Stochastic cycle selection in active flow networks Francis G. Woodhousea,1, Aden Forrowb, Joanna B. Fawcettc, and Jörn Dunkelb aDepartment of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom; bDepartment of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139-4307; and cCentre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved June 1, 2016 (received for review March 1, 2016) Active biological flow networks pervade nature and span a wide lattice of bacterial vortices (41). In this paper, we generalize this idea range of scales, from arterial bloodvesselsandbronchialmucus by constructing a generic lattice field model for an incompressible transport in humans to bacterial flow through porous media or active medium flowing in an arbitrary network of narrow channels. plasmodial shuttle streaming in slime molds. Despite their ubiquity, Combining concepts from transition rate theory and graph theory, little is known about the self-organization principles that govern flow we show how the competition between incompressibility, noise, and statistics in such nonequilibrium networks. Here we connect concepts spontaneous flow can trigger stochastic switching between states from lattice field theory, graph theory, and transition rate theory to comprising cycles of flowing edges separated by acyclic sets of understand how topology controls dynamics in a generic model for nonflowing edges. As a main result, we find that the state transi- actively driven flow on a network. Our combined theoretical and tion rates for individual edges can be related to one another via numerical analysis identifies symmetry-based rules that make it the cycle structure of the underlying network, yielding a topolog- possible to classify and predict the selection statistics of complex ical heuristic for predicting these rates in arbitrary networks. We flow cycles from the network topology. The conceptual framework conclude by establishing a mapping between incompressible active developed here is applicable to a broad class of biological and – nonbiological far-from-equilibrium networks, including actively con- flow networks and generalized ice-type or loop models (42 44). trolled information flows, and establishes a correspondence between Model active flow networks and generalized ice-type models. Lattice ϕ6 Field Theory for Active Flow Networks. Our network is a ∈ ∈ networks | active transport | stochastic dynamics | topology setofverticesv V connected by edges e E, forming an un- directed loop-free graph Γ. (We use graph theoretic terminology BIOPHYSICS AND throughout, where a loop is a single self-adjacent edge and a cycle is COMPUTATIONAL BIOLOGY iological flow networks, such as capillaries (1), leaf veins (2) and a closed vertex-disjoint walk.) To describe signed flux, we construct slime molds (3), use an evolved topology or active remodeling B the directed graph Γ^ by assigning an arbitrary orientation to each to achieve near-optimal transport when diffusion is ineffectual or ϕ ϕ > – edge.Now,let e be the flux along edge e,where e 0 denotes inappropriate (2, 4 7). Even in the absence of explicit matter flux, Γ^ ϕ < flow in the direction of the orientation of e in and e 0 denotes living systems often involve flow of information currents along – physical or virtual links between interacting nodes, as in neural the opposite. To model typical active matter behavior (31 33, 41), PHYSICS networks (8), biochemical interactions (9), epidemics (10), and we assume that fluxes either spontaneously polarize into flow states ϕ ≈ ± ϕ ≈ traffic flow (11). The ability to vary the flow topology gives network- e 1, or adopt some other nonflowing mode e 0. We for- ϕ = − = ϕ4 + based dynamics a rich phenomenology distinct from that of equiv- malize this by imposing a bistable potential Vð eÞ ð1 4Þ e = ϕ6 alent continuum models (12). Identical local rules can invoke ð1 6Þ e on each flux variable. This form, of higher order than in a dramatically different global dynamical behaviors when node con- typical Landau theory, ensures that incompressible potential min- ϕ − + nectivities change from nearest-neighbor interactions to the broad ima are polarized flows with every e in the set f 1,0, 1g,rather distributions seen in many networks (13–16). Certain classes of than the continuum of fractional flow states that a typical ϕ4 po- interacting networks are now sufficiently well understood to be able tential would yield (SI Local Potential). to exploit their topology for the control of input–output relations (17, 18), as exemplified by microfluidic logic gates (19, 20). How- Significance ever, when matter or information flow through a noisy network is not merely passive but actively driven by nonequilibrium constitu- Nature often uses interlinked networks to transport matter or ents (3), as in maze-solving slime molds (6), there are no over- information. In many cases, the physical or virtual flows between arching dynamical