Numerical algorithm for optimal control development for annealing stage of polymerase chain reaction https://doi.org/10.33108/visnyk_tntu2019.01.147
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Abstract
In the work the general methodology of control is used for obtaining the solution of the problem of optimal annealing stage in a polymerase chain reaction in order to effectively conduct the study and
the possibility of providing a multi-stage cyclic regime of temperature change. The annealing stage should occur at certain temperatures and over time, because otherwise the necessary transformations of DNA molecules may
not occur. The developed model of annealing stage of the polymerase chain reaction, which takes into account the ratio of the number of single-stranded DNA, primer, single-stranded DNA bound to the primer, direct and reverse
reaction rate for annealing, was used. In the model under study, the Arrhenius equation is used, which takes into account the dependence of the reaction rate on absolute temperature. The principle of Pontryagin's maximum is
applied to the problem of optimal control of annealing stage and the necessary optimality condition is formulated. The direct method of numerical solving of the problem of optimal annealing control, which is implemented in the
package of Java classes, is developed. In the form of graphs are presented the results of numerical simulation of the problem of optimal control of the annealing stage polymerase chain reaction. The results of numerical
modeling of the optimal control of the annealing stage of polymerase chain reaction for changing the number of single-stranded DNA, the number of primers, changes in the number of single-stranded DNA that are connected
to the primer and the optimal temperature value of the investigated stage are constructed. The obtained results of numerical simulation of the problem of optimal control of the annealing stage of polymerase chain reaction will
help to minimize the necessary time for the implementation of this stage. The scheme of temperature setting thus constructed minimizes the required time of implementation of the annealing stage, which in the general case will
ensure the achievement of the minimum time for polymerase chain reaction
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References
1. Sverstiuk A. S., Bihuniak T. V., Pereviznyk B. O. Ohliad metodiv ta modelei polimerazno-lantsiuhovoi reaktsii. Medychna informatyka ta inzheneriia. № 3. 2014. Р. 97−100.
2. Martseniuk V. P., Sverstiuk A. S., Kuchvara O. M. Zadacha optymalnoho keruvannia stadiieiu vidpalu polimerazno-lantsiuhovoi reaktsii. Klynycheskaia ynformatyka y telemedytsyna. № 12. 2015. Р. 47−51.https://doi.org/10.31071/kit2015.12.04
3. Aach J., Church G. M. Mathematical models of diffusion-constrained polymerase chainreactions: basis of high-throughput nucleic acid assays and simple self-organizing systems. Journal of Theoretical Biology. 2004. Vol. 228. Pp. 31−46. https://doi.org/10.1016/j.jtbi.2003.12.003
4. Pfaffl M. W. A new mathematical model for relative quantification in real-time RT–PCR. Oxford Journals Science & Mathematics Nucleic Acids Research. Vol. 29. No. 900. Pp. 45−51. https://doi.org/10.1093/nar/29.9.e45
5. Xiangchun X., Sinton D., Dongqing L. Thermal end effects on electroosmotic flow in capillary. Int. J. of Heat and Mass transfer. 2004. Vol. 47. Iss. 14−16, pp. 3145−3157. https://doi.org/10.1016/j.ijheatmasstransfer.2004.02.023
6. Stone E., Goldes J., Garlick M. A multi-stage model for quantitative PCR. Mathematical biosciences and engineering. 2000. Vol. 00. No. 0. Pp. 1−17.
7. Garlick M., Powell J., Eyre D., Robbins T. Mathematically modeling PCR: an asymptotic approximation with potential for optimization Mathematical biosciences and engineering. 2010. Vol. 07. No. 2. Pp. 363−384. https://doi.org/10.3934/mbe.2010.7.363
8. Lukes D. L. Differential Equations: Classical to Controlled. Academic Press. New York. 1982. Vol. 162. 322 p.
9. Piccinini L. C., Stampacchia G., Vidossich G. Ordinary Differential Equations in Rn. Problems and Methods Ordinary. Berlin-Heidelberg-New York-Tokyo. Springer-Verlag. 1984. Vol. XII. 385 p. https://doi.org/10.1007/978-1-4612-5188-0
10. Macki J., Strauss A. Introduction to Optimal Control Theory. Springer-Verlag. New York. 1982. Vol. XIV. 168 p. https://doi.org/10.1007/978-1-4612-5671-7
11. Fleming W. H., Rishel R. W. Deterministic and Stochastic Optimal Control. Springer Verlag. New York. 1975. Vol. XIII. 222 p. https://doi.org/10.1007/978-1-4612-6380-712. Kamien M. I., Schwartz N. L. Dynamic Optimization. North-Holland. Amsterdam. 1991. Vol. 3. 272 p.
13. Pontriahyn L. S., Boltianskyi V. H., Hamkrelydze R. V., Myshchenko E. F. Matematycheskaia teoryia optymalnykh protsessov. M.: 1983. 393 c.
14. Kelly K., Kostin M. Non-Arrhenius rate constants involving diffusion and reaction. Journal of Chemical Physics. 1986. Vol. 85. Іss. 12. Рp. 7318−7335. https://doi.org/10.1063/1.451371
15.John T. Betts. Practical Methods for Optimal Control Using Nonlinear Programming Society for Industrial and Applied Mathematics. 2001. 190 p.
16. O. von Stryk, R. Bulirsch. Direct and indirect methods for trajectory optimization. Annals of Operations Research. 1992. No. 2. Vol. 37. Іss. 1−4. Рр. 357−373. https://doi.org/10.1007/BF02071065
17. Arthur E. Bryson, Jr. and Yu-Chi Ho. Applied optimal control. Hal-sted Press, New York, 1975. 481 p.
18.B. C. Fabien Some tools for the direct solution of optimal control problems. Advances in Engineering Software. 1998. Vol. 29. Рp. 45−61. https://doi.org/10.1016/S0965-9978(97)00025-2