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Xiao R, Gao Q, Azaele S, Sun Y. Effects of noise on the critical points of Turing instability in complex ecosystems. Phys Rev E 2023; 108:014407. [PMID: 37583214 DOI: 10.1103/physreve.108.014407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2023] [Accepted: 07/01/2023] [Indexed: 08/17/2023]
Abstract
Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.
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Affiliation(s)
- Rui Xiao
- School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
| | - Qingyu Gao
- College of Chemical Engineering, China University of Mining and Technology, Xuzhou 221116, China
| | - Sandro Azaele
- Department of Physics and Astronomy "G. Galileo," University of Padova, Padova Via Francesco Marzolo 8, 35131 Padova, Italy
| | - Yongzheng Sun
- School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China
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2
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Gao S, Chang L, Perc M, Wang Z. Turing patterns in simplicial complexes. Phys Rev E 2023; 107:014216. [PMID: 36797896 DOI: 10.1103/physreve.107.014216] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/18/2022] [Accepted: 12/06/2022] [Indexed: 02/18/2023]
Abstract
The spontaneous emergence of patterns in nature, such as stripes and spots, can be mathematically explained by reaction-diffusion systems. These patterns are often referred as Turing patterns to honor the seminal work of Alan Turing in the early 1950s. With the coming of age of network science, and with its related departure from diffusive nearest-neighbor interactions to long-range links between nodes, additional layers of complexity behind pattern formation have been discovered, including irregular spatiotemporal patterns. Here we investigate the formation of Turing patterns in simplicial complexes, where links no longer connect just pairs of nodes but can connect three or more nodes. Such higher-order interactions are emerging as a new frontier in network science, in particular describing group interaction in various sociological and biological systems, so understanding pattern formation under these conditions is of the utmost importance. We show that a canonical reaction-diffusion system defined over a simplicial complex yields Turing patterns that fundamentally differ from patterns observed in traditional networks. For example, we observe a stable distribution of Turing patterns where the fraction of nodes with reactant concentrations above the equilibrium point is exponentially related to the average degree of 2-simplexes, and we uncover parameter regions where Turing patterns will emerge only under higher-order interactions, but not under pairwise interactions.
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Affiliation(s)
- Shupeng Gao
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China.,School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, China
| | - Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, China.,Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, China
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia.,Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 404332, Taiwan.,Alma Mater Europaea, Slovenska ulica 17, 2000 Maribor, Slovenia.,Complexity Science Hub Vienna, Josefstädterstraße 39, 1080 Vienna, Austria.,Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an 710072, China.,School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi'an 710072, China
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3
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Reaction-diffusion models in weighted and directed connectomes. PLoS Comput Biol 2022; 18:e1010507. [DOI: 10.1371/journal.pcbi.1010507] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/03/2022] [Revised: 11/23/2022] [Accepted: 08/22/2022] [Indexed: 11/07/2022] Open
Abstract
Connectomes represent comprehensive descriptions of neural connections in a nervous system to better understand and model central brain function and peripheral processing of afferent and efferent neural signals. Connectomes can be considered as a distinctive and necessary structural component alongside glial, vascular, neurochemical, and metabolic networks of the nervous systems of higher organisms that are required for the control of body functions and interaction with the environment. They are carriers of functional epiphenomena such as planning behavior and cognition, which are based on the processing of highly dynamic neural signaling patterns. In this study, we examine more detailed connectomes with edge weighting and orientation properties, in which reciprocal neuronal connections are also considered. Diffusion processes are a further necessary condition for generating dynamic bioelectric patterns in connectomes. Based on our high-precision connectome data, we investigate different diffusion-reaction models to study the propagation of dynamic concentration patterns in control and lesioned connectomes. Therefore, differential equations for modeling diffusion were combined with well-known reaction terms to allow the use of connection weights, connectivity orientation and spatial distances.
Three reaction-diffusion systems Gray-Scott, Gierer-Meinhardt and Mimura-Murray were investigated. For this purpose, implicit solvers were implemented in a numerically stable reaction-diffusion system within the framework of neuroVIISAS. The implemented reaction-diffusion systems were applied to a subconnectome which shapes the mechanosensitive pathway that is strongly affected in the multiple sclerosis demyelination disease. It was found that demyelination modeling by connectivity weight modulation changes the oscillations of the target region, i.e. the primary somatosensory cortex, of the mechanosensitive pathway.
In conclusion, a new application of reaction-diffusion systems to weighted and directed connectomes has been realized. Because the implementation were performed in the neuroVIISAS framework many possibilities for the study of dynamic reaction-diffusion processes in empirical connectomes as well as specific randomized network models are available now.
