β-adrenergic effects on cardiac myofilaments and contraction in an integrated rabbit ventricular myocyte model

Abstract

A five-state model of myofilament contraction was integrated into a well-established rabbit ventricular myocyte model of ion channels, Ca²⁺ transporters and kinase signaling to analyze the relative contribution of different phosphorylation targets to the overall mechanical response driven by β-adrenergic stimulation (β-AS). β-AS effect on sarcoplasmic reticulum Ca²⁺ handling, Ca²⁺, K⁺ and Cl⁻ currents, and Na⁺/K⁺-ATPase properties were included based on experimental data. The inotropic effect on the myofilaments was represented as reduced myofilament Ca²⁺ sensitivity (XBCa) and titin stiffness, and increased cross-bridge (XB) cycling rate (XBcy). Assuming independent roles of XBCa and XBcy, the model reproduced experimental β-AS responses on action potentials and Ca²⁺ transient amplitude and kinetics. It also replicated the behavior of force-Ca²⁺, release-restretch, length-step, stiffness-frequency and force-velocity relationships, and increased force and shortening in isometric and isotonic twitch contractions. The β-AS effect was then switched off from individual targets to analyze their relative impact on contractility. Preventing β-AS effects on L-type Ca²⁺ channels or phospholamban limited Ca²⁺ transients and contractile responses in parallel, while blocking phospholemman and K⁺ channel (I_Ks) effects enhanced Ca²⁺ and inotropy. Removal of β-AS effects from XBCa enhanced contractile force while decreasing peak Ca²⁺ (due to greater Ca²⁺ buffering), but had less effect on shortening. Conversely, preventing β-AS effects on XBcy preserved Ca²⁺ transient effects, but blunted inotropy (both isometric force and especially shortening). Removal of titin effects had little impact on contraction. Finally, exclusion of β-AS from XBCa and XBcy while preserving effects on other targets resulted in preserved peak isometric force response (with slower kinetics) but nearly abolished enhanced shortening. β-AS effects on XBCa vs. XBcy have greater impact on isometric vs. isotonic contraction, respectively.

1. Introduction

β-adrenergic stimulation (β-AS) is an important physiological modulator of cardiac activity, increasing peak force (inotropy) and the rate of force development and relaxation (lusitropy). β-AS raises intracellular cAMP levels, activating protein kinase A (PKA), which in turn phosphorylates key proteins involved in excitation-contraction (E–C) coupling. These include the L-type Ca²⁺ channel [1, 2] and phospholamban (PLB), respectively increasing Ca²⁺ entry into the cytoplasm and sarcoplasmic reticulum (SR) Ca²⁺ reuptake by the SR Ca²⁺-ATPase2a (SERCA2a) [3, 4], and these effects contribute to increased SR Ca²⁺ content, inotropy and lusitropy. High SR Ca²⁺ load also increases Ca²⁺ leak and fractional Ca²⁺ release during E-C coupling [5]. β-AS can increase the probability of ryanodine receptor Ca²⁺ release channel (RyR2) opening in response to cytosolic Ca²⁺ [6], but that may be mediated more by Ca²⁺/calmodulin-dependent protein kinase II (CaMKII) activation, downstream of nitric oxide synthase (nNOS) or Epac (exchange protein activated by cAMP) [7–11]. K⁺ channels are also targets of PKA phosphorylation and can shorten action potential (AP) duration (APD) [12, 13] during β-AS.

At the myofilaments, PKA phosphorylates troponin I (TnI) and myosin binding protein-C (MyBP-C) [14–17]. Although experimental studies have attempted to link specific contractile effects of β-AS to specific TnI and MyBP-C phosphorylation sites, precise roles in altering myofilament Ca²⁺ sensitivity (XBCa) and cross-bridge (XB) cycling rate (XBcy) are still debated. Some studies have suggested that PKA phosphorylation of TnI predominates in both decreased XBCa and enhanced XBcy [18], and the optimization of cardiac function at different heart rate, load and inotropic state [19]. Conversely, it was postulated that TnI and MyBP-C phosphorylation have separate effects on XBCa and XBcy [20, 21], a shared responsibility in PKA-dependent decrease of XBCa [22, 23], or a central role of MyBP-C in the PKA-dependent increased XBcy kinetics [22]. However, the relative PKA effects of these two proteins on twitch contractions, Ca²⁺ transients and APs is lacking. A computational model, as we describe here, can be a valuable complementary tool for further analysis of the impact of XBCa and XBcy and their interaction during β-AS induced inotropy.

Myocyte models have evolved from mathematical descriptions of ion channels to detailed formulations of Ca²⁺ transport mechanisms [24, 25], cell compartmentalization [25, 26], cell signaling pathways [27–30], metabolic processes [31], and pH regulation [32]. Even though mechanical models of the cardiac muscle have been developed [33], they have typically been divorced from electrical/Ca²⁺ handling models [34–37]. Some studies have begun to fill this gap, including descriptions of length and force-dependent processes of Ca²⁺-induced XB formation [38] and length-dependent activation [39], but most have been limited to myocyte ion handling and electrophysiological descriptions [27, 29, 40].

The purpose of this study is three-fold: (1) to help clarify unresolved questions of how altered Ca²⁺ sensitivity and XB cycling rate interact during β-AS in ventricular myocytes (and how putative TnI and MyBP-C effects may interact), (2) to provide a new quantitative framework to study the complex interaction of β-AS dynamic effects on Ca²⁺ transient, AP, and contractile properties, which are not well resolved, and (3) to provide a freely available new tool for all investigators to further study these aspects. We have built the β-AS effect on our prior detailed contraction model [37] and incorporated that into an updated Soltis-Saucerman model [29], which integrates the Shannon-Bers Ca²⁺-electrophysiological myocyte model [25] with dynamic descriptions of CaMKII and PKA signaling pathways [28, 41] and dynamic target phosphorylation.

2. Methods

2.1 Generation of XB force

Our previously described mechanical model [37] has been slightly modified. It represents a muscle or myocyte as a series arrangement of units, each defined by the half-sarcomere length and the muscle cross-sectional area. The half-sarcomere is composed of inextensible thick and thin filaments in parallel with an internal elastic load (Fig. 1A). The thick (ThF) and thin filaments (Tf) can slide past each other defining a zone of overlap where XBs can attach to the thin filament. According to the XB theory, attached XBs act as independent force generators, occupying different states in the XB cycle [42]. Individual XB force is f_i = a × h_i, where a is the elastic constant of one attached XB and hi = XB elongation. Then, total XB force (Ft) is:

$$\begin{array}{c}{\mathrm{F}}_{\mathrm{t}}=\sum {\mathrm{f}}_{\mathrm{i}}=\mathrm{a}\sum {\mathrm{h}}_{\mathrm{i}}=\mathrm{N}\times \mathrm{a}\times (\sum {\mathrm{h}}_{\mathrm{i}}/\mathrm{N})\\ {\mathrm{F}}_{\mathrm{t}}=\mathrm{N}\times \mathrm{a}\times \mathrm{h}\end{array}$$ (1)

where N is number of attached XBs, N × a represents the elastic constant corresponding to the pool of attached XBs and h is the mean elongation of all XBs. Thus, Ft is the force of an equivalent XB, representing all attached XBs, and h is the equivalent XB length.

Figure 1.

Model. A: Constitutive muscle unit, consisting of half-sarcomere length (L) composed of thick (ThF) and thin (Tf) filaments in parallel with an elastic element. The equivalent cross-bride (XB) representing all attached cross-bridges is part of the ThF. It attaches to the Tf by the mobile end of its elastic structure, with h elongation, defining an inextensible half-sarcomere length X = L − h. A series elastic element with length L_s accounts for compliant muscle ends and together with L make up total muscle length (L_m). B: Cross-bridge dynamics showing steadystate XB elongation (h_r), its decrease due to ΔL and later return to h_r at shorter half-sarcomere length (L-ΔL). C: Four compartment myocyte model: Bulk Cytosol, Cleft, sarcoplasmic reticulum (SR) and subsarcolemmal space (Sub-SL) [25, 29] with ion currents as described by these authors. This model is coupled to the myofilament force development model consisting of 5-state troponin systems (TS) with Ca²⁺ binding. Each TS is composed of three adjacent troponin–tropomyosin regulatory units able to act cooperatively to bind Ca²⁺ in three successive steps. Troponin systems are: free TS; Ca²⁺ bound to TS without attached XBs (TSCa₃), Ca²⁺ bound to TS with attached XBs in the weak state (TSCa₃~), Ca²⁺ bound to TS with attached XBs in the power state (TSCa₃*), and TS without Ca²⁺ with attached XBs in the power state (TS*). Baseline mechanical parameters and all changes used for ISO are in Table 1.

