December  2011, 4(6): 1457-1464. doi: 10.3934/dcdss.2011.4.1457

The dynamics of zeroth-order ultrasensitivity: A critical phenomenon in cell biology

1. 

College of Mathematics, Jilin University, Changchun 130012, China

2. 

Department of Applied Mathematics, University of Washington, Seattle, WA 98195, United States

Received  March 2009 Revised  September 2009 Published  December 2010

It is well known since the pioneering work of Goldbeter and Koshland [Proc. Natl. Acad. Sci. USA, vol. 78, pp. 6840-6844 (1981)] that cellular phosphorylation- dephosphorylation cycle (PdPC), catalyzed by kinase and phosphatase under saturated condition with zeroth order enzyme kinetics, exhibits ultrasensitivity, sharp transition. We analyse the dynamics aspects of the zeroth order PdPC kinetics and show a critical slowdown akin to the phase transition in condensed matter physics. We demonstrate that an extremely simple, though somewhat mathematically "singular" model is a faithful representation of the ultrasentivity phenomenon. The simplified mathematical model will be valuable, as a component, in developing complex cellular signaling netowrk theory as well as having a pedagogic value.
Citation: Qingdao Huang, Hong Qian. The dynamics of zeroth-order ultrasensitivity: A critical phenomenon in cell biology. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1457-1464. doi: 10.3934/dcdss.2011.4.1457
References:
[1]

D. A. Beard and H. Qian, "Chemical Biophysics: Quantitative Analysis of Cellular Systems," Cambridge Texts Biomed. Engr., Cambridge Univ. Press, London, 2008.

[2]

O. G. Berg, J. Paulsson and M. Ehrenberg, Fluctuations and quality of control in biological cells: Zero-order ultrasensitivity reinvestigated, Biophys. J., 79 (2000), 1228-1236. doi: 10.1016/S0006-3495(00)76377-6.

[3]

C. Domb, The critical point: A historical introduction to the modern theory of critical phenomena, Taylor & Francis, London, 1996.

[4]

E. H. Fischer and E. G. Krebs, Conversion of phosphorylase b to phosphorylase a in muscle extracts, J. Biol. Chem., 216 (1955), 121-133.

[5]

H. Ge and M. Qian, Sensitivity amplification in the phosphorylation-dephosphorylation cycle: Nonequilibrium steady states, chemical master equation, and temporal cooperativity, J. Chem. Phys., 129 (2008), 015104. doi: 10.1063/1.2948965.

[6]

A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, Proc. Natl. Acad. Sci. USA, 78 (1981), 6840-6844. doi: 10.1073/pnas.78.11.6840.

[7]

J. Gunawardena, Multisite protein phosphorylation makes a good threshold but can be a poor switch, Proc. Natl. Acad. Sci. USA, 102 (2005), 14617-14622. doi: 10.1073/pnas.0507322102.

[8]

Q. Huang and H. Qian, Ultrasensitive dual phosphorylation dephosphorylation cycle kinetics exhibits canonical competition behavior, Chaos, 19 (2009), 033109. doi: 10.1063/1.3187790.

[9]

N. I. Markevich, J. B. Hoek and B. N. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell Biol., 164 (2004), 353-359. doi: 10.1083/jcb.200308060.

[10]

J. D. Murray, "Mathematical Biology I: An Introduction," 3rd Ed., Springer-Verlag, New York, 2002.

[11]

D. Poland and H. A. Scheraga, "Theory of Helix-Coil Transitions," Academic Press, New York, 1970.

[12]

R. Phillips, J. Kondev and J. Theriot, "Physical Biology of the Cell," Garland Science, New York, 2008.

[13]

H. Qian, Thermodynamic and kinetic analysis of sensitivity amplification in biological signal transduction, Biophys. Chem., 105 (2003), 585-593. doi: 10.1016/S0301-4622(03)00068-1.

[14]

H. Qian, Phosphorylation energy hypothesis: Open chemical systems and their biological functions, Ann. Rev. Phys. Chem., 58 (2007), 113-142. doi: 10.1146/annurev.physchem.58.032806.104550.

[15]

H. Qian and J. A. Cooper, Temporal cooperativity and sensitivity amplification in biological signal transduction, Biochem., 47 (2008), 2211-2220. doi: 10.1021/bi702125s.

[16]

E. R. Stadtman and P. B. Chock, Superiority of interconvertible enzyme cascades in metabolic regulation: Analysis of monocyclic systems, Proc. Natl. Acad. Sci. USA, 74 (1977), 2761-2765. doi: 10.1073/pnas.74.7.2761.

