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A Quantitative Analysis of Cardiac Myocyte Relaxation: A Simulation Study

Bioengineering Institute and Department of Engineering Science, The University of Auckland, Auckland, New Zealand

Correspondence: Address reprint requests to S. A. Niederer, E-mail: s.niederer{at}auckland.ac.nz.

	ABSTRACT

TOP ABSTRACT INTRODUCTION TROPONIN TROPOMYOSIN TENSION DEVELOPMENT MODEL SUMMARY VALIDATION RELAXATION FACTORIAL ANALYSIS DISCUSSION SUMMARY ACKNOWLEDGEMENTS REFERENCES

 
The determinants of relaxation in cardiac muscle are poorly understood, yet compromised relaxation accompanies various pathologies and impaired pump function. In this study, we develop a model of active contraction to elucidate the relative importance of the [Ca²⁺]_i transient magnitude, the unbinding of Ca²⁺ from troponin C (TnC), and the length-dependence of tension and Ca²⁺ sensitivity on relaxation. Using the framework proposed by one of our researchers, we extensively reviewed experimental literature, to quantitatively characterize the binding of Ca²⁺ to TnC, the kinetics of tropomyosin, the availability of binding sites, and the kinetics of crossbridge binding after perturbations in sarcomere length. Model parameters were determined from multiple experimental results and modalities (skinned and intact preparations) and model results were validated against data from length step, caged Ca²⁺, isometric twitches, and the half-time to relaxation with increasing sarcomere length experiments. A factorial analysis found that the [Ca²⁺]_i transient and the unbinding of Ca²⁺ from TnC were the primary determinants of relaxation, with a fivefold greater effect than that of length-dependent maximum tension and twice the effect of tension-dependent binding of Ca²⁺ to TnC and length-dependent Ca²⁺ sensitivity. The affects of the [Ca²⁺]_i transient and the unbinding rate of Ca²⁺ from TnC were tightly coupled with the effect of increasing either factor, depending on the reference [Ca²⁺]_i transient and unbinding rate.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION TROPONIN TROPOMYOSIN TENSION DEVELOPMENT MODEL SUMMARY VALIDATION RELAXATION FACTORIAL ANALYSIS DISCUSSION SUMMARY ACKNOWLEDGEMENTS REFERENCES

 
With each beat, the heart pumps blood around the body. The cyclical activation and relaxation of tension that takes place occurs at the sarcomere spatial scale and is controlled by cellular mechanisms. Each sarcomere is made up of interdigitated protein filaments of actin and myosin. Crossbridges protruding from myosin bind to actin, whereupon they undergo a conformational change, causing the bound crossbridges to pull each filament in opposite directions, producing tension. The process is controlled by both the local free Ca²⁺ and the intrinsic properties of the sarcomeres themselves.

Contraction is initiated by an increase in local [Ca²⁺]_i. Ca²⁺ binds to troponin C (TnC) and the resulting shift of tropomyosin reveals the actin binding sites, allowing crossbridges to bind and generate tension. After the removal of [Ca²⁺]_i, bound Ca²⁺ unbinds from TnC, tropomyosin blocks the actin binding sites, crossbridges detach, and tension returns to zero. The above processes producing the initiation of contraction in cardiac muscle are extensively quantified; however, the equally important mechanisms governing relaxation after contraction are poorly characterized. Thus although the steps are known in the process of relaxation, what controls each step and which step is the most important is unknown.

Relaxation is often quantified by the half-time to relaxation (RT₅₀ the time for tension to decay by 50% from the peak value). RT₅₀ is determined by both the [Ca²⁺]_i transient and the intrinsic properties of the myofilaments. The [Ca²⁺]_i transient influences RT₅₀ when altered pharmacologically (1,2) or by increasing the stimulation frequency (3). The myofilament properties implicated as determinants of RT₅₀ include the tension-dependent binding of Ca²⁺ to TnC (4), inhomogeneous sarcomere shortening (5), crossbridges inhibiting tropomyosin returning to its resting state (4), phosphorylation of troponin I (6), and sarcomere length (SL) (7). Unraveling the relative influences of each mechanism has proven challenging experimentally. In this study, we address this issue by unifying experimental data into a mathematical model to analyze the significant factors determining relaxation.

Building on the successful approach of cardiac electrophysiology models, cardiac contraction models are now beginning to quantify numerous phenomena, which have previously been poorly defined or only understood and characterized in isolation. Models of cardiac contraction have quantified cooperative mechanisms (8), the effects of contraction in the forward problem of electrocardiography (9), and ventricular pacing on contraction (10), for example. However, the parameters used in active contraction models are often derived from limited sets of experimental results. Here, each parameter was rationalized from numerous sources, and where possible, multiple experimental modalities, through an extensive review of the literature. The sources of each parameter and a brief description of the experimental conditions under which it was obtained are provided in the Tables.

