## Abstract

Energy liberation rate (*Ė*) during steady muscle shortening is a monotonic increasing or biphasic function of the shortening velocity (*V*). The study examines three plausible hypotheses for explaining the biphasic *Ė-V* relationship (EVR): *1*) the cross-bridge (XB) turnover rate from non-force-generating (weak) to force-generating (strong) conformation decreases as *V* increases; *2*) XB kinetics is determined by the number of strong XBs (XB*-*XB cooperativity); and *3*) the affinity of troponin for calcium is modulated by the number of strong XBs (XB*-*Ca cooperativity). The relative role of the various energy-regulating mechanisms is not well defined. The hypotheses were tested by coupling calcium kinetics with XB cycling. All three hypotheses yield identical steady-state characteristics: *1*) hyperbolic force-velocity relationship; *2*) quasi-linear stiffness-force relationship; and *3*) biphasic EVR, where *Ė* declines at high *V* due to decrease in the number of cycling XBs or in the weak-to-strong transition rate. The hypotheses differ in the ability to describe the existence of both monotonic and biphasic EVRs and in the effect of intracellular free calcium concentration ([Ca^{2+}]_{i}) on the EVR peak. Monotonic and biphasic EVRs with a shift in EVR peak to higher velocity at higher [Ca^{2+}]_{i} are obtained only by XB*-*Ca cooperativity. XB*-*XB cooperativity provides only biphasic EVRs. A direct effect of *V* on XB kinetics predicts that EVR peak is obtained at the same velocity independently of [Ca^{2+}]_{i}. The study predicts that measuring the dependence of the EVR on [Ca^{2+}]_{i} allows us to test the hypotheses and to identify the dominant energy-regulating mechanism. The established XB*-*XB and XB*-*Ca mechanisms provide alternative explanations to the various reported EVRs.

- muscle energetics
- cross bridge
- excitation-contraction coupling
- cardiac mechanics
- cooperativity

the dominant mechanism underlying the modulation of energy liberation from muscle by the loading conditions is not well defined (12, 29). The rate of energy liberation (*Ė*), i.e., mechanical power plus heat production rate, is higher in steady shortening than in isometric contraction (*Ė*_{0}*)* (29). Hill's original study (17) has demonstrated that *Ė* increased monotonically with shortening velocity (*V*). Later studies by Hill (16) and others (1, 29) have established the existence also of a biphasic *Ė-V* relationship (EVR*)* with a decrease of *Ė* as *V* approaches the unloaded shortening velocity (*V*_{u}). This biphasic phenomenon and an apparent plateau at velocities about one-half the maximal velocity were reported in fast-twitch [frog sartorius (16), mouse extensor digitorum longus (2, 3), and dogfish white myotomal fibers (8)] and slow-twitch [tortoise rectus femoris (28) and mouse soleus (1)] muscles of various vertebrates. There is a lot of evidence suggesting that the EVR cannot be described by one typical function, and various shapes of the EVR were reported in different muscle types [slow vs. fast (1–3)], physiological conditions [as fatigue (1)], and activation levels (4, 5).

The aim of this study is to suggest and explore possible regulatory mechanisms underlying both monotonic and biphasic EVRs reported in the literature. The study tests the prevalent hypothesis that the biphasic EVR is derived from modulation of XB kinetics (2, 7, 21, 27) and suggests two additional novel hypotheses, utilizing established cooperativity mechanisms, that relate the biphasic EVR to the regulation of XB recruitment. These later two mechanisms modulate the number of cycling XBs and decrease the number of available XBs as the velocity increases, without assuming any changes in the rate of XB attachment. The tested hypotheses are as follows.

*1*) The apparent rate constant (*f*) of XB turnover from non-force-generating (weak) to force-generating (strong) conformation is a decreasing function of V. Huxley (21) suggested that at high shortening velocities the probability of XB attachment decreases. This apparent decrease of *f* at high shortening velocities could also be explained by XB detachment without ATP hydrolysis (2, 7, 21).

*2*) The interactions between attached XBs and regulatory proteins determine adjacent XB recruitment rate (XB*-*XB cooperativity) (14). Cooperative interactions between regulatory units may occur not only along the thin filament but also across the thin filament (i.e., between its 2 intertwining strands) (13). Thus *f* is a function of the number of strong XBs, and faster *V* indirectly decreases *f*.

