arXiv subjects

Ismail Belgacem

Publications and source records attributed to Ismail Belgacem.

9 recordsLinked to original sources

arXiv2026

Logistic Gene Regulatory Networks: A Modelling Framework Beyond Hill Functions

Boolean networks model gene regulatory networks, but extracting quantitative dynamics requires translating their logical rules into differential equations, and the sigmoidal kernel chosen carries direct biological consequences. The near-universal choice, the Hill function, sets production to exactly zero when an activator is absent, creating a spurious absorbing off-state with no biological counterpart. We develop a product-of-logistics framework in which increasing logistic functions represent activation, decreasing logistic functions represent repression, and a recursive De Morgan product formula translates an arbitrary Boolean rule into a continuous regulatory function. The translation is automatic, confines every regulatory function to the unit interval, and retains a strictly positive basal rate. Our central result is a recovery theorem: every steady state of the Boolean network reappears, for sufficiently steep response, as an exponentially stable equilibrium of the continuous model, so the translation provably refines rather than distorts the Boolean analysis. We establish global well-posedness, forward invariance, and an explicit Lipschitz constant, and prove, for the two canonical two-gene motifs, global asymptotic stability of the negative-feedback oscillator and a closed-form bistability threshold for the toggle switch. Every threshold remains a positive, measurable concentration, unlike weighted-sum logistic formulations that place repressor thresholds at meaningless negative values. The eleven-gene Traynard mammalian cell-cycle network is translated automatically: in the proliferative regime its trajectories settle onto a sustained limit cycle reproducing the Boolean cyclic attractor. Because it is purely structural, the translation applies unchanged to existing Boolean models and supports exact feedback linearisation for control.

math.DS↗

arXiv2026

State-Feedback Control of Logistic-Based Gene Regulatory Networks: Closed-Form Lyapunov Certificates, Monostabilization, and Delay-Uniform Stability

Gene regulatory networks (GRNs) are high-value targets for therapeutic and synthetic-biology control. Classical Hill models carry a structural defect: the production term vanishes when the activator is absent, causing loss of controllability under multiplicative actuation and a collapse of network coupling under additive actuation -- precisely where biological operation is most common. Building on logistic functions as robust Hill alternatives, we develop an additive state-feedback framework for logistic-based GRNs, with two companion scalar results. A feedforward-plus-proportional law turns any positive setpoint into a closed-loop equilibrium, regardless of the uncontrolled dynamics. We prove local exponential stability under a Gershgorin gain bound and, via a common quadratic Lyapunov function built on the logistic sector bound, global exponential stability under the explicit condition $(\gamma_1+K_1)(\gamma_2+K_2)>\kappa_1\kappa_2\lambda^2/64$, with a closed-form rate. A diagonal Lyapunov certificate $P=\mathrm{diag}(B,A)$ yields an explicit settling-time bound (within $1\%$ of simulation) and an ISS ultimate-bound estimate. We further establish a parameter-uniform monostabilization budget $K^{*}=\kappa\lambda/4-\gamma$ for bistable self-activation switches, and a Halanay-type delay-uniform global exponential stability theorem under $\gamma+K>\kappa\lambda/4$, with two-sided closed-form rate bounds. Numerical comparison with the Hill counterpart confirms robust tracking in the nominal range while exposing the structural divergence near the boundary of the positive orthant: the Hill coupling $|[J_f]_{21}|=\Theta(x_{d,1}^{n-1})\to0$ as $x_{d,1}\to0$, whereas the logistic coupling stays strictly positive.

math.DS↗

arXiv2026

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 $1/2^{m_i}$ of the product-of-logistics model and sharing its canonical equilibrium $x_i^*=\kappa_i/(\gamma_i 2^{m_i})$ under self-consistency. The bcw system inherits global $C^\infty$ 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 $\sigma''=0$ for $m_i\ge2$; 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 $J^{bcw}=D(J^{prod}+\Gamma)-\Gamma$ with $D=\mathrm{diag}(2-2^{1-m_i})$, giving a stability dichotomy. In the contractive regime $\gamma_i>\kappa_i\lambda m_i/4$ both systems converge globally and exponentially to the shared equilibrium at rate $\alpha=\min_i(\gamma_i-\kappa_i\lambda m_i/4)$. A two-gene self-activating negative-feedback motif illustrates the theory: the bcw form undergoes Hopf bifurcation at $2/3$ of the product threshold and sustains limit cycles where the product is linearly stable, with eigenvalue amplification factor $D=3/2$.

