In order to determine the properties of our designed protein pairs that cannot be directly measured, like the binding affinities/dissociation constants and cooperativity of our designed orthogonal pairs, we have developed a model based on mass-action and Michealis-Menten kinetics.

Our readouts will be done with our system in a quasi-steady state, so that is what our model will also be based on: It will be divided in 2 separate segments, the first will determine the quasi-steady state concentrations of all the proteins and protein-complexes in our system, the second will simulate the total activity based on the combined concentrations of the active complexes (both scaffold and non-scaffold mediated) in our system.

In order to find the quasi-steady state approximation of the system we must analyze the enter cascade of complex-formations starting from the base proteins/ligands (see figure 1). The steady state assumption allows us to determine the mass-action kinetics between the proteins with dissociation constants. Dissociation constants present in our system are between T14-3-3 and fusicoccin (K_{D,Fc})and between the T14-3-3-FC complex and the CT52s (K_{D,C1} and K_{D,C2}), The Dissociation constants must be separated since they are not identical and thus do not have the same binding affinity to their respective T14-3-3 monomers, and the Dissociation constant of the non scaffold mediated active complex (K_{D,NoB}). Besides Dissociation constants, a cooperativity parameter σ is introduced to simulate the stimulating (σ > 1) or inhibitory (σ < 1) effects of the first CT52 on the scaffold on the binding of the second CT52.

From figure 1 the mass-action equations can be derived. For the ease of notation the T14-3-3 scaffold will be represented as B, fusicoccin as F, and CT52 by C, followed by a number (1 or 2) if necessary. The following equations can be derived:

Using the law of mass conservation we can setup equations for the total concentrations of the separate proteins and ligands in our system, which we can control during our assay:

By substituting equations 1-13 in equations 14-17, we get (ridiculously lengthy) equations describing the free concentrations of the system:

In order to solve equations 18 – 21 which contain multiple unknown variables, fixed point iteration is used using the initial conditions we use during our assays. Once equations 18-21 do not significantly change anymore we can assume the system is in a steady state and by using the results of the fixed point iteration we can calculate the concentrations of the complex species using equations 1-13, and finish the first segment of our model.

In order to determine the activity of the active complexes, we first state that concentration of enzyme [E] is the sum concentrations of scaffold (BFFCC) and non-scaffold (CC) mediated complexes. Next we use Michaelis-Menten kinetics to determine the initial concentration of the enzyme-substrate complex:

Using this initial enzyme-substrate concentration and the catalytic rate (kcat) of the active enzyme, the initial reaction rate can be calculated. To convert this reaction rate (which is calculated in μM/s) into the activity which is measured in the assay and then calibrated using a substrate specific calibration curve, the following formula is used:

With Mw the molecular weight of the fused C9-CT52 protein (the molecular weight of different variants of CT-52 are assumed to be the same), the resulting activity is expressed in U/mg.