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Model Mathematics

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Component: environment

Component: NAn

\frac{d}{d time} (NAn) = Vn_leak_Na + Vn_stim - (3 Vn_pump)

Component: GLCn

\frac{d}{d time} (GLCn) = V_en_GLC - Vn_hk

Component: G6Pn

\frac{d}{d time} (G6Pn) = Vn_hk - Vn_pgi

Component: F6Pn

\frac{d}{d time} (F6Pn) = Vn_pgi - Vn_pfk

Component: GAPn

\frac{d}{d time} (GAPn) = 2 Vn_pfk - Vn_pgk

Component: PEPn

\frac{d}{d time} (PEPn) = Vn_pgk - Vn_pk

Component: PYRn

\frac{d}{d time} (PYRn) = Vn_pk - (Vn_ldh + Vn_mito)

Component: LACn

\frac{d}{d time} (LACn) = Vn_ldh - Vne_LAC

Component: NADHn

\frac{d}{d time} (NADHn) = Vn_pgk - (Vn_ldh + Vn_mito)

Component: ATPn

\frac{d}{d time} (ATPn) = (Vn_pgk + Vn_pk + nOP Vn_mito + Vn_ck - (Vn_hk + Vn_pfk + Vn_ATPase + Vn_pump)) {(1 - dAMP_dATPn)}^{-1}

Component: PCrn

\frac{d}{d time} (PCrn) = - Vn_ck

Component: O2n

\frac{d}{d time} (O2n) = Vcn_O2 - (NAero Vn_mito)

Component: GLUn

\frac{d}{d time} (GLUn) = Vg_gs \frac{1}{Rng} - Vn_stim_GLU

Component: NAg

\frac{d}{d time} (NAg) = Vg_leak_Na + 3 Veg_GLU - (3 Vg_pump)

Component: GLCg

\frac{d}{d time} (GLCg) = Vcg_GLC + Veg_GLC - Vg_hk

Component: G6Pg

\frac{d}{d time} (G6Pg) = Vg_hk + Vg_glyp - (Vg_pgi + Vg_glys)

Component: F6Pg

\frac{d}{d time} (F6Pg) = Vg_pgi - Vg_pfk

Component: GAPg

\frac{d}{d time} (GAPg) = 2 Vg_pfk - Vg_pgk

Component: PEPg

\frac{d}{d time} (PEPg) = Vg_pgk - Vg_pk

Component: PYRg

\frac{d}{d time} (PYRg) = Vg_pk - (Vg_ldh + Vg_mito)

Component: LACg

\frac{d}{d time} (LACg) = Vg_ldh - (Vge_LAC + Vgc_LAC)

Component: NADHg

\frac{d}{d time} (NADHg) = Vg_pgk - (Vg_ldh + Vg_mito)

Component: ATPg

\frac{d}{d time} (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

\frac{d}{d time} (PCrg) = - Vg_ck

Component: O2g

\frac{d}{d time} (O2g) = Vcg_O2 - (NAero Vg_mito)

Component: GLYg

\frac{d}{d time} (GLYg) = Vg_glys - Vg_glyp

Component: GLUg

\frac{d}{d time} (GLUg) = Veg_GLU - Vg_gs

Component: GLCe

\frac{d}{d time} (GLCe) = Vce_GLC - (Veg_GLC \frac{1}{Reg} + V_en_GLC \frac{1}{Ren})

Component: LACe

\frac{d}{d time} (LACe) = Vne_LAC \frac{1}{Ren} + Vge_LAC \frac{1}{Reg} - Vec_LAC

Component: GLUe

\frac{d}{d time} (GLUe) = Vn_stim_GLU \frac{1}{Ren} - (Veg_GLU \frac{1}{Reg})

Component: O2c

\frac{d}{d time} (O2c) = Vc_O2 - (Vcn_O2 \frac{1}{Rcn} + Vcg_O2 \frac{1}{Rcg})

Component: GLCc

\frac{d}{d time} (GLCc) = Vc_GLC - (Vce_GLC \frac{1}{Rce} + Vcg_GLC \frac{1}{Rcg})

Component: LACc

\frac{d}{d time} (LACc) = Vc_LAC + Vec_LAC \frac{1}{Rce} + Vgc_LAC \frac{1}{Rcg}

Component: CO2c

\frac{d}{d time} (CO2c) = Vnc_CO2 \frac{1}{Rcn} + Vgc_CO2 \frac{1}{Rcg} - Vc_CO2

Component: Vv

\frac{d}{d time} (Vv) = Fin_t - Fout_t

Component: dHb

\frac{d}{d time} (dHb) = Fin_t (O2a - O2c) - (Fout_t \frac{dHb}{Vv})

Component: Vn_leak_Na

Vn_leak_Na = \frac{Sm_n}{Vn} \frac{gn_NA}{F} (\frac{RT}{F} \ln (\frac{NAe}{NAn}) - Vm)

