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Optimal geometric control applied to the protein misfolding cyclic amplification process. (English) Zbl 1309.93111

Summary: Protein Misfolding Cyclic Amplification is a procedure used to accelerate the prion-replication process involved during the incubation period of transmissible spongiform encephalopathies. This technique could be used to design an efficient diagnosis test detecting the abnormally-shaped protein responsible of the decease before the affected person is at an advanced stage of the illness. In this paper, we investigate the open problem to determine what is the optimal strategy to produce maximum replication in a fixed time. Primarily, we expand on prior attempt to answer this question in general, and provide results under some specific assumptions.

MSC:

93C95 Application models in control theory
70Q05 Control of mechanical systems
70E60 Robot dynamics and control of rigid bodies
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[1] Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989) · doi:10.1007/978-1-4757-2063-1
[2] Balagué, D., Cañizo, J., Gabriel, P.: Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinet. Relat. Models 6(2), 219-243 (2013) · Zbl 1270.35095 · doi:10.3934/krm.2013.6.219
[3] Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Springer, Berlin (2003) · Zbl 1022.93003
[4] Bonnard, B., Caillau, J.-B., Trélat, E.: Second order optimality conditions in the smooth case and applications in optimal control. ESAIM Control Optim. Calc. Var. 13, 207-236 (2007) · Zbl 1123.49014 · doi:10.1051/cocv:2007012
[5] Bonnard, B., Chyba, M., Jacquemard, A., Marriott, J.: Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance. Math. Control Relat. Fields 3(4), 397-432 (2013). Special issue in the honor of Bernard Bonnard. Part II · Zbl 1273.94128 · doi:10.3934/mcrf.2013.3.397
[6] Caillau, J.-B., Cots, O., Gergaud, J.: Differential continuation for regular optimal control problems. Optim. Methods Softw. 27(2), 177-196 (2012) · Zbl 1248.49025 · doi:10.1080/10556788.2011.593625
[7] Calvez, V., Doumic, M., Gabriel, P.: Self-similarity in a general aggregation-fragmentation problem. Application to fitness analysis. J. Math. Pures Appl. 98(9), 1-27 (2012) · Zbl 1259.35151 · doi:10.1016/j.matpur.2012.01.004
[8] Calvez, V., Gabriel, P., Gaubert, S.: Non-linear eigenvalue problems arising from growth maximization of positive linear dynamical systems (2014), submitted. arXiv:1404.1868
[9] Castilla, J., Saa, P., Soto, C.: Detection of prions in blood. Nat. Med. 11, 982-985 (2005)
[10] Cohen, F.E., Prusinier, S.B.: Pathologic conformations of prion proteins. Annu. Rev. Biochem. 67, 793-819 (1998) · doi:10.1146/annurev.biochem.67.1.793
[11] Collinge, J.: Prion diseases of humans and animals: their causes and molecular basis. Annu. Rev. Neurosci. 24, 519-555 (2001) · doi:10.1146/annurev.neuro.24.1.519
[12] Coron, J.-M., Gabriel, P., Shang, P.: Optimization of an Amplification Protocol for Misfolded Proteins by using Relaxed Control. J. Math. Biol. Published online: February 25, 2014. doi:10.1007/s00285-014-0768-9 · Zbl 1306.49003
[13] Cots, O.: Contrôle optimal géométrique: méthodes homotopiques et applications. Ph.D. Thesis, Institut Mathématiques de Bourgogne, Dijon, France (2012)
[14] Doumic, M., Gabriel, P.: Eigenelements of a general aggregation-fragmentation model. Math. Models Methods Appl. Sci. 20, 757-783 (2010) · Zbl 1201.35086 · doi:10.1142/S021820251000443X
[15] Fernandez-Borges, N., Castilla, J.: PMCA. A decade of in vitro prion replication. Curr. Chem. Biol. 4, 200-207 (2010)
