A Quantitative Analysis of Cardiac Myocyte Relaxation: A Simulation Study

Abstract

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

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.

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.

The model was validated using rapid length-step experiments, caged Ca²⁺ tension transients, clamped and unclamped SL tension traces, and RT₅₀ as a function of SL.

TROPONIN

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.

TABLE 1.

Binding affinities of Ca²⁺ to site II of cardiac troponin C

Species	Temp (°C)	Troponin complex	Bound Ca2+ measure	Mg (mM)	Kd (M−1)	K (μM)	Ref.
Human	30	NTnC	NMRs	—	4 × 105	2.5	(18)
Human	15	TnC	F27W	None	4.2 × 104	24	(25)
Human	15	TnC	F27W	3	1.4 × 105	7.1	(25)
Human	30	TnC	NMRs	—	5 × 104	20	(24)
Bovine	4	TnC	SC	None	2.5 × 105	4.0	(13)
Bovine	4	TnC	SC	4	2.5 × 105	4.0	(13)
Bovine	—	TnC	IAANS	3	7 × 105	1.4	(35)
Bovine	7	TnC	F27W	—	9.3 × 104	11	(17)
Bovine	21	TnC	F27W	—	1.9 × 105	5.3	(17)
Bovine	37	TnC	F27W	—	2.6 × 105	3.9	(17)
Mammal	RT	TnC	IAANS	3	2.5 × 105	4.0	(23)
Mammal	RT	TnC	IAANS	None	4.5 × 105	2.2	(23)
Mammal	21	TnC	F27W	None	2 × 105	5.0	(19)
Rat	4	TnC	IAANS (84)	3	3.2 × 105	3.1	(20)
R/B	23	TnC	IAANS	3	7.2 × 105	1.4	(21)
Chicken	23	TnC	IAANS (84)	None	2.9 × 105	3.5	(22)
Chicken	23	TnC	IAANS	None	3.6 × 105	2.8	(22)
Chicken	23	TnC-TnI	IAANS (84)	5	8 × 105	1.3	(22)
Bovine	—	TnC-TnI	IAANS	3	1.5 × 107	0.07	(35)
Rat	—	TnC-TnI	IAANS	3	1.7 × 106	0.59	(26)
R/B	23	TnC-TnI	IAANS	3	1.5 × 106	0.67	(21)
Mammal	RT	TnC-TnI	IAANS	None	3 × 106	0.33	(23)
Bovine	4	TnC-TnI	SC	4	1 × 106	1.0	(13)
Bovine	4	TnC-TnI	SC	None	1 × 106	1.0	(13)
Chicken	23	Tn	IAANS	5	1.2 × 106	0.83	(22)
Bovine	4	Tn	SC	None	2.5 × 106	0.40	(13)
Bovine	4	Tn	SC	4	2.5 × 106	0.40	(13)
Bovine	25	Tn-Tm	IAANS	2.5	1.2 × 106	0.83	(27)
P/C*	RT	SP	IAANS (84)	1	6.3 × 105	1.6	(29)
R/C	RT	SP	IAANS (84)	1	6.3 × 105	1.6	(29)
Bovine	25	SP	SC	5	4 × 106†	0.25	(33)
Bovine	25	SP	SC	5	2 × 106†	0.50	(32)
Bovine‡	25	Tn-Tm-A	SC	2.5	4 × 105	2.5	(27)
Bovine	25	Tn-Tm-A	SC	2.5	9.6 × 105	1.0	(27)
Bovine	25	Tn-Tm-A	IAANS	2.5	1.1 × 106	0.91	(27)
Bovine	25	SP	SC	5	2 × 106†	0.5	(34)
R/C	23	SP	IAANS	1	2 × 105	0.5	(22)
R/C	23	SP	IAANS (84)	1	4.7 × 105	2.1	(22)
Canine	25	SP	SC	2, 10	1.2 × 106	0.83	(28)
Canine	25	SP	SC	2	2.36 × 105	4.2	(31)

IAANS is IAANS-labeled TnC; IAANS (84) is IAANS-labeled TnC, with Cys amino acids at residue 84; NMRs is NMR spectroscopy; None = <1 × 10–3 mM; P/C is porcine fiber with chicken TnC; R/B is rat fiber/bovine Tn-Tm; R/C is rat fiber with chicken TnC; SC is scintillation counting; and SP is skinned preparation. RT is room temperature.

^*BDM added.

^†Affinity for sites II, III, and IV combined.

^‡IAANS bound, but not used to measure affinity.

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); × is IAANS (Cys-35); ▾is F27W; and □ is MRI spectroscopy.

Ca²⁺ binding kinetics

Equation 2 defines the kinetic binding of Ca²⁺ to site II in TnC was proposed by Robertson et al. (14). The value k_on is the rate of binding and k_off is the tension-dependent rate of unbinding of Ca²⁺ from TnC. Equation 1 is the steady-state solution to Eq. 2 with K = k_off/k_on:

(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.

TABLE 2.

Binding and unbinding rates of Ca²⁺ to site II in TnC

Species	Muscle	Temp (°C)	Troponin complex	Bound Ca2+ measure	Kon (μM−1 s−1)	Koff (s−1)	Ref.
Rabbit	S	4	TnC	F29W	100–200	—	(50)
Rabbit	S	4	TnC	DANZ	100–200	—	(50)
Rabbit	S	15	TnC	F29W	100	340	(51)
Rabbit	S	22	TnC	F29W	—	350	(50)
Rabbit	S	22	TnC	DANZ	—	551	(50)
Rabbit	S	22	TnC	Quin-2	290	462	(50)
Human	C	4	TnC	IAANS	—	73	(52)
Human	C	15	TnC	F27W	170	1159–1263	(25)
Human	C	20	TnC	IAANS	100	483	(52)
Human	C	30	TnC	NMRs	250	5000	(24)
Bovine	C	4	TnC	IAANS	51	11	(48)
Bovine	C	15	TnC	Quin-2	—	136.5	(49)
Chicken	C	4	TnC	Quin-2, BAPTA	200–400	700–800	(53)
Rat	C	4	TnC	IAANS	140	437	(20)
Bovine	C	4	TnC-TnI	IAANS	59	12	(48)
Bovine	C	15	Tn-Tm	IANDB	—	16.2–18.2	(49)
Bovine	C	15	Tn-Tm	Quin-2	—	23	(49)
Chicken	S	4	Tn-Tm	IANDB	10	1.3	(47)
Chicken	S	20	Tn-Tm	IANDB	65	13	(47)
Model	C	N/A	SP	—	39	19.6	(14)

C is cardiac; S is skeletal.

Tension-dependent Ca²⁺ unbinding rate from Troponin C

The tension-dependent binding of Ca²⁺ to TnC has been elucidated via a number of discrete experimental techniques. The affinity of Ca²⁺ for TnC decreases during rapid step-length reduction experiments on intact muscle (55–57). The concentration of Ca²⁺ bound to TnC decreases when tension development is inhibited with vanadium (33,40). Modeling experiments using Ca²⁺-sensitizing drugs indicate that Ca²⁺ binding to TnC is likely to be tension-dependent (58). The tension-dependent unbinding rate from Eq. 2 is defined by Eq. 3, below. The form of Eq. 3 captures tension-dependent components of Ca²⁺ binding to site II of TnC as well as the length-dependent components, as discussed below.

(3)

where k_refoff is the unbinding rate in the absence of tension, γ 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 γ 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 γ 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 γ 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); γ was determined for each measurement, and the average γ-value was 1.9.

TABLE 3.

