell, resisted by the pedal glue is equal to the
L, is used as a measure of the animal's size. Y [=R(l
force exerted by the weight of the snail
— cos a/2)] is the maximum distance that the mucus (proportional to L3) times the distance from
extends away from the shell's point of contact with
the rock to the center of gravity (proporthe rock.
tional to L). Thus the moment increases as
L4. Y increases in proportion to L, and / in
proportion to L3. Thus
serves water. At the onset of a daytime low
tide these animals glue their shells to the
a,, oc L4L/L3 oc L2
(33)
rock using pedal mucus, pull their body
into the shell and tightly seal the opercuaa = 2,580L3 0.27L 0.27L
lum. In this manner the snail minimizes its
contact with the rock surface (which, in the
tropics can reach temperatures > 60°C),
1 - cos ^
thereby minimizing one aspect of its thermal load.
This behaviour however, relies on the
2(0.27L)3 2.5 X 10"
strength of pedal mucus, and thus potentially limits the size of these organisms. The
a
a
mechanics of the system are shown in Fig3
4 sin ure 8. The snail is attached to the surface
2
2
by pedal mucus along an arc of the apera . a
ture lip of approximately 1.8 radians. If
(34)
+ cos - sin the snail grows isometrically (and to all
appearances they do) the radius of the
aperture increases directly with the length The breaking strength of the pedal mucus
of the shell and consequently the arc length of L. aspera collected in Panama was deterincreases directly with shell length. The mined as follows. The snails were allowed
mucus layer is laid down by the foot as it to attach to unglazed ceramic tiles. The
is pulled into the shell. This glue layer is tiles were placed in the sun, and in response
extremely thin and therefore difficult to the snails quickly glued the edge of their
measure, however the thickness measured aperture to the tile. The mucus holding
for 10 specimens of L. aspera collected in the snail in place was allowed to dry in the
Panama did not vary measurably for snails sun for approximately 2 hr; the temperaranging in length from approximately 0.3 ture of the tile surface was typically 50to 1.0 cm, each sample having a thickness 60°C. Snails thus attached were brought
that varied between 2 and 3 ixm. I there- into the lab and were dislodged by being
34
MARK W. DENNY
struck by a small pendulum. A force trans- Maximum locomotory size
ducer mounted on the end of the penduThe method outlined above for calculum measured the force of dislodgement. lating the maximum crawling speed (Eqs.
The distance above the tile surface at which 4 to 20) also provides a means for calcuthe pendulum contacted the snail's shell lating the maximum practical size for a
was measured; this distance, when multi- crawling gastropod. The calculations made
plied by the force of dislodgement, gives a above assume that the only force placed on
measure of the moment required to break the interwaves is due to the forward motion
the mucus glue. The breaking stress was of the foot, a situation which is true only
then calculated using Eq. 32. The mean if the animal is crawling horizontally. If,
breaking strength measured in this man- however, the animal attempts to crawl verner was 1.03 x 10 8 N/m 2 . Using this value tically a force due to its weight is also placed
and Eq. 34 the maximum size for littorine on the interwaves. Equation 9 can thus be
snails is calculated to be 8.6 cm. This value rewritten as
is somewhat larger than the littorines found
in nature (Littorina zebra reaches 3.2 cm,
(36)
Fn = Fm
Fw
Keen [1971]), but given the approximate
(37)
Fn =
Ar)
nature of the calculation, is reasonably
close. It is also quite likely that in addition where Kb(L)d defines Fw the force due to
to withstanding a force due to their own the animal's weight. Measurements made
weight these snails must resist the fluid on A. columbianus and Limax maximus show
dynamic force due to the first wave which that Kb = 136.2 (in units to render Fu. in
washes over them as the tide rises; they do N) and d = 2.92 when L is the slug length
not extend the foot until they are wetted. in m. From data presented by Denny (1981)
This additional, but for the present uncal- the wave and rim areas may be expressed
culable, force would lower the predicted as functions of slug length; Au. = 0.35L2,
size.
