Kinetics of Oxygen Consumption after a Single Isometric Tetanus of Frog Sartorius Muscle at 20~ MICHAEL

MAHLER

From the Department o f Physiology, School of Medicine, University of California at Los Angeles, Los Angeles, California 90024. Dr. Mahler's present address is the Department of Pharmacology, School of Medicine, University of Southern California, Los Angeles, California 90033.

n B S T R A C T The time-course of the rate of oxygen consumption ((202) has been measured in the excised frog sartorius muscle after single isometric tetani of 0.11.0 s at 20~ To measure A(2o2(t), the change in (202 from its basal level, a novel method was devised, based on the validity in this tissue of the one-dimensional diffusion equation fi)r oxygen, established in the preceding paper. After a tetanus, A(2o2 reached a peak within 45-90 s, then declined exponentially, and could be well fit by AQo2(t) = O.0 + 0a(e -#lt - e-k2t), r2 (= l/k2), which characterized the rise of ~Qo2, was a decreasing function of tetanus duration (range: from 1.1 -+ 0.28 min [n = 5] for a 0.l-s tetanus, to 0.34 + 0.05 min[n = 8] for a 1.0-sec tetanus).rT (= 1/ kl), which characterized the decline of ~ o 2 , was not dependent on tetanus duration, with mean 3.68 -+ 0.24 min (n = 46). A fi)rthcoming paper in this series shows that these kinetics of ~ o 2 are the responses to impulse-like changes in the rate of A T P hydrolysis. The variation ofT2 with tetanus duration thus indicates the involvement of a nonlinear process in the coupling of 02 consumption to ATP hydrolysis. However, the monoexponential decline of ~o~(t), with time constant independent of tetanus duration, suggests that during this phase, the coupling is rate-limited by a single reaction with apparent first order kinetics. INTRODUCTION

T w o f u n d a m e n t a l tenets o f m u s c l e energetics a r e t h a t the f r e e e n e r g y f o r cell f u n c t i o n a n d m a i n t e n a n c e is e n t i r e l y p r o v i d e d by t h e hydrolysis o f a d e n o s i n e t r i p h o s p h a t e ( A T P ) , a n d t h a t the p r i m a r y s o u r c e o f A T P is t h e o x i d a t i o n o f substrates by m o l e c u l a r o x y g e n . A c o u p l i n g b e t w e e n the hydrolysis o f A T P a n d its resynthesis via oxidative m e t a b o l i s m thus a p p e a r s essential f o r n o r m a l muscle f u n c t i o n , a n d the e l u c i d a t i o n o f the m e c h a n i s m s by w h i c h this o c c u r s is a central p r o b l e m in t h e s t u d y o f m e t a b o l i c c o n t r o l in m u s c l e ( C h a n c e et al., 1962; J6bsis, 1964; J a c o b u s a n d L e h n i n g e r , 1973; O w e n a n d Wilson, 1974; Saks et al., 1974, 1976). T h e s e e f f o r t s a r e h a n d i c a p p e d , h o w e v e r , by the a b s e n c e o f a g e n e r a l quantitative d e s c r i p t i o n o f the d y n a m i c s o f this c o u p l i n g as it exists in an intact muscle. F r o m the p o i n t o f view o f systems analysis, it is n a t u r a l to c o n s i d e r the events which link o x y g e n c o n s u m p t i o n to A T P hydrolysis as a system, f o r w h i c h the J. G~N. PHYSIOL. 9 The Rockefeller University Press 9 0022-1295/78/0501-055951.00

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input is the rate o f A T P hydrolysis, and the o u t p u t is the rate o f o x y g e n c o n s u m p t i o n (Qo2), b o t h c o n s i d e r e d as functions o f time. From m e a s u r e m e n t s o f the kinetics o f (2o2 elicited by impulse changes in the rate o f A T P hydrolysis, it is possible to decide w h e t h e r the system is linear, and if it is, to formulate a single system equation, valid at all times, which relates the (2o2 to the rate o f A T P hydrolysis (Milsum, 1966; Riggs, 1970). I n the experiments described in this p a p e r , it was i n t e n d e d that stimulation o f the excised sartorius muscle o f Rana pipiens for 0.1-1.0 s at 20~ would p r o d u c e a time-course o f c h a n g e in its rate o f A T P hydrolysis which, on the time scale o f oxidative recovery metabolism, would satisfactorily a p p r o x i m a t e an impulse. D u r i n g the stimulation, the nmscle p e r f o r m e d a maximal isometric contraction. T h e e x p e r i m e n t s were designed to quantify the time-course o f c h a n g e in (2o2 f r o m its basal level (AQo2) after the tetanus, and it was thus intended that these kinetics would r e p r e s e n t an impulse response o f the system. T h e kinetics o f recovery o x y g e n c o n s u m p t i o n in an isolated muscle at 20~ d o not a p p e a r to have been previously r e p o r t e d . T h e m e t h o d used here to measure A(2o2(t) takes advantage o f the fact that (202 and intramuscular Poz in the excised f r o g sartorius are linked by the one-dimensional diffusion equation for oxygen (Eq. 1 below; Gore a n d Whalen, 1968; Mahler, 1978b). T h e time-course o f Po2 at a closed surface o f a muscle was m e a s u r e d before, d u r i n g , and after an isometric tetanus; in terms o f Eq. 1, this was P(0, t). Given P(0, t), techniques o f systems t h e o r y were used to solve the diffusion equation for AQo~(t). MATERIALS

AND

METHODS

Measurement of Po2 at the Muscle Surface The technique used to measure the time-course of Po2 at a closed muscle surface during and after an isometric contraction was identical to that described in the preceding paper (Mahler, 1978b). For the present experiments, a pair of stimulating electrodes and a strain gauge were incorporated into the muscle chamber. The stimulating electrodes were situated on the chamber floor, and lay perpendicular to long axis of a muscle, between the pelvic bone and the oxygen electrode. The strain gauge was attached to a clamp which held the pelvic bone. The experimental protocol before the stimulation of a muscle was essentially the same as that for the method I experiments described previously. A drained muscle was mounted in the chamber at its in vivo length, with the oxygen electrode recessed, and the chamber was then immersed in the water bath. The chamber gas composition was usually 75.2% 02, 3.0% CO2, and 21.8% N2, but in a few experiments was 95% 02, 5% CO2. A few minutes after the muscle temperature had reached 20~ the oxygen electrode was brought into contact with the lower surface of the muscle. The subsequent time-course of the Po2 at the muscle surface was generally similar to that in the method I experiments described in the preceding paper, but was not formally analyzed. In the present context, the purpose of this period was to ensure that before the muscle was stimulated, the surface Po~, and by implication, the Qo2 and the intramuscular Po2 profile, had become constant. If the Po2 trace did not eventually become level, the muscle was discarded; this occurred in about 10% of experiments. If the Po~ trace became level, the muscle was stimulated for 0.1-2.0 s at a just supramaximal voltage, with stimuli of duration 0.6 ms and frequency 70 Hz. In all cases, the contraction was isometric. During the contraction, and in some cases for several seconds thereafter, the electrode

MAHLER Kineticsof 02 Consumption in StimulatedFrog Sartorius

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current rose above its precontraction level. Usually, this rise had the form of a spike of small amplitude, which appeared to be caused at least in part by pressure exerted on the electrode d u r i n g the tetanus, and possibly in part by electrical p h e n o m e n a . After the spike, the electrode current was in most cases at its original level, and then began a gradual decline due to the change in Po2 at the muscle surface (cf. Fig. 1). Occasionally, the c u r r e n t immediately after the spike was marginally higher or lower than its precontraction level, and this was attributed to a slight repositioning of the muscle as a result of the tetanus, with an attendant small change in the muscle thickness over the cathode. In cases when the electrode current was still elevated above its precontraction level after ~10 s from the beginning of the stimulation, records were discarded. As explained in the following section, the first reading from acceptable records was usually not taken until 24 s after the contraction. Records obtained after an isometric tetanus were analyzed only if they eventually r e t u r n e d to a steady level, indicating that the Qo2 had become constant. After the last suitable record of P(0, t) had been obtained, the oxygen electrode was withdrawn from the muscle surface, and again exposed to the chamber gas; this made it possible to measure the drift of the recording system d u r i n g the entire experimental period. This averaged - 1 % per h. T h e drift was assumed to have occurred at a constant rate, and the Po2 records were corrected accordingly.