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Kato Y, Nakao H. Turing instability in quantum activator–inhibitor systems. Sci Rep 2022; 12:15573. [PMID: 36114210 PMCID: PMC9481611 DOI: 10.1038/s41598-022-19010-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 08/23/2022] [Indexed: 11/26/2022] Open
Abstract
Turing instability is a fundamental mechanism of nonequilibrium self-organization. However, despite the universality of its essential mechanism, Turing instability has thus far been investigated mostly in classical systems. In this study, we show that Turing instability can occur in a quantum dissipative system and analyze its quantum features such as entanglement and the effect of measurement. We propose a degenerate parametric oscillator with nonlinear damping in quantum optics as a quantum activator–inhibitor unit and demonstrate that a system of two such units can undergo Turing instability when diffusively coupled with each other. The Turing instability induces nonuniformity and entanglement between the two units and gives rise to a pair of nonuniform states that are mixed due to quantum noise. Further performing continuous measurement on the coupled system reveals the nonuniformity caused by the Turing instability. Our results extend the universality of the Turing mechanism to the quantum realm and may provide a novel perspective on the possibility of quantum nonequilibrium self-organization and its application in quantum technologies.
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5
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Chang L, Guo L, Liu C, Wang Z, Sun G. The qualitative and quantitative relationships between pattern formation and average degree in networked reaction-diffusion systems. CHAOS (WOODBURY, N.Y.) 2022; 32:093129. [PMID: 36182400 DOI: 10.1063/5.0107504] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2022] [Accepted: 08/26/2022] [Indexed: 06/16/2023]
Abstract
The Turing pattern is an important dynamic behavior characteristic of activator-inhibitor systems. Differentiating from traditional assumption of activator-inhibitor interactions in a spatially continuous domain, a Turing pattern in networked reaction-diffusion systems has received much attention during the past few decades. In spite of its great progress, it still fails to evaluate the precise influences of network topology on pattern formation. To this end, we try to promote the research on this important and interesting issue from the point of view of average degree-a critical topological feature of networks. We first qualitatively analyze the influence of average degree on pattern formation. Then, a quantitative relationship between pattern formation and average degree, the exponential decay of pattern formation, is proposed via nonlinear regression. The finding holds true for several activator-inhibitor systems including biology model, ecology model, and chemistry model. The significance of this study lies that the exponential decay not only quantitatively depicts the influence of average degree on pattern formation, but also provides the possibility for predicting and controlling pattern formation.
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Affiliation(s)
- Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shanxi, China
| | - Luyao Guo
- School of Mathematics, Southeast University, Nanjing 210096, China
| | - Chen Liu
- School of Ecology and Environment Science, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Zhen Wang
- Center for Optical Imagery Analysis and Learning, Northwestern Polytechnical University, Xi'an 710072, Shaanxi, China
| | - Guiquan Sun
- Department of Mathematics, North University of China, Taiyuan 030051, Shanxi, China
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6
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Zheng Q, Shen J, Zhou L, Guan L. Turing pattern induced by the directed ER network and delay. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:11854-11867. [PMID: 36653978 DOI: 10.3934/mbe.2022553] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/17/2023]
Abstract
Infectious diseases generally spread along with the asymmetry of social network propagation because the asymmetry of urban development and the prevention strategies often affect the direction of the movement. But the spreading mechanism of the epidemic remains to explore in the directed network. In this paper, the main effect of the directed network and delay on the dynamic behaviors of the epidemic is investigated. The algebraic expressions of Turing instability are given to show the role of the directed network in the spread of the epidemic, which overcomes the drawback that undirected networks cannot lead to the outbreaks of infectious diseases. Then, Hopf bifurcation is analyzed to illustrate the dynamic mechanism of the periodic outbreak, which is consistent with the transmission of COVID-19. Also, the discrepancy ratio between the imported and the exported is proposed to explain the importance of quarantine policies and the spread mechanism. Finally, the theoretical results are verified by numerical simulation.
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Affiliation(s)
- Qianqian Zheng
- School of Science, Xuchang University; Henan Joint International Research Laboratory of High Performance Computation for Complex Systems, Xuchang 461000, China
| | - Jianwei Shen
- School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
| | - Lingli Zhou
- School of Mathematical Sciences, Soochow University, Suzhou 215006, China
| | - Linan Guan
- School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
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7
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Gao S, Chang L, Romić I, Wang Z, Jusup M, Holme P. Optimal control of networked reaction-diffusion systems. J R Soc Interface 2022; 19:20210739. [PMID: 35259961 PMCID: PMC8905157 DOI: 10.1098/rsif.2021.0739] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2021] [Accepted: 02/08/2022] [Indexed: 12/16/2022] Open
Abstract
Patterns in nature are fascinating both aesthetically and scientifically. Alan Turing's celebrated reaction-diffusion model of pattern formation from the 1950s has been extended to an astounding diversity of applications: from cancer medicine, via nanoparticle fabrication, to computer architecture. Recently, several authors have studied pattern formation in underlying networks, but thus far, controlling a reaction-diffusion system in a network to obtain a particular pattern has remained elusive. We present a solution to this problem in the form of an analytical framework and numerical algorithm for optimal control of Turing patterns in networks. We demonstrate our method's effectiveness and discuss factors that affect its performance. We also pave the way for multidisciplinary applications of our framework beyond reaction-diffusion models.
