[论文解读] Rate-Independent Computation in Continuous Chemical Reaction Networks

Ho-Lin Chen, David Doty+2 more|arXiv (Cornell University)|Jul 29, 2021

Advanced biosensing and bioanalysis techniques参考文献 53被引用 44

一句话总结

本论文在连续化学反应网络(CRNs)中定义了对时间无关的计算概念，并利用分段可达性框架和负值的双轨表示，准确地刻画在任意时变速率律下能够计算的实值函数。

ABSTRACT

Coupled chemical interactions in a well-mixed solution are commonly formalized as chemical reaction networks (CRNs). However, despite the widespread use of CRNs in the natural sciences, the range of computational behaviors exhibited by CRNs is not well understood. Here we study the following problem: what functions $f:\mathbb{R}^k o \mathbb{R}$ can be computed by a CRN, in which the CRN eventually produces the correct amount of the "output" molecule, no matter the rate at which reactions proceed? This captures a previously unexplored, but very natural class of computations: for example, the reaction $X_1 + X_2 o Y$ can be thought to compute the function $y = \min(x_1, x_2)$. Such a CRN is robust in the sense that it is correct no matter the kinetic model of chemistry, so long as it respects the stoichiometric constraints. We develop a reachability relation based on "what could happen" if reaction rates can vary arbitrarily over time. We define *stable computation* analogously to probability 1 computation in distributed computing, and connect it with a seemingly stronger notion of rate-independent computation based on convergence under a wide class of generalized rate laws. We also consider the "dual-rail representation" that can represent negative values as the difference of two concentrations and allows the composition of CRN modules. We prove that a function is rate-independently computable if and only if it is piecewise linear (with rational coefficients) and continuous (dual-rail representation), or non-negative with discontinuities occurring only when some inputs switch from zero to positive (direct representation). The many contexts where continuous piecewise linear functions are powerful targets for implementation, combined with the systematic construction we develop for computing these functions, demonstrate the potential of rate-independent chemical computation.

研究动机与目标

• 激发对 CRN 在不受反应速率影响的情况下能够可靠执行哪些计算的理解。
• 将计量化学计量的计算能力与速率定律在 CRNs 中分离出来。
• 开发一个可达性框架，以捕捉在任意速率变化下“可能发生的事情”。
• 在连续性和分段线性方面表征速率无关的可计算函数。
• 展示双轨编码以处理负值并实现模块的组合。

提出的方法

• 基于尊重计量关系和因果约束的段（直线）轨迹，定义广义可达性关系。
• 形式化稳定计算，类似于分布式系统中的概率计算。
• 在广义速率律下引入公平计算，并证明与前馈 CRN 的稳定计算之间的联系。
• 证明速率无关可计算性对应于正连续的分段线性函数（直接表示）或连续分段线性函数（双轨表示）。
• 使用双轨表示以允许负值并实现 CRN 模块的组合。

实验结果

研究问题

• RQ1 当反应速率随时间任意变化时，CRN 在速率无关意义上能计算哪些实值函数？
• RQ2 我们如何在广义速率律下形式化可达性，以捕捉 CRNs 中可能发生的情况？
• RQ3 在直接表示和双轨表示下，速率无关可计算函数的精确定义是什么？
• RQ4 速率无关模块能否在不损失可计算性的情况下进行组合？
• RQ5 稳定计算与公平计算在连续 CRN 中如何相关，特别是对于前馈网络？

主要发现

• 基于广义可允速率律的可达性关系等价于非负性、反应物可用性和物种因果产生的直观约束。
• 对于前馈CRN，稳定计算和公平计算一致，将鲁棒收敛与速率无关联系起来。
• 恰好速率无关可计算的函数是正连续的分段线性（直接表示）或连续分段线性（双轨表示）。
• 双轨编码实现 CRN 模块的无干扰组合，并支持负值输出。
• 在非输入物种具有非零初始浓度时，速率无关 CRN 在表达能力上类似于 ReLU 神经网络。
• 该框架在统一的可达性模型下同时容纳质量作用速率律及其他速率律，包括米氏动力学和Hill函数动力学。