Antibody Avidity (2D): Bivalent Antigen Binding

Binding Kinetics

Association rate k_on[Ag] 0.90 /s

Per-arm dissociation k_off 0.60 /s

Local re-engagement k_reb 1.20 /s

Bivalent IgG (vs. Fab)

Live Readouts

Bound fraction (particles)

Bound fraction (ODE)

Measured k_eff (particle regression)

Measured k_eff (ODE regression)

Theoretical k_eff & avidity gain

How it works

This is the same bivalent-antibody avidity model as the 3D biosensor scene, rebuilt as an independent 2D engine: a top-down view of the sensor surface plus a genuine second computation running alongside it. Each antibody is a discrete particle cycling through three Markov states with per-arm rate constants: unbound U, singly bound S1, and doubly bound S2 (bivalent IgG only). A released arm sitting beside its still-attached partner has high local effective concentration and can rebind before the molecule diffuses away:

U  --kon[Ag]-->  S1        S1 --koff-->  U
S1 --k_reb-->    S2        S2 --2·koff--> S1

In parallel with the per-particle stochastic simulation, a continuum ODE for the population fractions is integrated every frame with 4th-order Runge–Kutta:

dS1/dt = -(koff+k_reb)·S1 + 2koff·S2
dS2/dt =  k_reb·S1 - 2koff·S2

During Wash (no free antigen), diagonalizing this 2×2 linear system gives the slow eigenvalue that governs the long-time decay of the bound fraction S1+S2:

k_eff = [ (3koff + k_reb) - sqrt((3koff + k_reb)^2 - 8koff^2) ] / 2

The chart above the readouts plots both the noisy particle-population curve and the smooth deterministic ODE curve during Wash, live-fit with a least-squares regression on ln(bound). They should converge to the same k_eff, which in turn should match the closed-form eigenvalue above -- three independent routes to the same number. When k_reb is large relative to koff (strong avidity) k_eff collapses far below the intrinsic koff; switch off "Bivalent IgG" to see k_eff reduce exactly to koff, since a monovalent Fab has no second arm to rebind with.

Inject / Wash — toggles whether free antigen is available (Inject) or removed so only dissociation and rebinding continue (Wash, used to measure k_eff).
Measured k_eff (particles) — least-squares fit of ln(bound fraction) vs. time over the stochastic particle population.
Measured k_eff (ODE) — the same regression applied to the deterministic continuum trajectory, which has no sampling noise.
Reset — clears every antibody and the ODE state back to unbound and restarts both decay histories.

Inject

particles (bound %) ODE (bound %)

Unbound antibody Singly bound (S1) Bivalent bound (S2) Antigen on surface