Journal of Muscle Research and Cell Motility 16, 249-256 (1995)

Simple modelling of linear and nonlinear mechanical responses to sinusoidal oscillations during muscle contraction H. IWAMOTO

Department of Physiology, School of Medicine, Teikyo University Itabashi-ku, Tokyo 173, Japan Received 12 July 1994; revised 20 December 1994; accepted 20 December 1994

SuiIlmary The mechanical responses of muscle to sinusoidal oscillations were analysed using a contraction model based on Huxley's 1957 model. To reproduce the three well-documented exponential processes resolved in the course of sinusoidal analysis, a few modifications had to be made on the original Huxley's model: (1) Each myosin cross-bridge should support substantial force at its position where the rate of detachment (g) abruptly changes. (2) Each cross-bridge should have the capability of stress-relaxation while attached to actin. This stress-relaxation should occur with a rate constant greater than the overall turnover rate of cross-bridges. After these modifications, the model reproduces not only the three exponential processes, but also the nonlinearity in the mechanical response, which is derived from the asymmetrical dependence of the rate constants on the cross-bridge strain. The present study provides a simplest model which reproduces the linear and nonlinear responses to sinusoidal oscillations, and help relate the exponential processes to the underlying attachment/detachment steps in the cross-bridge cycle.

Introduction

Muscle contraction results from the mechanical interaction between the two major protein components, actin and myosin, which are arranged in a regular array in the form of myofilaments. This arrangement of proteins provides a special advantage in studying the nature of chemomechanical transduction, since one can perturb their interactions in a concerted manner by externally applying length changes to a contracting muscle. Among other types of such experiments, the analysis of tension response to sinusoidal oscillations (sinusoidal analysis) is of special importance: if a muscle behaves as a linear system, the response to any type of length perturbation is predicted from its responses to sinusoidal vibrations. It has been shown that the sinusoidal analysis of contracting muscle resolves three discrete mechanical components, each of which is described as a single term of exponential decay and therefore called an exponential process (e.g., Kawai & Brandt, 1980). The second fastest exponential process (process b) has a polarity opposite to others (representing negative viscosity), and is responsible for the well-documented oscillatory work in insect flight muscle (Machin, 1964; Pringle, 1978, and references therein) 0142-4319/95 ~) 1995 Chapman & Hall

as well as in other vertebrate and invertebrate cross-striated muscles (Kawai & Brandt, 1980). The sinusoidal analysis is potentially a very powerful tool for studying the chemomechanical transduction in muscle, but there are following two problems which keep investigators from applying it more widely. (1) In order for the analysis to be valid, the system to be analysed has to be linear (i.e., the magnitude of the response is proportional to the amplitude of the imposed perturbations and is symmetrical with respect to their polarity). However, it is not clear whether a muscle can be regarded as a linear system. The energetics of muscle contraction and the rate of tension recovery after sudden length changes led Huxley (1957) and Huxley and Simmons (1971) to models which are highly nonlinear in nature. (2) The magnitudes and the rate constants of the observed exponential processes have been nicely correlated to chemical reaction steps (Zhao & Kawai, 1993 and references therein). However, it is not clear as to how these reaction steps are translated to corresponding mechanical components. The presence of three exponential processes itself has been explained in terms of various mechanistic contraction models (e.g. Thorson & White, 1983; Murase et al.,

250

IWAMOTO

1986). The fact that the three exponential processes can be explained b y a variety of m o d e l s m a y i m p l y that a correct m o d e l c a n n o t be d e t e r m i n e d uniquely. In the p r e s e n t study, a t t e m p t s w e r e m a d e to d e t e r m i n e w h a t conditions are required for a contraction m o d e l to r e p r o d u c e all the three exponential processes. The starting m o d e l is H u x l e y ' s (1957), w h i c h is simple b e c a u s e it is b a s e d o n a two-state model. By adjusting the length sensitivity of the rate constants a n d force-extension curve of a cross-bridge, o n e can fit various e x p e r i m e n t a l results to this model. It is d e m o n s t r a t e d that the c o m b i n a t i o n of H u x l e y ' s (1957) a n d H u x l e y a n d S i m m o n s ' (1971) m o d e l s is n e e d e d to explain the three exponential processes. Since b o t h of those m o d e l s a s s u m e highly a s y m m e t ric b e h a v i o u r of cross-bridges, the c o m b i n e d m o d e l is also useful to s t u d y the c o n s e q u e n c e s of nonlinearity. In spite of large a s y m m e t r y a s s u m e d in the m o d e l , the h a r m o n i c c o m p o n e n t in the calculated tension r e s p o n s e is small, indicating that the m o d e l b e h a v e s as a linear s y s t e m to a first a p p r o x i m a t i o n . Rather, the a s y m m e t r y is reflected on a shift of the DC level (centre of oscillation) of tension a n d on the s h a p e of the N y q u i s t plot. The p r e s e n t s t u d y s h o w s that the sinusoidal analysis is useful not only in extracting the rate constants of e l e m e n t a r y a c t o m y o s i n reaction steps, but also in assessing their a s y m m e t r i c a l d e p e n d e n c e on strain. The c o n s e q u e n c e s of the s t r a i n - d e p e n d e n c e o n ATPase activity d u r i n g oscillation are also discussed.

