Overview

Fluxion is a mechanistic simulation: every reaction carries kinetic and thermodynamic constants, and the behaviour on screen is the consequence of those numbers rather than a scripted animation. This page records where those numbers come from — which are measured constants taken from the literature, and which are modelling choices made to keep a reduced-order model of one muscle cell physiological. It is written so that a student or reviewer can check the model against its sources.

How to read this page

Each value is given beside the reaction it belongs to, using the model's own names (for example ComplexI, ATPSynthase). Three kinds of number appear:

• Vf — a forward velocity, in mM/s. It represents how much active catalyst is

present, so it is the number an expression change, inhibitor, or knockdown moves. It is not a measured turnover number; it is a model-scaled capacity.

• Keq — an equilibrium constant, derived from thermodynamics (standard free

energies or midpoint potentials) rather than fitted. Reverse velocities follow from Vf and Keq through the Haldane relationship, so the equilibrium position is fixed by chemistry, not chosen.

• Km — a half-saturation concentration, in mM, from measured enzyme affinities

where these are known.

Citations. Each value is followed by a bracketed reference number — for example [1] — that links to the numbered References list at the foot of the page. A value with no bracketed number is a modelling choice rather than a measured quantity, and is labelled as such in the text; where the literature gives a different value, the entry says so and gives the reason.

Concentrations are in millimolar (mM); the mitochondrial membrane potential is a voltage in millivolts (mV). Compartments are marked by suffix: _c cytosol, _i intermembrane space, _m matrix, _b blood.

A note on matrix velocities. The model treats the cell as compartments with real relative volumes — cytosol 0.70, matrix 0.07, intermembrane space 0.02 of the cell. Because a matrix reaction is stated per litre of matrix, its velocity is multiplied by MATRIX_CAPACITY_SCALE = 0.70 / 0.07 = 10 to keep the same capacity per cell. Where a matrix Vf is quoted below as, say, 0.018 × 10, the first number is the per-litre value in the source and the second is that scaling.

Electron transport chain and oxidative phosphorylation

The respiratory chain is modelled as what it physically is: redox reactions that move charge across the inner membrane, and an ATP synthase that lets that charge back in to make ATP. The membrane potential is a real voltage that the complexes build and ATP synthase spends, not a fitted parameter.

Redox midpoint potentials and equilibrium constants

Each complex's equilibrium constant is derived from the standard midpoint potentials of its redox couples at pH 7 and 37 °C [5][3], using

ΔG = −n F ΔE and Keq = exp(−ΔG / RT), RT = 2577 J/mol at 310 K

with n = 2 electrons. The midpoint potentials used (volts) are the standard reference values [5]:

• NAD⁺/NADH −0.320
• FAD/FADH₂ +0.031 (the succinate-dehydrogenase-bound flavin)
• CoQ/CoQH₂ +0.045
• cytochrome c +0.254
• O₂/H₂O +0.816

which give:

• ComplexI — ΔE = 0.365 V, ΔG = −70.4 kJ/mol, Keq = 7.4 × 10¹¹ [5]
• ComplexII — ΔE = 0.014 V, ΔG = −2.7 kJ/mol, Keq = 2.85 [5]. Barely favourable, which is correct: the succinate flavin sits almost level with the quinone pool, and that narrow margin is why succinate yields less ATP than NADH.
• ComplexIII — ΔE = 0.209 V, ΔG = −40.3 kJ/mol, Keq = 6.3 × 10⁶ [5]
• ComplexIV — ΔE = 0.562 V, ΔG = −108.4 kJ/mol, Keq = 1.9 × 10¹⁸ [5]. Far from equilibrium at any physiological potential, which is why cytochrome c oxidase is effectively one-way and is the committed step.

Proton and charge stoichiometry

Per two electrons, which totals the accepted ten protons per NADH oxidised [5]:

• ComplexI — 4 H⁺ to the intermembrane space, 5 from the matrix (4 pumped, 1 for ubiquinone reduction); 4 charges [5].
• ComplexIII — 4 H⁺ to the intermembrane space, 2 from the matrix; 2 charges — two protons leave ubiquinol on the outer face and cross no membrane [5].
• ComplexIV — 2 H⁺ to the intermembrane space, 4 from the matrix (2 pumped, 2 forming water); 4 charges [5].

