反应网络的 lumping：一般参数与临界参数
Lumping of reaction networks: Generic and critical parameters
AI总结：
研究参数依赖的质量作用反应网络的线性 lumping，区分一般参数与临界参数，证明一般参数下精确线性 lumping 仅产生“明显”简化，并开发算法识别临界参数集以实现非平凡 lumping。
摘要：
我们研究了参数依赖的质量作用反应网络的线性 lumping，区分了一般参数和临界参数区间。对于一般参数——即在参数空间的某个非空开子集中取值的参数——我们证明精确线性 lumping 仅产生“明显”的简化：消除非反应物物种或沿化学计量首次积分的投影。这一刻画扩展到具有乘积形式动力学的反应网络，包括 Michaelis-Menten 和 Hill 型速率定律。对于质量作用系统，我们进一步开发了一种算法方法来识别临界参数集——参数空间中的代数子簇，其中非平凡 lumping 变得可用。该过程将 lumping 映射的确定简化为有限多个多项式方程的系统。它也适用于受约束的 lumping 场景（通常由化学考虑驱动）。然后我们回顾并扩展了关于适当 lumping 的结果。最后，我们讨论了自复制系统和双途径酶机制的 lumping，以证明我们的方法在相关场景中的可行性。我们的结果阐明了结构（参数无关）和精细调谐（参数相关）简化之间的关系，对系统参数接近临界值时的近似 lumping 具有启示意义。
英文摘要：
We investigate linear lumping for parameter-dependent mass action reaction networks, distinguishing between generic and critical parameter regimes. For generic parameters -- those ranging in some non-empty open subset of parameter space -- we prove that exact linear lumping yields only "obvious" reductions: elimination of non-reactant species or projections along stoichiometric first integrals. This characterization extends to reaction networks with product-form kinetics, including Michaelis-Menten and Hill-type rate laws. For mass action systems we proceed to develop an algorithmic approach to identify critical parameter sets -- algebraic subvarieties in parameter space where non-trivial lumpings become available. This procedure reduces the determination of lumping maps to a system of finitely many polynomial equations. It also applies to constrained lumping scenarios (which are frequently motivated by chemical considerations). We then review and extend results about proper lumpings. Finally, we discuss lumpings of a self-replicator system, and of a two-pathway enzyme mechanism, to document the viability of our methods in relevant scenarios. Our results clarify the relationship between structural (parameter-independent) and fine-tuned (parameter-dependent) reductions, with implications for approximate lumping when system parameters lie near critical values
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