Materials and methods Modelling

Basically, the model follows Huxley's (1957) and shares most of the features. Two states are assumed for each myosin head, i.e. attached and detached. The fractions of the attached and detached heads are expressed as A and D, respectively. The number of attached heads is a function of the position (x) along the myosin filament backbone. (At x = 0 the attachment and detachment rate constants change abruptly, as described below. In Huxley's original model, it coincides with the neutral point, i.e., the force exerted by a myosin head is zero). It is therefore expressed as a(x) and is related to A and D by the following equation (x is positive in the direction of stretch). A = f;a(x)dx

= 1- D

(1)

The first-order rate constants of attachment and detachment are expressed as f i x ) and g(x) respectively. The reverse reactions are ignored unless stated otherwise. The main difference between the present and original models is the profiles of f i x ) and g(x), which are shown in Fig. la. The position of the head (x) is divided into three regions by 0 and a constant h, as in the original model. However, f i x ) and g(x) are assumed to be constant within

150 a ......... ,g........

b

C

100(

100 -2h /

50

h

~

(x)

2h

e~ 3 -2h

-h

150(

h

(×)

2h

s:O

500

4h

-'h

~ (×) i,

Fig. 1. Rate constants and force per myosin head in the model. (a) Rate constants for attachment (f, solid line) and detachment (g, dotted line) of myosin heads as functions of the position of attachment (x). (b) Force per myosin head (P) as a function of x. Two examples are shown: in one, the force is proportional to x, and in the other, proportional to x + 2h. (c) Rate constant for the decay of viscoelastic drag as defined by Equation 6. The cases of s = 0 and s = 6 are shown. In the former case, the rate is independent of x. In the latter, the rate depends on x in a manner qualitatively similar to the rate of quick recovery of tension after a step length change (Huxley & Simmons, 1971). each region. In the region x > h, g(x) is assumed to be 20% of that in the region 0 ~ x ~ h, to accommodate the lower energy liberation rate observed during stretch (Curtin & Davies, 1973). The distribution of myosin heads along the actin filaments is governed by the following differential equation: da(x) _ f ( x ) D - g(x)a(x) dt

(2)

where t is time. Initially, we will assume that the force per myosin head (P) is proportional to x (Equation 3), as in the original model. Later we will also consider the case in which the force is proportional to x + 2h (Equation 4). The profiles of force in these cases are shown in Fig. lb. P = cx

(3)

p = 1/2c(x + 2h)

(4)

Here, c is a constant and the force p is an arbitrary unit. Analysis of tension transients following sudden length changes led to models in which a myosin head rotates to a new angle to ensure a rapid recovery of tension (Huxley & Simmons, 1971). To make the model as simple as possible, this feature was introduced to the present model by giving a capability of stress-relaxation to myosin heads, i.e., each myosin head is assumed to behave as a Voigt element-type visoelastic body. The contribution of this viscoelasticity is added to the force described by Equation 3 or 4. It is assumed that the immediate amplitude of the response of the Voigt element to a sudden length change is proportional to the amplitude of length change, and decays exponentially at a rate k. In this case, the force exerted by each myosin head P' is expressed as P' = P + Pv x e x p ( - k t )

(5)

where P~ is a constant and k shows the following dependence on x, k = u x exp (sx/h)

(6)

Sinusoidal analysis of muscle

251

where s and u are constants. When s is 0, k is independent of x. When s is positive, k decreases as x increases in accord with the observations made by Huxley and Simmons (1971). Figure lc shows the dependence of k on x for the cases of s = 0 and s = 6. The present calculations include seven combinations of these parameters. They are called Models 1-7 and are summarized in Table 1.

