  Brownian clocks are biomolecular networks that can count time. A paradigmatic
example are proteins that go through a cycle thus regulating some oscillatory
behaviour in a living system. Typically, such a cycle requires free energy
often provided by ATP hydrolysis. We investigate the relation between the
precision of such a clock and its thermodynamic costs. For clocks driven by a
constant thermodynamic force, a given precision requires a minimal cost that
diverges as the uncertainty of the clock vanishes. In marked contrast, we show
that a clock driven by a periodic variation of an external protocol can achieve
arbitrary precision at arbitrarily low cost. This result constitutes a
fundamental difference between processes driven by a fixed thermodynamic force
and those driven periodically. As a main technical tool, we map a periodically
driven system with a deterministic protocol to one subject to an external
protocol that changes in stochastic time intervals, which simplifies
calculations significantly. In the non-equilibrium steady state of the
resulting bipartite Markov process, the uncertainty of the clock can be deduced
from the calculable dispersion of a corresponding current.

Statistical Mechanics;Biological Physics