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Affiliation(s)
- Shupeng Gao
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
- School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
| | - Lili Chang
- Complex Systems Research Center, Shanxi University, Taiyuan 030006, People’s Republic of China
- Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis for Disease Control and Prevention, Taiyuan 030006, People’s Republic of China
| | - Ivan Romić
- School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
- Statistics and Mathematics College, Yunnan University of Finance and Economics, Kunming 650221, People’s Republic of China
- Graduate School of Economics, Osaka City University, Osaka 558-8585, Japan
| | - Zhen Wang
- School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
- School of Artificial Intelligence, Optics, and Electronics (iOPEN), Northwestern Polytechnical University, Xi’an 710072, People’s Republic of China
| | - Marko Jusup
- Tokyo Tech World Hub Research Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 152-8550, Japan
| | - Petter Holme
- Tokyo Tech World Hub Research Initiative (WRHI), Institute of Innovative Research, Tokyo Institute of Technology, Yokohama 152-8550, Japan
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8
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Anderson KE, Fahimipour AK. Body size dependent dispersal influences stability in heterogeneous metacommunities. Sci Rep 2021; 11:17410. [PMID: 34465802 PMCID: PMC8408130 DOI: 10.1038/s41598-021-96629-5] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/19/2021] [Accepted: 08/12/2021] [Indexed: 11/25/2022] Open
Abstract
Body size affects key biological processes across the tree of life, with particular importance for food web dynamics and stability. Traits influencing movement capabilities depend strongly on body size, yet the effects of allometrically-structured dispersal on food web stability are less well understood than other demographic processes. Here we study the stability properties of spatially-arranged model food webs in which larger bodied species occupy higher trophic positions, while species’ body sizes also determine the rates at which they traverse spatial networks of heterogeneous habitat patches. Our analysis shows an apparent stabilizing effect of positive dispersal rate scaling with body size compared to negative scaling relationships or uniform dispersal. However, as the global coupling strength among patches increases, the benefits of positive body size-dispersal scaling disappear. A permutational analysis shows that breaking allometric dispersal hierarchies while preserving dispersal rate distributions rarely alters qualitative aspects of metacommunity stability. Taken together, these results suggest that the oft-predicted stabilizing effects of large mobile predators may, for some dimensions of ecological stability, be attributed to increased patch coupling per se, and not necessarily coupling by top trophic levels in particular.
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Affiliation(s)
- Kurt E Anderson
- Department of Evolution, Ecology, and Organismal Biology, University of California, Riverside, CA, USA.
| | - Ashkaan K Fahimipour
- Department of Computer Science, University of California, Davis, CA, USA.,Institute of Marine Sciences, University of California, Santa Cruz, CA, USA
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9
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Tourigny DS. Cooperative metabolic resource allocation in spatially-structured systems. J Math Biol 2021; 82:5. [PMID: 33479850 DOI: 10.1007/s00285-021-01558-6] [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: 12/05/2019] [Revised: 06/30/2020] [Accepted: 10/27/2020] [Indexed: 10/22/2022]
Abstract
Natural selection has shaped the evolution of cells and multi-cellular organisms such that social cooperation can often be preferred over an individualistic approach to metabolic regulation. This paper extends a framework for dynamic metabolic resource allocation based on the maximum entropy principle to spatiotemporal models of metabolism with cooperation. Much like the maximum entropy principle encapsulates 'bet-hedging' behaviour displayed by organisms dealing with future uncertainty in a fluctuating environment, its cooperative extension describes how individuals adapt their metabolic resource allocation strategy to further accommodate limited knowledge about the welfare of others within a community. The resulting theory explains why local regulation of metabolic cross-feeding can fulfil a community-wide metabolic objective if individuals take into consideration an ensemble measure of total population performance as the only form of global information. The latter is likely supplied by quorum sensing in microbial systems or signalling molecules such as hormones in multi-cellular eukaryotic organisms.
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Affiliation(s)
- David S Tourigny
- Columbia University Irving Medical Center, 630 West 168th Street, New York, NY, 10032, USA.
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10
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Nicoletti S, Fanelli D, Zagli N, Asllani M, Battistelli G, Carletti T, Chisci L, Innocenti G, Livi R. Resilience for stochastic systems interacting via a quasi-degenerate network. CHAOS (WOODBURY, N.Y.) 2019; 29:083123. [PMID: 31472518 DOI: 10.1063/1.5099538] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/11/2019] [Accepted: 08/04/2019] [Indexed: 06/10/2023]
Abstract
A stochastic reaction-diffusion model is studied on a networked support. In each patch of the network, two species are assumed to interact following a non-normal reaction scheme. When the interaction unit is replicated on a directed linear lattice, noise gets amplified via a self-consistent process, which we trace back to the degenerate spectrum of the embedding support. The same phenomenon holds when the system is bound to explore a quasidegenerate network. In this case, the eigenvalues of the Laplacian operator, which governs species diffusion, accumulate over a limited portion of the complex plane. The larger the network, the more pronounced the amplification. Beyond a critical network size, a system deemed deterministically stable, hence resilient, can develop seemingly regular patterns in the concentration amount. Non-normality and quasidegenerate networks may, therefore, amplify the inherent stochasticity and so contribute to altering the perception of resilience, as quantified via conventional deterministic methods.
