## Abstract

The intracellular control mechanism leading to the well-known linear relationship between energy consumption by the sarcomere and the generated mechanical energy is analyzed here by coupling calcium kinetics with cross-bridge cycling. A key element in the control of the biochemical-to-mechanical energy conversion is the effect of filament sliding velocity on cross-bridge cycling. Our earlier studies have established the existence of a negative mechanical feedback mechanism whereby the rate of cross-bridge turnover from the strong, force-generating conformation to the weak, non-force-generating conformation is a linear function of the filament sliding velocity. This feedback allows the analytic derivation of the experimentally established Hill's equation for the force-velocity relationship. Moreover, it allows us to derive the transient length response to load clamps and the transient force response to sarcomere shortening at constant velocity. The results are in agreement with experimental studies. The mechanical feedback regulates the generated power, maintains the linear relationship between energy liberated by the actomyosin-ATPase and the generated mechanical energy, and determines the efficiency of biochemical-to-mechanical energy conversion. The mechanical feedback defines three elements of the mechanical energy:*1*) external work done; *2*) pseudopotential energy, required for cross-bridge recruitment; and *3*) energy dissipation caused by the viscoelastic property of the cross bridge. The last two elements dissipate as heat.

- cross-bridge dynamics
- force-length area
- excitation-contraction coupling
- efficiency
- mechanical feedback
- potential energy

one of the most interesting aspects of muscle physiology is the regulation of energy conversion from biochemical to mechanical energy (24, 25). Extensive experimental studies (26, 28) at the level of the global left ventricle (LV) have established the existence of a linear relationship between oxygen consumption (V˙o
_{2}) and the generated mechanical energy, which is quantified by the pressure-volume area (PVA) in the LV pressure-volume plane as
Equation 1where*a* and *b* are constants. The constant *b*represents the energy consumed by the Ca-ATPase, the Na-K pumps, and the basal metabolic energy consumption. The mechanical energy, PVA, which corresponds to the mechanical energy generated by the cross bridges, is the sum of the external work (W) done by the LV and the mechanical potential energy (PE) as given by
Equation 2Suga (28) and Sagawa et al. (26) have suggested that PE represents the elastic energy generated during the contraction and stored at end systole in the LV wall.

Using ferret papillary muscle fibers, Hisano and Cooper (15) have shown that the force-length area (FLA), the cardiac fiber analog of the ventricular PVA, is also closely correlated withV˙o
_{2} as
Equation 3where*a** and *b** are constants. The FLA is defined, similar to the ventricular PVA, as the sum of W and PE. Mast and Elzinga (23) have similarly shown that the FLA correlates with the tension-dependent heat in an isometric contraction. The linear relationship between the FLA and V˙o
_{2} at the muscle fiber level (*Eq. 3
*) suggests that the linear relationship between PVA and V˙o
_{2} at the LV level (*Eq. 1
*) is not due to some unique characteristics of the LV but is an integrated basic characteristic of the myocytes. The following question arises: What cellular mechanism sustains the linear relationship between energy consumption and the generated mechanical energy?

Our hypothesis that the global LV mechanics and energetics can be interpreted in terms of the underlying control mechanisms of the contraction of the sarcomere (18-22) is supported by the studies of Campbell et al. (4, 5) and Beyar and Sideman (2). Campbell et al. have examined the pressure response of the LV to quick volume changes in the tetanized isolated heart (4) and in the beating heart (5). The dynamics of the pressure-volume-flow characteristics are remarkably identical to the dynamics of the tension changes observed at the level of the isolated cardiac fiber undergoing similar quick length changes. Hence, both the mechanics (2, 4, 5) and the energetics (15) of the LV can be interpreted in terms of the underlying sarcomere function.

The present study concentrates on the role of the negative mechanical feedback (19, 21) in the regulation of biochemical-to-mechanical energy conversion. The mechanical feedback suggests that the filament sliding velocity determines the biochemical rate of cross-bridge weakening, i.e., the rate of cross-bridge transition from the “strong” force-generating conformation to the “weak” non-force-generating conformation. The existence of this mechanism is substantiated (21) by the analytic derivation of the force-velocity relationship (FVR), in agreement with the experimentally derived Hill's equation (14).

The FVR relates only to steady-state conditions. However, the cardiac muscle is never at steady state during the twitch, and the free calcium during the twitch changes with time. Therefore, the FVR cannot describe cardiac muscle mechanics during physiological contraction. Here we demonstrate the ability of the mechanical feedback concept to describe the real physiology by presenting its ability to simulate the transient responses to load changes.

