Glucose stimulation and Mitochondrial dynamics
Difference from the original Fridlyand’s model ¶ We added an dilution term for pyruvate and a ADP Michaelis constant in Glyceraldehyde 3-phosphate dehydrogenase to increase the robustness in the ODE systems.
We added adenylate kinase (Adk) equilibrium for the ATP-ADP-AMP pool to address the high energy phosphate distribution in the cytosol. Also, the adenylate pool size was bumped up to keep the steady state ATP/ADP ratio the same as before (~5) at the resting glucose level of 5mM.
We altered the mathematical expressions of mitochondrial sodium calcium exchanger (NCLX). The specific concentrations of sodium and calcium were on the exponent in the original model. We moved the specific concentrations down to multipliers was inconsistent with the original model of Nguyen et al.
The mitochondrial fission rate is fixed to once per 10 minutes, and the fusion rate depends on the proton leak flux and the ATP synthase flux.
Current Limitations ¶ Glycolysis, ETC, ATP synthesis, and cytosolic calcium dynamics were expressed in an empirical fashion rather than fully mechanistic. No signal transduction nor complex calcium dynamics.
There is no role for reactive oxygen species (ROS), potential signals for bioenergetics and mitochondrial dynamics.
Mitochondrial content was assumed conserved and homogenous.
General parameters ¶ Parameter Value Description V i V_{i} V i 0.53 Relative cytoplasmic volume V m V_{m} V m 0.06 Relative mitochondrial volume V m t x V_{mtx} V m t x 0.0144 Relative mitochondrial matrix volume (Adjustable) C m i t o C_{mito} C mi t o 1.812 mM/V Mitochondrial membrane capacitance F F F 96484.6 C/mol Faraday’s constant δ C a \delta_{Ca} δ C a 0.0003 Fraction of free Ca in mitochondria [ N a + ] c [Na^+]_c [ N a + ] c 10 mM Cytoplasmic Na concentration [ N a + ] m [Na^+]_m [ N a + ] m 5 mM Mitochondrial Na concentration T v T_v T v 26.73 mV Thermal voltage (RT/F) ( Σ A c \Sigma A_c Σ A c 4.5 mM Cellular adenine nucleotides concentration (Adjustable) Σ N m \Sigma N_m Σ N m 2.2 mM Free pyridine nucleotides concentration in Σ N c \Sigma N_c Σ N c 2.0 mM Free pyridine nucleotides concentration in k g p d k_{gpd} k g p d 0.01/s Consumption rate of G3P k N A D H m k_{NADHm} k N A DH m 0.1/s Consumption rate of mito NADH k N A D H c k_{NADHc} k N A DHc 0.1/s Consumption rate of cyto NADH k A T P k_{ATP} k A TP 0.04/s Basal consumption rate of ATP k A T P C a k_{ATPCa} k A TPC a 90/mM/s Consumption rate of ATP activated by calcium
Conservation relationships ¶ Σ N m = [ N A D + ] m + [ N A D H ] m Σ N c = [ N A D + ] c + [ N A D H ] c Σ A c = [ A T P ] c + [ A D P ] c + [ A M P ] c 1 = X 1 + 2 X 2 + 3 X 3 \begin{aligned}
\Sigma N_{m} &= [NAD^+]_m + [NADH]_m \\
\Sigma N_{c} &= [NAD^+]_c + [NADH]_c \\
\Sigma A_{c} &= [ATP]_c + [ADP]_c + [AMP]_c \\
1 &= X_1 + 2X_2 + 3X_3
\end{aligned} Σ N m Σ N c Σ A c 1 = [ N A D + ] m + [ N A DH ] m = [ N A D + ] c + [ N A DH ] c = [ A TP ] c + [ A D P ] c + [ A MP ] c = X 1 + 2 X 2 + 3 X 3 Adenylate kinase ¶ J A D K = k f ( [ A D P ] c 2 − [ A T P ] c [ A M P ] c / K e q A K ) \begin{aligned}
J_{ADK} &= k_f ([ADP]_c^2 - [ATP]_c [AMP]_c / K_{eq}^{AK})
\end{aligned} J A DK = k f ([ A D P ] c 2 − [ A TP ] c [ A MP ] c / K e q A K ) Parameter Value Description k f k_f k f 1000 mM/s Forward (AMP-forming) rate constant of adenylate kinase. K e q A K K_{eq}^{AK} K e q A K 0.931 Equilibrium constant of adenylate kinase (AMP-forming).
