148
S. Yuvan and M. Bier
the quorum is met, a mean-field approach applies and a phase transition with a critical
exponent of 1/2 once again occurs.
Acknowledgements We are grateful to Katja Lindenberg, Hans Westerhoff, and Bartek Lisowski
for very useful feedback on earlier versions of this manuscript.
References
1. S. Arajs, B.L. Tehan, E.E. Anderson, A.A. Stelmach, Critical magnetic behavior of nickel near
the Curie point. Phys. Stat. Sol. 41(2), 639–648 (1970)
2. E. Batschelet, Circular Statistics in Biology (Academic Press, New York, 1981), pp. 54–58
3. M. Bier, B. Teusink, B.N. Kholodenko, H.V. Westerhoff, Control analysis of glycolytic oscillations. Biophys. Chem. 62(1), 15–24 (1996)
4. M. Bier, B.M. Bakker, H.V. Westerhoff, How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment. Biophys. J. 78(3), 1087–1093 (2000)
5. M. Bier, B. Lisowski, E. Gudowska-Nowak, Phase transitions and entropies for synchronizing
oscillators. Phys. Rev. E 93(1), 012143 (2016)
6. B. Blocken, T. van Druenen, Y. Toparlar, F. Malizia, P. Mannion, T. Andrianne, T. Marchal,
G.J. Maas, J. Diepens, Aerodynamic drag in cycling pelotons: new insights by CFD simulation
and wind tunnel testing. J. Wind Eng. Ind. Aerod. 179, 319–337 (2018)
7. E. Borysewicz, Bicycle Road Racing: The Complete Program for Training and Competition
(Velo-News, Boulder, CO, 2005)
8. W. Boyce, R. DiPrima, Elementary Differential Equations, 10th edn. (Wiley, Hoboken, NJ,
2012)
9. J.P. Broker, C.R. Kyle, E.R. Burke, Racing cyclist power requirements in the 4000-m individual
and team pursuits. Med. Sci. Sports Exerc. 31(11), 1677 (1999)
10. P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University
Press, Cambridge, UK, 1995)
11. K. Dhahri, N. Dhahri, J. Dhahri, K. Taibi, E.K. Hlil, Critical phenomena and estimation of
the spontaneous magnetization from a mean field analysis of the magnetic entropy change in
La 0.7 Ca 0.1 Pb 0.2 Mn 0.95 Al 0.025 Sn 0.025 O 3 . RSC Adv. 8(6), 3099–3107 (2018)
12. B. Eckhardt, E. Ott, S.H. Strogatz, D.M. Abrams, A. McRobie, Modeling walker synchronization on the Millennium Bridge. Phys. Rev. E 75(2), 021110 (2007)
13. E.A. Guggenheim, The principle of corresponding states. J. Chem. Phys. 13(7), 253–261 (1945)
14. R. Heinrich, T.A. Rapoport, A linear steady-state treatment of enzymatic chains. Eur. J.
Biochem. 42(1), 89–95 (1974)
15. E.A. Jackson, Perspectives of Nonlinear Dynamics, vol. 2 (Cambridge University Press, New
York, 1991)
16. H. Kacser, The control of flux. Symp. Soc. Exp. Biol. 27, 65–104 (1973)
17. L.P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. Palciauskas, M. Rayl, J.
Swift, D. Aspnes, J. Kane, Static phenomena near critical points: theory and experiment. Rev.
Mod. Phys. 39(2), 395–431 (1967)
18. P. Läuger, Electrogenic Ion Pumps (Sinauer Associates Inc., Sunderland, MA, 1991)
19. W. Moore, Physical Chemistry, 5th edn. (Prentice-Hall, Englewood, New Jersey, 1974)
20. G. Nicolis, I. Prigogine, Self-organization in Non-Equilibrium Systems: From Dissipative Structures to Order Through Fluctuations (Wiley, New York, 1977)
21. B. Novák, J.J. Tyson, Design principles of biochemical oscillators. Nat. Rev. Mol. Cell Biol.
9(12), 981–991 (2008)
22. M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd edn. (World Scientific Publishing, Singapore, 1994)
S. Yuvan and M. Bier
the quorum is met, a mean-field approach applies and a phase transition with a critical
exponent of 1/2 once again occurs.