*3*) The shortening velocity decreases the number of available cycling XBs through the effect of the number of strong XBs on calcium affinity (XB*-*Ca cooperativity) (14, 25). The number of available cycling XBs is determined by the amount of bound calcium. An increase in *V* decreases the number of strong XBs, as well as the amount of bound calcium, through this XB*-*Ca cooperativity.

Our four-state model of the sarcomere (25), which couples calcium kinetics with XB dynamics, is used here to explain the biphasic phenomenon and to test these three regulatory mechanisms. These plausible mechanisms were used to simulate sarcomere mechanics and energetics during ramp-length releases at constant velocities while the muscle is tetanically stimulated. All three hypotheses correctly describe a hyperbolic force-velocity relationship (FVR) and a quasi-linear stiffness-force relationship. The hypotheses differ in their ability to describe both monotonic and biphasic EVR and in the dependence of their EVRs on free intracellular calcium concentration ([Ca^{2+}]_{i}).

The study suggests that further experimental characterization of the dependence of EVR on calcium may allow identifying the dominant mechanism that regulates XB recruitment and may highlight the sarcomere control of biochemical-to-mechanical energy conversion. We demonstrate that XB recruitment modulating mechanism, via XB*-*Ca cooperativity kinetics, predicts a biphasic EVR and a wider spectrum of phenomena, such as the dependence of the EVR peak on the activation level and the existence of both monotonic and biphasic EVR. This theoretical study provides valuable predictions and may inspire further experimental studies.

## METHODS

The basic assumptions are derived from the Landesberg and Sideman model (25) that couples the kinetics of calcium binding to troponin with the regulation of XB cycling and simulates cardiac sarcomere control of contraction. The assumptions of the present model are as follows.

*1*) The regulatory unit includes a single troponin-tropomyosin complex with seven neighboring actin monomers and the adjacent myosin heads.

*2*) The XB cycle is a repeated oarlike cycle between weak and strong conformations (10). XB transitions between weak and strong conformations relate to the biochemical rate-limiting steps of nucleotide binding and release. ATP hydrolysis and phosphate release are required for XB weak-to-strong transition (10). There is a loose coupling between the XB biochemical transition between strong and weak conformations and the myosin mechanical interaction with the actin binding site. During fast shortening, a XB may perform multiple stroke steps while being in strong biochemical conformation.

*3*) The rate of energy liberation during sarcomere shortening, at steady-state conditions, is equal to the rate of energy consumption (16, 20, 29).

*4*) The active force is proportional to the number of XBs in the strong conformation in the single overlap region. The fraction of XBs in the strong conformation depends on calcium and XB kinetics.

*5*) The individual XB acts like a Newtonian pseudoviscous element (9, 15), i.e., the average force per XB is a linear function of *V,* i.e., *F*_{XB} = *F̄* − η*V*, where *F̄* is the unitary force per XB at isometric regime and *η* is the XB pseudoviscosity coefficient (9, 15, 25). [At normal physiological conditions the XB does not move at velocities faster than the maximal unloaded velocity (*V*_{U} = *F̄*/*η*), and thus the XB generates only a positive force.]

*6*) The major role of calcium binding to troponin is to regulate ATPase activity and XB weak-to-strong transition (6). The activity of the ATPase is inhibited, and only an insignificant number of weak-to-strong transitions occur in the absence of bound calcium (6).

*7*) Calcium may dissociate from troponin before XB turnover to the weak conformation, i.e., XBs can generate force without having adjacent troponin with bound calcium (25).

*8*) The rate of XB turnover from strong to weak conformation (*g*), denoted as the XB weakening rate, linearly depends on *V* (25), i.e., *g* = *g*_{0} + *g*_{1}*V*, where *g*_{0} represents the rate of XB weakening at isometric contraction and *g*_{1} describes the effect of *V* on the weakening rate and is denoted as the mechanical feedback coefficient. The mechanical feedback decreases the number of strong XBs as *V* increases. The assumption that the rate of XB weakening is a linear function of *V* is required (24) to derive Hill's well-established equation for the FVR (25).

