Geodesic
A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping
C. N. Angstmann1 ; I. C. Donnelly1 ; B. I. Henry1 ; T. A. M. Langlands2
1School of Mathematics and Statistics UNSW Australia
2School of Agricultural, Computational and Environmental Sciences University of Southern Queensland, Australia
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 142-156
Cet article a éte moissonné depuis la source EDP Sciences

Voir la notice de l'article

There is growing evidence that many neurodegenerative disease processes involve the proliferation, accumulation and spread of pathogenic proteins. The transport of proteins in the brain is typically hindered on small scales by micro-domain traps and binding sites but it may be enhanced on larger scales by neuronal pathways identified as white matter transport networks. We have introduced a mathematical network model to simulate a pathogenic protein neurodegenerative disease in the brain taking into account the anomalous transport. The proliferation and accumulation of pathogenic proteins is modelled using a set of reaction kinetics equations on the nodes of a network. Transport of the proteins on the network is modelled as a continuous time random walk with power law distributed waiting times on the nodes. This power law waiting time distribution is shown to be consistent with anomalously slowed diffusion on local scales but transport is enhanced on larger scales by the jumps between nodes. The model reveals that the disease spreads as a propagating front throughout the brain. The anomalous behaviour leads to a lessor variation in the concentration of pathogenic proteins. The enhanced transport on the network ensures that the approach to equilibrium is dominated by the short time behaviour of the waiting time density, hence the effects of subdiffusion are not as pronounced as in a spatial continuum.
DOI : 10.1051/mmnp/20161139

C. N. Angstmann  1   ; I. C. Donnelly  1   ; B. I. Henry  1   ; T. A. M. Langlands  2

1 School of Mathematics and Statistics UNSW Australia
2 School of Agricultural, Computational and Environmental Sciences University of Southern Queensland, Australia 
@article{10_1051_mmnp_20161139,
     author = {C. N. Angstmann and I. C. Donnelly and B. I. Henry and T. A. M. Langlands},
     title = {A {Mathematical} {Model} for the {Proliferation,} {Accumulation} and {Spread} of {Pathogenic} {Proteins} {Along} {Neuronal} {Pathways} with {Locally} {Anomalous} {Trapping}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {142--156},
     year = {2016},
     volume = {11},
     number = {3},
     doi = {10.1051/mmnp/20161139},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20161139/}
}
TY  - JOUR
AU  - C. N. Angstmann
AU  - I. C. Donnelly
AU  - B. I. Henry
AU  - T. A. M. Langlands
TI  - A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping
JO  - Mathematical modelling of natural phenomena
PY  - 2016
SP  - 142
EP  - 156
VL  - 11
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20161139/
DO  - 10.1051/mmnp/20161139
LA  - en
ID  - 10_1051_mmnp_20161139
ER  - 
%0 Journal Article
%A C. N. Angstmann
%A I. C. Donnelly
%A B. I. Henry
%A T. A. M. Langlands
%T A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping
%J Mathematical modelling of natural phenomena
%D 2016
%P 142-156
%V 11
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20161139/
%R 10.1051/mmnp/20161139
%G en
%F 10_1051_mmnp_20161139
C. N. Angstmann; I. C. Donnelly; B. I. Henry; T. A. M. Langlands. A Mathematical Model for the Proliferation, Accumulation and Spread of Pathogenic Proteins Along Neuronal Pathways with Locally Anomalous Trapping. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 3, pp. 142-156. doi: 10.1051/mmnp/20161139

[1] M. Jucker, L. C. Walker Ann Neurol 532 540 2011

[2] S. B. Prusiner Science 136 144 1982

[3] ——, Prion biology and diseases.Cold Spring Harbor Laboratory Press, 2004.