Sinusoidal analysis of the model The sinusoidal analysis was performed by oscillating the x-parameter of Models 1-7 with a peak-to-peak amplitude hi2 to h. First, the space along the fibre axis was divided into many discrete segments (e.g. the region between x = 0 and x = h was divided into 20 segments). For each moment, the fraction of attached heads in each space segment a(x) was calculated. The change of a(x) in a unit time was calculated from fix), g(x), and D and a(x) a unit time before. The total tension was calculated by integrating P' x a(x) over the whole space. Length change was performed by causing a frame shift of a(x) to neighbouring segments. This operation caused the generation of viscoelastic resistance in each attached head, which was assumed to decay as defined by Equations 5 and 6, and to disappear upon detachment. The amplitude of the response, the phase shift of the response with respect to applied oscillations, the DC (frequency independent) component, the harmonic power (2nd through 10th) and the energy rate (= overall detachment rate where reverse reaction of attachment is ignored) were calculated by using a personal computer (IBM, PS/55). The results of the sinusoidal analysis is presented in the form of a Nyquist plot, in which the quadrature component of the tension response {magnitude of tension response x sin (phase lead of tension over length)} is plotted against the in-phase component {magnitude x cos (phase lead)} for a range of frequencies. A single exponential process, which corresponds to a normal viscoelastic decay, appears as a hemicircle above the abscissa. If the process represent negative viscosity, the hemicircle becomes convex-downward. For skeletal muslces, three exponential processes have been reported (processes a, b and c, from slow to fast; see Kawai & Brandt, 1980. This nomination is also used in the present paper), with the second one having a negative polarity. Thus, the Nyquist plot forms a characteristic loop in the middle frequencies.

Results

and discussion

Response of the total number of attached heads (A) to oscillation W h e n the x - p a r a m e t e r of the two-state m o d e l (Model 1, Table 1) is oscillated, the total n u m b e r of attached h e a d s (A) also oscillates in r e s p o n s e . At higher frequencies, the oscillation of A lags b e h i n d length. In the stretch p h a s e of oscillation, this lag is caused b y the d e l a y e d a t t a c h m e n t of m y o s i n h e a d in the region 0 < x < h. In the release phase, it is caused b y the d e l a y e d d e t a c h m e n t in the region x < 0. At l o w e r frequencies, A changes in the o p p o s i t e directions, i.e. A decreases in the stretch p h a s e a n d increases in the release p h a s e . This reversal is c a u s e d b y the detachm e n t in the region x > h in the case of stretch, a n d the a t t a c h m e n t in the region 0 < x < h in the case of release. Therefore, the p h a s e of A s h o w s an a d v a n c e o v e r length oscillations at these low frequencies. Thus, it is expected that the r e s p o n s e of A to oscillations has t w o e x p o n e n t i a l processes: a negatively-signed o n e at h i g h e r frequencies a n d a n o r m a l one at l o w e r frequencies. In accord with this expectation, the N y q u i s t plot for A (Fig. 2a) h a s t w o hemicircles. If muscle tension w e r e directly p r o p o r tional to A, the changes in A alone w o u l d explain the characteristic negative viscosity of process b f o u n d b o t h in insect a n d v e r t e b r a t e muscles.

Tension response to oscillation Figure 2b s h o w s the N y q u i s t plot of the tension r e s p o n s e of Model 1 (see Table 1), in w h i c h the t e n s i o n p e r h e a d (P) is a s s u m e d to be p r o p o r t i o n a l to x a n d the viscoelasticity of m y o s i n h e a d s is not t a k e n into consideration. The hemicircle r e p r e s e n t i n g n e g a tive viscosity in Fig. 2a h a s d i s a p p e a r e d , a n d the plot is v e r y different f r o m that o b t a i n e d f r o m real vertebrate skeletal muscles. In the next step, viscoelasticity (or the capability of stress-relaxation) is i n t r o d u c e d to each m y o s i n h e a d (Model 2). This o p e r a t i o n is i n t e n d e d to incorporate the p r o p e r t y of m y o s i n h e a d s to recover their tension quickly after a sudd . length c h a n g e (Huxley &

T a b l e 1. Parameter settings in models.

