# Coherence of Biochemical Oscillations is Bounded by Driving Force and Network Topology

## TL;DR

This paper demonstrates that the maximum number of coherent biochemical oscillations is universally limited by the system's thermodynamic driving force and network topology, with broad implications for biological oscillators.

## Contribution

It establishes a universal bound on biochemical oscillation coherence based on thermodynamics and network structure, applicable to arbitrary Markov processes.

## Key findings

- Bound on coherence increases with thermodynamic force
- Network topology constrains oscillation precision
- Universal eigenvalue relation conjectured for stochastic matrices

## Abstract

Biochemical oscillations are prevalent in living organisms. Systems with a small number of constituents cannot sustain coherent oscillations for an indefinite time because of fluctuations in the period of oscillation. We show that the number of coherent oscillations that quantifies the precision of the oscillator is universally bounded by the thermodynamic force that drives the system out of equilibrium and by the topology of the underlying biochemical network of states. Our results are valid for arbitrary Markov processes, which are commonly used to model biochemical reactions. We apply our results to a model for a single KaiC protein and to an activator-inhibitor model that consists of several molecules. From a mathematical perspective, based on strong numerical evidence, we conjecture a universal constraint relating the imaginary and real parts of the first non-trivial eigenvalue of a stochastic matrix.