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Balankin AS, Elizarraraz BE. Reply to "Comment on 'Hydrodynamics of fractal continuum flow' and 'Map of fluid flow in fractal porous medium into fractal continuum flow'". PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:057002. [PMID: 24329395 DOI: 10.1103/physreve.88.057002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/12/2013] [Indexed: 06/03/2023]
Abstract
The aim of this Reply is to elucidate the difference between the fractal continuum models used in the preceding Comment and the models of fractal continuum flow which were put forward in our previous articles [Phys. Rev. E 85, 025302(R) (2012); 85, 056314 (2012)]. In this way, some drawbacks of the former models are highlighted. Specifically, inconsistencies in the definitions of the fractal derivative, the Jacobian of transformation, the displacement vector, and angular momentum are revealed. The proper forms of the Reynolds' transport theorem and angular momentum principle for the fractal continuum are reaffirmed in a more illustrative manner. Consequently, we emphasize that in the absence of any internal angular momentum, body couples, and couple stresses, the Cauchy stress tensor in the fractal continuum should be symmetric. Furthermore, we stress that the approach based on the Cartesian product measured and used in the preceding Comment cannot be employed to study the path-connected fractals, such as a flow in a fractally permeable medium. Thus, all statements of our previous works remain unchallenged.
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Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," ESIME-Zacatenco, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
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52
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Jiang H, Cheng Y, Yuan L, An F, Jin K. A fractal theory based fractional diffusion model used for the fast desorption process of methane in coal. CHAOS (WOODBURY, N.Y.) 2013; 23:033111. [PMID: 24089947 DOI: 10.1063/1.4813597] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/02/2023]
Abstract
Based on the realistic property of the pore structure in coal, we established a fractal theory based Fractional diffusion model (FFDModel) by introducing the fractal dimension df to the Fick's classical model and changing the first-order partial differential equation about time into a υ fractional-order partial differential equation. Then, the solution of the FFDModel was obtained with separation variables technique. In order to verify the correctness of the solution, three coal samples with different rank from China were collected to do the methane desorption experiment of the fast desorption stage. The results indicate that the fractal dimension (df) of the coking coal is the lowest of the three coal ranks. By comparing the FFDModel with Fick's classical model, we can see that the FFDModel fits better with the three measured samples.
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Affiliation(s)
- Haina Jiang
- National Engineering Research Center of Coal Gas Control, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China
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53
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Wang S, Ma ZF, Yao HQ, Liu FS. Fractal model used for heterogeneously isothermal nth-order reaction. Chem Eng Sci 2013. [DOI: 10.1016/j.ces.2013.07.012] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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54
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Bianco F, Chibbaro S, Vergni D, Vulpiani A. Reaction spreading on percolating clusters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:062811. [PMID: 23848733 DOI: 10.1103/physreve.87.062811] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2013] [Indexed: 06/02/2023]
Abstract
Reaction-diffusion processes in two-dimensional percolating structures are investigated. Two different problems are addressed: reaction spreading on a percolating cluster and front propagation through a percolating channel. For reaction spreading, numerical data and analytical estimates show a power-law behavior of the reaction product as M(t)~t(d(l)), where d(l) is the connectivity dimension. In a percolating channel, a statistically stationary traveling wave develops. The speed and the width of the traveling wave are numerically computed. While the front speed is a low-fluctuating quantity and its behavior can be understood using a simple theoretical argument, the front width is a high-fluctuating quantity showing a power-law behavior as a function of the size of the channel.
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Affiliation(s)
- Federico Bianco
- Dipartimento di Fisica, Università La Sapienza, Piazzale Aldo Moro 2, I-00185 Roma, Italy
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55
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Abstract
A mathematical model is devised to study the diffusion of mRNA in the nucleus from the site of synthesis to a nuclear pore where it is exported to the cytoplasm. This study examines the role that nuclear structure can play in determining the kinetics of export by considering models in which elements of the nuclear skeleton and confinement by chromatin direct the mRNA movement. As a rule, a dense chromatin layer favours rapid export by reducing the effective volume for diffusion. However, it may also result in a heavy tail in the export time distribution because of the low mobility of molecules that accidentally find their way deep into the dense layer. An anisotropic solid-state transport system can also assist export. There exist both an optimal ratio of the anisotropy and an optimal depth of the solid-state transport layer that favour rapid export.
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Affiliation(s)
- M R Roussel
- Department of Chemistry and Biochemistry, University of Lethbridge, Lethbridge, Canada.
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56
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Cherstvy AG, Metzler R. Population splitting, trapping, and non-ergodicity in heterogeneous diffusion processes. Phys Chem Chem Phys 2013; 15:20220-35. [DOI: 10.1039/c3cp53056f] [Citation(s) in RCA: 97] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/19/2023]
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57
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Calcagni G. Diffusion in multiscale spacetimes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:012123. [PMID: 23410299 DOI: 10.1103/physreve.87.012123] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2012] [Indexed: 06/01/2023]
Abstract
We study diffusion processes in anomalous spacetimes regarded as models of quantum geometry. Several types of diffusion equation and their solutions are presented and the associated stochastic processes are identified. These results are partly based on the literature in probability and percolation theory but their physical interpretation here is different since they apply to quantum spacetime itself. The case of multiscale (in particular, multifractal) spacetimes is then considered through a number of examples, and the most general spectral-dimension profile of multifractional spaces is constructed.
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Affiliation(s)
- Gianluca Calcagni
- Albert Einstein Institute, Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476 Golm, Germany.
