# Model Mathematics

### Component: NAn

$ddtime NAn = Vn_leak_Na + Vn_stim - 3 ⁢ Vn_pump$

### Component: GLCn

$ddtime GLCn = V_en_GLC - Vn_hk$

### Component: G6Pn

$ddtime G6Pn = Vn_hk - Vn_pgi$

### Component: F6Pn

$ddtime F6Pn = Vn_pgi - Vn_pfk$

### Component: GAPn

$ddtime GAPn = 2 ⁢ Vn_pfk - Vn_pgk$

### Component: PEPn

$ddtime PEPn = Vn_pgk - Vn_pk$

### Component: PYRn

$ddtime PYRn = Vn_pk - Vn_ldh + Vn_mito$

### Component: LACn

$ddtime LACn = Vn_ldh - Vne_LAC$

$ddtime NADHn = Vn_pgk - Vn_ldh + Vn_mito$

### Component: ATPn

$ddtime ATPn = Vn_pgk + Vn_pk + nOP ⁢ Vn_mito + Vn_ck - Vn_hk + Vn_pfk + Vn_ATPase + Vn_pump ⁢ 1 - dAMP_dATPn -1$

### Component: PCrn

$ddtime PCrn =- Vn_ck$

### Component: O2n

$ddtime O2n = Vcn_O2 - NAero ⁢ Vn_mito$

### Component: GLUn

$ddtime GLUn = Vg_gs ⁢ 1 Rng - Vn_stim_GLU$

### Component: NAg

$ddtime NAg = Vg_leak_Na + 3 ⁢ Veg_GLU - 3 ⁢ Vg_pump$

### Component: GLCg

$ddtime GLCg = Vcg_GLC + Veg_GLC - Vg_hk$

### Component: G6Pg

$ddtime G6Pg = Vg_hk + Vg_glyp - Vg_pgi + Vg_glys$

### Component: F6Pg

$ddtime F6Pg = Vg_pgi - Vg_pfk$

### Component: GAPg

$ddtime GAPg = 2 ⁢ Vg_pfk - Vg_pgk$

### Component: PEPg

$ddtime PEPg = Vg_pgk - Vg_pk$

### Component: PYRg

$ddtime PYRg = Vg_pk - Vg_ldh + Vg_mito$

### Component: LACg

$ddtime LACg = Vg_ldh - Vge_LAC + Vgc_LAC$

$ddtime NADHg = Vg_pgk - Vg_ldh + Vg_mito$

### Component: ATPg

$ddtime ATPg = Vg_pgk + Vg_pk + nOP ⁢ Vg_mito + Vg_ck - Vg_hk + Vg_pfk + Vg_ATPase + Vg_pump + Vg_gs ⁢ 1 - dAMP_dATPg -1$

### Component: PCrg

$ddtime PCrg =- Vg_ck$

### Component: O2g

$ddtime O2g = Vcg_O2 - NAero ⁢ Vg_mito$

### Component: GLYg

$ddtime GLYg = Vg_glys - Vg_glyp$

### Component: GLUg

$ddtime GLUg = Veg_GLU - Vg_gs$

### Component: GLCe

$ddtime GLCe = Vce_GLC - Veg_GLC ⁢ 1 Reg + V_en_GLC ⁢ 1 Ren$

### Component: LACe

$ddtime LACe = Vne_LAC ⁢ 1 Ren + Vge_LAC ⁢ 1 Reg - Vec_LAC$

### Component: GLUe

$ddtime GLUe = Vn_stim_GLU ⁢ 1 Ren - Veg_GLU ⁢ 1 Reg$

### Component: O2c

$ddtime O2c = Vc_O2 - Vcn_O2 ⁢ 1 Rcn + Vcg_O2 ⁢ 1 Rcg$

### Component: GLCc

$ddtime GLCc = Vc_GLC - Vce_GLC ⁢ 1 Rce + Vcg_GLC ⁢ 1 Rcg$

### Component: LACc

$ddtimeLACc=Vc_LAC+Vec_LAC⁢1Rce+Vgc_LAC⁢1Rcg$

### Component: CO2c

$ddtime CO2c = Vnc_CO2 ⁢ 1 Rcn + Vgc_CO2 ⁢ 1 Rcg - Vc_CO2$

### Component: Vv

$ddtime Vv = Fin_t - Fout_t$

### Component: dHb

$ddtime dHb = Fin_t ⁢ O2a - O2c - Fout_t ⁢ dHb Vv$

### Component: Vn_leak_Na

$Vn_leak_Na=Sm_nVn⁢gn_NAF⁢RTF⁢ln⁡NAeNAn-Vm$

### Component: Vn_pump

$Vn_pump = Sm_n Vn ⁢ kpump ⁢ ATPn ⁢ NAn ⁢ 1 + ATPn Km_pump -1$

### Component: Vn_stim

$Vn_stim = v_stim$

### Component: V_en_GLC

$V_en_GLC = Vm_en_GLC ⁢ GLCe GLCe + Km_en_GLC - GLCn GLCn + Km_en_GLC$

### Component: Vn_hk

$Vn_hk = Vmax_n_hk ⁢ ATPn ⁢ GLCn GLCn + Km_GLC ⁢ 1 - 1 1 +ⅇ- aG6P_inh_hk ⁢ 1 ⁢ G6Pn - G6P_inh_hk$

