Anomalous diffusion in branched elliptical structure

SuleimanKheder; ZhangXuelan; WangErhui; LiuShengna; ZhengLiancun

doi:10.1088/1674-1056/ac5c39

Anomalous diffusion in branched elliptical structure

doi: 10.1088/1674-1056/ac5c39

1.
School of Energy and Environmental Engineering, University of Science and Technology Beijing,Beijing 100083,China
2.
School of Mathematics and Physics, University of Science and Technology Beijing,Beijing 100083,China

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出版历程
- 收稿日期: 2022-01-16
- 网络出版日期: 2023-05-16
- 刊出日期: 2023-01-01

摘要

- anomalous diffusion /
- Fokker-Planck equation /
- branched elliptical structure
Abstract: Diffusion in narrow curved channels with dead-ends as in extracellular space in the biological tissues, e.g., brain, tumors, muscles, etc. is a geometrically induced complex diffusion and is relevant to different kinds of biological, physical, and chemical systems. In this paper, we study the effects of geometry and confinement on the diffusion process in an elliptical comb-like structure and analyze its statistical properties. The ellipse domain whose boundary has the polar equation $ρ (θ) = \frac{b}{\sqrt{1 - e^{2} \cos^{2} θ}}$ with 0 < e < 1, θ ∈ [0,2 π], and b as a constant, can be obtained through stretched radius r such that Υ = r ρ ( θ) with r ∈ [0,1]. We suppose that, for fixed radius r = R, an elliptical motion takes place and is interspersed with a radial motion inward and outward of the ellipse. The probability distribution function (PDF) in the structure and the marginal PDF and mean square displacement (MSD) along the backbone are obtained numerically. The results show that a transient sub-diffusion behavior dominates the process for a time followed by a saturating state. The sub-diffusion regime and saturation threshold are affected by the length of the elliptical channel lateral branch and its curvature.

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1 Schematic picture of an elliptical comb-like structure, where the radii are continuously distributed over the ellipse of radius R = ρ( θ) with θ ∈ [0,2 π].

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2 The comparison of the exact solution via Eq.(16) with numerical solution for b = 1, e = 0.5, $D = \tilde{D} = 1$ , τ = 10 ⁻⁴, h _r = 10 ⁻², and h _θ = π/20 at T = 1.

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3 Time evolution of PDF in the backbone of classical comb, along x direction, for $D = \tilde{D} = 1$ , L = 5, and (a) R = 1.5, (b) R = 6.

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4 Time evolution of PDF in the backbone of classical comb, along x direction, for $D = \tilde{D} = 1$ , R = 3, and (a) L = 5, (b) L = 10.

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5 Time evolution of probability distribution in the structure for different value of the parameter b at T = 1: (a) $D = \tilde{D} = 1$ , e = 0.5, and b = 2, (b) $D = \tilde{D} = 1$ , e = 0.5, and b = 3, (c) $D = \tilde{D} = 1$ , e = 0.5, and b = 4, (d) $D = \tilde{D} = 1$ , e = 0.5, and b = 5.

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6 Time evolution of PDF in the backbone of elliptical comb, along direction θ, for $D = \tilde{D} = 1$ , e = 0.5, and (a) b = 2, (b) b = 8.

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7 Time evolution of PDF in the backbone of elliptical comb, along direction θ, for $D = \tilde{D} = 1$ , b = 4, and (a) e = 0.3, (b) e = 0.8.

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8 Temporal evolution of MSD in the backbone of classical comb, along x direction for $D = \tilde{D} = 1$ , R = {1.5, 3, 6}, and (a) L = 5, (b) L = 10.

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9 Temporal evolution of MSD in the backbone of classical comb, along x direction for $D = \tilde{D} = 1$ , L = {5, 7.5, 10}, and (a) R = 1.5, (b) R = 6.

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10 Temporal evolution of MSD in the backbone of classical comb, along θ direction for $D = \tilde{D} = 1$ , b = {2, 4, 5, 6, 8, 10}, and (a) e = 0, (b) e = 0.2, (c) e = 0.5, (d) e = 0.8.

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11 MSD along θ direction for short time for $D = \tilde{D} = 1$ , e = {0, 0.3, 0.5, 0.8}, and (a) b = 4, (b) b = 7.

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12 MSD along elliptical channel without constrained geometries for short time for $D = \tilde{D} = 1$ , and (a) e = 0, 0.5, 0.8, and b = 10, (b) e = 0.5 and b = 3, 6, 8.

