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Copyright © 2006 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 90, Issue 5, 1697-1722, 1 March 2006

doi:10.1529/biophysj.105.069534

Muscle and Contractility

A Quantitative Analysis of Cardiac Myocyte Relaxation: A Simulation Study

S.A. Niederer^,  , P.J. Hunter and N.P. Smith

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

Address reprint requests to S. A. Niederer.

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:

Display large version of this figure

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.

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 70μM 15,16:

(1)

Display large version of this figure

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.

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)

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 55,56,57. 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)

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 2μM, 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 1.8–2.3μm 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.

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)

(5)

(6)

(7)

(8)

The Ca²⁺ affinity for TnC has also been shown to vary with SL32,40,44. 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 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)

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).

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 ≈10–12s⁻¹ and ≈2–4s⁻¹, respectively, at 12–15°C for rat and guinea pig 60,62,64,65; and ≈16–18s⁻¹ and ≈1s⁻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 73.5s⁻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 53.4ms, approximately the same as the control rat measurements at 15°C from Fitzimons et al. 62, which had a half-time to relaxation of 64.5ms. 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 SL6,64, 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 2s⁻¹, 1.75s⁻¹, 0.15, and 3, respectively.