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Model description

TCA cycle rates

Conservation relationship

$$ \begin{align} \Sigma_{CAC} = [CIT] + [ISOC] + [\alpha KG] + [SCoA] + [SUC] + [FUM] + [MAL] + [OAA] \end{align} $$

Parameter Value Unit Description
$\Sigma_{CAC}$ 1.300 mM Sum of TCA cycle intermediates

Citrate synthase (CS)

$$ \begin{align} J_{CS} = \frac{k_{cat}^{CS} E_T^{CS} ([AcCoA] / K_m^{AcCoA})([OAA] / K_m^{OAA})}{(1+[AcCoA] / K_m^{AcCoA})(1+[OAA] / K_m^{OAA})} \end{align} $$

Parameter Value Unit Description
$k_{cat}^{CS}$ 0.23523 Hz Catalytic constant
$E_T^{CS}$ 0.4 mM Enzyme concentration of CS
$K_m^{AcCoA}$ 12.6 μM Michaelis constant for AcCoA
$K_m^{OAA}$ 0.64 μM Michaelis constant for OAA
$[AcCoA]$ 0.1 mM Acetyl CoA concentration

Aconitase (ACO)

$$ J_{ACO} = k_f^{ACO} ([CIT] - [ISOC] / K_{eq}^{ACO}) $$

Parameter Value Unit Description
$k_f^{ACO}$ 0.1 Hz Forward rate constant of ACO
$K_{eq}^{ACO}$ 2.22 - Equilibrium constant of ACO

Isocitrate dehydrogenase, NADH-producing (IDH3)

$$ \begin{align} J_{IDH3} &= \frac{k_{cat}^{IDH3} E_T^{IDH3} AB}{f_H AB + f_i B + f_a A + f_a f_i} \ f_H & = 1 + \frac{[H^+]m}{K{H1}^{IDH3}} + \frac{K_{H2}^{IDH3}}{[H^+]m} \ A &= [NAD] / K{NAD}^{IDH3} \ B &= ([ISOC] / K_{ISOC}^{IDH3})^n_{IDH3} \ f_a &= \frac{K_A^{IDH3}}{K_A^{IDH3} + [ADP]m} \frac{K{CA}^{IDH3}}{K_{CA}^{IDH3} + [Ca^{2+}]m} \ f_i &= 1 + \frac{[NADH]}{K{NADH}^{IDH3}} \ \end{align} $$

Parameter Value Unit Description
$k_{cat}^{IDH3}$ 535 Hz Rate constant of IDH3
$E_T^{IDH3}$ 0.109 mM Concentration of IDH3
$K_{H1}^{IDH3}$ 1 nM Ionization constant of IDH3
$K_{H2}^{IDH3}$ 900 nM Ionization constant of IDH3
$K_{NAD}^{IDH3}$ 0.923 mM Michaelis constant for NAD
$K_{ISOC}^{IDH3}$ 1.520 mM Michaelis constant for isocitrate
$n_{IDH3}$ 2 - Cooperativity for isocitrate
$K_A^{IDH3}$ 0.62 mM Activation constant by ADP
$K_{CA}^{IDH3}$ 0.5 μM Activation constant for calcium
$K_{NADH}^{IDH3}$ 0.19 mM Inhibition constant by NADH

Alpha-ketoglutarate dehydrogenase (KGDH)

$$ \begin{align} J_{KGDH} &= \frac{k_{cat}^{KGDH} E_T^{KGDH} AB}{f_H AB + f_a (A + B)} \ f_H & = 1 + \frac{[H^+]m}{K{H1}^{KGDH}} + \frac{K_{H2}^{KGDH}}{[H^+]m} \ A &= [NAD] / K{NAD}^{KGDH} \ B &= ([\alpha KG] / K_{AKG}^{KGDH})^n_{KGDH} \ f_a &= \frac{K_{MG}^{KGDH}}{K_{MG}^{KGDH} + [Mg^{2+}]m} \frac{K{CA}^{KGDH}}{K_{CA}^{KGDH} + [Ca^{2+}]_m} \ \end{align} $$

Parameter Value Unit Description
$k_{cat}^{KGDH}$ 17.9 Hz Rate constant of KGDH
$E_T^{KGDH}$ 0.5 mM Concentration of KGDH
$K_{H1}^{KGDH}$ 40 nM Ionization constant of KGDH
$K_{H2}^{KGDH}$ 70 nM Ionization constant of KGDH
$K_{NAD}^{KGDH}$ 38.7 mM Michaelis constant for NAD
$K_{AKG}^{KGDH}$ 30 mM Michaelis constant for αKG
$n_{KGDH}$ 1.2 - Hill coefficient for αKG
$K_{MG}^{KGDH}$ 30.8 μM Activation constant for Mg
$K_{CA}^{KGDH}$ 0.15 μM Activation constant for Ca

Succinate-CoA ligase (SL)

$$ \begin{align} J_{SL} &= k_f^{SL} ([SCoA][ADP]m[Pi]m - [SUC][ATP]m[CoA]/K{eq}^{app}) \ K{eq}^{app} &= K{eq}^{SL} \frac{P_{SUC}P_{ATP}}{P_{Pi}P_{ADP}} \ \end{align} $$

Parameter Value Unit Description
$k_f^{SL}$ 28.4 1Hz/mM² Forward rate constant of SL
$K_{eq}^{SL}$ 3.11 - Equilibrium constant of SL
[CoA] 0.020 mM Coenzyme A concentration

Succinate dehydrogenase (SDH)

See OXPHOS part: complex II (Succinate dehydrogenase).

Fumarate hydratase (FH)

$$ J_{FH} = k_f^{FH} ([FUM] - [MAL] / K_{eq}^{FH}) $$

Parameter Value Unit Description
$k_f^{FH}$ 8.3 Hz Forward rate constant
$K_{eq}^{FH}$ 1.0 - Equilibrium constant

Malate dehydrogenase (MDH)

$$ \begin{align} J_{MDH} &= \frac{k_{cat}^{MDH} E_T^{MDH} AB f_a f_i}{(1+A)(1+B)} \ A &= \frac{[MAL]}{K_{MAL}^{MDH}}\frac{K_{OAA}^{MDH}}{K_{OAA}^{MDH} + [OAA]} \ B &= [NAD] / K_{NAD}^{MDH} \ f_a &= k_{offset}^{MDH} + \left( 1 + \frac{[H^+]m}{K{H1}^{MDH}} (1 + \frac{[H^+]m}{K{H2}^{MDH}}) \right)^{-1} \ f_i &= \left( 1 + \frac{K_{H3}^{MDH}}{[H^+]m} (1 + \frac{K{H4}^{MDH}}{[H^+]_m}) \right)^{2} \ \end{align} $$

Parameter Value Units Description
$k_{cat}^{MDH}$ 125.9 Hz Rate constant
$E_T^{MDH}$ 154 μM
$K_{H1}^{MDH}$ 11.31 nM Ionization constant
$K_{H2}^{MDH}$ 26.7 mM Ionization constant
$K_{H3}^{MDH}$ 6.68 pM Ionization constant
$K_{H4}^{MDH}$ 5.62 nM Ionization constant
$k_{offset}^{MDH}$ 0.0399 - Offset of MDH pH activation factor
$K_{NAD}^{MDH}$ 224.4 μM Michaelis constant for NAD
$K_{MAL}^{MDH}$ 1.493 mM Michaelis constant for malate
$K_{OAA}^{MDH}$ 31 μM Inhibition constant for oxaloacetate

Aspartate aminotransferase (AAT)

$$ \begin{align} J_{AAT} = k_f^{AAT} [OAA][GLU] \frac{k_{ASP}^{AAT} K_{eq}^{AAT}}{k_{ASP}^{AAT} K_{eq}^{AAT} + k_f[\alpha KG]} \end{align} $$

Parameter Value Units Description
$k_f^{AAT}$ 21.7 Hz/mM Forward rate constant
$k_{ASP}^{AAT}$ 0.0015 Hz Rate constant of aspartate consumption
$K_{eq}^{AAT}$ 6.6 - Equilibrium constant
[GLU] 30 mM Glutamate concentration

