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

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

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

Karl Peter Hadeler ^1,

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

Received January 2013 Revised March 2013 Published August 2013

Abstract
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Under a Creative Commons license

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).

Keywords: enzyme kinetics, nonlinear least squares, Parameter identification, operator repressor kinetics..

Mathematics Subject Classification: Primary: 92C45, 93E24; Secondary: 34C12, 34D1.

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), 338-339.
[2]	K. P. Hadeler and D. Jukić, K. Sabo, Least squares problems for Michaelis-Menten kinetics, Math. Meth. Appl. Sci., 30 (2007), 1231-1241. doi: 10.1002/mma.835.
[3]	M. W. Hirsch and H. L. Smith, Monotone dynamical systems, in "Handbook of Differential Equations, Vol. 2," Elsevier B. V., Amsterdam, (2006), 239-357
[4]	M. C. Mackey, M. Tyran-Kaminska and R. Yvinec, Molecular distributions in gene regulatory dynamics, J. Theoretical Biology, 274 (2011), 84-96. doi: 10.1016/j.jtbi.2011.01.020.
[5]	L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.
[6]	L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945-961. doi: 10.3934/dcdsb.2011.16.945.
[7]	L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, Thermochim. Acta, 31 (1989), 446-477. doi: 10.1137/1031091.
[8]	Gad Yail and Ezra Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis, Biophysical Journal, 11 (1971), 11-27.

show all references

References:

[1]	G. E. Briggs and J. B. S. Haldane, A note on the kinematics of enzyme action, Biochem. J., 19 (1925), 338-339.
[2]	K. P. Hadeler and D. Jukić, K. Sabo, Least squares problems for Michaelis-Menten kinetics, Math. Meth. Appl. Sci., 30 (2007), 1231-1241. doi: 10.1002/mma.835.
[3]	M. W. Hirsch and H. L. Smith, Monotone dynamical systems, in "Handbook of Differential Equations, Vol. 2," Elsevier B. V., Amsterdam, (2006), 239-357
[4]	M. C. Mackey, M. Tyran-Kaminska and R. Yvinec, Molecular distributions in gene regulatory dynamics, J. Theoretical Biology, 274 (2011), 84-96. doi: 10.1016/j.jtbi.2011.01.020.
[5]	L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.
[6]	L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945-961. doi: 10.3934/dcdsb.2011.16.945.
[7]	L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, Thermochim. Acta, 31 (1989), 446-477. doi: 10.1137/1031091.
[8]	Gad Yail and Ezra Yagil, On the relation between effector concentration and the rate of induced enzyme synthesis, Biophysical Journal, 11 (1971), 11-27.

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