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Cavendish Laboratory, Cambridge CB3 0HE, United Kingdom
Correspondence: Address reprint requests to Andrej Vilfan, J. Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia. Tel.: 386-1-4773-900; Fax: 386-1-4263269; E-mail: andrej.vilfan{at}ijs.si.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSMOTORS ACTING BETWEEN A...MOTORS IN MUSCLE SARCOMERESDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSMOTORS ACTING BETWEEN A...MOTORS IN MUSCLE SARCOMERESDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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, experimental data on transients has informed theoretical models of the interaction between myosin and actin (Huxley and Simmons, 1971; Hill, 1974; Eisenberg et al., 1980; Chen and Brenner, 1993; Huxley, 2000), providing a more detailed picture than could be obtained from the force-velocity relation (Hill, 1939; Huxley, 1957) alone. The reason is that the actomyosin interaction involves several processes which occur on different timescales, and these individual components can be resolved during the transient response (Huxley and Simmons, 1971; Ford et al., 1977, 1985, 1986; Brenner, 1991; Brenner et al., 1995). The quickest process is the elastic deformation of the myosin cross-bridges that link the thick and thin filaments. Rapid transitions between two or more different bound states of the myosin molecule are thought to be the next fastest events, whereas detachment and reattachment of myosin heads occur on a slower timescale.
In an experiment to determine the isometric transient response, a muscle fiber is held at both ends to prevent it from contracting. The muscle is then suddenly shortened (or stretched) by a fixed amount, and the tension T that it generates is measured. Immediately after the imposed change of length, the tension shifts from the isometric value T0 to a new value, which is termed T1. But shortly afterwards (typically within 2 ms), the tension adjusts to a new value, termed T2. Subsequently, it gradually reverts to the original isometric value T0, and the entire transient response is usually completed in a fraction of a second. It is generally accepted that the initial response T1 corresponds to the mechanical deformation of cross-bridges and provides a direct measure of their elasticity (Huxley and Simmons, 1971). The interpretation of T2 is rather more controversial. It is often attributed to force generation by the working stroke of bound myosin molecules (Huxley and Simmons, 1971; Hill, 1974; Eisenberg et al., 1980; Huxley and Tideswell, 1996; Brenner et al., 1995; Duke, 1999, 2000) and this interpretation has recently gained support from x-ray interference techniques applied to shortening fibers (Irving et al., 2000; Piazzesi et al., 2002b). But alternative models suggest that the force regeneration might be due, in part, to the rapid binding of new myosin heads to the thin filament (Brenner, 1991; Howard, 2001), or that it might involve the activation of the second myosin head (Huxley and Tideswell, 1997).
In this article, we wish to address a fundamental problem connected with the interpretation of force transients. The present theories are all based on the consideration of a single pair of filaments, i.e., one filament containing myosin molecules, interacting with one actin filament. The dynamics of this filament pair is generalized to that of a whole muscle fiber by assuming that all filament pairs in a fiber behave in exactly the same way. This assumption is certainly justified as long as there are no static or dynamic instabilities in the system. However, the possibility of such instabilities has been known for a long time (Hill, 1974). Moreover, a stochastic model of the actomyosin cycle, based on the swinging lever-arm hypothesis, has shown that instabilities do arise when values of parameters such as the lever-arm displacement and the cross-bridge elasticity are chosen to provide effective energy transduction (Duke, 1999, 2000). Such instabilities would give rise to a region of negative slope in the T2 curve of a single filament pair. Several reasons have been advanced for the absence of any negative slope in the experimentally determined T2 curve. Huxley and Simmons (1971) argued that the power stroke is subdivided into several small steps, and fixed the step size so that the T2 curve had zero slope for limitingly small changes of length. In the model proposed by Eisenberg et al. (1980), the flatness of the T2 curve was explained by a broad distribution of cross-bridge strain after attachment, combined with a specific strain dependence of the transition rates to ensure the proper occupancies of the two bound states. A further explanation involved the compliance of the filaments and the distribution of binding sites on the thin filament in addition to a subdivided power stroke (Huxley and Tideswell, 1996). Duke (1999) has suggested that the flat T2 curve of a muscle fiber can arise despite an instability in the dynamics of a single pair of filaments, owing to the symmetry of a sarcomere. We investigate this possibility further in this article.
