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Li SN, Cao BY. On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations. ENTROPY 2019; 21:e21020204. [PMID: 33266919 PMCID: PMC7514686 DOI: 10.3390/e21020204] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 01/22/2019] [Revised: 02/16/2019] [Accepted: 02/18/2019] [Indexed: 11/30/2022]
Abstract
Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using the relaxation time approximation and power series expansion, macroscopic approximations are derived for these entropic concepts. For the standard BTE, our results can recover the entropic frameworks of classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT) as if there exists a well-defined effective thermal conductivity. For the fractional BTEs corresponding to the generalized Cattaneo equation (GCE) class, the entropy flux and entropy production rate will deviate from the forms in CIT and EIT. In these cases, the entropy flux and entropy production rate will contain fractional-order operators, which reflect memory effects.
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Sánchez Ó, Calvo J, Ibáñez C, Guerrero I, Soler J. Modeling Hedgehog Signaling Through Flux-Saturated Mechanisms. Methods Mol Biol 2015; 1322:19-33. [PMID: 26179036 DOI: 10.1007/978-1-4939-2772-2_3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/16/2023]
Abstract
Hedgehog (Hh) molecules act as morphogens directing cell fate during development by activating various target genes in a concentration dependent manner. Hitherto, modeling morphogen gradient formation in multicellular systems has employed linear diffusion, which is very far from physical reality and needs to be replaced by a deeper understanding of nonlinearities. We have developed a novel mathematical approach by applying flux-limited spreading (FLS) to Hh morphogenetic actions. In the new model, the characteristic velocity of propagation of each morphogen is a new key biological parameter. Unlike in linear diffusion models, FLS modeling predicts concentration fronts and correct patterns and cellular responses over time. In addition, FLS considers not only extracellular binding partners influence, but also channels or bridges of information transfer, such as specialized filopodia or cytonemes as a mechanism of Hh transport. We detect and measure morphogen particle velocity in cytonemes in the Drosophila wing imaginal disc. Indeed, this novel approach to morphogen gradient formation can contribute to future research in the field.
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Affiliation(s)
- Óscar Sánchez
- Departamento de Matemática Aplicada, Universidad de Granada, Campus de Fuentenueva, Granada, 18071, Spain
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On flux-limited morphogenesis. Phys Life Rev 2013. [DOI: 10.1016/j.plrev.2013.10.005] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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Comment on "Morphogenetic action through flux-limited spreading" by Verbeni, Sánchez, Mollica, Siegl-Cachedenier, Carleton, Guerrero, Ruiz i Altaba, and Soler. Phys Life Rev 2013; 10:493-4; discussion 495-7. [PMID: 24216138 DOI: 10.1016/j.plrev.2013.10.006] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2013] [Accepted: 10/10/2013] [Indexed: 11/24/2022]
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Mathematical models for morphogenesis: linear or nonlinear diffusion: comment on "Morphogenetic action through flux-limited spreading" by Verbeni, Sánchez, Mollica, Siegl-Cachedenier, Carleton, Guerrero, Ruiz i Altaba, Soler. Phys Life Rev 2013; 10:485-6; discussion 495-7. [PMID: 24140027 DOI: 10.1016/j.plrev.2013.10.002] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2013] [Accepted: 10/02/2013] [Indexed: 11/20/2022]
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Caselles V. Saturated spreading, an alternative to Fick's law: comment on "Morphogenetic action through flux-limited spreading", by M. Verbeni, O. Sánchez, E. Mollica, I. Siegl-Cachedenier, A. Carleton, I. Guerrero, A. Ruiz i Altaba, J. Soler. Phys Life Rev 2013; 10:491-2; discussion 495-7. [PMID: 24054725 DOI: 10.1016/j.plrev.2013.08.004] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2013] [Accepted: 08/21/2013] [Indexed: 11/27/2022]
Affiliation(s)
- Vicent Caselles
- Departament de Tecnologies de la Informació i les Comunicacions, Universitat Pompeu-Fabra, Barcelona, Spain.
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Mazón JM. Flux-limited diffusion equation: comment on "Morphogenetic action through flux-limited spreading", by Verbeni M., Sánchez O., Mollica E., Siegl-Cachedernier I., Carleton A., Guerrero I., Ruiz i Altaba A., Soler J. Phys Life Rev 2013; 10:482-4; discussion 495-7. [PMID: 23973005 DOI: 10.1016/j.plrev.2013.07.028] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2013] [Accepted: 07/29/2013] [Indexed: 10/26/2022]
Affiliation(s)
- J M Mazón
- Departamento de Análisis Matemático, Universitat de València, Valencia, Spain.
