AMER. ZOOL., 36:710-722 (1996)
Coupling Internal and External Mechanics to Predict Swimming Behavior:
A General Approach?1
CHRIS E. JORDAN
EPO Biology, Campus Box 334, University of Colorado,
Boulder, Colorado 80309-0334
SYNOPSIS.
The biological world is full of systems whose component
parts interact in a coupled non-linear fashion. As a result, studying any
component of the system in isolation may not be representative of its
natural behavior due to the coupling, and predicting the behavior of the
entire system as a function of variation in any one parameter may be
quite difficult due to the non-linear nature of the interactions. Swimming
with whole body undulations is just such a system. The component parts
of the swimming-system (muscle, skeleton, soft-tissue, and the surrounding fluid), are mechanically and physiologically coupled in a strongly nonlinear manner. Therefore, to predict the outcome of the entire system, i.e.,
swimming behavior, or to understand the role any one component plays
as a determinant of the outcome, a mechanistic approach encompassing
the form of the component's interactions is required. This approach is
essential for developing scaling arguments, or discussing the consequences of morphological and physiological variation on behavioral and
evolutionary "performance." Below I outline an example of this method:
a simplistic model of the mechanical interactions between the swimmingsystem components of a leech. The model is based on in vitro characterizations of these components and first principle descriptions of their interactions. Solving the model's governing equations generates swimming
behavior in the model organism. In addition, the model can predict the
behavior of the swimming-system's component parts, allowing calculations of swimming performance and parameter variation not possible with
other approaches.
INTRODUCTION
The "performance paradigm"
The "performance paradigm" (Arnold,
1983) is often posed as an underlying tenet
of functional morphology and physiological
ecology. This construct posits that some aspect of an organism's phenotype is a strong
determinant of the organism's "performance," which in turn is a determinant of
the organism's fitness. For example, the
height of fins along the body of a fish may
play a major role in this animal's ability to
1
execute a rapid-start, thereby affecting its
survival and ultimately its fitness. Such a
scenario is heuristically pleasing but is it
reasonable to expect that something as
complex as an organism's fitness is determined by a single performance measure
(with the possible exception of life-history
traits such as number of gametes produced),
or that "performance" is a simple function
of a single phenotypic character? In reality,
both fitness and "performance" are emergent properties of a series of interdependent
processes {e.g., inter- and intra-specific interactions, behavior, physiological and biochemical processes, and mechanical or
From the Symposium Aquatic Locomotion: New physical constraints), and as Such, variation
for Integrative and Comparative Biology, 27-30 De- ber of outcomes depending on the form of
cember 1995, at Washington, D.c.
these interdependencies.
710
A MODEL OF UNDULATORY SWIMMING
There are animal systems whose behavior is dominated by a single morphological
or physiological parameter (e.g., Wainwright, 1987, 1988), but these are the exceptions rather than the norm; in general
natural selection is weak, acting on many
characters to affect an evolutionary response (Grant and Grant, 1995). In the
cases where numerous characters have been
identified as determinants of organismal
performance their importance is often demonstrated in a correlative manner via multivariate analyses (e.g., Garland, 1984,
Lauder et al, 1986). Unfortunately, this approach cannot elucidate the form of the interactions between characters, that is, it cannot be used to assess mechanisms, and ultimately it is mechanisms that determine the
role any character plays as a determinant of
organismal performance.
Invoking the performance paradigm implies a mechanistic approach, requiring an
explicit accounting of these interactions to
understand fully the role each element plays
in determining organismal performance. At
every level of biology, each quantity of interest is the result of a series of interactions
at both lower and higher levels of organization. In general, the form of these interactions is not the subject of study, is not
usually known, and is often implied by the
method of data analysis (e.g., linear regressions). While the correlational approach
may be adequate in some cases, it can be
quite problematical when used in situations
where the role of these interactions is fundamental to the conclusions reached. Therefore, establishing a causal relationship between some aspect of an organism's phenotype and its fitness requires two things—
a performance that affects fitness, and
morphological or behavioral variation that
affects the performance. The former can be
demonstrated experimentally, and the latter
requires a mechanistic relationship between
the morphological/behavioral characters
and the performance.
While most biological systems must fit
the performance paradigm to some degree,
demonstrating the link between phenotypic
characters and fitness will be difficult in the
majority of cases. However, I think that
aquatic locomotion presents the best oppor-
711
tunity for closing the loop between morphology, behavior and evolutionary performance. Performance measures for swimming such as cost of transport, efficiency,
steady-state swimming speed, and fast-start
behavior are easily identified, and at least
some of these criteria have demonstrable
benefits for the swimming organism (e.g.,
fast-starts as predator avoidance and predatory strike; Watkins, 1996, Webb, 1976,
1984). Additionally, with the ever increasing understanding of the physics of swimming, general rules for the role morphological and behavioral characters play in determining swimming performance are within reach.
