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Langmuir 1:1 — Mass Transport Limited

Same 1:1 chemistry as the basic Langmuir model, but the analyte must diffuse through a thin boundary layer to reach the surface. When binding is fast compared to diffusion the curves bend — the apparent rate constants are no longer the true rate constants.

Reaction
bulk: A (constant)boundary layerA_surf (transport rate kₜ)diffusionsurface: ligand L + complex A·L(kₒₙ, kₒff)
dRdt=kt (kon A (Rmax−R)−koff R)kt+kon (Rmax−R)\frac{dR}{dt} = \frac{k_t \, (k_{on}\, A\, (R_{max} - R) - k_{off}\, R)}{k_t + k_{on}\, (R_{max} - R)}

The two-compartment quasi-steady-state form of the three-state diffusion-limited Langmuir model (Marquart 14.3). kt: mass-transport coefficient. As kt → ∞ the equation collapses to the basic Langmuir 1:1.

Assumptions
  • Quasi-steady state for the surface concentration Asurf.
  • One-to-one stoichiometry, fully reversible chemistry.
  • Mass-transport rate kt is constant (geometry, viscosity, flow rate).
  • Onsager L = kon·Rmax/kt. L > 1 means transport-limited; L< 1 means kinetics-limited.
Parameters

kon (M⁻¹ s⁻¹)

koff (s⁻¹)

kt (pg/mm²·M⁻¹·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 (k_off/k_on = 1.0e-3 / 1.0e6)

Req (kinetic limit, highest C) = 100 pg/mm²

Onsager L = 1.0e0

Regime: kinetics-limited

Diagnostic cues
  • Linear-looking association at high concentrations is the signature of transport limitation.
  • Apparent koff looks slower than truth — analyte rebinds before it can diffuse away.
  • Push kt up by an order of magnitude — the curves should collapse onto the basic Langmuir 1:1.
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