### Component: Vn_pgi

$Vn_pgi = Vmaxf_n_pgi ⁢ G6Pn G6Pn + Km_G6P - Vmaxr_n_pgi ⁢ F6Pn F6Pn + Km_F6P_pgi$

### Component: Vn_pfk

$Vn_pfk = kn_pfk ⁢ ATPn ⁢ F6Pn F6Pn + Km_F6P_pfk ⁢ 1 + ATPn Ki_ATP nH -1$

### Component: Vn_pgk

$Vn_pgk = kn_pgk ⁢ GAPn ⁢ ADPn ⁢ NADn NADHn$

### Component: Vn_pk

$Vn_pk = kn_pk ⁢ PEPn ⁢ ADPn$

### Component: Vn_ldh

$Vn_ldh = kfn_ldh ⁢ PYRn ⁢ NADHn - krn_ldh ⁢ LACn ⁢ NADn$

### Component: Vn_mito

$Vn_mito=Vmax_n_mito⁢O2nO2n+Km_O2⁢ADPnADPn+Km_ADP⁢PYRnPYRn+Km_PYR⁢1-11+ⅇ-aATP_mito⁢1⁢ATPnADPn-1⁢rATP_mito$

### Component: Vne_LAC

$Vne_LAC=Vmax_ne_LAC⁢LACnLACn+Km_ne_LAC-LACeLACe+Km_ne_LAC$

### Component: Vn_ATPase

$Vn_ATPase = Vmax_n_ATPase ⁢ ATPn ATPn + 0.001$

### Component: Vn_ck

$Vn_ck = kfn_ck ⁢ PCrn ⁢ ADPn - krn_ck ⁢ CRn ⁢ ATPn$

### Component: Vcn_O2

$Vcn_O2 = PScapn Vn ⁢ Ko2 ⁢ HbOP O2c - 1.0 -1.0 nh_O2 - O2n$

### Component: Vg_leak_Na

$Vg_leak_Na=Sm_gVg⁢gg_NAF⁢RTF⁢ln⁡NAeNAg-Vm$

### Component: Vg_pump

$Vg_pump=Sm_gVg⁢kpump⁢ATPg⁢NAg⁢1+ATPgKm_pump-1$

### Component: Veg_GLC

$Veg_GLC = KO1 ⁢ Vm_eg_GLC ⁢ GLCe GLCe + Km_eg_GLC - GLCg GLCg + Km_eg_GLC$

### Component: Vcg_GLC

$Vcg_GLC = Vm_cg_GLC ⁢ GLCc GLCc + Km_cg_GLC - GLCg GLCg + Km_cg_GLC$

### Component: Vg_hk

$Vg_hk = Vmax_g_hk ⁢ ATPg ⁢ GLCg GLCg + Km_GLC ⁢ 1 - 1 1 +ⅇ- aG6P_inh_hk ⁢ 1 ⁢ G6Pg - G6P_inh_hk$

### Component: Vg_pgi

$Vg_pgi = Vmaxf_g_pgi ⁢ G6Pg G6Pg + Km_G6P - Vmaxr_g_pgi ⁢ F6Pg F6Pg + Km_F6P_pgi$

### Component: Vg_pfk

$Vg_pfk = kg_pfk ⁢ ATPg ⁢ F6Pg F6Pg + Km_F6P_pfk ⁢ 1 + ATPg Ki_ATP nH -1$

### Component: Vg_pgk

$Vg_pgk = kg_pgk ⁢ GAPg ⁢ ADPg ⁢ NADg NADHg$

### Component: Vg_pk

$Vg_pk = kg_pk ⁢ PEPg ⁢ ADPg$

### Component: Vg_ldh

$Vg_ldh = kfg_ldh ⁢ PYRg ⁢ NADHg - krg_ldh ⁢ LACg ⁢ NADg$

### Component: Vg_mito

$Vg_mito=Vmax_g_mito⁢O2gO2g+Km_O2⁢ADPgADPg+Km_ADP⁢PYRgPYRg+Km_PYR⁢1-11+ⅇ1⁢-aATP_mito⁢ATPgADPg-1⁢rATP_mito$