The troponin regulatory units are distributed between four states (Fig. 1) that are determined by two kinetics: *1*) calcium binding and dissociation from troponin and *2*) XB cycling between the weak and the strong conformations. *State R* represents the rest state, in which the XBs 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 the adjacent XBs are still weak. ATP hydrolysis and phosphate release allows XB turnover from the weak to the strong conformation and leads to *state T*, where the regulatory unit is with bound calcium and adjacent strong XBs. Calcium dissociation from *state T* leads to *state U*, in which the XBs are still in the strong conformation but without bound calcium. Note that the sum of the regulatory states is constant: *R + A + T + U = Tro*, where the constant *Tro* denotes the total density of the troponin regulatory units along the single-overlap length.

The model that is based on the above assumptions, wherein the rate of XB turnover to the strong state (*f*) and calcium dissociation rate (*k*_{−l}) are constants, is denoted as the basic model. Three different variants of the basic model were developed here; each includes only one of the following plausible mechanisms.

*1*) The first variant, *f*(*V*), simulates the suggested (21) decrease in the weak-to-strong XB transition rate (*f*) as *V* increases. According to Huxley's suggestion, force-generating XBs do not have the time to form if the sliding speed is too high (21). A linear function is used for the formulation of this hypothesis, *f* = *f*_{0} − *f*_{1}·*V*, where *f*_{0} is the rate of weak-to-strong transition in isometric contraction and *f*_{1} describes the dependence of *f* on *V*.

*2*) The second variant, *f*(*S*), alludes to the role of XB*-*XB cooperativity mechanism (14) and is simulated by using a sigmoidal dependence of *f* on the number of strong XBs (*N*_{XB}), *f* = *f*_{0} + *f*_{1}·*N*/(*f*+ *N*), where *N*_{XB} *= L*_{s}·(*T + U*) and *L*_{s} is the single-overlap length, *f*_{0} is the minimal rate of *f,* and *f*_{1} is the maximal change of *f* due to existence of strong XBs. The parameter *n* determines the steepness of this sigmoidal function, and *f*_{m} is the number of strong XBs that increases *f* by half of *f*_{1}.

*3*) The last variant, *k*(*S*), describes the XB*-*Ca cooperativity (14, 25). A sigmoidal function is used to describe the decrease in calcium dissociation rate (*k*_{−l}) as the number of strong XBs (*N*_{XB}) increases, *k*_{−l} = *k*_{0} − *k*_{1}·*N*/(*k*+ *N*), where *k*_{0} is the rate of calcium dissociation in the absence of strong XBs and *k*_{1} determines the maximal change of *k*_{−l} due to the existence of strong XBs. The parameter *n* determines the steepness of the sigmoidal function. The parameter *k*_{m} is the number of strong XBs required for decreasing *k*_{−l} by half of *k*_{1}.

Sarcomere dynamics (Fig. 1) is described by a set of four first-order differential equations: (1) where *k*_{l} ≜ *k*_{L}·[Ca^{2+}]_{i} and the rate of calcium association, *k*_{L}, is a constant (see Table 2). The rate of XB strong-to-weak transition, *g*, is a linear function of *V* (*assumption 8*). *I*_{in} and *I*_{out}, the inward and outward calcium currents through the sarcolemma and sarcoplasmic reticulum, are deterministic functions of time (25) that do not depend on *V*.

The generated force is determined by the sarcomere single-overlap length (*L*_{s}), the density of strong regulatory units (*T + U*), and the force per XB (*assumption 5*): (2)

The dynamic stiffness, measured by small and fast sarcomere length oscillations (23), is proportional to the strong regulatory unit density. To obtain a first approximation for the stiffness, we do not include here the possible contribution to the stiffness of attached XBs that do not consume ATP (see end of discussion and *Limitation*): (3)

The sarcomere energy consumption is determined by the number of XB transitions from *state A* to *state T* because each transition requires hydrolysis of one ATP. Thus *Ė* is determined by two quantities: the density of regulatory units in *state A* and the rate constant of XB turnover from weak to strong conformation (*f*), (4)

The basic model and the three hypotheses *f*(*S*)*, f*(*V*), and *k*(*S*) were subjected to isovelocity shortenings within the plateau region of the FLR. The system of differential equations was integrated by utilizing the fourth-order Runge-Kutta algorithm, using the commercially available MATLAB software, on an IBM-compatible personal computer. The parameters of the tested models are specified in Tables 1 and 2.

