Mast cells display IgE antibodies bound to high-affinity Fcε receptors (FcεRI). A single allergen molecule binding one receptor does nothing — degranulation needs a multivalent allergen to physically bridge two IgE-occupied receptors, cross-linking them:
Receptor + IgE-Ag -> mono-bound receptor (one arm free)
mono-bound + free receptor -> cross-linked pair (bridge closed)
ρ = (cross-linked pairs) / (total receptors / 2)
P(degranulate) = ρ^n / (ρ^n + Kd^n), n = 3
Each frame the simulator rolls this Hill-function probability against the live cross-link density ρ, plotted continuously on the strip chart below the membrane view against the Kd threshold line. Once enough bridges accumulate, the threshold is crossed, calcium influx is triggered, and the granule payload — histamine, tryptase, leukotrienes — is dumped in one burst, after which every receptor and antigen resets to free.
This 2D version renders the identical closed-form binding/unbinding kinetics as the 3D source (same capture radius, search rate, off-rates and Hill exponent — only the rendering changed): receptors sit on a real sphere in an internal (x,y,z) coordinate space, and dragging the membrane view rotates that sphere about two axes exactly like an orbit camera, projected orthographically with depth-based size and opacity so the far hemisphere reads as behind translucent cytoplasm.
- Allergen dose — how many allergen molecules are circulating near the cell. Counter-intuitively, cranking this to maximum does not always degranulate fastest: at very high dose, most receptors get grabbed singly by different allergen molecules before either arm finds a second free receptor, so cross-linking is starved of partners — a real phenomenon called the prozone / high-dose hook effect. Watch the amber (mono-bound) population balloon while red (cross-linked) stays low at extreme dose, and the ρ trace on the strip chart plateau below Kd.
- Receptor density — more FcεRI sites per cell makes finding a second partner easier at a given dose, lowering the effective threshold dose.
- Kd threshold — the cross-link density needed for a 50% chance of firing; this stands in for individual sensitivity (a highly sensitized, anaphylaxis-prone cell has a low Kd). The dashed line on the strip chart tracks it live.
- Inject Allergen Pulse — simulates a sting or food exposure: a burst of allergen molecules appears near the membrane immediately.
- Drag the membrane view — orbit the sphere manually; auto-rotate speed and view zoom are independently adjustable, and Reset view snaps the camera back to the default angle.
Numerical check on the source model: the density formula ρ = cross-linked pairs / (receptors/2) and the Hill exponent n = 3 used here were verified against the 3D engine's own update loop — both are internally consistent with the "50% chance to fire at ρ = Kd" description in its theory text, so no correction was needed; this 2D engine reuses the same constants (capture radius 0.17, search-partner reach 0.55, off-rates 0.35/s mono and 0.06/s cross-linked) rather than re-deriving them.