Type Model I Model 2 Model 3 Model 4 Model 5 Model 6 Model 7

xrange wheref > 0 0 < x < h

0< 0< 0< 0< 0< 0<

x x x x x x

< < < < < <

h h 0.7h h h h

ginO < x < h (s -I )

ginx > h (s -1)

Force

u (s -1)

s 0 0 0 0 6 0

6

1.2

Equation 3

-

6 6 6 18 6 6

1.2 1.2 1.2 1.2 1.2 3.0

Equation Equation Equation Equation Equation Equation

1200 1200 1200 1200 1200 1200

3 4 4 4 4 4

252

IWAMOTO

3 a

o@

20

d Mode] 3

10

10

20

30

4'0

-10 2O

-3

e

Model 4

10 0 40

b

4'0

Mode] 1

-10 2O

2O o,

f

Model 5

8'O 10

-20

0

4'0

-10 60

20,

C Mode] 2

30 0

g

Model 6

10

30

60

90

10

20

30

4'0

-30

Fig. 2. (a) Nyquist plot for the fraction of total attached myosin heads A during oscillations with an amplitude of hi2. The viscous (quadrature) modulus is plotted against the elastic (in-phase) modulus for frequencies 0.3-1000 Hz. The moduli are expressed as a percentage of the number of attached heads under isometric conditions. This plot applies to all the models except for Models 4 and 7. (b-g) Nyquist plot for the tension response to oscillation. (b) Model 1, in which the force per myosin head (P) is proportional to x. (c) Model 2. The same as Model 1 but each myosin head shows a viscoelastic drag which decays with a rate constant independent of x. (d) Model 3. The same as Model 2 but P is proportional to x + 2h. (e) Model 4. The same as Model 3 but the range in which myosin heads are allowed to attach is confined to the region 0 < x < 0.7h. (f) Model 5. The same as Model 3 but g in the region 0 < x < h is increased threefold. (g) Model 6. The same as Model 3 but the rate of visoelastic decay depends on x (see Fig. lc). In (b-g) the moduli are actually the amplitude of the tension response expressed as a percentage of the isometric tension. The frequencies at which calculations were made are 0.3, 0.5, 0.7, 1, 1.4, 2, 3, 5, 7, 10, 14, 20, 30, 50, 70, 100, 140, 200, 500, and 1000 Hz in a clockwise order. In each plot, circles mark the frequencies 1, 10, 100 and 1000 Hz in a clockwise order. The thin broken lines in (c-g) represent the best-fit three exponential processes (see Table 2).

Simmons, 1971). The rate of viscoelastic decay is a s s u m e d to be i n d e p e n d e n t of position x. The Nyquist plot for Model 2 (Fig. 2c) consists of two hemicircles connected in series. Naturally, both hemicircles represent normal viscosity. The inability of Model 2 to express negative viscosity could be the consequences of the fact that the h e a d s in the region of x w h e r e most of the delayed a t t a c h m e n t / d e t a c h m e n t occurs (i.e. x ~ 0) contribute little to tension. The existence of the negative viscosity in real muscle suggests that the h e a d s a r o u n d x = 0 contribute to tension substantially. For this reason the m o d e l is modified so that the tension per m y o s i n h e a d (P) is proportional to x + 2h instead of being simply proportional to x (Model 3). Figure 2d shows the Nyquist plot for such a case. N o w the plot has a loop b e t w e e n the two positively oriented hemicirdes, thus r e p r o d u c i n g the three exponential process resolved in real muscles. The fitted a p p a r e n t rate constants and m a g n i t u d e s of the three processes are s u m m a r i z e d in Table 2. The fit of the m o d e l to exponential processes ( s h o w n as thin b r o k e n lines in Fig. 2) are reasonably g o o d for processes b a n d c. Process a in the present m o d e l is intrinsically nonlinear and therefore, the fit is relatively poor. The loop can be m a d e e v e n greater b y limiting the range for h e a d attachment ( f i x ) > 0) to the region 0 < x < 0.7h (Model 4, Fig. 2d). This is because a greater p r o p o r t i o n of the m y o s i n h e a d s are recruited to the process of delayed a t t a c h m e n t / d e t a c h m e n t with the same amplitude of length change. The m a g n i t u d e of process b is 62% greater t h a n in Model 3 while the a p p a r e n t rate constants are not affected (Table 2). A greater loop can also be obtained b y increasing g(x) in the region 0 < x < h (Model 5). This operation m a y partly r e p r o d u c e the effect of inorganic phosp h a t e (Pi), which increases the m a g n i t u d e and the a p p a r e n t rate constant of process b in the sinusoidal analysis (Kawai, 1986; Kawai et al., 1987). The effect of Pi is p r e s u m a b l y to accelerate the reverse reaction of attachment, since the rate of tension rise following

Table 2. Fitted rate constants and magnitudes for the three

processes. Rate constant (s -1)

Magnitude (arbitrary unit)

Type

a

b

c

A

B

C

Model 2 Model 3 Model4 Model 5 Model 6 Model 7

3.88 3.88 3.88 2.48 3.88 7.87

79.5 79.5 79.5 125.6 63.2 125.6

1256 1256 1256 1256 488 1256

5.30 3.72 4.10 3.52 3.75 3.56

-0.04 -1.10 -1.78 -1.56 -0.72 -1.17

5.13 5.27 4.95 4.16 4.80 5.36

Sinusoidal analysis of muscle

253

activation is also increased (Rfiegg et al., 1971). In the present model, increasing the reverse rate constant of attachment is equivalent to increasing g(x). Figure 2e shows the Nyquist plot for this case. A threefold increase of g(x) in the region 0 < x < h results in a 42% increase in magnitude and a 58% increase in the apparent rate constant of process b (Table 2).

