  A model of coupled molecular oscillators is proposed to study nonequilibrium
thermodynamics of synchronization. We find that synchronization of
nonequilibrium oscillators costs energy even when the oscillator-oscillator
coupling is conservative. By solving the steady state of the many-body system
analytically, we show that the system goes through a nonequilibrium phase
transition driven by energy dissipation, and the critical energy dissipation
depends on both the frequency and strength of the exchange reactions. Moreover,
our study reveals the optimal design for achieving maximum synchronization with
a fixed energy budget. We apply our general theory to the Kai system in
Cyanobacteria circadian clock and predict a relationship between the KaiC
ATPase activity and synchronization of the KaiC hexamers. The theoretical
framework can be extended to study thermodynamics of collective behaviors in
other extended nonequilibrium active systems.
