PaperlayerView exact revision on arXiv

Source-linked AI summary

A note on bistability of a two-gene competitive system

Eduardo D. Sontag

arXiv:2609.27270v2math.DScs.SYeess.SY

TL;DR

The paper asks when three positive equilibria in a two-gene competitive system imply deterministic bistability. It analyzes nullcline intersections and competitive dynamics, showing that three equilibria have a stable-node/saddle/stable-node structure and positive trajectories converge to equilibria. The global basin decomposition is not a claim that either stable equilibrium attracts the entire positive quadrant.

  • Problem

    The note asks whether three distinct positive equilibria alone force the expected bistable phase portrait, without explicitly solving for equilibria or imposing a separate transversality condition.

    Source: §1 · p. 1

  • Method

    The proof combines restricted nullcline geometry and a degree-at-most-four intersection polynomial with strong competition and planar convergence.

    Source: §1 · p. 2

  • Results

    With exactly three positive equilibria, the outer equilibria are hyperbolic asymptotically stable nodes and the middle one is a hyperbolic saddle; every positive solution converges to an equilibrium.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The positive quadrant is partitioned into the basins of the two stable nodes and the saddle’s one-dimensional stable manifold, yielding global bistability in this sense.

    Source: §1 · p. 2

    “Global” does not mean either stable equilibrium is globally asymptotically stable on the entire positive quadrant.

    Source: §3 · p. 11

Abstract

from arXiv · show

Self-regulation together with mutual promoter competition provides a simple mechanism for bistability in gene-regulatory models. We study a two-gene system with regulatory terms of Hill exponent one, allowing distinct basal production rates and distinct degradation rates. Each gene product, when bound to its own promoter, may enhance or reduce production relative to the basal rate, while the two products compete through promoter occupancy. We show that the system has at least one and at most three equilibria in the positive quadrant. Exactly two positive equilibria can occur only if one nullcline intersection is degenerate; consequently, a configuration in which all positive nullcline intersections are transverse has either one or three positive equilibria. If there are exactly three distinct positive equilibria, then all three are automatically hyperbolic: the two outer equilibria are asymptotically stable nodes and the middle equilibrium is a saddle. Moreover, every positive solution converges to an equilibrium. Hence the positive quadrant is the disjoint union of the basins of attraction of the two stable nodes and the one-dimensional stable manifold of the saddle, yielding global bistability.

1 Introduction

This note analyzes a two-gene model with self-regulation and mutual promoter competition, asking when its positive equilibria produce deterministic bistability. It shows that three distinct positive equilibria force a stable-node/saddle/stable-node configuration and global convergence to those equilibria.

1 Introduction

  • The model uses Hill-exponent-one regulatory terms, distinct basal production and degradation rates, and promoter competition, while self-binding may activate, leave unchanged, or repress production.Three positive steady states can occur with strong self-activation or without it when cross-binding is sufficiently tighter than self-binding.Source: §1 · pp. 1–2
  • The system has at least one and at most three positive equilibria; exactly two require a degenerate nullcline intersection, so transverse configurations have one or three.The two-equilibrium case has one transverse intersection and one tangency.Source: §1 · p. 1
  • With exactly three distinct positive equilibria, the outer equilibria are hyperbolic asymptotically stable nodes and the middle equilibrium is a hyperbolic saddle.All three are nondegenerate.Source: §1 · p. 2
  • Every positive solution converges to an equilibrium, and the basins of the two stable nodes are separated by the saddle’s one-dimensional stable manifold.The basins are nonempty and open; this is the paper’s sense of global bistability.Source: §1 · p. 2
  • The proof combines decreasing nullclines and a degree-at-most-four intersection polynomial with strong competition and planar convergence.Endpoint signs constrain the possible nullcline intersections.Source: §1 · p. 2
  • With only qualitative hypotheses, three equilibria can instead include two tangencies and fail to yield bistability; the model’s degree-four bound excludes that alternative.The paper gives an explicit polynomial example of the degenerate alternative.Source: §1 · p. 3
  • The deterministic exclusive-switch special case has exactly one positive equilibrium, while the synthetic toggle-switch model uses a different equation class.The exclusive-switch model’s reported bistability without cooperative binding is stochastic, not deterministic bistability of its rate equations.Source: §1 · p. 3

2 Nullclines and the number of positive equilibria

The positive nullclines are decreasing graphs whose intersections determine the system’s positive equilibria. Their structure and a degree bound constrain the number of equilibria and identify when intersections are degenerate.

2 Nullclines and the number of positive equilibria

  • The first and second positive nullclines are decreasing graphs, and their endpoint behavior guarantees at least one intersection.Distinct equilibria have distinct first coordinates, with second coordinates ordered oppositely.Source: §2 · p. 4
  • The positive equilibria correspond exactly to zeros of a nonzero polynomial of degree at most four along the first positive nullcline.Only first coordinates in the nullcline’s positive interval are included.Source: §2 · pp. 4–5
  • A positive equilibrium is degenerate exactly when the corresponding nullcline-intersection function has a zero of order greater than one.At the intersection, degeneracy is equivalent to the two nullclines having equal slopes.Source: §2 · pp. 6–7
  • The system has one, two, or three distinct positive equilibria; if there are two, one intersection is transverse and the other is a tangency, while three intersections are all nondegenerate.Thus configurations with only transverse intersections have either one or three positive equilibria.Source: §2 · p. 7
  • The degree bound and endpoint signs rule out four distinct positive equilibria and force three distinct equilibria to be nondegenerate.For exactly two equilibria, one zero is simple and the other double.Source: §2 · p. 7
  • Examples show that three positive equilibria can occur with strong self-activation and with no change in production upon self-binding.Thus Hill exponent one alone does not preclude deterministic bistability in this model class.Source: §2 · pp. 7–8

3 Proof of Theorem 1

With three positive equilibria, the outer points are stable nodes and the middle point is a saddle. Every positive solution converges to an equilibrium, producing two open sink basins and the saddle’s stable manifold.

