Molecular recognition

Molecular recognition is the specific noncovalent interaction between molecules with complementary shape and chemistry: antibodies bind antigens, enzymes recognize substrates, receptors engage ligands. The interaction is held together by many weak forces — hydrogen bonds, electrostatics, hydrophobic contacts, van der Waals — that together produce strong, specific binding.

Throughout this page we use a single reaction scheme: an analyte$A$ binds a ligand $B$ to form a complex$AB$. The reaction is reversible — complexes form and dissociate continuously.

Kinetics — how fast does it bind?

Binding is dynamic. Two rate constants describe it:

• $k_{on}$ — the association rate constant. Units $M^{-1} s^{-1}$. The forward rate is $v_{assoc} = k_{on}[A][B]$.
• $k_{off}$ — the dissociation rate constant. Units $s^{-1}$. The reverse rate is $v_{diss} = k_{off}[AB]$.

For typical proteins in solution, $k_{on}$ is in the $10^{4}$–$10^{7}\ M^{-1}s^{-1}$ range (the diffusion limit for proteins is higher, about $10^{8}$–$10^{9}\ M^{-1}s^{-1}$), and $k_{off}$ spans many orders of magnitude — from $10^{-1}\ s^{-1}$ for loose binders down to $10^{-6}\ s^{-1}$ for very stable complexes (long residence time).

The complex half-life — set entirely by $k_{off}$ via $t_{1/2} = \ln 2 / k_{off}$ — sets how long the dissociation phase needs to be. A reliable $k_{off}$ fit requires enough measured decay to separate dissociation from noise and drift; the fit here applies a 5% drop from the association plateau as its acceptance threshold, which is this software’s criterion rather than a general law; the durations in the table below aim higher, at enough decay to resolve the rate rather than merely detect it. If the measured dissociation does not constrain $k_{off}$, extend the dissociation where practical, or report the rate as unresolved. Equilibrium analysis is an alternative only when the association phases reach, or reliably constrain, their steady-state responses — a very slow $k_{off}$ also means slow equilibration, so it does not automatically make equilibrium the easier measurement.

A dissociation too slow to measure does not make equilibrium the easier measurement. The same small $k_{off}$ means the approach to equilibrium is slow too, most of all at the low concentrations that carry the information. Equilibrium is the alternative only when the association phases actually reach, or reliably constrain, their steady-state responses — otherwise extend the experiment or report the rate as unresolved.

Equilibrium — how strongly does it bind?

At equilibrium the forward and reverse rates balance: $k_{on}[A][B] = k_{off}[AB]$. Rearranging gives the equilibrium dissociation constant

$K_D$ is the concentration of free analyte at which half the binding sites are occupied. Lower means tighter binding: nanomolar $K_D$ is tight, micromolar is moderate, millimolar is weak.

On a surface with a finite number of sites, the fractional occupancy follows a hyperbola:

In sensor units, multiplying by the surface ceiling $R_{max}$ gives the steady-state response measured in a titration:

Two practical numbers govern whether an isotherm fit will succeed. The injection has to last long enough for the response to settle: the time to reach 95% of equilibrium is approximately $t_{95\%} \approx 3 / (k_{on}[A] + k_{off})$, so low $[A]$ equilibrates more slowly. With unknown rates, start with 5–10 minute injections and confirm the trace has plateaued. The concentration series meant to be fitted — as opposed to a first scout for an unknown $K_D$, which is coarser and is covered under choosing concentrations — should span roughly $0.1\,K_D$ to $10\,K_D$ with at least six non-zero points plus a blank, which brackets the transition from about 9% to 91% occupancy. The most informative 20–80% part of a 1:1 isotherm lies around $0.25\,K_D$ to $4\,K_D$, so keep several points inside that band.

Kinetics vs equilibrium at a glance

Both views are needed. Kinetics is the richer measurement; equilibrium depends less on resolving the time course, which is its main advantage. It is not artifact-free — surface decay, heterogeneous ligand activity, mass transport and plateaus that never form all bias an equilibrium result too. The table below summarizes when each is preferable.

Common binding models

The 1:1 Langmuir model is the reference. Real systems sometimes deviate — and most deviations have a name and a fit equation. The sections below cover the deviations the simulators support.

