MECHANISMS RELATING FORCE AND HIGH-FREQUENCY STIFFNESS IN SKELETAL MUSCLE J. BOBET,*

R. B. STErNt

and M. N. OGUZTC)R&LI

Departments of Physiology and Mathematics. University of Alberta. Edmonton, Canada Absrraet-Muscle stiffnessincreasesfaster than muscle force during the rising phase ofa tetanic contraction. and decreases more slowly during the falling phase. DitTerenl models of the stiffness arising from series, paralle!. and crossbridge elasticity were compared lo determine whether they could account quantitatively for the, observedtime course of force and stitmess. Data for slow and fast twitch mouse mucles at temper#tures from 6 to 37°C (Stein and Gordon. Can. J. Physiol. Pharmucol. 64, 12361244, 1986) and for single frog muscle fibers (Cecchi CI ul.. Contractile Mechanisms in Mu&, pp. 641655. Plenum, New York, 1984) were compared. The results showed that a good fit to Ihe data for mouse muscles could be obtained with a model in which: (I) a nonlinear serieselastici(y contributed significantly to stitiness;(2) the attached crossbrjdges went from a stilT,force-generating state lo a stiR, non-force-generating state; and (3) the rate of transition between these two states increased abruptly al the onset of relaxation. The increased transition rate probably arises from the internal rearrangement in which some sarcomeres shorten at the expense of other sr+rcomeres,once the muscle begins lo relax. A significant serieselasticity was nol required for rhe frog data. but a pre-tension state was then needed IO obtain a good fit.

1. INTRODUCTION

conditions might more properly be called ‘impedance’. because of its dependance upon the frequency of the When small, step-like stretches or releases (Icss than applied length changes, but the term ‘high-frequency about 0.5% of muscle length) arc applied to a con- stiffness’ is now generally used in the literature. tracting musc~c. the force produced by the muscle By applying, for example, a I kHz sinusoidal length shows a complex time-varying pattern consisting of change to a muscle during a time-varying contraction, several phases(Ford et ul.. 1986). The initial part of the changes in stiffnesscan be measured with a resolution response is fairly simple, however; the force produced of I ms. Typically, muscle stifTnessappears to develop is proportional lo the applied length change and is in more quickly and decay more slowly than force (see phase with it. This part of the response is typical of such rcccnt studies as Ford et ul., 1986; Stein and that of a purely elastic structure and can be quantified Gordon, 1986; Cecchi et ul., 1987; Bressleret al.. 1988; by calculating the ‘stifTness’of the muscle, the ratio of Hatta et al.. 1988). This phenomenon may have imthe resulting initial force change to the length change portant implications for understanding the role of that produced it. In single fibers or small bundles in muscles in movement, for it indicates that the muscle’s which tendon qompliance has been minimized. most of ability to resist a perturbation has a different time this stiffness arises from crossbridges between the course than its abiIity to develop force. thick and thih filaments. The slower time-varying Under normal conditions muscle contraction is pattern has also been shown to arise, not from inertial accompanied by significant internal shortening and or viscous sources, which are minimal in these prep- lengthening of sarcomeres (Huxley and Simmons, arations, but ‘from detachment and attachment of 1970; Edman and Flitney, 1982). This internal recrossbridges (Ford et al., 1986). arrangement can be minimized by isolating individual A similar calculation can also be made when the muscle fibers and using length feedback (‘length applied length change is a high-frequency sinusoid clamp’) lo maintain the length of a portion of the fiber rather than a step. At sufficiently high frequencies approximately constant. Under these conditions the (from several hundred lo several thousand Hz, de- time course of stiffness and force changes have been pehding on experimental conditions) the muscle again attributed to specific states in the crossbridge cycle displays a purely elastic response, with the force (Cecchi et al.. 1984, 1987). Whether the relative time changes being. proportional to the applied length course of stiffness and force in whole, unclamped changes and in phase with them. The ratio of the muscles can be similarly explained is not clear, since change in force to the change in length under these factors such as tendinous compliance and internal shortening could presumably dominate the responses. Several hypotheses have been proposed lo explain the relative time course of force and stiffnesschanges. *Present Address: Department of Physical Education (I) Cecchi et al. (1984, 1987) suggested that the lead University of Alberta, Edmonton AB. and Sport Stud and lag of stiffnesson force in isolated frog fibers were Canada T6G 2 Js9.’ due to the existence of stiffness-generatingstates, both tAddress eorrkspondence to: Dr R. B. Stein, Division of prior to and subsequent to the force-generating state Neuroscience, dniversity of Alberta, Edmonton. Canada in the crossbridge cycle. (2) Julian (1969) analysed a T6G 2S2. M

