## Abstract

Length-dependent steady-state and dynamic responses of five models of isometric force generation in cardiac myofilaments were compared with similar experimental data from the literature. The models were constructed by assuming different subsets of three putative cooperative mechanisms. *Cooperative mechanism 1* holds that cross-bridge binding increases the affinity of troponin for Ca^{2+}. In the models, *cooperative mechanism 1*can produce steep force-Ca^{2+}(F-Ca) relations, but apparent cooperativity is highest at midlevel Ca^{2+} concentrations. During twitches, *cooperative mechanism 1* has the effect of increasing latency to peak as the magnitude of force increases, an effect not seen experimentally.*Cooperative mechanism 2* holds that the binding of a cross bridge increases the rate of formation of neighboring cross bridges and that multiple cross bridges can maintain activation of the thin filament in the absence of Ca^{2+}. Only*cooperative mechanism 2* can produce sarcomere length (SL)-dependent prolongation of twitches, but this mechanism has little effect on steady-state F-Ca relations.*Cooperativity mechanism 3* is designed to simulate end-to-end interactions between adjacent troponin and tropomyosin. This mechanism can produce steep F-Ca relations with appropriate SL-dependent changes in Ca^{2+} sensitivity. With the assumption that tropomyosin shifting is faster than cross-bridge cycling, *cooperative mechanism 3*produces twitches where latency to peak is independent of the magnitude of force, as seen experimentally.

- myofilaments
- modeling
- twitches
- force-calcium relations
- isometric relaxation

in cardiac muscle, steady-state force-Ca^{2+} (F-Ca) relations exhibit apparent cooperativity much higher than that predicted by the single regulatory Ca^{2+} binding site on troponin (reviewed in Ref. 2). There is considerable debate as to the exact mechanisms of this high cooperativity. One hypothesis assumes that attached cross bridges increase the affinity with which troponin binds Ca^{2+}(*hypothesis 1*). Experimental evidence for this type of cooperativity derives mainly from studies showing that the affinity of troponin for Ca^{2+} is decreased by blocking cross-bridge cycling with the phosphate analog vanadate (27). Also, manipulations that alter cross-bridge attachment states can produce changes in the cytosolic Ca^{2+}concentration ([Ca^{2+}]) transients, presumably by modifying Ca^{2+} binding to troponin (1, 2,30).

A second hypothesis is that attachment of one cross bridge increases the rate of formation of neighboring cross bridges (*hypothesis 2*). Such an interaction may occur if the formation of the first cross bridge holds tropomyosin in a permissive state, facilitating the formation of nearby cross bridges. The experimental evidence for this type of cooperativity derives from studies in which the number of available cross bridges is modulated. For example, in vitro studies of the binding of isolated myosin heads to thin filament show steep (highly cooperative) binding functions (see e.g., Ref. 44). A second line of evidence derives from studies showing that the addition of subfragment 1 (S1) rigor-bound myosin heads can produce activation in the absence of activator Ca^{2+} (7, 50, 53).

A third hypothesis is that end-to-end interactions between adjacent troponin and tropomyosin molecules increase apparent cooperativity (*hypothesis 3*). This hypothesis holds that once a tropomyosin becomes permissive to cross-bridge cycling, it can facilitate the transition of its nearest neighbors into permissive states. In such a scheme, activation may spread end to end along the thin filament, thus increasing cooperativity. The evidence for end-to-end interactions comes from a number of experimental approaches. Partial extraction of troponin complexes in muscle preparations produces decreased apparent cooperativity, as shown by less steep F-Ca relations (43). Likewise, apparent cooperativity is reduced by end-to-end interactions that are disrupted by truncating tropomyosin with proteolytic enzymes (46, 52).

Researchers have proposed a number of models of thin-filament activation and force generation that are based on the cooperativity hypotheses described above. A model proposed by Landesberg and Sideman (39), in which cross-bridge binding increases the affinity of troponin for Ca^{2+}, is similar to*hypothesis 1*. A Monte Carlo model proposed by Zou and Phillips (61) incorporates*hypothesis 2* in that cross-bridge formation increases the rate of formation of nearby cross bridges. This model also incorporates representations of the other two cooperative hypotheses. Models focusing on *hypothesis 3* have been proposed by Hill and co-workers (25, 26) and by Dobrunz et al. (14). In these models, activated troponin/tropomyosin units help to activate their nearest neighbors.

Most of these models exhibit highly cooperative behavior as indicated by steady-state F-Ca relations, indicating that simple F-Ca relations are inadequate to distinguish between different models of cooperativity. This study attempts to further resolve the potential contribution of alternative cooperative mechanisms by testing model responses against more demanding experimental responses. First, F-Ca relationships are strongly modulated by sarcomere length (SL), showing increases in both Ca^{2+} sensitivity and maximum force (for review, see Ref. 2). These changes cannot be explained by changes in thick and thin filament overlap alone, suggesting that the underlying cooperative mechanisms are enhanced as SL increases. In this modeling study, steady-state F-Ca relations are simulated for a range of SLs for comparison with experimentally determined changes in apparent cooperativity.

A second set of tests involves simulation of twitches. These dynamically changing force transients represent a more natural activation pattern than steady-state F-Ca relations. Twitches also provide a more demanding test for models because the dynamic behavior of muscle during a twitch cannot be predicted simply from the F-Ca relation but appears to have additional contributions from the dynamics of thin-filament activation and/or cross-bridge cycling (5). Changes in SL also modulate dynamic behavior. For example, longer SLs both increase peak force and decrease relaxation rate, thus prolonging twitch force (31, 51). This relatively complex dynamic behavior provides more critical tests for distinguishing between alternative models.

With the use of these tests, this paper explores the behaviors of five models of force generation in cardiac muscle. The first two models in this study are derived from existing published models and provide a baseline of performance for comparison. The next three models incorporate novel approaches to modeling cooperative activation in cardiac muscle. These models are developed to progressively incorporate more cooperative mechanisms that include end-to-end troponin-tropomyosin interactions, neighboring cross-bridge interactions, and feedback on troponin affinity for Ca^{2+}. It is hypothesized that multiple mechanisms of cooperativity may coexist and contribute to the responses of cardiac muscle.

## MODEL CONSTRUCTION

The responses of five models are explored in this paper. All models are similar in that they are structured around a functional unit of troponin, tropomyosin, and actin. Tropomyosin is assumed to exist in either permissive or nonpermissive states. Permissive states refer to tropomyosin for which the accompanying actin binding sites have been made available for cross bridges to bind and generate force. Depending on the model version, one or more cross bridges are also assumed to exist in the functional unit. These are assumed to be either weakly bound (non-force generating) or strongly bound (force generating). In the first three models, four states (*N0*,*N1*,*P0*, and*P1*) are needed to describe the functional units, as shown in Table 1. The nonpermissive and permissive tropomyosin states are represented by*Nx* and*Px*, respectively, where*x* denotes the number of strongly bound cross bridges (either 0 or 1 in *models 1–3*).

#### Model 1.

This model is shown schematically in Fig.1
*A*. Its activation and cross-bridge dynamics were formulated by Peterson et al. (48). Whereas the full derivation is given in the original work, a brief description of the rationale is also provided here (note that a full set of rate constants is provided in the
). The rest state for this model is *N0* (nonpermissive tropomyosin with no strongly bound cross bridge). Binding of Ca^{2+} to troponin is assumed to produce an immediate shift of tropomyosin to a permissive conformation (*P0*) that allows a cross bridge to become strongly bound (*P1*). There is also a “residual” cross-bridge state (*N1*) in which the cross bridge remains in a strongly bound state even after Ca^{2+} has dissociated from troponin. The “on” rate constants,*k*
_{on} and
, are assumed to be second order, depending on [Ca^{2+}]. In*model 2*,
is set 40 times larger than *k*
_{on}. This is done to capture *hypothesis 1*in which the presence of strongly bound cross bridges increases the affinity of troponin for Ca^{2+}.

