Department of Biomedical Engineering, Julius Silver Institute,
Heart System Research Center, Haifa 32000, Israel
The well-known
linear relationship between oxygen consumption and force-length area or
the force-time integral is analyzed here for isometric contractions.
The analysis, which is based on a biochemical model that couples
calcium kinetics with cross-bridge cycling, indicates that the change
in the number of force-generating cross bridges with the change in the
sarcomere length depends on the force generated by the cross bridges.
This positive-feedback phenomenon is consistent with our reported
cooperativity mechanism, whereby the affinity of the troponin for
calcium and, hence, cross-bridge recruitment depends on the number of
force-generating cross bridges. Moreover, it is demonstrated that a
model that does not include a feedback mechanism cannot describe the
dependence of energy consumption on the loading conditions. The
cooperativity mechanism, which has been shown to determine the
force-length relationship and the related Frank-Starling law, is shown
here to provide the basis for the regulation of energy consumption in
the cardiac muscle.
 |
INTRODUCTION |
THE EXISTENCE OF a linear relationship between oxygen
consumption (
O2) and the
mechanical energy generated by the left ventricle (LV) has been
demonstrated by Suga et al. (28, 32). The total mechanical energy
generated by the LV is quantified by the pressure-volume area (PVA) in
the LV pressure-volume plane (Fig.
1A),
where PVA is the sum of the external work (EW) done by the LV and the
mechanical potential energy (PE) given by
|
(1)
|
PE is defined as the area bounded by the end-systolic pressure-volume
relationship (ESPVR) curve, the passive end-diastolic pressure-volume
relationship, and the end-systolic volume (21, 32).
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Fig. 1.
Schematic description of left ventricular (LV) function in
pressure-volume plane. A: potential
energy (PE) is area bounded by end-systolic pressure-volume
relationship (ESPVR), passive end-diastolic pressure-volume
relationship, and end-systolic volume. EW, external work.
B: force-length area (FLA) defined as
sum of PE and EW.
|
|
Suga's experimental data yields (32)
|
(2)
|
where
PVA, the mechanical energy, corresponds to the energy consumed by the
actomyosin ATPase and b represents the
basal metabolic energy consumption, i.e., the energy used by the
sodium-potassium pumps and by the calcium pumps. The parameters
a and
b are constants (31).
Using ferret papillary muscle fibers, Hisano and Cooper (16) have shown
that the force-length area (FLA), the cardiac fiber analog of the
ventricular PVA for the mechanical energy, is also closely correlated
with oxygen consumption as
|
(3)
|
where
k1 and
k0 are constants.
The FLA is defined, similarly to the ventricular PVA, as the sum of EW
and PE (Fig. 1B). As was
experimentally established at the LV level, the relationship between
the FLA and O2 is independent
of the mode of contraction, either isometric or shortening. Similarly,
Mast and Elzinga (26) have shown, using rabbit papillary muscles, that
the FLA correlates with the tension-dependent heat in isometric contraction.
Alpert et al. (3), Hisano and Cooper (16), and Wannenburg et al. (35)
have demonstrated the existence of a linear relationship between the
energy consumption and the force-time integral (FTI), i.e., the
integral over the time course of the force, for isometric contractions.
Thus
|
(4)
|
where
C1 and
C0 are constants.
Both the FLA and the FTI correlate with energy consumption for
isometric contractions (16, 26, 35).
The linear relationship between the FLA and
O2 at the level of the muscle
fiber (Eq. 3) (16, 26) suggests that
the linear relationship between PVA and
O2 at the LV level
(Eq. 2) (28, 32) is not due to some
unique characteristics of the LV but is an integrated result of the
basic characteristics of the myocytes. The cellular mechanism
underlying this tight biochemical-mechanical coupling is still not
fully understood.
Here, we analyze the correlation between the energy consumption and the
FLA for isometric contraction and relate it to our current model of the
biochemical-mechanical coupling by the sarcomere.
Our earlier studies (22-25) describe the intracellular control
mechanisms of the contractile filaments and couple the kinetics of
calcium binding to troponin with the regulation of cross-bridge cycling
by the troponin regulatory proteins. The analyses of skinned (24) and
intact muscle function (25) suggest the existence of two intracellular
feedback mechanisms: the positive feedback, i.e., the cooperativity
mechanism (22, 24), and the negative feedback, i.e., mechanical
feedback (25).
The cooperativity mechanism (24), the dominant positive-feedback
mechanism in the sarcomere, relates to the dependence of the affinity of troponin for calcium on the number of cross bridges in
the strong, force-generating conformation. This mechanism determines the force-length relationship (FLR) in the cardiac muscle (22, 24) and
the related Frank-Starling law.
The negative mechanical feedback (25) assures that the rate of
cross-bridge weakening, i.e., the rate of transition from the
"strong" force-generating conformation to the "weak"
non-force-generating conformation, is linearly dependent on the
cross-bridge strain rate and, thus, on the filament shortening
velocity. The negative mechanical feedback determines the
force-velocity relationship (FVR) and the ability of the muscle to
generate power. The analytic derivation of the FVR in the cardiac
muscle (25) is in agreement with the well-established, experimentally
derived Hill's equation. A detailed description of the dependence of
the shortening velocity on the sarcomere length, the free calcium
concentration, the time during the contraction, and other parameters
are given elsewhere (25). Also, the negative mechanical feedback
provides the basis for the linear relationship between energy
consumption and the generated mechanical energy (23).
The intracellular control model (21-24), which is based on
biochemical and physiological assumptions and includes these two feedback loops, successfully describes a wide variety of experimental observations and well-established phenomena at the cardiac fiber level
as well as at the global LV level. These include the FLR (24, 25), the
FVR (25), and, at the LV level, the time-varying elastance (TVE), the
effect of shortening velocity on the generated force, and the positive
effect of ejection on the end-systolic pressure (22).
The study considers isometric contractions and the related mechanical
potential energy (32). The paper highlights the role of the
cooperativity mechanism in the regulation of energy consumption by the
sarcomere. The cooperativity mechanism determines the amount of calcium
bound to troponin and, hence, the rate of energy consumption by the
actomyosin ATPase. It provides the adaptive mechanism whereby the
loading conditions affect energy consumption.
