Resilience of the positive gene autoregulation loop

Gene expression in response to stimuli is regulated by transcription factors (TFs) through feedback loop motifs, aimed at maintaining the desired TF concentration despite uncertainties and perturbations. In this work, we consider a stochastic model of the positive gene autoregulating feedback loop and we probabilistically quantify its resilience, \textit{i.e.}, its ability to preserve the equilibrium associated with a prescribed concentration of TFs, and the corresponding basin of attraction, in the presence of noise. We show that the formation of larger oligomers, corresponding to larger Hill coefficients of the regulation function, and thus to sharper non-linearities, improves the system resilience, even close to critical concentrations of TFs. We also explore a complementary definition of resilience that can be assessed within a stochastic formulation relying on the Fokker-Planck equation. Our formal results are accompanied by numerical simulations.

💡 Research Summary

The paper investigates the resilience of a positive gene autoregulation feedback loop by constructing both deterministic and stochastic mathematical models and analyzing their behavior under parameter variations and noise.
In the deterministic setting, the authors start from a biochemical description of a transcription factor (TF) that activates its own gene expression. After nondimensionalization, the dynamics are captured by the ordinary differential equation

 dx/dt = a xⁿ/(1 + xⁿ) − x + r,

where x ≥ 0 denotes the normalized TF concentration, a > 0 is a scaled maximal transcription rate, r > 0 is a basal transcription term, and n ∈ ℕ is the Hill coefficient reflecting the oligomerization degree of the TF. The right‑hand side is written as f(x) = f₁(x) − f₂(x) with f₁(x)=a xⁿ/(1 + xⁿ) (a sigmoidal Hill function) and f₂(x)=x − r (a linear degradation‑plus‑basal term). By studying the intersections of f₁ and f₂, the authors show that for n = 1 the system has a unique globally asymptotically stable equilibrium, whereas for n ≥ 2 the system can exhibit one, two, or three equilibria depending on the values of a and r. When three equilibria exist, the outer two are stable (low‑ and high‑expression states) and the middle one is unstable, giving rise to classic bistability. The transition between the different equilibrium regimes occurs through two fold (saddle‑node) bifurcations at critical values a_c,1 and a_c,2 (or equivalently r_c,1 and r_c,2). The authors derive analytical expressions for these critical points and map the dynamics near them to the normal form dz/dt = β ± z², confirming the fold‑bifurcation nature. Bifurcation diagrams for various n (2, 3, 5, 8) illustrate that increasing n sharpens the Hill function, reduces the distance between the stable and unstable equilibria, and thus modifies the shape of the “resilience profile”.

To address stochastic perturbations, the deterministic model is augmented with additive Gaussian white noise, leading to the stochastic differential equation

 dx = f(x) dt + λ dW(t),

where λ ≥ 0 quantifies noise intensity and W(t) is a standard Wiener process. Positivity of x is enforced (e.g., by reflecting boundary conditions) because TF concentrations cannot become negative. The authors adopt the “practical resilience” framework introduced in previous work: given an attractor A (a stable equilibrium of the deterministic system) and its basin of attraction B(A), a system is (τ, γ, δ, ε)‑practically resilient if, for every initial condition within an ε‑neighbourhood of A and inside B(A), the probability that the stochastic trajectory stays within a δ‑neighbourhood of A for the whole time interval