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Bunimovich L, Skums P. Fractal networks: Topology, dimension, and complexity. CHAOS (WOODBURY, N.Y.) 2024; 34:042101. [PMID: 38598678 DOI: 10.1063/5.0200632] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/27/2024] [Accepted: 03/24/2024] [Indexed: 04/12/2024]
Abstract
Over the past two decades, the study of self-similarity and fractality in discrete structures, particularly complex networks, has gained momentum. This surge of interest is fueled by the theoretical developments within the theory of complex networks and the practical demands of real-world applications. Nonetheless, translating the principles of fractal geometry from the domain of general topology, dealing with continuous or infinite objects, to finite structures in a mathematically rigorous way poses a formidable challenge. In this paper, we overview such a theory that allows to identify and analyze fractal networks through the innate methodologies of graph theory and combinatorics. It establishes the direct graph-theoretical analogs of topological (Lebesgue) and fractal (Hausdorff) dimensions in a way that naturally links them to combinatorial parameters that have been studied within the realm of graph theory for decades. This allows to demonstrate that the self-similarity in networks is defined by the patterns of intersection among densely connected network communities. Moreover, the theory bridges discrete and continuous definitions by demonstrating how the combinatorial characterization of Lebesgue dimension via graph representation by its subsets (subgraphs/communities) extends to general topological spaces. Using this framework, we rigorously define fractal networks and connect their properties with established combinatorial concepts, such as graph colorings and descriptive complexity. The theoretical framework surveyed here sets a foundation for applications to real-life networks and future studies of fractal characteristics of complex networks using combinatorial methods and algorithms.
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Affiliation(s)
- L Bunimovich
- School of Mathematics, Georgia Institute of Technology, 686 Cherry St NW, Atlanta, Georgia 30332, USA
| | - P Skums
- School of Computing, University of Connecticut, 371 Fairfield Way, Storrs, Connecticut 06269, USA
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2
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Le Verge-Serandour M, Alim K. Active fluids navigate networks by solving sudoku-like problems. Nature 2024; 627:39-40. [PMID: 38321159 DOI: 10.1038/d41586-024-00277-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/08/2024]
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3
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Chen S, Alim K. Network topology enables efficient response to environment in Physarum polycephalum. Phys Biol 2023; 20. [PMID: 37190961 DOI: 10.1088/1478-3975/accef2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/24/2022] [Accepted: 04/20/2023] [Indexed: 05/17/2023]
Abstract
The network-shaped body plan distinguishes the unicellular slime mouldPhysarum polycephalumin body architecture from other unicellular organisms. Yet, network-shaped body plans dominate branches of multi-cellular life such as in fungi. What survival advantage does a network structure provide when facing a dynamic environment with adverse conditions? Here, we probe how network topology impactsP. polycephalum's avoidance response to an adverse blue light. We stimulate either an elongated, I-shaped amoeboid or a Y-shaped networked specimen and subsequently quantify the evacuation process of the light-exposed body part. The result shows that Y-shaped specimen complete the avoidance retraction in a comparable time frame, even slightly faster than I-shaped organisms, yet, at a lower almost negligible increase in migration velocity. Contraction amplitude driving mass motion is further only locally increased in Y-shaped specimen compared to I-shaped-providing further evidence that Y-shaped's avoidance reaction is energetically more efficient than in I-shaped amoeboid organisms. The difference in the retraction behaviour suggests that the complexity of network topology provides a key advantage when encountering adverse environments. Our findings could lead to a better understanding of the transition from unicellular to multicellularity.
