ECME-RIRR-DOX model description
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 | \(\mathrm{μF/cm^2}\) | Plasma membrane capacitance |
| \(C_{mito}\) | 1.812 | mM/V | Mitochondrial inner membrane capacitance |
| \(\delta_{Ca}\) | 0.0003 | - | Mitochondrial free calcium fraction |
| \(V_{myo}\) | \(25.84\) | 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 |
| \(V_{SS}\) | 4.95\(\cdot 10^{-4}\) | pL | Subspace volume |
| \(A_{cap}\) | 1.534 \(\cdot 10^{-4}\) | cm² | Cell capacitance area |
| \(\mathrm{pH}_i\) | 7 | - | Cytosolic pH |
| \(\mathrm{pH}_m\) | 7.6 | - | 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 |
| \([Pi]_m\) | 8.6512 | mM | Mitochondrial inorganic phosphate |
| \([K^+]_{o}\) | \(5.4\) | mM | Extracellular potassium |
| \([Na^+]_{o}\) | \(140\) | mM | Extracellular sodium |
| \([Ca^{2+}]_{o}\) | \(2\) | mM | Extracellular calcium |
| \(\Sigma[N]\) | 3 | mM | Sum of mitochondrial NAD and NADH |
| \(\Sigma[A]_m\) | 1.5 | mM | Sum of mitochondrial ATP and ADP |
| \(\Sigma[A]_i\) | 8 | mM | Sum of cytosolic ATP and ADP |
| \([H^+]_i\) | 100 | nM | Cytosolic proton (pH = 7) |
| \([H^+]_m\) | 25.12 | nM | Mitochondrial proton (pH = 7.6) |
Electrophysiology
The GHK current equation is defined as.
\[ \begin{align} \Phi_s(P_s, z_s, V_m, [S]_i, [S]_o) := P_sz^2_s\frac{V_mF^2}{RT}\frac{[S]_i - [S]_o\exp(-z_sV_mF/RT)}{1-\exp(-z_sV_mF/RT)} \end{align} \]
Differential equations for electrophysiology:
\[ \begin{align} \frac{d[Na^+]_i}{dt} &= -(I_{Na} + 3I_{NaCa} + 3I_{NaK})\frac{A_{cap}}{V_{myo}F} \\ \frac{d[K^+]_i}{dt} &= -(I_{Ks} + I_{Kr} + I_{K1} + I_{Kp} + I_{Ca,K}-2I_{NaK})\frac{A_{cap}}{V_{myo}F} \\ C_m\frac{dV_m}{dt} &= -(I_{Na} + I_{CaL} + I_{Kr} + I_{Ks} + I_{K1} + I_{Kp} + I_{NaCa} + I_{NaK} + I_{pCa} + I_{Ca, b} + I_{K_{ATP}} + I_{stim}) \\ \frac{d[Ca^{2+}]_i}{dt} &= \beta_i(J_{xfer}\frac{V_{ss}}{V_{myo}} - J_{up} - J_{trpn} - (I_{Ca,b} -2I_{NaCa} + I_{pCa})\frac{A_{cap}}{2V_{myo}F} + (V_{NaCa} - V_{uni})\frac{V_{mito}}{V_{myo}}) \\ \frac{d[Ca^{2+}]_{NSR}}{dt} &= (J_{up}\frac{V_{myo}}{V_{NSR}} - J_{tr}\frac{V_{JSR}}{V_{NSR}}) \\ \frac{d[Ca^{2+}]_{JSR}}{dt} &= \beta_{SR}(J_{tr} - J_{rel}) \\ \frac{d[Ca^{2+}]_{SS}}{dt} &= \beta_{SS} (\frac{V_{JSR}J_{rel} - V_{myo}J_{xfer}}{V_{SS}} - \frac{I_{CaL}A_{cap}}{2V_{SS}F} ) \\ β_i &= \frac{(K_m^{CMDN} + [Ca^{2+}]_i)^2}{ (K_m^{CMDN} + [Ca^{2+}]_i)^2 + K_m^{CMDN} \cdot [CMDN]_{tot}} \\ β_{SR} &= \frac{(K_m^{CSQN} + [Ca^{2+}]_{JSR})^2}{(K_m^{CSQN} + [Ca^{2+}]_{JSR})^2 + K_m^{CSQN} \cdot [CSQN]_{tot}} \\ β_{SS} &= \frac{(K_m^{CMDN} + [Ca^{2+}]_{SS})^2}{(K_m^{CMDN} + [Ca^{2+}]_{SS})^2 + K_m^{CMDN} \cdot [CMDN]_{tot}} \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(K_{m}^{CMDN}\) | \(2.38\) | μM | Ca2+ half saturation constant for calmodulin |
| \(K_{m}^{CSQN}\) | \(800\) | μM | Ca2+ half saturation constant for calsequestrin |
| \(\Sigma[CMDN]\) | \(50\) | μM | Total myoplasmic calmodulin concentration |
| \(\Sigma[CQSN]\) | \(15\) | mM | Total NSR calsequestrin concentration |
Ryanodine receptor (Jrel)
\[ \begin{align} J_{tr} &= r_{tr} ([Ca^{2+}]_{NSR} - [Ca^{2+}]_{JSR}) \\ J_{xfer} &= r_{xfer}([Ca^{2+}]_{ss} - [Ca^{2+}]_{i}) \\ J_{rel} &= r_{ryr} (P_{O1} + P_{O2})([Ca^{2+}]_{JSR} - [Ca^{2+}]_{ss}) \\ 1 &= P_{C1} + P_{O1} - P_{O2} - P_{C2} \\ v_{o1c1} &= -k_a^-P_{O1} + k_a^+ [Ca^{2+}]_{ss}^4 P_{C1} \\ v_{o1o2} &= k_b^+ [Ca^{2+}]_{ss}^3 P_{O1} - k_b^- P_{O2} \\ v_{o1c2} &= k_c^+ P_{O1} - k_c^- P_{C2} \\ \dot{P_{O1}} &= -v_{o1c1} - v_{o1o2} - v_{o1c2} \\ \dot{P_{O2}} &= v_{o1o2} \\ \dot{P_{C2}} &= v_{o1c2} \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| \(r_{RyR}\) | 3600 | Hz | RyR flux channel activity |
| \(n_{RyR}\) | 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 |
| \(r_{tr}\) | \(110\) | Hz | Calcium transfer rate from from NSR to JSR |
| \(r_{xfer}\) | \(1740\) | Hz | Calcium transfer rate from subspace to cytoplasm |
SERCA (Jup)
\[ \begin{align} J_{up} &= \frac{V_{f}^{up}f_b-V_{r}^{up}r_b}{(1 + f_b + r_b)f_{a}} \\ f_b &= \left( \frac{[Ca^{2+}]_i}{K_{fb}} \right)^{N_{fb}} \\ r_b &= \left( \frac{[Ca^{2+}]_{NSR}}{K_{rb}} \right)^{N_{rb}} \\ f_{a} &= \frac{K_{m}^{up}}{[ATP]_i} ( \frac{[ADP]_i}{K_{i1}^{ up}} + 1) + \frac{[ADP]_i}{K_{i2}^{up}} + 1 \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| \(V_{f}^{up}\) | 0.2989 | Hz*mM | SERCA forward rate constant |