Of the different techniques that can be used to evaluate the FVR, we have concentrated on the isometric-to-isotonic changeover (load clamps) method (8, 30) and the isovelocity (sarcomere length control) technique (7, 8). After completing earlier studies of the FVR (18, 21), we have proceeded to describe the transient length response to load clamp and the transient force response to sarcomere shortening at a constant velocity. Note that the transient response is prolonged in the isometric-isotonic changeover technique and is shorter in the quick-release technique (7, 8). The present analysis aims to provide an explanation for the empirical observations obtained in these studies.

The mechanical feedback determines the generated power (18, 21), and, as shown here, it is the key element in the understanding of the regulation of energy conversion from biochemical to mechanical energy by the sarcomere. The mechanical feedback explains the linear relationship between energy consumption and the generation of mechanical energy, i.e., the external work and liberated heat. Moreover, the mechanical feedback reveals that the generated mechanical energy sustains three components: external work, energy dissipation caused by the viscoelastic property of the cross bridge, and the pseudopotential energy, which is determined by the force-time integral (FTI) (21, 22).

## THE PHYSIOLOGICAL MODEL

### Four-State Model

Sarcomere mechanics and energetics depend on calcium kinetics and cross-bridge cycling. The basic assumptions underlying our four-state model are detailed elsewhere (18, 20, 21), and only those relevant to the control of the shortening velocity and energy consumption are briefly summarized here for coherency and convenience.

#### Assumption 1.

The regulatory unit is a single regulatory troponin-tropomyosin complex with the fourteen neighboring actin molecules and the adjacent heads of the myosin.

#### Assumption 2.

The cross bridge cycles between the weak, non-force-generating conformation and the strong, force-generating conformation because of nucleotide binding and release. The hydrolysis of ATP occurs as the cross bridge turns from the weak to the strong conformation (3, 9). Thus energy consumption by the sarcomere is proportional to the total amount of cross-bridge turnover from the weak to the strong conformation.

#### Assumption 3.

The individual cross bridges act like a Newtonian viscoelastic element: the FVR of a single cross bridge is linear (7) on the basis of simultaneous measurements of the generated force, shortening velocity, and the dynamic stiffness.

#### Assumption 4.

Calcium binding to the low-affinity troponin sites regulates the actomyosin-ATPase activity (6) 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 energy consumption by the sarcomere.

Calcium binding and dissociation from troponin and cross-bridge cycling between the weak and the strong conformation characterize the four different states of the troponin regulatory units (Fig.1). *State R *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 A. State A* denotes a regulatory unit “activated” by calcium binding but in which the adjacent cross bridges are still in the weak conformation. Thus *state A*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. Cross-bridge turnover from the weak to the strong conformation leads to *state T,* in which calcium is bound to the low-affinity sites and the cross bridges are in the strong conformation. Calcium dissociation at *state T* leads to *state U, *in which the cross bridges are still in the strong conformation but are already without bound calcium. The significance of*state U* is discussed in detail elsewhere (20, 22).

### Feedback Mechanisms

Two feedback mechanisms regulate the biochemical-mechanical coupling in the sarcomere: a positive cooperativity mechanism (20, 22) and a negative mechanical feedback mechanism (21). The cooperativity feedback mechanism is based on the analysis of the force-length-free calcium relationships in skinned cardiac fibers (20). The cooperativity mechanism relates the affinity of troponin for calcium to the number of cross bridges in the force-generating (strong) conformation. It determines the amount of calcium bound to troponin, and hence the rates of force generation and energy consumption. The cooperativity mechanism explains the force-length relationship (FLR) and the related Frank-Starling law (18-20). It also explains (22) the linear relationship between energy consumption and FTI (1, 15, 29) as well as the linear relationship between energy consumption and the FLA (15, 26) for isometric contractions.

The present study concentrates on the role of the negative mechanical feedback, which ensures that the rate of cross-bridge weakening is linearly dependent on the filament shortening velocity (*V*). The rate of cross-bridge transition from the strong to the weak confirmation (*g*) is given by
Equation 4where*g*
_{0} is the rate of cross-bridge weakening under the isometric regime and *g*
_{1} is the mechanical feedback coefficient; this rate describes the effect of the filament sliding velocity on the rate of cross-bridge weakening (in units of 1/m). The existence of this mechanism is substantiated (21) by the analytic derivation of the FVR, in agreement with the experimentally derived Hill's equation (14). A detailed description of the dependence of the shortening velocity on various physiological parameters, such as the rate of cross-bridge cycling, internal load, calcium kinetics, sarcomere length, and the time during the contraction, is given elsewhere (21) and is in agreement with experimental observations (7, 8).

