A MODEL OF EXCITATION-CONTRACTION COUPLING IN FROG CARDIAC MUSCLE*? ALAN
K. WONG:
Y.
Biophysics and Bioengineering Research Laboratory, Faculty of Medicine. Dalhousie University. Halifax, Nova Scotia, Canada B3H 4H7 Abstract-A model for the excitation-contraction (EC) coupling of frog heait muscle was developed. It includes the slow calcium current attributable to phasic tension, and an internal, voltage-dependent, calcium source that gives rise to tonic tension. This internal calcium source is implicitly linked to the Ca-Na competition scheme. The model’s predictions qualitatively agreed with the behavior of frog heart muscle (under voltage-clamp condition) for the following electrical-mechanical events: tension-voltage-duration relation; phasic and tonic contraction; influence of repetitive depolarization on contraction, and influence of membrane potential on restoration of contraction; and tension development and extracellular sodium concentration. Without further assumptions or complexity, the present model inadequately describes EC coupling of mammalian heart muscle.
INTRODUCI’ION
Excitation-contraction (EC) coupling muscle has been extensively investigated
of
rise to developed tension, and the model of Tritthart et al. (1973) described a Ca pool additional to Ic.,.
cardiac
during the past decade (see review by Langer, 1973), and much is now known about the correlation between mechanical and electrical events. It is generally accepted that calcium plays a major role in contraction and recently it was shown that sodium ions are equally impottant (Vassort, 1973), but little is known of the mechanism by which intracellular calcium and transmembrane calcium fluxes determine development of tension. Various models of this mechanism have been proposed. Those described by Kaufmann et al. (1971); Langer (1971); Vassort (1973), are merely descriptive and schematic, and the quantitative models (Ashley and Moisescu, 1972; Bassingthwaighte and Reuter, 1972) emphasize the movement of calcium without relating electrophysiological events to the mechanical activities of muscle fibers. The mathematical models (Johnson and Kuohung, 1968; Johnson and Shepherd, 1971; Manring and Hollander, 1971), dealing mainly with peak tensionfrequency relation, do not delineate EC coupling. Experiments with voltage-clamp or sucrose-gap techniques have shown that the myocardium contains specific Ca pools or compartments that play discrete functional roles in EC coupling. However, the magnitude of tension development is not proportional to the slow inward current attributable to Ca (I& i.e. the tension staircase does not directly relate to Ic. (Beeler and Reuter, 1970; Ochi and Trautwein, 1971). The investigation by Ong and Bailey (1972) evidenced at least two Ca pools, a slow Ca current and activator Ca that activates the contractile element and gives * Receiwd 2 September 1975. 7 Supported by grants from the Medical Research Council of Canada and the Nova Scotia Heart Foundation. $ MRC Research Scholar.
In recent studies by Vassort and Rougier (1972) and Vassort (1973) on frog atria1 tissue. tension could be elicited by low or very high depolarizations in the absence of Ic, or extracellular Ca ions or the presence of Mn ions, and Einwachter et al. (1972). using D600 to suppress Icr, elicited tension in frog atria1 fiber with both short and long depolarization pulses. From these studies one can postulate that, in additlon to I,,, an ‘internal store’ is another Ca source for contraction. Ca sources from this internal store and the Ca current trigger the release of activator calcium, initiating contraction. This activator calcium then regulates the interaction of myosin and actin in the muscle model based on Huxley’s sliding-filament theory (Wang, 1971, 1972).
THE MODEL
The proposed model for EC coupling (Fig. 1) is comprised of the following compartments. 1. Ni, the internal store from which Ca is released upon depolarization. It is assumed that the amount released depends upon the potential amplitude, that the store is refilled from other Ca sources; and that after Ni is filled to its resting level, the excessive Ca is pumped out. 2. Ic,, the inward calcium current, which contributes directly to the activator Ca compartment. (The tail current is not included in this study.) 3. N,, the compartment that sequesters Ca from myofibril and recycles it to Ni. 4. y, the activator Ca compartment: its involvement in the contractile process consists in regulating the rate of connection between myosin and actin filaments.
319
ALAN
320 Ni > Ni
Y. K. WONG dN,ldt = -riNi,
o
Jr’ [~‘I0b2+l I
dN,/dr = aFy
o
I
(3)
dy/dt = -ay + aiNi + UQIC,
M
J = 0.
The above equations indicate the influx of Ca (i.e. I,-,) and the release of Ca by Ni and y; some Ca is also accumulated by N, during the depolarizing period. Since it has been assumed that Ca efflux occurs when the Ca level in Ni exceeds its resting level, J = 0 during depolarization. And during and after repolarization: dNi/dt = a,N, - J dN,/dt = -a,N, dy/dr = -ay
The kinetics of the above compartments dNJdt = -aiN,
+ cN~ - J ,
dy/dt = -ay + aiNi + u&, dN,/dt = -a,N,
are:
+ ay
(1)
i
where ai, q, a, and ac. may be constants or function of voltage. Two possibilities neither with sufficient data support, can be assigned to N, : (a) that N, can continuously recycle Ca from y during depolarization and repolarization. (b) that N, recycles Ca from y during repolarization only. For simplicity, we used assumption (b); the proposed scheme for the a’s is as follows: at E = E,; aI 6: [exp@E$ - 11, a, = 0, a = a at E = 0;
ai = 0, a, = a,, a = a.
(2)
The potential E can be considered as the potential level above the threshold, such as resting potential. Equation (2) implies that Ni releases Ca to y during depolarization, but y constantly provides Ca for contraction, depending on the amount of Ca available. Since N, is assumed to transfer Ca only during and after repolarization, this compartment will accumulate Ca during depolarization. If, as assumed, only a fraction (F) of the Ca used for contraction is accumulated, the rest is distributed elsewhere. During depolarization, equation (1) becomes;
(4)
i
J=O Fig. 1. Scheme of the proposed model. The arrows represent Ca fluxes. The contractile process is regulated by activator Ca (y), which receives Ca sources from inward Ca current (I& and the interoai store (N,). A fraction of y is accumulated by N, during depolarization. Upon re+rization, Ca is sequestered from myofibrils by N, and is transferred to*N,. When N, is tilled to resting level, excessive Ca is pumped out of the membrane. M, membrane; [Ca]Q’ and [Na]: are external concentration of Ca and Na. J is the Ca e&x.
,
+ ay
Ni I Ni.0.
J=a,N,
Ni.0
T.
where v(r), N,(T), N,(T) are the values of 7, N, and Ni at the end of depolarization. EXCITATION-CONTRACTION
COUPLING
Ca released from the activator compartment y probably moves to troponin sites. The hypothesis of Ebashi and Endo (1968) assigns to activator Ca the specific role of removing troponin-dependent inhibition of actin-myosin interaction. The more Ca binds per unit time, the more force-generating bridges are formed per unit time. If A and M represent the actin and myosin filament sites, the formation and breakage of actomyosin are:
A+MLAM AM+XPA
MXP+A.