ODEs in the citric acid cycle

$$ \begin{align} \frac{d [ISOC]}{dt} &= J_{ACO} -J_{IDH3} -J_{IDH2} \\ \frac{d [\alpha KG]}{dt} &= J_{IDH3} + J_{IDH2} - J_{KGDH} + J_{AAT} \\ \frac{d [SCoA]}{dt} &= J_{KGDH} - J_{SL} \\ \frac{d [SUC]}{dt} &= J_{SL} - J_{SDH} \\ \frac{d [FUM]}{dt} &= J_{SDH} - J_{FH} \\ \frac{d [MAL]}{dt} &= J_{FH} - J_{MDH} \\ \frac{d [OAA]}{dt} & = J_{MDH} - J_{CS} - J_{AAT} \\ \end{align} $$

Endoplasmic reticulum

Ryanodine receptor (Jrel)

$$ \begin{align} P_{C1} &= 1 - P_{O1} - P_{O2} - P_{C2} \ v_{o1c1} &= -k_a^-P_{O1} + k_a^+[Ca^{2+}]{ss}^n P{C1} \ v_{o1o2} &= k_b^+ [Ca^{2+}]{ss}^m{RyR} P_{O1} - k_b^- P_{O2} \ v_{o1c2} &= k_c^+ P_{O1} - k_c^- P_{C2} \ \dot{P_{O1}} &= -v_{o1c1} - v_{o1o2} - v_{o1c2} \ \dot{P_{O2}} &= v_{o1o2} \ \dot{P_{C2}} &= v_{o1c2} \ J_{rel} &= r_{ryr} (P_{O1} + P_{O2})([Ca^{2+}]{JSR} - [Ca^{2+}]{ss}) \ \end{align} $$

Parameter Value Units Description
$r_{RyR}$ 3600 Hz RyR flux channel constant
$n_{RyR}$ 4 - Cooperativity parameter
$m_{RyR}$ 3 - Cooperativity parameter
$k_a^+$ 12.15 Hz/μM⁴ RyR rate constant
$k_a^-$ 576 Hz RyR rate constant
$k_b^+$ 0.00405 Hz/μM³ RyR rate constant
$k_b^-$ 1930 Hz RyR rate constant
$k_c^+$ 100 Hz RyR rate constant
$k_c^-$ 0.8 Hz RyR rate constant

SERCA (Jup)

Michaelis-Menten dependence of enzyme activity with respect to ATP and mixed-type inhibition of the enzyme by ADP. Reversible during diastole with low cytoplasmic calcium levels.

$$ \begin{align} J_{up} &= \frac{V_{f}^{up}f_b-V_{r}^{up}r_b}{(1 + f_b + r_b)f_{ATP}^{SERCA}} \ f_b &= \left( \frac{[Ca^{2+}]i}{K{fb}} \right)^{N_{fb}} \ r_b &= \left( \frac{[Ca^{2+}]{NSR}}{K{rb}} \right)^{N_{rb}} \ f_{ATP}^{SERCA} &= \frac{K_{m,up}^{ATP}}{[ATP]_i} ( \frac{[ADP]i}{K{i1, up}} + 1) + \frac{[ADP]i}{K{i2, up}} + 1 \ \end{align} $$

Parameter Value Units Description
$V_{max, f}^{SERCA}$ 0.2989 Hz*mM SERCA forward rate parameter
$V_{max, b}^{SERCA}$ 0.3179 Hz*mM SERCA reverse rate parameter
$K_{f}^{SERCA}$ 0.24 μM Forward Ca2+ half-saturation constant of SERCA
$K_{r}^{SERCA}$ 1.64269 mM Reverse Ca2+ half-saturation constant of SERCA
$N_{f}^{SERCA}$ 1.4 - Forward cooperativity constant of SERCA
$N_{r}^{SERCA}$ 1.0 - Reverse cooperativity constant of SERCA
$K_{ATP}^{SERCA}$ 10 μM ATP half-saturation constant for SERCA
$K_{ADP1}^{SERCA}$ 140 μM ADP first inhibition constant for SERCA
$K_{ADP2}^{SERCA}$ 5.1 mM ADP second inhibition constant for SERCA

Sarcoplasmic ion currents

GHK current equation

$$ \begin{align} \Phi_s(P_s, z_s, V_m, [S]_i, [S]_o) := P_sz^2_s\frac{V_mF^2}{RT}\frac{[S]_i - [S]_o\exp(-z_sV_mF/RT)}{1-\exp(-z_sV_mF/RT)} \end{align} $$

Time-dependent delayed rectifier potassium current (IK)

$$ \begin{align} I_K &= \bar G_K X_1 X_K^2 (V - E_K) \ E_K &= \frac{RT}{F} \ln \frac{[K^+]o + P{Na,K}[Na^+]_o}{ [K^+]i + P{Na,K}[Na^+]_i} \ \bar G_K &= 0.282 (mS/cm^2)\sqrt{[K^+]_o /5.4mM} \ X_1 &= (1+ e^{(V_m-40)/40})^{-1} \ \frac{dX_k}{dt} &= \alpha_X - X_k (\alpha_X + \beta_X) \ \alpha_X &= \frac{V_m+30}{1 - e^{-0.148(V_m+30)}} * 0.0719Hz \ \beta_X &= \frac{V_m+30}{e^{0.0687(V_m+30)} -1} * 0.131Hz \ \end{align} $$

Time-independent potassium current (IK1)

$$ \begin{align} \Delta V &= V_m - E_{K1} \ I_{K1} &= \bar G_{K1}K_{1 \infty}\Delta V \ E_{K1} &= \frac{RT}{F} \ln \frac{[K^+]o}{[K^+]i} \ \bar G{K1} &= 0.748(mS/cm^2)\sqrt{[K^+]o / 5.4mM} \ K{1 \infty} &= \frac{\alpha{K_1}}{\alpha_{K_1} + \beta_{K_1}} \ \alpha_{K_1} &= \frac{1.02}{1 + e^{0.2385(\Delta V -59.215)}} * \text{kHz} \ \beta_{K_1} &= \frac{0.4912e^{0.28032(\Delta V + 5.476)} + e^{0.06175(\Delta V -594.31)}}{1 + e^{-0.5143(\Delta V + 4.753)}} * \text{kHz} \end{align} $$

Plateau potassium current (IKp)

$$ \begin{align} E_{Kp} &= \frac{RT}{F} \ln \frac{[K^+]o}{[K^+]i} \ I{Kp} &= \frac{\bar G{Kp} (V - E_{Kp})}{1 + e^{(7.488-V_m) / 5.98}} \ \end{align} $$

Fast Na current (INa)

$$ \begin{align} I_{Na} &= \bar G_{Na} m_{Na}^{3} h_{Na} j_{Na} (V_m-E_{Na}) \ E_{Na} &= \frac{RT}{F} \ln \frac{[Na^{+}]o}{[Na^{+}]i} \ \frac{dm{Na}}{dt} &= \alpha{m} - m_{Na}(\alpha_{m} + \beta_{m}) \ \frac{dh_{Na}}{dt} &= \alpha_{h} - h_{Na}(\alpha_{h} + \beta_{h}) \ \frac{dj_{Na}}{dt} &= \alpha_{j} - m_{Na}(\alpha_{j} + \beta_{j}) \ \alpha_{m} &= 0.32kHz \frac{V_m + 47.13}{1 - e^{-0.1(V_m+47.13)}} \ \beta_{m} &= 0.08kHz \times e^{-V_m / 11} \ \ For \ V_m & \ge -40mV \ \alpha_{h} &= \alpha_{j} = 0 \ \beta_{h} &= (0.13 ms (1+e^{-(V_m+10.66)/11.1}))^{-1} \ \beta_{j} &= 0.3kHz\frac{e^{-2.535 \times 10^{-7}V_m}}{1 + e^{-0.1(V_m + 32)}} \ \ For \ V_m & < -40mV \ \alpha_{h} &= 0.135kHz * e^{-(V_m+80)/6.8} \ \alpha_{j} &= (-127140e^{0.2444 V_m}-3.474 \times 10^{-5}e^{-0.04391 V_m})\frac{V_m + 37.78}{1+e^{0.311( V_m +79.23)}} \times \text{kHz} \ \beta_{h} &= (3.56e^{0.079 V_m} + 3.1 \times 10^{5}e^{0.35 V_m}) \times \text{kHz} \ \beta_{j} &= \frac{0.1212e^{-0.01052 V_m}}{1+e^{-0.1378(V_m + 40.14)}} \times \text{kHz} \ \end{align} $$