Rate law: exchange currents, not a maximum velocity

Complexes I, III, and IV use Butler–Volmer / Marcus electron-transfer kinetics rather than the reversible Michaelis–Menten law, so that Vf is an exchange current — a small internal resistance — rather than a saturating maximum. This is a modelling choice of rate-law form (no literature turnover number is implied): the small exchange current (Vf = 0.018 × 10) keeps the resting chain below its thermodynamic ceiling, which is what lets the membrane potential fall smoothly from rest to hard work. ComplexII (Vf = 0.85 × 10) is an ordinary reversible step, as it is not electrogenic. Km values for the electron carriers are 0.05 mM, within measured ranges [2].

ATP synthase

Reaction ATPSynthase: ADP + Pi + (8/3) H⁺ → ATP + H₂O + (8/3) H⁺. The mammalian enzyme has a ring of eight c-subunits and makes three ATP per revolution, so 8/3 protons per ATP [5]. The equilibrium constant comes from the free energy of ATP synthesis, ΔG = +30.5 kJ/mol, giving Keq = 7.3 × 10⁻⁶ [3] — driven entirely by the proton-motive force. Calcium activation (half-effect near 1 µM matrix Ca²⁺) is the demand half of parallel activation [6].

P/O ratio

The chain separates ten charges per NADH; ATP synthase spends 8/3 per ATP and the translocase one more, so 10 / (8/3 + 1) ≈ 2.7 ATP per NADH, against the accepted P/O ratio of about 2.5 once proton leak is included [12].

Proton leak

Reaction ProtonLeak: protons returning to the matrix outside ATP synthase. This accounts for roughly 20–25 % of resting mitochondrial oxygen consumption in skeletal muscle [13], and stops the potential running away when demand is low. It is modelled as a Hill function of the membrane potential (half-point 175 mV, coefficient 4, up to 25-fold), which reproduces the observed non-ohmic shape [13]; the exact Hill constants are a modelling choice fitted to that shape.

Membrane potential

Resting Dpsi_m settles near 180 mV (a respiring mitochondrion runs at roughly 150–190 mV) and falls toward the working range under load [5]. The pH gradient across the inner membrane adds a further ~25 mV, so the total proton-motive force is roughly Δψ + 25 mV [5].

Modelling choices called out honestly

• Ubiquinone midpoint potential (+0.045 V). This couple is the least certain of the five — reported values run from about +0.04 to +0.11 V [5] — and it is also the one the model is most sensitive to, because it sets how the driving force divides between Complex I and Complex III. +0.045 V is chosen because it is the value at which both near-equilibrium complexes still run forward; this is a modelling choice within the reported range.
• Phosphate carrier stoichiometry. The carrier is written with two protons per phosphate, where the literature usually gives one [5]. This is a compromise, made because the model lacks the potassium cycling that normally partitions the proton-motive force; with one proton the modelled matrix runs to pH 11. The total proton-motive force is correct either way — only its split between voltage and pH depends on it.

Glycolysis

Equilibrium constants (reversible steps)

Set to the apparent equilibrium constants at pH 7, which match the standard tabulated values [1][3]:

• PGI (glucose-6-P ⇌ fructose-6-P) — Keq = 0.35 [1]
• Aldolase (fructose-1,6-bisP ⇌ DHAP + GAP) — Keq = 0.08 [1]
• TPI (DHAP ⇌ GAP) — Keq = 0.045, favouring DHAP ~22:1 [1]
• GAPDH — Keq = 0.08 [1]
• PGK (1,3-bisphosphoglycerate + ADP ⇌ 3-PG + ATP) — Keq = 3300 [1]
• PGM (3-PG ⇌ 2-PG) — Keq = 0.17 [1]
• Enolase (2-PG ⇌ PEP) — Keq = 0.50, derived from ΔG°′ ≈ +1.8 kJ/mol [3]

Regulated irreversible steps

• Hexokinase — Km(glucose) = 0.05 mM, the measured low-Km hallmark of hexokinase (versus liver glucokinase) [2]. Product inhibition by glucose-6-phosphate (Ki = 0.05 mM) is its classic feedback [3].
• PFK1 — inhibited by ATP and by protons, activated by AMP: the direction of all three controls matches the textbook regulation of the committed, pH-sensitive step [4][3]. The AMP activation strength (up to 10-fold) has no specific literature value — the code marks it a placeholder chosen to exceed the engine's former 2-fold ceiling, and it should be traced to a source.
• PyruvateKinase — feed-forward activation by fructose-1,6-bisphosphate and inhibition by ATP; both match the established regulation of the M-type isozyme [3].