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Affiliation(s)
- Sara Nicoletti
- Dipartimento di Fisica e Astronomia, CSDC and INFN, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
| | - Duccio Fanelli
- Dipartimento di Fisica e Astronomia, CSDC and INFN, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
| | - Niccolò Zagli
- Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
| | - Malbor Asllani
- MACSI, Department of Mathematics and Statistics, University of Limerick, Limerick V94 T9PX, Ireland
| | - Giorgio Battistelli
- Dipartimento di Ingegneria dell'Informazione, Università di Firenze, Via S. Marta 3, 50139 Florence, Italy
| | - Timoteo Carletti
- naXys, Namur Institute for Complex Systems, University of Namur, 8 Rempart de la Vierge, B5000 Namur, Belgium
| | - Luigi Chisci
- Dipartimento di Ingegneria dell'Informazione, Università di Firenze, Via S. Marta 3, 50139 Florence, Italy
| | - Giacomo Innocenti
- Dipartimento di Ingegneria dell'Informazione, Università di Firenze, Via S. Marta 3, 50139 Florence, Italy
| | - Roberto Livi
- Dipartimento di Fisica e Astronomia, CSDC and INFN, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
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11
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Gyorgy A, Arcak M. Pattern Formation over Multigraphs. IEEE TRANSACTIONS ON NETWORK SCIENCE AND ENGINEERING 2018; 5:55-64. [PMID: 29520363 PMCID: PMC5839348 DOI: 10.1109/tnse.2017.2730261] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 03/14/2024]
Abstract
Two of the most common pattern formation mechanisms are Turing-patterning in reaction-diffusion systems and lateral inhibition of neighboring cells. In this paper, we introduce a broad dynamical model of interconnected modules to study the emergence of patterns, with the above mentioned two mechanisms as special cases. Our results do not restrict the number of modules or their complexity, allow multiple layers of communication channels with possibly different interconnection structure, and do not assume symmetric connections between two connected modules. Leveraging only the static input/output properties of the subsystems and the spectral properties of the interconnection matrices, we characterize the stability of the homogeneous fixed points as well as sufficient conditions for the emergence of spatially non-homogeneous patterns. To obtain these results, we rely on properties of the graphs together with tools from monotone systems theory. As application examples, we consider patterning in neural networks, in reaction-diffusion systems, and contagion processes over random graphs.
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Affiliation(s)
- Andras Gyorgy
- Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720 USA
| | - Murat Arcak
- Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94720 USA
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12
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Petit J, Lauwens B, Fanelli D, Carletti T. Theory of Turing Patterns on Time Varying Networks. PHYSICAL REVIEW LETTERS 2017; 119:148301. [PMID: 29053314 DOI: 10.1103/physrevlett.119.148301] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2017] [Indexed: 06/07/2023]
Abstract
The process of pattern formation for a multispecies model anchored on a time varying network is studied. A nonhomogeneous perturbation superposed to an homogeneous stable fixed point can be amplified following the Turing mechanism of instability, solely instigated by the network dynamics. By properly tuning the frequency of the imposed network evolution, one can make the examined system behave as its averaged counterpart, over a finite time window. This is the key observation to derive a closed analytical prediction for the onset of the instability in the time dependent framework. Continuously and piecewise constant periodic time varying networks are analyzed, setting the framework for the proposed approach. The extension to nonperiodic settings is also discussed.
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Affiliation(s)
- Julien Petit
- naXys, Namur Institute for Complex Systems, University of Namur, B5000 Namur, Belgium
- Department of Mathematics, Royal Military Academy, B1000 Brussels, Belgium
| | - Ben Lauwens
- Department of Mathematics, Royal Military Academy, B1000 Brussels, Belgium
| | - Duccio Fanelli
- Dipartimento di Fisica e Astronomia and CSDC, Università degli Studi di Firenze, 50019 Sesto Fiorentino, Italy
- INFN Sezione di Firenze, 50019 Sesto Fiorentino, Italy
| | - Timoteo Carletti
- naXys, Namur Institute for Complex Systems, University of Namur, B5000 Namur, Belgium
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13
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Expert P, de Nigris S, Takaguchi T, Lambiotte R. Graph spectral characterization of the XY model on complex networks. Phys Rev E 2017; 96:012312. [PMID: 29347091 DOI: 10.1103/physreve.96.012312] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2016] [Indexed: 06/07/2023]
Abstract
There is recent evidence that the XY spin model on complex networks can display three different macroscopic states in response to the topology of the network underpinning the interactions of the spins. In this work we present a way to characterize the macroscopic states of the XY spin model based on the spectral decomposition of time series using topological information about the underlying networks. We use three different classes of networks to generate time series of the spins for the three possible macroscopic states. We then use the temporal Graph Signal Transform technique to decompose the time series of the spins on the eigenbasis of the Laplacian. From this decomposition, we produce spatial power spectra, which summarize the activation of structural modes by the nonlinear dynamics, and thus coherent patterns of activity of the spins. These signatures of the macroscopic states are independent of the underlying network class and can thus be used as robust signatures for the macroscopic states. This work opens avenues to analyze and characterize dynamics on complex networks using temporal Graph Signal Analysis.