Total XB force is normalized for cross-sectional area (Fb) measured at a defined reference state:

$${\mathrm{F}}_{\mathrm{b}}={\mathrm{F}}_{\mathrm{t}}/\text{reference area}$$ (2)

where reference area = vol_r/L_r with vol_r = half-sarcomere volume of myoplasmic compartment (Bulk Cytosol) containing the mechanical elements and L_r = half-sarcomere slack length, i.e, resting conditions for force close to zero. Then, substituting Eq. 1 into Eq. 2 gives:

$${\mathrm{F}}_{\mathrm{b}}=\mathrm{N}\times \mathrm{a}\times \mathrm{h}\times {\mathrm{L}}_{\mathrm{r}}/{\text{vol}}_{\mathrm{r}}$$ (3)

Since N = Avogadro’s number (A) × (moles of attached XBs), it can be expressed as:

$$\mathrm{N}=\mathrm{A}\times [\mathrm{XB}]\times {\text{vol}}_{\mathrm{r}}$$

where [XB] is total attached XB concentration (see Ca²⁺ kinetics). Replacing N in Eq. 3:

$${\mathrm{F}}_{\mathrm{b}}=\mathrm{A}\times [\mathrm{XB}]\times \mathrm{a}\times \mathrm{h}\times {\mathrm{L}}_{\mathrm{r}}$$

Grouping A × a × L_r and changes in concentration units into A_e, results in:

$${\mathrm{F}}_{\mathrm{b}}={\mathrm{A}}_{\mathrm{e}}\times [\mathrm{XB}[\times \mathrm{h}$$ (4)

The equivalent XB is considered to be situated at the free end of the half myosin filament. Consequently, in the model, half-sarcomere length (L) has two components: an elastic component (h) and a non-elastic component (X) formed by the half-sarcomere thick filament and the parts of the thin filament not overlapping either with the thick filament or h (Fig. 1A). Then:

$$\mathrm{L}=\mathrm{X}+\mathrm{h}$$ (5)

We assume that individual XBs attach and detach cyclically and asynchronously from the thin filament, but the equivalent XB remains attached while there is at least one attached XB. In steady state conditions, the equivalent XB has a constant length h_r and during a quick sarcomere shortening equal to ΔL, h instantly absorbs this length change (Fig. 1B), so: h = h_r − ΔL.

The equivalent XB then detaches and reattaches in a different position along the thin filament to return to h_r. This is represented as sliding of the point of attachment of the equivalent XB to reproduce the asynchronous behavior of individual XBs. Thus, the return to h_r produces a change in X expressed as the differential equation:

$$\mathrm{dX}/\mathrm{dt}=\mathrm{B}\times (\mathrm{h}-{\mathrm{h}}_{\mathrm{r}})$$ (6)

and substitution by Eq. 5 gives:

$$\mathrm{dX}/\mathrm{dt}=\mathrm{B}\times (\mathrm{L}-\mathrm{X}-{\mathrm{h}}_{\mathrm{r}})$$ (7)

where dX/dt is the velocity of motion of the mobile equivalent XB end and B is the involved parameter [43]. The sliding movement can also be expressed from the derivation of Eq. 5 and substitution by Eq. 7 as:

$$\mathrm{dL}/\mathrm{dt}=\mathrm{B}\times (\mathrm{h}-{\mathrm{h}}_{\mathrm{r}})+\mathrm{dh}/\mathrm{dt}.$$ (8)

Thus, once shortening ends, dL/dt = 0 and in Eq. 8 returns h exponentially to h_r. Similar criteria apply to the description of muscle stretching and concomitant XB elongation.

Similarly to our previous model [37], two states of attached XBs are defined: XB_w (weak state, with h = h_w), a state in which the equivalent XB is attached to the filament but has not yet developed the power stroke, and XB_p (power state, with h = h_p), a state in which the equivalent XB can develop the power stroke [36]. Contrary to XB_p, XB_w elongation h_wr is assumed close to zero in steady state conditions. However, when the muscle shortens or stretches, the XB_w elastic element is passively stretched, h_w ≠ 0, and XB_w respectively contributes negatively or positively to F_b. Then, according to Eq. 4, total XB force becomes:

$${\mathrm{F}}_{\mathrm{b}}={\mathrm{A}}_{\mathrm{w}}\times [{\mathrm{XB}}_{\mathrm{w}}]\times {\mathrm{h}}_{\mathrm{w}}+{\mathrm{A}}_{\mathrm{p}}\times [{\mathrm{XB}}_{\mathrm{p}}]\times {\mathrm{h}}_{\mathrm{p}}$$ (9)

where A_w and A_p are elastic constants for the weak and power states. According to this assumption Eq. 5 becomes:

$$\mathrm{L}={\mathrm{X}}_{\mathrm{w}}+{\mathrm{h}}_{\mathrm{w}}$$

and

$$\mathrm{L}={\mathrm{X}}_{\mathrm{p}}+{\mathrm{h}}_{\mathrm{p}}$$

showing that for the same L, the inextensible X lengths and the equivalent XB elastic lengths differ according to the XB state. Similarly, from Eq. 8, the two equivalent XB kinetic equations are:

$$\mathrm{dL}/\mathrm{dt}={\mathrm{B}}_{\mathrm{w}}\times ({\mathrm{h}}_{\mathrm{w}}-{\mathrm{h}}_{\mathrm{wr}})+{\mathrm{dh}}_{\mathrm{w}}/\mathrm{dt}$$ (10)

$$\mathrm{dL}/\mathrm{dt}={\mathrm{B}}_{\mathrm{p}}\times ({\mathrm{h}}_{\mathrm{p}}-{\mathrm{h}}_{\mathrm{pr}})+{\mathrm{dh}}_{\mathrm{p}}/\mathrm{dt}$$ (11)

where B_w and B_p are the weak and power parameters.

2.2 Parallel and series elasticity

As shown in Fig. 1A, the muscle unit includes an undamped parallel elastic element that symbolizes the sum of the non-linear titin and the linear cellular passive element components attributed to intermediate filaments (Fig. S1); then, parallel elastic force (F_p) results as:

$${\mathrm{F}}_{\mathrm{p}}={\mathrm{K}}_{\mathrm{e}}{(\mathrm{L}-{\mathrm{L}}_{\mathrm{o}})}^5+{\mathrm{L}}_{\mathrm{e}}(\mathrm{L}-{\mathrm{L}}_{\mathrm{o}})$$ (12)

where K_e and L_e, respectively represent titin and cellular element elasticities for F_p normalized by cross-sectional area, and L_o is slack length (L for F_p = 0). Then, total muscle force (F_m) is the sum of F_b plus the force needed to stretch F_p:

$${\mathrm{F}}_{\mathrm{m}}={\mathrm{F}}_{\mathrm{b}}+{\mathrm{F}}_{\mathrm{p}}.$$

An elastic element in series with the muscle (Fig. 1A) is added to describe compliant muscle ends, producing elastic force F_s:

$${\mathrm{F}}_{\mathrm{s}}={\mathrm{F}}_{\mathrm{m}}=\mathrm{\alpha }\times ({\mathrm{e}}^{{\mathrm{\beta L}}_{\mathrm{S}}}-1)$$ (13)

where α and β are parameters defining the elastic relation and L_s is series elastic length, showing that changes in F_m produced by F_b elicit changes in L_s. Then, total muscle length (L_m) results as:

$${\mathrm{L}}_{\mathrm{m}}={\mathrm{L}}_{\mathrm{s}}+\mathrm{L}.$$

Therefore, according to the model, in isometric conditions, changes in F_m produced by F_b generation trigger changes in L_s and hence in L because L_m is constant. Variations in L then lead to h changes whose return to h_r is achieved through Eq. 6. In isotonic conditions, where F_m is constant, F_b and F_p change due to L variations, and again these L changes lead to h adjustments with the ensuing return to h_r through Eq. 6. Finally, in isosarcometric contractions L is constant, L_s = 0, and consequently L_m = L [37].

