生化计算模块的收敛速率下界
On the Convergence Rate Lower Bound of Biochemical Computational Modules
AI总结：
本研究针对生化反应网络计算精度问题，提出多输出生化计算模块收敛速率的简洁表征，证明其受雅可比矩阵最大实部特征值约束，为生化计算模块设计提供实用指标。
AI中文摘要：
生化反应网络已成为实现分子计算的核心理论框架，关键挑战在于有限时间计算精度，因为计算输出编码于物质浓度的极限稳态（LSS），而实际实现仅能运行有限时间。本研究对具有多个输出物质的生化计算模块的收敛速率提出了简洁表征，并严格证明其受雅可比矩阵实部最大（最接近零）的特征值约束。两个数值示例说明该理论下界如何塑造收敛速率范围，并揭示其对反应速率常数的依赖关系。该公式可用于系统评估生化计算速度，为构建高精度、误差可控的生化计算模块提供实用设计指标。
英文摘要：
Biochemical reaction networks have become a central theoretical framework for implementing molecular computation. A key challenge is finite time computational accuracy, as computation outputs are encoded in limiting steady states (LSSs) of species concentrations while practical implementations operate for only finite time. This work proposes a concise characterization of convergence rate for biochemical computational modules with multiple output species, and rigorously establishes it as being bounded by the eigenvalue with largest (least negative) real part of the Jacobian matrix. Two numerical examples illustrate how the theoretical lower bound shapes the convergence rate range and reveals its dependence on reaction rate constants. This formulation enables systematic evaluation of biochemical computation speed and provides a practical design measure for constructing high-accuracy and error-controlled biochemical computational modules.