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Affiliation(s)
- Siyu Chen
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Karen Alim
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- TUM School of Natural Sciences, Department of Bioscience, Center of Protein Assemblies (CPA), Technical University of Munich, Garching 85748, Germany
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Elek O, Burchett JN, Prochaska JX, Forbes AG. Monte Carlo Physarum Machine: Characteristics of Pattern Formation in Continuous Stochastic Transport Networks. ARTIFICIAL LIFE 2022; 28:22-57. [PMID: 34905603 DOI: 10.1162/artl_a_00351] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
We present Monte Carlo Physarum Machine (MCPM): a computational model suitable for reconstructing continuous transport networks from sparse 2D and 3D data. MCPM is a probabilistic generalization of Jones's (2010) agent-based model for simulating the growth of Physarum polycephalum (slime mold). We compare MCPM to Jones's work on theoretical grounds, and describe a task-specific variant designed for reconstructing the large-scale distribution of gas and dark matter in the Universe known as the cosmic web. To analyze the new model, we first explore MCPM's self-patterning behavior, showing a wide range of continuous network-like morphologies-called polyphorms-that the model produces from geometrically intuitive parameters. Applying MCPM to both simulated and observational cosmological data sets, we then evaluate its ability to produce consistent 3D density maps of the cosmic web. Finally, we examine other possible tasks where MCPM could be useful, along with several examples of fitting to domain-specific data as proofs of concept.
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Affiliation(s)
- Oskar Elek
- University of California, Santa Cruz, Computational Media, Creative Coding Lab.
| | | | - J Xavier Prochaska
- University of California, Santa Cruz, Astronomy and Astrophysics
- The University of Tokyo, Kavli Institute for the Physics and Mathematics of the Universe.
| | - Angus G Forbes
- University of California, Santa Cruz, Computational Media, Creative Coding Lab.
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Fleig P, Kramar M, Wilczek M, Alim K. Emergence of behaviour in a self-organized living matter network. eLife 2022; 11:62863. [PMID: 35060901 PMCID: PMC8782570 DOI: 10.7554/elife.62863] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2020] [Accepted: 12/20/2021] [Indexed: 12/16/2022] Open
Abstract
What is the origin of behaviour? Although typically associated with a nervous system, simple organisms also show complex behaviours. Among them, the slime mold Physarum polycephalum, a giant single cell, is ideally suited to study emergence of behaviour. Here, we show how locomotion and morphological adaptation behaviour emerge from self-organized patterns of rhythmic contractions of the actomyosin lining of the tubes making up the network-shaped organism. We quantify the spatio-temporal contraction dynamics by decomposing experimentally recorded contraction patterns into spatial contraction modes. Notably, we find a continuous spectrum of modes, as opposed to a few dominant modes. Our data suggests that the continuous spectrum of modes allows for dynamic transitions between a plethora of specific behaviours with transitions marked by highly irregular contraction states. By mapping specific behaviours to states of active contractions, we provide the basis to understand behaviour’s complexity as a function of biomechanical dynamics.
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Affiliation(s)
- Philipp Fleig
- Department of Physics & Astronomy, University of Pennsylvania
- Max Planck Institute for Dynamics and Self-Organization
| | - Mirna Kramar
- Max Planck Institute for Dynamics and Self-Organization
| | | | - Karen Alim
- Max Planck Institute for Dynamics and Self-Organization
- Physik-Department and Center for Protein Assemblies, Technische Universität München
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Kotwal T, Moseley F, Stegmaier A, Imhof S, Brand H, Kießling T, Thomale R, Ronellenfitsch H, Dunkel J. Active topolectrical circuits. Proc Natl Acad Sci U S A 2021; 118:e2106411118. [PMID: 34349024 PMCID: PMC8364202 DOI: 10.1073/pnas.2106411118] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The transfer of topological concepts from the quantum world to classical mechanical and electronic systems has opened fundamentally different approaches to protected information transmission and wave guidance. A particularly promising emergent technology is based on recently discovered topolectrical circuits that achieve robust electric signal transduction by mimicking edge currents in quantum Hall systems. In parallel, modern active matter research has shown how autonomous units driven by internal energy reservoirs can spontaneously self-organize into collective coherent dynamics. Here, we unify key ideas from these two previously disparate fields to develop design principles for active topolectrical circuits (ATCs) that can self-excite topologically protected global signal patterns. Realizing autonomous active units through nonlinear Chua diode circuits, we theoretically predict and experimentally confirm the emergence of self-organized protected edge oscillations in one- and two-dimensional ATCs. The close agreement between theory, simulations, and experiments implies that nonlinear ATCs provide a robust and versatile platform for developing high-dimensional autonomous electrical circuits with topologically protected functionalities.