Proof of Theorem 1

  • With three equilibria, all are nondegenerate: the outer equilibria have positive determinant and negative trace, while the middle has negative determinant, yielding stable nodes and a saddle, respectively.The outer equilibria are hyperbolic asymptotically stable nodes; the middle equilibrium is a hyperbolic saddle.Source: §3 · pp. 8–9
  • Every positive solution converges to exactly one of the three equilibria, after positive semiorbits remain bounded away from the axes and have compact closure.The proof establishes forward invariance and bounds, then applies planar competitive-system convergence.Source: §3 · pp. 8–10
  • The two stable-node basins are nonempty and open, while the saddle has a one-dimensional stable manifold; initial conditions outside that manifold converge to a stable node.Together these sets give the disjoint basin decomposition and global bistability.Source: §3 · p. 10

Remark 2 (The symmetric exclusive switch is monostable)

  • The symmetric exclusive-switch special case has exactly one positive equilibrium for every r, k, d > 0, so its reported bistability is stochastic rather than deterministic.Symmetry rules out exactly two equilibria, and the three-equilibrium case would contradict the Jacobian determinant at the symmetric equilibrium.Source: §3 · p. 10

Remark 3 (Transversality)

  • A transverse configuration has either one or three positive equilibria; the two-equilibrium case is exceptional and requires a degenerate nullcline intersection.Transversality means the positive nullclines meet transversely at every positive equilibrium, equivalently det J ≠ 0 there.Source: §3 · p. 11

Remark 4 (Index-theoretic viewpoint)

  • For three nondegenerate planar equilibria, the boundary index is +1, so their indices comprise two +1 stable nodes and one −1 saddle; the middle equilibrium is the saddle.Source: §3 · p. 11

Remark 5

  • Here, “global” means every positive trajectory converges to an equilibrium, not that either stable equilibrium attracts the entire positive quadrant.The only positive initial conditions that do not converge to a sink lie on the saddle’s stable manifold.Source: §3 · p. 11

4 The general case

This section examines how degeneracy affects stability and bistability when a system has three positive equilibria, including the two-gene model and conditions for extending the general theorem to other Hill-exponent models.

4.1 Qualitative hypotheses

  • Under (A1)–(A3), positive equilibria correspond one-to-one with zeros of H, and degeneracy occurs exactly when H′ vanishes at the corresponding zero.(A1) specifies strong competition, while (A2)–(A3) require decreasing graph-shaped positive nullclines; (A4) supplies endpoint signs, and (A5) is needed only for global conclusions.Source: §4.1 · pp. 12–13

4.2 A dichotomy for three equilibria

  • With exactly three positive equilibria, either H changes sign at all three or at only one; in the latter case, at least two equilibria are degenerate.If all three are nondegenerate, the first alternative holds strictly: the outer equilibria are asymptotically stable nodes and the middle one is a hyperbolic saddle. Under (A5), every positive solution converges to an equilibrium, and the basins and saddle’s one-dimensional stable manifold partition the positive quadrant.Source: §4.2 · pp. 13–14

4.3 The degenerate alternative occurs

  • The polynomial system (34) satisfies (A1)–(A4) and has three positive equilibria, but only the middle equilibrium is stable, so the system is not bistable.Its equilibria are (1, 3), (2, 2), and (3, 1); the outer equilibria are degenerate and unstable, demonstrating that the qualitative hypotheses alone do not exclude the one-sign-change alternative.Source: §4.3 · pp. 15–16
  • In example (34), the outer equilibria are degenerate and unstable, while the middle equilibrium is the unique stable equilibrium, despite a forward-invariant box containing all three.The failure is not caused by unbounded solutions; condition (A5) fails globally, but this does not affect the local instability argument.Source: §4.3 · p. 18

4.4 The two-gene system as a special case

  • With exactly three positive equilibria, the two-gene system cannot realize the degenerate alternative, and all three equilibria are nondegenerate, so Theorem 1 follows from Theorem 2.The orders of vanishing at the three equilibria sum to at most four, whereas the degenerate alternative requires a sum of at least five.Source: §4.4 · p. 18
  • The degree-four bound on the polynomial’s distinct-root multiplicities separates the two-gene model from example (34), whose equilibrium orders are (2, 1, 2), totaling five.For the two-gene model the total is at most four; the example’s total is exactly five.Source: §4.4 · p. 18
  • For higher integer Hill exponents the degree obstruction disappears, while noninteger exponents yield a nonpolynomial equilibrium condition; Theorem 2 applies when (A1)–(A4) hold.Three positive equilibria give stable node–saddle–stable node once nondegeneracy is established, and global bistability follows if (A5) also holds; the degenerate alternative is not claimed to occur.Source: §4.4 · pp. 18–19
Loading 2609.27270v2…