The model is depicted in Fig. 1 and was based on the framework proposed by Hunter et al. (11). The equations and the parameters are described in the following stages:

1. The kinetics of Ca²⁺ binding to TnC in the absence, and then presence, of tension was defined using steady-state and transient experimental data.
   
2. The shift in tropomyosin (Tm) to reveal the actin binding sites resulting from Ca²⁺ binding to TnC was characterized using light-activated Ca²⁺ chelator experiments and force-Ca²⁺ (F-pCa) curves.
   
3. Active tension was defined by the product of the available actin sites, the maximum isometric tension, and a sarcomere velocity-dependent scalar. Available actin sites were calculated from Tm kinetics. Maximum isometric tension was described by a linear function of SL, with parameters for this component defined by the maximum velocity, rapid length step, and sinusoidal perturbation experimental results.
   

View larger version (11K): [in this window] [in a new window]   FIGURE 1  Flow diagram depicting the relationships of the active contraction framework proposed by Hunter et al. (11). The model is driven by SL and sarcomere velocity, and intracellular [Ca2+]i. Inputs are in bold, algebraic length dependencies are in italics, processes described by differential equations are standard font.

	TROPONIN

TOP ABSTRACT INTRODUCTION TROPONIN TROPOMYOSIN TENSION DEVELOPMENT MODEL SUMMARY VALIDATION RELAXATION FACTORIAL ANALYSIS DISCUSSION SUMMARY ACKNOWLEDGEMENTS REFERENCES

 
Steady-state Ca²⁺ binding to troponin
The ternary cardiac troponin complex (Tn) consists of three subunits: Troponin I, T, and C. TnC contains the regulatory Ca²⁺ binding site, where the binding of Ca²⁺ initiates contraction. Troponin I (TnI) inhibits actin-myosin interaction. Troponin T (TnT) plays a structural role binding to TnC, TnI, and Tropomyosin (Tm) (12). TnC consists of N- and C-terminal globular lobes and is connected by a long central helix. The C-terminal contains binding sites III and IV, which bind Ca²⁺ and magnesium competitively. Under physiological conditions, both sites III and IV are saturated. The N-terminal contains Ca²⁺specific or low-affinity binding sites I and II. In skeletal TnC, both sites are active; but in cardiac TnC, only site II is active, due to an increased positive charge near site I prohibiting the binding of Ca²⁺ (13). In cardiac TnC, site II regulates muscle contraction and is the focus of a large number of studies, due to its potential as a target for Ca²⁺sensitizing drugs and its fundamental role in excitation-contraction coupling.

Steady-state Ca²⁺ binding to TnC can be described by a Hill equation with a Hill coefficient of 1 (Eq. 1) (14). Defining [Ca²⁺]_Trpn as the concentration of Ca²⁺ bound to TnC site II, [Ca²⁺]_TrpnMax, as the maximum concentration of ions that can bind to site II, [Ca⁺²]_i is the concentration of free Ca²⁺ and K is the tension-dependent affinity of Ca²⁺ for TnC. K was determined initially from experimental results with zero or minimal tension (T) and the tension dependence is considered below. [Ca²⁺]_TrpnMax was set to 70 µM (15,16):

(1)

Experimental measurements of Ca²⁺ affinity to site II are performed on a range of species and Tn subunits, under varying chemical conditions at different temperatures (see Table 1). The combinations of Tn subunits, Tm, and actin fundamentally affect the Ca²⁺ affinity of site II, as shown in Fig. 2. TnC in isolation has an affinity of 1 µM (13,17–25). TnC-TnI, Tn, and Tn-Tm have an affinity of 0.1 µM (13,21–23,26,27) Tn-Tm-Actin and skinned fibers have an affinity of 1 µM (22,27–31). However, outliers do exist in the literature: Fuchs and co-workers (32–34) did not differentiate between sites III and IV and site II affinities. Li and co-workers (18,24) measured affinities of 2.5 µM and 20 µM with whole TnC and the N-terminal of TnC containing site II, respectively, and found no evidence to rationalize the significant variation. Ball et al. (35) measured a higher affinity of Ca²⁺ to TnC, yet there does not appear to be a reason for this discrepancy. The effect of magnesium is varied between experiments, with magnesium having both minimal (13,23,36,37) and significant (25,29) effects; this may be due to differences in muscle or species types, as a significant difference is seen between rabbit skeletal and porcine cardiac muscle (29). Temperature, however, has only a minimal effect (17,30). Fuchs and co-workers (32–34,38–44) have shown that Ca²⁺ binding to Tn is dependent on active tension. In skinned preparations, bound crossbridges may play a role in determining the binding affinity. However, the majority of K-values were measured at low [Ca²⁺] (<2.5 µM) and so tension was assumed to be minimal (45,46). Ca²⁺ affinity values for Tn bound to actin lie between 0.83 µM (28) and 5 µM (22) using scintillation counting and IAANS florescence with cysteine (Cys) 35 in place, respectively. It has been suggested that IAANS results where the Cys amino acid located at residue 35 has been removed are more accurate than when Cys-35 is present (22). IAANS results with Cys-35 removed record affinities of 1.6–2.3 µM (22,29) and scintillation counting recorded affinities of 2.5 µM (27) and 2.0 µM (47) for TnC in whole fibers; therefore the binding affinity of TnC (K) contained in whole fibers was set to be 2 µM in the absence of tension. This value was used in the model and the allosteric affects, if any, of magnesium binding were assumed to be minimal.

