🔄 Two-State Reaction (Conformational Change)

Following academy convention, A = analyte (in solution) and L = ligand (immobilised on the surface). The analyte binds the ligand (1:1), and the encounter complex then undergoes a conformational change to a more stable state. Mental model: the complex first forms, then locks in.

The same kinetic scheme is consistent with both induced fit (the most common interpretation) and conformational selection (where L pre-equilibrates between two conformations and A binds the productive one). "Induced fit" is one mechanistic reading of two-state kinetics, not a synonym for it.

The Model

The first step is bimolecular 1:1 binding (k_a, k_d). The second step is a unimolecular conformational rearrangement of the complex (k_a2, k_d2) — both directions are first-order in surface species.

Symbol Definitions

The ODEs

The total measured response is R + R*. Free binding sites are R_max − R − R* — both states occupy a ligand site.

Overall Apparent K_D

Define the dimensionless equilibrium constant of the second (unimolecular) step as K_2 ≡ k_a2 / k_d2. Then at equilibrium:

When the conformational step is favourable (K_2 ≫ 1) the apparent affinity is much tighter than K_D1 alone. When the step is unfavourable (K_2 ≪ 1) the apparent affinity collapses back to K_D1. The steady-state response vs [A] still traces a hyperbola — just with K_D,app in place of K_D — which is why two-state behaviour is easy to miss if you only fit equilibrium data.

One subtlety: the hyperbola does not saturate at R_max. The equilibrium amplitude is

so the apparent R_max is reduced by a factor K_2/(1+K_2). When K_2 is small, fitting a 1:1 hyperbola yields a K_D close to K_D1 and systematically underestimates R_max relative to the true site density.

Try increasing K_2 = k_a2/k_d2 in the simulator and watch the dissociation curve become visibly biphasic.

Key Signatures

• Biphasic dissociation: A fast initial phase (AL → A + L) is followed by a slow phase (AL* → AL → A + L). Formally, the dissociation matrix has two eigenvalues; when k_d2 ≪ k_d the slow eigenvalue is exactly k_d · k_d2 / (k_d + k_a2) — proportional to k_d2, modulated by the partitioning factor k_d/(k_d+k_a2) (the chance that AL exits to free A rather than re-locking to AL*).
• Density-independent pattern: The biphasic shape does NOT change with ligand density — this distinguishes two-state from heterogeneous ligand.
• k_d2 is typically slow: Conformational changes are often rate-limiting, leading to very slow off-rates.

Fitting Caveats

Two-state fits are notoriously under-determined. k_a2 and k_d2 are often strongly correlated, and on short dissociation phases k_d2 is unidentifiable — it floats. As a rule of thumb, dissociation should be observed for at least 5 × 1/k_d2 before k_d2 can be trusted. Always report parameter confidence intervals, not just point estimates.

When to Use

Use the two-state model when you have independent evidence of conformational change (structural data, HDX-MS) or observe biphasic dissociation that is independent of ligand density and not explainable by rebinding.

Common Examples

• Antibody–antigen interactions with induced fit
• Enzyme–inhibitor complexes with conformational change
• Receptor–ligand interactions where binding triggers structural rearrangement

Distinguishing from Heterogeneous Ligand (and Rebinding)

Biphasic dissociation has at least three common causes — two-state kinetics, a heterogeneous ligand population, and mass-transport-limited rebinding. The diagnostic tests:

• Vary ligand density. Two-state gives the same kinetics at different densities. Heterogeneous ligand changes — the fast/slow ratio shifts because you change the proportion of active vs degraded sites.
• Vary flow rate (and check at low density). Mass-transport-limited rebinding can mimic a slow second phase that looks indistinguishable from AL* in fits; higher flow and lower density suppress rebinding and should collapse the biphasic shape if rebinding is the cause.

For SPR data on IgG, biphasic dissociation is more often avidity or heterogeneous ligand than a true conformational rearrangement — two-state should not be the default interpretation for antibody data.