| \(V_{r}^{up}\) | 0.3179 | Hz*mM | SERCA reverse rate constant |
| \(K_{fb}\) | 0.24 | μM | Forward Ca2+ half-saturation constant of SERCA |
| \(K_{rb}\) | 1.64269 | mM | Reverse Ca2+ half-saturation constant of SERCA |
| \(N_{fb}\) | 1.4 | - | Forward cooperativity constant of SERCA |
| \(N_{rb}\) | 1.0 | - | Reverse cooperativity constant of SERCA |
| \(K_{m}^{up}\) | 10 | μM | ATP half-saturation constant for SERCA |
| \(K_{i1}^{up}\) | 140 | μM | ADP first inhibition constant for SERCA |
| \(K_{i2}^{up}\) | 5.1 | mM | ADP second inhibition constant for SERCA |
Time-dependent delayed rectifier potassium current
\[ \begin{align} I_K &= \bar G_K X_1 X_K^2 (V - E_K) \\ E_K &= \frac{RT}{F} \ln \frac{[K^+]_o + P_{Na,K}[Na^+]_o}{ [K^+]_i + P_{Na,K}[Na^+]_i} \\ \bar G_K &= G_{K,0}\sqrt{[K^+]_o / 5.4mM} \\ X_1 &= 1 / (1+ e^{(V_m-40)/40}) \\ \frac{dX_k}{dt} &= \alpha_X - X_k (\alpha_X + \beta_X) \\ \alpha_X &= \frac{V_m+30}{1 - e^{-0.148(V_m+30)}} * 0.0719Hz \\ \beta_X &= \frac{V_m+30}{e^{0.0687(V_m+30)} -1} * 0.131Hz \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(G_{K,0}\) | 0.282 | \(mS/cm^2\) | Maximal IK conductance |
| \(P_{NaK}\) | 0.01833 | - | Na+ permeability ratio of K+ channel |
Time-independent potassium current
\[ \begin{align} \Delta V_{k1} &= V_m - E_{K1} \\ I_{K1} &= \bar G_{K1}K_{1 \infty}\Delta V_{k1} \\ E_{K1} &= \frac{RT}{F} \ln \frac{[K^+]_o}{[K^+]_i} \\ \bar G_{K1} &= G_{K1,0} \sqrt{[K^+]_o / 5.4mM} \\ K_{1 \infty} &= \frac{\alpha_{K_1}}{\alpha_{K_1} + \beta_{K_1}} \\ \alpha_{K_1} &= \frac{1.02}{1 + e^{0.2385(\Delta V_{k1} -59.215)}} \times \text{kHz} \\ \beta_{K_1} &= \frac{0.4912e^{0.28032(\Delta V_{k1} + 5.476)} + e^{0.06175(\Delta V_{k1} -594.31)}}{1 + e^{-0.5143(\Delta V_{k1} + 4.753)}} \times \text{kHz} \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(G_{K1,0}\) | \(0.748\) | \(mS/cm^2\) | Maximal IK1 conductance |
Plateau potassium current
\[ \begin{align} E_{Kp} &= \frac{RT}{F} \ln \frac{[K^+]_o}{[K^+]_i} \\ I_{Kp} &= \frac{\bar G_{Kp} (V - E_{Kp})}{1 + \exp((7.488-V_m) / 5.98)} \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(G_{Kp}\) | \(0.00828\) | \(mS/cm^2\) | Maximal plateau K channel conductance |
Fast Na current
\[ \begin{align} I_{Na} &= G_{Na} m_{Na}^{3} h_{Na} j_{Na} (V_m-E_{Na}) \\ E_{Na} &= \frac{RT}{F} \ln \frac{[Na^{+}]_o}{[Na^{+}]_i} \\ \frac{dm_{Na}}{dt} &= \alpha_{m} - m_{Na}(\alpha_{m} + \beta_{m}) \\ \frac{dh_{Na}}{dt} &= \alpha_{h} - h_{Na}(\alpha_{h} + \beta_{h}) \\ \frac{dj_{Na}}{dt} &= \alpha_{j} - m_{Na}(\alpha_{j} + \beta_{j}) \\ \alpha_{m} &= 0.32 \frac{V_m + 47.13}{1 - \exp(-0.1(V_m+47.13))} \times \text{kHz} \\ \beta_{m} &= 0.08 \exp(-V_m / 11) \times \text{kHz}\\ \\ For \ V_m & \ge -40mV \\ \alpha_{h} &= \alpha_{j} = 0 \\ \beta_{h} &= (0.13 \text{ms} (1+e^{-(V_m+10.66)/11.1}))^{-1} \\ \beta_{j} &= 0.3 \frac{e^{-2.535 \times 10^{-7}V_m}}{1 + e^{-0.1(V_m + 32)}} \times \text{kHz} \\ \\ For \ V_m & < -40mV \\ \alpha_{h} &= 0.135 \times \exp(-(V_m + 80)/6.8) \times \text{kHz} \\ \alpha_{j} &= (-127140\exp(0.2444 V_m)-3.474 \times 10^{-5}\exp(-0.04391 V_m))\frac{(V_m + 37.78) }{1+ \exp(0.311( V_m +79.23))} \frac{\text{kHz}}{\text{mV}} \\ \beta_{h} &= (3.56\exp(0.079 V_m) + 3.1 \times 10^{5}\exp(0.35 V_m)) \times \text{kHz} \\ \beta_{j} &= \frac{0.1212 \exp(-0.01052 V_m)}{1+\exp(-0.1378(V_m + 40.14))} \times \text{kHz} \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(G_{Na}\) | 12.8 | \(mS/cm^2\) | Maximal Na channel conductance |
Sodium-calcium exchanger current
\[ \begin{align} I_{NaCa} &= k_{NaCa} \cdot f_{Nao } \cdot f_{Cao }\frac{\exp(V_mF/RT)\phi_{Na}^3 - \phi_{Ca}}{\exp((1 - \eta) V_mF/RT ) + k_{sat}} \\ f_{Nao} &= \frac{([Na^+]_o)^3}{([Na^+]_o)^3 + (K_{M,Na}^{NaCa})^3} \\ f_{Cao} &= \frac{[Ca^+]_o}{[Ca^+]_o + K_{M,Ca}^{NaCa}} \\ \phi_{Na} &= \frac{[Na^+]_i}{ [Na^+]_o} \\ \phi_{Ca} &= \frac{[Ca^{2+}]_i}{[Ca^{2+}]_o} \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(K_{NaCa}\) | \(9000\) | \(\mu A/cm^2\) | 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 |
Background calcium and sodium currents
