Thermodynamic Bounds on the Range and Sensitivity of Covalent Switching

Abstract

A recurring motif in biological signaling consists of a modifying group being added to and removed from a substrate via distinct chemical pathways catalyzed by different enzymes. In vivo such a cycle is typically driven by a non-zero thermodynamic force and thus entails a continuous dissipation of energy; for this reason it was at one time referred to as a “futile cycle”. It is now understood that the maintenance of a covalent modification cycle away from equilibrium has a functional payoff: the fraction of substrate in the modified state can thus be kept at a steady state away from its thermodynamic equilibrium value. Moreover, this steady-state fraction is sensitive to changes in the concentrations of the modifying enzymes, which allows the covalent modification cycle to act as a switch. Intuition and numerical analysis have supported the conjecture that the larger the thermodynamic force driving the modification/demodification cycle, the greater the possible sensitivity. We now present analytic expressions for the maximum sensitivity of the switch in terms of the thermodynamic force in two regimes: the high-substrate limit and the low-substrate limit. We also show that the magnitude of the thermodynamic force limits the range over which the fraction of modified substrate can vary, and present an analytic expression for this relationship that is valid in all cases.