In this paper I present a mathematical
model that addresses the second of the two
criteria proposed above: predicting swimming performance by modeling the interactions between the components of the
swimming-system. By this approach, swimming behavior is an emergent property of
the physical and physiological properties of
muscle, soft tissue, skeletal elements, and
the fluid-body interactions. Because the
model is based explicitly on the interactions
between the components of the swimmingsystem, it predicts the effect morphological
and behavioral variation has on swimming
performance, and thus is one step towards
determining the adaptive value of a swimming organism's phenotype.
Why a model of animal swimming?
The issue of how the size and shape of
an aquatic organism affects its locomotory
performance has been at the center of much
of the research on the mechanics and physiology of animal locomotion. This issue, of
both physiological and ecological interest,
has been approached by both experimentalists and theoreticians. The theoretical studies that began with the classic works of
Gray and Hancock (1955), with later work
by Taylor (1951), Wu (1971) and Lighthill
(1975) have shown important consequences
of size variation to swimming performance
that result, in part, from fluid dynamic issues of size (i.e., Reynolds number) and, in
part, from size-dependent efficiency terms
in calculations of thrust. The experimental
work, pioneered by Webb (e.g., 1977, 1978)
CHRIS E. JORDAN
712
Mass
Hinge
Rigid Elements
B.
3 Dimensional Shape
Active and Passive
Tissue
D«1*1
1+1
713
A MODEL OF UNDULATORY SWIMMING
has been directed towards testing many of
the predictions and assumptions of the classic slender-body approaches.
More recently, several investigators have
shown that the internal dynamics of thrust
production may be extremely strong determinants of locomotor performance or efficiency (van Leeuwen et al., 1990, Rome et
al, 1993, Altringham and Johnston, 1990).
Much of this recent work arises in part from
Alexander's (1969) pioneering work on
muscle fiber trajectories in swimming fish.
He showed that power output and speed for
swimming are strongly correlated with fiber
architecture. Others have gone on to measure such parameters following Josephson's
work-loop method (Josephson, 1985) but
also have shown regional variation along
the length of the body (Rome et al., 1993).
All of these studies point to a rather significant limit to existing approaches to the
study of aquatic locomotion. In the hydrodynamic models (slender body theory and
coefficient techniques), there is no a priori
accounting of the internal dynamics of
force production. As such, when these theories are used to explore the effect of morphological variation they may assume kinematic conditions that violate known internal physiology (Jordan, 1992). Conversely,
generating rules for muscle force production in vitro neglects changes in animal
size, shape and behavior, i.e., neglects the
resistive and reactive load imposed on the
in vivo muscle by the body and fluid. In
reality, the swimming organism must simultaneously satisfy the force-velocity relationship for both the muscle and the fluiddynamic forces. Therefore, what is missing
in current models of aquatic locomotion is
the explicit simultaneous accounting of
both internal (muscle and soft tissue) forces
and external (fluid dynamic and inertial)
forces that is both mechanically and biologically realistic. The model I present below
attempts this accounting in a mechanically
simplistic manner. Nonetheless, the model
is able to generate swimming behaviors and
predict swimming performance for undulatory locomotion based on measured mechanical and physiological properties of the
leech swimming-system.
THEORY
A 2-dimensional model of an undulatory
swimmer
The model represents the body of an undulatory swimmer as a series of rigid elements joined by hinges (Fig. 1). Each element has mass, as well as a rigid crossmember to which I connect muscles and
soft tissue. I use jointed rigid elements as
an approximation of a continuously flexible
body to simplify the mathematical description, however, for a segmented animal like
a leech, each element can be thought of as
a body segment. This approach is similar to
that used to model lamprey swimming by
Bowtell and Williams (1991), and Ekeberg
(1993). Unlike lamprey, leeches do not posses a rigid axial skeleton, and so may appear to be a poor candidate for modeling in
this manner. However, a structure is required against which the longitudinal muscles act and to maintain body length during
swimming. In a leech this "structure" is the
hydrostatic skeleton formed by the bodywall, connective tissue, and hemocoel. As
a "first-cut" at modeling the swimming behavior of a leech I chose to represent the
hydrostatic skeleton with a rigid endoskel-
FIG. 1. Mechanical form of the model. A. The mathematical model represents a mechanical framework made
up of rigid elements with a point mass connected by hinges. Rigid cross-members of adjacent elements are
connected with a material that generates contractile forces and resists deformation. B. The model has the 3-d
shape of an elliptical cylinder for the fluid resistive and reactive terms. C. A free-body diagram showing the
forces and moments acting on a segment. FD and F[ are the fluid resistive and reactive forces (Hoerner, 1965),
where the n and t subscripting indicates normal and tangential components. Fmus and Fpas are the tissue forces,
active and passive, with separate accounting for the dorsal and ventral surfaces. The hinge forces, Fh, are due
to the interactions of adjacent segments. The moments, Mm, and Ma, are due to the fluid interactions with a
rotating body. Other forces also contribute to the rotational component of each element, for example Fmus X f
gives the moment induced by the muscle acting on the element's mass.