Component: Vn_pump

Vn_pump = \frac{Sm_n}{Vn} kpump ATPn NAn {(1 + \frac{ATPn}{Km_pump})}^{-1}

Component: Vn_stim

Vn_stim = v_stim

Component: V_en_GLC

V_en_GLC = Vm_en_GLC (\frac{GLCe}{GLCe + Km_en_GLC} - (\frac{GLCn}{GLCn + Km_en_GLC}))

Component: Vn_hk

Vn_hk = Vmax_n_hk ATPn \frac{GLCn}{GLCn + Km_GLC} (1 - (\frac{1}{1 + ⅇ^{(- aG6P_inh_hk) 1 (G6Pn - G6P_inh_hk)}}))

Component: Vn_pgi

Vn_pgi = Vmaxf_n_pgi \frac{G6Pn}{G6Pn + Km_G6P} - (Vmaxr_n_pgi \frac{F6Pn}{F6Pn + Km_F6P_pgi})

Component: Vn_pfk

Vn_pfk = kn_pfk ATPn \frac{F6Pn}{F6Pn + Km_F6P_pfk} {(1 + {(\frac{ATPn}{Ki_ATP})}^{nH})}^{-1}

Component: Vn_pgk

Vn_pgk = kn_pgk GAPn ADPn \frac{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 \frac{O2n}{O2n + Km_O2} \frac{ADPn}{ADPn + Km_ADP} \frac{PYRn}{PYRn + Km_PYR} (1 - (\frac{1}{1 + ⅇ^{(- aATP_mito) 1 (\frac{ATPn}{ADPn} - (1 rATP_mito))}}))

Component: Vne_LAC

Vne_LAC = Vmax_ne_LAC (\frac{LACn}{LACn + Km_ne_LAC} - (\frac{LACe}{LACe + Km_ne_LAC}))

Component: Vn_ATPase

Vn_ATPase = Vmax_n_ATPase \frac{ATPn}{ATPn + 0.001}

Component: Vn_ck

Vn_ck = kfn_ck PCrn ADPn - (krn_ck CRn ATPn)

Component: Vcn_O2

Vcn_O2 = \frac{PScapn}{Vn} (Ko2 {(\frac{HbOP}{O2c} - 1.0)}^{\frac{-1.0}{nh_O2}} - O2n)

Component: Vg_leak_Na

Vg_leak_Na = \frac{Sm_g}{Vg} \frac{gg_NA}{F} (\frac{RT}{F} \ln (\frac{NAe}{NAg}) - Vm)

Component: Vg_pump

Vg_pump = \frac{Sm_g}{Vg} kpump ATPg NAg {(1 + \frac{ATPg}{Km_pump})}^{-1}

Component: Veg_GLC

Veg_GLC = KO1 Vm_eg_GLC (\frac{GLCe}{GLCe + Km_eg_GLC} - (\frac{GLCg}{GLCg + Km_eg_GLC}))

Component: Vcg_GLC

Vcg_GLC = Vm_cg_GLC (\frac{GLCc}{GLCc + Km_cg_GLC} - (\frac{GLCg}{GLCg + Km_cg_GLC}))

Component: Vg_hk

Vg_hk = Vmax_g_hk ATPg \frac{GLCg}{GLCg + Km_GLC} (1 - (\frac{1}{1 + ⅇ^{(- aG6P_inh_hk) 1 (G6Pg - G6P_inh_hk)}}))

Component: Vg_pgi

Vg_pgi = Vmaxf_g_pgi \frac{G6Pg}{G6Pg + Km_G6P} - (Vmaxr_g_pgi \frac{F6Pg}{F6Pg + Km_F6P_pgi})

Component: Vg_pfk

Vg_pfk = kg_pfk ATPg \frac{F6Pg}{F6Pg + Km_F6P_pfk} {(1 + {(\frac{ATPg}{Ki_ATP})}^{nH})}^{-1}

Component: Vg_pgk

Vg_pgk = kg_pgk GAPg ADPg \frac{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 \frac{O2g}{O2g + Km_O2} \frac{ADPg}{ADPg + Km_ADP} \frac{PYRg}{PYRg + Km_PYR} (1 - (\frac{1}{1 + ⅇ^{1 (- aATP_mito) (\frac{ATPg}{ADPg} - (1 rATP_mito))}}))

Component: Vge_LAC

Vge_LAC = Vmax_ge_LAC (\frac{LACg}{LACg + Km_ge_LAC} - (\frac{LACe}{LACe + Km_ge_LAC}))

Component: Vgc_LAC

Vgc_LAC = Vmax_gc_LAC (\frac{LACg}{LACg + Km_gc_LAC} - (\frac{LACc}{LACc + Km_gc_LAC}))

Component: Vg_ATPase

Vg_ATPase = Vmax_g_ATPase \frac{ATPg}{ATPg + 0.001}

Component: Vg_ck

Vg_ck = kfg_ck PCrg ADPg - (krg_ck CRg ATPg)

Component: Vcg_O2

Vcg_O2 = \frac{PScapg}{Vg} (Ko2 {(\frac{HbOP}{O2c} - 1.0)}^{\frac{-1.0}{nh_O2}} - O2g)

