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October  2017, 14(5&6): 1499-1514. doi: 10.3934/mbe.2017078

Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

Qiuyan Zhang 1,2, , Lingling Liu 1,3, and  Weinian Zhang 1,3,,

1. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2. 

College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China
3. 

School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China

* Corresponding author: Weinian Zhang

Received  July 02, 2016 Accepted  January 2017 Published  May 2017

Fund Project: Supported by NSFC # 11221101, # 11231001, # 11501475 and SPDEF 16ZB0080

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

Keywords: Enzyme-catalyzed reaction, cusp, Bogdanov-Takens bifurcation, normal form.
Mathematics Subject Classification: 34C23, 92C45.
Citation: Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
References:
[1]

A. Betz and E. Sel'kov, Control of phosphofructokinase [PFK] activity in conditions simulating those of glycolysing yeast extract, FEBS Lett., 3 (1969), 5-9.  doi: 10.1016/0014-5793(69)80082-7.  Google Scholar

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.  Google Scholar

[3]

F. A. Davidson and J. Liu, Global stability of the attracting set of an enzyme-catalysed reaction system, Math. Comput. Model., 35 (2002), 1467-1481.  doi: 10.1016/S0895-7177(02)00098-5.  Google Scholar

[4]

F. A. Davidson, R. Xu and J. Liu, Existence and uniqueness of limit cycles in an enzyme-catalysed reaction system, Appl. Math. Comput., 127 (2002), 165-179.  doi: 10.1016/S0096-3003(01)00065-0.  Google Scholar

[5]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. Dyn. Syst., 7 (1987), 375-413.  doi: 10.1017/S0143385700004119.  Google Scholar

[6]

D. Erle, Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions, J. Math. Anal. Appl., 82 (1981), 386-391.  doi: 10.1016/0022-247X(81)90203-1.  Google Scholar

[7]

D. Erle, K. H. Mayer and T. Plesser, The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208.  doi: 10.1016/0025-5564(79)90081-6.  Google Scholar

[8]

A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608193.  Google Scholar

[9]

A. Goldbeter and G. Dupont, Allosteric regulation, cooperativity and biochemical oscillations, Biophy. Chem., 37 (1990), 341-353.  doi: 10.1016/0301-4622(90)88033-O.  Google Scholar

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1990. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[11]

B. Hassard and K. Jiang, Unfolding a point of degenerate Hopf bifurcation in an enzyme-catalyzed reaction model, SIAM J. Math. Anal., 23 (1992), 1291-1304.  doi: 10.1137/0523072.  Google Scholar

[12]

X. Hou, R. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree n in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.  doi: 10.1016/S0898-1221(02)00108-6.  Google Scholar

[13]

J. P. Kernévez, G. Joly, M. C. Duban, B. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immmobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56.  doi: 10.1007/BF00276413.  Google Scholar

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[15]

Z. Leng, B. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reaction, Math. Comput. Model., 49 (2009), 556-562.  doi: 10.1016/j.mcm.2008.03.006.  Google Scholar

[16]

J. Liu, Coordination restriction of enzyme-catalysed reaction systems as nonlinear dynamical systems, Proc. R. Soc. Lond. A, 455 (1999), 285-298.  doi: 10.1098/rspa.1999.0313.  Google Scholar

[17]

A. G. Marangoni, Enzymes Kinetics: A Modern Approach, Wiley-Interscience, Hoboken, NJ, 2003. doi: 10.1002/0471267295.  Google Scholar

[18]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.   Google Scholar

[19]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, Interdisciplinary Applied Mathematics 17, Springer, Berlin, 2002.  Google Scholar

[20]

H. G. Othmer and J. A. Aldridge, The effects of cell density and metabolite flux on cellular dynamics, J. Math. Biol., 5 (1978), 169-200.  doi: 10.1007/BF00275897.  Google Scholar

[21]

I. Stoleriu, F. A. Davidson and J. Liu, Effects of priodic input on the quasi-steady state assumptions for enzyme-catalyzed reactions, J. Math. Biol., 50 (2005), 115-132.  doi: 10.1007/s00285-004-0282-6.  Google Scholar

[22]

Y. Tang, D. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.  doi: 10.1093/imamat/hxr005.  Google Scholar

[23]