714
CHRIS E. JORDAN
eton. (The future incorporation of internal
fluid dynamics is discussed below, but see
Wilson et ah, 1995, for a full 3-dimensional
pressurized model of leech crawling behavior.)
The fundamental governing equations for
the model are the equations of motion for
a body segment free to translate and rotate
in two dimensions (dorso—ventral and anterior-posterior axes). Ignoring the third dimension assumes left-right symmetry and
saves computational time since including
the third dimension would greatly increase
the complexity of the system by roughly
doubling the number of state equations.
For each body segment the equations of
motion are as follows:
(1)
= m
d2
y
Fy = m —
c
(2)
(3)
which say that the sum of forces in the x
and y directions equal the object's mass (m)
times the acceleration in the x and y directions, (1) and (2) respectively. And that the
sum of the moments (Mz) is balanced by
the angular acceleration times I, the moment of inertia. Therefore, the position in
space of a body segment through time is
given by the solution to (1, 2, 3), a system
of 2nd order ordinary differential equations.
In the model, the animal's body is represented by n segments (generally 10), each
with mass m,, connected by hinges that
transmit force, but not moments {i.e., are
frictionless). The equations of motion are
satisfied for each segment, resulting in a
system of 3n 2nd order ordinary differential
equations in 3n unknowns, (x,, y,, <}),; position and angle in a global frame of reference) for each segment.
I incorporate the physiology and mechanics of each body segment through the
force terms in the above system of equations. In addition, I represent the hydrodynamic interactions of the fluid and the body
at the level of each segment. For example,
expanding the sum of forces in (1) and (2)
using vector notation (over bar) for simplicity, the balance of rectilinear forces for
the \'h segment becomes,
i x, = FIt + FDi + F,,
" F,
h +1
(4)
with F, and FDi the reactive and resistive
fluid forces respectively, FmuS| the active
muscle force component, FpaS| the passive
tissue force, and Fh. and F h+| the hinge forces from the adjacent elements (over-dotting
indicates temporal derivatives). Therefore,
to account for measurable parameters that
pertain to a swimming animal such as a
leech, I need to describe each component in
such a way that it can be included directly
in (4).
Parameterizing the model
Hydrodynamic forces.—Just as the motions of a free-body are described by a set
of governing equations, so too are the motions of the fluid surrounding the body. The
equations of motion for a fluid, the NavierStokes equations, describe the stress distribution within a fluid. However, there are
only a limited number of situations for
which explicit solutions to these equations
exist, and given the present state of computers, rather strong constraints on the cases
for which numerical solutions can be
reached. Classically, simplifying the Navier-Stokes equations can be done in one of
two manners: ignore inertial terms (low
Reynolds number or Stokes flow), or ignore
viscous terms (high Reynolds number or
ideal flow). These two assumptions aid
greatly in determining the fluid stress distribution, but in the case of a swimming
organism they imply either very small slow
swimmers, or large fast swimmers, respectively. Unfortunately, the size and speed
range of most swimming organisms places
them in the problematic intermediate Reynolds number range where these assumptions
do not apply, and no simple rules exist to
predict fluid stresses.
Body-fluid interactions are included in
the model with a sectional force coefficient
form of resistive and reactive slender body
theory (Gray and Hancock [1955], Lighthill
[1975], respectively; known as the Morri-
A MODEL OF UNDULATORY SWIMMING
son Equation, [Sarpkaya and Storm, 1985]).
The fluid forces are approximated by the
linear superposition of these local resistive
and reactive forces. This method assumes
that each section of the body sees undisturbed fluid, and that the force exerted on
each section is the sum of the fully-developed drag and added mass forces given the
instantaneous velocity and acceleration of
the section. The obvious limitations of this
approach are the neglected segment-segment interactions through the fluid, the assumption of fully-developed flow over each
segment, and the linear combination of resistive and reactive components. Nonetheless, in the circumstances where these assumptions are not violated (Williams,
1994a, b; Jordan, 1992), this approach predicts segmental fluid forces very well.
The drag force on a body moving
through a fluid of density pf is proportional
to the product of the object's size (Sp, projected area) and speed (U) squared,
FD = %CdPfSpU|U|
(5)
while the added mass force on a body accelerating through a fluid is proportional to
the product of the object's size and acceleration.