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58
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Balankin AS, Mena B, Martínez-González CL, Matamoros DM. Random walk in chemical space of Cantor dust as a paradigm of superdiffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:052101. [PMID: 23214828 DOI: 10.1103/physreve.86.052101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2012] [Revised: 07/31/2012] [Indexed: 06/01/2023]
Abstract
We point out that the chemical space of a totally disconnected Cantor dust K(n) [Symbol: see text E(n) is a compact metric space C(n) with the spectral dimension d(s) = d(ℓ) = n > D, where D and d(ℓ) = n are the fractal and chemical dimensions of K(n), respectively. Hence, we can define a random walk in the chemical space as a Markovian Gaussian process. The mapping of a random walk in C(n) into K(n) [Symbol: see text] E(n) defines the quenched Lévy flight on the Cantor dust with a single step duration independent of the step length. The equations, describing the superdiffusion and diffusion-reaction front propagation ruled by the local quenched Lévy flight on K(n) [Symbol: see text] E(n), are derived. The use of these equations to model superdiffusive phenomena, observed in some physical systems in which propagators decay faster than algebraically, is discussed.
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Affiliation(s)
- Alexander S Balankin
- Grupo Mecánica Fractal, Instituto Politécnico Nacional, México Distrito Federal 07738, Mexico
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59
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Burioni R, Chibbaro S, Vergni D, Vulpiani A. Reaction spreading on graphs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:055101. [PMID: 23214833 DOI: 10.1103/physreve.86.055101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/02/2012] [Indexed: 06/01/2023]
Abstract
We study reaction-diffusion processes on graphs through an extension of the standard reaction-diffusion equation starting from first principles. We focus on reaction spreading, i.e., on the time evolution of the reaction product M(t). At variance with pure diffusive processes, characterized by the spectral dimension d{s}, the important quantity for reaction spreading is found to be the connectivity dimension d{l}. Numerical data, in agreement with analytical estimates based on the features of n independent random walkers on the graph, show that M(t)∼t{d{l}}. In the case of Erdös-Renyi random graphs, the reaction product is characterized by an exponential growth M(t)e{αt} with α proportional to ln(k), where (k) is the average degree of the graph.
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Affiliation(s)
- Raffaella Burioni
- Dipartimento di Fisica and INFN, Università di Parma, Parco Area delle Scienze 7/A, 43100 Parma, Italy
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60
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Balankin AS, Elizarraraz BE. Map of fluid flow in fractal porous medium into fractal continuum flow. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:056314. [PMID: 23004869 DOI: 10.1103/physreve.85.056314] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2012] [Indexed: 06/01/2023]
Abstract
This paper is devoted to fractal continuum hydrodynamics and its application to model fluid flows in fractally permeable reservoirs. Hydrodynamics of fractal continuum flow is developed on the basis of a self-consistent model of fractal continuum employing vector local fractional differential operators allied with the Hausdorff derivative. The generalized forms of Green-Gauss and Kelvin-Stokes theorems for fractional calculus are proved. The Hausdorff material derivative is defined and the form of Reynolds transport theorem for fractal continuum flow is obtained. The fundamental conservation laws for a fractal continuum flow are established. The Stokes law and the analog of Darcy's law for fractal continuum flow are suggested. The pressure-transient equation accounting the fractal metric of fractal continuum flow is derived. The generalization of the pressure-transient equation accounting the fractal topology of fractal continuum flow is proposed. The mapping of fluid flow in a fractally permeable medium into a fractal continuum flow is discussed. It is stated that the spectral dimension of the fractal continuum flow d(s) is equal to its mass fractal dimension D, even when the spectral dimension of the fractally porous or fissured medium is less than D. A comparison of the fractal continuum flow approach with other models of fluid flow in fractally permeable media and the experimental field data for reservoir tests are provided.
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Affiliation(s)
- Alexander S Balankin
- Grupo "Mecánica Fractal," Instituto Politécnico Nacional, México, Distrito Federal, Mexico
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61
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Özarslan E, Shepherd TM, Koay CG, Blackband SJ, Basser PJ. Temporal scaling characteristics of diffusion as a new MRI contrast: findings in rat hippocampus. Neuroimage 2012; 60:1380-93. [PMID: 22306798 PMCID: PMC3303993 DOI: 10.1016/j.neuroimage.2012.01.105] [Citation(s) in RCA: 34] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2011] [Revised: 01/18/2012] [Accepted: 01/20/2012] [Indexed: 11/21/2022] Open
Abstract
Features of the diffusion-time dependence of the diffusion-weighted magnetic resonance imaging (MRI) signal provide a new contrast that could be altered by numerous biological processes and pathologies in tissue at microscopic length scales. An anomalous diffusion model, based on the theory of Brownian motion in fractal and disordered media, is used to characterize the temporal scaling (TS) characteristics of diffusion-related quantities, such as moments of the displacement and zero-displacement probabilities, in excised rat hippocampus specimens. To reduce the effect of noise in magnitude-valued MRI data, a novel numerical procedure was employed to yield accurate estimation of these quantities even when the signal falls below the noise floor. The power-law dependencies characterize the TS behavior in all regions of the rat hippocampus, providing unique information about its microscopic architecture. The relationship between the TS characteristics and diffusion anisotropy is investigated by examining the anisotropy of TS, and conversely, the TS of anisotropy. The findings suggest the robustness of the technique as well as the reproducibility of estimates. TS characteristics of the diffusion-weighted signals could be used as a new and useful marker of tissue microstructure.
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Affiliation(s)
- Evren Özarslan
- Section on Tissue Biophysics and Biomimetics, PPITS, NICHD, National Institutes of Health, Bethesda, MD 20892, USA.
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62
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Weigel AV, Ragi S, Reid ML, Chong EKP, Tamkun MM, Krapf D. Obstructed diffusion propagator analysis for single-particle tracking. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041924. [PMID: 22680515 DOI: 10.1103/physreve.85.041924] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/30/2011] [Indexed: 06/01/2023]
Abstract
We describe a method for the analysis of the distribution of displacements, i.e., the propagators, of single-particle tracking measurements for the case of obstructed subdiffusion in two-dimensional membranes. The propagator for the percolation cluster is compared with a two-component mobility model against Monte Carlo simulations. To account for diffusion in the presence of obstacle concentrations below the percolation threshold, a propagator that includes the transient motion in finite percolation clusters and hopping between obstacle-induced compartments is derived. Finally, these models are shown to be effective in the analysis of Kv2.1 channel diffusive measurements in the membrane of living mammalian cells.