Sodium-calcium exchanger current (INaCa)

$$ \begin{align} I_{NaCa} &= k_{NaCa} \cdot f_{Nao } \cdot f_{Cao }\frac{exp(V_mF/RT)\phi_{Na}^3 - \phi_{Ca}}{exp((1 - \eta) V_mF/RT ) + k_{sat}} \ f_{Nao} &= \frac{([Na^+]o)^3}{([Na^+]o)^3 + (K{M,Na}^{NaCa})^3} \ f{Cao} &= \frac{[Ca^+]o}{[Ca^+]o + K{M,Ca}^{NaCa}} \ \phi{Na} &= \frac{[Na^+]_i}{ [Na^+]o} \ \phi{Ca} &= \frac{[Ca^{2+}]_i}{[Ca^{2+}]_o} \ \end{align} $$

Background calcium ($I_{Ca,b}$) and sodium currents ($I_{Na,b}$)

$$ \begin{align} I_{Ca,b} &= \bar G_{Ca,b} (V_m - \frac{RT}{2F} \ln \frac{[Ca^{2+}]o}{[Ca^{2+}]i}) \ I{Na,b} &= \bar G{Na,b} (V_m - \frac{RT}{F} \ln \frac{[Na^{+}]_o}{[Na^{+}]_i}) \ \end{align} $$

Non-specific calcium-activated current (InsCa)

$$ \begin{align} f_{Ca} &= \frac{([Ca^{2+}]i)^3}{([Ca^{2+}]i)^3 + (K{m}^{nsCa})^3}\ I{nsNa} &= 0.75 \cdot f_{Ca} \cdot \Phi_{Na}(P_{nsNa}, z_{Na}, V_m, [Na^+]i, [Na^+]o) \ I{nsK} &= 0.75 \cdot f{Ca} \cdot \Phi_{K}(P_{nsK}, z_{K}, V_m, [K^+]_i, [K^+]_o) \ \end{align} $$

Sodium-potassium ATPase current (INaK)

The Na+/K+ ATPase activity depends on the ATP concentration, as well as the competitive inhibition by ADP.

$$ \begin{align} I_{NaK} &= \bar I_{NaK} \cdot f_{ATP} \cdot f_{Na} \cdot f_{K} \cdot f_{NaK} \ \sigma &= \frac{e^{[Na^+]o / 67.3mM}-1}{7} \ f{NaK} &= (1 + 0.1245 \cdot \exp(-0.1V_m F / RT) + 0.0365 \sigma \cdot \exp(-V_m F / RT))^{-1} \ f_{Na} &= \frac{([Na^+]i)^{1.5}}{([Na^+]i)^{1.5} + (K{m, Na_i})^{1.5}} \ f{K} &= \frac{[K^+]o}{[K^+]o + K{m, K_o}} \ f{ATP} &= \frac{[ATP]i}{[ATP]i + K{M,ATP}^{NaK} / f{ADP}} \ f_{ADP} &= \frac{K_{i,ADP}^{NaK}}{K_{i,ADP}^{NaK} + [ADP]_i} \ \end{align} $$

L-type Ca current (ICa & ICaK)

"Common pool" subspace calcium model.

$$ \begin{align} \alpha &= 0.4 e^{(V_m+2) / 10} \ \beta &= 0.4 e^{-(V_m+2) / 13} \ \alpha^\prime &= a \alpha \ \beta^\prime &= \beta / b \ \gamma &= \gamma_0 [Ca^{2+}]{ss} \ C_0 &= 1 - C_0 - C_1 - C_2 - C_3 - C_4 - O - C{Ca0} - C_{Ca1} - C_{Ca2} - C_{Ca3} - C_{Ca4} \ v_{01} &= 4\alpha C_0 - \beta C_1 \ v_{12} &= 3\alpha C_1 - 2\beta C_2 \ v_{23} &= 2\alpha C_2 - 3\beta C_3 \ v_{34} &= \alpha C_3 - 4\beta C_4 \ v_{45} &= f C_4 - g O \ v_{67} &= 4\alpha^\prime C_{Ca0} - \beta^\prime C_{Ca1} \ v_{78} &= 3\alpha^\prime C_{Ca1} - 2\beta^\prime C_{Ca2} \ v_{89} &= 2\alpha^\prime C_{Ca2} - 3\beta^\prime C_{Ca3} \ v_{910} &= \alpha^\prime C_{Ca3} - 4\beta^\prime C_{Ca4} \ v_{06} &= \gamma C_0 - \omega C_{Ca0} \ v_{17} &= a \gamma C_1 - \omega C_{Ca1} / b \ v_{28} &= a^2 \gamma C_2 - \omega C_{Ca2} / b^2 \ v_{39} &= a^3 \gamma C_3 - \omega C_{Ca3} / b^3 \ v_{410} &= a^4 \gamma C_4 - \omega C_{Ca4} / b^4 \ \end{align} $$

$$ \begin{align} \frac{dC_0}{dt} &= -v_{01} -v_{06} \ \frac{dC_1}{dt} &= v_{01} - v_{12} - v_{17} \ \frac{dC_2}{dt} &= v_{12} - v_{23} - v_{28} \ \frac{dC_3}{dt} &= v_{23} - v_{34} - v_{34} \ \frac{dC_4}{dt} &= v_{34} - v_{45} - v_{410} \ \frac{dO}{dt} &= v_{45} \ \frac{dC_{Ca0}}{dt} &= v_{06} - v_{67} \ \frac{dC_{Ca1}}{dt} &= v_{17} + v_{67} - v_{78} \ \frac{dC_{Ca2}}{dt} &= v_{28} + v_{78} - v_{89} \ \frac{dC_{Ca3}}{dt} &= v_{39} + v_{89} - v_{910} \ I_{Ca}^{max} &= \Phi_{Ca}(P_{Ca}, z_{Ca}, V_m, 0.001, 0.341[Ca^{2+}]o) \ I{Ca} &= 6 I_{Ca}^{max} \cdot y_{Ca} \cdot O \ I_{Ca,K} &= y_{Ca} \cdot O \cdot \Phi_{Ca}(P_{K}, z_{K}, V_m, [K^+]i, [K^+]o) \ P{K} &= P{K}^{max} \frac{I_{Ca}^{half}}{I_{Ca}^{half} + I_{Ca}^{max}} \ y_\infty &= \frac{1}{1 + e^{(V_m + 55) / 7.5}} + \frac{0.5}{1 + e^{(-V_m + 21) / 6}} \ \tau_y &= 20ms + \frac{600ms}{1 + e^{(V_m + 30) / 9.5}} \ \frac{dy_{Ca}}{dt} &= \frac{y_\infty - y_{Ca}}{\tau_y} \ \end{align} $$

Parameter Value Units Description
$A$ 2 Mode transition parameter
$B$ 2 Mode transition parameter
$\gamma_0$ 187.5 Hz/μM Mode transition parameter
$\omega$ 10 Hz Mode transition parameter
$f$ 300 Hz Transition rate into open state
$g$ 2000 Hz Transition rate into open state
$P_{Ca}^{LCC}$ $1.24 \cdot 10^{-3}$ cm/s L-type Ca2+ channel permeability to Ca2+
$P_{K}^{LCC}$ $1.11 \cdot 10^{-11}$ cm/s L-type Ca2+ channel permeability to K+
$I_{Ca, half}$ $-0.4583$ $\mu A / cm^{2}$ ICa level that reduces equation Pk by half

Plasma membrane calcium ATPase (PMCA) current (IpCa)

Modified rate expression incorporating the ATP-dependence of pump activity. Plasma membrane calcium ATPase (PMCA) rate exhibits two different K0.5 values for ATP.

$$ \begin{align} I_{pCa} &= I_{max}^{PMCA} \times \frac{[Ca^{2+}]i}{[Ca^{2+}]i + K{M, Ca}^{PMCA}} \times f{ATP} \ f_{ATP} &= \frac{[ATP]i}{[ATP]i + K{M2,ATP}^{PMCA}} + \frac{[ATP]i}{[ATP]i + K{M1,ATP}^{PMCA} / f{ADP}} \ f{ADP} &= \frac{K{i,ADP}^{PMCA}}{K_{i,ADP}^{PMCA} + [ADP]_i} \ \end{align} $$

Parameter Value Units Description
$I_{max}^{PMCA}$ $0.575$ $\mu A/cm^2$ Maximum sarcolemmal Ca2+ pump current
$K_{Ca}^{PMCA}$ $0.5$ $uM$ Ca2+ half-saturation constant for sarcolemmal Ca2+ pump
$K_{ATP1}^{PMCA}$ $0.012$ $mM$ First ATP half-saturation constant for sarcolemmal Ca2+ pump
$K_{ATP2}^{PMCA}$ $0.23$ $mM$ Second ATP half-saturation constant for sarcolemmal Ca2+ pump
$K_{ADP}^{PMCA}$ $1.0$ $mM$ ADP inhibition constant for sarcolemmal Ca2+ pump