Glucokinase (GK) ¶ J g l u = V m [ A T P ] c [ A T P ] c + K A T P [ G l c ] n [ G l c ] n + K G l c n J_{glu} = V_{m} \frac{[ATP]_c}{[ATP]_c + K_{ATP}} \frac{[Glc]^n}{[Glc]^n + K_{Glc}^{n}} J g l u = V m [ A TP ] c + K A TP [ A TP ] c [ Gl c ] n + K Gl c n [ Gl c ] n Parameter Value Description V m V_{m} V m 0.011 mM/s Max rate of glucokinase K A T P K_{ATP} K A TP 0.5 mM Michaelis constant for ATP K G l c K_{Glc} K Gl c 7 mM Michaelis constant for glucose n 1.7 Cooperativity for glucose
Glyceraldehyde 3-phosphate dehydrogenase (GPD) ¶ J g p d = V m [ G 3 P ] [ G 3 P ] + K G 3 P [ N A D + ] c [ N A D + ] c + K N A D [ N A D H ] c J_{gpd} = V_m \frac{[G3P]}{[G3P] + K_{G3P}} \frac{[NAD^+]_c}{[NAD^+]_c + K_{NAD}[NADH]_c} J g p d = V m [ G 3 P ] + K G 3 P [ G 3 P ] [ N A D + ] c + K N A D [ N A DH ] c [ N A D + ] c Parameter Value Description V m V_{m} V m 0.5 mM/s Max rate of GPD (Adjustable) K G 3 P K_{G3P} K G 3 P 0.2 mM Michaelis constant for G3P K N A D K_{NAD} K N A D 0.09 Activation constant for cytosolic NAD/NADH ratio
Lactate production by lactate dehydrogenase (LDH) ¶ The rate of lactate output is approximately 5% of the rate of glucose consumption when the glucose level is 8 mM.
J L D H = V m [ P y r ] [ P y r ] + K P y r [ N A D H ] c [ N A D H ] c + K N A D H [ N A D + ] c J_{LDH} = V_m \frac{[Pyr]}{[Pyr] + K_{Pyr}} \frac{[NADH]_c}{[NADH]_c + K_{NADH} [NAD^+]_c} J L DH = V m [ P yr ] + K P yr [ P yr ] [ N A DH ] c + K N A DH [ N A D + ] c [ N A DH ] c Parameter Value Description V m V_{m} V m 1.2 mM/s Max rate of LDH (Adjustable) K P y r K_{Pyr} K P yr 0.0475 mM Michaelis constant for pyruvate K N A D H K_{NADH} K N A DH 1 Activation constant for cytosolic NADH/NAD ratio
Steady-state cytosolic calcium levels ¶ [ C a 2 + ] c = [ C a 2 + ] R + k A C a ( [ A T P ] c ) n ( [ A T P ] c ) n + ( K A T P [ A D P ] c ) n [Ca^{2+}]_c = [Ca^{2+}]_R + k_{A}^{Ca} \frac{([ATP]_c)^n}{([ATP]_c)^n + (K_{ATP} [ADP]_c)^n} [ C a 2 + ] c = [ C a 2 + ] R + k A C a ([ A TP ] c ) n + ( K A TP [ A D P ] c ) n ([ A TP ] c ) n Parameter Value Description [ C a 2 + ] R [Ca^{2+}]_R [ C a 2 + ] R 90 nM Resting cytoplasmic calcium concentration k A C a k_{A}^{Ca} k A C a 250 nM Maximal activated calcium concentration K A T P K_{ATP} K A TP 25 Activation constant for ATP/ADP ratio n n n 4 Cooperativity for ATP/ADP ratio
Oscillating calcium levels ¶ Use in simulations for Fig. 4 : oscillating calcium on mitochondrial bioenergetics and dynamics only.