Calculation of Qq ) from P(O, t) For these calculations, it was assumed that the intramuscular Po2 profile was related to the Qo2 by the one-dimensional Fick diffusion equation:

02P

OP

D ~ ~- ~ (x, t) - ,~ ~

(x, t) = Q(t),

(1)

where P is the partial pressure of oxygen (Po2), {2 is the {202, x is the distance perpendicular to the muscle surface, t is time, ot is the solubility of oxygen in muscle, and D is the diffusion coefficient for oxygen. This equation tacitly asssumes that Qo2 is uniform throughout the muscle, and thus varies only with time. If so, Eq. 1 implies that for each time-course of change by the surface Po2 from its initial steady-state value, denoted A P(0, t), there is a unique A Q(t). The idea of calculating the kinetics of Qo2 in an isolated tissue from the kinetics of Po2 at its surface, via the one-dimensional diffusion equation for O~, appears to have originated with the work of Connelly et al. (1953) on isolated nerve. However, the mathematical techniques used by these authors were accurate only in special cases. For the present paper, a numerical method derived from the theory of linear systems has been used, which allows the calculation of A Q(t) from an arbitrary A P(0, t). T h e diffusion equation has the form of a system differential equation, with Q(t) as the input, and P(x, t) the output. As shown in Appendix I, for a muscle oxygenated only from one surface, and there by a constant Po2, for the case x = 0 the transfer function for this system has the form: sech(/v~) H(~) =

ors

- 1 ,

(2)

where l is the thickness of the muscle above the platinum cathode, and s is a d u m m y complex variable. AQ(t) can be expressed in terms of AP(0, t) by the equation: hQ(t) = ..~-' { ~[AP(O,How)t)](co)_'[j(t),

(3)

where ,~ and o%-~ denote the direct and inverse Fourier transforms, o is a d u m m y

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complex frequency, and./ = ~/---1. T h e derivation o f Eq. 3 is also given in A p p e n d i x I. In the present case, AP(0, t) is not known in closed fi~rm, but only as a series o f points. In practice, therefi)re, the exact Fourier transfi)rms in Eq. 3 were a p p r o x i m a t e d by their discrete fi)rms, as c o m p u t e d by a Fast Fourier Transfi)rm (FFT) routine (Brigham. 1974). T h e resulting AQ(t) was thus also given as a series o f points, with the same time base as AP(0, t). T h e values used fi)rD and a in Eq. 2 were 1.34 • 10-:' cm2/s and 0.0307 ~! 02/ g. mm Hg, respectively (Mahler, 1978b). T o d e t e r m i n e the errors inherent in the use of the discrete Fourier transfi~rm rather than its exact fi)rm, and as a general test o f the computational method, an experiment was simulated analytically. T h e muscle was assumed to be initially in steady state, and Qo2 was then allowed to change from its basal level with the time course AQ(t) = e -t e-l~ T h e diffusion equation was then solved to yield an exact expression fi)r the c o r r e s p o n d i n g change in P(0, t); cf. A p p e n d i x II. As required by the F F T routine, this curve was then sampled at a fixed time interval, and using this data, ~Q(t) was calculated via the method outlined above. With as few as 64 o u t p u t points, the calculated values matched the known, exact values with 1% accuracy; the total a m o u n t o f extra oxygen consumed, f~ AQ(t) dt, was a p p r o x i m a t e d to about 2% accuracy by numerically integrating the calculated points with the trapezoidal rule. For higher sampling rates, the calculated and exact values o f AQ(t) could be made virtually identical. This method can be generally applied to recover the input to any stable linear system fi)r which the output and transfer function are known, and is considerably more accurate than the "unit response" method traditionally used fi)r the correction o f myothermal records (Hill, 1966, C h a p t e r 13). T h e transfi)rm method is discussed in greater detail in a separate p a p e r (Mahler, 1978a). In practice, because o f the large amounts of c o m p u t e r storage necessary to process records o f > 128 points, the experimental records of P(0, t) were usually sampled at 24-s intervals, and either 64 or 128 points were used in the F F T routines. Curve fitting to the calculated points fi)r AQo2(t) was done with a nonlinear least squares method (Brown and Dennis, 1970). With a few exceptions, all experiments were done d u r i n g the months of December and January. R E S U L T S

Kinetics o f (202 after an Isometric Tetanus

Fig. 1 s h o w s a t y p i c a l r e c o r d i n g o f t h e P o 2 at t h e l o w e r s u r f a c e o f a m u s c l e a f t e r a n i s o m e t r i c t e t a n u s . I n e v e r y case, t h e s e r e c o r d s d i s p l a y e d a n initial d e c l i n e a l o n g a n S - s h a p e d p a t h , w i t h i n f l e c t i o n p o i n t at - 1 m i n , to a m i n i m a l level r e a c h e d a f t e r - 3 - 5 m i n , f o l l o w e d b y a g r a d u a l rise b a c k to a s t e a d y level, a g a i n w i t h a n S - s h a p e d t i m e - c o u r s e , with t h e f i n a l level r e a c h e d a f t e r 2 0 - 5 0 r a i n . O n t h e a v e r a g e , f o r c o n t r a c t i o n s o f 0 . 1 - 0 . 4 s, this final P02 m a t c h e d t h e p r e c o n t r a c t i o n v a l u e ; f o r l o n g e r t e t a n i , t h e final P02 t e n d e d to b e s l i g h t l y l o w e r t h a n t h e initial level. T h e l o n g e r t h e d u r a t i o n o f t h e t e t a n u s , t h e l a r g e r was t h e t r a n s i e n t in t h e s u r f a c e P02. T h e r e l a t i v e t i m e c o u r s e o f P 0 2 s h o w e d little v a r i a t i o n with t h e t e t a n u s d u r a t i o n in a g i v e n m u s c l e , at l e a s t f o r c o n t r a c t i o n s o f 0 . 1 - 1 . 0 s, a l t h o u g h f r o m o n e m u s c l e to t h e n e x t , t h e t i m e s p a n s i n v o l v e d c o u l d v a r y considerably. I t s h o u l d b e e m p h a s i z e d t h a t a c c o r d i n g to t h e d i f f u s i o n e q u a t i o n , t h e i n t r a m u s c u l a r P o 2 was at its lowest at t h e s u r f a c e w h e r e it was b e i n g m e a s u r e d ,