Specifically, this study aims to 1)
prove, analytically, that the experimentally observed linear
relationship between O2 and
mechanical energy indicates that cross-bridge recruitment depends on
the generated force, which corresponds to the role of the positive
"cooperativity" feedback mechanism in the regulation of
cross-bridge recruitment; 2) show
that a model that does not include a positive cooperativity feedback
cannot explain the control of energy consumption; and
3) show that utilizing the
cooperativity mechanism enables the derivation and simulation of the
experimentally established linear relationship between the energy
consumption and FTI for isometric contraction.
| |
PHYSIOLOGICAL MODEL |
The model utilized here describes the intracellular control mechanisms
of contractile filament contraction and couples the kinetics of calcium
binding to troponin with the regulation of cross-bridge cycling. The
basic assumptions underlying the model are detailed elsewhere (24, 25),
and the mathematical description is summarized in the
APPENDIX. Only the assumptions
relevant to the control of the energy conversion are briefly summarized
here for coherency and convenience.
Assumption 1.
The cross bridge cycles between the weak, non-force-generating
conformation and the strong, force-generating conformation due to
nucleotide binding and release (9). The hydrolysis of ATP occurs as the
cross bridge turns from the weak to the strong conformation (7, 9).
Thus energy consumption is proportional to the total amount of
cross-bridge turnover from the weak to the strong conformation.
Assumption 2.
Calcium binding to the low-affinity troponin sites regulates the
actomyosin ATPase activity (8) and the rate of phosphate dissociation
from the myosin-ADP-P complex, which is required for the transition of
the cross bridges to the strong conformation (9). Thus calcium binding
to troponin regulates cross-bridge recruitment and the energy
consumption by the sarcomere.
Assumption 3.
The sarcomere contains three regions of overlap between the thin
(actin) and the thick (myosin) filament: a nonoverlap region, a
single-overlap region, and a double-overlap region. The double overlap
of the actin filaments in the double-overlap region does not interfere
with cross-bridge cycling (29). However, the net force generated by the
cycling cross bridges in the double-overlap region is zero (29). This
assumption is discussed further in Activation level
and energy consumption.
The troponin regulatory units are divided into four different
physiological states defined by their calcium-binding and cross-bridge conformation, i.e., the states are characterized by the biochemical kinetics of calcium binding and dissociation from troponin and by the
cross-bridge cycling between the weak and the strong conformations (24,
25). The different states in the single-overlap region at isometric
regime are depicted in Fig. 2. State
Rs represents the
rest state; the cross bridges are in the weak conformation, and no
calcium is bound to the troponin. Calcium binding to troponin leads to
state As. State
As denotes a
regulatory unit "activated" by calcium binding, but in which the
adjacent cross bridges are still in the weak conformation. State
As represents the
level of the mechanical activation, i.e., the number of available cross bridges in the weak conformation that can turn to the strong, force-generating conformation. (The importance of this definition to
the calculation of energy consumption is discussed in
Activation level and energy
consumption.) Cross-bridge cycling leads
to state Ts, a
state in which calcium is bound to the low-affinity sites and the cross
bridges are in the strong conformation. Calcium dissociation at state
Ts leads to state
Us, in which the
cross bridges are still in the strong conformation but without bound calcium.
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Fig. 2.
Diagram showing transitions between four states of troponin regulatory
units. Rs, rest state;
As, bound calcium
non-force-generating state that describes activation level;
Ts, bound calcium
force-generating state;
Us, unbound calcium
force-generation state;
kl and
k l,
association and dissociation rates of calcium binding to low-affinity
troponin sites; f, rate of
cross-bridge turnover;
g0, isometric
rate of cross-bridge weakening.
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|
The loose coupling in state
Us suggests that
calcium can dissociate from troponin while the adjacent cross bridge is
still in the strong conformation and, as shown by Peterson et al. (27), calcium dissociation from troponin can precede the cross-bridge weakening and force relaxation. The loose-coupling component in Fig. 2
is essential for the description of the time course of force
development and the force-calcium relationship. This is discussed in
detail elsewhere (24, 25). The cross bridges at states
Ts and
Us generate the
same average force, and the rate of cross-bridge weakening is identical
in these states. Hence, the linear relationship between the energy
consumption and FTI or the FLA is not related to the loose-coupling
concept of the biochemical model.
We define 
s,
s,
s, and
s as the
densities of the four troponin states existing within the
single-overlap region. For example,
s = Rs/Ls,
where Ls is the
length of the single overlap. The isometric force generated by the
sarcomere is proportional to the number of regulatory troponin units
associated with force-generating cross bridges in the single-overlap region, states Ts
and
Us.
With 
cb
denoting the unitary isometric force developed by each regulatory
unit, the isometric force generated by the contractile element per unit
filament cross section (Fce) is
given by
|
(5)
|
where
Lm is the thick
filament length that is carrying myosin heads and 
is the overlap
ratio ( = Ls/Lm).
The force generated by the fiber per unit cross section is given by F = Fce + Fpe, where
Fpe is the internal load
(force/unit cross section) and is simulated by the parallel element
(24, 25), which has a passive elastic property.
Activation level and energy consumption.
The analysis of cardiac muscle mechanics is commonly based on two
fundamental mechanical characteristics: the FLR and FVR (11). These
phenomenological relationships are usually related to the activation
level (11) and the prevailing mechanical conditions. Clearly, the
ability to describe these fundamental characteristics and to simulate
cardiac muscle dynamics depends on a reasonable quantitative
description of the activation function and the complex intracellular
relationship between calcium kinetics and cross-bridge cycling.
Following the method of Ford (11), we have defined the mechanical
activation level as the ability of the muscle to generate new
force-producing cross bridges (24, 25). On the basis of reported
biochemical studies (8, 9), the activation level is defined here as the
number of available cross bridges in the weak conformation that can
turn to the strong, force-generating conformation, i.e., the activation
level is described here by state
As in Fig. 2.