F, =
dU
f—
(6)
The constants of proportionality, Cd (the
drag coefficient) and a (the added mass coefficient) are explicit functions of the object's size and shape. In addition, the use of
sectional force coefficients allows me to include morphological variation along the
length of the organism since Sp, Cd, and a
in (5) and (6) can be defined differently for
each segment. Thus, knowing the object's
speed, acceleration and the two force coefficients I can calculate the fluid forces on
the body. The advantage of using this approximation lies in the ease of implementation and in the explicit shape dependence
of the force coefficients. The disadvantages
include the limitations outlined above and
the fact that force coefficients for arbitrary
shapes and arbitrary motions cannot be calculated or predicted, rather they must be experimentally determined. Therefore, the external mechanics are coupled to the internal
715
mechanics via the fluid force terms since it
is the muscles, skeletal and soft tissue mechanics that generate the motions that determine the fluid forces. However, the fluid
forces feed back on the internal mechanics
by resisting movement of the body through
the fluid; hence the need to account simultaneously for the internal and external forces.
Muscle mechanics.—The behavior of
striated muscle from a wide range of organisms can be represented by similarly
shaped force-velocity and length-tension relationships (Prosser, 1973). Therefore, it is
possible to generalize the force production
of striated muscle from most organisms as
a function of fiber-strain and contractionvelocity, with between-species differences
captured by the magnitude of the characteristic velocity (Vmax) and tension (To). I
model the active mechanical properties of
muscle as a material that generates force
proportional to its cross sectional area,
strain and strain-rate. The classical relationships describing the mechanical properties
of muscle are the hyperbolic force-velocity
curve (Hill equation [Hill, 1938]), and the
hump-shaped length-tension curve (Gordon
et al, 1966). However, more recent work
has called into question the validity of representing the dynamical properties of muscle with its 'static' isometric and isotonic
behavior (Altringham and Johnston, 1990).
Regardless of experimental protocol, the
non-linear behavior of muscle force production {i.e., not a simple function of length
or contraction velocity) can be included in
the model by mathematically characterizing
in vitro experimental data with contractile
force as a function of fiber strain and strainrate. I give the longitudinal muscle of the
virtual leech a constant length-tension relationship since very flat length-tension
curves have been measured in other annelids {Lumbricus, Tashiro, 1971; Haemopis,
Miller, 1975), but represent the dynamic behavior as the product of the Hill equation
and a term of the form:
Activation(Lel) = 1.0 - ViErfc(aVt^) (7)
or
Deactivation(t,.e|) = be"1"1
(8)
716
CHRIS E. JORDAN
Which say that muscle force output is a
function of strain rate (Hill equation) times
a constant that varies from 0 to 1 depending
on the time since the onset or end of muscle
excitation (t^ in (7) and (8), respectively).
For the sigmoidal rise in tension, I used a
complimentary error function (Erfc) with
an argument proportional to the square root
of time, and for the decay in tension, I used
a series of negative exponentials. The parameters were determined by non-linear
curve-fitting to force records from in vitro
muscle stimulation experiments on sections
of leech body wall muscle (The full form
of the parameterization is not important for
this discussion. Details are given in Jordan
[1994].)
Soft tissue mechanics.—Biological materials in general, and soft tissue in particular, exhibit non-linear visco-elastic behavior (Fung, 1993; Wainwright et al., 1982).
That is, they resist deformation both as a
function of strain (elastic component) and
strain-rate (viscous component), with the
added complexity that the coefficients of
proportionality, elastic and viscous moduli,
are themselves functions of strain and
strain-rate (non-linear aspects). To capture
this behavior I model the passive behavior
of soft tissue as a set of standard linear solids in parallel, but with elastic and viscous
coefficients as functions of strain. Standard
linear solids (SLS) are series viscous and
elastic elements in parallel with an elastic
element. My choice of strain dependent coefficients represents the length dependent
non-linearity, while using a number of
SLS's in parallel captures the rate dependence of the non-linearity. For example, a
SLS is described by the following set of
coupled equations:
Fpai, = A,(E0€,(t) + E,(e,(t)
1
M-i
M-l,
v
,(t)))
(9)
(10)
cients estimated. I parameterized the softtissue mechanics by curve-fitting the solution to a pair of SLSs in parallel to the
stress-strain curve generated by stress-relaxation of sections of leech body wall
(Jordan, 1994). As a result, the stress in the
soft tissue is known given its strain and
strain-rate.
Note that the strong non-linear behavior
of biological material precludes predicting
tissue stresses from simple scaling arguments. For example, geometrically scaling
up a swimming organism (constant morphological and kinematic parameters as
scaled to body length) results in unpredictable tissue stresses, i.e., stress does not
scale linearly with body length, and therefore the two situations would not be mechanically equivalent. It is only by explicitly accounting for soft-tissue and muscle
stresses as a function of strain and strainrate that the mechanical consequences of
changes in morphology and kinematics can
be elucidated.
Activation.—Solving the force balance at
this point would predict the behavior of a
very relaxed leech; what remains is to activate the muscles. To generate propulsive
forces, undulatory swimmers use a retrograde wave of body bending corresponding
to a retrograde wave of muscle excitation
(Kristan et al., 1974; Williams and Sigvart,
1992; van Leeuwen et al, 1990). To include this behavior in the model, I switch
the muscles between active and inactive
states with a square wave moving along the
body. The speed and period of the activation wave are taken from extracellular recordings of swimming-muscle motoneurons
in a leech (Jordan, 1994). While the activation—deactivation transition is sharp, the
change in muscle tension is gradual, as governed by (7) and (8). Thus, the activation
wave is meant to mimic the electrical excitation of the swimming-muscles, while
the force production of a given muscle segment is a complex function of the segment's
velocity and acceleration relative to the adjacent segments and the surrounding fluid.