Y. Tang and W. Zhang, Bogdanov-Takens bifurcation of a polynomial differential system in biochemical reaction, Comput. Math. Appl., 48 (2004), 869-883.  doi: 10.1016/j.camwa.2003.05.012.  Google Scholar

[24]

R. Varón, M. García-Moreno, F. García-Molina, M. E. Fuentes, E. Arribas, J. M. Yago, M. Ll. Amo-Saus and E. Valero, Two new regulatory properties arising from the transient phase kinetics of monocyclic enzyme cascades, J. Math. Chem., 38 (2005), 437-450.  doi: 10.1007/s10910-004-6895-6.  Google Scholar

[25]

Y. -Q. Ye et al., Theory of Limit Cycles, Transl. Math. Monogr. 66 American Mathematical Society, Providence, RI, 1986.  Google Scholar

[26]

Z. -F. Zhang, T. -R. Ding, W. -Z. Huang and Z. -X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., 101 Amer. Math. Soc., Providence, RI, 1992.  Google Scholar

[27]

Q. Zhang, L. Liu and W. Zhang, Local bifurcations of the enzyme-catalyzed reaction comprising a branched network, Int. J. Bifur. Chaos, 25 (2015), 155081 (26 pages). doi: 10.1142/S0218127415500819.  Google Scholar

show all references

References:
[1]

A. Betz and E. Sel'kov, Control of phosphofructokinase [PFK] activity in conditions simulating those of glycolysing yeast extract, FEBS Lett., 3 (1969), 5-9.  doi: 10.1016/0014-5793(69)80082-7.  Google Scholar

[2]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.  Google Scholar

[3]

F. A. Davidson and J. Liu, Global stability of the attracting set of an enzyme-catalysed reaction system, Math. Comput. Model., 35 (2002), 1467-1481.  doi: 10.1016/S0895-7177(02)00098-5.  Google Scholar

[4]

F. A. Davidson, R. Xu and J. Liu, Existence and uniqueness of limit cycles in an enzyme-catalysed reaction system, Appl. Math. Comput., 127 (2002), 165-179.  doi: 10.1016/S0096-3003(01)00065-0.  Google Scholar

[5]

F. Dumortier, R. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergod. Theor. Dyn. Syst., 7 (1987), 375-413.  doi: 10.1017/S0143385700004119.  Google Scholar

[6]

D. Erle, Nonuniqueness of stable limit cycles in a class of enzyme catalyzed reactions, J. Math. Anal. Appl., 82 (1981), 386-391.  doi: 10.1016/0022-247X(81)90203-1.  Google Scholar

[7]

D. Erle, K. H. Mayer and T. Plesser, The existence of stable limit cycles for enzyme catalyzed reactions with positive feedback, Math. Biosci., 44 (1979), 191-208.  doi: 10.1016/0025-5564(79)90081-6.  Google Scholar

[8]

A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511608193.  Google Scholar

[9]

A. Goldbeter and G. Dupont, Allosteric regulation, cooperativity and biochemical oscillations, Biophy. Chem., 37 (1990), 341-353.  doi: 10.1016/0301-4622(90)88033-O.  Google Scholar

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1990. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[11]

B. Hassard and K. Jiang, Unfolding a point of degenerate Hopf bifurcation in an enzyme-catalyzed reaction model, SIAM J. Math. Anal., 23 (1992), 1291-1304.  doi: 10.1137/0523072.  Google Scholar

[12]

X. Hou, R. Yan and W. Zhang, Bifurcations of a polynomial differential system of degree n in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.  doi: 10.1016/S0898-1221(02)00108-6.  Google Scholar

[13]

J. P. Kernévez, G. Joly, M. C. Duban, B. Bunow and D. Thomas, Hysteresis, oscillations, and pattern formation in realistic immmobilized enzyme systems, J. Math. Biol., 7 (1979), 41-56.  doi: 10.1007/BF00276413.  Google Scholar

[14]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci., 112, Springer, New York, 1995. doi: 10.1007/978-1-4757-2421-9.  Google Scholar

[15]

Z. Leng, B. Gao and Z. Wang, Qualitative analysis of a generalized system of saturated enzyme reaction, Math. Comput. Model., 49 (2009), 556-562.  doi: 10.1016/j.mcm.2008.03.006.  Google Scholar

[16]