View larger version (10K): [in this window] [in a new window]   FIGURE 2  Affinity of Ca2+ to TnC contained in various components of mammalian cardiac thin filaments, from studies listed in Table 1. The plus-symbol (+) is scintillation counting; is IAANS (Cys-35, Cys-84); x is IAANS (Cys-35); is F27W; and is MRI spectroscopy.

(2)

Large variations are seen in the reported unbinding rates of Ca²⁺ from site II (k_off). Temperature does not appear to have a significant affect on the unbinding rate. The rate lies between 1.3 s^–1 and 750 s^–1 at 4°C, 17 s^–1, and 1200 s^–1 at 15°C, and 13 s^–1 and 900 s^–1 at room temperature. It is important to again note that there is significant variation in the binding affinity (K) of Ca²⁺ for TnC for different combinations of Tn subunits, Tm, and actin, as outlined in the above section (see Fig. 2). The variation in K will be reflected in the kinetics by a change in the ratio of unbinding and binding rates with different combinations of Tn complexes. Unbinding rates for TnC bound to Tn-Tm, Tn-TnI, and Tn are in the order of 10 s^–1 (47–49), which coincide with the higher reported affinities for the same Tn complexes (center column of Fig. 2). TnC in isolation has varied unbinding rates between 11 s^–1 and 5000 s^–1 and a lower binding affinity (illustrated in the right column of Fig. 2). The majority of binding rates lie within a factor of 2 of 100 µM^–1 s^–1 (20,25,48,50–53) (see Table 2) with no apparent variation between temperatures or combinations of Tn subunits and Tm. Binding and unbinding rates have not been measured in whole fiber preparations. To determine the rates in whole fibers, it was assumed that it was primarily the unbinding rate, and not the binding rate, which changes for different TnC complexes, resulting in the changes in binding affinity (K) reported above. This assumption is consistent with the hypothesis that binding of Ca²⁺ to site II is diffusion-limited (54). The binding rate was set to 100 µM^–1 s^–1, and using the affinity of Ca²⁺ for TnC derived above from Table 1 of 2 µM, the unbinding rate for site II of TnC contained in Tn-Tm-actin was k_off = K · k_on= 200 s^–1.

(3)

where k_refoff is the unbinding rate in the absence of tension, is a measure of the affect of tension on the unbinding rate, T is the active tension, and T_ref is the reference tension (described below). The value is not measured directly experimentally but can be calculated using results from Ca²⁺ affinity and length-step experiments in skinned and intact preparations, respectively. As now discussed, calculating in both intact and skinned preparations confirms that the tension-dependent binding of Ca²⁺ to TnC is not affected by the skinning process.

Fuchs and co-workers have shown that the concentration of bound Ca²⁺ is both tension- (33,40,44) and length-dependent (32,38,40,42,44) in skinned preparations. However, it is likely that these two dependencies are related by the number of attached crossbridges (55), which increases with length (59) and has been shown to increase the binding affinity of Ca²⁺ for TnC (47). The tension dependence of Ca²⁺ binding to TnC is consistently supported by the results within individual experiments. Comparing results between experiments, however, reveals inconsistencies. In the absence of bound crossbridges due to the addition of vanadate, between 75% (33,40) and 50% (44) of site IIs are occupied at pCa = 5. It was then expected that, if tension were the only factor determining the binding of Ca²⁺ to TnC, then at pCa = 5, no less than 50–75% of sites should be occupied in the presence of tension, yet measurements of 35% (32), 49% (44), and 69% (42) of site IIs occupied at pCa 5 in the presence of tension have been reported. The variation can be rationalized by three potential mechanisms. Firstly, that some mechanism other than bound crossbridges affects affinity. Secondly, variations between preparations affected the results. Thirdly, the method used here to calculate the number of ions bound to site II introduces or increases variation in the measurements. Fuchs and co-workers measured the total concentration of Ca²⁺ bound to all three sites of TnC. As sites III and IV are known to have a higher affinity than site II it was assumed that they are both saturated at pCa 5. Therefore the fraction of site IIs occupied by Ca²⁺ is equal to the total number of ions bound per TnC molecule less two. As a result, the fraction of site IIs occupied by Ca²⁺ is sensitive to experimental noise. If, on average, a total of 2.8 ions are bound to TnC at pCa = 5, then a variation of 3.6% (32) in the total number of ions bound to TnC corresponds to a variation of 0.1 ions. Subtracting the two ions bound to the saturated sites III and IV from the total number of ions bound (2.8 ions) means that there is a 0.1 ion variation in the remaining 0.8 ions bound to site II, which corresponds to a 12.5% variation in the number of ions bound to site II. This amplification of the experimental noise potentially explains the variation in experimental results observed. The affinity of Ca²⁺ for TnC in the absence of tension derived from Table 1 was 2 µM, which corresponds to 83% of site IIs being occupied at pCa 5 in the absence of tension, comparable with measurements of 75% (33,40). The value was determined using the K-value in the absence of tension defined above and a subset of results from Table 3. Results from experiments where Dextran or Vanadate were added, when the average SL was outside the physiological range of 1.8–2.3 µm or when the fraction of bound Ca²⁺ is <83% at pCa 5 (the value defined by Eq. 1 at zero tension), were excluded from the subset. The final subset took results from Fuchs and co-workers (32,33,38,42,44); was determined for each measurement, and the average -value was 1.9.