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METHODS |
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We assumed complete mechanical relaxation of the system in each step, i.e., the strain of all elastic elements is equilibrated before the next transition takes place. The structure involving sarcomeres, filaments, and myosin heads was described as a circuit of elements with given resting lengths and compliances. The strain of every cross-bridge was calculated, given the constraint of fixed total length of the system (isometric conditions), or of fixed force acting on the ends of the system (isotonic conditions).
The T2 transients were always determined 2 ms after the stretch/release. All other parameter values are summarized in Table 1.
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MOTORS ACTING BETWEEN A SINGLE PAIR OF FILAMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSMOTORS ACTING BETWEEN A...MOTORS IN MUSCLE SARCOMERESDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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, 2000) and shown in Fig. 1. A molecule can exist in either a detached state (D) or in one of the attached states, A1 (with ADP·Pi) or A2 (with ADP). The state A1 is also referred to as the weak-binding state (Brenner, 1991). The chemical transition between states A1 and A2 is concomittant with a conformational change of the molecule, in which the lever arm moves through displacement d. From state A2 the head undergoes another conformational change with a lever-arm shift 
, associated with ADP release. In the ATP-rich physiological environment, this step is quickly followed by detachment of the head and we therefore skip the transitional state (A3) in our model. The attachment and detachment rates, which are summarized in Table 1, determine the shape of the force-velocity relation, but have little influence on the transient response to length steps, which mainly depend on the power-stroke displacement d and the elastic constant K of the myosin cross-bridge. We assume that the transitions A1 
A2 take place on a faster timescale than the detachment and reattachment of heads. Therefore, once a head has bound with strain 
in the state A1, its state can be described as a statistical ensemble of the states A1 and A2 with probabilities given by the Boltzmann factors
 | (1) |
) = 
-P2(), where Gstroke is the free energy change associated with phosphate release.
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() (normalized to 1), then the isometric force that they generate is
 | (2) |
x is suddenly applied to the pair of filaments, the force will first change to
 | (3) |
+ x). As a result the force adjusts to
 | (4) |
As a first approximation, we can neglect the distribution of strains and assume = 0 for all myosin heads, which allows us to calculate the T2 curves analytically. Their shape depends on two dimensionless parameters,
 | (5) |
 | (6) |
A2). The parmeter 
0 measures this energy relative to the thermal energy, while 1 compares it to the magnitude of the chemical free-energy change Gstroke that accompanies the power-stroke transition. Note that if 1 < 1, the power stroke can occur immediately after the myosin head binds to the thin filament; but if 1 > 1, the conformational change is energetically inhibited initially, and it is only likely to occur once the thin filament has been moved forward by the action of other motors. On grounds of efficiency of energy transduction, we expect the value of 1 to be as high as possible, while ensuring that there is sufficient chemical energy to drive the power stroke, i.e., 1 
1 (Duke, 1999). The power-stroke displacement d has been measured directly in a number of single-molecule experiments, and the values obtained range between 5 and 10 nm (reviewed by Tyska and Warshaw, 2002). Data on the cross-bridge elasticity K are less reliable, because there are inevitably other sources of compliance in the system (Veigel et al., 1998). But indirect evidence is provided by the energetic efficiency of muscle, which peaks at 
50% (Kushmerick and Davies, 1969; Barclay, 1998; Piazzesi et al., 2002a). Thus a lower estimate for the energy stored in the elastic element after the power stroke is one-half the free-energy change accompanying the hydrolysis of an ATP molecule GATP 20 kBT, which leads to 0 
10. Taking into account that not all energy stored in the spring can be converted to mechanical work one obtains a better estimate 0 20 (Duke, 1999) (which is consistent with 1 1 if most of the energy of hydrolysis is used to power the stroke Gstroke GATP). This corresponds to an elastic constant K = 2.5 pN/nm if a power-stroke distance of d = 8 nm is assumed.
The form of the T2 curve depends critically on the values of these dimensionless parameters. If 0 > 2, an interval with a negative slope (a hysteresis) exists (Hill, 1974). This means that under isotonic (constant load) conditions the system can be bistable. The location of the interval of negative slope depends on the value of 1. If the hysteresis spans the origin, then the state with x = 0 will be unstable under isotonic conditions. Putting 
we see that this occurs if
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1 1 the T2 curve always has a negative slope around the stationary point. For example, with 0 = 20, Eq. 7 yields 0.85 < 1 < 1.22. However, it should be borne in mind that this calculation does not take into account the distribution of strains on cross-bridges within the ensemble, and so the actual range might deviate slightly from this estimate. An example of the T2 transient for a group of myosin heads between two firmly clamped filaments is shown in Fig. 2.