Acknowledgements We are grateful to Katja Lindenberg, Hans Westerhoff, and Bartek Lisowski
for very useful feedback on earlier versions of this manuscript.
References
1. S. Arajs, B.L. Tehan, E.E. Anderson, A.A. Stelmach, Critical magnetic behavior of nickel near
the Curie point. Phys. Stat. Sol. 41(2), 639–648 (1970)
2. E. Batschelet, Circular Statistics in Biology (Academic Press, New York, 1981), pp. 54–58
3. M. Bier, B. Teusink, B.N. Kholodenko, H.V. Westerhoff, Control analysis of glycolytic oscillations. Biophys. Chem. 62(1), 15–24 (1996)
4. M. Bier, B.M. Bakker, H.V. Westerhoff, How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment. Biophys. J. 78(3), 1087–1093 (2000)
5. M. Bier, B. Lisowski, E. Gudowska-Nowak, Phase transitions and entropies for synchronizing
oscillators. Phys. Rev. E 93(1), 012143 (2016)
6. B. Blocken, T. van Druenen, Y. Toparlar, F. Malizia, P. Mannion, T. Andrianne, T. Marchal,
G.J. Maas, J. Diepens, Aerodynamic drag in cycling pelotons: new insights by CFD simulation
and wind tunnel testing. J. Wind Eng. Ind. Aerod. 179, 319–337 (2018)
7. E. Borysewicz, Bicycle Road Racing: The Complete Program for Training and Competition
(Velo-News, Boulder, CO, 2005)
8. W. Boyce, R. DiPrima, Elementary Differential Equations, 10th edn. (Wiley, Hoboken, NJ,
2012)
9. J.P. Broker, C.R. Kyle, E.R. Burke, Racing cyclist power requirements in the 4000-m individual
and team pursuits. Med. Sci. Sports Exerc. 31(11), 1677 (1999)
10. P.M. Chaikin, T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University
Press, Cambridge, UK, 1995)
11. K. Dhahri, N. Dhahri, J. Dhahri, K. Taibi, E.K. Hlil, Critical phenomena and estimation of
the spontaneous magnetization from a mean field analysis of the magnetic entropy change in
La 0.7 Ca 0.1 Pb 0.2 Mn 0.95 Al 0.025 Sn 0.025 O 3 . RSC Adv. 8(6), 3099–3107 (2018)
12. B. Eckhardt, E. Ott, S.H. Strogatz, D.M. Abrams, A. McRobie, Modeling walker synchronization on the Millennium Bridge. Phys. Rev. E 75(2), 021110 (2007)
13. E.A. Guggenheim, The principle of corresponding states. J. Chem. Phys. 13(7), 253–261 (1945)
14. R. Heinrich, T.A. Rapoport, A linear steady-state treatment of enzymatic chains. Eur. J.
Biochem. 42(1), 89–95 (1974)
15. E.A. Jackson, Perspectives of Nonlinear Dynamics, vol. 2 (Cambridge University Press, New
York, 1991)
16. H. Kacser, The control of flux. Symp. Soc. Exp. Biol. 27, 65–104 (1973)
17. L.P. Kadanoff, W. Götze, D. Hamblen, R. Hecht, E.A.S. Lewis, V.V. Palciauskas, M. Rayl, J.
Swift, D. Aspnes, J. Kane, Static phenomena near critical points: theory and experiment. Rev.
Mod. Phys. 39(2), 395–431 (1967)
18. P. Läuger, Electrogenic Ion Pumps (Sinauer Associates Inc., Sunderland, MA, 1991)
19. W. Moore, Physical Chemistry, 5th edn. (Prentice-Hall, Englewood, New Jersey, 1974)
20. G. Nicolis, I. Prigogine, Self-organization in Non-Equilibrium Systems: From Dissipative Structures to Order Through Fluctuations (Wiley, New York, 1977)
21. B. Novák, J.J. Tyson, Design principles of biochemical oscillators. Nat. Rev. Mol. Cell Biol.
9(12), 981–991 (2008)
22. M. Plischke, B. Bergersen, Equilibrium Statistical Physics, 2nd edn. (World Scientific Publishing, Singapore, 1994)