Modeling the effects of thin filament near-neighbor cooperative interactions in mammalian myocardium

Abstract

The mechanisms underlying cooperative activation and inactivation of myocardial force extend from local, near-neighbor interactions involving troponin-tropomyosin regulatory units (RU) and crossbridges (XB) to more global interactions across the sarcomere. To better understand these mechanisms in the hearts of small and large mammals, we undertook a simplified mathematical approach to assess the contribution of three types of near-neighbor cooperative interactions, i.e., RU-induced, RU-activation (RU-RU), crossbridge-induced, crossbridge-binding (XB-XB), and XB-induced, RU-activation (XB-RU). We measured the Ca2+ and activation dependence of the rate constant of force redevelopment in murine- and porcine-permeabilized ventricular myocardium. Mathematical modeling of these three near-neighbor interactions yielded nonlinear expressions for the RU-RU and XB-RU rate coefficients (kon and koff) and XB-XB rate coefficients describing the attachment of force-generating crossbridges (f and f'). The derivation of single cooperative coefficient parameters (u = RU-RU, w = XB-RU, and v = XB-XB) permitted an initial assessment of the strength of each near-neighbor interaction. The parameter sets describing the effects of discrete XB-XB or XB-RU interactions failed to adequately fit the in vitro contractility data in either murine or porcine myocardium. However, the Ca2+ dependence of ktr in murine and porcine ventricular myocardium was well fit by parameter sets incorporating the RU-RU cooperative interaction. Our results indicate that a significantly stronger RU-RU interaction is present in porcine ventricular myocardium compared with murine ventricular myocardium and that the relative strength of the near-neighbor RU-RU interaction contributes to species-specific myocardial contractile dynamics in small and large mammals.

Figure 1.

Schematic of the model. The model integrates three states of a single RU, consisting of blocked (B), closed (C), and open (M), and two states of an attached XB, including strongly bound pre-power stroke (M₁) and strongly bound postpower stroke (M₂), into a coupled four-state system. A thin filament RU is represented by a Tm bar associated with the seven-circle actin chain, with a myosin XB represented by the ellipse with a tail. A RU is blocked when the Tm bar is below the chain. A RU is in the closed state when Ca²⁺ binds to TnC (represented by a small red star) and the Tm bar is above the chain, with a XB weakly bound. A RU is in the open state when a XB is strongly bound. A XB may be detached (C) or attached (M₁ and M₂) to the thin filament. In isometric conditions, force generation occurs in the strongly bound postpower stroke state, M₂.

Figure 2.

Nearest neighbor cooperative interactions - Schematic of RU–RU interactions. Left panel: The transition from a blocked to a closed state of an RU and the transition from a closed to an open state of an RU can be favored when its neighbor RUs are either in closed state or in open state. Right panel: The reversed transition from a closed to a blocked state of an RU and the reversed transition from an open to a closed state can be facilitated by its blocked neighbor RUs.

Figure 3.

Nearest neighbor cooperative interactions - Schematic of XB-XB interactions. Left panel: A strongly bound force generating XB facilitates the transition from a weakly bound to a strongly bound state of a neighbor XB. Right panel: A non-force generating XB (B, C, M₁) eases the reversed transition from a strongly bound to a weakly bound state of a neighbor XB.

Figure 4.

Nearest neighbor cooperative interactions - Schematic of XB-RU interactions. Left panel: A strongly bound force generating XB postively influences the transition of a near neighbor RU from the blocked to the closed state. Right panel: A non-force generating XB (B,C, M₁) helps facilitate the reverse transition from a closed to a blocked state of a neighbor RU.

Figure S1.

Steady-state tension development in mammalian-permeabilized myocardium. Ca²⁺-activated tension–pCa relationship was measured in murine- (black circle) and porcine- (blue circle) permeabilized ventricular myocardium. All values represent means, and error bars represent ± SEM.

Figure S2.

Ca ²⁺ and activation dependence of the rate of force redevelopment in mammalian-skinned myocardium. The rate constant of force redevelopment (ktr) following a rapid release/restretch maneuver was measured in permeabilized myocardial preparations isolated de novo from murine ventricular myocardium (black-filled circles) and porcine ventricular myocardium (blue-filled circles, from Patel et al. [2023]). Data points represent the means, and the error bars are the SEM (Table S1). (A and B) The Ca²⁺ dependence of the rate constant of force redevelopment (A) and relative ktr (B). (C and D) The activation dependence of the rate constant of force redevelopment (C) and relative ktr (D).

Figure S3.