## THE MATHEMATICAL MODEL

### Interaction Between State Variables

We define , , , and as the density (per unit length) of the various troponin states existing within the single-overlap region between the actin and myosin filament. The transitions between the density state variables (Fig. 1) within the single-overlap region are given by (19, 20)

Equation 5where [Ca] denotes the free calcium concentration. The rate coefficients *k*
_{l} and*k*
_{−}
_{l} represent the rate constants of calcium binding to and calcium dissociation from, respectively, the low-affinity sites of troponin. Note that the rate coefficient*k*
_{−}
_{l} is not constant because the cooperativity mechanism dictates the dependence of this coefficient on the state variables (20, 22). Cross-bridge cycling is described by*f, g*
_{0}, and *g*
_{1}, where *f*denotes the rate of cross-bridge turnover from the weak to the strong conformation.

The force (F) generated by the sarcomere is a product of the density of the force-generating cross bridges in the single-overlap region (27) (
+
), the length of the single overlap (*L*
_{s}), and the average force generated by each cross bridge. The individual cross bridge acts like a Newtonian viscoelastic element (a*ssumption 3*). Hence, the generated force is given by
Equation 6whereF̅ is the unitary isometric force developed by each cross bridge and η represents the viscous property of the cross bridge (7).

### Energy Consumption and the Activation Level

Ford (11) defined the mechanical activation level as the ability of the muscle to generate new force-producing cross bridges. The change in the density of force-generating cross bridges (
+
) is derived from *Eq.5
* as
Equation 7As shown in *Eq. 7
*, the ability of the muscle to generate new force-producing cross bridges is determined by *state*
, the activation level, which represents the number of available cross bridges in the weak conformation that can turn to the strong, force-generating conformation. The transition from *state*
to *state*
describes cross-bridge cycling from the weak to the strong conformation, which requires one ATP hydrolysis and phosphate release (3, 6) per each cross-bridge turnover from the weak to the strong conformation. Thus the rate of energy consumption by the actomyosin-ATPase (*E˙*) is determined by *state*
and *f,* the rate of cross-bridge turnover from the weak to the strong conformation, as
Equation 8where
_{ATP} denotes the free energy liberated from the hydrolysis of a single ATP molecule. Clearly, *state *
, i.e., the activation level, determines the rate of force generation (*Eq. 7
*) as well as the rate of energy consumption (*Eq.8
*).

Finally, the relationship between the rate of force generation and the rate of change in the sarcomere shortening velocity is derived from*Eq. 6
*, utilizing *Eq. 7
*, and is given by
Equation 9

## RESULTS

### Isometric-Isotonic Changeover (Load Clamps) Studies

The muscle in the load-clamp technique (8, 30) is allowed to contract in the isometric regime, and the transition from isometric to isotonic regime occurs at the time of peak isometric force (F_{m}). The transient length response presents an initial rapid decrease in muscle length during the force step, followed by a slow monotonical decrease in the length, which fits an exponential time course and ends with an almost constant shortening velocity.

For isometric contraction [change in shortening velocity (d*V*/d*t*) = *V *= 0] and at peak isometric force [F = F_{m}, load change (dF/d*t*) = 0], *Eq. 9
* reduces to
Equation 10
*Equation10
* allows us to approximate the activation level
that prevails just before the quick release and during the short time interval during which the length response is measured. Utilizing *Eq. 10
*, we reduced*Eq. 9
* to
Equation 11where *V*
_{u} =F̅/η is the unloaded shortening velocity (21).

During the isotonic contraction, dF/d*t *= 0. For the asymptotic steady-state shortening velocity (d*V*/d*t* = 0), *Eq.11
* reduces to Hill's equation (14) for the FVR (21)
Equation 12where*V*
_{H} denotes the steady-state shortening velocity, and Hill's constants *a*
_{H} and *b*
_{H}are given by
Equation 13
*Equations12
* and *
13
* provide the physiological meaning for the experimentally derived Hill's constants *a*
_{H} and*b*
_{H} based on cross-bridge dynamics, i.e.,*g*
_{0}, *g*
_{1}, and*V*
_{u}.

Figure 2 describes the FVR and the power-load relationship for three different amplitudes of the mechanical feedback coefficient. The generated power is defined by the load and the shortening velocity, and hence it depends on the curvature of the FVR and on the mechanical feedback coefficient ( *g*
_{1}).