2.3 Myofilament Ca²⁺ kinetics

Ca²⁺ kinetics is represented by five states of troponin systems (TS) (Fig. 1C), which is a slight simplification of the prior six state model [37], by condensing an intermediate state (TS^~) into the TS* to TS transition. Each TS is defined as 3 adjacent troponin-tropomyosin regulatory units (RU) [36], so that each can bind one Ca²⁺ ion (TSCa, TSCa₂, TSCa₃). In this scheme, Ca²⁺ binding to the three RU within one TS is cooperative, and we assume that all three Ca²⁺ bind in one step (so only TSCa₃ is indicated) driven by the on- and off-rate constants (Y_b and Z_b, Fig. 1C), where [Ca²⁺]i is free Ca²⁺ concentration in the bulk cytosol. It is further postulated that XB attachment takes place when Ca²⁺ RUs are fully activated and driven by the forward (f) and reverse (g) rate constants. The five TS states are thus: TS (free); TSCa₃ (Ca²⁺ bound to TS, without XB attachment), TSCa₃~ (weak binding XB (XB_w) with Ca²⁺ bound to TS), and TS* (attached XBs in the power state (XB_p) involving TS without Ca²⁺). Because force depends on attached [XB], Eq. 9 results as:

$${\mathrm{F}}_{\mathrm{b}}={\mathrm{A}}_{\mathrm{w}}(3[{{\mathrm{TSCa}}_3}^{~}]){\mathrm{h}}_{\mathrm{w}}+{\mathrm{A}}_{\mathrm{p}}(3[{\mathrm{TSCa}}_3\ast ])+3[\mathrm{TS}\ast ]{\mathrm{h}}_{\mathrm{p}}$$ (14)

and redefining A_p and A_w to include the common factor 3 gives:

$${\mathrm{F}}_{\mathrm{b}}={\mathrm{A}}_{\mathrm{w}}([{{\mathrm{TSCa}}_3}^{~}]){\mathrm{h}}_{\mathrm{w}}+{\mathrm{A}}_{\mathrm{p}}([{\mathrm{TSCa}}_3\ast ])+[\mathrm{TS}\ast ]{\mathrm{h}}_{\mathrm{p}}.$$ (15)

Rates of change of TS states (Fig 1C) are described by the following differential equations:

$$\mathrm{d}[{\mathrm{TSCa}}_3]/\mathrm{dt}={\mathrm{Y}}_{\mathrm{b}}[\mathrm{TS}][{\mathrm{Ca}}^{2+}]{\mathrm{i}}^3-{\mathrm{Z}}_{\mathrm{b}}[{\mathrm{TSCa}}_3]+{{\mathrm{g}[\mathrm{TSCa}}_3}^{~}]-\mathrm{f}[{\mathrm{TSCa}}_3]$$ (16)

$$\mathrm{d}[{{\mathrm{TSCa}}_3}^{~}]/\mathrm{dt}=\mathrm{f}[{\mathrm{TSCa}}_3]-\mathrm{g}[{{\mathrm{TSCa}}_3}^{~}]+{\mathrm{Z}}_{\mathrm{p}}[{\mathrm{TSCa}}_3\ast ]-{\mathrm{Y}}_{\mathrm{p}}[{{\mathrm{TSCa}}_3}^{~}]$$ (17)

$$\mathrm{d}[{\mathrm{TSCa}}_3\ast ]/\mathrm{dt}={\mathrm{Y}}_{\mathrm{p}}[{{\mathrm{TSCa}}_3}^{~}]-{\mathrm{Z}}_{\mathrm{p}}[{\mathrm{TSCa}}_3\ast ]+{\mathrm{Z}}_{\mathrm{r}}[\mathrm{TS}\ast ][{\mathrm{Ca}}^{2+}]{\mathrm{i}}^3-{\mathrm{Y}}_{\mathrm{r}}[{\mathrm{TSCa}}_3\ast ]$$ (18)

$$\mathrm{d}[\mathrm{TS}\ast ]/\mathrm{dt}={\mathrm{Y}}_{\mathrm{r}}[{\mathrm{TSCa}}_3\ast ]-{\mathrm{Z}}_{\mathrm{r}}[\mathrm{TS}\ast ][{\mathrm{Ca}}^{2+}]{\mathrm{i}}^3-\mathrm{gd}[\mathrm{TS}\ast ]$$ (19)

and introducing [TSt] as total concentration, the five TS can be expressed as:

$$[\mathrm{TS}]=[\mathrm{TSt}]-[{\mathrm{TSCa}}_3]-[{{\mathrm{TSCa}}_3}^{~}]-[{\mathrm{TSCa}}_3\ast ]-[\mathrm{TS}\ast [,$$ (20)

where Y_b, Z_b, f, g, Y_p, Z_p, Y_r, Z_r, and g_d are reaction rate parameters with subscripts b, p, r and d respectively related to binding of Ca²⁺, power state transition, reversal of Ca²⁺ binding and dissociation of XB (Fig 1C).

The model utilizes three length-dependent reaction rate parameters. The first is the sarcomere attachment rate function which accounts for XB attachment in the zone of overlap between thick and thin filaments [42]:

$$\mathrm{f}={\mathrm{Y}}_{\mathrm{a}}\times {\mathrm{e}}^{-{\mathrm{R}}_{\mathrm{a}}{(\mathrm{L}-{\mathrm{L}}_{\mathrm{a}})}^2}$$ (21)

where L_a is the optimal overlap length for maximum XB attachment, Y_a is maximum f for L = L_a and R_a commands the kurtosis of the curve, changing the effective zone of overlap between thin and thick filaments [37, 43]. The second is the redefined velocity-dependent XB detachment rate constant g, with the asymmetrical detachment behavior postulated by Huxley [42] (Fig. 2A):

$$\mathrm{g}={\mathrm{Z}}_{\mathrm{a}}+{\mathrm{Y}}_{\mathrm{h}}$$ (22)

where Y_h is the detachment function characterized as

Figure 2.

Asymmetrical detachment function validated by its effect on afterloaded contractions. A: Normalized detachment as a function of the difference between weak (h_w) and steady state weak (h_wr) XB length. The previous symmetrical detachment function [37] based on the proposition of Slawnych et al. [44] was corrected, incorporating the asymmetry (New function) postulated by Huxley [42]. B: Experimental afterloaded contractions from Brutsaert et al. [86]. C: Simulation of myocyte afterloaded contractions with Negroni et al. detachment function [37] and the new detachment function, resembling more accurately the experimental reports.

$${\mathrm{Y}}_{\mathrm{h}}={\mathrm{F}}_{\mathrm{h}}\times {\mathrm{Y}}_{\mathrm{v}}\times (1-{\mathrm{e}}^{-\gamma {({\mathrm{h}}_{\mathrm{w}}-{\mathrm{h}}_{\mathrm{wr}})}^2}).$$ (23)

In Eq. 23, (1 – e^{−γ(hw – hwr)}) represents the symmetrical detachment function used by Slawnych et al. [44] where γ defines the kurtosis of the curve and sarcomere L velocity dependence through (h_w-h_wr), F_h provides asymmetry [42] (with F_h = 0.1 or F_h = 1 for h_w > h_wr or h_w < h_wr, respectively), Y_v amplifies the detachment effect, and Z_a is a constant representing the value of g when h_w = h_wr. This more accurate g description resulted in improved simulation of afterloaded contractions (Fig. 2B). The third is the sarcomere length-dependent rate function of XB detachment from TS* to TS given by:

$${\mathrm{g}}_{\mathrm{d}}={\mathrm{Y}}_{\mathrm{d}}\times {\mathrm{e}}^{-{\mathrm{Y}}_{\mathrm{C}}(\mathrm{L}-{\mathrm{L}}_{\mathrm{C}})}$$ (24)

where L_c determines the range of sarcomere length around which g_d varies, Y_c defines the rate of change of g_d, and Y_d is the value of g_d for L = L_c. This equation represents the lattice space effect on the rate of XB detachment whereby g_d decreases at larger lengths reducing lattice space [45].

To express Ca²⁺ binding and buffering by myofilaments, the low affinity troponin C (TnC_low) Ca²⁺ buffer in the Soltis-Saucerman model [29] (from the Shannon-Bers model [25]) was interpreted as the sum of change of all TS bound to Ca²⁺:

$$\mathrm{d}[{\mathrm{TnC}}_{\text{low}}\cdot \mathrm{Ca}]/\mathrm{dt}=3(\mathrm{d}[{\mathrm{TSCa}}_3]/\mathrm{dt}+\mathrm{d}[{{\mathrm{TSCa}}_3}^{~}]/\mathrm{dt}+\mathrm{d}[{\mathrm{TSCa}}_3\ast ]/\mathrm{dt})$$ (25)

expressing buffer action dependence on the contractile machinery and hence on L_m and F_m and the mode of contraction.

Table 1 shows the classification of contractile parameters. One group of parameters is structural (Strct). A second group of parameters corresponds to XBCa, involving the rate constants Y_b, Z_b, Y_r and Z_r acting in transitions where there are only changes in Ca²⁺ binding to TS with no modifications in the XB force generating capacity. The third group of parameters comprises XBcy rate constants Z_p and Y_p, and parameters related to rate functions g, f, and g_d, acting in transitions of XB attachment and detachment, altering the force generating capacity and cycling rate without producing changes in Ca²⁺ binding to TS.

Table 1.

Mechanical parameter classification and values.

