AI 中文总结
研究随机爆发网络涨落，通过构建双组件网络证2019年猜想一般为假，又证明正几何爆发的猜想，还得出任意调节结构和特定爆发律下的权衡及不可能性结果。
AI 中文摘要
随机反应网络中低拷贝数组件的生产和降解事件会产生显著涨落。2019年Yan等人猜想，对于线性降解和任意交叉调节生产率的网络，反馈不能将每个组件的平稳涨落抑制到其恒定速率对应物的涨落以下。该猜想最近在单位出生（$K_i\equiv1$）的情况下得到证明。本文通过构建一个双组件网络表明该猜想一般是错误的，其生产率和爆发大小有界且在{1,2}中，两个平稳Fano因子都低于其共同的恒定速率基线。然后证明了随机基因表达中典型爆发模型——正几何爆发的猜想。对于任意调节结构和非线性交叉调节生产率，证明了一个精确的加权权衡，排除了同时将每个组件抑制到其几何爆发基线以下的情况。还证明了具有有限二阶矩的任意正整数值爆发律的互补结构不可能性结果：如果激活和抑制相互作用具有全局一致的符号结构，即调节相互作用图的每个循环包含偶数个负相互作用，那么每个组件单独满足$F_{X_i}\ge B_i$，其中$F_{X_i}$是其平稳Fano因子，$B_i$是其恒定速率爆发基线。
英文摘要
Stochastic reaction networks often involve components at low copy number, where individual production and degradation events generate substantial fluctuations. Yan et al.\ conjectured in 2019 that for networks with linear degradation and arbitrary cross-regulatory production rates, feedback cannot suppress the stationary fluctuations for each component below the fluctuations of its constant-rate counterpart. Their formulation allows random burst sizes $K_i$: the unit-birth case has $K_i\equiv1$, the biologically important burst model takes $K_i$ to be geometrically distributed, but more general positive integer-valued burst laws are also permitted. The conjecture was recently proved for unit births, $K_i\equiv1$. We show here that the conjecture is \textit{false in general} by constructing a two-component network with bounded production rates and burst sizes in $\{1,2\}$ for which both stationary Fano factors lie below their common constant-rate baseline. We then prove the conjecture for positive geometric bursts, the canonical burst model in stochastic gene expression. For arbitrary regulatory architecture and nonlinear cross-regulatory production rates, we prove an exact weighted tradeoff that rules out simultaneous suppression of every component below its geometric-burst baseline. We also prove a complementary structural impossibility result for arbitrary positive integer-valued burst laws with finite second moments: if the activating and inhibiting interactions have a globally consistent sign structure, in the sense that every cycle of the regulatory interaction graph contains an even number of negative interactions, then every component individually satisfies $F_{X_i}\ge B_i$, where $F_{X_i}$ is its stationary Fano factor and $B_i$ is its constant-rate burst baseline.