[ C a 2 + ] c = [ C a 2 + ] R + k A C a ( A x e 1 − A x ) B x = t T − ⌊ t T ⌋ \begin{aligned}
[Ca^{2+}]_c &= [Ca^{2+}]_R + k_{A}^{Ca}(Axe^{1-Ax})^B \\
x &= \frac{t}{T} - \lfloor \frac{t}{T} \rfloor
\end{aligned} [ C a 2 + ] c x = [ C a 2 + ] R + k A C a ( A x e 1 − A x ) B = T t − ⌊ T t ⌋ Parameter Value Description [ C a 2 + ] R [Ca^{2+}]_R [ C a 2 + ] R 90 nM Resting cytoplasmic calcium concentration k A C a k_{A}^{Ca} k A C a 250 nM Maximal activated calcium concentration A A A 5 Asymmetric factor B B B 4 Steepness factor T T T 2 minute Period of calcium oscillations.
Pyruvate dehydrogenase (PDH) ¶ We assume that pyruvate diffuses freely and fast across the inner mitochondrial membrane (IMM). Therefore, pyruvate is the same concentration in the cytosol and in the mitochondrial matrix.
J P D H = V m [ P y r ] [ P y r ] + K P y r [ N A D + ] m [ N A D + ] m + K N A D ( 1 + C ) 2 ( 1 + C ) 2 ( 1 + u 2 ) + u 2 u 1 C = [ C a 2 + ] m / K C a \begin{aligned}
J_{PDH} &= V_m \frac{ [Pyr] }{ [Pyr] + K_{Pyr}} \frac{[NAD^+]_m}{[NAD^+]_m + K_{NAD}} \frac{(1 + C)^2}{(1 + C)^2 (1 + u_2) + u_2 u_1} \\
C &= [Ca^{2+}]_m / K_{Ca}
\end{aligned} J P DH C = V m [ P yr ] + K P yr [ P yr ] [ N A D + ] m + K N A D [ N A D + ] m ( 1 + C ) 2 ( 1 + u 2 ) + u 2 u 1 ( 1 + C ) 2 = [ C a 2 + ] m / K C a Parameter Value Description V m V_{m} V m 0.3 mM/s Max rate of PDH K P y r K_{Pyr} K P yr 0.0475 mM Michaelis constant for pyruvate K N A D K_{NAD} K N A D 81 Activation constant for mitochondrial NAD/NADH ratio K C a K_{Ca} K C a 50 nM Activation constant for mitochondrial Ca u 1 u_1 u 1 1.5 Factor for calcium activation u 2 u_2 u 2 1.1 Factor for calcium activation
Electron transport chain (ETC) ¶ J h r = V m [ N A D H ] m [ N A D H ] m + K N A D H 1 + k A Δ Ψ m 1 + k B Δ Ψ m F O 2 \begin{aligned}
J_{hr} &= V_m \frac{[NADH]_m}{[NADH]_m + K_{NADH}} \frac{1 + k_A \Delta \Psi_m}{1 + k_B \Delta \Psi_m} F_{O_2}
\end{aligned} J h r = V m [ N A DH ] m + K N A DH [ N A DH ] m 1 + k B Δ Ψ m 1 + k A Δ Ψ m F O 2 Parameter Value Description V m V_{m} V m 22 mM/s Max rate of ETC K N A D H K_{NADH} K N A DH 3 mM Michaelis constant for NADH k A k_A k A -4.92 /Volt thermodynamic potential factor k B k_B k B -4.43 /Volt thermodynamic potential factor F O 2 F_{O_2} F O 2 1 Oxygen availability
F1Fo ATPase (ATP synthase) ¶ ATP synthase was lumped with ANT and depended on cytosolic ADP.
