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Langmuir 1:1 — Decaying Surface

Same 1:1 chemistry as the basic Langmuir model, but the active surface ligand population decays exponentially with rate kdecay (denaturation, hydrolysis of a labile capture, leaching). Both association and dissociation phases see fewer and fewer binding sites over time.

Reaction
bulk: A (constant)surface ligand L(t) = L₀·e^(-k_decay·t)complex A·L (kₒₙ, kₒff)effective R_max(t) = R_max·e^(-k_decay·t)
Rassoc(t)=konA RmaxkonA+koff(e−kdecayt−e−(konA+koff+kdecay)t)R_{assoc}(t) = \frac{k_{on} A \, R_{max}}{k_{on} A + k_{off}} \left( e^{-k_{decay} t} - e^{-(k_{on} A + k_{off} + k_{decay}) t} \right)
Rdiss(t)=Rend⋅e−(koff+kdecay)(t−tassoc)R_{diss}(t) = R_{end} \cdot e^{-(k_{off} + k_{decay})(t - t_{assoc})}

Closed-form solution of the linear first-order ODE dR/dt = kon·A·(Rmax·e^(-kdecay·t) − R) − koff·R, where the total ligand pool decays as Ltot(t) = Rmax·e^(-kdecay·t). kdecay → 0 collapses to the basic Langmuir 1:1.

Assumptions
  • One-to-one stoichiometry, fully reversible chemistry.
  • Surface ligand decays first-order with rate kdecay.
  • Decay is irreversible and independent of binding state.
  • Mass transport is fast (no diffusion limitation).
  • Decay half-life ln(2) / kdecay sets the timescale beyond which signal is dominated by surface loss, not chemistry.
Parameters

kon (M⁻¹ s⁻¹)

koff (s⁻¹)

kdecay (s⁻¹)

Concentration series (nM)
Kinetic mode
Single-cycle: concentrations are injected sequentially on the same surface, no regeneration.

Noise σ (pg/mm²)

Derived

KD = 1 nM

Req (no-decay reference, highest C) = 100 pg/mm²

Decay half-life = 1400 s

Surface remaining at end of association = 86%

Surface remaining at end of run = 11%

Diagnostic cues
  • Association curves can rise then fall before the injection ends — surface is being lost faster than complex accumulates.
  • Apparent koff looks faster than truth — both unbinding and surface loss empty the signal.
  • Cycle-to-cycle drift in Rmax on a real instrument is the fingerprint to watch for.
  • Drop kdecay by an order of magnitude — the curves should collapse onto the basic Langmuir 1:1.
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