Exchange Models

Six exchange models are available, selected interactively at the start of the analysis workflow. Three describe intramolecular exchange between conformations of a single species; three describe bimolecular binding, where populations are derived from a binding equation and the total concentrations of the interacting molecules.

New models are added by defining a subtype of AbstractModel — see Extending Exchange1D if you need a mechanism not covered here.

Parameter fitting conventions

Model parameters are not fitted directly in the units you enter them in:

• Rate constants and dissociation constants (kex, koff, Kd, kclose, kopen, …) are fitted internally in log space (stored as logkex, logKd, etc.), which keeps them positive during optimisation without needing bounds. You still view and enter them as ordinary linear values (s⁻¹, or concentration units) in the parameter editor.
• Populations for the purely intramolecular models (pB, pC) are fitted internally via a free-energy-like parameter dGB = ln(pA/pB) (and dGC similarly), which keeps them between 0 and 1 for any real fitted value. Again, you only ever see and enter the population fraction itself.

Binding-model populations are not fitted parameters at all — they are derived at each titration point from Kd/koff (and, for induced fit, kclose/kopen) together with the total concentrations of the interacting molecules, via the equations below.

Intramolecular exchange models

These apply to a single observed molecule exchanging between internal conformations or states; no sample concentration information is needed.

No exchange

A single non-exchanging state — the null model, useful as a fitting baseline or when no exchange contribution is expected.

Two-state exchange

Exchange between two conformations, A ⇌ B, parameterised by the total exchange rate kex and the minor-state population pB:

\[\mathbf{K} = \begin{pmatrix} -k_\text{ex} p_B & k_\text{ex} p_A \\ k_\text{ex} p_B & -k_\text{ex} p_A \end{pmatrix}\]

with $p_A = 1 - p_B$.

Three-state exchange

A linear three-state system with a shared ground state, A ⇌ B and A ⇌ C (no direct B ⇌ C exchange), parameterised by the two off-rates koffB/koffC and populations pB/pC:

\[\mathbf{K} = \begin{pmatrix} -k_{\text{off},B}\frac{p_B}{p_A} - k_{\text{off},C}\frac{p_C}{p_A} & k_{\text{off},B} & k_{\text{off},C} \\ k_{\text{off},B}\frac{p_B}{p_A} & -k_{\text{off},B} & 0 \\ k_{\text{off},C}\frac{p_C}{p_A} & 0 & -k_{\text{off},C} \end{pmatrix}\]

with $p_A = 1 - p_B - p_C$.

Binding models

Binding models describe a two-molecule system: an observed species (role :A, e.g. the protein being followed) and a titrant (role :X, e.g. the ligand). After selecting a binding model you are prompted to assign these roles to the sample components found in the experiment metadata — see the molecule mapping step of the analysis workflow. Total concentrations for each titration point are read from the sample metadata attached to each experiment (falling back to a manually entered value if that metadata is missing).

Two-state binding

Simple 1:1 binding, A + X ⇌ AX, parameterised by the dissociation constant Kd and off-rate koff. Free titrant concentration $X_\text{free}$ is obtained from the standard quadratic binding equation, given total concentrations $A_0$ (observed) and $X_0$ (titrant):

\[X_\text{free} = \tfrac{1}{2}\left(X_0 - A_0 - K_d + \sqrt{(K_d + A_0 + X_0)^2 - 4A_0 X_0}\right)\]

The bound population is then $p_B = (X_0 - X_\text{free})/A_0$, and the pseudo-first-order on-rate is $k_\text{on,eff} = k_\text{off}\, p_B / p_A$.

Three-state binding

Two independent, non-competing binding sites on the same observed molecule, A + X ⇌ B and A + X ⇌ C, each with its own Kd/koff (Kd1/koff1 and Kd2/koff2). There is no direct B ⇌ C exchange. Free titrant concentration is obtained from the corresponding quadratic equation for two parallel 1:1 sites, and populations follow from detailed balance across each independent binding step. Use this when a single observed resonance reports on two chemically distinct, mutually exclusive binding events (e.g. two ligands competing for the same molecule, or two non-interacting sites) rather than a single two-step mechanism — for the latter, see induced fit binding below.

Induced fit binding

A two-step binding mechanism in which the initial bimolecular complex undergoes a subsequent conformational change:

\[A + L \underset{k_\text{off}}{\overset{k_\text{on}}{\rightleftharpoons}} B \underset{k_\text{open}}{\overset{k_\text{close}}{\rightleftharpoons}} C\]

Free ligand and equilibrium species concentrations are obtained analytically from the coupled mass-action/mass-balance equations (ported from the TITAN bmInducedFit model), given the dissociation constant Kd, off-rate koff for the bimolecular step, and kclose/kopen for the conformational step. There is no direct A ⇌ C exchange:

\[\mathbf{K} = \begin{pmatrix} -k_{AB} & k_\text{off} & 0 \\ k_{AB} & -k_\text{off}-k_\text{close} & k_\text{open} \\ 0 & k_\text{close} & -k_\text{open} \end{pmatrix}, \qquad k_{AB} = \frac{k_\text{off}}{K_d}\, L_\text{free}\]

Use this model — rather than three-state binding — when states B and C represent the same bound complex before and after a conformational change (e.g. an "open" encounter complex isomerising to a "closed", conformationally selected complex), so that all ligand binds through a single bimolecular step.