MAHCER Kinetics of O~ Consumption in Stimulated Frog Sartorius

563

a n d w h e r e it was t y p i c a l l y at least 200 m m H g ( e . g . , cf. Fig. 1), well in e x c e s s o f t h e r e p o r t e d "critical P o 2 " f o r this m u s c l e ( H i l l , 1948; G o r e a n d W h a l e n , 1968). I t s e e m s safe to i n f e r t h a t t h e (2o2 was n e v e r l i m i t e d by 0 2 d e l i v e r y . Fig. 2 shows examples of the time-course of the suprabasal rate of oxygen c o n s u m p t i o n , AQ(t), c a l c u l a t e d f r o m t h e r e c o r d s o f s u r f a c e Po2; f o r t h e s a k e o f c l a r i t y , t h e d i s c r e t e v a l u e s o f AQ(t) h a v e b e e n c o n n e c t e d b y c o n t i n u o u s lines. T h e o s c i l l a t i o n s e v i d e n t in t h e s e r e c o r d s , w h i l e n o d o u b t d u e in p a r t to b o t h v a r i a t i o n in t h e s u r f a c e P o 2 a n d to e r r o r s in s a m p l i n g it, c a n also b e a t t r i b u t e d in p a r t to t h e u s e o f d i s c r e t e r a t h e r t h a n e x a c t F o u r i e r t r a n s f o r m s in t h e c a l c u l a t i o n o f 2~Q(t) via Eq. 3 ( B r i g h a m , 1974). I n a l m o s t all cases, it was a p p a r e n t t h a t t h e d e s c e n d i n g l i m b o f hQ(t) c o u l d b e well a p p r o x i m a t e d b y a s i n g l e e x p o n e n t i a l ; a c c o r d i n g l y , t h e b e s t - f i t t i n g c u r v e o f t h e f o r m a + be-et was c a l c u l a t e d f o r e a c h r e c o r d . A l t h o u g h t h e g o o d n e s s o f fit was d i f f i c u l t to q u a n t i f y ? it was n e v e r t h e l e s s e v i d e n t f r o m visual i n s p e c t i o n t h a t

t 200 rnm ~ "l 0j

3rnin

FIGURE I. Photograph o f a typical recording o f the time-course o f Po2 at the lower surface o f a muscle befiwe, during, and after an isometric tetanus at 20~ T e t a n u s duration 0.8 s, Inset: p h o t o g r a p h of the recording o f tension developed d u r i n g the tetanus. One division on the abscissa corresponds to 40 ms, and on the ordinate, tf~ 2 g. f o r t e t a n i o f 0 . 1 - 1 . 0 s, t h e fits c o u l d b e c o n s i d e r e d q u i t e g o o d in a b o u t 90% o f all cases. C u r v e s a a n d b o f Fig. 2 a r e e x a m p l e s o f a v e r a g e fits, c h o s e n as m u c h to i l l u s t r a t e t y p i c a l d e v i a t i o n s f r o m a n e x p o n e n t i a l t i m e - c o u r s e , as a strict a d h e r e n c e to it: in c u r v e a , t h e o s c i l l a t i o n s in t h e l a t t e r p a r t o f t h e c u r v e a r e u n u s u a l l y l a r g e ; in c u r v e b, a f t e r AQ(t) h a s d r o p p e d to a b o u t 10% o f its p e a k v a l u e , it remains slightly but consistently higher than the best-fitting exponential for a b o u t l 0 m i n ; as d i s c u s s e d b e l o w , this t e n d e n c y was a c c e n t u a t e d f o r t e t a n i o f l o n g e r t h a n 1.0 s. T h e fit s h o w n in Fig. 2c was a m o n g t h e b e s t o b s e r v e d . The parameter r=

erro L

I

(~Q[ti])z t

|

which approximates the average relative error, did not appear to be a particularly useful index of the goodness of tic. In cases for which the oscillations ar(mnd the fitted curve were relatively large, but still quite uniform, so that the fit would be subjectively judged very good, the values o f r were typically relatively large; in comparison, smaller values ofr were often obtained when the deviations were smaller, but less uniformly distributed about the fitted curves, so that the fits appeared by inspection to be considerably worse. The average value for r was about 0.15; for records a, b, and c of Fig. 2, the values of r are 0.147, 0.107, and 0.098, respectively.

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THE .JOURNAL OF GENERAL PHYSIOLOGY " VOLUME 71 9 1978

/'~

a

1/29 r=

0.2s = 2.48min

0.6

0.4

0.2 (p.l/g.min)

v ~

0 0

200

400

600

800

tO00

1200

1400

t(s) AQot(t] (p.lIcj.min)

3.0

1/29

b

08s

T I : 2l~

225

rain

1.5

075

0 0

4(30

800

1200 1600

2000 2400 2800

t(s) A QO/t) (H'I/cJ'min)

20

12/23 0 5 s

c

15

"{'1 = 7.97 rain

I0

05

0 O

400

800

1200

1 6 0 0 2000

2400

2800

t (s) FIGURE 2. T y p i c a l r e c o r d s o f t h e c a l c u l a t e d t i m e - c o u r s e o f AQo2 a f t e r i s o m e t r i c t e t a n i (>f 0 . 1 - 1 . 0 s at 20~ T e t a n u s d u r a t i o n s (a) 0.2 s, (b) 0.8 s, a n d (c), 0.5 s. C u r v e b is d e r i v e d f r o m t h e P o 2 r e c o r d s h o w n in Fig. 1. S m o o t h c u r v e s : b e s t - f i t t i n g c u r v e s o f t h e f i ) r m a + b e -t'lt .

565

MAHLER Kinetics of 02 Consumption in Stimulated Frog Sartorius

It was d e t e r m i n e d in early e x p e r i m e n t s that, after a 2-s tetanus, the Po~ at the closed surface o f the muscles sometimes fell to zero and r e m a i n e d there for several minutes; this o c c u r r e d in muscles 1.0-1.2 m m thick, despite a Po2 o f about 540 m m H g at the u p p e r surface. T h e s e e x p e r i m e n t s p r o v e d incidentally useful, by p r o v i d i n g f u r t h e r evidence that, for a Pou o f zero, the electrode c u r r e n t was the same w h e t h e r the external m e d i u m was a muscle or a test gas, an assumption that was m a d e routinely in the calibration o f the electrode currents r e c o r d e d f r o m muscle (cf. Methods in Mahler, 1978b). For the main body o f e x p e r i m e n t s , tetani o f 1 s or less were used, to e n s u r e that the muscles were well o x y g e n a t e d . However, several e x p e r i m e n t s were d o n e in which muscles were adequately o x y g e n a t e d after tetani o f 1.2-2.0 s. Fig. 3 shows the results o f one such e x p e r i m e n t . In general, the time-course o f AiQo2 was at first 8

~ Q02(t) (~.1/g.min) 11/25 20 S T~-- 2.35rain

0

600

1200

1 8 0 0 2400

3000

3600

4200

4800

t(s) FIGURE 3. T h e c a l c u l a t e d t i m e - c o u r s e o f AQo2 a f t e r a n i s o m e t r i c t e t a n u s o f 2.0 s. S m o o t h c u r v e : b e s t - f i t t i n g c u r v e o f t h e f o r m a + be -~'t .

similar to that described above for shorter tetani; once it had fallen to - 2 0 % o f its peak value, however, the subsequent decline to a steady value was markedly slower than for the shorter tetani, so that the entire d e s c e n d i n g limb o f ~ o 2 was poorly fit by a single exponential. Fig. 4 illustrates the d e p e n d e n c e on tetanus duration o f the time constant o f the descending limb o f AQoz(t); for reasons m a d e clear below, this time constant will be designated T T. T h e values plotted r e p r e s e n t all e x p e r i m e n t s with tetani o f 0.1-1.0 s (n = 46). Linear regression showed no d e p e n d e n c e ofT1 on tetanus d u r a t i o n (r = - 0 . 0 4 1 , P > 0.7); this conclusion is clouded, however, by the scatter in the values OfT1 (range, 1.9-8.0 min). An alternative a p p r o a c h was to consider only cases in which two or m o r e e x p e r i m e n t s had been d o n e with a single muscle, and to c o m p a r e , for all possible pairs o f such experiments, the value of~'~ for the longer tetanus o f the pair to that for the shorter. T h e results o f these comparisons are shown graphically in Fig. 5. W h e n the values ofT1 were plotted against each o t h e r pairwise, the points clustered along the line o f identity; for the 59 possible pairs, based on 45 e x p e r i m e n t s in 14 muscles, the

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m e a n value o f (~'longert~tanush'shorter t~tanus) was 0.999 -+ 0.022. Clearly, any d e p e n d ence of~'l on the tetanus duration must be slight over the r a n g e 0.1-1.0 s. T h e m e a n value ofT1 for all experiments was 3.68 + 0.24 min (n = 46); inasmuch as the distribution was somewhat skewed, it is also o f interest that the median value was 3.09 min. T h e final value o f A(2o2 after a tetanus, d e n o t e d (AQo2)0, represents the c h a n g e in the basal level o f Qo2 f r o m its precontraction value. For tetani o f 0.10.4 s, (AQo2)0 was negligible (pooled m e a n = 0.0012 -+ 0.0048 t~l/g, rain, n = 19). For tetani o f 0.5 s a n d h m g e r , the average value o f (A(2o2)0 r a n g e d f r o m 0.02 to 0.11 /~l/g. rain; in c o m p a r i s o n , the resting (2o2 at 20~ was previously observed to be about 0.5/.d/g. rain (Mahler, 1978b). (AQo2)b was quite variable for tetani o f 0.5 s and longer: it was sometimes near zero, even for tetani o f 2.0 s (cf. Fig. 6.0

mn

,5.0 40

3.0

.

