# American Institute of Mathematical Sciences

• Previous Article
Towards a long-term model construction for the dynamic simulation of HIV infection
• MBE Home
• This Issue
• Next Article
Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland
2007, 4(3): 471-488. doi: 10.3934/mbe.2007.4.471

## A theoretic control approach in signal-controlled metabolic pathways

 1 Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, AR 72035, United States, United States 2 Department of Biology, University of Central Arkansas, 201 Donaghey Avenue, Conway, AR 72035, United States

Received  August 2006 Revised  February 2007 Published  May 2007

Cells use a signal transduction mechanism to regulate certain metabolic pathways. In this paper, the regulatory mechanism is analyzed mathematically. For this analysis, a mathematical model for the pathways is first established using a system of differential equations. Then the linear stability, controllability, and observability of the system are investigated. We show that the linearized system is controllable and observable, and that the real parts of all eigenvalues of the linearized system are nonpositive using Routh's stability criterion. Controllability and observability are structural properties of a dynamical system. Thus our results may explain why the metabolic pathways can be controlled and regulated. Finally observer-based and proportional output feedback controllers are designed to regulate the end product to its desired level. Applications to the regulation of blood glucose levels are discussed.
Citation: Ramesh Garimella, Uma Garimella, Weijiu Liu. A theoretic control approach in signal-controlled metabolic pathways. Mathematical Biosciences & Engineering, 2007, 4 (3) : 471-488. doi: 10.3934/mbe.2007.4.471
 [1] Peter W. Bates, Yu Liang, Alexander W. Shingleton. Growth regulation and the insulin signaling pathway. Networks and Heterogeneous Media, 2013, 8 (1) : 65-78. doi: 10.3934/nhm.2013.8.65 [2] Wing-Cheong Lo, Ching-Shan Chou, Kimberly K. Gokoffski, Frederic Y.-M. Wan, Arthur D. Lander, Anne L. Calof, Qing Nie. Feedback regulation in multistage cell lineages. Mathematical Biosciences & Engineering, 2009, 6 (1) : 59-82. doi: 10.3934/mbe.2009.6.59 [3] Richard L Buckalew. Cell cycle clustering and quorum sensing in a response / signaling mediated feedback model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 867-881. doi: 10.3934/dcdsb.2014.19.867 [4] Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control and Optimization, 2016, 6 (2) : 127-151. doi: 10.3934/naco.2016004 [5] Jian Chen, Tao Zhang, Ziye Zhang, Chong Lin, Bing Chen. Stability and output feedback control for singular Markovian jump delayed systems. Mathematical Control and Related Fields, 2018, 8 (2) : 475-490. doi: 10.3934/mcrf.2018019 [6] Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control and Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275 [7] Orit Lavi, Doron Ginsberg, Yoram Louzoun. Regulation of modular Cyclin and CDK feedback loops by an E2F transcription oscillator in the mammalian cell cycle. Mathematical Biosciences & Engineering, 2011, 8 (2) : 445-461. doi: 10.3934/mbe.2011.8.445 [8] Jaroslaw Smieja, Marzena Dolbniak. Sensitivity of signaling pathway dynamics to plasmid transfection and its consequences. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1207-1222. doi: 10.3934/mbe.2016039 [9] David S. Ross, Christina Battista, Antonio Cabal, Khamir Mehta. Dynamics of bone cell signaling and PTH treatments of osteoporosis. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2185-2200. doi: 10.3934/dcdsb.2012.17.2185 [10] N. U. Ahmed. Existence of optimal output feedback control law for a class of uncertain infinite dimensional stochastic systems: A direct approach. Evolution Equations and Control Theory, 2012, 1 (2) : 235-250. doi: 10.3934/eect.2012.1.235 [11] Ta T.H. Trang, Vu N. Phat, Adly Samir. Finite-time stabilization and $H_\infty$ control of nonlinear delay systems via output feedback. Journal of Industrial and Management Optimization, 2016, 12 (1) : 303-315. doi: 10.3934/jimo.2016.12.303 [12] Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375 [13] Gonzalo Robledo. Feedback stabilization for a chemostat with delayed output. Mathematical Biosciences & Engineering, 2009, 6 (3) : 629-647. doi: 10.3934/mbe.2009.6.629 [14] Pep Charusanti, Xiao Hu, Luonan Chen, Daniel Neuhauser, Joseph J. DiStefano III. A mathematical model of BCR-ABL autophosphorylation, signaling through the CRKL pathway, and Gleevec dynamics in chronic myeloid leukemia. Discrete and Continuous Dynamical Systems - B, 2004, 4 (1) : 99-114. doi: 10.3934/dcdsb.2004.4.99 [15] Yacine Chitour, Frédéric Grognard, Georges Bastin. Equilibria and stability analysis of a branched metabolic network with feedback inhibition. Networks and Heterogeneous Media, 2006, 1 (1) : 219-239. doi: 10.3934/nhm.2006.1.219 [16] Krzysztof Fujarewicz, Marek Kimmel, Andrzej Swierniak. On Fitting Of Mathematical Models Of Cell Signaling Pathways Using Adjoint Systems. Mathematical Biosciences & Engineering, 2005, 2 (3) : 527-534. doi: 10.3934/mbe.2005.2.527 [17] Ruth F. Curtain, George Weiss. Strong stabilization of (almost) impedance passive systems by static output feedback. Mathematical Control and Related Fields, 2019, 9 (4) : 643-671. doi: 10.3934/mcrf.2019045 [18] Ahmadreza Argha, Steven W. Su, Lin Ye, Branko G. Celler. Optimal sparse output feedback for networked systems with parametric uncertainties. Numerical Algebra, Control and Optimization, 2019, 9 (3) : 283-295. doi: 10.3934/naco.2019019 [19] Alexei Pokrovskii, Dmitrii Rachinskii. Effect of positive feedback on Devil's staircase input-output relationship. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1095-1112. doi: 10.3934/dcdss.2013.6.1095 [20] Zhao-Han Sheng, Tingwen Huang, Jian-Guo Du, Qiang Mei, Hui Huang. Study on self-adaptive proportional control method for a class of output models. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 459-477. doi: 10.3934/dcdsb.2009.11.459

2018 Impact Factor: 1.313