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Affiliation(s)
- Tejas Kotwal
- Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
- Division of Applied Mathematics, Brown University, Providence, RI 02912
| | - Fischer Moseley
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
| | - Alexander Stegmaier
- Institut für Theoretische Physik und Astrophysik, Universität Würzburg, D-97074 Würzburg, Germany
| | - Stefan Imhof
- Physikalisches Institut der Universität Würzburg, Universität Würzburg, D-97074 Würzburg, Germany
| | - Hauke Brand
- Physikalisches Institut der Universität Würzburg, Universität Würzburg, D-97074 Würzburg, Germany
| | - Tobias Kießling
- Physikalisches Institut der Universität Würzburg, Universität Würzburg, D-97074 Würzburg, Germany
| | - Ronny Thomale
- Institut für Theoretische Physik und Astrophysik, Universität Würzburg, D-97074 Würzburg, Germany
| | - Henrik Ronellenfitsch
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139;
- Physics Department, Williams College, Williamstown, MA 01267
| | - Jörn Dunkel
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139;
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Bernardi A, Swanson JM. CycFlowDec: A Python module for decomposing flow networks using simple cycles. SOFTWAREX 2021; 14:100676. [PMID: 34703873 PMCID: PMC8545271 DOI: 10.1016/j.softx.2021.100676] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
New algorithms for determining the expected flow through simple cycles in a closed network are presented. Current network analysis software do not implement algorithms for expected cyclic flow decomposition, despite its potential value. Decomposing networks into expected cycle flows provides a quantitative characterization of network cycles that can be further analyzed for sensitivity and correlative behavior. An efficient, general algorithm has been coded into CycFlowDec, an open source Python module available at https://github.com/austenb28/CycFlowDec.
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Skums P, Bunimovich L. Graph fractal dimension and the structure of fractal networks. JOURNAL OF COMPLEX NETWORKS 2020; 8:cnaa037. [PMID: 33251012 PMCID: PMC7673317 DOI: 10.1093/comnet/cnaa037] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2020] [Accepted: 09/28/2020] [Indexed: 06/12/2023]
Abstract
Fractals are geometric objects that are self-similar at different scales and whose geometric dimensions differ from so-called fractal dimensions. Fractals describe complex continuous structures in nature. Although indications of self-similarity and fractality of complex networks has been previously observed, it is challenging to adapt the machinery from the theory of fractality of continuous objects to discrete objects such as networks. In this article, we identify and study fractal networks using the innate methods of graph theory and combinatorics. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colourings and graph descriptive complexity, and analyse the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework in evolutionary biology and virology by analysing networks of viral strains sampled at different stages of evolution inside their hosts. Our methodology revealed gradual self-organization of intra-host viral populations over the course of infection and their adaptation to the host environment. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.