322
ALAN Y. K. WONG
where XP is a high-energy phosphate compound e.g. adenosine triphosphate) and f; g are the rate constants. Based on Huxley’s sliding-filament theory, the time rate of change in n (the proportion of contraction sites at which M combines with A) is given by: an/at
= (I - rl)f - r7g
(9
and the rate constants f and g vary in the following manner : x PcE, AL was negative, n was shifted to the positive u direction, and SE was shortened an amount AL. This process
Excitation-zontraction
coupling in frog cardiac muscle
was repeated until equilibrium was attained. At the end of At, n became n, for the next AL No attempt was made to confirm experimental data on EC coupling; instead, mechanical phenomena of isometric contractions in response to various depolarizations were simulated. The parameters used in the computation were those for papillary muscle (Wong, 1971). and their values are listed in Nomenclature. RESULTS
The tension development by muscle fiber depends on its length and physical characteristics such as the passive and active elasticity (i.e. PE and SE in HillMaxwell or Voigt Model). The tension is also determined by extracellular ionic concentration (Ca’+, Na+, etc.), depolarization amplitude, duration and probably temperature. Therefore it is difficult to fit the simulated results to any reported experimental data. To demonstrate the mechanical phenomena of muscle contraction under various electrical activities, the depolarization amplitude and Ic. were assigned arbitrary units; and the calcium concentration in Ni, N, and y together with developed tension were normalized to unity in order to show their relative contribution and importance. The rate constants cli a, and a were chosen to give maximal tension for one second depolarization at E = 2. The ai = {@I?a2]‘/o[Na]~2 exp (ZFE/RT)1) was assumed to be 1 at E = 2 (K[Ca]~‘/~a]O+* was taken as 1). ThiIs these numerical values of rate constants give appreciable tension (in normal&d unit) even with depolarizing duration of 2 sec. However, these values are too large for prolonged depolarization of 10 set such as in Kaveler’s experiment (1974). Probably, the ‘corrected’ values of these rate constants should be reduced tenfold. While it seems desirable to estimate the Ca transfer from one compartment to another, difficulty is encountered because of the number of unknown parameters. Nevertheless, a crude approximation of Ic, and Ni,o is given in the following. It is supposed that 40 mV is the threshold for activating Ica and 160mV is the maximal voltage at which Ic. is negligible (Vassort, 1972), a depolarizing pulse of 2 units (E = 2) corresponds to 70mV. At 70mV with 500 msec duration, the simulated peak tension occurs at about 3OOmsec after stimulation; but the peak y (i.e. y = 0.52) occurs at 150 msec. This peak value represents the maximal amount of calcium bound to the myofibrils. Using the numerical values of parameters (listed in Nomenclature), the contribution of N, and Ic. to y is in the ratio of 1 to 4 (i.e. 0.10 to 0.43 I,, where lp = 1 at E = 2). Further, it is assumed that the amount of Ca needed for full activation of the contractile protein is 0.05 PM/~ of heart muscle (Katz, 1970). y = 0.52 is assumed to correspond to 0.025 PM/g. The contri-
bution from Icl at 70 mV is about 0.02 /lM/g. Using a tissue volume of 0.04 mm3. (0.2 mm in length and 0.5 mm in thickness, Einwachter et al.. 1972) and the density of 1 g/cm3 for the muscle. and 0.2 x lo6 C/M of calcium are assumed, the tissue mass is about 40 x 10-6 g. Then 0.431, = 0.020 x 40 x 10mb x 0.2 x 10”pA. I, L 0.4/G% At 70 mV, the peak Ca current is 0.4 PA. a value comparable to experimental observation (Einwachter et al., 1972). Since a normalised value of 0.10 of Ni contribution corresponds to 0.005 PM/~, when 7 = 0.52 the resting value of Ni (Ni,o = 1.0) is estimated to be 0.05 pM:g which is sufficient to fully activate the contractile protein if it can be completely utilized. Tension voltage relation Figure 3 depicts developed tension elicited at three depolarization amplitudes (1, 2, and 5). With short pulses (50 and 250msec) the tension curves are twitch-like, whether Icp is included (at E = 2) or not (at E = 1,5). When the pulse duration is prolonged, tension is increased and maintained. With small. long depolarization pulse (E = 1, duration = IOOOmsec), at which I,, is assumed inactivated, peak tension occurs at the end of depolarization. Isometric tension at various potentials was simulated; peak developed tensions (preload excluded) were plotted against depolarization amplitude (Fig. 4A). Comparison of these sigmoid curves with the experimental data obtained by Vassort and Rougier (1972) showed (Fig. 4B) that at voltages with Ic, inactivated (E 5 1 and E 2 5), tension (tonic component) is elicited and is dependent on the depolarization amplitude and duration. Both sets of data suggest that tension in frog heart muscle can be elicited in the absence of Ic.. Relation between duration of depolarization
and
con-
traction
To determine the influence of depolarization duration on tension development, tension was computed at a certain voltage level with various durations (Fig. 5). As in experiments by Morad and Orkand (1971) and Vassort (1972), the time course and amplitude of tension depended directly on duration of depolarization. The initial rate of tension rise for potentials of different durations appears independent of depolarization amplitude. The relation between depolarization duration and maximal developed tension at three potential levels (Fig. 6A) was compared with a reconstruction of Vassort’s experimental data (Fig. 6B). Phasic and tonic contraction
It is widely accepted that in frog cardiac muscle, phasic tension is attributable to I, and tonic tension is caused by a Ca source other than lc. (Vassort and
ALANY. K. WONG
324
0.5
T
Y 0.0
t!L 2 ’
2
2
0.5 2 z 0
5
!i “: 1 0.0
It! *
500’
0
TIME,MSEC
Fig. 3. The simulated time course of Ir,, 7, and tension, in response to three depolarization amplitudes (1, 2, and 5) with pulse duration of 50 msec (A), 250msec (B), and 1OOOmsec(C).
1.0
% iii e s=
0.6
0.4
0.2
Fig. 4. Simulated peak tension vs depolarization amplitude (in arbitrary units) with duration of 10, 50, and 250msec. The lower curve is peak Ic, (fraction of 16) vs voltage. (B). The tension voltage current relation of frog heart muscle. These curves were reconstructed from the experimental data of Vassort and Rougier (1972). with permission of the authors.
Excitationcontraction
325
coupling in frog cardiac muscle
Ls___ 0
I
co
-1 f
0.5
&A-
Y i
o.ok-----
Fig. 5. The time course of I,,, y, and tension, in response to depolarizing pulse of: (A) amplitude of 1, and durations of 50, 200 and 4OOmsec; (B) amplitude of 2, with 10, 100, 3OOmsec; (C) amplitude of 5, with 50, 150 and 4OOmsec.
Rougier, 1972). Both the phasic and tonic components were simulated in the present study (Fig. 7). According to the present model, the activator Ca for a resulting tension curve is the sum of those for phasic and tonic components. Total tension is not equal to the sum of the two components, because of its dependence at any time point on the number of actin (A) and myosin (M) formations and their distribution. Doubling the activator Ca level does not necessarily double AM formation or maintain its distribution. Influence of repetitive depolarization
on contraction
Simulation of the contractile response to repetitive depolarization (Fig. 8) showed that pulse duration was too short to activate Ic. fully, because of the numerical values chosen for a and b in (equation 6). (This figure compares to Fig. 16 in the report by Einwachter et al., 1972). At low frequency (l/O.7 xc), two contractions are distinguishable; higher tension is developed in the second contraction. This is because the AM formation is not completely broken and the summation effect is apparent. As the frequency increases (B-D), assuming I,, remains unaltered with ensuing pulses, the accumulation of activator Ca causes mechanical responses to individual pulses to blend into one ‘tetanus-like’
contraction.
A D.6
sMgB 4
1
D
/--lvmv
250 DURATION
MO
1%
t msec I
Fig. 6. (A) Simulated tension duration relation at E = 2, fc8 is maximal; E = 5, (tonic tension only (A -A) (O--O); E = 2, phasic tension only and I,, is maximal (S--W); E = I, tonic tension only (O--O). (B). Experimental tension duration relation at 25, 45, 90 and 130 mV. These curves were reconstructed from the data of Vassort (1972) with permission of the author.
ALAN Y. K. WONG
326
1960) in dog myocardium, (Reeler and Reuter, 1970) in sheep Purkinje fiber (Gibbons and Fozzard, 1971) in frog atria (Einwachter er al.. 1972) and Vassort (1973) and in cat ventricular fiber (Tritthart et al., 1973). This phenomenon was simulated (Fig. 9) by assuming that Ic. is reduced when the potential level between two pulses is elevated. As shown in Fig. 10 (which compares to Fig. 17 in the report by Einwachter et ul.. 1972 and Fig. 2 of Vassort’s 1973 paper) the contractile system recovered fully when the intrainterval potential is near resting value. As in the experiments of Einwachter et al. (1972) and Vassort (1973), relaxation was reduced during the interval at increased potential level. Einwachter et al. interpreted this as due to the development of tonic tension-as is predicted by the model. In accordance with the model, compartment Ni is refilled to its resting level when the potential during the interval is zero, providing the same amount of Ca to the activator compartment during both first and second depolarizations. As the potential level between periods is elevated, Ni continuously releases Ca to y, which activates the contractile system. Application of the second pulse limits the availability of Ca from Ni, so that less tonic tension is elicited. Phasic tension also is reduced during the second pulse, due to the decreased Ic, (Vassort, 1973). Horowitz,
Time,
msec
Fig. 7. Tonic (--) and phasic (---) components of resulting tension (-). yr is the Ni contribution to activator compartment, giving rise to tonic tension; y,, is due to 10, attributed to phasic tension. y = yr + yp. Depolarizing pulse = 2 units; duration, 1tWOmsec.
Influence
of membrane
potential
on restoration
of con-
traction The
ability
of muscle
fiber
to contract
upon
a
second depolarization is dependent on the potential level between the first and second depolarizations as demonstrated in frog skeletal muscle (Hodgkin and
0.5
Y 0.0
[Ca],,/[Na],,
on tension development
Dependence of tension on [CaJ,,/[Na]i is well known (Niedergerke, 1963; Orkand, 1968); Vassort
M
Fig. 8. Superimposition of tonic and phasic contraction upon repetitive depolarization (E = 2; duration, 50 msec). The pulse frequency is (A) l/O.7 set, (B) Z/set, (C) 3/set and (D) 4/set.
