Oxidative phosphorylation: ATP-Synthetase [1, 2] (ATP -> ADP + Pi) ⎛ ⎞ syn ATPmito v= V ⋅⎜ ADP ⋅ P − ⎟ syn max ⎜ mito mito ⎟ ⎝ K eq− syn ⎠ k syn ⎛ dG0 ⎞ ⎛ hcyt ⎞ ⎜ ⎟ ⎜ ⎟ Keq− syn = exp⎜ −k ⋅ U ⎟⋅⎜ ⎟ ⎝ RT⋅ ⎠ ⎝ hmito ⎠ syn dG0 = 30500 k = 3 ATP-exchanger: [3] (ATP_mito + ADP_cyt <-> ADP_mito + ATP_cyt) ⎛ ATP⋅ ADP ⎞ ⎜ 1− in mito exp()U ⎟ ⎜ ⎟ ATP−exchanger ADPin ⋅ ATPmito vATP−exchanger = Vmax ⎜ ⎟ ⎛ ATP ⎛ ADP ⎞⎞ ⎜ ⎜1+ in exp()SU⋅⎜1 + mito ⎟⎟ ⎟ ⎜ ⎜ ADP Vmm ⎜ ATP ⎟⎟ ⎟ ⎝ ⎝ in ⎝ mito ⎠⎠ ⎠ S = 0.3 Vmm ATP-consumption ATP consumption is modeled as a Michaelis-Menten equation with low Km-value: ATPin vATP− use = Vmax−ATP − use ATP ATPin+ K m ATP K m = 0.1 Electrophysiology: VF⋅ U= mm 1000⋅RT ⋅ C F =96490.0 mol J R =8.314 K⋅ mol T = 310 K Electro diffusion: The passive efflux of sodium ions, potassium ions, chloride ions and protons is modeled by the Goldman-Hodgkin equation. Potassium [4] ed ⎛ KKUin −mito ⋅exp( )⎞ IAPUF= ⋅ ⋅ ⋅ ⋅⎜ ⎟ Potmito m Potmito ⎜ ⎟ ⎝ 1− exp()U ⎠ −10 m P =2 ⋅ 10 Potmito s Sodium ed ⎛ Nain − Namito ⋅ exp(U )⎞ IAPUF= ⋅ ⋅ ⋅ ⋅⎜ ⎟ Na mito m Namito ⎜ ⎟ ⎝ 1− exp()U ⎠ −11 m P =1 ⋅ 10 Namito s Protons: [5], [6] ed ⎛ HHUin −mito ⋅exp( )⎞ IAPUF= − ⋅ ⋅ ⋅ ⋅⎜ ⎟ Hmito m Hmito ⎜ ⎟ ⎝ 1− exp()U ⎠ −4 m P =4 ⋅ 10 H mito s Pumps: [7] The pumping of sodium, potassium and phosphate is modeled as electro-neutral proton driven antiport: Phos−exchanger vPhos−exchanger = Vmax ()PHPHin⋅ in −mito ⋅ mito pump Na− pump I= V() Na⋅ H − Na ⋅ H Namito max in mito mito in pump Pot− pump IVKHKH= ()⋅ − ⋅ Potmito max in mito mito in Complex III: [8], [9], [10] The state of complex III is described as an array consisting of two numbers, a letter, two numbers and another letter. The numbers denote the reduction state, 0 for oxidized, 1 for reduced, of the corresponding complex III elements, the letter the binding, b, or not binding, n, of the ubisemiquinone. From left to right the array position denotes the reduction state of the cytochrome c1 (c1), the reduction state of the Iron sulfur cluster (Fe-S), the binding of the ubisemiquinone at p site (SQp), the reduction state of the low b- type heme (bL), the reduction state of the high b-type heme (bH) and the binding of the ubiseiquinone at n site (SQn). c1 Fe− S SQp b L b H SQn 0 0 n 0 0 n 1 1 b 1 1 b As examples 00n00n describes the fully oxidized complex III, with no bound ubisemiquinone, while 11b11b describes the fully reduced complex III with ubisemiquinone bound at n and p site, and 10n11b describes