\[ \begin{align} I_{Ca,b} &= \bar G_{Ca,b} (V_m - \frac{RT}{2F} \ln \frac{[Ca^{2+}]_o}{[Ca^{2+}]_i}) \\ I_{Na,b} &= \bar G_{Na,b} (V_m - \frac{RT}{F} \ln \frac{[Na^{+}]_o}{[Na^{+}]_i}) \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(G_{Ca,b}\) | \(5.45\cdot 10^{-4}\) | \(mS/cm^2\) | Maximum background current Ca2+ conductance |
| \(G_{Na,b}\) | \(3.22\cdot 10^{-3}\) | \(mS/cm^2\) | Maximum background current Na+ conductance |
Non-specific calcium-activated current
\[ \begin{align} f_{Ca} &= \frac{([Ca^{2+}]_i)^3}{([Ca^{2+}]_i)^3 + (K_{m}^{nsCa})^3}\\ I_{nsNa} &= 0.75 \cdot f_{Ca} \cdot \Phi_{Na}(P_{nsNa}, z_{Na}, V_m, [Na^+]_i, [Na^+]_o) \\ I_{nsK} &= 0.75 \cdot f_{Ca} \cdot \Phi_{K}(P_{nsK}, z_{K}, V_m, [K^+]_i, [K^+]_o) \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(P_{ns,Na}\) | \(1.75 \cdot 10^{-7}\) | cm/s | Nonspecific channel current Na permeability |
| \(P_{ns,K}\) | 0 | cm/s | Nonspecific channel current K permeability |
| \(K_{m}^{nsCa}\) | 1.2 | μM | Ca2+ half-saturation constant for nonspecific current |
Sodium-potassium ATPase (NKA) current
The maximal activity of Sodium-potassium ATPase (NKA) was increased to limit cytosolic sodium concentration uner 1Hz pacing.
\[ \begin{align} I_{NaK} &= \bar I_{NaK} \cdot f_{ATP} \cdot f_{Na} \cdot f_{K} \cdot f_{NaK} \\ \sigma &= \frac{e^{[Na^+]_o / 67.3mM}-1}{7} \\ f_{NaK} &= (1 + 0.1245 \cdot \exp(-0.1V_m/ V_T) + 0.0365 \sigma \cdot \exp(-V_m / V_T))^{-1} \\ f_{Na} &= \frac{([Na^+]_i)^{1.5}}{([Na^+]_i)^{1.5} + (K_{m, Na_i})^{1.5}} \\ f_{K} &= \frac{[K^+]_o}{[K^+]_o + K_{m, K_o}} \\ f_{ATP} &= \frac{[ATP]_i}{[ATP]_i + K_{M,ATP}^{NaK} / f_{ADP}} \\ f_{ADP} &= \frac{K_{i,ADP}^{NaK}}{K_{i,ADP}^{NaK} + [ADP]_i} \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(\bar I_{NaK}\) | 4.5 | \(\mu A/cm^2\) | Maximal NKA current(*) |
| \(K_{m, Na_i}\) | 10 | mM | Na half-saturation constant of NKA |
| \(K_{m, K_o}\) | 1.5 | mM | K half-saturation constant of NKA |
| \(K_{m, K_o}\) | 1.5 | mM | K half-saturation constant of NKA |
| \(K_{M,ATP}^{NaK}\) | 8 | μM | ATP half-saturation constant of Na-K ATPase |
| \(K_{i,ADP}^{NaK}\) | 0.1 | mM | ADP half-saturate constant of Na-K ATPase |
L-type Ca current
\[ \begin{align} \alpha &= 0.4 e^{(V_m+2) / 10} \\ \beta &= 0.4 e^{-(V_m+2) / 13} \\ \alpha^\prime &= a \alpha \\ \beta^\prime &= \beta / b \\ \gamma &= \gamma_0 [Ca^{2+}]_{ss} \\ C_0 &= 1 - C_0 - C_1 - C_2 - C_3 - C_4 - O - C_{Ca0} - C_{Ca1} - C_{Ca2} - C_{Ca3} - C_{Ca4} \\ v_{01} &= 4\alpha C_0 - \beta C_1 \\ v_{12} &= 3\alpha C_1 - 2\beta C_2 \\ v_{23} &= 2\alpha C_2 - 3\beta C_3 \\ v_{34} &= \alpha C_3 - 4\beta C_4 \\ v_{45} &= f C_4 - g O \\ v_{67} &= 4\alpha^\prime C_{Ca0} - \beta^\prime C_{Ca1} \\ v_{78} &= 3\alpha^\prime C_{Ca1} - 2\beta^\prime C_{Ca2} \\ v_{89} &= 2\alpha^\prime C_{Ca2} - 3\beta^\prime C_{Ca3} \\ v_{910} &= \alpha^\prime C_{Ca3} - 4\beta^\prime C_{Ca4} \\ v_{06} &= \gamma C_0 - \omega C_{Ca0} \\ v_{17} &= a \gamma C_1 - \omega C_{Ca1} / b \\ v_{28} &= a^2 \gamma C_2 - \omega C_{Ca2} / b^2 \\ v_{39} &= a^3 \gamma C_3 - \omega C_{Ca3} / b^3 \\ v_{410} &= a^4 \gamma C_4 - \omega C_{Ca4} / b^4 \\ \end{align} \]
\[ \begin{align} \frac{dC_0}{dt} &= -v_{01} -v_{06} \\ \frac{dC_1}{dt} &= v_{01} - v_{12} - v_{17} \\ \frac{dC_2}{dt} &= v_{12} - v_{23} - v_{28} \\ \frac{dC_3}{dt} &= v_{23} - v_{34} - v_{34} \\ \frac{dC_4}{dt} &= v_{34} - v_{45} - v_{410} \\ \frac{dO}{dt} &= v_{45} \\ \frac{dC_{Ca0}}{dt} &= v_{06} - v_{67} \\ \frac{dC_{Ca1}}{dt} &= v_{17} + v_{67} - v_{78} \\ \frac{dC_{Ca2}}{dt} &= v_{28} + v_{78} - v_{89} \\ \frac{dC_{Ca3}}{dt} &= v_{39} + v_{89} - v_{910} \\ I_{Ca}^{max} &= \Phi_{Ca}(P_{Ca}, z_{Ca}, V_m, 0.001\mathrm{mM}, 0.341[Ca^{2+}]_o) \\ I_{Ca} &= 6 I_{Ca}^{max} \cdot y_{Ca} \cdot O \\ I_{Ca,K} &= y_{Ca} \cdot O \cdot \Phi_{Ca}(P_{K}, z_{K}, V_m, [K^+]_i, [K^+]_o) \\ P_{K} &= P_{K}^{max} \frac{I_{Ca}^{half}}{I_{Ca}^{half} + I_{Ca}^{max}} \\ y_\infty &= \frac{1}{1 + e^{(V_m + 55) / 7.5}} + \frac{0.5}{1 + e^{(-V_m + 21) / 6}} \\ \tau_y &= 20ms + \frac{600ms}{1 + e^{(V_m + 30) / 9.5}} \\ \frac{dy_{Ca}}{dt} &= \frac{y_\infty - y_{Ca}}{\tau_y} \\ \end{align} \]
| 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 |
| \(P_{Ca}^{LCC}\) | \(8 \cdot 10^{-4}\) | 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 |
(*): adjusted for proper calcium transients and diastolic cytosolic calcium concentrations.
