The Disease State ODE System
We model the ME/CFS disease state as a system of coupled ordinary differential equations governing the evolution of key state variables. The state vector \(\mathbf{x}(t) = (\alpha_\text{CI}, S, \mathcal{T}, C_\text{pro}, V, \mathbf{\mathcal{M}})\) captures:
- \(\alpha_\text{CI}\): Complex I (mitochondrial) function, normalized to \([0, 1]\)
- \(S\): Safe mode activation level, \(S \in [0, 1]\)
- \(C_\text{pro}\): Pro-inflammatory cytokine burden
- \(V\): Viral reactivation load
- \(\mathcal{T}\): Composite threat signal driving safe mode engagement
- \(\mathbf{\mathcal{M}} = (m_1, m_2, ..., m_n)\): Per-locus methylation vector. Each \(m_i \in [0, 1]\) represents fractional CpG methylation at locus \(i\), with \(m_i^\text{baseline}\) the healthy reference level. Loci span three functional classes: (1) \(\mathbf{\mathcal{M}}_\text{ProB} = {m_i \in \text{ProB RepSeqs}}\) — pericentromeric satellites, young LINE-1s, selected ERVs where methylation maintains heterochromatin compartment identity (Bonnet et al. 2026); (2) \(\mathbf{\mathcal{M}}_\text{ProA} = {m_i \in \text{ProA RepSeqs}}\) — euchromatic repeat elements where methylation regulates gene expression; (3) \(\mathbf{\mathcal{M}}_\text{gene} = {m_i \in \text{gene promoters, enhancers, gene bodies}}\) — protein-coding and regulatory regions including loci such as PTPRN2 (hypomethylated in ME/CFS (Chalder et al. 2026)), glucocorticoid receptor NR3C1 Vega, Vernon, and McGowan (2021), and immune regulatory genes. Consolidation depth is \(||\mathbf{\mathcal{M}} - \mathbf{\mathcal{M}}^\text{baseline}||\), capturing deviation in either direction (hyper- or hypomethylation). This vector formulation contains both the gain and loss models as special cases (gain-dominant and loss-dominant subsets of the locus vector) and resolves the gain-vs-loss tension of the earlier scalar formulation (see Speculation Consolidation as Loss of Methylation). At DNMT3B target sites, separate entries track CpG \(m_i^\text{CpG}\) and CAG \(m_i^\text{CAG}\) methylation (DNMT3B methylates CAG trinucleotides in early development and neurons; tissue-specific relevance).
1 Epigenetic Consolidation Formal Treatment
DNMT3B zero-sum constraint. De novo methyltransferase activity is finite per cell. In cancer, redistribution of DNMT3B away from ProB repeats toward ProA/gene regions — driven by chronic inflammation or oxidative stress — produces a coupled change: \(\sum_i \Delta m_i = 0\) at the enzyme-allocation level. Loss at ProB repeats and gain at gene promoters (and at Alu sequences) are not independent observations; they represent one mechanism (enzyme redistribution) with two manifestations. This DNMT3B global redistribution model is proposed by Bonnet, Hulo, Fourel et al. for cancer and likely chronic stimulation, but is not proposed by those authors for ME/CFS (Geneviève Fourel, pers. comm. May 2026). For ME/CFS, the proposed mechanism is more targeted: specific loss-meCpG at HSAT2 (and possibly HSAT3) pericentromeric repeats, with loss of DNMT3B activity at these loci being a consequence of HSAT2 transcriptional activation, not the initial cause. The zero-sum constraint remains a structural feature of the global cancer model; its applicability to locus-specific ME/CFS methylation changes has not been established. Direct experimental evidence for DNMT3B redistribution kinetics between heterochromatin and euchromatin compartments was not retrieved in the current literature cycle (rate-limited); the constraint is mechanistically grounded in finite enzyme concentration but remains an unmeasured structural claim pending ChIP-seq or locus-specific binding data.
Irreversibility threshold. At each locus \(i\), methylation maintenance requires a critical density \(m_i^\text{crit}\) of methyl-CpG binding proteins (MeCP2, MBD1-4) that recruit DNMT1 for replication-coupled maintenance. If methylation falls below \(m_i^\text{crit}\), the self-reinforcing maintenance loop breaks. Recovery at that locus then requires de novo remethylation by DNMT3A/3B — kinetically unfavourable and SAM-dependent — rather than passive DNMT1-mediated maintenance. Experimental support: Tiedemann et al. (2024) demonstrated that low-density CpG methylation is particularly vulnerable to disruption of UHRF1 ubiquitin ligase activity and DNMT1 ubiquitin reading — low-CpG-density regions are harder to maintain (Tiedemann et al. 2024). DNMT1’s allosteric binary switch further corroborates: the enzyme is processive on highly methylated substrates but becomes inactivated at poorly methylated sites, creating a density-dependent maintenance barrier (Kimura and Sasaki 2012). This creates a qualitative distinction in the bold(cal(M)) landscape: loci with \(m_i > m_i^\text{crit}\) can recover passively after disease-state signals are removed; loci with \(m_i < m_i^\text{crit}\) require active, targeted remethylation and may remain hypomethylated indefinitely without intervention.