Role	Symbol	Defining Eq.	Units	Value	Effect of 100 nM [ISO]
Strct	Ap	9, 14, 15	mN mm−2 μm−1 μM−1	2700
Strct	Aw	9, 14, 15	mN mm−2 μm−1 μM−1	540
Strct	α	13	mN mm−2	0.5
Strct	β	13	μm−1	80
Strct	Ke	12	mN mm−2 μm−5	105000	50% decrease [56]
Strct	Le	12	mN mm−2 μm−1	10
Strct	Lo	12	μm	0.97
Strct	[TSt]	20	μM	23
XBCa	Yb	16	μM−3 ms−1	0.1816
XBCa	Yr	18, 19	ms−1	0.1397	2.2-fold increase [18–22, 63]
XBCa	Zb	16	ms−1	0.1397	4.2-fold increase [18–22, 63]
XBCa	Zr	18, 19	μM−3 ms−1	7.2626	80% increase [18–22, 63]
XBcy	Bw	10	ms−1	0.35	3.4-fold increase [18–22, 63]
XBcy	Bp	11	ms−1	0.5	3.4-fold increase [18–22, 63]
XBcy	hpr	11	μm	0.006
XBcy	hwr	10, 23	μm	0.0001
XBcy	Fh	23	adimensional	0.1 or 1
XBcy	γ	23	μm−2	28000
XBcy	La	21	μm	1.15
XBcy	Lc	24	μm	1.05
XBcy	Ra	21	μm−2	20	40% decrease (model fit)
XBcy	Ya	21	ms−1	0.0023	24% increase [18–22, 63]
XBcy	Yc	24	μm−1	4	40% decrease (model fit)
XBcy	Yd	24	ms−1	0.028	2.2-fold increase (model fit)
XBcy	Yp	17, 18	ms−1	0.1397	2.2-fold increase [18–22, 63]
XBcy	Yv	23	ms−1	0.9	60% increase (model fit)
XBcy	Za	22	ms−1	0.0023	24% increase [18–22, 63]
XBcy	Zp	17, 18	ms−1	0.2095	2.2-fold increase [18–22, 63]

2.4 Integrated ionic, contractile, and β-AS model

We embedded the contraction model with the Soltis-Saucerman model [29], and updated the formulation of target phosphorylation by PKA in several ways (detailed in the Supplement): (1) I_Ks phosphorylation kinetics were slowed, as done previously [46] in accordance with recent experimental observations [47]; (2) phosphorylation of phospholemman (PLM) to increase Na⁺–K⁺-ATPase (NKA) affinity for internal Na⁺ [48] was added as done previously [30, 46, 49, 50]; (3) PKA-dependent modulation of I_Kr was introduced based on experiments by Harmati et al. [51] with kinetics that we used for I_Ks phosphorylation previously [52]; (4) formulation of PKA-dependent modulation of I_Cl(Ca) [53] was added with the same kinetics as for CFTR phosphorylation. Figure S2 shows the responses of ionic PKA targets to isoproterenol (ISO), and Table 2 describes the effects of 100 nM [ISO] administration; (5) PKA-dependent effects on contraction were added, with phosphorylation extent and kinetics as was formulated for TnI in the Soltis-Saucerman model. Table 1 reports all changes to myofilament model parameters induced by 100 nM [ISO] administration. PKA phosphorylation has been reported to reduce stiffness (attributed to titin) [54] and XBCa [55], in addition to increasing XBcy [18, 20–22]. The maximal ISO-induced reduction of titin stiffness was represented by decreasing the parallel element elasticity (K_e) by 50% to explain sarcomere passive force reduction with β-AS [56] (see Supplement and Fig. S3). To reproduce PKA effects on contractile targets, XBCa was adjusted by increasing the rate constants Zb, Yr and Zr to reduce Ca²⁺ affinity and alter the ratio of Ca²⁺ binding over unbinding rate parameters (Yb/Zb and Yr/Zr) favoring Ca²⁺ release. The ISO effect on XBcy was represented by modifying the rate constants Zp, Yp and parameters involved in g, f, and g_d to increase the rate constants implicated in XB attachment and detachment (Fig. 1C).

Table 2.

Effect of β-AS on ionic model parameters.

Target	Effect of β-AS	Effect of 100 nM [ISO]
ICaL	Enhanced channel opening probability and increased fraction of available channels	8% of channels in gating mode 2 and 29% increase in channel availability
RyR	Enhanced channel opening probability	53% increase in koSRCa
PLB	Enhanced Ca2+-sensitivity of SERCA	68% reduction of forward mode Kmf
PLM	Enhanced NKA activity due to increased affinity for [Na+]i	KmNaip reduced from 11 to 8 mM
IKs	Enhanced channel availability and left-shifted current-voltage relationship	2.8-fold increase in channel availability and 15-mV shift
IKr	Enhanced channel availability and left-shifted current-voltage relationship	30% increase in channel availability and 10-mV shift
ICFTR	Graded current activation	Total conductance gCFTR increased to 0.0043 A/F
ICl(Ca)	Increased Ca2+-affinity of the channel	30% reduction in KdClCa

All simulations were performed in MATLAB (The MathWorks, Natick, MA, USA) using the stiff ordinary differential equation solver ode15s. The model code is available for download at: https://somapp.ucdmc.ucdavis.edu/Pharmacology/bers/.

3. Results

3.1 β-AS behavior at imposed [Ca²⁺]

Figures 3 and 4 and Table 3 show results of control and β-AS at fixed [Ca²⁺]_i, to assess PKA effects on XBCa and XBcy separately, independent of [Ca²⁺]_i changes. We compared simulations to experiments done in transgenic animals in which PKA phosphorylation on TnI or MyBP-C sites was prevented. To this end, abrogation of the ISO-induced XBCa desensitization was simulated by eliminating ISO effects on parameters involved in XBCa (ISO-XBCa), and to suppress ISO-increased XBcy we removed ISO effects on XBcy parameters (ISO-XBcy). Simulations were compared with skinned fiber data in the literature. Fig. 3A illustrates experimental (top) [20] and model (bottom) for XB force (F_b) vs. [Ca²⁺]_i relationships at halfsarcomere length of 1.05 μm. Skinned fibers often exhibit higher half-activating [Ca²⁺]_i ([Ca²⁺]₅₀) vs. measurements in intact fibers or myocytes, due partly to supra-physiological free [Mg²⁺] in the former [57], which would prevent contraction for normal myocyte Ca²⁺ transients (below 1 μM). Our control half-activating [Ca²⁺]₅₀ = 0.79 μM and apparent Hill coefficient (3.6) agrees with baseline data in intact muscle. From this baseline, ISO produced a rightward shift of the force-[Ca²⁺] relationship, as seen experimentally. ISO-XBcy behavior was comparable to that of ISO, as reported in transgenic mice expressing non-phosphorylatable MyBP-C [20]. ISO-XBCa reversed the ISO relationship towards control as observed by Kentish et al. in mice expressing non phosphorylatable TnI [18]. In simulation-generated values fitted to the Hill equation [37, 58], ISO caused a rightward shift of 51% in [Ca²⁺]₅₀ with only slight changes in Hill coefficient (nH) and F_b(max), in agreement with experimental studies in mouse skinned cardiac muscle [18, 20], ferret papillary muscle [59] and rat trabeculae [60].

Figure 3.

Experimentally reported (upper panels) and model simulated (lower panels) contractile response at fixed [Ca²⁺]_i. A: Model simulated XB force (F_b) – [Ca²⁺]_i relationships depict rightward shift in ISO (100 nM) and ISO-XBcy, but comparable to control in ISO-XBCa, as seen in experimental results with PKA in WT or non phosphorylatable MyBPC (cMyoBP-C(t3SA)) mice [20]. B: Stiffness-frequency relationship simulations exhibit a shift of minimum stiffness towards higher frequency in ISO (see inset), as seen in experimental data with ISO in WT mice [18]. This effect is partially abolished in ISO-XBCa, though the effect is less than in TG experiments, and is eliminated in ISO-XBcy. C: force-velocity curves built from isotonic shortening velocities at imposed loads starting at isometric force (L_m = 1.05 μm). Shifts with ISO are comparable to experimental responses [63]. ISO-XBCa was similar to ISO and ISO-XBcy analogous to control, suggesting force-velocity dependence mainly on XBcy (see Table 3 for simulation numerical values).

Figure 4.