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1 = 1.33, dashed line) the difference between this T2 curve and the one in the strictly isometric state is only quantitative. The most important result is that the range of negative slope in T2 remains. On the other hand, in the oscillating case (1 = 0.8, dot-dashed line) the negative slope is flattened out because of the broader distribution of strains on cross-bridges.
Effect of filament elasticity
The compliance of the thick and thin filaments is also expected to affect the T2 curves. We model it by introducing a linear elasticity in the backbone connecting the heads (which is, to first order, equivalent to an elasticity in the track the motors are running on) (Vilfan et al., 1998). With 
being the linear modulus of the filaments, which has the measured value = 44,000 pN (Kojima et al., 1994), the spring constant of a filament segment of length L is Kf = L-1. In (Vilfan et al., 1998) a linear two-state model was used to study the effect of filament elasticity on force-velocity relations, showing that they reduce the isometric force if the compliance of a filament segment between two bound motors becomes comparable to the motor spring constant K. Thin filaments are just stiff enough to prevent significant losses due to this effect. However, the filament compliance can have a significantly bigger effect on the transient response. The result of a simulation is shown in Fig. 4. Although the filament compliance almost halves the total stiffness of the system, it leaves the major part of the interval of negative slope unaffected.
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38 nm (7 monomers). We therefore propose that the binding rate to a particular site is proportional to the Boltzmann factor determined by the spring distortion energy, composed of a longitudinal and a lateral component
 | (8) |
= x - (i + 7j)c and 
where j denotes the repeat on the helix and i the consecutive number of the actin monomer. The total binding rate at a given position x then reads
 | (9) |
These curves have been measured experimentally using S1 myosin heads by Steffen et al. (2001) and the data were fitted with KA = 15 pN/nm, the value which we use here. The position dependence of the binding rate for these parameters is shown in Fig. 5. A similar distribution has also been measured for myosin V (Veigel et al., 2002).
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, both the filament compliance and the discrete binding sites contribute to the flattening of the T2 curve. However, our simulation indicates that their combined effect is not sufficient to cancel the negative slope with the parameters used here. We therefore conclude that the absence of negative slope in the measured T2 curves cannot be explained in terms of a single-filament model, but requires taking into account the action of serially connected sarcomeres. |
MOTORS IN MUSCLE SARCOMERES |
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, 0.4 ATP molecules per second per head (with 25% of the heads attached this gives 1.6 s-1 per attached head), is compatible with the first scenario. However, other measurements give higher values (Ebus and Stienen, 1996), more consistent with the second case. We therefore conclude that from the experimental values of the ATPase rate in the isometric state neither scenario can be excluded.
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1, the model shows an interesting feature. In the isometric state the number of cross-bridges in the state A2 can be very low, although the generated force per attached cross-bridge is as high as 5 pN, in agreement with single-filament measurements (Kawai et al., 2000). This allows the muscle to support a load under isometric conditions with little ATP consumption (Duke, 2000). It might at first sound paradoxical that most of the isometric force is generated by the prepower-stroke state A1. This is possible because the myosin heads bind with a stochastic distribution of strains. Because those with a negative strain are more likely to undergo the power stroke and then detach, the remaining ensemble produces a positive force. In this aspect, the power-stroke actually serves to eliminate negatively strained cross-bridges rather than direct generation of positive force. This notion is in agreement with recent experiments, which have shown that high phosphate concentration does reduce the isometric force significantly (Cooke and Pate, 1985), but it does not have a visible effect on the conformation of the myosin heads or even their catalytic domains (Baker et al., 1999). But let us stress again that this holds only for the isometric state. The force in a contracting muscle originates mainly from the state A2.
Isotonic response
Another important class of experiments which provides information on the actomyosin interation involves the isotonic transient. Here the applied force is initially set at the value of the stalling load T0, so that the fiber is prevented from contracting. The force is then suddenly changed and held constant at a different value, while the length of the fiber is recorded. Some early experiments showed that after a small step change of load, damped oscillations were imposed on the steady contraction or extension of the fiber (Podolsky, 1960; Granzier et al., 1990). Such oscillations are particularly clear in recent experiments on single muscle fibers (Edman and Curtin, 2001).
