/* * A Biophysical Model of the Mitochondrial Respiratory System * and Oxidative Phosphorylation, Beard, 2005 * * Model Status * * This model has been edited to correct dimensional errors and * checked for other problems. It runs in COR and OpenCell to replicate * results from the original matlab code. It was based on the original * matlab code provided by the model authors. * * Model Structure * * ABSTRACT: A computational model for the mitochondrial respiratory * chain that appropriately balances mass, charge, and free energy * transduction is introduced and analyzed based on a previously * published set of data measured on isolated cardiac mitochondria. * The basic components included in the model are the reactions * at complexes I, III, and IV of the electron transport system, * ATP synthesis at F1F0 ATPase, substrate transporters including * adenine nucleotide translocase and the phosphate\u2013hydrogen * co-transporter, and cation fluxes across the inner membrane * including fluxes through the K+/H+ antiporter and passive H+ * and K+ permeation. Estimation of 16 adjustable parameter values * is based on fitting model simulations to nine independent data * curves. The identified model is further validated by comparison * to additional datasets measured from mitochondria isolated from * rat heart and liver and observed at low oxygen concentration. * To obtain reasonable fits to the available data, it is necessary * to incorporate inorganic-phosphate-dependent activation of the * dehydrogenase activity and the electron transport system. Specifically, * it is shown that a model incorporating phosphate-dependent activation * of complex III is able to reasonably reproduce the observed * data. The resulting validated and verified model provides a * foundation for building larger and more complex systems models * and investigating complex physiological and pathophysiological * interactions in cardiac energetics. * * The original paper reference is cited below: * * A Biophysical Model of the Mitochondrial Respiratory System * and Oxidative Phosphorylation, Daniel A. Beard, 2005, PLoS Computational * Biology , September 9, 1(4). PubMed ID: 16163394 * * Illustration of the Components Included in the Model of Mitochondrial * Oxidative Phosphorylation: * * model diagram * * [[Image file: beard_2005a.png]] * * A) Schematic diagram of the major components of the electron * transport system, which transfers reducing potential from NADH * to oxygen, and the F1F0 ATPase, which transduces energy from * proton motive force to ATP, are illustrated. Complexes I, III, * and IV are labeled C1, C3, and C4, respectively. * * model diagram * * [[Image file: beard_2005b.png]] * * B) Schematic diagram of the substrate transport process included * in the model, including the ANT and PiHt on the inner membrane, * and passive permeation of ATP, ADP, AMP, and phosphate across * the outer membrane. The AK reaction in the IM space is shown. * * model diagram * * [[Image file: beard_2005c.png]] * * C) Schematic