1
|
Leite D, De Bacco C. Similarity and economy of scale in urban transportation networks and optimal transport-based infrastructures. Nat Commun 2024; 15:7981. [PMID: 39266572 PMCID: PMC11393328 DOI: 10.1038/s41467-024-52313-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2022] [Accepted: 09/01/2024] [Indexed: 09/14/2024] Open
Abstract
Designing and optimizing the structure of urban transportation networks is a challenging task. In this study, we propose a method inspired by optimal transport theory and the principle of economy of scale that uses little information in input to generate structures that are similar to those of public transportation networks. Contrarily to standard approaches, it does not assume any initial backbone network infrastructure but rather extracts this directly from a continuous space using only a few origin and destination points, generating networks from scratch. Analyzing a set of urban train, tram and subway networks, we find a noteworthy degree of similarity in several of the studied cases between simulated and real infrastructures. By tuning one parameter, our method can simulate a range of different subway, tram and train networks that can be further used to suggest possible improvements in terms of relevant transportation properties. Outputs of our algorithm provide naturally a principled quantitative measure of similarity between two networks that can be used to automatize the selection of similar simulated networks.
Collapse
Affiliation(s)
- Daniela Leite
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen, Germany
| | - Caterina De Bacco
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen, Germany.
| |
Collapse
|
2
|
Ibrahim AA, Muehlebach M, De Bacco C. Optimal Transport with Constraints: From Mirror Descent to Classical Mechanics. PHYSICAL REVIEW LETTERS 2024; 133:057401. [PMID: 39159100 DOI: 10.1103/physrevlett.133.057401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2023] [Accepted: 07/01/2024] [Indexed: 08/21/2024]
Abstract
Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is limited by the lack of principled and flexible ways to incorporate realistic constraints. We propose a principled physics-based approach to impose constraints flexibly in optimal transport problems. Constraints are included in mirror descent dynamics using the D'Alembert-Lagrange principle from classical mechanics. This results in a sparse, local and linear approximation of the feasible set leading in many cases to closed-form updates.
Collapse
|
3
|
Waszkiewicz R, Shaw JB, Lisicki M, Szymczak P. Goldilocks Fluctuations: Dynamic Constraints on Loop Formation in Scale-Free Transport Networks. PHYSICAL REVIEW LETTERS 2024; 132:137401. [PMID: 38613264 DOI: 10.1103/physrevlett.132.137401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 02/26/2024] [Indexed: 04/14/2024]
Abstract
Adaptive transport networks are known to contain loops when subject to hydrodynamic fluctuations. However, fluctuations are no guarantee that a loop will form, as shown by loop-free networks driven by oscillating flows. We provide a complete stability analysis of the dynamical behavior of any loop formed by fluctuating flows. We find a threshold for loop stability that involves an interplay of geometric constraints and hydrodynamic forcing mapped to constant and fluctuating components. Loops require fluctuation in the relative size of the flux between nodes, not just a temporal variation in the flux at a given node. Hence, there is both a minimum and a maximum amount of fluctuation relative to the constant-flux component where loops are supported.
Collapse
Affiliation(s)
- Radost Waszkiewicz
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - John Burnham Shaw
- Department of Geosciences, University of Arkansas, Fayetteville, Arkansas, USA
| | - Maciej Lisicki
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| | - Piotr Szymczak
- Institute of Theoretical Physics, Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
| |
Collapse
|
4
|
Qi Y, Chang SS, Wang Y, Chen C, Baek KI, Hsiai T, Roper M. Hemodynamic regulation allows stable growth of microvascular networks. Proc Natl Acad Sci U S A 2024; 121:e2310993121. [PMID: 38386707 PMCID: PMC10907248 DOI: 10.1073/pnas.2310993121] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2023] [Accepted: 01/16/2024] [Indexed: 02/24/2024] Open
Abstract
How do vessels find optimal radii? Capillaries are known to adapt their radii to maintain the shear stress of blood flow at the vessel wall at a set point, yet models of adaptation purely based on average shear stress have not been able to produce complex loopy networks that resemble real microvascular systems. For narrow vessels where red blood cells travel in a single file, the shear stress on vessel endothelium peaks sharply when a red blood cell passes through. We show that stable shear-stress-based adaptation is possible if vessel shear stress set points are cued to the stress peaks. Model networks that respond to peak stresses alone can quantitatively reproduce the observed zebrafish trunk microcirculation, including its adaptive trajectory when hematocrit changes or parts of the network are amputated. Our work reveals the potential for mechanotransduction alone to generate stable hydraulically tuned microvascular networks.
Collapse
Affiliation(s)
- Yujia Qi
- Department of Mechanical Engineering, University of California, Los Angeles, CA90095
| | - Shyr-Shea Chang
- Department of Mathematics, University of California, Los Angeles, CA90095
| | - Yixuan Wang
- Department of Mathematics, University of California, Los Angeles, CA90095
| | - Cynthia Chen
- Department of Bioengineering, University of California, Los Angeles, CA90095
| | - Kyung In Baek
- Department of Bioengineering, University of California, Los Angeles, CA90095
| | - Tzung Hsiai
- Department of Bioengineering, University of California, Los Angeles, CA90095
| | - Marcus Roper
- Department of Mathematics, University of California, Los Angeles, CA90095
- Department of Computational Medicine, University of California, Los Angeles, CA90095
| |
Collapse
|
5
|
Lonardi A, De Bacco C. Bilevel Optimization for Traffic Mitigation in Optimal Transport Networks. PHYSICAL REVIEW LETTERS 2023; 131:267401. [PMID: 38215368 DOI: 10.1103/physrevlett.131.267401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2023] [Revised: 09/21/2023] [Accepted: 11/22/2023] [Indexed: 01/14/2024]
Abstract
Global infrastructure robustness and local transport efficiency are critical requirements for transportation networks. However, since passengers often travel greedily to maximize their own benefit and trigger traffic jams, overall transportation performance can be heavily disrupted. We develop adaptation rules that leverage optimal transport theory to effectively route passengers along their shortest paths while also strategically tuning edge weights to optimize traffic. As a result, we enforce both global and local optimality of transport. We prove the efficacy of our approach on synthetic networks and on real data. Our findings on the international European highways suggest that thoughtfully devised routing schemes might help to lower car-produced carbon emissions.
Collapse
Affiliation(s)
- Alessandro Lonardi
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen 72076, Germany
| | - Caterina De Bacco
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen 72076, Germany
| |
Collapse
|
6
|
Klemm K, Martens EA. Bifurcations in adaptive vascular networks: Toward model calibration. CHAOS (WOODBURY, N.Y.) 2023; 33:093135. [PMID: 37748484 DOI: 10.1063/5.0160170] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2023] [Accepted: 08/30/2023] [Indexed: 09/27/2023]
Abstract
Transport networks are crucial for the functioning of natural and technological systems. We study a mathematical model of vascular network adaptation, where the network structure dynamically adjusts to changes in blood flow and pressure. The model is based on local feedback mechanisms that occur on different time scales in the mammalian vasculature. The cost exponent γ tunes the vessel growth in the adaptation rule, and we test the hypothesis that the cost exponent is γ=1/2 for vascular systems [D. Hu and D. Cai, Phys. Rev. Lett. 111, 138701 (2013)]. We first perform bifurcation analysis for a simple triangular network motif with a fluctuating demand and then conduct numerical simulations on network topologies extracted from perivascular networks of rodent brains. We compare the model predictions with experimental data and find that γ is closer to 1 than to 1/2 for the model to be consistent with the data. Our study, thus, aims at addressing two questions: (i) Is a specific measured flow network consistent in terms of physical reality? (ii) Is the adaptive dynamic model consistent with measured network data? We conclude that the model can capture some aspects of vascular network formation and adaptation, but also suggest some limitations and directions for future research. Our findings contribute to a general understanding of the dynamics in adaptive transport networks, which is essential for studying mammalian vasculature and developing self-organizing piping systems.
Collapse
Affiliation(s)
- Konstantin Klemm
- Instituto de Física Interdisciplinar y Sistemas Complejos (IFISC, CSIC-UIB), Campus Universitat de les Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Erik A Martens
- Centre for Mathematical Sciences, Lund University, Sölvegatan 18B, 22100 Lund, Sweden
| |
Collapse
|
7
|
Bhattacharyya K, Zwicker D, Alim K. Memory capacity of adaptive flow networks. Phys Rev E 2023; 107:034407. [PMID: 37073018 DOI: 10.1103/physreve.107.034407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2022] [Accepted: 02/09/2023] [Indexed: 04/20/2023]
Abstract
Biological flow networks adapt their network morphology to optimize flow while being exposed to external stimuli from different spatial locations in their environment. These adaptive flow networks retain a memory of the stimulus location in the network morphology. Yet, what limits this memory and how many stimuli can be stored are unknown. Here, we study a numerical model of adaptive flow networks by applying multiple stimuli subsequently. We find strong memory signals for stimuli imprinted for a long time into young networks. Consequently, networks can store many stimuli for intermediate stimulus duration, which balance imprinting and aging.