As described by Woledge (31) and Alpert et al. (1), the curvature of the FVR is determined by the ratio of *a*
_{H} to F_{m} (*a*
_{H}/F_{m}). The power increases with the increase in *a*
_{H}/F_{m}. However, *Eq. 13
* suggests that*a*
_{H}/F_{m} is a function of the mechanical feedback coefficient *g*
_{1} because*a*
_{H}/F_{m} ≅*g*
_{0}/*g*
_{1}
*V*
_{U}. The smaller the mechanical feedback coefficient *g*
_{1}, the smaller the curvature of the FVR and the higher the power at any given load (Fig. 2).

### Transient Length Response (Load Clamps)

A transient length response of sarcomere shortening is observed in the load-clamp technique before steady-state velocity is reached. Because the load is constant (dF/d*t* = 0), *Eq. 11
* reduces to
Equation 14Utilizing *Eq. 12
*, we obtained
Equation 15 Rearranging*Eq. 15
* gives
Equation 16 Integration yields
Equation 17 Substitution of the expression for *V*
_{H} in *Eq. 12
* gives
Equation 18 where the rate constant *G*
_{max} = *g*
_{0} + (*g*
_{1} + 1/*L*
_{s})*V*
_{u} (in units of 1/s). The constant *C* in *Eq. 18
* is calculated from the initial condition, i.e., the shortening velocity at *t *= 0 (*V*
_{0}). The transient shortening velocity*V*(*t*) for isometric-to-isotonic changeover is finally given by
Equation 19 The transient shortening velocity approaches the steady-state shortening velocity *V*
_{H} exponentially, and because*g*
_{1} >> 1/*L*
_{s}, the rate constant *G*
_{max} depends on the maximal rate of cross-bridge weakening, i.e., *G*
_{max} ≅*g*
_{0} + *g*
_{1}
*V*
_{u}.

Integrating *Eq. 19
* yields the transient length changes*L*(*t*) given by
Equation 20Note that when the initial shortening velocity after the quick release to the isotonic force level equals the steady-state shortening velocity, i.e., *V*
_{0} = *V*
_{H}, the shortening velocity remains constant without any transient velocity changes. This conclusion is consistent with experimental observations (8, 30).

### Transient Force Response (Sarcomere Length Control)

In the isovelocity technique used by Daniels et al. (7) and de Tombe and ter Keurs (8), the sarcomere length is initially kept isometric until a selected moment during the twitch. The fiber is then rapidly released, and the quick length release is followed by a controlled constant velocity shortening. The magnitude of the quick release and the velocity of the isovelocity phase are selected empirically so as to shorten the duration and magnitude of the transient force response.

For isovelocity shortening (d*V*/d*t* = 0) starting at the time of peak isometric force (F = F_{m}), the transient force response is derived from *Eq. 11
* as
Equation 21 where*G* = *g*
_{0} + (*g*
_{1} + 1/*L*
_{s})*V*, and the transient force is given by
Equation 22 where F_{0} is the initial force level after the quick release and F_{H} is the steady-state force at constant shortening velocity. F_{H} is derived from *Eq. 21
* when dF/d*t* = 0, leading to Hill's equation (14)
Equation 23When the initial force level after the quick release equals the steady-state force, i.e., F_{0} = F_{H}, the force remains constant, without a transient force response. This is consistent with experimental observations (7, 8). Otherwise, an exponential transient force response is predicted by *Eq. 22
*. Note that the time constant of the force response in the length control experiments (*G*) is slower than the length response at the load clamp experiments (*G*
_{max}) and depends on the sarcomere shortening velocity (*Eq. 21
*).

