J. theor. Biol. (1997) 184, 253–259
Optimum Muscle Design for Oscillatory Movements R. MN. A Department of Biology, University of Leeds, Leeds LS2 9JT, U.K. (Received on 11 January 1996, Accepted on 1 August 1996)
A simple model represents oscillatory movements such as many animals use for running, swimming or flight. A plate is oscillated in a fluid by a pair of muscles that exert the forces needed to overcome its inertia and hydrodynamic drag. The muscles have spring-like tendons. Empirically-based equations that take account of the force exerted by a muscle and the rate at which it is shortening are used to estimate the metabolic energy cost of the oscillation. The maximum shortening speed (vmax ) of the muscles and the elastic compliance of their tendons are varied to find the optimum combination that minimizes metabolic cost. If hydrodynamic forces predominate (as in swimming), the cost is highly sensitive to muscle speed (which should be relatively high) but less sensitive to compliance. If inertial forces predominate (as in running) the cost is highly sensitive to tendon compliance, and less sensitive to muscle speed (which should preferably be low). The muscles that power the locomotion of various animals are discussed in the light of these conclusions. 7 1997 Academic Press Limited
1. Introduction Many animals make oscillatory movements in locomotion: flying birds and insects beat their wings, fishes move their tails from side to side and running animals swing their legs forward and back. This paper discusses how their muscles should be designed, to minimize energy costs. As well as the contractile component of the muscles, it discusses tendons and other structures that can serve as energy-saving springs. Weis-Fogh (1972) discussed the hovering flight of insects and hummingbirds. In every wing stroke the muscles must do work against the aerodynamic drag on the wings. In addition, work is needed to accelerate the wings at the beginning of each stroke and negative work (as performed by brakes) is needed to decelerate them at the end of the stroke. Metabolic energy is used both when muscles do positive work and when they do negative work. The positive and negative work requirements for flight muscles cannot be calculated simply by adding together the aerodynamic and the inertial work, because aerodynamic drag helps to decelerate the wings at the end of each stroke, but the total work requirement is generally greater than the aerodynamic work alone. 0022–5193/97/030253 + 07 $25.00/0/jt960271
Weis-Fogh (1972) pointed out that the inertial work requirement could be eliminated by elastic mechanisms. Suitably arranged springs could halt the wings at the end of a stroke (doing negative work and storing elastic strain energy), then recoil doing positive work to accelerate the wings for the next stroke. He suggested that the rubber-like protein resilin in the thoraxes of insects might serve this function. Weis-Fogh’s (1972) discussion implies that the springs are arranged in parallel with the muscles. Bennett et al. (1987) considered the possible energy-saving role of springs in series with muscles, such as the tendons of the muscles that power swimming in dolphins. They calculated the compliance that the tendons should have to minimize the work required of the muscles. With tendons of this compliance the muscles would do only positive work. These studies implied the assumption that if work was minimized, the metabolic energy cost of movement would also be minimized. This paper shows that the assumption may be false. Metabolic energy costs will be calculated for muscles powering oscillatory movement. The properties of the muscles 7 1997 Academic Press Limited
. c.
254
and their tendons will be varied to find the combination that minimizes metabolic cost.
the square of velocity. This assumption is realistic for all the cases we will consider. Thus the force the muscles must exert to overcome drag can be written:
2. The Model
Fhydro = 2D(ds/dt)2 = 2Da 2 v 2 sin2 vt,
Figure 1 shows a generalized model of oscillatory motion powered by muscles. A plate of mass m (including the added mass of moving fluid) is oscillated in a fluid with amplitude a and circular frequency v. The muscles that drive it have tendons of compliance Cs (subscript s identifies this as the series compliance). In addition, a spring of compliance Cp acts in parallel to the muscles. The muscles move the mass, giving it displacement s at time t:
[using eqn (2)], where D is a constant. We will consider only a half cycle in which the plate moves in the positive direction, so Fhydro will be positive. The total force that must be exerted by one or other of the muscles is:
s = −a cos vt.
(1)
(The negative sign makes the plate move in the positive direction, in the first half cycle.) By differentiation: ds/dt = av sin vt.
(2)
d2s/dt 2 = av 2 cos vt.