Collapse
Affiliation(s)
- Komal Bhattacharyya
- Max Planck Institute for Dynamics and Self-Organisation, 37077 Göttingen, Germany
| | - David Zwicker
- Max Planck Institute for Dynamics and Self-Organisation, 37077 Göttingen, Germany
| | - Karen Alim
- Max Planck Institute for Dynamics and Self-Organisation, 37077 Göttingen, Germany
- Center for Protein Assemblies and Department of Bioscience, School of Natural Sciences, Technische Universität München, 85748 Garching, Germany
| |
Collapse
|
8
|
Lonardi A, Facca E, Putti M, De Bacco C. Infrastructure adaptation and emergence of loops in network routing with time-dependent loads. Phys Rev E 2023; 107:024302. [PMID: 36932530 DOI: 10.1103/physreve.107.024302] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2022] [Accepted: 01/24/2023] [Indexed: 02/05/2023]
Abstract
Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work we study a dynamical systems where the edge conductivities of a network are regulated by time-varying mass loads injected on nodes. Motivated by empirical observations, we assume that conductivities adapt slowly with respect to the characteristic time of the loads. Furthermore, assuming the loads to be periodic, we derive a dynamics where the evolution of the system is controlled by a matrix obtained with the Fourier coefficients of the input loads. Remarkably, we find a sufficient condition on these coefficients that determines when the resulting network topologies are trees. We show an example of this on the Bordeaux bus network where we tune the input loads to interpolate between loopy and tree topologies. We validate our model on several synthetic networks and provide an expression for long-time solutions of the original conductivities.
Collapse
Affiliation(s)
- Alessandro Lonardi
- Max Planck Institute for Intelligent Systems, Cyber Valley, 72076 Tübingen, Germany
| | - Enrico Facca
- Laboratoire Paul Painlevé, UMR No. 8524, CNRS, Inria, Université Lille, 59000 Lille, France
| | - Mario Putti
- Department of Mathematics "Tullio Levi-Civita," University of Padua, Via Trieste 63, 35131 Padua, Italy
| | - Caterina De Bacco
- Max Planck Institute for Intelligent Systems, Cyber Valley, 72076 Tübingen, Germany
| |
Collapse
|
9
|
Bohr T, Hjorth PG, Holst SC, Hrabětová S, Kiviniemi V, Lilius T, Lundgaard I, Mardal KA, Martens EA, Mori Y, Nägerl UV, Nicholson C, Tannenbaum A, Thomas JH, Tithof J, Benveniste H, Iliff JJ, Kelley DH, Nedergaard M. The glymphatic system: Current understanding and modeling. iScience 2022; 25:104987. [PMID: 36093063 PMCID: PMC9460186 DOI: 10.1016/j.isci.2022.104987] [Citation(s) in RCA: 94] [Impact Index Per Article: 47.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
We review theoretical and numerical models of the glymphatic system, which circulates cerebrospinal fluid and interstitial fluid around the brain, facilitating solute transport. Models enable hypothesis development and predictions of transport, with clinical applications including drug delivery, stroke, cardiac arrest, and neurodegenerative disorders like Alzheimer's disease. We sort existing models into broad categories by anatomical function: Perivascular flow, transport in brain parenchyma, interfaces to perivascular spaces, efflux routes, and links to neuronal activity. Needs and opportunities for future work are highlighted wherever possible; new models, expanded models, and novel experiments to inform models could all have tremendous value for advancing the field.
Collapse
Affiliation(s)
- Tomas Bohr
- Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark
| | - Poul G. Hjorth
- Department of Applied Mathematics and Computer Science, Technical University of Denmark, Richard Petersens Plads, 2800 Kgs. Lyngby, Denmark
| | - Sebastian C. Holst
- Neuroscience and Rare Diseases Discovery and Translational Area, Roche Pharmaceutical Research and Early Development, Roche Innovation Center Basel, Grenzacherstrasse 124, 4070 Basel, Switzerland
| | - Sabina Hrabětová
- Department of Cell Biology and The Robert Furchgott Center for Neural and Behavioral Science, State University of New York Downstate Medical Center, Brooklyn, NY, USA
| | - Vesa Kiviniemi
- Oulu Functional NeuroImaging, Department of Diagnostic Radiology, MRC, Oulu University Hospital, Oulu, Finland
- Medical Imaging, Physics and Technology, the Faculty of Medicine, University of Oulu, Oulu, Finland
| | - Tuomas Lilius
- Department of Pharmacology, Faculty of Medicine, University of Helsinki, Helsinki, Finland
- Individualized Drug Therapy Research Program, Faculty of Medicine, University of Helsinki, Helsinki, Finland
- Center for Translational Neuromedicine, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark
- Department of Emergency Medicine and Services, Helsinki University Hospital and University of Helsinki, Helsinki, Finland
| | - Iben Lundgaard
- Department of Experimental Medical Science, Lund University, Lund, Sweden
- Wallenberg Centre for Molecular Medicine, Lund University, Lund, Sweden
| | - Kent-Andre Mardal
- Department of Mathematics, University of Oslo, Oslo, Norway
- Simula Research Laboratory, Department of Numerical Analysis and Scientific Computing, Oslo, Norway
| | | | - Yuki Mori
- Center for Translational Neuromedicine, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark
| | - U. Valentin Nägerl
- Instítut Interdisciplinaire de Neurosciences, Université de Bordeaux / CNRS UMR 5297, Centre Broca Nouvelle-Aquitaine, 146 rue Léo Saignat, CS 61292 Case 130, 33076 Bordeaux Cedex France
| | - Charles Nicholson
- Department of Neuroscience and Physiology, New York University Grossman School of Medicine, New York, NY, USA
- Department of Cell Biology, SUNY Downstate Health Sciences University, Brooklyn, NY, USA
| | - Allen Tannenbaum
- Departments of Computer Science/ Applied Mathematics and Statistics, Stony Brook University, Stony Brook, NY, USA
| | - John H. Thomas
- Department of Mechanical Engineering, University of Rochester, Rochester, 14627 NY, USA
| | - Jeffrey Tithof
- Department of Mechanical Engineering, University of Minnesota, Minneapolis, USA
| | - Helene Benveniste
- Department of Anesthesiology, Yale School of Medicine, New Haven, CT, USA
- Department of Biomedical Engineering, Yale School of Medicine, New Haven, CT, USA
| | - Jeffrey J. Iliff
- VISN 20 Mental Illness Research, Education and Clinical Center, VA Puget Sound Health Care System, Seattle, WA, USA
- Department of Psychiatry and Behavioral Sciences, University of Washington School of Medicine, Seattle, WA, USA
- Department of Neurology, University of Washington School of Medicine, Seattle, WA, USA
| | - Douglas H. Kelley
- Department of Mechanical Engineering, University of Rochester, Rochester, 14627 NY, USA
| | - Maiken Nedergaard
- Center for Translational Neuromedicine, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark
- Center for Translational Neuromedicine, University of Rochester Medical Center, Rochester, 14642 NY, USA
| |
Collapse
|
10
|
Bhattacharyya K, Zwicker D, Alim K. Memory Formation in Adaptive Networks. PHYSICAL REVIEW LETTERS 2022; 129:028101. [PMID: 35867448 DOI: 10.1103/physrevlett.129.028101] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/27/2021] [Revised: 06/14/2022] [Accepted: 06/15/2022] [Indexed: 06/15/2023]
Abstract
The continuous adaptation of networks like our vasculature ensures optimal network performance when challenged with changing loads. Here, we show that adaptation dynamics allow a network to memorize the position of an applied load within its network morphology. We identify that the irreversible dynamics of vanishing network links encode memory. Our analytical theory successfully predicts the role of all system parameters during memory formation, including parameter values which prevent memory formation. We thus provide analytical insight on the theory of memory formation in disordered systems.