Excitation-contraction
coupling in frog cardiac
E
327
muscle
______________ .____ ___i L-_--_-_---_--1 8
'Ca -1
I= CN
0.6
--N
1.0
0.8
0.6 0.4 AE I E
0.2
0.0
Fig. 10. Influence of potential level between two depolarizing pulses on restoration of contraction. Pulse duration. 250msec: amplitude, 2 units; interpulse interval. 2 sec. Potential during the interval: 0 (--), 1.6 (-----). and 2 (-) units.
Fig. 9. Ratio of peak tension T,/T,) (O---O) and peak Its (1,/I,) (-) as a function of interpulse potential level. I, was calculated with equation (6). in which E6 was varied from 1 to 2. E = 2. E, = 5. and Rougier, (1972) showed that when the ratio is increased or decreased by varying [Cal,, slow inward current and tension are increased or reduced; but if [Cal0 is modified, keeping the ratio constant, neither parameter alters. When the ratio is changed by diminishing wa],, to l/,/3 of its normal value, slow inward current decreases slightly or remains constant but tension increases. Since Ic. is apparently not modified or even slightly reduced, presumably reduction in [Na]e enhances the development of tonic tension. This is consistent with the scheme indicated by equation (St_i.e. Ui x [Ca]e/[Na]i exp (ZFE/RT)-implying that increase in the ratio will increase the rate of Ca release from Ni to 7 compartment, thus elevating tonic tension Figure 11(B) shows the simulation of tension development due to a reduction of [Na]6 to 1/,,‘3 of its normal value. The peak tension vs [Na]6 is depicted in Fig. 11(A). Peak tension apparently levels off as [Na],, is reduced, reaching a maximal level of 0.62 (E = 4, duration = 250 msec) when [Na]6 approaches zero. The increased ratio due to increased [Cal6 was not simulated because the variation in Ic, is not known. Continuous sequestration The model indicates that the voltage tension duration relation at a depolarization pulse > 500 or 250msec are not significantly different. This is because the activator Ca level, without the contribution of Ic,. falls before repolarization, thereby de-
0.62
A
e
2 Y a54 g 0.58 a50 i,, 1.0
,
,
,
0.8
0.6
0.4
FRACTIONOF [Na], 1.0
B 1 *
a6 -
0” p a-4e a2 -
0.0:.
,
.
,
0
,
,
,
500 TlM
.
.
, lMl0
( msec 1
Fig. 11. Simulation of the influence of [Narc on tension. With normal [Cal0 and [Na], (e.g. in Ringer’s solution, the ratio [Ca],,/[Na]$ is taken as 1). By assuming [Ca], unchanged, the increase in ratio simulates the decrease in [Na]e. (A) Peak tension vs fraction of normal [Na]6. (I?) Isometric tension with normal [Na]6 and l/,/3 of normal CNalo. =. is assumed unaltered by change in [Na]c. Depolarising pulse = 4 units; duration = 250 msec.
AWN Y. K.
328
WONG
1.0
creasing both the number of AM formations and tension. Since peak tension increases as pulse duration 0.8 n is prolonged (Fig. 6B), especially at higher voltage (E 2 E,,,), failure of the model to conform to this observation was thought due to the assumption of Ca sequestration during and after repolarization. Hence, the model was tested for continuous sequestration. (i.e. a, # 0 during depolarizing phase). Figure 12 depicts the time courses of Ni, y, and tension due to continuous sequestration (a, # 0), and due to sequestration during repolarisation only (a, = 0). For short pulse, these curves do not differ significantly. 1.0 For long pulse, however, the tension plateau is maintained with continuous sequestration, the release of Ca from Ni being in relative equilibrium with the return of Ca from N, ; consequently, the y level is constant until repolarization. Thus, continuous sequestration does not appreciably alter the voltagea2 tension-duration relation (Figs. 3 and 5). With the present model, when continuous sequesta0 ration is assumed the tension is sustained as long as 0 200 400 600 800 lal0 1zaI 1400 1600 the pulse. If the interpulse. potential level is elevated TIME ( mssc I as high as the depolarizing pulse, tension elicited by the second pulse will be high instead of zero (Fig. Fig. 12. Ca level in NI y and the tension development due 10). The influence of membrane potential on the to continuous sequestration (-) and sequestration relaxation of contraction cannot be simulated when occurring upon repolarization only (----). E = 2; duration = 1000msec. continuous sequestration is assumed.
a6 0.4
r a2 QO
TIM f mssc 1 Fig. 13. Simulation of EC coupling in mammalian cardiac muscle. The internal storage compartment N, (--) is assumed to be filled above the resting level (1.0) upon repolarimtion; consequently, the activator calcium y level increases with ensuing clamp pulses but peak tension increases only slightly at the fifth beat. Depolarization amplitude = 2 units; duration = 5OOmsec; repolarization = 1sec.
Excitation-contraction
coupling m frog cardiac muscle
EC coupling of‘ mammalian cardiac muscle
Because simulation agreed reasonably with experimental observations in frog heart, simulation of the EC coupling process of mammalian heart muscle was attempted by varying the parameter values only. Mammalian myocardium usually relaxes fully before the end of a clamp pulse of longer duration than a normal action potential, say SOOmsec. To simulate this mechanical response, CLwas made equal to a,; u was increased from 5 to 22.5. and ui from 1 to 22.5. so that y peaked early and decayed before repolarization. Unlike frog heart muscle (in which contractile response to a given step depolarization is almost constant regardless of the number of pulses applied), mammalian myocardium exhibits a gradual increase in twitch tension upon depolarization, maximal tension being attained after 5-8 identical clamps applied at intervals of a few seconds. (Beeler and Reuter, 1970). To simulate the mechanical activity of mammalian cardiac muscle, the internal store was assumed to be filled above its resting level upon repolarization, while the contribution of I,, remained unchanged. After a few beats the stored CYahad reached maximal level, contributing maximal amount to the activator compartment: despite steady increase in I’, peak tension increased only slightly after the fifth beat (Fig. 13). The Ca level (1.3) in Ni at the end of the fifth beat remained maximal, and even the repolarizing period was prolonged. This implies that maximal tension can be elicited when the muscle is restimulated after a long resting period. as opposed to the gradual increase with ensuing beats observed experimentally. Therefore, the present model is unlikely to be applicable for mammalian cardiac muscle.
DISCUSSION
The model proposes not only that the arbitrary expression for I,, is voltage-dependent but also that the coefficient cli for the ‘internal store’ is a function of depolarization. As a result, depolarization influences the availability of activator Ca for activating the contractile protein, or the muscle’s mechanical activities. Thus the model supports the concept that Ca influx and mechanical activities in frog heart muscle are directly controlled by membrane potential (Langer, 1973). It has been tacitly assumed that the rate of Ca release from the ‘internal store’ Ni to the activator compartment y depends on depolarization amplitude, but not duration, despite the increase in peak tension with duration of depolarizing pulse that releases more calcium to y. providing more activator calcium for the formation of cross-bridges. The rate of Ca release from Ni may be a function of both potential amplitude and duration-although duration seems an unlikely primary cause for increased tension, since the muscle fiber can ‘detect’ only the potential applied
32’)
and does not ‘know’ in advance the extent of its duration. Evidence strongly indicates that with two exceptions-dog ventricular bundles and sheep Purkinje fibers (see editorial by Coraboeuf. 1974)--two components of tension are manifested in many species by cardiac muscle. One is phasic tension. attributable to Ic,,--and the other is tonic tension. due to an intracellular Ca source other than I E,-,) of long duration, Zc. was presumably negligible and the peak tonic tension increased with higher voltage. The model’s inability to predict this phenomenon may be. due to the following factors: (a) The coefficient ai (of Ni) is not a function of duration; (b) The value of a(of y) is too large; and (c) The kinetics of Ca movement (equation 1) may be second-order or even nonlinear. Factor (a) has been ruled out on grounds discussed earlier. It is difficult to reject factor (c), although data
on the Ca transient in cardiac muscle are so rare that a kinetic process, whether linear or non-linear, is conjectural. However, no advantage is gained by introducing a further, complex process. Factor (b) is analysed in the following. With the present numerical value for a, at E > 5, and Zc. is zero, y reaches maximal value before repolarization, so that tension declines before the end of depolarization. To have y increased or maintained as long as the depolarization pulse requires the following scheme : atE=Ej;
aioC[exp(lE,)-l]
a,=O,
a=0
atE=O
ai=O
a, = 5L,, ?I = a.