J. Liu, Coordination restriction of enzyme-catalysed reaction systems as nonlinear dynamical systems, Proc. R. Soc. Lond. A, 455 (1999), 285-298.  doi: 10.1098/rspa.1999.0313.  Google Scholar

[17]

A. G. Marangoni, Enzymes Kinetics: A Modern Approach, Wiley-Interscience, Hoboken, NJ, 2003. doi: 10.1002/0471267295.  Google Scholar

[18]

L. Michaelis and M. L. Menten, Die kinetik der invertinwirkung, Biochem. Z., 49 (1913), 333-369.   Google Scholar

[19]

J. D. Murray, Mathematical Biology Ⅰ: An Introduction, Interdisciplinary Applied Mathematics 17, Springer, Berlin, 2002.  Google Scholar

[20]

H. G. Othmer and J. A. Aldridge, The effects of cell density and metabolite flux on cellular dynamics, J. Math. Biol., 5 (1978), 169-200.  doi: 10.1007/BF00275897.  Google Scholar

[21]

I. Stoleriu, F. A. Davidson and J. Liu, Effects of priodic input on the quasi-steady state assumptions for enzyme-catalyzed reactions, J. Math. Biol., 50 (2005), 115-132.  doi: 10.1007/s00285-004-0282-6.  Google Scholar

[22]

Y. Tang, D. Huang and W. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.  doi: 10.1093/imamat/hxr005.  Google Scholar

[23]

Y. Tang and W. Zhang, Bogdanov-Takens bifurcation of a polynomial differential system in biochemical reaction, Comput. Math. Appl., 48 (2004), 869-883.  doi: 10.1016/j.camwa.2003.05.012.  Google Scholar

[24]

R. Varón, M. García-Moreno, F. García-Molina, M. E. Fuentes, E. Arribas, J. M. Yago, M. Ll. Amo-Saus and E. Valero, Two new regulatory properties arising from the transient phase kinetics of monocyclic enzyme cascades, J. Math. Chem., 38 (2005), 437-450.  doi: 10.1007/s10910-004-6895-6.  Google Scholar

[25]

Y. -Q. Ye et al., Theory of Limit Cycles, Transl. Math. Monogr. 66 American Mathematical Society, Providence, RI, 1986.  Google Scholar

[26]

Z. -F. Zhang, T. -R. Ding, W. -Z. Huang and Z. -X. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., 101 Amer. Math. Soc., Providence, RI, 1992.  Google Scholar

[27]

Q. Zhang, L. Liu and W. Zhang, Local bifurcations of the enzyme-catalyzed reaction comprising a branched network, Int. J. Bifur. Chaos, 25 (2015), 155081 (26 pages). doi: 10.1142/S0218127415500819.  Google Scholar

Figure 1.  Reaction scheme
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Figure 2.  Bifurcation surfaces projection on the $(a, \kappa)$-plane
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Figure 3.  Bifurcation diagrams of system (4) for the case that $c>1$ and $b=(c+1)^2/4$
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Figure 4.  An attracting limit cycle
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Table 4.  Dynamics of system (4) in various cases of parameter $(a, \kappa)$
Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram
$E_0$ $E_1$ $E_2$ $E_*$
$ \mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$
$ \mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$
$\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$
$\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$
$\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$
$ \mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$
$ \mathcal{D}_{IV}$ $\mathcal{D}_{IV}$
$(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$
$ \mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
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Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram
$E_0$ $E_1$ $E_2$ $E_*$
$ \mathcal{D}_{I}$ saddle unstable focus saddle $\mathcal{D}_{I}$
$ \mathcal{L}$ saddle unstable focus saddle one homoclinic rrbit $\mathcal{L}$
$\mathcal{D}_{II}$ saddle unstable focus saddle one limit cycle $\mathcal{D}_{II}$
$\mathcal{H}$ saddle stable focus saddle $\mathcal{H}$
$\mathcal{D}_{III}$ saddle stable focus saddle $\mathcal{D}_{III}$
$ \mathcal{SN}^+$ saddle-node $\mathcal{SN}^+$
$ \mathcal{D}_{IV}$ $\mathcal{D}_{IV}$
$(a_*, \kappa_*)$ cusp $(a_*, \kappa_*)$
$ \mathcal{SN}^-$ saddle-node $\mathcal{SN}^-$
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