Electrophysiology ODEs

$$ \begin{align} \frac{d[Na^+]i}{dt} &= -(I{Na} + 3I_{NaCa} + 3I_{NaK})\frac{A_{cap}}{V_{myo}F} + (V_{NHE} - 3V_{NaCa}) \frac{V_{mito}}{V_{myo}} \ \frac{d[K^+]i}{dt} &= -(I{Ks} + I_{Kr} + I_{K1} + I_{Kp} + I_{Ca,K}-2I_{NaK})\frac{A_{cap}}{V_{myo}F} \ C_m\frac{dV_m}{dt} &= -(I_{Na} + I_{CaL} + I_{Kr} + I_{Ks} + I_{K1} + I_{Kp} + I_{NaCa} + I_{NaK} + I_{pCa} + I_{Ca, b} + I_{K_{ATP}} + I_{stim}) \ β_i &= \frac{(K_m^{CMDN} + [Ca^{2+}]i)^2}{ (K_m^{CMDN} + [Ca^{2+}]i)^2 + K_m^{CMDN} \cdot [CMDN]{tot}} \ β{SR} &= \frac{(K_m^{CSQN} + [Ca^{2+}]{SR})^2}{(K_m^{CSQN} + [Ca^{2+}]{SR})^2 + K_m^{CSQN} \cdot [CSQN]{tot}} \ \frac{d[Ca^{2+}]i}{dt} &= \beta_i(J{xfer}\frac{V{ss}}{V_{myo}} - J_{up} - J_{trpn} - (I_{Ca,b} -2I_{NaCa} + I_{pCa})\frac{A_{cap}}{2V_{myo}F} + (V_{NaCa} - V_{uni})\frac{V_{mito}}{V_{myo}}) \ \frac{d[Ca^{2+}]{SR}}{dt} &= \beta{SR}(J_{up}\frac{V_{myo}}{V_{SR}} - J_{rel}\frac{V_{ss}}{V_{SR}}) \ \end{align} $$

Symbol Value Units Description
$G_{Na}$ $12.8$ $mS/cm^2$ Maximal Na channel conductance
$G_{Kp}$ $0.00828$ $mS/cm^2$ Maximal plateau K channel conductance
$G_{K,0}$ $0.282$ $mS/cm^2$ IK conductance
$G_{K1,0}$ $0.748$ $mS/cm^2$ IK1 conductance
$P_{NaK}$ $0.01833$ Na+ permeability ratio of K+ channel
$K_{NaCa}$ $9000$ $\mu A/cm^2$ NCX current
$K_{Na}^{NCX}$ $87.5$ $mM$ Dissociation constant of sodium for NCX
$K_{Ca}^{NCX}$ $1.38$ $mM$ Dissociation constant of calcium for NCX
$K_{sat}^{NCX}$ $0.1$ NCX saturation factor at negative potentials
$\eta^{NCX}$ $0.35$ Voltage dependence of NCX
$P_{ns,Na}$ $1.75 \cdot 10^{-7}$ $cm/s$ Nonspecific channel current Na permeability
$P_{ns,K}$ $0$ $cm/s$ Nonspecific channel current K permeability
$K_{ca}^{ns}$ $1.2$ $\mu M$ Ca2+ half-saturation constant for nonspecific current
$G_{Ca,b}$ $0.003217$ $mS/cm^2$ Maximum background current Ca2+ conductance
$G_{Na,b}$ $0.003217$ $mS/cm^2$ Maximum background current Na+ conductance
$\tau_{tr}$ $574.7$ Hz Time constant for transfer from subspace to myoplasm
$\tau_{xfer}$ $9090$ Hz Time constant for transfer from NSR to JSR
$K_{m}^{CMDN}$ $2.38$ $\mu M$ Ca2+ half saturation constant for calmodulin
$K_{m}^{CSQN}$ $800$ $\mu M$ Ca2+ half saturation constant for calsequestrin
$\Sigma[HTRPN]$ $140$ $\mu M$ Total troponin high-affinity sites
$\Sigma[LTRPN]$ $70$ $\mu M$ Total troponin low-affinity sites
$\Sigma[CMDN]$ $50$ $\mu M$ Total myoplasmic calmodulin concentration
$\Sigma[CQSN]$ $15$ $mM$ Total NSR calsequestrin concentration

Force generation

The rate of ATP hydrolysis associated with force generation through actomyosin ATPase depends explicitly on both ATP and ADP. 1

$$ \begin{align} f_{01} &= 3f_{XB} \ f_{12} &= 10f_{XB} \ f_{23} &= 7f_{XB} \ g_{01} &= g_{XB}^{min} \ g_{12} &= 2g_{XB}^{min} \ g_{23} &= 3g_{XB}^{min} \ g_{01,SL} &= \phi \cdot g_{01} \ g_{12,SL} &= \phi \cdot g_{12} \ g_{23,SL} &= \phi \cdot g_{23} \ g_{01,SL, off} &= \phi \cdot g_{off} \ \phi &= 1 + \frac{2.3-SL}{(2.3-1.7)^{1.6}} \ K_{Ca}^{trop} &= \frac{k^-{ltrpn}}{k^+{ltrpn}} \ K_{1/2}^{trop} &= \left( 1 + \frac{K_{Ca}^{trop}}{1.7 \cdot 10^{-3} - 0.8 \cdot 10^{-3}\frac{(SL-1.7)}{0.6}} \right)^{-1} \ N_{trop} &= 3.5 \cdot SL - 2.0 \ k_{np}^{trop} &= k_{pn}^{trop} \left( \frac{[LTRPNCa]}{K_{1/2}^{trop}[LTRPN]{tot}} \right) ^{N{trop}} \ \Sigma PATHS &= g_{01}g_{12}g_{23} + f_{01}g_{12}g_{23} + f_{01}f_{12}g_{23} + f_{01}f_{12}f_{23} \ P1_{max} &= \frac{f_{01}g_{12}g_{23}}{\Sigma PATHS} \ P2_{max} &= \frac{f_{01}f_{12}g_{23}}{\Sigma PATHS} \ P3_{max} &= \frac{f_{01}f_{12}f_{23}}{\Sigma PATHS} \ Force &= \zeta \frac{[P_1] + 2[P_2] + 3[P_3] + [N_1]}{P1_{max} + 2P2_{max} + 3P3_{max}} \ Force_{norm} &= \frac{[P_1] + [P_2] + [P_3] + [N_1]}{P1_{max} + P2_{max} + P3_{max}} \ \end{align} $$

$$ \begin{align} v_{01} &= f_{01} [P_0] - g_{01(SL)} [P_1] \ v_{12} &= f_{12} [P_1] - g_{21(SL)} [P_2] \ v_{23} &= f_{23} [P_2] - g_{23(SL)} [P_3] \ v_{04} &= k_{pn}^{trop} [P_0] - k_{np}^{trop} [N_0] \ [N_0] &= 1 - [P_0] - [P_1] - [P_2] - [P_3] - [N_1] \ v_{15} &= k_{pn}^{trop} [P_1] - k_{np}^{trop} [N_1] \ v_{54} &= g_{01,off} [N_1] \ [HTRPN] &= [HTRPN]{tot} - [HTRPNCa] \ [LTRPN] &= [LTRPN]{tot} - [LTRPNCa] \ f_{ATP}^{AM} &= \frac{[ATP]i}{[ATP]i + K{m,AM}^{ATP}/f{ADP}^{AM} } \ f_{ADP}^{AM} &= \frac{K_{i,AM}^{ADP}}{[ADP]i + K{i,AM}^{ADP}} \ V_{AM} &= V_{max}^{AM} \cdot f_{ATP}^{AM} \cdot \frac{f_{01}[P_0] + f_{12}[P_1] + f_{23}[P_2]}{f_{01} + f_{12} + f_{23}} \ J_{trpn} &= \frac{d[HTRPNCa]}{dt} + \frac{d[LTRPNCa]}{dt} \ \frac{d[HTRPNCa]}{dt} &= k^{+}{htrpn}[Ca^{2+}]i[HTRPN] - k^{-}{htrpn}[HTRPNCa] \ \frac{d[LTRPNCa]}{dt} &= k^{+}{ltrpn}[Ca^{2+}]i[LTRPN] - k^{-}{ltrpn}(1-\frac{2}{3}Force_{norm})[LTRPNCa] \ \frac{d[P_0]}{dt} &= - v_{01} - v_{04} \ \frac{d[P_1]}{dt} &= v_{01} - v_{12} - v_{15} \ \frac{d[P_2]}{dt} &= v_{12} - v_{23} \ \frac{d[P_3]}{dt} &= v_{23} \ \frac{d[N_1]}{dt} &= v_{15} - v_{54} \ \end{align} $$