.

.

.

.

+

20

.

.

.

tL t .

.

.

.

.

t

1.0

0

0.2

04

0.6

0.8

1.0

tetanus duration (s)

FIGURe: 4. The relationship between the duration of an isometric tetanus and the time constant 7~ of AQo2(t) after the tetanus, based on pooled data from all experiments with tetani of 0.1-1.0 s. Dashed line: ~'1 = 3.68 min, the mean value. The actual line of best fit is Y = (3.79 +- 0.48) + (0.21 +- 0.77)X. (r = - 0.041). 3), and was significantly different f r o m zero only for 1.0 s tetani. However, a small increase in the basal Qo2 does a p p e a r to have o c c u r r e d in some experiments. Except fi)r the fact that AQo2 reaches its peak value only after 45-90 s, its general time-course as r e p o r t e d here is well described as the response o f a first o r d e r system to an impulse input. An exact first o r d e r response would have the form; AQo2(t)

=

A(~o2(0).

e -kt ,

(4)

and this time-course was in fact p r o p o s e d by Kushmerick and Paul (1976) to describe their results at 0~ This raises the question o f w h e t h e r the present e x p e r i m e n t s can distinguish with certainty between the observed kinetics and those o f Eq. 4. A related but m o r e f u n d a m e n t a l question is whether the calculated 45-90-s rise in AQo2 is a methodological artifact. With r e g a r d to the second question, two factors can be identified that might cause the r e c o r d e d

MAHLER

567

Kineticsof Oz Consumption in Stimulated Frog Sartorius

time-course o f Po2 to c h a n g e m o r e slowly t h a n that actually o c c u r r i n g at the lower surface o f the muscle; this would in t u r n cause the calculated A(2o2(t) to lag b e h i n d the t r u e curve. T h e first factor is a lag in the r e s p o n s e o f the system used to r e c o r d Po~; the second is the existence o f a layer o f connective tissue a n d R i n g e r fluid b e t w e e n the muscle fibers a n d the o x y g e n electrode. T h e i r effects were d e d u c e d by m a k i n g a p p r o p r i a t e modifications in the equations by which AQo~(t) is linked to the o b s e r v e d P(0, t), calculating the new values for AQo2(t) f r o m a s a m p l e e x p e r i m e n t a l r e c o r d o f P(0, t), a n d then c o m p a r i n g these new values to the original calculated time-course o f AQo2. T h e r e s p o n s e time o f the r e c o r d i n g system was i n c o r p o r a t e d into Eq. 3 as an additional first o r d e r

(Tf) longer tetanus (rain)

o

?s

/

so//

6,0

,

/d / / /

4,5

30

o Ogo~ ~ ~ ~%~

1.5

/

/

/ 0

i

0

,

1,5

,

,

50

,

,

4.5

,

,

,

60

,

,

7.5

(TI) shorter tetanus ( rain )

FIGURE 5. The relationship between the value ofT, for the longer tetanus and that for the shorter tetanus fi)r all possible pairs of measurements of A()o2(t) after tetani of different durations on a single muscle. Dashed line: line of identity. The actual line of best fit is Y = (-0.012 --+ 0.222) + (0.997 --- 0.058)X. (r = 0.92). system, with a time constant o f 2.0 s (Mahler, 1978b), a n d as e x p e c t e d , the resulting c h a n g e s in AQo~(t) were negligible. T o estimate the effect on the calculated AQo2 o f the layer o f connective tissue and Ringer solution lying between the muscle fibers a n d the o x y g e n electrode, it was a s s u m e d that the Po2 profile within this layer was d e t e r m i n e d by the onedimensional diffusion equation (Eq. 1), that its rate o f oxygen c o n s u m p t i o n was zero, and that D a n d c~ had the same values there as in water. T h e existence o f the layer necessitates a c h a n g e in the b o u n d a r y condition for Po2 at the lower surface o f the muscle; the derivation o f the modified system t r a n s f e r function relating AQo2(t) to the o b s e r v e d P(0, t) is given in A p p e n d i x I V o f Mahler (1976). It was a s s u m e d that 25 /xm was a g e n e r o u s u p p e r limit for the thickness o f the n o n c o n s u m i n g layer. According to Hill (1949), the excised f r o g sartorius has at its "outer" surface, which was originally next to the skin, a layer o f connective tissue with a v e r a g e thickness a b o u t 6 /xm, whereas at the "inner" surface, no such layer is evident. In the present e x p e r i m e n t s , it was the i n n e r surface which

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was in contact with the o x y g e n electrode, so the n o n c o n s u m i n g layer was probably c o m p o s e d simply o f Ringer solution. Hill (1949) estimated the thickness o f the layer o f Ringer solution between a d r a i n e d muscle a n d a t h e r m o p i l e to be a b o u t 1 or 2 /zm. In the present context, it seems realistic to accept this only as a lower limit; using the fact that a muscle weighs - 6 % m o r e w h e n d r a i n e d than when blotted, it can be calculated that if in a d r a i n e d muscle this extra weight is due to a layer o f Ringer solution s p r e a d u n i f o r m l y o v e r the entire p e r i p h e r y , then the thickness o f the layer will be a b o u t 2 0 / z m for muscles o f the size used in the present work. Accordingly, in calculating the effect on AQo2(t) o f the n o n c o n s u m i n g layer, it was a s s u m e d that the layer was no m o r e t h a n 25 /~m thick. Even w h e n this figure was used, the calculated time-course o f AQo2 was only slightly c h a n g e d : d u r i n g the first 1-2 min, individual values were generally increased by 2-10%, but the time at which the peak in AQo2 o c c u r r e d was the same; in the latter portion o f the curve, the values were in general slightly smaller; the area u n d e r the curve a n d the time constant ~'1 were smaller by only a few percent. Because these effects were small, and no m e a s u r e m e n t s were m a d e o f the thickness o f the n o n c o n s u m i n g layer, corrections o f this type were not i n c o r p o r a t e d into the results. It r e m a i n s to be c o n s i d e r e d w h e t h e r the m e t h o d o l o g y used h e r e m a k e s it possible to rule out a m o n o e x p o n e n t i a l time-course for A(2o2 d u r i n g the entire recovery period. T h i s question was answered by a s s u m i n g that AQo2(t) did in fact have the f o r m p r o p o s e d in Eq. 4, calculating the c o r r e s p o n d i n g time-course o f Po2 at the muscle surface p r e d i c t e d by the diffusion equation, a n d c o m p a r i n g this r e c o r d to those observed e x p e r i m e n t a l l y (for details, cf. A p p e n d i x II). T h e results are shown graphically in Fig. 6. A(2o2(t) can, fi)r the p u r p o s e o f these calculations, be well a p p r o x i m a t e d by the function e - k i t - e - ~ 2 t , w h e r e kl a n d k2 have the values 0.29 m i n -2 a n d 2.0 min -~ , c o r r e s p o n d i n g to time constants o f 3.5 rain and 30 s, respectively. T h e s e o b s e r v e d kinetics are shown in the inset o f Fig. 6 as c u r v e a. T h e c o r r e s p o n d i n g m o n o e x p o n e n t i a l time-course, e-k1t, a p p e a r s as c u r v e b. C u r v e c is the time-course o f Apo2 at the muscle surface predicted by Eq. 1 w h e n AQo2(t) is specified by curve a; it closely matches the e x p e r i m e n t a l records (cf. Fig. 1). C u r v e d shows, on the same scale, the timecourse o f Apo2 which would occur if AQo2(t) was given by b r a t h e r t h a n a. Given exact values f r o m curve c, the transfl)rm m e t h o d can recover the i n p u t e-k~t - e -k2t with a high d e g r e e o f accuracy (cf. Methods, and Mahler, 1978a); given points f r o m curve d, its i n p u t e-k1r can also be a p p r o x i m a t e d with good accuracy. 2 In practice, points f r o m e x p e r i m e n t a l records were r e a d by eye; this was p r e s u m a b l y d o n e with small r a n d o m errors, b u t these can be shown to have a negligible effect on the accuracy o f the transfl)rm m e t h o d . It fl)llows that if the micro-O2-electrode a n d r e c o r d i n g system were sufficiently stable, responsive, a n d sensitive that a Po2 transient described by curve d would not be r e c o r d e d as curve c , it will t h e n be d e m o n s t r a t e d that the m e t h o d used h e r e was a d e q u a t e to distinguish a (2o2 transient o f the f o r m e -k~t - e-k2t f r o m o n e o f the f o r m e-k,t. T h i s a p p a r a t u s was stable to within 1% p e r h, a n d h a d a time 2 Mahler, M. Unpublished observation.

