RIRR-DOX 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)

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

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)

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

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)

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

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)

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

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

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

Endoplasmic reticulum

Ryanodine receptor (Jrel)

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

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.

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

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)

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

Time-independent potassium current (IK1)

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

Plateau potassium current (IKp)

\[\begin{split} \begin{align} E_{Kp} &= \frac{RT}{F} \ln \frac{[K^+]_o}{[K^+]_i} \\ I_{Kp} &= \frac{\bar G_{Kp} (V - E_{Kp})}{1 + e^{(7.488-V_m) / 5.98}} \\ \end{align} \end{split}\]

Fast Na current (INa)

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

Sodium-calcium exchanger current (INaCa)

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

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

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

Non-specific calcium-activated current (InsCa)

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

Sodium-potassium ATPase current (INaK)

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

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

L-type Ca current (ICa & ICaK)

“Common pool” subspace calcium model.

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

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

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.

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

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

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

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]

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

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

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.

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

\[\begin{split} \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} \end{split}\]

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

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)

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

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

\[\begin{split} \begin{align} f_{hi} & = [H^+]_{i} / 10^{-7}M \\ v_{1} &= v_{Q}^{C1} + v_{Q}^{C2} \\ v_2 &= k_d([QH_2]_{n} - [QH_2]_{p}) \\ k_{3} &= k_{03}K_{eq3}f_{hi} \\ k_{-3} &= k_{03} \\ v_{3} &= k_3[QH_2]_{p} [FeS]_{ox} - k_{-3}[Q^-]_p [FeS]_{rd} \\ k_{4, ox} &= k_{04}K_{eq4, ox} \exp(-\alpha\delta_1\Delta\Psi_m F/ RT) \\ k_{4, rd} &= k_{04}K_{eq4, rd} \exp(-\alpha\delta_1\Delta\Psi_m F/ RT) \\ k_{-4, ox} &= k_{04} \exp(\alpha(1-\delta_1)\Delta\Psi_m F/ RT) \\ k_{-4, rd} &= k_{04} \exp(\alpha(1-\delta_1)\Delta\Psi_m F/ RT) \\ v_{4, ox} &= k_{4, ox}[Q^-]_p [cytb_1] - k_{-4, ox}[Q]_{p} [cytb_2] \\ v_{4, rd} &= k_{4, rd}[Q^-]_p [cytb_3] - k_{-4, rd}[Q]_{p} [cytb_4] \\ v_{5} &= k_d([Q]_{p} - [Q]_{n}) \\ k_{6} &= K_{06}K_{eq6} \exp( -\beta\delta_2\Delta\Psi_m / V_T) \\ k_{-6} &= k_{06} \exp( \beta(1-\delta_2)\Delta\Psi_m / V_T) \\ v_{6} &= k_{6} [cytb_2] - k_{-6} [cytb_3] \\ k_{7, ox} &= k_{07, ox}K_{eq7, ox}\exp(-\gamma\delta_3\Delta\Psi_m F/ RT) \\ k_{7, rd} &= k_{07, rd}K_{eq7, rd}\exp(-\gamma\delta_3\Delta\Psi_m F/ RT) \\ k_{-7, ox} &= k_{07, ox} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ k_{-7, rd} &= k_{07, rd} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ v_{7, ox} &= (k_{7, ox}[Q]_{n}[cytb_3] - k_{-7, ox}[Q^-]_n [cytb_1])C3_{inhib} \\ v_{7, rd} &= (k_{7, rd}[Q]_{n}[cytb_4] - k_{-7, rd}[Q^-]_n [cytb_2])C3_{inhib} \\ \end{align} \end{split}\]

