Abstract
Constructing systems that exhibit timescales much longer than those of the underlying components, as well as emergent dynamical and collective behavior, is a key goal in fields such as synthetic biology and materials self-assembly. Inspiration often comes from living systems, in which robust global behavior prevails despite the stochasticity of the underlying processes. Here, we present two-dimensional stochastic networks that consist of minimal motifs representing out-of-equilibrium cycles at the molecular scale and support chiral edge currents in configuration space. These currents arise in the topological phase because of the bulk-boundary correspondence and dominate the system dynamics in the steady state, further proving robust to defects or blockages. We demonstrate the topological properties of these networks and their uniquely non-Hermitian features such as exceptional points and vorticity, while characterizing the edge-state localization. As these emergent edge currents are associated with macroscopic timescales and length scales, simply tuning a small number of parameters enables varied dynamical phenomena, including a global clock, dynamical growth and shrinkage, and synchronization. Our construction provides a novel topological formalism for stochastic systems and fresh insights into non-Hermitian physics, paving the way for the prediction of robust dynamical states in new classical and quantum platforms.
- Received 13 November 2020
- Revised 12 April 2021
- Accepted 25 May 2021
DOI:https://doi.org/10.1103/PhysRevX.11.031015
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. Open access publication funded by the Max Planck Society.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
The interior of a cell is a complex place, full of small molecules randomly bumping against each other. This raises the question of how cellular behavior manages to remain so robust, exhibiting reliable patterns and oscillations that spread over length scales and timescales much longer than those of the individual molecules that support them. Here, we show how certain simple motifs, which represent random processes on the microscopic scale, can give rise to robust oscillations on the macroscopic scale that display many of the features typical of biological systems.
Our models are inspired by a feature first discovered in quantum systems known as topological protection. In topological materials, electronic transport avoids the materials’ interior and becomes concentrated along its boundary. Instead, we consider biochemical processes that take place in a 2D configuration space, for example, the growth of a protein made of subunits of one type and subunits of another. By repeating simple motifs that represent microscopic processes (such as the addition or removal of one subunit), we construct networks that support oscillations along their boundaries, independently of their size or shape. The robustness of these oscillations can be understood as arising from topological protection, just like in quantum systems.
In the future, we expect that specific biochemical processes can be mapped onto our versatile framework. Moreover, our results will feed back into the design of topological states in quantum and classical systems with new features from our biologically inspired models.