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Affiliation(s)
- Aubrey V Weigel
- School of Biomedical Engineering, Colorado State University, Fort Collins, Colorado 80523, USA
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63
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Swiergiel J, Jadżyn J. Phase transition from liquid to disordered solid phase of cyclooctanone studied with dielectric methods. J Phys Chem B 2012; 116:3789-94. [PMID: 22394069 DOI: 10.1021/jp3007798] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Abstract
The dielectric studies performed for cyclooctanone in its static dielectric regime have shown that at the phase transition from liquid to disordered (plastic) solid phase of the compound, the following singular phenomena occur: (a) the static permittivity (ε(s)) exhibits a small increase at the transition, instead of that usually observed for polar liquids, a strong decrease of the permittivity at the transition to the crystalline solid phase; (b) temperature dependence of the permittivity, ε(s)(T), is practically the same in both phases, reflecting the same dipolar orientational entropy increment induced by the probing electric field; (c) a distinct jump occurs in the slope of the frequency dependence (in log-log scale) of the dielectric losses due to an ionic current, from the "ohmic" value -1 in the liquid phase to about -0.9 just after the transition to the plastic phase; (d) a similar jump is observed in the shape of the electric modulus spectra of cyclooctanone. The results clearly show the liquid-like freedom in the molecular dynamics in the plastic phase as well as similar intermolecular interactions in both phases of cyclooctanone. The differences in the dielectric losses frequency behavior may reflect the change in the ions dynamics: from the normal translational Brownian diffusion in the liquid phase to the subdiffusional dynamics in the plastic phase.
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Affiliation(s)
- Jolanta Swiergiel
- Institute of Molecular Physics, Polish Academy of Sciences, Poznań, Poland.
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64
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Srokowski T. Anomalous diffusion in systems driven by the stable Lévy noise with a finite noise relaxation time and inertia. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:021118. [PMID: 22463164 DOI: 10.1103/physreve.85.021118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2011] [Revised: 11/23/2011] [Indexed: 05/31/2023]
Abstract
Dynamical systems driven by a general Lévy stable noise are considered. The inertia is included and the noise, represented by a generalized Ornstein-Uhlenbeck process, has a finite relaxation time. A general linear problem (the additive noise) is solved: the resulting distribution converges with time to the distribution for the white-noise, massless case. Moreover, a multiplicative noise is discussed. It can make the distribution steeper and the variance, which is finite, depends sublinearly on time (subdiffusion). For a small mass, a white-noise limit corresponds to the Stratonovich interpretation. On the other hand, the distribution tails agree with the Itô interpretation if the inertia is very large. An escape time from the potential well is calculated.
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Affiliation(s)
- Tomasz Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland
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65
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Tateishi AA, Lenzi EK, da Silva LR, Ribeiro HV, Picoli S, Mendes RS. Different diffusive regimes, generalized Langevin and diffusion equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:011147. [PMID: 22400552 DOI: 10.1103/physreve.85.011147] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2011] [Revised: 10/28/2011] [Indexed: 05/31/2023]
Abstract
We investigate a generalized Langevin equation (GLE) in the presence of an additive noise characterized by the mixture of the usual white noise and an arbitrary one. This scenario lead us to a wide class of diffusive processes, in particular the ones whose noise correlation functions are governed by power laws, exponentials, and Mittag-Leffler functions. The results show the presence of different diffusive regimes related to the spreading of the system. In addition, we obtain a fractional diffusionlike equation from the GLE, confirming the results for long time.
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Affiliation(s)
- A A Tateishi
- Departamento de Física, Universidade Estadual de Maringá Avenida Colombo, 5790-87020-900 Maringá-PR, Brazil
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66
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Lenzi EK, Fernandes PRG, Petrucci T, Mukai H, Ribeiro HV. Anomalous-diffusion approach applied to the electrical response of water. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041128. [PMID: 22181108 DOI: 10.1103/physreve.84.041128] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/27/2011] [Indexed: 05/31/2023]
Abstract
We investigate the electrical response of Milli-Q deionized water by using a fractional diffusion equation of distributed order with the interfaces (i.e., the boundary conditions at the electrodes limiting the sample) governed by integrodifferential equations. We also consider that the positive and negative ions have the same mobility and that the electric potential profile across the sample satisfies Poisson's equation. In addition, the good agreement between the experimental data and this approach evidences the presence of anomalous diffusion due to the surface effects in this system.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá, Paraná, Brazil
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67
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Meyer B, Chevalier C, Voituriez R, Bénichou O. Universality classes of first-passage-time distribution in confined media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:051116. [PMID: 21728499 DOI: 10.1103/physreve.83.051116] [Citation(s) in RCA: 25] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/10/2010] [Indexed: 05/31/2023]
Abstract
We study the first-passage time (FPT) distribution to a target site for a random walker evolving in a bounded domain. We show that in the limit of large volume of the confining domain, this distribution falls into universality classes indexed by the walk dimension d(w) and the fractal dimension d(f) of the medium, which have been recently identified previously [Bénichou et al., Nat. Chem. 2, 472 (2010)]. We present in this paper a complete derivation of these universal distributions, discuss extensively the range of applicability of the results, and extend the method to continuous-time random walks. This analysis puts forward the importance of the geometry, and in particular the position of the starting point, in first-passage statistics. Analytical results are validated by numerical simulations, applied to various models of transport in disordered media, which illustrate the universality classes of the FPT distribution.
