📒 Tran 2009

A thermodynamic model of the cardiac sarcoplasmic/endoplasmic Ca(2+) (SERCA) pump1



  • The SERCA pump is a P-type ATPase enzyme found in all muscle cells, and functions to sequester Ca2+ into the sarcoplasmic reticulum (SR) to facilitate muscle relaxation
  • existing models are not designed to be energetically sensitive and consistent with the fundamental principles of free energy transduction or too complex to be integrated into a CMC model
  • Here we present a mathematical model of the cardiac SERCA pump that consolidates a variety of experimental data, collected from a range of modalities, and the fundamental principles of free energy transduction into a consistent single framework


  • based on the E1-E2 model for Ca2+ transport with the addition of H+ binding steps
  • When the pump operates in the forward (clockwise) direction, two Ca2+ are bound from the cytoplasm and released into the SR lumen
  • Ca2+ binding mechanism: partially cooperative
  • Ca2+ binding mechanism: fully cooperative

Ca2+/H+ counter-transport

  • The ejection of H+ from SR vesicles after the addition of Ca2+ and ATP has been observed by a number of experimental groups, due to the counter-transport of H+ for Ca2+
  • The stoichiometry of H+ binding in the SERCA model was left as an adjustable parameter in the fitting process as it has been suggested that the number of H+ extruded per cycle depends on pH

Competitive H+ binding

  • Protons also directly compete with Ca2+ in binding to the two Ca2+ -binding sites on the SERCA enzyme
  • modeled as a side reaction at states P2 and P3

Metabolite binding

  • In the exp, MgATP and Ca can bind in any order.

Thermodynamic constraints

  • Briefly, the free energy of ATP hydrolysis depends on the ratio of MgATP to its hydrolysis products as $$ \Delta G_{\mathrm{MgATP}}=\Delta G_{\mathrm{MgATP}}^{o}+R T \ln \frac{[\mathrm{MgADP}][\mathrm{Pi}]\left[\mathrm{H}^{+}\right]}{[\mathrm{MgATP}]} $$
  • The energy required to move two Ca2+ against a concentration gradient is given by $$ \Delta G_{\mathrm{SERCA}}=2 R T \ln \frac{\left[\mathrm{Ca}^{2+}\right]_{\mathrm{sr}}}{\left[\mathrm{Ca}^{2+}\right]_{\mathrm{i}}} $$
  • The SERCA pump will cycle in the forward direction when ΔGMgATP + ΔGSERCA < 0, as is the case under normal physiological conditions and Ca2+ will be transported back into the SR
  • a constraint on the kinetic rate constants of the SERCA pump model $$ \frac{k_{1}^{+} k_{2}^{+} k_{3}^{+} K_{\mathrm{d}, \mathrm{Casr} 1} K_{\mathrm{d}, \mathrm{Casr} 2} K_{\mathrm{d}, \mathrm{Hsr}} K_{\mathrm{d}, \mathrm{H}}}{k_{1}^{-} k_{2}^{-} k_{3}^{-} K_{\mathrm{d}, \mathrm{Cail}} K_{\mathrm{d}, \mathrm{Cai} 2} K_{\mathrm{d}, \mathrm{Hi}}}=e^{\Delta G_{\mathrm{MgATP}}^{\mathrm{o}} / R T} $$