The transition from state
As to state
Ts describes
cross-bridge turnover from the weak to the strong conformation, which
requires one ATP hydrolysis and phosphate release (9) for each
cross-bridge turnover. Thus the rate of energy consumption, i.e., the
rate of ATP hydrolysis by the actomyosin ATPase in the single-overlap region
(
s) is
determined by the amount of available cross bridges in the weak
conformation that can turn to the strong conformation, i.e., by the
activation level, state
As
(As = s · · Lm)
and by the rate of cross-bridge turnover
(f) from the weak to the strong
conformation, and is given by
|
(6)
|
where
cb denotes
the free energy of hydrolysis of a single ATP molecule.
Cross-bridge cycling occurs in the double-overlap region as well as in
the single-overlap region (29). With the assumption that the
distribution of the myosin heads along the whole thick filament is
uniform (24), the rate of energy consumption in the double-overlap
region
(d) is
given by
|
(7)
|
Combining Eqs. 6 and 7 gives the rate of energy consumption
() and total amount of energy consumed
(E) by the whole sarcomere as
|
(8)
|
and
|
(9)
|
where
T is the time during the twitch.
| |
RESULTS |
Analysis of experimental observations.
The total amount of energy consumed by the cross bridges in isometric
contraction is given by
|
(10)
|
where
Ncb is the total
number of cross bridges that are in the strong conformation during the
twitch in the single- and double-overlap regions.
Consider Fig. 3, in which we depict
isometric contractions at two different fiber lengths
(L1 and
L2).
The total number of force-generating cross bridges increases as the
muscle fiber length increases (by
L) from
L2 to
L1. On the basis
of experimental data (16) and Eq. 3,
the increase in the energy consumption (E), when going from
L2 to
L1, is given by
|
(11)
|
Utilizing
Eqs. 10 and 11 gives the change in the number of
cross bridges involved with changing the sarcomere length as
|
(12)
|
The
change in the FLA (FLA) in Fig. 3 is defined as
|
(13)
|
where
Fm(L)
is the peak isometric force at sarcomere length
L.
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Fig. 3.
Schematic presentation of 2 isometric contractions.
A: force-length plane. FLA, change
in force-length area as muscle fiber length increases from
L2 to
L1.
B: force-time plane. FTI, change in
force-time integral.
|
|
Combining Eqs. 12 and 13 for a differential length step
dL yields the relationship between the
increase in the number of cycling cross bridges
(dNcb) and the
increase in the sarcomere length (dL) as
|
(14)
|
Because k1 and
cb are
constants, Eq. 14 indicates that the
increase in the number of force-producing cross bridges (dNcb) with an
increase in the sarcomere length of dL
depends on the generated peak isometric force. Intuitively, the number of cycling cross bridges can depend on the sarcomere length and the
free calcium concentration. However, Eq. 14 suggests that the sarcomere length and the free
calcium affect cross-bridge recruitment through their effect on the
magnitude of the generated force. Thus, whereas the same magnitude of
isometric force can be obtained at different combinations of sarcomere
lengths and free calcium concentrations, the dependence of cross-bridge
recruitment on the sarcomere length is mediated by the prevailing force
and not directly by the free calcium or the sarcomere length.
The tight correlation between the experimentally derived FLA and energy
consumption in isometric contractions (16, 26, 35) suggests that the
effect of the sarcomere length on cross-bridge recruitment depends on
the prevailing number of force-generating cross bridges. This
conclusion is in accordance with our suggested cooperativity mechanism
(24), whereby calcium affinity and, hence, cross-bridge recruitment
depend on the magnitude of the generated isometric force, i.e., on the
number of force-generating cross bridges in the single-overlap region.
Note that
Fm(L)
in Eq. 14 relates to the peak
isometric force but that
dNcb relates to
the change in the total number of force-generating cross bridges
throughout the contraction cycle.
Ncb is
proportional to the FTI for isometric contraction, but the time course
of the force is affected by the sarcomere length (1). Clearly, there is
no simple relationship between
Ncb and the peak
isometric force.
Comparison with TVE model.
It is instructive at this point to consider two basic concepts of the
TVE model (28) and its analogs for the isolated cardiac muscle.
1) The time course of the elastance
[e(t)]
is independent of the preload (28), and, hence, the time to peak
isometric force is also unaffected by the muscle length (28).
2) The instantaneous isovolumic
pressure [P(t)]is
determined by the instantaneous elastance [e(t)]
and the LV volume (V) as (21)
|
(15)
|
Accordingly, the instantaneous isometric force
[F(t)] at the isolated
fiber level is given by
|
(16)
|
where
efiber(t)
is the analogous form of the
e(t)
expression for the entire LV, and
(L) represents the geometric
transformation from the LV volume to the fiber length.
For isometric contraction, the energy consumed by the cycling cross
bridges is proportional to the FTI (3) and is calculated by integrating
Eq. 16 as
|
(17)
|
The peak isometric force derived from Eq. 16 is given by
|
(18)
|
where
emax is the
maximal elastance analog at the cardiac fiber level.
Combining Eqs. 17 and 18 yields
|
(19)
|
Equation 19 is based on the TVE
concepts (Eq. 16) and states that
the ratio of energy consumption to peak isometric force is constant.
However, returning to Eqs. 11 and 12, which are based on experimental
data, and rearranging yields
|
(20)
|
where
(L) = dFm(L)/dL
and is the local elastance at sarcomere length
L.
Equation 20, which is derived from
experimental data (16, 26, 35), shows that the relationship between the
energy consumption and the peak isometric force is not constant. The
relationship will be constant, and Eqs.
20 and 19 will be
equivalent, only if the cardiac muscle FLR is exponential, i.e., if
|
(21)
|
However,
the cardiac FLR is not exponential (1). Therefore, Eq. 21 is not valid for the general case, and
Eqs. 20 and 19 are not equivalent. Consequently,
the TVE concepts cannot explain the linear relationship between energy
consumption and the FTI for isometric contractions
(Eq. 4).
Force-time integral.