## RESULTS

#### FVR.

The general equation for the steady-state force (*F*), at constant shortening velocity and activation, is determined by calculating the density of strong XBs (*T + U*) from *Eq. 1*, when d*A/*d*t =* d*T/*d*t =* d*U/*d*t =* 0: (5)

Under physiological conditions, (*g* + *f*) ≫ *f*·*k*_{−l}/(*g* + *k*_{l} + *k*_{−l}) in the denominator, leading to: (6)

Substitution of *Eq. 6* into *Eq. 2* and rearrangement yield Hill's hyperbolic FVR: (7) where the isometric force (*F*_{0}) and Hill's parameters (*a*_{H} and *b*_{H}) are given by: (8)

The normalized FVR, derived from *Eq. 7*, describes the relation between the normalized force (*F/F*_{0}) and the normalized shortening velocity (*V̄* = *V/V*_{u}): (9) where (10)

Thus the normalized FVR (Fig. 2*A*) is determined by a single constant, *a*_{N}.

The parameters of the three variants (Tables 1 and 2) were adjusted to obtain identical normalized FVR with *a*_{N} = 02, as observed in some slow skeletal muscles (Table 2.II in Ref. 29) (Fig. 2*A*), and biphasic EVR (Fig. 2*D*), as reported for slow skeletal muscle (1). The exerted mechanical power is a product of the force (*F*) and *V* (Fig. 2*B*). Similar parabolic power*-*velocity relationships are obtained for the basic model and its three variants. The parabolas are skewed toward the low shortening velocities region, and the peak power is at about one-third of *V*_{u}.

#### Stiffness-force relationship.

The normalized stiffness (*K*_{N}), defined as the dynamic stiffness divided by the stiffness at isometric contraction, is determined by the ratio between the strong XB density during steady shortening and the isometric strong XB density: *K*_{N} ≜ (*T* + *U*)/(*T* + *U*)_{0}, based on *Eq. 3*. Utilization of *Eqs. 6* and *10* gives the following: (11)

All the models provide similar quasi-linear stiffness-force relationships (Fig. 2*C*). At high shortening velocities the stiffness predicted by the three variants *f*(*V*), *f*(*S*), and *k*(*S*) deviates from that derived for the basic model. The relatively lower stiffness predicted by the three variants is readily explained by considering the lower XB weak-to-strong transition rate postulated by models *f*(*V*) and *f*(*S*) or the decrease in the number of available XBs for weak-to-strong transition postulated by model *k*(*S*).

#### Rate of energy liberation.

At saturating free calcium levels, the behavior of the four-state system (Fig. 1) can be approximated and simulated by relating only to the two states with bound calcium (*A* and *T*). The rate of energy liberation (*Ė*) normalized by the isometric rate of energy liberation (*Ė*_{0}), for this simplified two-state system, is given by: (12) where (13)

Note that *g*_{N} (*Eq. 13*) is determined exclusively by the XB strong-to-weak transition parameters (*g*_{0} and *g*_{1}), whereas *a*_{N} (*Eq. 10*) depends also on the XB weak-to-strong transition rate (*f*). The EVR is a monotonic increasing function of *V*, because the derivative of *Eq. 12* is always positive and equal to *f*·*a*_{N}/[*g*_{0}·(*V̄* + *a*_{N})^{2}].

All four states of the model should be considered in the general case of partial calcium activation. *Ė* is determined by *state A* (*Eq. 4*), which is derived from *Eq. 1*: (14) where *g = g*_{0} *+ g*_{1}*V* (*assumption 8*). Substitution of *Eq. 14* in *Eq. 4* yields the EVR: (15)

To test whether the basic model can describe a biphasic EVR, the derivative of *Eq. 15* is calculated. A derivative of a polynomial function as in *Eq. 15* is always positive, when the denominator and the numerator are of the same order (2 in *Eq. 15*), all the coefficients are positive, the polynomials have the same coefficient for the highest order (*g* in *Eq. 15*), and the coefficients of the lower orders are larger in the denominator than in the numerator. It follows that the derivative d*Ė*/d*V* is always positive, and *Ė* for the basic model always increases as *V* increases (Fig. 2*D*).