One possible cause of this oscillatory response has previously been suggested on the basis of the stochastic model of the actomyosin interaction used in this article. When 1 > 1, the chemical cycles of myosin motors on the same filament can become synchronized at loads close to the stalling force (Duke, 1999, 2000). A pair of filaments then slides in a step-wise fashion under isotonic conditions. But during steady shortening, the motors on different filaments within the same muscle fiber operate out of phase, so that there is no macroscopic manifestation of the steps. However, an abrupt change in the load can cause the synchronization of a large fraction of the bound motors, whereupon the steps do become observable (Duke, 1999). Because the correlation of the motors soon decays, the macroscopic steps fade and a damped oscillation is seen.
A much stronger oscillatory response is seen in the regime where individual pairs of filaments perform oscillations in the near-isometric state (Fig. 3). The synchronized oscillations can then be very pronounced after a small decrease in the load, as shown in Fig. 11 A. On the other hand, no damped oscillations are observed after a larger drop in the load, e.g. to T0/3 (Fig. 11 B), because the individual filament pairs immediately move out of the hysteretic regime. These properties are in agreement with recent experiments on the isotonic response of single muscle fibers carried out by Edman and Curtin (2001).
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) and subsequently in a kinetic cross-bridge model with a strain-dependent detachment rate (Vilfan et al., 1999). With the parameters we use, this hysteresis covers a rather small range of velocities and only results in a small inflexion in the force-velocity curve (Duke, 1999, 2000). |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSMOTORS ACTING BETWEEN A...MOTORS IN MUSCLE SARCOMERESDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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); it typically has either a positive or a negative slope for limitingly small step displacements. Thus, in order to reproduce the experimental T2 curve of muscle, some of the parameters have needed to be finely adjusted (Huxley and Simmons, 1971; Eisenberg et al., 1980), and limitations have been imposed on the size d of the power stroke and the rigidity K of the cross-bridges (Huxley and Tideswell, 1996). In this article we have investigated the isometric transient response using a stochastic model of the actomyosin cycle. When values of the parameters d and K are chosen to explain other characteristics of muscle, like the force-velocity relation and the efficiency of energy transduction, the T2 curve of a single pair of filaments displays a region of negative slope, which cannot be eliminated by factors including filament compliance and the discrete nature of the binding sites on the actin filament. However, we argue that the symmetric structure of sarcomeres must be taken into consideration when computing the isometric transient response of muscle. After a step change of length of a muscle fiber, a redistribution of sarcomere lengths occurs within the fiber. Some half sarcomeres contract, whereas others extend. This redistribution always eliminates the negative slope, leading to a flat T2 curve. It is a generic feature of unstable elements connected in series, and does not require any special values of model parameters.
Our model shows that there are two different regimes of the microscopic dynamics in near-isometric conditions. For a range of values of the parameter 1 close to unity, the isometric point falls in the interval where the slope of the T2 curve is negative. In this case individual filament pairs oscillate with small amplitude. In the other regime, where the T2 curve has positive slope at the isometric point, the individual filaments are stationary, apart from stochastic fluctuations. The macroscopic manifestations of these two regimes differ in few respects. Because the oscillations of different filament pairs have different phases, oscillatory motion is not normally observable on the scale of a whole muscle fiber in steady conditions. However, a sudden change of load can synchronize the oscillations and thereby make them visible. The existence of damped oscillations in the isotonic transient response of single muscle fibers (Edman and Curtin, 2001) therefore argues in favor of the oscillating regime. We note, however, that damped oscillations can also be a manifestation of step-wise shortening (Duke, 1999), which can exist in both regimes. Because the efficient transduction of energy demands 1 1, which is close to value of this parameter at the boundary of the two regimes, it is possible that both regimes exist depending on conditions such as the myosin isoform, phosphate concentration, pH, ionic strength and temperature. Indeed, measurements by Edman and Curtin (2001) show a dependence of the oscillation decay on the solution pH and on muscle fatigue. Further experiments, in which conditions are systematically varied, could shed more light on the mechanism of oscillation.
As a final remark, we emphasize that according to our model, the T2 curve of an individual pair of filaments differs from that of a muscle fiber. Recently, assays have been developed to measure the force-velocity relation of a single filament within a half sarcomere (Kawai et al., 2000). In order to test our predictions, it would worthwhile to develop such high-precision techniques to measure the transient response of a single filament.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSMOTORS ACTING BETWEEN A...MOTORS IN MUSCLE SARCOMERESDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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FOOTNOTES |
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Submitted on November 5, 2002;
Revision received April 22, 2003.
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REFERENCES |
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