Ca²⁺ bound to bovine cardiac troponin C site II

Temp (°C)	Additives	Sarcomere length (μm)	Fraction of site II with bound Ca2+	Ref.
22	None	1.5–1.6	0.49	(44)
22	None	1.7	0.83	(38)
—	None	1.7	0.69	(42)
RT	None	1.74	0.25	(40)
25	None	1.81	0.35	(32)
22	None	1.9	0.80	(43)
22	None	2.2–2.3	1	(44)
22	None	2.2–2.3	1	(44)
22	None	2.3	1	(38)
—	None	2.3	0.94	(42)
25	None	2.34	1	(32)
25	None	2–2.5	1	(33)
RT	None	2.45	1	(42)
22	2% dextran	2	1	(38)
22	5% dextran	1.7	1	(38)
—	5% dextran	1.7	1.0	(42)
—	5% dextran	2.3	1.0	(42)
22	5% dextran	1.9	1.0	(43)
22	10% dextran	1.9	0.80	(43)
22	15% dextran	1.9	0.26	(43)
25	1 mM Vi	2–2.5	0.74	(33)
RT	1 mM Vi	2.28–2.34	0.72	(40)
RT	1 mM Vi	1.52–1.78	0.72	(40)
22	1 mM Vi	2.2–2.3	0.50	(44)

RT is room temperature. Bound calcium is calculated using 0.34 μmol troponin/g of fiber. Bovine cardiac muscle at pCa5 is used in all cases. It is assumed that, at pCa 5.0, the high-affinity sites are saturated. If total calculated bound Ca²⁺ was >3, it is assumed that site II is saturated. It is assumed T ≈T₀ at pCa5, in the absence of any additives.

To ensure that the skinning procedure did not affect the tension dependence of K, skinned results were compared with intact values. In intact preparations, length-step experiments elucidate the value of γ. Results by Allen and Kentish (56), using the Ca²⁺ released during length step experiments, estimated that the affinity of Ca²⁺ for TnC (K) would halve when tension was dropped to zero during a length step corresponding to a γ-value of 2. Komukai et al. (57) found a linear relationship between Δ[Ca⁺²]_i/[Ca⁺²]_i (the ratio of the quantity of Ca²⁺ released during a step change to the free Ca²⁺ before a length change) with the tension before by the length change (T₁) as tension increased (see Eq. 4). The size of the length steps were defined such that the tension after the length change was equal to zero:

(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 ≫ Δ[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, γ is equal to 2.6, using T_ref = 56.2 kPa. The calculated γ-value of 1.9 from Fuchs and co-workers for skinned preparations is close to calculated γ-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 γ -value required to capture this phenomenon is close to the γ-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 γ will be set to 2.

TROPOMYOSIN

Tropomyosin is a highly extended α-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 α_r1 and α_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, α₀, 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 α₀ 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 (α_r1, α_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, α₀ 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 α_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 α_r1, α_r2, K_z, and n_r equal to 2 s⁻¹, 1.75 s⁻¹, 0.15, and 3, respectively.

TABLE 4.

Light-activated calcium chelator tension relaxation transients

Species	Prep	Temp (°C)	Initial pCa	SL (μm)	T1/T0	T2/T0	P1	k1(s−1)	P2	k2(s−1)	Ref.
Rat	T	14–15	5.8	2.2	0.5	0.05	18.9	73.5	85.3	9.8	(61)
Rat	T	14–15	6.1	2.2	—	—	93	12.1	8	2	(60)
Rat	T	14–15	5.8	2.2	—	—	93	12.1	8	2	(60)
Rat	T	14–15	5.6	2.2	—	—	93	12.1	8	2	(60)
GP	T	12	—	2.2	1	0	49	10.0	51	4.23	(65)
GP	T	20	—	2.2	1	0	92	18.4	15	2.52	(65)
Rat	T	22	—	2.1	0.4	0.1	—	16.1	—	0.97	(66)
GP	T	22	—	2.1	0.61	0.33	—	2.99	—	—	(66)
GP	T	12	—	—	1	0	45	11.2	57	2.9	(64)
Rat	V	15	5.5	2.3	0.48	0	—	10.8*	—	—	(62)
Pig	V	—	—	—	0.8	0	—	6.3*	—	—	(6)
Rat	T	22	5.56	2.1	—	0	46	16.6	54	1.1	(63)
Rat	T	22	5.56	2.1	—	0	48	17.3	52	1.3	(63)
GP†‡	V	10	4.5	2.25–2.4	1	0	—	12	—	0.3	(5)
Mouse‡	V	10	4.5	2.25–2.4	1	0	—	18	—	1.8	(145)
GP‡	T	10	4.5	2.25–2.4	1	0	—	10.8	—	0.6	(145)
Human‡	T	10	4.5	2.25–2.4	1	0	—	4.6	—	0.15	(145)

SL is sarcomere length. GP is guinea pig. T is trabeculae; V is ventricle.

^*Calculated from half-time of relaxation. [Ca²⁺] was rapidly decreased using diazo-2.

^†Stehle et al. (5,145) fitted a piecewise function containing a linear region with a slope equal to k₂ and an exponential region with rate k₁.

^‡Used rapid solution changes to change [Ca²⁺].

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.

Activation parameters

Having defined the parameters corresponding to relaxation, the remaining parameters could be fitted using caged Ca²⁺ tension transient experiments or F-pCa curves. Caged Ca²⁺ experiments are principally performed over a narrow range of SL, inhibiting their ability to characterize any length dependence. F-pCa curves, which are performed over a wide range of SLs, are used to characterize the remaining parameters.

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, α₀ 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.

TABLE 5.

Rat tension-free calcium sensitivity

Species	Muscle	Temp (°C)	Ionic strength	Mg2+ (mM)	ATP (mM)	Sarcomere length (μm)	pCa50	n1	n2	Ref.
Rat	T	22–24	200	3	5	1.65	5.02	4.35	—	(45)
Rat	T	15	200	1	5	1.85	5.35	4.8	8.6	(46)
Rat	V	15	180	—	4.74	1.85	5.41	—	3.4	(71)
Rat	T	22–24	200	3	5	1.75	5.11	2.82	—	(45)
Rat	T	22–24	200	3	5	1.85	5.17	3.85	—	(45)
Rat	V	22–26	160	3	3	1.9	5.5	3.2	—	(86)
Rat	T	15	180	1	5	1.95	5.47	6.5	—	(83)
Rat	T	15	180	1	5	1.95	5.47	6.34	—	(74)
Rat	T	15	200	1	5	1.95	5.39	4.7	11	(46)
Rat	T	22–24	200	3	5	1.95	5.27	3.91	—	(45)
Rat	T	20	200	0.3	3	1.9–2.04	5.26	2.37	—	(77)
Rat	T	23.8	200	1	5	2.0	5.36	—	—	(78)
Rat	T	23.8	200	1	5	2.0	5.36	—	—	(82)
Rat	T	15	200	1	5	2.05	5.43	5.4	9.9	(46)
Rat	T	22–24	200	3	5	2.05	5.36	4.5	—	(45)
Rat	V	RT	160	3	3	2.0–2.1	5.73	3.1	—	(87)
Rat	T	15	180	1	5	2.1	5.5	6	—	(74)
Rat	T	15	180	1	5	2.1	5.51	6.01	—	(83)
Rat	T	20	230	1.5	5	2.1	5.54	5.98	—	(80)
Rat	T	23.8	200	1	5	2.1	5.47	—	—	(82)
Rat	T	23.8	200	1	5	2.1	5.47	—	—	(78)
Rat	T	22	200	1	6.3	2.14	5.6	3.05	—	(66)
Rat	T	15	200	1	5	2.15	5.45	5.6	8.8	(46)
Rat	T	22–24	200	3	5	2.15	5.42	4.54	—	(45)
Rat	V	15	180	—	4.74	2.2	5.51	—	3.2	(71)
Rat	V	15	182	1	5	2.2	5.62	3.1	—	(88)
Rat	V	22	180	1	4	2.2	5.74	2.33	—	(84)
Rat	T	23.8	200	1	5	2.2	5.63	—	—	(82)
Rat	T	23.8	200	1	5	2.2	5.63	—	—	(78)
Rat	T	15	180	1	5	2.25	5.56	5.34	—	(74)
Rat	T	15	200	1	5	2.25	5.49	4.7	7.3	(46)
Rat	T	15	180	1	5	2.25	5.56	5.4	—	(83)
Rat	T	20–22	—	0.5	5	2.2–2.3	6.0	3.75	—	(89)
Rat	T	20–22	—	1.2	5	2.2–2.3	5.65	2.72	—	(89)
Rat	V	22–26	160	3	3	2.3	5.89	4.01	—	(86)
Rat	V	10	200	1	4	2.31	5.36	—	—	(79)
Rat	V	15	200	1	4	2.31	5.68	1.85	3.5	(79)
Rat	V	22	200	1	4	2.31	5.84	—	—	(79)
Rat	T	20	200	0.3	3	2.3–2.5	5.48	2.7	—	(77)
Rat*	T	20–22	—	0.72	—	2.2–2.3	6.2	4.87	—	(89)
Rat*	T	20–22	—	—	—	2.2	6.2	—	—	(95)
Rat*	T	20–22	—	—	—	2.1–2.3	6.19	5.2	—	(2)

T is trabeculae; V is ventricle. RT is room temperature.