[16] Gabriel, P.: Equations de transport-fragmentation et applications aux maladies à prions (Transport-fragmentation equations and applications to prion deceases). Ph.D. Thesis, Paris (2011)
[17] Gonzalez-Montalban, N., Makarava, N., Ostapchenko, V.G., Savtchenk, R., Alexeeva, I., et al.: Highly efficient protein misfolding cyclic amplification. PLoS Pathog. 7(2), e1001277 (2011). doi:10.1371/journal.ppat.1001277 · doi:10.1371/journal.ppat.1001277
[18] Greer, M.L., Pujo-Menjouet, L., Webb, G.F.: A mathematical analysis of prion proliferation. J. Theor. Biol. 242(3), 598-606 (2006) · Zbl 1447.92089 · doi:10.1016/j.jtbi.2006.04.010
[19] Jarrett, J.T., Lansbury, P.T.: Seeding one-dimensional crystallization of amyloid: a pathogenic mechanism in Alzheimer’s disease and scrapie? Cell 73(6), 1055-1058 (1993) · doi:10.1016/0092-8674(93)90635-4
[20] Krener, A.J.: The high order maximal principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256-293 (1977) · Zbl 0354.49008 · doi:10.1137/0315019
[21] Kupka, I., The ubiquity of Fuller’s phenomenon Nonlinear controllabillity and optimal control, No. 133, 313-350 (1990), New York · Zbl 0739.49001
[22] Ledzewicz, U., Schättler, H.: Analysis of a cell-cycle specific model for cancer chemotherapy. J. Biol. Syst. 10(03), 183-206 (2002) · Zbl 1099.92035 · doi:10.1142/S0218339002000597
[23] Ledzewicz, U., Schättler, H.: Drug resistance in cancer chemotherapy as an optimal control problem. Discrete Contin. Dyn. Syst., Ser. B 6(1), 129-150 (2002) · Zbl 1088.92040
[24] Ledzewicz, U., Schättler, H.: Analysis of models for evolving drug resistance in cancer chemotherapy. Dyn. Contin. Discrete Impuls. Syst. 13B(suppl.)(03), 291-304 (2006)
[25] Lee, E.B., Markus, L.: Fondations of Optimal Control Theory, 2nd edn. Krieger, Melbourne (1986)
[26] Masel, J., Jansen, V.A.A., Nowak, M.A.: Quantifying the kinetic parameters of prion replication. Biophys. Chem. 77(2-3), 139-152 (1999) · doi:10.1016/S0301-4622(99)00016-2
[27] Mays, C.E., Titlow, W., Seward, T., Telling, G.C., Ryou, C.: Enhancement of protein misfolding cyclic amplification by using concentrated cellular prion protein source. Biochem. Biophys. Res. Commun. 388, 306-310 (2009) · doi:10.1016/j.bbrc.2009.07.163
[28] Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) (Results in Mathematics and Related Areas (2)), vol. 34. Springer, Berlin (1994) · Zbl 0797.14004 · doi:10.1007/978-3-642-57916-5
[29] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Wiley, New York (1962) · Zbl 0102.32001
[30] Roos, R., Gajdusek, D.C., Gibbs, C.J. Jr.: The clinical characteristics of transmissible Creutzfeldt-Jakob disease. Brain 96, 1-20 (1973) · doi:10.1093/brain/96.1.1
[31] Saa, P., Castilla, J., Soto, C.: Ultra-efficient replication of infectious prions by automated protein misfolding cyclic amplification. J. Biol. Chem. 281(16), 35245-35252 (2006) · doi:10.1074/jbc.M603964200
[32] Saborio, G.P., Permanne, B., Soto, C.: Sensitive detection of pathological prion protein by cyclic amplification of protein misfolding. Nature 411, 810-813 (2001) · doi:10.1038/35081095
[33] Serre, D., Theory and applications, No. 16 (2002), New York
[34] Zaslavski, J.: Turnpike Properties in the Calculus of Variations and Optimal Control. Nonconvex Optimization and Its Applications, vol. 80. Springer, New York (2006) · Zbl 1100.49003
[35] Zelikin, M.I., Borisov, V.F.: Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering. Birkhäuser, Basel (1994) · Zbl 0820.70003
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