Collapse
Affiliation(s)
- Komal Bhattacharyya
- Max Planck Institute for Dynamics and Self-Organisation, Göttingen 37077, Germany
| | - David Zwicker
- Max Planck Institute for Dynamics and Self-Organisation, Göttingen 37077, Germany
| | - Karen Alim
- Max Planck Institute for Dynamics and Self-Organisation, Göttingen 37077, Germany
- Center for Protein Assemblies (CPA), Physik-Department, Technische Universität München, Garching 85748, Germany
| |
Collapse
|
11
|
Zareei A, Pan D, Amir A. Temporal Evolution of Erosion in Pore Networks: From Homogenization to Instability. PHYSICAL REVIEW LETTERS 2022; 128:234501. [PMID: 35749180 DOI: 10.1103/physrevlett.128.234501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2021] [Revised: 12/22/2021] [Accepted: 04/12/2022] [Indexed: 06/15/2023]
Abstract
We study the dynamics of flow networks in porous media using two and three dimensional pore-network models. We consider a class of erosion dynamics for a single phase flow with no deposition, chemical reactions, or topology changes assuming a constitutive law depending on flow rate, local velocities, or shear stress at the walls. We show that depending on the erosion law, the flow may become uniform and homogenized or become unstable and develop channels. By defining an order parameter capturing these different behaviors we show that a phase transition occurs depending on the erosion dynamics. Using a simple model, we identify quantitative criteria to distinguish these regimes and correctly predict the fate of the network, and discuss the experimental relevance of our result.
Collapse
Affiliation(s)
- Ahmad Zareei
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02148, USA
| | - Deng Pan
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02148, USA
| | - Ariel Amir
- School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02148, USA
| |
Collapse
|
12
|
Ibrahim AA, Leite D, De Bacco C. Sustainable optimal transport in multilayer networks. Phys Rev E 2022; 105:064302. [PMID: 35854570 DOI: 10.1103/physreve.105.064302] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2022] [Accepted: 05/18/2022] [Indexed: 11/07/2022]
Abstract
Traffic congestion is one of the major challenges faced by the transportation industry. While this problem carries a high economic and environmental cost, the need for an efficient design of optimal paths for passengers in multilayer network infrastructures is imperative. We consider an approach based on optimal transport theory to route passengers preferably along layers that are more carbon-efficient than the road, e.g., rails. By analyzing the impact of this choice on performance, we find that this approach reduces carbon emissions considerably compared to shortest-path minimization. Similarly, we find that this approach distributes traffic more homogeneously, thus alleviating the risk of traffic congestion. Our results shed light on the impact of distributing traffic flexibly across layers guided by optimal transport theory.
Collapse
Affiliation(s)
| | - Daniela Leite
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen 72076, Germany
| | - Caterina De Bacco
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen 72076, Germany
| |
Collapse
|
13
|
Lonardi A, Putti M, De Bacco C. Multicommodity routing optimization for engineering networks. Sci Rep 2022; 12:7474. [PMID: 35523923 PMCID: PMC9076927 DOI: 10.1038/s41598-022-11348-9] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/20/2021] [Accepted: 04/21/2022] [Indexed: 11/12/2022] Open
Abstract
Optimizing passengers routes is crucial to design efficient transportation networks. Recent results show that optimal transport provides an efficient alternative to standard optimization methods. However, it is not yet clear if this formalism has empirical validity on engineering networks. We address this issue by considering different response functions-quantities determining the interaction between passengers-in the dynamics implementing the optimal transport formulation. Particularly, we couple passengers' fluxes by taking their sum or the sum of their squares. The first choice naturally reflects edges occupancy in transportation networks, however the second guarantees convergence to an optimal configuration of flows. Both modeling choices are applied to the Paris metro. We measure the extent of traffic bottlenecks and infrastructure resilience to node removal, showing that the two settings are equivalent in the congested transport regime, but different in the branched one. In the latter, the two formulations differ on how fluxes are distributed, with one function favoring routes consolidation, thus potentially being prone to generate traffic overload. Additionally, we compare our method to Dijkstra's algorithm to show its capacity to efficiently recover shortest-path-like graphs. Finally, we observe that optimal transport networks lie in the Pareto front drawn by the energy dissipated by passengers, and the cost to build the infrastructure.
Collapse
Affiliation(s)
- Alessandro Lonardi
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen, 72076, Germany.
| | - Mario Putti
- Department of Mathematics "Tullio Levi-Civita", University of Padua, Via Trieste 63, Padua, Italy
| | - Caterina De Bacco
- Max Planck Institute for Intelligent Systems, Cyber Valley, Tübingen, 72076, Germany
| |
Collapse
|
14
|
Sugishita K, Abdel-Mottaleb N, Zhang Q, Masuda N. A growth model for water distribution networks with loops. Proc Math Phys Eng Sci 2021; 477:20210528. [PMID: 35153598 PMCID: PMC8610702 DOI: 10.1098/rspa.2021.0528] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/30/2021] [Accepted: 10/26/2021] [Indexed: 11/12/2022] Open
Abstract
Water distribution networks (WDNs) expand their service areas over time. These growth dynamics are poorly understood. One facet of WDNs is that they have loops in general, and closing loops may be a functionally important process for enhancing their robustness and efficiency. We propose a growth model for WDNs that generates networks with loops and is applicable to networks with multiple water sources. We apply the proposed model to four empirical WDNs to show that it produces networks whose structure is similar to that of the empirical WDNs. The comparison between the empirical and modelled WDNs suggests that the empirical WDNs may realize a reasonable balance between cost, efficiency and robustness in terms of the network structure. We also study the design of pipe diameters based on a biological positive feedback mechanism. Specifically, we apply a model inspired by Physarum polycephalum to find moderate positive correlations between the empirical and modelled pipe diameters. The difference between the empirical and modelled pipe diameters suggests that we may be able to improve the performance of WDNs by following organizing principles of biological flow networks.
Collapse
Affiliation(s)
- Kashin Sugishita
- Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA
- Department of Transdisciplinary Science and Engineering, Tokyo Institute of Technology, 152-8550 Tokyo, Japan
| | - Noha Abdel-Mottaleb
- Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL 33620, USA
| | - Qiong Zhang
- Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL 33620, USA
| | - Naoki Masuda
- Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, USA
- Computational and Data-Enabled Science and Engineering Program, State University of New York at Buffalo, Buffalo, NY 14260-5030, USA
- Faculty of Science and Engineering, Waseda University, 169-8555 Tokyo, Japan
| |
Collapse
|
15
|
Abstract
Modeling traffic distribution and extracting optimal flows in multilayer networks is of the utmost importance to design efficient, multi-modal network infrastructures. Recent results based on optimal transport theory provide powerful and computationally efficient methods to address this problem, but they are mainly focused on modeling single-layer networks. Here, we adapt these results to study how optimal flows distribute on multilayer networks. We propose a model where optimal flows on different layers contribute differently to the total cost to be minimized. This is done by means of a parameter that varies with layers, which allows to flexibly tune the sensitivity to the traffic congestion of the various layers. As an application, we consider transportation networks, where each layer is associated to a different transportation system, and show how the traffic distribution varies as we tune this parameter across layers. We show an example of this result on the real, 2-layer network of the city of Bordeaux with a bus and tram, where we find that in certain regimes, the presence of the tram network significantly unburdens the traffic on the road network. Our model paves the way for further analysis of optimal flows and navigability strategies in real, multilayer networks.
Collapse
|
16
|
Ronellenfitsch H. Optimal Elasticity of Biological Networks. PHYSICAL REVIEW LETTERS 2021; 126:038101. [PMID: 33543959 DOI: 10.1103/physrevlett.126.038101] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/04/2020] [Accepted: 12/24/2020] [Indexed: 06/12/2023]
Abstract
Reinforced elastic sheets surround us in daily life, from concrete shell buildings to biological structures such as the arthropod exoskeleton or the venation network of dicotyledonous plant leaves. Natural structures are often highly optimized through evolution and natural selection, leading to the biologically and practically relevant problem of understanding and applying the principles of their design. Inspired by the hierarchically organized scaffolding networks found in plant leaves, here we model networks of bending beams that capture the discrete and nonuniform nature of natural materials. Using the principle of maximal rigidity under natural resource constraints, we show that optimal discrete beam networks reproduce the structural features of real leaf venation. Thus, in addition to its ability to efficiently transport water and nutrients, the venation network also optimizes leaf rigidity using the same hierarchical reticulated network topology. We study the phase space of optimal mechanical networks, providing concrete guidelines for the construction of elastic structures. We implement these natural design rules by fabricating efficient, biologically inspired metamaterials.