The three variants *f*(*V*), *f*(*S*), and *k*(*S*) yield a biphasic EVR (Fig. 2*D*), although they describe similar FVRs (Fig. 2*A*) as the basic model. It does not imply that the three variants present larger maximal thermodynamic efficiency. The normalized thermodynamic efficiency, *ρ*_{N}, is defined as the ratio between normalized mechanical power, i.e., (*F*·*V*)/(*F*_{0}·*V*_{u}), and the normalized rate of energy liberation (*Eq. 12*). With the use of *Eq. 9* and *Eq. 12*, the normalized thermodynamic efficiency of all the models is given by: (16)

The normalized thermodynamic efficiency (*Eq. 16*), according to the basic model and its variants, depends only on a single normalized parameter, *g*_{N}. The higher *g*_{N} is, the lower is *ρ*_{N}. Because all of the models studied here have the same *g*_{N}, they also have the same *ρ*_{N}.

The isometric ATPase rate is derived from *Eq. 15* (for *V = 0*): (17)

For full activation, saturating free calcium concentration (*k*_{l} = *k*_{L}[Ca^{2+}]_{i} ≫ *f*, *k*_{−l}, *g*_{0}), this equation is reduced to the commonly used equation: (18)

#### Effects of [Ca^{2+}]_{i} on the EVR.

Aside from the mechanical feedback (*g*), the EVR (*Eq. 15*) is determined by three other parameters: *1*) [Ca^{2+}]_{i}, in *k*_{l} *= k*_{L}·[Ca^{2+}]_{i}, *2*) calcium dissociation rate (*k*_{−l}), and *3*) XB weak-to-strong turnover rate (*f*). Figure 3 depicts the dependence of the EVR on [Ca^{2+}]_{i} for the three variants. The EVRs were normalized by the isometric energy consumption rate at the same [Ca^{2+}]_{i}.

The higher the calcium level, the higher is the normalized *Ė* (Fig. 3). The EVRs are always biphasic for *models f*(*V*) and *f*(*S*), at physiological and full-saturated calcium activations (Fig. 3, *A* and *B*). In contrast, the biphasic EVRs observed at a low [Ca^{2+}]_{i} turn to a monotonic relationship at high calcium level only in model *k*(*S*) (Fig. 3*C*), where shortening affects the number of available XBs. Note that the EVR peak is obtained at higher velocities as the level of [Ca^{2+}]_{i} increases in all the models, except for the *f*(*V*) model (Fig. 3*A*). For the *f*(*V*) model the EVR peak is obtained at the same shortening velocity, which is independent of [Ca^{2+}]_{i}.

## DISCUSSION

The basis of our analysis of the EVR is the four-state model of the sarcomere developed by Landesberg and Sideman (25) that couples calcium kinetics with XB dynamics. The Landesberg-Sideman model describes basic sarcomere characteristics, such as the FVR, force-length relationship, force–pCa relationship, stiffness-force relationship, rate of force redevelopment, and the regulation of biochemical to mechanical energy conversion (25, 26, 30). It is used here to test three suggested hypotheses for explaining the biphasic EVRs.

The steady-state muscle characteristics simulated here (Fig. 2), FVR, mechanical power, stiffness, and EVR, are in agreement with experimental data. The maximum exerted mechanical power (Fig. 2*B*) is ∼8% of the product of the isometric force and the unloaded shortening velocity (*F*_{0}·*V*_{u}) (29). The peak of mechanical power (Fig. 2*B*) is obtained at a velocity close to one-third of *V*_{u} (29). Fiber stiffness at *V*_{u} ranges between one-sixth to one-third of the stiffness at isometric contraction, measured by imposing small-length oscillations at frequencies up to 2,000 Hz (9, 11, 23), depending on *a*_{N} (*Eq. 11*; Fig. 2*C*). The maximum value of *Ė* in our simulations, ∼3.5 (Fig. 2*D*), is within the range (2–7) of reported values (1, 16, 29).

