A sigmoidal dose-response curve that plots biological response against drug concentration on a logarithmic x-axis, fitted using a four-parameter logistic (4PL) model. This visualization is essential for determining drug potency metrics such as EC50 (half-maximal effective concentration) or IC50 (half-maximal inhibitory concentration), Hill slope steepness, and upper/lower response asymptotes. It enables rapid visual comparison of compound efficacy and is a standard tool in pharmacological analysis.
""" anyplot.ai
curve-dose-response: Pharmacological Dose-Response Curve
Library: altair 6.2.2 | Python 3.13.14
Quality: 91/100 | Updated: 2026-06-24
"""
import os
import altair as alt
import numpy as np
import pandas as pd
from PIL import Image
from scipy.optimize import curve_fit
from scipy.stats import t as t_dist
# Theme-adaptive chrome tokens (Imprint palette)
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
INK_MUTED = "#6B6A63" if THEME == "light" else "#A8A79F"
# Imprint palette — positions 1 and 2 for two-series categorical
IMPRINT_PALETTE = ["#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477", "#99B314"]
# Data
np.random.seed(42)
concentrations = np.logspace(-9, -4, 10)
def logistic_4pl(x, bottom, top, ec50, hill):
return bottom + (top - bottom) / (1 + (ec50 / x) ** hill)
compound_params = {
"Atorvastatin": {"bottom": 5, "top": 95, "ec50": 1e-7, "hill": 1.2},
"Simvastatin": {"bottom": 8, "top": 88, "ec50": 3e-6, "hill": 1.8},
}
rows = []
for name, params in compound_params.items():
true_response = logistic_4pl(concentrations, params["bottom"], params["top"], params["ec50"], params["hill"])
noise = np.random.normal(0, 3, size=(5, len(concentrations)))
replicates = true_response + noise
means = replicates.mean(axis=0)
sems = replicates.std(axis=0, ddof=1) / np.sqrt(5)
for c, m, s in zip(concentrations, means, sems, strict=True):
rows.append(
{
"concentration": c,
"log_conc": np.log10(c),
"response": m,
"sem": s,
"response_upper": m + s,
"response_lower": m - s,
"compound": name,
}
)
df = pd.DataFrame(rows)
# Fit 4PL curves, compute smooth fit lines and 95% CI via delta method
fit_rows = []
ref_rows = []
ci_rows = []
for name, group in df.groupby("compound"):
xdata = group["concentration"].values
ydata = group["response"].values
params_init = compound_params[name]
p0 = [params_init["bottom"], params_init["top"], params_init["ec50"], params_init["hill"]]
popt, pcov = curve_fit(logistic_4pl, xdata, ydata, p0=p0, maxfev=10000)
bottom_fit, top_fit, ec50_fit, hill_fit = popt
x_smooth = np.logspace(-9.5, -3.5, 200)
y_smooth = logistic_4pl(x_smooth, *popt)
for xs, ys in zip(x_smooth, y_smooth, strict=True):
fit_rows.append({"log_conc": np.log10(xs), "response": ys, "compound": name})
n = len(xdata)
dof = n - len(popt)
t_val = t_dist.ppf(0.975, dof)
jacobian = np.zeros((len(x_smooth), 4))
for i, xs in enumerate(x_smooth):
ratio = (ec50_fit / xs) ** hill_fit
denom = 1 + ratio
jacobian[i, 0] = 1 - 1 / denom
jacobian[i, 1] = 1 / denom
jacobian[i, 2] = -(top_fit - bottom_fit) * hill_fit * ratio / (ec50_fit * denom**2)