This implies that the ‘gates’ of compartment y and N, are opened only upon repolarization. The Ca level in y compartment during depolarization is: Y=
NLOU
e -““)
-
1
5.
Within the potential range of 1-5, the rate of Ca release from y is dyfdt = a,Ni + cc~.Zc,, or y = N,,,(l - e-““) + ccc. ZC,,dt. where Zc. = Z,eSO’(l - embr). The contribution
of Z, (i.e. second term) is then e-“’
y1 =
@cJ, -
a
e-(a+b)r
+-
a+b
+ c,
1 where c is a constant to be determined. If c is so chosen that at t = 0, y1 = 0, y1 level is maintained as long as the depolarizing pulse and phasic tension due to y1 are not twitch-like but sustaining. When y1 is made zero after a certain time t = T (say 200 msec), the y , level at r = 0 is
The non-zero y, at t = 0 implies that, within the potential range of l-5, tension is possible even before a stimulating pulse is applied. This is impossibleunless the y, (t = 0) can be considered the resting Ca level, only above which contraction is initiated. The model can simulate most mechanical activities observable in experiments and predicts one process that cannot be reproduced experimentally-the activator Ca transient and the actual contractile process (AM formation). How realistic is the proposed activator calcium? No data of Ca transient in cardiac muscle are available, but Blinks (1973), using Caaequorin, has sensitive bioluminescent protein demonstrated it in striated muscle cells. Comparison of the time course of y with the Ca transient in striated muscle during comparable contractile events, such as isometric contraction or contractions at
Excitation+ontraction
coupling m frog cardiac muscle
various frequencies of stimulation, shows that y is similar to the observed time course of Ca transient.
The concept of activator Ca is not new. Julian (1969) incorporated an activation factor, or the time course of Ca binding, in his model of skeletal muscle, although he did not explicitly include EC coupling in his calculation. Manring and Hollander (1971), adopting the postulate of Wood et al. (1969), considered that a fixed fraction of the intracellular calcium was used by the contractile mechanism. In the analog computer model of Kaufmann et al. (1975) for the EC coupling of mammalian heart muscle, the c compartment (sarcoplasm free Ca) was similar to activator Ca in concept and function. If activator Ca does regulate the rate of cross-bridge formation, and thus tension development, it is reasonable and logical to consider it “active state”; but this would not solve the problem of measuring y, or elucidate the independent contribution of each Ca pool to y. Acknowledgement-The author thanks Drs. H. Tritthart and G. Vassort for the v&able discussion concerning the EC coupling of cardiac muscle. He is indebted to Miss Ursula Matthews of the Faculty of Medicine’s Editorial Service, for help with his manuscripts. REFERENCES
Ashley, C. C. and Moisescu, D. G. (1972) Model for the action of calcium in muscle. Nature 237, 208-211. Baker, P. F., Blaustein, M. P., Hodgkin, A. L. and Steinhardf R. A. (1969). The influence of calcium on sodium efflux in squid axon. J. Physiol. 200, 431-458. Baker, P. F. (1972) Transport and metabolism of calcium ions and nerve. Progr. Biophys. 24, 177-223. Bassingthwaighte, J. V. and Reuter, H. (1972) Calcium movements and excitation contraction coupling in cardiac cells. In Electrical Phenomena in the Heart (Edited by De.Mello, W. C.). Academic Press. New York. Beeler, G. W., Jr. and Reuter. H. (1970) The relation between membrane potential, membrane currents and activation of contraction in ventricular myocardial fibers. .!. Physiol. 207. 21 l-229. Blinks, J. R. (1973) Calcium transients in striated muscle cell. Eur. J. Cardiol. 1, 135-142. Coraboeuf, E. (1971) Membrane ionic permeabilities and contractile phenomena in myocardium. Cardiouasc. Res. Suppl. 1, 55-63. Coraboeuf, E. (1974) Membrane electrical activity and double component contraction in cardiac tissue. J. Molec. cell. Cardiol. 6, 215-225. Ebashi. S. and Endo. M. (1968) Calcium ion and muscle contraction. Progr.’ Eiodhys. ‘Molec. Biol. 18, 123-183. Einwachter, H. M., Haas, H. G. and Kern, R. (1972) Membrane current and contraction in frog atrial fibers. J. Physiol. 227, 141-171. Gibbons, W. R. and Fozzard, H. A. (1971) Voltage dependence and time dependence of contraction in sheep cardiac Purkinje fibers. Circulation Res. 28, 446-460. Glitsch, H. G., Reuter, H. and Scholz. H. (1970) The effect of the internal sodium concentration on calcium fluxes in isolated guinea-pig auricles. J. Physiol. 209, 25-43. Hodgkin. A. L. and Horowitz, P. (1960) The effect of sudden changes in ionic concentrations on the membrane potential of single muscle fibers. J. Physiol. bnd. 153, 370-385. Horackova, M. and Vassort, G. (1974) Excitation-Contraction coupling in frog heart: effect of veratrine. Pjuegers Arch. 352, 291-302.
331
Johnson, E. A. and Kuohung, P. W. (1968) The Tri-gamma system: a model of the intrinsic mechanism of control of cardiac contractility. Math. Biosci. 3, 65-89.
Johnson, E. A. and Shepherd, N. (1971) Models of the force frequency relationship of rabbit papillary muscle. Cardiocasc. Rex Suppl. 1, 101-108. Julian, F. J. (1969) Activation in a skeletal muscle contraction model with a modification for insect fibrillar muscle. Biophys. J. 9, 547-570. Katz, A. M. (1970) Contractile proteins of the heart. Physiol. Rev. 50, 63-158. Kaufmann, R. L.. Antoni, H.. Hennekes. R.. Jacob, R.. Kohlhardt, M. and Lab. M. J. (1971) Mechanical response of the mammalian myocardium to modifications of the action potential. Cnrdiouasc. Res. Suppl. 1, 64-70. Kaufmann, R.. Bayer. R., Fumess, T., Krause, H. and Tritthart, H. (1975) Calcium movement controlling cardiac contractility II. Analog Computation of cardiac excitation-contraction coupling on the basis of calcium kinetics in multicompartment model. J. Molec. cell. Cardiol. 6, 545-560. Kavaler, F. (1974) Electromechanical time course in frog ventricle: manipulation of calcium level during voltage clamps. J. Molec. cell. Cnrdiol. 6. 575-580. Langer, G. A. (1971) Coupling calcium in mammalian ventricle: its source and factors regulating its quantity. Cardiovasc. Rex Suppl. 1, 71-75. Langer, G. A. (1973) Heart: excitation contraction coupling. A. Rev. Physiol. 35, 55-86. Leoty, C. and Raymond, G. (1972) Mechanical activity and ionic currents in frog trabeculae. Pfluegers Arch. 334, 114-128. Manring, A. and Hollander, P. B. (1971) The intervalstrength relationship in mammalian atrium: a calcium exchange model. Bbphys. J. 11, 483-500.
Morad. M. and Orkand. R. K. 11971)Excitation contraction ‘coupling in frog’ ventricle: eiidence from voltage clamp studies. J. Physiol. 219, 167-189. Niedergerke, R. (1963) Movements of Ca in frog heart ventricles at rest and during contractures. J. Physiol. 167, 515-550. Ochi, R. and Trautwein, W. (1971) The dependence of cardiac contraction on depolarization and slow inward current. PjIuegers Arch. 323, 187-203. Ong, S. K. and Bailey, L. E. (1972) Two functionally distinct calcium pools in the excitation contraction coupling process. Experientia 8, 14461447.
Orkand, R. K. (1968) Facilitation of heart muscle contraction and its dkpenhence on external calcium and sodium. J. Physiol. I%, 311-325. Sommer, J. R. and Johnson, E. A. (1969) Cardiac Muscle: a comparative ultrastructural study with special reference to frog and chicken hearts. 2. Zellforsch. Mikrosk. Anat. 98, 437-468. Staley, N. A. and Benson, E. S. (1968) The ultrastructure
of frog ventricular cardiac muscle and its relationship to mechanisms of excitation-contraction coupling. J. cell. Biol. 38, 99-114. Tritthart, H., Kaufmann, R., Volkmer, H.-P., Bayer, R. and Krause, H. (1973) Ca-movement controlling myocardial contractility. Pfiuegers Arch. 338, 207-231. Vassort, G. (1972) Couplage excitation-contraction. These de Docteur es-Sciences Naturelles, L’Universitk de Paris. Vassort, G. and Rougier, 0. (1972) Membrane potential and slow inward current dependence of frog cardiac mechanical activity. Pjuegers Arch. 331, 191-203. Vassort, G. (1973) Existence of two components in frog cardiac mechanical activities: influence of Na ions. Eur. J. Cardiol. 1, 163-168. Vassort, G. (1974) Evidence for two components m frog cardiac mechanical activity. In Myocardial Biology. (Edited by Dhalla, N. S.), pp. 671-675. University Park Press. Baltimore, MD.
