Abstract
Constructing systems that exhibit time-scales 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 due to 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 to 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.
Highlights
Why are biological functions carried out so robustly, even when the underlying components are stochastic in time and randomly distributed in space? Living systems can have stable properties that endure for timescales much longer than the lifetime of the underlying constituents, which contribute to memory and adaptive processes [1,2]
The two-dimensional networks we introduce are constructed from simple repetitive motifs, which correspond to out-of-equilibrium cycles at the molecular scale, and form the analog of cyclotron orbits in the quantum Hall system
The emergence of edge currents can be understood as a topological transition showing unique non-Hermitian properties, which highlight the nonequilibrium character of the system
Summary
Why are biological functions carried out so robustly, even when the underlying components are stochastic in time and randomly distributed in space? Living systems can have stable properties that endure for timescales much longer than the lifetime of the underlying constituents, which contribute to memory and adaptive processes [1,2]. The system transport is dominated by the propagating edge states, which are exponentially localized at the boundaries and protected from disorder and perturbations The robustness of these edge currents makes them a desirable feature for the support of stable emergent phenomena. Beyond the analogy to quantum Hall physics, the chiral edge currents in our system imply motion along the boundaries of configuration space rather than real space As such, they enable oscillations (e.g., cyclical conformational changes of a protein complex, or assembly and disassembly of a biopolymer) governed by physical constraints in the system rather than the specific timescales of the underlying microscopic transitions, which do not need to be fine-tuned [1,2,7]. We end with a discussion of the implications and future directions for our work
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