Collapse
Affiliation(s)
- Henrik Ronellenfitsch
- Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, Massachusetts 02139, USA
- Physics Department, Williams College, 33 Lab Campus Drive, Williamstown, Massachusetts 01267, USA
| |
Collapse
|
17
|
Kaiser F, Ronellenfitsch H, Witthaut D. Discontinuous transition to loop formation in optimal supply networks. Nat Commun 2020; 11:5796. [PMID: 33199688 PMCID: PMC7670464 DOI: 10.1038/s41467-020-19567-2] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2020] [Accepted: 10/19/2020] [Indexed: 11/09/2022] Open
Abstract
The structure and design of optimal supply networks is an important topic in complex networks research. A fundamental trait of natural and man-made networks is the emergence of loops and the trade-off governing their formation: adding redundant edges to supply networks is costly, yet beneficial for resilience. Loops typically form when costs for new edges are small or inputs uncertain. Here, we shed further light on the transition to loop formation. We demonstrate that loops emerge discontinuously when decreasing the costs for new edges for both an edge-damage model and a fluctuating sink model. Mathematically, new loops are shown to form through a saddle-node bifurcation. Our analysis allows to heuristically predict the location and cost where the first loop emerges. Finally, we unveil an intimate relationship among betweenness measures and optimal tree networks. Our results can be used to understand the evolution of loop formation in real-world biological networks. Supply networks with optimal structure do not contain loops but these can arise as a result of damages or fluctuations. Here Kaiser et al. uncover the mechanisms of loop formation, predict their location and draw analogies with loop formation in biological networks such as plants and animal vasculature.
Collapse
Affiliation(s)
- Franz Kaiser
- Forschungszentrum Jülich, Institute for Energy and Climate Research (IEK-STE), 52428, Jülich, Germany. .,Institute for Theoretical Physics, University of Cologne, 50937, Köln, Germany.
| | - Henrik Ronellenfitsch
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA.,Physics Department, Williams College, 33 Lab Campus Drive, Williamstown, MA, 01267, USA
| | - Dirk Witthaut
- Forschungszentrum Jülich, Institute for Energy and Climate Research (IEK-STE), 52428, Jülich, Germany. .,Institute for Theoretical Physics, University of Cologne, 50937, Köln, Germany.
| |
Collapse
|
18
|
Mao M, Bei HP, Lam CH, Chen P, Wang S, Chen Y, He J, Zhao X. Human-on-Leaf-Chip: A Biomimetic Vascular System Integrated with Chamber-Specific Organs. SMALL (WEINHEIM AN DER BERGSTRASSE, GERMANY) 2020; 16:e2000546. [PMID: 32329575 DOI: 10.1002/smll.202000546] [Citation(s) in RCA: 31] [Impact Index Per Article: 7.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/28/2020] [Revised: 03/14/2020] [Accepted: 03/19/2020] [Indexed: 05/24/2023]
Abstract
The vascular network is a central component of the organ-on-a-chip system to build a 3D physiological microenvironment with controlled physical and biochemical variables. Inspired by ubiquitous biological systems such as leaf venation and circulatory systems, a fabrication strategy is devised to develop a biomimetic vascular system integrated with freely designed chambers, which function as niches for chamber-specific vascularized organs. As a proof of concept, a human-on-leaf-chip system with biomimetic multiscale vasculature systems connecting the self-assembled 3D vasculatures in chambers is fabricated, mimicking the in vivo complex architectures of the human cardiovascular system connecting vascularized organs. Besides, two types of vascularized organs are built independently within the two halves of the system to verify its feasibility for conducting comparative experiments for organ-specific metastasis studies in a single chip. Successful culturing of human hepatoma G2 cells (HepG2s) and mesenchymal stem cells (MSCs) with human umbilical vein endothelial cells (HUVECs) shows good vasculature formation, and organ-specific metastasis is simulated through perfusion of pancreatic cancer cells and shows distinct cancer encapsulation by MSCs, which is absent in HepG2s. Given good culture efficacy, study design flexibility, and ease of modification, these results show that the bioinspired human-on-leaf-chip possesses great potential in comparative and metastasis studies while retaining organ-to-organ crosstalk.
Collapse
Affiliation(s)
- Mao Mao
- Department of Biomedical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049, China
| | - Ho Pan Bei
- Department of Biomedical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
| | - Chun Hei Lam
- Department of Biomedical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
| | - Pengyu Chen
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049, China
| | - Shuqi Wang
- State Key Laboratory for Diagnosis and Treatment of Infectious Diseases, National Clinical Research Center for Infectious Diseases, National Medical Center for Infectious Diseases, Collaborative Innovation Center for Diagnosis and Treatment of Infectious Diseases, The First Affiliated Hospital, College of Medicine, Zhejiang University, Hangzhou, Zhejiang, 310003, China
- Institute for Translational Medicine, Zhejiang University, Hangzhou, Zhejiang, 310029, China
| | - Ying Chen
- Guangdong Provincial Key Laboratory of Functional Soft Condensed Matter, School of Materials and Energy, Guangdong University of Technology, Guangzhou, Guangdong, 510000, China
| | - Jiankang He
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an, 710049, China
| | - Xin Zhao
- Department of Biomedical Engineering, The Hong Kong Polytechnic University, Hung Hom, Hong Kong, China
| |
Collapse
|
19
|
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.
Collapse
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
| |
Collapse
|
20
|
Kirkegaard JB, Sneppen K. Optimal Transport Flows for Distributed Production Networks. PHYSICAL REVIEW LETTERS 2020; 124:208101. [PMID: 32501061 DOI: 10.1103/physrevlett.124.208101] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2019] [Accepted: 04/23/2020] [Indexed: 06/11/2023]
Abstract
Network flows often exhibit a hierarchical treelike structure that can be attributed to the minimization of dissipation. The common feature of such systems is a single source and multiple sinks (or vice versa). In contrast, here we study networks with only a single source and sink. These systems can arise from secondary purposes of the networks, such as blood sugar regulation through insulin production. Minimization of dissipation in these systems leads to vascular shunting, a single vessel connecting the inlet and outlet. We show instead how optimizing the transport time yields network topologies that match those observed in the insulin-producing pancreatic islets. These are patterns of periphery-to-center and center-to-periphery flows. The obtained flow networks are broadly independent of how the flow velocity depends on the flow flux, but continuous and discontinuous phase transitions appear at extreme flux dependencies. Lastly, we show how constraints on flows can lead to buckling of the branches of the network, a feature that is also observed in pancreatic islets.
Collapse
Affiliation(s)
| | - Kim Sneppen
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| |
Collapse
|
21
|
Li B. Global Existence and Decay Estimates of Solutions of a Parabolic–Elliptic–Parabolic System for Ion Transport Networks. RESULTS IN MATHEMATICS 2020; 75:45. [DOI: 10.1007/s00025-020-1172-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2018] [Accepted: 02/13/2020] [Indexed: 09/01/2023]
|
22
|
Impact of global structure on diffusive exploration of organelle networks. Sci Rep 2020; 10:4984. [PMID: 32188905 PMCID: PMC7080787 DOI: 10.1038/s41598-020-61598-8] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2019] [Accepted: 02/25/2020] [Indexed: 01/08/2023] Open
Abstract
We investigate diffusive search on planar networks, motivated by tubular organelle networks in cell biology that contain molecules searching for reaction partners and binding sites. Exact calculation of the diffusive mean first-passage time on a spatial network is used to characterize the typical search time as a function of network connectivity. We find that global structural properties — the total edge length and number of loops — are sufficient to largely determine network exploration times for a variety of both synthetic planar networks and organelle morphologies extracted from living cells. For synthetic networks on a lattice, we predict the search time dependence on these global structural parameters by connecting with percolation theory, providing a bridge from irregular real-world networks to a simpler physical model. The dependence of search time on global network structural properties suggests that network architecture can be designed for efficient search without controlling the precise arrangement of connections. Specifically, increasing the number of loops substantially decreases search times, pointing to a potential physical mechanism for regulating reaction rates within organelle network structures.
Collapse
|
23
|
Ronellenfitsch H, Katifori E. Phenotypes of Vascular Flow Networks. PHYSICAL REVIEW LETTERS 2019; 123:248101. [PMID: 31922876 DOI: 10.1103/physrevlett.123.248101] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Indexed: 06/10/2023]
Abstract
Complex distribution networks are pervasive in biology. Examples include nutrient transport in the slime mold Physarum polycephalum as well as mammalian and plant venation. Adaptive rules are believed to guide development of these networks and lead to a reticulate, hierarchically nested topology that is both efficient and resilient against perturbations. However, as of yet, no mechanism is known that can generate such networks on all scales. We show how hierarchically organized reticulation can be constructed and maintained through spatially correlated load fluctuations on a particular length scale. We demonstrate that the network topologies generated represent a trade-off between optimizing transport efficiency, construction cost, and damage robustness and identify the Pareto-efficient front that evolution is expected to favor and select for. We show that the typical fluctuation length scale controls the position of the networks on the Pareto front and thus on the spectrum of venation phenotypes.