ALAN Y. K. WONO
332
Wang, A. Y. K. (1971) Mechanics of muscle, based on Huxley’s model: Mathematical simulation of isometric contraction. J. Biomechanics 4, 529-540. Wong, A. Y. K. (1972) Mechanics of cardiac muscle based on Huxley’s model: simulation of active state and forcevelocity relation. J. Biomechanics 5, 107-117. Wood, E. H., Heppner, R. L. and Weidmann, S. (1969) Inotropic effects of electric currents-l. Positive and negative effects of constant electric currents or current pulses applied during cardiac action potentials-II. Hypotheses: calcium movements, excitation-contraction coupling and inotropic effects. Circulation Res. 24, 409445. NOMENCLATURE
Unless defined spectically, used throughout the study: PE SE CE PP6
the following notations
parallel elastic element series elastic element contractile element tension in PE tension in SE PSC tension in CE PCE preload PL constant characterizing PE PO muscle tension PM resting muscle length at zero preload Lo L muscle length L ma” muscle length at which the developed
are
time at which developed tension is maximal nH distribution of n at tM PL(Lrmx) preload when L = L,... peak tension is maximum AL extension of SE constant characterizing PE and K,vK, SE, respectively KM stiffness of side-piece M K SE stiffness of SE = dP,s/d(AL) t time (msec) after stimulus tM
Parameters Lo = 6.8 mm or 1.67 pm At=2msec f,(L,.,) = 117.3 Sl(L_,) = 21.5 g2 = 27.5 Lmu = 9.2mm or 2.20~ PO= 0.0317 K, = 0.9155/mm K, = 0.4254/h h = 130”A a = 30 b=5 a = 5.0 = 7.5 I”,’= 3.102 crc. = 140. A=03466 F = 0.25.
K. WONG:
Y.
Biophysics and Bioengineering Research Laboratory, Faculty of Medicine. Dalhousie University. Halifax, Nova Scotia, Canada B3H 4H7 Abstract-A model for the excitation-contraction (EC) coupling of frog heait muscle was developed. It includes the slow calcium current attributable to phasic tension, and an internal, voltage-dependent, calcium source that gives rise to tonic tension. This internal calcium source is implicitly linked to the Ca-Na competition scheme. The model’s predictions qualitatively agreed with the behavior of frog heart muscle (under voltage-clamp condition) for the following electrical-mechanical events: tension-voltage-duration relation; phasic and tonic contraction; influence of repetitive depolarization on contraction, and influence of membrane potential on restoration of contraction; and tension development and extracellular sodium concentration. Without further assumptions or complexity, the present model inadequately describes EC coupling of mammalian heart muscle.
INTRODUCI’ION
Excitation-contraction (EC) coupling muscle has been extensively investigated
of
rise to developed tension, and the model of Tritthart et al. (1973) described a Ca pool additional to Ic.,.
cardiac
during the past decade (see review by Langer, 1973), and much is now known about the correlation between mechanical and electrical events. It is generally accepted that calcium plays a major role in contraction and recently it was shown that sodium ions are equally impottant (Vassort, 1973), but little is known of the mechanism by which intracellular calcium and transmembrane calcium fluxes determine development of tension. Various models of this mechanism have been proposed. Those described by Kaufmann et al. (1971); Langer (1971); Vassort (1973), are merely descriptive and schematic, and the quantitative models (Ashley and Moisescu, 1972; Bassingthwaighte and Reuter, 1972) emphasize the movement of calcium without relating electrophysiological events to the mechanical activities of muscle fibers. The mathematical models (Johnson and Kuohung, 1968; Johnson and Shepherd, 1971; Manring and Hollander, 1971), dealing mainly with peak tensionfrequency relation, do not delineate EC coupling. Experiments with voltage-clamp or sucrose-gap techniques have shown that the myocardium contains specific Ca pools or compartments that play discrete functional roles in EC coupling. However, the magnitude of tension development is not proportional to the slow inward current attributable to Ca (I& i.e. the tension staircase does not directly relate to Ic. (Beeler and Reuter, 1970; Ochi and Trautwein, 1971). The investigation by Ong and Bailey (1972) evidenced at least two Ca pools, a slow Ca current and activator Ca that activates the contractile element and gives * Receiwd 2 September 1975. 7 Supported by grants from the Medical Research Council of Canada and the Nova Scotia Heart Foundation. $ MRC Research Scholar.
In recent studies by Vassort and Rougier (1972) and Vassort (1973) on frog atria1 tissue. tension could be elicited by low or very high depolarizations in the absence of Ic, or extracellular Ca ions or the presence of Mn ions, and Einwachter et al. (1972). using D600 to suppress Icr, elicited tension in frog atria1 fiber with both short and long depolarization pulses. From these studies one can postulate that, in additlon to I,,, an ‘internal store’ is another Ca source for contraction. Ca sources from this internal store and the Ca current trigger the release of activator calcium, initiating contraction. This activator calcium then regulates the interaction of myosin and actin in the muscle model based on Huxley’s sliding-filament theory (Wang, 1971, 1972).
THE MODEL
The proposed model for EC coupling (Fig. 1) is comprised of the following compartments. 1. Ni, the internal store from which Ca is released upon depolarization. It is assumed that the amount released depends upon the potential amplitude, that the store is refilled from other Ca sources; and that after Ni is filled to its resting level, the excessive Ca is pumped out. 2. Ic,, the inward calcium current, which contributes directly to the activator Ca compartment. (The tail current is not included in this study.) 3. N,, the compartment that sequesters Ca from myofibril and recycles it to Ni. 4. y, the activator Ca compartment: its involvement in the contractile process consists in regulating the rate of connection between myosin and actin filaments.
319
ALAN
320 Ni > Ni
Y. K. WONG dN,ldt = -riNi,
o
Jr’ [~‘I0b2+l I
dN,/dr = aFy
o
I
(3)
dy/dt = -ay + aiNi + UQIC,
M
J = 0.
The above equations indicate the influx of Ca (i.e. I,-,) and the release of Ca by Ni and y; some Ca is also accumulated by N, during the depolarizing period. Since it has been assumed that Ca efflux occurs when the Ca level in Ni exceeds its resting level, J = 0 during depolarization. And during and after repolarization: dNi/dt = a,N, - J dN,/dt = -a,N, dy/dr = -ay
The kinetics of the above compartments dNJdt = -aiN,
+ cN~ - J ,
dy/dt = -ay + aiNi + u&, dN,/dt = -a,N,
are:
+ ay
(1)
i
where ai, q, a, and ac. may be constants or function of voltage. Two possibilities neither with sufficient data support, can be assigned to N, : (a) that N, can continuously recycle Ca from y during depolarization and repolarization. (b) that N, recycles Ca from y during repolarization only. For simplicity, we used assumption (b); the proposed scheme for the a’s is as follows: at E = E,; aI 6: [exp@E$ - 11, a, = 0, a = a at E = 0;
ai = 0, a, = a,, a = a.
(2)
The potential E can be considered as the potential level above the threshold, such as resting potential. Equation (2) implies that Ni releases Ca to y during depolarization, but y constantly provides Ca for contraction, depending on the amount of Ca available. Since N, is assumed to transfer Ca only during and after repolarization, this compartment will accumulate Ca during depolarization. If, as assumed, only a fraction (F) of the Ca used for contraction is accumulated, the rest is distributed elsewhere. During depolarization, equation (1) becomes;
(4)
i
J=O Fig. 1. Scheme of the proposed model. The arrows represent Ca fluxes. The contractile process is regulated by activator Ca (y), which receives Ca sources from inward Ca current (I& and the interoai store (N,). A fraction of y is accumulated by N, during depolarization. Upon re+rization, Ca is sequestered from myofibrils by N, and is transferred to*N,. When N, is tilled to resting level, excessive Ca is pumped out of the membrane. M, membrane; [Ca]Q’ and [Na]: are external concentration of Ca and Na. J is the Ca e&x.
,
+ ay
Ni I Ni.0.
J=a,N,
Ni.0
T.
where v(r), N,(T), N,(T) are the values of 7, N, and Ni at the end of depolarization. EXCITATION-CONTRACTION
COUPLING
Ca released from the activator compartment y probably moves to troponin sites. The hypothesis of Ebashi and Endo (1968) assigns to activator Ca the specific role of removing troponin-dependent inhibition of actin-myosin interaction. The more Ca binds per unit time, the more force-generating bridges are formed per unit time. If A and M represent the actin and myosin filament sites, the formation and breakage of actomyosin are:
A+MLAM AM+XPA
MXP+A.