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Affiliation(s)
- Paul Expert
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
- EPSRC Centre for Mathematics of Precision Healthcare, Imperial College London, London SW7 2AZ, United Kingdom
- Centre for Neuroimaging Sciences, Institute of Psychiatry, King's College London, London SE5 8AF, United Kingdom
| | - Sarah de Nigris
- NaXys, Département de Mathématique, Université de Namur, 8 Rempart de la Vierge, 5000 Namur, Belgium
- Université Lyon, CNRS, ENS de Lyon, Inria, UCB Lyon 1, LIP UMR No. 5668, F69342, Lyon, France
| | - Taro Takaguchi
- National Institute of Information and Communications Technology, Tokyo 184-8795, Japan
- National Institute of Informatics, Tokyo 101-8430, Japan
- JST, ERATO, Kawarabayashi Large Graph Project, Tokyo 101-8430, Japan
| | - Renaud Lambiotte
- NaXys, Département de Mathématique, Université de Namur, 8 Rempart de la Vierge, 5000 Namur, Belgium
- Complexity Science Hub Vienna, Josefstaedter Strasse 39, A 1080 Vienna, Austria
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14
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Zheng MM, Shao B, Ouyang Q. Identifying network topologies that can generate turing pattern. J Theor Biol 2016; 408:88-96. [DOI: 10.1016/j.jtbi.2016.08.005] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2016] [Revised: 07/15/2016] [Accepted: 08/08/2016] [Indexed: 12/29/2022]
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15
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Contemori S, Di Patti F, Fanelli D, Miele F. Multiple-scale theory of topology-driven patterns on directed networks. Phys Rev E 2016; 93:032317. [PMID: 27078376 DOI: 10.1103/physreve.93.032317] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2015] [Indexed: 11/07/2022]
Abstract
Dynamical processes on networks are currently being considered in different domains of cross-disciplinary interest. Reaction-diffusion systems hosted on directed graphs are in particular relevant for their widespread applications, from computer networks to traffic systems. Due to the peculiar spectrum of the discrete Laplacian operator, homogeneous fixed points can turn unstable, on a directed support, because of the topology of the network, a phenomenon which cannot be induced on undirected graphs. A linear analysis can be performed to single out the conditions that underly the instability. The complete characterization of the patterns, which are eventually attained beyond the linear regime of exponential growth, calls instead for a full nonlinear treatment. By performing a multiple time scale perturbative calculation, we here derive an effective equation for the nonlinear evolution of the amplitude of the most unstable mode, close to the threshold of criticality. This is a Stuart-Landau equation the complex coefficients of which appear to depend on the topological features of the embedding directed graph. The theory proves adequate versus simulations, as confirmed by operating with a paradigmatic reaction-diffusion model.
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Affiliation(s)
- Silvia Contemori
- Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
| | - Francesca Di Patti
- Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, CSDC and INFN, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
| | - Duccio Fanelli
- Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, CSDC and INFN, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
| | - Filippo Miele
- Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
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Challenger JD, Burioni R, Fanelli D. Turing-like instabilities from a limit cycle. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022818. [PMID: 26382465 DOI: 10.1103/physreve.92.022818] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2015] [Indexed: 05/03/2023]
Abstract
The Turing instability is a paradigmatic route to pattern formation in reaction-diffusion systems. Following a diffusion-driven instability, homogeneous fixed points can become unstable when subject to external perturbation. As a consequence, the system evolves towards a stationary, nonhomogeneous attractor. Stable patterns can be also obtained via oscillation quenching of an initially synchronous state of diffusively coupled oscillators. In the literature this is known as the oscillation death phenomenon. Here, we show that oscillation death is nothing but a Turing instability for the first return map of the system in its synchronous periodic state. In particular, we obtain a set of approximated closed conditions for identifying the domain in the parameter space that yields the instability. This is a natural generalization of the original Turing relations, to the case where the homogeneous solution of the examined system is a periodic function of time. The obtained framework applies to systems embedded in continuum space, as well as those defined on a networklike support. The predictive ability of the theory is tested numerically, using different reaction schemes.
