Topological phases in discrete stochastic systems

TL;DR

This work addresses how to extend topological concepts to discrete, non-equilibrium stochastic systems found in biology and synthetic materials. It develops and surveys models where edge states and global cycles arise from topology in configuration space, including 1D and 2D stochastic lattices and their mappings to quantum-like formalisms, especially under non-Hermitian dynamics. Key contributions include demonstrations of robust edge localization, dimensional reduction, and scalable motif building blocks, with concrete applications to circadian rhythms, sensory adaptation, gene transcription, microtubule dynamics, microfluidic networks, and microswimmer design. The findings provide design principles for robust function in noisy, non-equilibrium settings and introduce tools such as transfer matrices and symmetry-based classifications, while outlining open theoretical and experimental directions for higher-dimensional and interacting systems.

Abstract

Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system response to a lower dimensional space and, in 2D systems, offer a mechanism for the emergence of global cycles within a large phase space. Topological invariants have been heavily studied in quantum electronic systems and have been observed in other classical platforms such as mechanical lattices. However, this framework largely describes equilibrium systems within an ordered crystalline lattice, whereas biological systems are often strongly non-equilibrium with stochastic components. We review recent developments in topological states in discrete stochastic models in 1D and 2D systems, and initial progress in identifying testable signature of topological states in molecular systems and ecology. These models further provide simple principles for targeted dynamics in synthetic systems and in the engineering of reconfigurable materials. Lastly, we describe novel theoretical properties of these systems such as the necessity for non-Hermiticity in permitting edge states, as well as new analytical tools to reveal these properties. The emerging developments shed light on fundamental principles for non-equilibrium systems and topological protection enabling robust biological function.

Figures (6)

• Figure 1: Topological invariants signal the emergence of edge states across diverse platforms. We depict an example of a topological invariant: a winding number. This can be defined in Fourier space, since most topological models are composed of repeated motifs (unit cells). The Fourier domain is periodic and shown as black circles. Transformed eigenvectors can have a phase (shown as blue arrows) that can wind by $2\pi$ as we move along the domain Obana2019TopologicalModel. Left: The eigenvector phase winds around the domain and thus has a nonzero winding number. This indicates a topological phase hence will correspond to the presence of localized states in the real system on its edges, shown as red dots at the ends of a beige line. Right: In contrast, here the eigenvector phase fluctuates but does not wind by $2\pi$, so this trivial phase will correspond to an absence of edge states in the real system (beige line). Topological models have been developed in a variety of systems. Top: Models were first developed in systems that can be described by the Schrödinger equation or mapped onto it, including quantum electrons Moore2010TheInsulatorsHasan2010Colloquium:Insulators, mechanical lattices Mao2018MaxwellMechanicsPedro2019TopologicalInteractionsKane2014, photonics Xiao2014SurfaceSystemsBenalcazar2017QuantizedInsulatorsYuce2019TopologicalModelPocock2018TopologicalEffects, and acoustics Ni_2017Liu2017PseudospinsLattice. Bottom: Discrete classical systems obey other dynamical equations such as the master equation, with various applications from molecular biology to synthetic systems Tang2021TopologySystemsMurugan2017TopologicallySystemsDasbiswasE9031, or the Lotka-Volterra equation in the case of population dynamics models Knebel2020TopologicalCyclesYoshida2021ChiralCycles.
• Figure 2: Proposals for topological models in stochastic systems. Non-equilibrium motifs (top) can be repeated to support a topological state. Left column: Futile cycles, such as phosphorylation-dephosphorylation cycles, appear in various biological systems from protein synthesis to muscular contraction, metabolism and sensory systems hopfield1974kineticsamoilov2005stochastic. (a) A series of biochemical reactions form a 1D chain that displays edge states at the interface of two different topological invariants; adapted from Ref. Murugan2017TopologicallySystems. (b) In a 2D lattice of interlinked futile cycles, a topological regime is found where probability accumulates at the edges of the system and a probability current with defined chirality spontaneously emerges, leading to system-spanning oscillations; adapted from Ref. Tang2021TopologySystems. Right column: The rock-paper-scissors interaction can model non-transitive evolutionary pressures in population dynamics. (c) In a 1D chain with such interactions, population can accumulate at one or the other end of the chain as a result of a topological phase transition; adapted from Ref. Knebel2020TopologicalCycles. (d) In a 2D Kagome lattice, excesses in population density are transported along the edges of the lattice in a chiral manner, due to a topological effect; adapted with permission from Ref. Yoshida2021ChiralCycles.
• Figure 3: Topological protection ensures robustness of the edge current to obstacles or missing components. When certain components of the system are missing or inaccessible (states with red crosses), the edge current will simply go around them to maintain the largest available phase space, when in the topological phase [${\gamma_\mathrm{ex}}\gg{\gamma_\mathrm{in}}$, see Fig. \ref{['extfig:models']}(b)]. This robustness of the edge state can shed light on how biological systems flexibly pivot in the presence of changing conditions or external stimuli.
• Figure 4: Topological model for emergent oscillations. (a) Macromolecules exhibit a large space of possible conformations, here illustrated with the KaiC hexamer that governs the circadian rhythm of cyanobacteria. Based on observations of autophosphorylation in the literature Brettschneider2010AClock.Kageyama2006CyanobacterialVitroLin2014MixturesClock, it is thought that monomers undergo phosphorylation and dephosphorylation cycles (black arrows, ${\gamma_\mathrm{ex}}$), as well as conformational changes between exposed and buried configurations, or binding with other proteins such as KaiA and KaiB (grey arrows, ${\gamma_\mathrm{in}}$). (b) These cycles can be laid out following the model of Fig. \ref{['extfig:models']}(b), with reversed rates included in this case. (c) KaiABC exhibits oscillations via a concerted global cycle of phosphorylation and dephosphorylation. During the day, all six KaiC monomers get phosphorylated at the T-sites, and then at the S-sites. This phosphorylation phase is promoted by interaction with KaiA molecules xu2003cyanobacterial. By night, fully phosphorylated KaiC binds to KaiB, which sequesters KaiA from the solution. In the absence of KaiA, all the T-sites get dephosphorylated, followed by the S-sites rust2007ordered. Since individual monomers can independently phosphorylate Brettschneider2010AClock., it is unclear why they would perform a concerted phosphorylation cycle that is robust. (d) A possible solution lies in the topological phase of the model, in which a global cycle emerges that recapitulates the experimentally observed phosphorylation sequence. Adapted from Ref. zheng2024a.
• Figure 5: A topological design for a microswimmer can flexibly navigate around malfunctions or obstacles. The velocity of a 3-sphere microswimmer is proportional to the area it encloses in shape space Shapere1987Self-propulsionNumbergolestanian2008mechanical. Topologically-protected edge cycles enclose all the available shape space and thus maximize the velocity of the microswimmer, even in the presence of component malfunctions or obstacles. Adapted from Ref. Tang2021TopologySystems.