322
ALAN Y. K. WONG
where XP is a high-energy phosphate compound e.g. adenosine triphosphate) and f; g are the rate constants. Based on Huxley’s sliding-filament theory, the time rate of change in n (the proportion of contraction sites at which M combines with A) is given by: an/at
= (I - rl)f - r7g
(9
and the rate constants f and g vary in the following manner : x PcE, AL was negative, n was shifted to the positive u direction, and SE was shortened an amount AL. This process
Excitation-zontraction
coupling in frog cardiac muscle
was repeated until equilibrium was attained. At the end of At, n became n, for the next AL No attempt was made to confirm experimental data on EC coupling; instead, mechanical phenomena of isometric contractions in response to various depolarizations were simulated. The parameters used in the computation were those for papillary muscle (Wong, 1971). and their values are listed in Nomenclature. RESULTS
The tension development by muscle fiber depends on its length and physical characteristics such as the passive and active elasticity (i.e. PE and SE in HillMaxwell or Voigt Model). The tension is also determined by extracellular ionic concentration (Ca’+, Na+, etc.), depolarization amplitude, duration and probably temperature. Therefore it is difficult to fit the simulated results to any reported experimental data. To demonstrate the mechanical phenomena of muscle contraction under various electrical activities, the depolarization amplitude and Ic. were assigned arbitrary units; and the calcium concentration in Ni, N, and y together with developed tension were normalized to unity in order to show their relative contribution and importance. The rate constants cli a, and a were chosen to give maximal tension for one second depolarization at E = 2. The ai = {@I?a2]‘/o[Na]~2 exp (ZFE/RT)1) was assumed to be 1 at E = 2 (K[Ca]~‘/~a]O+* was taken as 1). ThiIs these numerical values of rate constants give appreciable tension (in normal&d unit) even with depolarizing duration of 2 sec. However, these values are too large for prolonged depolarization of 10 set such as in Kaveler’s experiment (1974). Probably, the ‘corrected’ values of these rate constants should be reduced tenfold. While it seems desirable to estimate the Ca transfer from one compartment to another, difficulty is encountered because of the number of unknown parameters. Nevertheless, a crude approximation of Ic, and Ni,o is given in the following. It is supposed that 40 mV is the threshold for activating Ica and 160mV is the maximal voltage at which Ic. is negligible (Vassort, 1972), a depolarizing pulse of 2 units (E = 2) corresponds to 70mV. At 70mV with 500 msec duration, the simulated peak tension occurs at about 3OOmsec after stimulation; but the peak y (i.e. y = 0.52) occurs at 150 msec. This peak value represents the maximal amount of calcium bound to the myofibrils. Using the numerical values of parameters (listed in Nomenclature), the contribution of N, and Ic. to y is in the ratio of 1 to 4 (i.e. 0.10 to 0.43 I,, where lp = 1 at E = 2). Further, it is assumed that the amount of Ca needed for full activation of the contractile protein is 0.05 PM/~ of heart muscle (Katz, 1970). y = 0.52 is assumed to correspond to 0.025 PM/g. The contri-
bution from Icl at 70 mV is about 0.02 /lM/g. Using a tissue volume of 0.04 mm3. (0.2 mm in length and 0.5 mm in thickness, Einwachter et al.. 1972) and the density of 1 g/cm3 for the muscle. and 0.2 x lo6 C/M of calcium are assumed, the tissue mass is about 40 x 10-6 g. Then 0.431, = 0.020 x 40 x 10mb x 0.2 x 10”pA. I, L 0.4/G% At 70 mV, the peak Ca current is 0.4 PA. a value comparable to experimental observation (Einwachter et al., 1972). Since a normalised value of 0.10 of Ni contribution corresponds to 0.005 PM/~, when 7 = 0.52 the resting value of Ni (Ni,o = 1.0) is estimated to be 0.05 pM:g which is sufficient to fully activate the contractile protein if it can be completely utilized. Tension voltage relation Figure 3 depicts developed tension elicited at three depolarization amplitudes (1, 2, and 5). With short pulses (50 and 250msec) the tension curves are twitch-like, whether Icp is included (at E = 2) or not (at E = 1,5). When the pulse duration is prolonged, tension is increased and maintained. With small. long depolarization pulse (E = 1, duration = IOOOmsec), at which I,, is assumed inactivated, peak tension occurs at the end of depolarization. Isometric tension at various potentials was simulated; peak developed tensions (preload excluded) were plotted against depolarization amplitude (Fig. 4A). Comparison of these sigmoid curves with the experimental data obtained by Vassort and Rougier (1972) showed (Fig. 4B) that at voltages with Ic, inactivated (E 5 1 and E 2 5), tension (tonic component) is elicited and is dependent on the depolarization amplitude and duration. Both sets of data suggest that tension in frog heart muscle can be elicited in the absence of Ic.. Relation between duration of depolarization
and
con-
traction
To determine the influence of depolarization duration on tension development, tension was computed at a certain voltage level with various durations (Fig. 5). As in experiments by Morad and Orkand (1971) and Vassort (1972), the time course and amplitude of tension depended directly on duration of depolarization. The initial rate of tension rise for potentials of different durations appears independent of depolarization amplitude. The relation between depolarization duration and maximal developed tension at three potential levels (Fig. 6A) was compared with a reconstruction of Vassort’s experimental data (Fig. 6B). Phasic and tonic contraction
It is widely accepted that in frog cardiac muscle, phasic tension is attributable to I, and tonic tension is caused by a Ca source other than lc. (Vassort and
ALANY. K. WONG
324
0.5
T
Y 0.0
t!L 2 ’
2
2
0.5 2 z 0
5
!i “: 1 0.0
It! *
500’
0
TIME,MSEC
Fig. 3. The simulated time course of Ir,, 7, and tension, in response to three depolarization amplitudes (1, 2, and 5) with pulse duration of 50 msec (A), 250msec (B), and 1OOOmsec(C).
1.0
% iii e s=
0.6
0.4
0.2
Fig. 4. Simulated peak tension vs depolarization amplitude (in arbitrary units) with duration of 10, 50, and 250msec. The lower curve is peak Ic, (fraction of 16) vs voltage. (B). The tension voltage current relation of frog heart muscle. These curves were reconstructed from the experimental data of Vassort and Rougier (1972). with permission of the authors.
Excitationcontraction
325
coupling in frog cardiac muscle
Ls___ 0
I
co
-1 f
0.5
&A-
Y i
o.ok-----
Fig. 5. The time course of I,,, y, and tension, in response to depolarizing pulse of: (A) amplitude of 1, and durations of 50, 200 and 4OOmsec; (B) amplitude of 2, with 10, 100, 3OOmsec; (C) amplitude of 5, with 50, 150 and 4OOmsec.
Rougier, 1972). Both the phasic and tonic components were simulated in the present study (Fig. 7). According to the present model, the activator Ca for a resulting tension curve is the sum of those for phasic and tonic components. Total tension is not equal to the sum of the two components, because of its dependence at any time point on the number of actin (A) and myosin (M) formations and their distribution. Doubling the activator Ca level does not necessarily double AM formation or maintain its distribution. Influence of repetitive depolarization
on contraction
Simulation of the contractile response to repetitive depolarization (Fig. 8) showed that pulse duration was too short to activate Ic. fully, because of the numerical values chosen for a and b in (equation 6). (This figure compares to Fig. 16 in the report by Einwachter et al., 1972). At low frequency (l/O.7 xc), two contractions are distinguishable; higher tension is developed in the second contraction. This is because the AM formation is not completely broken and the summation effect is apparent. As the frequency increases (B-D), assuming I,, remains unaltered with ensuing pulses, the accumulation of activator Ca causes mechanical responses to individual pulses to blend into one ‘tetanus-like’
contraction.
A D.6
sMgB 4
1
D
/--lvmv
250 DURATION
MO
1%
t msec I
Fig. 6. (A) Simulated tension duration relation at E = 2, fc8 is maximal; E = 5, (tonic tension only (A -A) (O--O); E = 2, phasic tension only and I,, is maximal (S--W); E = I, tonic tension only (O--O). (B). Experimental tension duration relation at 25, 45, 90 and 130 mV. These curves were reconstructed from the data of Vassort (1972) with permission of the author.
