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Angelani L, Garra R. g-fractional diffusion models in bounded domains. Phys Rev E 2023; 107:014127. [PMID: 36797940 DOI: 10.1103/physreve.107.014127] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/20/2022] [Accepted: 01/04/2023] [Indexed: 01/19/2023]
Abstract
In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive regimes. In this paper we study the problem of g-fractional diffusion in a bounded domain with absorbing boundaries. We find the explicit solution for the initial boundary value problem, and we study the first-passage time distribution and the mean first-passage time (MFPT). The main outcome is the proof that with a particular choice of the function g it is possible to obtain a finite MFPT, differently from the anomalous diffusion described by a fractional heat equation involving the classical Caputo derivative.
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Affiliation(s)
- L Angelani
- ISC-CNR, Institute for Complex Systems, Piazzale A. Moro 2, 00185 Rome, Italy.,Dipartimento di Fisica, Sapienza Università di Roma, Piazzale A. Moro 2, 00185 Rome, Italy
| | - R Garra
- Institute of Marine Sciences, National Research Council (CNR), Via del Fosso del Cavaliere, I-00133 Rome, Italy
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Runfola C, Vitali S, Pagnini G. The Fokker-Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm. ROYAL SOCIETY OPEN SCIENCE 2022; 9:221141. [PMID: 36340511 PMCID: PMC9627453 DOI: 10.1098/rsos.221141] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 09/21/2022] [Accepted: 10/11/2022] [Indexed: 06/16/2023]
Abstract
By collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules' displacement and we derive the corresponding Fokker-Planck equation. Molecules' distribution emerges to be related to the Krätzel function and its Fokker-Planck equation to be a fractional diffusion equation in the Erdélyi-Kober sense. The irreducibility of the derived Fokker-Planck equation to those of other literature models is also discussed.
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Affiliation(s)
- C. Runfola
- Department of Physics and Astronomy, University of Bologna, Viale Berti Pichat 6/2, I-40127 Bologna, Italy
- BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country, Spain
| | - S. Vitali
- BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country, Spain
- Eurecat, Centre Tecnológic de Catalunya, Unit of Digital Health, Data Analytics in Medicine, E-08005 Barcelona, Catalunya, Spain
| | - G. Pagnini
- BCAM – Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E-48009 Bilbao, Basque Country, Spain
- Ikerbasque – Basque Foundation for Science, Plaza Euskadi 5, E-48009 Bilbao, Basque Country, Spain
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The Fate of Molecular Species in Water Layers in the Light of Power-Law Time-Dependent Diffusion Coefficient. Symmetry (Basel) 2022. [DOI: 10.3390/sym14061146] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023] Open
Abstract
In the present paper, we propose two methods for tracking molecular species in water layers via two approaches of the diffusion equation with a power-law time-dependent diffusion coefficient. The first approach shows the species densities and the growth of different species via numerical simulation. At the same time, the second approach is built on the fractional diffusion equation with a time-dependent diffusion coefficient in the sense of regularised Caputo fractional derivative. As an illustration, we present here the species densities profiles and track the normal and anomalous growth of five molecular species OH, H2O2, HO2, NO3-, and NO2- via the calculation of the mean square displacement using the two methods.
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Batra A, Sharma R. A near analytic solution of a stochastic immune response model considering variability in virus and T-cell dynamics. J Chem Phys 2021; 154:195104. [PMID: 34240889 DOI: 10.1063/5.0047442] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/30/2022] Open
Abstract
Biological processes at the cellular level are stochastic in nature, and the immune response system is no different. Therefore, models that attempt to explain this system need to also incorporate noise or fluctuations that can account for the observed variability. In this work, a stochastic model of the immune response system is presented in terms of the dynamics of T cells and virus particles. Making use of the Green's function and the Wilemski-Fixman approximation, this model is then solved to obtain the analytical expression for the joint probability density function of these variables in the early and late stages of infection. This is then also used to calculate the average level of virus particles in the system. Upon comparing the theoretically predicted average virus levels to those of COVID-19 patients, it is hypothesized that the long-lived dynamics that are characteristics of such viral infections are due to the long range correlations in the temporal fluctuations of the virions. This model, therefore, provides an insight into the effects of noise on viral dynamics.