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Affiliation(s)
- Joseph D Challenger
- Department of Infectious Disease Epidemiology, Imperial College London, London, W2 1PG, United Kingdom
- Dipartimento di Fisica e Astronomia, Università di Firenze, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy
| | - Raffaella Burioni
- Dipartimento di Fisica e Scienza della Terra and INFN, Università di Parma, viale G. P. Usberti 7/A 43124, Parma, Italy
| | - Duccio Fanelli
- Dipartimento di Fisica e Astronomia, Università di Firenze, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Firenze, Italy
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Kouvaris NE, Hata S, Guilera AD. Pattern formation in multiplex networks. Sci Rep 2015; 5:10840. [PMID: 26042606 PMCID: PMC4455352 DOI: 10.1038/srep10840] [Citation(s) in RCA: 79] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2014] [Accepted: 04/17/2015] [Indexed: 11/24/2022] Open
Abstract
The advances in understanding complex networks have generated increasing interest in dynamical processes occurring on them. Pattern formation in activator-inhibitor systems has been studied in networks, revealing differences from the classical continuous media. Here we study pattern formation in a new framework, namely multiplex networks. These are systems where activator and inhibitor species occupy separate nodes in different layers. Species react across layers but diffuse only within their own layer of distinct network topology. This multiplicity generates heterogeneous patterns with significant differences from those observed in single-layer networks. Remarkably, diffusion-induced instability can occur even if the two species have the same mobility rates; condition which can never destabilize single-layer networks. The instability condition is revealed using perturbation theory and expressed by a combination of degrees in the different layers. Our theory demonstrates that the existence of such topology-driven instabilities is generic in multiplex networks, providing a new mechanism of pattern formation.
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Affiliation(s)
- Nikos E. Kouvaris
- Department of Physics, University of Barcelona, Martí i Franqués 1, E-08028, Barcelona, Spain
| | - Shigefumi Hata
- Department of Mathematical Science and Advanced Technology, Japan Agency for Marine-Earth Science and Technology, 236-0001 Kanagawa, Japan
| | - Albert Díaz- Guilera
- Department of Physics, University of Barcelona, Martí i Franqués 1, E-08028, Barcelona, Spain
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Asllani M, Busiello DM, Carletti T, Fanelli D, Planchon G. Turing patterns in multiplex networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042814. [PMID: 25375556 DOI: 10.1103/physreve.90.042814] [Citation(s) in RCA: 42] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2014] [Indexed: 05/03/2023]
Abstract
The theory of patterns formation for a reaction-diffusion system defined on a multiplex is developed by means of a perturbative approach. The interlayer diffusion constants act as a small parameter in the expansion and the unperturbed state coincides with the limiting setting where the multiplex layers are decoupled. The interaction between adjacent layers can seed the instability of a homogeneous fixed point, yielding self-organized patterns which are instead impeded in the limit of decoupled layers. Patterns on individual layers can also fade away due to cross-talking between layers. Analytical results are compared to direct simulations.
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Affiliation(s)
- Malbor Asllani
- Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, via Valleggio 11, 22100 Como, Italy and Dipartimento di Fisica e Astronomia, University of Florence, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy
| | - Daniel M Busiello
- Dipartimento di Fisica e Astronomia, University of Florence, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy
| | - Timoteo Carletti
- Department of Mathematics and Namur Center for Complex Systems - naXys, University of Namur, rempart de la Vierge 8, B 5000 Namur, Belgium
| | - Duccio Fanelli
- Dipartimento di Fisica e Astronomia, University of Florence, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy
| | - Gwendoline Planchon
- Dipartimento di Fisica e Astronomia, University of Florence, INFN and CSDC, Via Sansone 1, 50019 Sesto Fiorentino, Florence, Italy and Department of Mathematics and Namur Center for Complex Systems - naXys, University of Namur, rempart de la Vierge 8, B 5000 Namur, Belgium
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The theory of pattern formation on directed networks. Nat Commun 2014; 5:4517. [DOI: 10.1038/ncomms5517] [Citation(s) in RCA: 81] [Impact Index Per Article: 8.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2014] [Accepted: 06/26/2014] [Indexed: 11/08/2022] Open
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Dispersal-induced destabilization of metapopulations and oscillatory Turing patterns in ecological networks. Sci Rep 2014; 4:3585. [PMID: 24394959 PMCID: PMC3882759 DOI: 10.1038/srep03585] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2013] [Accepted: 12/02/2013] [Indexed: 11/16/2022] Open
Abstract
As shown by Alan Turing in 1952, differential diffusion may destabilize uniform distributions of reacting species and lead to emergence of patterns. While stationary Turing patterns are broadly known, the oscillatory instability, leading to traveling waves in continuous media and sometimes called the wave bifurcation, remains less investigated. Here, we extend the original analysis by Turing to networks and apply it to ecological metapopulations with dispersal connections between habitats. Remarkably, the oscillatory Turing instability does not lead to wave patterns in networks, but to spontaneous development of heterogeneous oscillations and possible extinction of species. We find such oscillatory instabilities for all possible food webs with three predator or prey species, under various assumptions about the mobility of individual species and nonlinear interactions between them. Hence, the oscillatory Turing instability should be generic and must play a fundamental role in metapopulation dynamics, providing a common mechanism for dispersal-induced destabilization of ecosystems.
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Muzika F, Schreiberová L, Schreiber I. Chemical computing based on Turing patterns in two coupled cells with equal transport coefficients. RSC Adv 2014. [DOI: 10.1039/c4ra08859j] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/21/2023] Open
Abstract
Two diffusively coupled reaction cells with a nonlinear reaction are used to perform chemical computing based on targeted perturbations switching between two Turing patterns defining two states of a logical device.