(3)
The force required to give the mass this acceleration is mav 2 cos vt, but this is supplied in part by the parallel spring, which exerts a force −s/ Cp = (a cos vt)/Cp . Thus the force that the muscles must exert to overcome the inertia of the plate is: Finert = [mv 2 − (1/Cp )]a cos vt =[1 − (vres /v)2 ]mav 2 cos vt,
F = Finert + Fhydro = av 2{m cos vt[1 − (vres /v)2 ] + Da sin2 vt} = Da 2v 2(mˆ cos vt + sin2 vt),
where vres = (mCp )−0.5, and is the natural circular frequency of the mass vibrating under the influence of the parallel spring. The expression [1 − (vres /v)2 ] m will be referred to as the effective mass of the plate. Note that it will be negative if the parallel spring is so stiff as to make vres greater than v. We will assume that the Reynolds number is high enough to make hydrodynamic drag proportional to Plate Cp Muscle Cs
(6)
where mx = (m/Da)[1 − (vres /v)2 ].
(7)
Note that mˆ is the ratio of the peak inertia force to the peak hydrodynamic force. It will be referred to as the effective mass parameter. We assume that at any instant, only one of the two muscles exerts force. The extension of this muscle’s tendon is FCs . Thus the length displacement x of the muscle (see Fig. 1) is: x = s + FCs = −a cos vt + Da 2v 2Cs (mˆ cos vt + sin2vt)] = a[−cos vt + C s (mˆ cos vt + sin2 vt)],
(4)
(5)
(8)
where C s = Dav 2Cs .
(9)
C s will be referred to as the series compliance parameter. It is the extension of the tendon due to the peak hydrodynamic force, divided by the amplitude. The rate of shortening, v, of the muscle can be obtained by differentiating eqn (8): v = av[sin vt − C s (mˆ sin vt − sin 2vt)].
(10)
Equations (6) and (10) give us the force being exerted by the active muscle and the rate at which it is shortening. The next section shows how these can be used to calculate the muscle’s metabolic rate. 3. Muscle Properties
x s F. 1. The model of oscillatory motion powered by muscles, which is analysed in the text.
Ma & Zahalak (1991) have shown that the metabolic power consumption Pmetab of a fully-activated muscle shortening at a rate v can be written: Pmetab = Fiso vmax F(v/vmax ),
(11)
Efficiency
(a)
255
0.4
0.2
0
1 v/vmax
2
Relative force –1
Metabolic cost function
(b)
(c)
1
0 v/vmax
1
–1
0.2
0.1
0 v/vmax
1
F. 2. Graphs showing how properties of a muscle depend on the rate at which it is shortening, according to the relationships assumed in the text. The horizontal axis in each case shows the rate of shortening as a fraction of the maximum (unloaded) rate. (a) shows efficiency (eqn (14)); (b) shows force as a multiple of the isometric force [eqn (13); and (c) shows the metabolic cost function F (eqn (12)].
where Fiso is the force it can exert in an isometric contraction and vmax is its maximum (unloaded) rate of shortening. The following equations for the function F have been fitted to the empirical data in Ma & Zahalak’s fig. 9 (giving preference to points derived from Hill (1964), over those from earlier experiments). For v Q 0 F(v/vmax ) = 0.01 − 0.11(v/vmax ) + 0.06 exp(23v/vmax ), (12a) and for 0 Q v Q vmax F(v/vmax ) = 0.23 − 0.16 exp(−8v/vmax ). (12b) By definition, v cannot exceed vmax . These equations are plotted in Fig. 2(c). Generally only some of the motor units of the muscle will be active. We will assume that in the course of each cycle of movement, motor units are activated and deactivated instantaneously as required to give F and v the values specified in eqns (6) and (10). The value of Fiso required for calculating metabolic power [eqn (11)] is the isometric force that
the currently active motor units are capable of exerting. Let the muscle have force–velocity properties as described by van Leeuwen (1992). For v Q 0 F/Fiso = 1.8 − 0.8[(vmax + v)/(vmax − 7.56Gv)],
(13a)
and for 0 Q v Q vmax F/Fiso = (vmax − v)/(vmax + Gv),
(13b)
where G is a constant for any particular muscle. We will take G = 4 as a typical value (see Woledge et al., 1985, in whose notation G is P0 /a). Equations (13a) and (13b) are plotted in Fig. 2(b). The efficiency, h, can be obtained from: h = Fv/Pmetab = (F/Fiso )(v/vmax )/F(v/vmax ).
(14)
This efficiency can be calculated from eqns (12), (13) and (14), for any given value v/vmax . Values are shown in Fig. 2(a). These are efficiencies of converting ATP energy to work. Note that if the force F calculated from eqn (6) is negative, the active muscle must be the one on the left
. c.
