Single-cell variability in gene expression is important for generating distinct cell types, but it is unclear how cells use the same set of regulatory molecules to specifically control similarly regulated genes. While combinatorial binding of transcription factors at promoters has been proposed as a solution for cell-type specific gene expression, we found that such models resulted in substantial information bottlenecks. We sought to understand the consequences of adopting sequential logic wherein the time-ordering of factors informs the final outcome. We showed that with noncommutative control, it is possible to independently control targets that would otherwise be activated simultaneously using combinatorial logic. Consequently, sequential logic overcomes the information bottleneck inherent in complex networks. We derived scaling laws for two noncommutative models of regulation, motivated by phosphorylation/neural networks and chromosome folding, respectively, and showed that they scale super-exponentially in the number of regulators. We also showed that specificity in control is robust to the loss of a regulator. Lastly, we connected these theoretical results to real biological networks that demonstrate specificity in the context of promiscuity. These results show that achieving a desired outcome often necessitates roundabout steps.
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