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Muzika F, Schreiber I. Control of Turing patterns and their usage as sensors, memory arrays, and logic gates. J Chem Phys 2013; 139:164108. [PMID: 24182005 DOI: 10.1063/1.4825379] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
We study a model system of three diffusively coupled reaction cells arranged in a linear array that display Turing patterns with special focus on the case of equal coupling strength for all components. As a suitable model reaction we consider a two-variable core model of glycolysis. Using numerical continuation and bifurcation techniques we analyze the dependence of the system's steady states on varying rate coefficient of the recycling step while the coupling coefficients of the inhibitor and activator are fixed and set at the ratios 100:1, 1:1, and 4:5. We show that stable Turing patterns occur at all three ratios but, as expected, spontaneous transition from the spatially uniform steady state to the spatially nonuniform Turing patterns occurs only in the first case. The other two cases possess multiple Turing patterns, which are stabilized by secondary bifurcations and coexist with stable uniform periodic oscillations. For the 1:1 ratio we examine modular spatiotemporal perturbations, which allow for controllable switching between the uniform oscillations and various Turing patterns. Such modular perturbations are then used to construct chemical computing devices utilizing the multiple Turing patterns. By classifying various responses we propose: (a) a single-input resettable sensor capable of reading certain value of concentration, (b) two-input and three-input memory arrays capable of storing logic information, (c) three-input, three-output logic gates performing combinations of logical functions OR, XOR, AND, and NAND.
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Affiliation(s)
- František Muzika
- Department of Chemical Engineering, Institute of Chemical Technology, Prague, 16628 Prague 6, Czech Republic
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Asslani M, Di Patti F, Fanelli D. Stochastic Turing patterns on a network. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:046105. [PMID: 23214650 DOI: 10.1103/physreve.86.046105] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2012] [Indexed: 06/01/2023]
Abstract
The process of stochastic Turing instability on a scale-free network is discussed for a specific case study: the stochastic Brusselator model. The system is shown to spontaneously differentiate into activator-rich and activator-poor nodes outside the region of parameters classically deputed to the deterministic Turing instability. This phenomenon, as revealed by direct stochastic simulations, is explained analytically and eventually traced back to the finite-size corrections stemming from the inherent graininess of the scrutinized medium.
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Affiliation(s)
- Malbor Asslani
- Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell'Insubria, via Valleggio 11, 22100 Como, Italy
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Breton J, Thomas D, Hervagault JF. Dissipative structures in a two-cell system: Numerical and experimental approaches. Proc Natl Acad Sci U S A 2010; 83:551-5. [PMID: 16593652 PMCID: PMC322901 DOI: 10.1073/pnas.83.3.551] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
It has been shown that the coupling between the photoreduction of the oxidized form of dichloroindophenol (an artificial electron acceptor) by thylakoids and the incident light intensity can lead to the appearance of multiple steady states when the system is operated under open conditions. In the present work, a numerical study and experimental evidence are presented on the occurrence of dissipative structures in an arrangement of two continuously stirred tank reactors with mutual mass exchange of dichloroindophenol through an inert membrane. The stable spatial structures are generated by the creation of transient internal and external asymmetries. A nontrivial hysteresis effect between symmetric and asymmetric stable steady states has been observed.
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Affiliation(s)
- J Breton
- Laboratoire de Technologie Enzymatique, Unité Associée No. 523, Centre National de la Recherche Scientifique, Université de Technologie de Compiègne, BP 233 60206 Compiègne, France
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Othmer HG. APPLICATIONS OF BIFURCATION THEORY IN THE ANALYSIS OF SPATIAL AND TEMPORAL PATTERN FORMATION. Ann N Y Acad Sci 2006. [DOI: 10.1111/j.1749-6632.1979.tb29458.x] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Saxena A, Jacobson J, Yamanashi W, Scherlag B, Lamberth J, Saxena B. A hypothetical mathematical construct explaining the mechanism of biological amplification in an experimental model utilizing picoTesla (PT) electromagnetic fields. Med Hypotheses 2003; 60:821-39. [PMID: 12699707 DOI: 10.1016/s0306-9877(03)00011-2] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
Abstract
We seek to answer the conundrum: What is the fundamental mechanism by which very weak, low frequency Electromagnetic fields influence biosystems? In considering the hydrophobicity of intramembranous protein (IMP) H-bonds which cross the phospholipid bilayer of plasma membranes, and the necessity for photonic recycling in cell surface interactions after dissipation of energetic states, present models lack structure and thermodynamic properties to maintain (DeltaE) sufficient energy sources necessary for amplifications by factors of 10(12). Even though one accepts that the ligand-receptor association alters the conformation of extracellular, extruding portions of IMP's at the cell surface, and that this change can be transmitted to the cytoplasm by the transmembranous helical segments by nonlinear vibrations of proteins with generation of soliton waves, one is still unable to account for repair and balanced function. Indeed, responses of critical molecules to certain magnetic field signals may include enhanced vibrational amplitudes, increased quanta of thermal energies and order