Contractile response to stretch at fixed [Ca²⁺]_i. A: Experimental (left, [21]) and simulated (right) force response to a length step. Simulations at expanded time scale show that the 100 nM [ISO] effect on k_rel is comparable to the experimental response in WT mice [21]. ISO-XBCa and ISO-XBcy simulations (insets) evidence a comparable response with non-phosphorylatable TnI (cTnI-Ala2, [21]) and non-phosphorylatable MyBPC (cMyoBP-C(t3SA), [20]) (experimental data not shown). B: Simulated force response following a release and restretch pulse (left panel), and simulated and experimental rates of force redevelopment (k_tr, right panel). k_tr increases in ISO and ISO-XBCa, equivalent to the behavior observed in WT and cTnI-Ala2 mice [23], and decreases in ISO-XBcy (see Table 3 for simulation numerical values).

Table 3.

Contractile response at imposed [Ca²⁺]_i in control, ISO, ISO-XBCa, ISO-XBcy, ISO-XBCa-XBcy, and ISO-titin.

Index	Units	Control	ISO	ISO-XBCa	ISO-XBcy	ISO-XBCa-XBcy	ISO-titin
nH	adimensional	4.03	4.00	4.22	3.74	4.03	4.00
[Ca2+]50	μM	0.765	1.154	0.801	1.141	0.765	1.154
Fb(max)	mN mm−2	92.5	97.2	97.2	92.5	92.5	97.2
krel	s−1	39.6	101.9	100.4	40.3	39.8	101.9
kdf	s−1	4.15	9.3	8.09	4.48	4.15	9.31
ktr	s−1	2.98	4.34	5.02	2.69	2.98	4.34
Stiff.min	mN mm−2 μm−1	70.6	22.2	37.6	36.9	70.6	22.2
Fr(Stiff.min)	Hz	0.85	1.20	1.12	0.85	0.85	1.20
Vmax	μs−1	1.01	1.62	1.67	0.93	1.01	1.53

The F_b-[Ca²⁺]_i relationship is characterized by nH, [Ca²⁺]₅₀ and F_b(max) of the Hill equation. k_rel and k_df describe the response to a length step, and k_tr depicts the reaction to a length pulse. Minimum stiffness (Stiff.min) and frequency (Fr) at minimum stiffness (Fr(Stiff.min)) describe the stiffness-frequency relationship, and V_max characterizes the response of the force-velocity relationship.

Figure 3B (top) shows an experimental stiffness-frequency relationship [18] reflecting actively cycling XB kinetics and Fig. 3B (bottom) depicts simulation results. Simulation of experimental stiffness-frequency relationships [61] has been previously described [37, 43]. These were obtained at steady-state with sinusoidal oscillations from a constant basal L (Lbas) to give an input L(t) = Lbas + ΔL sin(2π freq t), where ΔL (< 1% of Lbas) is the amplitude and freq the frequency of the applied oscillations. The stiffness modulus is defined as ΔF/ΔL, where ΔF is the amplitude of the resulting force oscillations. Stiffness simulations showed a minimum at 0.85 Hz frequency in control conditions. ISO produced a 41% rightward shift of this relationship, within the range experimentally reported (19% [62] and 63% [18]). ISO-XBCa only slightly limited the ISO effect (32% vs. 41% rightward shift vs. control), a change that was less pronounced than that seen in non-phosphorylatable TnI mice [18]. Conversely, ISO-XBcy completely abrogated the shift in stiffness minimum, suggesting that XBcy is more important than XBCa in the dynamic stiffness response to ISO.

Figure 3C shows the effect of β-AS on XB kinetics assessed by force-velocity relationships. Similarly to experimental results (top panel) [63], simulated isotonic shortening velocities during force clamps at constant [Ca²⁺]_i (2 μM) (bottom panel) showed that ISO enhanced shortening velocity at every force value. Maximal velocity at F_m = 0 increased by 60%, comparable to experiments in cat papillary muscle with norepinephrine [64] and rat myocytes with PKA stimulation [63]. ISO-XBCa produced a similar result (65% increase with respect to control) while ISO-XBcy abolished the ISO-induced response, suggesting that, like the dynamic stiffness behavior, XBcy predominates on the force-velocity reaction in enhanced inotropic state.

Figure 4A (top, left) shows experimental muscle force (F_m) response to 1% step increase in muscle length (L_m) [21]. Following the length step the apparent rate constant of force decay (k_rel) indicates rates of detachment/reattachment of strongly bound XBs, and the apparent rate constant of force redevelopment (k_df) indicates XB recruitment into the force-generating state. Simulation results (Fig. 4A, right) show F_m responses to a similar 1% stretch. The insets illustrate more clearly that ISO increased k_rel by 157% and k_df by 124% (Table 3). As expected, a similar response was observed for ISO-XBCa, comparable to reported effects in wild type and non-PKA phosphorylatable TnI transgenic mice [21]. On the other hand, ISO-XBcy reversed the response towards control, suggesting major XBcy involvement in the k_rel response, in accordance with the behavior observed in mice with non-phosphorylatable MyBP-C sites [20].

XB kinetics were also assessed during release-re-stretch (20%) from half-sarcomere length = 1.05 μm, lasting 20 ms at [Ca²⁺]_i = 0.75 μM, as done experimentally [23, 65] (Fig. 4B, bottom right). The rate of force redevelopment (k_tr) following the length pulse (determined at half-time force recovery [66], upper panel) was 2.98 s⁻¹ in control, similar to the linear slope of two phase k_tr-pCa Hill plots (3.21–1.83 s⁻¹) reported in similar conditions in isolated rat myocytes [65]. ISO increased k_tr by 46% and further enhanced it by 68% in ISO-XBCa, comparable to experiments on PKA-treated wild type and non-phosphorylatable TnI rat ventricular trabeculae Fig 4B, right) [23]. Preventing the cycling rate effect (ISO-XBcy) abolished the ISO-induced increase in k_tr, which was decreased even below control. Taken together, these results suggest that both XBCa and XBcy influence k_tr.

3.2 β-AS effect on action potentials, Ca²⁺ transients and contractions

Figures 5 and 6 and Tables 4 and S1 show effects of β-AS on 1 Hz steady state AP, Ca²⁺ transients and contractions for intact myocyte twitches. We separately assessed the impact of each PKA target. ISO (100 nM) decreased APD only slightly (from 211 to 204 ms), consistent with some experimental data [67, 68]. This is due to a balance of effects on I_CaL and PLM which would prolong APD vs. I_Ks which would shorten APD. That is, in Fig. 5A, switching off the I_CaL effect (blue AP) allows ISO to shorten APD, whereas disabling the I_Ks effect allows ISO to prolong APD (Fig 5D). Removing the PLM effect allows [Na⁺]_i to rise and increases outward (repolarizing) Na⁺-pump current Fig 5C). ISO effects on PLB (Fig. 5B), RyR, I_Kr, I_CFTR and I_Cl(Ca) (Fig. S5) or myofilaments (Fig. 6A) had no appreciable effects on APD.

Figure 5.

Effects of PKA-phosphorylation on individual ion channel and transporter targets. Time courses of membrane potential (top panels), Ca²⁺ transient (middle) and isometric force (bottom) during steady state 1-Hz stimulation are shown before (black), with 100 nM [ISO] (red) an with ISO, but with PKA effects on the indicated transporter disabled, for I_CaL (A), PLB (B), PLM (C), and I_Ks (D).

Figure 6.

Simulation of AP, [Ca²⁺]_i and isometric or isotonic twitches. Results are shown for control, ISO (100 nM), and ISO with suppression of different target effects (ISO-XBCa, ISO-XBcy and ISO-XBCa-XBcy). A: ISO slightly reduces APD, a response seen in all β-AS conditions. B: ISO and ISO-XBcy increase peak [Ca²⁺]_i, its velocity of rise and faster decline (TCa₅₀ and TCa₉₀), while abrogation of conditions involving XBCa decrease the peak [Ca²⁺]_i. C–D: ISO increases isometric force and isotonic shortening which is further enhanced upon eliminating ISO effect from XBCa (ISO-XBCa). Abrogation of ISO effect on XBcy (ISO-XBcy) blunts inotropy for force and shortening, while removing ISO effect on both contractile targets (ISO-XBCa-XBcy) allows the inotropic increase in force but not shortening (see also Table 4).

Table 4.

Isometric and isotonic twitch responses to ISO, ISO-XBCa, ISO-XBcy, ISO-XBCa-XBcy, ISO-titin, and ISO-Cytofl.