Plasma membrane calcium ATPase (PMCA) current
\[ \begin{align} I_{pCa} &= I_{max}^{PMCA} \frac{[Ca^{2+}]_i}{[Ca^{2+}]_i + K_{M, Ca}^{PMCA}} f_{ATP} \\ f_{ATP} &= \frac{[ATP]_i}{[ATP]_i + K_{M2,ATP}^{PMCA}} + \frac{[ATP]_i}{[ATP]_i + K_{M1,ATP}^{PMCA} / f_{ADP}} \\ f_{ADP} &= \frac{K_{i,ADP}^{PMCA}}{K_{i,ADP}^{PMCA} + [ADP]_i} \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| \(I_{max}^{PMCA}\) | \(0.575\) | \(\mu A/cm^2\) | Maximum sarcolemmal Ca2+ pump current |
| \(K_{Ca}^{PMCA}\) | \(0.5\) | μM | 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 |
Force generation
\[ \begin{align} f_{01} &= 3f_{XB} \\ f_{12} &= 10f_{XB} \\ f_{23} &= 7f_{XB} \\ g_{01} &= g_{XB}^{min} \\ g_{12} &= 2g_{XB}^{min} \\ g_{23} &= 3g_{XB}^{min} \\ g_{01,SL} &= \phi \cdot g_{01} \\ g_{12,SL} &= \phi \cdot g_{12} \\ g_{23,SL} &= \phi \cdot g_{23} \\ g_{01,SL, off} &= \phi \cdot g_{off} \\ \phi &= 1 + \frac{2.3-SL/ \mathrm{μm}}{(2.3-1.7)^{1.6}} \\ K_{Ca}^{trop} &= \frac{k^-_{ltrpn}}{k^+_{ltrpn}} \\ K_{1/2}^{trop} &= \left( 1 + \frac{K_{Ca}^{trop}}{1.7 \cdot 10^{-3} - 0.8 \cdot 10^{-3}\frac{(SL-1.7\mathrm{μm})}{0.6\mathrm{μm}}} \right)^{-1} \\ N_{trop} &= 3.5 \cdot SL / \mathrm{μm} - 2.0 \\ k_{np}^{trop} &= k_{pn}^{trop} \left( \frac{[LTRPNCa]}{K_{1/2}^{trop}[LTRPN]_{tot}} \right) ^{N_{trop}} \\ \Sigma PATHS &= g_{01}g_{12}g_{23} + f_{01}g_{12}g_{23} + f_{01}f_{12}g_{23} + f_{01}f_{12}f_{23} \\ P1_{max} &= \frac{f_{01}g_{12}g_{23}}{\Sigma PATHS} \\ P2_{max} &= \frac{f_{01}f_{12}g_{23}}{\Sigma PATHS} \\ P3_{max} &= \frac{f_{01}f_{12}f_{23}}{\Sigma PATHS} \\ Force &= \zeta \frac{[P_1] + 2[P_2] + 3[P_3] + [N_1]}{P1_{max} + 2P2_{max} + 3P3_{max}} \\ Force_{norm} &= \frac{[P_1] + [P_2] + [P_3] + [N_1]}{P1_{max} + P2_{max} + P3_{max}} \\ \end{align} \]
\[ \begin{align} v_{01} &= f_{01} [P_0] - g_{01(SL)} [P_1] \\ v_{12} &= f_{12} [P_1] - g_{21(SL)} [P_2] \\ v_{23} &= f_{23} [P_2] - g_{23(SL)} [P_3] \\ v_{04} &= k_{pn}^{trop} [P_0] - k_{np}^{trop} [N_0] \\ [N_0] &= 1 - [P_0] - [P_1] - [P_2] - [P_3] - [N_1] \\ v_{15} &= k_{pn}^{trop} [P_1] - k_{np}^{trop} [N_1] \\ v_{54} &= g_{01,off} [N_1] \\ [HTRPN] &= \Sigma[HTRPN] - [HTRPNCa] \\ [LTRPN] &= \Sigma[LTRPN] - [LTRPNCa] \\ f_{ATP}^{AM} &= \frac{[ATP]_i}{[ATP]_i + K_{m,AM}^{ATP}/f_{ADP}^{AM} } \\ f_{ADP}^{AM} &= \frac{K_{i,AM}^{ADP}}{[ADP]_i + K_{i,AM}^{ADP}} \\ V_{AM} &= V_{max}^{AM} \cdot f_{ATP}^{AM} \cdot \frac{f_{01}[P_0] + f_{12}[P_1] + f_{23}[P_2]}{f_{01} + f_{12} + f_{23}} \\ J_{trpn} &= \frac{d[HTRPNCa]}{dt} + \frac{d[LTRPNCa]}{dt} \\ \frac{d[HTRPNCa]}{dt} &= k^{+}_{htrpn}[Ca^{2+}]_i[HTRPN] - k^{-}_{htrpn}[HTRPNCa] \\ \frac{d[LTRPNCa]}{dt} &= k^{+}_{ltrpn}[Ca^{2+}]_i[LTRPN] - k^{-}_{ltrpn}(1-\frac{2}{3}Force_{norm})[LTRPNCa] \\ \frac{d[P_0]}{dt} &= - v_{01} - v_{04} \\ \frac{d[P_1]}{dt} &= v_{01} - v_{12} - v_{15} \\ \frac{d[P_2]}{dt} &= v_{12} - v_{23} \\ \frac{d[P_3]}{dt} &= v_{23} \\ \frac{d[N_1]}{dt} &= v_{15} - v_{54} \\ \end{align} \]
| Symbol | Value | Units | Description |
|---|---|---|---|
| \(k_{pn}^{trop}\) | 40 | Hz | Transition rate from tropomyosin permissive to non-permissive |
| \(\text{SL}\) | 2.15 | μm | Sarcomere length |
| \(f_{XB}\) | 50 | Hz | Transition rate from weak to strong crossbridge |
| \(g_{XB}^{min}\) | 100 | 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 | Hz * mM | Conversion factor normalizing to physiological force |
| \(K_{ATP}^{AM}\) | 0.03 | mM | ATP half-saturation constant of AM ATPase |
| \(K_{ADP}^{AM}\) | 0.26 | mM | ADP inhibition constant of AM ATPase |
| \(h_{trpn}^{+}\) | 100 | Hz/μM | Ca2+ on-rate for troponin high-affinity sites |
| \(h_{trpn}^{-}\) | 0.33 | Hz | Ca2+ off-rate for troponin high-affinity sites |
| \(l_{trpn}^{+}\) | 100 | Hz/μM | Ca2+ on-rate for troponin low-affinity sites |
| \(l_{trpn}^{-}\) | 40 | Hz | Ca2+ off-rate for troponin low-affinity sites |
| \(\Sigma[HTRPN]\) | 140 | μM | Total troponin high-affinity sites |
| \(\Sigma[LTRPN]\) | \(70\) | μM | Total troponin low-affinity sites |
Citric acid cycle
ODEs in the citric acid cycle.