Where FpaSi is the stress in the system, e^t),
ew (t), the strain of the whole system and the
viscous element, respectively. The three coefficients (Eo, E,, (JL,) are the two elastic
APPLICATIONS OF THE MODEL
moduli and the coefficient of viscosity. For
Using experimentally determined parama known stress-strain history, this system of
equations can be solved, and the coeffi- eters for the muscle, soft-tissue and exci-
A MODEL OF UNDULATORY SWIMMING
-OS
0
0.5
Body Length
-0.5
0
0.5
Body Length
FIG. 2. Swimming kinematics of a real and a virtual
leech. In both cases the animal is already swimming
at a steady speed when entering the frame from the
upper right. Each successive frame (0.033 seconds
apart) is displaced downward for clarity, and the body
length is normalized to 1.0. The arrow indicates the
direction of travel for the body, while the propulsive
wave moves rearward along the body. A. The swimming kinematics of an 8.0 cm leech. B, The swimming
kinematics predicted by the model for a 10.0 cm leech.
tation terms, and literature values for fluid
force coefficients, the model predicts the
swimming behavior of a whole body undulator (Fig. 2). A bending wave passes
along the body from head to tail, growing
in amplitude towards the posterior end
while the whole body is propelled forward.
717
Thus, from a simplistic mathematical and
mechanical description of the swimmingsystem components, swimming behavior is
recovered. However, what the model has
succeeded in doing is to predict something
that is more easily measured with the real
organism itself. It is true that, given all the
assumptions made, I have shown something
approaching the minimum level of complexity required to describe the mechanics
of undulatory swimming, but that has little
biological relevance. Where this approach
becomes very useful lies in two abilities of
the model: (i) its predictive power when
presented with novel physical or physiological situations, and (ii) calculations or analyses of existing swimming-systems that are
not presently possible.
As discussed above, invoking the performance paradigm implies the need for a
mechanistic understanding of the link between phenotypic characters and performance. This modeling approach is fundamentally mechanistic as it predicts swimming behavior from the mode of interaction
between the components of the swimmingsystem. As such, the model can predict
changes in swimming performance resulting from variation in any of the input parameters. Using a model in this manner is,
in some sense, equivalent to performing an
experiment. The advantage of this 'experimental' approach is that parameters not free
to vary in vivo can be manipulated 'in virtue' For example, Figure 3 shows the results of a number of simulations where I
varied two characteristic properties of the
swimming-muscle, maximum isometric
tension (To), and maximum velocity of contraction (Vmax) to see their effect on swimming performance. In general, such an experiment would not be possible as these two
parameters are not easily manipulated.
However, these parameters certainly have
changed through evolutionary time, and
may even vary within a single fiber type
(Altringham et al., 1993). Thus, without a
modeling approach of this type I could not
fully explore questions of organismal design for swimming since whole suites of
characters that naturally are free to vary
could not be assayed.
Assessing the role that morphological
718
CHRIS E. JORDAN
o.o
2.0
3.0
4.0
5.0
6.0
7.0
Maximum Isometric Tension (lOkPa)
4.0
o.o
o.o
4.0
o.o
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Maximum Velocity of Contraction (BL/s)
1.0
1.1
FIG. 3. The predicted effect of variation in two characteristic muscle parameters on two measures of swimming
performance. The two performance measures, swimming speed and overall mechanical efficiency are shown as
relative values, scaled to the swimming speed or efficiency predicted at the measured values of To and V,,^ (as
indicated by the arrows), respectively. The swimming speed (triangles) reported is the mean steady-state speed,
and the efficiency (squares) is the mean over-all mechanical efficiency for steady swimming (Pou,/Pln = [F,^, •
UbodyVtFmus • VmilJ). A. Varying the maximum isometric tension (To) of the swimming-muscle. B. Varying the
maximum velocity of contraction (Vmas) of the swimming-muscle.
and physiological characters play in gener- logical importance of the performance meaating swimming behavior requires a choice sure is conferred by assertion. Granted, perof performance measure. Often no rationale formance measures like fast-start ability
is provided for the choice of performance and steady-state swimming speed do have
measure, or if it is, the evolutionary or eco- easily imagined biological significance,
A MODEL OF UNDULATORY SWIMMING
however, evolutionary constructs such as
the "performance paradigm" require an established link between behavioral and evolutionary "performance." Thus, traditional,
easily measured swimming performance
criteria may be inappropriate if they are not
strong determinants of fitness.