Model resolution matrices generated by fitted results for modeling the Ca ²⁺ dependence of the rate of force redevelopment in mammalian myocardium. Numerical identification of rate coefficients, cooperative coefficients (u₁, u₂, z₁, z₂, v, and w) and nearest neighbor interaction factors (α, α, β, and β) are summarized in Table S5. Upper panels represent the model resolution matrices for murine myocardium for parameter set 1 (left), parameter set 2 (middle), and parameter set 8 (right). Lower panels represent the model resolution matrices for porcine myocardium for parameter set 1 (left), parameter set 2 (middle), and parameter set 8 (right).

Figure S4.

Modeling the effect of near-neighbor cooperative interactions on the force–pCa relationship in mammalian myocardium. Ca²⁺-activated relative force (P/P₀) was measured in murine- (black circles) and porcine- (blue circles) permeabilized ventricular myocardium. All values represent means, and error bars represent ± SEM (from Table S1). (A–C)Eqs. 2, 3, and 4 were used in conjunction with parameter set 1 (A), parameter set 2 (B), and parameter set 8 (C) to fit to the force–pCa data. Fitted parameters are shown in Table S4.

Figure 5.

Modeling the effect of RU–RU, XB–XB, and XB–RU cooperative interactions on the Ca ²⁺ dependence of the rate of force redevelopment in mammalian myocardium. (A and B) Eqs. 2, 3, and 4 were used in conjunction with parameter set 1 (Table 2) to fit to the contractile data derived from the (A) ktr versus pCa and (B) relative ktr–pCa relationships. The rate constant of force redevelopment was measured in murine- and porcine-permeabilized ventricular myocardium (from Fig. S2). The data points (filled circles) represent the means, and the error bars are the SEM (from Table S1).

Figure 6.

Modeling the effects of RU-RU, XB-XB, and XB-RU cooperative interactions on the Ca ²⁺ dependence of the rate of force redevelopment in mammalian myocardium. (A and B) Eqs. 2, 3, and 4 were used in conjunction with parameter set 2 (Table 2) to fit to the contractile data derived from the (A) ktr versus pCa and (B) relative ktr versus pCa relationships. The rate of constant of force redevelopment was measured in murine and porcine permeabilized ventricular myocardium (from Fig. S2). The data points (filled circles) represent the means and the error bars are the SEM (from Table S1).

Figure 7.

Modeling the effects of RU-RU, XB-XB, and XB-RU cooperative interactions on the Ca²⁺ dependence of the rate of force redevelopment in mammalian myocardium. (A and B) Eqs. 2, 3, and 4 were used in conjunction with parameter set 8 (Table 2) to fit to the contractile data derived from the (A) ktr versus pCa and (B) relative ktr versus pCa relationships. The rate constant of force redevelopment was measured in murine and porcine permeabilized ventricular myocardium (from Fig. S2). The data points (filled circles) represent the means and the error bars are the SEM (from Table S1).

Figure 8.

Possible near neighbor effects on a thin filament regulatory unit. The probability of a central RU transitioning from the blocked to closed to open state is influenced by the states of its neighboring RUs: both neighbors blocked (BB); one neighbor blocked, one neighbor closed (BC, CB); one neighbor blocked, one neighbor open (BM, MB); bothe neighbors closed (CC); one neighbor closed, one neighbor open (CM, MC); and both neighbors open (MM). The success frequency of a central RU transitioning away from the blocked state is highest when both neighbors are in the open state (i.e., activation energy is lowest) and lowest when both neighbors are in the blocked state (i.e., activation energy is the highest). Figure adapted from Razumova et al. (2000).

Figure 9.

The effects of altering the strength of RU-RU cooperative coefficients on the activation-dependencies of the rate of force development. The rate constant of force redevelopment was measured in permeabilized myocardial preparations isolated from murine and porcine ventricular myocardium. The data points (filled circles) represent the means and the error bars are the SEM (from Table S1). The model, comprising Eqs. 2, 3, and 4, is fitted to the data of P/P_o versus pCa and the data of ktr versus pCa in murine and porcine myocardium using the parameters derived from the ensemble RU-RU, XB-XB, and XB-RU interactions (Parameter Set 8 - Table 4). (A and B) The effect of altering the RU-RU blocked-to-closed cooperative coefficient u₂ on the activation-dependence of ktr in murine (A) and porcine (B) myocardium. (C and D) The effect of altering the RU-RU closed-to-open cooperative coefficient z₂on the activation-dependence of ktr in murine (C) and porcine (D) myocardium.