### Generated Mechanical Energy

#### Energy conversion without accounting for cross-bridge viscoelasticity.

The energy consumption by the sarcomere and the generated mechanical energy are derived from cross-bridge dynamics. For clarity and simplicity of the analysis, we first disregard the cross-bridge viscous property and assume that the average force generated by each cross bridge is constant and independent of the shortening velocity. In this case, the generated force equals the number of cross bridges in the strong conformation multiplied by the average force generated by each cross bridge, and *Eq. 6
* reduces to
Equation 24 The instantaneous change in the number of cross bridges that are in the strong conformation is given by *Eq. 7
*. Integrating both sides of *Eq. 7
* over the twitch (i.e., from *t* = 0 to *t*= *t*
_{D}, where *t*
_{D} is twitch duration) and using *Eq. 24
*, we obtain
Equation 25The left-hand side of *Eq. 25
* is zero because both
and
return to their initial values at the end of twitch. The first term on the right-hand side of *Eq.25
* represents the amount of ATP consumed during the twitch, and hence the energy consumption by the cross bridges during the twitch (*E),* as defined by *Eq. 8
*. Rearranging *Eq. 25
*yields
Equation 26or
Equation 27 *Equation27
* states that the amount of ATP consumed during the twitch (*E*/
_{ATP}) is proportional to the FTI and the generated external work. The proportionality coefficients are determined by the rates of cross-bridge weakening in the isometric regime, *g*
_{0}, and the mechanical feedback coefficient, *g*
_{1}, respectively. Dividing *Eq. 27
* by*g*
_{1}/F̅ yields the simplified, approximated relationship between chemical energy consumption and the generated mechanical energy
Equation 28 where
ρ^{0}is the approximated efficiency of the biochemical-to-mechanical energy conversion and *E*
is the approximated pseudopotential energy. Note that the efficiency is inversely proportional to the mechanical feedback coefficient,*g*
_{1}.

#### Energy conversion accounting for cross-bridge viscoelasticity.

de Tombe and ter Keurs (7) have shown that cross bridges exhibit a Newtonian viscous property and that the average force generated by each cross bridge is linearly proportional to the shortening velocity (*assumption 3*). Substituting the number of regulatory units in the strong conformation from *Eq. 6
*, (
+
) = F/*L*
_{s}(F̅ − η*V*), into *Eq. 7
*yields
Equation 29because *V*
_{u} =F̅/η (7). Using the equality 1/(1 −*x*) = 1 + *x* + *x*
^{2}/(1 −*x*) gives
Equation 30Opening the parentheses and rearranging provides the general equation for energy conversion
Equation 31Dividing both sides of *Eq. 31
* by (1/F̅)(*g*
_{1} +*g*
_{0}/*V*
_{U}) gives
Equation 32 where
and W is as defined by *Eq. 28. E*
_{PP} is the pseudopotential energy and is proportional to the FTI.*Q*
_{η} represents energy dissipation caused by the viscous property of the cross bridge. *Q*
_{η}represents the effect of the viscous component because it is proportional to η and is given by the integral over the force multiplied by the square of the velocity and by even higher degrees of the velocity [since *x*
^{2}/(1 − *x*) = *x*
^{2} + *x*
^{3} +*x*
^{4}…; *x* =*V*/*V*
_{max}]. ρ is the efficiency of the biochemical-to-mechanical energy conversion and is inversely proportional to the mechanical feedback coefficient,*g*
_{1}.

### Simulated Results

The simulations presented here aim to demonstrate the effect of the mechanical feedback coefficient, *g*
_{1}, on the end-systolic stress-length relationship and on the efficiency of the biochemical-to-mechanical energy conversion. Figure3
*A* depicts one set of isometric contractions at different sarcomere lengths and one set of physiological contractions, starting from the same preload but with different afterloads. The afterload is described by the windkessel model. When the mechanical feedback coefficient *g*
_{1}equals 7.5, the end-systolic stress (ESS) of the shortening beat exceeds the ESS of the isometric beat at the same end-systolic length, in agreement with Hunter's finding (16) at the whole heart level. A linear relationship is obtained between the energy consumption and the generated mechanical energy (Fig. 3
*B*). The mechanical energy was calculated here by utilizing the FLA concept (i.e., without the viscous term) according to Suga (28) and Hisano and Cooper (15). The calculated efficiency is 71%.

Figures 4 and 5depict similar simulations of isometric and physiological contractions, but with larger mechanical feedback coefficients, i.e.,*g*
_{1} = 8.25 and 10.5, respectively. As shown in Fig.4, the ESS of the isometric contractions falls on the imaginary curve that can be drawn through the ESS of the shortening beats. The single ESS-length relationship shown in this simulation (Fig. 4) is independent of the loading conditions, consistent with the elastance model (26). The increase in the negative mechanical feedback coefficient decreases the stroke length in Fig. 4 relative to Fig. 3, and the efficiency decreases from 71% to 66%.

Increasing the mechanical feedback coefficient (*g*
_{1}= 10.5) further decreases the stroke length, as shown in Fig. 5. Here the ESS of isometric contraction exceeds the ESS of the shortening beat at the same end-systolic length, a phenomenon that was denoted as the “shortening deactivation” (26). The efficiency decreases to 54%.