J h f = V m f A D P f Ψ f C a J A N T = J h f / H A T P f A D P = [ M g A D P ] c n A [ M g A D P ] c n A + K A D P n A f Ψ = Δ Ψ m n Ψ Δ Ψ m n Ψ + K Ψ n Ψ f C a = 1 − exp ( − [ C a 2 + ] m / K C a ) [MgADP] c = 0.055 [ A D P ] c \begin{aligned}
J_{hf} &= V_m f_{ADP} f_{\Psi} f_{Ca} \\
J_{ANT} &= J_{hf} / H_{ATP} \\
f_{ADP} &= \frac{[MgADP]_c^{n_A}}{[MgADP]_c^{n_A} + K_{ADP}^{n_A}} \\
f_{\Psi} &= \frac{\Delta\Psi_m^{n_\Psi}}{\Delta\Psi_m^{n_\Psi} + K_{\Psi}^{n_\Psi}} \\
f_{Ca} &= 1 - \exp(-[Ca^{2+}]_m / K_{Ca} ) \\
\text{[MgADP]}_c &= 0.055[ADP]_c
\end{aligned} J h f J A NT f A D P f Ψ f C a [MgADP] c = V m f A D P f Ψ f C a = J h f / H A TP = [ M g A D P ] c n A + K A D P n A [ M g A D P ] c n A = Δ Ψ m n Ψ + K Ψ n Ψ Δ Ψ m n Ψ = 1 − exp ( − [ C a 2 + ] m / K C a ) = 0.055 [ A D P ] c Parameter Value Description V m V_{m} V m 8 mM/s Max rate of ATP synthase (Adjustable) K A D P K_{ADP} K A D P 20 μM Apparent Michaelis constant for cytosolic MgADP n A n_A n A 2 Cooperativity for MgADP n Ψ n_\Psi n Ψ 8 Cooperativity for mitochondrial potential K Ψ K_{\Psi} K Ψ 131.4 mV Mid-activity constant for mitochondrial potential K C a K_{Ca} K C a 0.165 μM Activation constant for mitochondrial calcium H A T P H_{ATP} H A TP 3 H:ATP ratio
Proton leak ¶ The basal leak approaches ~20% of the electron transport rate at Δ Ψ m \Delta\Psi_m Δ Ψ m
J h l = P H exp ( k l p Δ Ψ m ) J_{hl} = P_{H}\exp(k_{lp} \Delta \Psi_m) J h l = P H exp ( k lp Δ Ψ m ) Parameter Value Description P H P_{H} P H 0.0024 mM/s leak coefficient k l p k_{lp} k lp 30.5/V membrane potential coefficient
NADH shuttles ¶ J T N A D H = T N A D H [ N A D H ] c [ N A D H ] c + [ N A D + ] c K c [ N A D + ] m [ N A D + ] m + [ N A D H ] m K m J_{TNADH} = T_{NADH} \frac{[NADH]_c}{[NADH]_c + [NAD^+]_c K_c} \frac{[NAD^+]_m}{[NAD^+]_m + [NADH]_m K_m} J TN A DH = T N A DH [ N A DH ] c + [ N A D + ] c K c [ N A DH ] c [ N A D + ] m + [ N A DH ] m K m [ N A D + ] m Parameter Value Description T N A D H T_{NADH} T N A DH 0.05 mM/s NADH transport rate K c K_c K c 0.002 Affinity coefficients for cytoplasmic NADH/NAD K m K_m K m 16.78 Affinity coefficients for mitochondrial NAD/NADH
Mitochondrial calcium uniporter (MCU) ¶ J u n i = P C a δ e δ − 1 ( e δ α i [ C a 2 + ] c − α m [ C a 2 + ] m ) δ = Z C a Δ Ψ m / V T \begin{aligned}
J_{uni} &= P_{Ca} \frac{\delta}{e^{\delta} - 1} (e^{\delta} \alpha_i [Ca^{2+}]_c - \alpha_m [Ca^{2+}]_m) \\
\delta &= Z_{Ca}\Delta\Psi_m / V_T
\end{aligned} J u ni δ = P C a e δ − 1 δ ( e δ α i [ C a 2 + ] c − α m [ C a 2 + ] m ) = Z C a Δ Ψ m / V T Parameter Value Description P C a P_{Ca} P C a 4 / s Permeability of calcium Z C a Z_{Ca} Z C a 2 Valence of calcium α i \alpha_i α i 0.341 Activity of cytoplasmic calcium α m \alpha_m α m 0.2 Activity of mitochondrial calcium
Mitochondrial Sodium-Calcium exchanger (NCLX) ¶ We used the electron-neutral descriptor of NCLX since this model generated smooth and monotonous increment of mitochondrial calcium levels upon increasing glucose levels.
