Transcriptional networks represent one of the most extensively studied types of systems in synthetic biology. Although the completeness of transcriptional networks for digital logic is well-established, *analog* computation plays a crucial role in biological systems and offers significant potential for synthetic biology applications. While transcriptional circuits typically rely on cooperativity and highly non-linear behavior of transcription factors to regulate *production* of proteins, they are often modeled with simple linear *degradation* terms. In contrast, general analog dynamics require both non-linear positive as well as negative terms, seemingly necessitating control over not just transcriptional (i.e., production) regulation but also the degradation rates of transcription factors. Surprisingly, we prove that controlling transcription factor production (i.e., transcription rate) without explicitly controlling degradation is mathematically complete for analog computation, achieving equivalent capabilities to systems where both production and degradation are programmable. We demonstrate our approach on several examples including oscillatory and chaotic dynamics, analog sorting, memory, PID controller, and analog extremum seeking. Our result provides a systematic methodology for engineering novel analog dynamics using synthetic transcriptional networks without the added complexity of degradation control and informs our understanding of the capabilities of natural transcriptional circuits. We provide a compiler, in the form of a Python package that can take any system of polynomial ODEs and convert it to an equivalent transcriptional network implementing the system *exactly*, under appropriate conditions.

翻译：转录网络是合成生物学中研究最为深入的系统类型之一。尽管转录网络在数字逻辑中的完备性已得到充分证实，但模拟运算在生物系统中扮演着关键角色，并为合成生物学应用提供了重大潜力。转录电路通常依赖转录因子的协同作用与高度非线性行为来调控蛋白质的*合成*，然而在建模时却常采用简单的线性*降解*项。相比之下，通用模拟动力学既需要非线性正项也需要非线性负项，这似乎要求不仅控制转录（即合成）调控，还需控制转录因子的降解速率。令人惊讶的是，我们证明：在无需显式控制降解的情况下，仅通过控制转录因子的生成（即转录速率）即可在数学上完备地实现模拟运算，其能力与可编程调控生成和降解的系统相当。我们通过若干示例（包括振荡与混沌动力学、模拟排序、记忆单元、PID控制器及模拟极值搜索）展示了该方法。本研究提供了一种系统化工程方案，可利用合成转录网络设计新型模拟动力学，无需增加降解控制的复杂性，同时加深了对天然转录电路能力的理解。我们还提供了编译器形式的Python代码包，可将任意多项式常微分方程系统在适当条件下精确转换为等效的转录网络实现。