ALAN Y. K. WONG
326
1960) in dog myocardium, (Reeler and Reuter, 1970) in sheep Purkinje fiber (Gibbons and Fozzard, 1971) in frog atria (Einwachter er al.. 1972) and Vassort (1973) and in cat ventricular fiber (Tritthart et al., 1973). This phenomenon was simulated (Fig. 9) by assuming that Ic. is reduced when the potential level between two pulses is elevated. As shown in Fig. 10 (which compares to Fig. 17 in the report by Einwachter et ul.. 1972 and Fig. 2 of Vassort’s 1973 paper) the contractile system recovered fully when the intrainterval potential is near resting value. As in the experiments of Einwachter et al. (1972) and Vassort (1973), relaxation was reduced during the interval at increased potential level. Einwachter et al. interpreted this as due to the development of tonic tension-as is predicted by the model. In accordance with the model, compartment Ni is refilled to its resting level when the potential during the interval is zero, providing the same amount of Ca to the activator compartment during both first and second depolarizations. As the potential level between periods is elevated, Ni continuously releases Ca to y, which activates the contractile system. Application of the second pulse limits the availability of Ca from Ni, so that less tonic tension is elicited. Phasic tension also is reduced during the second pulse, due to the decreased Ic, (Vassort, 1973). Horowitz,
Time,
msec
Fig. 7. Tonic (--) and phasic (---) components of resulting tension (-). yr is the Ni contribution to activator compartment, giving rise to tonic tension; y,, is due to 10, attributed to phasic tension. y = yr + yp. Depolarizing pulse = 2 units; duration, 1tWOmsec.
Influence
of membrane
potential
on restoration
of con-
traction The
ability
of muscle
fiber
to contract
upon
a
second depolarization is dependent on the potential level between the first and second depolarizations as demonstrated in frog skeletal muscle (Hodgkin and
0.5
Y 0.0
[Ca],,/[Na],,
on tension development
Dependence of tension on [CaJ,,/[Na]i is well known (Niedergerke, 1963; Orkand, 1968); Vassort
M
Fig. 8. Superimposition of tonic and phasic contraction upon repetitive depolarization (E = 2; duration, 50 msec). The pulse frequency is (A) l/O.7 set, (B) Z/set, (C) 3/set and (D) 4/set.
Excitation-contraction
coupling in frog cardiac
E
327
muscle
______________ .____ ___i L-_--_-_---_--1 8
'Ca -1
I= CN
0.6
--N
1.0
0.8
0.6 0.4 AE I E
0.2
0.0
Fig. 10. Influence of potential level between two depolarizing pulses on restoration of contraction. Pulse duration. 250msec: amplitude, 2 units; interpulse interval. 2 sec. Potential during the interval: 0 (--), 1.6 (-----). and 2 (-) units.
Fig. 9. Ratio of peak tension T,/T,) (O---O) and peak Its (1,/I,) (-) as a function of interpulse potential level. I, was calculated with equation (6). in which E6 was varied from 1 to 2. E = 2. E, = 5. and Rougier, (1972) showed that when the ratio is increased or decreased by varying [Cal,, slow inward current and tension are increased or reduced; but if [Cal0 is modified, keeping the ratio constant, neither parameter alters. When the ratio is changed by diminishing wa],, to l/,/3 of its normal value, slow inward current decreases slightly or remains constant but tension increases. Since Ic. is apparently not modified or even slightly reduced, presumably reduction in [Na]e enhances the development of tonic tension. This is consistent with the scheme indicated by equation (St_i.e. Ui x [Ca]e/[Na]i exp (ZFE/RT)-implying that increase in the ratio will increase the rate of Ca release from Ni to 7 compartment, thus elevating tonic tension Figure 11(B) shows the simulation of tension development due to a reduction of [Na]6 to 1/,,‘3 of its normal value. The peak tension vs [Na]6 is depicted in Fig. 11(A). Peak tension apparently levels off as [Na],, is reduced, reaching a maximal level of 0.62 (E = 4, duration = 250 msec) when [Na]6 approaches zero. The increased ratio due to increased [Cal6 was not simulated because the variation in Ic, is not known. Continuous sequestration The model indicates that the voltage tension duration relation at a depolarization pulse > 500 or 250msec are not significantly different. This is because the activator Ca level, without the contribution of Ic,. falls before repolarization, thereby de-
0.62
A
e
2 Y a54 g 0.58 a50 i,, 1.0
,
,
,
0.8
0.6
0.4
FRACTIONOF [Na], 1.0
B 1 *
a6 -
0” p a-4e a2 -
0.0:.
,
.
,
0
,
,
,
500 TlM
.
.
, lMl0
( msec 1
Fig. 11. Simulation of the influence of [Narc on tension. With normal [Cal0 and [Na], (e.g. in Ringer’s solution, the ratio [Ca],,/[Na]$ is taken as 1). By assuming [Ca], unchanged, the increase in ratio simulates the decrease in [Na]e. (A) Peak tension vs fraction of normal [Na]6. (I?) Isometric tension with normal [Na]6 and l/,/3 of normal CNalo. =. is assumed unaltered by change in [Na]c. Depolarising pulse = 4 units; duration = 250 msec.
AWN Y. K.
328
WONG
1.0
creasing both the number of AM formations and tension. Since peak tension increases as pulse duration 0.8 n is prolonged (Fig. 6B), especially at higher voltage (E 2 E,,,), failure of the model to conform to this observation was thought due to the assumption of Ca sequestration during and after repolarization. Hence, the model was tested for continuous sequestration. (i.e. a, # 0 during depolarizing phase). Figure 12 depicts the time courses of Ni, y, and tension due to continuous sequestration (a, # 0), and due to sequestration during repolarisation only (a, = 0). For short pulse, these curves do not differ significantly. 1.0 For long pulse, however, the tension plateau is maintained with continuous sequestration, the release of Ca from Ni being in relative equilibrium with the return of Ca from N, ; consequently, the y level is constant until repolarization. Thus, continuous sequestration does not appreciably alter the voltagea2 tension-duration relation (Figs. 3 and 5). With the present model, when continuous sequesta0 ration is assumed the tension is sustained as long as 0 200 400 600 800 lal0 1zaI 1400 1600 the pulse. If the interpulse. potential level is elevated TIME ( mssc I as high as the depolarizing pulse, tension elicited by the second pulse will be high instead of zero (Fig. Fig. 12. Ca level in NI y and the tension development due 10). The influence of membrane potential on the to continuous sequestration (-) and sequestration relaxation of contraction cannot be simulated when occurring upon repolarization only (----). E = 2; duration = 1000msec. continuous sequestration is assumed.
a6 0.4
r a2 QO
TIM f mssc 1 Fig. 13. Simulation of EC coupling in mammalian cardiac muscle. The internal storage compartment N, (--) is assumed to be filled above the resting level (1.0) upon repolarimtion; consequently, the activator calcium y level increases with ensuing clamp pulses but peak tension increases only slightly at the fifth beat. Depolarization amplitude = 2 units; duration = 5OOmsec; repolarization = 1sec.
Excitation-contraction
coupling m frog cardiac muscle
EC coupling of‘ mammalian cardiac muscle
Because simulation agreed reasonably with experimental observations in frog heart, simulation of the EC coupling process of mammalian heart muscle was attempted by varying the parameter values only. Mammalian myocardium usually relaxes fully before the end of a clamp pulse of longer duration than a normal action potential, say SOOmsec. To simulate this mechanical response, CLwas made equal to a,; u was increased from 5 to 22.5. and ui from 1 to 22.5. so that y peaked early and decayed before repolarization. Unlike frog heart muscle (in which contractile response to a given step depolarization is almost constant regardless of the number of pulses applied), mammalian myocardium exhibits a gradual increase in twitch tension upon depolarization, maximal tension being attained after 5-8 identical clamps applied at intervals of a few seconds. (Beeler and Reuter, 1970). To simulate the mechanical activity of mammalian cardiac muscle, the internal store was assumed to be filled above its resting level upon repolarization, while the contribution of I,, remained unchanged. After a few beats the stored CYahad reached maximal level, contributing maximal amount to the activator compartment: despite steady increase in I’, peak tension increased only slightly after the fifth beat (Fig. 13). The Ca level (1.3) in Ni at the end of the fifth beat remained maximal, and even the repolarizing period was prolonged. This implies that maximal tension can be elicited when the muscle is restimulated after a long resting period. as opposed to the gradual increase with ensuing beats observed experimentally. Therefore, the present model is unlikely to be applicable for mammalian cardiac muscle.
