A Bias-Corrected Weighted Logistic Model for Gene Regulatory Networks: Functional Equivalence with the Product-of-Logistics and Comparison with Weighted-Sum Formulations

We introduce a bias-corrected weighted-logistic (bcw) formulation for ODE models of gene regulatory networks. Each gene’s regulatory function is a single sigmoid $\sigma (\lambda S_i+\lambda b_i)$ of the signed weighted regulator sum $S_i$, with a combinatorial bias $b_i=-{\lambda }^{-1}\log (2^{m_i}-1)$ depending only on the in-degree $m_i$ and the steepness $\lambda$. We prove this is the unique single-sigmoid correction recovering the critical-point value $1\mathrm{/}{2}^{m_i}$ of the product-of-logistics model and sharing its canonical equilibrium $x_i^{\ast }={\kappa }_i\mathrm{/}({\gamma }_i{2}^{m_i})$ under self-consistency. The bcw system inherits global $C^{\mathrm{\infty }}$ well-posedness, an explicit Lipschitz bound, a positively invariant box, and strictly positive basal output. We derive an exact algebraic identity for the discrepancy $f^{prod}-f^{bcw}$ as a sum over proper non-empty regulator subsets; a slope ratio $2-2^{1-m_i}$ at the critical point; a curvature mismatch breaking the symmetry ${\sigma }^{\mathrm{\prime }\mathrm{\prime }}=0$ for $m_i\ge 2$; a no-go result excluding global equivalence by any state-independent constant; and a mean-field reading of the bias as a zero-order log-sum-exp approximation. At a shared equilibrium the Jacobians satisfy $J^{bcw}=D(J^{prod}+\mathrm{\Gamma })-\mathrm{\Gamma }$ with $D=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(2-2^{1-m_i})$, giving a stability dichotomy. In the contractive regime ${\gamma }_i>{\kappa }_i\lambda m_i\mathrm{/}4$ both systems converge globally and exponentially to the shared equilibrium at rate $\alpha ={\min }_i({\gamma }_i-{\kappa }_i\lambda m_i\mathrm{/}4)$. A two-gene self-activating negative-feedback motif illustrates the theory: the bcw form undergoes Hopf bifurcation at $2\mathrm{/}3$ of the product threshold and sustains limit cycles where the product is linearly stable, with eigenvalue amplification factor $D=3\mathrm{/}2$.