### Component: Vge_LAC

$Vge_LAC=Vmax_ge_LAC⁢LACgLACg+Km_ge_LAC-LACeLACe+Km_ge_LAC$

### Component: Vgc_LAC

$Vgc_LAC=Vmax_gc_LAC⁢LACgLACg+Km_gc_LAC-LACcLACc+Km_gc_LAC$

### Component: Vg_ATPase

$Vg_ATPase = Vmax_g_ATPase ⁢ ATPg ATPg + 0.001$

### Component: Vg_ck

$Vg_ck = kfg_ck ⁢ PCrg ⁢ ADPg - krg_ck ⁢ CRg ⁢ ATPg$

### Component: Vcg_O2

$Vcg_O2 = PScapg Vg ⁢ Ko2 ⁢ HbOP O2c - 1.0 -1.0 nh_O2 - O2g$

### Component: Vc_O2

$Vc_O2 = 2.0 ⁢ Fin_t Vc ⁢ O2a - O2c$

### Component: Vc_GLC

$Vc_GLC = 2.0 ⁢ Fin_t Vc ⁢ GLCa - GLCc$

### Component: Vce_GLC

$Vce_GLC = Vm_ce_GLC ⁢ GLCc GLCc + Km_ce_GLC - GLCe GLCe + Km_ce_GLC$

### Component: Vc_LAC

$Vc_LAC = 2.0 ⁢ Fin_t Vc ⁢ LACa - LACc$

### Component: Vec_LAC

$Vec_LAC = Vm_ec_LAC ⁢ LACe LACe + Km_ec_LAC - LACc LACc + Km_ec_LAC$

### Component: Vnc_CO2

$Vnc_CO2 = 3.0 ⁢ Vn_mito$

### Component: Vgc_CO2

$Vgc_CO2 = 3.0 ⁢ Vg_mito$

### Component: Vn_stim_GLU

$Vn_stim_GLU = Vn_stim ⁢ R_GLU_NA ⁢ KO2 ⁢ GLUn GLUn + Km_GLU$

### Component: Vg_gs

$Vg_gs=Vmax_g_gs⁢GLUgGLUg+Km_GLU⁢ATPgATPg+Km_ATP$

### Component: Veg_GLU

$Veg_GLU = Vmax_eg_GLU ⁢ GLUe GLUe + Km_GLU$

### Component: Vc_CO2

$Vc_CO2 = 2.0 ⁢ Fin_t Vc ⁢ CO2c - CO2a$

### Component: Vg_glys

$Vg_glys = Vmax_glys ⁢ G6Pg G6Pg + Km_G6P_glys ⁢ 1 - 1 1 +ⅇ- aGLY_inh ⁢ 1 ⁢ GLYg - GLY_inh$

### Component: Vg_glyp

$Vg_glyp = Vmax_glyp ⁢ GLYg GLYg + Km_GLY ⁢ deltaVt_GLY$$deltaVt_GLY=1+stim⁢delta_GLY⁢KO3⁢11+ⅇ1⁢-sr_GLY⁢time-to+to_GLY⁢1-unitstepSB2$

### Component: Fin_t

$Fin_t = CBF0 + stim ⁢ CBF0 ⁢ deltaf ⁢ 1 1 +ⅇ 1 ⁢- sr ⁢ time - to + t1 - 3 - stim ⁢ CBF0 ⁢ deltaf ⁢ 1 1 +ⅇ 1 ⁢- sr ⁢ time - to + tend + t1 + 3$

### Component: Fout_t

$Fout_t=CBF0⁢VvVv02+tv⁢VvVv0-0.5⁢Fin_tVv01+CBF0⁢tv⁢VvVv0-0.5⁢1Vv0$

$NADn = NADH_n_tot - NADHn$

$NADg = NADH_g_tot - NADHg$

### Component: CRn

$CRn = PCrn_tot - PCrn$

### Component: CRg

$CRg = PCrg_tot - PCrg$

$ADPn=ATPn2⁢-qak+qak2+4⁢qak⁢ATPtotATPn-1$

$ADPg=ATPg2⁢-qak+qak2+4⁢qak⁢ATPtotATPg-1$

### Component: u_n

$u_n = qak 2 + 4 ⁢ qak ⁢ ATPtot ATPn - 1$

### Component: u_g

$u_g = qak 2 + 4 ⁢ qak ⁢ ATPtot ATPg - 1$

### Component: dAMP_dATPn

$dAMP_dATPn = qak 2 + qak ⁢ ATPtot ATPn ⁢ u_n - 1 + 0.5 ⁢ u_n$

### Component: dAMP_dATPg

$dAMP_dATPg = qak 2 + qak ⁢ ATPtot ATPg ⁢ u_g - 1 + 0.5 ⁢ u_g$

### Component: AMPn

$AMPn=ATPtot-ATPn+ADPn$

### Component: AMPg

$AMPg=ATPtot-ATPg+ADPg$

### Component: BOLD

$BOLD = Vv0 ⁢ k1 + k2 ⁢ 1 - dHb dHb0 - k2 + k3 ⁢ 1 - Vv Vv0$

### Component: v_stim

$v_stim=stim⁢v1_n+v2_n⁢time-tot_n_stim⁢ⅇ-time-to⁢unitpulseSBt_n_stim⁢unitpulseSB$$unitpulseSB = 1 if time ≥ to ∧ time ≦ to + tend 0 otherwise$

### Component: unitstepSB

$unitstepSB = 1 if time - tend + to ≥ 0 0 otherwise$

### Component: unitstepSB2

$unitstepSB2=1iftime-tend_GLY+to+to_GLY≥00otherwise$
Source
Derived from workspace Cloutier 2009 at changeset c47e7ddfa3a4.
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