Bistability, Absolute Concentration Robustness, and Hysteresis in Dual-Site Futile Cycles with Bifunctional Enzymes

TL;DR

This paper provides a comprehensive mathematical analysis of dual-site futile cycles with bifunctional enzymes, revealing complex behaviors like bistability, hysteresis, and absolute concentration robustness in biochemical networks.

Contribution

It characterizes the steady states and bifurcation structures of four dual-site futile cycle networks with bifunctional enzymes, highlighting novel dynamical phenomena.

Findings

All four networks admit boundary steady states, unlike non-bifunctional cases.

Networks vary in the number and stability of steady states, with some showing bistability.

One network exhibits simultaneous bistability and absolute concentration robustness in the final product.

Abstract

Bifunctional enzymes, which catalyze both the forward and reverse steps of a substrate modification reaction, arise naturally in bacterial two-component signaling systems and metabolic regulation. Beyond their well-known role in conferring absolute concentration robustness (ACR) on substrate species, bifunctional enzymes profoundly shape the dynamical landscape of the networks in which they appear. We study a class of dual-site futile cycles in which the reverse modification steps are carried out by bifunctional enzyme-substrate compounds, and provide a complete mathematical analysis of all four such networks, characterizing the existence, number, and stability of steady states, as well as the bifurcation structure as total substrate is varied. All four networks admit boundary steady states, in contrast to the non-bifunctional case. The networks differ in the number and stability of boundary steady states, in the maximum number of positive steady states (ranging from two to four), and in whether bistability is present. In two networks, a transcritical bifurcation connects the boundary and positive steady state branches; in one case this is a backward bifurcation, producing hysteresis. Perhaps the most striking phenomenon occurs in one of the four networks, which simultaneously exhibits bistability and ACR in the final modification state, where the system can settle into either of two stable steady states with different intermediate concentrations yet identical final product concentration.