- Select a model (Single-site, Hill, Competitive, or Two-site).
- Enter parameters (Kd, n, competitor values, etc.) with correct units.
- For a curve, enable batch and provide a list or generate a range.
- Click Calculate. Results show above the form, and exports become available.
- Single-site: θ = [L]/([L]+Kd)
- Hill: θ = [L]n / ([L]n + Kdn)
- Competitive: θA = ([A]/KA) / (1 + [A]/KA + [B]/KB)
- Two-site: θ = f·[L]/([L]+Kd1) + (1−f)·[L]/([L]+Kd2)
- Depletion (1:1 totals): Bound = ((Lt+Rt+Kd) − √((Lt+Rt+Kd)² − 4LtRt)) / 2, then θ = Bound/Rt
- Cheng–Prusoff: Ki = IC50 / (1 + [Lrad]/Kdrad)
1. Receptor occupancy in physical terms
Receptor occupancy is the fraction of binding sites that are ligand‑bound at equilibrium. It links concentration to site filling, so it helps with dose planning, sensor response, and adsorption modeling. The calculator reports occupancy as a fraction and percent. It also helps compare experiments performed at different concentrations and temperatures.
2. Single‑site binding and Kd
In the single‑site (Langmuir) model, occupancy follows θ = [L]/([L]+Kd). Kd is the concentration that gives 50% occupancy, so Kd sets the midpoint. For example, Kd = 10 nM implies θ ≈ 0.50 at 10 nM, θ ≈ 0.09 at 1 nM, and θ ≈ 0.91 at 100 nM.
3. Cooperativity with the Hill model
Some systems show cooperative effects where binding is steeper or shallower than single‑site theory. The Hill model uses θ = [L]^n/([L]^n+Kd^n). When n > 1, the transition around Kd sharpens; when n < 1, it broadens. This tool lets you explore how n changes the slope while the midpoint stays near Kd.
4. Competitive binding under a shared site
When two ligands compete for the same receptors, occupancy depends on both concentrations and affinities. The model is θA = ([A]/KA)/(1 + [A]/KA + [B]/KB). It supports inhibition studies and displacement assays, where a fixed competitor level reduces occupancy of the primary ligand. By scanning [A] in batch mode, you can see how potency changes with [B].
5. Two‑site binding for heterogeneous receptors
Materials and receptors can have heterogeneous sites. The two‑site model blends two Langmuir terms: θ = f·θ1 + (1−f)·θ2, where f is the fraction of site‑type 1. If Kd1 ≪ Kd2, you often see an early rise from high‑affinity sites, then a slower approach to saturation.
6. Ligand depletion and finite receptor effects
At low volumes or high receptor densities, free ligand can be depleted by binding, so the simple formula overestimates occupancy. The depletion option solves the 1:1 equilibrium using total ligand Lt, total receptor Rt, and Kd, then reports bound and θ = Bound/Rt. This matters in tight‑binding regimes and microfluidic assays.
7. Curves, targets, and log spacing
Batch mode generates a curve from either a typed list or a defined range. Log‑spaced ranges are often best because binding spans orders of magnitude. The summary can show approximate concentrations for 10%, 50%, and 90% occupancy. Use these targets to choose points that bracket the transition region.
8. Units, reporting, and export
The calculator converts between pM, nM, µM, mM, and M so inputs stay consistent. Rounding can be fixed‑decimal or significant‑figure based. After a run, export a CSV table for plotting, or a PDF summary for sharing. The on‑page plot offers a quick visual check. Saved outputs include the full curve table and key targets for reporting.
FAQs
What does Kd represent in this calculator?
Kd is the dissociation constant and equals the ligand concentration giving 50% occupancy in the single‑site model. Smaller Kd means higher affinity and higher occupancy at the same concentration.
When should I use the Hill model?
Use the Hill model when data show a steeper or flatter transition than single‑site binding. The Hill coefficient n adjusts curve steepness: n>1 suggests positive cooperativity; n<1 suggests negative cooperativity or heterogeneity.
How does competitive binding differ from single‑site occupancy?
Competitive binding accounts for a second ligand competing for the same sites. Occupancy for ligand A decreases as competitor B increases, depending on their affinities KA and KB and their concentrations.
What is the purpose of the two‑site model?
It approximates heterogeneous receptors or surfaces by mixing two affinities. The fraction f sets how much of the population follows Kd1 versus Kd2, producing a curve with two characteristic regions.
When should I enable ligand depletion correction?
Enable it when receptor concentration is not negligible relative to ligand, or when binding is tight. The tool uses total ligand and total receptor to solve bound ligand, then computes occupancy as Bound/Rt.
Why do you recommend log‑spaced ranges for curves?
Binding transitions often span orders of magnitude around Kd. Log spacing places more points near the steep part of the curve, giving smoother plots and better estimates of 10%, 50%, and 90% targets.
What do the CSV and PDF downloads include?
CSV contains the summary plus the full curve table for external plotting. PDF provides a compact text summary and a preview of the first rows, useful for sharing results without extra software.