Feature	Units	Control	ISO	ISO-XBCa	ISO-XBcy	ISO-XBCa-XBcy	ISO-titin	ISO-Cytofl
Isometric contractions (Lm = 1.05 μm)
APD90	Ms	211.3	204.3	201.2	204.9	201.9	204.2	212.1
[Ca2+]CL(max)	μM	77.0	228.4	277.7	223.5	272.1	228.4	69.7
[Ca2+]SL(max)	μM	5.94	10.00	11.60	9.84	11.41	10.00	5.61
[Ca2+]i(max)	μM	0.577	1.110	0.814	1.111	0.816	1.110	0.780
[Ca2+]i(min)	μM	0.091	0.031	0.033	0.031	0.032	0.031	0.090
TCa(max)	Ms	77.3	75.4	73.5	75.3	76.6	74.0	75.2
TCa50	Ms	114.6	105.6	125.7	109.0	125.6	107.3	88.0
TCa90	Ms	213.2	157.2	192.7	154.0	186.5	159.4	167.6
Fm(max)	mN mm−2	14.5	34.3	49.3	22.9	31.5	34.3	12.8
Fm(min)	mN mm−2	0.91	0.76	0.77	0.76	0.77	0.83	0.78
TP50	Ms	107.3	87.4	89.5	109.3	110.6	88.0	85.2
RT50	Ms	93.4	66.2	82.2	46.7	65.4	65.8	68.5
T90	Ms	378.5	280.7	314.2	325.1 87	363.1	279.8	275.4
Isotonic contraction (Fm = 0.87 mN mm−2)
APD90	Ms	211.9	205.5	202.2	206.0	202.2	205.4	212.7
[Ca2+]i(max)	μM	0.580	1.109	0.813	1.106	0.810	1.110	0.787
[Ca2+]i(min)	μM	0.091	0.031	0.032	0.031	0.032	0.031	0.090
TCa(max)	Ms	79.6	77.5	77.3	77.9	77.8	77.2	76.3
TCa50	Ms	115.8	108.6	128.0	109.6	129.4	108.7	90.0
TCa90	Ms	203.6	148.9	182.8	146.7	181.7	149.8	162.2
% short	%	12.1	22.7	24.3	13.8	15.9	21.0	18.3
RT50	Ms	73.9	73.2	105.3	49.5	63.8	78.6	70.6

APD₉₀, APD at 90% repolarization; [Ca²⁺]_CL(max) and [Ca²⁺]_SL(max), peak in cleft and sub-sarcolemmal [Ca²⁺]; [Ca²⁺]_i(max) and [Ca²⁺]_i(min), peak and minimum in bulk cytosolic [Ca²⁺]; TCa(max), time to [Ca²⁺]_i(max); TCa₅₀ and TCa₉₀, time to 50% and 90% [Ca²⁺]_i decay; TP₅₀, time from the stimulus to 50% peak force; RT₅₀, time from peak force or peak shortening to 50% relaxation; T₉₀, time from the stimulus to 90% muscle force (F_m) relaxation; F_m(max) and F_m(min), maximum and minimum F_m; % short, percent shortening expressed as the percentage of the difference between maximum and minimum muscle length divided by maximum muscle length.

ISO had the expected effects on Ca²⁺ transients and contraction, increasing the amplitude and rising rate of both and also greatly accelerating [Ca²⁺]_i decline and relaxation (TCa₅₀ and TCa₉₀ in Table 4) [59, 62]. The ISO effects on I_CaL and PLB were important to the increase in Ca²⁺ transients and contraction amplitude, because preventing either (especially I_CaL) reduced peak [Ca²⁺]i and inotropy (Fig. 5A–B). Conversely, the ISO effects on PLM (by limiting [Na⁺]_i) and I_Ks (by limiting APD) both diminish Ca²⁺ loading, Ca²⁺ transient increases and the inotropic effect seen with ISO (Fig. 5C–D). As expected, PLB effects are important for the ISO-induced acceleration of [Ca²⁺]_i decline and relaxation. None of the other ion transporters (RyR, I_Kr, CFTR, I_Cl(Ca)) contribute substantially to the effects of ISO on [Ca²⁺]_i or contraction (Fig. S5).

3.3 β-AS effect on twitch contractions driven by action potentials

Figure 6B shows that the ISO-induced increase in peak [Ca²⁺]_i and kinetics of the Ca²⁺ transient was unaltered in ISO-XBcy. However, preventing the ISO-induced decrease in myofilament Ca²⁺ affinity (ISO-XBCa and ISO-XBCa-XBcy) reduced peak [Ca²⁺]_i in response to ISO by 27%, and thus blunted the ISO-induced increase in [Ca²⁺]_i by 55%. This emphasizes the quantitative impact of myofilament Ca²⁺ buffering during the normal Ca²⁺ transient [69, 70], and that part of the normal ISO-induced increase in Ca²⁺ transients is caused by the reduction in myofilament Ca²⁺ buffering. Thus greater myofilament Ca²⁺ binding in ISO-XBCa (vs. ISO) results in higher force development (Fig. 6C).

Rapid buffering of cytosolic Ca²⁺ is dominated by TnC and SERCA, which are present at 70 and 47 μM respectively [69]. Notably, ISO reduces TnC affinity and increases SERCA affinity for Ca²⁺. So, with respect to overall cytosolic Ca²⁺ buffering these two effects would tend to partially offset each other, as shown experimentally [70]. Consequently, ISO may have little net effect on cytosolic Ca²⁺ buffering. But with only the SERCA effect (ISO-XBCa) net Ca²⁺ buffering is increased so the same amount of total SR Ca²⁺ release will produce less rise in free [Ca²⁺]_i (ISO-XBCa vs. ISO in Fig. 6B).

APs were almost identical in ISO vs. ISO-XBCa (APD₉₀ = 204 vs. 201 ms; Table 4), despite the much smaller Ca²⁺ transient in ISO-XBCa. Given the importance of Ca²⁺-dependent currents (e.g. I_CaL, Na⁺-Ca²⁺ exchanger (NCX) current (I_NCX), and I_Cl(Ca)) the superimposable APs are surprising. The reason is that the change in myofilament Ca²⁺ binding is relatively isolated from the cleft and subsarcolemmal space (Sub-SL, Fig. 1C). During I_CaL and SR Ca²⁺ release, the cleft [Ca²⁺] ([Ca²⁺]_CL) is almost uninfluenced by the difference in TnC Ca²⁺ buffering, but rather depends mainly on Ca²⁺ fluxes into the cleft, local buffering in the cleft and diffusion out to the Sub-SL (see Supplement and Fig. S4). Note that the amount and kinetics of Ca²⁺ released in ISO and ISO-XBCa are almost identical. The Sub-SL (where all non-cleft Ca²⁺-sensitive channels and transporters are) is only somewhat less isolated than the cleft from the myofilaments. That is, during release [Ca²⁺]_CL drives up [Ca²⁺] in the Sub-SL ([Ca²⁺]_SL, which is unaltered by TnC buffering), while diffusion from the Sub-SL to bulk cytosol is only slightly slowed by the higher [Ca²⁺]_i in ISO vs. ISO-XBCa. Most of the diffusional driving force ([Ca²⁺]_SL – [Ca²⁺]_i) comes from the high [Ca²⁺]_SL. The result is that the altered myofilament buffering has no effect on cleft Ca²⁺ channels (I_CaL and RyR) and only slight effects on non-cleft Ca²⁺-dependent currents like I_NCX. If we double the diffusion constants for Ca²⁺ and Na⁺ from Sub-SL to cytosol, then [Ca²⁺]_SL is still almost unchanged. In contrast, if we reduce SR Ca²⁺ release by 50%, that produces strong and relatively proportional decreases in [Ca²⁺]_CL, [Ca²⁺]_SL and [Ca²⁺]_i (and of course I_CaL inactivation, I_NCX and APD).

Figure 6C shows isometric twitch force (F_m) at 1 Hz stimulation and L_m = 1.05 μm. ISO increased peak F_m 2.4 fold, and accelerated relaxation, reducing RT₅₀ (time from peak to 50% decline) by 29% and T₉₀ (time from stimulus to 90% relaxation) by 26%, similar to results obtained in rat trabeculae [71]. As noted above, peak F_m was further enhanced in ISO-XBCa vs. ISO, but the ISO-induced lusitropy was partially reversed, suggesting that ~59% of the lusitropic effect may be due to XBCa effects, while the remaining 41% may be due to PLB-SERCA effects. That is consistent with the 11% acceleration of [Ca²⁺]_i decline with ISO-XBCa vs. Control, and the slowing of [Ca²⁺]_i decline and relaxation when PLB phosphorylation is blocked (Fig. 6B). Thus, both myofilament Ca²⁺ desensitization and enhanced SERCA pumping contribute to ISO-induced lusitropy.

Preventing the ISO-induced increase in XB cycling (ISO-XBcy) attenuated the ISO-induced inotropy, despite the normal increase in peak [Ca²⁺]_i. ISO-XBcy also exhibited faster relaxation. This emphasizes not only the critical role of enhanced XB cycling to ISO-induced inotropy, but also that it may slightly limit lusitropy (which seems counter-intuitive). When both cycling and Ca²⁺ binding were suppressed (ISO-XBCa-XBcy) peak force was similar to that with ISO (2.2-fold increase), but contraction duration (both rise and fall) was substantially delayed vs. ISO (TP₅₀ was near the control value). The result is that the negative inotropic effect of reduced myofilament Ca²⁺ sensitivity with ISO is compensated by the faster XB cycling, allowing larger and faster isometric contraction, and both factors contribute to faster relaxation with ISO.