With permissive tropomyosin, the rate constants for cross-bridge cycling between weakly (*P0*) and strongly bound (*P1*) states are given by *f* and*g*. These constants were chosen so that this model would match myosin ATPase rates. Note that, with nonpermissive tropomyosin, there is a single “off” rate,*g*′, with no corresponding on rate. This off rate is about seven times larger than*g*, as determined by matching data on relaxation rates during perturbed twitches (48). This simple two-state model of cross-bridge cycling lacks explicit biochemical detail of ATP hydrolysis and force productions. For this reason, the attachment and detachment rates should be more properly referred to as apparent rates (i.e., *f*
_{app} and*g*
_{app}). However, in this and subsequent models, we will use*f* and*g* with the assumption that they refer to apparent rates, not strict biochemical reaction rates.

In the present paper, force (F) is reported as a normalized value between 0 and 1. A value of 1 corresponds to the case in which the maximum possible number of force generators are strongly bound. For example, assuming full activation in *model 1*, all units distribute between*P0* and*P1*. The fraction in*P1*(F_{max}) is computed as
Equation 1where*f* and*g* are as described above.

Although not considered in the original model description, the effect of sarcomere geometry is added to *model 1* in this paper. Because of the physical structure of thick and thin filaments within a sarcomere, zones can exist with no, single, or double overlap (39). Of these, only the single-overlap zone is assumed to contribute to force generation. To describe this effect, an overlap ratio (α) is defined (39). This ratio gives the fraction of thick filament myosin heads in the single-overlap conformation. For α = 1, all myosin heads are able to interact with actin in the single-overlap zone, whereas α < 1 when some of the myosin heads are in the double- or no-overlap zones. For *model 1*, α as a function of SL is fit to the classic data of Gordon et al. (19), as shown in Fig.2
*A*. An alternate interpretation of α is that it corresponds to the maximal normalized force that can be generated by assuming full activation of the muscle. With the contribution of the SL included, the normalized force is computed as
Equation 2where*P1* and*N1* are the fractions of functional units in force-generating states (i.e., with strongly bound cross bridges).

#### Model 2.

*Model 2* is shown in Fig.1
*B*. The activation and cross-bridge dynamics were formulated by Landesberg and Sideman (38, 39). As in the previous model, *hypothesis 1* is assumed to be the major cooperative mechanism. However, there are important differences between this model and the previous one as to how force-generating cross bridges affect the affinity of troponin for Ca^{2+}. In *model 1*, there is a change in the on rate (
) when a cross bridge is strongly bound (force generating). In contrast, in*model 2*, the off rate of Ca^{2+} from troponin (*k*
_{off}) is assumed to be a decreasing function of the fraction of units with cross bridges in strongly bound states (*P1* +*N1*). The effect is to increase the apparent Ca^{2+}-binding constant of troponin with increasing developed force (see
for details). Developed force affects the off rates for both *P0* and*P1* equally (in *model 1*, only the on rate for*P1* is affected). Although a functional unit in the *P0* state does not have a strongly bound cross bridge, the off rate for this unit is assumed to be affected by neighboring functional units in force-generating states (36, 37). The method of changing the off rate of Ca^{2+} from troponin used in*model 2* is called*cooperative mechanism 1* to distinguish it from the more rudimentary cooperativity of having a “residual” cross-bridge state (as in *model 1*). Another difference between the models is in the sarcomere-overlap function (α). As shown in Fig. 2
*A*, for *model 2* α is assumed to be a monotonically increasing function of SL throughout the range from 1.7 to 2.3 μm.

#### Model 3.

*Model 3* is constructed with the premise that *hypothesis 3* (end-to-end interaction of troponin and tropomyosin molecules) is the most important cooperative mechanism controlling force generation. The major difference between this model and the previous two is the manner in which Ca^{2+} binding to troponin affects tropomyosin shifting. In the previous models, these events are directly coupled (i.e., binding of Ca^{2+} to troponin produced an immediate shift in tropomyosin). In *model 3*, these events are assumed to be coupled indirectly, as represented by the dashed arrows in Fig.1
*C*. This construct, called*cooperative mechanism 3*, allows the binding of Ca^{2+} to troponin to be uncooperative while producing Ca^{2+}-dependent shifting of tropomyosin that shows high apparent cooperativity.*Cooperative mechanism 3* is a phenomenological approach to the representation of*hypothesis 3* (end-to-end troponin-tropomyosin interactions) that produces a low-order system of equations. There is no attempt to explicitly model end-to-end interaction because this would require Monte Carlo approaches (25). This point is addressed further in thediscussion.

The uncoupling of Ca^{2+} binding from tropomyosin shifting required two sets of states. The first set of states governs only Ca^{2+} binding to troponin. In *model 3*, the first state (*T*) represents troponin with no Ca^{2+} bound to the regulatory (low affinity) site.*T*
_{Ca} represents troponin with Ca^{2+} bound to the regulatory site. All functional units are assumed to be in one of these two states, such that
Equation 3where*T* and*T*
_{Ca} refer to the probabilities of being in each state. The binding of Ca^{2+} is assumed to be simple and uncooperative with rate constants*k*
_{on} = 40 μM^{−1} ⋅ s^{−1}and *k*
_{off} = 20 s^{−1} (49). Thus the [Ca^{2+}] for 50% binding to troponin (*K*
_{Ca}) is
Equation 4The second set of states, described in Table 1, involves tropomyosin shifting and cross-bridge formation. In *model 3*, the shifting of tropomyosin is assumed to produce all of the apparent cooperativity observed in steady-state F-Ca relations in cardiac muscle. To achieve this behavior, troponin shifting must be a highly cooperative function of the fraction of troponin with Ca^{2+} bound to the regulatory site (*T*
_{Ca}). This relationship is illustrated in Fig. 2
*B*in which the abscissa shows the fraction of units in*T*
_{Ca}, whereas the ordinate shows the resulting fraction of units with tropomyosin in permissive conformation (*P0* +*P1*). The increasing steepness and leftward shift of these traces as a function of SL is assumed to be the source of the SL-dependent increases in Ca^{2+} sensitivity and apparent cooperativity seen in F-Ca relations in cardiac muscle.

The relations in Fig. 2
*B* are Hill functions with the properties that*1*) cooperativity (*N*) increases with SL and*2*) the value of*T*
_{Ca} producing half-maximal shifting (*K*
_{1/2}) decreases. To produce this behavior, the forward rate of tropomyosin shifting (*k*
_{1}) is assumed to be a function of both*T*
_{Ca} and SL, as shown below
Equation 5where
Equation 6
Equation 7where SL_{norm} is a dimensionless factor that ranges from 0 to 1 (see *Eqs. 8
* and *
9
*). The rate of tropomyosin shifting from permissive to nonpermissive (*k*
_{−1}) is estimated from experimental data in reconstituted thin filament (42). With the removal activator Ca^{2+}, the thin filament shifts with a rate of ∼43 s^{−1}, as assessed by a decrease in fluorescence resonance energy.

With the relations in *Eqs. 8 and 9
*, the steady-state fraction of units with tropomyosin in permissive conformation is a Hill function of*T*
_{Ca}.
Equation 8
Equation 9A more familiar representation is obtained by the substitution of*Eq. 5
* for*k*
_{1}
Equation 10
The plots of “fraction permissive” in Fig.2
*B* show increasing steepness and a leftward shift as SL increases. The steepness is a result of increasing*N* (in *Eq.8
*) from 7 at SL = 1.7 μm to 10 at SL = 2.3 μm. The leftward shift results from the decrease in*K*
_{1/2} (in*Eq. 7
*) at longer SLs. The increasing steepness and leftward shift are preserved when the fraction of units in permissive states are plotted versus [Ca^{2+}] in Fig.2
*C*. Note that the uncooperative binding of Ca^{2+} to troponin (Fig.2
*C*, dashed line) is much less steep. In *model 3*, Ca^{2+} binding is simple, with no dependence on SL or force, so that there is only one binding curve for all SLs.

As in previous models, the cross-bridge cycling rates are assumed to be fixed. The cross-bridge attachment rate*f* is set to 10 s^{−1}, and the detachment rate*g* is set to 20 s^{−1}. Similar attachment and detachment rates of 12 and 22 s^{−1}, respectively, have been reported using fluctuation analysis for small numbers of myosin heads in near-isometric conditions (29).