MAHLER

569

Kinetics of 02 Consumption in Stimulated Frog Sartorius

constant o f about 2 s (cf. Mahler, 1978b). Although no specific e x p e r i m e n t s were d o n e to d e t e r m i n e the limits o f its sensitivity, its excellent linearity implies that changes in Po2 on the o r d e r o f a few m m H g can be m e a s u r e d , and previous workers have routinely used similar instruments for this p u r p o s e (e.g. Connelly et al. 1953; G o r e and Whalen, 1968; Kawashiro et al., 1975; K u s h m e r ick and Paul, 1976). In contrast, Figs. 1 and 6 illustrate that u n d e r the conditions o f the present experiments, the observed records describe large changes in electrode c u r r e n t , with m a x i m u m deflections on the o r d e r o f 50%, and occur gradually over many minutes; m o r e o v e r , on a realistic scale, a curve o f f o r m d would differ f r o m one o f f o r m c by as much as 30-40 m m Hg. It follows that according to the m e t h o d o l o g y e m p l o y e d in this paper, it can be concluded that after a tetanus o f 0 . 1 - 1 . 0 s, AQo~(t) does not have the f o r m Q a e - k i t . A p02(t ) (arbitrary units)

0

-0.4 c -0.8

,o d

o.Ts

-12

AQ0=(t) b

0,60

-16

01

,

o ~ 0

2

4

6

8

I0

12

"

"

~

z~ 3~ ,~ 14

16

t (rnin)

FIGURE 6. Inset: curve a, AQo2(t) = e -'~'~t - e - k 2 t ; curve b . AQo2(t) = e - k i t . Curve c is the time-cnurse of APo2 at a muscle surface predicted by the diffusion equation (Eq. 1) when A(2o2(t) is given by curve a. Curve d is the corresponding time-course of APoz when AQo2(t) is given by curve b. A somewhat m o r e subtle question is w h e t h e r any constants C and k can be f o u n d for which a m o n o e x p o n e n t i a l AQoz(t) given by C e - k t can p r o d u c e a timecourse o f haPo2 which will match, within e x p e r i m e n t a l e r r o r , the observed kinetics given by curve c. Given the validity o f Eq. 1, this possibility can also be ruled out. For a m o n o e x p o n e n t i a l AQo2(t), Apo2 will always have a cusp at t = 0, and a monotonically increasing slope d u r i n g the initial phase o f recovery (cf. curve d); in contrast, the e x p e r i m e n t a l records invariably followed an S-shaped path d u r i n g this time, with a p r o n o u n c e d "shoulder" at the start (cf. Fig. 1 and curve c). This initial discrepancy could be lessened, and the t r o u g h points in APo2 m a d e to coincide, only by choosing k considerably smaller than kl, with a p p r o p r i a t e scale factor C; this resulted in a wide divergence between the ascending limbs o f the predicted and observed curves o f APo2. T h e s e results a r g u e that the initial 45-90-s rise in AQo2 calculated by the present m e t h o d is a real p h e n o m e n o n , and it thus appears justified to i n c o r p o r a t e these early kinetics into the quantification o f A(2o2(t).

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T h e entire time-course o f AQo2 could be a p p r o x i m a t e d by the expression: Qo + Ql(e - k ' t - e-k=t);

(5)

cf. Fig. 7. Q0 is the final value o f Eq. 5 after a tetanus, and Q0 + Q1 is the value obtained by extrapolating its m o n o e x p o n e n t i a l descending limb back to t = 0. For the d e t e r m i n a t i o n o f the best-fitting values o f k2 and QI, AQo2(t) was fit by Eq. 5 with k~ and Q0 held constant at the values previously d e t e r m i n e d by fitting the declining limb o f A(2o2(t) only. Reasonable fits could sometimes be obtained with o t h e r functions: e.g., a linear rise fi)llowed by a m o n o e x p o n e n t i a l fall (cf. Fig. 2c); however, most records showed a gradual r o u n d i n g in the vicinity o f the peak value which was consistent with Eq. 5. Nevertheless, the values o f AQo2(t) d u r i n g its rise and early fall were generally not as well fit by Eq. 5 as was its d e s c e n d i n g limb, d u r i n g which Eq. 5 was essentially m o n o e x p o n e n t i a l . It AQoz(t)(k~lIg'min]

2.0

1/29 0.5s T, = 302 min : '

/~ 1.5

0.5

. . . . 0

^^ ~

0

200

400

600

800

I000

1200

A/~A

-vvy 1 1400

t (s)

FIGURE 7. A fit of AQo2(t) after an isometric tetanus by the function Q0 + Q1(e-~',, - e-~'2,). seems a p p r o p r i a t e to consider the rate constant k2 simply as a phenomenological p a r a m e t e r which, via Eq. 5, provides a good first a p p r o x i m a t i o n to the early kinetics o f AQo2(t), but which does not necessarily deserve a strict mechanistic interpretation. Unlike rl, ~'2 (= l/k2) was d e p e n d e n t on the tetanus d u r a t i o n (cf. Fig. 8). W h e n values of~'2 ft)r d i f f e r e n t tetanus durations in a single muscle were c o m p a r e d pairwise, ~'2 for the longer tetanus o f the pair was smaller than that fi)r the s h o r t e r tetanus in 53 o f 59 cases (P < 0.001 if72 were i n d e p e n d e n t o f duration). T h e p a r a m e t e r Q1 increased in curvilinear fashion with the tetanus duration. For tetani o f 1.2-2.0 s, and occasionally fi)r shorter tetani, AQo2(t) could not be well fit by Eq. 5: c o m p o n e n t s o f the fi)rm e-kit and e-k2t were still a p p a r e n t , but it was necessary to also include in the description o f AQo~(t) an approximately linear c o m p o n e n t substantially slower than e-~',t (cf. Fig. 3). As shown in Fig. 9, the maximal value o f AQo2 after a tetanus was related to the tetanus duration by a slightly curvilinear function. After a 2-s tetanus, it was about 6.5 /~l/g. min, or about 13 times the resting Qoz. For a given tetanus

571

MAHLER Kinetics of 02 Consumption in Stimulated Frog Sartorius

duration, this peak value o f AQ02 was linearly related to the rate constant (k~) which characterized its subsequent fall. Inasmuch as both parameters might be expected to depend on the num ber of mitochondria in a muscle, this result suggests a high level of internal consistency in these measurements o f AQo2(t). 1.5

1.2

0.9

0.6

"r'= (min) 0.3

0

0

02

04

0.6

08

10

tetonus durotion (s)

FIGURE 8. T h e r e l a t i o n s h i p b e t w e e n t h e d u r a t i o n o f a n i s o m e t r i c t e t a n u s a n d t h e t i m e c o n s t a n t rz o f Qoz(t) a f t e r t h e t e t a n u s .