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Affiliation(s)
| | - Leonid Bunimovich
- School of Mathematics, Georgia Institute of Technology, 686 Cherry St NW, Atlanta, GA 30313, USA
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Karschau J, Scholich A, Wise J, Morales-Navarrete H, Kalaidzidis Y, Zerial M, Friedrich BM. Resilience of three-dimensional sinusoidal networks in liver tissue. PLoS Comput Biol 2020; 16:e1007965. [PMID: 32598356 PMCID: PMC7351228 DOI: 10.1371/journal.pcbi.1007965] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2019] [Revised: 07/10/2020] [Accepted: 05/19/2020] [Indexed: 12/19/2022] Open
Abstract
Can three-dimensional, microvasculature networks still ensure blood supply if individual links fail? We address this question in the sinusoidal network, a plexus-like microvasculature network, which transports nutrient-rich blood to every hepatocyte in liver tissue, by building on recent advances in high-resolution imaging and digital reconstruction of adult mice liver tissue. We find that the topology of the three-dimensional sinusoidal network reflects its two design requirements of a space-filling network that connects all hepatocytes, while using shortest transport routes: sinusoidal networks are sub-graphs of the Delaunay graph of their set of branching points, and also contain the corresponding minimum spanning tree, both to good approximation. To overcome the spatial limitations of experimental samples and generate arbitrarily-sized networks, we developed a network generation algorithm that reproduces the statistical features of 0.3-mm-sized samples of sinusoidal networks, using multi-objective optimization for node degree and edge length distribution. Nematic order in these simulated networks implies anisotropic transport properties, characterized by an empirical linear relation between a nematic order parameter and the anisotropy of the permeability tensor. Under the assumption that all sinusoid tubes have a constant and equal flow resistance, we predict that the distribution of currents in the network is very inhomogeneous, with a small number of edges carrying a substantial part of the flow-a feature known for hierarchical networks, but unexpected for plexus-like networks. We quantify network resilience in terms of a permeability-at-risk, i.e., permeability as function of the fraction of removed edges. We find that sinusoidal networks are resilient to random removal of edges, but vulnerable to the removal of high-current edges. Our findings suggest the existence of a mechanism counteracting flow inhomogeneity to balance metabolic load on the liver.
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Affiliation(s)
| | - André Scholich
- Max Planck Institute for the Physics of Complex Systems, Dresden, Germany
| | - Jonathan Wise
- cfaed, TU Dresden, Dresden, Germany
- Univ. Grenoble Alpes, CNRS, LPMMC, Grenoble, France
| | | | - Yannis Kalaidzidis
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
| | - Marino Zerial
- Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany
- Cluster of Excellence ‘Physics of Life’, TU Dresden, Dresden, Germany
| | - Benjamin M. Friedrich
- cfaed, TU Dresden, Dresden, Germany
- Cluster of Excellence ‘Physics of Life’, TU Dresden, Dresden, Germany
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10
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Svenšek D, Pleiner H, Brand HR. A dynamic preferred direction model for the self-organization dynamics of bacterial microfluidic pumping. SOFT MATTER 2019; 15:2032-2042. [PMID: 30724307 DOI: 10.1039/c9sm00023b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
It is known that some flagellated bacteria like Serratia marcescens, when deposited and affixed onto a surface to form a "bacterial carpet", self-organize in a collective motion of the flagella that is capable of pumping fluid through microfluidic channels. We set up a continuum model comprising two macroscopic variables that is capable of describing this self-organization mechanism as well as quantifying it to the extent that an agreement with the experimentally observed channel width dependence of the pumping is reached. The activity is introduced through a collective angular velocity of the helical flagella rotation, which is an example of a dynamic macroscopic preferred direction. Our model supports and quantifies the view that the self-coordination is due to a positive feedback loop between the bacterial flagella and the local flow generated by their rotation. Moreover, our results indicate that this biological active system is operating close to the self-organization threshold.
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Affiliation(s)
- Daniel Svenšek
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia.
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11
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Mathijssen AJTM, Guzmán-Lastra F, Kaiser A, Löwen H. Nutrient Transport Driven by Microbial Active Carpets. PHYSICAL REVIEW LETTERS 2018; 121:248101. [PMID: 30608743 DOI: 10.1103/physrevlett.121.248101] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2018] [Indexed: 06/09/2023]
Abstract
We demonstrate that active carpets of bacteria or self-propelled colloids generate coherent flows towards the substrate, and propose that these currents provide efficient pathways to replenish nutrients that feed back into activity. A full theory is developed in terms of gradients in the active matter density and velocity, and applied to bacterial turbulence, topological defects and clustering. Currents with complex spatiotemporal patterns are obtained, which are tunable through confinement. Our findings show that diversity in carpet architecture is essential to maintain biofunctionality.