#### Model 4.

As in *model 3*, *model 4* is constructed under the premise that*cooperative mechanism 3* is the important cooperative mechanism controlling force generation. However, in *model 4*, up to three cross bridges are assumed to exist in the vicinity of each functional unit. As shown in Fig.3
*A*, up to six states are associated with tropomyosin and cross bridges in the functional unit (Table 2).*Model 4* also incorporates three new features: *1*) an SL-dependent detachment rate for cross bridges,*2*) maintenance of the tropomyosin in a permissive conformation by strongly bound cross bridges, and*3*) cooperative formation of cross bridges within a functional unit.

The SL-dependent detachment rate for cross bridges is suggested by experiments on skeletal muscle (32, 60). Including this feature in*model 4* allows for more realistic SL-dependent changes in plateau force at saturating [Ca^{2+}]. Recall that the troponin-shifting construct of *model 3* is designed to produce steep Ca^{2+} sensitivity, with proper SL dependence. However, this construct alone does not produce SL-dependent increases in plateau force that are as large as those measured experimentally (i.e., see examples in Fig.5
*A* and see data in Ref. 12). In*model 3*, the fraction of tropomyosin in the permissive conformation exceeds 85% for all SLs (see Fig.2
*C*). Therefore, the effects of tropomyosin shifting alone can increase maximum force by <15% as SL increases from 1.7 to 2.3 μm. A second feature that affects plateau force in *model 3* is the sarcomere-overlap function in Fig. 2
*A*. This feature produces an increase in force with length up to SL = 2.0 μm; however, the effect eventually saturates and decreases for further SL increases. In contrast, the experimental data show larger and generally monotonic increases.

Consequently, in *model 4* the off rate of cross-bridge binding is assumed to be a decreasing function of SL
Equation 11where*g** is the minimal detachment rate and SL_{norm} is a dimensionless quantity between 0 and 1, as described in *Eq.9
*. Thus the cross-bridge detachment rate increases by a factor of two as SL decreases from 2.3 to 1.7 μm. Three versions of*model 4*, with one, two, or three cross bridges in the functional unit, have been constructed. The minimal detachment rate ( *g**) changes for each of the three versions of *model 4* (1 cross bridge: *g** = 20 s^{−1}; 2 cross bridges:*g** = 27.5 s^{−1}; and 3 cross bridges:*g** = 35 s^{−1}). The reason for the higher detachment rates is to promote faster relaxation as the number of cross bridges increases. This is necessary to counteract the effect of slowing relaxation by having more cross bridges per functional unit.

The second important feature of *model 4* is the cooperative action of multiple cross bridges to maintain tropomyosin in a permissive conformation. Notice in Fig.3
*A* that there are no direct transitions from multiple cross bridge-bound states (*P2*,*P3*) to nonpermissive tropomyosin states. The rationale for this feature is that two or more bound cross bridges within the same functional unit are assumed to hold tropomyosin in its permissive state. This construction is suggested by experimental evidence from Ishii and Lehrer (28) showing that one to two bound SL heads per tropomyosin can trap the thin filament in an activated state (or “on” state in the authors’ terminology). In the same study, activation of regulated actin (as assessed by fluorescence) decays only after all cross bridges have dissociated (as assessed by light scattering). Similar results are obtained with and without Ca^{2+}, suggesting that the presence of strongly bound cross bridges can maintain the thin filament in an activated state without Ca^{2+} bound to troponin.

The third important feature incorporated into the multiple cross-bridge construct is cooperative formation. That is, the rates of cross-bridge formation are assumed to increase progressively as more cross bridges form. To understand this construct, consider first the case for three cross bridges in which formation is not cooperative. Figure 4
*A*shows all the permutations of possible states for three cross bridges (states arranged vertically with 0 = detached and 1 = attached). The cross bridges are assumed to act independently, and each individual cross bridge has on rate *f* and off rate *g*. This explicit system can be represented as a composite system, as shown in Fig.4
*B*. For instance, the net transition rate from one to two cross bridges is 2*f*, which arises from the two separate paths that form a double-attached composite from each configuration with one cross bridge attached. Figure4
*B* shows the complete kinetics diagram for the composite system in which forward rates are 3*f*, 2*f*, and*f*. Likewise, the reverse rates are*g*, 2*g*, and 3*g*. The integral multiples of the basic rate constants are derived from the multiple pathways for association/dissociation of cross bridges, not from any cooperative binding of individual cross bridges.

Next, consider the system shown in Fig.4
*C* in which cooperative formation is assumed. These formation rates are 3*f*, 14*f*, and 10*f*, implying that the formation rates of the second and third cross bridges are effectively 7 and 10 times greater than in the uncooperative system. Note that, as shown in Fig.4
*C*, only cross-bridge on rates are cooperative, and off rates are not cooperative. The net result of this cooperative formation is to produce a system in which increased levels of force can be attained if activation is prolonged. This effect works synergistically with the previous cooperative effect described above for *model 4* (i.e., no direct transition to nonpermissive states when two or more cross bridges are present) to increase the duration of force generation at high levels of force. Together, these features, collectively referred to as*cooperative mechanism 2*, are designed to simulate the cooperativity between neighboring cross bridges (*hypothesis 2*). Hence,*model 4* contains both*cooperative mechanism 2*, described here, and *cooperative mechanism 3*, carried over from the tropomyosin shifting functions of*model 3* (see *Eqs.5-9
* and Fig. 2*C*).

One final issue is the reporting of normalized force. Because of the multiple-cross-bridge structure, the maximum force (F_{max}) is no longer given by*Eq. 1
* but is instead determined by a more complicated function using the King-Altman rule (35). However, the normalization factor remains conceptually the same in that F_{max} is computed by assuming full activation (i.e.,*k*
_{1} is assumed to be very large so that all units distribute among*P0* through*P3*). The steady-state values of*P1*,*P2*, and*P3* are then multiplied by the weighing factors 1, 2, and 3, respectively, to account for the number of force-generating cross bridges represented by each state. Full details of the calculation of normalized force are provided in the
.

#### Model 5.

*Model 5* is a refinement of*model 4*, obtained by adding a feedback pathway in which attached cross bridges increase the affinity of troponin for Ca^{2+}. Hence,*model 5* adds*cooperative mechanism 1* to*model 4*, which includes both*cooperative mechanisms 2* and*3*. *Cooperative mechanism 1* is shown schematically by the dashed arrow directed from the force-generating cross bridge states (*P1*,*N1*,*P2*,*P3*) to the off rate of Ca^{2+} from troponin. Similarly to*model 2*, this feature is designed to simulate experimentally measured increases in troponin Ca^{2+} affinity in the presence of cycling cross bridges (*hypothesis 1*). In *model 5*,
, the off rate for troponin, decreases linearly with increasing normalized force
Equation 12where*k*
_{off} is the same as in *models 3* and*4* and F is normalized by the maximum value (F_{max}). F_{max} is computed by assuming full activation (i.e.,*k*
_{1} is assumed to be very large so that all units distribute among states*P0* through*P3*).

