# American Institute of Mathematical Sciences

May  2012, 17(3): 775-799. doi: 10.3934/dcdsb.2012.17.775

## Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation

 1 Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France, France, France 2 Université de Lyon, CNRS UMR 5208, INSA-Lyon, Institut Camille Jordan, 21, avenue Jean Capelle, F-69621 Villeurbanne cedex, France

Received  June 2011 Revised  October 2011 Published  January 2012

The Greer, Pujo-Menjouet and Webb model [Greer et al., J. Theoret. Biol., 242 (2006), 598--606] for prion dynamics was found to be in good agreement with experimental observations under no-flow conditions. The objective of this work is to generalize the problem to the framework of general polymerization-fragmentation under flow motion, motivated by the fact that laboratory work often involves prion dynamics under flow conditions in order to observe faster processes. Moreover, understanding and modelling the microstructure influence of macroscopically monitored non-Newtonian behaviour is crucial for sensor design, with the goal to provide practical information about ongoing molecular evolution. This paper's results can then be considered as one step in the mathematical understanding of such models, namely the proof of positivity and existence of solutions in suitable functional spaces. To that purpose, we introduce a new model based on the rigid-rod polymer theory to account for the polymer dynamics under flow conditions. As expected, when applied to the prion problem, in the absence of motion it reduces to that in Greer et al. (2006). At the heart of any polymer kinetical theory there is a configurational probability diffusion partial differential equation (PDE) of Fokker-Planck-Smoluchowski type. The main mathematical result of this paper is the proof of existence of positive solutions to the aforementioned PDE for a class of flows of practical interest, taking into account the flow induced splitting/lengthening of polymers in general, and prions in particular.
Citation: Ionel Sorin Ciuperca, Erwan Hingant, Liviu Iulian Palade, Laurent Pujo-Menjouet. Fragmentation and monomer lengthening of rod-like polymers, a relevant model for prion proliferation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 775-799. doi: 10.3934/dcdsb.2012.17.775
##### References:
 [1] R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory," J. Wiley & Sons, New York, 1987. [2] V. Calvez, N. Lenuzza, D. Oelz, J. P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Size distribution dependence of prion aggregates infectivity, Mathematical Biosciences, 217 (2009), 88-99. doi: 10.1016/j.mbs.2008.10.007. [3] B. Caughey, G. S. Baron, B. Chesebro and M. Jeffrey, Getting a grip on prions: Oligomers, amyloids, and pathological membrane interactions, Annu. Rev. Biochem., 78 (2009), 177-204. doi: 10.1146/annurev.biochem.78.082907.145410. [4] M. Doumic, T. Goudon and T. Lepoutre, Scaling limit of a discrete prion dynamics model, Comm. in Math. Sci., 7 (2009), 839-865. [5] H. Engler, J. Prüss and G. F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl., 324 (2006), 98-117. doi: 10.1016/j.jmaa.2005.11.021. [6] M. L. Greer, L. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol., 242 (2006), 598-606. doi: 10.1016/j.jtbi.2006.04.010. [7] M. L. Greer, P. van den Driessche, L. Wang and G. F. Webb, Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation, SIAM J. Appl. Math., 68 (2007), 154-170. doi: 10.1137/06066076X. [8] R. R. Huilgol and N. Phan-Thien, "Fluid Mechanics of Viscoelasticity," Elsevier, Amsterdam, 1997. [9] J. G. Kirkwood, "Macromolecules," ed. P. L. Auer, Gordon and Breach, 1968. [10] P. T. Lansbury and B. Caughey, The chemistry of scrapie infection: Implications of the 'ice 9' metaphor, Chemistry & Biology, 2 (1995), 1-5. doi: 10.1016/1074-5521(95)90074-8. [11] P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics, J. Evol. Equ., 7 (2007), 241-264. doi: 10.1007/s00028-006-0279-2. [12] J. Masel, V. A. Jansen and M. A. Nowak, Quantifying the kinetic parameters of prion replication, Biophys. Chem., 77 (1999), 139-152. doi: 10.1016/S0301-4622(99)00016-2. [13] F. Otto and A. E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules, Commun. Math. Phys., 277 (2008), 729-758. doi: 10.1007/s00220-007-0373-5. [14] S. B. Prusiner, Prions, Proc. Natl. Acad. Sci. USA, 95 (1998), 13363-13383. doi: 10.1073/pnas.95.23.13363. [15] J. Prüss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 225-235. [16] T. Scheibel, A. S. Kowal, J. D. Bloom and S. L. Lindquist, Bidirectional amyloid fiber growth for a yeast prion determinant, Curr. Biol., 11 (2001), 366-369. doi: 10.1016/S0960-9822(01)00099-9. [17] G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl., 324 (2006), 580-603. doi: 10.1016/j.jmaa.2005.12.036. [18] C. Walker, Prion proliferation with unbounded polymerization rates, in "Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations," Electron. J. Diff. Eqns. Conference, 15, Southwest Texas State Univ., San Marcos, TX, (2007), 387-397. [19] V. Zamoza-Signoret, J.-D. Arnaud, P. Fontes, M.-T. Alvarez-Martinez and J.-P. Liautard, Physiological role of the cellular prion protein, Vet. Res, 39 (2008).