\[ \begin{align} \frac{d [ISOC]}{dt} &= J_{ACO} -J_{IDH3} \\ \frac{d [\alpha KG]}{dt} &= J_{IDH3} - J_{KGDH} + J_{AAT} \\ \frac{d [SCoA]}{dt} &= J_{KGDH} - J_{SL} \\ \frac{d [SUC]}{dt} &= J_{SL} - J_{SDH} \\ \frac{d [FUM]}{dt} &= J_{SDH} - J_{FH} \\ \frac{d [MAL]}{dt} &= J_{FH} - J_{MDH} \\ \frac{d [OAA]}{dt} & = J_{MDH} - J_{CS} - J_{AAT} \\ \Sigma_{CAC} &= \mathrm{[CIT]} + \mathrm{[ISOC]} + \mathrm{[\alpha KG]} + \mathrm{[SCoA]} + \mathrm{[SUC] }+ \mathrm{[FUM]} + \mathrm{[MAL]} + \mathrm{[OAA]} \\ \Sigma{[N]} &= [NAD] + [NADH] \end{align} \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(\Sigma_{CAC}\) | 1.3 | mM | Sum of TCA cycle intermediates |
Citrate synthase (CS)
\[ \begin{align} J_{CS} &= k_{cat}^{CS} E_T^{CS} f_{accoa} f_{OAA} \\ f_{accoa} &= \frac{[AcCoA]}{[AcCoA] + K_m^{AcCoA}} \\ f_{OAA} &= \frac{[OAA]}{[OAA] + K_m^{OAA}} \\ \end{align} \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(k_{cat}^{CS}\) | 50 | Hz | Catalytic constant of CS |
| \(E_T^{CS}\) | 400 | μM | Enzyme concentration of CS |
| \(K_m^{AcCoA}\) | 12.6 | μM | Michaelis constant for AcCoA |
| \(K_m^{OAA}\) | 0.64 | μM | Michaelis constant for OAA |
| \(\mathrm{[AcCoA]}\) | 1 | mM | Acetyl CoA concentration |
Aconitase (ACO)
\[ J_{ACO} = k_f^{ACO} (\mathrm{[CIT]} - \mathrm{[ISOC]} / K_{eq}^{ACO}) \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(k_f^{ACO}\) | 12.5 | Hz | Forward rate constant of ACO |
| \(K_{eq}^{ACO}\) | 2.22 | - | Equilibrium constant of ACO |
Isocitrate dehydrogenase, NADH-producing (IDH3)
\[ \begin{align} J_{IDH3} &= \frac{k_{cat}^{IDH3} E_T^{IDH3} AB}{f_H AB + f_i B + f_a A + f_a f_i} \\ f_H & = 1 + \frac{[H^+]_m}{K_{H1}^{IDH3}} + \frac{K_{H2}^{IDH3}}{[H^+]_m} \\ A &= [NAD] / K_{NAD}^{IDH3} \\ B &= ([ISOC] / K_{ISOC}^{IDH3})^2 \\ f_a &= \frac{K_A^{IDH3}}{K_A^{IDH3} + [ADP]_m} \frac{K_{CA}^{IDH3}}{K_{CA}^{IDH3} + [Ca^{2+}]_m} \\ f_i &= 1 + \frac{[NADH]}{K_{NADH}^{IDH3}} \\ \end{align} \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(k_{cat}^{IDH3}\) | 50 | Hz | Rate constant of IDH3 |
| \(E_T^{IDH3}\) | 109 | μM | Concentration of IDH3 |
| \(K_{H1}^{IDH3}\) | 1 | nM | Ionization constant of IDH3 |
| \(K_{H2}^{IDH3}\) | 900 | nM | Ionization constant of IDH3 |
| \(K_{NAD}^{IDH3}\) | 923 | μM | Michaelis constant for NAD |
| \(K_{ISOC}^{IDH3}\) | 1520 | μM | Michaelis constant for isocitrate |
| \(K_A^{IDH3}\) | 620 | μM | Activation constant by ADP |
| \(K_{CA}^{IDH3}\) | 500 | nM | Activation constant for calcium |
| \(K_{NADH}^{IDH3}\) | 190 | μM | Inhibition constant by NADH |
Alpha-ketoglutarate dehydrogenase (KGDH)
\[ \begin{align} J_{KGDH} &= \frac{k_{cat}^{KGDH} E_T^{KGDH} AB}{f_H AB + f_a (A + B)} \\ f_H & = 1 + \frac{[H^+]_m}{K_{H1}^{KGDH}} + \frac{K_{H2}^{KGDH}}{[H^+]_m} \\ A &= [NAD] / K_{NAD}^{KGDH} \\ B &= ([\alpha KG] / K_{AKG}^{KGDH})^{1.2} \\ f_a &= \frac{K_{MG}^{KGDH}}{K_{MG}^{KGDH} + [Mg^{2+}]_m} \frac{K_{CA}^{KGDH}}{K_{CA}^{KGDH} + [Ca^{2+}]_m} \\ \end{align} \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(k_{cat}^{KGDH}\) | 50 | 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 |
| \(K_{AKG}^{KGDH}\) | 1.94 | 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)
\[ \begin{align} J_{SL} &= k_f^{SL} ([SCoA][ADP]_m[Pi]_m - [SUC][ATP]_m[CoA]/K_{eq}^{SL}) \\ \end{align} \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(k_f^{SL}\) | 28 | Hz/mM² | Forward rate constant of SL |
| \(K_{eq}^{SL}\) | 3.115 | - | Equilibrium constant of SL |
| [CoA] | 20 | μM | Coenzyme A concentration |
Succinate dehydrogenase (SDH)
See OXPHOS part: complex II (Succinate dehydrogenase).
