Normalization with respect to the $A-B_i$ dissociation constants

Supposing the $A-B_i$ dimerization reactions are at equilibrium (a reasonable assumption given the generally-fast rate of protein association by random 3D diffusion), and that $a_t \ll D_i$ for all $i$ (see Cinquin6, for a relaxation of that assumption), using the law of mass action one gets

$$[AB_i]= \frac{a_t [B_i] /D_i} {1+\Sigma_j [B_j]/D_j},$$

where $D_i$ is the dissociation constant for each $A-B_i$ complex, and $a_t$ is the total quantity of A. If the synthesis of $B_i$ depends on the concentration of the $A-B_i$ complex, in a non-linear fashion described by a Hill function of degree 2 with maximal value $\sigma_i$ and half-maximal synthesis for $[AB_i]=K_2$, and $B_i$ has a degradation rate $d_i$, writing $x_i=[B_i]$ one gets

$$\frac{\mathrm{d}x_i}{\mathrm{d}t}=-d_i x_i + \sigma_i \frac{[AB_i]^2}{K_2^2+[AB_i]^2}$$

Now let $y_i=x_i/D_i$. Then

$$\frac{\mathrm{d}y_i}{\mathrm{d}t}=\frac{1}{D_i} \frac{\mathrm{d}x... ...rm{d}t} = - d_i y_i + \frac{\sigma_i} {D_i} \frac{y_i^2} {\alpha D^2 + y_i^2},$$

with $D=1+\Sigma_{i=1}^ny_i$, $\alpha=K_{2}^2/a_{t}^2 \in {\sf R\hspace*{-0.9ex}\rule{0.15ex}{1.5ex}\hspace*{0.9ex}}_{*}^{+}$.

The normalization with respect to the dissociation constants $D_i$ has thus led to the replacement of each maximal synthesis rate $\sigma_i$ by $\sigma_i/D_i$.