(4)

Equations 5 and 6 define the concentration of bound Ca²⁺ before and after the length-step change, respectively, using Eq. 1, and the Ca²⁺ affinity is an unknown function of tension K(T),

(5)

(6)

Combining Eqs. 5 and 6, and assuming [Ca²⁺]_TrpnMax >> [Ca²⁺]_i (56),

(7)

Now the Ca²⁺ tension relationship defined in Eq. 4 can be used to transform Eq. 7 from a Ca²⁺-dependence of K to a tension-dependence of K. Using Eq. 4 proposed by Komukai and K(T = 0) is equal to the affinity of Ca²⁺ to site II in the absence of tension (K):

(8)

Setting Eq. 8 equal to Eq. 3 divided by k_on, is equal to 2.6, using T_ref = 56.2 kPa. The calculated -value of 1.9 from Fuchs and co-workers for skinned preparations is close to calculated -values in length-step experiments in intact preparations and suggests that the skinning process has a minimal affect on the tension dependence of Ca²⁺ binding to TnC.

The Ca²⁺ affinity for TnC has also been shown to vary with SL (32,40,44). In length-step experiments, the affinity of TnC drops significantly but the SL remains largely unchanged. The -value required to capture this phenomenon is close to the -value required to model the change in calcium affinity at varying fixed SL values. Hence, the SL dependence of the affinity is accounted for by the tension dependence, as maximum tension and tension-based Ca²⁺ sensitivity increase with increasing SL (discussed below). This hypothesis coincides with experiments, where crossbridge heads (myosin subfragment 1) bound to actin in the absence of myosin or any reference length increased the binding affinity of Ca²⁺ for TnC (47). In the model, the length dependence is accounted for by the tension dependence and the form of the equation is validated by Komukai's results. Considering these results will be set to 2.

	TROPOMYOSIN

TOP ABSTRACT INTRODUCTION TROPONIN TROPOMYOSIN TENSION DEVELOPMENT MODEL SUMMARY VALIDATION RELAXATION FACTORIAL ANALYSIS DISCUSSION SUMMARY ACKNOWLEDGEMENTS REFERENCES

 
Tropomyosin is a highly extended -helical coil situated in the actin groove, with each tropomyosin molecule spanning seven actin monomers (27). When tropomyosin is shifted out of the actin groove, the steric hindrance preventing actin binding to myosin is removed, allowing tension to develop. In the model, tropomyosin was characterized by z, the fraction of actin sites available for crossbridge binding. In this study, it is assumed that crossbridges bind rapidly relative to thin filament kinetics and that not all actin sites are available at full activation. Thus, tension is proportional to z and the ratio of z to the fraction of actin sites available at full activation for a given SL (z/z_Max) is equal to the ratio of the isometric tension to the maximum tension at full activation for the same SL (T₀/T_0Max). The value z is defined by Eq. 9, below. The fraction of actin sites available at full activation (z_Max) is defined by Eq. 14, the steady-state solution to Eq. 9 at full activation ([Ca²⁺]_Trpn = [Ca²⁺]_TrpnMax). The value T₀ is the isometric tension at a given [Ca²⁺]_i and SL (see Eq. 16). The value T_0Max is the isometric tension at full activation for a given SL (see Eq. 15).