256
side of Fig. 1, which pulls in the negative direction. In that case, vmax should be given a negative sign. The following procedure was adopted to calculate the metabolic cost of driving the model. F and v were calculated for successive time increments during one half cycle. Equations (13a) and (13b) were used to calculate for each (F, v) pair the required value of Fiso (showing how much muscle must be active). This was used in eqn (11) to obtain the metabolic power, Pmetab , which was integrated over the half cycle to obtain the energy cost. The results will be expressed as the efficiency of doing work against hydrodynamic drag. The hydrodynamic power is Fhydro ·ds/dt so this efficiency is: hhydro =
g
g
p/v
p/v
Fhydro (ds/dt)·dt/
0
Pmetab ·dt.
(15)
0
Only the hydrodynamic work is considered, in calculating this efficiency, because the requirement for inertial work can be eliminated by adjustment of the model’s parallel or series compliances. By substituting eqns (2) and (5) in (15) and integrating:
0g
hhydro = 4Da 3v 2/ 3
p/v
1
Pmetab dt .
0
(16)
A computer program performed all these calculations. 4. Results In addition to the hydrodynamic efficiency that has just been defined, three dimensionless parameters will be used in presenting results. As already explained, the effective mass parameter (mˆ , eqn (7)) is the ratio of peak inertia force to peak hydrodynamic force, and the series compliance parameter (C s , eqn (9) is the tendon extension due to the peak hydrodynamic force, as a fraction of the amplitude. In addition, I will use a muscle speed parameter: vˆmax = vmax /av.
(17)
This is the maximum shortening speed of the muscle divided by the peak speed of the plate. Figure 3(a) to (d) show results for four different values of the effective mass parameter. In each case, the axes are the series compliance parameter and the muscle speed parameter, so the graphs presents a range of possible tendon and muscle properties. Hatched areas are not feasible because, in them, the muscle would at some stage have to shorten at a rate faster than vmax . The contours show the efficiency with which work is performed against hydrodynamic drag.
The maxima, marked by stars, represent the optimum combination of muscle and tendon properties, which minimize metabolic energy cost. For negative and zero effective masses [Figs 3(a) and (b)] the optima have zero series compliance and relatively high muscle speeds. For an effective mass parameter of 1 [Fig. 3(c)] substantial compliance is needed to maximize efficiency, and the muscles should still be quite fast. With further increase in the effective mass [Fig. 3(d)] the optimum values both of the series compliance parameter and of the muscle speed parameter fall. The falling compliance (increasing stiffness) of the tendons is appropriate to the larger inertial forces that act when effective mass is high. Figure 4(a) shows the best obtainable efficiencies (those corresponding to the optima in Fig. 3) for different effective masses. For all positive effective masses, efficiencies in the range 0.43–0.45 are attainable, but optimum efficiency is lower for negative effective masses. Figures 4(b) and 4(c) show the muscle speeds and series compliances corresponding to the optima in Fig. 3. The broken line in Fig. 4(c) shows the series compliance parameter that would eliminate the need for muscles to perform negative work at any stage of the cycle, calculated as explained by Bennett et al. (1987). Note that the broken and continuous lines converge only at high effective masses. When effective mass is low it is not optimal to eliminate negative work. However, the contours in Figs 3(b) and 3 (c) show that at low effective masses, efficiency is reduced only a little by quite large deviations of series compliance from the optimum. In contrast, at high effective masses [Fig. 3(d)] efficiency is highly sensitive to deviations of compliance from the optimum. 5. Discussion Effective mass parameters have been estimated for some examples of oscillatory motion, described in the literature. The corresponding optimal series compliance and muscle speed parameters, taken from Fig. 4, are shown in Fig. 5. These examples will be discussed in turn. Scallops (Pecten and related genera) swim by jet propulsion by repeated opening and closing of the shell. An elastic hinge ligament functions as a parallel spring, more or less eliminating the need for the adductor muscle to exert inertial forces (De Mont, 1990; Marsh et al., 1992). Thus we can estimate the effective mass parameter as zero. Figure 4 tells us that the optimum series compliance is zero. Appropriately,
257
the muscle has no tendon. Figure 4 also tells us that the optimum value for the muscle speed parameter is 3.9. The actual muscle speed parameter can be estimated as about 5 from the data of Alexander (1979) and 3.8 from Olson & Marsh (1993).
the need for muscles to perform negative work, but this paper has shown how a lower series compliance can give a slightly higher efficiency. There seem to be no empirical data that would enable us to compare the predicted optimum muscle speed parameter with actual values.