^*These preparations were not skinned.

Length-dependent activation

[Ca²⁺]₅₀ is dependent on SL, which is seen as a leftward shift in the F-pCa curve as SL increases (67–70). The increased sensitivity has been associated with axial stretch (46), interfilament spacing (44), bound crossbridges kinetics (71), and titin (72). Fuchs et al. (38,42–44) and McDonald and Moss (73) found that fiber width predominantly determined length-dependent Ca²⁺ sensitivity. Then, using fiber width as an indicator of interfilament spacing, they concluded that interfilament spacing determined length-dependent Ca²⁺ sensitivity. Recently, Konhilas et al. (74) measured interfilament spacing using x-ray diffraction, and showed that fiber width is a poor indicator of interfilament spacing and that interfilament spacing does not affect length-dependent Ca²⁺ sensitivity. Moss and Fitzsimons (75), in their critical review of the results of Konhilas et al. (74), acknowledged the technical difficulties of the experiments and the potential for experimental error, but also believe that this does not detract from the fundamental finding that myofilament spacing is not the predominant cause of the length dependence of Ca²⁺ sensitivity. Unfortunately few other hypothesis exist to account for this phenomenon (75,76), though modeling results from Rice and de Tombe (76) suggest the possibility that cooperative effects between the regulatory units (troponin and tropomyosin) could account for increased Ca²⁺ sensitivity at longer SLs.

[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 β₁ = −4.0 in skinned preparations,

(12)

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(λ−1)).

Backx et al. (2) reported significant differences in Ca²⁺ sensitivity of skinned muscle compared to intact muscle—half-activation is 2–10-fold larger and the Hill coefficient is 2–3 times smaller in skinned muscle. The change in steady-state calcium sensitivity can potentially be rationalized by three different mechanisms:

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, β₁ 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 λ = 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 λ = 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 λ = 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 α₀ 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 α₀ 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 λ = 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.

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 λ = 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.

Maximum activation

In this study, it is assumed that, at maximum activation, not all actin sites are available. However, to ensure that when TnC was saturated the maximum tension reported experimentally was achieved, z was normalized by z_Max, such that when [Ca²⁺]_Trpn = [Ca²⁺]_TrpnMax,, z/z_Max = 1 and T₀ = T_0Max. The value z_Max was calculated by solving Eq. 9 with dz/dt = 0 and [Ca²⁺]_Trpn = [Ca²⁺]_TrpnMax. This is a nonlinear equation and requires an iterative solution method. To maintain an explicit formulation for the model, the nonlinear component was linearized around a point z_p using a Taylor expansion to equal zK₁+K₂, where K₁ and K₂ are defined by Eq. 13. The value z_p was set to 0.85 and ensures an error of <1% for z ∈ [0.6,1.0],

(13)

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

(14)

TENSION DEVELOPMENT

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 λ is equal to SL over the resting SL of 2 μm (46,101,102), T_ref is the maximum tension at resting SL, and β₀ is the slope of the λ−T_0Max relationship,

(15)

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 β₀ = 4.9, which is close to the β₀ 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.

TABLE 6.

Isometric tension

Species	Muscle	Prep	Temp (°C)	SL (μm)	Mg+2 (mM)	ATP (mM)	Tmax (kPa)	Ref.
Rat	T	SP	23	1.65	3	5	46.2	(45)
Rat	T	SP	23	1.75	3	5	50	(45)
Rat	T	SP	23	1.85	3	5	63.2	(45)
Mouse	P	SP	RT	1.9	1	5	38.4	(104)
Rat	T	SP	20	1.9	1	4	44.6	(72)
Rats	V	SP	15	1.95	1	5	56.8	(46)
Rat	T	SP	23	1.95	3	5	69.2	(45)
Rat	V	SP	20	1.98	3	3–5	36.25	(77)
Cat	T	IP	25	2.0	—	—	108	(105)
Rat	T	SP	20	2	1	4	56.3	(72)
Rats	V	SP	15	2.05	1	5	60.6	(46)
Rat	T	SP	23	2.05	3	5	75	(45)
Rat	T	SP	20	2.1	1	4	68.8	(72)
Rat	T	SP	20	2.1	1.5	5	37	(80)
Rat	T	SP	20	2.1	3	5	37.5	(128)
GP	T	SP	22	2.13	1	6.3	50.6	(66)
Rat	T	SP	22	2.14	1	6.3	78.9	(66)
Rat	T	SP	23	2.15	3	5	86.3	(45)
Rats	V	SP	15	2.15	1	5	65.3	(46)
Rat*	T	IP	22	2.2	—	—	88.5	(133)
Rat†	T	IP	22	2.07–2.2	—	—	45.1	(133)
Rat	V	SP	15	2.2	1	5	39	(88)
Rat	T	SP	20	2.2	1	4	82.1	(72)
Rat	T	IP	20–22	2.2	—	—	102.4	(95)
Rat	T	IP	20–22	2.1–2.3	1.2	5	121	(2)
Rat	T	SP	20	2.2–2.3	0.5	5	90	(89)
Rat	T	IP	20	2.2–2.3	0.72	—	93	(89)
Rats	V	SP	15	2.25	1	5	72	(46)
Rat	V	SP	20	2.35	3	3–5	48.8	(77)
Rat	T	SP	20	2.3	1	4	92.0	(72)
Mouse	P	SP	RT	2.3	1	5	65.3	(104)
Rats	V	SP	13	2.29–2.36	1	4	21	(110)
Rat	T	SP	20	2.4	1	4	96.4	(72)
Cat	V	IP	30	—	—	—	18	(55)
Ferret	P	IP	30	—	—	—	54.4	(57)

SL is sarcomere length. GP is guinea pig; P is papillary; T is trabeculae; V is ventricle. IP is intact preparation. RT is room temperature. SP is skinned preparation.

^*Controlled SL.

^†Uncontrolled SL.

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(λ−1)).

Combing T₀/T_0Max = z/z_max and Eq. 15 gives Eq. 16, which defines isometric tension as a function of both SL and thin filament kinetics,

(16)

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 α_i and A_i are the exponential rate constants and associated weighting coefficients; respectively, λ 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)

(18)

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.

(19)

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.

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.

During constant velocity experiments dλ/dt = −V, and the linear component of the fading memory model (right-hand side of Eq. 17) is equal to

(20)

Setting the sum of A_i/α_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.

(21)

The linear time-dependent component of the fading memory was described by three exponential processes (11,114) and requires the definition of three rate constants and three amplitude coefficients. Although the effect of temperature on crossbridge kinetics is significant, there was insufficient data to characterize crossbridge kinetics at physiological temperatures and, as such, the parameter set defined here will be for room temperature, where room temperature was taken as 20–23°C. Sinusoidal analysis provides two rate constants explicitly. The four remaining parameters were fitted using the two defined rate constants and four constraints derived from experimental results. These constraints are outlined in detail in the following sections. First, sinusoidal analysis was used to find the minimum stiffness frequency, maximum phase shift frequency, and the high frequency stiffness. Secondly, the instantaneous step change required to drop tension to zero was determined, which required the high frequency stiffness values from the sinusoidal analysis. Thirdly, the maximum velocity was defined. The remaining four fading memory model parameters were then fitted using the four constraints and the two defined rate constants.

