Which binding model should you use to fit FIDA affinity data?

Different binding models are used in the binding curve module of the Fida data analysis software. In general, the binding models are derived assuming that the apparent diffusivity of the indicator is a weighted average of the fractions bound and unbound of the indicator. The most commonly used models for extracting the affinity are the 1:1 model and the excess indicator model. With Flow Induced Dispersion Analysis (FIDA), you can measure simple 1:1 binding as well as more complicated binding stoichiometries, ternary complexes and kinetics. The exported data can then be treated in generic spreadsheet-based analysis software, and custom graphics can be designed for presentations and publications.

In the Fida software, a detailed description of every model is under Help, Description of Fida models, in the binding curve section. The software does the calculations; they are shown here to explain the models.

Can I assess different binding stoichiometry with FIDA?

Yes. Different fitting models are available in the Fida software. They can be applied to the binding curve depending on whether the interaction is 1:1, 1:2 or 1:3. Of these models, 1:3 is the highest stoichiometry.

The 1-1 binding model

Interaction:

AI = A + I   K_D   (1)

Assumptions:

This model assumes a 1-1 interaction between the analyte A and the indicator I. The indicator is a fluorescently labelled, or intrinsically fluorescent, species. It is further assumed that the total concentration of A is equal to the free concentration of A. In practice, this approximation is valid when the indicator concentration is less than the K_D.

Binding isotherm based on hydrodynamic radius (R):

R_app = (1 + [A]/K_D) / [ (R_I)⁻¹ − (R_IA)⁻¹ + (1 + [A]/K_D)(R_IA)⁻¹ ]   (2)

R_app is the apparent hydrodynamic radius measured by the Fida instrument, K_D is the equilibrium dissociation constant defined in eq 1, R_I is the hydrodynamic radius of the unbound indicator, R_IA is the hydrodynamic radius of the 1-1 complex and [A] is the analyte concentration (approximated by the total analyte concentration).

The 1-2 binding model

Interaction:

A₂I = 2A + I   K_D   (3)

Assumptions:

This model assumes a 1-2 interaction between the analyte A and the indicator I: two analytes bind one indicator. The indicator is a fluorescently labelled, or intrinsically fluorescent, species. It is further assumed that the total concentration of A is equal to the free concentration of A, and that each of the binding events is independent.

Binding isotherm:

R_app = (1 + [A]²/K_D) / [ (R_I)⁻¹ − (R_IA)⁻¹ + (1 + [A]²/K_D)(R_IA)⁻¹ ]   (4)

R_app is the apparent hydrodynamic radius measured by the Fida instrument, K_D is the equilibrium dissociation constant defined in eq 3, R_I is the hydrodynamic radius of the unbound indicator, R_IA is the hydrodynamic radius of the 1-2 complex and [A] is the analyte concentration (approximated by the total analyte concentration).

The 1-3 binding model

Interaction:

A₃I = 3A + I   K_D   (5)

Assumptions:

This model assumes a 1-3 interaction between the analyte A and the indicator I: three analytes bind one indicator. The indicator is a fluorescently labelled, or intrinsically fluorescent, species. It is further assumed that the total concentration of A is equal to the free concentration of A, and that each of the binding events is independent.

Binding isotherm:

R_app = (1 + [A]³/K_D) / [ (R_I)⁻¹ − (R_IA)⁻¹ + (1 + [A]³/K_D)(R_IA)⁻¹ ]   (6)

R_app is the apparent hydrodynamic radius measured by the Fida instrument, K_D is the equilibrium dissociation constant defined in eq 5, R_I is the hydrodynamic radius of the unbound indicator, R_IA is the hydrodynamic radius of the 1-3 complex and [A] is the analyte concentration (approximated by the total analyte concentration).

The excess indicator model

Interaction:

AI = A + I   K_D   (7)

Assumptions:

This model assumes a 1-1 interaction between the analyte A and the indicator I, and covers the case where the indicator concentration is higher than the affinity. The indicator is a fluorescently labelled, or intrinsically fluorescent, species. There are no approximations regarding free versus total A concentration, but the model will be most accurate when [A] is in the interval 0.05 to 0.5 K_D or higher. A clear sign of working in excess indicator mode is a binding curve that is much steeper, resembling a step function rather than a sigmoidal curve.

