F. C. Sheldon | Searcharxiv

arXiv2023

Number of attractors in the critical Kauffman model is exponential

The Kauffman model is the archetypal model of genetic computation. It highlights the importance of criticality, at which many biological systems seem poised. In a series of advances, researchers have honed in on how the number of attractors in the critical regime grows with network size. But a definitive answer has proved elusive. We prove that, for the critical Kauffman model with connectivity one, the number of attractors grows at least, and at most, as $(2/\!\sqrt{e})^N$. This is the first proof that the number of attractors in a critical Kauffman model grows exponentially.

q-bio.MN↗

arXiv2023

Insights from number theory into the critical Kauffman model with connectivity one

The Kauffman model of genetic computation highlights the importance of criticality at the border of order and chaos. The model with connectivity one is of special interest because it is exactly solvable. But our understanding of its behavior is incomplete, and much of what we do know relies on heuristic arguments. Here, we show that the key quantities in the model are intimately related to aspects of number theory. Using these links, we derive improved bounds for the number of attractors as well as the mean attractor length, which is harder to compute. Our work suggests that number theory is the natural language for deducing many properties of the critical Kauffman model with connectivity one, and opens the door to further insight into this deceptively simple model.

q-bio.MN↗