Dynamic stiffness

Sinusoidal analysis provides two of the rate constants explicitly and the minimum dynamic stiffness frequency, the maximum phase shift frequency, and the high frequency stiffness are used to constrain the remaining parameters. The dynamic stiffness of muscle has been effectively measured by imposing a small sinusoidal or random noise perturbation while the muscle is in isometric contraction. The dynamic stiffness is equal to the tension over the applied length change and can be described in the frequency domain using Eq. 22 as proposed by Kawai and Brandt (115) or in the time domain using the fading memory model (Eq. 17). In Eq. 22, a < b < c < d are the rate constants and are equal to the corresponding α_i values from Eq. 17 over 2π. A, B, C, and D are coefficients and equal to the product of the corresponding A_i values in Eq. 17 and T_ref(1+a). Using standard nomenclature, the negative coefficient is B. In cardiac muscle, B has the lowest rate and no A process is seen. H is the linear passive stiffness, which was not considered in the model of active contraction described here:

(22)

Rossmanith et al. (116) showed that the myosin isoform present in the preparation played a prominent role in determining crossbridge kinetics and results from sinusoidal analysis. Ventricular myosin in mammals appears in three different forms—i.e., V₁, V₂, and V₃. The predominant forms are V₁ and V₃ or α and β, respectively, and their presence is determined by species and thyroid state. In ferret and rabbit, V₁ is commonly expressed whereas in bovine, humans, and rats, the V₃ type is predominant (116,117). Considering this, crossbridge kinetic results will be taken preferentially from rat, human, and bovine preparations.

Sinusoidal perturbation experiments are commonly performed in skinned preparations at full activation (pCa < 4) and λ > 1.1. At full activation, [Ca⁺²]_Trpn/[Ca⁺²]_TrpnMax ≈ 1 and z/z_Max ≈ 1, such that dynamic stiffness (E) is equal to (11):

(23)

Measurements of coefficients and rate constants from sinusoidal perturbation experiments are presented in Table 7. Experiments by Kawai et al. (118), Saeki et al. (114), and Shibata et al. (119,120) in ferret and rabbit preparations recorded similar results for V₁ myosin isoforms. Results by Wannenburg et al. (82) in rat myocardium are considerably different. However, Wannenburg et al. (82) did not report rates or coefficients with process D fitted, although they commented that the exclusion of process D had minimal affect on the rates of processes B and C (α₁ and α₂). The converted rate constants b and c (i.e., α₁ = 30.78 s⁻¹ and α₂ = 132.6 s⁻¹) reported by Wannenburg et al. (82) are comparable to measurements by Ruf et al. (121) and Campbell et al. (122) for human and rat myocardium, respectively, and will be used in the model.

TABLE 7.

Sinusoidal analysis results

Species	Temp (°C)	pCa	Frequency range (Hz)	B	2πb	C	2πc	D	2πd	Ref.
Ferret*	20	Ba2+†	0.13–135	0.82	7.6	2.09	26	0.90	682	(114)
Ferret	20	4.55	0.13–135	0.72	8.6	1.57	30	0.59	703	(114)
Ferret	20	4.82	0.13–135	0.58	9.9	1.31	23.6	0.42	552	(118)
Rat	23	6.11	0.5–70	1.09	5.97	3.49	47	—	—	(82)
Rat	23	5.92	0.5–70	2.92	30.78	11.3	133	—	—	(82)
Human‡	35	5	0.125–100	0.376	25	0.525	70.37	—	—	(123)
Rat	—	4.3	0.1–40	—	33.6	—	190.4	—	—	(122)
Ferret	—	—	0.1–40	—	30.41	—	70.56	—	—	(122)

^*These preparations were not skinned.

^†Barium contractures.

^‡Human patients on numerous medications before the preparations were sampled; patients had an average age of 61.2 years.

Dynamic stiffness data is commonly represented in two plots—the dynamic modulus plot of the absolute value of dynamic stiffness against frequency, and the phase shift against frequency plot. The Nyquist diagram plots phase against amplitude. The dynamic modulus plot exhibits a characteristic dip in stiffness at the minimum stiffness frequency (f_min). The phase shift plot has a single maximum at the maximum phase shift frequency (Θ_max) (115). These two frequencies provide two points of reference and ensure that the model produces characteristic plots. Table 8 lists reported f_min and Θ_max values. The value f_min was calculated using a range of species and temperatures. The value f_min increases significantly with temperature from 1 Hz to 17.25 Hz at 20°C and 40°C, respectively. There was no definitive relationship between myosin isoform and f_min. Rabbit and ferret have reported f_min values of 0.9–2 Hz at room temperature. However, the 2 Hz result was recorded with ATP at 1 mM; excluding this value reduces the range to 0.9–1.44 Hz. Rat and bovine f_min values range between 1 and 3.2 Hz. The 3.2 Hz reported by Wannenburg et al. (82) was for partially activated preparations, but Wannenburg et al. (82) did note that this had a minimal affect on f_min. Rossmanith et al. (116) reported that V₁ and V₃ myosin isoforms had an f_min value of 1.2 Hz and 2.2 Hz, respectively, at 25°C. The value f_min for the model was taken to be 2 Hz. The value Θ_max was reported less often than f_min, and ranges between 2.1 and 8 Hz. It was not known how partial activation affects Θ_max, so the Wannenburg et al. (82) results are not considered here. The value Θ_max for human data is 4.3 Hz (123), which is comparable to rat (4 Hz) (116), rabbit (4 Hz) (119,124), and ferret (3.5 Hz) (118) myocardium. Considering these results, the Θ_max for the model was taken as 4 Hz.

TABLE 8.

Maximum phase shift and minimum dynamic stiffness

Species	Temp (°C)	Θmax (Hz)	Fmin (Hz)	Comments	Ref.
Human	35	4.3	9		(123)
Bovine	40	—	17.25		(127)
Bovine	30	—	5.21		(127)
Bovine	20	—	1		(127)
Bovine	25	7.6	2	5 mM ATP	(151)
Rat	25	4	1.8		(116)
Rat	23	8	3.2	pCa 5.92	(82)
Rabbit	24	4.0	1.2		(119)
Rabbit	24	4	1.44		(124)
Ferret	20.2	2.1	1.3	Skinned	(114)
Ferret	20.2	1.7	0.9	Intact	(114)
Ferret	20	3	2	1 mM ATP	(118)
Ferret	20	3.5	2	1 mM ATP	(118)

The high-frequency stiffness (HFS) is the stiffness calculated using the model at pCa < 4 and f→∞ (Eq. 24). Experimentally, the HFS is the stiffness at the highest frequency recorded:

(24)

The HFS varies significantly between authors. Campbell et al. (122) measured HFS of ≈0.2–1 M Nm⁻² at varying isometric tensions using a low maximum frequency range (40 Hz). Wannenburg et al. (82) reported the HFS to be ≈7.25 M Nm⁻², again using a low maximum frequency (70 Hz). The experiments were performed at only partial activation but recorded viable isometric tensions (87 kPa) at full activation. Kawai et al. (118) recorded HFS values of ≈0.9 M Nm⁻² using a maximum frequency of 135 Hz, but measured low isometric tension values (18.3 kPa) and inorganic phosphate concentrations were high (8 mM) in skinned preparations. Mulieri et al. (123) recorded an HFS of ≈0.59 M Nm⁻² in human hearts of patients (average age 61.6) at a maximum frequency of 100 Hz. Saeki et al. (114) recorded HFS values of 2.50 and 1.44 M Nm⁻² for intact and skinned preparations, respectively, using a maximum frequency of 135 Hz, but recorded isometric tensions of 49 kPa and 26 kPa for intact and skinned preparations, respectively, at L_max (SL ≈ 2.3 μm). The isometric tension values recorded at full activation in sinusoidal analysis experiments were characteristically low when compared with static isometric tension data from Table 6. The large variations in HFS values meant that these data were not used directly to fit model parameters but instead were used in the following section to define the length step required to drop tension to zero.

