πŸ“’ Tran 2010

A metabolite-sensitive, thermodynamically constrained model of cardiac cross-bridge cycling: implications for force development during ischemia1



  • A thermodynamically consistent and metabolite-sensitive model of cardiac myosin cross-bridge cycling, driven by the free energy released from the hydrolysis of ATP.
  • Thermodynamically consistent model as defined by Hill and explicitly incorporate the regulatory effects of MgATP and its hydrolysis products, MgADP, Pi, and H+.
  • Mean-field models have been proposed that ignore the microscopic details of cross-bridge population distributions, removing the need for solving PDEs.
  • The model is based on the work by Rice and coworkers2

The model

  • Parameterized for guinea pig myocardium, setting xbmodspecies = 0.2, effectively slowing the kinetics by five folds from that of in the rat.

  • Add a reversible transition between AM1 and Pxb to account for the full (reversible) thermodynamic cycle. The value of fxbT is determined by the thermodynamic constraint that links the rate constants in the cycle to the hydrolysis of MgATP

  • The reaction cycle need to satisfy the following $$ \frac{\prod \alpha_{\mathrm{i}}^{+}}{\prod \alpha_{\mathrm{i}}^{-}}=e^{\frac{-\Delta G_{\mathrm{ATP}}}{R T}} $$ $$ \Delta G_{\mathrm{ATP}}=\Delta G_{\mathrm{ATP}}^{0}+R T \ln \frac{[\mathrm{MgADP}][\mathrm{Pi}]\left[\mathrm{H}^{+}\right]}{[\mathrm{MgATP}]} $$

  • Assuming normal physiological metabolic conditions are given by MgATP = 5 mM, MgADP = 36 ΞΌM, Pi = 2 mM, and pH = 7.15

MgATP reactions

  • Measurements of isometric force have shown a descending trend with increasing concentrations of MgATP
  • MgATP binding step is therefore placed in the transition from state XBPostR to state PXB (gxbT)

$$ g_{\mathrm{xbT}}^{\prime}=\frac{g_{\mathrm{xbT}}}{\mathrm{MgATP}^{\prime}} $$

  • gβ€²xbT represents the true first-order rate constant, MgATPβ€² is the physiological MgATP concentration, and gxbT is the pseudo-first order rate constant from the Rice et al. Under normal metabolic conditions, when [MgATP] = MgATPβ€², this gives gβ€²xbT = gxbT

Inorganic phosphate reactions

  • increasing [Pi] leads to a decrease in isometric force
  • Pi-binding step at gappT produces a better quantitative fit to the data

$$ g^{\prime}{\mathrm{appT}}=\frac{g{\mathrm{appT}}}{P i^{\prime}} $$

pH influence

  • Decrease in Ca2+ sensitivity with decreasing pH is due to the competitive binding of H+ to the single Ca2+ binding site on troponin C
  • Rapid equilibrium reaction in troponin C => shift in Ca2+ sensitivity

$$ k_{\mathrm{onT}}^{\prime}=\frac{k_{\mathrm{dHCa}}^{\mathrm{m}}+H^{\mathrm{m}}}{k_{\mathrm{dHCa}}^{\mathrm{m}}+\left[\mathrm{H}^{+}\right]^{\mathrm{m}}} \times k_{\mathrm{onT}} $$

  • the (reverse) rate constants in the cross-bridge cycle to be associated with the release of the metabolic H+

$$ h_{\mathrm{bT}}^{\prime}=\frac{h_{\mathrm{bT}}}{H^{\prime}} $$


  • MgADP binds to attached cross-bridges that remain attached after the binding event.
  • Increasing MgADP concentrations also leads to an increase in steady-state isometric force development in both skeletal and cardiac muscle preparations
  • The original four-state model therefore contains an insufficient number of states to represent the binding of MgADP
  • MgADP is assumed to bind to state AM1. states AM1 and AM2 are in rapid equilibrium
  • The complete H+ and MgADP-dependent transition from XBPostR to XBPreR is determined as

$$ \left[\mathbf{H}^{+}\right] \times h_{\mathrm{bT}}^{\prime} \times\left(\frac{k_{\mathrm{dADP}}+M g A D P^{\prime}}{M g A D P^{\prime}} \times \frac{[\mathrm{MgADP}]}{k_{\mathrm{dADP}}+[\mathrm{MgADP}]}\right) $$

Parameter estimation and model selection

  • The authors found that the Extended Model best fits the data when the Pi binding step is placed at gappT
  • Parameters governing the competitive binding of H+ to the Ca2+ binding sites on troponin C, we use the guinea pig force-Ca data of Orchard and Kentish. The best fit to the data is obtained at m = 1 and kdHCa = 2 Γ— 10βˆ’5 mM.
  • A decrease of pH leads to a decrease in maximum steady-state force. However, quantitatively, the placement of the H+ release step at hbT produces better agreement with the data from Orchard and Kentish


Force redevelopment and MgATP

  • force-MgATP curve is sensitive to low levels of MgADP concentration

Sinusoidal length perturbations

  • at different MgATP and Pi concentrations

Force development during ischemia: sensitivity analysis

  • the maximum force is most sensitive to changes in [Pi] and pH. Changes in the sensitivity to [MgATP] and [MgADP] over the ischemic time

Model limitations

  • This model of Ca2+ activation and cross-bridge kinetics is a mean-field representation

Inversion of the force-MgADP relationship

  • A reversal point at [MgATP] = 0.9 mM where the MgADP-dependent curves intersect


  • To ensure that the cycle is thermodynamically consistent
  • The cross-bridge cycle can reverse, albeit at a very low rate, and this is primarily due to the kinetics of the cross-bridge cycle

Implications for ischemia

  • Exponential-like decrease in force after ischemia, consistent with experimental data
  • Phosphate and pH are the primary contributors to the rapid fall in maximum tension during ischemia
  • However, falling pH directly inhibits the ryanodine receptors and indirectly leads to a rise in intracellular sodium, a rise in the diastolic Ca2+ concentration and the amplitude of the Ca2+ transient although the force production decreases

  1. Tran K, Smith NP, Loiselle DS, Crampin EJ. A metabolite-sensitive, thermodynamically constrained model of cardiac cross-bridge cycling: implications for force development during ischemia. Biophys. J. 2010;98(2):267-276. doi:10.1016/j.bpj.2009.10.011. ↩︎

  2. Rice JJ, Wang F, Bers DM, de Tombe PP. Approximate model of cooperative activation and crossbridge cycling in cardiac muscle using ordinary differential equations. Biophys J. 2008;95(5):2368-90. ↩︎