diagram of the transporters for hydrogen and potassium * ions on the inner membrane, including K+/H+ antiporter and passive * proton and potassium fluxes. It is assumed that these cations * rapidly equilibrate across the outer membrane. */ import nsrunit; unit conversion on; unit cell = fundamental; unit cytoplasm = fundamental; unit mitochondria = fundamental; unit cyto_per_cell=1 cell^(-1)*cytoplasm^1; unit mito_per_cell=1 cell^(-1)*mitochondria^1; unit l_cyto_per_l_mito=1 cytoplasm^1*mitochondria^(-1); unit l_water_per_l_mito=1 mitochondria^(-1); unit mg_per_mitochondria=1E-6 kilogram^1*mitochondria^(-1); // unit molar predefined unit molar_per_second=1E3 meter^(-3)*second^(-1)*mole^1; unit molar_squared=1E6 meter^(-6)*mole^2; unit molar_cubed=1E9 meter^(-9)*mole^3; // unit micromolar predefined unit per_molar=.001 meter^3*mole^(-1); unit micron_per_second=1E-6 meter^1*second^(-1); unit per_micron=1E6 meter^(-1)*mitochondria^(-1); unit kilojoule_per_mole=1E3 kilogram^1*meter^2*second^(-2)*mole^(-1); // unit millivolt predefined unit kilojoule_per_mole_per_millivolt=1E6 second^1*ampere^1*mole^(-1); unit molar_per_cyto=1E3 meter^(-3)*mole^1*mitochondria^(-1); unit molar_per_mito=1E3 meter^(-3)*mole^1*mitochondria^(-1); unit mole_per_second_per_l_cyto=1E3 meter^(-3)*second^(-1)*mole^1*mitochondria^(-1); unit mole_per_second_per_l_mito=1E3 meter^(-3)*second^(-1)*mole^1*mitochondria^(-1); unit mole_per_second_per_l_mito_per_molar=1 second^(-1)*mitochondria^(-1); unit mole_per_second_per_l_mito_per_molar_per_half_molar=.03162278 meter^1.5*second^(-1)*mole^(-0.5)*mitochondria^(-1); unit mole_per_second_per_l_mito_per_molar_per_molar=.001 meter^3*second^(-1)*mole^(-1)*mitochondria^(-1); unit mole_per_l_mito_per_millivolt=1E6 kilogram^(-1)*meter^(-5)*second^3*ampere^1*mole^1*mitochondria^(-1); unit mole_per_second_per_l_mito_per_molar_per_millivolt=1E3 kilogram^(-1)*meter^(-2)*second^2*ampere^1*mitochondria^(-1); math main { realDomain t second; t.min=0; extern t.max; extern t.delta; real Mg_tot molar; Mg_tot=0.005; real Pi_e molar; Pi_e=0.000125; real ADP_e molar; ADP_e=0; real RT kilojoule_per_mole; RT=2.4734; real F kilojoule_per_mole_per_millivolt; F=0.096484; real n_A dimensionless; n_A=3; real dG_C1o kilojoule_per_mole; dG_C1o=-69.37; real dG_C3o kilojoule_per_mole; dG_C3o=-32.53; real dG_C4o kilojoule_per_mole; dG_C4o=-122.94; real dG_F1o kilojoule_per_mole; dG_F1o=36.03; real pH_e dimensionless; pH_e=7.1; real H_e molar; real K_e molar; K_e=0.15; real ATP_e molar; ATP_e=0; real AMP_e molar; AMP_e=0; real k_dHPi molar; real k_dHatp molar; real k_dHadp molar; real K_DT molar; K_DT=2.4e-5; real K_DD molar; K_DD=3.47e-4; real K_AK dimensionless; K_AK=0.4331; real W_m l_water_per_l_mito; W_m=0.72376; real W_x l_water_per_l_mito; real W_i l_water_per_l_mito; real gamma per_micron; gamma=5.99; real Ctot molar; Ctot=0.0027; real Qtot molar; Qtot=0.00135; real NADtot molar; NADtot=0.00297; real H_i molar; real K_i molar; real k_Pi1 molar; k_Pi1=1.3413e-4; real k_Pi2 molar; k_Pi2=6.7668e-4; real