A force-dependent troponin affinity for Ca^{2+}, as described above, increases the overall cooperativity in the model. To maintain steady-state F-Ca relations with similar apparent cooperativity, a compensatory modification is required to make tropomyosin shifting less sensitive to Ca^{2+} than in*model 4*. This modification is implemented by decreasing *N* and increasing *K*
_{1/2}. Specifically, *Eqs. 7
* and *
8
* are modified to become
Equation 13
Equation 14where SL_{norm} is as defined in*Eq. 9
*. A second version of*model 5* with stronger feedback on the off rate for troponin is also developed. In this version, the equation for
, the off rate for troponin, is given as
Equation 15The compensatory modifications required in this case are
Equation 16
Equation 17

## RESULTS

#### Steady-state F-Ca relations.

Cooperativity is most directly quantified by the steady-state F-Ca relation. Effects of SL on experimentally determined F-Ca relations are shown in Fig.5
*A*. Increasing SL changes three key features:*1*) plateau force,*2*) half-activation point ([Ca]_{50}), and*3*) Hill coefficient (*N*
_{H}). Plateau force refers to the maximal force at saturating levels of [Ca^{2+}]. Variation of SL over the indicated range nearly doubles plateau force. [Ca]_{50} refers to the [Ca^{2+}] producing half the plateau force value. In Fig. 5
*A*, [Ca]_{50} points are connected by a dashed line. These points show a leftward shift with increasing SL. Note that the data in Fig.5
*A* are from a skinned muscle preparation in which [Ca]_{50} ranged from ∼3.59 to 13.4 μM. These values are about one order of magnitude larger than those measured in intact muscle (5, 17, 59). These data do, however, exhibit the expected SL dependence of [Ca]_{50} (i.e., [Ca]_{50} increased by 3.7 times as SL decreased from 2.15 to 1.65 μm). These experimental data are also fit to Hill functions. Estimated*N*
_{H} increases from 3.3 to 5.4 as SL increases from 1.65 to 2.15 μm. Intact preparations are generally thought to show higher*N*
_{H} values near 6 for medium-range SLs (5, 14, 17). The data shown in Fig.5
*A*, along with other data sets (13), show increasing*N*
_{H} with SL. However, other experimental results show little change in*N*
_{H} with increasing SL (23, 36, 40).

The correspondence between the simulated F-Ca relations and experimental results are assessed using plateau force, [Ca]_{50}, and*N*
_{H}. For simulated data, *N*
_{H} is estimated by plotting the F-Ca relations on a logarithmic scale and then using the relationship
Equation 18where*S*
_{50} is the slope at [Ca]_{50} and F_{p} is the plateau force.*Equation 18
* holds for true Hill functions for which maximum slope occurs exactly at [Ca]_{50}. In the simulated data, the F-Ca relations show minor deviation from true Hill functions. However, the maximum slope always occurs at or near [Ca]_{50}, so*N*
_{H} as computed above yields a reasonable estimate of the maximum steepness of F-Ca functions.

Simulated F-Ca relations for *model 1*are shown in Fig. 5
*B*. These differ from experimental results in two ways. First, the changes in plateau force with SL are too small and do not increase monotonically. The changes in plateau force are a direct reflection of α, the nonmonotonic sarcomere-overlap function (see Fig.2
*A* and *Eq. 2*). Second, the simulated relations are less steep than the experimental results (note that the abscissa in Fig.5
*B* covers 4 orders of magnitude). Also, there is no dependence of [Ca]_{50} and*N*
_{H} on SL (*N*
_{H} = 1.1 and [Ca]_{50} = 0.59 μM for all SLs). There is only rudimentary cooperativity provided by an increase in the on rate of Ca^{2+}from troponin when a cross bridge is strongly bound (
>*k*
_{on} in Fig.1
*A*). Therefore, this feature does not provide sufficient cooperativity to produce steep F-Ca relations.

The F-Ca relations for *model 2* are shown in Fig. 5
*C*. These are much closer to the experimental results. Plateau force showed incremental (termed “graded”) changes with SL. As in*model 1*, the changes in plateau force are a direct result of changes in α, the sarcomere-overlap function. Slope and sensitivity of the F-Ca relations for *model 2* are also more similar to the experimental data than for *model 1*. Note that the F-Ca relations are steepest for intermediate [Ca^{2+}] values (0.6 to 1 μM) and less steep at high and low [Ca^{2+}]. The maximum slopes, as quantified by*N*
_{H}, are the largest for all models, but the changes in both*N*
_{H} and [Ca]_{50} follow the trends in the experimental data (see Table3 for summary of*N*
_{H} and [Ca]_{50} values).

The main difference between *model 2*(which does produce steep F-Ca relations) and *model 1* (which does not) is in a mechanism by which force affects the Ca^{2+} affinity of troponin. In *model 2*, increasing force reduces the off rate of Ca^{2+} from troponin (*k*
_{off}) for both the *P0* and*P1* states. In contrast, in*model 1*, the presence of a strongly bound cross bridge increases the on rate of Ca^{2+}(*k*
_{on}) for the*P1* state only.

The F-Ca relations for *model 3* are shown in Fig. 5
*D*. Plateau forces are more graded for *model 3* than for*model 1*, even though the same sarcomere-overlap function is used in both models. Despite the increase in gradation, the SL-dependent changes in plateau force are smaller than those seen in the experimental data. Also, the longest SL produces a nonmonotonic plateau force, a feature not observed in the experimental data. However, other experimental data show a decline in maximum developed force when cardiac muscle is stretched past the optimal length of 2.2–2.3 μm (15). Although force declines at these long lengths, Ca^{2+}sensitivity continues to increase (16). This observation is consistent with the behavior of *model 3* in that*N*
_{H} increases and [Ca]_{50} decreases despite a lower plateau force at the longest SL.

As SL is increased, changes in [Ca]_{50} and*N*
_{H} produced by*model 3* are more similar to the experimental findings than those for *models 1* and *2*. The SL-dependent changes in steepness and Ca^{2+} sensitivity reflect the way in which tropomyosin is assumed to shift in response to changes in [Ca^{2+}] or SL (see*Eqs. 5-9
* and Fig. 2*C*). This construct produces F-Ca relations that more closely resemble Hill relations in that there is not pronounced steepness for intermediate [Ca^{2+}] values, as seen for *model 2*. In *model 2*, the increased steepness in the midrange of [Ca^{2+}] arises from the force-dependent changes in affinity of troponin (*hypothesis 1*). Results for*model 3* suggest that another cooperative mechanism (*hypothesis 3*) may play an important role in shaping F-Ca relations.

Figure 6 shows steady-state F-Ca relations for *models 4* and*5*. Two versions of*model 4* are shown, a one-cross-bridge model (Fig. 6
*A*) and a three-cross- bridge model (Fig. 6
*B*). The data show that Ca^{2+} sensitivity of both versions of *model 4* is similar to that of *model 3*. The similarity results from the incorporation of *cooperative mechanism 3* in each of these models. One difference is that*model 4* produces larger and more graded changes in plateau force with SL. This effect is a consequence of SL-dependent detachment rate (see *Eq.11
*).

Figure 6, *C* and*D*, shows the steady-state F-Ca relations for *model 5*. The F-Ca relations resemble a hybrid of the responses produced by*model 2* (Fig.5
*C*) and by the version of*model 4* with three cross bridges (Fig.6
*B*). The explanation is that*model 5* contains*cooperative mechanism 3*(tropomyosin-shifting cooperativity) similar to that of model 4 and*cooperative mechanism 1* (feedback on Ca^{2+} binding) similar to that of*model 2*. When force feedback on troponin Ca^{2+} binding is weaker, F-Ca relations are generally similar to *model 4* but exhibit a slight increase for intermediate [Ca^{2+}] values. As the degree of feedback increases, the response of *model 5* becomes closer to that of *model 2* with more marked steepness for intermediate [Ca^{2+}] values (e.g., compare SL = 2.3 μm in traces in Figs.5
*C* and6
*D*).

The results thus far have shown that *model 1* does not adequately reproduce steady-state F-Ca relations. The rudimentary cooperativity assumed in this model is unable to produce sufficiently steep steady-state F-Ca relations, and there are no SL-dependent changes in apparent cooperativity.*Model 2* produces steeper F-Ca relations, more similar to the experimental results, as a consequence of much stronger feedback of force on Ca^{2+} binding to troponin (*cooperative mechanism 1*). A side effect, inconsistent with experimental results, is that F-Ca relations are most steep for intermediate [Ca^{2+}] values.*Model 3* produced steep F-Ca relations by the method by which Ca^{2+}binding to troponin is assumed to shift tropomyosin in a highly cooperative manner (*cooperative mechanism 3*). However, resulting F-Ca relations do not show graded changes in plateau force as seen in the experimental data.*Model 4* expands on*model 3* with the addition of two features: *1*) variable cross-bridge detachment rates and *2*) multiple cross bridges (*cooperative mechanism 2*). Addition of variable cross-bridge detachment rate produces more graded changes in plateau force in F-Ca relations. The multiple-cross-bridge formation had little effect on F-Ca relations. Finally, *model 5* expands on*model 4* by adding feedback of force on Ca^{2+} binding to troponin (*cooperative mechanism 1*).*Model 5* therefore contains representations of each hypothesized cooperative mechanism. However, only a modest degree of feedback of force on Ca^{2+} binding to troponin may be added without causing responses that are inconsistent with experimental data. Specifically, large amounts of feedback cause the F-Ca relations to become very steep for intermediate [Ca^{2+}] values, similar to that seen for *model 2*.

