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Tikhonov's theorem and quasi-steady state
1. | Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany, Germany |
References:
[1] |
P. Atkins and J. de Paula, "Physical Chemistry,'' 8th edition, Oxford Univ. Press, Oxford, 2006. |
[2] |
J. M. Berg, J. L. Tymovzko and L. Stryer, "Biochemistry: International Edition,'' 6th edition, Palgrave Macmillan, New York, 2006. |
[3] |
Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,'' Lecture Notes in Mathematics, 702, Springer, Berlin, 1979. |
[4] |
J. A. M. Borghans, R. J. de Boer and L. A. Segel, Extending the quasi-steady state approximation by changing variables, Bull. Math. Biol., 58 (1996), 43-63.
doi: 10.1007/BF02458281. |
[5] |
S. Cunha Orfao, G. Jank, K. Mottaghy, S. Walcher and E. Zerz, Qualitative properties and stabilizability of a model for blood thrombin formation, J. Math. Anal. Appl., 346 (2008), 218-226.
doi: 10.1016/j.jmaa.2008.05.060. |
[6] |
P. Duchêne and P. Rouchon, Kinetic scheme reduction via geometric singular perturbation techniques, Chem. Engineering Sci., 51 (1996), 4461-4472. |
[7] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqs., 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[8] |
S. J. Fraser and M. R. Roussel, Phase plane geometries in enzyme kinetics, Can. J. Chem., 72 (1993), 800-812.
doi: 10.1139/v94-107. |
[9] |
W. Gröbner and H. Knapp (Eds.), "Contributions to the Method of Lie Series,'' Bibliographisches Institut, Mannheim, 1967 |
[10] |
F. C. Hoppensteadt, Singular perturbations on the infinite interval, Trans. Amer. Math. Soc., 123 (1966), 521-535.
doi: 10.1090/S0002-9947-1966-0194693-9. |
[11] |
C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems, Montecatini Terme, 1994,'' (ed. R. Johnson), Lecture Notes in Mathematics, 1609, Springer, Berlin, (1995), 44 - 118. |
[12] |
H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165 (2002), 66-93.
doi: 10.1016/S0167-2789(02)00386-X. |
[13] |
J. Keener and J. Sneyd, "Mathematical Physiology,'' Springer, New York, 1998. |
[14] |
S. Lang, "Algebra,'' 2nd edition, Addison-Wesley, Reading, 1984. |
[15] |
L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. |
[16] |
J. D. Murray, "Mathematical Biology,'' 2nd edition Springer, New York, 1993.
doi: 10.1007/b98869. |
[17] |
L. Noethen, "Quasi-Stationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen,'' Ph.D. thesis, RWTH Aachen, 2008. |
[18] |
L. Noethen and S. Walcher, Quasi-steady state in the Michaelis-Menten system, Nonlin. Analysis Real World Appl., 8 (2007), 1512-1535.
doi: 10.1016/j.nonrwa.2006.08.004. |
[19] |
L. Noethen and S. Walcher, Quasi-steady state and nearly invariant sets, SIAM J. Appl. Math., 70 (2009), 1341-1363.
doi: 10.1137/090758180. |
[20] |
M. Schauer and R. Heinrich, Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction, J. Theor. Biol., 79 (1979), 425-442.
doi: 10.1016/0022-5193(79)90235-2. |
[21] |
M. Schauer and R. Heinrich, Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks, Math. Biosci., 65 (1983), 155-170.
doi: 10.1016/0025-5564(83)90058-5. |
[22] |
S. Schnell and P. K. Maini, Enzyme kinetics at high enzyme concentration, Bull. Math. Biol., 62 (2000), 483-499.
doi: 10.1006/bulm.1999.0163. |
[23] |
L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Review, 31 (1989), 446-477.
doi: 10.1137/1031091. |
[24] |
M. Seshadri and G. Fritzsch, Analytical solutions of a simple enzyme kinetic problem by a perturbative procedure, Biophys. Struct. Mech., 6 (1980), 111-123.
doi: 10.1007/BF00535748. |
[25] |
M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593-609.
doi: 10.1007/s002850050116. |
[26] |
A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb., 31 (1952), 575-586. |
[27] |
A. R. Tzafriri and E. R. Edelman, The total quasi-steady-state approximation is valid for reversible enzyme kinetics, J. Theoret. Biol., 226 (2004), 303-313.
doi: 10.1016/j.jtbi.2003.09.006. |
[28] |
A. B. Vasil'eva, Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russ. Math. Surveys, 18 (1963), 13-84.
doi: 10.1070/RM1963v018n03ABEH001137. |
[29] |
F. Verhulst, "Methods and Applications of Singular Perturbations,'' Springer, Berlin, 2005.
doi: 10.1007/0-387-28313-7. |
show all references
References:
[1] |
P. Atkins and J. de Paula, "Physical Chemistry,'' 8th edition, Oxford Univ. Press, Oxford, 2006. |
[2] |
J. M. Berg, J. L. Tymovzko and L. Stryer, "Biochemistry: International Edition,'' 6th edition, Palgrave Macmillan, New York, 2006. |
[3] |
Yu. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations,'' Lecture Notes in Mathematics, 702, Springer, Berlin, 1979. |
[4] |
J. A. M. Borghans, R. J. de Boer and L. A. Segel, Extending the quasi-steady state approximation by changing variables, Bull. Math. Biol., 58 (1996), 43-63.