Dudley & Ellington (1990) investigated the flight of bumblebees (Bombus), finding that the inertial power was about twice the aerodynamic power over a range of speeds. The inertial work from which they calculated this power is the peak kinetic energy of the plate, 1/2 ma2v 2. This work is done over one quarter cycle. From the discussion leading to eqn (16), the hydrodynamic work done over a quarter cycle is 2/3 Da3v 2. Thus the ratio of powers is 0.75 m/Da, which is 0.75 mˆ if there is no parallel spring. A power ratio of 2, as calculated by Dudley & Ellington (1990) corresponds to a mass parameter of 2.7. This effective mass parameter has been used to locate the bee point
Dolphins swim by beating their tail flukes up and down. Bennett et al. (1987) suggested that energy might be saved by the elastic compliance of the tendons of the tail muscles and of the intervertebral discs. Their hydrodynamic analysis was inadequate but, in an improved analysis, Blickhan & Cheng (1994) showed that the effective mass parameter is about 0.3. (Their parameter K is the reciprocal of mˆ .) Hence, from Fig. 4, the optimum value for the series compliance parameter would be 0.22. The actual value, calculated from the measurements of Bennett et al. (1987) is 0.67. Blickhan & Cheng (1994) point out that this is close to the value that would eliminate
Muscle speed parameter (v^max)
10
10 (a)
(b)
5
5 0.3
0.1
0.4
0.2
2
2
1 0.5
1
0.5
0
1.0
10 Muscle speed parameter (v^max)
0.3 0.2
0.5
0.5
0
1.0
10 (c)
(d) 0.2
5
0.1
5 0.3
2
0.4
0.3 0.2
1 0.5
2
0.4
0
0.5 ^ Compliance parameter (Cs)
1
1.0
0.5
0
0.5 ^ Compliance parameter (Cs)
1.0
F. 3. Graphs showing how the efficiency, with which the muscles of the mode work against hydrodynamic forces, depends on the properties of the muscles and tendons. In each graph, the axes represent the series compliance parameter, Cs , and the muscle speed parameter, vmax (the latter on a logarithmic scale). The contours show the efficiency, hhydro (eqn (16)), with stars marking the maxima. Hatched areas are not feasible because the muscles would have to contract faster than their maximum shortening speed. Each graph refers to a different effective parameter mx , as follows: (a) −1; (b) 0; (c) +1; (d) +5.
. c.
258
Optimum efficiency
(a) 0.45
0.40
–1
–1
0
1 2 3 Effective mass parameter
4
0.6
5 Compliance parameter
Optimum muscle speed parameter
(b)
(c)
4
3
2
Optimum compliance
0.4
Compliance for no negative work 0.2
1
0
1 2 3 Effective mass parameter
4
5
–1
0
2 3 1 Effective mass parameter
4
5
F. 4. Graphs showing how optimum muscle and tendon properties for the model depend on the effective mass parameter. (a) Shows the efficiency, hhydro , obtainable with the optimum combination of muscle speed and series compliance; (b) shows the optimum muscle speed parameter; and (c) shows the optimal series compliance parameter. In addition, a broken line in (c) shows the series compliance parameter that would eliminate the need for the muscles to do negative work.
in Fig. 5. However, the true value may be lower if the thorax functions as a parallel spring. Wells (1993) calculated that the inertial power for a hovering hummingbird was four times the aerodynamic power. By the argument used above for bees the effective mass parameter (assuming, as seems to be the case, that there is no parallel spring) is about 5. The tendons of the wing muscles and the feathers may function as series compliances but no data are available to discover whether either the series compliance or the muscle speed are close to the values that the theory predicts to be optimal. The model analysed in this paper may be regarded as representing the forward and backward swinging of a running mammal’s legs. Alternatively, the deceleration and re-acceleration of the trunk that occur during each footfall can be represented by an isolated half cycle of the model’s movement. In neither case is there any parallel spring, and in both cases the aerodynamic forces are trivial, in comparison with the inertial forces. Thus the effective mass
parameter is very high. Figure 3(d) shows that to maximize efficiency, a well adjusted series compliance is required together with relatively slow muscles. Maximizing the efficiency of aerodynamic work may seem an eccentric viewpoint on running, but it has the effect of minimizing metabolic costs. The importance of series compliances in the running of mammals is well known (Alexander, 1988). In quadrupedal mammals, the aponeurosis of the longissimus muscle and probably the fascia lata (Bennett, 1989) serve as series compliances for the forward and backward swinging of the legs. Both in quadrupeds and in bipeds, the Achilles tendon and other tendons in the distal parts of the legs serve as series compliances for the deceleration and re-acceleration of the trunk in each stance phase. In humans, the arch of the foot serves as an additional series compliance (Ker et al., 1987). In two cases, humans and kangaroos, we have data to compare empirical compliances with the theoretical optima. When the effective mass is very high, optimum series compliances will store all of the energy lost and regained in each oscillation. Ker et al. (1987) estimated for humans that the Achilles tendon and the arch of the foot together store half the kinetic plus potential
Optimum muscle speed parameter
10
259
have short fascicles and long tendons while more proximal muscles have long fascicles that can shorten fast.
