Protein complexes, typically made up of a small number of identical subunits, are very common in biology. These subunits can additionally undergo post-translational modifications, such as phosphorylation and dephosphorylation, resulting in a high dimensional state space for the protein complex. Importantly, such modifications are catalyzed by enzymes that are driven out of equilibrium by the consumption of a fuel such as ATP. I will discuss, from a theoretical perspective, how simple enzyme-catalyzed operations at the single subunit level can result in emergent behaviour at the level of the entire protein complex. First, I will discuss how topologically-protected edge currents emerge and become enhanced in arbitrarily high-dimensional stochastic systems representing the state of the complex, extending previous results for two-dimensional stochastic systems [1]. Second, I will discuss how enzymes that act on a subunit in a context-dependent manner provide a molecular implementation of stochastic cellular automata, that can be exploited to engineer molecular-scale computing devices, such as an error-tolerant memory or a finite-state machine [2].

[1] E. Tang, J. Agudo-Canalejo, and R. Golestanian, Phys. Rev. X 11, 031015 (2021)

[2] J. Kocka, K. Husain, and J. Agudo-Canalejo, PRX Life 4, 013036 (2026)