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Affiliation(s)
- B Meyer
- Laboratoire de Physique Théorique de la matière Condensée (UMR 7600), Université Paris 6, Paris, France
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68
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Ergodic and nonergodic processes coexist in the plasma membrane as observed by single-molecule tracking. Proc Natl Acad Sci U S A 2011; 108:6438-43. [PMID: 21464280 DOI: 10.1073/pnas.1016325108] [Citation(s) in RCA: 352] [Impact Index Per Article: 27.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2023] Open
Abstract
Diffusion in the plasma membrane of living cells is often found to display anomalous dynamics. However, the mechanism underlying this diffusion pattern remains highly controversial. Here, we study the physical mechanism underlying Kv2.1 potassium channel anomalous dynamics using single-molecule tracking. Our analysis includes both time series of individual trajectories and ensemble averages. We show that an ergodic and a nonergodic process coexist in the plasma membrane. The ergodic process resembles a fractal structure with its origin in macromolecular crowding in the cell membrane. The nonergodic process is found to be regulated by transient binding to the actin cytoskeleton and can be accurately modeled by a continuous-time random walk. When the cell is treated with drugs that inhibit actin polymerization, the diffusion pattern of Kv2.1 channels recovers ergodicity. However, the fractal structure that induces anomalous diffusion remains unaltered. These results have direct implications on the regulation of membrane receptor trafficking and signaling.
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69
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Roberts AP, Haynes CP. Electrostatic approximation of source-to-target mean first-passage times on networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:031113. [PMID: 21517460 DOI: 10.1103/physreve.83.031113] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2010] [Indexed: 05/30/2023]
Abstract
We show that the distance dependence of the source-to-target mean-first-passage time (MFPT) on a finite network with M links is approximately given by 2M times the target-to-shell resistance. For networks on which a random walker is transient the long-range MFPT is well approximated by the site-dependent resistance from the target to infinity. The result extends a recent scaling result for the MFPT to site-inhomogeneous lattices where the MFPT depends on the location of the source and targets and can be highly source-target asymmetric.
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Affiliation(s)
- Anthony P Roberts
- School of Mathematics and Physics, University of Queensland, Brisbane 4072, Australia
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70
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Bénichou O, Chevalier C, Meyer B, Voituriez R. Facilitated diffusion of proteins on chromatin. PHYSICAL REVIEW LETTERS 2011; 106:038102. [PMID: 21405302 DOI: 10.1103/physrevlett.106.038102] [Citation(s) in RCA: 46] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2010] [Indexed: 05/30/2023]
Abstract
We present a theoretical model of facilitated diffusion of proteins in the cell nucleus. This model, which takes into account the successive binding and unbinding events of proteins to DNA, relies on a fractal description of the chromatin which has been recently evidenced experimentally. Facilitated diffusion is shown quantitatively to be favorable for a fast localization of a target locus by a transcription factor and even to enable the minimization of the search time by tuning the affinity of the transcription factor with DNA. This study shows the robustness of the facilitated diffusion mechanism, invoked so far only for linear conformations of DNA.
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Affiliation(s)
- O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée CNRS-UPMC, 4 Place Jussieu, 75255 Paris Cedex, France
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71
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Calvo-Muñoz EM, Selvan ME, Xiong R, Ojha M, Keffer DJ, Nicholson DM, Egami T. Applications of a general random-walk theory for confined diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:011120. [PMID: 21405674 DOI: 10.1103/physreve.83.011120] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2010] [Indexed: 05/30/2023]
Abstract
A general random walk theory for diffusion in the presence of nanoscale confinement is developed and applied. The random-walk theory contains two parameters describing confinement: a cage size and a cage-to-cage hopping probability. The theory captures the correct nonlinear dependence of the mean square displacement (MSD) on observation time for intermediate times. Because of its simplicity, the theory also requires modest computational requirements and is thus able to simulate systems with very low diffusivities for sufficiently long time to reach the infinite-time-limit regime where the Einstein relation can be used to extract the self-diffusivity. The theory is applied to three practical cases in which the degree of order in confinement varies. The three systems include diffusion of (i) polyatomic molecules in metal organic frameworks, (ii) water in proton exchange membranes, and (iii) liquid and glassy iron. For all three cases, the comparison between theory and the results of molecular dynamics (MD) simulations indicates that the theory can describe the observed diffusion behavior with a small fraction of the computational expense. The confined-random-walk theory fit to the MSDs of very short MD simulations is capable of accurately reproducing the MSDs of much longer MD simulations. Furthermore, the values of the parameter for cage size correspond to the physical dimensions of the systems and the cage-to-cage hopping probability corresponds to the activation barrier for diffusion, indicating that the two parameters in the theory are not simply fitted values but correspond to real properties of the physical system.
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Affiliation(s)
- Elisa M Calvo-Muñoz
- Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA
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72
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Tejedor V, Bénichou O, Voituriez R, Jungmann R, Simmel F, Selhuber-Unkel C, Oddershede LB, Metzler R. Quantitative analysis of single particle trajectories: mean maximal excursion method. Biophys J 2010; 98:1364-72. [PMID: 20371337 DOI: 10.1016/j.bpj.2009.12.4282] [Citation(s) in RCA: 167] [Impact Index Per Article: 11.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2009] [Revised: 11/19/2009] [Accepted: 12/02/2009] [Indexed: 11/26/2022] Open
Abstract
An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss what we believe are new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions (MMEs), i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean-squared displacement we show that the MME analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the MME method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.