Collapse
Affiliation(s)
- Henrik Ronellenfitsch
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Eleni Katifori
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| |
Collapse
|
24
|
Meigel FJ, Cha P, Brenner MP, Alim K. Robust Increase in Supply by Vessel Dilation in Globally Coupled Microvasculature. PHYSICAL REVIEW LETTERS 2019; 123:228103. [PMID: 31868401 DOI: 10.1103/physrevlett.123.228103] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2019] [Indexed: 06/10/2023]
Abstract
Neuronal activity induces changes in blood flow by locally dilating vessels in the brain microvasculature. How can the local dilation of a single vessel increase flow-based metabolite supply, given that flows are globally coupled within microvasculature? Solving the supply dynamics for rat brain microvasculature, we find one parameter regime to dominate physiologically. This regime allows for robust increase in supply independent of the position in the network, which we explain analytically. We show that local coupling of vessels promotes spatially correlated increased supply by dilation.
Collapse
Affiliation(s)
- Felix J Meigel
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| | - Peter Cha
- John A. Paulson School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Michael P Brenner
- John A. Paulson School of Engineering and Applied Sciences and Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts 02138, USA
| | - Karen Alim
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
- Physik-Department, Technische Universität München, 85748 Garching, Germany
| |
Collapse
|
25
|
Haskovec J, Jönsson H, Kreusser LM, Markowich P. Auxin transport model for leaf venation. Proc Math Phys Eng Sci 2019; 475:20190015. [PMID: 31824212 PMCID: PMC6894547 DOI: 10.1098/rspa.2019.0015] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2019] [Accepted: 10/22/2019] [Indexed: 11/12/2022] Open
Abstract
The plant hormone auxin controls many aspects of the development of plants. One striking dynamical feature is the self-organization of leaf venation patterns which is driven by high levels of auxin within vein cells. The auxin transport is mediated by specialized membrane-localized proteins. Many venation models have been based on polarly localized efflux-mediator proteins of the PIN family. Here, we investigate a modelling framework for auxin transport with a positive feedback between auxin fluxes and transport capacities that are not necessarily polar, i.e. directional across a cell wall. Our approach is derived from a discrete graph-based model for biological transportation networks, where cells are represented by graph nodes and intercellular membranes by edges. The edges are not a priori oriented and the direction of auxin flow is determined by its concentration gradient along the edge. We prove global existence of solutions to the model and the validity of Murray's Law for its steady states. Moreover, we demonstrate with numerical simulations that the model is able connect an auxin source-sink pair with a mid-vein and that it can also produce branching vein patterns. A significant innovative aspect of our approach is that it allows the passage to a formal macroscopic limit which can be extended to include network growth. We perform mathematical analysis of the macroscopic formulation, showing the global existence of weak solutions for an appropriate parameter range.
Collapse
Affiliation(s)
- Jan Haskovec
- Mathematics, CEMSE, KAUST, Thuwal 23955-6900, KSA
| | - Henrik Jönsson
- Sainsbury Laboratory, University of Cambridge, Bateman Street, Cambridge CB2 1LR, UK
- DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
| | | | - Peter Markowich
- Mathematics, CEMSE, KAUST, Thuwal 23955-6900, KSA
- Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, Vienna 1090, Austria
| |
Collapse
|
26
|
Li B. Long time behavior of the solution to a parabolic–elliptic system. COMPUTERS & MATHEMATICS WITH APPLICATIONS 2019; 78:3345-3362. [DOI: 10.1016/j.camwa.2019.05.005] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
|
27
|
Jiang J, Wang X, Lai YC. Optimizing biologically inspired transport networks by control. Phys Rev E 2019; 100:032309. [PMID: 31640064 DOI: 10.1103/physreve.100.032309] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2019] [Indexed: 11/07/2022]
Abstract
Transportation networks with intrinsic flow dynamics governed by the Kirchhoff's current law are ubiquitous in natural and engineering systems. There has been recent work on designing optimal transportation networks based on biological principles with the goal to minimize the total dissipation associated with the flow. Despite being biologically inspired, e.g., adaptive network design based on slime mold Physarum polycephalum, such methods generally lead to suboptimal networks due to the difficulty in finding a global or nearly global optimum of the nonconvex optimization function. Here we articulate a design paradigm that combines engineering control and biological principles to realize optimal transportation networks. In particular, we show how small control signals applied only to a fraction of edges in an adaptive network can lead to solutions that are far more optimal than those based solely on biological principles. We also demonstrate that control signals, if not properly designed, can lead to networks that are less optimal. Incorporating control principle into biology-based optimal network design has broad applications not only in biomedical science and engineering but also in other disciplines such as civil engineering for designing resilient infrastructure systems.
Collapse
Affiliation(s)
- Junjie Jiang
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
| | - Xingang Wang
- School of Physics and Information Technology, Shaanxi Normal University, Xi'an 710062, China
| | - Ying-Cheng Lai
- School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA.,Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
| |
Collapse
|
28
|
A Priori Estimates for a Nonlinear System with Some Essential Symmetrical Structures. Symmetry (Basel) 2019. [DOI: 10.3390/sym11070852] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022] Open
Abstract
In this paper, we are concerned with a nonlinear system containing some essential symmetrical structures (e.g., cross-diffusion) in the two-dimensional setting, which is proposed to model the biological transport networks. We first provide an a priori blow-up criterion of strong solution of the corresponding Cauchy problem. Based on this, we also establish a priori upper bounds to strong solution for all positive times.
Collapse
|
29
|
Kang MY, Berthelot G, Tupikina L, Nicolaides C, Colonna JF, Sapoval B, Grebenkov DS. Morphological organization of point-to-point transport in complex networks. Sci Rep 2019; 9:8322. [PMID: 31171797 PMCID: PMC6554344 DOI: 10.1038/s41598-019-44701-6] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2019] [Accepted: 05/21/2019] [Indexed: 11/18/2022] Open
Abstract
We investigate the structural organization of the point-to-point electric, diffusive or hydraulic transport in complex scale-free networks. The random choice of two nodes, a source and a drain, to which a potential difference is applied, selects two tree-like structures, one emerging from the source and the other converging to the drain. These trees merge into a large cluster of the remaining nodes that is found to be quasi-equipotential and thus presents almost no resistance to transport. Such a global “tree-cluster-tree” structure is universal and leads to a power law decay of the currents distribution. Its exponent, −2, is determined by the multiplicative decrease of currents at successive branching points of a tree and is found to be independent of the network connectivity degree and resistance distribution.
Collapse
Affiliation(s)
- Min-Yeong Kang
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, IP Paris, 91128, Palaiseau, France
| | - Geoffroy Berthelot
- Centre de Mathématiques Appliquées, CNRS - Ecole Polytechnique, IP Paris, 91128, Palaiseau, France.,Research Laboratory for Interdisciplinary Studies (RELAIS), 75012, Paris, France.,Institut National du Sport, de l'Expertise et de la Performance (INSEP), 75012, Paris, France
| | - Liubov Tupikina
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, IP Paris, 91128, Palaiseau, France.,The Center for Research and Interdisciplinarity (CRI), University Paris Descartes, INSERM, Paris, France
| | - Christos Nicolaides
- Department of Business and Public Administration, University of Cyprus, 1 Panepistimiou Av., 2109, Aglantzia, Nicosia, Cyprus.,MIT Sloan School of Management, 100 Main Street, Cambridge, MA, 02142, USA
| | - Jean-Francois Colonna
- Centre de Mathématiques Appliquées, CNRS - Ecole Polytechnique, IP Paris, 91128, Palaiseau, France
| | - Bernard Sapoval
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, IP Paris, 91128, Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, IP Paris, 91128, Palaiseau, France.
| |
Collapse
|
30
|
Mondal A, Nguyen C, Ma X, Elbanna AE, Carlson JM. Network models for characterization of trabecular bone. Phys Rev E 2019; 99:042406. [PMID: 31108725 DOI: 10.1103/physreve.99.042406] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2018] [Indexed: 06/09/2023]
Abstract
Trabecular bone is a lightweight, compliant material organized as a web of struts and rods (trabeculae) that erode with age and the onset of bone diseases like osteoporosis, leading to increased fracture risk. The traditional diagnostic marker of osteoporosis, bone mineral density (BMD), has been shown in ex vivo experiments to correlate poorly with fracture resistance when considered on its own, while structural features in conjunction with BMD can explain more of the variation in trabecular bone strength. We develop a network-based model of trabecular bone by creating graphs from micro-computed tomography images of human bone, with weighted links representing trabeculae and nodes representing branch points. These graphs enable calculation of quantitative network metrics to characterize trabecular structure. We also create finite element models of the networks in which each link is represented by a beam, facilitating analysis of the mechanical response of the bone samples to simulated loading. We examine the structural and mechanical properties of trabecular bone at the scale of individual trabeculae (of order 0.1 mm) and at the scale of selected volumes of interest (approximately a few mm), referred to as VOIs. At the VOI scale, we find significant correlations between the stiffness of VOIs and 10 different structural metrics. Individually, the volume fraction of each VOI is most strongly correlated to the stiffness of the VOI. We use multiple linear regression to identify the smallest subset of variables needed to capture the variation in stiffness. In a linear fit, we find that node degree, weighted node degree, Z-orientation, weighted Z-orientation, trabecular spacing, link length, and the number of links are the structural metrics that are most significant (p<0.05) in capturing the variation of stiffness in trabecular networks.
