Complete Model Equations
1 Energy Metabolism Model
The energy metabolism model (Chapter Energy Metabolism Models) comprises 8 state variables. The complete ODE system is:
\[ \begin{aligned} \frac{d[\text{ATP}]}{d t} &= 2 J_\text{glyc} + J_\text{ATPsyn} - J_\text{demand}(t) \\ \frac{d[\text{ADP}]}{d t} &= J_\text{demand}(t) - 2 J_\text{glyc} - J_\text{ATPsyn} + J_{\text{AK,fwd}} - J_{\text{AK,rev}} \\ \frac{d[\text{AMP}]}{d t} &= J_{\text{AK,rev}} - J_{\text{AK,fwd}} \\ \frac{d[\text{NADH}]}{d t} &= 2 J_\text{glyc} + 3 J_\text{Krebs} - J_\text{CI} \\ \frac{d[\text{Pyr}]}{d t} &= J_\text{glyc} - J_\text{PDH} \\ \frac{d \Delta \Psi}{d t} &= \frac{1}{C_m}(4 J_\text{CI} + 4 J_\text{CIII} + 2 J_\text{CIV} - n_\text{ATP} J_\text{ATPsyn} - J_\text{leak}) \\ \frac{d[\text{ROS}]}{d t} &= J_\text{ROS} - k_\text{SOD}[\text{SOD}][\text{ROS}] - k_\text{GPx}[\text{GPx}][\text{ROS}] \\ \frac{d D}{d t} &= k_D [\text{ROS}] \cdot \mathbb{1}_{[\text{ROS}] > \text{ROS}_\text{thr}} - k_\text{rep} \frac{[\text{ATP}]}{K_\text{rep} + [\text{ATP}]} \end{aligned} \tag{1}\]
where the proton stoichiometry coefficients (4, 4, 2) are the number of protons pumped per electron pair by Complexes I, III, and IV respectively (total 10 H⁺ per NADH), \(n_\text{ATP} \approx 8/3\) is the H⁺/ATP ratio of ATP synthase, \(J_\text{AK}\) denotes the adenylate kinase equilibrium ($ 2 arrows.lr + $, \(K_\text{eq} \approx 0.44\)), and \(J_\text{PDH}\) is the pyruvate dehydrogenase flux linking glycolysis to the Krebs cycle. The conservation constraints \([\text{ATP}] + [\text{ADP}] + [\text{AMP}] = A_\text{total}\) and \([\text{NAD}]^+ + [\text{NADH}] = N_\text{total}\) reduce the independent variables to 6. The adenylate kinase equilibrium is maintained by setting \(J_{\text{AK,fwd}} - J_{\text{AK,rev}} = k_\text{AK}(K_\text{eq}[\text{ADP}^2] - [\text{ATP}][\text{AMP}])\).
2 Energy Metabolism Parameter Table
Table Complete Model Equations lists all parameters with healthy baseline values, ME/CFS ranges, units, and sources. Parameter values are drawn from published enzymology, metabolomics, and CPET data where available; parameters without direct experimental constraints are indicated.
| Parameter | Healthy | ME/CFS | Units | Source |
|---|---|---|---|---|
| \(V_{max, \text{PFK-1}}\) | 1.2 | 1.2 | mM/min | Enzymology |
| \(K_{0.5}^{\text{F6P}}\) | 0.1 | 0.1 | mM | Enzymology |
| \(n_H\) (PFK-1) | 2.5 | 2.5 | – | Enzymology |
| \(K_i^{\text{ATP}}\) | 1.0 | 1.0 | mM | Enzymology |
| \(V_{max, \text{CI}}\) | 1.0 | 0.5–0.8 | mM/min | (Tomas et al. 2017) |
| \(K_m^{\text{NADH}}\) | 0.05 | 0.05 | mM | Enzymology |
| \(\Delta \Psi_\text{threshold}\) | 110 | 110 | mV | Biophysics |
| \(J_{\text{leak,0}}\) | 0.05 | 0.075–0.15 | mM/min | Estimated |
| \(A_\text{total}\) | 8.0 | 8.0 | mM | Physiology |
| \(N_\text{total}\) | 1.0 | 0.6–0.9 | mM | (Heng et al. 2025) (PBMC data; tissue value predicted) |
| \(k_\text{ROS}\) | 0.01 | 0.01 | min⁻¹ | Estimated |
| \(k_\text{SOD}\) | 10.0 | 10.0 | mM⁻¹min⁻¹ | Enzymology |
| \(k_D\) | 0.001 | 0.001 | min⁻¹ | Estimated |
| \(k_\text{rep}\) | 0.005 | 0.005 | min⁻¹ | Estimated |
3 Immune System Model
The immune model (Chapter Immune System Models) comprises 12 state variables for cell populations and 6 for cytokines. The NK cell, T cell, and B cell dynamics are specified by Equations nk dynamics and bcell dynamics. The cytokine network is defined by the production and degradation terms in Equation cytokine general. Full parameter tables for the immune model follow the same format as Table Complete Model Equations; key parameters are listed below.
| Parameter | Healthy | ME/CFS | Units | Source |
|---|---|---|---|---|
| \(s_N\) (NK production) | 50 | 50 | cells/\(\mu\)L/day | Physiology |
| \(k_{\text{act,0}}\) (NK) | 0.2 | 0.2 | day⁻¹ | Estimated |
| \(k_\text{exh}\) (NK) | 0.05 | 0.1–0.2 | day⁻¹ | (Hardcastle et al. 2016) |
| \(k_\text{recov}\) (NK) | 0.1 | 0.03–0.07 | day⁻¹ | Estimated |
| \(\delta_{\text{IL-6}}\) | 0.5 | 0.5 | h⁻¹ | Pharmacokinetics |
| \(\delta_\text{TNF}\) | 0.7 | 0.7 | h⁻¹ | Pharmacokinetics |
| \(d_P\) (plasma cell) | 0.003 | 0.003 | day⁻¹ | Immunology |
| \(\lambda_0\) (reactivation) | 0.001 | 0.005–0.02 | day⁻¹ | Estimated |
4 Neuroendocrine Model
The HPA axis (Equation hpa axis), autonomic (Equation ans balance), neurotransmitter (Equations tryptophan and catecholamines), and sleep–wake (Equation sleep wake) models contribute 14 additional state variables. Key parameters:
| Parameter | Healthy | ME/CFS | Units | Source |
|---|---|---|---|---|
| \(\sigma_H\) (CRH) | 10 | 10 | pg/mL/h | Endocrinology |
| \(a_c\) (circadian) | 0.4 | 0.15–0.25 | – | (Cambras et al. 2018) |
| \(n_F\) (feedback) | 2.0 | 3.0–4.0 | – | (Cleare et al. 1999) |
| \(\delta_F\) (cortisol) | 0.08 | 0.08 | h⁻¹ | Pharmacokinetics |
| \(G_S\) (baroreflex) | 0.5 | 0.2–0.4 | bpm/mmHg | (Newton et al. 2007) |
| \(r_\text{decay}\) (sleep) | 0.05 | 0.02–0.04 | h⁻¹ | Estimated |
| \(v_{\text{IDO,basal}}\) | 0.01 | 0.01 | mM/h | (Kavyani et al. 2022) |