(9)

where the relaxation kinetics in Eq. 9 are described by _r1 and _r2, K_z and n_r, which correspond to the slow and fast relaxation rates, respectively, observed in light-activated Ca²⁺ chelator experiments. The tension transients produced in light-activated Ca²⁺ chelator experiments are potentially defined by three mechanisms—the [Ca²⁺]_i bound to TnC, crossbridge kinetics, or the intrinsic properties of tropomyosin. The rate that tension decreases is significantly slower than the rate that calcium disassociates from TnC and, as such, the rate of relaxation is unlikely to be defined by [Ca²⁺]_i bound to TnC. The rate that tension decreases after a step decrease in Ca²⁺ is similar between species (rat (60–63) and guinea pig (5,64,65)) exhibiting different myosin isoforms, suggesting that crossbred kinetics do not define relaxation rates. Palmer and Kentish observed significant differences between guinea pig and rat preparation relaxation rates of 16.1 s^–1 and 2.99 s^–1, respectively. However, their results are inconsistent with other experimental observations, which have consistently reported guinea pig relaxation rates of 10 s^–1 and above (5,64,65). As such, the intrinsic properties of tropomyosin are assumed to be defined by the tension transients after step decreases in calcium. The length-dependent activation kinetics are described by [Ca²⁺]_Trpn50, ₀, and n. The value [Ca²⁺]_Trpn50 is the Ca²⁺ bound to TnC at half-activation and was derived from the free Ca²⁺ at half-activation. The value ₀ describes the monoexponential activation rate seen in caged Ca²⁺ experiments. The value n is analogous to the Hill coefficient in the steady-state force Ca²⁺ curve (F-pCa) and provides a phenomenological representation of the high cooperativity, due to nearest-neighbor interactions between tropomyosin and/or crossbridges, seen in the activation of tension in cardiac muscle.

Tropomyosin kinetics are described in four stages, which were cyclically iterated through. Briefly, first, the relaxation parameters (_r1, _r2, K_z, and n_r) are defined using step changes in Ca²⁺ experiments. Secondly, length-dependent activation ([Ca²⁺]_Trpn50) is defined using half-activation values from skinned preparations and the resulting equation is scaled to match intact preparation data. Thirdly, ₀ and n are defined for skinned and intact preparations by fitting the steady-state solution of Eq. 9 to the respective F-pCa curves. Finally, z_Max—the fraction of actin sites available at maximum activation—is calculated using the steady-state solution to Eq. 9.

Relaxation parameters
The relaxation kinetics described by Eq. 9 propose two stages for relaxation, as seen experimentally. It was found that a linear component involving _r1 characterized the slow process and a nonlinear component in the form of a Hill relation was required to model the fast component. Using the combined linear and nonlinear off-rates, the biphasic nature of relaxation was captured. To compare model simulations with experimental results, the relative tension T₀/T_0Max (or equivalently, T/T₀ in experimental nomenclature) was calculated using T₀/T_0Max= z/z_Max, where z_Max is defined below by Eq. 14.

The relaxation components of Eq. 9 were fitted using the tension transient after a step decrease in free Ca²⁺ using the light-activated Ca²⁺ chelator diazo-2. Ca²⁺ disassociates rapidly from TnC after a step decrease in and was therefore expected to have a minimal affect, such that the relaxation kinetics of tropomyosin solely determine the tension transient. In Ca²⁺ step experiments the muscle is often removed from the bathing solution and exposed to a pulse of light, which greatly increases the affinity of diazo-2 for Ca²⁺, causing a reduction of free Ca²⁺ on a millisecond timescale. Data from rat and guinea pig preparations were used with both similarities and dissimilarities between species being observed. Experiments are commonly performed in air at the due temperature to reduce the affects of evaporation or condensation. As such, most experiments are performed at 12–15°C with the exception of results from Kentish and Palmer (63,66), where the muscle was kept in the bathing solution at 20–22°C. Results from diazo-2 experiments are summarized in Table 4. A biphasic tension transient is observed in most experiments, which is fitted with two exponentials. The rates are 10–12 s^–1 and 2–4 s^–1, respectively, at 12–15°C for rat and guinea pig (60,62,64,65); and 16–18 s^–1 and 1 s^–1 at 22°C for rat in two other studies, respectively (63,66). The sole anomaly was recorded by Saeki et al. (61), who reported a fast transient of 73.5 s^–1, six times larger than any other experiment. The subsequent article by the same group using similar methods did not record the higher transient rate, and made no reference to their earlier results (60). The tension transients recorded by Simnett et al. (65) at 20°C had a half-time to relaxation of 53.4 ms, approximately the same as the control rat measurements at 15°C from Fitzimons et al. (62), which had a half-time to relaxation of 64.5 ms. However, removing the muscle from the bath would result in a drop in temperature of 2–3°C (65), and during activation, ADP and P_i can build up in preparations removed from the bath (63), both of which would affect relaxation. Palmer and Kentish (63,66) recorded relaxation times using rat trabeculae contained in the bathing solution, reducing any buildup of P_i or ADP and attaining data at 22°C, but characteristic tension traces published by Palmer and Kentish had a maximum tension of 0.3 T₀. The relaxation parameters were chosen to fit experimental results from Saeki et al. (60) and Simnett et al. (65) at 12–15°C, since an accurate fit to all data was not possible with limited information on SL (6,64), relative amplitudes of fast and slow processes (62,66), and initial tension (63). Fig. 3 shows results from Saeki et al. (60) (points) and model simulations (lines) with _r1, _r2, K_z, and n_r equal to 2 s^–1, 1.75 s^–1, 0.15, and 3, respectively.

