INTRODUCTION

TOP ABSTRACT INTRODUCTION THE PHYSIOLOGICAL MODEL THE MATHEMATICAL MODEL RESULTS DISCUSSION REFERENCES

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 (O₂) and the generated mechanical energy, which is quantified by the pressure-volume area (PVA) in the LV pressure-volume plane as

(1)

where 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

(2)

Suga (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 with O₂ as

(3)

where 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 O₂ at the muscle fiber level (Eq. 3) suggests that the linear relationship between PVA and O₂ 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

TOP ABSTRACT INTRODUCTION THE PHYSIOLOGICAL MODEL THE MATHEMATICAL MODEL RESULTS DISCUSSION REFERENCES

Four-State Model

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.

View larger version (41K): [in this window] [in a new window]   Fig. 1.   Transitions between four states of troponin regulatory units are defined by calcium kinetics and cross-bridge cycling rates. R, rest state; A, bound calcium non-force-generating state that describes activation level; T, bound calcium force-generating state; U, unbound calcium force-generating state; kl and kl, 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; g1, mechanical feedback coefficient; V, filament shortening velocity; [Ca++], free calcium concentration.

Feedback Mechanisms

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

(4)

where g₀ is the rate of cross-bridge weakening under the isometric regime and g₁ 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

TOP ABSTRACT INTRODUCTION THE PHYSIOLOGICAL MODEL THE MATHEMATICAL MODEL RESULTS DISCUSSION REFERENCES

Interaction Between State Variables

(5)

where [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, g₀, and g₁, 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 (assumption 3). Hence, the generated force is given by

(6)

where is the unitary isometric force developed by each cross bridge and  represents the viscous property of the cross bridge (7).

Energy Consumption and the Activation Level

(7)

(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

(9)

	RESULTS

TOP ABSTRACT INTRODUCTION THE PHYSIOLOGICAL MODEL THE MATHEMATICAL MODEL RESULTS DISCUSSION REFERENCES

Isometric-Isotonic Changeover (Load Clamps) Studies

For isometric contraction [change in shortening velocity (dV/dt) = V = 0] and at peak isometric force [F = F_m, load change (dF/dt) = 0], Eq. 9 reduces to

(10)

Equation 10 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

(11)

where V_u = / is the unloaded shortening velocity (21).

During the isotonic contraction, dF/dt = 0. For the asymptotic steady-state shortening velocity (dV/dt = 0), Eq. 11 reduces to Hill's equation (14) for the FVR (21)

(12)

where V_H denotes the steady-state shortening velocity, and Hill's constants a_H and b_H are given by

(13)

Equations 12 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., g₀, g₁, 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 ( g₁).

View larger version (14K): [in this window] [in a new window]   Fig. 2.   Curvature of force-velocity relationship (FVR) (A) and generated power output (B) depend on magnitude of mechanical feedback (g1). F, generated force; F0, initial force after quick release; Vmax, maximal shortening velocity.

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 g₁ because a_H/F_m  g₀/g₁V_U. The smaller the mechanical feedback coefficient g₁, the smaller the curvature of the FVR and the higher the power at any given load (Fig. 2).

Transient Length Response (Load Clamps)

(14)

(15)

(16)

(17)

(18)

(19)

Integrating Eq. 19 yields the transient length changes L(t) given by

(20)

Note that when the initial shortening velocity after the quick release to the isotonic force level equals the steady-state shortening velocity, i.e., V₀ = 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)

For isovelocity shortening (dV/dt = 0) starting at the time of peak isometric force (F = F_m), the transient force response is derived from Eq. 11 as

(21)

where G = g₀ + (g₁ + 1/L_s)V, and the transient force is given by

(22)

where F₀ 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/dt = 0, leading to Hill's equation (14)

(23)

When the initial force level after the quick release equals the steady-state force, i.e., F₀ = 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

(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

(25)

The 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

(26)

or

(27)

Equation 27 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, g₀, and the mechanical feedback coefficient, g₁, respectively. Dividing Eq. 27 by g₁/ yields the simplified, approximated relationship between chemical energy consumption and the generated mechanical energy

(28)

where


⁰ is the approximated efficiency of the biochemical-to-mechanical energy conversion and E⁰_pp is the approximated pseudopotential energy. Note that the efficiency is inversely proportional to the mechanical feedback coefficient, g₁.

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(  V), into Eq. 7 yields

(29)

because V_u = / (7). Using the equality 1/(1  x) = 1 + x + x²/(1  x) gives

(30)

Opening the parentheses and rearranging provides the general equation for energy conversion

(31)

Dividing both sides of Eq. 31 by (1/)(g₁ + g₀/V_U) gives

(32)

where


and W is as defined by Eq. 28. E_PP is the pseudopotential energy and is proportional to the FTI. Q represents energy dissipation caused by the viscous property of the cross bridge. Q represents the effect of the viscous component because it is proportional to  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 x²/(1  x) = x² + x³ + x⁴...; x = V/V_max].  is the efficiency of the biochemical-to-mechanical energy conversion and is inversely proportional to the mechanical feedback coefficient, g₁.

Simulated Results

View larger version (27K): [in this window] [in a new window]   Fig. 3.   A: isometric contraction at different sarcomere lengths (indicated by thick vertical lines) and physiological contraction from same preload but with different afterload. B: linear relationship between energy consumption and generated mechanical energy for both isometric () and shortening beats (+). Magnitude of mechanical feedback: g1 = 7.5.

Figures 4 and 5 depict similar simulations of isometric and physiological contractions, but with larger mechanical feedback coefficients, i.e., g₁ = 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%.

View larger version (28K): [in this window] [in a new window]   Fig. 4.   A: isometric contraction at different sarcomere lengths (indicated by thick vertical lines) and physiological contraction from same preload but with different afterload. B: linear relationship between energy consumption and generated mechanical energy for both isometric () and shortening beats (+). Magnitude of mechanical feedback: g1 = 8.25.

View larger version (28K): [in this window] [in a new window]   Fig. 5.   A: isometric contraction at different sarcomere lengths (indicated by thick vertical lines) and physiological contraction from same preload but with different afterload. B: linear relationship between energy consumption and generated mechanical energy for both isometric () and shortening beats (+). Magnitude of mechanical feedback: g1 = 10.5.

Increasing the mechanical feedback coefficient (g₁ = 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 g₁ determines the efficiency of energy conversion from biochemical to mechanical energy. The smaller the mechanical feedback coefficient, the higher the efficiency of energy conversion.

View larger version (24K): [in this window] [in a new window]   Fig. 6.   Mechanical feedback determines efficiency (Eff) of biochemical-to-mechanical energy conversion.

	DISCUSSION

TOP ABSTRACT INTRODUCTION THE PHYSIOLOGICAL MODEL THE MATHEMATICAL MODEL RESULTS DISCUSSION REFERENCES

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

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

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 (g₀ + g₁V or g₀ + g₁V_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, g₀, i.e., f + g₀ (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 (F₀ = 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 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 = 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 consum