[17]

H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Rev. Mod. Phys., 71 (1999), S358-S366. doi: 10.1103/RevModPhys.71.S358.

[18]

M. Thomson and J. Gunawardena, Unlimited multistability in multisite phosphorylation systems, Nature, 460 (2009), 274-277. doi: 10.1038/nature08102.

[19]

Z.-X. Wang, B. Zhou, Q. M. Wang and Z.-Y. Zhang, A kinetic approach for the study of protein phosphatase-catalyzed regulation of protein kinase activity, Biochem., 41 (2002), 7849-7857. doi: 10.1021/bi025776m.

show all references

References:
[1]

D. A. Beard and H. Qian, "Chemical Biophysics: Quantitative Analysis of Cellular Systems," Cambridge Texts Biomed. Engr., Cambridge Univ. Press, London, 2008.

[2]

O. G. Berg, J. Paulsson and M. Ehrenberg, Fluctuations and quality of control in biological cells: Zero-order ultrasensitivity reinvestigated, Biophys. J., 79 (2000), 1228-1236. doi: 10.1016/S0006-3495(00)76377-6.

[3]

C. Domb, The critical point: A historical introduction to the modern theory of critical phenomena, Taylor & Francis, London, 1996.

[4]

E. H. Fischer and E. G. Krebs, Conversion of phosphorylase b to phosphorylase a in muscle extracts, J. Biol. Chem., 216 (1955), 121-133.

[5]

H. Ge and M. Qian, Sensitivity amplification in the phosphorylation-dephosphorylation cycle: Nonequilibrium steady states, chemical master equation, and temporal cooperativity, J. Chem. Phys., 129 (2008), 015104. doi: 10.1063/1.2948965.

[6]

A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, Proc. Natl. Acad. Sci. USA, 78 (1981), 6840-6844. doi: 10.1073/pnas.78.11.6840.

[7]

J. Gunawardena, Multisite protein phosphorylation makes a good threshold but can be a poor switch, Proc. Natl. Acad. Sci. USA, 102 (2005), 14617-14622. doi: 10.1073/pnas.0507322102.

[8]

Q. Huang and H. Qian, Ultrasensitive dual phosphorylation dephosphorylation cycle kinetics exhibits canonical competition behavior, Chaos, 19 (2009), 033109. doi: 10.1063/1.3187790.

[9]

N. I. Markevich, J. B. Hoek and B. N. Kholodenko, Signaling switches and bistability arising from multisite phosphorylation in protein kinase cascades, J. Cell Biol., 164 (2004), 353-359. doi: 10.1083/jcb.200308060.

[10]

J. D. Murray, "Mathematical Biology I: An Introduction," 3rd Ed., Springer-Verlag, New York, 2002.

[11]

D. Poland and H. A. Scheraga, "Theory of Helix-Coil Transitions," Academic Press, New York, 1970.

[12]

R. Phillips, J. Kondev and J. Theriot, "Physical Biology of the Cell," Garland Science, New York, 2008.

[13]

H. Qian, Thermodynamic and kinetic analysis of sensitivity amplification in biological signal transduction, Biophys. Chem., 105 (2003), 585-593. doi: 10.1016/S0301-4622(03)00068-1.

[14]

H. Qian, Phosphorylation energy hypothesis: Open chemical systems and their biological functions, Ann. Rev. Phys. Chem., 58 (2007), 113-142. doi: 10.1146/annurev.physchem.58.032806.104550.

[15]

H. Qian and J. A. Cooper, Temporal cooperativity and sensitivity amplification in biological signal transduction, Biochem., 47 (2008), 2211-2220. doi: 10.1021/bi702125s.

[16]

E. R. Stadtman and P. B. Chock, Superiority of interconvertible enzyme cascades in metabolic regulation: Analysis of monocyclic systems, Proc. Natl. Acad. Sci. USA, 74 (1977), 2761-2765. doi: 10.1073/pnas.74.7.2761.

[17]

H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Rev. Mod. Phys., 71 (1999), S358-S366. doi: 10.1103/RevModPhys.71.S358.

[18]

M. Thomson and J. Gunawardena, Unlimited multistability in multisite phosphorylation systems, Nature, 460 (2009), 274-277. doi: 10.1038/nature08102.

[19]

Z.-X. Wang, B. Zhou, Q. M. Wang and Z.-Y. Zhang, A kinetic approach for the study of protein phosphatase-catalyzed regulation of protein kinase activity, Biochem., 41 (2002), 7849-7857. doi: 10.1021/bi025776m.

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