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.
1---2name: tooluniverse-dose-response3description: Dose-response / concentration-response curve fitting — IC50, EC50, Hill slope, Emax/Emin efficacy, and relative potency from paired concentration vs response data (enzyme/cell assays, drug screening, agonist/antagonist pharmacology). Fits the 4-parameter logistic (Hill sigmoidal) model. Use when you have concentrations + responses and need a potency value, to compare two compounds' potency, or to judge curve quality. NOT for image-derived dose-response (use tooluniverse-image-analysis) and NOT for survival/regression (use tooluniverse-statistical-modeling).4---5
6# Dose-Response / Concentration-Response Analysis
7
8Turn 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.
9
10## When to use this
11
12- Enzyme inhibition / activation assays, cell viability, reporter assays, radioligand binding, agonist/antagonist pharmacology.
13- You have a list of concentrations and the response at each, and want IC50/EC50 + Hill slope.
14- You want to say "compound A is N-fold more potent than B."
15
16The model is the 4-parameter logistic (4PL) / Hill sigmoidal:
17`f(x) = Emin + (Emax − Emin) / (1 + (EC50/x)^n)` — where `n` is the Hill slope.
18
19## Step 1 — Prepare the data (where results go wrong)
20
21| Issue | What to do |
22|---|---|
23| **Concentration units** | Pick ONE unit (µM, nM, M) and use it for every point. The IC50 comes back in that unit. Don't mix. |
24| **Log vs linear concentrations** | Pass concentrations on the **linear** scale (e.g. `0.01, 0.1, 1, 10`), not log10. The fitter logs internally. |
25| **Zero/control concentration** | Drop a literal `0` concentration (log(0) is undefined). Keep it only as the Emin/Emax reference if normalizing. |
26| **Direction** | Inhibition curves go high→low (IC50); activation curves go low→high (EC50). The tools handle both; just be consistent. |
27| **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. |
28| **Replicates** | Average technical replicates per concentration before fitting, or pass all points (the fit weights them equally). |
29
30**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).
31
32## Step 2 — Fit / get the potency
33
34Single curve → IC50 or EC50 (same math; "IC50" for inhibition, "EC50" for activation):
35
36```bash
37tu run DoseResponse_calculate_ic50 '{"operation":"calculate_ic50",
38 "concentrations":[0.001,0.01,0.1,1,10,100],
39 "responses":[98,95,80,45,12,3]}'
40```
41
42Returns `ic50`, `ic50_95_confidence_interval`, `hill_slope`, `emax`, `emin`, `r_squared`, `log_ic50`.
43
44Full 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`).
45
46For non-standard needs (constrained plateaus, weighting, plotting), `scripts/fit_dose_response.py` runs a scipy 4PL fit from a CSV and matches the tool.
47
48## Step 3 — Interpret the four parameters
49
50| Parameter | Meaning | Sanity check |
51|---|---|---|
52| **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). |
53| **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. |
54| **Emax** | Maximal response (top plateau) = efficacy. | For % data, full agonist ≈100; a **partial agonist** plateaus well below 100 even at saturating dose. |
55| **Emin** | Bottom plateau (baseline/floor). | For % inhibition data, ≈0 for a complete inhibitor. |
56| **r²** | Fit quality. | ≥0.95 good; <0.90 → inspect for outliers, wrong model, or incomplete curve before trusting the IC50. |
57
58**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.
59
60## Step 4 — Quality gotchas (state these, don't hide them)
61
62- **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.
63- **<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.
64- **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.
65- **IC50 vs Ki.** IC50 depends on assay conditions (substrate/ligand concentration). Don't report IC50 as an affinity (Ki) without a Cheng-Prusoff correction.
66- **Units.** The IC50 is only as correct as the concentration unit you fed in — always state the unit.
67
68## Honest limitations
69
70- 4PL assumes a monotonic sigmoid; it cannot describe biphasic, bell-shaped, or steep all-or-none responses.
71- A confident IC50 from a poor or incomplete curve is the most common error — let r² and curve coverage gate how you report it.
72- Potency (IC50/EC50) is not efficacy (Emax) — a more potent compound can be a weaker (partial) agonist; report both.
73
74## Related skills
75- `tooluniverse-image-analysis` — dose-response on image-derived measurements (.tif, colony, fluorescence).
76- `tooluniverse-gpcr-structural-pharmacology` / `tooluniverse-network-pharmacology` — receptor pharmacology context.
77- `tooluniverse-statistical-modeling` — general regression, EC50 via spline, power analysis.
mims-harvard