Symbol Value Units Description
$k_{pn}^{trop}$ $40$ $\text{Hz}$ Transition rate from tropomyosin permissive to non-permissive
$\text{SL}$ $2.15$ $\mu \text{m}$ Sarcomere length
$f_{XB}$ $50$ $\text{Hz}$ Transition rate from weak to strong crossbridge
$g_{XB}^{min}$ $100$ $\text{Hz}$ Minimum transition rate from strong to weak crossbridge
$\zeta$ $0.1$ $\text{N/mm}^2$ Conversion factor normalizing to physiological force
$V_{AM}^{max}$ $7.2$ $\text{mM/s}$ Conversion factor normalizing to physiological force
$K_{ATP}^{AM}$ $0.03$ $\text{mM}$ ATP half-saturation constant of AM ATPase
$K_{ADP}^{AM}$ $0.26$ $\text{mM}$ ADP inhibition constant of AM ATPase
$h_{trpn}^{+}$ $100000$ $\text{Hz/mM}$ Ca2+ on-rate for troponin high-affinity sites
$h_{trpn}^{-}$ $0.33$ $\text{Hz}$ Ca2+ off-rate for troponin high-affinity sites
$l_{trpn}^{+}$ $100000$ $\text{Hz/mM}$ Ca2+ on-rate for troponin low-affinity sites
$l_{trpn}^{-}$ $40$ $\text{Hz}$ Ca2+ off-rate for troponin low-affinity sites

OXPHOS

Complex I

Assuming single electron transfer for each redox reaction.

$$ \begin{align} \nu &= \exp((\Delta\Psi_m - \Delta\Psi_B) F/ RT) \ a_{12} &= k_{12} ([H^+]m)^2 \ a{21} &= k_{21} \ a_{65} &= k_{65} ([H^+]i)^2 \ a{56} &= k_{56} \ a_{61} &= k_{61} / \nu \ a_{16} &= k_{16} \nu \ a_{23} &= k_{23} \sqrt{[NADH]} \ a_{32} &= k_{32} \ a_{34} &= k_{34} \ a_{43} &= k_{43} \sqrt{[NAD^+]} \ a_{47} &= C1_{inhib} \cdot K_{47} \sqrt{[Q_n][H^+]m} \ a{74} &= k_{74} \ a_{57} &= C1_{inhib} \cdot K_{57} \sqrt{[QH_2]} \ a_{75} &= k_{75} \ k_{42}^\prime &= k_{42} \ a_{42} &= k_{42}^\prime [O_2] \ K_{eq}^{ROS} &= \exp((E_{FMN} - E_{sox}) F / RT) \ a_{24} &= a_{42} K_{eq}^{ROS} [O_2^{ \cdot -}]m \ a{25} &= a_{52} = 0 \ \end{align} $$

$$ \begin{align} e_{1} &= a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{61} \cdot a_{74} + a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{21} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{21} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} \\ e_{2} &= a_{12} \cdot a_{32} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{32} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{12} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{32} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{32} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{12} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{12} \cdot a_{34} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{34} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{12} \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{16} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} \ &+ a_{16} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} \\ e_{3} &= a_{12} \cdot a_{23} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{23} \cdot a_{42} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{23} \cdot a_{42} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{12} \cdot a_{23} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{23} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{23} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{12} \cdot a_{23} \cdot a_{43} \cdot a_{57} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{23} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{23} \cdot a_{47} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{12} \cdot a_{24} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{24} \cdot a_{43} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{43} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{12} \cdot a_{24} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{23} \cdot a_{42} \cdot a_{57} \cdot a_{65} \cdot a_{74} \ &+ a_{16} \cdot a_{23} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{24} \cdot a_{43} \cdot a_{57} \cdot a_{65} \cdot a_{74} \\ e_{4} &= a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{57} \cdot a_{61} \cdot a_{74} \ &+ a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{56} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{56} \cdot a_{61} \cdot a_{75} \ &+ a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{57} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{56} \cdot a_{61} \cdot a_{74} \ &+ a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{56} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{57} \cdot a_{61} \cdot a_{74} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{57} \cdot a_{65} \cdot a_{74} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{23} \cdot a_{34} \cdot a_{57} \cdot a_{65} \cdot a_{74} \ &+ a_{16} \cdot a_{24} \cdot a_{32} \cdot a_{57} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{24} \cdot a_{34} \cdot a_{57} \cdot a_{65} \cdot a_{74} \\ e_{5} &= a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{65} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{61} \cdot a_{75} \ &+ a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{65} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{61} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{65} \cdot a_{75} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{65} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{65} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{65} \cdot a_{74} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{65} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{47} \cdot a_{65} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{65} \cdot a_{74} \ &+ a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{65} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{47} \cdot a_{65} \cdot a_{75} + a_{16} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{65} \cdot a_{75} \ &+ a_{16} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{65} \cdot a_{75} + a_{16} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{65} \cdot a_{75} \\ e_{6} &= a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{75} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{75} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{56} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{56} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{74} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{56} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{56} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{74} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{75} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{56} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{56} \cdot a_{75} \ &+ a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{74} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{75} + a_{16} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{75} \ &+ a_{16} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{75} + a_{16} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{75} \\ e_{7} &= a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{61} + a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{61} + a_{12} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{65} \ &+ a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{56} \cdot a_{61} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{57} \cdot a_{61} + a_{12} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{57} \cdot a_{65} \ &+ a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{56} \cdot a_{61} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{61} + a_{12} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{65} \ &+ a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{42} \cdot a_{57} \cdot a_{65} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{43} \cdot a_{57} \cdot a_{65} + a_{16} \cdot a_{21} \cdot a_{32} \cdot a_{47} \cdot a_{57} \cdot a_{65} \ &+ a_{16} \cdot a_{21}choco \cdot a_{34} \cdot a_{42} \cdot a_{57} \cdot a_{65} + a_{16} \cdot a_{21} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{65} + a_{16} \cdot a_{23} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{65} \ &+ a_{16} \cdot a_{24} \cdot a_{32} \cdot a_{47} \cdot a_{57} \cdot a_{65} + a_{16} \cdot a_{24} \cdot a_{34} \cdot a_{47} \cdot a_{57} \cdot a_{65} \\ \end{align} $$

$$ \begin{align} Δ &= e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6} + e_{7} \\ \rho_{C1}^\prime &= \rho_{C1} \cdot mt_{prot} / \Delta \\ J_{Hres}^{C1} &= 2\rho_{C1}^\prime (e_{6}a_{61} - e_{1}a_{16}) \\ J_{Q}^{C1} &= 0.5\rho_{C1}^\prime (e_{4}a_{47} - e_{7}a_{74}) \\ J_{NADH}^{C1} &= 0.5\rho_{C1}^\prime (e_{3}a_{34} - e_{4}a_{43}) \\ J_{ROS}^{C1} &= \rho_{C1}^\prime (e_{4}a_{42} - e_{2}a_{24}) \\ \end{align} $$

Parameter Value Units Desc.
$\rho_{C1}$ 5 mM Concentration of complex I
(Adjustable)
$\Delta\Psi_B$ 50 mV Phase boundary potential
$k_{12}$ 6.3396E11 $Hz/mM^2$
$k_{21}$ 5 Hz
$k_{56}$ 100 Hz
$k_{65}$ 2.5119E13 $Hz/mM^2$
$k_{61}$ 1E7 Hz
$k_{16}$ 130 Hz
$k_{23}$ 3886.7 $Hz/mM^{1/2}$
$k_{32}$ 9.1295E6 Hz
$k_{34}$ 639.1364 Hz
$k_{43}$ 3.2882 $Hz/mM^{1/2}$
$k_{47}$ 1.5962E7 Hz/mM
$k_{74}$ 65.2227 Hz
$k_{75}$ 24615 Hz
$k_{57}$ 1166.7 $Hz/mM^{1/2}$
$k_{42}$ 6.0318 Hz/mM
$E_{FMN}$ -375 mV Midpoint potential of flavin mononucleotide
$E_{sox}$ -150 mV Midpoint potential of superoxide