#### Dynamic response: twitches.

Figure 7 presents normalized force transients at different SLs. Experimental data are shown in Fig.7
*A*; responses of*models 1–3* are shown in Fig. 7,*B–D*. The experimental data exhibit three important changes with SL. First, peak force increases considerably with SL, more than doubling (0.4 to 1.0 in normalized units) for the range shown (SL = 1.9–2.2 μm). Second, rate of force development is approximately proportional to peak force. This observation is more evident after each trace is individually normalized by its peak value, as shown in Fig.8
*A*. The overlap of force traces during the rising phase implies that the rate of rise is approximately proportional to the peak force. Because of this proportionality, the time to reach peak force is approximately the same for each SL (although there is a slight increase at longer SLs). Finally, as SL increases, the time required for relaxation also increases. The twitch is fully relaxed by 0.4 s for SL = 1.9 μm, whereas the twitch is not completely relaxed until 0.6 s at SL = 2.2 μm. This increase in relaxation time is evident by the spread of traces between the opposing arrows in Fig. 8, indicating the points of 50% relaxation in individually normalized twitch responses.

Simulation data for *model 1* are shown in Fig. 7
*B*. The model is driven by the simulated Ca^{2+} transient shown by the dashed trace. The most important difference between simulated and experimental data is that simulated twitches are simply scaled versions of each other (see Fig. 8
*B*), with the scaling provided by the sarcomere-overlap function (Fig.2
*A*). The sarcomere-overlap function can generate only modest changes in peak force and has no effect on dynamics in *model 1*, so there are no length-dependent changes in the relaxation times, as seen in the experimental data. The results of *model 1* show clearly that the experimentally determined changes in force transients involve more than simply scaling by means of a sarcomere-overlap function.

Simulated data for *model 2* are shown in Figs. 7
*C* and8
*C*. The magnitude of peak force shows large changes with SL, similar to the experimental data. However,*model 2* does not reproduce the experimentally observed changes in twitch time course. For example, the rate of force onset does not increase as fast as the peak force. This causes a progressive increase in the time to peak force, as shown by the rightward shift of the rising phases in Fig.8
*C*. The time to peak increases by 0.064 s.

Results from *model 2* and the experimental data also differ in the relaxation phases.*Model 2* produces an SL-dependent prolongation of the twitch duration, as seen in the experimental data (i.e., the traces in Fig. 5
*C* show incremental rightward shifts much like those in Fig.5
*A*). However, the rightward shifts in the simulated data result mainly from differences in the time to peak force. SL delays the 50% relaxation times by 0.073 s (lower arrows) and the times to peak by 0.064 s (upper arrows). In contrast, the experimental results show a relatively constant time to peak but increasing twitch durations. Another difference is that*model 2* produces a slow final phase of relaxation (as force decays below 25% normalized force) with little dependence on SL. The experimental data shown in Fig.5
*A*, and also additional data reported elsewhere (51), indicated that final relaxation is faster as SL decreases.

Results for *model 3* are shown in Figs.7
*D* and8
*D*. This model produces twitches that better match the experimental data in two respects:*1*) the peak force shows large changes with SL, and *2*) the rising phases of force and the time to peak force are relatively independent of SL. Total twitch duration and the relaxation rate are similar to the experimental data, at least for shorter SLs in Fig.7
*A*. However, the responses of*model 3* differ from the experimental results in that there is little SL-dependent prolongation of twitch force. Another difference is that *model 3* produced a final phase of relaxation with a time constant that exhibited little dependence on SL.

The deficiencies of relaxation timing in *model 3* are substantially alleviated by the new mechanism introduced in *model 4*. Figure9 shows data for the three versions of*model 4* with one, two, or three cross bridges per functional unit. Assumption of a single cross bridge produces twitches that are similar to those of *model 3* in that there is little SL-dependent prolongation of twitch force. Whereas *model 4* also incorporates SL-dependent cross-bridge detachment rate (*Eq. 11
*), this feature alone does not produce dramatic prolongation, as indicated by the 50% relaxation times. Figure 9, *C* and*D*, shows twitches for the two-cross-bridge model. The additional cross bridge produces prolongation of twitch duration with increasing SL (compare Fig. 9,*B* and*D*). Further prolongation can be achieved by adding additional cross bridges. As shown in Fig. 9,*E* and*F*, the three-cross-bridge model produces larger and more graded prolongation, as shown by the 50% relaxation times.

To quantify the final relaxation rate, the relaxation is fit to an exponential. The time constants (τ) are shown for traces corresponding to SLs of 1.7 and 2.3 μm. For the one-cross-bridge version of *model 4*, the time constant increases only slightly (from 0.042 to 0.054 s) over the SL range. This increase is mainly the result of the decreasing cross-bridge detachment rate as SL increases. For the two-cross-bridge version of*model 4*, there is almost a doubling of the time constants (from 0.055 to 0.094 s). In the three-cross-bridge version, there is little further change in the time constants (compare Fig. 9, *D* and*F*). Therefore, the third cross bridge produces a larger and more graded prolongation in twitch force but little change in time constant of the final relaxation phase.

The mechanism by which multiple cross bridges prolong force is shown in Fig. 10. The fraction of functional units in the force-producing states (*P1*,*N1*,*P2*, and*P3*) are shown for SL = 2.0 μm (Fig. 10
*A*) and 2.3 μm (Fig.10
*B*). These data correspond to the 2.0-μm (closed circles) and 2.3-μm (asterisks) traces in Fig.9
*E*. First, consider the 2.0-μm case (Fig. 10
*A*). The fraction of units in the *P1*,*P2*, and*P3* states peaks at progressively later times. For example, *P1* (solid trace) peaks at 0.9 s, whereas *P3* (dot-dash trace) peaks at 1.2 s. The last to peak is*N1* (long-dash trace) at 1.4 s, because this state becomes most highly populated during relaxation. As [Ca^{2+}] falls and the population of *T*
_{Ca}decreases, there is a corresponding decrease in*k*
_{1}. With a small*k*
_{1}, the nonpermissive states (*Nx*) are favored over permissive states (*Px*) (see Fig. 3
*A*). This decreases the probability of being in *P1*,*P2*, or*P3* states, with a corresponding increase in the probability of being in the*N1* state.

The mechanism by which force is prolonged at longer SLs is illustrated with data for SL = 2.3 μm (Fig.10
*B*). The fraction of units in strongly bound states (*P1*,*N1*,*P2*, and*P3*) is now larger, as would be expected for the greater total force generation. Except for*P1*, the time to the peak of each state is also later than the corresponding data for 2.0 μm (Fig.10
*A*). The proportion of units in the different states also changes. For example, at the shorter SL, the peak of *P1* exceeds the peak of*P3*, whereas the opposite occurs at the longer SL. At both lengths, the *N1*state peaks later than the other force-generating states. However, at the longer length, the peak of *N1* is more delayed as the persistence of the other force-generating states is fed more slowly into the *N1* state.