J N C L X = V m ( A B − P Q ) / D D = 1 + A + B + P + Q + A B + P Q A = ( [ N a + ] c / K N a ) 2 B = [ C a 2 + ] m / K C a P = ( [ N a + ] m / K N a ) 2 Q = [ C a 2 + ] c / K C a \begin{aligned}
J_{NCLX} &= V_{m} (AB - PQ ) / D \\
D &= 1 + A + B + P + Q + AB + PQ \\
A &= ([Na^+]_c / K_{Na})^2 \\
B &= [Ca^{2+}]_m / K_{Ca} \\
P &= ([Na^+]_m / K_{Na})^2 \\
Q &= [Ca^{2+}]_c / K_{Ca} \\
\end{aligned} J NC L X D A B P Q = V m ( A B − PQ ) / D = 1 + A + B + P + Q + A B + PQ = ([ N a + ] c / K N a ) 2 = [ C a 2 + ] m / K C a = ([ N a + ] m / K N a ) 2 = [ C a 2 + ] c / K C a Parameter Value Description V m V_m V m 0.075 mM/s Max rate of NCLX K C a K_{Ca} K C a 8 μM Dissociation constant of Ca K N a K_{Na} K N a 8.2 mM Dissociation constant of Na
Mitochondrial Dynamics ¶ k f u s e , 1 = k 0 f u s e J A N T J H L k f i s s , 1 = k 0 f i s s k f u s e , 2 = 0.1 k f u s e , 1 k f i s s , 2 = 1.5 k f i s s , 1 \begin{align}
k_{fuse, 1} &= k_{0}^{fuse} \frac{ J_{ANT} }{ J_{HL} } \\
k_{fiss, 1} &= k_{0}^{fiss} \\
k_{fuse, 2} &= 0.1k_{fuse, 1} \\
k_{fiss, 2} &= 1.5k_{fiss, 1} \\
\end{align} k f u se , 1 k f i ss , 1 k f u se , 2 k f i ss , 2 = k 0 f u se J H L J A NT = k 0 f i ss = 0.1 k f u se , 1 = 1.5 k f i ss , 1 Parameter Value Description k 0 f u s s k_0^{fuss} k 0 f u ss 1 600 \frac{1}{600} 600 1 The basal fusion rate k 0 f i s s k_0^{fiss} k 0 f i ss 1 600 \frac{1}{600} 600 1 The basal fission rate
Ordinary differential equations ¶ d d t [ G 3 P ] = 1 V i ( 2 J g l u − J G P D ) − k g 3 p [ G 3 P ] d d t [ P y r ] = 1 V i + V m t x ( J G P D − J L D H − J P D H ) − k p y r [ P y r ] d d t [ N A D H ] c = 1 V i ( J G P D − J L D H − J N A D H T ) − k n a d h c [ N A D H ] c d d t [ N A D H ] m = 1 V m t x ( J N A D H T + 4.6 J P D H − 0.1 J h r ) − k n a d h m [ N A D H ] m d d t Δ Ψ m = 1 C m i t o ( J h r − J h f − J A N T − J h l − 2 J u n i ) d d t [ C a 2 + ] m = f m V m t x ( J u n i − J N C L X ) d d t [ A T P ] c = 1 V i ( J A N T − 2 J G l u + 2 J G P D + J A d K ) − ( k A T P + k C a A T P [ C a 2 + ] c ) [ A T P ] c d d t [ A D P ] c = − d d t [ A T P ] c − J A d K V i d d t X 2 = k f u s e , 1 X 1 2 − ( k f i s s , 1 + k f u s e , 2 X 1 ) X 2 + k f i s s , 2 X 3 d d t X 3 = k f u s e , 2 X 1 X 2 − k f i s s , 2 X 3 \begin{aligned}