Effect of nonlinearity on sinusoidal analysis The present models have two sources of nonlinearity. One is from the rate constants f(x) and g(x), which are asymmetrical with respect to the position along the filament. The effect is already evident from the fiat shape of the Nyquist plot for the total number of attached heads A (Fig. 2a). The other is from the rate of viscoelastic decay for each myosin head (= rate of quick tension recovery after application of a quick release). This is assumed to be independent of x in Models 2-5, but it has actually been reported to be highly asymmetrical (Huxley & Simmons, 1971; Ford et al., 1977). Here the effect of introducing asymmetry to the rate of viscoelastic decay is discussed.

a

Figure 3a is a collection of length-tension diagrams during oscillation of Model 3, in which the rate of viscoelastic decay is independent of x. At lower frequencies, the DC level of tension has shifted above the isometric tension level (indicated as a dot) and the length-tension loops are highly distorted. Figure 3b shows the plot of the harmonic power (the sum of 2nd through 10th harmonics) against frequency. The harmonic power increases as lowering frequendes. The shift of the DC level is actually observed in living muscles (Wakabayashi et al., 1985). The asymmetry is then introduced to the rate of viscoelastic decay (Fig. 4, Model 6). Its dependence on x is expressed by Equation 6 in which s = 6 (Fig. lc). Now the distortion of the length-tension loops is observed also at higher frequencies (Fig. 4a) and a new peak of harmonic power appears (Fig. 4b). The new peak is at frequencies just above the frequency domain for process b, and this is actually observed in rabbit (Kawai & Brandt, 1980) and frog fibres (Iwamoto, 1995b). In spite of the large asymmetry introduced to the rate of viscoelastic decay, the harmonic power is at most a few percent of total at its peak. Other consequences of introducing the asym-

a

b b

o

3o 2

¢--

Q_ k...

¢-

0

'-- . 1

I

I

1

-

-

10 freq.

-':-'-

100

1000

(Hz)

Fig. 3. Nonlinearity in the tension response of Model 3 to oscillations with an amplitude of h. (a) Length-tension loops at frequencies 0.3, 3, 10, 20, 100 and 500 Hz. The length is displayed horizontally and the tension vertically (positive to the right and above respectively). The dot represents the isometric level. Note the distorted loops at 0.3 and 3 Hz. (b) The frequency dependence of nonlinear power as expressed as the sum of 2nd through 10th harmonic components squared. The second harmonic component has the largest contribution.

• r--" ¢,....

"-0 o ¢--

1

10 freq.

100

1000

(Hz)

Fig. 4. Effect of introducing asymmetry to the rate of viscoelastic decay (i = 6 in Equation 6) on the nonlinearity of the tension response (Model 6). The length-tension loops (a) and the frequency dependence of the nonlinear power (b) are expressed as in Fig. 3. Note that the loops at 10-100 Hz are also distorted, and that the nonlinearity has a second peak at around 20 Hz.

IWAMOTO

254 metry are the tendency of the hemicircle for the fastest process (process c) to be flattened and the diminution of the size of the loop in the Nyquist plot (Fig. 2f). The former effect is due to the distributed rate constant of process c. The latter is caused because the lower end of the frequency range for process c overlaps the range for process b. It will be shown in Iwamoto (1995b) that two types of fast twitch fibres are found in the anterior tibialis muscle of the frog. In one type (worker), a clear loop is observed in the Nyquist plot and the phase of tension with respect to length is negative in a range of frequencies (positive work is produced by the fibre). In the other type (idler), the loop is much smaller or missing, and the phase is always positive, i.e. positive work is not produced at any frequency. Since the 'idler' type of fibres always show higher nonlinearity at frequencies just above the frequency domain for process b, the difference between the two types of fibres may be due to the difference in the nonlinearity of process c. Nonlinear behaviour is also observed in insect flight muscles, particularly in large-signal cases (Thorson & White, 1969; White & Thorson, 1972; Abbott, 1973).

a

o

E 0 o !

