# American Institute of Mathematical Sciences

2009, 6(1): 173-188. doi: 10.3934/mbe.2009.6.173

## Solution of the Michaelis-Menten equation using the decomposition method

 1 Department of Radiological Sciences, University of Oklahoma Health Sciences Center, Oklahoma City, OK 73190, United States 2 Cell Culture Development, Global Biologics Development, Bayer HealthCare, 800 Dwight Way, Berkeley, CA 94710, United States

Received  December 2007 Revised  August 2008 Published  December 2008

We present a low-order recursive solution to the Michaelis-Menten equation using the decomposition method. This solution is algebraic in nature and provides a simpler alternative to numerical approaches such as differential equation evaluation and root-solving techniques that are currently used to compute substrate concentration in the Michaelis-Menten equation. A detailed characterization of the errors in substrate concentrations computed from decomposition, Runge-Kutta, and bisection methods over a wide range of $s_{0}$:$K_{m}$ values was made by comparing them with highly accurate solutions obtained using the Lambert $W$ function. Our results indicated that solutions obtained from the decomposition method were usually more accurate than those from the corresponding classical Runge-Kutta methods. Moreover, these solutions required significantly fewer computations than the root-solving method. Specifically, when the stepsize was 0.1% of the total time interval, the computed substrate concentrations using the decomposition method were characterized by accuracies on the order of 10$^-8$ or better. The algebraic nature of the decomposition solution and its relatively high accuracy make this approach an attractive candidate for computing substrate concentration in the Michaelis-Menten equation.
Citation: Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173
 [1] Karl Peter Hadeler. Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1541-1560. doi: 10.3934/mbe.2013.10.1541 [2] Urszula Ledzewicz, Helen Moore. Optimal control applied to a generalized Michaelis-Menten model of CML therapy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 331-346. doi: 10.3934/dcdsb.2018022 [3] Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097 [4] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020259 [5] Yangjin Kim, Khalid Boushaba. An enzyme kinetics model of tumor dormancy, regulation of secondary metastases. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1465-1498. doi: 10.3934/dcdss.2011.4.1465 [6] Jeng-Huei Chen. An analysis of functional curability on HIV infection models with Michaelis-Menten-type immune response and its generalization. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2089-2120. doi: 10.3934/dcdsb.2017086 [7] Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143 [8] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [9] Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65 [10] Dan Stanescu, Benito Chen-Charpentier. Random coefficient differential equation models for Monod kinetics. Conference Publications, 2009, 2009 (Special) : 719-728. doi: 10.3934/proc.2009.2009.719 [11] Fang Zeng, Pablo Suarez, Jiguang Sun. A decomposition method for an interior inverse scattering problem. Inverse Problems & Imaging, 2013, 7 (1) : 291-303. doi: 10.3934/ipi.2013.7.291 [12] Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 [13] Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006 [14] Nobu Kishimoto. Resonant decomposition and the $I$-method for the two-dimensional Zakharov system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4095-4122. doi: 10.3934/dcds.2013.33.4095 [15] Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297 [16] Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems & Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523 [17] Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2019034 [18] Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 [19] Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 [20] Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45

2018 Impact Factor: 1.313