Binding isotherm:

R_app = ( (R_IA)⁻¹ · [ (C_I + C_A + K_D) − √((C_I + C_A + K_D)² − 4C_AC_I) ] / 2C_I + (R_I)⁻¹ · [ (C_I − C_A − K_D) + √((C_I + C_A + K_D)² − 4C_AC_I) ] / 2C_I )⁻¹   (8)

R_app is the apparent hydrodynamic radius measured by the Fida instrument, K_D is the equilibrium dissociation constant defined in eq 7, R_I is the hydrodynamic radius of the unbound indicator, R_IA is the hydrodynamic radius of the 1-1 complex, C_A is the total analyte concentration, and C_I is the diluted indicator concentration at the point of detection.

The ternary complex model

The ternary complex model describes the formation of a ternary complex (three binding partners). One of the binding partners is the fluorescent indicator. The model is for example relevant in describing interactions of PROTACs, BiDACs and molecular glues with a target protein (protein of interest) and a ligase.

Interactions:

P1_fl-D = P1_fl + D   K₁   (9)

P1_fl-D-P2 = P1_fl + D-P2   K_1b   (10)

D-P2 = D + P2   K₂   (11)

P1_fl-D-P2 = P1_fl-D + P2   K_2b   (12)

Here, P1_fl is fluorescently labelled protein P1, P2 is a non-labelled protein and D is the drug compound. P1_fl-D is the complex between P1_fl and D, D-P2 is the complex between D and P2, and P1_fl-D-P2 is the ternary complex.

The cooperativity, α, is given by:

α = K₁/K_1b = K₂/K_2b   (13)

The total concentration of drug, D_tot, and total concentration of P2, P2_tot, are given by:

D_tot = [D] + [D-P2] + [P1_fl-D] + [P1_fl-D-P2]   (14)

P2_tot = [P2] + [D-P2] + [P1_fl-D-P2]   (15)

Assumptions:

In the model, the following approximations are made.

D_tot ≈ [D] + [D-P2]   (16)

P2_tot ≈ [P2] + [D-P2]   (17)

In practice, this assumption is valid for low concentrations of P1_fl and/or small fractions of P1_fl bound to D and in the ternary complex. In practice this holds when [P1_fl] < K_D1. For higher concentrations of P1_fl, the model may still give good estimates (there is a 10x dilution of P1_fl in the capillary), but it is advised to check if there is a concentration dependence on the parameters under such circumstances.

Under these assumptions, expressions for the fraction free, fraction binary complex and fraction ternary complex of P1_fl can be derived. As with the simpler models, the measured apparent hydrodynamic radius is then linked to the fraction bound and free of the fluorescently labelled indicator. In the Fida data analysis software it is possible to type in guesses of the parameters in the model, as well as to fix some of the parameters. It is for example often possible to measure protein size in independent experiments. If no parameters are known beforehand, it is advisable to make drug titrations at several protein concentrations to get the most reliable parameters.

How to study protein cooperativity?

Using FIDA, you can study ternary complex formation, where the cooperativity is one of the parameters extracted. On top of the cooperativity, you also gain information about the size of the complex and the affinity of the interactions. Ternary complex measurements for targeted protein degradation are described in How to measure ternary complex formation for targeted protein degradation?

Frequently asked questions

Which binding model should I use for a simple affinity measurement?

The 1:1 model, when the indicator concentration is below the K_D. When the indicator concentration is higher than the affinity, use the excess indicator model.

How do I know if I am in excess indicator mode?

The binding curve is much steeper, resembling a step function rather than a sigmoidal curve.

What is the highest stoichiometry the FIDA binding models cover?

1:3, where three analytes bind one indicator.

Related resources

See how the ternary complex model is used in practice in the application note Targeted Protein Degradation: Full Characterization of Ternary complexes and in the posters Fully automated characterization of ternary complex formation using flow induced dispersion analysis and Characterization of Molecular Glues: an automated in-solution platform.

New to binding affinity? What is K_D (dissociation constant)? explains the affinity the models report, How to perform a binding affinity assay? explains how the binding curve is measured, the assay formats page covers Premix, Capmix and the other set-ups, and How to measure binding kinetics in solution? covers the kinetic model. The binding affinity readout and the targeted protein degradation page show the applications. The full Binding Models course is on the Fidabio Learning Hub: register for the Fidabio Learning Hub.