Hormesis is a well-documented toxicological phenomenon in which a substance that is harmful at high doses produces a mild beneficial or stimulatory effect at low doses — the dose-response curve is biphasic (an inverted-U for stimulatory endpoints) rather than the simple monotonic decline assumed by a threshold model. It is quantified with the Brain–Cousens hormesis model, an extension of the standard log-logistic dose-response curve:
y(x) = c + (d − c + f·x) / (1 + (x/e)^b)
x = dose d = control response (x→0)
c = minimum response (high-dose plateau)
e = ED50, dose giving the midpoint of inhibition
b = Hill slope (steepness of the toxic decline)
f = hormetic coefficient — the low-dose stimulatory term
At very low x, the (d − c + f·x) numerator rises slightly faster than the denominator suppresses it, lifting y(x) above the control baseline — mild stress activates adaptive repair pathways (e.g. Nrf2 antioxidant response, heat-shock proteins) that overcorrect. As dose keeps climbing, the sigmoid denominator dominates and viability collapses toward the toxic plateau c.
- Dose slider — moves a real toxin concentration (log scale) through the curve; the marker plane tracks it live.
- Hormetic strength f — how strong the low-dose overcorrection is; f = 0 recovers an ordinary monotonic dose-response curve.
- Potency (slope b) and ED50 (e) — set how steep and how far right the toxic decline sits, exactly as they do for real LD50/EC50 curves.
- Each sphere in the population grid is an individual organism with a fixed random sensitivity draw; it "dies" (dims to grey) once the current viability probability drops below its personal threshold, so the population's own count of survivors is what drives the viability readout — not just the formula.
Real-world relevance: hormesis is documented for many chemical stressors, drugs (e.g. low-dose statins, some chemotherapeutics) and — controversially — ionizing radiation, where it sits opposite the linear-no-threshold (LNT) model used for regulatory radiation protection. Toxicologists led by Edward Calabrese have argued hormetic dose-response curves are far more common in the literature than classical threshold models assume.