Exploring Logistic Functions as Robust Alternatives to Hill Functions in Genetic Network Modeling

TL;DR

This paper proposes replacing Hill functions with logistic functions in gene regulatory network models to avoid analytical issues, providing a more robust and mathematically tractable framework.

Contribution

The authors introduce logistic functions as alternatives to Hill functions, resolving common pathologies and establishing a globally well-posed, bounded solution framework for GRN modeling.

Findings

Logistic functions eliminate derivative singularities at zero for non-integer Hill coefficients.

The logistic-based model admits globally unique, smooth, bounded solutions with explicit Lipschitz constants.

The logistic framework offers structural advantages over Hill functions, applicable under various hypotheses.

Abstract

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 $h^+(x,\theta,n)=x^n/(x^n+\theta^n)$ and repression $h^-(x,\theta,n)=\theta^n/(x^n+\theta^n)$ with the logistic counterparts $f^+(x,\theta,\lambda)=1/(1+e^{-\lambda(x-\theta)})$ and $f^-(x,\theta,\lambda)=1/(1+e^{\lambda(x-\theta)})$. The matching $\lambda=n/\theta$ 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 $|\partial f^\pm/\partial x|\le\lambda/4$, the closed-form logit inverse, an elementary antiderivative, and the nonzero basal output $f^+(0)=1/(1+e^{\lambda\theta})>0$. We prove the product-of-logistics GRN model admits globally unique, smooth, uniformly bounded solutions with explicit Lipschitz constant $L_F\le M=\max_i(\kappa_i\sum_j L_i^j+\gamma_i)$. The identity $h^+(x,\theta,n)=\sigma(n\ln(x/\theta))$ shows the Hill is a logistic of the log-ratio, but the change of variable $s=\ln(x/\theta)$ 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.