The isometric force is determined in our four-state model (24, 25) by
the number of cross bridges in the strong conformation in the
single-overlap region (Eq. 6), i.e.,
states TS and
US in Fig. 2. The
temporal change in the density of cross bridges in the strong
conformation is given by
![<FR><NU>d[<OVL><IT>T</IT></OVL><SUB>s</SUB>(<IT>t</IT>) + <OVL><IT>U</IT></OVL><SUB>s</SUB>(<IT>t</IT>)]</NU><DE>d<IT>t</IT></DE></FR> = <IT>f</IT> · <OVL><IT>A</IT></OVL><SUB>s</SUB>(<IT>t</IT>) − <IT>g</IT> · [<OVL><IT>T</IT></OVL><SUB>s</SUB>(<IT>t</IT>) + <OVL><IT>U</IT></OVL><SUB>s</SUB>(<IT>t</IT>)]](/content/vol276/issue3/fulltext/H998/img028.gif)
|
(22)
|
Integrating both sides of Eq. 22 over
the twitch, i.e., from t = 0 to
t = T, gives
![− <LIM><OP>∫</OP><LL>0</LL><UL><IT>T</IT></UL></LIM> <IT>g</IT> · [<OVL><IT>T</IT></OVL><SUB>s</SUB>(<IT>t</IT>) + <OVL><IT>U</IT></OVL><SUB>s</SUB>(<IT>t</IT>)] · d<IT>t</IT> = 0](/content/vol276/issue3/fulltext/H998/img030.gif)
|
(23)
|
The left-hand side of Eq. 23 is zero,
because both Ts
and Us return to
their initial values at the end of the twitch. The first term on the
right-hand side of Eq. 23 represents
the energy consumption by the cross bridges, as defined by
Eq. 9. Rewriting Eq. 23 and utilizing Eq. 5
for the whole cross section of the sarcomere gives the FTI
as
|
(24)
|
and by utilizing Eq. 9 we obtain
|
(25)
|
Equation 25 states that FTI is
proportional to the product of E, the
energy consumption by the cross bridges, and , the single-overlap ratio, and is inversely related to the isometric rate of cross-bridge weakening, g0.
The proportionality coefficients correspond to the unitary force per
cross bridge and the free energy per hydrolysis of one ATP.
The linear relationship between FTI and
E provided by this model is in
agreement with the experimental studies of Alpert et al. (3), Hisano
and Cooper (16), and Wannenburg et al. (35), who have shown that the
FTI correlates linearly with the energy consumption for isometric
contraction. The effect of on the FTI is relatively small for the
cardiac muscle compared with the changes in
E (see
DISCUSSION).
Examining the effect of the cooperativity mechanism.
As emphasized in Activation level and energy
consumption, state
As represents the
activation level, and the product
As × f gives the rate of cross-bridge
turnover from the weak to the strong conformation and determines the
rate of ATP hydrolysis (see Assumption 1 and Eq. 9). The rate of change of the
activation level is given by
![<FR><NU>d<OVL><IT>A</IT></OVL><SUB>s</SUB></NU><DE>d<IT>t</IT></DE></FR> = (<IT>k</IT><SUB>l</SUB> · [Ca] · <OVL><IT>R</IT></OVL><SUB>s</SUB>) − [(<IT>f</IT> + <IT>k</IT><SUB>−l</SUB>) · <OVL><IT>A</IT></OVL><SUB>s</SUB>] + (<IT>g</IT><SUB>0</SUB> · <OVL><IT>T</IT></OVL><SUB>s</SUB>)](/content/vol276/issue3/fulltext/H998/img035.gif)
|
(26)
|
from
Eq. A3 in the
APPENDIX.
In the absence of a feedback-cooperativity mechanism, the coefficients
kl,
kl,
f, and
g0 in
Eq. 26 are constants. Because the
state variables are distributed uniformly along the sarcomere, the
right-hand side of Eq. 26 is length
independent. Equation 26 thus suggests
that the activation level
(As) and,
hence, the energy expenditure, calculated without a cooperativity
mechanism, are independent of the sarcomere length
|
(27)
|
However,
the isometric force and, hence, the mechanical energy depend on the
sarcomere length (Eq. 5). Thus the
analytic expressions for energy consumption (Eq. 9) and force generation (Eq. 5) imply that without a cooperativity mechanism there
is a dissociation between energy consumption and the generated force. The energy expenditure is then constant and independent of the muscle
length, whereas the force is length dependent.
Figure 4 presents simulations of isometric
contractions of various sarcomere lengths, in which no cooperativity
mechanism is assumed to exist and the affinity of troponin for calcium
(K[Ca]) is assumed to be constant
(K[Ca] = 100,000 M1). Figure
4A depicts the time course of force
development at various sarcomere lengths. Figure
4B demonstrates that the time course of the activation level, state
As, is identical
for the entire range of sarcomere lengths. As shown in Fig.
4C, the developed peak force and
energy consumption are not interrelated. The energy expenditure is
independent of the muscle length and, consequently, as shown in Fig.
4D, the energy consumed by the cross
bridges is independent of the FLA and the FTI. The excess energy is
consumed by the double-overlap region. Also note the very moderate
slope of the FLR in Fig. 4B, which is
inconsistent with the steep FLR observed in the cardiac muscle (1).
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Fig. 4.
Simulated isometric contractions at various sarcomere lengths without
incorporation of the cooperativity mechanism.
A: time course of active force
development. B: time course of
activation level (state
As)
C: peak force and total energy
consumption. D: FLA and FTI versus
energy consumption by cross bridges.
|
|
It is noteworthy that the relationships observed in Fig. 4 can be
obtained in the skeletal muscle when the skeletal muscle is forced to
contract in the ascending limb of the FLR (29). Indeed, the effect of
the cooperativity mechanism diminishes at high and constant free
calcium concentrations. Moreover, the depicted results of the
simulation in Fig. 4 are also in agreement with experimental data
obtained with skinned cardiac muscle at constant and full activation
(19), where pCa = 4.3.
The effect of incorporating the cooperativity mechanism is seen in Fig.
5, in which cardiac fiber isometric
contractions are simulated at various sarcomere lengths. Figure
5A depicts the time course of the
force at various sarcomere lengths. Figure 5B shows the corresponding time course
of the activation level, state
As. Figure
5C describes peak force and the total
energy consumption at the different sarcomere lengths. As shown,
accounting for the cooperativity mechanism yields the steep FLR
observed in the cardiac muscle (1). Figure
5D demonstrates the tight linkage
between the energy consumption and the FLA or the FTI in isometric
contractions.
