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First published online December 28, 2007
Journal of Experimental Biology 211, 180-186 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.013466
Research Article, Motor Systems |
A spatially explicit model of muscle contraction explains a relationship between activation phase, power and ATP utilization in insect flight
1 Department of Biology, University of Washington, Seattle, WA 98195, USA
2 Department of Bioengineering, University of Washington, Seattle, WA 98195,
USA
* Author for correspondence (e-mail: bcwt{at}u.washington.edu)
Accepted 4 October 2007
Summary
Using spatially explicit, stochastically kinetic, molecular models of muscle force generation, we examined the relationship between mechanical power output and energy utilization under differing patterns of length change and activation. A simulated work loop method was used to understand prior observations of sub-maximal power output in the dominant flight musculature of the hawkmoth Manduca sexta L. Here we show that mechanical work output and energy consumption (via ATP) vary with the phase of activation, although they do so with different phase sensitivities. The phase relationship for contraction efficiency (the ratio of power output to power input) differs from the phase relationships of energy consumption and power output. To our knowledge, this is the first report to suggest that ATP utilization by myosin cross-bridges varies strongly with the phase of activation in muscle undergoing cyclic length changes.
Key words: spatially explicit modeling, work loop, phase, flight, muscle power, Manduca sexta
Introduction
Over the past several decades, studies of insect flight have attempted to
understand how the aerodynamics of wing motions determine the forces and
energy required to power flight. Much of this effort focused on the
engineering aspects of wing motions, using approaches that include detailed
and elegant reconstruction of wing motions in scale models (for a review, see
Sane, 2003
), theoretical and
computational aspects of flapping flight
(Hedrick and Daniel, 2006),
measurements of wing motions, and the subsequent patterns of fluid motion.
Processes such as wake capture, delayed stall, wing rotation and potential
unsteady fluid dynamic flow have all been implicated as important factors
affecting the timing and magnitude of wing muscle forces generated during
flight. These factors contribute to the overall energy requirements, as well
as underlying control mechanisms, for insect flight.
A variety of physiological experiments complemented these engineering
studies of flight. For example, direct measurement of metabolic gas exchange
(Dickinson and Lighton, 1995;
Ellington, 1985), as well as
the in vivo timing and magnitude of muscle activity, have been used
to assess the power output in insect flight
(Tu and Daniel, 2004). A
unifying approach pioneered by Josephson
(Josephson, 1985;
Josephson, 1989) measured the
temporal patterns of muscle force and the associated changes in muscle length
to quantify mechanical work generated by muscle. This work loop method relates
the timing of muscle activation, relative to its length change, to the overall
mechanical power output. Presumably, the power required for flight, as
estimated by the host of aerodynamic studies, agrees with the power measured
by these physiological methods.
Hawkmoths, among the largest insect fliers, power their flight with two
sets of synchronously activated muscles connected to the thorax. The
dorso-longitudinal muscles (dlm) compress the thorax in the antero-posterior
axis to power the wing downstroke while the dorso-ventral muscles (dvm)
compress the thorax in the orthogonal plane to power the wing upstroke. Work
loop analyses of the dlm showed a strong phase dependence of muscle mechanical
power output that was consistent with prior findings for insect muscles
(Josephson, 1997).
Enigmatically, however, the physiological (in vivo) dlm phase of
activation generated only 40% of the maximal realizable power (max=100 W
kg–1) (Tu and Daniel,
2004).
Operating at sub-maximal power seems advantageous, assuming that the physiological phase of activation affords ample power reserves required for maneuvering flight. Therefore, any additional energy required for turns, elevations and accelerations becomes available by simply shifting activation phases to produce higher mechanical power output. With no clear evidence that the phase of activation affects the energy derived from myosin cross-bridge ATPase activity, we have a paradoxical situation: muscles operating at phases generating sub-maximal power would operate inefficiently unless ATPase rate varies with phase.
Because neuro-muscular control of locomotion inextricably links multiple
levels of organization (from protein forces to aerodynamic forces), the
dynamics and kinetics of protein level processes affect organ and organism
function. This paper attempts to resolve the above paradox, drawing on a
series of molecular models of muscle contraction
(Daniel et al., 1998;
Chase et al., 2004;
Tanner et al., 2007) to
examine the coupled relationship between activation phase, mechanical work and
ATP utilization. Specifically, we ask whether the mechanics and kinetics of
myosin cross-bridges lead to a phase-dependent ATPase behavior at the
sarcomeric level. Our simulations show that mechanical power output (work) and
ATP utilization (energy input) depend upon phase of activation (each having
different sensitivities to phase). These results also show that some phases
maximizing work do so at large energy costs (high ATPase rates). Combining our
computational results with prior experimental studies
(Tu and Daniel, 2004) shows
that physiological phases of activation generating sub-maximal power output
may do so more efficiently (by optimizing the ratio of mechanical power output
to ATPase rate).