Figure 6D shows isotonic contractions (at F_m = 0.87 mN.mm⁻²), where ISO increased maximal shortening (by 88%) and caused faster relaxation, as seen experimentally [72]. Compared to the isometric case, ISO-XBCa only slightly enhanced shortening vs. ISO, and the main effect was a prolongation of contraction time. Thus, the ISO-induced reduction in myofilament Ca²⁺ sensitivity is less limiting for isotonic shortening (at low load) than it is for isometric force. Failure to boost XB cycling (ISO-XBcy) blunted shortening rate and extent vs. ISO (as with F_m), but in this case the combination ISO-XBCa-XBcy was unable to appreciably restore maximum shortening to the ISO level (unlike the isometric case).

Elimination of ISO effects on titin stiffness while preserving it on the other targets (ISO-titin) (Tables 3 and 4) only affected passive force F_m(min) in isometric twitches, with only very slight slowing of force development and decline compared with ISO. In isotonic conditions where force was the imposed variable (and was constant), ISO-titin slowed relengthening by ~7%. These results are consistent with a contribution of titin phosphorylation to restoring force as seen in experimental and simulation restoring force studies [56] (see Supplement and Fig. S6).

Removing the effect of ISO from all myocyte ion fluxes while preserving its effect on myofilaments (ISO-Cytofl) abolished inotropy; in fact, peak force was 12% lower than control. This emphasizes the importance of enhanced Ca²⁺ transients for ISO-induced inotropy. It also shows that the reduced XB Ca²⁺ affinity with ISO slightly supersedes the enhanced XB cycling with respect to isometric force development. Note also that peak [Ca²⁺]_i is higher than control for ISO-Cytofl, mainly because of the lower Ca²⁺ buffering by myofilaments. For the isotonic case, the ISO-enhanced shortening (88%) was only partially reduced (51%) in ISO-Cytofl. This again emphasizes that enhanced XB cycling is particularly important for ISO effects on isotonic shortening at low load. Obviously as the load increases the isotonic results would gradually shift to be more like the isometric case.

3.4 Kinetics and dose-dependence of β-AS effect on twitch contractions

Figure 7 illustrates the time course (panels A–D) and steady-state (panels E–H) cell responses to instantaneous application of 20 and 100 nM [ISO], where the former is roughly half-maximally activating for most of the PKA targets. As previously shown experimentally [73] and in models [49], we observed a rise of Ca²⁺ transient amplitude and force to a peak that subsided to a slightly lower steady state level during maintained ISO exposure. The secondary sag from the maximum was concurrent with the gradual decline in [Na⁺]_i which is more evident at higher [ISO]. Notably, APD rapidly prolonged upon ISO (due to faster enhancement of I_CaL vs. I_Ks [46]), and subsequently shortened to a variable extent. APD shortening was less marked at higher [ISO], in part because the lower [Na⁺]_i drives smaller outward I_NKA (repolarizing) compared to 100 nM [ISO]. There is also higher Na⁺ influx rate at higher [ISO]. That is because the Ca²⁺ influx and transients are larger, and in the steady state that greater Ca²⁺ influx must be balanced by greater Ca²⁺ efflux via NCX, resulting in greater Na⁺ influx.

Figure 7.

Time-course and graded effect of β-AS. Simulated action potential duration (A), [Ca²⁺]i (B), [Na⁺]i (C), and isometric force (D) in virtual myocyte paced at 1 Hz during 4 min exposure to ISO (20 and 100 nM; black and blue). Expanded time scales for membrane potential (E), [Ca²⁺]i (F), [Na⁺]i (G), and isometric force (H) before (solid) and after (dashed) 20 nM (left) and 100 nM (right) [ISO] administration.

4. Discussion

We incorporated a new contractile model into a robust electrophysiological, Ca²⁺ handling, and β-AS signaling myocyte model [25, 29]. This provides a new quantitative framework to dynamically study complex interactions of β-AS effects on Ca²⁺ transient, AP and contractile properties. This allowed us to start to clarify unresolved questions about the interaction of altered XB Ca²⁺ sensitivity and XB cycling rate during β-AS in ventricular myocytes. This may also provide a useful resource for other investigators for related studies (the model is freely available).

The contractile model exhibited appropriate fundamental mechanical properties with respect to experimental data (Figs. 2–4). This includes asymmetric XB detachment (which improved fit to experimental afterloaded contraction kinetics), force-[Ca²⁺]_i relationship, frequency-dependence of stiffness, force-velocity curves and kinetic responses to abrupt stretch and release-stretch protocols. These properties and the effects of PKA were validated against critical experimental results, including data from mice with either non-phosphorylatable TnI or MyBP-C, to help analyze relative roles of XBCa and XBcy (as well as TnI and MyBP-C) in the response of twitch isometric and isotonic contractions. When embedded in a well-established electrophysiology/Ca²⁺ handling/signaling model [25, 29] the composite model produced normal APs, Ca²⁺ transients and isometric and isotonic contractions, as well as ISO responses for each.

4.1 Model assumptions

Three key changes were incorporated into our prior 6-element Ca²⁺ kinetic model [37]. First, we simplified this to a recently used 5-element version [74]. Because [TS~] in the original model, which was in the transition from TS* to TS (weak XB, Ca²⁺-free), was <5% of the total attached XBs, the irreversible pathway from TS* to TS was combined into a single step (eliminating TS^~; Fig. 1C). Second, we changed the velocity-dependent detachment function from a symmetrical to an asymmetrical function, a behaviour originally postulated by Huxley [42]. Third, we changed the sarcomere length-dependent lattice spacing effect on the rate function g_d from a parabolic description to an exponential representation [74], better accounting for g_d changes at any L. These changes preserved results from the previous model [37] (<10% difference in index values of F_b-[Ca²⁺]i, stiffness-frequency and F_m-velocity, not shown). The newer model here has fewer differential equations and improved simulation of afterloaded contractions.

Incorporating the contractile model into the Soltis-Saucerman model [29] allowed replacing the passive Ca²⁺-buffering by TnC with the bi-directionally coupled Ca²⁺-XB model (Fig. 1C). Thus, length-dependent force generation is coupled to Ca²⁺ kinetics, and incorporates cooperativity between nearest-neighbor troponin complexes (within a TS). This coupling also provides inherent feedback from length and force to Ca²⁺ binding (and cytosolic buffering) by influencing transitions between states that have different Ca²⁺ affinity. Moreover, the model reproduces the classical observation [75] that abrupt shortening step during the AP causes release of Ca²⁺ from myofilaments due to the length-dependent decrease in Ca²⁺ affinity (Fig. S7). The ISO-induced reduction in TnC Ca²⁺ affinity also reduces myofilament Ca²⁺ binding, and allows peak [Ca²⁺]i to reach a significantly higher level for almost the same amount of Ca²⁺ supplied by ion channels (Fig. 6B). Indeed, a potentially surprising emergent result from the model was that changes in cytosolic (or myofilament) Ca²⁺ buffering alone had almost no effect on the AP shape, ionic currents or Ca²⁺ release and extrusion function. As discussed above, this is because the cleft and Sub-SL regions (where all Ca²⁺ transporters and channels are located) are partially isolated from the cytosolic buffering. Also the most rapid changes in [Ca²⁺]_CL and [Ca²⁺]_SL, which drive current changes, occur very early in the AP, and although high [Ca²⁺]_i may reduce inward I_CaL in the Sub-SL, it also increases inward I_NCX eliminating Ca²⁺. This raises the intriguing possibility that therapeutic approaches aimed at selective alteration of cytoplasmic buffering to accelerate twitch [Ca²⁺]_i decline [76], may have relatively modest effects on the cardiac AP and arrhythmias. This, of course, is fundamentally different for primary changes in ICaL or SR Ca²⁺ release, which will have large and relatively proportional effects on Ca²⁺-dependent currents, their electrophysiological consequences and on contractile properties (apparent in the ISO cases in Fig. 6 and Table 4).