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Affiliation(s)
- Vincent Tejedor
- Physics Department, Technical University of Munich, Garching, Germany
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73
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Kimmich R, Gneiting T, Kotitschke K, Schnur G. Fluctuations, exchange processes, and water diffusion in aqueous protein systems: A study of bovine serum albumin by diverse NMR techniques. Biophys J 2010; 58:1183-97. [PMID: 19431772 DOI: 10.1016/s0006-3495(90)82459-0] [Citation(s) in RCA: 62] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022] Open
Abstract
Experimental frequency, concentration, and temperature dependences of the deuteron relaxation times T(1) and T(2) of D(2)O solutions of bovine serum albumin are reported and theoretically described in a closed form without formal parameters. Crucial processes of the theoretical concept are material exchange, translational diffusion of water molecules on the rugged surfaces of proteins, and tumbling of the macromolecules. It is also concluded that, apart from averaging of the relaxation rates in the diverse deuteron phases, material exchange contributes to transverse relaxation by exchange modulation of the Larmor frequency. The rate limiting factor of macromolecular tumbling is determined by the free water content. In a certain analogy to the classical free-volume theory, a "free-water-volume theory" is presented. There are two characteristic water mass fractions indicating the saturation of the hydration shells (C(s) approximately 0.3) and the onset of protein tumbling (C(0) approximately 0.6). The existence of the translational degrees of freedom of water molecules in the hydration shells has been verified by direct measurement of the diffusion coefficient using an NMR field-gradient technique. The concentration and temperature dependences show phenomena indicating a percolation transition of clusters of free water. The threshold water content was found to be C(c) (w) approximately 0.43.
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Affiliation(s)
- R Kimmich
- Sektion Kernresonanzspektroskopie, Universität Ulm, D-7900 Ulm, Federal Republic of Germany
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74
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Haugh JM. Analysis of reaction-diffusion systems with anomalous subdiffusion. Biophys J 2009; 97:435-42. [PMID: 19619457 DOI: 10.1016/j.bpj.2009.05.014] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2009] [Revised: 05/07/2009] [Accepted: 05/08/2009] [Indexed: 12/21/2022] Open
Abstract
Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.
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Affiliation(s)
- Jason M Haugh
- Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, North Carolina, USA.
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75
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Beatus T, Tlusty T, Bar-Ziv R. Burgers shock waves and sound in a 2D microfluidic droplets ensemble. PHYSICAL REVIEW LETTERS 2009; 103:114502. [PMID: 19792377 DOI: 10.1103/physrevlett.103.114502] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/06/2009] [Indexed: 05/28/2023]
Abstract
We investigate the collective motion of a two-dimensional disordered ensemble of droplets in a microfluidic channel far from equilibrium and at Reynolds number approximately 10(-4). The ensemble carries ultraslow shock waves and sound, propagating at approximately 100 microm s(-1) and superposed on diffusive droplets motion. These modes are induced by long-range hydrodynamic dipolar interactions between droplets, the result of the symmetry breaking flow. The modes obey the Burgers equation due to a local coupling between droplets velocity and number density. This stems from a singular effect of the channel sidewall boundaries upon summation of the hydrodynamic interaction in two dimensions.
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Affiliation(s)
- Tsevi Beatus
- Department of Materials and Interfaces, The Weizmann Institute of Science, Rehovot, Israel
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76
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Lenzi EK, Evangelista LR, Lenzi MK, da Silva LR. Fokker-Planck equation in a wedge domain: anomalous diffusion and survival probability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021131. [PMID: 19792101 DOI: 10.1103/physreve.80.021131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2009] [Revised: 06/27/2009] [Indexed: 05/28/2023]
Abstract
We obtain exact solutions and the survival probability for a Fokker-Planck equation subjected to the two-dimensional wedge domain. We consider a spatial dependence in the diffusion coefficient and the presence of external forces. The results show an anomalous spreading of the solution and, consequently, a nonusual behavior of the survival probability which can be connected to anomalous diffusion.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, 87020-900 Maringá, PR, Brazil
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77
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Haynes CP, Roberts AP. Generalization of the fractal Einstein law relating conduction and diffusion on networks. PHYSICAL REVIEW LETTERS 2009; 103:020601. [PMID: 19659194 DOI: 10.1103/physrevlett.103.020601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2009] [Indexed: 05/28/2023]
Abstract
We settle a long-standing controversy about the exactness of the fractal Einstein and Alexander-Orbach laws by showing that the properties of a class of fractal trees violate both laws. A new formula is derived which unifies the two classical results by showing that if one holds, then so must the other, and resolves a puzzling discrepancy in the properties of Eden trees and diffusion-limited aggregates. We also conjecture that the result holds for networks which have no fractal dimension. The failure of the classical laws is attributed to anisotropic exploration of the network by a random walker. The occurrence of this newly revealed behavior means that the conventional laws must be checked if they, or numerous results which depend on them, are to be applied accurately.
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Affiliation(s)
- Christophe P Haynes
- School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia
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78
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Affan H, Friedrich R, Eule S. Anomalous diffusion in a field of randomly distributed scatterers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:011137. [PMID: 19658683 DOI: 10.1103/physreve.80.011137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/29/2009] [Indexed: 05/28/2023]
Abstract
We consider the motion of particles which are scattered by randomly distributed obstacles. In between scattering events the particles move uniformly. The governing master equation is obtained by mapping the problem onto a master equation which was previously devised for the description of anomalous diffusion of particles with inertia [R. Friedrich, Phys. Rev. Lett. 96, 230601 (2006)]. We show that for a scale-free distance distribution of scatterers a time-fractional master equation arises. The corresponding diffusion equation which exhibits a power-law diffusion coefficient is solved in d dimensions via the method of subordination.