Materials and methods
The spatially explicit computational model used for these work loop
analyses follows structural, kinetic and mechanical analyses described more
fully in a series of prior studies (Daniel
et al., 1998; Daniel and Tu,
1999; Chase et al.,
2004; Tanner et al.,
2007). These modeling studies incorporated a half-sarcomere-length
lattice of compliant thick and thin filaments to determine the mechanics and
energetics of force generation in muscle. A recent advancement for these
models included kinetics for calcium activated thin filaments
(Tanner et al., 2007). This
refinement permits simultaneous control of activation and muscle length, while
assessing force, work and energy consumption. Here we briefly review elements
of the models and indicate how we modified prior approaches to generate work
loop predictions for muscle.
Geometry and mechanics
Rather than specifically modeling the details of Manduca sexta
flight muscle per se, we focus on examining how phase of activation
may influence the energetics of contraction. The model is based on geometry of
a half-sarcomere consisting of four thick and eight thin filaments packed in a
hexagonal array (Tanner et al.,
2007). Each thick filament bears 120 myosin cross-bridges arranged
in a two-start helix. Cross-bridges directly face thin filaments bearing 90
binding sites per filament. These geometries reflect those used in previous
spatially explicit models (Chase et al.,
2004; Tanner et al.,
2007) and generally approximate the cross-bridge and binding site
pattern observed in striated muscle. As no published accounts of the
myofilament ultrastructure from Manduca flight muscle exist, this
geometric model serves as an approximation for the thick and thin filament
arrangement.
Estimates of the myofilament mechanical properties are layered upon the
geometric arrangement described above
(Daniel et al., 1998;
Chase et al., 2004;
Tanner et al., 2007). Each
cross-bridge has a spring constant (kxb) of 5 pN
nm–1, somewhat stiffer than the values reported in the
literature from single molecule studies, but more consistent with recent
estimates from cellular experiments
(Howard, 2001;
Piazzesi et al., 2002;
Linari et al., 2007). Thick
and thin filament spring constants (km and
ka, respectively) were set at 6060 and 5230 pN
nm–1 for resting (unbound) thick and thin filament lengths of
14.3 and 12.3 nm, respectively, consistent with parameter values used in
previous studies (Daniel et al.,
1998; Chase et al.,
2004; Tanner et al.,
2007).
Kinetics of cross-bridges and calcium activation
Ca2+-binding proteins regulate the binding of force-generating
motor proteins (cross-bridges) within this myofilament network. The kinetics
of these two coupled protein systems are described with a three-state
[Ca2+]-sensitive, thin-filament regulatory cycle coupled to a
three-state cross-bridge cycle (for details, see
Tanner et al., 2007). Briefly,
the thin-filament regulatory states (Fig.
1A) describe Ca2+ binding to troponin (a transition
from state T1 to T2), which triggers
movement of tropomyosin (into state T3) and exposes
myosin-binding sites along the thin filament
(McKillop and Geeves, 1993;
Xu et al., 1999;
Pirani et al., 2006).
Transition rates for the thin filament kinetics are as specified elsewhere
(Tanner et al., 2007), with
the exception of the rates describing Ca2+ dissociation from the
thin filament (rt,21 and rt,31, which
were set to 100 s–1; Fig.
1A). These modified rates produced faster relaxation dynamics to
simulate the time-course of a Manduca twitch more effectively
(Tu and Daniel, 2004). The
second order rates associated with activating troponin are functions of
intracellular [Ca2+].