View larger version (14K): [in this window] [in a new window]   FIGURE 3  Relaxation after activation of Ca2+ chelator; data points from Saeki et al. (60), and lines are from model simulations. The data points and dashed line, data points and dotted line, and data points and solid line had initial relative tensions of 0.5, 0.451, and 0.0925, respectively, and the pCa values after the activation of the calcium chelator were 5.78, 5.92, and 6.3, respectively. The pCa values were calculated using the final relative tensions and the skinned F-pCa curve defined below with a Hill coefficient of n = 3 and half-activation value of pCa50 = 5.6. SL = 2.2 µm.

F-pCa curves were used to define steady-state z in the solution of Eq. 9 using the commonly fitted experimental relationship in Eq. 10. The F-pCa relationship was described by a Hill curve with half-activation [Ca²⁺]₅₀ and Hill coefficient nH. These two variables are commonly recorded experimentally (see Table 5),

(10)

The activation parameters in Eq. 9 were defined using F-pCa curves in two parts. First, the length-dependent activation was defined by determining the length dependence of [Ca²⁺]_Trpn50, which was calculated from the length-dependent [Ca²⁺]₅₀ values defined in Eq. 10 from skinned and intact preparations by rearranging the equations outlined above. Secondly, ₀ and n were determined by fitting the steady-state solution to Eq. 9 to F-pCa curves. The F-pCa curves were defined by the half-activation values used to calculate [Ca²⁺]_Trpn50 and Hill coefficients for skinned and intact preparations.

[Ca²⁺]_Trpn50 is defined by Eq. 11, derived from Eq. 15 and Eq. 2, as a function of [Ca²⁺]₅₀ values and strain. Eq. 15 is defined below, and describes the length dependence of isometric tension. Here it was used to define the ratio between T/T_ref from Eq. 2 as a function of strain:

(11)

[Ca²⁺]₅₀ values contained in Eqs. 11 and 10 were defined as a linear function of strain (see Eq. 12). Fig. 4 plots [Ca²⁺]₅₀ values for measurements taken under physiological conditions at room temperature in Table 5 against strain. The length-dependent Ca²⁺ sensitivity was defined in terms of a linear fit to the Ca₅₀ values. Data was taken predominantly from rats at room temperature under physiological conditions (45,66,77–84). Data was excluded on the grounds of extreme (SL < 1.8 µm, SL > 2.3 µm) or unstated SLs (57,77,85), low levels of ATP (<4 mM) (80), low ionic strength (86,87) (<180 mM), and low or unstated temperatures (46,79,88) (<20°C). Intact muscle experiments (2,3,89) are considered separately below. The relationship between [Ca²⁺]₅₀ and strain is described by Eq. 12 with [Ca²⁺]_50ref = 4.72 µM and ß₁ = –4.0 in skinned preparations,

(12)

View larger version (10K): [in this window] [in a new window]   FIGURE 4  Calcium value at half-activation as a function of strain with resting SL = 2 µm. (Solid line) Points fitted to [Ca2+]50 = 4.72(1–4.0(–1)).

First, due to intracellular proteins and ions, the spatially averaged Ca²⁺ concentration measured in fluorescence studies does not represent the true concentration of Ca²⁺ in the vicinity of the myofibrils and therefore the concentration of Ca²⁺ at the myofibrils is 2–10 times larger at half-activation (89,90). In this case, the concentration of Ca²⁺ presented to TnC can be scaled by a factor of 2–10, relative to the spatially averaged measures.

Second, the skinning and storage process may reduce the concentration of TnC (91), change the chemical environment (90), or alter the degree of phosphorylation of TnI (92) resulting in a change in the affinity of Ca²⁺ to TnC. This can be represented by increasing the affinity of Ca²⁺ to TnC.

Third, skinning, storage, and a change in chemical environment may cause changes to myosin, actin, tropomyosin, or titin, and the concentration of Ca²⁺ bound to TnC required to cause half-activation may be significantly less in intact preparations and the cooperative mechanisms represented by the Hill coefficient may be degraded (93,94).

Assuming that the affects of skinning on TnC does not alter the capacity of fluorescence data to represent a scalable weighted average of bound Ca²⁺, free Ca²⁺ in the vicinity of the myofilaments is independent of skinning, assuming the myofilament charge remains unchanged. Then most of the observed changes in Ca²⁺ sensitivity will be a result of the protein–protein interactions after binding of Ca²⁺ to TnC. Hence the affect of skinning can be represented by changing the [Ca²⁺]_50ref. Three studies on intact rat preparations with SLs of 2.2 µm calculated the pCa₅₀ value to be 6.2 (2,89,95), which compares with a value of 5.6 for skinned preparations (71,78,82,84,88). [Ca²⁺]_50ref for intact preparations was scaled to account for the variation between intact and skinned preparations, resulting in a [Ca²⁺]_50ref value of 1.05 µM in intact preparations. This definition of [Ca²⁺]_50ref takes into consideration the affects of changes in the chemical environment or degradation of the myofilaments. Insufficient data was available to accurately characterize the length dependence of activation in intact cells, and as a result, ß₁ was assumed independent of the skinning process.