math.DS↗

arXiv2026

Sustained Limit Cycles in the Logistic Two-Gene Genetic Oscillator: A Delay-Driven Hopf Bifurcation

The logistic two-gene negative-feedback oscillator is locally asymptotically stable for all biological parameter values, since the trace of the Jacobian is uniformly negative. Real biological oscillators (circadian rhythms, the segmentation clock, Hes1, p53) nevertheless rely on delays. We extend the logistic two-gene model to a delay-differential system with transcriptional delays $\tau_1$ and $\tau_2$, and prove that the equilibrium loses stability through a Hopf bifurcation as the total delay $\tau=\tau_1+\tau_2$ crosses an explicit critical value $\tau_c$. The Hopf frequency $\omega_c$ and $\tau_c$ are computed in closed form from the logistic derivatives; the loop-gain condition $AB>\gamma_1\gamma_2$ is necessary and sufficient; the transversality $\mathrm{Re}(d\mu/d\tau)|_{\tau_c}>0$ admits a parameter-uniform positive lower bound; and the bifurcation persists globally. A sum-of-delays symmetry reduces the analysis to the scalar parameter $\tau$. Numerical simulations confirm three regimes (damped, small limit cycle, relaxation), the supercritical amplitude scaling $A\sim c\sqrt{\tau-\tau_c}$, and the deep-relaxation period asymptote $T\sim 2\tau+C_\infty$ with closed-form offset $C_\infty$. For the symmetric-threshold loop, supercriticality is proved by a Lindstedt--Poincar\'e reduction yielding closed-form amplitude and frequency laws; for the general asymmetric loop it delivers a closed-form first Lyapunov coefficient and an explicit criticality criterion. Calibrated to p53--Mdm2 data, the closed-form Hopf period matches the observed oscillation within $3\%$, and the standard Hill-function model within a few percent. The analysis extends to cyclic $N$-gene loops, with a closed-form transversality rate valid for every $N$ and -- in the symmetric case -- an explicit delay-induced-Hopf window $\gamma^N<\Lambda<\gamma^N\sec^N(\pi/N)$.

math.DS↗

arXiv2026

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

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 $0 0$. 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.

math.DS↗

arXiv2026

Numerical Reliability of Logistic Gene Regulatory Network Models: Preventing Expression Shutdown and Robust Integration of Boolean-Derived ODE Systems

Gene regulatory networks are routinely translated from Boolean update rules into large continuous ODE systems integrated numerically for attractor identification, sensitivity analysis, and control design. The reliability of that integration depends critically on the sigmoidal kernel representing regulation. This simulation study shows that the Hill function -- the near-universal choice -- is a generically unreliable kernel, while the logistic function is a robust replacement. Two failure modes are demonstrated. First, because the Hill function vanishes at zero input, bistable circuits acquire an absorbing off-state: with experimentally grounded \textit{E. coli} galactose-operon autoregulation parameters, a Hill model stays trapped below the unstable separatrix, whereas the logistic model -- whose basal rate is strictly positive by construction -- escapes in about $44$~minutes through basal production alone, matching an analytical estimate of ${\approx}58$~min. A saddle-node analysis characterises the bistable window via an explicit transcendental equation and identifies the threshold $\lambda\theta=2$ separating monostable from bistable regimes. Second, when the Hill exponent is non-integer -- as in dose-response fits -- the power law $x^n=e^{n\ln x}$ turns complex-valued whenever a solver overshoots into negative concentrations. On an $80$-gene Boolean-derived benchmark with $n\approx3.509$, the Hill solver is silently contaminated by complex values from $t\approx52.64$, yielding smooth but spurious trajectories, whereas the logistic formulation completes $t\in[0,200]$ without a single warning. Because the logistic vector field is globally Lipschitz with explicit constant, we further prove an a priori global-error bound of classical order -- a guarantee structurally unavailable to the Hill formulation.