4.0 (AQ02} peak

(yllg.min)

§

3.0

2.0 +

/0

0

0

02

04

0.6

0.8

1.0

~ r ~ s durofion (s ) FIGURE 9. T h e r e l a t i o n s h i p b e t w e e n t h e d u r a t i o n o f a n i s o m e t r i c t e t a n u s a n d t h e p e a k v a l u e o f AQo2(t) a f t e r t h e t e t a n u s . DISCUSSION

Spatial Dependence of Qo2 in the Frog Sartorius T h e method presented here for the measurement of AQo2 is based on the validity o f the diffusion Eq. 1, which assumes that the rate of oxygen consumption is the same throughout the muscle. Several lines of evidence support this assumption. Histochemical investigation showed that cross sections of the sartorius of R. pipiens stain uniformly for myosin ATPase and succinate dehydrogenase, indicating that the sites o f ATP hydrolysis and oxygen con-

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sumption are u n i f o r m l y distributed. Single fiber cross sections typically cut 40100 mitochondria, and the average area "served" by a m i t o c h o n d r i o n is roughly 20 ftm2. 3 T h e validity o f Eq. 1 in the resting sartorius (Mahler, 1978b) indicates that the resting Qo2 is u n i f o r m t h r o u g h o u t the muscle. Finally, results consistent with the uniformity o f muscle oxygen consumption were provided by experiments designed to test the effect o f the placement o f the oxygen electrode on the m e a s u r e d AQ(t) for a tetanus o f 0.2 s. In the first o f a lpair o f experiments, a muscle was m o u n t e d in the usual way, with the platinum cathode ~ 8 inm from the pelvic bone, and approximately equidistant f r o m the lateral edges o f the muscle (cf. Fig. 1 o f Mahler, 1978b). For the second e x p e r i m e n t , the positions o f the ends o f the muscle were e x c h a n g e d , so that the cathode now lay 8 m m from the distal e n d , and 20-25 m m f r o m the pelvic bone. O n average, Q0, Q1, r , , ~'2, and the total o x y g e n c o n s u m p t i o n f~AQ.(t) agreed to within 10% for the two electrode placements (n = 6).

Comparison with Previous Work TIME-COURSE OF A Q o 2 T h e results p r e s e n t e d h e r e show that after a single brief isometric tetanus o f 20~ the time-course o f AQo~ in the excised frog sartorius quickly becomes m o n o e x p o n e n t i a l . T h e s e kinetics are consistent with virtually all previous observations in this field, m a d e on skeletal muscles o f the frog at 0~ for which ~- -~ 10-20 min (Hill, 1940a; Kushmerick, and Paul, 1976), and at 12~ (r -~ 10 min, Baskin and Gaffin, 1965); the d o g (~- = 24 s at 36~ Piiper et al., 1968); and m a n (~- -~ 45 s; for review cf. Berg, 1947, and Casaburi et al., 1977). T h e kinetics o f o x y g e n c o n s u m p t i o n in frog skeletal muscle at 20~ have a p p a r e n t l y not been previously described; however, G o d f r a i n d - d e B e c k e r (1972, 1973) r e p o r t e d that after isometric tetani o f 0.5-1.5 s by excised toad and frog sartorii at 20~ the rate o f heat p r o d u c t i o n became m o n o e x p o n e n t i a l after 2-3 rain, with a time constant o f 3.3-4.0 rain, and that N A D H fluorescence in the toad sartorius had essentially the same kinetics d u r i n g this time span. Both o f these processes can be e x p e c t e d to occur in parallel with AQo2 (Hill, 1940a, b; J6bsis and Duffield, 1967). T h e results o f G o d f r a i n d - d e B e c k e r (1972, 1973) thus seem quantitatively consistent with those r e p o r t e d here. According to the present results, the time constant o f the d e s c e n d i n g limb o f AQo2(t) after a tetanus is i n d e p e n d e n t o f the tetanus duration over the range 0.1-1.0 s. This conclusion is also consistent with virtually all previous results, except those o f Kushmerick and Paul (1976), who r e p o r t e d that after single tetani in the sartorius o f R. pipiens at 0~ the exponential time constant fi)r AQo2 increased markedly with the tetanus d u r a t i o n over the range 1-30 s. In contrast, Hill (1940a) c o n c l u d e d on the basis o f similar e x p e r i m e n t s with R. temporaria that the time constant for AQo2 was invariant for tetani o f u p to 20 s. Moreover, the rate o f aerobic recovery heat p r o d u c t i o n , which according to Hill (1940a, b) parallels AQoz, has also been r e p o r t e d to have a time constant which does not vary with the tetanus duration, in excised sartorii o f the frog at 0~ (Hill, 1940b) and the toad at 20~ ( G o d f r a i n d - d e Becker, 1973). Analogous results have been r e p o r t e d for canine and h u m a n skeletal muscle. Piiper et al. 3 Eisenberg, B., and A. Kuda. Unpublished observation.

MAHLEa Kineticsof Oz Consumption in Stimulated Frog Sartorius

573

(1968) used the Fick principle to approximate transient kinetics of oxygen uptake by the in situ dog gastrocnemius during series of tetani of fixed duration and rate; these series presumably caused approximately stepwise increases in the rate of ATP utilization by the muscle, in contrast to the impulse increase which presumably accompanies a single tetanus (cf. Introduction). The transients in oxygen uptake measured by Piiper et al. could be approximated by curves of the form (1 - e-trY), with ~" independent of the steady state Qo2 over a wide range in the latter. In the context of a systems analysis of the link between ATP splitting and oxygen consumption, these results are consistent with the independence of~'l from tetanus duration reported by Hill (1940a) and in this paper. Moreover, numerous studies on the kinetics of oxygen uptake in man during work suggest that there is a wide range of conditions for which AQo2 has exponential kinetics in human skeletal muscle after a step change in work rate, with time constant again independent of step size (cf. Casaburi et al., 1977 for review). According to the present results, A(2o2 in the frog sartorius does not have exclusively monoexponential kinetics after a single tetanus at 20~ an initial rapid component is present which results in a delayed rise to the peak value. The only strictly comparable published evidence appears to be that reported by Hill (1940a) and Kushmerick and Paul (1976) for the excised frog sartorius at 0~ Hill's results were similar to those reported here, in that differentiation of his records of cumulative suprabasal oxygen consumption after a tetanus indicates that AQo2(t) reached its peak value only after 2-3 rain. In contrast, Kushmerick and Paul concluded that AQo2(t) was at its peak by the end of a tetanus. Examination of their experimental records shows that suprabasal oxygen uptake was essentially zero for several minutes after a tetanus; because the rate of 02 consumption is the sum of the rate of uptake and the rate at which the muscle 02 store is decreasing, it fi)llows that for AQo2(t) to have been maximal by the end of a tetanus, the calculated rate of depletion of the 02 store of the muscle must have been relatively large during the early phase of recovery. This rate was calculated via the diffusion equation (Eq. 1), with the diffusion coefficient for oxygen in muscle at 0~ assumed to have the value 2.75 x 10-4 cm2/min (Hill, 1966). However, as reported in the preceding paper (Mahler, 1978b), although Eq. 1 does appear to be valid in the excised frog sartorius, D has the value 4.94 (--- 0.16) • 10-4 cm2/min at 0~ from which it follows that the actual changes in the 02 store of the muscle wer eonly about half as large as those calculated by Kushmerick and Paul (1976), and that the initial phase of AQo2(t) in their experiments may have been similar to that reported here and by Hill (1940a). A similar criticism in fact applies to the results of Hill (1940a) as well, and implies that the peak values of A(2o2 in his experiments occurred somewhat later than is evident from his corrected records. Indirect measures of the early kinetics of AQo2 in amphibian skeletal muscle after single tetani also appear consistent with the results reported here. A delayed rise to a maximum has been reported for the NADH fluorescence change in the toad sartorius at 12~ 06bsis and Duffield, 1967) and 20~ (Godfraind-deBecker, 1972, 1973), and fi)r the rate of aerobic recovery heat production in the frog sartorius at 0~

