关于双基因竞争系统双稳定性的注记
A note on bistability of a two-gene competitive system
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- Northeastern University（东北大学）
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中文总结 AI 辅助
本文研究双基因正自调控与相互竞争系统，证明其正平衡点数目为1或3，且在3个平衡点时外侧为稳定结点、中间为鞍点，所有正解收敛至平衡点，实现全局双稳定性。
中文摘要 AI 辅助
正自调控与相互竞争相结合是基因调控模型中能够产生双稳定性的最简单机制之一。我们研究了一个双基因系统，其中每个基因激活自身的表达，且两个基因通过Hill指数为1的调控项进行竞争。我们首先证明该系统在正象限中至少有一个且至多有三个平衡点。恰好两个正平衡点只能出现在退化的零斜线相切处；因此，在非退化情形下，正平衡点的数目为一个或三个。如果恰好存在三个不同的正平衡点，则无需非退化假设：这三个平衡点自动都是双曲的，两个外侧平衡点是渐近稳定的结点，中间平衡点是鞍点。此外，每个正解都收敛到一个平衡点。因此，正象限是两个稳定结点的吸引域与鞍点的一维稳定流形的不交并，从而产生全局双稳定性。
英文摘要
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.