inducing interactions. We may accept that microtrabecular reticulum-receptor is associated with actin filaments and ATP molecules which contribute to the activation of the cyclase enzyme system through piezoelectricity. Magnetic fields will pass through the membrane which sharply attenuates the electric field component of an EM field, due to its high impedance. Furthermore, EM oscillations are converted to mechanical vibrations; i.e., photon-phonon transduction, to induce molecular vibrations of frequencies specifically responsible for bioamplifications of weak triggers at the membrane surface, as well as GAP junctions. The hydrogen bonds of considerable importance are those in proteins (10(12)Hz) and DNA (10(11)Hz) and may be viewed as centers of EM radiation emission in the range from the mm microwaves to the far IR. However, classical electrodynamical theory does not yield a model for biomolecular resonant responses which are integrated over time and account for the connection between the phonon field and photons. Jacobson Resonance does supply an initial physical mechanism, as equivalencies in energy to that of Zeeman Resonance (i.e., zero-order magnetic resonance) and cyclotron resonance may be derived from the DeBroglie wave particle equation. For the first time, we view the introduction of Relativity Theory to biology in the expression, mc(2)=BvLq, where m is the mass of a particle in the 'box' or 'string' (molecule in a biosystem), c is the velocity of electromagnetic field in space, independent of its inertial frame of reference, B is the magnetic flux density,v is the velocity of the carrier or 'string' (a one or two dimensional 'box') in which the particle exists, L is its dimension (length) and q represents a unit charge q=1C, by defining electromotive force as energy per unit charge. Equivalencies suggest that qvBL is one of the fundamental expressions of energy of a charged wave-particle in magnetic fields, just as Zeeman and cyclotron resonance energy expressions, gbetaB and qhB/2pim, and is applicable to all charged particles (molecules in biological systems). There may exist spontaneous, independent and incessant interactions of magnetic vector B and particles in biosystems which exert Lorentz forces. Lorentz forces may be transmitted from EM field to gravitational field as a gravity wave which return to the phonon field as microgravitational fluctuations to therein produce quantum vibrational states that increase quanta of thermal energies integrated over time. This may account for the differential of 10(12) between photonic energy of ELF waves and the Boltzman energy kT. Recent data from in vivo controlled studies are included as empirical support for the various hypotheses presented.
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Affiliation(s)
- Anjali Saxena
- Department of Biological Sciences, Fairleigh Dickinson University, 1000 River Road, Teaneck, New Jersey 07666, USA
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Kevrekidis IG. Matrices are forever: On applied mathematics and computing in chemical engineering. Chem Eng Sci 1995. [DOI: 10.1016/0009-2509(95)00274-x] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Ecological Interactions in Patchy Environments: From Patch-Occupancy Models to Cellular Automata. LECTURE NOTES IN BIOMATHEMATICS 1993. [DOI: 10.1007/978-3-642-50155-5_8] [Citation(s) in RCA: 28] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/06/2023]
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Hunt KLC, Kottalam J, Hatlee MD, Ross J. Multiple steady states in coupled flow tank reactors. J Chem Phys 1992. [DOI: 10.1063/1.462535] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Marmillot P, Kaufman M, Hervagault J. Multiple steady states and dissipative structures in a circular and linear array of three cells: Numerical and experimental approaches. J Chem Phys 1991. [DOI: 10.1063/1.461151] [Citation(s) in RCA: 35] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Peusner L. Integration of membrane equations in the presence of sources and directional biases: on the consistent definition of kinetic potentials in diffusion-reaction-convection coupling and related transport problems. J Memb Sci 1988. [DOI: 10.1016/s0376-7388(00)80300-2] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Levin SA. Population Models and Community Structure in Heterogeneous Environments. MATHEMATICAL ECOLOGY 1986. [DOI: 10.1007/978-3-642-69888-0_12] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
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Sahimi M, Hughes BD, Scriven LE, Davis HT. Stochastic transport in disordered systems. J Chem Phys 1983. [DOI: 10.1063/1.444631] [Citation(s) in RCA: 167] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Othmer HG, Aldridge JA. The effects of cell density and metabolite flux on cellular dynamics. J Math Biol 1978; 5:169-200. [PMID: 366054 DOI: 10.1007/bf00275897] [Citation(s) in RCA: 34] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Abstract
Density-dependent regulation of cell growth in tissue culture is a well-known phenomenon but the mechanism of regulation remains obscure. Here we explore the effects of cell density and metabolite flux on the collective dynamics of a cell population. The intracellular dynamics are modelled by positive feedback kinetic mechanisms of the kind known to apply to yeast cells. Several experimental observations related to glycolytic oscillations are predicted and it is suggested that the general conclusions may be applicable in a broader context.
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Abstract
During Xenopus embryogenesis and early growth, somite number stays close to a species-typical value for each morphological stage. This remains true even after operations on blastulae which lead to the development of abnormally small but otherwise complete early embryos, involving reduction in number of cells assigned to each somite. Evidence presented suggests that a body-position gradient may be involved, but in rather different ways at different stages, in controlling total somite number.
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