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J. BOBET et al.

model which combined the Huxley (1957&zrqssbridge scheme with a simple representation of excitationcontraction coupling and of the series elasticity. This model can be used to predict the time course of force and stiffness during twitches, tetani or other contractions. (3) Stein and Gordon (1986) proposed a simple model containing a series and an internal stiffness, both of which increased with force. They showed that the nonlinear interaction of these two could produce both an apparent lead and lag of stiffness with respect to force. (4) Stein et al. (1988) found that to fit the relationship between free calcium and force in a variety of muscles, transition rates between different attached states in the crossbridge cycle had to be allowed to vary. Such variation could obviously affect the relationship between force and stiffness and needs further consideration. The first four sections of the Results (3.1-3.4) treat each of these hypotheses in turn in relation to their ability to predict the time course of stiffness changes. Overall, a broad range of conditions in whole mouse muscles have been studied. Slow and last twitch muscles were examined at various temperatures and for a variety of activation patterns (twitches, doublets and tetani). Data from single frog fibers wcrc also considered. By combining ideas from scvcral of these hypotheses, we developed fairly simple models. based on likely physiological mechanisms, which arc quite successful in fitting the data studied.

several attached states, for example in the sequence AM.ADP.P,

-, AM.ADP

-. AM

where A = actin, M = myosin, ADP = adenosine diphosphate and P, = inorganic phosphate. If these represent a pre-tension state (x,). a tension-producing state (x2) and a post-tension or rigor state (x,). then we have the kinetic scheme shown in Fig. 1. where x0 and xb represent one or more unattached states that do not contribute to observed force and stiffness and the J; and g, represent forward and backward rate constants. The total contribution to stiffness of the bonds in the ith state, K,(t), will be the product of the stiffness of each bond and the number of bonds in that state. For simplicity, we have assumed here that the stiffness of a bond is the same for all states and is equal to unity. Under these conditions, we can write from standard chemical kinetics that dKl(t)/dl=/tK,(t)+8*K,(I)-(j~+g,)K2(t)

(1)

dK,(t)ldl=/,K,(I)-(l,+g,)K,(t).

(2)

f -X xO0

fZ,

1,

f3

, --x2-x3---‘X’ 91 92 F S

0 S

0

2. MlTilODS The data used for whole mouse soleus and extensor digitorum longus (EDL) muscles were those of Stein and Gordon (1986). Data for frog fibers from Cecchi et of. (1984. their Fig. 2) were digitized and interpolated using a cubic spline. All models except the Huxley-Julian model of Section 3.2 were described by equations with unknown parameters. The values for the parameters were varied systematically using a gradient-search algorithm until the best-fitting values were obtained (see Stein et 01.. 1988). Goodness of fit was quantified by the sum of squared deviations between the predicted and observed stiffness values, divided by the standard deviation of the observed stifl’ness. This ratio was multiplied by 100 to obtain a root mean square (RMS) error in per cent. For the Huxley-Julian model, this approach was not practical because of the computing time involved. It was implemented as described by Stein and Wong (1974) and its behavior was examined qualitatively.

3. RESULTS 3.1. Crossbridge auached

models with pre- and post-tension

stales

In vitro studies (reviewed in Eisenberg and Hill, 1985) have shown that the crossbridge cycle contains

A

2 ‘..

“0

REUTIVE

mcor. Model

stiffness 1

Model 3

FORCE

1

Fig. I. Top: Three-attached state model of crossbridgc cycle. Lcttcrs S. F. and D denote stilhxss-generating. forcc-gcncrsting, and detached states. Ttio detached states may or may not be the same state. The /1 and gr indicate forward and backward rate constants. Middle and bottom: time history and phase-plane representation ol force and stiffnessfor frog TA fiber at 4°C. Data from Cccchi et al. (1984). Mcasurcd stinircss shows a ‘lead’ over force on the rising phase and a ‘lag’ during the falling phase, but little or no hysteresis. The stilTncsspredicted by both a three-state crossbridge model (Model I) and a simple model with two seriessprings (Model 3) agree well with the data.

Modelling mude

stifhss

I5

HUXLEY-JULIAN

The total stiffnessK(t) and the force F(t) are&n

MODEL

by K(t)=K*(r)+Kz(t)+K,(t)

(3)

F(t)= &0)/b

(4)

where b is a constant relating the stiffnessto the force of attached crossbridges. Solving equations (1) and (2) for I