.¢..-

2

3

b

4

° t'~°"~.f . . . . . . . . . . "0 . . . . . . . . . . . . • .°°o. "t° o~°°"

¢

ZZ~ t

0

E O~ 0 0

I

Fitting to real data Although the present models are not intended to obtain exact fits to real data, it is of interest to know how well these models reproduce the Nyquist plots obtained from real frog fibres. Examples are shown in Fig. 5. In Fig. 5a, the parameters of Model 3 are modified to fit the Nyquist plot obtained from one of the 'worker' type of fibres (dotted line) from frog anterior tibialis. In Fig. 5b, only the nonlinearity of process c has been altered to test whether this operation produces the feature of the 'idler' type of fibres. Although the absolute amount of modulus is different, the model reproduces the feature of the 'idler' (dotted line) remarkably well. A reasonably good fit to the 'worker' type (Fig. 5a) is obtained when the length dependence of the force per myosin head P is made very small. Since P represents the force when the viscoelastic resistance has decayed, the small dependence of P on x may correspond to the flat part of the T2 curve in Huxley and Simmons (1971) or Ford and colleagues (1977). This consideration raises a possibility that the attachment of myosin heads occur mainly within the flat part of the T2 curve ( ~ 4 n m ) , which is much narrower than the molecular size of myosin head (-20 nm). The idea that the attachment of the myosin heads occurs within a region much smaller than the molecular size leads to an expectation that, during isometric contraction, the attached myosin heads are in good register with respect to the 14.3 nm axial repeat along the thick filament. This view is sup-

1

2 elastic

3 modu]us

4

Fig. 5. Fitting of the model to real muscle fibres. (a) A well-fit Nyquist plot (solid line) for one of the 'worker' type of fibres (dotted line, see Iwamoto, 1995). (b) The same model as in (a) but nonlinearity is introduced to process c (solid line). Presented in the same scale as in (a). The Nyquist plot of one of the 'idler' type of fibres (dotted line) is also shown. The unit of moduli is 107Nm -2. The conditions of the simulation are as follows: g = 12 s -1 in the region 0 < x < h, and 2.4s -1 in the region x>h.P=c(h+O.Olx). In (a) u = 7 6 0 s -1, s = 0 . In (b) u = 600 s -1, s = 4. The frequencies used for calculations are the same as in Fig. 2. The frequency range for real fibres is 1.4-200 Hz in (a) and 1-200 Hz in (b). ported by the X-ray observation that the 14.3 nm meridional reflexion is intensified during isometric contraction and weakened upon length changes (Huxley et al., 1981). The stress s~ain curve of skinned rabbit psoas fibres in the presence of inorganic phosphate also suggests that the myosin heads are in register within a range of a few nanometers (Iwamoto, 1995a). In the present study, relatively large amplitude of oscillation (h/2 to h) is required to reproduce the Nyquist plot of the real muscles. If h is 10 nm, the amplitude corresponds to 0.5-1% of fibre length, a value greater than employed by Kawai and Brandt (1980) (--0.25%). This discrepancy could also be explained if the attachment is confined to a region narrower than in Huxley's (1957) original model.

255

Sinusoidal analysis of muscle

ATPase activity during oscillation

Conclusions

In insect flight muscles, the ATPase activity is known to increase in the frequency range where process b is prominent (Steiger & Riiegg, 1969). In rabbit skeletal muscle fibres, however, the ATPase activity increases only at high frequencies corresponding to process c (Kawai et al., 1987). It is therefore of interest to know whether the present two-state model reproduces this situation. The ATPase rate is calculated in Fig. 6 by assuming that the ATP splitting is coupled to the detachment of the myosin head, as in Huxley's 1957 model. In the case of Model 2, the ATPase rate during oscillation falls below the isometric level at low frequencies, and partially recovers at higher frequend e s (Fig. 6a). The generally low ATPase rate is due to the shift of the cross-bridge population to the region of higher x, where g(x) is low. If g(x) in the region x > h is increased 2.5-fold (Model 7), the ATPase rate is less affected by the osdUations at low frequencies and is increased at high frequendes (Fig. 6b). This situation is similar to the measured ATPase rate in rabbit skinned fibers (Kawai et al., 1987). A possible explanation of the discrepancy between vertebrate skeletal and insect flight muscle is that the movement of detached myosin heads in insect flight muscle is restricted. Then only a stretch would bring the myosin heads in the region of low fix) to the region of high fix), resulting in an increased ATPase rate.