Complex II (Succinate dehydrogenase)

$$ \begin{align} f_Q &= \frac{[Q]n}{[Q]n + [QH_2]n} \ f{OAA} &= \frac{K{i, OAA}}{[OAA] + K{i, OAA}} \ f_{FUM} &= \frac{K_{i, FUM}}{[FUM] + K_{i, FUM}} \ f_{SUC} &= \frac{[SUC]}{[SUC] + K_{m, SUC} / f_{OAA} / f_{FUM}} \ J_{SDH} &= V_{SDH} C2_{inhib} f_{SUC} \frac{f_Q}{f_Q + K_{m, Q}} \ J_{c2} &= J_{SDH} \ \end{align} $$

Parameter Value Units Desc.
$V_{SDH}$ 250 mM / minute Maximum rate of SDH
$K_{i, OAA}$ 0.150 mM Inhibition constant for oxaloacetate
$K_{m, Q}$ 0.6 - Michaelis constant for CoQ
$K_{i, FUC}$ 0.150 mM Inhibition constant for fumarate
$K_{m, SUC}$ 0.6 - Michaelis constant for succinate

Complex III

$$ \begin{align} f_{hi} & = [H^+]{i} / 10^{-7}M \ v{1} &= v_{Q}^{C1} + v_{Q}^{C2} \ v_2 &= k_d([QH_2]{n} - [QH_2]{p}) \ k_{3} &= k_{03}K_{eq3}f_{hi} \ k_{-3} &= k_{03} \ v_{3} &= k_3[QH_2]{p} [FeS]{ox} - k_{-3}[Q^-]p [FeS]{rd} \ k_{4, ox} &= k_{04}K_{eq4, ox} \exp(-\alpha\delta_1\Delta\Psi_m F/ RT) \ k_{4, rd} &= k_{04}K_{eq4, rd} \exp(-\alpha\delta_1\Delta\Psi_m F/ RT) \ k_{-4, ox} &= k_{04} \exp(\alpha(1-\delta_1)\Delta\Psi_m F/ RT) \ k_{-4, rd} &= k_{04} \exp(\alpha(1-\delta_1)\Delta\Psi_m F/ RT) \ v_{4, ox} &= k_{4, ox}[Q^-]p [b1] - k{-4, ox}[Q]{p} [b2] \ v{4, rd} &= k_{4, rd}[Q^-]p [b3] - k{-4, rd}[Q]{p} [b4] \ v{5} &= k_d([Q]{p} - [Q]{n}) \ k_{6} &= K_{06}K_{eq6} \exp( -\beta\delta_2\Delta\Psi_m / V_T) \ k_{-6} &= k_{06} \exp( \beta(1-\delta_2)\Delta\Psi_m / V_T) \ v_{6} &= k_{6} [b2] - k_{-6} [b3] \ k_{7, ox} &= k_{07, ox}K_{eq7, ox}\exp(-\gamma\delta_3\Delta\Psi_m F/ RT) \ k_{7, rd} &= k_{07, rd}K_{eq7, rd}\exp(-\gamma\delta_3\Delta\Psi_m F/ RT) \ k_{-7, ox} &= k_{07, ox} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \ k_{-7, rd} &= k_{07, rd} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \ v_{7, ox} &= (k_{7, ox}[Q]{n}[b3] - k{-7, ox}[Q^-]n [b1])C3{inhib} \ v_{7, rd} &= (k_{7, rd}[Q]{n}[b4] - k{-7, rd}[Q^-]n [b2])C3{inhib} \ \end{align} $$

$$ \begin{align} f_{hm} & = [H^+]{m} / 10^{-7}M \ k{8, ox} &= k_{08, ox}K_{eq8, ox}\exp(-\gamma\delta_3\Delta\Psi_m F/ T)(f_{hm})^2 \ k_{8, rd} &= k_{08,rd}K_{eq8, rd}\exp(-\gamma\delta_3\Delta\Psi_m F/ T)(f_{hm})^2 \ k_{-8, ox} &= k_{08, ox} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \ k_{-8, rd} &= k_{08, rd} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \ v_{8, ox} &= (k_{8, ox}[Q^-]{n}[b3] - k{-8, ox}[QH_2]{n}[b1])C3{inhib} \ v_{8, rd} &= (k_{8, rd}[Q^-]{n}[b4] - k{-8, rd}[QH_2]{n}[b2])C3{inhib} \ k_9 &= k_{09}K_{eq9} \ k_{-9} &= k_{09} \ v_{9} &= k_{9}[FeS]{rd}[cytc1]{ox} - k_{-9}[FeS]{ox}[cytc1]{rd}\ k_{10} &= k_{010}K_{eq10} \ k_{-10} &= k_{010} \ v_{10} &= k_{10}[Q^-]p[O_2] - k{-10}[Q]p[O_2^-] \ v{10b} &= v_{10} \ v_{33} &= k_{33}(K_{eq}[cytc1]{rd}[cytc]{ox} - [cytc]{rd}[cytc1]{ox}) \ \rho_{C3}^{\prime} &= \rho_{C3} \cdot mt_{prot} \ \rho_{C4}^{\prime} &= \rho_{C4} \cdot mt_{prot} \ FeS_{rd} &= \rho_{C3}^{\prime} - FeS_{ox} \ cytc1_{rd} &= \rho_{C3}^{\prime} - cytc1_{ox} \ cytc_{rd} &= \rho_{C4}^{\prime} - cytc_{ox} \ [b4] &= \rho_{C3}^{\prime} - [b1] - [b2] - [b3] \ [QH_2]p &= \Sigma Q - [Q]n - [Q]p - [QH_2]n - [Q^-]p - [Q^-]n \ J{hRes}^{C3} &= 2v{3} \ J{ROS, m}^{C3} &= v{10} \ J{ROS, i}^{C3} &= v{10b} \ \end{align} $$

$$ \begin{align} \frac{d[Q]n}{dt} &= v_5 - v{7,ox}- v_{7,rd} - v_1  \ \frac{d[Q^-]n}{dt} &= v{7,ox} + v_{7,rd} - v_{8,ox}- v_{8,rd}  \ \frac{d[QH_2]n}{dt} &= v{8,ox} + v_{8,rd} + v_1 - v_2   \ \frac{d[QH_2]p}{dt} &= v_2 -v_3 \ \frac{d[Q^-]p}{dt} &= v_3 - v{10} - v{10b} - v_{4,ox} - v_{4,rd}   \ \frac{d[Q]p}{dt} &= v{10} + v_{10b} + v_{4,ox} + v_{4,rd} - v_5   \ \frac{d[b1]}{dt} &= v_{7,ox} + v_{8,ox} - v_{4,ox}    \ \frac{d[b2]}{dt} &= v_{4,ox} + v_{7,rd} - v_{8,rd} - v_6   \ \frac{d[b3]}{dt} &= v_6 - v_{4,rd} + v_{7,ox} - v_{8,ox}    \ \frac{d[b4]}{dt} &= v_{4,rd} - v_{7,rd} - v_{8,rd}   \ \frac{d[FeS]{ox}}{dt} &= v_9 - v_3      \ \frac{d[cytc1]{ox}}{dt} &= v_{33} - v_9   \ \frac{d[cytc]{ox}}{dt} &= V_e - v{33}   \ \end{align} $$

Parameter Value Unit Desc.
$k_{03}$ 1,666.63 Hz/mM Reverse rate constant for reaction 3
$K_{eq3}$ 0.6877 - Equilibrium constant for reaction 3
$k_{04}$ 60.67 Hz/mM Reverse rate constant for reaction 4
$K_{eq4,ox}$ 129.9853 - Equilibrium constant for reaction 4
(bH oxidized)
$K_{eq4,rd}$ 13.7484 - Equilibrium constant for reaction 4
(bH reduced)
$\delta_1$ 0.5 -
$\alpha$ 0.2497 -
$k_d$ 22000 Hz Diffusion rate of ubiquinone across the membrane
$k_{06}$ 166.67 Hz/mM Reverse rate constant for reaction 6
$K_{eq6}$ 9.4596 - Equilibrium constant for reaction 6
$\delta_2$ 0.5 -
$\beta$ 0.5006 -
$k_{07,ox}$ 13.33 Hz/mM Reverse rate constant for reaction 7
(bL oxidized)
$K_{eq7,ox}$ 3.0748 - Equilibrium constant for reaction 7
(bL oxidized)
$k_{07,rd}$ 1.667 Hz/mM Reverse rate constant for reaction 7
(bL reduced)
$K_{eq7,rd}$ 29.0714 - Equilibrium constant for reaction 7
(bL reduced)
$\delta_3$ 0.5 -
$\gamma$ 0.2497 - $\alpha + \beta + \gamma = 1$
$k_{08,ox}$ 83.33 Hz/mM Reverse rate constant for reaction 8
(bL oxidized)
$K_{eq8,ox}$ 129.9853 - Equilibrium constant for reaction 8
(bL oxidized)
$k_{08,rd}$ 8.333 Hz/mM Reverse rate constant for reaction 8
(bL reduced)
$K_{eq8,rd}$ 9.4596 - Equilibrium constant for reaction 8
(bL reduced)
$k_{09}$ 833 Hz/mM Reverse rate constant for reaction 9
$K_{eq9}$ 0.2697 - Equilibrium constant for reaction 9
$k_{010}$ 0.8333 Hz/mM Reverse rate constant for reaction 10
$K_{eq10}$ 1.4541 - Equilibrium constant for reaction 10
$k_{33}$ 2469.13 Hz/mM Reverse rate constant for reaction 33
$K_{eq33}$ 2.1145 - Equilibrium constant for reaction 33
$\rho_{C3}$ 0.325 mM Complex III content