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Affiliation(s)
- Abhilasha Batra
- Department of Chemistry, Indian Institute of Science Education and Research (IISER) Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, India
| | - Rati Sharma
- Department of Chemistry, Indian Institute of Science Education and Research (IISER) Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, India
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Lanoiselée Y, Moutal N, Grebenkov DS. Diffusion-limited reactions in dynamic heterogeneous media. Nat Commun 2018; 9:4398. [PMID: 30353010 PMCID: PMC6199324 DOI: 10.1038/s41467-018-06610-6] [Citation(s) in RCA: 80] [Impact Index Per Article: 13.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2018] [Accepted: 09/11/2018] [Indexed: 12/02/2022] Open
Abstract
Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule. “Diffusing diffusivity” concept has been recently put forward to account for rapid structural rearrangements in soft matter and biological systems. Here the authors propose a general mathematical framework to compute the distribution of first-passage times in a dynamically heterogeneous medium.
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Affiliation(s)
- Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France
| | - Nicolas Moutal
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France.
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Korosak D, Cvikl B, Kramer J, Jecl R, Prapotnik A. Fractional calculus applied to the analysis of spectral electrical conductivity of clay-water system. JOURNAL OF CONTAMINANT HYDROLOGY 2007; 92:1-9. [PMID: 17175061 DOI: 10.1016/j.jconhyd.2006.11.005] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/20/2005] [Revised: 11/09/2006] [Accepted: 11/13/2006] [Indexed: 05/13/2023]
Abstract
The analysis of the low-frequency conductivity spectra of the clay-water mixtures is presented. The frequency dependence of the conductivity is shown to follow the power-law with the exponent n=0.67 before reaching the frequency-independent part. When scaled with the value of the frequency-independent part of the spectrum the conductivity spectra for samples at different water content values are shown to fit to a single master curve. It is argued that the observed conductivity dispersion is a consequence of the anomalously diffusing ions in the clay-water system. The fractional Langevin equation is then used to describe the stochastic dynamics of the single ion. The results indicate that the experimentally observed dielectric properties originate in anomalous ion transport in clay-water system characterized with time-dependent diffusion coefficient.
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Affiliation(s)
- Dean Korosak
- Applied Physics, Faculty of Civil Engineering, University of Maribor, Smetanova 17, SI-2000 Maribor, Slovenia.
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Demontis P, Suffritti GB. Fractional diffusion interpretation of simulated single-file systems in microporous materials. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051112. [PMID: 17279882 DOI: 10.1103/physreve.74.051112] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2006] [Indexed: 05/13/2023]
Abstract
The single-file diffusion of water in the straight channels of two different crystalline microporous aluminosilicates (zeolites bikitaite and Li-ABW) was studied by comparing the results of molecular dynamics computer simulations with the predictions of anomalous diffusion theory modeled by using fractional diffusion equations. At high coverage, the agreement is reasonably good, in particular for sufficiently large displacements and sufficiently long times. At low coverage, interesting phenomena appear in the simulation results, such as multimodal propagators, which could be interpreted on the basis of fractional Fokker-Planck equations. The results are discussed also in view of different theories that have been proposed to model the single-file diffusion process.
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Affiliation(s)
- Pierfranco Demontis
- Dipartimento di Chimica, Università di Sassari and Consorzio Interuniversitario Nazionale per la Scienza e Tecnologia dei Materiali (INSTM), Unità di ricerca di Sassari, Via Vienna, 2, I-07100 Sassari, Italy.
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Fa KS. Generalized Langevin equation with fractional derivative and long-time correlation function. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:061104. [PMID: 16906806 DOI: 10.1103/physreve.73.061104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2006] [Indexed: 05/11/2023]
Abstract
We investigate the motion of a particle governed by a generalized Langevin equation with fractional derivative, nonlocal dissipative force, and long-time correlation function. We derive general expressions for the variances with a linear external force. We also analyze their asymptotic behaviors for the power-law correlation function without external force.
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Affiliation(s)
- Kwok Sau Fa
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá PR, Brazil
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