q-bio.MN↗

arXiv2026

Beyond Linear Additive and Hill Functions: A General Logistic Reformulation of Delay-Coupled Gene Regulatory Networks with Equilibrium Analysis, Hopf Bifurcation, and Lipschitz Stability

Hill functions, dominant in gene regulatory network modeling, carry fundamental limitations: at non-integer cooperativity exponents, routine when fitting dose-response data, derivatives diverge at the origin, complex arithmetic corrupts ODE trajectories, and zero output at zero activation traps models in off-states. This paper employs logistic-based models that are globally $C^\infty$, real-valued, and strictly positive at zero concentration, resolving all three pathologies while preserving sigmoidal dynamics. Using the delay-coupled two-gene mutual-activation and self-repression network of Vinoth et al.\ as a concrete model, we analyze two reformulations: linear additive activation with logistic self-repression, and a fully sigmoidal form with both terms logistic. A closed-form matching relation $λ= n/θ$ follows from equating slopes at half-maximal points. Closed-form parameters of the weighted logistic formulation are derived by matching basal rates and local slopes to the Hill-linear hybrid model. The unique biologically feasible equilibrium is computed in each case; it is lower in the weighted logistic case, the reduction arising from saturation of the bounded activation term. In the delay-free case ($τ=0$), local asymptotic stability holds in both formulations since the Jacobian trace is strictly negative for all positive parameters; stability persists for $τ\in(0,τ_c)$ and is lost via Hopf bifurcation at the critical delay $τ_c$. Numerical solution of the full transcendental system locates $τ_c$, with higher-order bifurcations characterised numerically in each case. Replacing linear additive with weighted logistic activation substantially reduces both the global Lipschitz constant of the right-hand side and that of its Jacobian, permitting larger integration steps.

math.DS↗

arXiv2025

Forward-Backward Binarization

Binarization of gene expression data is a \textbf{critical prerequisite} for the synthesis of Boolean gene regulatory network (GRN) models from omics datasets. Because Boolean networks encode gene activity as binary variables, the accuracy of binarization directly conditions whether the inferred models can faithfully reproduce biological experiments, capture regulatory dynamics, and support downstream analyses such as controllability and therapeutic strategy design. In practice, binarization is most often performed using thresholding methods that partition expression values into two discrete levels, representing the absence or presence of gene expression. However, such approaches oversimplify the underlying biology: gene-specific functional roles, measurement uncertainty, and the scarcity of time-resolved experimental data render thresholding alone insufficient. To overcome these limitations, we propose a novel \textbf{regulation-based binarization method} tailored to snapshot data. Our approach combines thresholding with functional binary value completion guided by the regulatory graph, propagating values between regulators and targets according to Boolean regulation rules. This strategy enables the inference of missing or uncertain values and ensures that binarization remains biologically consistent with both regulatory interactions and Boolean modeling principles of the gene regulation. Validation against ODE simulations of artificial and established Boolean GRNs demonstrates that the method achieves accurate and robust binarization, thereby strengthening the reliability of Boolean network synthesis.

cs.DM↗

arXiv2024

Computer-aided analysis of high-dimensional Glass networks: periodicity, chaos, and bifurcations in a ring circuit

Glass networks model systems of variables that interact via sharp switching. A body of theory has been developed over several decades that, in principle, allows rigorous proof of dynamical properties in high dimensions that is not normally feasible in nonlinear dynamical systems. Previous work has, however, used examples of dimension no higher than 6 to illustrate the methods. Here we show that the same tools can be applied in dimensions at least as high as 20. An important application of Glass networks is to a recently-proposed design of a True Random Number Generator that is based on an intrinsically chaotic electronic circuit. In order for analysis to be meaningful for the application, the dimension must be at least 20. Bifurcation diagrams show what appear to be periodic and chaotic bands. Here we demonstrate that the analytic tools for Glass networks can be used to rigorously show where periodic orbits are lost, and the types of bifurcations that occur there. The main tools are linear algebra and the stability theory of Poincaré maps. All main steps can be automated, and we provide computer code. The methods reviewed here have the potential for many other applications involving sharply switching interactions, such as artificial neural networks.

nlin.CD↗