*Model 4* demonstrates clearly that multiple cross bridges prolong the duration of force as SL increases. Therefore, *model 4* can simulate, at least qualitatively, the experimental data considered so far (both F-Ca relations and twitches). *Cooperative mechanism 1* (attached cross bridges modify the Ca^{2+} affinity of troponin) is not included in *model 4*. However, there is considerable experimental evidence that the presence of cycling cross bridges can increase the affinity of troponin for Ca^{2+} (21, 27).*Model 5* adds this feature to the three-cross-bridge version of *model 4*. The off rate of Ca^{2+} from troponin is decreased in proportion to the number of strongly bound cross bridges (see *Eq. 12
*). Figure11, *A*and *B*, shows the twitch responses for*model 5* with a relatively modest feedback. The responses of this model are similar to those of*model 4*, but the latency to peak force increases as SL increases. This increasing delay results from the feedback of force on Ca^{2+} binding to troponin. Recall that this behavior is also evident in the responses of *model 2* (see Fig.8
*C*), which had feedback of force on Ca^{2+} binding to troponin as its major cooperative mechanism.

As the degree of feedback increases, the responses of*model 5* (Fig. 11,*C* and*D*) more closely resemble those of*model 2* (Figs.7
*C* and8
*C*). The most prominent feature is an increasing latency to peak force as SL increases, as was also seen for *model 2* (Fig.8
*C*). A more subtle effect is a slowing of the final relaxation phase at short SLs. This effect is indicated by the time constant of the final relaxation phase for SL = 1.7 μm. The time constant increases from 0.064 to 0.085 s when the degree of feedback is increased (compare Fig. 11,*B* and*D*). In general, slowing of the final relaxation phase occurs when the steady-state F-Ca relations have low apparent cooperativity at short SLs. For example, the trace at SL = 1.7 μm for *model 5* (Fig.6
*D*) indicates that there is small but nonnegligible force at [Ca^{2+}] ≈ 0.3 μm, compared with the same trace for *model 4* (Fig. 6
*B*) with higher apparent cooperativity. The low apparent cooperativity results in a tendency to maintain force production late in the twitches, when [Ca^{2+}] is slowly returning to diastolic levels. Such an effect is more apparent in the second version of *model 5* because it is less cooperative at short SLs and low [Ca^{2+}]. The two effects of feedback of force on Ca^{2+} binding to troponin described here (increased latency to peak force and decreased relaxation rate) are both counter to the effect observed in experimental data. Hence, the results with *model 5* suggest that there may be only a modest amount of feedback of force on Ca^{2+} binding to troponin.

## DISCUSSION

The responses of the five models described are closely tied to the cooperative mechanisms assumed for each model. Clearly, all the cooperative mechanisms are not equal in how they affected the steady-state and dynamic responses of the models. The following discussion will focus in more detail on these cooperative mechanisms and will also cover cross-bridge cycling and limitations of the models.

#### Cooperative mechanism 1.

*Cooperative mechanism 1* holds that the presence of strongly bound cross bridges increases the affinity of troponin for Ca^{2+}. This cooperativity is incorporated in *models 2* and *5*, and the effects are similar in both models. This mechanism of cooperativity increases Ca^{2+} sensitivity by increasing steepness in F-Ca relations, especially in the midlevel ranges of force (see Fig. 5
*C* and 6,*C* and*D*). Note, however, that this mechanism does not produce length-dependent changes in plateau force in F-Ca relations. In the simulations, Ca^{2+} can always be made large enough so that full activation is achieved with or without feedback. Therefore, other features are necessary to control plateau force (e.g., the sarcomere-overlap function in *model 2*).

During twitches, feedback on Ca^{2+}binding can produce the length-dependent increases in peak force (i.e., Fig. 7
*C*). However, such feedback also increases latency to peak force (see Figs.8
*C* and11
*D*). The mechanism of the increase in latency can be explained intuitively. We assume that thin-filament activation (i.e., making actin sites available for cross-bridge binding) is fast relative to cross-bridge cycling (this assumption is considered later). As [Ca^{2+}] rises during a transient, the thin filament becomes activated rapidly. Cross bridges then become strongly bound after a delay as governed by their own slower kinetics. After cross-bridge attachment, troponin affinity increases, producing more activation, which in turn allows more cross bridges to form at their slower attachment rate. As a consequence of this feedback, the rate of activation is slowed so that it becomes similar to that of cross-bridge cycling.

Several researchers have suggested that the feedback of cycling cross bridges on the troponin affinity for Ca^{2+} may play a role in prolongation of force at high force levels (2). The modeling results presented here agree with this suggestion in part. When feedback on troponin affinity is the major cooperative mechanism (*model 2*), there is no dramatic prolongation of force duration. In contrast, *model 5* produces a prolongation of force. Here, changes in troponin affinity appear to work synergistically with the other cooperative mechanisms to produce prolongation of force. Feedback of force on the affinity of troponin for Ca^{2+}, in conjunction with other cooperative mechanisms, may play a role in prolongation of the force transient, but this mechanism alone does not seem capable of producing significant prolongation of force.

An important finding of this study is that *cooperative mechanism 1* is not crucial to reproduce any of the experimental results. In *model 5*,*cooperative mechanism 1* is added, but only a small degree of feedback of force on the affinity of Ca^{2+} binding to troponin could be added before simulated results became inconsistent with experimental findings (i.e., latency to peak force increased too greatly with SL, see Fig. 11
*D*). These results suggest that feedback of force on Ca^{2+}binding may not play a major role in controlling force generation in cardiac muscle. This conclusion is surprising given that feedback of force on Ca^{2+} binding has been presumed to be an important cooperativity mechanism (2). Contrary to this finding, we note that not all researchers have reported that active contraction has a large effect on Ca^{2+} binding to troponin (see e.g., Ref. 47). Also, force generation generally produces small changes in the free Ca^{2+} transient (3,30). Such an observation may suggest little change in Ca^{2+} bound to troponin, especially because changes in bound Ca^{2+}should be amplified to produce a relatively larger change in free Ca^{2+} (20). The fraction of free intracellular Ca^{2+} is estimated to be only a small percentage of the much larger pool of bound Ca^{2+} (20, 47).

#### Cooperative mechanism 2.

*Cooperative mechanism 2* holds that cross-bridge binding increases the rate of formation of neighboring cross bridges and that multiple cross bridges can maintain activation of the thin filament. This mechanism is important for the prolongation of force at long SLs. For example, *model 4* only exhibited prolongation with multiple cross bridges per functional unit (see Figs. 9 and 10). Surprisingly, the*N1* state has little effect on prolongation. Intuitively, this feature should help to prolong force because cross bridges can remain attached even after Ca^{2+} has dissociated and activation levels have fallen. For example,*N1* is preferentially populated late in the twitch (see Fig. 9, *E* and*F*). However, the presence of*N1* alone does not appear to produce much prolongation at longer SLs. For example, twitch duration shows little prolongation in *models 1–3*, all of which have the*N1* state (see Fig. 8,*B–D*). In these models, the*N1* state is not preferentially populated or more delayed at long SLs, as is the case for*model 4*.

*Cooperative mechanism 2* does not appear to strongly influence the steepness of F-Ca relations. This is evident from the similarity of the F-Ca relations for the one- and three-cross-bridge versions of *model 4*(compare Fig. 6, *A* and*B*). The steepness of the F-Ca relations is most influenced by earlier activation events such as Ca^{2+} binding to troponin/tropomyosin shifting. Put another way, the regulatory proteins appear to be the important switch. Once the regulatory proteins are switched “on,” *cooperativity mechanism 2* influences the dynamics of cross-bridge formation. For example, in*model 4*, regulatory proteins affect the shift from nonpermissive (*Nx*) states to permissive (*Px*) states (see Fig. 3
*A*), which in turn controls the steepness of F-Ca relations. Once tropomyosin is in the permissive conformation, the second cooperativity mechanism controls the time-dependent distribution among states*P0* through*P3*, as shown in Fig. 10.

Intuitively, cooperativity in cross-bridge formation should not by itself lead to force generation. Consider a model in which activation occurs via a mechanism by which one cross bridge facilitates formation of neighboring cross bridges. Such a scheme may produce a realistic onset of force, but there is no mechanism to terminate force production. This implies that cooperativity between cross bridges is a “double-edged sword,” increasing force for a given level of activation but also decoupling force production from control by the regulatory proteins and, ultimately, Ca^{2+}. The manner in which the second type of cooperativity is incorporated into the structure of*models 4* and*5* reflects the need for ultimate control of force to lie within the regulatory proteins. In these models, the decline of Ca^{2+} drove tropomyosin to the nonpermissive states. Two or more strongly bound cross bridges can maintain the permissive conformation, prolonging force for a short while. However, tropomyosin eventually returns to nonpermissive states, and relaxation occurs.

