ATP Production Models
1 Glycolysis Kinetics
Glycolysis converts glucose to pyruvate through ten enzymatic steps. For modeling purposes, the pathway is reduced to three lumped reactions capturing the key regulatory points: hexokinase (HK), phosphofructokinase-1 (PFK-1), and pyruvate kinase (PK). Each enzyme is modeled with Michaelis–Menten kinetics modified by allosteric regulation:
\[ v_{\text{PFK-1}} = V_{max, \text{PFK-1}} \cdot \frac{[\text{F6P}]^{n_H}}{K_{0.5}^{n_H} + [\text{F6P}]^{n_H}} \cdot \frac{K_i^{\text{ATP}}}{K_i^{\text{ATP}} + [\text{ATP}]} \cdot \frac{[\text{AMP}]}{K_a^{\text{AMP}} + [\text{AMP}]} \tag{1}\]
where \([\text{F6P}]\) is fructose-6-phosphate concentration, \(n_H \approx 1.5\)–$ 4$ is the Hill coefficient reflecting cooperative kinetics, \(K_i^{\text{ATP}}\) is the ATP inhibition constant, and \(K_a^{\text{AMP}}\) is the AMP activation constant. PFK-1 is the committed step of glycolysis and the primary site of metabolic regulation: it is inhibited by ATP (the end product) and activated by AMP (a signal of energy deficit). This regulatory logic is central to ME/CFS modeling because it determines how rapidly glycolysis accelerates when oxidative phosphorylation is impaired.
The net glycolytic flux determines pyruvate production and, consequently, the substrate supply to the Krebs cycle. In healthy tissue, the glycolytic rate is tightly coupled to mitochondrial demand. In ME/CFS, evidence of increased lactate production at lower workloads (Keller et al. 2024) suggests that this coupling is disrupted, with glycolysis operating at higher flux relative to oxidative capacity—a pattern consistent with reduced \(V_{max}\) of downstream mitochondrial enzymes.
2 Krebs Cycle Model
The Krebs cycle is modeled as a four-reaction lumped system capturing citrate synthase (CS), isocitrate dehydrogenase (IDH), \(\alpha\)-ketoglutarate dehydrogenase (\(\alpha\)-KGDH), and succinate dehydrogenase (SDH, also Complex II of the ETC). The overall cycle flux depends on acetyl-CoA supply, \(\text{NAD}^\text{+}\) availability, and product inhibition:
\[ J_{\text{Krebs}} = V_{max, \text{CS}} \cdot \frac{[\text{AcCoA}]}{K_m^{\text{AcCoA}} + [\text{AcCoA}]} \cdot \frac{[\text{OAA}]}{K_m^{\text{OAA}} + [\text{OAA}]} \cdot \frac{[\text{NAD}^+]}{[\text{NAD}^+] + [\text{NADH}]} \tag{2}\]
where the last factor captures the \(\text{NAD}^\text{+}\)/NADH ratio dependence. Metabolomic studies in ME/CFS have identified accumulation of specific Krebs cycle intermediates (citrate, isocitrate) consistent with reduced flux through IDH and \(\alpha\)-KGDH (Germain et al. 2020) (Naviaux et al. 2016). In the model, this is represented by reduced \(V_{max}\) values for these enzymes, which can arise from oxidative damage to iron–sulfur clusters, substrate competition from inflammatory metabolites, or epigenetic downregulation of enzyme expression.
3 Electron Transport Chain Model
The electron transport chain is modeled as a four-complex proton pump driving ATP synthase. The rate of proton pumping by Complex I is:
\[ J_{\text{CI}} = V_{max, \text{CI}} \cdot \frac{[\text{NADH}]}{K_m^{\text{NADH}} + [\text{NADH}]} \cdot \frac{[\text{UQ}]}{K_m^{\text{UQ}} + [\text{UQ}]} \cdot (1 - \frac{\Delta \Psi}{\Delta \Psi_{max}}) \tag{3}\]
where \([\text{UQ}]\) is the ubiquinone (oxidized CoQ10) concentration, \(\Delta \Psi\) is the mitochondrial membrane potential, and the last factor captures thermodynamic backpressure: as the proton gradient builds, further pumping becomes energetically unfavorable. The membrane potential evolves according to:
\[ C_m \frac{d \Delta \Psi}{d t} = \sum_{k \in {\text{CI,CIII,CIV}}} n_k J_k - J_{\text{ATP synthase}} - J_{\text{leak}} \tag{4}\]
where \(C_m\) is the membrane capacitance, \(n_k\) is the number of protons pumped per electron pair by complex \(k\), and \(J_{\text{leak}}\) represents proton leak across the inner mitochondrial membrane. ATP synthase flux follows:
\[ J_{\text{ATP synthase}} = V_{max, \text{F}_1 \text{F}_0} \cdot \frac{[\text{ADP}][\text{P}_i]}{K_m^{\text{ADP}} K_m^{\text{P}_i} + [\text{ADP}] K_m^{\text{P}_i} + [\text{P}_i] K_m^{\text{ADP}} + [\text{ADP}][\text{P}_i]} \cdot \frac{\Delta \Psi - \Delta \Psi_{\text{threshold}}}{\Delta \Psi} \tag{5}\]
The threshold term ensures that ATP synthesis requires a minimum proton-motive force (\(\Delta \Psi_{\text{threshold}} \approx 100\)–$ 120$ mV). This is significant for ME/CFS because any reduction in ETC complex activity lowers \(\Delta \Psi\), and once \(\Delta \Psi\) approaches the threshold, ATP synthesis drops precipitously—a nonlinear “cliff” effect that may explain the disproportionate symptom severity relative to modest biochemical abnormalities.
4 Integrated Energy Production
The complete energy production model couples glycolysis, the Krebs cycle, and the ETC through shared metabolite pools. The central state variables are:
- \([\text{ATP}]\), \([\text{ADP}]\), \([\text{AMP}]\) (adenine nucleotide pool, conserved: \([\text{ATP}] + [\text{ADP}] + [\text{AMP}] = A_{\text{total}}\))
- \([\text{NAD}^+]\), \([\text{NADH}]\) (nicotinamide nucleotide pool, conserved)
- \([\text{Pyruvate}]\), \([\text{AcCoA}]\), \([\text{OAA}]\) (carbon intermediates)
- \(\Delta \Psi\) (mitochondrial membrane potential)
The ATP balance equation integrates all production and consumption:
\[ \frac{d[\text{ATP}]}{d t} = 2 J_{\text{glycolysis}} + J_{\text{ATP synthase}} - J_{\text{demand}}(t) \tag{6}\]
where \(J_{\text{demand}}(t)\) represents cellular ATP consumption, which varies with activity level. The factor of 2 reflects the net ATP yield per glucose through glycolysis (after subtracting the two ATP invested in the preparatory phase).