k_Pi3 molar; k_Pi3=1.9172e-4; real k_Pi4 molar; k_Pi4=0.02531; real k_PiH molar; k_PiH=4.5082e-4; real r dimensionless; r=4.5807; real x_DH mole_per_second_per_l_mito_per_molar; x_DH=0.09183; real x_C1 mole_per_second_per_l_mito_per_molar; x_C1=0.36923; real x_C3 mole_per_second_per_l_mito_per_molar; x_C3=0.091737; real x_C4 mole_per_second_per_l_mito_per_molar; x_C4=3.2562e-5; real x_F1 mole_per_second_per_l_mito_per_molar_per_molar; x_F1=150.93; real x_ANT mole_per_second_per_l_mito; x_ANT=0.0079204; real x_Pi1 mole_per_second_per_l_mito_per_molar; x_Pi1=339430; real x_KH mole_per_second_per_l_mito_per_molar_per_molar; x_KH=2.9802e7; real x_Hle mole_per_second_per_l_mito_per_molar_per_millivolt; x_Hle=250; real x_K mole_per_second_per_l_mito_per_molar_per_millivolt; x_K=0; real k_mADP molar; k_mADP=3.5e-6; real x_AK mole_per_second_per_l_mito_per_molar_per_molar; x_AK=0; real p_A micron_per_second; p_A=85; real k_O2 molar; k_O2=1.2e-4; real x_buff per_molar; x_buff=100; real x_MgA mole_per_second_per_l_mito_per_molar_per_molar; x_MgA=1000000; real x_Pi2 micron_per_second; x_Pi2=327; real dG_H(t) kilojoule_per_mole; real dPsi(t) millivolt; when(t=t.min) dPsi=160; real H_x(t) molar; when(t=t.min) H_x=6.30957344480193e-8; real J_DH(t) mole_per_second_per_l_mito; real NAD_x(t) molar; real NADH_x(t) molar; when(t=t.min) NADH_x=0.0015; real Pi_x(t) molar; when(t=t.min) Pi_x=0.001; real J_C1(t) mole_per_second_per_l_mito; real dG_C1op(t) kilojoule_per_mole; real Q(t) molar; real QH2(t) molar; when(t=t.min) QH2=8e-4; real J_C3(t) mole_per_second_per_l_mito; real dG_C3op(t) kilojoule_per_mole; real Cox(t) molar; real Cred(t) molar; when(t=t.min) Cred=0.001; real J_C4(t) mole_per_second_per_l_mito; real dG_C4op(t) kilojoule_per_mole; real O2(t) molar; when(t=t.min) O2=2.6e-5; real J_F1(t) mole_per_second_per_l_mito; real ADP_mx(t) molar; when(t=t.min) ADP_mx=0; real ATP_mx(t) molar; when(t=t.min) ATP_mx=0; real J_ANT(t) mole_per_second_per_l_mito; real Psi_x(t) millivolt; real Psi_i(t) millivolt; real ADP_fi(t) molar; real ATP_fi(t) molar; real ADP_fx(t) molar; real ATP_fx(t) molar; real mincond molar; mincond=1e-12; real J_MgATPx(t) mole_per_second_per_l_mito; real ATP_x(t) molar; when(t=t.min) ATP_x=0; real Mg_x(t) molar; when(t=t.min) Mg_x=0.005; real J_MgADPx(t) mole_per_second_per_l_mito; real ADP_x(t) molar; when(t=t.min) ADP_x=0.01; real J_MgATPi(t) mole_per_second_per_l_mito; real ATP_i(t) molar; when(t=t.min) ATP_i=0; real ATP_mi(t) molar; when(t=t.min) ATP_mi=0; real Mg_i molar; real J_MgADPi(t) mole_per_second_per_l_mito; real ADP_i(t) molar; when(t=t.min) ADP_i=0; real ADP_mi(t) molar; when(t=t.min) ADP_mi=0; real J_ATP(t) mole_per_second_per_l_mito; real J_ADP(t) mole_per_second_per_l_mito; real J_AMP(t) mole_per_second_per_l_mito; real AMP_i(t) molar; when(t=t.min) AMP_i=0; real J_Pi2(t) mole_per_second_per_l_mito; real Pi_i(t) molar; when(t=t.min) Pi_i=0.001; real J_Pi1(t) mole_per_second_per_l_mito; real H2PIi(t) molar; real H2PIx(t) molar; real