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参考文献(83)

[1]	Stein W D 1986 Transport and diffusion across cell membranes (Academic Press)
[2]	Bressloff P C 2014 Stochastic Processes in Cell Biology (Springer)
[3]	Sibatov R T, Uchaikin V V 2015 J. Comput. Phys. 293 409 10.1016/j.jcp.2015.01.022 doi: 10.1016/j.jcp.2015.01.022
[4]	Mendez V, Fedotov S, Horsthemke W 2010 Reaction-Transport Systems (Springer)
[5]	Viswanathan G M, Luz M, Raposo E P, Stanley H E 2011 The Physics of Foraging (Cambridge University Press)
[6]	Dix J A, Verkman A S 2008 Annu. Rev. Biophys. 37 247 10.1146/annurev.biophys.37.032807.125824 doi: 10.1146/annurev.biophys.37.032807.125824
[7]	Weiss M, Hashimoto H, Nilsson T 2003 Biophys. J. 84 4043 10.1016/S0006-3495(03)75130-3 doi: 10.1016/S0006-3495(03)75130-3
[8]	Höfling F, Franosch T 2013 Rep. Prog. Phys. 76 46602 10.1088/0034-4885/76/4/046602 doi: 10.1088/0034-4885/76/4/046602
[9]	Sun H G, Sheng H, Chen Y Q, Chen W, Yu Z B 2013 Chin. Phys. Lett. 30 46601 10.1088/0256-307X/30/4/046601 doi: 10.1088/0256-307X/30/4/046601
[10]	Wu W X, Zheng Z G, Song Y L, Han Y R, Sun Z C, Li C P 2020 Chin. Phys.B 29 90503 10.1088/1674-1056/ab99b7 doi: 10.1088/1674-1056/ab99b7
[11]	Wang B, Kuo J, Bae S C, Granick S 2015 Nat. Mater. 11 481 10.1038/nmat3308 doi: 10.1038/nmat3308
[12]	Metzler R, Jeon J H, Cherstvy A G 2016 Biochim. Biophys. Acta - Biomembr. 1858 2451 10.1016/j.bbamem.2016.01.022 doi: 10.1016/j.bbamem.2016.01.022
[13]	Metzler R 2020 Eur. Phys. J. Spec. Top. 229 711 10.1140/epjst/e2020-900210-x doi: 10.1140/epjst/e2020-900210-x
[14]	Metzler R 2017 Biophys. J. 112 413 10.1016/j.bpj.2016.12.019 doi: 10.1016/j.bpj.2016.12.019
[15]	Jeon J H, Javanainen M, Martinez-Seara H, Metzler R, Vattulainen I 2016 Phys. Rev.X 6 21006
[16]	Szymanski J, Weiss M 2009 Phys. Rev. Lett. 103 38102 10.1103/PhysRevLett.103.038102 doi: 10.1103/PhysRevLett.103.038102
[17]	Gupta S, Mel J, Perera R M, Zolnierczuk P, Schneider G J 2018 J. Phys. Chem. Lett. 9 2956 10.1021/acs.jpclett.8b01008 doi: 10.1021/acs.jpclett.8b01008
[18]	Tabei S, Burov S, Kim H Y, Kuznetsov A, Huynh T, Jureller J, Philipson L H, Dinner A R, Scherer N F 2013 Proc. Natl. Acad. Sci. USA 110 4911 10.1073/pnas.1221962110 doi: 10.1073/pnas.1221962110
[19]	Xiao F, Hrabe J, Hrabetova S 2015 Biophys. J. 108 2384 10.1016/j.bpj.2015.02.034 doi: 10.1016/j.bpj.2015.02.034
[20]	Goeppert N, Goldscheider N, Berkowitz B 2020 Water Res. 178 115755 10.1016/j.watres.2020.115755 doi: 10.1016/j.watres.2020.115755
[21]	Caspi A, Granek R, Elbaum M 2000 Phys. Rev. Lett. 85 5655 10.1103/PhysRevLett.85.5655 doi: 10.1103/PhysRevLett.85.5655
[22]	Reverey J F, Jeon J H, Bao H, Leippe M, Metzler R, Selhuber-Unkel C 2015 Sci. Rep. 5 11690
[23]	Rammal R, Toulouse G 1983 J. Phys. Lett. 44 13 10.1051/jphyslet:0198300440101300 doi: 10.1051/jphyslet:0198300440101300
[24]	Sokolov I M 2012 Soft Matter 8 9043 10.1039/c2sm25701g doi: 10.1039/c2sm25701g
[25]	Metzler R, Jeon J H, Cherstvy A G, Barkai E 2014 Phys. Chem. Chem. Phys. 16 24128 10.1039/C4CP03465A doi: 10.1039/C4CP03465A