Forward velocities throughout glycolysis are model-scaled capacities, not measured turnover numbers (see How to read this page).

Glycogen storage and mobilisation

• GlycogenPhosphorylase — the reciprocal allosteric pattern is textbook [4]: activated by AMP, inhibited by ATP and by glucose-6-phosphate. Km(Pi) = 5 mM is genuinely millimolar, consistent with the measured phosphate dependence [2]. The specific inhibition/activation constants (and the 15-fold AMP activation) are modelling choices; no exact literature fold-activation was identified.
• PhosphoglucoMutase — Keq = 17 (glucose-1-P → glucose-6-P), matching the standard ~17–19 [1].
• GlycogenSynthase — modelled as a single step (G6P + ATP → glycogen + ADP + 2 Pi). Real synthesis runs through UDP-glucose and costs two ATP-equivalents per unit; this differs from the literature pathway [3] because the model deliberately collapses the UDP-glucose route into one step, so its stoichiometry and velocity are a simplification rather than a measured enzyme rate. Its reciprocal regulation (inhibited by AMP, activated by G6P) is qualitatively literature-grounded [3].

Anaerobic glycolysis

• LactateDehydrogenase_c (pyruvate + NADH + H⁺ ⇌ lactate + NAD⁺) — the apparent equilibrium constant at pH 7 is ~3.6 × 10⁴ favouring lactate, represented as Keq = 3.6 × 10⁸ once the proton is explicit; this matches the standard value [3][1]. Km values are within measured ranges [2]; the mild product inhibition by lactate (Ki = 20 mM) is a documented feature [2].

Pyruvate oxidation and the citric acid cycle

Matrix velocities are per-cell values, i.e. the per-litre-of-matrix figure ×10 (see the Overview).

Equilibrium constants (reversible steps)

Set from thermodynamics and consistent with the standard tabulated values [1]:

• Aconitase (citrate ⇌ isocitrate) — Keq = 0.067, favouring citrate ~15:1 [1]
• SuccinylCoASynthetase — Keq = 3.8 [1]
• SuccinateDehydrogenase — Keq = 1.0, set to reflect the near-level succinate flavin / ubiquinone couple (see the ETC section) [5]
• Fumarase (fumarate ⇌ malate) — Keq = 4.4 [1]
• MalateDehydrogenase (malate + NAD⁺ ⇌ OAA + NADH) — Keq = 2.6 × 10⁻⁵, matching the standard ~2.8 × 10⁻⁵ [1]

Regulation of the dehydrogenases

PyruvateDehydrogenase, IsocitrateDehydrogenase, and AlphaKGDehydrogenase are inhibited by NADH (and their acyl-CoA / ATP products) and activated by matrix calcium; isocitrate dehydrogenase is additionally activated by ADP. This pattern matches the literature — the three dehydrogenases are the classic calcium- and energy-charge-sensitive control points [6]. The calcium half-activation (Ka ≈ 1 µM matrix Ca²⁺) is in the measured range for these enzymes [6]. The exact product-inhibition constants (for example the NADH Ki of 0.02 mM) are modelling choices tuned so the resting matrix stays oxidised — faithful in direction and steepness but not taken from a single measured value.

Anaplerosis (a deliberate departure)

• PyruvateCarboxylase — its capacity is cut roughly 60-fold below a generic setting. This differs from a literature enzyme rate deliberately: skeletal muscle carries very little pyruvate carboxylase (it is chiefly a liver/kidney gluconeogenic enzyme) [3], and the model has no cataplerotic exit, so anaplerosis is held minimal to keep the matrix organic-acid pools physiological. The inhibition by matrix malate (Ki = 0.5 mM) is a modelling device with no direct literature counterpart.

The NADH shuttles

Cytosolic NADH is handed across the inner membrane by two shuttles, both textbook [3]. Their equilibrium constants are set from thermodynamics: cytosolic malate dehydrogenase (Keq = 2.6 × 10⁻⁵) [1], the aspartate aminotransferases (Keq = 0.147, matching the standard ~0.15) [1], and the cytosolic glycerol-3-phosphate dehydrogenase (Keq = 1 × 10⁴) [1].