Steady-state F-pCa curve
F-pCa curves were defined by [Ca²⁺]₅₀ and the Hill coefficient. The pCa₅₀ values defined above at = 1.1 were 5.6 and 6.2 for skinned and intact preparations, respectively. The Hill coefficient has been shown to be both length-dependent (45) and independent (46,73) experimentally. Recent studies by Dobesh et al. (46) showed no length dependence of the Hill coefficient when SL was accurately controlled during contraction. Kentish et al. (45) reported a strong length-dependence of the Hill coefficient, but also reported significant sarcomere shortening during contraction, which potentially distorted their results. Here, we assumed that the Hill coefficient is independent of length. A Hill coefficient of 5 was taken from the F-pCa curve for = 1.1 observed by Gao et al. (89), Backx et al. (2), and You et al. (96) for intact rat preparations at room temperature. In skinned preparations, Hill coefficients values range between 1.9 and 10.6. At room temperature, for = 1.1, the Hill coefficient was 3. It is possible that these curves underestimate the Hill coefficient due to uncontrolled sarcomere shortening as proposed by Dobesh et al. (46), but Dobesh used skinned preparations at 15°C and it is hard to quantify the effect these conditions would have on the Hill coefficients.

The values ₀ and n were found by fitting the steady-state solution of Eq. 9 to F-pCa curves using Eq. 10. The steady-state solution of Eq. 9 was solved iteratively as a function of bound Ca²⁺. The bound Ca²⁺ was calculated using tension from F-pCa curves defined by half-activation values and Hill coefficients defined above. The resulting values for ₀ and n were 6 s^–1 and 3.4 and 8 s^–1 and 3 for skinned and intact preparations, respectively. Fig. 5 shows the fitted F-pCa and idealized F-pCa curves at = 1.1. The dashed and dash-dot lines were calculated from pCa₅₀ values 5.6 and 6.2 and Hill coefficients 5 and 3 for skinned and intact preparations, defined above. Solid and dotted lines were generated by the model using the skinned and intact parameter sets.

View larger version (23K): [in this window] [in a new window]   FIGURE 5  Relative steady-state tension with respect to free pCa value. The dashed and dot-dashed lines are calculated from Eq. 10, with pCa50 values 5.6 and 6.2 and Hill coefficients 5 and 3 for skinned and intact preparations, respectively, at = 1.1. Solid and dotted lines are generated by the model using the steady-state solution to Eq. 9, with the intact and skinned parameters, respectively.

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The value z_Max is then defined by solving the linearized Eq. 9 with dz/dt = 0 and [Ca²⁺]_Trpn = [Ca²⁺]_TrpnMax, giving

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	TENSION DEVELOPMENT

TOP ABSTRACT INTRODUCTION TROPONIN TROPOMYOSIN TENSION DEVELOPMENT MODEL SUMMARY VALIDATION RELAXATION FACTORIAL ANALYSIS DISCUSSION SUMMARY ACKNOWLEDGEMENTS REFERENCES

 
Isometric tension
Isometric tension is described in two parts. First, the maximum tension at full activation for a given SL (T_0Max) is defined using experimentally reported tensions at pCa 4. Secondly, the isometric tension as a function of SL and the fraction of available sites is defined by combining T₀/T_0Max = z/z_max and Eq. 15. The maximum steady-state isometric tension were defined solely by SL, as at maximal activation it was assumed that thin filament kinetics would play a minimal role. During SL increases, neither crossbridge kinetics (82) or the force generated per bound crossbridge (59) are altered. Gao et al. (89) reported that isometric tension remained the same before and after skinning, whereas interfilament spacing has been shown to increase in skinned cells compared to intact cells (97), indicating that interfilament spacing does not affect the length dependence of isometric tension. The higher maximum tensions at longer SL can potentially be explained by an increase in the number of crossbridges that can attach at longer SL due to a decrease in the double overlap between actin filaments (76,98–100). The value T_0Max is defined by Eq. 15, where is equal to SL over the resting SL of 2 µm (46,101,102), T_ref is the maximum tension at resting SL, and ß₀ is the slope of the –T_0Max relationship,

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Limited isometric tension data at varying SL values at physiological temperatures was available so the parameters of Eq. 15 were fitted to room temperature results from Table 6. Numerous isometric tension values were reported for varying temperatures, magnesium, and ATP concentrations and preparation types. Fig. 6 plots isometric tensions taken from Table 6 against SL. The trend line for isometric tension values recorded under physiological conditions at room temperature (less four outliers) in Fig. 6 was described by Eq. 15 with T_ref = 56.2 kPa and ß₀ = 4.9, which is close to the ß₀ value reported for human myocardium of 4.27 (103). The four excluded results are indicated by the plus, up-triangle, asterisk, and square symbols, three from mouse (plus-symbol) (104), cat (up-triangle) (105), and guinea pig (asterisk) (66), which may indicate a species difference as the favored animal in muscle studies is rat. The remaining anomaly (square symbol) (95) is for intact rat muscle but was excluded due to the large standard deviation of 24.1 kPa.