Model simplification: rapid equilibrium assumptions

  • It is assumed in the model that the ionic reactions (binding of Ca2+ and H+) occur at a faster rate than the reactions involving the metabolites (MgATP, MgADP and Pi)
  • the clockwise cycle rate per pump at steady state $$ \begin{aligned} J_{up} &= 2 v_{cycle} \rho_{SR} A_{SR} = 2 k_{up} v_{cycle} \cr v_{cycle} &= \frac{\alpha_1^+\alpha_2^+\alpha_3^+ - \alpha_1^-\alpha_2^-\alpha_3^-}{\Delta_3} \cr \alpha_1^+ &= k_1^+ [MgATP] \cr \alpha_2^+ &= \frac{k_2^+ \phi_{Ca_i}^2}{\phi_{Ca_i}^2 (1 + \phi_{H_i}) + \phi_{H_i}(1 + \phi_{H_1})} \cr \alpha_3^+ &= \frac{k_3^+ \phi_{H_{sr}}}{\phi_{H} (1 + \phi_{Casr}^2) + \phi_{H_{sr}}(1 + \phi_{H})} \cr \alpha_1^- &= \frac{k_1^- \phi_{H_i}}{\phi_{Ca_i}^2 (1 + \phi_{H_i}) + \phi_{H_i}(1 + \phi_{H_1})} \cr \alpha_2^- &= \frac{k_2^- [MgADP] \phi_{Ca_{sr}}^2 \phi_{H_{sr}}}{\phi_{H} (1 + \phi_{Ca_{sr}}^2) + \phi_{H_{sr}}(1 + \phi_{H})} \cr \alpha_3^- &= k_3^- [Pi] \cr \phi_{Cai} &= [Ca^{2+}]_i / K_{d, Cai} \cr \phi_{Casr} &= [Ca^{2+}]_{sr} / K_{d, Casr} \cr \phi_{Hi} &= [H^+]_i^n / K_{d, Hi} \cr \phi_{H1} &= [H^+]_i / K_{d, H1} \cr \phi_{Hsr} &= [H^+]_i^n / K_{d, Hsr} \cr \phi_{H} &= [H^+]_i / K_{d, H} \cr k_1^- &= \frac{k_1^+k_2^+k_3^+K_{d, Casr}^2K_{d, Hi}K_{d, H}}{k_2^-k_3^-K_{d, Cai}^2K_{d, Hsr}e^{\Delta G^0_{MgATP} / RT}} \cr \Delta_3 &= \alpha_{2}^{+} \alpha_{3}^{+}+\alpha_{1}^{-} \alpha_{3}^{+}+\alpha_{1}^{-} \alpha_{2}^{-}+\alpha_{1}^{+} \alpha_{3}^{+}+\alpha_{2}^{-} \alpha_{1}^{+} +\alpha_{2}^{-} \alpha_{3}^{-}+\alpha_{1}^{+} \alpha_{2}^{+} +\alpha_{3}^- \alpha_{1}^- + \alpha_{3}^- \alpha_{2}^{+} \end{aligned} $$
  • ΔGoMgATP = 11.9 kJ / mol, R = 8.314 J / K mol

Data set to constrain parameters

  • The aim was to use whole pump flux data as a function of Ca2+, H+, and each of the metabolites and, where possible
  • The entire data set was normalized to a maximum cycle rate of 5Hz at optimal conditions, defined as [Ca2+]sr = 0, [Ca2+]i = 10 μM, [ATP] = 5 mM, [ADP] = 0, [Pi] = 0, and pH = 7.2
    • rabbit: 4.2 Hz; guinea pig: 7.5 Hz @ 37 ; canine: 5Hz @ 25

Parameter optimization

  • estimated using the particle swarm optimization method
  • then used as initial conditions in a Levenberg-Marquardt least-squares fitting procedure to refine the solution


Ca2+ binding

  • The fully cooperative Ca2+ binding model is favored over the partially cooperative Ca2+ binding model as it achieves a similar quality of fit while using three fewer parameters.
    The fit of the three-state model with fully cooperative Ca2+ binding

Ca2+/ H+ countertransport

  • The pump operates optimally at a pH of ∼7.1
  • The optimal fit produced a value of 2.0, which is consistent with experimental observations of 2

Metabolite dependence
vSERCA vs Pi

  • MgATP binding first is better than Ca binding first

Thermodynamic properties of the model
The effect of changes in the net free energy on the cycle pump rate at different luminal Ca2+ concentrations

  • The pump stops completely at the reversal point where the net free energy is zero
The effect of increasing MgADP vs Pi, to progressively reduce the available free energy

  • The kinetics of MgATP binding therefore limits the maximum reversal rate of the pump