As demonstrated in Fig. 6, the mechanical feedback coefficient *g*
_{1} determines the efficiency of energy conversion from biochemical to mechanical energy. The smaller the mechanical feedback coefficient, the higher the efficiency of energy conversion.

## DISCUSSION

The cooperativity mechanism and the mechanical feedback that regulate sarcomere dynamics and determine the FLR (18, 20, 22) and the FVR (21) also regulate the energy consumption (22) and the generated mechanical energy. The present study relates only to the role of the mechanical feedback in the regulation of the sarcomere shortening velocity and the biochemical-to-mechanical energy conversion.

### Weak/Strong Versus Detached/Attached Conformation

Both our model and Huxley's classic model of cross-bridge attachment-detachment (17) describe the FVR and the muscle energetics phenomena. There is in fact an apparent similarity between the two models: both attribute the FVR and the regulation of energy consumption to cross-bridge cycling between two conformations, a force-generating conformation that is strong (ours) or “attached” (Huxley's) and a non-force-generating conformation that is weak (ours) or “detached” (Huxley's). This apparent symmetry is where the similarity ends.

There are two main differences between our model and the classic Huxley model. *1*) Huxley's model (17) links the biochemical process of ATP hydrolysis to the physical process of cross-bridge attachment-detachment, with one molecule of ATP hydrolyzed for each stroke work. Huxley's model assumes that the kinetics that describe the dynamics of cross-bridge attachment-detachment are the same as the kinetics of nucleotide (ATP, ADP) association and dissociation and energy consumption. Consequently, there is a 1:1 relationship between the number of ATP molecules consumed and the number of cross-bridge stroke steps. Our model suggests that cross-bridge dynamics are determined by two different kinetics of two interrelated processes: the biochemical kinetics of cross-bridge cycling between two biochemical conformations (weak and strong) and the physical kinetics that relate to the actin-myosin interaction (attachment and detachment). The physical kinetics relate to the viscoelastic properties of the cross bridges (7). The present model allows multiple stroke steps per single ATP consumption compared with Huxley's 1:1 relationship. *2*) Huxley's model (17) assumes that the rate of cross-bridge attachment and the rate of cross-bridge detachment are functions of the strain, i.e., the displacement of the cross bridge from the equilibrium state. Our model suggests that the rate kinetics are a function of the strain rate, i.e., the velocity. Moreover, the dependence of the rate constant on the strain rate is quite simple: the rate of cross-bridge turnover from the weak to the strong conformation is constant. The rate of cross-bridge weakening is a simple linear function of the filament sliding velocity (*Eq. 4
*).

### Force-Velocity Relationship

The mechanical feedback concept discussed here was inspired by the biochemical studies of Eisenberg and Hill (9), who suggested that the filament sliding velocity affects the rate of cross-bridge weakening. The magnitude of the mechanical feedback is described by the parameter*g*
_{1}, which determines the effect of the filament sliding velocity on the rate of cross-bridge weakening. The existence of this mechanism was substantiated in our earlier studies (18, 21). The analytically derived Hill's equation (*Eq. 12
*) for the steady-state FVR verifies the ability of the mechanical feedback concept to describe muscle mechanics. Moreover, Hill's parameters (14), *a*
_{H} and *b*
_{H} were derived (17) on the basis of cross-bridge kinetics and are inversely dependent on the mechanical feedback coefficient.

Campbell et al. (5) studied the short time response of the LV pressure to quick and small amplitude changes in the volume at various flow rates and various volumes. They fitted their data to a two-state model of pressure generators, as in Huxley's theory of muscle contraction (17), and found that the rate of turnover from the strong to the weak state increases with the increase in the shortening velocity. Moreover, the rate of weakening depends only on the flow rate and is independent of the magnitude of the volume changes, i.e., the rate of weakening depends only on the shortening velocity and is independent of the displacement itself. These results are in agreement with the mechanical feedback employed here.

