Optimal control problem of the elongation stage in the polymerase chain reaction
Abstract
The general methodology of optimal control for obtaining the solution of the optimal flow problem in the elongation stage of a polymerase chain reaction is applied. The examined model of the elongation stage takes into account the dependence of reaction’s rate on the absolute temperature, which is described by the Arrhenius equation. This equation can be used in the investigation of the elongation stage of a polymerase chain reaction, since the temperature controls the process. A scheme of the temperature control in the process of a polymerase chain reaction is examined. Pontryagin’s maximum principle for the optimal control problem is used and the necessary condition for optimality is formulated. The obtained results are required for the numerical calculations of optimal control of the examined stage and help to minimize the required duration of the elongation stage, which will allow to minimize the duration of a polymerase chain reaction in general.References
Sverstyuk A.S., Bihunyak T.V., Pereviznyk B.O. Ohlyad metodiv ta modeley polimerazno-lantsyuhovoyi reaktsiyi // Medychna informatyka ta inzheneriya. — # 3. — S. 97–100.
Putyntseva H.Y. Medychna henetyka: pidruchnyk. — K.: Medytsyna, 2008. — S. 81–87.
Bulyanitsa А.L. Matematicheskoye modelirovaniye tsiklicheskikh rezhimov upravleniya temperaturoy pri realizatsii polimerazno tsepnoy reaktsii (PTSR) dlya metoda molekulyarnykh koloniy (MMK) na mikrochipe // Nauchnoye priborostroyeniye. — 2011. — 21, № 1. — S. 87–96.
Arnheim N. Polymerase chain reaction strategy // Annual review of biochemistry. — 1992. — 61, XIV+1359P. — P. 131–156.
Xiangchun X., Sinton D., Dongqing L. Thermal end effects on electroosmotic flow in capillary // Int. J. of Heat and Mass transfer. — 2004. — 47. — P. 3145–3157.
Stone E., Goldes J., Garlick M. A multi-stage model for quantitative PCR // Mathematical biosciences and engineering. — http://hs.umt.edu/math/research/technicalreports/documents/2009/10_pcrpaper4.pdf
Lukes D.L. Differential Equations: Classical to Controlled // Academic Press, New York. — 1982. — 162. — P. 322.
Piccinini L. C., Stampacchia G., Vidossich G. Ordinary Differential Equations in Rn. Problems and Methods Ordinary // Berlin-Heidelberg-New York-Tokyo, Springer-Verlag. — 1984. — XII. — P. 385.
Macki J., Strauss A. Introduction to Optimal Control Theory, Springer-Verlag // New York. — 1982. — XIV. — P. 168.
Fleming W.H., Rishel R.W. Deterministic and Stochastic Optimal Control, Springer Verlag // New York. — 1975. — XIII. — P. 222.
Kamien M.I., Schwartz N.L. Dynamic Optimization // North-Holland, Amsterdam. — 1991. — 3. — P. 272.
Pontryagin L.S., Boltyanskiy V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal’nykh protsessov. — M.: Nauka, 1983. — 393 s.
Kelly K., Kostin M. Non-Arrhenius rate constants involving diffusion and reaction // Journal of Chemical Physics. —1986. — 85, issue 12. — P. 7318.