Length-step experiments

The step response of the fading memory model is found by setting λ(t) = Δλ × H(t), where H(t) is the Heaviside step function, or equivalently dλ/dt = Δλ × δ(t), where δ(t) is the Dirac δ-function (11). This gives

(25)

The length step required to drop tension to zero immediately after a length change is found by setting t = 0 in Eq. 25 and is equal to Eq. 26:

(26)

Instantaneous length steps are not possible in real experiments due to the inertia of testing equipment and muscle. To overcome this, various experimental methods have been proposed to elucidate the magnitude of Δλ. The value T₁ is the tension after a rapid length step over a finite duration of time. For length steps over a finite duration of time (Δt) with T₁ = 0, there is a linear relationship between Δλ and Δt (100). Hunter et al. (11) exploited this relationship to determine Δλ of 0.5%, as Δt tends to zero using frog and ferret data (at 0°C and 27°C, respectively). Backx and ter Keurs (125) used a similar method to determine a Δλ value of 1.2% in rat muscle (at 5°C). An alternate method is to assume a linear relationship between tension and small length changes, such that Δλ is equal to the ratio of the high frequency stiffness (HFS) and isometric tension. The HFS value is not used in isolation, as it varies considerably between authors as mentioned above. However, the ratio of HFS to isometric tension varies less and appears to be less sensitive to myosin isoform, temperature, and muscle viability than HFS alone. At an SL of 2.1 μm, Wannenburg et al. (82) measured the ratio of HFS to maximum isometric tension as 1.2%, which is comparable to HFS measurements at L_max (the muscle length, which achieves maximum tension or SL ≈ 2.3 μm) of 1.6–2.0% by Shibata et al. (120), Saeki et al. (114), and Kawai et al. (118). Considering the results of both modalities, the value of Δλ required to drop tension to zero in this model will be set to 1.2%.

Maximum velocity experiments

The maximum velocity (V₀) of contraction is directly related to ATPase (111,126), and as such, indicative of crossbridge kinetics. The maximum velocity occurs under zero load, and is independent of SL for SL values between 1.9 and 2.2 μm (111). The fading memory model parameters can be constrained using Eq. 27 and results from maximum velocity experiments (11),

(27)

Maximum velocity experiments are performed over a range of temperatures as reported in Table 9, but can be adjusted to room temperature using a Q₁₀ of 4.6 (111). A maximum velocity per sarcomere of ≈19 μm s⁻¹ or a strain rate of 9.5 s⁻¹ at room temperature with a resting SL of 2 μm was set to constrain the parameters in Eq. 23.

TABLE 9.

Maximum muscle velocity at 23°C; strain rates adjusted to 23°C using a Q₁₀ value of 4.6

Species	Original temp (°C)	Strain rate (s−1)	Strain rate at 23°C (s−1)	Ref.
Rat	25	13.3	9.8	(149)
Rat	20	6.13	9.6	(111)
Rat	25	12.68	9.34	(111)
Rat	30	23.44	8	(111)
Rat	12	3.45*	18.5	(150)
Cat	25	9.8	7.2	(105)
Rat	22	9.82*	11.4	(84)

^*Calculated using ML s⁻¹ and SL.

Fading memory model parameters

The linear time-dependent component of the fading memory model has three rate (α_i) values and three coefficient (A_i) values. The frequency values of α₁ = 30 s⁻¹ and α₂ = 130 s⁻¹ were set directly to those reported by Wannenburg et al. (82), as explained above. The remaining four parameters were calculated by determining the best fit to the four independent constraints outlined above. These are, the maximum velocity (Eq. 27), length step experiments (Eq. 26), and f_min (2 Hz) and Θ_max (4 Hz) frequencies. The fitting process was begun by setting A₁ and A₂, then calculating A₃ using Eq. 26, then α₃ was determined using Eq. 27 and finished using Eq. 23 to determine f_min and Θ_max numerically. Thus, by searching the parameter space of A₁ and A₂, the best parameter set for the four constraints was determined. The resulting values for the process magnitudes A₁, A₂, and A₃ are −29, 138, and 129, respectively. The process rates α₁, α₂, and α₃ are 30 s⁻¹, 130 s⁻¹, and 625 s⁻¹, respectively. The resulting f_min and Θ_max values are 0.9 Hz and 3.75 Hz, respectively. The value f_min was smaller than the f_min value determined above, from V₃ myosin experimental results; however, it was still comparable with V₃ myosin results from Fujita and Kawai (127) (f_min = 1 Hz) and V₁ myosin results.

MODEL SUMMARY

The complete model is summarized, with the set of parameters for (primarily intact rat ventricular) cardiac muscle at room temperature. The model can be adapted to represent skinned preparations by changing α₀, n, and [Ca²⁺]_50ref to 6 s⁻¹, 3.4, and 4.72 μM, respectively.

Parameters

TnC-Ca binding

Equations 2 and 3, where k_on = 100 μM⁻¹ s⁻¹; k_offref = 200 s⁻¹; γ = 2.0; and [Ca²⁺]_TrpnMax = 70 μM.

Tropomyosin kinetics

Equations 9, 11, and 14, where α_r1 = 2 s⁻¹; α_r2 = 1.75 s⁻¹; K_z = 0.15; n_r = 3; β₁ = −4.0; α₀ = 8 s⁻¹; n = 3; z_p = 0.85; and [Ca²⁺]_50ref = 1.05 μM.

Tension development

Equations 15, 18, and 21, where T_ref = 56.2; β₀ = 4.9; a = 0.35; A₁ = −29; A₂ = 138; A₃ = 129; α₁ = 30 s⁻¹; α₂ = 130 s⁻¹; and α₃ = 625 s⁻¹.

VALIDATION

The three model components, Ca²⁺ binding to TnC kinetics, tropomyosin kinetics, and crossbridge kinetics, have been constructed and parameterized in isolation using a wide range of preparation types and spatial scales. Comparison has already been made with experimental data isolating the function of each component from data using the parameter set allocated in previous sections. To validate tropomyosin kinetics, model simulations were compared with caged Ca²⁺ experiments. Crossbridge kinetics were validated using tension transients after rapid length increases. No obvious experimental results were available to validate Ca²⁺ binding to TnC kinetics, but the functional form of Ca²⁺ binding to TnC (Eq. 2) is a generally accepted model and the tension dependence of the unbinding rate can be derived from length-step experiments (Eqs. 4–8). To ensure that when the ensemble of model components were integrated together the resulting simulations matched experimental results, model simulations were compared with SL clamped and SL unclamped tension traces. Finally, to test that the complete model was able to characterize RT₅₀, a comparison was made between simulated RT₅₀ and experimental measurements at increasing muscle length.

Caged calcium activation curves

To verify tropomyosin kinetics, which were defined by tension relaxation and F-pCa curves, the model was compared with caged calcium release experiment results. In caged Ca²⁺ experiments nitrophenyl-EGTA is saturated with Ca²⁺ and then exposed to a flash of light resulting in a rapid decrease in the Ca²⁺ affinity, causing a rapid increase in the concentration of free Ca²⁺ (60). This provides a means of measuring the rate of tension development after a step change in Ca²⁺. Tropomyosin activation kinetics were defined using the relaxation kinetics and F-pCa curves. Fig. 8 plots tension traces for guinea-pig caged Ca²⁺ experiments with SL = 2.18 μm at 23°C, taken from Martin et al. (81) against simulated results using the skinned preparation parameter set. The Ca²⁺ concentrations after the release of caged Ca²⁺ was calculated using the final relative tensions and F-pCa curves for skinned preparations described above with pCa₅₀ = 5.6 and n = 3.