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( H a r t r e e a n d Hill, 1922; Hill, 1940b) a n d 20~ ( H a r t r e e and Hill, 1922; Hill, 1966). A c c o r d i n g to the results s u m m a r i z e d in Fig. 9, the longer the d u r a t i o n o f an isometric tetanus, the faster is the rise o f A0-o2(t) to its peak value after the tetanus. A l t h o u g h no c o m p a r a b l e analysis o f the early kinetics o f AQo2(t) after a contraction has been published, similar b e h a v i o r has been r e p o r t e d for the rate o f aerobic heat p r o d u c t i o n in the frog sartorius at 20~ ( H a r t r e e and Hill, 1922; Hill, 1966) and for N A D H fluorescence in the toad sartorius at 12~ (J6bsis and Duffield, 1967). SENSITIVITY O x y g e n u p t a k e by an excised muscle has previously been m e a s u r e d directly, as the a m o u n t o f Oz r e m o v e d f r o m a well-stirred c h a m b e r (for m e t h o d s cf. Fenn, 1927; Gemmill, 1936; Hill, 1940a; K u s h m e r i c k and Paul, 1976). A potential difficulty with such m e t h o d s is that the rate at which oxygen disappears f r o m the c h a m b e r is often quite small in c o m p a r i s o n with the a m o u n t actually present. For e x a m p l e , O2-filled c h a m b e r s typically contain roughly 5 ml 02 (Fenn, 1927; Gemmill, 1936; Hill, 1940a; Baskin a n d Gaffin, 1964); even a Ringer-filled c h a m b e r ( K u s h m e r i c k and Paul, 1976) o f 5 ml, if bubbled with 95% 02 at 20~ would contain 140 /~1 Oz. In c o m p a r i s o n , according to the present results, the total suprabasal oxygen c o n s u m p t i o n by a 60-rag muscle a f t e r a 1.0 s tetanus is only - 1 /M, or for a 0.1-s tetanus, 0.2 /xl; m o r e o v e r , these volumes are c o n s u m e d only over a period o f 20-40 rain. With the m e t h o d described in this p a p e r , the kinetics o f muscle oxygen c o n s u m p t i o n are not m e a s u r e d directly, but d e d u c e d f r o m the kinetics o f Po2 at a muscle surface. A l t h o u g h this technique u n d o u b t e d l y entails a m o r e complicated set o f assumptions t h a n previous m e t h o d s , it a p p e a r s to o f f e r considerably g r e a t e r sensitivity. For an a p p r o p r i a t e choice o f the c h a m b e r gas, the c h a n g e in surface Po2 after a tetanus can be m a d e to constitute a large, easily m e a s u r e d fraction o f the initial value. For e x a m p l e , with a c h a m b e r gas containing 5-10% 02. it is possible to m e a s u r e AQo2(t) in the f r o g sartorius after a single isometric twitch at 0oC.2

Implications for Control of O.o2 T h e overall aim o f this and the a c c o m p a n y i n g papers (Mahler, 1978 b, c) has been to provide data which would m a k e it possible to quantify the dynamic c o u p l i n g between the rates o f A T P hydrolysis and 02 c o n s u m p t i o n in well o x y g e n a t e d muscle cells. I f a system is d e f i n e d by specifying these rates as its input and o u t p u t , respectively, the p r o b l e m at hand becomes one o f system identification. Evidence p r e s e n t e d in a f o r t h c o m i n g p a p e r in this series indicates that d u r i n g and after a single isometric tetanus o f 1 s or less at 20~ the timecourse o f the suprabasal rate o f A T P hydrolysis in the sartorius o f R . pipiens can be well described as an impulse, a n d it follows that in the present context, the kinetics o f A0.o2 r e p o r t e d here for tetani o f 1 s or less are those o f impulse responses. T h e application o f these results to the d e t e r m i n a t i o n o f general equations linking the rates o f A T P hydrolysis and 02 c o n s u m p t i o n in this muscle is i n t e n d e d to be the topic o f a separate p a p e r . H o w e v e r , some f u n d a m e n t a l conclusions can be noted here. First, the system is nonlinear. I f it were linear, then for tetani o f increasing d u r a t i o n , which p r o d u c e impulse-like inputs o f

MAnLER

Kinetics of O= Consumption in Stimulated Frog Sartorius

575

increasing a r e a , the time constants rl a n d r= o f AQo2(t) would r e m a i n fixed. T h e variation o f rz with tetanus d u r a t i o n (cf. Fig. 8) thus indicates the i n v o l v e m e n t o f a nonlinear process. Second, the m o n o e x p o n e n t i a l decline o f AQoz(t) f r o m its p e a k value a f t e r a tetanus, with time constant (rl) i n d e p e n d e n t o f tetanus d u r a t i o n (cf. Figs. 4 and 5), suggests that d u r i n g this period, the sequence o f events linking A T P hydrolysis a n d 02 c o n s u m p t i o n is rate-limited by a single reaction with a p p a r e n t first o r d e r kinetics. An essentially similar hypothesis was a d v a n c e d by Hill (1940a) to explain his results at 0~ ADDENDUM

For the calculation o f A Q(t), the t r a n s f e r function H ( s ) w h i c h links A P(0, t ) a n d A Q(t) has been evaluated via Eq. 2, using an average value o f D based on previous e x p e r i m e n t s , and individually m e a s u r e d values o f / . D u r i n g the review o f this p a p e r , it was pointed out that if an e x p e r i m e n t o f the type described here is a c c o m p a n i e d by one d o n e with the muscle in the resting state, by either o f m e t h o d s I or I I described in the p r e c e d i n g p a p e r , a simpler r e p r e s e n t a t i o n is possible for H(s). In the m e t h o d I a n d I I e x p e r i m e n t s , the time-course o f Po2 at the muscle surface b e c o m e s m o n o e x p o n e n t i a l , with rate constant k = rr2D/4l 2. H(s) can thus be e x p r e s s e d as:

H(s) =

sech(1.571x/~)-

1

OlS

(6)

Eq. 6 a p p e a r s p r e f e r a b l e to Eq. 2 for two reasons. First, it is potentially m o r e accurate, because it involves o n e less m e a s u r e d p a r a m e t e r ; m o r e o v e r , k can usually be m e a s u r e d with excellent accuracy, especially with m e t h o d II (cf. Fig. 5 o f Mahler, 1978b). Second, if Eq. 6 is used, the e x p e r i m e n t a l p r o c e d u r e for m e a s u r i n g A Q(t) can be simplified considerably, because the muscle thickness I n e e d not be m e a s u r e d (cf. Methods). APPENDIX

I

Derivation o f Eqs. 2 a n d 3 The one-dimensional diffusion equation.

02P Dol ~ ( x ,

OP t) -- a ~ ( x , t) = Q(t),

(1.1 a)

has the form of a system diferential equation, for which Q(t) is the input, and P(x, t) is the output. For the conditions of the present experiments, the initial and boundary conditions on P(x, t) are:

Q0

P(x, 0) = P0 - 2~a(/~ - xZ);

(1.2a)

P(l, t) = P0;

(l.3a)

~--(0, p

0.4a)

and

i~x

t) = 0;

576

THE

JOURNAL

OF

GENERAL

PHYSIOLOGY

9 VOLUME

71 9 1978

where P0 denotes the Po2 o f the c h a m b e r gas. Q0 is the basal rate o f o x y g e n c o n s u m p t i o n . a n d l is the muscle thickness. For a c h a n g e in Q(t) from Q0, d e n o t e d AQ(t), the c o r r e s p o n d i n g forced o u t p u t is A P ( x , t), where: AQ(t) = Q(t) - Qo,

(l.5a)

A P ( x , t) = P(x, t) - P(x, 0).