• The electrogenic aspartate/glutamate carrier (net one negative charge exported) is what makes the malate-aspartate shuttle effectively one-way; its electrogenicity is well established [5].
• The malate-aspartate carriers and dehydrogenases are scaled to roughly 5 % of a full capacity. This differs from a literature rate by design: skeletal muscle relies chiefly on the glycerophosphate shuttle [3], so the malate-aspartate route is kept minor — which also prevents the matrix dicarboxylate pool from over-filling. The 5 % figure is a modelling choice with no single literature value.

Mitochondrial transport and the adenine nucleotide translocase

• AdenineNucleotideTranslocase — electrogenic ATP⁴⁻/ADP³⁻ exchange (net one negative charge out). Its electrogenicity, and the quarter of the proton-motive force it costs to export each ATP, are literature-grounded [5] and set the P/O ratio correctly [12]. The velocity is a model capacity.
• PyruvateCarrier — electroneutral proton symport, the established mechanism of the mitochondrial pyruvate carrier [9]. Its capacity is reduced about threefold from a naïve setting; this differs from a literature rate deliberately, so the mitochondria do not drain cytosolic pyruvate to near-zero and prevent lactate dehydrogenase engaging under heavy demand.
• PhosphateCarrier — modelled with two protons per phosphate, where the literature usually gives one [5]; see the ETC modelling-choices note for why.
• MatrixProtonHomeostat — a single electroneutral proton exchange holding the matrix ~0.4 pH units alkaline. It is a lumped stand-in for the aggregate K⁺/H⁺ and Na⁺/H⁺ antiporters [10], so it has no single literature rate; the 0.4-unit gradient is set below the ~0.5–0.7 often quoted [5] so more of the proton-motive force sits in the membrane potential.
• OM_* (outer-membrane porins) — fast, near-equilibrating exchange reflecting the freely permeable outer membrane [5]; the large velocity is deliberate, not a measured rate.
• AdenylateKinase_c / _m (2 ADP ⇌ ATP + AMP) — Keq = 0.44 (written as its reciprocal, 2.27), which matches the standard value [7]; its near-equilibrium operation is what makes AMP a sensitive energy-stress signal.

Calcium signalling

Calcium is the feed-forward matching mitochondrial output to demand ("parallel activation"), a well-established mechanism [6].

• CalciumUniporter — the membrane-potential-driven MCU (net two positive charges in). Its low calcium affinity — it engages in the micromolar range — is literature-grounded [6]; the exact Km and velocity are model capacities.
• CalciumEffluxMito (lumped NCLX / mHCX) and CalciumReuptake (SERCA-like) — lumped devices with no single literature rate, sized to hold resting cytosolic calcium near 0.1 µM and matrix calcium near 0.2 µM, the measured resting levels [3].

The phosphocreatine buffer

• CreatineKinase_c — the apparent equilibrium constant near 150 at pH 7 (represented as Keq = 1.5 × 10⁶ once the proton is explicit) matches the standard value [7]. Km(PCr) = 0.72 mM and Km(ADP) = 0.035 mM are within measured ranges [2]. The high velocity reflects the enzyme's fast near-equilibrium buffering — a literature-grounded property [3] — though the exact velocity is a model capacity.

Lactate export and the blood compartment

• LactateTransport_c_to_b — monocarboxylate transport carrying one proton per lactate, the established MCT mechanism [9]; exporting lactate is therefore also how a working muscle exports acid. Km ≈ 1 mM is broadly consistent with reported MCT affinities [9].
• BloodLactateConsumption — an artificial systemic sink standing in for lactate disposal by other tissues (the Cori cycle). It is a boundary term rather than an enzyme, so it has no literature rate by construction.

Proton buffering

• ProtonBuffer_c / _i / _m — reversible buffers with pKa ≈ 7.3, the phosphate/histidine range that dominates muscle cytosolic buffering [3]. The total capacity is ≈ 51 mmol H⁺ per pH unit per litre, at the upper edge of the 30–50 range reported for muscle [3] — a modest, deliberate choice for numerical stability.
• ProtonBuffer_b — the blood buffer uses pKa ≈ 7.1, a bicarbonate-weighted value, sized to ~24 mmol titratable base so plasma holds near pH 7.4 and acidifies only modestly, consistent with measured blood buffering [3].

Boundary conditions: supply, demand, and gas exchange

These are source/sink terms representing the cell's environment, not enzymes.