Fumarate hydratase (FH)
\[ J_{FH} = k_f^{FH} (\mathrm{[FUM]} - \mathrm{[MAL]} / K_{eq}^{FH}) \]
| Parameter | Value | Unit | Description |
|---|---|---|---|
| \(k_f^{FH}\) | 8.3 | Hz | Forward rate constant |
| \(K_{eq}^{FH}\) | 1.0 | - | Equilibrium constant |
Malate dehydrogenase (MDH)
\[ \begin{align} J_{MDH} &= \frac{k_{cat}^{MDH} E_T^{MDH} f_{mal}f_{nad} f_a f_i}{(1+f_{mal})(1+f_{nad})} \\ f_{mal} &= \frac{[MAL]}{K_{MAL}^{MDH}}\frac{K_{OAA}^{MDH}}{K_{OAA}^{MDH} + [OAA]} \\ f_{nad} &= [NAD] / K_{NAD}^{MDH} \\ f_a &= k_{offset}^{MDH} + \left( 1 + \frac{[H^+]_m}{K_{H1}^{MDH}} (1 + \frac{[H^+]_m}{K_{H2}^{MDH}}) \right)^{-1} \\ f_i &= \left( 1 + \frac{K_{H3}^{MDH}}{[H^+]_m} (1 + \frac{K_{H4}^{MDH}}{[H^+]_m}) \right)^{2} \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| \(k_{cat}^{MDH}\) | 126 | Hz | Rate constant of MDH |
| \(E_T^{MDH}\) | 154 | μM | Activity of MDH |
| \(K_{H1}^{MDH}\) | 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)
\[ \begin{align} J_{AAT} = k_f^{AAT} \mathrm{[OAA]} \mathrm{[GLU]} \frac{k_{ASP}^{AAT} K_{eq}^{AAT}}{k_{ASP}^{AAT} K_{eq}^{AAT} + k_f^{AAT}\mathrm{[\alpha KG]}} \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| \(k_f^{AAT}\) | 0.644 | Hz/mM | Forward rate constant of AAT |
| \(k_{ASP}^{AAT}\) | 0.0015 | Hz | Rate constant of aspartate consumption |
| \(K_{eq}^{AAT}\) | 6.6 | - | Equilibrium constant of AAT |
| [GLU] | 10 | mM | Glutamate concentration |
Oxidative phosphorylation (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, SDH)
New, reversible formulation.
\[ \begin{align} i_{OAA} &= \frac{K_{i, OAA}}{K_{i, OAA} + [OAA]} \\ J_{SDH} &= k_{SDH} C2_{inhib} i_{OAA} ([SUC][Q]_n - [FUM][QH_2]_n / K_{eq}^{C2} ) \\ K_{eq}^{C2} &= \exp(2(E_{m, Q}^{C2} - E_{m, SUC})/V_T) \end{align} \]
| Parameter | Value | Units | Desc. |
|---|---|---|---|
| \(k_{SDH}\) | 250 | 1 / (mM * minute) | Activity of SDH |
| \(K_{i, OAA}\) | 0.150 | mM | Inhibition constant for oxaloacetate |
| \(E_{m, Q}^{C2}\) | 100 | mV | Midpoint potential of ubiquinone in complex II |
| \(E_{m, SUC}\) | 40 | mV | Midpoint potential of succinate/fumarate pair |
Complex III (bc1 complex)
Updated to semireverse mechanism.
\[ \begin{align} f_{hi} & = [H^+]_{i} / 10^{-7}M \\ f_{hm} & = [H^+]_{m} / 10^{-7}M \\ v_{1} &= v_{QH_2}^{C1} + J_{SDH} \\ v_2 &= k_d([QH_2]_{n} - [QH_2]_{p}) \\ k_{3} &= k_{03}K_{eq3}f_{hi} \\ k_{-3} &= k_{03} \\ v_{3} &= k_3[QH_2]_{p} [FeS]_{ox} - k_{-3}[Q^-]_p [FeS]_{rd} \\ k_{4, ox} &= k_{04}K_{eq4, ox} \exp(-\alpha\delta_1\Delta\Psi_m F/ RT) \\ k_{4, rd} &= k_{04}K_{eq4, rd} \exp(-\alpha\delta_1\Delta\Psi_m F/ RT) \\ k_{-4, ox} &= k_{04} \exp(\alpha(1-\delta_1)\Delta\Psi_m F/ RT) \\ k_{-4, rd} &= k_{04} \exp(\alpha(1-\delta_1)\Delta\Psi_m F/ RT) \\ v_{4, ox} &= k_{4, ox}[Q^-]_p [cytb_1] - k_{-4, ox}[Q]_{p} [cytb_2] \\ v_{4, rd} &= k_{4, rd}[Q^-]_p [cytb_3] - k_{-4, rd}[Q]_{p} [cytb_4] \\ v_{5} &= k_d([Q]_{p} - [Q]_{n}) \\ k_{6} &= K_{06}K_{eq6} \exp( -\beta\delta_2\Delta\Psi_m / V_T) \\ k_{-6} &= k_{06} \exp( \beta(1-\delta_2)\Delta\Psi_m / V_T) \\ v_{6} &= k_{6} [cytb_2] - k_{-6} [cytb_3] \\ k_{7, ox} &= k_{07, ox}K_{eq7, ox}\exp(-\gamma\delta_3\Delta\Psi_m F/ RT) \\ k_{7, rd} &= k_{07, rd}K_{eq7, rd}\exp(-\gamma\delta_3\Delta\Psi_m F/ RT) \\ k_{-7, ox} &= k_{07, ox} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ k_{-7, rd} &= k_{07, rd} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ v_{7, ox} &= (k_{7, ox}[Q]_{n}[cytb_3] - k_{-7, ox}[Q^-]_n [cytb_1])C3_{inhib} \\ v_{7, rd} &= (k_{7, rd}[Q]_{n}[cytb_4] - k_{-7, rd}[Q^-]_n [cytb_2])C3_{inhib} \\ \end{align} \]