The integrative mechanical modeling approach allows me to calculate performance
measures not easily measured in vivo, making it a powerful tool for assessing the effects of morphological and physiological
variation. For example, the cost of transport
may be of more interest to the organism
than distance traveled per unit time. In this
case, some measure of the efficiency by
which chemical energy is converted to mechanical work may be the performance criterion of choice. Unfortunately, traditional
models of aquatic locomotion cannot provide this information. Slender-body theory
gives the power output of swimming at either very low, or very high Reynolds number (Lighthill, 1975), but says nothing
about the power required to generate the
body deformations, nor the hydrodynamic
cost of swimming at intermediate Reynolds
numbers. However, because I am accounting for the mechanical interactions at the
level of the muscle and the fluid, I can estimate both the power output of the muscle
and power expended to translate the body.
Figure 3 shows the comparison of two performance measures, swimming speed and
mechanical efficiency, over a range of
swimming-muscle To and Vmax. For variation in Vmax (Fig. 3b) both performance
measures show similar behavior, however,
for variation in To (Fig. 3a), different "optimal" muscle characteristics are supported.
LIMITATIONS AND IMPROVEMENTS
As I have demonstrated, swimming behavior can be modeled as an emergent
property of the mechanical interactions between the swimming-system's components.
The success of this approach is not an end
in itself, rather it is the starting point for
understanding the determinants of swimming performance. At issue is the affect of
morphological and physiological variation
on swimming behavior in the context of the
evolutionary consequences of performance.
719
How well the model performs this task is a
function of two things: (i) the accuracy of
its framing, and (ii) the robustness of the
system it models. To conclude I introduce
several avenues I am currently pursuing to
improve the model's physical representation
and its generality (i.e., its ability to model
the swimming behavior of non-leechlike organisms).
Refining the fluid force coefficients
Sectional fluid force coefficients provide
a reasonable first approximation of the interaction between an organism and the surrounding fluid. However, the linear superposition of resistive and reactive sectional
forces cannot accurately represent all flow
conditions. This limitation exists for two
reasons: (i) sectional force coefficients do
not account for intersegmental interactions
via the fluid, and (ii) for large amplitude
motions and possibly for complex shapes,
the fluid force on a body is not simply the
sum of the added mass and drag forces. Additionally, in those situations where the approach is appropriate, experimental data on
the magnitude of sectional resistive and reactive force coefficients in non-rectilinear
flow for non-engineering shapes are generally lacking (with the exception of rough
circular cylinders in harmonic cross-flow,
Sarpkaya and Storm, 1985). Therefore, the
first step to improving the model is to measure sectional force coefficients from a
physical model of an undulating swimmer.
Sectional fluid forces for biological shapes
and motions can be measured with a mechanical undulator made of independent but
closely opposed segments. The shape of the
segments can be varied to replicate the
overall morphology of a variety of axial locomotors. For example, a series of uniformly shaped segments mimics the body form
of a leech, whereas the addition of fins (stiff
or flexible) on posterior segments and variation of body depth and width represents
other body forms. This approach cannot explicitly account for intersegmental interactions through the fluid since the fluid motion is not known, but it implicitly accounts
for the interactions by measuring the resultant forces in a situation where the interactions are present.
720
CHRIS E. JORDAN
Improving the over-all body mechanics
The 2-dimensional force balance outlined
above is not sufficient to describe the mechanical structure of all axial locomotors.
The 2-dimensional description may be inadequate due to extensive changes in body
length {e.g., a soft bodied organism), or, despite the presence of an axial skeleton, pressure or local volume changes may be mechanically important (Videler, 1993; Wilson, et ah, 1996). Either case requires a
3-dimensional force balance to capture
these mechanical interactions. It is possible
to represent a 3-dimensional structure with
a framework of extensible hinged elements
enclosing a constant volume (e.g., Niebur
and Erdos, 1991). Such a construct is capable of forming a highly plastic body but
can be very computationally intensive. In
addition, no refinement of the fluid-body
interaction is gained by this method. Fortunately a novel numerical representation of
fluid-body interaction, the immersed
boundary method (introduced by Peskin,
1977 to model blood flow in the heart) is
ideally suited to this problem.
In the immersed boundary method, the
organism is represented by a distribution of
forces applied to the fluid (see Fauci in this
volume). This approach is analogous in
spirit to the distribution of Stokeslets used
to model the hydrodynamics of flagellar
propulsion (Gray and Hancock, 1955), but
there is no restriction to zero Reynolds
number flow and individual segments of the
organism do interact through the fluid. The
layer of force that represents the organism
results in fluid motion (from solution of the
N-S equations at each time step), and the
representation of the organism moves with
the velocity and acceleration of the fluid
points corresponding to its boundary (by
the no-slip condition). The lack of a material representation of the organism overcomes the primary hurdle to solving the full
incompressible Navier-Stokes equations for
arbitrary motions at arbitrary sizes, that of
time dependent boundary conditions.