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Affiliation(s)
- H Affan
- Institute for Theoretical Physics, University of Münster, Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany
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79
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Loverdo C, Bénichou O, Voituriez R, Biebricher A, Bonnet I, Desbiolles P. Quantifying hopping and jumping in facilitated diffusion of DNA-binding proteins. PHYSICAL REVIEW LETTERS 2009; 102:188101. [PMID: 19518914 DOI: 10.1103/physrevlett.102.188101] [Citation(s) in RCA: 72] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2008] [Indexed: 05/27/2023]
Abstract
Facilitated diffusion of DNA-binding proteins is known to speed up target site location by combining three dimensional excursions and linear diffusion along the DNA. Here we explicitly calculate the distribution of the relocation lengths of such 3D excursions, and we quantify the short-range correlated excursions, also called hops, and the long-range uncorrelated jumps. Our results substantiate recent single-molecule experiments that reported sliding and 3D excursions of the restriction enzyme EcoRV on elongated DNA molecules. We extend our analysis to the case of anomalous 3D diffusion, likely to occur in a crowded cellular medium.
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Affiliation(s)
- C Loverdo
- Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex France
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80
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Wang S, Ma ZF, Yao HQ. Fractal diffusion model used for diffusion in porous material within limited volume of stiff container. Chem Eng Sci 2009. [DOI: 10.1016/j.ces.2008.11.016] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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81
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Brault P, Josserand C, Bauchire JM, Caillard A, Charles C, Boswell RW. Anomalous diffusion mediated by atom deposition into a porous substrate. PHYSICAL REVIEW LETTERS 2009; 102:045901. [PMID: 19257443 DOI: 10.1103/physrevlett.102.045901] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2008] [Indexed: 05/27/2023]
Abstract
Constant flux atom deposition into a porous medium is shown to generate a dense overlayer and a diffusion profile. Scaling analysis shows that the overlayer acts as a dynamic control for atomic diffusion in the porous substrate. This is modeled by generalizing the porous diffusion equation with a time-dependent diffusion coefficient equivalent to a nonlinear rescaling of time.
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Affiliation(s)
- Pascal Brault
- Groupe de Recherches sur l'Energétique des Milieux Ionisés, UMR6606, Université d'Orléans, France
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82
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Bénichou O, Meyer B, Tejedor V, Voituriez R. Zero constant formula for first-passage observables in bounded domains. PHYSICAL REVIEW LETTERS 2008; 101:130601. [PMID: 18851430 DOI: 10.1103/physrevlett.101.130601] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2008] [Revised: 08/22/2008] [Indexed: 05/26/2023]
Abstract
In this Letter, we develop an analytical approach which provides an explicit determination of mean first-passage times (MFPTs) for random walks in bounded domains for a wide class of transport processes. In particular, we derive for the first time explicit expressions of MFPTs for emblematic models of transport in complex media, such as diffusion on deterministic and random fractals. This approach relies on a scale-invariance hypothesis and a large volume expansion of the MFPT, which actually proves to be very accurate even for small system sizes as shown by numerical simulations. This explicit determination of MFPTs can be straightforwardly generalized to further useful first-passage observables such as occupation times and splitting probabilities.
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Affiliation(s)
- O Bénichou
- Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), Université Pierre et Marie Curie, 4 Place Jussieu, 75255 Paris Cedex, France
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83
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Arkhincheev VE. Random walks on the Comb model and its generalizations. CHAOS (WOODBURY, N.Y.) 2007; 17:043102. [PMID: 18163766 DOI: 10.1063/1.2772179] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/25/2023]
Abstract
Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to "effective reducing" of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed.
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Affiliation(s)
- V E Arkhincheev
- Buryat Science Center, Siberian Branch of Russian Academy of Sciences, 670047, str. Sakhyanovoi 6, Ulan-Ude, Russia
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84
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Srokowski T. Non-Markovian Lévy diffusion in nonhomogeneous media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:051105. [PMID: 17677020 DOI: 10.1103/physreve.75.051105] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2006] [Revised: 02/02/2007] [Indexed: 05/16/2023]
Abstract
We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The resulting distributions have the form of the Lévy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the system.
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Affiliation(s)
- T Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, Kraków, Poland
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85
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Anomalous Stochastic Processes in the Fractional Dynamics Framework: Fokker-Planck Equation, Dispersive Transport, and Non-Exponential Relaxation. ADVANCES IN CHEMICAL PHYSICS 2007. [DOI: 10.1002/9780470141762.ch3] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/13/2023]
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86
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Sellers S, Barker JA. Generalized diffusion equation for anisotropic anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:061103. [PMID: 17280034 DOI: 10.1103/physreve.74.061103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2006] [Indexed: 05/13/2023]
Abstract
Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.
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Affiliation(s)
- S Sellers
- Mechanical and Aerospace Engineering, Washington University, St Louis, Missouri 63119, USA
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87
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Kiessling V, Crane JM, Tamm LK. Transbilayer effects of raft-like lipid domains in asymmetric planar bilayers measured by single molecule tracking. Biophys J 2006; 91:3313-26. [PMID: 16905614 PMCID: PMC1614489 DOI: 10.1529/biophysj.106.091421] [Citation(s) in RCA: 180] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/18/2023] Open
Abstract
Cell membranes have complex lipid compositions, including an asymmetric distribution of phospholipids between the opposing leaflets of the bilayer. Although it has been demonstrated that the lipid composition of the outer leaflet of the plasma membrane is sufficient for the formation of raft-like liquid-ordered (l(o)) phase domains, the influence that such domains may have on the lipids and proteins of the inner leaflet remains unknown. We used tethered polymer supports and a combined Langmuir-Blodgett/vesicle fusion (LB/VF) technique to build asymmetric planar bilayers that mimic plasma membrane asymmetry in many ways. We show that directly supported LB monolayers containing cholesterol-rich l(o) phases are inherently unstable when exposed to water or vesicle suspensions. However, tethering the LB monolayer to the solid support with the lipid-anchored polymer 1,2-dimyristoyl phophatidylethanolamine-N-[poly(ethylene glycol)-triethoxysilane] significantly improves stability and allows for the formation of complex planar-supported bilayers that retain >90% asymmetry for 1-2 h. We developed a single molecule tracking (SPT) system for the study of lipid diffusion in asymmetric bilayers with coexisting liquid phases. SPT allowed us to study in detail the diffusion of individual lipids inside, outside, or directly opposed to l(o) phase domains. We show here that l(o) phase domains in one monolayer of an asymmetric bilayer do not induce the formation of domains in the opposite leaflet when this leaflet is composed of palmitoyl-oleoyl phosphatidylcholine and cholesterol but do induce domains when this leaflet is composed of porcine brain phosphatidylcholine, phosphatidylethanolamine, phosphatidylserine, and cholesterol. The diffusion of lipids is similar in l(o) and liquid-disordered phase domains and is not affected by transbilayer coupling, indicating that lateral and transverse lipid interactions that give rise to the domain structure are weak in the biological lipid mixtures that were employed in this work.