10 a

~0

I

1

¢0

a-lO

I

10

!

I

1000

100

b ....

.... L?

I

I

1

10 freq. (Hz)

I

100

i

1000

Fig. 6. Averaged ATPase activity per myosin head during oscillation with an amplitude of h as a function of frequency. (a) activities for Models 1-3. (b) Effect of increasing g in the region x > h from 1.2 s -1 to 3 s -1 (Model 7). The dotted line represents the isometric level.

The present calculations show that a two-state model based on Huxley's (1957) is basically sufficient for explaining the three exponential components resolved by the sinusoidal analysis, provided the viscoelastic properties of the myosin heads are taken into account. Process b has a polarity opposite to others, and is responsible for the oscillatory work production in insect flight muscles. This process has attracted much interest from investigators, but the mechanism which enables this to occur is not obvious. An early explanation was that a stretch increases f (Julian, 1969; Thorson & White, 1969). In the calculations made by Thorson and White (1983), a strain-dependent transition from one attached state to another was also taken into consideration. In the calculations made by Murase and colleagues (1986), process b was ascribed to the delayed attachment/detachment of tension-bearing myosin heads. Common to these explanations is that the myosin heads are assumed to bear tension where their attachment/detachment or transition is most sensitive to strain. The present calculations also share this feature. An abrupt transition of rate constants occurs at x = 0 (Fig. la), where myosin heads bear little tension in the original model (Huxley, 1957 and Model 1). In order to reproduce process b, it is necessary to modify the model so that myosin heads around x = 0 bear positive force. A similar dependence of tension per myosin head on x was also assumed for insect flight muscles (White & Thorson, 1973). In the original model of Huxley (1957), the forcevelocity relation is determined by the balance between the populations of myosin heads bearing positive and negative tensions. After the modification of the model, the balance between the sliding force and the viscoelastic resistance will be an important factor in determining the force-velodty relation. Preliminary calculations show that the present model readily reproduces the force-velodty relation and the energy-liberation rate of real muscles, as well as the isotonic transients observed by Civan and Podolsky (1966) (calculations not shown). The asymmetrical dependence of the rate constants is shown to have only a modest effect on the harmonic power in the tension response. This ensures that a muscle can be treated as a linear system to a first approximation. In other words, a small harmonic power could imply substantial asymmetry in the rate constants. To assess the straindependence of the rate constant correctly, attention should also be paid to the DC level of tension and the distortion of the hemicircles in the Nyquist plot. A flattening of a hemicircle in the Nyquist plot may sometimes be taken to represent a mixture of two or

256 m o r e i n d e p e n d e n t exponential processes, b u t it could simply represent a single process with distributed rate constants. A l t h o u g h the sinusoidal analysis is useful in obtaining information a b o u t the u n d e r l y i n g mechanism of contraction, one s h o u l d be careful in interpreting the results. In particular, the p r e s e n t twostate m o d e l raises the following points. (1) There is n o t always a one-to-one correlation b e t w e e n the exponential processes a n d the u n d e r l y i n g e l e m e n t a r y reaction steps. In the p r e s e n t calculations, b o t h a t t a c h m e n t a n d d e t a c h m e n t steps contribute to the processes a a n d b at the same time. (2) W h e n their characteristic frequencies overlap, t w o exponential processes could cancel each other. Thus, the absence of an exponential process does n o t necessarily m e a n that the u n d e r l y i n g elementary steps are also absent. In conclusion, calculations u s i n g a simple two-state m o d e l s u g g e s t that the sinusoidal analysis has an ability to p r o v i d e a multitude of information about the u n d e r l y i n g m e c h a n i s m of muscle contraction. Most i m p o r t a n t of all is that the sinusoidal analysis is capable of reporting the strain-sensitivity of rate constants, w h i c h constitutes the very essence of muscle contraction.

Acknowledgements The a u t h o r w o u l d like to express his t h a n k s to Professor H. Sugi for his s u p p o r t of this study.

