🚚 Mass Transport Model

Describes a 1:1 interaction where the supply of analyte to the surface is slower than the binding event itself.

The Problem

When binding is fast relative to diffusion, analyte depletes near the surface faster than it's replenished. The surface "sees" a lower effective concentration than what you injected, distorting your kinetics.

The ODE with Transport

Where k_t is the mass transport coefficient, which depends on flow rate, flow cell geometry, and the analyte's diffusion coefficient. [A_s] is the effective surface concentration under the quasi-steady-state assumption — i.e. the diffusive flux into the depletion layer balances the net reactive flux into the surface, so [A_s] tracks the binding state without its own transient.

During association the k_t × [A_bulk] term dominates the numerator and [A_s] → [A_bulk] when transport is fast (k_t ≫ k_a × (R_max − R)). During dissociation [A_bulk] = 0 but the k_d × R term does not vanish — that residual is the back-flux of dissociated analyte that has not yet diffused out of the depletion layer (i.e. rebinding), and it is what biases the apparent k_d downward at high Da. The same expression therefore explains both the k_a and k_d distortions under MTL.

Damköhler Number

This is the initial Damköhler number, evaluated at R = 0 when all sites are free. The instantaneous Da scales with the free-site density, k_a × (R_max − R) / k_t, matching the (R_max − R) factor in the [A_s] expression above. Using R_max gives a conservative upper bound on transport bias.

• Da ≪ 1: Reaction-limited (good) — analyte is replenished faster than it binds. Normal 1:1 kinetics apply.
• Da ≫ 1: Transport-limited (bad) — kinetics are artifacts of diffusion, not true binding rates.

How to Diagnose Mass Transport Limitation

1. Linear association phase — instead of the expected exponential curvature, the binding phase looks straight. The surface consumes analyte as fast as it arrives.
2. Flow-rate dependence — kinetics change when you change the flow rate. True 1:1 kinetics are flow-rate independent. Note that the Lévêque expression gives k_t ∝ Q^1/3, so a flow-rate test typically needs to span at least a 5–10× range to be conclusive; a 2–3× change only moves k_t by ~25–45%.
3. Artificially similar k_a values — different interactions yield suspiciously similar on-rates because you're measuring diffusion, not binding.
4. Density-dependent k_d — drop the ligand density and the fitted k_d accelerates. This is widely regarded as the single cleanest test for MTL because it isolates rebinding from instrument artifacts.

How to Fix It

• Lower ligand density — reduces R_max, which reduces Da. This is the most effective fix.
• Increase flow rate — increases k_t, improving analyte replenishment at the surface.
• Use smaller flow cell or microfluidic geometry — improves mass transport characteristics.
• Use mass transport model for fitting — adds k_t as a parameter. This is a last resort; it's better to fix the experiment. Note that at Da ≫ 1 the two-compartment fit can look excellent (low χ²) while the recovered k_a has no chemical meaning — it is fixed by flow-cell geometry rather than the interaction. k_t and k_a are also strongly correlated, so a tight fit is not evidence of a reliable on-rate.

Who Is Most at Risk?

Mass transport limitation disproportionately affects certain experimental setups. If you recognize your situation below, pay extra attention to your association phase shape and consider a flow-rate test.

• Amine-coupled surfaces with high R_max — the most common culprit. Amine coupling is easy to over-immobilize, especially with small ligands that give low RU per molecule.
• Small-molecule analytes — lower diffusion coefficients than antibodies, so k_t is inherently lower. Combined with fast binding (common for fragments), Da rises quickly.
• Very fast binders (k_a > 10⁶ M^-1s^-1) — even moderate ligand density can push Da above 1 when the on-rate is this high.
• Low flow rates (< 30 µL/min) — older protocols or reagent-saving strategies sometimes use 5–10 µL/min, which dramatically reduces k_t.

How Big Are the Errors?

The practical consequence of mass transport limitation is that your fitted k_a is an underestimate of the true k_a. To first order, k_a,app ≈ k_a / (1 + Da), so:

• Da < 0.05: A few percent error in k_a. Safe to ignore.
• Da ~ 0.1: ~9% underestimation of k_a. Generally tolerable for screening, marginal for quantitative work.
• Da ~ 0.1–1: ~9–50% underestimation of k_a. Significant for quantitative work. Lower the density or increase flow rate.
• Da > 1: The fitted k_a converges to a transport-limited plateau (≈ k_t / R_max) and no longer reflects the intrinsic on-rate. The plateau is still informative — it sets a lower bound on the true k_a and gives you k_t — but it should not be reported as a binding on-rate.

Importantly, k_d is also affected when Da is high: dissociated analyte rebinds before it can diffuse away, producing an apparent k_d that is slower than the true off-rate. The net effect on K_D can go either direction, but the individual rate constants are unreliable.

When to Suspect It

The classic recipe for mass transport limitation: high ligand density + fast binder + low flow rate. If all three are present, check your data carefully before trusting the fitted k_a.