The basic model and the three variants describe two identical basic sarcomere characteristics at steady-state shortening (Fig. 2): a hyperbolic FVR (Fig. 2*A*; *Eq. 9*) (29) and a quasi-linear stiffness-force relationship (Fig. 2*C*; *Eq. 11*) (9, 23). However, the basic model describes only a monotonic increasing EVR (Fig. 2*D*; *Eq. 15*), whereas the other three variants can describe also biphasic EVR (Fig. 2*D* and Fig. 3), with a decline of *Ė*(*V*) at high shortening velocities. The decline in *Ė* in *models f*(*V*) and *f*(*S*) is mediated by a decrease in the rate of XB cycling, whereas the cause of this decline in *model k*(*S*) is a decrease in the number of available XBs for the weak-to-strong transition.

The normalized FVR (*Eq. 9*) and the normalized stiffness-force relationship (*Eq. 11*) can be described by using only a single parameter (*a*_{N}), whereas the expression for the normalized EVR (*Eq. 12*) depends on two parameters (*a*_{N} and *g*_{N}). Thus the inherent information in the two graphs, EVR and FVR, is different, and a wider spectrum of EVR relative to FVR is expected. Muscles with identical normalized FVR are expected to have the same normalized stiffness-force relationship. However, a given normalized FVR or normalized stiffness-force relationship does not dictate the normalized EVR.

[Ca^{2+}]_{i} has significant effects on the EVR (Fig. 3). For a given shortening velocity, *Ė* increases as [Ca^{2+}]_{i} increases because more XBs are available for the energy-consuming transition from *state A* to *state T* in all models. However, two additional effects of [Ca^{2+}]_{i} on the EVR allow a distinction between the various models: *1*) transition from monophasic to biphasic EVR at lower [Ca^{2+}]_{i} and *2*) shift in the peak of the EVRs toward lower velocities as [Ca^{2+}]_{i} decreases.

Interestingly, only *model k*(*S*) (Fig. 3*C*), in contrast to *models f*(*V*) and *f*(*S*) (Fig. 3, *A* and *B*), presents a biphasic *Ė*(*V*) at low [Ca^{2+}]_{i} and a monotonic *Ė*(*V*) at high [Ca^{2+}]_{i}. In *models f*(*V*) and *f*(*S*), the EVR decreases at high velocity due to the decrease in *f*, either through the direct effect of *V* on *f* in *model f*(*V*) or the decrease in the number of strong XBs, at high *V*, in *model* *f*(*S*). The biphasic EVR in *model k*(*S*) results from the dependence of calcium affinity and the bound calcium on the number of strong XBs. At high velocities the number of available XBs decreases due to the decrease in the bound calcium. However, as [Ca^{2+}]_{i} increases, the bound calcium approaches saturation and is less sensitive to changes in the affinity. Thus the number of available XBs, dictated by the bound calcium, is less sensitive to the velocity as [Ca^{2+}]_{i} increases, yielding monotonic EVR at high [Ca^{2+}]_{i}.

Only in *model f*(*V*) is the peak of the normalized EVRs independent of [Ca^{2+}]_{i}, whereas the other two model variants, *f*(*S*) and *k*(*S*), show a shift in the peaks of the normalized EVRs toward lower velocities as [Ca^{2+}]_{i} decreases (Fig. 3, *B* and *C*). In *models f*(*S*) and *k*(*S*), the shape of the EVR depends on [Ca^{2+}]_{i} through the effect of [Ca^{2+}]_{i} on the number of strong XBs. The lack of a peak shift in *model f*(*V*) is explained by the independence of *f* on the number of strong XBs and hence on the activation level, which is unique to this model.

The study suggests that a detailed description of the effect of [Ca^{2+}]_{i} on EVR allows us to determine which of the discussed hypotheses play the key role in modulating the EVR. An observed shift of the velocity at peak EVR negates the hypothesis that the direct dependence of *f* on *V* is the dominant mechanism. There is lack of knowledge, and experimental data pertaining to a shift of the EVR peak with changes in [Ca^{2+}]_{i} are scarce. Buschman et al. (5) compared the EVRs of a fast-skeletal muscle fiber during fused tetanic contraction and unfused tetanus. They have found no evidence of change in the shape of the FVR at different activation levels. However, they observed significant change in the rate of heat production between high and low activation, with no increase in the rate of heat production above the isometric heat rate at low activation. Thus different EVR slopes, the sum of power and heart rate, are expected at different activation levels (Fig. 3 in Ref. 5), with a shift in the peak EVR to higher velocity as the activation level increases. These observations (5) establish the effect of the activation level on the EVR profile and strengthen the predictions of *k*(*S*) and *f*(*S*) models. However, with the scattered data and the difficulties in measuring energy liberation at high shortening velocity and partial activation level, more studies should be conducted to substantiate the role of the activation level.