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Fig. 5.
Simulated isometric contractions at various sarcomere lengths with
incorporation of cooperativity mechanism.
A: time course of force development.
B: time course of activation level
(state As)
C: peak force and total energy
consumption. D: FLA and FTI versus
energy consumption by cross bridges.
|
|
As shown in Fig. 5C, the decrease in
the generated force at shorter sarcomere lengths is somewhat steeper
than the decrease in energy consumption. The noted increase in the
difference between the normalized peak force and the normalized energy
consumption, as the sarcomere length is shortened, is due to the
increase in the double-overlap region at shorter sarcomere lengths and
the consequent increase of the energy consumption by the cycling cross bridges in the double-overlap region. However, because the
cooperativity mechanism determines cross-bridge recruitment along the
whole sarcomere, in both the single- and double-overlap regions the tight relation between the FTI and the energy consumption is preserved, despite the increased energy consumption at the double-overlap region
with the decrease in the sarcomere length.
| |
DISCUSSION |
Validating the cooperativity mechanism.
The basic hypothesis underlining the present study is that it is
possible to describe the mechanical performance and the regulation of
biochemical to mechanical energy conversion in the cardiac muscle on
the basis of the intracellular control of calcium kinetics and
cross-bridge cycling.
The linear relationship between the energy consumption and the
generated mechanical energy represents a fundamental characteristic of
the cardiac muscle. Using ferret papillary fibers, Hisano and Cooper
(16) have shown a linear relationship between the FLA or FTI and energy
consumption in isometric contraction with correlation coefficients of
0.965 ± 0.05 and 0.970 ± 0.06, respectively. The linear
relationship between the FLA and the energy consumption is independent
of the mode of contraction, whether isometric or shortening (16). Mast
and Elzinga (26) have also obtained a linear relationship between the
FLA and the energy consumption by the cross bridges for isometric
contractions of rabbit papillary muscle. This linear relationship is
simulated here for isometric contraction in Fig. 5 by utilizing the
cooperativity mechanism.
The cooperativity mechanism provides the feedback loop whereby the
afterload affects the kinetics of calcium binding to troponin and the
amount of calcium bound to troponin. The cooperativity mechanism
relates the dependence of the affinity of troponin for calcium to the
number of cross bridges in the strong conformation and is the dominant
feedback mechanism that regulates calcium binding to troponin (24, 25).
This mechanism is responsible for the "length-dependent calcium
sensitivity" (1), i.e., the increase in calcium sensitivity with
increasing sarcomere length (24, 25). Lengthening the sarcomere
increases the number of available cycling cross bridges in the
single-overlap region and, thus, increases the activation level
As. Elevation of
the activation level increases the number of cross bridges in the
strong conformation and, through the cooperativity feedback, elevates
the affinity of troponin for calcium (23). Thus the cooperativity
feedback mechanism has two major components: the number of
force-generating cross bridges (input) and the affinity of troponin for
calcium (output). Consequently, the cooperativity mechanism determines the FLR and provides the basic intracellular mechanism for the Frank-Starling law (22).
The present study suggests that the cooperativity mechanism regulates
the energy consumption by the sarcomere and explains the linear
correlation between energy consumption and the FLA or the FTI for
isometric contractions. Interestingly, the cooperativity mechanism also
provides the cellular basis for the phenomenological TVE model (22). In
this context, the TVE model can be viewed as a simple presentation of
the more complex intracellular control of contraction and, as such, is
insufficient to explain the wide spectrum of experimental observations.
Calcium binding to troponin regulates cross-bridge recruitment (24) and
determines the amount of the activated actomyosin ATPase. The ATP
hydrolysis mediates cross-bridge turnover from the weak to the strong
conformation (9), i.e., force development and mechanical work
generation. Thus the rate of cross-bridge recruitment is linked to the
rate of ATP consumption by actomyosin ATPase (7). The quantitative
expressions of these phenomena are given by our four-state control
model, which couples the kinetics of calcium binding to troponin with
cross-bridge cycling kinetics.
It is instructive at this point to return to the simulation of the
four-state control model without a feedback mechanism (Fig. 4). In this
case the affinity of troponin for calcium is constant, and a constant
amount of calcium will bind to the troponin for the different sarcomere
lengths. Consequently, the energy consumption will be length
independent, and the experimentally observed linear relationship
between energy consumption and the generated mechanical energy cannot
be analytically reconstructed (Fig.
4C).
The observed increase of the generated force and energy consumption
with the increase in the sarcomere length cannot be explained without
the suggested cooperativity loop, unless it is argued that the amount
of calcium released from the sarcoplasmic reticulum is length dependent
(1). Alternatively, we can argue that the length-dependent calcium
sensitivity of the sarcomere is due to the dependence of the affinity
of troponin for calcium on the sarcomere length (1). However, these two
arguments are inconsistent with reality.
Measurements of energy consumption at various loading conditions
suggest that the PVA-independent energy consumption is constant at the
same "contractility" (32). Higashiyama et al. (15) have used
2,3-butanedione monoxime to inhibit cross-bridge cycling and to
quantify the nonmechanical energy cost at different LV volumes. They
have shown that the LV end-diastolic volume does not affect the
nonmechanical energy consumption. Therefore, the energy expenditure for
calcium sequestration from the cytoplasm, which is the major
constituent of the PVA-independent energy consumption, is length
independent (15, 28, 32). These studies suggest that the amount of
calcium released from the sarcoplasmic reticulum is practically
independent of the sarcomere length. This is consistent with Allen and
Kentish (1), who have suggested that the main mechanism for the FLR is
"length-dependent calcium sensitivity of the sarcomere" and not
the length-dependent calcium release from the sarcoplasmic reticulum.
Moreover, if the length-dependent calcium release from the sarcoplasmic
reticulum regulates the FLR in the cardiac muscle, an increase in the
peak free calcium transient with the increase in sarcomere length is
expected. However, no significant effect of the sarcomere length on the
free calcium transient has been observed experimentally (17).