|
We summarize these two kinetic processes by a set of coupled differential
equations that are solved by Monte-Carlo simulation. The governing equations
for each interaction between a cross-bridge and a potential thin filament
binding site are described with three states of thin filament activation
(T1, T2, T3) and
three states associated with the cross-bridge cycle (X1,
X2, X3):
 | (1) |
). The equations describing interactions between these
two protein systems are coupled by the interaction between unbound
cross-bridges (X1) and available binding sites
(T3). The availability of binding sites is regulated by
Ca2+ binding to troponin. The system of equations above can be
further reduced to four equations with:
 | (2) |
As described more fully elsewhere
(Daniel et al., 1998;
Tanner et al., 2007), we
implement an instantaneous force balance for a system of linear springs that
characterize each cross bridge, thin filament and thick filament
(Fig. 1B). The total force
generated by the half-sarcomere is determined by the number of attached
cross-bridges and their local distortion. The force on each thick filament is
proportional to the distortion of the spring element closest to the
M-line (
x–xrest) times the
thick filament spring constant (km). Similarly, the force
on each thin filament is proportional to the distortion of the spring element
closest to the Z-disk
(y–yrest) times the thin filament
spring constant (ka). The total force borne by the model
half-sarcomere (FT) is the sum of the forces for all four
thick and eight thin filaments, respectively:
 | (3) |
Controlling motion and activation
Initial half-sarcomere length (L0) was 1200 nm
(Tanner et al., 2007), from
which sinusoidal length oscillations varied the length (L) as a
function of time (t):
L(t)=–Acos(2
ft)+L0,
where f represents wing beat frequency for the hawkmoth
(f=25 Hz). Length oscillated at amplitudes (A) of 30 and 15
nm per half-sarcomere, yielding normalized peak-to-peak strain amplitudes
(
=2A/L0) of 0.05 and 0.025. The frequencies
and amplitudes of length change were chosen to be close to those values
reported previously (Tu and Daniel,
2004).
The modeled activation pulse represents a Ca2+ transient within
the cell (following a neural stimulus) that leads to muscle contraction, or
twitch (in Manduca and other insects there is a one-to-one
correspondence between evoked potential and muscle twitch forces). This
activation transient was modeled with a square wave of binary amplitude,
representing intracellular [Ca2+] of 10–4 or
10–9 mol l–1 for activated or inactivated
contraction, respectively. While the actual temporal pattern of
[Ca2+] is more likely represented as a skewed Gaussian function,
this simple square wave provided a sufficient model for probing phase
relationships and duty cycle. Two parameters determined the duration of this
wave: period (P), set to 40 ms (=1/f); and duty cycle
(
Ca), the fraction of P that
[Ca2+]=10–4 mol l–1.
Following with the experimental work loop analysis of Tu and Daniel
(Tu and Daniel, 2004), raw
phase (
) equals zero when the onset of force generation and muscle
lengthening are co-incident in time. To adjust their phases accordingly, Tu
and Daniel (Tu and Daniel,
2004) measured the temporal delay between external stimulus and
muscle force generation (tep was 
8 ms).
Several processes may contribute to this delay, including the timing of
Ca2+ release from the sarcoplasmic reticulum and subsequent
diffusion into the cytosol. As we do not know the actual transient, we
simulated a number [Ca2+] transients by changing their duty cycle
(Ca) and phase (Ca). We used three values
for Ca: 0.1, 0.2 and 0.5, each representing an increasing
duration of activation (4, 8 and 20 ms, respectively). To compensate for the
variable square wave duration affecting phase of activation, we adjusted the
Ca2+ sensitive phase of activation (Ca) with
respect to muscle lengthening:
Ca=–Ca/2. This adjustment aligned
the center of the [Ca2+] activation pulse with the onset of muscle
lengthening (when =0) even though the duration of activation changes with
Ca. Consistent with the experimental work loop analysis
(Tu and Daniel, 2004), these
simulations examine the relationship between work output and phase of
activation.
Implementation
Full details about the simulation algorithm are described elsewhere
(Tanner et al., 2007).
Briefly, simulations were implemented in Matlab (The MathWorks, Natick, MA,
USA) using the Matlab distributed computing engine on a 16 node IBM cluster
running Linux. Thin filament and cross-bridge kinetics were stochastically
driven using a uniform random number generator and state transitions were
accepted using Monte Carlo algorithms. Simulations at each parameter set ran
for 10 s using time steps of 1 ms, yielding 250 work loops. At each time step
in the simulation we record force, ATP utilization, and kinetic state of thin
filament binding sites and cross-bridges. Average values were calculated from
245 individual work loop cycles, as the initial five `start up' cycles were
omitted from averaging.
Results
Simulations of simultaneous periodic calcium activation and muscle length
change result in an oscillating, twitch-like force response as well as an
oscillating pattern of ATP consumption
(Fig. 2). Interestingly, the
force response was shifted in phase with respect to the Ca2+
activation transient, and ATP utilization was further phase shifted relative
to force. At these particularly high activation rates [relative to those in
Tanner et al. (Tanner et al.,
2007)] cross-bridges remained bound after the Ca2+
activation transient ended. These results largely followed from temporal
delays associated with thin filament de-activation, cross-bridge binding
(force generation) and cross-bridge cycling (ATP consumption).
|
|
). With a
density of 1000 kg m–3, this would lead to a mass of
6.7x10–18 kg and specific power output of approximately
4 W kg–1. Over the full set of simulations, we would predict
a range of 1–6 W kg–1 for specific power output. These
predictions are less than the operating power output of 45 W
kg–1 reported previously
(Tu and Daniel, 2004), but
well within the range of power output values reported (–20 to 80 W
kg–1).