Both XB*-*XB cooperativity and XB*-*Ca cooperativity mechanisms were substantiated experimentally (14), whereas the suggested direct dependence of the XB weak-to-strong transition rate (*f*) on *V* (21) has not been confirmed experimentally. The study establishes the importance of the XB*-*XB cooperativity and XB*-*Ca cooperativity mechanisms, which modulate XB recruitment, compared with the direct effect of *f* on *V*, on the regulation of energy consumption.

Few other models in the literature have succeeded in describing the biphasic EVR phenomenon (2, 7, 21, 27). These Huxley-type models assume that a negative force is exerted at normal physiological functioning by negatively strained XBs. To the best of our knowledge, there is no direct experimental support for this assumption still (24). In addition, the contribution of calcium kinetics to the regulation of XB cycling is ignored in these studies (2, 7, 21, 27). Note also that in all the models discussed here, calcium kinetics is incorporated.

#### Limitations.

*1*) Our analysis does not preclude the possibility that several regulatory mechanisms act in parallel but assumes the existence of only one dominant mechanism and allows defining this dominant mechanism. *2*) This study assumes that force generation and energy liberation during shortening at constant velocity have reached a steady state. Though experimentally valid when the sarcomere shortens at a moderate shortening velocity within the plateau region of the force-length relationship, steady state may not be reached at *V*_{u} or when there is a significant change in the single-overlap length (18, 19). *3*) The models describe muscle stiffness when the stiffness is measured by small oscillations up to 2,000 Hz (9, 22, 23). The basic model predicts (*Eq. 11*) that the relative stiffness while shortening at *V*_{u} would amount to *a*_{N}*/*(1 + *a*_{N}). The normalized stiffness for *a*_{N} = 0.2 (Fig. 2*C*) is 0.17. Normalized stiffness well below 0.2 was found in cardiac muscles (9). Julian and Morgan (22) used frog tibialis anterior with *a*_{N} = 0.39 (Table 2.II in Ref. 29); the predicted (*Eq. 11*) stiffness at *V*_{u} is 0.28. Interestingly, the experimental value is also ≃0.28, when measured with small sinusoidal vibrations at frequencies ranging up to 2,000 Hz (Fig. 4 in Ref. 22). The stiffness measurements depend on the oscillation frequency, and higher normalized stiffness was observed at higher frequencies (4,000 Hz) (22). This phenomenon was not included in the current presentation.

In conclusion, the study examines three plausible hypotheses for explaining the biphasic EVR. The relative role of these three energy-regulating mechanisms is not well defined. The present results provide tools for identifying the relative role of the plausible mechanisms. All three mechanisms can explain the biphasic EVR but differ in two aspects: *1*) the ability to describe the existence of both monotonic and biphasic EVRs and *2*) the effect of [Ca^{2+}]_{i} on the EVR peak. XB*-*Ca cooperativity, whereby calcium affinity is determined by the number of strong XBs, allows the existence of both monotonic and biphasic EVR and the shift in the EVR peak to higher velocity at higher [Ca^{2+}]_{i}. XB*-*XB cooperativity cannot describe monotonic EVR. A direct effect of *V* on XB kinetics does not allow monotonic EVR, similar to the XB*-*XB cooperativity mechanism. Moreover, a direct effect of *V* on XB kinetics predicts that the EVR peak is obtained at the same velocity, independently of [Ca^{2+}]_{i}, unlike the XB*-*XB cooperativity mechanism. A direct effect of *V* on the XBs weak-to-strong transition rate was not corroborated experimentally. The established XB recruitment modulating mechanisms (XB*-*Ca cooperativity and XB*-*XB cooperativity), operating in the cardiac muscle, provide an explanation to the modulation of the EVR by [Ca^{2+}]_{i}.

## GRANTS

This study was supported by the fund for the promotion of research at the Technion and a grant from the United States-Israel-Binational Science Foundation (research project no. 2003399).

## Acknowledgments

O. Tchaicheeyan thanks Marie and Moshe Tchaicheeyan for financial support.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 2005 by the American Physiological Society