Collapse
Affiliation(s)
- Avik Mondal
- Department of Physics, University of California, Santa Barbara, Santa Barbara, California 93106, USA
| | - Chantal Nguyen
- Department of Physics, University of California, Santa Barbara, Santa Barbara, California 93106, USA
| | - Xiao Ma
- Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Ahmed E Elbanna
- Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
| | - Jean M Carlson
- Department of Physics, University of California, Santa Barbara, Santa Barbara, California 93106, USA
| |
Collapse
|
31
|
Chang SS, Roper M. Microvscular networks with uniform flow. J Theor Biol 2019; 462:48-64. [PMID: 30420333 PMCID: PMC6599712 DOI: 10.1016/j.jtbi.2018.10.049] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2018] [Revised: 10/11/2018] [Accepted: 10/25/2018] [Indexed: 02/03/2023]
Abstract
Within animals, oxygen exchange occurs within vascular transport networks containing potentially billions of microvessels that are distributed throughout the body. By comparison, large blood vessels are theorized to minimize transport costs, leading to tree-like networks that satisfy Murray's law. We know very little about the principles underlying the organization of healthy micro-vascular networks. Indeed capillary networks must also perfuse tissues with oxygen, and efficient perfusion may be incompatible with minimization of transport costs. While networks that minimize transport costs have been well-studied, other optimization principles have received much less scrutiny. In this work we derive the morphology of networks that uniformize blood flow distribution, inspired by the zebrafish trunk micro-vascular network. To find uniform flow networks, we devise a gradient descent algorithm able to optimize arbitrary differentiable objective functions on transport networks, while exactly respecting arbitrary differentiable constraint functions. We prove that in a class of networks that we call stackable, which includes a model capillary bed, the uniform flow network will have the same flow as a uniform conductance network, i.e., in which all edges have the same conductance. This result agrees with uniform flow capillary bed network found by the algorithm. We also show that the uniform flow completely explains the observed radii within the zebrafish trunk vasculature. In addition to deriving new results on optimization of uniform flow in micro-vascular networks, our algorithm provides a general method for testing hypotheses about possible optimization principles underlying real microvascular networks, including exposing tradeoffs between flow uniformity and transport cost.
Collapse
Affiliation(s)
- Shyr-Shea Chang
- Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, USA.
| | - Marcus Roper
- Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, USA; Department of Biomathematics, University of California Los Angeles, Los Angeles, CA 90095, USA
| |
Collapse
|
32
|
Ronellenfitsch H, Dunkel J, Wilczek M. Optimal Noise-Canceling Networks. PHYSICAL REVIEW LETTERS 2018; 121:208301. [PMID: 30500224 DOI: 10.1103/physrevlett.121.208301] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2018] [Revised: 10/03/2018] [Indexed: 06/09/2023]
Abstract
Natural and artificial networks, from the cerebral cortex to large-scale power grids, face the challenge of converting noisy inputs into robust signals. The input fluctuations often exhibit complex yet statistically reproducible correlations that reflect underlying internal or environmental processes such as synaptic noise or atmospheric turbulence. This raises the practically and biophysically relevant question of whether and how noise filtering can be hard wired directly into a network's architecture. By considering generic phase oscillator arrays under cost constraints, we explore here analytically and numerically the design, efficiency, and topology of noise-canceling networks. Specifically, we find that when the input fluctuations become more correlated in space or time, optimal network architectures become sparser and more hierarchically organized, resembling the vasculature in plants or animals. More broadly, our results provide concrete guiding principles for designing more robust and efficient power grids and sensor networks.
Collapse
Affiliation(s)
- Henrik Ronellenfitsch
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
| | - Jörn Dunkel
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307, USA
| | - Michael Wilczek
- Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
| |
Collapse
|
33
|
Papadopoulos L, Blinder P, Ronellenfitsch H, Klimm F, Katifori E, Kleinfeld D, Bassett DS. Comparing two classes of biological distribution systems using network analysis. PLoS Comput Biol 2018; 14:e1006428. [PMID: 30192745 PMCID: PMC6145589 DOI: 10.1371/journal.pcbi.1006428] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/23/2018] [Revised: 09/19/2018] [Accepted: 08/11/2018] [Indexed: 11/18/2022] Open
Abstract
Distribution networks-from vasculature to urban transportation pathways-are spatially embedded networks that must route resources efficiently in the face of pressures induced by the costs of building and maintaining network infrastructure. Such requirements are thought to constrain the topological and spatial organization of these systems, but at the same time, different kinds of distribution networks may exhibit variable architectural features within those general constraints. In this study, we use methods from network science to compare and contrast two classes of biological transport networks: mycelial fungi and vasculature from the surface of rodent brains. These systems differ in terms of their growth and transport mechanisms, as well as the environments in which they typically exist. Though both types of networks have been studied independently, the goal of this study is to quantify similarities and differences in their network designs. We begin by characterizing the structural backbone of these systems with a collection of measures that assess various kinds of network organization across topological and spatial scales, ranging from measures of loop density, to those that quantify connected pathways between different network regions, and hierarchical organization. Most importantly, we next carry out a network analysis that directly considers the spatial embedding and properties especially relevant to the function of distribution systems. We find that although both the vasculature and mycelia are highly constrained planar networks, there are clear distinctions in how they balance tradeoffs in network measures of wiring length, efficiency, and robustness. While the vasculature appears well organized for low cost, but relatively high efficiency, the mycelia tend to form more expensive but in turn more robust networks. As a whole, this work demonstrates the utility of network-based methods to identify both common features and variations in the network structure of different classes of biological transport systems.
Collapse
Affiliation(s)
- Lia Papadopoulos
- Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
| | - Pablo Blinder
- Sagol School of Neuroscience, TelAviv University, Tel Aviv, Israel
- Department of Neurobiology, George S. Wise Faculty of Life Sciences, Tel Aviv University, Tel Aviv, Israel
| | - Henrik Ronellenfitsch
- Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
- Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States of America
| | - Florian Klimm
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
- Systems Approaches to Biomedical Science Doctoral Training Centre, University of Oxford, Oxford, United Kingdom
| | - Eleni Katifori
- Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
| | - David Kleinfeld
- Department of Physics, University of California San Diego, La Jolla, California, United States of America
- Section of Neurobiology, University of California San Diego, La Jolla, California, United States of America
| | - Danielle S. Bassett
- Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
- Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
- Department of Electrical & Systems Engineering, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
- Department of Neurology, University of Pennsylvania, Philadelphia, Pennsylvania, United States of America
| |
Collapse
|
34
|
Li B. Uniqueness of classical solution to an elliptic-parabolic system in biological network formation. JOURNAL OF PHYSICS: CONFERENCE SERIES 2018; 1053:012024. [DOI: 10.1088/1742-6596/1053/1/012024] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/01/2023]
|
35
|
The pial vasculature of the mouse develops according to a sensory-independent program. Sci Rep 2018; 8:9860. [PMID: 29959346 PMCID: PMC6026131 DOI: 10.1038/s41598-018-27910-3] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/18/2018] [Accepted: 06/12/2018] [Indexed: 12/15/2022] Open
Abstract
The cerebral vasculature is organized to supply the brain’s metabolic needs. Sensory deprivation during the early postnatal period causes altered neural activity and lower metabolic demand. Neural activity is instructional for some aspects of vascular development, and deprivation causes changes in capillary density in the deprived brain region. However, it is not known if the pial arteriole network, which contains many leptomeningeal anastomoses (LMAs) that endow the network with redundancy against occlusions, is also affected by sensory deprivation. We quantified the effects of early-life sensory deprivation via whisker plucking on the densities of LMAs and penetrating arterioles (PAs) in anatomically-identified primary sensory regions (vibrissae cortex, forelimb/hindlimb cortex, visual cortex and auditory cortex) in mice. We found that the densities of penetrating arterioles were the same across cortical regions, though the hindlimb representation had a higher density of LMAs than other sensory regions. We found that the densities of PAs and LMAs, as well as quantitative measures of network topology, were not affected by sensory deprivation. Our results show that the postnatal development of the pial arterial network is robust to sensory deprivation.