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Affiliation(s)
- Arnold J T M Mathijssen
- Department of Bioengineering, Stanford University, 443 Via Ortega, Stanford, California 94305, USA
| | - Francisca Guzmán-Lastra
- Facultad de Ciencias, Universidad Mayor, Av. Manuel Montt 367, Providencia, Santiago 7500994, Chile
- Departamento de Física, FCFM Universidad de Chile, Beauchef 850, Santiago 8370448, Chile
| | - Andreas Kaiser
- Department of Biomedical Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität, D-40225 Düsseldorf, Germany
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12
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Wilson DB, Baker RE, Woodhouse FG. Topology-dependent density optima for efficient simultaneous network exploration. Phys Rev E 2018; 97:062301. [PMID: 30011429 DOI: 10.1103/physreve.97.062301] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2017] [Indexed: 11/07/2022]
Abstract
A random search process in a networked environment is governed by the time it takes to visit every node, termed the cover time. Often, a networked process does not proceed in isolation but competes with many instances of itself within the same environment. A key unanswered question is how to optimize this process: How many concurrent searchers can a topology support before the benefits of parallelism are outweighed by competition for space? Here, we introduce the searcher-averaged parallel cover time (APCT) to quantify these economies of scale. We show that the APCT of the networked symmetric exclusion process is optimized at a searcher density that is well predicted by the spectral gap. Furthermore, we find that nonequilibrium processes, realized through the addition of bias, can support significantly increased density optima. Our results suggest alternative hybrid strategies of serial and parallel search for efficient information gathering in social interaction and biological transport networks.
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Affiliation(s)
- Daniel B Wilson
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford OX2 6GG, United Kingdom
| | - Ruth E Baker
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford OX2 6GG, United Kingdom
| | - Francis G Woodhouse
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
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Forrow A, Woodhouse FG, Dunkel J. Mode Selection in Compressible Active Flow Networks. PHYSICAL REVIEW LETTERS 2017; 119:028102. [PMID: 28753360 DOI: 10.1103/physrevlett.119.028102] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2016] [Indexed: 06/07/2023]
Abstract
Coherent, large-scale dynamics in many nonequilibrium physical, biological, or information transport networks are driven by small-scale local energy input. Here, we introduce and explore an analytically tractable nonlinear model for compressible active flow networks. In contrast to thermally driven systems, we find that active friction selects discrete states with a limited number of oscillation modes activated at distinct fixed amplitudes. Using perturbation theory, we systematically predict the stationary states of noisy networks and find good agreement with a Bayesian state estimation based on a hidden Markov model applied to simulated time series data. Our results suggest that the macroscopic response of active network structures, from actomyosin force networks to cytoplasmic flows, can be dominated by a significantly reduced number of modes, in contrast to energy equipartition in thermal equilibrium. The model is also well suited to study topological sound modes and spectral band gaps in active matter.
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Affiliation(s)
- Aden Forrow
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA
| | - Francis G Woodhouse
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
| | - Jörn Dunkel
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA
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Woodhouse FG, Dunkel J. Active matter logic for autonomous microfluidics. Nat Commun 2017; 8:15169. [PMID: 28440273 PMCID: PMC5414041 DOI: 10.1038/ncomms15169] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/02/2016] [Accepted: 03/06/2017] [Indexed: 01/24/2023] Open
Abstract
Chemically or optically powered active matter plays an increasingly important role in materials design, but its computational potential has yet to be explored systematically. The competition between energy consumption and dissipation imposes stringent physical constraints on the information transport in active flow networks, facilitating global optimization strategies that are not well understood. Here, we combine insights from recent microbial experiments with concepts from lattice-field theory and non-equilibrium statistical mechanics to introduce a generic theoretical framework for active matter logic. Highlighting conceptual differences with classical and quantum computation, we demonstrate how the inherent non-locality of incompressible active flow networks can be utilized to construct universal logical operations, Fredkin gates and memory storage in set-reset latches through the synchronized self-organization of many individual network components. Our work lays the conceptual foundation for developing autonomous microfluidic transport devices driven by bacterial fluids, active liquid crystals or chemically engineered motile colloids.
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Affiliation(s)
- Francis G. Woodhouse
- Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | - Jörn Dunkel
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA
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