Component: Vc_O2

Vc_O2 = 2.0 \frac{Fin_t}{Vc} (O2a - O2c)

Component: Vc_GLC

Vc_GLC = 2.0 \frac{Fin_t}{Vc} (GLCa - GLCc)

Component: Vce_GLC

Vce_GLC = Vm_ce_GLC (\frac{GLCc}{GLCc + Km_ce_GLC} - (\frac{GLCe}{GLCe + Km_ce_GLC}))

Component: Vc_LAC

Vc_LAC = 2.0 \frac{Fin_t}{Vc} (LACa - LACc)

Component: Vec_LAC

Vec_LAC = Vm_ec_LAC (\frac{LACe}{LACe + Km_ec_LAC} - (\frac{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 \frac{GLUn}{GLUn + Km_GLU}

Component: Vg_gs

Vg_gs = Vmax_g_gs \frac{GLUg}{GLUg + Km_GLU} \frac{ATPg}{ATPg + Km_ATP}

Component: Veg_GLU

Veg_GLU = Vmax_eg_GLU \frac{GLUe}{GLUe + Km_GLU}

Component: Vc_CO2

Vc_CO2 = 2.0 \frac{Fin_t}{Vc} (CO2c - CO2a)

Component: Vg_glys

Vg_glys = Vmax_glys \frac{G6Pg}{G6Pg + Km_G6P_glys} (1 - (\frac{1}{1 + ⅇ^{(- aGLY_inh) 1 (GLYg - GLY_inh)}}))

Component: Vg_glyp

Vg_glyp = Vmax_glyp \frac{GLYg}{GLYg + Km_GLY} deltaVt_GLY

deltaVt_GLY = 1 + stim delta_GLY KO3 \frac{1}{1 + ⅇ^{1 (- sr_GLY) (time - (to + to_GLY))}} (1 - unitstepSB2)

Component: Fin_t

Fin_t = CBF0 + (stim CBF0 deltaf \frac{1}{1 + ⅇ^{1 (- sr) (time - (to + t1 - 3))}} - (stim CBF0 deltaf \frac{1}{1 + ⅇ^{1 (- sr) (time - (to + tend + t1 + 3))}}))

Component: Fout_t

Fout_t = CBF0 \frac{{(\frac{Vv}{Vv0})}^{2} + tv {(\frac{Vv}{Vv0})}^{-0.5} \frac{Fin_t}{Vv0}}{1 + CBF0 tv {(\frac{Vv}{Vv0})}^{-0.5} \frac{1}{Vv0}}

Component: NADn

NADn = NADH_n_tot - NADHn

Component: NADg

NADg = NADH_g_tot - NADHg

Component: CRn

CRn = PCrn_tot - PCrn

Component: CRg

CRg = PCrg_tot - PCrg

Component: ADPn

ADPn = \frac{ATPn}{2} ((- qak) + \sqrt{{qak}^{2} + 4 qak (\frac{ATPtot}{ATPn} - 1)})

Component: ADPg

ADPg = \frac{ATPg}{2} ((- qak) + \sqrt{{qak}^{2} + 4 qak (\frac{ATPtot}{ATPg} - 1)})

Component: u_n

u_n = {qak}^{2} + 4 qak (\frac{ATPtot}{ATPn} - 1)

Component: u_g

u_g = {qak}^{2} + 4 qak (\frac{ATPtot}{ATPg} - 1)

Component: dAMP_dATPn

dAMP_dATPn = \frac{qak}{2} + qak \frac{ATPtot}{ATPn \sqrt{u_n}} - (1 + 0.5 \sqrt{u_n})

Component: dAMP_dATPg

dAMP_dATPg = \frac{qak}{2} + qak \frac{ATPtot}{ATPg \sqrt{u_g}} - (1 + 0.5 \sqrt{u_g})

Component: AMPn

AMPn = ATPtot - (ATPn + ADPn)

Component: AMPg

AMPg = ATPtot - (ATPg + ADPg)

Component: BOLD

BOLD = Vv0 ((k1 + k2) (1 - (\frac{dHb}{dHb0})) - ((k2 + k3) (1 - (\frac{Vv}{Vv0}))))

Component: v_stim

v_stim = stim (v1_n + v2_n \frac{time - to}{t_n_stim} ⅇ^{- ((time - to) \frac{unitpulseSB}{t_n_stim})}) unitpulseSB

unitpulseSB = \{\begin{matrix} 1 if time \geq to \land time ≦ to + tend \\ 0 otherwise \end{matrix}

Component: unitstepSB

unitstepSB = \{\begin{matrix} 1 if time - (tend + to) \geq 0 \\ 0 otherwise \end{matrix}

Component: model_parameters

Component: unitstepSB2

unitstepSB2 = \{\begin{matrix} 1 if time - (tend_GLY + to + to_GLY) \geq 0 \\ 0 otherwise \end{matrix}

Source: Derived from workspace Cloutier 2009 at changeset c47e7ddfa3a4.

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An integrative dynamic model of brain energy metabolism using in vivo neurochemical measurements