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Affiliation(s)
- Volker Kiessling
- Department of Molecular Physiology and Biological Physics, University of Virginia, Charlottesville, 22908-0736, USA
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88
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Srokowski T, Kamińska A. Diffusion equations for a Markovian jumping process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:021103. [PMID: 17025389 DOI: 10.1103/physreve.74.021103] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2006] [Indexed: 05/12/2023]
Abstract
We consider a Markovian jumping process which is defined in terms of the jump-size distribution and the waiting-time distribution with a position-dependent frequency, in the diffusion limit. We assume the power-law form for the frequency. For small steps, we derive the Fokker-Planck equation and show the presence of the normal diffusion, subdiffusion, and superdiffusion. For the Lévy distribution of the step size, we construct a fractional equation, which possesses a variable coefficient, and solve it in the diffusion limit. Then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance. We also solve the master equation numerically and demonstrate that there are deviations from the Lévy stable distribution for large wave numbers.
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Affiliation(s)
- T Srokowski
- Institute of Nuclear Physics, Polish Academy of Sciences, PL-31-342 Kraków, Poland
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89
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Kuriakose J, Ghosh A, Ravi Kumar V, Kulkarni BD. Isometric graphing and multidimensional scaling for reaction-diffusion modeling on regular and fractal surfaces with spatiotemporal pattern recognition. J Chem Phys 2006; 120:5432-43. [PMID: 15267417 DOI: 10.1063/1.1647046] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Heterogeneous surface reactions exhibiting complex spatiotemporal dynamics and patterns can be studied as processes involving reaction-diffusion mechanisms. In many realistic situations, the surface has fractal characteristics. This situation is studied by isometric graphing and multidimensional scaling (IGMDS) of fractal surfaces for extracting geodesic distances (i.e., shortest scaled distances that obtain edges of neighboring surface nodes and their interconnections) and the results obtained used to model effects of surface diffusion with nonlinear reactions. Further analysis of evolved spatiotemporal patterns may be carried out by IGMDS because high-dimensional snapshot data can be efficiently projected to a transformed subspace with reduced dimensions. Validation of the IGMDS methodology is carried out by comparing results with reduction capabilities of conventional principal component analysis for simple situations of reaction and diffusion on surfaces. The usefulness of the IGMDS methodology is shown for analysis of complex patterns formed on both regular and fractal surfaces, and using generic nonlinear reaction-diffusion systems following FitzHugh Nagumo and cubic reaction kinetics. The studies of these systems with nonlinear kinetics and noise show that effects of surface disorder due to fractality can become very relevant. The relevance is shown by studying properties of dynamical invariants in IGMDS component space, viz., the Lyapunov exponents and the KS entropy for interesting situations of spiral formation and turbulent patterns.
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Affiliation(s)
- Jainy Kuriakose
- Chemical Engineering Division, National Chemical Laboratory, Pune 411008, India
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90
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Xu M, Tan W. Intermediate processes and critical phenomena: Theory, method and progress of fractional operators and their applications to modern mechanics. ACTA ACUST UNITED AC 2006. [DOI: 10.1007/s11433-006-0257-2] [Citation(s) in RCA: 68] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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91
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Denisov SI, Vitrenko AN, Horsthemke W, Hänggi P. Anomalous diffusion for overdamped particles driven by cross-correlated white noise sources. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:036120. [PMID: 16605611 DOI: 10.1103/physreve.73.036120] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2005] [Indexed: 05/08/2023]
Abstract
We study the statistical properties of overdamped particles driven by two cross-correlated multiplicative Gaussian white noises in a time-dependent environment. Using the Langevin and Fokker-Planck approaches, we derive the exact probability distribution function for the particle positions, calculate its moments, and find their corresponding long-time, asymptotic behaviors. The generally anomalous diffusive regimes of the particles are classified, and their dependence on the friction coefficient and the characteristics of the noises is analyzed in detail. The asymptotic predictions are confirmed by exact solutions for two examples.
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Affiliation(s)
- S I Denisov
- Institut für Physik, Universität Augsburg, Universitätsstrasse 1, D-86135 Augsburg, Germany
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92
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Assis PC, de Souza RP, da Silva PC, da Silva LR, Lucena LS, Lenzi EK. Non-Markovian Fokker-Planck equation: solutions and first passage time distribution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:032101. [PMID: 16605577 DOI: 10.1103/physreve.73.032101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2005] [Indexed: 05/08/2023]
Abstract
We investigate the solutions and first passage time distribution for an anomalous diffusion process governed by a generalized non-Markovian Fokker-Planck equation. In our analysis, we also consider the presence of external forces and absorbent (source) terms. In addition, we show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.