Our simulations used lower strain amplitudes than in the earlier
experiments (Tu and Daniel,
2004). This constraint followed from a computational limitation in
which we were restricted to time steps greater that those required for large
amplitude strains. Because strain strongly determines predicted power output
values, we rescaled predictions with the assumption that the filament lattice
is linearly elastic. This assumption allows us to compare simulation results
with experiments via normalizing power output to the square of strain
amplitude (2). This calculation yields predicted values
ranging from 1860 to 2640 W kg–1 (=0.05) and 1190 to
4761 W kg–1 (=0.025) for mass-specific power output per
2 (Fig. 4A).
These values are similar to estimates of power output for Manduca at
the measured physiological strain (=0.09 yields about 5000 W
kg–1 –2)
(Tu and Daniel, 2004). Thus,
our half-sarcomere model predicts physiological levels of mechanical work and
power, previously measured in whole muscle studies.
|
),
also depends on phase of activation. While these efficiencies show a very
similar phase dependence, the magnitudes remains highly strain dependent (with
different ordinate scales between the two panels,
Fig. 4C). At both levels of
strain, the peak value for mechanical work output, ATP use and efficiency all
occur at different phases. Thus, muscles operating at phases that may maximize
work may do so at some considerable reduction in efficiency. Adjusting phase
of activation could modulate the muscle function, moving the performance from
high power output to high efficiency.
Although mechanical work output depends on activation phase, this response
is influenced by compliance of the filament network and duty cycle
(Ca) of calcium activation. The effects of these parameters
are summarized in Fig. 5, where
we simulated work loops at three different duty cycles (0.1, 0.2 and 0.5) and
two levels of filament compliance (the standard values listed above and one
tenth the standard values). Under these standard filament conditions (at
either strain), decreasing duty cycle leads to decreased mechanical work
output and ATP use (broken lines in Fig.
5). Decreasing duty cycle also shifted the phase maximizing
mechanical work output and ATP use. These phase shifts differed between
mechanical work output and ATP use and their sensitivities varied with strain.
The correlation between decreasing Ca and decreased ATP use
follows directly from the smaller fraction of time during which cross-bridges
may cycle each period. Because duty cycle yields different phase sensitivities
between work output and ATP use, the ratio of these two (efficiency) resulted
in even a greater (and unique) sensitivity to phase as duty cycle
decreased.
|
Under these standard filament conditions (at either strain), decreasing
duty cycle shifted the phase maximizing mechanical work output and ATP use
(broken lines in Fig. 5).
Interestingly, the range of mechanical work output across all phases was
unchanged by duty cycle (at standard filament conditions). In contrast,
decreasing duty cycle decreased the maximal ATP use value as well as the range
of ATP (broken, colored lines in Fig.
5). The correlation between greater ATP use and increased
Ca follows directly from the greater fraction of time during
which cross-bridges may cycle each period. Furthermore, decreasing duty cycle
leads to greater sensitivity of efficiency to phase.
In general, increasing filament compliance (the thick and thin filaments
becoming more flexible) decreased work output and increased ATP use (solid
lines, Fig. 5), compared to
standard filament conditions. These observations agree with prior results
showing that cross-bridge cycling rate increases with increased cross-bridge
binding via greater filament movement (compliant realignment of
binding sites) (Daniel et al.,
1998; Tanner et al.,
2007). As with the standard filament conditions (at either
strain), decreasing duty cycle shifted the phase maximizing mechanical work
output and ATP use (solid lines, Fig.
5). Interestingly, the range of mechanical work output across all
phases was unchanged by duty cycle (different from the standard filament
condition results). In contrast with these mechanical work output results,
decreasing duty cycle decreased the maximal ATP use value and the shift in
this peak (toward higher phases at lower duty cycles) became more pronounced
for the more compliant filament values. Lower values for work with greater ATP
utilization (for these more compliant myofilament values versus
standard values) result in lower efficiency at the more compliant filament
values. The sensitivity of efficiency to phase is also lower when filament
compliance increases. Together, these results demonstrate that the coupled
relationship between filament compliance, duty cycle and activation phase
determines the energetics of muscle contraction.
Discussion
The phase dependence of mechanical power output predicted by our model is
consistent with the studies of whole muscle function (e.g.