Collapse
|
36
|
Cheng L, Karney BW. Organization and scaling in water supply networks. Phys Rev E 2018; 96:062317. [PMID: 29347385 DOI: 10.1103/physreve.96.062317] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/14/2017] [Indexed: 11/07/2022]
Abstract
Public water supply is one of the society's most vital resources and most costly infrastructures. Traditional concepts of these networks capture their engineering identity as isolated, deterministic hydraulic units, but overlook their physics identity as related entities in a probabilistic, geographic ensemble, characterized by size organization and property scaling. Although discoveries of allometric scaling in natural supply networks (organisms and rivers) raised the prospect for similar findings in anthropogenic supplies, so far such a finding has not been reported in public water or related civic resource supplies. Examining an empirical ensemble of large number and wide size range, we show that water supply networks possess self-organized size abundance and theory-explained allometric scaling in spatial, infrastructural, and resource- and emission-flow properties. These discoveries establish scaling physics for water supply networks and may lead to novel applications in resource- and jurisdiction-scale water governance.
Collapse
Affiliation(s)
- Likwan Cheng
- Physical Science Department, City Colleges of Chicago, Chicago, Illinois 60601, USA
| | - Bryan W Karney
- Department of Civil Engineering, University of Toronto, Toronto, Ontario M5S 1A4, Canada
| |
Collapse
|
37
|
Chang SS, Tu S, Baek KI, Pietersen A, Liu YH, Savage VM, Hwang SPL, Hsiai TK, Roper M. Optimal occlusion uniformly partitions red blood cells fluxes within a microvascular network. PLoS Comput Biol 2017; 13:e1005892. [PMID: 29244812 PMCID: PMC5747476 DOI: 10.1371/journal.pcbi.1005892] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2017] [Revised: 12/29/2017] [Accepted: 11/26/2017] [Indexed: 12/29/2022] Open
Abstract
In animals, gas exchange between blood and tissues occurs in narrow vessels, whose diameter is comparable to that of a red blood cell. Red blood cells must deform to squeeze through these narrow vessels, transiently blocking or occluding the vessels they pass through. Although the dynamics of vessel occlusion have been studied extensively, it remains an open question why microvessels need to be so narrow. We study occlusive dynamics within a model microvascular network: the embryonic zebrafish trunk. We show that pressure feedbacks created when red blood cells enter the finest vessels of the trunk act together to uniformly partition red blood cells through the microvasculature. Using mathematical models as well as direct observation, we show that these occlusive feedbacks are tuned throughout the trunk network to prevent the vessels closest to the heart from short-circuiting the network. Thus occlusion is linked with another open question of microvascular function: how are red blood cells delivered at the same rate to each micro-vessel? Our analysis shows that tuning of occlusive feedbacks increase the total dissipation within the network by a factor of 11, showing that uniformity of flows rather than minimization of transport costs may be prioritized by the microvascular network. Arterial trees shuttle red blood cells from the heart to billions of capillaries distributed throughout the body. These trees have long been thought to be organized to minimize transport costs. Yet red blood cells are tightly squeezed within the finest vessels, meaning that these vessels account for as much as half of the total transport costs within the arterial network. It is unclear why vessel diameters and red blood cell diameters are so closely matched in a network that is presumed to optimize transport. Here, we use mathematical modeling and direct observations of red blood cell movements in embryonic zebrafish to show that occlusive feedbacks—the pressure feedbacks that alter the flows into a vessel when it is nearly blocked by a red blood cell—can optimally distribute red blood cells through microvessels. In addition to revealing an adaptive function for the matching of vessel and red blood cell diameters, this work shows that uniformity of red blood cell fluxes can be a unifying principle for understanding the elegant hydraulic organization of microvascular networks.
Collapse
Affiliation(s)
- Shyr-Shea Chang
- Department of Mathematics, University of California Los Angeles, Los Angeles, California, United States of America
- * E-mail:
| | - Shenyinying Tu
- Department of Mathematics, University of California Los Angeles, Los Angeles, California, United States of America
| | - Kyung In Baek
- Department of Bioengineering, School of Engineering & Applied Science, University of California Los Angeles, Los Angeles, California, United States of America
| | - Andrew Pietersen
- Department of Bioengineering, School of Engineering & Applied Science, University of California Los Angeles, Los Angeles, California, United States of America
| | - Yu-Hsiu Liu
- Department of Life Science, National Taiwan University, Taipei, Taiwan, Republic of China
| | - Van M. Savage
- Department of Biomathematics, University of California Los Angeles, Los Angeles, California, United States of America
- Department of Ecology and Evolutionary Biology, University of California Los Angeles, Los Angeles, California, United States of America
- Santa Fe Institute, Santa Fe, New Mexico, United States of America
| | - Sheng-Ping L. Hwang
- Institute of Cellular and Organismic Biology, Academia Sinica, Nankang, Taipei, Taiwan, Republic of China
| | - Tzung K. Hsiai
- Department of Bioengineering, School of Engineering & Applied Science, University of California Los Angeles, Los Angeles, California, United States of America
- Division of Cardiology, Department of Medicine, University of California Los Angeles, Los Angeles, California, United States of America
| | - Marcus Roper
- Department of Mathematics, University of California Los Angeles, Los Angeles, California, United States of America
- Department of Biomathematics, University of California Los Angeles, Los Angeles, California, United States of America
| |
Collapse
|
38
|
Akita D, Schenz D, Kuroda S, Sato K, Ueda KI, Nakagaki T. Current reinforcement model reproduces center-in-center vein trajectory of Physarum polycephalum. Dev Growth Differ 2017; 59:465-470. [PMID: 28707306 DOI: 10.1111/dgd.12384] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2017] [Revised: 05/08/2017] [Accepted: 06/07/2017] [Indexed: 11/30/2022]
Abstract
Vein networks span the whole body of the amoeboid organism in the plasmodial slime mould Physarum polycephalum, and the network topology is rearranged within an hour in response to spatio-temporal variations of the environment. It has been reported that this tube morphogenesis is capable of solving mazes, and a mathematical model, named the 'current reinforcement rule', was proposed based on the adaptability of the veins. Although it is known that this model works well for reproducing some key characters of the organism's maze-solving behaviour, one important issue is still open: In the real organism, the thick veins tend to trace the shortest possible route by cutting the corners at the turn of corridors, following a center-in-center trajectory, but it has not yet been examined whether this feature also appears in the mathematical model, using corridors of finite width. In this report, we confirm that the mathematical model reproduces the center-in-center trajectory of veins around corners observed in the maze-solving experiment.
Collapse
Affiliation(s)
- Dai Akita
- Graduate School of Life Science, Hokkaido University, N10 W8, Sapporo, Japan
| | - Daniel Schenz
- Graduate School of Life Science, Hokkaido University, N10 W8, Sapporo, Japan
| | - Shigeru Kuroda
- Mathematical and Physical Ethology Laboratory, Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido, 001-0020, Japan
| | - Katsuhiko Sato
- Mathematical and Physical Ethology Laboratory, Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido, 001-0020, Japan
| | - Kei-Ichi Ueda
- Graduate School of Science and Engineering, University of Toyama, Toyama, Japan
| | - Toshiyuki Nakagaki
- Mathematical and Physical Ethology Laboratory, Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, Sapporo, Hokkaido, 001-0020, Japan
| |
Collapse
|
39
|
Taylor-King JP, Basanta D, Chapman SJ, Porter MA. Mean-field approach to evolving spatial networks, with an application to osteocyte network formation. Phys Rev E 2017; 96:012301. [PMID: 29347066 DOI: 10.1103/physreve.96.012301] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2017] [Indexed: 11/07/2022]
Abstract
We consider evolving networks in which each node can have various associated properties (a state) in addition to those that arise from network structure. For example, each node can have a spatial location and a velocity, or it can have some more abstract internal property that describes something like a social trait. Edges between nodes are created and destroyed, and new nodes enter the system. We introduce a "local state degree distribution" (LSDD) as the degree distribution at a particular point in state space. We then make a mean-field assumption and thereby derive an integro-partial differential equation that is satisfied by the LSDD. We perform numerical experiments and find good agreement between solutions of the integro-differential equation and the LSDD from stochastic simulations of the full model. To illustrate our theory, we apply it to a simple model for osteocyte network formation within bones, with a view to understanding changes that may take place during cancer. Our results suggest that increased rates of differentiation lead to higher densities of osteocytes, but with a smaller number of dendrites. To help provide biological context, we also include an introduction to osteocytes, the formation of osteocyte networks, and the role of osteocytes in bone metastasis.