self-organization principles known. In such an network nodes are actively driven, consuming energy to achieve active network, noise and flow may conspire to produce behavior transport along different links. However, there are currently very radically different from that of a classical forced network. This raises few elementary principles known to predict the behavior of this the general question of how path selection and flow statistics in an broad class of nonequilibrium systems. Our work develops a active flow network depend on its interaction topology. generic foundational understanding of mass-conserving active Flow networks can be viewed as approximations of a complex flow networks. By merging previously disparate concepts from physical environment, using nodes and links to model intricate mathematical graph theory and physicochemical reaction rate geometric constraints (21–23). These constraints can profoundly – – theory, we relate the self-organizing behavior of actively driven affect matter transport (24 27), particularly for active systems (28 flows to the fundamental topological symmetries of the un- 30) where geometric confinement can enforce highly ordered col- – derlying network, resulting in a class of predictive models with lective dynamics (20, 31 40). In symmetric geometries like discs applicability across many biological scales. and channels, active flows can often be effectively captured by a ϕ single variable ðtÞ, such as angular velocity (32, 41) or net flux Author contributions: F.G.W., A.F., J.B.F., and J.D. designed research, performed research, ±ϕ (37), that tends to adopt one of two preferred states 0.External and wrote the paper. or intrinsic fluctuations can cause ϕðtÞ to diffuse in the vicinity of, The authors declare no conflict of interest. −ϕ ϕ say, 0, and may occasionally trigger a fast transition to 0 and This article is a PNAS Direct Submission. vice versa (37, 41). Geometrically coupling together many such 1To whom correspondence should be addressed. Email: [email protected]. confined units then results in a lattice field theory, reducing a ac.uk. nonequilibrium active medium to a discrete set of variables obeying This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. pseudoequilibrium physics, as was recently demonstrated for a 1073/pnas.1603351113/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1603351113 PNAS Early Edition | 1of6 Downloaded by guest on September 24, 2021 Incompressibility, appropriate to dense bacterial suspensions or state has, the lower its energy and therefore the longer-lived that Pactive liquid crystals, is imposed as follows. The net flux into vertex v is state will be; thus, in K4, for example, 4-cycles, which are global D ϕ , where the discrete negative divergence operator minima, persist longer than 3-cycles (Fig. 1 B and C). e∈E ve e ^ D = ðDve Þ is the jVj × jEj incidence matrix of Γ such that Dve is −1ife A graph possessing an Eulerian cycle—a nonrepeating tour of is directed out of v, +1ife is directed into v,and0ife is not incident to all edges starting and ending at one vertex—has global energy v (45). Exact incompressibility corresponds to the constraint DΦ = 0 minima with all edges flowing, which exists if and only if all vertices Φ = ϕ ∈ RjEj on the global flow configuration ð eÞ .Toallowforsmall are of even degree. By contrast, a graph possessing many vertices fluctuations, modeling variability in the microscopic flow structure, we of odd degree will have minimum energy states with nonflowing apply this as a soft constraint via an interaction potential ∝ jDΦj2.The edges, because edges flowing into and out of such a vertex pair up total energy HðΦÞ of the active flow network then reads to leave an odd number of zero-flow edges. Such “odd” networks are particularly interesting dynamically, as they are more suscep- X 1 tible to noise-induced state switches than graphs with even degree HðΦÞ = λ Vðϕ Þ + μjDΦj2, [1] e vertices (SI Model). For this reason, we specialize from now on to e∈ 2 E cubic (or 3-regular) graphs where all vertices have degree three. with coupling constants λ and μ. This energy is comparable to that of a lattice spin field theory, but with interactions given by higher- Waiting Times and Graph Symmetries. The cycle-swapping behavior can be quantified by the distribution of the waiting time for an edge dimensional quadratic forms akin to a spin theory on the vertices − of a hypergraph (SI Model). to transition between states in f 1,0,1g.Forsomeedges,de- pendent on Γ, this distribution will be identical: the interactions in 1 Network Dynamics. Appealing to recent results showing that bac- the energy in Eq. are purely topological, with no reference to an Γ — terial vortex lattices obey equilibrium-like physics (41), we impose embedding of , implying that only topological properties in par- — that Φ obeys the overdamped Langevin equation ticular, graph symmetries can influence the dynamics.