Ar = 0.55L2. Thus
Stephen Garrity of the Smithsonian
0.055L2)
Fn =
292
Tropical Research Institute has shown that
+ 136.2L
(38)
littorines orient themselves so as to minimize their desiccation, presumably by min- For A. columbianus the number of waves
imizing the shell area exposed to the sun does not vary as the slug grows; conse(personal communication). This orienta- quently the interwave length is some fixed
tion is accomplished as the snails glue fraction, B, of the slug length. Equation 18
themselves in place, and must be main- can be rewritten as
tained through the 6-12 hr that they are
Fs = K4AXy,,,axy(0.5BLy-<S'-b (39)
emersed; thus requiring that the mucus
2
glue not creep excessively. The creep mea- For A. columbianus A, = 0.060L . The maxsurements made on dehydrated A. colum- imum length is reached when F, = Fa
bianus mucus (Fig. 1) can be coupled with
0.055L2)
the stress calculated in Eq. 34 and used to
estimate the creep occurring in the mucus
+ 136.2L292 - A:40
of a typical (1 cm length) littorine
(40)
•(0.SBL)b-'Sr-h = 0
2
7, = aj, = (1.41 x 1O'°L )
For fixed values of Ku K2, ymax, thickness,
•(3.85 x i o - ¥ " ° ) = 0.16 (35) B, and speed the maximum value of L may
For a mucus layer 0.1-0.2 mm in original be determined. Using the values shown in
length this amounts to less than 0.02 mm Table 2 the maximum length determined
of total creep in 12 hr, a negligible amount. by numerically solving Eq. 40 is shown in
Further, these values are probably over- Figure 9 as a function of crawling speed.
estimates since the littorine mucus is drier Actual maximum vertical crawling speeds
than the slug mucus used for these calcu- ranged from 0.8 to 2.3 mm/sec depending
on the individual slug, which by these callations.
35
GASTROPOD STRUCTURE AND PERFORMANCE
TABLE 2. Values used in calculating length.
a = 0.440
b = 0.214
c = 0.492
T = 2 x 10-s m
B = 0.033
K, = 72.1 kg m" 1 s""2 (see note)
K, = 345 kg m" 1 s'-*-2
7-« = 6
Note: 76% of K, (Table 1) to adjust horizontal
crawling speed to that observed in nature.
SPEED
culations would limit slugs to lengths of
6.5-8 cm. The maximum lengths observed
for A. columbianus and L. maximus were 812 cm. The prediction made here thus
somewhat underestimates the size limits for
slug crawling vertically, but are accurate
within a factor of 2. As for the calculations
of maximum speed, minor changes in the
coefficients used could bring this prediction more closely in line with the observed
size.
CONCLUSIONS
The examples cited here are intended as
an exploration of the ways in which the
mechanical properties of pedal mucus
potentially affect the limits to size and locomotory performance in gastropods. In each
case the predictions made using the best
available approximations of the various
constants and coefficients are close to (but
in no case precisely equal to) the values
actually observed in nature. I conclude
from this that pedal mucus may well be
important in determining the limitations
of gastropod structure and performance,
and suggest that this is an area ripe for
further study.
This conclusion is not as strong as might
be desired. One might be tempted to argue,
for instance, that because the predicted
maximum size of littorine snails is 2.5 times
that observed in nature that the strength
of the pedal glue is unimportant, and that
in this case some other factor is the critical
one which limits these organisms' size. But
this would be missing the point. The accuracy of the values on which these calculations are based does not at present permit
this sort of fine scale argument to be
(UU/S)
FIG. 9. Predicted maximum slug length as a function
of vertical crawling speed. A graphical representation
of the solutions to Eq. 40 in the text.
decided one way or the other. Indeed it
will be some time before our knowledge of
the complex mechanical properties of a
material such as pedal mucus will be sufficiently complete and precise to allow such
decisions to be made. For the present these
calculations should serve to demonstrate
that this sort of analysis is feasible for a
material such as pedal mucus, and that such
mechanical factors must be considered
along with the various biological alternatives (defense against predators, ontological constraints, etc.) when examining the
structure and function of gastropods. It will
be interesting to watch the interactions
among ecologists, physiologists, and biomechanics as their interpretations become
more refined concerning which selective
factors (if any) have been predominant in
the evolution of gastropod structure. At
present I am content with having provided
a mechanical argument for why slugs crawl
at a rate of mm/sec rather than m/sec and
why, abalone the size of beef cattle unfortunately do not exist.
ACKNOWLEDGMENTS
I thank Susan Denny for moral support
and M. Koehl and T. Daniel for stimulating
discussions of the ideas presented here.
Various portions of this work were supported by N.S.E.R.C. grant 67-6934 to J.
M. Gosline, ONR contract N00014-79-C0611 and NSF grant OCE 80-25578 to R.
T. Paine, and the Smithsonian Tropical
Research Institute.
36
MARK W. DENNY
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