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##### References:
 [1] R. B. Bird, C. F. Curtiss, R. C. Armstrong and O. Hassager, "Dynamics of Polymeric Liquids, Vol. 2: Kinetic Theory," J. Wiley & Sons, New York, 1987. [2] V. Calvez, N. Lenuzza, D. Oelz, J. P. Deslys, P. Laurent, F. Mouthon and B. Perthame, Size distribution dependence of prion aggregates infectivity, Mathematical Biosciences, 217 (2009), 88-99. doi: 10.1016/j.mbs.2008.10.007. [3] B. Caughey, G. S. Baron, B. Chesebro and M. Jeffrey, Getting a grip on prions: Oligomers, amyloids, and pathological membrane interactions, Annu. Rev. Biochem., 78 (2009), 177-204. doi: 10.1146/annurev.biochem.78.082907.145410. [4] M. Doumic, T. Goudon and T. Lepoutre, Scaling limit of a discrete prion dynamics model, Comm. in Math. Sci., 7 (2009), 839-865. [5] H. Engler, J. Prüss and G. F. Webb, Analysis of a model for the dynamics of prions II, J. Math. Anal. Appl., 324 (2006), 98-117. doi: 10.1016/j.jmaa.2005.11.021. [6] M. L. Greer, L. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theoret. Biol., 242 (2006), 598-606. doi: 10.1016/j.jtbi.2006.04.010. [7] M. L. Greer, P. van den Driessche, L. Wang and G. F. Webb, Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation, SIAM J. Appl. Math., 68 (2007), 154-170. doi: 10.1137/06066076X. [8] R. R. Huilgol and N. Phan-Thien, "Fluid Mechanics of Viscoelasticity," Elsevier, Amsterdam, 1997. [9] J. G. Kirkwood, "Macromolecules," ed. P. L. Auer, Gordon and Breach, 1968. [10] P. T. Lansbury and B. Caughey, The chemistry of scrapie infection: Implications of the 'ice 9' metaphor, Chemistry & Biology, 2 (1995), 1-5. doi: 10.1016/1074-5521(95)90074-8. [11] P. Laurençot and C. Walker, Well-posedness for a model of prion proliferation dynamics, J. Evol. Equ., 7 (2007), 241-264. doi: 10.1007/s00028-006-0279-2. [12] J. Masel, V. A. Jansen and M. A. Nowak, Quantifying the kinetic parameters of prion replication, Biophys. Chem., 77 (1999), 139-152. doi: 10.1016/S0301-4622(99)00016-2. [13] F. Otto and A. E. Tzavaras, Continuity of velocity gradients in suspensions of rod-like molecules, Commun. Math. Phys., 277 (2008), 729-758. doi: 10.1007/s00220-007-0373-5. [14] S. B. Prusiner, Prions, Proc. Natl. Acad. Sci. USA, 95 (1998), 13363-13383. doi: 10.1073/pnas.95.23.13363. [15] J. Prüss, L. Pujo-Menjouet, G. F. Webb and R. Zacher, Analysis of a model for the dynamics of prions, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 225-235. [16] T. Scheibel, A. S. Kowal, J. D. Bloom and S. L. Lindquist, Bidirectional amyloid fiber growth for a yeast prion determinant, Curr. Biol., 11 (2001), 366-369. doi: 10.1016/S0960-9822(01)00099-9. [17] G. Simonett and C. Walker, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl., 324 (2006), 580-603. doi: 10.1016/j.jmaa.2005.12.036. [18] C. Walker, Prion proliferation with unbounded polymerization rates, in "Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations," Electron. J. Diff. Eqns. Conference, 15, Southwest Texas State Univ., San Marcos, TX, (2007), 387-397. [19] V. Zamoza-Signoret, J.-D. Arnaud, P. Fontes, M.-T. Alvarez-Martinez and J.-P. Liautard, Physiological role of the cellular prion protein, Vet. Res, 39 (2008).
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