jacobian[i, 3] = -(top_fit - bottom_fit) * ratio * np.log(ec50_fit / xs) / denom**2
pred_var = np.array([j @ pcov @ j for j in jacobian])
pred_se = np.sqrt(np.maximum(pred_var, 0))
ci_upper = logistic_4pl(x_smooth, *popt) + t_val * pred_se
ci_lower = logistic_4pl(x_smooth, *popt) - t_val * pred_se
for xs, cu, cl in zip(x_smooth, ci_upper, ci_lower, strict=True):
ci_rows.append({"log_conc": np.log10(xs), "ci_upper": cu, "ci_lower": cl, "compound": name})
half_response = (bottom_fit + top_fit) / 2
ec50_sci = f"{ec50_fit:.1e}"
ref_rows.append(
{
"compound": name,
"ec50_log": np.log10(ec50_fit),
"half_response": half_response,
"bottom_fit": bottom_fit,
"top_fit": top_fit,
"ec50_label": f"EC₅₀ = {ec50_sci} M",
"x_left": -9.5, # left edge of x-domain for clipped hline
"y_bottom": 0.0, # bottom of y-domain for clipped vline
}
)
df_fit = pd.DataFrame(fit_rows)
df_ci = pd.DataFrame(ci_rows)
df_ref = pd.DataFrame(ref_rows)
# Color scale — Imprint positions 1 (#009E73) and 2 (#C475FD)
color_scale = alt.Scale(domain=["Atorvastatin", "Simvastatin"], range=[IMPRINT_PALETTE[0], IMPRINT_PALETTE[1]])
# Shared encoding shortcuts for multi-layer color consistency
color_no_legend = alt.Color("compound:N", scale=color_scale, legend=None)
color_with_legend = alt.Color("compound:N", scale=color_scale, legend=alt.Legend(title="Compound"))
# Nearest-point hover selection
nearest = alt.selection_point(nearest=True, on="pointerover", fields=["log_conc"], empty=False)
base_x = alt.X(
"log_conc:Q",
title="log₁₀ Concentration (M)",
scale=alt.Scale(domain=[-9.5, -3.5]),
axis=alt.Axis(values=list(range(-9, -3))),
)
base_y = alt.Y(
"response:Q", title="Response (%)", scale=alt.Scale(domain=[0, 105]), axis=alt.Axis(values=[0, 20, 40, 60, 80, 100])
)
# 95% CI shaded bands
ci_band = (
alt.Chart(df_ci)
.mark_area(opacity=0.22)
.encode(x=alt.X("log_conc:Q"), y=alt.Y("ci_lower:Q"), y2="ci_upper:Q", color=color_no_legend)
)
# Fitted 4PL curves — legend anchor layer
fitted_lines = (
alt.Chart(df_fit).mark_line(strokeWidth=2.5).encode(x=alt.X("log_conc:Q"), y=base_y, color=color_with_legend)
)
# SEM error bars
error_bars = (
alt.Chart(df)
.mark_rule(strokeWidth=1.5)
.encode(x=alt.X("log_conc:Q"), y=alt.Y("response_lower:Q"), y2="response_upper:Q", color=color_no_legend)
)
# Invisible hover capture layer
select_layer = (
alt.Chart(df)
.mark_point(size=200, opacity=0)
.encode(x=alt.X("log_conc:Q"), y=alt.Y("response:Q"))
.add_params(nearest)
)
# Hover crosshair
hover_rule = (
alt.Chart(df)
.mark_rule(strokeWidth=1, color=INK_MUTED, strokeDash=[3, 3])
.encode(x=alt.X("log_conc:Q"))
.transform_filter(nearest)
)
# Data points with hover-size interaction and tooltips
data_points = (
alt.Chart(df)
.mark_point(filled=True, stroke="white", strokeWidth=1)
.encode(
x=base_x,
y=base_y,
color=color_no_legend,
size=alt.condition(nearest, alt.value(100), alt.value(50)),
tooltip=[
alt.Tooltip("compound:N", title="Compound"),
alt.Tooltip("log_conc:Q", title="log₁₀ [C]", format=".2f"),
alt.Tooltip("response:Q", title="Response (%)", format=".1f"),
alt.Tooltip("sem:Q", title="SEM", format=".2f"),
],
)
)
