I. Introduction
Transcriptional regulatory networks (TRN) are collections of gene regulations that are mediated by transcriptional factors (TF). Variation in TRN contributes to diverse biological functions inside the cells. Adaptation is a biological function concerning the temporal dynamics of gene expression. It consists of a response phase, where the expression level of a gene responds transiently to an external stimulus, and a recovery phase, where the expression level adapts gradually to the initial value (Fig. 1) [1], [2]. Examples of adaptation include signal transduction [3]–[6], bacteria chemotaxis [7]–[9], and homeostasis [10]. It is well known that TRN with certain topologies such as incoherent feedforward loops (IFFL) and negative feedback loops (NFBL) can mediate adaptations robustly. In other words, even though chemical kinetic rates vary significantly across a cell population, networks like IFFL are more likely to execute adaptations than most other networks. Many studies used the term adaptive robustness to describe the ability of a network to achieve adaptations regardless of the parameters [2], [11], [12]. In synthetic biology, investigating the adaptive robustness of various networks is especially important, as it is impossible to achieve precise control of chemical kinetic rates in synthetic TRN due to complex cellular environments. However, a standard quantitative definition of adaptive robustness has not been formalized.