5
Scallop
REFERENCES
Dolphin
2
Bee
1
Hummingbird 0.5
Runners 0
0.5 Optimum compliance parameter
1.0
F. 5. A graph with the same axes as shown for Fig. 3, showing optimal combinations of series compliance and muscle speed corresponding to estimates of the effective mass parameter for various oscillatory movements. Further explanation is given in the text.
energy lost and regained during each step. Ker et al. (1986) and Biewener & Baudinette (1995) estimated for wallabies that leg tendons store and return up to one third of the kinetic plus potential energy that is lost and regained. In each case, the series compliances seem to be smaller than would be optimal. The theory predicts that as well as having appropriate series compliances, muscles used for running should be relatively slow. Extreme examples of this are found in camels and horses (Alexander et al., 1982; Dimery et al., 1986) in which some of the more distal leg muscles have rudimentary muscle bellies with fascicles only a few millimetres long. In the extreme case of the camel plantaris, muscle fascicles only 1–3 mm long connect to a 1300 mm tendon. No measurements have been made of the shortening speeds of these fascicles but they must be very low: even a speed of many lengths per second would result only in slow movement of the muscle insertion. Very slow leg muscles may be optimal while a mammal is running at constant speed, but faster muscles are needed for acceleration and jumping. In the legs of running mammals, many distal leg muscles
A, R. MN. (1979). The Invertebrates. Cambridge: Cambridge University Press. A, R. MN. (1988). Elastic Mechanisms in Animal Movement. Cambridge: Cambridge University Press. A, R. MN., M, G. M. O., K, R. F., J, A. S. & W, C. N. (1982). The role of tendon elasticity in the locomotion of the camel Camelus dromedarius J. Zool. (Lond.) 198, 293–313. B, M. B. (1989). A possible energy-saving role for the major fascia of the thigh in running quadrupedal mammals. J. Zool. (Lond.) 219, 221–230. B, M. B., K, R. F. & A, R. MN. (1987). Elastic properties of structures in the tails of cetaceans (Phocaena and Lagenorhynchus) and their effect on the energy cost of swimming. J. Zool. (Lond.) 211, 177–192. B, A. A. & B, R. V. (1995). In vivo muscle force and elastic energy storage during steady-speed hopping of tammar wallabies (Macropus eugenii ). J. Exp. Biol. 198, 1829–1841. B, R. & C, J.-Y. (1994). Energy storage by elastic mechanisms in the tail of large swimmers – a reevaluation. J. theor. Biol. 168, 315–321. D M, M. E. (1990). Tuned oscillations in the swimming scallop Pecten maximus. Can. J. Zool. 68, 786–791. D, N. J., A, R. MN. & K, R. F. (1986). Elastic extensions of leg tendons in the locomotion of horses (Equus caballus). J. Zool. (Lond.) A 210, 415–425. D, R. & E, C. P. (1990). Mechanics of forward flight in bumble bees. II. Quasi-steady lift and power requirements. J. Exp. Biol. 148, 53–88. H, A. V. (1964). The effect of load on the heat of shortening of muscle. Proc. Roy. Soc. Lond. B 159, 297–318. K, R. F., D, N. J. & A, R. MN. (1986). The role of tendon elasticity in hopping in a wallaby (Macropus rufogriseus). J. Zool. (Lond.) A 208, 417–428. K, R. F., B, M. B., B, S. R., K, R. C. & A, R. MN. (1987). The spring in the arch of the human foot. Nature 325, 147. M, S. & Z, G. I. (1991). A distribution-moment model of energetics in skeletal muscle. J. Biomechan. 24, 21–35. M, R. L., O, J. M. G. S, K. (1992). Mechanical performance of scallop adductor muscle during swimming. Nature 357, 411–413. O, J. M. & M, R. L. (1993). Contractile properties of the striated adductor muscle in the bay scallop Argopecten irradians at several temperatures, J. Exp. Biol. 176, 175–193 V L, J. L. (1992). Muscle function in locomotion. In Mechanics of Animal Locomotion (Alexander, R. McN., ed.) pp. 191. Berlin: Springer-Verlag. W-F, T. (1972). Energetics of hovering flight in hummingbirds and Drosophila. J. Exp. Biol. 56, 79–104. W, D. J. (1993). Muscle performance in hovering hummingbirds. J. Exp. Biol. 178, 39–56. W, R., C, N. A. & H, E. (1985). Energetic Aspects of Muscle Contraction. London: Academic Press.