Complex IV

$$ \begin{align} f_{H_{m}} &= \exp(-\delta_5\Delta\Psi_m F/ RT) ([H^+]m /10^{-7}M) \ f{H_{i}} &= \exp((1-\delta_5)\Delta\Psi_m F/ RT) ([H^+]i /10^{-7}M) \ f{C_{rd}} &= [cytc]{rd} \ f{C_{ox}} &= \text{exp}((1-\delta_5)\Delta\Psi_m F/ RT) [cytc]{ox} \ a{12} &= k_{34} f_{C_{rd}}^3 f_{H_{m}}^4 \ a_{14} &= k_{-37} f_{H_{i}} \ a_{21} &= k_{-34} f_{C_{ox}}^3 f_{H_{i}} \ a_{23} &= k_{35} [O_2] C4_{inhib} \ a_{34} &= k_{36} f_{C_{rd}} f_{H_{m}}^3 \ a_{41} &= k_{37} f_{H_{m}} \ a_{43} &= k_{-36} f_{C_{ox}} f_{H_{i}}^2 \ e_1 &= a_{21}a_{41}a_{34} + a_{41}a_{34}a_{23} \ e_2 &= a_{12}a_{41}a_{34} \ e_3 &= a_{23}a_{12}a_{41} + a_{43}a_{14}a_{21} + a_{23}a_{43}a_{12} + a_{23}a_{43}a_{14} \ e_4 &= a_{14}a_{34}a_{21} + a_{34}a_{23}a_{12} + a_{34}a_{23}a_{14} \ \Delta &= e_1 + e_2 + e_3+ e_4 \ Y &= e_1 / \Delta \ Yr &= e_2 / \Delta \ YO &= e_3 / \Delta \ YOH &= e_4 / \Delta \ v_{34} &= \rho_{C4}^\prime (Y \cdot a_{12} - Yr \cdot a_{21}) \ v_{35} &= \rho_{C4}^\prime Yr \cdot a_{23} \ v_{36} &= \rho_{C4}^\prime (YO \cdot a_{34} - YOH \cdot a_{43}) \ v_{37} &= \rho_{C4}^\prime (YOH \cdot a_{41} - Y \cdot a_{14}) \ V_e &= 3v_{34} + v_{35} \ J_{hRes}^{C4} &= v_{34} + 2v_{36} + v_{37} \ J_{O_2} &= v_{35} \ J_{hRes} &= J_{hRes}^{C1} + J_{hRes}^{C3} + J_{hRes}^{C4} \ \rho_{C4}^\prime &= \rho_{C4} \cdot mt_{prot} \end{align} $$

Parameter Value Unit Desc.
$\Sigma cytc$ 0.325 mM Cytochrome c pool
$\rho_{C4}$ 0.325 mM Complex IV content
$k_{34}$ 2.9445E10 Hz/mM^3 Rate constant @ pH = 7
$k_{-34}$ 290.03 Hz/mM^3 Rate constant @ pH = 7
$k_{35}$ 45000 Hz/mM
$k_{36}$ 4.826E11 Hz/mM Rate constant @ pH = 7
$k_{-36}$ 4.826 Hz/mM Rate constant @ pH = 7
$k_{37}$ 1.7245E8 Hz Rate constant @ pH = 7
$k_{-37}$ 17.542 Hz Rate constant @ pH = 7

Complex V (ATP synthase)

$$ \begin{align} J_{F1Fo} &= -\rho^{F1} ((100 p_a + p_{c1} v_B) v_a - (p_a + p_{c2} v_a) v_h) / \Delta \\ J_H^{F1Fo} &= -3\rho^{F1} (100p_a(1 + v_a) - (p_a + p_b)v_h) / \Delta \\ \Delta &= (1 + p_1 v_a)v_B + (p_2 + p_3 v_a)v_h \\ v_B &= \text{exp}(3\Delta\Psi_B / V_T) \\ v_h &= \text{exp}(3\Delta p / V_T) \\ v_a &= \frac{K_{eq}^{'} \cdot \Sigma[ATP]_m}{ \Sigma[Pi]_m \cdot \Sigma[ADP]_m } \\ \end{align} $$

Parameter Value Unit Desc.
$\rho_{F1}$ 5 mM Concentration of F1-Fo ATPase
$K_{eq}^{'}$ 6.47E5 M Apparent equilibrium constant for ATP hydrolysis2
$\Delta\Psi_B$ 50 mV Phase boundary potential
$p_{a}$ 1.656E-5 Hz Sum of products of rate constants
$p_{b}$ 3.373E-7 Hz Sum of products of rate constants
$p_{c1}$ 9.651E-14 Hz Sum of products of rate constants
$p_{c2}$ 4.585E-14 Hz Sum of products of rate constants
$p_{1}$ 1.346E-4 - Sum of products of rate constants
$p_{2}$ 7.739E-7 - Sum of products of rate constants
$p_{3}$ 6.65E-15 - Sum of products of rate constants

Reactive oxygen species (ROS) scavenging and transport

Catalase (CAT)

Includes inhibition by high levels of hydrogen peroxide

$$ \begin{align} V_{CAT} = 2k_1E_T[H_2O_2]_i \cdot e^{-fr[H_2O_2]_i} \\ \end{align} $$

Parameter Value Unit Desc.
$k_1$ 17 1/(mM*ms) Rate constant of catalase
$E_T$ 0.01 mM Extra-matrix concentration of catalase
$fr$ 0.05 1/mM Hydrogen peroxide inhibition factor

Superoxide dismutase (SOD)

Based on (McADAM, 1976) model.

$$ \begin{align} J_{SOD} &= \frac{2 k_5 E_T f_{sox} (k_1 + k_3^\prime)}{ k_5 (2 k_1 + k_3^\prime) + k_3^\prime f_{sox}} \\ k_3^\prime &= k_3 (1 + \frac{[H_2O_2]}{K_{H_2O_2}}) \\ f_{sox} &= k_1^{SOD} [O_2^-] \end{align} $$

Parameter Value Unit Desc.
$k_1$ 1200 1/(mM*ms) Rate constant for EA -> EB
$k_3$ 24 1/(mM*ms) Rate constant for EB -> EC
$k_5$ 0.24 1/s Rate constant for EC -> EA
$K_{i}$ 500 μM Inhibition constant for H2O2
$E_{T}$ 3 μM Concentration of Cu,ZnSOD (cytosolic)

Glutathione peroxidase (GPX)

Dalziel type Ping-pong mechanism.

$$ \begin{align} J_{GPX} &= \frac{E_T}{A + B} \\ A &= \frac{\Phi_1}{[H_2O_2] } \\ B & = \frac{\Phi_2}{[GSH] } \\ \end{align} $$

Parameter Value Unit Desc.
$E_T$ 10 μM GPX content
$\Phi_1$ 5 mM/s Dalziel coefficient
$\Phi_2$ 75 mM/s Dalziel coefficient

Glutathione reductase (GR)

Michaelis-Menten kinetics.

$$ \begin{align} J_{GR} &= k_1^{GR} E_T \frac{[GSSG]}{[GSSG] + K_{GSSG}} \frac{[NADPH]}{[NADPH] + K_{NADPH}} \\ \Sigma [GSH] &= [GSH] + 2 [GSSG] \end{align} $$