Following up on our earlier study of the steady-state FVR and the unloaded shortening velocity (21), we address here the transient length response to load clamps and the transient force response to controlled shortening velocity. The analysis provides a theoretical explanation for the various empirical observations obtained in the study of the FVR. It explains the transient exponential length changes in response to quick load changes (load clamps) (8, 30) and the transient exponential force responses observed in the isovelocity control method (7, 8). Note that the transient exponential responses observed in these two methods depend only on the rate of cross-bridge weakening (*g*
_{0} +* g*
_{1}
*V *or*g*
_{0} + *g*
_{1}
*V*
_{u}). The analysis predicts that precise measurements of these time constants will provide a direct way of quantifying the rate of cross-bridge weakening. This is unlike the rate of isometric force redevelopment after a quick release, which depends on the rate of cross-bridge turnover from the weak to the strong conformation, *f*, as well as the rate of cross-bridge weakening, *g*
_{0}, i.e.,*f + g*
_{0} (20). Whereas the time constant of the transient force response in the isovelocity method depends on the sarcomere shortening velocity, the time constant of the transient length response in the load clamps experiment is constant and is determined by the mechanical feedback coefficient and the unloaded shortening velocity. Consequently, the transient response is faster and maximal in the load-clamp method compared with the isovelocity method.

The analysis provides the theoretical basis for the empirical observation that a quick release imposed between the isometric and isovelocity phases in the isovelocity method abbreviates the transient response (7, 8). Moreover, no transient force change will be observed when the initial force after the imposed quick release will equal the steady-state force (F_{0} = F_{H}). Similarly, no transient velocity change will be observed when the initial velocity after the load change in the load-clamp method equals the steady-state velocity. However, this is impossible to perform technically. Consequently, it is easier to reach the steady-state FVR by the isovelocity method.

### Mechanical Energy and Energy Conversion

The negative mechanical feedback mechanism regulates the generated power (18-22) and leads to the linear relationship between energy consumption and the generated mechanical energy (*Eq. 32
*). Moreover, the mechanical feedback defines three parts of the generated mechanical energy: *1*) external work, *2*) pseudopotential energy (22), and *3*) energy dissipation as heat due to the viscous property of the cross bridges. As shown in *Eqs. 28
* and *
32
*, the efficiency (ρ) of the biochemical-to-mechanical energy conversion is inversely proportional to the mechanical feedback coefficient, *g*
_{1}.

The term “mechanical energy” used here is analogous to the one in the PVA model (26, 28), where the mechanical energy is defined as the sum of the external work and the potential energy (*Eq. 2
*). However, there are several differences between the elastance concept and the present model, especially in explaining the underlying mechanisms and the meaning of the term “potential energy.” Also, the energy dissipation due to the viscous element is not included in the commonly accepted PVA model (26). Indeed, the contribution of the viscous element is relatively small at slow shortening velocities. However, it is not negligible when the shortening velocity approaches the unloaded velocity. Hence, the present definition of the generated mechanical energy, *Eq. 32
*, which includes the energy dissipation due to the viscous property of the cross bridges, can also describe the quick-release experiments that are not described properly by the PVA concept (15, 26).

The term “pseudopotential energy” was coined (22) by analogy to the potential energy term in Suga's elastance model (28). As shown in*Eq. 32
*, the energy consumption in isometric contraction (*V*= 0, W = 0, *Q*
_{η} = 0) is proportional to the FTI. Consistently, the potential energy defined by Suga (28) for isometric contraction is proportional to the FTI (22). According to the elastance model, all the energy consumed in isometric contractions is stored as potential energy in the LV wall as elastic energy. Consequently, the elastance model suggests that when a quick release is imposed at the time of peak isometric force, it does not affect the PVA and has no effect on the energy consumption. PVA and the energy consumption. However, it was shown (13, 15) that quick releases imposed after end systole at isometric contraction reduced the oxygen consumption. This finding is inconsistent with the elastance theory and with the stipulation that the potential energy is stored in passive elastic components. Moreover, it implies that energy is also consumed during the relaxation phase of the isometric contraction, which cannot be described by the PVA concept but is well described by utilizing the FTI. Therefore, it seems more appropriate to quantify the energy consumption for cross-bridge recruitment by the FTI, as suggested by the present model, than by the “classic' PVA concept.

Moreover, the elastance concept and the related potential energy suggest that there is a physical conserving field in the contraction phenomena. In contrast, the pseudopotential energy in the present study is consumed during the contraction for cross-bridge recruitment (22), and only part of it can be utilized to generate external work. The pseudopotential energy is defined here by *Eqs. 28
* and *
32
* to be a function of the FTI multiplied by*g*
_{0}/*g*
_{1}. The energy stored in the cross bridges can either turn to heat, as occurs in isometric contraction, or can be used for generating external work. The rate constants *g _{0} *and

*g*

_{1}define the rate of energy conversion to heat and work, respectively. A more detailed comparison between the potential energy in the PVA concept and the present pseudopotential energy is given elsewhere (22).