\frac{d}{dt}[G3P] &= \frac{1}{V_i}(2J_{glu} - J_{GPD}) - k_{g3p}[G3P] \\
\frac{d}{dt}[Pyr] &= \frac{1}{V_i + V_{mtx}}(J_{GPD} - J_{LDH} - J_{PDH}) - k_{pyr}[Pyr] \\
\frac{d}{dt}[NADH]_c &= \frac{1}{V_i}(J_{GPD} - J_{LDH} - J_{NADHT}) - k_{nadhc}[NADH]_c \\
\frac{d}{dt}[NADH]_m &= \frac{1}{V_{mtx}}(J_{NADHT} + 4.6J_{PDH} - 0.1J_{hr}) - k_{nadhm}[NADH]_m \\
\frac{d}{dt}\Delta\Psi_m &= \frac{1}{C_{mito}} (J_{hr} - J_{hf} - J_{ANT} - J_{hl} - 2J_{uni}) \\
\frac{d}{dt}[Ca^{2+}]_m &= \frac{f_m}{V_{mtx}} (J_{uni} - J_{NCLX}) \\
\frac{d}{dt}[ATP]_c &= \frac{1}{V_i} (J_{ANT} - 2J_{Glu} + 2J_{GPD} + J_{AdK}) - (k_{ATP} + k_{CaATP}[Ca^{2+}]_c)[ATP]_c \\
\frac{d}{dt}[ADP]_c &= -\frac{d}{dt}[ATP]_c - \frac{J_{AdK}}{V_i} \\
\frac{d}{dt} X_2 &= k_{fuse, 1} X_1^2 - (k_{fiss, 1} + k_{fuse, 2}X_1) X_2 + k_{fiss, 2}X_3 \\
\frac{d}{dt} X_3 &= k_{fuse, 2}X_1X_2- k_{fiss, 2}X_3
\end{aligned} d t d [ G 3 P ] d t d [ P yr ] d t d [ N A DH ] c d t d [ N A DH ] m d t d Δ Ψ m d t d [ C a 2 + ] m d t d [ A TP ] c d t d [ A D P ] c d t d X 2 d t d X 3 = V i 1 ( 2 J g l u − J GP D ) − k g 3 p [ G 3 P ] = V i + V m t x 1 ( J GP D − J L DH − J P DH ) − k p yr [ P yr ] = V i 1 ( J GP D − J L DH − J N A DH T ) − k na d h c [ N A DH ] c = V m t x 1 ( J N A DH T + 4.6 J P DH − 0.1 J h r ) − k na d hm [ N A DH ] m = C mi t o 1 ( J h r − J h f − J A NT − J h l − 2 J u ni ) = V m t x f m ( J u ni − J NC L X ) = V i 1 ( J A NT − 2 J Gl u + 2 J GP D + J A d K ) − ( k A TP + k C a A TP [ C a 2 + ] c ) [ A TP ] c = − d t d [ A TP ] c − V i J A d K = k f u se , 1 X 1 2 − ( k f i ss , 1 + k f u se , 2 X 1 ) X 2 + k f i ss , 2 X 3 = k f u se , 2 X 1 X 2 − k f i ss , 2 X 3 Initial conditions ¶ State variable Value Description [ G 3 P ] [G3P] [ G 3 P ] 2.8μM Glyceraldehyde-3-phosphate [ P y r ] [Pyr] [ P yr ] 8.5μM Pyruvate [ N A D H ] c [NADH]_c [ N A DH ] c 1μM Cytosolic NADH [ N A D H ] m [NADH]_{m} [ N A DH ] m 60μM Mitochondrial NADH [ A T P ] c [ATP]_c [ A TP ] c 4mM Cytosolic ATP concentration [ A D P ] c [ADP]_c [ A D P ] c 0.5mM Cytosolic ADP concentration [ C a 2 + ] m [Ca^{2+}]_{m} [ C a 2 + ] m 0.250μM Mitochondrial calcium concentration Δ Ψ m \Delta\Psi_{m} Δ Ψ m 100mV Mitochondrial membrane potential X 2 X_2 X 2 0.20 Population of degree-2 mitochondrial nodes X 3 X_3 X 3 0.05 Population of degree-3 mitochondrial nodes