[26]	Ben-Avraham D, Havlin S 2000 Diffusion and reactions in fractals and disordered systems (Cambridge university press)
[27]	Gao S L, Zhong S C, Wei K, Ma H 2012 Acta Phys. Sin. 61 6 (in Chinese)
[28]	Goodknight R C, Klikoff W A, Fatt I 1960 J. Phys. Chem. 64 1162 10.1021/j100838a014 doi: 10.1021/j100838a014
[29]	Pinner A, Nye P H 1982 J. Soil Sci. 33 25 10.1111/ejs.1982.33.issue-1 doi: 10.1111/ejs.1982.33.issue-1
[30]	Rao P, Jessup R E, Addiscott T M 1982 Soil Sci. 133 154
[31]	Tao A, Tao L, Nicholson C 2005 J. Theor. Biol. 234 525 10.1016/j.jtbi.2004.12.009 doi: 10.1016/j.jtbi.2004.12.009
[32]	Hrabe J, HrabeTová S, Segeth K 2004 Biophys. J. 87 1606 10.1529/biophysj.103.039495 doi: 10.1529/biophysj.103.039495
[33]	Ernest P, Si B R 1964 J. Gen. Physiol. 47 1129 10.1085/jgp.47.6.1129 doi: 10.1085/jgp.47.6.1129
[34]	Safford R E, Bassingthwaighte E A, Bassingthwaighte J B 1978 J. Gen. Physiol. 72 513 10.1085/jgp.72.4.513 doi: 10.1085/jgp.72.4.513
[35]	Santamaria F, Wils S, De Schutter E, Augustine G J 2006 Neuron 52 635 10.1016/j.neuron.2006.10.025 doi: 10.1016/j.neuron.2006.10.025
[36]	Iomin A, Mendéz V, Horsthemke W 2018 Fractional dynamics in comb-like structures (World Scientific)
[37]	Weiss G H, Havlin S 1986 Phys.A 134 474 10.1016/0378-4371(86)90060-9 doi: 10.1016/0378-4371(86)90060-9
[38]	Weiss G H, Havlin S 1987 Philos. Mag.B 56 941 10.1080/13642818708215329 doi: 10.1080/13642818708215329
[39]	Arkhincheev V E, Baskin E M 1991 Sov. Phys. JETP 73 161
[40]	Méndez V, Iomin A, Horsthemke W, Campos D 2017 J. Stat. Mech. Theory Exp. 2017
[41]	Méndez V, Iomin A 2013 Chaos Soliton. Fract. 53 46
[42]	Iomin A 2012 Phys. Rev.E 86 32101 10.1103/PhysRevE.86.032101 doi: 10.1103/PhysRevE.86.032101
[43]	Arkhincheev V E, Kunnen E, Baklanov M R 2011 Microelectron. Eng. 88 694 10.1016/j.mee.2010.08.028 doi: 10.1016/j.mee.2010.08.028
[44]	Liu L 2018 J. Comput. Math. 36 563 10.4208/jcm doi: 10.4208/jcm
[45]	Liu L, Zheng L, Liu F 2017 J. Mol. Liq. 233 326 10.1016/j.molliq.2017.03.034 doi: 10.1016/j.molliq.2017.03.034
[46]	Iomin A, Méndez V 2016 Chaos Soliton. Fract. 82 142
[47]	Dzhanoev A R, Sokolov I M 2018 Chaos Soliton. Fract. 106 330
[48]	Havlin S, Kiefer J E, Weiss G H 1987 Phys. Rev.A 36 1403 10.1103/PhysRevA.36.1403 doi: 10.1103/PhysRevA.36.1403
[49]	Yuste S B, Abad E, Baumgaertner A 2016 Phys. Rev.E 94 012118 10.1103/PhysRevE.94.012118 doi: 10.1103/PhysRevE.94.012118
[50]	Iomin A 2011 Phys. Rev.E 83 052106
[51]	Sandev T, Iomin A, Kantz H 2015 Phys. Rev.E 91 032108 10.1103/PhysRevE.91.032108 doi: 10.1103/PhysRevE.91.032108
[52]	Sandev T, Iomin A, Kantz H 2017 Phys. Rev.E 95 52107 10.1103/PhysRevE.95.052107 doi: 10.1103/PhysRevE.95.052107
[53]	Fan Y, Liu L, Zheng L, Li X 2019 Sci. Numer. Simul. 77 225
[54]	Wang Z, Zheng L 2020 Phys.A 549 123889
[55]	Iomin A 2021 Chaos Soliton. Fract. 142 110488
[56]	Suleiman K, Zheng L, Liu C, Zhang X, Wang E 2020 Int. Commun. Heat Mass Transf. 117 104733 10.1016/j.icheatmasstransfer.2020.104733 doi: 10.1016/j.icheatmasstransfer.2020.104733
[57]	Suleiman K, Song Q, Zhang X, Liu S, Zheng L 2022 Chaos Soliton. Fract. 155 111742