\[ \begin{align} k_{8, ox} &= k_{08, ox}K_{eq8, ox}\exp(-\gamma\delta_3\Delta\Psi_m F/ T)(f_{hm})^2 \\ k_{8, rd} &= k_{08,rd}K_{eq8, rd}\exp(-\gamma\delta_3\Delta\Psi_m F/ T)(f_{hm})^2 \\ k_{-8, ox} &= k_{08, ox} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ k_{-8, rd} &= k_{08, rd} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ v_{8, ox} &= (k_{8, ox}[Q^-]_{n}[cytb_3] - k_{-8, ox}[QH_2]_{n}[cytb_1])C3_{inhib} \\ v_{8, rd} &= (k_{8, rd}[Q^-]_{n}[cytb_4] - k_{-8, rd}[QH_2]_{n}[cytb_2])C3_{inhib} \\ k_9 &= k_{09}K_{eq9} \\ k_{-9} &= k_{09} \\ v_{9} &= k_{9}[FeS]_{rd}[cytc1]_{ox} - k_{-9}[FeS]_{ox}[cytc1]_{rd}\\ k_{10} &= k_{010}K_{eq10} \\ k_{-10} &= k_{010} \\ v_{10} &= k_{10}[Q^-]_p[O_2] - k_{-10}[Q]_p[O_2^-] \\ v_{10b} &= v_{10} \\ v_{33} &= k_{33}(K_{eq}[cytc1]_{rd}[cytc]_{ox} - [cytc]_{rd}[cytc1]_{ox}) \\ \rho_{C3}^{\prime} &= \rho_{C3} \cdot mt_{prot} \\ \rho_{C4}^{\prime} &= \rho_{C4} \cdot mt_{prot} \\ FeS_{rd} &= \rho_{C3}^{\prime} - FeS_{ox} \\ cytc1_{rd} &= \rho_{C3}^{\prime} - cytc1_{ox} \\ cytc_{rd} &= \rho_{C4}^{\prime} - cytc_{ox} \\ [cytb_4] &= \rho_{C3}^{\prime} - [cytb_1] - [cytb_2] - [cytb_3] \\ [QH_2]_p &= \Sigma Q - [Q]_n - [Q]_p - [QH_2]_n - [Q^-]_p - [Q^-]_n \\ J_{hRes}^{C3} &= 2v_{3} \\ J_{ROS, m}^{C3} &= v_{10} \\ J_{ROS, i}^{C3} &= v_{10b} \\ \end{align} \]
\[ \begin{align} \frac{d[Q]_n}{dt} &= v_5 - v_{7,ox}- v_{7,rd} - v_1 \\ \frac{d[Q^-]_n}{dt} &= v_{7,ox} + v_{7,rd} - v_{8,ox}- v_{8,rd} \\ \frac{d[QH_2]_n}{dt} &= v_{8,ox} + v_{8,rd} + v_1 - v_2 \\ \frac{d[QH_2]_p}{dt} &= v_2 -v_3 \\ \frac{d[Q^-]_p}{dt} &= v_3 - v_{10} - v_{10b} - v_{4,ox} - v_{4,rd} \\ \frac{d[Q]_p}{dt} &= v_{10} + v_{10b} + v_{4,ox} + v_{4,rd} - v_5 \\ \frac{d[cytb_1]}{dt} &= v_{7,ox} + v_{8,ox} - v_{4,ox} \\ \frac{d[cytb_2]}{dt} &= v_{4,ox} + v_{7,rd} - v_{8,rd} - v_6 \\ \frac{d[cytb_3]}{dt} &= v_6 - v_{4,rd} + v_{7,ox} - v_{8,ox} \\ \frac{d[cytb_4]}{dt} &= v_{4,rd} - v_{7,rd} - v_{8,rd} \\ \frac{d[FeS]_{ox}}{dt} &= v_9 - v_3 \\ \frac{d[cytc1]_{ox}}{dt} &= v_{33} - v_9 \\ \frac{d[cytc]_{ox}}{dt} &= V_e - v_{33} \\ \end{align} \]
| 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
\[ \begin{align} f_{H_{m}} &= \exp(-\delta_5\Delta\Psi_m F/ RT) ([H^+]_m /10^{-7}M) \\ f_{H_{i}} &= \exp((1-\delta_5)\Delta\Psi_m F/ RT) ([H^+]_i /10^{-7}M) \\ f_{C_{rd}} &= [cytc]_{rd} \\ f_{C_{ox}} &= \text{exp}((1-\delta_5)\Delta\Psi_m F/ RT) [cytc]_{ox} \\ a_{12} &= k_{34} f_{C_{rd}}^3 f_{H_{m}}^4 \\ a_{14} &= k_{-37} f_{H_{i}} \\ a_{21} &= k_{-34} f_{C_{ox}}^3 f_{H_{i}} \\ a_{23} &= k_{35} [O_2] C4_{inhib} \\ a_{34} &= k_{36} f_{C_{rd}} f_{H_{m}}^3 \\ a_{41} &= k_{37} f_{H_{m}} \\ a_{43} &= k_{-36} f_{C_{ox}} f_{H_{i}}^2 \\ e_1 &= a_{21}a_{41}a_{34} + a_{41}a_{34}a_{23} \\ e_2 &= a_{12}a_{41}a_{34} \\ e_3 &= a_{23}a_{12}a_{41} + a_{43}a_{14}a_{21} + a_{23}a_{43}a_{12} + a_{23}a_{43}a_{14} \\ e_4 &= a_{14}a_{34}a_{21} + a_{34}a_{23}a_{12} + a_{34}a_{23}a_{14} \\ \Delta &= e_1 + e_2 + e_3+ e_4 \\ Y &= e_1 / \Delta \\ Yr &= e_2 / \Delta \\ YO &= e_3 / \Delta \\ YOH &= e_4 / \Delta \\ v_{34} &= \rho_{C4}^\prime (Y \cdot a_{12} - Yr \cdot a_{21}) \\ v_{35} &= \rho_{C4}^\prime Yr \cdot a_{23} \\ v_{36} &= \rho_{C4}^\prime (YO \cdot a_{34} - YOH \cdot a_{43}) \\ v_{37} &= \rho_{C4}^\prime (YOH \cdot a_{41} - Y \cdot a_{14}) \\ \rho_{C4}^\prime &= \rho_{C4} \cdot mt_{prot} \\ J_{O_2} &= v_{35} \\ J_{hRes}^{C4} &= 8J_{O_2} \\ J_{hRes} &= J_{hRes}^{C1} + J_{hRes}^{C3} + J_{hRes}^{C4} \\ \end{align} \]
| 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)
\[ \begin{align} J_{F1Fo} &= -\rho^{F1} ((100 p_a + p_{c1} v_B) v_a - (p_a + p_{c2} v_a) v_h) / \Delta_{F1} \\ J_H^{F1Fo} &= -3\rho^{F1} (100p_a(1 + v_a) - (p_a + p_b)v_h) / \Delta_{F1} \\ \Delta_{F1} &= (1 + p_1 v_a)v_B + (p_2 + p_3 v_a)v_h \\ v_B &= \text{exp}(3\Delta\Psi_B / V_T) \\ v_h &= \text{exp}(3\Delta p / V_T) \\ v_a &= \frac{K_{eq}^{'} \cdot \Sigma[ATP]_m}{ \Sigma[Pi]_m \cdot \Sigma[ADP]_m } \\ \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(\rho_{F1}\) | 5 | mM | Concentration of F1-Fo ATPase |
| \(K_{eq}^{'}\) | 2E5 | M | Apparent equilibrium constant for ATP hydrolysis 1 |
| \(\Delta\Psi_B\) | 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.