[4] K. Brown, J. A. Mastrianni J. Geriatr. Psychiatry Neurol. 277 298 2010

[5] A. Aguzzi, L. Rajendran Neuron 783 790 2009

[6] M. Cushman, B. S. Johnson, O. D. King, A. D. Gitler, J. Shorter J. Cell Sci. 1191 1201 2010

[7] C. Duyckaerts, B. Delatour, M.-C. Potier Acta Neuropathol. 5 36 2009

[8] H. Braak, K. Del Tredici Neurol. 1916 1925 2008

[9] L. C. Walker, H. Levine J. Bio. Chem. 33109 33115 2012

[10] M. Jucker, L. C. Walker Nature 45 51 2013

[11] H. Baker, R. Ridley, L. Duchen, T. Crow, C. Bruton Int. J Exp. Pathol. 441 1993

[12] R. Ridley, H. Baker, C. Windle, R. Cummings J Neural Transm. 1243 1251 2006

[13] J. Goudsmit, C. H. Morrow, D. M. Asher, R. T. Yanagihara, C. L. Masters, C. J. Gibbs, D. C. Gajdusek Neurology 945 945 1980

[14] S. B. Prusiner Proc. Nat. Aca. Sci. 13363 13383 1998

[15] S.-J. Lee, P. Desplats, C. Sigurdson, I. Tsigelny, E. Masliah Nature Reviews Neurology 702 706 2010

[16] C. Saper, B. Wainer, D. German Neuroscience 389 398 1987

[17] H. Braak, E. Braak Acta Neuropathol. 239 259 1991

[18] D. R. Thal, U. Rüb, M. Orantes, H. Braak Neurology 1791 1800 2002

[19] H. Braak, K. Del Tredici, U. Rüb, R. A. De Vos, E. N. J. Steur, E. Braak Neurobiology of aging 197 211 2003

[20] J. M. Ravits, A. R. Laspada Neurology 805 811 2009

[21] J. Brettschneider, K. Del Tredici, J. B. Toledo, J. L. Robinson, D. J. Irwin, M. Grossman, E. Suh, V. M. Deerlin, E. M. Wood, Y. Baek Annals of neurology 20 38 2013

[22] M. Goedert, M. G. Spillantini, K. Del Tredici, H. Braak Nature Reviews Neurology 13 24 2013

[23] J. Liu, L. Zhao, J. Nan, G. Li, S. Xiong, K. M. Von Deneen, Q. Gong, F. Liang, W. Qin, J. Tian Experimental neurology 196 204 2013

[24] D. Le Bihan, J. F. Mangin, C. Poupon, C. Clark, S. Pappata, N. Molko, H. Chabriat J. Magn. Reson. Im. 534 546 2001

[25] A. Raj, A. Kuceyeski, M. Weiner Neuron 1204 1215 2012

[26] E. Syková, C. Nicholson Physio. Rev. 1277 1340 2008

[27] E. Özarslan, P. J. Basser, T. M. Shepherd, P. E. Thelwall, B. C. Vemuri, S. J. Blackband J. Magnet. Reson. 315 323 2006

[28] E. Özarslan, T. M. Shepherd, C. G. Koay, S. J. Blackband, P. J. Basser Neuroimage 1380 1393 2012

[29] D. Le Bihan, E. Breton, D. Lallemand, P. Grenier, E. Cabanis, M. Laval-Jeantet Radiology 401 407 1986

[30] R. Sener Comput. Med. Imag. Grap. 299 326 2001

[31] R. V. Mulkern, H. Gudbjartsson, C.-F. Westin, H. P. Zengingonul, W. Gartner, C. R. Guttmann, R. L. Robertson, W. Kyriakos, R. Schwartz, D. Holtzman NMR Biomed. 51 62 1999

[32] K. M. Bennett, K. M. Schmainda, D. B. Rowe, H. Lu, J. S. Hyde Magnet. Reson. Med. 727 734 2003

[33] M. G. Hall, T. R. Barrick Magnet. Reson. Med. 447 455 2008

[34] S. De Santis, A. Gabrielli, M. Bozzali, B. Maraviglia, E. Macaluso, S. Capuani Magnet. Reson. Med. 1043 1052 2011

