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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.14325 |
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Table of Contents:
- Hill functions dominate gene regulatory network (GRN) modeling, but their fractional exponents create analytical pathologies when the Hill coefficient $n$ is non-integer -- a ubiquitous occurrence in experimental fits. We replace the Hill activation $h^+(x,θ,n)=x^n/(x^n+θ^n)$ and repression $h^-(x,θ,n)=θ^n/(x^n+θ^n)$ with the logistic counterparts $f^+(x,θ,λ)=1/(1+e^{-λ(x-θ)})$ and $f^-(x,θ,λ)=1/(1+e^{λ(x-θ)})$. The matching $λ=n/θ$ preserves the slope at the half-maximal concentration. Four families of Hill pathologies appear for non-integer $n$: derivative singularities at the origin ($h^{+\prime}(x)\to\infty$ as $x\to 0^+$ for $0<n<1$; higher-order derivatives diverging for $n\in(k,k+1)$); integrals requiring hypergeometric functions; multivalued fractional-power inversions; and logarithmic small-$n$ approximations diverging at low expression. Each is resolved by a structural property of the logistic: the uniform bound $|\partial f^\pm/\partial x|\leλ/4$, the closed-form logit inverse, an elementary antiderivative, and the nonzero basal output $f^+(0)=1/(1+e^{λθ})>0$. We prove the product-of-logistics GRN model admits globally unique, smooth, uniformly bounded solutions with explicit Lipschitz constant $L_F\le M=\max_i(κ_i\sum_j L_i^j+γ_i)$. The identity $h^+(x,θ,n)=σ(n\ln(x/θ))$ shows the Hill is a logistic of the log-ratio, but the change of variable $s=\ln(x/θ)$ introduces a state-dependent factor $e^{-s}$ on the production side, so the two ODE models are nonequivalent. They encode different hypotheses -- multiplicative-increment versus additive-threshold sensitivity -- and the structural advantages of the logistic framework hold under either.