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. 2022 Jun 21;12:10490. doi: 10.1038/s41598-022-14362-z

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© The Author(s) 2022

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Schematics of our model circuits and characteristics of a protein complex formation system. (A–D) Diagrams of the four assessed circuits. (A) Circuit 1 (open loop without cooperative stability). (B) Circuit 2 (open loop with cooperative stability). (C) Circuit 3 (negative feedback loops without cooperative stability). (D) Circuit 4 (negative feedback loops with cooperative stability). (E–H) Schematics of the characteristics of a protein complex formation system. (E–G) Equilibrium properties, i.e. (E) efficiency, (F) dimer ratio, and (G) controllability, of a system at steady-state. (H) Dynamic behaviors of the system, i.e., response time to upward regulation, response time to downward regulation, and recovery time after a shot noise.