B compartment strength. The 3D genome integrity depends on methylation at ProB repeats. We define a derived quantity:
\[ B_\text{strength} = f(\mathbf{\mathcal{M}}_\text{ProB}) \]
where \(f\) is a monotonically increasing function of per-locus methylation at ProB RepSeqs. In cancer, DNMT3B redistribution → \(\mathbf{\mathcal{M}}_\text{ProB}\) falls → \(B_\text{strength}\) falls → A/B compartment identity weakens → genome unfolds → genes normally silenced by heterochromatin proximity become derepressed. This derived variable connects per-locus methylation to the structural genome effect that Fourel et al. identify as the mechanism of methylation-loss pathology in cancer (Bonnet et al. 2026). For ME/CFS, the applicable mechanism is more targeted: HSAT2-specific loss-meCpG is proposed rather than global redistribution (Geneviève Fourel, pers. comm. May 2026).
Functional form of \(f\). HP1-driven heterochromatin formation operatingally via liquid-liquid phase separation (Strom et al. 2017) (Larson et al. 2017). Both Drosophila HP1a and human HP1α nucleate into phase-separated droplets only above a critical concentration threshold — below this threshold, droplets fail to nucleate and the compartment dissolves. This biophysics predicts that \(f\) is sigmoidal rather than linear: ProB repeat methylation must exceed a nucleation threshold to nucleate the HP1/H3K9me3 condensate that maintains B compartment identity. Near the threshold, small additional methylation loss produces disproportionate compartment weakening (the steep region of the sigmoid). Formally: \(f(x) = 1 / (1 + e^(-k(x - x_0)))\) where \(x_0\) is the nucleation threshold and \(k\) the cooperativity parameter. \(B_\text{strength}\) is a candidate parameter for linking the epigenomic subsystem to downstream gene expression models; both \(x_0\) and \(k\) are currently unconstrained for ProB repeat loci in ME/CFS.
Provisionality. The ProA/ProB repeat framework (Bonnet, Hulo, Fourel et al. 2026 preprint) is a computational genomics hypothesis developed in cancer; its extension to ME/CFS is our extrapolation, not endorsed by those authors, and the framework itself is not yet peer-reviewed. The vector model inherits this uncertainty: it provides a better mathematical container for the ProA/ProB hypothesis than the scalar model, but the hypothesis content within the container remains provisional.
2 Formal Test of the ProA/ProB Hypothesis: Compartment Bistability Model
The ProA/ProB hypothesis makes a core quantitative claim: ProB repeat sequences actively determine B compartment identity, and their methylation-dependent silencing is the rate-limiting step. This claim can be formalized and tested using existing Hi-C and methylation data.
Compartment bistability. For each genomic bin \(j\) in a Hi-C experiment, define the net compartment-driving signal as the weighted sum of methylated ProB repeats minus unmethylated ProA repeats in 3D proximity (contact weight \(w_{i j}\)):
\[ S_j = \sum_{i \in \text{ProB}} w_{i j} m_i - \sum_{i \in \text{ProA}} w_{i j} (1 - m_i) \]
Compartment score \(C_j\) is B (negative) when \(S_j\) is below a threshold, A (positive) when above it, and undetermined in between. As methylation at ProB repeats falls (DNMT3B redistribution away from heterochromatin), the ProB contribution weakens and the ProA contribution strengthens — producing a categorical B-to-A compartment flip rather than a gradual shift, because the threshold creates a discontinuity.
HP1 nucleation strengthens bistability. HP1 binding to H3K9me3 is cooperative, and HP1 concentration must exceed a critical threshold for liquid droplet nucleation (Strom et al. 2017) (Larson et al. 2017). The local HP1 concentration \(H_j\) is proportional to the weighted sum of methylated ProB repeats near bin \(j\). Once \(H_j\) exceeds the nucleation threshold, the phase-separated HP1 condensate adds a cooperative term \(\gamma\) to the compartment-driving signal, reinforcing B identity. This creates hysteresis: the HP1 concentration needed to nucleate a new B compartment exceeds the concentration needed to maintain an existing one — the biophysical analogue of the epigenetic hysteresis captured in the ODE model (Extended Subsystem Couplings).
Falsifiable predictions. The compartment bistability model generates testable predictions from existing data:
Compartment identity correlates with local ProB repeat methylation. Hi-C bins with higher weighted ProB repeat methylation should have more negative PC1 values (stronger B identity). Testable with existing Hi-C + WGBS data — no new experiment needed.