A length-dependent calcium affinity of the troponin cannot, by itself,
explain the linear relation between the energy consumption and the
mechanical energy. If we assume that the amount of calcium bound to
troponin depends on the preload, then the same amount of calcium will
be bound to the troponin at the same preload but at different
afterloads, whereas the mechanical work and the mechanical energy
depend on the loading conditions, including the afterload. Hence, the
relationship between the energy consumption and the generated
mechanical energy will not be linear. This inconsistency is resolved
here by showing that the length-dependent calcium sensitivity of the
sarcomere is consistent with, and can be explained by, the
cooperativity feedback mechanism (24).
As shown in Fig. 5, the cooperativity mechanism regulates the energy
consumption of the sarcomere and, for the isometric contraction, provides the linear relation between energy consumption and the generated mechanical energy, which is quantified by the FLA or the FTI.
Moreover, this mechanism is responsible for the ability of the muscle
to adapt to the loading conditions, because elevation in the afterload
will increase the rate of ATP hydrolysis and afterload reduction will
reduce energy expenditure.
The experimental evidence supports the hypothesis that the isometric
force, or the number of force-generating cross bridges, and not the
length by itself, is the activating parameter. Allen and Kentish (2)
have measured the force and the free calcium response to length
perturbation in skinned ventricular ferret muscles. They have shown
that 1) the magnitude of the changes in the free calcium concentration correlates with that of the changes
in the tension rather than with the changes in length; and
2) the time course of the changes in
the free calcium concentration correlates with the time course of the
tension changes rather than with the time course of the length changes.
Rapid stretches of the muscle length (10 ms) resulted in a slow
increase in tension (
= 410 ms) and a slow decrease in the
fluorescent light of the calcium dye ( = 330 ms). These findings are
consistent with Gordon and Ridgway (13), who have shown in barnacle
muscle that the decrease in the affinity of troponin for calcium and
the extra calcium release after quick release relate to the decrease in the force. Kurihara and Komukai (21) have studied the effect of
muscle-length perturbation on the free calcium and the generated force
in the ferret ventricle muscle. They have shown that the changes in the
free calcium concentration, produced by quick length releases, depend
on the magnitude of the fall in the force rather than the length change.
These studies on the effects of length and force on the free calcium
transient are supported by the study of Hofmann and Fuchs (17). Using
45Ca2+,
they have shown that the amount of calcium bound to the troponin increases with the increase in sarcomere length. However, the addition
of vanadate, an ATP analog that interferes with cross-bridge cycling,
reduces the amount of bound calcium. Moreover, the amount of calcium
bound to troponin was independent of the muscle length in the presence
of vanadate.
Kentish et al. (20) have observed a steep force-free calcium
relationship in the intact and the skinned rat trabeculae. At a
constant sarcomere length of 2.15 µm, Hill's coefficient for the
force-free calcium relationship was 4.5. This large Hill coefficient
cannot be explained by a cooperativity between the calcium regulatory
binding sites, and, because the sarcomere length was constant, the only
remaining variable is the number of force-generating cross bridges. Our
analysis of this data (24) and other data from skinned cardiac muscle
supports this hypothesis.
Wang and Fuchs (34) have studied the effect of sarcomere length on the
sensitivity of the myofilaments to calcium and tested the hypothesis
that the calcium sensitivity is not a function of the length per se but
of the spacing between the actin and myosin filaments. They measured
force generation and calcium binding at different sarcomere lengths and
calcium concentrations in skinned bovine cardiac fibers while the
fibers were exposed to varying concentrations of Dextran T-500. Osmotic
compression by 5% Dextran T-500 at a sarcomere length of 1.7 µm
produced a reduction in the fiber width equivalent to sarcomere
stretching from 1.7 to 2.3 µm. This osmotic compression also produced
an increase in calcium sensitivity of ~0.25 pCa units so that the
normalized force-pCa relation at a sarcomere length of 1.7 µm with
the given osmotic compression was almost identical to the normalized
force-pCa relation at a sarcomere length of 2.3 µm. Fuchs and Wang
(12) were able to determine calcium sensitivity and the calcium binding as a function of sarcomere length under the condition of (almost) constant interfilament spacing. Both calcium binding and calcium sensitivity were found to correlate more closely with changes in
filament spacing than with changes in sarcomere length. These studies
(12, 13) strengthen the importance of the actin-myosin filament
interactions in modulation of the FLR.
The suggested effect of the sarcomere length on interfilament spacing
cannot explain the entire complex force-length-calcium relationship
observed in the cardiac fiber. A fundamental characteristic of the
cardiac fiber is a steep force-pCa relationship with a large Hill
coefficient at constant sarcomere length. Consequently, at constant
length (and hence, presumably, constant filament spacing), some other
important mechanism, such as the suggested cooperativity mechanism,
must dictate the force-pCa relationship. Indeed, the cooperativity
mechanism provides the steep force-pCa relation and explains the
significant sensitivity of the generated force to the number of
available cross bridges, i.e., to the filament length and interfilament
spacing. Consequently, we suggest that the cooperativity mechanism is
the dominant feedback loop that determines the FLR in the cardiac
muscle, whereas the sarcomere length, or the filament interspacing,
only dictates the initial conditions, i.e., the number of available
cross bridges for force generation.
All the above-mentioned studies suggest that the cooperativity
mechanism requires a close interaction between the troponin regulatory
complex and the cross bridges. Babu et al. (4) have suggested that
troponin C alone has an intrinsic property that enables it to sense the
muscle length and to modulate the sensitivity of the sarcomere for
calcium. They have substituted the native cardiac troponin C with a
skeletal troponin C and observed a decline of the steepness of the FLR
and a smaller length-induced shift of the force-pCa relationship, i.e.,
a significant decline of the muscle sensitivity to length changes.
However, the studies mentioned above suggest that the changes in the
affinity of troponin for calcium are determined by the force, or the
number of force-generating cross bridges, and not by the length.
Consequently, the troponin complexes bind calcium and determine the
activity of the actomyosin ATPase and cross-bridge recruitment. They
also interact with the cross bridges, and the affinity for calcium is
modulated by the number of cross bridges in the strong conformation.
