Unlimited ultrasensitivity in a kinase/phosphatase “futile cycle” has been a paradigmatic example of collective behaviour in multi-enzyme systems. However, its analysis has relied on the Michaelis-Menten reaction mechanism, which remains widely used despite a century of new knowledge. Modifying and demodifying enzymes accomplish different biochemical tasks; the donor that contributes the modifying group is often ignored without the impact of this time-scale separation being taken into account; and new forms of reversible modification are now known. We exploit new algebraic methods of steady-state analysis to reconcile the analysis of multi-enzyme systems with single-enzyme biochemistry using zero-order ultrasensitivity as an example. We identify the property of “strong irreversibility”, in which product re-binding is disallowed. We show that unlimited ultrasensitivity is preserved for a class of complex, strongly-irreversible reaction mechanisms and determine the corresponding saturation conditions. We show further that unlimited ultrasensitivity arises from a singularity in a novel “invariant” that summarises the algebraic relationship between modified and unmodified substrate. We find that this singularity also underlies knife-edge behaviour in allocation of substrate between modification states, which has implications for the coherence of futile cycles within an integrated tissue. When the enzymes are irreversible, but not strongly so, the singularity disappears in the form found here and unlimited ultrasensitivity may no longer be preserved. The methods introduced here are widely applicable to other reversible modification systems.
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