DISCUSSION
The model proposes not only that the arbitrary expression for I,, is voltage-dependent but also that the coefficient cli for the ‘internal store’ is a function of depolarization. As a result, depolarization influences the availability of activator Ca for activating the contractile protein, or the muscle’s mechanical activities. Thus the model supports the concept that Ca influx and mechanical activities in frog heart muscle are directly controlled by membrane potential (Langer, 1973). It has been tacitly assumed that the rate of Ca release from the ‘internal store’ Ni to the activator compartment y depends on depolarization amplitude, but not duration, despite the increase in peak tension with duration of depolarizing pulse that releases more calcium to y. providing more activator calcium for the formation of cross-bridges. The rate of Ca release from Ni may be a function of both potential amplitude and duration-although duration seems an unlikely primary cause for increased tension, since the muscle fiber can ‘detect’ only the potential applied
32’)
and does not ‘know’ in advance the extent of its duration. Evidence strongly indicates that with two exceptions-dog ventricular bundles and sheep Purkinje fibers (see editorial by Coraboeuf. 1974)--two components of tension are manifested in many species by cardiac muscle. One is phasic tension. attributable to Ic,,--and the other is tonic tension. due to an intracellular Ca source other than I E,-,) of long duration, Zc. was presumably negligible and the peak tonic tension increased with higher voltage. The model’s inability to predict this phenomenon may be. due to the following factors: (a) The coefficient ai (of Ni) is not a function of duration; (b) The value of a(of y) is too large; and (c) The kinetics of Ca movement (equation 1) may be second-order or even nonlinear. Factor (a) has been ruled out on grounds discussed earlier. It is difficult to reject factor (c), although data
on the Ca transient in cardiac muscle are so rare that a kinetic process, whether linear or non-linear, is conjectural. However, no advantage is gained by introducing a further, complex process. Factor (b) is analysed in the following. With the present numerical value for a, at E > 5, and Zc. is zero, y reaches maximal value before repolarization, so that tension declines before the end of depolarization. To have y increased or maintained as long as the depolarization pulse requires the following scheme : atE=Ej;
aioC[exp(lE,)-l]
a,=O,
a=0
atE=O
ai=O
a, = 5L,, ?I = a.
This implies that the ‘gates’ of compartment y and N, are opened only upon repolarization. The Ca level in y compartment during depolarization is: Y=
NLOU
e -““)
-
1
5.
Within the potential range of 1-5, the rate of Ca release from y is dyfdt = a,Ni + cc~.Zc,, or y = N,,,(l - e-““) + ccc. ZC,,dt. where Zc. = Z,eSO’(l - embr). The contribution
of Z, (i.e. second term) is then e-“’
y1 =
@cJ, -
a
e-(a+b)r
+-
a+b
+ c,
1 where c is a constant to be determined. If c is so chosen that at t = 0, y1 = 0, y1 level is maintained as long as the depolarizing pulse and phasic tension due to y1 are not twitch-like but sustaining. When y1 is made zero after a certain time t = T (say 200 msec), the y , level at r = 0 is
The non-zero y, at t = 0 implies that, within the potential range of l-5, tension is possible even before a stimulating pulse is applied. This is impossibleunless the y, (t = 0) can be considered the resting Ca level, only above which contraction is initiated. The model can simulate most mechanical activities observable in experiments and predicts one process that cannot be reproduced experimentally-the activator Ca transient and the actual contractile process (AM formation). How realistic is the proposed activator calcium? No data of Ca transient in cardiac muscle are available, but Blinks (1973), using Caaequorin, has sensitive bioluminescent protein demonstrated it in striated muscle cells. Comparison of the time course of y with the Ca transient in striated muscle during comparable contractile events, such as isometric contraction or contractions at
Excitation+ontraction
coupling m frog cardiac muscle
various frequencies of stimulation, shows that y is similar to the observed time course of Ca transient.
The concept of activator Ca is not new. Julian (1969) incorporated an activation factor, or the time course of Ca binding, in his model of skeletal muscle, although he did not explicitly include EC coupling in his calculation. Manring and Hollander (1971), adopting the postulate of Wood et al. (1969), considered that a fixed fraction of the intracellular calcium was used by the contractile mechanism. In the analog computer model of Kaufmann et al. (1975) for the EC coupling of mammalian heart muscle, the c compartment (sarcoplasm free Ca) was similar to activator Ca in concept and function. If activator Ca does regulate the rate of cross-bridge formation, and thus tension development, it is reasonable and logical to consider it “active state”; but this would not solve the problem of measuring y, or elucidate the independent contribution of each Ca pool to y. Acknowledgement-The author thanks Drs. H. Tritthart and G. Vassort for the v&able discussion concerning the EC coupling of cardiac muscle. He is indebted to Miss Ursula Matthews of the Faculty of Medicine’s Editorial Service, for help with his manuscripts. REFERENCES
Ashley, C. C. and Moisescu, D. G. (1972) Model for the action of calcium in muscle. Nature 237, 208-211. Baker, P. F., Blaustein, M. P., Hodgkin, A. L. and Steinhardf R. A. (1969). The influence of calcium on sodium efflux in squid axon. J. Physiol. 200, 431-458. Baker, P. F. (1972) Transport and metabolism of calcium ions and nerve. Progr. Biophys. 24, 177-223. Bassingthwaighte, J. V. and Reuter, H. (1972) Calcium movements and excitation contraction coupling in cardiac cells. In Electrical Phenomena in the Heart (Edited by De.Mello, W. C.). Academic Press. New York. Beeler, G. W., Jr. and Reuter. H. (1970) The relation between membrane potential, membrane currents and activation of contraction in ventricular myocardial fibers. .!. Physiol. 207. 21 l-229. Blinks, J. R. (1973) Calcium transients in striated muscle cell. Eur. J. Cardiol. 1, 135-142. Coraboeuf, E. (1971) Membrane ionic permeabilities and contractile phenomena in myocardium. Cardiouasc. Res. Suppl. 1, 55-63. Coraboeuf, E. (1974) Membrane electrical activity and double component contraction in cardiac tissue. J. Molec. cell. Cardiol. 6, 215-225. Ebashi. S. and Endo. M. (1968) Calcium ion and muscle contraction. Progr.’ Eiodhys. ‘Molec. Biol. 18, 123-183. Einwachter, H. M., Haas, H. G. and Kern, R. (1972) Membrane current and contraction in frog atrial fibers. J. Physiol. 227, 141-171. Gibbons, W. R. and Fozzard, H. A. (1971) Voltage dependence and time dependence of contraction in sheep cardiac Purkinje fibers. Circulation Res. 28, 446-460. Glitsch, H. G., Reuter, H. and Scholz. H. (1970) The effect of the internal sodium concentration on calcium fluxes in isolated guinea-pig auricles. J. Physiol. 209, 25-43. Hodgkin. A. L. and Horowitz, P. (1960) The effect of sudden changes in ionic concentrations on the membrane potential of single muscle fibers. J. Physiol. bnd. 153, 370-385. Horackova, M. and Vassort, G. (1974) Excitation-Contraction coupling in frog heart: effect of veratrine. Pjuegers Arch. 352, 291-302.
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ALAN Y. K. WONO
332
Wang, A. Y. K. (1971) Mechanics of muscle, based on Huxley’s model: Mathematical simulation of isometric contraction. J. Biomechanics 4, 529-540. Wong, A. Y. K. (1972) Mechanics of cardiac muscle based on Huxley’s model: simulation of active state and forcevelocity relation. J. Biomechanics 5, 107-117. Wood, E. H., Heppner, R. L. and Weidmann, S. (1969) Inotropic effects of electric currents-l. Positive and negative effects of constant electric currents or current pulses applied during cardiac action potentials-II. Hypotheses: calcium movements, excitation-contraction coupling and inotropic effects. Circulation Res. 24, 409445. NOMENCLATURE
Unless defined spectically, used throughout the study: PE SE CE PP6
the following notations
parallel elastic element series elastic element contractile element tension in PE tension in SE PSC tension in CE PCE preload PL constant characterizing PE PO muscle tension PM resting muscle length at zero preload Lo L muscle length L ma” muscle length at which the developed
are
time at which developed tension is maximal nH distribution of n at tM PL(Lrmx) preload when L = L,... peak tension is maximum AL extension of SE constant characterizing PE and K,vK, SE, respectively KM stiffness of side-piece M K SE stiffness of SE = dP,s/d(AL) t time (msec) after stimulus tM
Parameters Lo = 6.8 mm or 1.67 pm At=2msec f,(L,.,) = 117.3 Sl(L_,) = 21.5 g2 = 27.5 Lmu = 9.2mm or 2.20~ PO= 0.0317 K, = 0.9155/mm K, = 0.4254/h h = 130”A a = 30 b=5 a = 5.0 = 7.5 I”,’= 3.102 crc. = 140. A=03466 F = 0.25.