The cooperativity mechanism seems to depend on the number of
force-generating cross bridges in the single overlap rather than on the
force itself. Our analysis (24) of the FLR in skinned fibers, in which
the force is proportional to the number of force-generating cross
bridges, cannot determine whether the input for the cooperativity mechanism is the number of force-generating cross bridges or the resulting force. Quick-release experiments (2, 21) have shown that
changes in the free calcium concentration have a slow time constant
that correlates with the changes in the isometric force. A rapid length
change of >1% of the muscle length causes cross-bridge detachment.
However, when quick releases are imposed at various amplitudes (5, 10, and 15% of the muscle length), the changes in the free calcium
correlate with the changes in the isometric force and no additional
component that relates to cross-bridge detachment is detected. This
suggests that the quick detachment of the cross bridges does not affect
the affinity for calcium. Consequently, the affinity of troponin for
calcium is regulated by the number of cross bridges in the strong
conformation and not by the force, which is determined by the number of
attached cross bridges.
Double-overlap and energy consumption.
Kentish and Stienen (19) have shown that there is little change in the
ATPase activity over the sarcomere length range of 2.0-2.4 µm,
whereas there is a significant fall in the ATPase activity at sarcomere
lengths <1.8 µm. Similar to the observation of Stephenson et al.
(29), the force decreases linearly as the sarcomere length is reduced
below 2.2 µm. The maximal difference between the normalized force and
the normalized ATP consumption, at full activation (pCa = 4.3), was
found at a sarcomere length of 1.8 µm, and the difference was 18.9 ± 2.8%. Kentish and Stienen (19) measured the sarcomere length
only during rest and could not measure the sarcomere length in the
active muscle because the laser-diffraction pattern was too diffuse.
This diffused laser pattern suggests that the sarcomeres were not
aligned uniformly along the fiber during contraction and that there was
a significant inhomogeneity in the sarcomere length along the fiber.
Sarcomere shortening at the center of the fiber may reach 10% (~0.2
µm) when the fiber length is held constant (isometric) (1). Hence, the sarcomere lengths in their study, which is based on measurements of
the resting length, are overestimated, and the significant fall in the
ATPase activity may be at a sarcomere length <1.8 µm.
Kentish and Stienen (19) have suggested that the decrease in the ATPase
activity results from the effect of the double overlap and that the
double overlap of the thin filament blocks some actin binding sites due
to steric effects. However, their hypothesis cannot explain the
deviation of the force from the ATPase activity at sarcomere lengths
>1.85 µm. At sarcomere lengths of 1.85-2.2 µm, which are
greater than the slack length for the rat trabeculae, there are no
significant restoring forces. If the double overlap reduces the ATPase
activity due to steric effects, the generated force should be parallel
to the ATPase activity at the sarcomere length range of 1.85-2.2
µm. The fall in the generated force relative to the ATPase activity
and the increase in the difference between these two parameters with a
decrease in the sarcomere length to 1.8 µm suggest that the double
overlap has different effects on the force and on the ATPase activity.
Moreover, if the double overlap blocks the actin binding sites, one
would expect a 20% decrease in the ATPase activity at a sarcomere
length of 1.8 µm, assuming that, for the rat, the actin length is
1.05 µm, the myosin length is 1.5 µm, and the bar zone and the Z
widths are ~0.1 µm. However, the measured decrease in the ATPase
activity was only ~10% and, as mentioned above, this may be
overestimated due to the inability to measure the sarcomere length
during the contraction.
Elzinga et al. (10) have measured the heat production and the force as
a function of the sarcomere length in two types of the frog skeletal
muscle, the sartorius and the extensor digitorum longus (EDL). The
energy consumption of the sartorius muscle, which is a slow-twitch
muscle, remains near its maximal value as the sarcomere length
decreases below 2.2 µm, whereas the force steadily declines. In
contrast, the energy consumption of the EDL muscle decreases in
parallel with the decline of the force as the sarcomere length
decreases below 2.2 µm.
The energy consumption in both the cardiac muscle and the slow-twitch
muscle remains constant for sarcomere lengths between 1.8 and 2.2 µm,
whereas the force decreases (11, 19). Contradicting observations have
been reported for the fast-twitch muscle (10, 29). A similarity between
the cardiac muscle and the slow skeletal muscle is also observed in
their force-calcium relationship. Thus the slow skeletal muscles and
the cardiac muscle have similar length-dependent sensitivity of the
filaments for calcium, whereas the fast skeletal muscle differs in its
the length-dependent sensitivity. Consistently, the troponin C of the
slow skeletal muscle is similar to the cardiac troponin C, and both
have only one regulatory binding site for calcium, whereas the fast
skeletal muscle has two sites (4). The difference in the dependence of
energy consumption on the sarcomere length between the fast skeletal
muscle and the cardiac muscle may be due to the difference in the mode
of cross-bridge recruitment (4) or due to basic differences in the
effect of the double overlap on the energy consumption. However, this
difference may also relate to the difference in the activation mode:
Stephenson et al. (29) have used skinned fiber and very high calcium
concentrations ([Ca] > 0.03 mM), whereas Elzinga et al.
(10) have used intact tetanized muscle and normal extracellular calcium
concentrations. Consequently, for the cardiac muscle at the sarcomere
lengths in the range of 1.8-2.3 µm simulated in this study, we
assume (see Assumption 3) that the
double overlap has different effects on the force and the ATPase
activity. At full activation, the ATPase activity remains constant
while the force declines with decreasing the sarcomere length.
Equation 24 predicts that the FTI
depends on the product of the single-overlap ratio and the energy
consumption of the cross bridges. Because the cycling cross bridges in
the double-overlap region also consume energy, Eq. 24 predicts that the discrepancy between the normalized
FTI and the normalized energy consumption will increase at short
sarcomere lengths. However, because the cooperativity mechanism
determines cross-bridge recruitment along the entire sarcomere,
including the single-overlap and double-overlap regions, the tight
relationship between the FTI and the energy consumption is preserved.
Indeed, the effect of is hardly of note in Fig.
5C. Note that the dynamic range of
sarcomere lengths is narrow; a 10% shortening of the sarcomere length
from Lmax, the
length at which maximal peak force is obtained, results in a
30-50% reduction in the generated force (1). Thus the energy consumption is determined mainly by the cooperativity mechanism that
modulates the FLR of the cardiac muscle. Obviously,
Eq. 24 should be further tested experimentally.