the state of complex III with reduced cytochrome c1, oxidized Fe-S, reduced b-type heme bL, reduced b-type heme bH and bound ubisemiquinone at n but not at p side. There are seven different reversible elementary processes with rate equation: • P site ubiquinol reacts with oxidized Fe-S to reduced Fe-S and p-site bound ubisemiquinone and release of two protons into the cytosol (QH2)p+ x0nxxx <-> x1bxxx + hin SQ ⎛ 1 ⎞ p ⎜ ⎟ vSQ = Vmax () QH2 ⋅ x0 nxxx − ⋅ x1 bxxx p ⎜ p SQp ⎟ ⎝ Keq ⎠ ⎛⎞SQp h SQp fVFV ⋅⋅ cyt K =⋅exp eq ⎜⎟ ⎝⎠R ⋅⋅Th1000 ref SQp fV = 1.0 −7 href = 10 • Reduction of the low b-type heme and release of ubiquinon at p site: x1b0xx <-> x1n1xx + Qp + hin Q ⎛ 1 ⎞ p ⎜ ⎟ vQ = Vmax x1 b 0 xx − ⋅x1 n 1 xx ⋅ Qp p ⎜ Qp ⎟ ⎝ Keq ⎠ ⎛⎞EfVFQQpp+⋅⋅ Q ()0 V h cyt K p =⋅exp⎜⎟ eq ⎜⎟R ⋅⋅Th1000 ⎝⎠ref Q p fV = 1.0 Q p E0 = 120 −7 href = 10 • Electrontransfer between the low b–type heme bL and the high b-typ heme bH: xxx10x <-> xxx01x hemes ⎛ 1 ⎞ v= V⎜ xxx10 x − ⋅ xxx01 x⎟ hemes max ⎜ hemes ⎟ ⎝ Keq ⎠ ⎛⎞EfVFhemes+⋅⋅ hemes hemes ()0 V K = exp⎜⎟ eq ⎜⎟RT⋅⋅1000 ⎝⎠ hemes fV = 8.0 hemes E0 = 80 • N site ubiqinone reacts with reduced high b-type heme to bound semiquinone at n site: xxxx1n + hmito+Qn <->xxxx0b ⎛ 1 ⎞ SQn ⎜ ⎟ vSQ = Vmax Qn ⋅ xxxx1 n − ⋅ xxxx0 b n ⎜ SQn ⎟ ⎝ Keq ⎠ SQ ⎛⎞SQn p EfVF0 +⋅⋅V SQ ()h mito K n =⋅exp⎜⎟ eq ⎜⎟R ⋅⋅Th1000 ⎝⎠ref SQn fV = 1.0 SQn E0 = 30 • Reduced high b-type heme reacts with n site bound ubisemiquinone: xxxx1b + hmito <-> xxxx0n + (QH2)n ⎛ ⎞ ()QH2 1 n ⎜ ⎟ v()QH = Vmax ⋅xxxx1 b −()QH ⋅xxx0 n ⋅() QH2 n 2 n ⎜ 2 n ⎟ ⎝ Keq ⎠ ()QH () QH ⎛⎞EfVF22nn+⋅⋅ QH ( 0 V ) h ()2 n ⎜⎟mito K =⋅exp eq ⎜⎟ ⎜⎟R ⋅⋅Th1000 ref ⎝⎠ ()QH2 n fV = 1.0 ()QH 2 n E0 = 130 −7 href = 10 • Reduced Fe-S reacts with oxidized cytochrome c1 : 01xxxx <-> 10xxxx cytc1⎛ 1 ⎞ v= V ⎜01xxxx − ⋅10xxxx⎟ cytc1 max ⎜ cytc1 ⎟ ⎝ Keq ⎠ cytc1 cytc1 ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 cytc1 E0 = −35 • Reduced cytochrome c1 reacts with oxidized cytochrom c: 1xxxxx + cytcox <-> 0xxxxx + cytcred cytc ⎛ 1 ⎞ v= V⎜1 xxxxx⋅ cytc − ⋅0xxxxx ⋅ cytc ⎟ cytc max ⎜ ox cytc1 red ⎟ ⎝ Keq ⎠ cytc cytc ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 cytc E0 −= 10 Complex I : [11], Hirst 2005 Biochemical Society Transaction vol 33, Rouslap 2010 nature (NADH + Q + 4 H_mito <-> NAD + QH2 + 4 H_in) Similar to complex III the state of complex I is described as an array of integer numbers of length 10, one number for the flavin, eight for the iron sulfur clusters, and one