#### Cooperative mechanism 3.

*Cooperative mechanism 3* is designed to simulate end-to-end interactions between adjacent troponin-tropomyosin units along the thin filament. This mechanism substantially affects the Ca^{2+} sensitivity and the apparent cooperativity in the steady-state F-Ca relations. Whereas there is a general consensus in the literature that increases in SL decrease the [Ca]_{50}, there is less certainty regarding the data for Hill coefficients. Several researchers have reported that Hill coefficients increase at longer SLs (13, 23,34), whereas others have reported that Hill coefficients show little change (36, 40, 41).

We designed *cooperative mechanism 3* to reproduce the former experimental results in which the Hill coefficient increases with SL (see Figs. 5
*D* and6). However, we also sought to replicate the latter experimental results by modifying *cooperative mechanism 3* so that *N* does not depend on SL (modification to *Eq. EA41
*in
). The modified model generates steady-state F-Ca relations with Hill coefficients that show little dependence on SL (data not shown). Otherwise, the modification produced only very minor changes in the twitch responses. Hence, the construction of *cooperative mechanism 3* allows for flexibility in the SL dependence of F-Ca relations so that the Hill coefficient can be either an increasing or a fixed function of SL.

The *cooperative mechanism 3* construct can produce a nearly constant time to peak force, independent of SL-dependent changes in peak magnitude. Such behavior is seen in the experimental data used here, as well as in other studies (6, 54). Alternatively, the constant time to peak force can be interpreted as proportionality between the peak force and the rate of force onset. Such proportionality appears to be an intrinsic property of cardiac muscle for isometric twitches. Experimental studies of interval-force relations showed virtually identical results for either peak force or the peak rate of force onset in isolated muscle (57, 58). Similarly, identical results are found for peak pressure or the peak rate of rise in pressure in whole ventricle (9).

In the models, an important assumption is necessary to reproduce the proportionality between the peak force and the rate of force onset. Recall that the construct of *cooperative mechanism 3* produces activation that is more rapid than the cross-bridge cycling rates during force onset. The experimental evidence for this assumption is as follows. Results from X-ray diffraction studies (37) in frog skeletal muscle suggest that thin-filament activation is rapidly produced by the addition of Ca^{2+}. Interestingly, the presence of activation did not require cross bridges, but activation could be maintained by attached cross bridges in the absence of Ca^{2+} (similar to*cooperative mechanism 2*). Other studies with fluorescence labeling of regulatory proteins also suggest that activation of the thin filament is rapid with rates on the order of ∼500 s^{−1} (18, 42, 53). Such high activation rates will produce a time constant for activation much smaller than that for strong binding of cross bridges. For the models developed in this paper, assumed rates are on the order of 10–40 s^{−1}, similar to some estimates from experimental and theoretical work (see below). Even lower estimates of cycling rates have been assumed elsewhere (e.g., Refs. 8 and 48) when measured ATPase rates were assumed to correspond to cross-bridge cycling.

#### Cross-bridge cycling rates.

Another unique feature of our modeling approach is the length dependence of cross-bridge detachment rate (see *Eq.11
*). There is some experimental evidence to support such a construct. Decreased lattice spacing can decrease the cross-bridge detachment rate, possibly by stabilizing the ADP-attached state or blocking the binding of Mg-ATP (32, 60). It has been suggested for some time that an increase in SL decreases the lattice spacing of muscle and may affect Ca^{2+}sensitivity (4, 41, 55). A change in lattice spacing could also likely change the orientation of myosin heads, which has been shown experimentally to greatly affect both velocity and force of cross-bridge interactions (29). A decrease in SL has also been shown to decrease the efficiency of force production, suggesting that the apparent cross-bridge off rate has increased (33). In contrast, a more recent study (56) did not show such a change in ATPase, suggesting that length-dependent changes in plateau force are not mediated by changes in cross-bridge off rate.

In the modeling work, the length dependence of cross-bridge detachment rate affects relaxation rates. Experimental evidence shows that relaxation rates are greater at shorter SLs than at longer SLs (31,51). No model that we tested could reproduce this behavior without changing cross-bridge off rates. This result makes some intuitive sense because late in the twitch, force depends more on slowly breaking cross bridges than on troponin/tropomyosin shifting or other activation events. For example, when experimentally measured twitches are plotted as loops in the F-Ca phase plane, the relaxation force generally lies above and to the left of the steady-state F-Ca relation (e.g., see phase-plane plots in Ref. 5). The Ca^{2+} dependence of relaxation rates suggests that cross-bridge kinetics, not the dissociation of Ca^{2+} from troponin, are rate limiting (45). Other experimental evidence derived from the breaking and reattaching of cross bridges also suggests that activation of the thin filament may decline more rapidly than force (48). These results suggest that attached cross bridges may produce the retention of force late in the twitch and imply that cross-bridge kinetics may be the most important factor affecting the final relaxation phase.

We hypothesize that the decreasing off rate at long SLs may also play an important role in producing graded changes in the plateau force in F-Ca relations. We have assumed that skeletal and cardiac myofilaments have similar sarcomere geometries, as suggested by some researchers (2). With this assumption, only *models 4* and *5* with SL-dependent detachment rates produce properly graded changes in plateau force. *Model 2* also produces properly graded changes, but this model incorporates a sarcomere-overlap function that is inconsistent with the classic literature on skeletal sarcomere geometry (19).

Although we assume a decreasing off rate as SL increases, other assumptions are also possible. The gradation of plateau force could also be produced by *1*) increased force per cross bridge, *2*) increased attachment rate, or *3*) a larger pool of cycling cross bridges as SL increases. In preliminary studies, these features were incorporated into models similar to those presented here. However, none of these models could reproduce the increased rate of relaxation as SL decreases (data not shown) that is seen in experimental data. In fact, an increase in attachment rate as SL increases has the opposite effect, slowing down relaxation as SL decreases.

One final issue on cross-bridge cycling involves the relationship to ATPase rates. There is a discrepancy between assumed cross-bridge attachment and detachment rates and experimentally measured ATPase rates. Our modeling results suggest that cross-bridge attachment and detachment rates on the order of 10–40 s^{−1} are required to correctly simulate the onset and relaxation rates seen in experimental twitch data. This corresponds to an ATPase turnover rate of ∼10 s^{−1}. This rate is in agreement with some data (55) but is much higher than other measurements in the same species (rat) and at a similar temperature (∼3 s^{−1}) (11,33).

Numerous modeling studies have demonstrated the need for fast cross-bridge cycling rates to reproduce dynamic responses. For example,*model 1*, with cross-bridge cycling rates derived from ATPase rates (∼1 s^{−1}), produces sluggish twitch responses. Likewise, the initial study by Landesberg and Sideman (38), focusing on simulating steady-state data, assumed slow cross-bridge kinetics derived from low ATPase rates (8). However, their subsequent model (39) (*model 2*in this paper), used to simulate twitches, assumed cycling rates that are about an order of magnitude higher, presumably to produce realistic twitch responses. Other modeling efforts to simulate rates of force redevelopment in cardiac muscle have also required cycling rates of ∼10 s^{−1} (10, 22).

If these cited low estimates of ATPase rates are correct, then there appears to be a discrepancy between the cycling rates derived from ATPase data and the cycling rates needed to replicate dynamic responses. The resolution could be that multiple interactions between actin and myosin occur per each ATP molecule hydrolyzed, as suggested by some data (24). Alternatively, the rapid dynamic responses could be produced by cross bridges losing force via a reversal of the force-generating step. As suggested by some authors (45), the backward strong-weak transition could be fast enough to account for the rapid relaxation rates measured experimentally.