• ATPConsumption — the basal ATP-demand rate, Vf = 0.026 mM/s, matches the measured resting ATP turnover of skeletal muscle (~0.02–0.05 mM/s) [3]. The guided exercises override this with their own workloads.
• GlucoseSupply — the default rate (5 × 10⁻⁵ mM/s) is consistent with measured skeletal-muscle glucose uptake (~2–5 µmol/min for a working muscle) [3]. Guided exercises set their own supply.
• OxygenSupply / CO2Removal — boundary gas-exchange terms. Oxygen delivery is self-limited so matrix O₂ settles near 0.012 mM, within the measured mitochondrial range of 0.005–0.03 mM [5]; the delivery rate itself is a boundary condition, not a measured enzyme rate.
• PhosphateSupply — a lumped homeostat standing in for the sodium-phosphate cotransporter and whole-body phosphate regulation. It is a placeholder device with no single literature rate.
• AMPRecovery_c / _m, NADHConsumption, FADH2Consumption — surrogate source/sink terms (the last two default to off). They are modelling devices, not measured reactions.

Starting concentrations

The initial concentrations are measured resting-skeletal-muscle values, and match the literature [11][3]:

• ATP 8 mM, phosphocreatine 30 mM, creatine 10 mM, cytosolic NAD⁺ 0.7 mM, free phosphate 3 mM, glycogen 80 mM glucosyl units [11].
• Free ADP (0.015 mM) and AMP (0.002 mM) are set far below total cellular content on purpose: most adenine nucleotide is protein-bound, and the low free concentrations are what enzymes sense — a literature-grounded distinction that makes AMP a sensitive stress signal [3].
• Matrix nucleotides sit near an ATP/ADP ratio of ~1 (much lower than the cytosol, because the electrogenic translocase drives export) [5] and a NAD⁺/NADH ratio near 5, both consistent with measured mitochondrial poise [8].
• Resting cytosolic calcium 0.1 µM, matrix calcium 0.2 µM, and blood pH 7.4 are standard resting values [3].

References

• [1] eQuilibrator — biochemical equilibrium constants (Flamholz, A., Noor, E., Bar-Even, A. & Milo, R., 2012, Nucleic Acids Research 40, D770–D775).
• [2] BRENDA — the comprehensive enzyme database; Michaelis constants (Schomburg, I. et al., Nucleic Acids Research).
• [3] Nelson, D.L. & Cox, M.M. Lehninger Principles of Biochemistry — pathway free energies, enzyme regulation, and standard resting concentrations.
• [4] Berg, J.M., Tymoczko, J.L. & Stryer, L. Biochemistry — allosteric regulation of glycolysis and glycogen metabolism.
• [5] Nicholls, D.G. & Ferguson, S.J. Bioenergetics — midpoint potentials, proton and charge stoichiometry, the electrogenic carriers, and the membrane potential.
• [6] McCormack, J.G., Halestrap, A.P. & Denton, R.M. (1990). Role of calcium ions in the regulation of mammalian intramitochondrial metabolism. Physiological Reviews 70, 391–425.
• [7] Lawson, J.W.R. & Veech, R.L. (1979). Effects of pH and free Mg²⁺ on the Keq of the creatine kinase reaction and other phosphate hydrolyses and transfers. Journal of Biological Chemistry 254, 6528–6537.
• [8] Williamson, D.H., Lund, P. & Krebs, H.A. (1967). The redox state of free nicotinamide-adenine dinucleotide in the cytoplasm and mitochondria of rat liver. Biochemical Journal 103, 514–527.
• [9] Halestrap, A.P. & Price, N.T. (1999). The proton-linked monocarboxylate transporter (MCT) family. Biochemical Journal 343, 281–299. (Mitochondrial pyruvate carrier: Halestrap, A.P., 1975; molecular identity, Bricker, D.K. et al., 2012, Science 337, 96–100.)
• [10] Garlid, K.D. & Paucek, P. (2003). Mitochondrial potassium transport: the K⁺ cycle. Biochimica et Biophysica Acta 1606, 23–41.
• [11] Kushmerick, M.J., Moerland, T.S. & Wiseman, R.W. (1992). Mammalian skeletal muscle fibers distinguished by contents of phosphocreatine, ATP, and Pi. Proceedings of the National Academy of Sciences 89, 7521–7525.
• [12] Hinkle, P.C. (2005). P/O ratios of mitochondrial oxidative phosphorylation. Biochimica et Biophysica Acta 1706, 1–11.
• [13] Rolfe, D.F.S. & Brand, M.D. (1996). Contribution of mitochondrial proton leak to skeletal-muscle respiration. American Journal of Physiology 271, C1380–C1389.