Simplifying to a two-state model

  • normal MgATP concentration in a healthy myocyte is ∼5 mM while it drops by 50% after ischemia for 15 mins
  • Using rapid equilibrium assumptions, the slow MgATP binding step is turned into a rapid binding step: Kd,ATP = k1− / k1+
  • the two models are essentially identical when MgATP is in the mM range but begin to diverge when MgATP drops to the μM range

$$ \begin{aligned} v_{cycle} &= \frac{\alpha_1^+\alpha_2^+ - \alpha_1^-\alpha_2^-}{\Delta_2} \cr \alpha_1^+ &= \frac{k_2^+\phi_{MgATP}\phi_{Cai}^2}{\phi_{MgATP}\phi_{Cai}^2 + \phi_{Hi}(1 + \phi_{MgATP}(1 + \phi_{H1} + \phi_{Cai}^2))} \cr \alpha_2^+ &= \frac{k_3^+\phi_{Hsr}}{\phi_{Hsr} (1 + \phi_{H}) + \phi_{H}(1 + \phi_{Casr}^2)} \cr \alpha_1^- &= \frac{k_2^-[MgADP]\phi_{Casr}^2\phi_{H}}{\phi_{Hsr} (1 + \phi_{H}) + \phi_{H}(1 + \phi_{Casr}^2)} \cr \alpha_2^- &= \frac{k_3^-[Pi]\phi_{Hi}}{\phi_{MgATP}\phi_{Cai}^2 + \phi_{Hi}(1 + \phi_{MgATP}(1 + \phi_{H1} + \phi_{Cai}^2))} \cr \Delta_2 &= \alpha_1^+ + \alpha_2^+ + \alpha_1^- + \alpha_2^- \cr \end{aligned} $$


  • Using the approach outlined in Smith and Crampin in Na-K ATPase, we have developed a mathematical model of the cardiac SERCA pump, which is sensitive to the metabolic state of the cell as well as the luminal and cytosolic concentrations of Ca2+ and H+
  • we have shown that this model is sufficient to describe a range of experimental data on the SERCA pump flux

Ca2+ binding

  • Ca2+ is known to bind in a positive cooperative manner. Mathematically, the fully cooperative mechanism has three fewer parameters.
  • For pH values close to 7, half-maximal binding of Ca2+ from the cytosol has been measured or calculated to be between 0.5 and 10 μM. Model here: 4.88 μM
  • SR Ca2+ binding constant of 3.6 mM, consistent with values from the literature of 1.7 mM.
  • The greater binding affinity of cytosolic Ca2+ over luminal Ca2+ of approximately three orders of magnitude is a key characteristic of the pump that facilitates the binding of Ca2+ from the cytosol

Variable stoichiometry of Ca2+ transport

Coupling ratio of 2 Ca2+ per ATP appropriate for CMC physiological settings

H+ binding

At neutral pH, the stoichiometry of the counter-transport between Ca2+ and H+ is measured to be 2:2, as in the model

Binding order of partial reactions

The order and placement of the partial reactions involving MgATP binding was shown to be an important determinant of the ability to fit the data set, where an acceptable fit was achieved only when the MgATP binding step was placed before the first Ca2+ binding step and after the release of Pi

Reversal of the pump

  • The SERCA pump has been shown experimentally to be reversible under conditions where the concentrations of MgADP and Pi are high and MgATP is kept very low
  • The thermodynamic constraint imposed on the model (see Methods) ensures that the pump will reverse when the net free energy is zero but the actual rate of cycling in the reverse direction is highly dependent on the kinetics of the model, particularly to the kinetics of MgATP binding
    An increase in the free [Ca2+]sr leads to a decrease in the pump rate

Data selection

  • The parameters of the cardiac SERCA model were constrained to a set of data which characterizes the dependence of the pump on the metabolites MgATP, MgADP, and Pi and the ions Ca2+ and H+
  • the pH curve and pH-dependent Ca2+ curves: mouse CMC with phospholamban effects
  • The ATP/ADP and Pi data: rabbit skeletal muscle expressing nonnative SERCA. The presence or absence of phospholamban is not relevant here
  • The density of cardiac SERCA pumps in mouse is estimated to be higher than that of rabbit. In the modeling process, this difference in Vmax between different data sets is accounted for by normalizing all the data to a single reference point
Parameter sensitivity

The parameter with the greatest influence in the model is n, the number of H+ that are counter-transported for each Ca2+. Taking together the high sensitivity of this parameter and the prediction of n = 2 (which is consistent with the literature). This provides us with a good degree of confidence in the optimized model parameters.

  1. Tran K, Smith NP, Loiselle DS, Crampin EJ. A thermodynamic model of the cardiac sarcoplasmic/endoplasmic Ca(2+) (SERCA) pump. Biophys J. 2009;96(5):2029-42.PMC2717298 ↩︎