FIGURE 8.

Tension transients after the release of caged Ca²⁺ by nitrophenyl-EGTA with SL = 2.18 μm and initial pCa = 7. Data points are from Martin et al. (81) and lines are from model simulations. The ▾data points and dashed line, • data points and dash-dot line, ▪ data points and dotted line, and ♦ data points and solid line had pCa values after the release of the caged calcium of 5.07, 5.41, 5.56, and 5.69, 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.

Finite duration length-step changes

For a 1-ms length step of 1% (SL = 2.1 μm) at room temperature at an inorganic phosphate concentration of 0 mM, the half-relaxation time is 6.0 ms (128), compared with a half-relaxation time of 6.2 ms predicted by the model (see Fig. 9). The model also exhibits the characteristic dip seen when muscle is rapidly stretched, but shows no evidence of further oscillations observed by Saeki et al. (129) and Berman et al. (124). The fading memory model predicts a significant rise in tension after a rapid length decrease, analogous to the dip in rapid length increase experiments. The rise predicted by the model after the length decrease was not seen in rapid length decrease experiments in cardiac muscle performed by Hancock et al. (130), but has been observed in both skeletal muscle (131) and cardiac muscle (132) for very small step sizes (Δλ ≤ 0.5%).

FIGURE 9.

Length steps of 1-ms duration. (A) 1% length-step increase at 23°C for SL = 2.1 μm and pCa = 4. The dashed line is the model prediction; ▪ is from Kentish (128), (tension scaled to model). (B) 1% length-step decrease at 23°C, for SL = 2.1 μm and pCa = 4.

Clamped and unclamped sarcomere length isometric twitches

Inhomogeneous SL shortening in muscle preparations during an isometric twitch has been shown to cause a significant decrease in developed tension. Janssen and de Tombe (133) showed that using iterative feedback SL control, the SL could be held constant, resulting in significantly increased developed tension and a decrease in the amplitude of the [Ca²⁺]_i transient. The tension traces for clamped and unclamped experiments recorded by Janssen et al. (7) were simulated by the model using the corresponding experimental [Ca²⁺]_i and SL traces. The fluorescent traces from Janssen and de Tombe were converted to [Ca²⁺]_i values using [Ca²⁺]_i = K′_d(R−R_min)/(R_max−R), where K_d′ = 2.95 μM, R_max = 10, R_min = 0.2, and R is the fluorescence ratio. K_d′ and R_max are taken from Gao et al. (3,89) and R_min is set such that resting [Ca²⁺]_i is ≈0.1 μM. The calculated [Ca²⁺]_i transient and the experimentally measured SLs are shown in Fig. 10 A. Cubic splines were fit through the data to produce smooth curves to input into the model and to compare with simulation results. The resulting simulated tension traces are compared with experimental measurements in Fig. 10, B and C. The simulated results are similar to experimental measurements, although the final stages in the unclamped simulations were notably slower and slightly slower in the clamped simulations. However, RT₅₀ values were well represented by the model.

FIGURE 10.

(A) Sarcomere lengths from unclamped SL (▪) and clamped SL (•) experiments from Janssen and de Tombe (133). [Ca²⁺]_i transients from unclamped SL (dashed line) and clamped SL (solid line) experiments. [Ca²⁺]_i transients were calculated from fluorescence data from Janssen and de Tombe (133) and converted to [Ca²⁺]_i values using [Ca²⁺]_i = K′_d(R−R_min)/(R_max−R), where K′_d = 2.95 μM, R_max = 10, R_min = 0.2 (4,89). (B and C) Clamped and unclamped experiments, respectively; experimental data are taken from Janssen and de Tombe (133) (solid lines), and lines are generated by the model (dashed lines).

Half-relaxation times

As cardiac muscle is stretched at room temperature RT₅₀ during isometric twitch increases (7). Fig. 11 B compares RT₅₀ for intact rat trabeculae at 22.5°C from Janssen et al. (7) with simulation results from the model. Results from Janssen and others were recorded at 10 equally spaced lengths between the resting length and maximum tension length, which are assumed to correspond to equally spaced SL values between 1.8 μm and 2.2 μm for simulations. The SL was assumed to remain constant during the isometric twitch, although SL was not controlled by feedback iterations in the study of Janssen et al. (7). Thus the [Ca²⁺]_i transient is held fixed for all simulations. Tension transients at each muscle length are shown in Fig. 11 B. The [Ca²⁺]_i transient is expected to vary at differing SLs, but limited experimental data is available to characterize this phenomenon.

FIGURE 11.

(A) Tension traces for increasing SL from 1.8 μm to 2.2 μm. (B) RT₅₀, points taken from Janssen et al. (7) at room temperature, with standard error. Line calculated from model simulations. Arbitrary units defined by Janssen et al. (7).

RELAXATION FACTORIAL ANALYSIS

The [Ca²⁺]_i transient, TnC kinetics, and length dependence have all being implicated in determining RT₅₀ (134). It is worth noting that many of the factors are coupled, via tension dependence, which often results in inconclusive experimental findings. To determine the relative importance of the factors listed in Table 10 on RT_50, a full factorial analysis (135) was performed. A factorial analysis provides a means to analyze the relative importance of a single factor and multiple factor interactions on a given output, by approximating the output as a linear function of all the factors and determining the coefficients. The magnitude of the coefficients indicates the relative influence of each factor, including multiple factor interactions. Although factorial analysis uses a linear approximation of a nonlinear output, which is not ideal, it provides a feasible means to identify significant factors, which can then be analyzed in detail. The output was defined as the RT₅₀ value for an isometric twitch with a prescribed [Ca²⁺]_i transient and clamped SL = 2.2 μm. Table 10 summarizes the factor definitions and values, and Fig. 12 A shows the relative influences. During the factorial analysis, the β₀- and γ-values used to calculate the concentration of calcium bound at half-activation (Eq. 11) were held constant, since they do not represent any mechanism in this equation. The factorial analysis identified the [Ca²⁺]_i transient magnitude and the unbinding rate of Ca²⁺ from TnC as the two most significant factors within the initial parameter space. A detailed analysis of the parameter space of the unbinding of Ca²⁺ from TnC and the magnitude of the [Ca²⁺]_i transient on RT₅₀ is shown in Fig. 12 B to illustrate coupled effects and to determine whether the effect of each factor is dependent on the parameter space.

TABLE 10.

Factors and variable descriptions and values for factorial experiment

Factor #	Variable	High value (+)	Low value (−)
1	Unbinding rate of Ca2+ from TnC	koff = 220 s−1	koff = 180 s−1
2	Length-dependent sensitivity	β0 = 5.4	β0 = 4.41
3	Length-dependent maximum tension	β1 = −1.3	β1 = −1.06
4	[Ca2+]i transient magnitude	110%	90%
5	Tension-dependent binding of Ca2+ to TnC	γ = 2.2	γ = 1.8

FIGURE 12.

(A) Relative effects of individual and two-factor interactions for prescribed [Ca²⁺]_i transient and SL = 2.2 μm. Single-digit factors correspond to the variables listed in Table 10 and two-digit factors indicate the coupled effect of the variables from Table 10 corresponding to each digit; for example, 45 corresponds to the coupled affect of the [Ca²⁺]_i transient and the tension dependence of Ca²⁺ binding to TnC . (B) The RT₅₀ calculated using the model with the [Ca²⁺]_i and the unbinding rate of Ca²⁺ from TnC varying between 50 and 150% of their original values.

DISCUSSION

In this article, we present an improved version of the HMT model for the active contraction of cardiac muscle and analyze its behavior, particularly during relaxation. Each model parameter was determined from a broad range of experimental results. The model components and the coupling between them were subsequently validated against a wide range of experimental protocols. After development and validation, the model was used to quantify the relative influence of the tension-dependent unbinding rate of Ca²⁺ from TnC, length-dependent sensitivity, length-dependent maximum tension, [Ca²⁺]_i transient, and the tension-dependent binding of Ca²⁺ to TnC using a factorial analysis. Here we discuss the validation results, the implications of the factorial analysis, the sensitivity of the model to specific parameters, and the limitations of the model.