# EC50 reference lines — clipped to form an "L" pointing at the EC50 intersection.
# Horizontal: left edge → EC50 x-position (avoids cluttering the right half)
ec50_hlines = (
alt.Chart(df_ref)
.mark_rule(strokeDash=[6, 4], strokeWidth=1.5, opacity=0.5)
.encode(x=alt.X("x_left:Q"), x2="ec50_log:Q", y=alt.Y("half_response:Q"), color=color_no_legend)
)
# Vertical: bottom → half-response (drops down to the x-axis)
ec50_vlines = (
alt.Chart(df_ref)
.mark_rule(strokeDash=[6, 4], strokeWidth=1.5, opacity=0.5)
.encode(x=alt.X("ec50_log:Q"), y=alt.Y("y_bottom:Q"), y2="half_response:Q", color=color_no_legend)
)
# Top and bottom asymptote guides
asymptote_top = (
alt.Chart(df_ref)
.mark_rule(strokeDash=[3, 3], strokeWidth=1, opacity=0.4)
.encode(y=alt.Y("top_fit:Q"), color=color_no_legend)
)
asymptote_bottom = (
alt.Chart(df_ref)
.mark_rule(strokeDash=[3, 3], strokeWidth=1, opacity=0.4)
.encode(y=alt.Y("bottom_fit:Q"), color=color_no_legend)
)
# EC50 value labels — offset above the intersection to avoid overlap with reference lines
ec50_labels = (
alt.Chart(df_ref)
.mark_text(fontSize=11, fontWeight="bold", align="left", dx=5, dy=-10)
.encode(x=alt.X("ec50_log:Q"), y=alt.Y("half_response:Q"), text=alt.Text("ec50_label:N"), color=color_no_legend)
)
# Compose — resolve_scale(color="independent") needed because layers span
# four distinct DataFrames (df, df_fit, df_ci, df_ref); each layer's explicit
# scale range is identical, so colors stay consistent across layers.
chart = (
(
ci_band
+ asymptote_top
+ asymptote_bottom
+ ec50_hlines
+ ec50_vlines
+ fitted_lines
+ error_bars
+ data_points
+ ec50_labels
+ select_layer
+ hover_rule
)
.resolve_scale(color="independent")
.properties(
width=620,
height=320,
background=PAGE_BG,
title=alt.Title(
"curve-dose-response · python · altair · anyplot.ai",
subtitle="4-Parameter Logistic Fit with 95% Confidence Intervals",
fontSize=16,
subtitleFontSize=11,
subtitleColor=INK_SOFT,
color=INK,
anchor="start",
),
)
.configure_view(fill=PAGE_BG, strokeWidth=0)
.configure_title(color=INK)
.configure_axis(
labelFontSize=10,
titleFontSize=12,
labelColor=INK_SOFT,
titleColor=INK,
gridColor=INK,
gridOpacity=0.12,
domainColor=INK_SOFT,
tickColor=INK_SOFT,
)
.configure_legend(
titleFontSize=10,
labelFontSize=10,
symbolSize=80,
labelColor=INK_SOFT,
titleColor=INK,
fillColor=ELEVATED_BG,
strokeColor=INK_SOFT,
)
)
# Save — scale_factor=4.0 with width=620, height=320 targets 3200×1800
TW, TH = 3200, 1800
chart.save(f"plot-{THEME}.png", scale_factor=4.0)
chart.save(f"plot-{THEME}.html")
# Pad PNG to exact target (vl-convert can land slightly short; never crop)
_img = Image.open(f"plot-{THEME}.png").convert("RGB")
_w, _h = _img.size
if _w > TW or _h > TH:
raise SystemExit(
f"altair vl-convert produced {_w}×{_h}, exceeds target {TW}×{TH}. "
f"Shrink chart .properties(width=, height=) values and re-render."
)
if _w < TW or _h < TH:
_canvas = Image.new("RGB", (TW, TH), PAGE_BG)
_canvas.paste(_img, ((TW - _w) // 2, (TH - _h) // 2))
_canvas.save(f"plot-{THEME}.png")
Part of Pharmacological Dose-Response Curve on anyplot.ai.