# American Institute of Mathematical Sciences

2013, 10(5&6): 1541-1560. doi: 10.3934/mbe.2013.10.1541

## Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches

 1 Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Received  January 2013 Revised  March 2013 Published  August 2013

The Michaelis-Menten (MM) function is a fractional linear function depending on two positive parameters. These can be estimated by nonlinear or linear least squares methods. The non-linear methods, based directly on the defect of the MM function, can fail and not produce any minimizer. The linear methods always produce a unique minimizer which, however, may not be positive. Here we give sufficient conditions on the data such that the nonlinear problem has at least one positive minimizer and also conditions for the minimizer of the linear problem to be positive.
We discuss in detail the models and equilibrium relations of a classical operator-repressor system, and we extend our approach to the MM problem with leakage and to reversible MM kinetics. The arrangement of the sufficient conditions exhibits the important role of data that have a concavity property (chemically feasible data).
Citation: 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
##### References:
 [1] G. E. Briggs and J. B. S. Haldane, A note on the kinematics of enzyme action,, Biochem. J., 19 (1925), 338. Google Scholar [2] K. P. Hadeler and D. Jukić, K. Sabo, Least squares problems for Michaelis-Menten kinetics,, Math. Meth. Appl. Sci., 30 (2007), 1231. doi: 10.1002/mma.835. Google Scholar [3] M. W. Hirsch and H. L. Smith, Monotone dynamical systems,, in, (2006), 239. Google Scholar [4] M. C. Mackey, M. Tyran-Kaminska and R. Yvinec, Molecular distributions in gene regulatory dynamics,, J. Theoretical Biology, 274 (2011), 84. doi: 10.1016/j.jtbi.2011.01.020. Google Scholar [5] L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung,, Biochem. Z., 49 (1913), 333. Google Scholar [6] L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945. doi: 10.3934/dcdsb.2011.16.945. Google Scholar [7] L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation,, Thermochim. Acta, 31 (1989), 446. doi: 10.1137/1031091. Google Scholar [8] Gad Yail and Ezra Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis,, Biophysical Journal, 11 (1971), 11. Google Scholar

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##### References:
 [1] G. E. Briggs and J. B. S. Haldane, A note on the kinematics of enzyme action,, Biochem. J., 19 (1925), 338. Google Scholar [2] K. P. Hadeler and D. Jukić, K. Sabo, Least squares problems for Michaelis-Menten kinetics,, Math. Meth. Appl. Sci., 30 (2007), 1231. doi: 10.1002/mma.835. Google Scholar [3] M. W. Hirsch and H. L. Smith, Monotone dynamical systems,, in, (2006), 239. Google Scholar [4] M. C. Mackey, M. Tyran-Kaminska and R. Yvinec, Molecular distributions in gene regulatory dynamics,, J. Theoretical Biology, 274 (2011), 84. doi: 10.1016/j.jtbi.2011.01.020. Google Scholar [5] L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung,, Biochem. Z., 49 (1913), 333. Google Scholar [6] L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945. doi: 10.3934/dcdsb.2011.16.945. Google Scholar [7] L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation,, Thermochim. Acta, 31 (1989), 446. doi: 10.1137/1031091. Google Scholar [8] Gad Yail and Ezra Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis,, Biophysical Journal, 11 (1971), 11. Google Scholar
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