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Affiliation(s)
- P C Assis
- International Center for Complex Systems and Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970 Natal-RN, Brazil
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93
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A review of anomalous diffusion phenomena at fractal interface for diffusion-controlled and non-diffusion-controlled transfer processes. J Solid State Electrochem 2006. [DOI: 10.1007/s10008-005-0084-9] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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94
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Fractional Diffusion Equation Approach to the Anomalous Diffusion on Fractal Lattices. B KOREAN CHEM SOC 2005. [DOI: 10.5012/bkcs.2005.26.11.1723] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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95
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Sanda F, Mukamel S. Multipoint correlation functions for continuous-time random walk models of anomalous diffusion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:031108. [PMID: 16241412 DOI: 10.1103/physreve.72.031108] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2005] [Indexed: 05/05/2023]
Abstract
Recursive relations are developed for computing the multipoint correlation functions of a particle undergoing a biased continuous-time random walk (CTRW) in an external potential. Two- and three-point correlation functions are calculated for waiting-time distributions with an anomalous power-law profile t(-alpha-1), 0 < alpha < 1, on intermediate time scales with a crossover to an exponential long time decay. Comparison of the CTRW with the Brownian harmonic oscillator model (Gaussian process) illustrates how higher-order correlation functions may be used to distinguish between dynamical models that have the same two-point correlation function.
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Affiliation(s)
- Frantisek Sanda
- Department of Chemistry, University of California, Irvine, California 92697, USA
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96
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Pedron IT, Mendes RS, Buratta TJ, Malacarne LC, Lenzi EK. Logarithmic diffusion and porous media equations: a unified description. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:031106. [PMID: 16241410 DOI: 10.1103/physreve.72.031106] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2005] [Indexed: 05/04/2023]
Abstract
In this work we present the logarithmic diffusion equation as a limit case when the index that characterizes a nonlinear Fokker-Planck equation, in its diffusive term, goes to zero. A linear drift and a source term are considered in this equation. Its solution has a Lorentzian form, consequently this equation characterizes a superdiffusion like a Lévy kind. In addition an equation that unifies the porous media and the logarithmic diffusion equations, including a generalized diffusion equation in fractal dimension, is obtained. This unification is performed in the nonextensive thermostatistics context and increases the possibilities about the description of anomalous diffusive processes.
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Affiliation(s)
- I T Pedron
- Universidade Estadual do Oeste do Paraná, Rua Pernambuco, 1777, 85960-000, Marechal Cândido Rondon, Paraná, Brazil
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97
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Fa KS. Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:020101. [PMID: 16196534 DOI: 10.1103/physreve.72.020101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2005] [Revised: 05/31/2005] [Indexed: 05/04/2023]
Abstract
We consider the Langevin equation with a multiplicative noise term that depends on time and space. The corresponding Fokker-Planck equation in the Stratonovich approach is investigated. Its exact solution is obtained for an arbitrary multiplicative noise term given by g (x,t) =D (x) T (t) , and the behaviors of probability distributions, for some specific functions of D (x) , are analyzed. We show that the asymptotic shape of the random-walk model and power-law decay obtained from other approaches can be reproduced from our solutions, by employing two simple functions for g (x,t) . In particular, for D(x) approximately /x/(-theta/2) , the physical solutions for the probability distribution in the Ito, Stratonovich, and postpoint discretization approaches can be obtained and analyzed.
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Affiliation(s)
- Kwok Sau Fa
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo, 5790, 87020-900, Maringá-Paraná, Brazil
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98
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Rauch EM, Bar-Yam Y. Estimating the total genetic diversity of a spatial field population from a sample and implications of its dependence on habitat area. Proc Natl Acad Sci U S A 2005; 102:9826-9. [PMID: 15998741 PMCID: PMC1174974 DOI: 10.1073/pnas.0408471102] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
The total genetic diversity of a species is a key factor in its persistence and conservation. Because realistic sample sizes are far smaller than the total population, it is impractical to exhaustively characterize diversity of most populations. Here, we demonstrate the possibility of calculating the genetic diversity of a spatial population from a sample using genealogical models. We trace the history of a population by simulating the locations of the ancestors of a particular sample of the population backwards in time. We use this method to estimate the genetic diversity of the global population of Pseudomonas bacteria. The same results are obtained whether using a global sample or a subsample restricted to a particular geographic region (California). The results are also validated by comparing additional predictions of the model to the data. Furthermore, we use these results to show that the level of genetic diversity in a population depends strongly on the size of its habitat, much more strongly than does biodiversity as measured by the number of species. The strong dependence of diversity on habitat area has significant implications for conservation strategies.
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Affiliation(s)
- Erik M Rauch
- Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.
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99
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Lenzi EK, Mendes RS, Andrade JS, da Silva LR, Lucena LS. N-dimensional fractional diffusion equation and Green function approach: spatially dependent diffusion coefficient and external force. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:052101. [PMID: 16089577 DOI: 10.1103/physreve.71.052101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2004] [Revised: 12/08/2004] [Indexed: 05/03/2023]
Abstract
We investigate an N-dimensional fractional diffusion equation with radial symmetry by using the Green function approach. We consider, in our analysis, the spatial dependence on the diffusion coefficient and the presence of an external force. In particular, we employ boundary conditions in a finite interval and after we extend it to a semi-infinite interval. We also show that a rich class of diffusive processes, including normal and anomalous ones, can be obtained from the solutions found here.
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Affiliation(s)
- E K Lenzi
- Departamento de Física, Universidade Estadual de Maringá, Avenida Colombo 5790, 87020-900 Maringá PR, Brazil
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100
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Fa KS, Lenzi EK. Anomalous diffusion, solutions, and first passage time: Influence of diffusion coefficient. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:012101. [PMID: 15697637 DOI: 10.1103/physreve.71.012101] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/20/2004] [Indexed: 05/24/2023]
Abstract
We investigate the solutions and the first passage time for anomalous diffusion processes governed by the usual diffusion equation. We consider a space- and time-dependent diffusion coefficient and the presence of absorbing boundaries. We obtain analytical results for the probability distribution and the first passage time distribution for finite and semi-infinite intervals. In addition, we compare our results for the first passage time distribution with the one obtained by the usual diffusion equation with constant diffusion coefficient.
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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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