A number of swimming organisms have
been modeled by the immersed boundary
method (Fauci and Peskin, 1988; Fauci,
1990, 1993). In these models the organisms
are represented by linearly elastic filaments
corresponding to the point sources of force
applied to the fluid. Traditionally, these
points have a time-dependent target configuration (in a sense a prescribed motion), but
due to the elasticity of the filaments, the
target configuration may not be reached. A
natural extension of this method would be
to relax the assumption of a target configuration by representing the body with a series of points joined in the non-linear coupled fashion outlined above. Herein lies the
overwhelming advantage of this approach;
an inherent coupling between the mechanical properties of the immersed boundary
and the surrounding fluid. With other approaches such a coupling results in either
computational intractability for models of
organisms as time dependent material-fluid
boundaries, or critical assumptions of fluid
realm (zero or infinite Reynolds number),
organism shape (spheres, infinite cylinders,
flat plates and Joukowski airfoils), or motions (harmonic or rectilinear) in order to
solve the full Navier-Stokes equations.
CONCLUSION
There is little doubt that some aspects of
behavioral "performance" strongly affect
an organism's evolutionary "performance."
However, of more use to evolutionary and
functional biology is the link between the
morphological and physiological determinants of behavior and evolutionary performance; in other words, the adaptive value
of a particular phenotype. To establish this
link I have argued the need for a context
specific, mechanism driven, approach to the
study of performance measures. To this
end, I have outlined a method by which it
is possible to explicitly demonstrate the determinants of a variety of performance measures in undulatory swimming. What remains then, is showing that a behavioral
performance measure is in fact a determinant of evolutionary performance. Only
then will the adaptive value of the phenotypic characters be known, and only because the mechanistic framework in which
they function forms the context for their
study.
A MODEL OF UNDULATORY SWIMMING
ACKNOWLEDGMENTS
This work benefited greatly from the assistance of S. G. Daniel and D. Griinbaum.
In addition, two anonymous reviewers provided very helpful comments and criticism
of the manuscript. This work was made
possible by NBTG:NIH GM07108, and a
donation from Silicon Graphics Inc.
REFERENCES
Alexander, R. McN. 1969. Orientation of muscle fibers in the myomeres of fishes. J. Mar. Biol. Ass.
U.K. 49:263-290.
Altringham, J. D., C. S. Wardle, and C. I. Smith. 1993.
Myotomal muscle function at different points on
the body of a swimming fish. J. Exp. Biol. 182:
191-206.
Altringham, J. D. and I. A. Johnston. 1990. Modelling
muscle power output in a swimming fish. J. Exp.
Biol. 148:395-402.
Arnold, S. J. 1983. Morphology, performance and fitness. Amer. Zool. 23:347-361.
Bowtell, G. and X L. Williams. 1991. Anguilliform
body dynamics: Modelling the interaction between muscle activation and body curvature. Phil.
Trans. R. Soc. Lond. 334:385-390.
Ekeberg, O. 1993. A combined neuronal and mechanical model of fish swimming. Bio. Cybernetics 69:
363-374.
Fauci, L. J. 1990. Interaction of oscillating filaments:
A computational study. J. Comput. Phys. 86:294313.
Fauci, L. J. 1993. Computational modeling of the
swimming of biflagellated algal cells. Contemp.
Math. 141:91-102.
Fauci, L. J. and C. S. Peskin. 1988. A computational
model of aquatic animal locomotion. J. Comput.
Phys. 77(l):85-101.
Fung, Y. C. 1993. Biomechanics. Springer Verlag,
New York.
Garland, X, Jr. 1984. Physiological correlates of locomotory performance in a lizard: An allometric
approach. Amer. J. Physiol. 247 (Reg. Integr.
Comp. Physiol. 16):R806-R815.
Gordon, A. M., A. F. Huxley, and F. J. Julian. 1966.
The variation inisometric tension with sarcomere
length in vertebrate muscle fibers. J. Physiol.,
London 184:170-192.
Gray, J. and G. J. Hancock. 1955. The propulsion of
sea-urchin spermatoza. J. Exp. Biol. 32:802-814.
Grant, P. R. and B. R. Grant. 1995. Predicting microevolutionary responses to directional selection
on heritable variation. Evolution 49(2):241-251.
Hill, A. V. 1938. The heat of shortening and the dynamic constants of muscle. Proc. Roy. Soc. B.
126:136-195.
Hoerner, S. F. 1965. Fluid-dynamic drag, p. 3-15.
Hoerner Fluid Dynamics, Brick Town, New Jersey.
Jordan, C. E. 1992. A model of rapid-start swimming
at intermediate reynolds number: Undulatory lo-
721
comotion in the chaetognath Sagilla elegans. J.
Exp. Biol. 163:119-137.
Jordan, C. E. 1994. Mechanics and dynamics of
swimming in the medicinal leech Hirudo medicinalis. Ph.D. Diss., University of Washington, Seattle.