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Verbeni M, Sánchez O, Mollica E, Siegl-Cachedenier I, Carleton A, Guerrero I, Ruiz i Altaba A, Soler J. Morphogenetic action through flux-limited spreading. Phys Life Rev 2013; 10:457-75. [PMID: 23831049 DOI: 10.1016/j.plrev.2013.06.004] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2013] [Accepted: 06/17/2013] [Indexed: 10/26/2022]
Abstract
A central question in biology is how secreted morphogens act to induce different cellular responses within a group of cells in a concentration-dependent manner. Modeling morphogenetic output in multicellular systems has so far employed linear diffusion, which is the normal type of diffusion associated with Brownian processes. However, there is evidence that at least some morphogens, such as Hedgehog (Hh) molecules, may not freely diffuse. Moreover, the mathematical analysis of such models necessarily implies unrealistic instantaneous spreading of morphogen molecules, which are derived from the assumptions of Brownian motion in its continuous formulation. A strict mathematical model considering Fick's diffusion law predicts morphogen exposure of the whole tissue at the same time. Such a strict model thus does not describe true biological patterns, even if similar and attractive patterns appear as results of applying such simple model. To eliminate non-biological behaviors from diffusion models we introduce flux-limited spreading (FLS), which implies a restricted velocity for morphogen propagation and a nonlinear mechanism of transport. Using FLS and focusing on intercellular Hh-Gli signaling, we model a morphogen gradient and highlight the propagation velocity of morphogen particles as a new key biological parameter. This model is then applied to the formation and action of the Sonic Hh (Shh) gradient in the vertebrate embryonic neural tube using our experimental data on Hh spreading in heterologous systems together with published data. Unlike linear diffusion models, FLS modeling predicts concentration fronts and the evolution of gradient dynamics and responses over time. In addition to spreading restrictions by extracellular binding partners, we suggest that the constraints imposed by direct bridges of information transfer such as nanotubes or cytonemes underlie FLS. Indeed, we detect and measure morphogen particle velocity in such cell extensions in different systems.
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Affiliation(s)
- M Verbeni
- Departamento de Matemática Aplicada, Universidad de Granada, 18071-Granada, Spain
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Abstract
In the presence of strong external fields and steep gradients the flux formulae are not linear. The relation between the flux and the diffusion coefficient must be modified. The different flux-limited theories are presented. The flux formulae for solid systems far from equilibrium are derived and different forms of phenomenological flux limiters are discussed. It is shown that in order to accurately compute diffusion flow that is generated by strong force fields and/or discontinuities, the flux-limited diffusion must be considered. The flux limiters improve the spatial accuracy and allow to avoid baseless oscillations in the solutions.
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Cisternas J, Gear CW, Levin S, Kevrekidis IG. Equation-free modelling of evolving diseases: coarse-grained computations with individual-based models. Proc Math Phys Eng Sci 2004. [DOI: 10.1098/rspa.2004.1300] [Citation(s) in RCA: 28] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- Jaime Cisternas
- Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
- Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
| | - C. William Gear
- Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
- NEC Research Institute, Princeton, NJ 08540, USA
| | - Simon Levin
- Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08544, USA
| | - Ioannis G. Kevrekidis
- Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA
- Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
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Kevrekidis PG, Kevrekidis IG, Bishop AR, Titi ES. Continuum approach to discreteness. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046613. [PMID: 12006053 DOI: 10.1103/physreve.65.046613] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2001] [Indexed: 05/23/2023]
Abstract
We study analytically and numerically continuum models derived on the basis of Padé approximations and their effectiveness in modeling spatially discrete systems. We not only analyze features of the temporal dynamics that can be captured through these continuum approaches (e.g., shape oscillations, radiation effects, and trapping) but also point out ones that cannot be captured (such as Peierls-Nabarro barriers and Bloch oscillations). We analyze the role of such methods in providing an effective "homogenization" of spatially discrete, as well as of heterogeneous continuum equations. Finally, we develop numerical methods for solving such equations and use them to establish the range of validity of these continuum approximations, as well as to compare them with other semicontinuum approximations.
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Affiliation(s)
- P G Kevrekidis
- Theoretical Division and Center for Nonlinear Studies, MS B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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