### Contractility

The presentation of the LV function on the pressure-volume plane (26,28) lacks the time dimension and does not properly describe the regulation of power generation. According to Suga's PVA approach, the LV contractility is quantified by the maximal elastance, i.e., the slope of the end-systolic pressure-volume relationship (ESPVR). Consequently, two different hearts that beat at different heart rates but generate the same maximal elastance (ESPVR) have the same ”contractility,“ on the basis of the elastance concept. Obviously, these hearts differ in their cardiac output and their ability to generate power. Therefore, the definition of cardiac ”contractility“ as based on pressure-volume loop, which is equivalent to the force-length relationship (FLR) at the isolated fiber level, is incomplete. Clearly, the definition of contractility should also include an index of the LV ability to generate power, which will relate to the FVR at the isolated fiber level.

The isolated muscle mechanics are characterized by two basic steady-state properties: the FLR and the FVR. The question rises as to what is the appropriate parameter that can properly describe the FVR characteristics of the cardiac fiber at the global LV level. Indeed, the present study suggests that the mechanical feedback coefficient,*g*
_{1}, which determines the FVR at the isolated fiber level, also provides the index that can characterize the capability of the whole heart to generate power.

The mechanical feedback mechanism explains the FVR and the linear relationship between energy consumption and the generated mechanical energy, whereas the cooperativity mechanism (20, 22) determines the FLR and provides the cellular basis for the Frank-Starling law. The cooperativity mechanism determines the amount of available free energy, whereas the mechanical feedback determines its conversion to mechanical energy. Consequently, cardiac muscle performance and the so-called contractility are determined by the interplay between the loading conditions and the intracellular control mechanisms, i.e., the mechanical feedback and the cooperativity (18).

The present description of the regulation of energy conversion, based on the intracellular control mechanism that couples calcium kinetics with cross-bridge cycling, has other advantages that were not described here. This description can explain the Fenn effect (10), the dependence of the magnitude of the Fenn effect on the activation level (12), and the observed positive effect of ejection on the generated pressure, where the end-systolic pressure of the shortening beat exceeds the isovolumic pressure at the same end-systolic volume (Fig. 3) (16). The description of the LV function based on the mechanical feedback provides the cellular parameters that determine the efficiency of muscle contraction and enriches our understanding of cardiac muscle and LV function.

In conclusion, our biochemically based intracellular control model, which describes the basic mechanical properties of the cardiac muscle, i.e., the FLR and the FVR, is extended here to describe analytically the transient length and force responses to load clamps and controlled sarcomere shortening, respectively, and the regulation of energy conversion in the cardiac muscle.

The analysis establishes the key role of the mechanical feedback mechanism in the regulation of sarcomere dynamics and energetics. The mechanical feedback, whereby the filament sliding velocity affects the rate of cross-bridge weakening, determines the FVR and the generated power. The mechanical feedback provides the explanation for the observed linear relationship between energy consumption and the generated mechanical energy. Moreover, it defines three components of the mechanical energy: external work production, energy dissipation as heat due to the viscoelastic property of the cross bridges, and a pseudopotential energy that is actually determined by the FTI. The pseudopotential energy describes the energy dissipation for cross-bridge recruitment. The efficiency of energy conversion is determined by the negative mechanical feedback coefficient,*g*
_{1}. The study indicates the potential of utilizing*g*
_{1} as a general and precise index of muscle capability to generate external work. It also provides the quantitative tool required to evaluate the efficiency of chemical-to-mechanical energy conversion.

In general, the present study suggests a new approach to the quantitative interpretation of physiological observations of muscle mechanics and describes cardiac muscle function on the basis of a better understanding of the sarcomere function and cross-bridge dynamics.

## Acknowledgments

The cooperation of Prof. Henk E. D. J. ter Keurs of the University of Calgary, Alberta, Canada, is noted with pleasure.

## Footnotes

Address for reprint requests and other correspondence: A. Landesberg, Dept. of Biomedical Engineering, Technion-IIT, Haifa 32000, Israel (E-mail: amir{at}biomed.technion.ac.il).

This program was initiated by the Levi-Eshkol Fellowship of the Ministry of Science and Arts (to A. Landesberg) and a grant from the Technion Vice President for Promotion of Research (to S. Sideman). This study was financed by a grant from the Germany-Israel Foundation (Research Project No. I-391-215.02/94). The highly distinguished Yigal Allon Grant (to A. Landesberg) is particularly noted. The continued support of Yochai Schneider of Las Vegas, NV, is greatly appreciated.

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- Copyright © 2000 the American Physiological Society