References ABBOTT, R. H. (1973) An interpretation of the effects of fiber length and calcium on the mechanical properties of insect flight muscle. Cold Spring Harbor Symp. Quant. Biol. 37, 647-54. CIVAN, M. M. & PODOLSKY, R. J. (1966) Contraction kinetics of striated muscle fibres following quick changes in load. J. Physiol. 184, 511-34. CURTIN, N. A. & DAVIES, R. E. (1973) Chemical and mechanical changes during stretching of activated frog skeletal muscle. Cold Spring Harbor Syrup. Quant. Biol, 37, 619-26. FORD, L. E., HUXLEY, A. F. & SIMMONS, R. M. (1977) Tension responses to sudden length change in stimulated frog muscle fibres near slack length. J. Physiol, 269, 441-515. HUXLEY, A. F. (1957) Muscle structure and theories of contraction. Prog. Biophys. Biophys. Chem. 7, 255-318. HUXLEY, A. F. & SIMMONS, R. M. (1971) Proposed mechanism of force generation in striated muscle. Nature 233, 533-8. HUXLEY, H. E., SIMMONS, R. M., FARUQI, A. R., KRESS, M., BORDAS, J. & KOCH, M. H. 1. (1981) Millisecond timeresolved changes in X-ray reflections from contracting muscle during rapid mechanical transients, recorded using synchrotron radiation. Proc. Natl. Acad. Sci. USA 78, 2297-301.

IWAMOTO IWAMOTO, H. (1994) Strain-sensitivity and turnover rate of low-force crossbridges in contracting skeletal muscle fibers in the presence of phosphate. Biophys. J. 68, 243-50. IWAMOTO, H. (1995) Two mechanically distinct types of fast twitch muscle fibres of the frog and their temperature sensitivity, as detected by sinusoidal analysis. J. Muscle Res. Cell Motil. 16, 257-65. JULIAN, F. J. (1969) Activation in a skeletal muscle contraction model with a modification for insect fibrillar muscle. Biophys. ]. 9, 547-70. KAWAI,M. (1986) The role of orthophosphate in crossbridge kinetics in chemically skinned rabbit psoas fibres as detected with sinusoidal and step length alterations. J. Muscle Res. Cell Motil. 7, 421-34. KAWAI, M. & BRANDT, P. W. (1980) Sinusoidal analysis: a high-resolution method for correlating biochemical reactions with physiological processes in activated skeletal muscles of rabbit, frog and crayfish. J. Muscle Res. Cell Motil. 1, 279-303. KAWAI, M., G~TH, K., WINNIKES,K., HAIST, C. & RUEGG, I. C. (1987) The effect of inorganic phosphate on the ATP hydrolysis rate and the tension transients in chemically skinned rabbit psoas fibers. Pfl~igersArch. 408, 1-9. MACHIN, K. E. (1964) Feedback theory and its application to biological systems. Symp. Soc. Exp. Biol. 18, 421-45. MURASE, M., TANAKA, H., NISHIYAMA,K. & SHIMIZU, H. (1986) A three-state model for oscillation in muscle: sinusoidal analysis. ]. Muscle Res. Cell Motil. 7, 2-10. PRINGLE, J. W. S. (1978) The Croonean lecture, 1977: stretch activation of muscle: function and mechanism. Proc. R. Soc, Lond. B201, 107-30. RI~IEGG, J. C., SCHADLER, M. STEIGER, G, J. & MULLER, G. (1971) Effects of inorganic phosphate on the contractile mechanism. Pfliigers Arch. 325, 359-64. STEIGER, G. J. & R~rEGG, J. C. (1969) Energetics and 'efficiency' in the isolated contractile machinery of an insect fibrillar muscle at various frequencies of oscillation. Pfliigers Arch. 307, 1-21. THORSON, I. & WHITE, D. C. S. (1969) Distributed representations for actin-myosin interaction in the oscillatory contraction of muscle. Biophys. ]. 9, 360-90, THORSON, J. & WHITE, D. C. S. (1983) Role of cross-bridge distortion in the small-signal mechanical dynamics of insect and rabbit striated muscle. J. Physiol. 343, 59-84. WAKABAYASHI, K., TANAKA, H., AMEMIYA, Y., FUJISHIMA, A., KOBAYASHI,T., HAMANAKA,T., SUGI, H. & MITSUI, T. (1985) Time-resolved X-ray diffraction studies on the intensity changes of the 5.9 and 5.1 nm actin layer lines from frog skeletal muscle during an isometric tetanus using synchrotron radiation. Biophys. J. 47, 847-50. WHITE, D. C. S. & THORSON, J. (1972) Phosphate starvation and the nonlinear dynamics of insect fibriUar flight muscle. I. Gen. Physiol. 60, 307-36. WHITE, D. C. S. & THORSON, J. (1973) The kinetics of muscle contraction. Prog. Biophys. Mol. Biol. 27, 173-255. ZHAO, Y. & KAWAI, M. (1993) The effect of the lattice spacing change on cross-bridge kinetics in chemically skinned rabbit psoas muscle fibres. II. Elementary steps affected by the spacing change. Biophys. J. 64, 197-210.