Optimal control of a fractional-order enzyme kinetic model

• Autorzy: Al-Basir F. , Elaiw A. M. , Kesh D. , Roy P. K.
• Języki publikacji: EN

Abstrakty

EN

Enzymes play a significant role in controlling the characteristics of various chemical and biochemical reactions. They act as catalysts that increase the rate of reaction without undergoing any change in quantity. Enzymatic reactions occur through the active sites, which combine with the substrates to form intermediate complexes, subsequently leading to products. An enzyme having two active sites can show cooperative phenomena. Against this background, an enzyme-kinetic mathematical model is formulated using fractional order derivatives. Optimal control mechanism has been incorporated into the fractional-order model system to maximize the product output. Euler-Lagrange optimality conditions are derived for the FOCP (fractional order control problem) using maximum principle. Numerical iterative schemes have been developed to solve the fractional order optimal control problem through Matlab.

Słowa kluczowe

EN

enzyme kinetics; cooperative phenomenon; fractional derivative; mathematical modeling; Hamiltonian; optimal control problem

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2015
• Tom: Vol. 44, no. 4
• Strony: 443--461
• Opis fizyczny: Bibliogr. 39 poz., rys., tab.

Twórcy

autor

Al-Basir F.

• Center for Mathematical Biology and Ecology Department of Mathematics, Jadavpur University, Kolkata - 700032, India

autor

Elaiw A. M.

• Department of Mathematics, Faculty of Science King Abdulaziz University P.O. Box 80203, Jeddah 21589, Saudi Arabia

autor

Kesh D.

• Center for Mathematical Biology and Ecology Department of Mathematics, Jadavpur University, Kolkata - 700032, India

autor

Roy P. K.

• Center for Mathematical Biology and Ecology Department of Mathematics, Jadavpur University, Kolkata - 700032, India