for the semi-ubiquinone. The flavin mononucleotide can be fully reduced (state=2), exist as flavin radical (state=1) or by fully oxidized (state=0). The iron-sulfur clusters of complex I can exist in oxidized form (represented by a zero) or in a reduced form (state=1). The last digit in the array represents the bound semi-ubiquinone, which can be absent (state=0) or bound (state=1). FMN N31 N a N 1 b N 456 N N a N 6 b N 2 SQ 2111111111 1000000000 0 This results in 1536 state variables and twelve elementary processes, generating 4480 reactions. This system of complex I was analyzed and reduced to a simplified system containing only six clusters. The iron sulfur clusters N1b, N4, N5, N6a, and N6b were lumped together to N* due to their similar midpoint potentials. The reduced system consists of 96 state variables, eight elementary processes and therewith 184 reactions. The occupation states of the remaining clusters are similar between the full and the reduced system. For the here developed model of the citric acid cycle and respiratory chain the simplified model was used. FMN N31 N a N * N 2 SQ 2 11111 100000 0 There are eight different reversible elementary processes: • Mitochondrial NADH reacts with oxidized FMN to NAD+ and frf. + NADHmito+ 0xxxxx <-> NAD + 2xxxxx frf ⎛⎞1 + v=⋅−⋅⋅ V⎜⎟ NADH02 xxxxx NAD xxxxx frf max ⎜⎟mito frf mito ⎝⎠Keq frf frf ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 frf Em0 =−32 V • One electron is transferred from the fully reduced flavin mononucleotide to N3. 20xxxx <-> 11xxxx frf− N 3 ⎛⎞1 v=−⋅ V⎜⎟20 xxxx 11xxxx frf− N 3max⎜⎟frf− N 3 ⎝⎠Keq frf− N 3 frf− N 3 ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 frf− N 3 E0 =12mV • Transfer from flavine-radical to N3. 10xxxx <-> 01xxxx fr− N 3 ⎛⎞1 v=−⋅ V⎜⎟10 xxxx 01xxxx fr− N 3max⎜⎟fr− N 3 ⎝⎠Keq fr− N 3 fr− N 3 ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 frf− N 3 Em0 =170 V • From N3 to N1A. x10xxx <-> x01xxx NNa31− ⎛⎞1 vVxxxxxxx=−⋅⎜⎟10 01 x NNa31− max ⎜⎟NNa31− ⎝⎠Keq NNa31− NNa31− ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 NNa31− Em0 =−135 V • From N1a to N*. xx10xx <-> xx01xx Na1*− N ⎛⎞1 v=−⋅ V⎜⎟ xx10 xx xx01 xx Na1*− N max ⎜⎟Na1*− N ⎝⎠Keq Na1*− N Na1*− N ⎛⎞EF0 ⋅ Keq = exp⎜⎟ ⎝⎠RT⋅⋅1000 Na1*− N Em0 =130 V • From N* to N2, thereby exporting one proton from the mitochondrial matrix. xxx10x + hmito <-> xxx01x + hin NN*2− ⎛⎞1 v=−⋅ V⎜⎟ xxx10 x xxx01 x NN*2− max ⎜⎟NN*2− ⎝⎠Keq ⎛⎞VE+⋅NN*2− F NN*2− ( mm 0 ) hmito K =⋅exp⎜⎟ eq ⎜⎟R ⋅⋅Th1000 ⎝⎠in NN*2− Em0 =150 V −7 href = 10 • Binding of ubichinon to complex I and transfer of one electron from N2 forming bound semi-ubichinon and exporting one proton from the mitochondrial matrix. xxxx10 + Q + hmito <-> xxxx01 + hin NSQ2− ⎛⎞1 v=⋅−⋅ V⎜⎟ Q xxxx10 xxxx01 NSQ2max− ⎜⎟NSQ2− ⎝⎠Keq ⎛⎞VE+⋅NSQ2− F 2 NSQ2− ( mm 0 ) ()hmito K =⋅exp⎜⎟ eq ⎜⎟R ⋅⋅Th1000 ⋅h ⎝⎠in ref NSQ2− E0 =185mV −7 href = 10 • Transfer of one electron from N2 to bound semi-ubichinon forming and releasing ubichinol, thereby exporting one proton from the mitochondrial matrix. xxxx11 + hmito <-> xxxx00 + QH2 + hin ⎛⎞ NQH2− 2 1 vNQH2m− =−⋅ Vax⎜⎟ xxxx11 xxxx00⋅ QH 2 2 ⎜⎟NQH2− 2 ⎝⎠Keq 2 ⎛⎞VE+⋅NQH2− 2 F NQH2− ()mm 0 ()hmito K 2 =⋅exp⎜⎟ eq ⎜⎟R ⋅⋅Th1000 ⋅h ⎝⎠in ref NSQ2− Em0 = 225 V −7 href = 10 Complex IV [12], [13], [14] (2 cyt_red + O2 + 4 H_mito -> H2O + 2 cyt_ox + 2 H_ in) n 6 Cx4 cytCred O2mito ⎛ hmito ⎞ v= V ⋅ ⎜ ⎟ C4 max n n O2 ⎜ ⎟ cytCred OK+ h cytCred + () Km 2mito m ⎝ in ⎠ O2 Km = 0.001 k ⎛ ATPmito ⎞ ⎜ ⎟ ⎝ ADPmito ⎠ n= A + B k k ⎛ ADP ⎞ ⎛ ATP ⎞ ⎜ mito ⎟ + ⎜ K ADP ⎟ ⎜ ⎟ ⎜ n ⎟ ⎝ ADPmito ⎠ ⎝ ⎠ A = 1.1 B= 8.0 ATP ADP Kn = 24 k = 3.2 6 ⎛ hmito ⎞ The factor ⎜ ⎟ was included to ensure proper activation of complex IV with increased demand, but complex IV activity ⎝ hin ⎠ might actually be regulated by the intra- and/or exrtramitochondrial ATP/ADP ratio. CAC Pyruvate exchanger [15] (Pyr_in + H_mito <-> Pyr_mito + H_in) Pyr−exchanger ()Pyrin⋅ H in − Pyrmito ⋅ H mito v pyr−exchanger = Vmax ⋅ ⎛ Pyr ⎞⎛ Pyr ⎞ ⎜1+ in ⎟⎜1+ mito ⎟ ⎜ Pyrin ⎟⎜ Pyrmito ⎟ ⎝ K m ⎠⎝ K m ⎠ Pyrin Km = 0.15 Pyrmito Km = 0.15 Pyruvatedehydrogenase complex [16], [17] (Pyr_mito + CoA + NAD -> AcCoA + Co2 + NADH) ⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞ pdhc Camito Camito Pyrmito NADmito CoAmito vpdhc = Vmax ⎜1 + Amax ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ Camito ⎟⎜ Pyr ⎟⎜ NADmito ⎟⎜ CoAmito ⎟ ⎝ Camito + Ka ⎠⎝ Pyrmtio + Km ⎠⎝ NADmito + Km ⎠⎝ CoAmito + Km ⎠ Camito Amax = 1.7 Camito −3 Ka = 10 Pyr Km = 0.090 NADmito K m = 0.036 CoAmito Km = 0.0047 Citrate synthase [18], (Oxa + AcCoA -> Cit) ⎛ ⎞⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ Oxa ⎟⎜ AcCoA ⎟ v= V cs mtio mito cs max ⎜ Cit ⎟⎜ CoA ⎟ ⎜ Oxa ⎛ mito ⎞ ⎟⎜ AcCoA ⎛ mito ⎞ ⎟ ⎜ Oxamito + K m ⎜1 + ⎟ ⎟⎜ AcCoAmito +K m ⋅⎜1 + ⎟ ⎟ ⎝ ⎝ 3.7 ⎠ ⎠⎝ ⎝ 0.025 ⎠ ⎠ Oxa Km = 0.0045 AcCoA K m = 0.005 Cit Ki = 3.7 CoA K i = 0.025 Aconitase [19], (Cit <-> IsoCit) IsoCitmito Citmito − Aco Aco Keq vAco = Vmax Citmito IsoCitmito