Validation

The model has been developed in components using specific experimental data, often recorded from experimental protocols specifically designed to isolate a particular mechanism or function. Model results have been successfully compared and validated against a combination of specific and physiological experimental results and normalized measures of tension kinetics.

Crossbridge kinetics were validated using rapid length-step experimental results from Kentish et al. (128) (see Fig. 9). The simulated results matched rapid length increases, but simulations for length-step decreases exhibited an unexpected rise in tension after the length step and no drop in steady-state tension after the length step was observed. The rise has been both present (132) and absent (130,136) in cardiac muscle experiments and has been observed in skeletal muscle (131). A potential explanation for the absence of the rise predicted by sinusoidal analysis (115) in quick release experiments is due to the buckling of the muscle during release (137). Buckling would not be observed in rapid stretch experiments and would be more likely to occur with larger length changes as Hancock et al. (130,136) used a 1% and 2% length change, compared to 0.05% used by Rossmanith and Tjokorda (131) and 0.5% used by de Beer et al. (132). Although no direct comparison can be made between model simulations and results by de Beer et al., who did not include the duration of the length step or provide a scale for tension. The rate constants used in the fading memory part of the HMT model are independent of [Ca²⁺]_i but the thin filament kinetics are not. So the effect of Ca²⁺ on the rate of tension redevelopment depends on [Ca²⁺]_i, which is not observed by Hancock et al. (130,136), but is seen by Backer et al. (138) and Wolff et al. (139). The second difference between simulations and experimental results was the absence of a decrease in steady-state tension after the length step. Significant decreases have been observed in ferret cardiac (130) and frog skeletal (140) muscle experimentally. In a critical review of this phenomenon, Rassier and Herzog (141) attributed the change in isometric tension to nonuniformity of SLs and stress-induced inhibition of crossbridge attachment after length steps. These observations are not accounted for by the model, due to the scarcity of specific data for cardiac tissue.

Ca²⁺ activation kinetics were defined using tension relaxation kinetics and F-pCa curves. Tension transients after the release of increasing concentrations of free Ca²⁺ were used to validate tropomyosin kinetics (see Fig. 8). The tension trace after the release of caged Ca²⁺ were closely followed by the simulated tension transients. The model was capable of representing a Ca²⁺-dependent monoexponential rise in tension, while at the same time having a biphasic relaxation, as seen experimentally (61).

The complete model was compared with isometric twitch data from Janssen and de Tombe (133) with clamped and unclamped SL values (see Fig. 10). The model was able to replicate the tension traces, although the final phase of relaxation predicted by the model for both the clamped and unclamped twitch were slower than observed experimentally. In both cases, the use of experimental data recorded at 15°C to determine the tropomyosin relaxation parameters may have contributed to the delay in relaxation. In the unclamped case, the approximation of the force-velocity curve may also have caused reduced relaxation rates, as discussed below. The oscillations in the late phase of relaxation in the unclamped case is due to the imposed SL, which results in a forced SL velocity and the uncoupling of the feedback between velocity and tension. If the active contraction model was coupled to a passive mechanics model then the feedback between the passive and active mechanics would result in a smoother relaxation phase.

The model also quantitatively matches the increase in RT₅₀ with increasing SL values, as seen experimentally by Janssen et al. (7) (see Fig. 11). An earlier study by Janssen and Hunter (4) using SL control to maintain the SL during contractions found lower RT₅₀ values than both the model predictions and experimental results from Janssen et al. (7). However, only one data set is available from Janssen and Hunter (4) with n = 1, and no corresponding Ca²⁺ transient is provided. As such, a direct comparison is not possible. Janssen and Hunter (4) also reported that as SL decreased, the rate that tension fell from 25 to 10% of maximum tension increased. This phenomenon is not explicitly captured by the model, which predicts an increase in relaxation rate with SL within this range, although the model does predict an increase in relaxation rates for decreasing SL for larger SL values as tension drops from 35 to 25%.

The ability of the model to replicate specific tension twitches and RT₅₀ trends supports the validity of the hypothesized model structure and parameter set.

Relaxation

The factorial analysis performed above highlighted the unbinding rate of Ca²⁺ from TnC and the [Ca²⁺]_i transient magnitude as the principle factors determining RT₅₀. The tension-dependent binding of Ca²⁺ to TnC and the length-dependent Ca²⁺ sensitivity played a secondary role and maximum tension and two-factor interactions did not significantly affect RT₅₀ values in the examined parameter space.

The effect of an increase in the unbinding rate of Ca²⁺ from TnC on the relaxation has been studied experimentally by phosphorylating troponin I using protein kinase A, causing ≈30% increase in the unbinding rate of Ca²⁺ from TnC (6,92). The increase in unbinding rate has been shown to cause a decrease in RT₅₀ in intact preparations (142) and both an increase (6,61) and no effect (64) in the rate of relaxation in skinned preparations, using light-activated Ca²⁺ chelators. The factorial analysis identified the unbinding of Ca²⁺ from TnC as a significant determinant of relaxation.

The effect of SL on RT₅₀ was separated into the length dependence of tension and calcium sensitivity. The SL tension dependence had a minimal affect on RT₅₀, whereas the length-dependent calcium sensitivity had a significant affect on RT₅₀. Experimental results from Janssen and Hunter (3) found tension and not SL correlated with relaxation times, suggesting that either the tension dependence of Ca²⁺ binding to TnC or that bound crossbridges inhibit tropomyosin returning to the off-state, are responsible for the increase in relaxation time with SL. Kentish and Wrzosek (143) and Janssen et al. (7) also recorded a decrease in RT₅₀ at shorter SLs, but the effect of SL would also be confounded by changes in the tension transient, which potentially causes changes in the binding of Ca²⁺ to TnC and thus has an effect on the relaxation time. The Ca²⁺ sensitivity affects both maximum tension and RT₅₀, coinciding with the observations of Janssen and Hunter (4). However, since no definitive mechanism currently exists to explain the length-dependent Ca²⁺ sensitivity as discussed above, the dependence of RT₅₀ on Ca²⁺ sensitivity cannot be determined.

Model simulations show that RT₅₀ increases with the [Ca²⁺]_i transient magnitude, when the [Ca²⁺]_i transient and the unbinding rate of Ca²⁺ from TnC are close to their reference values. Increasing the extracellular Ca²⁺ concentration, which increases the [Ca²⁺]_i transient magnitude, has been observed to cause an increase (3,4), a decrease (3), and no change (144) in relaxation rates. Fig. 12 B shows the combined effects of the unbinding rate and the [Ca²⁺]_i transient magnitude on RT₅₀. The [Ca²⁺]_i transient causes both a decrease and an increase in RT₅₀, as seen by Gao et al. (3), depending on the [Ca²⁺]_i transient magnitude and the unbinding rate of Ca²⁺ from TnC.

The unbinding of Ca²⁺ from TnC and the [Ca²⁺]_i transient are the most significant factors in determining RT₅₀. The coupled effects of TnC and the [Ca²⁺]_i transient on RT₅₀ were determined using a broader range of parameter values (see Fig. 12 B) and showed that the minimum RT₅₀ was a function of both the [Ca²⁺]_i transient and the unbinding rate of Ca²⁺ from TnC, such that the effect of the [Ca²⁺]_i transient and the unbinding rate of Ca²⁺ from TnC could both either increase or decrease RT₅₀ depending on the reference value of each parameter. For example, for the base [Ca²⁺]_i transient, increasing the unbinding rate of Ca²⁺ from TnC increase or decreases RT₅₀ when the unbinding rate is more or less than 130%, respectively.