#### Limitations of modeling approach.

The five models described are quite similar in approach and general design. The important differences between the models lie in the cooperative mechanisms that are employed. Thus the discussion above focused on how the cooperative mechanisms govern model response. Because of the large number of parameters even in these relatively simple models, an exhaustive search of the parameter space is not practical. However, the analysis has emphasized qualitative behaviors that are generic in that they are unaffected by small parameter changes. The following discussion describes some of the limitations of our modeling approach and how these may effect our conclusions.

In our modeling approach, the implementation of cooperative mechanisms is phenomenological to produce relatively low-order models that can reproduce experimental data. For example, *cooperative mechanism 3* is designed to capture essential features of end-to-end interactions of troponin and tropomyosin along the thin filament without explicitly modeling molecular interactions. Explicitly modeling these end-to-end interactions for a dynamic system requires a Monte Carlo approach (25) that would be much different from the other models considered. Whereas our simpler approach is clearly a compromise, we suggest that it still provides insight into a complex biological system. We note that representing actin-myosin interactions as *f*
_{app} and*g*
_{app} is often a useful compression of a system for which there is much fuller biochemical detail available. Likewise, we chose a fairly phenomenological formulation to describe the complicated end-to-end interactions thought to occur along the thin filament.

Similar to our findings with *cooperative mechanism 3*, we cannot claim that *cooperative mechanisms 1* and *2* are based on first principles, and we chose a simple two-state representation for cross-bridge cycling. Also, we have constructed models with functions that depend directly on SL, which is clearly a convenient simplification. A more mechanistic model would likely consider the effect of SL on lattice spacing, which has been shown to be an important determinant of Ca^{2+} sensitivity (41, 55). Although one generally prefers the highest level of detail possible, modeling all molecular events in cardiac muscle may not be feasible at this time. As discussed in the introduction, much controversy still exists as to the exact molecular mechanisms of cooperative activation and cross-bridge cycling. Even in the absence of full biomolecular detail, we believe that the models described here can help to illuminate the essential cooperative interactions required to reproduce experimental data for cardiac muscle under isometric conditions.

In conclusion, the responses of five models of activation and force generation are explored in this paper. *Model 1* fails in that the F-Ca relations are not steep enough and there is little change with SL. *Model 2* incorporates more cooperativity with the allowance of force generation to greatly increase the affinity of troponin for Ca^{2+}. This construct produces steeper F-Ca relations, but most of the steepness is in the midlevel [Ca^{2+}] range. During twitches, this type of cooperativity increases the latency to peak force at long SLs, an effect not seen in experimental data.*Model 3* is formulated to simulate end-to-end interactions along the thin filament as the major cooperative mechanism. This model produces steep F-Ca relations with SL-dependent Ca^{2+} sensitivity that are similar to relations of the experimental data. During twitches, this model produced SL-dependent increases in peak force without an increase in latency to peak. The main failure is that this model does not produce prolongation of twitches at long SLs.*Model 4* corrects this failure with the addition of multiple cross bridges that form cooperatively as well as assuming that two or more strongly bound cross bridges can hold tropomyosin in a permissive conformation. Also, *model 4* assumes that cross-bridge detachment rates decreased with SL. Together, these features act to prolong twitch duration and decrease final relaxation rates as SL increases. *Model 5* extends *model 4* with the addition of feedback of force on Ca^{2+} affinity of troponin, similar to that employed in *model 2*. However, such feedback can be added only to a limited degree before the dynamic behavior becomes inconsistent with experimental results.

## Acknowledgments

This work was supported by National Heart, Lung, and Blood Institute Grant HL-30552, National Science Foundation Grant BIR-9117874, and The Whitaker Foundation.

## Appendix

#### Model 1.

Rate constants are
Equation A1
Equation A2
Equation A3
Equation A4
Equation A5
Equation A6Normalized force is computed as
Equation A7where α is the sarcomere-overlap function as given in Fig.2
*A*,*P1* and*N1* are the fractions of functional units in the two states, respectively, and F_{max} is defined as
Equation A8

#### Model 2.

Rate constants are
Equation A9
Equation A10
Equation A11The dissociation rate of Ca^{2+} from troponin is a function of force
Equation A12where*K*
_{app} is a function of F as defined in *Eq. EA7
*. Note, however, that a different α must be used in*Eq. EA7
* because a different sarcomere-overlap function is assumed in *model 2* (see Fig. 2
*A*). The function *K*
_{app} is shown in Fig. 3 of Ref. 39. A third-order polynomial fit to this function (units in μM^{−1}) is used in this paper as follows
Equation A13

#### Model 3.

Rate constants are
Equation A14
Equation A15
Equation A16As in the previous models, the cross-bridge cycling rates are assumed to be fixed constants
Equation A17
Equation A18The rate of tropomyosin shifting from a nonpermissive to a permissive state is a function of both the fraction of troponin with Ca^{2+} bound (*T*
_{Ca}) and the SL
Equation A19where
Equation A20
Equation A21
Equation A22In*model 3*, the normalized force is computed in the same way as in *model 1*using *Eqs. EA7
* and *
EA8
*.

#### Model 4.

*Model 4* retains most of the structure of *model 3* but has multiple cross bridges that can form cooperatively. One-, two-, and three-cross-bridge versions of this model have been developed. Rate constants for formation for a one-cross bridge model are
Equation A23
Equation A24Rate constants for formation for a two-cross-bridge model are
Equation A25
Equation A26
Equation A27
Equation A28 Rate constants for formation for a three-cross-bridge model are
Equation A29
Equation A30
Equation A31
Equation A32
Equation A33
Equation A34The on rate *f* is unchanged from*model 3* (*Eq.EA17
*). The rate of cross-bridge dissociation is assumed to be a function of SL
Equation A35where SL_{norm} is unchanged from*Eq. EA22
* and*g** gives the minimum cross-bridge detachment rate. The minimum cross-bridge detachment rate changes with the number of cross bridges as
Equation A36
Equation A37
Equation A38The increased minimum off rate (*g**) is necessary to speed relaxation to counteract the prolongation caused by the multiple cross bridges.

In *model 4*, the variable off rate [*g*(SL)] decreases the plateau force in F-Ca relations as SL decreases. Plateau force is affected by two other features: *1*) sarcomere overlap (Fig. 2
*A*) and*2*) tropomyosin shifting (*Eq. EA19
*). The sarcomere-overlap function remains unchanged from *model 3*. The effects of the second feature, tropomyosin shifting, are decreased in *model* 4 by changing troponin binding rate constants to
Equation A39
Equation A40This modification produces gradation of the plateau force in F-Ca relations similar to that of experimental results. One final change is that the Hill coefficient of tropomyosin shifting is modified to be slightly less cooperative (compare with *Eq.EA21
*)
Equation A41In*model 4*, normalized force is computed by weighing the relative contribution of the force-generating states as
Equation A42where α is the sarcomere-overlap function as given in Fig.2
*A* and*P1*,*N1*,*P2*, and*P3* are the fractions of functional units in the respective force-generating states. F_{max} is calculated as
Equation A43These terms for maximum values are found using the King-Altman rule (see Ref.35) as follows
Equation A44
Equation A45
Equation A46where
Equation A47
The off rates are shown with an asterisk to indicate that they are at their minimum values corresponding to the longest SL (see*Eq. EA35
*).

#### Model 5.

*Model 5* differs from the three-cross bridge version of *model 4* in that the off rate of Ca^{2+} from troponin is a function of the total force developed. In the first version, there is a modest degree of feedback on the off rate, as given by
Equation A48A second version of the model has stronger feedback on binding Ca^{2+} to troponin, as given by
Equation A49where*k*
_{off} is unchanged from before (see *Eq. EA40
*) and F is calculated as in *model 4*(*Eqs. EA42-EA47
*). Because this modification increases cooperativity, there is a modification to make the tropomyosin shifting less sensitive to [Ca^{2+}]. In the first version, which has a modest degree of feedback, decreased sensitivity is incorporated by modifying *Eqs.EA20-EA21
* to become
Equation A50
Equation A51where SL_{norm} is as defined in*Eq. EA22
*. In the second version, which has stronger feedback, the cooperativity of tropomyosin shifting is further reduced by replacing *Eq. EA51
*with
Equation A52

## Footnotes

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