# How are the fitting models calculated?

The 1-1 binding model

Interaction:

AI = A + I Kd (1)

Assumptions:

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

Binding isotherm based on hydrodynamic radius (R):

𝑅app=1+1𝐾d[A]((𝑅I)−1−(𝑅IA)−1)+(1+1𝐾d[A])(𝑅IA)−1 (2)

Rapp is the apparent hydrodynamic radius measured by the Fida 1, Kd is the equilibrium dissociation constant defined in eq 1, RI is the hydrodynamic radius of the unbound indicator, RIA 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 Kd (3)

Assumptions:

This model assumes a 1-2 interaction between the analyte A and the indicator I. The indicator is a fluorescently labeled, or intrinsically fluorescent species. It is further assumed that the total concentration of A is equal to the free concentration of A.

Binding isotherm:

𝑅app=1+1𝐾d[A]2((𝑅I)−1−(𝑅IA)−1)+(1+1𝐾d[A]2)(𝑅IA)−1 (4)

Rapp is the apparent hydrodynamic radius measured by the Fida 1, Kd is the equilibrium dissociation constant defined in eq 3, RI is the hydrodynamic radius of the unbound indicator, RIA 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 Kd (5)

Assumptions:

This model assumes a 1-3 interaction between the analyte A and the indicator I. The indicator is a fluorescently labeled, or intrinsically fluorescent species. It is further assumed that the total concentration of A is equal to the free concentration of A.

Binding isotherm:

𝑅app=1+1𝐾d[A]3((𝑅I)−1−(𝑅IA)−1)+(1+1𝐾d[A]3)(𝑅IA)−1 (6)

Rapp is the apparent hydrodynamic radius measured by the Fida 1, Kd is the equilibrium dissociation constant defined in eq 5, RI is the hydrodynamic radius of the unbound indicator, RIA 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 Kd (7)

Assumptions:

This model assumes a 1-1 interaction between the analyte A and the indicator I. The indicator is a fluorescently labeled, 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 0.05-0.5  Kd or higher.

Binding isotherm:

𝑅app=( (𝑅IA)−1∙((𝐶I + 𝐶A + 𝐾d) − √(𝐶I + 𝐶A + 𝐾d)2 − 4𝐶A · 𝐶I)2𝐶I)+(𝑅I)−1∙((𝐶I − 𝐶A − 𝐾d) + √(𝐶I + 𝐶A + 𝐾d)2 − 4𝐶A · 𝐶I))2𝐶I)) −1 (8)

Rapp is the apparent hydrodynamic radius measured by the Fida 1, Kd is the equilibrium dissociation constant defined in eq 7, RI is the hydrodynamic radius of the unbound indicator, RIA is the hydrodynamic radius of the 1-1 complex, CA is the total analyte concentration, and CI 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:

P1fl-D = P1fl + D K1 (9)

P1fl-D-P2 = P1fl + D-P2 K1b (10)

D-P2 = D + P2 K2 (11)

P1fl-D-P2 = P1fl-D + P2 K2b (12)

Here, P1fl is fluorescently labeled protein P1, P2 is a non-labeled protein and D is the drug compound. P1fl-D is the complex between P1fl and D, D-P2 is the complex between D and P2 and P1fl-D-P2 is the ternary complex.

The cooperativity, , is given by:

 = K1/K1b = K2/K2b (13)

The total concentration of drug, Dtot, and total concentration of P2, P2tot, are given by:

Dtot = [D] + [D-P2] + [P1fl-D] + [P1fl-D-P2] (14)

P2tot = [P2] + [D-P2] + [P1fl-D-P2] (15)

Assumptions:

In the model, the following approximations are made.

Dtot  [D] + [D-P2] (16)

P2tot  [P2] + [D-P2] (17)

In practice, this assumptions is valid for low concentrations of P1fl and/or small fractions P1fl bound to [D] and in the ternary complex. In praxis this holds when [P1fl] < Kd1. For higher concentrations of [P1fl], the model may still give good estimates (there is a 10x dilution of P1fl 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 P1fl 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 labeled 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 the 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.