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):

bash
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 |n| (>4) usually means a bad fit or too few transition points, not real cooperativity. | | 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.