Josephson, 1985). However, to
our knowledge, no prior study has shown that myosin cross-bridge ATPase rates
vary with phase of activation. In this regard, our simulations show three key
features: (1) mechanical power output and power input (ATP consumption rate)
both vary with phase, but their respective maxima occur at different phases
(Figs 4,
5); (2) therefore the maximum
efficiency of contraction (the ratio power output to power input) occurs at
yet a third, distinct phase (Figs
4,
5); and (3) the specific phases
leading to maximum power output, minimum energy consumption or maximum
efficiency vary with the duty cycle of activation and the filament lattice
compliance (Fig. 5). Moreover,
our simulation results suggest that these changes in muscle function could
occur over relatively small adjustments in phase of activation.
The notion that efficiency and power may not share a phase relationship has
not been a central focus of muscle functional studies, possibly because there
has not been a compelling case made for changes in ATPase rate (and thus
efficiency) with activation phase. There is good evidence, however, that these
metrics should co-vary. A trade-off between power and efficiency was shown in
dogfish muscle (Curtin and Woledge,
1996), and a modified Hill model
(Lichtwark and Wilson, 2005)
demonstrated the co-variance of efficiency and power output during oscillatory
length changes over a range of frequencies.
Periodic length changes produce periodic velocity changes. With strong
evidence that ATPase rate varies with shortening velocity, we might expect the
energy consumption to depend on phase. Despite these observations, no one has
proposed a molecular mechanism that would explain the co-variance of phase and
energetics. Our simulations show that some phases of activation may be more
advantageous for the efficiency of contraction, whereas other phases may lead
to higher levels of power output. Thus one reason moths may operate at a phase
of activation leading to sub-maximal power output
(Tu and Daniel, 2004) may
reside in the potential trade-off between power and efficiency with respect to
phase. As suggested (Tu and Daniel,
2004), operating at sub-maximal power output affords power
reserves for use in extreme behaviors such as escape, load carrying, or
take-off. However, this only makes sense if ATP consumption also varies with
phase, so that sub-maximal power output reduces energy consumption rate.
The relationship between power and efficiency is complicated by the strong dependence of these parameters on duty cycle (Fig. 5). Very short duty cycles lead to lower mechanical power output and lower ATP use. At the same time, the phases producing peak power output and peak ATP use increase with decreasing duty cycle (each with different sensitivities to phase; Fig. 5). One component of this process that we have not considered is the possible correlation between short duty cycles (at a given frequency) and the energetics of Ca2+ pumping into the sarcoplasmic reticulum of flight muscle. Hence, short duty cycles may require greater ATP use simply because the faster rates at which Ca2+ sequestering must occur become energetically expensive. Adding these ATP costs at decreased duty cycles may yield more similar values of ATP use per oscillatory cycle, thereby further amplifying the differences in efficiency of muscle contraction (work/ATP) shown in Fig. 5.
Filament compliance also affected the magnitudes of power output and power
input (Fig. 5). Interestingly,
the phases at which these components generate their respective maxima and
minima are somewhat independent of the mechanical properties of the
myofilament lattice. Our result showing higher ATP use with greater filament
compliance is consistent with previous work
(Daniel et al., 1998;
Daniel and Tu, 1999;
Tanner et al., 2007). Thus,
these basic mechanical characteristics may underlie varied performance in
muscles among diverse taxa. The extent to which the mechanical properties of
filaments vary among species is not well understood, although the lattice
geometry (myofilament ultrastructure) varies considerably
(Hoyle, 1983). To our knowledge
studies of the mechanical properties of isolated filaments and cross-bridges
have focused on vertebrate striated muscle.
Combining our existing ability to calculate work loops from measured force
and length changes (Tu and Daniel,
2004) with simultaneous measurements of oxygen consumption and
carbon dioxide production allows one to measure the phase dependence of power
and efficiency relationships in synchronous insect flight muscle. While
instantaneous measurements of ATPase activity in muscle during oscillatory
length changes and concomitant phase changes may be challenging, the
relatively poor capacity of insect flight muscle for anaerobic metabolism
allows one to use oxygen consumption as a proxy for ATP use. As shown
previously (Tu and Daniel,
2004), work loop preparations are stable for a sufficiently long
time to facilitate this experimental approach (approximately 1 h).
Acknowledgments
We thank Jessica L. Fox and Drs Michael S. Tu and Johan van Leeuwen for their comments on the manuscript. This work was supported by HL65497 to M.R. and T.D., funds from the Komen Endowed Chair to T.D., and an NIH pre-doctoral training grant T32 EB001650 to B.T.
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