Josephson, R. K. 1985. Mechanical power output
from striated muscle during cyclic contration. J.
Exp. Biol. 114:493-512.
Kristan, W. B Jr., G. S. Stent, and C. A. Ort. 1974.
Neuronal control of swimming in the medicinal
leech. III. Impulse patterns of the motor neurons.
J. Comp. Physiol. 94:155-176.
Lauder, G. V., P. C. Wainwright, and E. Findeis. 1986.
Physiological mechanisms of aquatic prey capture
in sunfishes: Functional determinants of buccal
pressure changes. Comp. Biochem. Physiol. 84A:
729-734.
Lighthill, M. J. 1975. Mathematical biofluiddynamics.
Regional Conference Series in Applied Mathematics. Vol. 17. SIAM, Philadelphia.
Miller, J. B. 1975. Xhe length-tension relationship of
the dorsal longitudinal muscle of a leech. J. Exp.
Biol. 62:42-53.
Niebur, E. and P. Erdos. 1991. Theory of the locomotion of nematodes: Dynamics of undulatory
progression on a surface. Biophys. J. 60:11321146.
Peskin, C. S. 1977. Numerical analysis of blood flow
in the heart. J. Comp. Phys. 25:220.
Prosser, L. C. 1973. Comparative animal physiology.
3rd ed., p. 738, W. B. Saunders Co., Philadelphia.
Rome, L. C, D. Swank, D. Corda. 1993. How fish
power swimming. Science 261:340—343.
Sarpkaya, T. and M. Storm. 1985. In-line force on a
cylinder translating in oscillatory flow. Appl.
Ocean Res. 7(4): 188-196.
Tashiro, N. 1971. Mechanical properties of the longitudinal and circular muscle in the earthworm. J.
Exp. Biol. 55:101-110.
Taylor, G. I. 1951. Analysis of swimming microscopic
organisms. Proc. R. Soc. London A. 209:447-461.
van Leeuwen, J. L., M. J. M. Lankheet, H. A. Akster
and J. W. M. Osse. 1990. Function of red axial
muscles of carp (Cyprinus carpio): Recruitment
and normalized power output during swimming in
different modes. J. Zool. London 220:123-145.
Videler, J. J. 1993. Fish swimming. Chapman & Hall,
London.
Wainwright, P. C. 1987. Biomechanical limits to ecological performance: Mollusc-crushing by the Caribbean hogfish, Lachnolaimus mximus (Labridae). J. Zool. (London) 213:283-297.
Wainwright, P. C. 1988. Morphology and ecology:
Functional basis of feeding constraints in Caribbean labrid fishes. Ecology 69:635-645.
Wainwright, S. A., W. D. Biggs, J. D. Currey, and J.
M. Gosline. 1982. Mechanical design in organisms. Princeton University Press, Princeton.
Watkins, T. B. 1996. Predator-mediated selection on
burst swimming performance in tadpoles of the
pacific tree frog, Pseudacris regilla. Phys. Zool.
69(1): 154-167.
Webb, P. W. 1976. The effect of size on the fast-start
722
CHRIS E. JORDAN
performance of rainbow trout Salmo gairdneri,
and a consideration of piscivorous predator-prey
interactions. J. Exp. Biol. 65:157-177.
Webb, P. W. 1977. Effects of size on performance and
energetics of fish. In X J. Pedley (ed.), Scale effects in animal locomotion. Academic Press, London.
Webb, P. W. 1978. Fast-start performance and body
form in seven species of teleost fish. J. Exp. Biol.
74:211-226.
Webb, P. W. 1984. Body form, locomotion, and foraging in aquatic vertebrates. Amer. Zool. 24:107120.
Williams, T. A. 1994a. Locomotion in developing Artemia larvae: Mechanical analysis of antennal propulsors based on large-scale models. Biol. Bui.
187(2):156-163.
Williams, T. A. 1994b. A model of rowing propulsion
and the ontogeny of locomotion in Artemia larvae.
Biol. Bui. 187(2):156-163.
Williams, T. and K. Sigvardt. 1992. How the lamprey
swims. NIPS 7:161-165.
Wilson, R. J. A., B. A. Skierczynski, J. K. Meyer, S.
Blackwood, R. Skalak, and W. B. Kristan, Jr.
1995. Predicting overt behavior from the pattern
of motor neuron activity. A biomechanical model
of a crawling leech. Proceedings of the Joint Symposium on Neural Computation 5:176—190.
Wilson, R. J. A., B. A. Skierczynski, S. Blackwood,
R. Skalak, and W. B. Kristan. 1996. Mapping
morot neuron activity to overt behavior in the
leech: Internal pressures produced during locomotion. J. Exp. Biol. 199:1415-1428.
Wu, T. Y.-T. 1971. Hydromechanics of swimming propulsion. Part 2. Some optimum shape problems. J.
Fluid Mech. 46(3):521-544.