The model characterizes the separate effect of β-AS on TnI and MyBP-C as independent changes in the Ca²⁺ kinetic parameters involving XBCa and XBcy, respectively. This assumption was based on data suggesting relative dominance of PKA-dependent TnI phosphorylation on XBCa and of MyBP-C phosphorylation on XBcy [20, 21]. However, the exclusivity of this target-function arrangement is not universally accepted. Although emerging evidence suggests that MyBP-C may be primarily responsible for changes in XBcy in response to phosphorylation [21, 22, 77, 78] it may also contribute to XBCa when regulatory light chain phosphorylation has been reduced [22], or when basal levels of phosphorylation are decreased by propranolol pretreatment [23]. Conversely, there is also some evidence that TnI phosphorylation by PKA is critical for XBcy effects [18]. While our model is consistent with a separate influence of β-AS on XBCa and XBcy, the model itself may be instrumental in further elucidation of the functional interaction between TnI and MyBP-C phosphorylation sites.

ISO effects on ion currents and Ca²⁺ transport were mostly as in the Soltis-Saucerman model [29], with modifications based on experimental data and our recent theoretical studies [46, 52] and indicated in Methods. In addition, I_Cl(Ca) was slightly enhanced by ISO [53], whereas I_NCX properties were not altered [79, 80], despite earlier data suggesting such modulation [81]. The set of myofilament parameters and ISO-induced shifts was adequate to mimic experimental findings in wild type and transgenic mice with non-phosphorylatable TnI or MyBP-C. That enabled predictions of relative XBCa and XBcy participation in isometric and isotonic twitch behavior. Although some relative shifts were inferred from transgenic mice, these were imbedded in the Soltis-Saucerman model that is based on the Shannon-Bers rabbit myocyte framework. This was motivated, in part, by the likely more direct extension to the human ventricular myocyte regarding electrophysiology and Ca²⁺ handling [82]. The model creates a template that will allow newer data to be added to improve its accuracy for both rabbit and human tissue. Also, our model allows simulating variable degrees of β-AS, as demonstrated in Fig. 7.

4.2 β-AS effects on contractility

We systematically switched off the β-AS effect from each of 11 individual targets to analyze their relative impact on contractility. Preventing β-AS effects on I_CaL or PLB limited Ca²⁺ transients and contractile responses in parallel, while blocking PLM and K⁺ channel (I_Ks) effects enhanced Ca²⁺ and inotropy. These results are all consistent with the expected effects of these individual targets. But the integrated model here allows a relative assessment of the impacts of each process to β-AS induced inotropy and lusitropy. Four other ion transporters (RyR, I_Kr, CFTR, I_Cl(Ca)) are modulated by PKA effects in the model, but their net effects on Ca²⁺ transients and contraction were minimal.

The satisfactory simulation of F_b-[Ca²⁺]_i relationships [18, 20, 83], release-restretch [20, 21] and length step [23] experiments at fixed [Ca²⁺]_i, demonstrates the adequate behavior of the present model. These results enabled the dissection of the ISO twitch response (Table 4 and Fig. 6) to understand the separate role of XBCa and XBcy on contractility. Results of Ca²⁺ transient and isometric and isotonic twitches suggest that enhanced XBcy is essential to explain the twitch inotropic response, and without that the larger Ca²⁺ transient appears only able to offset the intrinsic negative inotropic effect of ISO-induced reduced myofilament Ca²⁺ sensitivity (XBCa). The larger and faster Ca²⁺ fluxes are also essential, because without them ISO would have a slightly negative or a reduced inotropic effect for isometric and isotonic contractions, respectively (Table 4; ISO-Cytofl). The absence of both XBCa and XBcy effects does not prevent inotropic and lusitropic effects of ISO, due to the larger and faster Ca²⁺ transients (although the lack of XBcy greatly limits inotropy in the isotonic case). The reduced XBCa contributes to faster relaxation and shorter time to peak, and the enhanced XBcy amplifies the inotropic effect (but also slows kinetics). Notably, the XBcy effect is especially critical for isotonic contraction during sarcomere shortening, because without that ISO reduces shortening vs. control despite the much larger Ca²⁺ transients. Looking at this another way, the ISO-induced increase in Ca²⁺ transient offsets the reduced XBCa, resulting in little difference in force or shortening vs. control, whereas enhanced XBcy drives the net enhancement of contraction.

For isometric force, enhanced XBcy and Ca²⁺ cycling contribute roughly equally to force development, whereas for isotonic contraction enhanced XBcy is more critical than enhanced Ca²⁺ cycling for maximal shortening. With respect to kinetics, the situation differs. Both faster Ca²⁺ kinetics and reduced XBCa contribute to the shorter time to peak and faster relaxation, and both work against the enhanced XBcy (which by itself slows relaxation).

4.3 Limitations

Our contraction model is based on cardiac data from multiple species (frog, guinea pig, rabbit, ferret and cat), whereas the ionic and Ca²⁺ handling model is based more explicitly on rabbit data. In both cases temperature-dependent parameters were adapted to 37°C [25, 37]. Ideally, all model parameters should be derived from rabbit at 37°C. However, as is often the case in model development [84], there is limited rabbit data at 37°C to constrain the mechanical model. So, we used available data from various species, including seminal rabbit data [35, 37] and from genetically modified mice, which are of unique value in distinguishing effects of TnI and MyBP-C phosphorylation sites. We think the model is a very good approximation, but this necessary compromise is a model limitation. Of course the model can be further tuned as additional relevant data becomes available with respect to any of the model assumptions.

The model also includes CaMKII signaling, and there is known to be some cross-talk between β-AS and CaMKII signaling. We tested how much CaMKII effects might contribute to the acute ISO effects reported here (Fig. S8 vs. Fig. 7). Turning off CaMKII effects had only slight effects on acute ISO effects, so the ISO effects we report here are almost entirely PKA-dependent. That agrees with data showing that ISO-induced inotropic and lusitropic effects are preserved in CaMKIIδ knockout mice [85].

4.4 Conclusions

We have developed a useful model to study the interplay of mechanical, Ca²⁺ handling and electrophysiological properties of cardiac myocytes. Here we have used it to improve our understanding of how XBCa and XBcy effects of ISO interact with each other during both isometric and isotonic contractions. Coupling the contractile and Ca²⁺-AP models allowed us to study how ion channel and Ca²⁺ changes influence contraction dynamically, and also how the contractile process feeds back to influence [Ca²⁺]_i. An emergent finding was that changes in myofilament Ca²⁺ buffering can influence [Ca²⁺]_i, but at the same time have little direct impact on ionic currents and the AP shape. Both increased XBcy and Ca²⁺ transients contribute to enhanced contractility during β-AS isometric systole, whereas XBcy may be more critical during cardiac ejection when sarcomere shortening occurs. The model may also be useful for both further study of these AP-Ca²⁺-XBCa-XBcy interactions and incorporation into multicellular or intact heart models.

Highlights.

• β-AS effects on cardiac contraction are analyzed with a new computer model.

• The model reproduces a range of experiments involving force and length changes.

• Impact of PKA targets to ®-AS effects on AP, Ca2+ and contraction is studied.

• ICa, PLB, PLM and I_Ks effects dominate in the ®-AS increase in Ca2+ transients.

• Increased XBcy compensates for lower myofilament Ca2+ sensitivity during inotropy.

Abbreviations

AP

Action potential

APD

AP duration

CaMKII

Ca²⁺/calmodulin-dependent protein kinase II

E–C

Excitation-contraction

F_m

Muscle force

I_CaL

L-type Ca²⁺ channel current

I_CFTR

CFTR-mediated Cl⁻ current

I_Cl(Ca)

Ca²⁺-dependent Cl⁻ current

I_Kr

Rapid delayed rectifier K⁺ current

I_Ks

Slow delayed rectifier K⁺ current

I_Na

Fast Na⁺ current

I_NCX

Na⁺-Ca²⁺ exchanger current

I_NKA

Na⁺-K⁺-ATPase current

ISO

Isoproterenol

ISO-Cytofl

ISO effect without β-AS on parameters involved in cytosolic fluxes

ISO-titin

ISO effect without β-AS on parameters involved in titin

ISO-XBCa

ISO effect without β-AS on parameters involved in XBCa

ISO-XBCa-XBcy

ISO effect without β-AS on parameters in XBCa and XBcy

ISO-XBcy

ISO effect without β-AS on parameters involved in XBcy

k_df

Rate of force redevelopment (response to a length step)

k_rel

Rate of force decay

k_tr

Rate of force redevelopment (response to a length pulse)

L_m

Muscle length

MyBP-C

Myosin binding protein-C

NCX

Na⁺-Ca²⁺ exchanger

NKA

Na⁺-K⁺-ATPase

PKA

Protein kinase A

PLB

Phospholamban

PLM

Phospholemman

RyR

Ryanodine receptor

SERCA

SR Ca²⁺-ATPase

SR

Sarcoplasmic reticulum

TnC

Troponin C

TnI

Troponin I

XB

Myofilament cross-bridge

XBCa

XB Ca²⁺ sensitivity

XBcy

XB cycling rate

β-AS

un β-adrenergic stimulation