(l.6a)

and

T h e p u r p o s e o f this A p p e n d i x is to derive the t r a n s f e r f u n c t i o n o f this system for the case x = O, i.e., to evaluate the function: Ap(0, t)](s) .~[ AQ(t)](s) "

H(s) -

(1.7a)

T h i s analysis can be simplified by settingQ0 = 0 = P0. T h e s e conditions d o not alter the form o f H(s), but allow Eqs. 1.2a, 1.3a, 1.Sa, a n d 1.6a to be rewritten: P(x, 0) = 0;

(1.8a)

P(I, t) = O;

(1.9a)

AQ(t) = Q(t);

(1.10a)

A P ( x , t) = P(x, t).

(1.11a)

and

It will be shown below that the explicit form o f Eq. 1.7a is:

H(s) =

sech(lx/J/O)

- 1

m

(1.12 a)

T h i s e q u a t i o n is derived below by first t a k i n g the Laplace transfi)rm o f Eq. 1.1 a with respect to t, t h e n solving the resulting o r d i n a r y differential e q u a t i o n in x. T h e Laplace t r a n s f o r m with respect to t o f Eq. 1.1 a is: 02p Da ~(x,

s) - o~P(x, s) = Q.(s),

(1.13a)

where P(x, s) denotes .~[P(x,t)](s), and (~(s), .~[Q(t)](s). Eq. 1.13a is an ordinary differential e q u a t i o n in x, which fi)r the sake of simplification can be written as: d2y dx 2

k2y = /3,

1.14a)

s);

1.15a)

where y(x) = p ( x ,

k2 = s/D;

1.16a)

/3 = O,(s)/Dc~.

(1.17a)

and

O n e way to solve Eq. 1.14a is to take its Laplace transfiwm with respect tox. which yields: u2.~(u) - k2~(u) = -# + /z .y(0) + y'(0),

(1.18a)

MAHLER Kinetics of Oz Consumption in Stimulated Frog Sartorins

577

where37(u) denotes LP[y(x)](u). T h e b o u n d a r y condition Eq. 1.4a on P(x, t) can be used to show thaty'(0) = 0, as follows: y(x) = ~ P ( x ,

t)](s);

(1.19 a)

d {Jct[P(x, t)](s)} = ~ t [ OP (x, t)](s);

y'(x) =

(x) = ~x

L Ox

y' (0) = ,5~t[ OP (0, ,)l ($) = k Ox J

t0](s) = 0

(1.20 a) (1.21 a)

It thus follows from Eq. 1.18a that: y(u) - / 3 +/x2y(0) (/z* - kS)it A

+

-

(1.22a) B

+

C

(1.23a)

Evaluating the partial fractions in Eq. 1.23a gives: A = 2 ~ + y(0).2,

(1.24a)

B = A;

(1.25a)

C = - O / k S.

(1.26a)

T h e solution of Eq 1.14a is thus: y(x) = A(e -Ax + e -hx) + C+ =

[/3

~ + y(0)

1 (e-~X + e+kX) 2

(1.27a) /3

(1.28a)

k 2'

= y(0)-cosh(kx) + ~ [cosh(kx) - 1].

(1.29a)

Using Eqs. 1.16a and 1.17a, y(x) = y ( 0 ) . c o s h ( x X / ~ )

+ t~(s)[cosh(xX/~)

- 1].

(1.30a)

ors

T h e object of this derivation is to express /6(0, s), or y(O), in terms of 12(s). This can be done by evaluating Eq. 1.30a at x = l. From b o u n d a r y condition Eq. 1.10a we have: y(l) = P(l, s) = ~[P(l, t)](s) = 0.

(1.31 a)

On the other hand, Eq. 1.30a implies that: y(/) = y(0).cosh(l s V ~ )

+ l~(s) [cosh(/X/sVsVsV~)- 1]. ors

(1.32a)

It follows that: y(0)

I~(s) [ 1 - c o s h ( l X / s / D ) ] = as c-osh(~-sX/~ / ; _

t~(s)[sech(/x/~) ors

- 1],

(1.33a) (1.34a)

578

T H E J O U R N A L OF G E N E R A L P H Y S I O L O G Y " V O L U M E 71 9 1 9 7 8

which is equivalent to Eq. 1.12a and to text Eq. 2. T e x t Eq. 3 follows from Eq. 1.7a. For any function f(t), d e f i n e d for 0 -< t < ~, if one formally d e f i n e s f ( t ) = 0 for 0o < t < 0, it follows from the definitions o f the Laplace and Fourier t r a n s f o r m s that: ~[f(t)](~o) = ~[/(t)](jco).

(1.35a)

T h e r e f o r e , it follows from Eqs. 1.7a and 1.35a that: H0"co) -

~:[A P(O, t)](cn) ~[A Q(t)](co) '

( 1.36 a)

which implies that:

~[A Q(t)](co) - ~[A P(0, t)](oJ)

(1.37 a)

H(flo) T a k i n g the inverse F o u r i e r t r a n s f o r m o f Eq. 1.37a yields text Eq. 3. APPENDIX

II

Derivation o f Exact Solution for A P(O, t) w h e n A Q(t) has the f o r m s e -k't and (e -kxt - e -k2t) T h e desired expression for AP(0, t) can be o b t a i n e d from the general formula: AP((),t)=-4

s

( - 1 ) " f(' -(2"+l)2"2D~tT, 0 (2n + 1) ) ~ ( ~ ' ) e 4~ " - "d~-.

(2.1 a)

Eq. 2.1a can be derived from the convolution integral: h(t - r)A Q(z)dr,

A P(O, t) =

(2.2 a)

where: h(t) = s

(2.3a)

l[H(s)](t),

and H(s) is the system t r a n s f e r function given in Eq. 2 of the text (for details, cf. A p p e n d i x V of Mahler, 1976).

H(s) =

sech(/k/~)-

1

,

(2.4 a)

OL$

and it can be shown that its inverse Laplace t r a n s f o r m is: - ( 2 n + l)m~/aD

h(t) = - 4 o~

a-~

(-1)" (2n + 1~ e

4t~

.t

(2.5a)

For A Q.(t) = e -kit, it follows directly from E l . 2.1a that:

~u~

t)=

e-k1' ~~ + (2n 0

1)(OZn-k0

0 (2n+ I)~-Z

kl ) ;

(2.6a)

where: a,, = (2n + 1)27r2D/412.

(2.7a)

MAHLER Kinetics of 02 Consumption in Stimulated Frog Sartorius

579

For ~Q(t) = e-~'lt - e-k2t, evaluating Eq. 2.1a gives: AP(0, t ) = - 4 { ~~ ( - I ) he-kIt ~ ( - 1 ) ne-~"t a--~ (2n + ~ ~ k,) - 0 (2n + 1)(a, - k0 ( _ 1),,e-k 2t -

(2.8 a) ~

+ 0E(2.

(_ l),e-,~t + - k,)j

T o obtain curves c and d in Fig. 6, computer programs were written to evaluate Eqs. 2.6a and 2.8a. I would like to thank Ivan Whitehorn, Nick Ricchiuti, and Bernard Tai for technical help, Robert Eisenberg, Arthur Peskoff, and Richard Mathias for stimulating discussions on mathematical matters, Chris Clausen for invaluable advice on APL programming, and Earl Homsher and Charles Kean for general discussions. This work was supported by training grant HL-05696 and Program Project grant HL-11351 from the U. S. Public Health Service.

Receivedfor publication 18 January 1978. REFERENCES

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