The olfactory transduction switch

Read as an R19 switch, the olfactory cascade ends in an all-or-none flip past the spinodal. In the model the cubic −s³ makes it a step; its steepness ratio to a graded control depends on the step size (229×, 48× or 24×). CNGB1 γ = 1.4357 equals the rod-vision value — a consistency check.

Each transduction gene's R19 field flips discontinuously past its spinodal; the cubic −s³ (order 3) is necessary in the model; its steepness ratio to a graded control is ≈229× at a ±1 % cubic vs ±10 % linear step and depends on those steps (code output). CNGB1 γ = 1.4357 equals the rod-vision value — a consistency check (same gene, same promoter window, same pipeline), not an independent verification.

All-or-none: the flip is discontinuous past the spinodal

The olfactory cascade (odorant → OR → Golf/GNAL → ADCY3/cAMP → CNG channel → ANO2) ends in the same R19 flip as vision. Settled from rest, each gene's field stays dark below its spinodal h* and snaps on above it with a finite, discontinuous jump.

On the principal CNG subunit CNGA2 the discontinuity is sharp: at 1.00·h* the field is still off (s = -0.6725), and one further quantum of drive at 1.01·h* flips the whole switch on (s = +1.2971). Less than a quantum does nothing.

The cooperativity is the cubic −s³, and it is necessary

The restoring term is third-order, so the cooperativity order is n = 3 — this is the model's cooperativity. Reported olfactory Hill coefficients are observations: the olfactory CNG channel's cAMP dose–response has a measured Hill coefficient of about 1.4–1.8 (Frings, Lynch & Lindemann 1992, J. Gen. Physiol. 100:45); linking them to the cubic is interpretation — the order of the normal form is not itself a Hill coefficient, and no mapping between the two is derived here. A structural control with the cubic struck out (ds/dt = −γ·s + h) loses the threshold, the basin, and the jump.

Across the fold the cubic switch slope is ≈ 177.2 while the graded control is ≈ 0.773 — a ratio of ≈ 229×. Removing the cubic makes the response a smooth proportional curve; the cubic makes it a step.

Code output, step-dependent: 177.2 is a finite difference over ±1 % of h* and 0.773 over ±10 %. With a ±5 % or ±10 % cubic step the same code gives ≈ 48× or ≈ 24×; at the fold the cubic slope formally diverges, so the ratio has no fixed value. The script's own grade line asserts only ≥ 20×.

CNGB1 — the same gene in both senses (a consistency check)

CNGB1, the CNG channel β subunit, is literally the same gene in rod vision and olfaction. Its olfactory γ = 1.4357 equals the rod-vision value 1.4357. This is consistency, not independent verification: the nose and eye volumes read the same promoter window (NC_000016.10, TSS−2000..+500, same strand) with the same pipeline, and the rod value is hard-coded in E2/run.py. That one gene serves both senses is an observation; reading it as one reused R19 switch is interpretation. Whether one promoter window covers both the rod (CNGB1a) and olfactory (CNGB1b) isoforms is [O].

Ordered by threshold the cascade reads CNGA2 → ANO2 → CNGA4 → GNAL → CNGB1 → ADCY3. Whether this spinodal order matches the real biochemical sequence (Golf → ADCY3 → cAMP → CNG → ANO2) is an open empirical question — it needs the measured cascade kinetics and is not assumed. The absolute odorant→drive→firing (Hz) scale is a separate calibration [O].