View larger version (13K): [in this window] [in a new window]   FIGURE 6  Isometric tensions from Table 6 at varying strains, assuming a resting SL of 2 µm. Solid data points (46,66,71,72,80,89,104,133) represent measurements taken under physiological conditions defined by [Mg+2] 0.5–1 mM (79,89), ATP 4–5 mM, and SL is 1.9–2.3 µm (7,73,88) at room temperature 20–22°C. The remaining data from Table 6 is plotted as open circles. Outlying measurements recorded under physiological conditions are marked and correspond to data with large SD. () (95) or taken from a species other than rat, namely mouse (+) (104), cat () (105), or guinea pig (*) (66). Line of best fit to solid circles (•) is T0 = 56.2(1 + 4.9(–1)).

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where the value z is the fraction of available actin sites determined by thin filament kinetics and z_max is the maximum fraction of available actin sites at a given SL as defined above by Eq. 14.

Crossbridge kinetics
The fading memory model (11) provides an efficient method to phenomenologically represent the tension development associated with crossbridge kinetics using a small number of parameters. The fading memory model describes the relationship between tension and sarcomere sliding velocity, by separating tension development into nonlinear static and linear time-dependent components. The linear time-dependent component of the fading memory model is described as the sum of three exponential processes (Eq. 17), where _i and A_i are the exponential rate constants and associated weighting coefficients; respectively, is the strain, g(T,T₀) is a static function of tension and isometric tension, Q_i is the value of the i^th integral, and n is the number of exponential processes. Cell models are commonly defined in terms of a system of ordinary differential equations. To adapt the fading memory model to fit this mold, the right-hand side of Eq. 17 was differentiated by time to give Eq. 18,

where

(17)

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Under conditions of steady-state shortening, Eq. 17 yields the well-known Hill force-velocity relation if g in Eq. 17 is chosen to be g(T/T₀) = (1–T/T₀)/(T/T₀+a). Choosing g as a single function of T/T₀ is also consistent with the idea that the force scales with the number of active crossbridges, reflected in T₀. Hill's force-velocity curve describes the shortening (positive velocities) but not the lengthening (negative velocities) of the muscle under a constant load. The velocity becomes negative when the constant load is greater than the isometric tension (T > T₀). Here, the Hill curve was extended to model negative velocities. In frog skeletal muscle at 0°C, the magnitude of the negative velocity increases significantly for T > 1.4 T₀, but no information was available for more negative velocities or larger tensions in cardiac muscle (100). It was assumed that the force v's velocity relationship for negative velocities was a reflection of the plot for positive velocities. The original Hill curve for positive velocities is equal to Eq. 19 for V > 0 and the extended curve for negative velocities is equal to Eq. 19 for V < 0. In Eq. 19, V is the sarcomere velocity, V₀ is the maximum velocity, T is the active tension, T₀ is the isometric tension, and a is a measure of the curvature of the force-velocity relation (the value a is equivalent to aT₀ in Hill's original equation). The original and extended Hill curves are depicted in Fig. 7. The resulting force-velocity relationship is C₁ (slope) continuous, which provides numerical stability for the computational solution method as described later.

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Experimental results from constant velocity experiments in cardiac muscle give values of a as 0.2 (106), 0.5 (1), 0.35 (107), and 0.25 (108). The value a is also dependent on the exercise regime (a 0.2–0.36) (109) and P_i concentration (a 0.32–0.36) (110). Considering these values, a was set to 0.35. Other authors have reported values for a ranging from 0.05 (111) to 0.8 (112) by fitting limited regions of the force-velocity curve. It has been noted that the Hill equation does not fit the force-velocity curve for small velocities (111–113), but the approximation error is relatively small.

View larger version (12K): [in this window] [in a new window]   FIGURE 7  The data points are calculated from the original Hill equation with a = 0.35. The solid line indicates adapted Hill equation for T/T0 > 1.

d

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Setting the sum of A_i/_i = 1/(aV₀) means that the sum of Q_i in Eq. 17 is equal to the right-hand side of Eq. 19. This forms the relationship between the linear time-dependent and nonlinear static components of the fading memory model. The final form of the fading memory model was described by Eq. 21. The extended Hill equation proposed in Eq. 19 ensures that the denominator of Eq. 21 is always greater than or equal to one, thereby removing the singularity that would be present if the standard Hill equation were used for both positive and negative velocities.

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The linear time-dependent component of the fading memory was described by three