[58]	Saxton M J 2007 Biophys. J. 92 1178 10.1529/biophysj.106.092619 doi: 10.1529/biophysj.106.092619
[59]	Baskin E, Iomin A 2004 Phys. Rev. Lett. 93 120603 10.1103/PhysRevLett.93.120603 doi: 10.1103/PhysRevLett.93.120603
[60]	Sandev T, Iomin A, Kocarev L 2020 Phys. Rev.E 102 042109 10.1103/PhysRevE.102.042109 doi: 10.1103/PhysRevE.102.042109
[61]	Singh R K, Sandev T, Iomin A, Metzler R 2021 J. Phys.A 54 404006 10.1088/1751-8121/ac20ed doi: 10.1088/1751-8121/ac20ed
[62]	Wang Z, Lin P, Wang E 2021 Chaos Soliton. Fract. 148 111009
[63]	Sandev T, Domazetoski V, Iomin A, Kocarev L 2021 Mathematics 9 1 10.3390/math9010001 doi: 10.3390/math9010001
[64]	Dagdug L, Berezhkovskii A M, Makhnovskii Y A, Zitserman V Y 2007 J. Chem. Phys. 127 224712 10.1063/1.2805068 doi: 10.1063/1.2805068
[65]	Berezhkovskii A M, Dagdug L 2011 J. Chem. Phys. 134 124109 10.1063/1.3567187 doi: 10.1063/1.3567187
[66]	Ogawa N 2010 Phys. Rev.E 81 61113 10.1103/PhysRevE.81.061113 doi: 10.1103/PhysRevE.81.061113
[67]	Ogawa N 2013 Phys. Lett.A 377 2465 10.1016/j.physleta.2013.07.054 doi: 10.1016/j.physleta.2013.07.054
[68]	Chávez Y, Chacón-Acosta G, Dagdug L 2018 J. Chem. Phys. 148 214106 10.1063/1.5030892 doi: 10.1063/1.5030892
[69]	Castro-Villarreal P, Villada-Balbuena A, Méndez-Alcaraz J M, Castañeda-Priego R, Estrada-Jiménez S 2014 J. Chem. Phys. 140 214115 10.1063/1.4881060 doi: 10.1063/1.4881060
[70]	Assenza S, Mezzenga R 2018 J. Chem. Phys. 148 54902 10.1063/1.5019224 doi: 10.1063/1.5019224
[71]	Dobrowolski T, Jarmoliński A 2018 Results Phys. 8 48 10.1016/j.rinp.2017.11.020 doi: 10.1016/j.rinp.2017.11.020
[72]	Burada P S, Hnggi P, Marchesoni F, Schmid G, Talkner P 2010 Chem. Phys. Chem. 10
[73]	Bénichou O, Illien P, Oshanin G, Sarracino A, Voituriez R 2018 J. Phys. Condens. Matter 30 443001 10.1088/1361-648X/aae13a doi: 10.1088/1361-648X/aae13a
[74]	Hrabĕtová S, Hrabe J, Nicholson C 2003 J. Neurosci. 23 8351 10.1523/JNEUROSCI.23-23-08351.2003 doi: 10.1523/JNEUROSCI.23-23-08351.2003
[75]	Nicholson C, Hrabĕtová S 2017 Biophys. J. 113 2133 10.1016/j.bpj.2017.06.052 doi: 10.1016/j.bpj.2017.06.052
[76]	Tao L, Nicholson C 2004 J. Theor. Biol. 229 59 10.1016/j.jtbi.2004.03.003 doi: 10.1016/j.jtbi.2004.03.003
[77]	Sykova E, Nicholson C 2008 Physiol. Rev. 88 1277 10.1152/physrev.00027.2007 doi: 10.1152/physrev.00027.2007
[78]	Bueno-Orovio A, Kay D, Grau V, Rodriguez B, Burrage K 2014 J. R. Soc. Interface 11 20140352 10.1098/rsif.2014.0352 doi: 10.1098/rsif.2014.0352
[79]	Wolak D J, Thorne R G 2013 Mol. Pharm. 10 1492 10.1021/mp300495e doi: 10.1021/mp300495e
[80]	Vendel E, Rottschfer V, Lange E 2019 Fluids Barriers CNS 16 12 10.1186/s12987-019-0133-x doi: 10.1186/s12987-019-0133-x
[81]	Caratelli D, Natalini P, Ricci P E, Yarovoy A 2010 Appl. Math. Comput. 216 556
[82]	Domazetoski V, Masó-Puigdellosas A, Sandev T, Méndez V, Iomin A, Kocarev L 2020 Phys. Rev. Res. 2 033027 10.1103/PhysRevResearch.2.033027 doi: 10.1103/PhysRevResearch.2.033027
[83]	Ghosh S K, Cherstvy A G, Metzler R 2014 Phys. Chem. Chem. Phys. 17 1847 10.1039/C4CP03599B doi: 10.1039/C4CP03599B