J_AKi(t) mole_per_second_per_l_mito; real J_Hle(t) mole_per_second_per_l_mito; real J_K(t) mole_per_second_per_l_mito; real K_x(t) molar; when(t=t.min) K_x=0.14; real J_KH(t) mole_per_second_per_l_mito; real ADP_me molar; real ADP_fe molar; real Mg_e molar; real C_im mole_per_l_mito_per_millivolt; C_im=6.756756756756757e-6; // // H_e=((1 molar)*10^((-1)*pH_e)); W_x=(.9*W_m); W_i=(.1*W_m); H_i=H_e; K_i=K_e; k_dHPi=((1 molar)*10^((-1)*6.75)); k_dHatp=((1 molar)*10^((-1)*6.48)); k_dHadp=((1 molar)*10^((-1)*6.29)); // // dG_H=(F*dPsi+RT*ln(H_i/H_x)); // J_DH=(x_DH*(r*NAD_x-NADH_x)*(1+Pi_x/k_Pi1)/(1+Pi_x/k_Pi2)); // dG_C1op=(dG_C1o-RT*ln(H_x/(1E-7 molar))-RT*ln(Q/QH2)); J_C1=(x_C1*(exp((-1)*(dG_C1op+4*dG_H)/RT)*NADH_x-NAD_x)); // dG_C3op=(dG_C3o+2*RT*ln(H_x/(1E-7 molar))-RT*ln(QH2/Q)); J_C3=(x_C3*(1+Pi_x/k_Pi3)/(1+Pi_x/k_Pi4)*(exp((-1)*(dG_C3op+4*dG_H-2*F*dPsi)/(2*RT))*Cox-Cred)); // dG_C4op=(dG_C4o-2*RT*ln(H_x/(1E-7 molar))-RT/2*ln(O2/(1 molar))); J_C4=(x_C4*1/(1+k_O2/O2)*Cred/Ctot*(exp((-1)*(dG_C4op+2*dG_H)/(2*RT))*Cred-Cox*exp(F*dPsi/RT))); // J_F1=(x_F1*(exp((-1)*(dG_F1o-n_A*dG_H)/RT)*K_DD/K_DT*ADP_mx*Pi_x-ATP_mx*(1 molar))); // Psi_x=((-1)*.65*dPsi); Psi_i=(.35*dPsi); J_ANT=(if ((ADP_fi>mincond) or (ATP_fi>mincond)) x_ANT*(ADP_fi/(ADP_fi+ATP_fi*exp((-1)*F*Psi_i/RT))-ADP_fx/(ADP_fx+ATP_fx*exp((-1)*F*Psi_x/RT)))*ADP_fi/(k_mADP+ADP_fi) else (0 mole_per_second_per_l_mito)); // ATP_fx=(ATP_x-ATP_mx); J_MgATPx=(x_MgA*(ATP_fx*Mg_x-K_DT*ATP_mx)); // ADP_fx=(ADP_x-ADP_mx); J_MgADPx=(x_MgA*(ADP_fx*Mg_x-K_DD*ADP_mx)); // ATP_fi=(ATP_i-ATP_mi); J_MgATPi=(x_MgA*(ATP_fi*Mg_i-K_DT*ATP_mi)); // ADP_fi=(ADP_i-ADP_mi); J_MgADPi=(x_MgA*(ADP_fi*Mg_i-K_DD*ADP_mi)); // J_ATP=(gamma*p_A*(ATP_e-ATP_i)); // J_ADP=(gamma*p_A*(ADP_e-ADP_i)); // J_AMP=(gamma*p_A*(AMP_e-AMP_i)); // J_Pi2=(gamma*x_Pi2*(Pi_e-Pi_i)); // H2PIi=(Pi_i*H_i/(H_i+k_dHPi)); H2PIx=(Pi_x*H_x/(H_x+k_dHPi)); J_Pi1=(x_Pi1*(H_x*H2PIi-H_i*H2PIx)/(H2PIi+k_PiH)); // J_AKi=(x_AK*(K_AK*ADP_i*ADP_i-AMP_i*ATP_i)); // J_Hle=(x_Hle*dPsi*(H_i*exp(F*dPsi/RT)-H_x)/(exp(F*dPsi/RT)-1)); // J_K=(x_K*dPsi*(K_i*exp(F*dPsi/RT)-K_x)/(exp(F*dPsi/RT)-1)); // J_KH=(x_KH*(K_i*H_x-K_x*H_i)); // NAD_x=(NADtot-NADH_x); // Q=(Qtot-QH2); // Cox=(Ctot-Cred); // ADP_fe=(ADP_e-ADP_me); ADP_me=((K_DD+ADP_e+Mg_tot-sqrt((K_DD+ADP_e+Mg_tot)^2-4*Mg_tot*ADP_e))/2); // Mg_e=(Mg_tot-ADP_me); Mg_i=Mg_e; // H_x:t=(x_buff*H_x*(J_DH-5*J_C1-2*J_C3-4*J_C4+(n_A-1)*J_F1+2*J_Pi1+J_Hle-J_KH)/W_x); // K_x:t=((J_KH+J_K)/W_x); // Mg_x:t=(((-1)*J_MgATPx-J_MgADPx)/W_x); // NADH_x:t=((J_DH-J_C1)/W_x); // QH2:t=((J_C1-J_C3)/W_x); // Cred:t=((2*J_C3-2*J_C4)/W_i); // ATP_x:t=((J_F1-J_ANT)/W_x); // ADP_x:t=(((-1)*J_F1+J_ANT)/W_x); // ATP_mx:t=(J_MgATPx/W_x); // ADP_mx:t=(J_MgADPx/W_x); // Pi_x:t=(((-1)*J_F1+J_Pi1)/W_x); // ATP_i:t=((J_ATP+J_ANT+J_AKi)/W_i); // ADP_i:t=((J_ADP-J_ANT-2*J_AKi)/W_i); // AMP_i:t=((J_AMP+J_AKi)/W_i); // ATP_mi:t=(J_MgATPi/W_i); // ADP_mi:t=(J_MgADPi/W_i); // Pi_i:t=(((-1)*J_Pi1+J_Pi2)/W_i); // dPsi:t=((4*J_C1+2*J_C3+4*J_C4-n_A*J_F1-J_ANT-J_Hle-J_K)/C_im); // O2:t=(0 molar_per_second); }