The science

This page explains the demand convention, the cycle-index readout, the mode classification, and the biological context the phase map recapitulates.

Supply/demand convention

The phase map is built on the integrated glycolysis + PPP kinetic model (the integrated model is required because it closes the pentose cycle: the F6P and GAP produced by the non-oxidative PPP re-enter glycolysis and regenerate G6P). Three demand levers drive the system, each implemented as a tunable maximal-rate sink:

• NADPH demand — redox / biosynthetic draw on NADPH,
• R5P demand — ribose / nucleotide draw (a PRPS1-anchored sink),
• ATP demand — energy draw / glycolytic pull (held as a fixed background per run, and varied by the explorer's ATP slider).

Each lever is set as a fraction φ of that species' sustainable supply ceiling S_X, so all three axes mean the same thing — "percent of what the pathway can actually make." The ceilings are anchored on the carbon bottleneck, ribose-5-phosphate isomerase (RPI):

with RPI_cap = RPI_Vmax · RPI_Conc. RPI is the bottleneck because it carries one third of the ribulose-5-phosphate in the full cycle, so the maximum sustainable oxidative flux is 3 · RPI_cap and the NADPH ceiling is 2 · (3 · RPI_cap) = 6 · RPI_cap (the oxidative enzymes are far over-provisioned relative to RPI/RPE clearance, so they are not the operative ceiling). R5P has two sources — RPI forward and the reverse non-oxidative ribose route — but the X5P co-produced by the reverse route must return through RPE → Ru5P → RPI, so two thirds of R5P still funnels through RPI and the sustainable R5P ceiling is 1.5 · RPI_cap. ATP demand anchors on the glycolytic ceiling 2 · HK1_cap.

The reversible non-oxidative reactions (RPI, RPE, the two transketolase reactions, and transaldolase) change sign between regimes; their net direction is the physical readout of linear ↔ pentose-cycle ↔ reverse PPP operation.

The pentose-cycle index

The single scalar summarizing each operating point is the pentose-cycle index

cycle index = (V_ox − V_R5Pase) / V_ox

where V_ox is the oxidative flux through G6PD and V_R5Pase is the R5P-export (demand) flux. It is the fraction of oxidative-PPP-generated pentose carbon that is recycled back into glycolysis rather than exported as ribose:

• ≈ 1 — full pentose cycle: oxidative carbon is recycled (the pathway runs to make NADPH),
• ≈ 0 — linear PPP: oxidative carbon leaves as R5P,
• < 0 — reverse PPP: R5P is produced non-oxidatively, faster than oxidative supply,
• NaN — undefined when V_ox ≈ 0.

Mode classification

Each cell is classified into one of four modes from its fluxes:

• :cycle — forward pentose cycle (index ≈ 1),
• :linear — linear oxidative-then-export operation (index ≈ 0),
• :reverse — reverse / non-oxidative R5P production (index < 0),
• :undetermined — oxidative flux too small to classify (index undefined).

Non-converged operating points are masked in the heatmaps (and counted), never silently dropped.

Biological context

The integrated glycolysis + PPP model is a kinetic (ODE) model of central carbon metabolism, so it can be used to predict how a cell re-routes glucose carbon as demand changes — in particular the two demands that set the oxidative/non-oxidative balance of the PPP:

• Oxidative stress raises the draw on NADPH: glutathione- and thioredoxin-based antioxidant systems regenerate their reduced cofactors by consuming NADPH, so a burst of reactive oxygen species reads out, metabolically, as a step up in NADPH demand.
• Ribose / proliferative demand raises the draw on ribose-5-phosphate for nucleotide (PRPP) synthesis, which the model carries as R5P demand.

Sweeping these two demands at a fixed ATP/glycolytic background and solving the model to steady state traces out the response surface shown in the phase map: as oxidative stress and ribose demand move, the model predicts whether the PPP runs linearly (oxidize, then export ribose), closes into a forward pentose cycle (recycle carbon back into glycolysis), or reverses its non-oxidative branch to make ribose without oxidation. Because the kinetic model is a full dynamical system, the same machinery can also be integrated for time-resolved trajectories — e.g. the transient flux redistribution following an acute oxidative insult — not just the steady states summarized here.