B-to-A switches occur first at sparse ProB repeat bins. Bins with few ProB repeats near the threshold should flip compartment earlier in cancer progression than bins with dense ProB repeats. This predicts an order of compartment loss determined by local ProB repeat composition, not by gene content.
The transition is discontinuous. As ProB methylation decreases, the fraction of B-compartment bins should decline sigmoidally (steep region) rather than linearly. The steepness parameter can be tested for deviation from linearity.
DNMT3B inhibition collapses B compartments at ProB-dense loci. Pharmacological DNMT3B inhibition should preferentially erase B identity at genomic locations with high ProB repeat density and Hi-C contact with satellites — not uniformly.
Testability. Prediction 1 requires only public Hi-C + methylation data. Prediction 3 requires 2 or more time points and a linearity test. Predictions 2 and 4 require perturbation experiments but are well-defined. The ProA/ProB framework is thus partially testable from existing data, unlike theories requiring inaccessible measurements.
Relationship to the vector model. The compartment bistability model bridges \(\mathbf{\mathcal{M}}\) to \(B_\text{strength}\) (The Disease State ODE System). \(B_\text{strength}\) is computable: the fraction of bins with \(C_j = -1\), or the mean PC1 value. DNMT3B redistribution → \(\mathbf{\mathcal{M}}_\text{ProB}\) falls → \(B_\text{strength}\) falls → compartment identity lost at weakest bins first → fraction of B-compartment bins declines along a sigmoidal nucleation-driven phase transition. The transition becomes irreversible if sufficient ProB repeat methylation falls below \(m_i^\text{crit}\) (The Disease State ODE System), because remethylation requires DNMT3B activity already redistributed to ProA sites — the zero-sum constraint prevents spontaneous recovery.
Certainty. 0.45 for the compartment bistability formulation (ProA/ProB framework is a preprint; HP1 nucleation threshold experimentally established but not linked to ProB repeats; Hi-C correlation prediction untested). 0.35 for hysteresis/bistability (consistent with HP1 phase separation (Strom et al. 2017) but no direct evidence). 0.25 for testability within 2 years (Hi-C + WGBS integration computationally feasible but not performed).
3 Threat Signal Composition
The composite threat signal \(\mathcal{T}\) that governs safe mode activation is a weighted sum:
\[ \mathcal{T} = w_\text{cyto} dot C_\text{pro} + w_\text{ROS} dot [\text{ROS}] + w_\text{LPS} dot [\text{LPS}] + w_V dot V \]
where \(w_\text{cyto} = 0.35\), \(w_\text{ROS} = 0.25\), \(w_\text{LPS} = 0.20\), \(w_V = 0.20\) are weights reflecting the relative contribution of each signal. Note that \([\text{ROS}]\) and \([\text{LPS}]\) are derived quantities (functions of the primary state variables) rather than independent state variables. These weights are not fixed constants across patients—genetic polymorphisms in oxidative stress sensing may alter \(w_\text{ROS}\) (see Metabolic Safe Mode as Trigger-Capable Root Cause), and the dominant signal source shifts as disease progresses.
4 Timescale Hierarchy
A critical structural feature of the ODE system is the separation of timescales across state variables. The response time \(\tau\) of each variable spans orders of magnitude:
| State Variable | Timescale \(\tau\) | Biological Basis |
|---|---|---|
| Calcium signaling (\(\text{Ca}^{2+}\) flux) | Minutes–hours | Ion channel kinetics, intracellular buffering |
| ATP / Complex I (\(\alpha_\text{CI}\)) | Hours–days | Mitochondrial biogenesis, substrate replenishment |
| Safe mode (\(S\)) | Days–weeks | Hypothalamic setpoint adjustment, cytokine clearance |
| Cytokines (\(C_\text{pro}\)) | Weeks–months | Immune cell reprogramming, clonal expansion/contraction |
| Autoantibodies | Months | Plasma cell lifespan, IgG half-life (\(\\approx\) 21 days) |
| Epigenetic consolidation (\(\mathbf{\mathcal{M}}\)) | Months–years | DNMT3A/B-driven methylation/de novo remethylation; DNMT3B redistribution ProB → ProA; TET-mediated erosion; passive demethylation. DNMT3B specificity: CpG + CAG trinucleotides (neurons, development). Loci below \(m_i^\text{crit}\) require active remethylation (SAM-dependent, slow). |
This timescale separation has profound implications for both disease dynamics and treatment strategy. Fast variables (calcium, ATP) respond quickly to perturbation but also revert quickly when the perturbation is removed. Slow variables (epigenetics) respond sluggishly but, once changed, persist long after the driving force has ceased. The slowest variable—epigenetic consolidation—effectively acts as the system’s “memory,” recording the cumulative disease history into the genome.