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广泛的系统之一。尽管转录网络在数字逻辑完备性方面已得到充分验证，但模拟计算在生物系统中起着关键作用，并为合成生物学应用提供了巨大潜力。虽然转录电路通常依赖转录因子的协同性和高度非线性行为来调控蛋白质的生成，但在建模时往往采用简单的线性降解项。相比之下，通用模拟动力学需要同时包含非线性正项和负项，这似乎要求不仅控制转录（即生成）调控，还需控制转录因子的降解速率。出乎意料的是，我们证明在不显式控制降解的情况下，仅控制转录因子生成（即转录速率）在数学上即可实现模拟计算的完备性，其能力等同于同时可编程调控生成与降解的系统。我们通过多个示例验证了该方法，包括振荡与混沌动力学、模拟排序、记忆、PID控制器以及模拟极值搜索。该结果提供了一种系统化方法，可在无需降解控制带来的额外复杂性的前提下，利用合成转录网络设计新型模拟动力学，并加深了对天然转录电路能力的理解。我们提供了一个编译器（以Python软件包形式），可将任意多项式常微分方程组在适当条件下精确转换为实现该系统的等效转录网络。