Abstract
Brownian clocks are biomolecular networks that can count time. A paradigmatic example are proteins that go through a cycle, thus regulating some oscillatory behavior 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 nonequilibrium steady state of the resulting bipartite Markov process, the uncertainty of the clock can be deduced from the calculable dispersion of a corresponding current.
1 More- Received 22 June 2016
DOI:https://doi.org/10.1103/PhysRevX.6.041053
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
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This article appears in the following collection:
Special Collection on Stochastic Thermodynamics
A Physical Review X special collection on stochastic thermodynamics.
Popular Summary
A crucial aspect of any clock is its precision. In the macroscopic world, clocks that have very high precision are the norm. But in the biological realm, clocks have to operate in a thermal environment, which exposes their constitutive parts to stochastic thermal noise, potentially affecting their precision. One might wonder whether for such “Brownian clocks” higher precision implies a higher energetic cost. Using the tools of stochastic thermodynamics, we investigate here the putative trade-off between precision and cost for two fundamentally different classes of Brownian clocks: clocks driven by a constant thermodynamic force and clocks driven by a periodic external energy source.
We model a Brownian clock as a random walk along a ring with states. For clocks driven by a constant thermodynamic force, such as the free energy liberated in ATP hydrolysis, there is indeed an inequality that establishes a minimal cost for a given precision. In contrast, for clocks driven by a time-periodic external protocol that modulates the energy of the states and the energy barriers between them, we show that by fine-tuning this energy landscape, one can indeed obtain a given precision with an arbitrarily low energy budget, provided the number of states is very large. Our findings provide a new perspective on biophysical and biochemical systems that are involved in “counting time.” We expect that future work may focus on determining how close to the theoretical limit real biological clocks operate and whether this trade-off has been a determining factor in their evolution.
Our results motivate the ongoing challenge of experimentally realizing a Brownian clock and will inform follow-up studies of the design of optimal artificial clocks operating on microscales and nanoscales.