The derived relationship between the energy consumption and the FTI
(Eq. 24) is further substantiated by
experiments in the fully activated skeletal and cardiac muscles, in
which the cooperativity mechanism is negligible. Furthermore, this
relationship represents the ability of the model to describe both the
cardiac and the skeletal muscle mechanics and energetics. The study of
Stephenson et al. (29) with the skeletal muscle shows that energy
consumption in the ascending limb of the FLR of the skeletal muscle is
constant and length independent, whereas the force decreases in
proportion to the decrease in the single-overlap length. Indeed, the
total amount of the cycling cross bridges in the ascending limb of the FLR at full activation is constant, and thus the energy consumption is
constant and independent of muscle length. However, the force is
produced only in the single-overlap region (29), and thus the generated
force is proportional to . Similar results were obtained by Kentish
and Stienen (19) with skinned cardiac muscle at steady-state full
activation. Only a small change in ATPase consumption was observed in
the fully activated skinned cardiac muscle at sarcomere lengths between
2.0 and 2.4 µm, whereas the force fell almost linearly as the
sarcomere lengths was decreased. Thus there is no linear relationship
in the skinned cardiac fibers between the energy consumption and the
peak force. This characteristic of the skinned cardiac cell resembles
the simulation presented in Fig. 4, because the cooperativity mechanism
plays an insignificant role at steady-state full activation. (pCa = 4.3).
TVE and pseudo-potential energy.
Suga and Sagawa (28, 31) relate their experimentally observed linear
relationship between the energy consumption and the mechanical energy
(Eq. 2) to their phenomenological
TVE model. Accordingly, the mechanical energy is
determined by the ESPVR, whereas the ESPVR is determined by
Emax, the maximal
value
E(t) can reach, i.e., the slope of the ESPVR. Thus the linear relationship between the energy consumption and the mechanical energy is presented as a consequence of the elastance concept and substantiates the stated
importance of the ESPVR as an index for LV contractility (28, 29).
However, the linear relationship between the mechanical energy and
oxygen consumption of the isolated cardiac muscle fiber (16, 26)
implies that the linear relationship between PVA and
O2 stems from the basic
characteristic of the myocytes and is not due to some unique
characteristic of the LV.
The TVE is a relatively simple phenomenological model that provides a
useful, convenient guide in the analysis of the LV function and is
quite accurate in the range of low ejection fractions (28), yet it
cannot explain a significant number of physiological observations. For
example, the TVE model cannot explain the close correlation between the
FTI and energy consumption (Eq. 4)
for isometric contractions at various sarcomere lengths. Note that the
FLA relates to the peak force, whereas the FTI relates to the entire
twitch. Moreover, the TVE does not correctly describe the regulation of
power generation. According to the TVE model, the LV contractility is
determined by
Emax; however,
Emax cannot
distinguish between two different hearts that generate the same
pressure-volume loops on the pressure-volume plane at two different
heart rates. Obviously, these hearts differ in their ability to
generate power, yet, according to the
Emax concept,
they exhibit the same "contractility." Consequently, the TVE
model cannot explain the positive effect of ejection on pressure
generation (18). Clearly, any definition of contractility should
include an index of the LV ability to generate power.
The TVE model predicts (Eq. 19) that
the ratio of the change in the energy consumption to the change in the
peak isometric force is constant, whereas experimental data
(Eq. 20) suggest that this relation
depends on the peak isometric force,
Fm(L).
The ratio is only constant if the cardiac muscle FLR is exponential (Eq. 21), but the cardiac muscle FLR
is not exponential (1).
The inability of the elastance model to explain the FLR stems from the
oversimplified nature of the TVE model and its inherent assumptions
that 1) the elastance
e(t)
is only a function of time and is independent of the loading
conditions, and 2) the isovolumic pressure is a product of two independent functions, time
[e(t)] and volume.
According to the TVE model, all the mechanical energy in an isometric
contraction is stored as potential energy (28). Thus the energy
consumption at isometric contraction is determined by the maximal
elastance and the preload. This approach leads to two conclusions.
1) The energy consumption is
independent of the work during the relaxation period, because it is
defined by the peak force and does not include the relaxation period.
Energy conversion from biochemical to mechanical energy occurs only
before the time of peak force, and mechanical perturbations after the peak force have no effect on the energy expenditure.
2) The entire potential energy can
be converted to external work, i.e., the potential energy is
energetically equivalent to external work.
Suga and co-workers (14, 30, 33, 36) have intensively studied the
mechanical work equivalent of the potential energy and the effect of
shortening after the peak of contraction on the energy consumption.
They suggest that the oxygen consumption is determined only by the
energy generated during systole and is independent of the external
mechanical work during diastole. However, the LV ejection velocity in
their experiment was <200 ml/s. Hata et al. (14) have shown that the
external mechanical work during the relaxation period does not
significantly affect the myocardial oxygen consumption. Moreover, they
have shown that 
93% of the potential energy, defined on the basis of
PVA, can be converted into external work. However, the maximal ejection velocity in their study was 118 ml/s, or 7 end-diastolic volumes (EDV)
per second. With the assumption of a spherical model of the heart for
simplicity, this ejection velocity corresponds to a sarcomere velocity
of ~2.3 µm/s. Additional evidence that they have used a very slow
ejection velocity is seen in their pressure recordings: there is hardly
any effect of the ejection on the pressure during the relaxation (Fig.
5, Ref. 14), and the pressure during the ejection period is almost
identical to that at the isovolumic contraction.
The importance of the ejection velocity in the energy conversion from
potential energy to external work, during relaxation, was addressed by
Suga (30). He has shown that there is an optimal, relatively slow
(50-100 ml/s) ejection velocity at which maximal energy is
converted from potential energy to external work. At higher ejection
velocities (~250 ml/s), only one-third of the calculated potential
energy can be converted into external work. Using higher ejection
rates, Yasumura et al. (36) have experimentally found substantial
changes in the energy consumption, which is inconsistent with the
elastance model. Linear relationships exist (36) between oxygen
consumption during isovolumic contraction (O2,i) and the PVA, as well
as between oxygen consumption during fast releases
(O2,q) and the
PVA
Top
Abstract
Introduction
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