\[ \begin{align} J_{SOD} &= \frac{2 k_5^{SOD} E_T f_{sox} (k_1^{SOD} + k_3^\prime)}{ k_5^{SOD} (2 k_1^{SOD} + k_3^\prime) + k_3^\prime f_{sox}} \\ k_3^\prime &= k_3^{SOD} (1 + \frac{[H_2O_2]}{K_{i}^{SOD}}) \\ f_{sox} &= k_1^{SOD} [O_2^-] \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(k_1^{SOD}\) | 1200 | 1/(mM*ms) | Rate constant for EA -> EB |
| \(k_3^{SOD}\) | 24 | 1/(mM*ms) | Rate constant for EB -> EC |
| \(k_5^{SOD}\) | 0.24 | Hz | Rate constant for EC -> EA |
| \(K_{i}^{SOD}\) | 500 | μM | Inhibition constant for H2O2 |
| \(E_{T}^{SOD}\) | 3 | μM | Concentration of Cu,ZnSOD (cytosolic) |
Glutathione peroxidase (GPX)
Dalziel type Ping-pong mechanism.
\[ \begin{align} J_{GPX} &= \frac{E_T}{\frac{\Phi_1}{[H_2O_2] } + \frac{\Phi_2}{[GSH] }} \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(E_T\) | 10 | μM | GPX content |
| \(\Phi_1\) | 5000 | μM/s | Dalziel coefficient |
| \(\Phi_2\) | 75000 | μM/s | Dalziel coefficient |
Glutathione reductase (GR)
Michaelis-Menten kinetics.
\[ \begin{align} J_{GR} &= k_1^{GR} E_T^{GR} \frac{[GSSG]}{[GSSG] + K_{GSSG}} \frac{[NADPH]}{[NADPH] + K_{NADPH}} \\ \Sigma [GSH] &= [GSH] + 2 [GSSG] \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(E_T^{GR}\) | 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 |
| \(\Sigma [GSH]\) | 1000 | μM | Cytosolic GSH pool |
Inner mitochondrial anion channel (IMAC)
\[ \begin{align} g_{IMAC} &= \left( a + b \frac{[O_2^-]_i}{[O_2^-]_i + K_{CC}} \right) \left( G_L + \frac{G_{max}}{1 + e^{κ(\Delta\Psi_m^b - \Delta\Psi_m)}} \right) \\ V_{IMAC} &= g_{IMAC}\Delta\Psi_m \\ J_{tr}^{ROS} &= j \cdot g_{IMAC} \left( \Delta\Psi_m + V_T ln \left( \frac{[O_2^-]_m}{[O_2^-]_i} \right) \right) \\ \end{align} \]
| 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 / (mV * s) | Integral conductance for IMAC |
| \(G_{max}\) | 3.9085 | μM / (mV * s) | Leak conductance of IMAC |
| \(\kappa\) | -0.07 | 1/mV | Steepness factor |
| \(\Delta\Psi_m^b\) | 4 | mV | Potential at half saturation |
| j | 0.5 | - | 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
\[ \begin{align} \frac{d [Ca^{2+}]_m}{dt} &=\delta_{Ca}( J_{uni} - J_{NCLX}) \\ C_{m}\frac{d \Delta \Psi_m}{dt} &= J_{Hres} - J_{Hu} - J_{ANT} - J_{Hleak} -J_{NCLX} - J_{uni} - J_{IMAC} \\ \end{align} \]
Adenine Nucleotide translocator (ANT)
\[ \begin{align} J_{ANT} &= V_{max}^{ANT}\frac{T_m D_i - \exp(-\Delta\Psi_m / V_T) T_i D_m}{(D_i + T_i\exp(- h_{ANT} \Delta\Psi_m / V_T))(T_m + D_m)} \\ T_m &= 0.025 [ATP]_m \\ D_i &= 0.45 [ADP]_i \\ T_i &= 0.25 [ATP]_i \\ D_m &= 0.17 [ADP]_m \\ \delta &= \exp(-\Delta\Psi_m / V_T) \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(V_{max}^{ANT}\) | 5000 | μM/s | Maximal rate of ANT |
| \(h_{ANT}\) | 0.5 | - | Fraction of MMP |
Mitochondrial calcium uniporter (MCU)
\[ \begin{align} J_{uni} &= V_{max}^{Uni} \frac{S (1+S)^3}{(1+S)^4 + L(1 + A)^n} \frac{\delta}{e^\delta-1} \\ S &= [Ca^{2+}]_i / K_{trans} \\ A &= [Ca^{2+}]_i / K_{act} \\ \delta &= -2 (\Delta\Psi_m - \Delta\Psi_0)/ V_T \\ \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(V_{max}^{Uni}\) | 4460 | μM/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)
\[ \begin{align} J_{NCLX} &= V_{max}^{NCLX} \exp(b\Delta\Psi_m /V_T) \phi_{ca} f_{na} f_{ca} \\ \phi_{ca} &= \frac{[Ca^{2+}]_m}{[Ca^{2+}]_i} \\ f_{na} &= \left( \frac{[Na^+]_i}{[Na^+]_i + K_{Na}^{NCLX}} \right)^n \\ f_{ca} &= \frac{[Ca^{2+}]_m}{[Ca^{2+}]_m + K_{Ca}^{NCLX}} \\ \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(V_{max}^{NCLX}\) | 46.65 | μM/s | Maximal rate of NCLX |
| b | 0.5 | - | Fraction of MMP |
| \(K_{Na}^{NCLX}\) | 9400 | μM | Dissociation constant for sodium |
| \(K_{Ca}^{NCLX}\) | 0.375 | μM | Dissociation constant for calcium |
| \(n\) | 3 | - | Cooperativity of NCLX |
Mitochondrial proton leak
\[ \begin{align} J_{hleak} = g_H\Delta\Psi_m \end{align} \]
| Parameter | Value | Unit | Desc. |
|---|---|---|---|
| \(g_{H}\) | 2 | μM / (mV * s) | Ionic conductance of the inner mitochondrial membrane |
Footnotes
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.↩︎