Sarcomere velocity and crossbridges inhibiting the movement of tropomyosin to the off-state were not included in this study. To attain realistic tension twitches, where the muscle contracts, requires the use of physiological force boundary conditions. To account for the affect of SL velocity under this protocol requires the definition of a passive stiffness model, which is beyond the scope of this article. Crossbridge inhibition is not accounted for explicitly, and could potentially affect a number of tropomyosin kinetics parameters (α_r2, n_r, K_z). Crossbridge inhibition may, in part, account for the significant effect of the [Ca²⁺]_i transient on RT₅₀, and may need to be considered at a later date.

Sensitivity

The model described above contains numerous parameters defined using a range of experimental data. To fully assess the validity of the model, the sensitivity of model results to individual parameters needed to be determined. The relevance and quality of data used to determine sensitive parameters was then assessed to attain a sense of the accuracy and degree of confidence in the model.

The sensitivity of the model to specific parameters was determined by calculating the Jacobean of the root-mean-square change in tension (for the unclamped sarcomere tension trace), with respect to the parameter set. The tension transient was most sensitive to Ca_Trpn50ref and K_z, and insensitive to changes in either the crossbridge kinetics or Ca²⁺ binding to TnC.

Ca_Trpn50ref was determined using the Ca²⁺ bound to TnC model and free Ca²⁺ (Eq. 12) at half-activation values from skinned and intact preparations. The Ca_Trpn50ref value for skinned preparations was calculated from 17 experiments at different SL values. However, adjusting Ca_Trpn50ref to account for the affects of skinning used only three measures at one SL and the effect of skinning on the length dependence of half-activation was assumed to be negligible. To confirm the validity of the value and length dependence of Ca_Trpn50ref, further experiments on intact preparations are required as, although many factors have been qualitatively implicated in determining the effects of skinning on the half-activation, there is currently no established relationship between skinned and intact half-activation values.

The value K_z was defined by fitting Eq. 14 to tension transients after step changes in Ca²⁺. The value K_z was determined by fitting a nonlinear equation to characteristic tension traces and our confidence in its value is therefore less than parameters directly reported in the literature, which are taken from numerous samples. The experiments used to determine K_z were also performed at 12–15°C in air and the tension reached during the experiment was commonly 50% of the maximum tension. Further experiments at more physiological temperatures over a broader range of tensions would assist in better defining K_z to increase the validity of the model and conclusions drawn from it.

Limitations

This model aims to account for a wide range of experimental results and mechanisms. Although the model is capable of representing a large proportion of available experimental data, it is an approximation, and as such, has limitations, which must be taken into consideration when considering results. In the model, certain components have a limited representation of experimental results and some potential mechanisms were not included. These include: the length- and tension-dependence of Ca²⁺ binding to TnC, the changes in half-activation due to skinning, relying on the approximation of the Hill equation, the models' approximation of the F-pCa curve, and inhomogeneous sarcomere shortening in relaxation, which are discussed below.

The tension-dependent binding of Ca²⁺ to TnC is only valid for the physiological range of SL values and fails to predict the concentration of bound Ca²⁺ during contraction at low free Ca²⁺ concentrations. The model is designed to operate within the physiological range of SL values for intact preparations between 1.8 < SL < 2.3 μm. Extensions beyond 2.3 μm do not occur under physiological conditions in vivo. Simulating isolated experimental protocols where SL > 2.3μm, the model will predict that more than one Ca²⁺ ion binding to each available site II in TnC as k_off becomes less than zero, which is not reported. For simulations where SL < 2.3 μm, k_off > 0 if T < γT_ref for all tension values, resulting in the condition that γ > 1 + β₀(λ−1), which is true for the proposed parameter set. Combining the active tension model described here with a model for passive tension (11) would constrain SLs to be below 2.3 μm and so avoid this problem. The fraction of Ca²⁺ bound to TnC at lower SL values reported by Fuchs et al. (44) is not accurately represented by the model. However, there is an amplification of experimental noise inherent in the method used here to extrapolate the fraction of Ca²⁺ bound to site II from the experimental results (discussed in the Tension-Dependent Ca²⁺ unbinding rate from Troponin C) which increases as bound Ca²⁺ decreases. This impedes the accurate measurement Ca²⁺ bound to site II using the total quantity of Ca²⁺ bound to all TnC sites as the sole source of experimental data.

The half-activation of cardiac muscle is often reduced after skinning. Here the half-activation is scaled to match intact preparation data, but the half-activation length dependence (β₀) is not defined for intact preparations and as such it is assumed to be independent of skinning. Further experiments on the length dependence of Ca²⁺ activation would help to elucidate which parts of the contractile machinery are affected by the skinning process.

Crossbridge kinetics after a length perturbation was described by the fading memory model. The static nonlinear component of the fading memory model is described by the Hill equation. The Hill equation exhibits two problems. Firstly, the Hill curve has been shown to be insufficient to represent experimental force-velocity results for small velocities (111). Secondly, the force-velocity relationship for negative velocities is not described by the Hill curve, which is therefore extended here to cover both positive and negative velocities. Further experimental evidence is required to appropriately characterize the force velocity curve for negative velocities.

The steady-state F-pCa curve (see Fig. 5) is steeper for midlevel [Ca²⁺] than low and high [Ca²⁺], which is a problem endemic in mean-field-averaged models as noted by Rice and de Tombe (76). A potential solution to this problem is the explicit representation of the regulatory proteins as proposed by Rice and de Tombe, but this method is not currently computationally feasible for the simulations performed here.

Recent tension transients after a rapid drop in local Ca²⁺ recorded by Stehele et al. (5,145) and Belus et al. (146) did not contain the fast and slow processes seen in light-activated Ca²⁺ chelator experiments. Stehele et al. and Belus et al. found an initial linear relaxation phase followed by exponential relaxation similar to that seen in skeletal muscle (100). The initial linear phase is most pronounced in the absence of P_i. However, the linear phase is muted when P_i is at physiological concentrations of 2 mM (146), and therefore is not considered in this model.

Our results reveal the importance of the [Ca²⁺]_i transient and unbinding of Ca²⁺ from TnC as the significant factors affecting RT₅₀. The tension dependence of Ca²⁺ binding to TnC and length-dependent Ca²⁺ have a lesser affect and length-dependent maximum tension appears to have a minimal influence on RT₅₀. A sensitivity analysis of the tension transient found simulation results to be most sensitive to tropomyosin kinetics, but this does not detract from the principal finding that the [Ca²⁺]_i transient and the unbinding of Ca²⁺ from TnC are the significant factors in determining the relaxation of cardiac muscle, based on the current understanding of cardiac contraction.

Applications

In this study, a model was developed to rationalize a wide range of experimental data and to rationalize the mechanisms that control relaxation. The efficient computational form of the model allows it to be applied to both single and multicellular simulations. For example, the active mechanics model described here has been incorporated into continuum models of tissue to determine large deformation mechanics solutions and, ultimately, coupled electromechanics interactions within the heart. This will ultimately provide the ability to link cellular mechanisms in health and pathological states to whole organ function and, in doing so, further the goals of integrative physiology efforts such as the IUPS physiome project (147).

SUMMARY

A wide range of experimental results has been surveyed to provide a parameter set for the HMT model, which coincides with the majority of reported experimental results. The model illustrates the importance of the coupled effects of the Ca²⁺ transient and the unbinding rate of Ca²⁺ from TnC on the rate of relaxation and the limited effects of SL on RT₅₀. The model tension traces are most sensitive to the bound Ca²⁺ at half-activation in intact preparations and K_z. Further experiments characterizing the length dependence of half-activation in intact preparations and relaxation kinetics performed under more physiological conditions over a wider range of tensions would assist in improving the confidence in model results. The model is designed to be applied in multiscale tissue and organ simulations, which is necessary to gain further understanding of the effects of tissue and passive mechanics on relaxation.

Data from experimental studies used to validate the model are listed. A full description of the model is available using the CellML ontology at www.cellml.org/examples/maths_pdf/NHS_maths.pdf (148).