Bibliografia

• [1] Abdullah F. A. (2011) Using fractional differential equations to model the Michaelis-Menten reaction in a 2-D region containing obstacles. Science Asia, 37(1), 75–78.
• [2] Al Basir F., Datta S., Roy P. K. (2015) Studies on Biodiesel Production from Jatropha Curcas Oil using Chemical and Biochemical methods - A Mathematical Approach. Fuel, 158, 503–511.
• [3] Antonini E., Brunori M. (1971) Hemoglobin and Myoglobin in their Reactions with Ligands. North-Holland Publishing Co, Amsterdam.
• [4] Agrawal O.P. (2008) A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control, 14 (9-10), 1291-1299.
• [5] Agrawal, O.P., Defterli, O., Baleanu, D. (2010) Fractional optimal control problems with several state and control variables. J. Vib. Control, 16 (13), 1967-1976.
• [6] Aguilar G. F., Francisco J., Razo-Hernandez J. R., Rosales-Garcia J., Gua-Calderon M. (2014) Fractional RC and LC Electrical Circuits. Ingeniera Investigacin y Tecnologa, 2, 311-319.
• [7] Ahmed A. A. (2013) Application of the Multistep Generalized Differential Transform Method to Solve a Time-Fractional Enzyme Kinetics. Discrete Dynamics in Nature and Society, 2013.
• [8] Basir, F. A., Roy, P. K. (2015) Effects of Temperature and Stirring on Mass Transfer to Maximize Biodiesel Production from Jatropha curcas Oil: A Mathematical Study. International Journal of Engineering Mathematics. DOI: 10.1155/2015/278275
• [9] Changpin L., Zeng F. (2015) Numerical Methods for Fractional Calculus. CRC Press, Taylor and Francis Group, 24.
• [10] Craiem D. O., Rojo, F. J., Atienza, J. M., Guinea, G. V., and Armentano, R. L. (2008) Fractional calculus applied to model arterial viscoelasticity. Latin American Applied Research, 38 (2), 141–145.
• [11] Ding Y., Wang Z., Ye H. (2012) Optimal Control of a Fractional-OrderHIVImmune System With Memory. IEEE Transactions On Control Systems Technology, 20(3).
• [12] Delavari H., Lanusse, P., Sabatier, J. (2013) Fractional order controller design for a flexible link manipulator robot. Asian J. Control, 15 (3), 783–795.
• [13] Du, M., Wang, Z., Hu, H. (2013) Measuring memory with the order of fractional derivative. Scientific Report 3, Nature Publishing Group, DOI: 10.1038/srep03431
• [14] Fleming W., Rishel R., (1975) Deterministic and Stochastic Optimal Controls. Springer Verlag.
• [15] Henry E. R., Bettati S., Hofrichter J., Eaton W. A. (2002) A tertiary two-state allosteric model for hemoglobin. Biophysical Chemistry, 98, 149– 164.
• [16] Kilbas A. A., Srivastava H. M., Trujillo J. J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, 204. Elsevier.
• [17] Lin W. (2007) Global existence theory and chaos control of fractional differential equations. JMAA 332, 709–726.
• [18] Moreau X., Ramus-Serment C., Oustaloup A. (2008) Fractional differentiation in passive vibration control. Nonlinear Dyn., 29, 343–362.
• [19] Magin R. (2008) Fractional Calculus in Bioengineering. Connecticut, USA. Begell House Inc. Publishers, Redding.
• [20] Murray J. D. (1989) Mathematical Biology. Springer.
• [21] Nandi S., Ghosh M. K., Bhattacharya R., Roy P. K. (2013) Mathematical modeling to optimize the product in enzyme kinetics. Control and Cybernetics, 42 (2)
• [22] Odibat, Z. M., Shawagfeh, N. T. (2007) Generalized Taylor’s formula. Applied Mathematics and Computation 186, 286–293.
• [23] Perutz M. F. (1970) Stereochemistry of cooperative effects in haemoglobin. Nature 228, 726–739.
• [24] Podlubny I., Chen Y. (2007) Adjoint fractional differential expressions and operators. Proc. ASME (2007) Int. Design Eng. Techn. Conf. & Comput. Inform. Eng. Conf., ASME, DETC2007–35005.
• [25] Rana S., Bhattacharya S., Pal J., Ngurkata G. M., Chattopadhyay J. (2013) Paradox of enrichment: A fractional differential approach with memory. Physica A 392, 610–3621.
• [26] Roberts D.V. (1977) Enzyme Kinetics. Cambridge.
• [27] Roy P.K., Mondal J., Rana S., Datta A. (2013) Host pathogen interactions with recovery rate using fractional-order derivative: A mathematical approach. Nonlinear Studies 20 (2) 251-261.
• [28] Roy P. K., Datta S., Nandi S., Basir F. A. (2014) Effect of mass transfer kinetics for maximum production of biodiesel from Jatropha Curcas oil: A mathematical approach. Fuel 134, 39-44.
• [29] Roy P. K., Nandi S., Ghosh M. K. (2013) Modeling of a control induced system for product formation in enzyme kinetics. Journal of Mathematical Chemistry, 51 (7), 2704–2717.
• [30] Rubinow Sol I. (1975) Introduction to Mathematical Biology. John Wiley, New York.
• [31] Szabo A., Karplus M. (1972) A mathematical model for structure-function relations in hemoglobin. J. Mol. Biol., 72, 163–197.
• [32] Segel L.A. (1980) Mathematical Models in Molecular and Cellular Biology. Cambridge University Press.
• [33] Sabatier J., Agrawal O.P., Teneiro-Machado J.A. (2007) Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Berlin.
• [34] Sun H.G., Chen W.,Wei H., Chen Y.Q. (2011) A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. The Eur. Phys. J. Spec. Top. 193, 185– 192.
• [35] Swan G. M. (1984) Application of Optimal Control Theory in Biomedicine. Chapman & Hall / CRC Pure and Applied Mathematics.
• [36] Sethi S.P., Thompson G.L. (2000) Optimal Control Theory, Applications to Management Science and Economics. Springer Verlag (2nd edition).
• [37] Stengel R. F. (1994) Optimal Control and Estimation. Dover Publications, Inc., New York.
• [38] Toledo-Hernandez R., Rico-Ramirez V., Gustavo A., Silva I., Diwekar U. M. (2014a) A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reactions. Chemical Engineering Science, 117, 217–228.
• [39] Toledo-Hernandez R., Rico-Ramirez V., Rico-Martinez R., HernandezCastro S., Diwekar U. M. (2014b) A fractional calculus approach to the dynamic optimization of biological reactive systems. Part II: Numerical solution of fractional optimal control problems. Chemical Engineering Science 117, 239–247.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.