doi: 10.1007/BF02458281. |
[5] |
S. Cunha Orfao, G. Jank, K. Mottaghy, S. Walcher and E. Zerz, Qualitative properties and stabilizability of a model for blood thrombin formation, J. Math. Anal. Appl., 346 (2008), 218-226.
doi: 10.1016/j.jmaa.2008.05.060. |
[6] |
P. Duchêne and P. Rouchon, Kinetic scheme reduction via geometric singular perturbation techniques, Chem. Engineering Sci., 51 (1996), 4461-4472. |
[7] |
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eqs., 31 (1979), 53-98.
doi: 10.1016/0022-0396(79)90152-9. |
[8] |
S. J. Fraser and M. R. Roussel, Phase plane geometries in enzyme kinetics, Can. J. Chem., 72 (1993), 800-812.
doi: 10.1139/v94-107. |
[9] |
W. Gröbner and H. Knapp (Eds.), "Contributions to the Method of Lie Series,'' Bibliographisches Institut, Mannheim, 1967 |
[10] |
F. C. Hoppensteadt, Singular perturbations on the infinite interval, Trans. Amer. Math. Soc., 123 (1966), 521-535.
doi: 10.1090/S0002-9947-1966-0194693-9. |
[11] |
C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems, Montecatini Terme, 1994,'' (ed. R. Johnson), Lecture Notes in Mathematics, 1609, Springer, Berlin, (1995), 44 - 118. |
[12] |
H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165 (2002), 66-93.
doi: 10.1016/S0167-2789(02)00386-X. |
[13] |
J. Keener and J. Sneyd, "Mathematical Physiology,'' Springer, New York, 1998. |
[14] |
S. Lang, "Algebra,'' 2nd edition, Addison-Wesley, Reading, 1984. |
[15] |
L. Michaelis and M. L. Menten, Die Kinetik der Invertinwirkung, Biochem. Z., 49 (1913), 333-369. |
[16] |
J. D. Murray, "Mathematical Biology,'' 2nd edition Springer, New York, 1993.
doi: 10.1007/b98869. |
[17] |
L. Noethen, "Quasi-Stationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen,'' Ph.D. thesis, RWTH Aachen, 2008. |
[18] |
L. Noethen and S. Walcher, Quasi-steady state in the Michaelis-Menten system, Nonlin. Analysis Real World Appl., 8 (2007), 1512-1535.
doi: 10.1016/j.nonrwa.2006.08.004. |
[19] |
L. Noethen and S. Walcher, Quasi-steady state and nearly invariant sets, SIAM J. Appl. Math., 70 (2009), 1341-1363.
doi: 10.1137/090758180. |
[20] |
M. Schauer and R. Heinrich, Analysis of the quasi-steady-state approximation for an enzymatic one-substrate reaction, J. Theor. Biol., 79 (1979), 425-442.
doi: 10.1016/0022-5193(79)90235-2. |
[21] |
M. Schauer and R. Heinrich, Quasi-steady-state approximation in the mathematical modeling of biochemical reaction networks, Math. Biosci., 65 (1983), 155-170.
doi: 10.1016/0025-5564(83)90058-5. |
[22] |
S. Schnell and P. K. Maini, Enzyme kinetics at high enzyme concentration, Bull. Math. Biol., 62 (2000), 483-499.
doi: 10.1006/bulm.1999.0163. |
[23] |
L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Review, 31 (1989), 446-477.
doi: 10.1137/1031091. |
[24] |
M. Seshadri and G. Fritzsch, Analytical solutions of a simple enzyme kinetic problem by a perturbative procedure, Biophys. Struct. Mech., 6 (1980), 111-123.
doi: 10.1007/BF00535748. |
[25] |
M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593-609.
doi: 10.1007/s002850050116. |
[26] |
A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Mat. Sb., 31 (1952), 575-586. |
[27] |
A. R. Tzafriri and E. R. Edelman, The total quasi-steady-state approximation is valid for reversible enzyme kinetics, J. Theoret. Biol., 226 (2004), 303-313.
doi: 10.1016/j.jtbi.2003.09.006. |
[28] |
A. B. Vasil'eva, Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russ. Math. Surveys, 18 (1963), 13-84.
doi: 10.1070/RM1963v018n03ABEH001137. |
[29] |
F. Verhulst, "Methods and Applications of Singular Perturbations,'' Springer, Berlin, 2005.
doi: 10.1007/0-387-28313-7. |
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