Langmuir 1:1 — the reference

One analyte, one ligand, one binding site. Mass-action kinetics:

With analyte at constant $[A]$, the response approaches equilibrium exponentially with time constant $\tau = 1/(k_{on}[A] + k_{off})$. After buffer wash, it decays with time constant $1/k_{off}$. If a real sensorgram fits this model cleanly, the extracted $k_{on}$, $k_{off}$, and $K_D$ are physically meaningful.

Heterogeneous Ligand — independent sites

The surface carries two distinct ligand populations — for example, correctly oriented and misoriented copies of the same protein, or a mixture of high- and low-affinity binders. The total response is the sum of two independent Langmuirs, each with its own $k_{on}$, $k_{off}$, and capacity. The signature is biphasic kinetics: a fast component that dominates early, a slow component that dominates later.

A 1:1 fit to such data tends to compromise — apparent rates fall between the two true ones, and residuals show systematic structure. Try the simulator to see how the two phases trade off.

Determine Kinetics offers one Heterogeneous binding — two components model for a sum of two independent Langmuir responses. The ligand and analyte examples below describe possible physical explanations, not separate fitting choices.

Heterogeneous Analyte — parallel reactions

Here the heterogeneity is in the injected sample rather than on the surface. Two analyte components — for example monomer and aggregate — contribute effective 1:1 response terms with different $k_{on}$, $k_{off}$, and amplitudes. A minor slow-dissociating component can dominate the late tail even when its early response is small.

The fitted component amplitudes are not direct mixture fractions: they also absorb molecular-mass response, activity, and accessible capacity. The parallel-reaction model does not explicitly represent competition for the shared ligand, so use it as an empirical or low-occupancy approximation and confirm the interpretation by changing sample purity or surface density.

Bivalent analyte and avidity

A different two-step model: one analyte (e.g. an IgG antibody) carries two equivalent binding sites and engages two surface ligands in series. The first step happens in solution with constants $k_{on,1}$ and $k_{off,1}$; the second step is intramolecular on the surface with $k_{on,2}$ and $k_{off,2}$ — typically faster and more favorable because the second binding partner is already nearby. Once both sites are bound, both bonds must break sequentially for the analyte to leave, so the apparent $k_{off}$ drops dramatically.

This model is physically distinct from heterogeneous ligand even though both produce two-phase sensorgrams. To recover the intrinsic single-site rates, lower the surface ligand density or use a monovalent fragment. The closest qualitative behavior can be explored with the heterogeneous-ligand 1:2 page above.

Langmuir 1:1 with Mass Transport

Analyte must diffuse from bulk solution to the sensor surface before it can bind. If the intrinsic $k_{on}$ is fast, analyte near the surface is consumed faster than it is replenished, and the observed binding rate is set by diffusion (the mass transport coefficient $k_t$) rather than by the chemistry. That regime is mass transport limitation, and this is the model that accounts for it.

Symptoms: the same analyte concentration gives different curves at different flow rates, and the early association looks linear instead of exponential. Mitigations: higher flow, lower ligand density, or fit with a mass-transport-inclusive model.

Decaying surface

The active ligand population shrinks during the run — through slow denaturation, leaching, photobleaching, or harsh regeneration. Identical injections give progressively smaller plateaus. A standard 1:1 fit will inflate $k_{off}$ to compensate. The decaying-surface simulator lets you set a first-order decay rate for the active sites and watch the kinetics deform.

Langmuir 1:1 — Partially Non-Dissociative

A fraction of bound analyte never leaves — perhaps because of avidity, covalent crosslinking, or surface-induced misfolding. Each cycle adds to a residual baseline, while the reversible fraction still washes off. The shared-fraction model applies the same non-dissociating fraction to every injection.

If the residual-to-peak ratio itself changes across injections, the multi-relation model can fit one fraction per injection. This extra flexibility costs one parameter per injection and should be supported by a repeatable pattern in the data.

Choosing a model

Choosing between these models is a judgement about a particular fit, so it lives with the rest of fit validation: Interpreting Results maps each symptom to the model that predicts it, and names the evidence a more complex model has to earn.

All simulators

Eight interactive simulators, one per model. Use them to sweep the model's parameters, compare sensorgram shapes, and add Gaussian noise to inspect fit robustness.