[35] M. G. Hall, T. R. Barrick NMR in Biomed. 286 294 2012

[36] J. H. Jensen, J. A. Helpern NMR in Biomed. 698 710 2010

[37] R. L. Magin, O. Abdullah, D. Baleanu, X. J. Zhou J. Magnet. Reson. 255 270 2008

[38] X. J. Zhou, Q. Gao, O. Abdullah, R. L. Magin Magnet. Reson. Med. 562 569 2010

[39] B. Fevrier, D. Vilette, F. Archer, D. Loew, W. Faigle, M. Vidal, H. Laude, G. Raposo P. Natl. Acad. Sci. USA 9683 9688 2004

[40] A. C. Magalhães, G. S. Baron, K. S. Lee, O. Steele-Mortimer, D. Dorward, M. A. Prado, B. Caughey J. Neurosci. 5207 5216 2005

[41] R. Hilfer, L. Anton Phys. Rev. E R848 1995

[42] R. Metzler, J. Klafter Phys. Rep. 1 77 2000

[43] E. Barkai, R. Metzler, J. Klafter Phys. Rev. E 132 2000

[44] I. M. Sokolov, J. Klafter Physical review letters 140602 1 2006

[45] B. I. Henry, T. A. M. Langlands, P. Straka Phys. Rev. Lett. 170602 2010

[46] B. I. Henry, T. A. M. Langlands, S. L. Wearne Phys. Rev. E 031116 2006

[47] S. Fedotov Physical Review E 011117 2010

[48] C. N. Angstmann, I. C. Donnelly, B. I. Henry Math. Model. Nat. Phenom. 17 27 2013

[49] A. A. Nepomnyashchy, V. A. Volpert Journal of Physics A: Mathematical and Theoretical 065101 2013

[50] V. A. Volpert, Y. Kanevsky, A. A. Nepomnyashchy Phys. Rev. E 012901 2014

[51] B. Henry, T. Langlands, S. Wearne Physical review letters 128103 2008

[52] T. A. M. Langlands, B. I. Henry, S. L. Wearne Journal of mathematical biology 761 808 2009

[53] T. Langlands, B. Henry, S. L. Wearne SIAM Journal on Applied Mathematics 1168 1203 2011

[54] C. Ingo, R. L. Magin, L. Colon-Perez, W. Triplett, T. H. Mareci Magnet. Reson. Med. 617 627 2014

[55] C. N. Angstmann, I. C. Donnelly, B. I. Henry Phys. Rev. E 032804 2013

[56] C. N. Angstmann, I. C. Donnelly, B. I. Henry, T. A. M. Langlands Phys. Rev. E 022811 2013

[57] F. E. Cohen, K.-M. Pan, Z. Huang, M. Baldwin, R. J. Fletterick, S. B. Prusiner Science (New York, NY) 530 1994

[58] T. A. M. Langlands, B. I. Henry J Comp. Phys. 719 736 2005

[59] A. L. Alexander, J. E. Lee, M. Lazar, A. S. Field Neurotherapeutics 316 329 2007

[60] A. Erickson, B. Henry, P. Klaase, C. Angstmann Biophysical Journal 164 172 2015

[61] H. J. Haubold, A. M. Mathai, R. K. Saxena Journal of Applied Mathematics 2011

[62] J. A. Brown, J. D. Rudie, A. Bandrowski, J. D. Vanhorn, S. Y. Bookheimer Frontiers in Neuroinformatics 2012

[63] J.D. Rudie, J. Brown, D. Beck-Pancer, L. Hernandez, E. Dennis, P. Thompson, S. Bookheimer, M. Dapretto NeuroImage: Clinical 79 94 2013

[64] K. B. Oldham and J. Spanier, The Fractional Calculus:Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, 1974.

Cité par Sources :