Collapse
Affiliation(s)
- Jake P Taylor-King
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom.,Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, Florida 33612, USA
| | - David Basanta
- Department of Integrated Mathematical Oncology, H. Lee Moffitt Cancer Center and Research Institute, Tampa, Florida 33612, USA
| | - S Jonathan Chapman
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom
| | - Mason A Porter
- Mathematical Institute, University of Oxford, Oxford, OX2 6GG, United Kingdom.,Department of Mathematics, University of California Los Angeles, Los Angeles, California 90095, USA.,CABDyN Complexity Centre, University of Oxford, Oxford, OX1 1HP, United Kingdom
| |
Collapse
|
40
|
Abstract
Random planar graphs appear in a variety of contexts and it is important for many different applications to be able to characterize their structure. Local quantities fail to give interesting information and it seems that path-related measures are able to convey relevant information about the organization of these structures. In particular, nodes with a large betweenness centrality (BC) display nontrivial patterns, such as central loops. We first discuss empirical results for different random planar graphs and we then propose a toy model which allows us to discuss the condition for the emergence of nontrivial patterns such as central loops. This toy model is made of a star network with N_{b} branches of size n and links of weight 1, superimposed to a loop at distance ℓ from the center and with links of weight w. We estimate for this model the BC at the center and on the loop and we show that the loop can be more central than the origin if w<w_{c} where the threshold of this transition scales as w_{c}∼n/N_{b}. In this regime, there is an optimal position of the loop that scales as ℓ_{opt}∼N_{b}w/4. This simple model sheds some light on the organization of these random structures and allows us to discuss the effect of randomness on the centrality of loops. In particular, it suggests that the number and the spatial extension of radial branches are the crucial ingredients that control the existence of central loops.
Collapse
Affiliation(s)
- Benjamin Lion
- Institut de Physique Théorique, CEA, CNRS-URA 2306, F-91191 Gif-sur-Yvette, France
| | - Marc Barthelemy
- Institut de Physique Théorique, CEA, CNRS-URA 2306, F-91191 Gif-sur-Yvette, France
| |
Collapse
|
41
|
Ronellenfitsch H, Katifori E. Global Optimization, Local Adaptation, and the Role of Growth in Distribution Networks. PHYSICAL REVIEW LETTERS 2016; 117:138301. [PMID: 27715085 DOI: 10.1103/physrevlett.117.138301] [Citation(s) in RCA: 58] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/01/2016] [Indexed: 06/06/2023]
Abstract
Highly optimized complex transport networks serve crucial functions in many man-made and natural systems such as power grids and plant or animal vasculature. Often, the relevant optimization functional is nonconvex and characterized by many local extrema. In general, finding the global, or nearly global optimum is difficult. In biological systems, it is believed that such an optimal state is slowly achieved through natural selection. However, general coarse grained models for flow networks with local positive feedback rules for the vessel conductivity typically get trapped in low efficiency, local minima. In this work we show how the growth of the underlying tissue, coupled to the dynamical equations for network development, can drive the system to a dramatically improved optimal state. This general model provides a surprisingly simple explanation for the appearance of highly optimized transport networks in biology such as leaf and animal vasculature.
Collapse
Affiliation(s)
- Henrik Ronellenfitsch
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| | - Eleni Katifori
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
| |
Collapse
|
42
|
Hunt D, Savage VM. Asymmetries arising from the space-filling nature of vascular networks. Phys Rev E 2016; 93:062305. [PMID: 27415278 DOI: 10.1103/physreve.93.062305] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/18/2015] [Indexed: 11/07/2022]
Abstract
Cardiovascular networks span the body by branching across many generations of vessels. The resulting structure delivers blood over long distances to supply all cells with oxygen via the relatively short-range process of diffusion at the capillary level. The structural features of the network that accomplish this density and ubiquity of capillaries are often called space-filling. There are multiple strategies to fill a space, but some strategies do not lead to biologically adaptive structures by requiring too much construction material or space, delivering resources too slowly, or using too much power to move blood through the system. We empirically measure the structure of real networks (18 humans and 1 mouse) and compare these observations with predictions of model networks that are space-filling and constrained by a few guiding biological principles. We devise a numerical method that enables the investigation of space-filling strategies and determination of which biological principles influence network structure. Optimization for only a single principle creates unrealistic networks that represent an extreme limit of the possible structures that could be observed in nature. We first study these extreme limits for two competing principles, minimal total material and minimal path lengths. We combine these two principles and enforce various thresholds for balance in the network hierarchy, which provides a novel approach that highlights the tradeoffs faced by biological networks and yields predictions that better match our empirical data.
Collapse
Affiliation(s)
- David Hunt
- Department of Biomathematics, University of California at Los Angeles, Los Angeles, California 90095, USA
| | - Van M Savage
- Department of Biomathematics, University of California at Los Angeles, Los Angeles, California 90095, USA.,Santa Fe Institute, Santa Fe, New Mexico 87501, USA.,Department of Ecology and Evolutionary Biology, University of California at Los Angeles, Los Angeles, California 90095, USA
| |
Collapse
|
43
|
Gräwer J, Modes CD, Magnasco MO, Katifori E. Structural self-assembly and avalanchelike dynamics in locally adaptive networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012801. [PMID: 26274219 DOI: 10.1103/physreve.92.012801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2014] [Indexed: 06/04/2023]
Abstract
Transport networks play a key role across four realms of eukaryotic life: slime molds, fungi, plants, and animals. In addition to the developmental algorithms that build them, many also employ adaptive strategies to respond to stimuli, damage, and other environmental changes. We model these adapting network architectures using a generic dynamical system on weighted graphs and find in simulation that these networks ultimately develop a hierarchical organization of the final weighted architecture accompanied by the formation of a system-spanning backbone. In addition, we find that the long term equilibration dynamics exhibit behavior reminiscent of glassy systems characterized by long periods of slow changes punctuated by bursts of reorganization events.
Collapse
Affiliation(s)
- Johannes Gräwer
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Goettingen, Germany
| | - Carl D Modes
- Center for Studies in Physics and Biology, The Rockefeller University, New York, New York 10065, USA
| | - Marcelo O Magnasco
- Center for Studies in Physics and Biology, The Rockefeller University, New York, New York 10065, USA
| | - Eleni Katifori
- Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Goettingen, Germany
- Department of Physics and Astronomy, University of Pennsylvania, 19104, Philadelphia, Pennsylvania 19104, USA
| |
Collapse
|
44
|
Farr RS, Harer JL, Fink TMA. Easily repairable networks: reconnecting nodes after damage. PHYSICAL REVIEW LETTERS 2014; 113:138701. [PMID: 25302922 DOI: 10.1103/physrevlett.113.138701] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2014] [Indexed: 06/04/2023]
Abstract
We introduce a simple class of distribution networks that withstand damage by being repairable instead of redundant. Instead of asking how hard it is to disconnect nodes through damage, we ask how easy it is to reconnect nodes after damage. We prove that optimal networks on regular lattices have an expected cost of reconnection proportional to the lattice length, and that such networks have exactly three levels of structural hierarchy. We extend our results to networks subject to repeated attacks, in which the repairs themselves must be repairable. We find that, in exchange for a modest increase in repair cost, such networks are able to withstand any number of attacks.
Collapse
Affiliation(s)
- Robert S Farr
- London Institute for Mathematical Sciences, 35a South Street, Mayfair, London W1K 2XF, United Kingdom and Unilever R&D, Colworth Science Park, MK44 1LQ Bedford, United Kingdom
| | - John L Harer
- Department of Mathematics, Duke University, Rayleigh-Durham, North Carolina 27708-0320, USA
| | - Thomas M A Fink
- London Institute for Mathematical Sciences, 35a South Street, Mayfair, London W1K 2XF, United Kingdom and Centre National de la Recherche Scientifique, Paris 75248, France
| |
Collapse
|