Dose-Response / Concentration-Response Analysis

Turn paired concentration vs response measurements into a potency (IC50/EC50), a Hill slope, an efficacy (Emax), and a quality judgment — and compare potency between compounds.

When to use this

• Enzyme inhibition / activation assays, cell viability, reporter assays, radioligand binding, agonist/antagonist pharmacology.
• You have a list of concentrations and the response at each, and want IC50/EC50 + Hill slope.
• You want to say "compound A is N-fold more potent than B."

The model is the 4-parameter logistic (4PL) / Hill sigmoidal: f(x) = Emin + (Emax − Emin) / (1 + (EC50/x)^n) — where n is the Hill slope.

Step 1 — Prepare the data (where results go wrong)

Issue	What to do
Concentration units	Pick ONE unit (µM, nM, M) and use it for every point. The IC50 comes back in that unit. Don't mix.
Log vs linear concentrations	Pass concentrations on the linear scale (e.g. 0.01, 0.1, 1, 10), not log10. The fitter logs internally.
Zero/control concentration	Drop a literal 0 concentration (log(0) is undefined). Keep it only as the Emin/Emax reference if normalizing.
Direction	Inhibition curves go high→low (IC50); activation curves go low→high (EC50). The tools handle both; just be consistent.
Normalization	Convert raw signal to % of control if you want Emax/Emin near 100/0: % = 100 × (raw − blank)/(control − blank). Raw values also fit, but plateaus are then in raw units.
Replicates	Average technical replicates per concentration before fitting, or pass all points (the fit weights them equally).

Coverage requirement: you need ≥4 points (the tools require it) and ideally 6–8 spanning both plateaus — points clearly above and clearly below the inflection. A curve that never plateaus gives an unreliable, extrapolated IC50 (see Step 4).

Step 2 — Fit / get the potency

Single curve → IC50 or EC50 (same math; "IC50" for inhibition, "EC50" for activation):

tu run DoseResponse_calculate_ic50 '{"operation":"calculate_ic50",
  "concentrations":[0.001,0.01,0.1,1,10,100],
  "responses":[98,95,80,45,12,3]}'

Returns ic50, ic50_95_confidence_interval, hill_slope, emax, emin, r_squared, log_ic50.

Full 4PL parameters only → DoseResponse_fit_curve (same inputs). Two compounds → DoseResponse_compare_potency with conc_a/resp_a/conc_b/resp_b (returns each IC50 + ic50_fold_shift_b_over_a + more_potent).

For non-standard needs (constrained plateaus, weighting, plotting), scripts/fit_dose_response.py runs a scipy 4PL fit from a CSV and matches the tool.

Step 3 — Interpret the four parameters

Parameter	Meaning	Sanity check
IC50 / EC50	Concentration giving half-maximal effect — the potency. Lower = more potent.	Should fall within your tested range; if it's at/beyond an endpoint, the curve is incomplete (Step 4).
Hill slope n	Steepness / apparent cooperativity. ~1 = simple one-site. >1.5 = steep/positive cooperativity (or non-specific). <0.5 = shallow/multiple sites or heterogeneity.	A wildly large `
Emax	Maximal response (top plateau) = efficacy.	For % data, full agonist ≈100; a partial agonist plateaus well below 100 even at saturating dose.
Emin	Bottom plateau (baseline/floor).	For % inhibition data, ≈0 for a complete inhibitor.
r²	Fit quality.	≥0.95 good; <0.90 → inspect for outliers, wrong model, or incomplete curve before trusting the IC50.

Potency comparison: report the fold-shift in IC50/EC50 (e.g. "A is 6.2× more potent than B"), and only call it meaningful if both fits are good (r²≥0.95) and the Hill slopes are comparable — a potency ratio between curves of very different slope is not a clean comparison.

Step 4 — Quality gotchas (state these, don't hide them)

• Incomplete curve / no plateau. If responses don't flatten at both ends, Emax/Emin (and thus IC50) are extrapolated and unstable. Report the IC50 as "approximate / right-shifted of the tested range" and recommend wider concentrations.
• <4–5 points or none near the inflection. The fit can converge to a nonsense IC50 with a high r². Check that points actually bracket the IC50.
• Biphasic / U-shaped data. A single 4PL is wrong for hormesis or two-site behavior — the fit will look poor (low r²); flag it rather than forcing one IC50.
• IC50 vs Ki. IC50 depends on assay conditions (substrate/ligand concentration). Don't report IC50 as an affinity (Ki) without a Cheng-Prusoff correction.
• Units. The IC50 is only as correct as the concentration unit you fed in — always state the unit.

Honest limitations

• 4PL assumes a monotonic sigmoid; it cannot describe biphasic, bell-shaped, or steep all-or-none responses.
• A confident IC50 from a poor or incomplete curve is the most common error — let r² and curve coverage gate how you report it.
• Potency (IC50/EC50) is not efficacy (Emax) — a more potent compound can be a weaker (partial) agonist; report both.

Related skills

• tooluniverse-image-analysis — dose-response on image-derived measurements (.tif, colony, fluorescence).
• tooluniverse-gpcr-structural-pharmacology / tooluniverse-network-pharmacology — receptor pharmacology context.
• tooluniverse-statistical-modeling — general regression, EC50 via spline, power analysis.