\[\begin{split} \begin{align} f_{hm} & = [H^+]_{m} / 10^{-7}M \\ k_{8, ox} &= k_{08, ox}K_{eq8, ox}\exp(-\gamma\delta_3\Delta\Psi_m F/ T)(f_{hm})^2 \\ k_{8, rd} &= k_{08,rd}K_{eq8, rd}\exp(-\gamma\delta_3\Delta\Psi_m F/ T)(f_{hm})^2 \\ k_{-8, ox} &= k_{08, ox} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ k_{-8, rd} &= k_{08, rd} \exp(\gamma(1-\delta_3)\Delta\Psi_m F/ RT) \\ v_{8, ox} &= (k_{8, ox}[Q^-]_{n}[cytb_3] - k_{-8, ox}[QH_2]_{n}[cytb_1])C3_{inhib} \\ v_{8, rd} &= (k_{8, rd}[Q^-]_{n}[cytb_4] - k_{-8, rd}[QH_2]_{n}[cytb_2])C3_{inhib} \\ k_9 &= k_{09}K_{eq9} \\ k_{-9} &= k_{09} \\ v_{9} &= k_{9}[FeS]_{rd}[cytc1]_{ox} - k_{-9}[FeS]_{ox}[cytc1]_{rd}\\ k_{10} &= k_{010}K_{eq10} \\ k_{-10} &= k_{010} \\ v_{10} &= k_{10}[Q^-]_p[O_2] - k_{-10}[Q]_p[O_2^-] \\ v_{10b} &= v_{10} \\ v_{33} &= k_{33}(K_{eq}[cytc1]_{rd}[cytc]_{ox} - [cytc]_{rd}[cytc1]_{ox}) \\ \rho_{C3}^{\prime} &= \rho_{C3} \cdot mt_{prot} \\ \rho_{C4}^{\prime} &= \rho_{C4} \cdot mt_{prot} \\ FeS_{rd} &= \rho_{C3}^{\prime} - FeS_{ox} \\ cytc1_{rd} &= \rho_{C3}^{\prime} - cytc1_{ox} \\ cytc_{rd} &= \rho_{C4}^{\prime} - cytc_{ox} \\ [cytb_4] &= \rho_{C3}^{\prime} - [cytb_1] - [cytb_2] - [cytb_3] \\ [QH_2]_p &= \Sigma Q - [Q]_n - [Q]_p - [QH_2]_n - [Q^-]_p - [Q^-]_n \\ J_{hRes}^{C3} &= 2v_{3} \\ J_{ROS, m}^{C3} &= v_{10} \\ J_{ROS, i}^{C3} &= v_{10b} \\ \end{align} \end{split}\]

\[\begin{split} \begin{align} \frac{d[Q]_n}{dt} &= v_5 - v_{7,ox}- v_{7,rd} - v_1  \\ \frac{d[Q^-]_n}{dt} &= v_{7,ox} + v_{7,rd} - v_{8,ox}- v_{8,rd}  \\ \frac{d[QH_2]_n}{dt} &= v_{8,ox} + v_{8,rd} + v_1 - v_2   \\ \frac{d[QH_2]_p}{dt} &= v_2 -v_3 \\ \frac{d[Q^-]_p}{dt} &= v_3 - v_{10} - v_{10b} - v_{4,ox} - v_{4,rd}   \\ \frac{d[Q]_p}{dt} &= v_{10} + v_{10b} + v_{4,ox} + v_{4,rd} - v_5   \\ \frac{d[cytb_1]}{dt} &= v_{7,ox} + v_{8,ox} - v_{4,ox}    \\ \frac{d[cytb_2]}{dt} &= v_{4,ox} + v_{7,rd} - v_{8,rd} - v_6   \\ \frac{d[cytb_3]}{dt} &= v_6 - v_{4,rd} + v_{7,ox} - v_{8,ox}    \\ \frac{d[cytb_4]}{dt} &= v_{4,rd} - v_{7,rd} - v_{8,rd}   \\ \frac{d[FeS]_{ox}}{dt} &= v_9 - v_3      \\ \frac{d[cytc1]_{ox}}{dt} &= v_{33} - v_9   \\ \frac{d[cytc]_{ox}}{dt} &= V_e - v_{33}   \\ \end{align} \end{split}\]

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

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

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)

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

Parameter	Value	Unit	Desc.
\(\rho_{F1}\)	5	mM	Concentration of F1-Fo ATPase
\(K_{eq}^{'}\)	6.47E5	M	Apparent equilibrium constant for ATP hydrolysis[2]
\(\Delta\Psi_B\)	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

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

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.

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

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.

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

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.

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

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)

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

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

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

Mitochondrial ion transport

Adenine Nucleotide translocator (ANT)

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

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)

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

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

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

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

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[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.