Parameter Value Unit Desc.
$E_T$ 10 μM GR content (cytosolic)
$k_1^{GR}$ 5 Hz Catalytic constant of GR
$K_{GSSG}$ 60 μM Michaelis constant for GSSG
$K_{NADPH}$ 15 μM Michaelis constant for NADPH
$\Sigma [GSH]$ 1 mM Cytosolic GSH pool

Inner mitochondrial anion channel (IMAC)

$$ \begin{align} g_{IMAC} &= \left( a + b \frac{[O_2^-]i}{[O_2^-]i + K{CC}} \right) \left( G_L + \frac{G{max}}{1 + e^{κ(\Delta\Psi_m^b + \Delta\Psi_m)}} \right) \ V_{IMAC} &= g_{IMAC}\Delta\Psi_m \ V_{tr}^{ROS} &= j \cdot g_{IMAC} \left( \Delta\Psi_m + V_T ln \left( \frac{[O_2^-]_m}{[O_2^-]_i} \right) \right) \ \end{align} $$

Parameter Value Unit Desc.
a 0.001 - Basal IMAC conductance
b 10000 - Activation factor by superoxide
$K_{CC}$ 10 μM Activation constant by superoxide
$G_L$ 0.035 μM * Hz / mV Integral conductance for IMAC
$G_{max}$ 3.9085 μM * Hz / mV Leak conductance of IMAC
$\kappa$ 0.07 1/mV Steepness factor
$\Delta\Psi_m^b$ 4 mV Potential at half saturation
j 0.1 - Fraction of IMAC conductance

ODEs for ROS transport and scavenging

$$ \begin{align} \frac{d [ O_{2}^{-}]{m}}{dt} &= J{ROS,m} - J^{Tr}{ROS} \ \frac{d [ O{2}^{-}]{i}}{dt} &= \frac{V{mito}}{V_{cyto}} J^{Tr}{ROS} -J{SOD,i} \ \frac{d[H_2O_2]i}{dt} &= 0.5J{SOD,i} -J_{GPX,i} - J_{CAT} \ \frac{d[GSH]i}{dt} &= J{GR,i} - J_{GPX,i} \ \end{align} $$

Mitochondrial ion transport

Adenine Nucleotide translocator (ANT)

$$ \begin{align} J_{ANT} &= V_{max}^{ANT}\frac{AB - \delta PQ}{(B + \delta^{h_{ANT}} P)(A + Q)} \\ A &= [ATP^{4-}]_m = 0.025 [ATP]_m \\ B &= [ADP^{3-}]_i = 0.45 [ADP]_i \\ P &= [ATP^{4-}]_i = 0.25 [ATP]_i \\ Q &= [ADP^{3-}]_m = 0.17 [ADP]_m \\ \delta &= \text{exp}(-\Delta\Psi_m F / RT) \end{align} $$

Parameter Value Unit Desc.
$V_{max}^{ANT}$ 5 mM/s Maximal rate of ANT
$h_{ANT}$ 0.5 - Fraction of MMP

Mitochondrial calcium uniporter (MCU)

$$ \begin{align} J_{uni} &= V_{max}^{Uni} \frac{S (1+S)^3}{(1+S)^4 + L(1 + A)^n} \frac{\delta}{e^\delta-1} \ S &= [Ca^{2+}]i / K{trans} \ A &= [Ca^{2+}]i / K{act} \ \delta &= -2 (\Delta\Psi_m - \Delta\Psi_0) F/RT \ \end{align} $$

Parameter Value Unit Desc.
$V_{max}^{Uni}$ 4.46 mM/s Maximal rate
$\Delta\Psi_0$ 91 mV Offset potential
$K_{act}$ 0.38 μM Activation constant for calcium
$K_{trans}$ 19 μM Dissociation constant for calcium
n -2.8 - Activation cooperativity
L 110 - Keq for conformational transitions

Mitochondrial sodium-calcium exchanger (NCLX)

$$ \begin{align} J_{NCLX} = V_{max}^{NCLX} \exp(b\Delta\Psi_m F/RT) \frac{[Ca^{2+}]_m}{[Ca^{2+}]_i} \left( \frac{[Na^+]_i}{[Na^+]i + K{Na}^{NCLX}} \right)^n \frac{[Ca^{2+}]_m}{[Ca^{2+}]m + K{Ca}^{NCLX}} \end{align} $$

Parameter Value Unit Desc.
$V_{max}^{NCLX}$ 0.04665 mM/s Maximal rate of NCLX
b 0.5 - Fraction of MMP
$K_{Na}^{NCLX}$ 9.4 mM Dissociation constant for sodium
$K_{Ca}^{NCLX}$ 0.375 μM Dissociation constant for calcium
$n$ 3 - Cooperativity

Mitochondrial proton leak

$$ J_{hleak} = g_H\Delta\Psi_m $$

Parameter Value Unit Desc.
$g_{H}$ 2 mM / (Volt * s) Ionic conductance of the inner mitochondrial membrane

ODEs for mitochondrial ion transport

$$ \begin{align} \frac{d [Ca^{2+}]m}{dt} &=\delta{Ca}( J_{uni} - J_{NCLX}) \ \frac{d [Na^+]m}{dt} &= J{NCLX} - J_{NaH} \ C_{m}\frac{d \Delta \Psi_m}{dt} &= J_{Hres} - J_{Hu} - J_{ANT} - J_{Hleak} -J_{NCLX} - J_{uni} - J_{IMAC} \ \end{align} $$

General parameters

Parameter Value Unit Desc.
F 96485 C/mol Faraday constant
T 310 K Absolute temperature
R 8.314 J/molK Universal gas constant
$V_T$ 26.71 mV Thermal voltage (=${RT}/{F}$)
$C_m$ 1.0 $\text{μF/cm}^2$ Plasma membrane capacitance
$C_{mito}$ 1.812 mM/V Mitochondrial inner membrane capacitance
$\delta_{Ca}$ 0.0003 - Mitochondrial free calcium fraction
$\delta_H$ 1E-5 - Mitochondrial proton buffering factor
$V_{myo}$ $25.84$ $pL$ Cytosolic volume
$V_{mito}$ $15.89$ $pL$ Mitochondrial volume
$V_{NSR}$ $1.4$ $pL$ Network SR volume
$V_{JSR}$ $0.16$ $pL$ Junctional SR volume
$V_{SS}$ $0.000495$ $pL$ Subspace volume
$A_{cap}$ $1.534 \cdot 10^{-4} $ $cm^{2}$ Capacitance area
$C_{m}$ $1.0$ $\mu F \cdot cm^{-2}$ Plasma membrane capacitance
$[K^+]_{o}$ $5.4$ $mM$ Extracellualr potassium
$[Na^+]_{o}$ $140$ $mM$ Extracellualr sodium
$[Ca^{2+}]_{o}$ $2$ $mM$ Extracellualr calcium
$C_{mito}$ $1.812 \cdot 10^{-3}$ $mM/mV$ Inner membrane capacitance
$g_{H}$ $1 \cdot 10^{-8}$ $mM/msmV$ Inner membrane conductance

Fixed concentrations

Parameter Value Unit Desc.
$pH_i$ 7 CytosoliWc pH
$pH_m$ 7.3-7.8 Mitochondrial pH
$[O_2]$ 0.006 mM Tissue oxygen concentration
$[Mg^{2+}]_i$ 1.0 mM Cytosolic magnesium concentration
$[Mg^{2+}]_m$ 0.4 mM Mitochondrial magnesium concentration
$\Sigma[Pi]_m$ 8.6512 mM Sum of mitochondrial inorganic phosphate
$\Sigma{[N]}$ 1 mM Sum of mitochondrial NAD and NADH
$\Sigma[A]_m$ 1.5 mM Sum of mitochondrial ATP and ADP
$\Sigma{[NADP]_m}$ 0.1 mM Sum of mitochondrial NADPH plus NADP
$[Ca^{2+}]_i$ 1E-4 mM Cytosolic calcium concentration

Initial conditions

Footnotes

  1. Rice JJ, Jafri MS, Winslow RL. Modeling short-term interval-force relations in cardiac muscle. Am J Physiol Heart Circ Physiol. 2000 Mar;278(3):H913-31. APS ↩

  2. Golding, E. M., Teague, W. E., & Dobson, G. P. (1995). Adjustment of K’ to varying pH and pMg for the creatine kinase, adenylate kinase and ATP hydrolysis equilibria permitting quantitative bioenergetic assessment. The Journal of Experimental Biology, 198(Pt 8), 1775–1782. ↩