Non-equilibrium phase transition in mesoscopic biochemical systems: From stochastic to nonlinear dynamics and beyond

Journal of the Royal Society Interface

Volumn 8, Issue 54, 2011, Pages 107-116

  Ge, Hao ^a   Qian, Hong ^a,b  

^a FUDAN UNIVERSITY   (China)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Bistability; Chemical master equation; Large deviation; Maxwell construction; Non equilibrium phase transition

Indexed keywords

BIFURCATION (MATHEMATICS); CHEMICAL ANALYSIS; CHEMICAL STABILITY; DYNAMICS; FUNCTION EVALUATION; MAXWELL EQUATIONS; NONLINEAR FEEDBACK; PHOSPHORYLATION; STOCHASTIC SYSTEMS; SUPERCONDUCTING MATERIALS;

BI-STABILITY; CHEMICAL MASTER EQUATION; LARGE DEVIATIONS; MAXWELL CONSTRUCTION; NONEQUILIBRIUM PHASE TRANSITIONS;

PHASE TRANSITIONS;

ANALYTIC METHOD; ARTICLE; BIOCHEMISTRY; CHEMICAL ANALYSIS; CHEMICAL REACTION; PHASE TRANSITION; STOCHASTIC MODEL;

EID: 78649786267     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2010.0202     Document Type: Article

References (64)

1. Goldenfeld, N. 1992 Lectures on phase transitions and the renormalization group. Reading, MA: Addison-Wesley.
2. Qian, H. 2006 Open-system nonequilibrium steady state: statistical thermodynamics, fluctuations, and chemical oscillations, J. Phys. Chem. B 110, 15063. (doi:10.1021/jp061858z)
3. Karsenti, E. 2008 Self-organization in cell biology: a brief history. Nat. Rev. Mol. Cell Biol. 9, 255-262. (doi:10.1038/nrm2357)
4. Phillips, R., Kondev, J. & Theriot, J. 2008 Physical biology of the cell. New York, NY: Garland Science.
5. Ferrell, J. E. & Xiong,W. 2001 Bistability in cell signaling: how to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11, 227-236 (doi:10.1063/1.1349894)
6. Qian, H. & Reluga, T. C. 2005 Nonequilibrium thermodynamics and nonlinear kinetics in a cellular signaling switch. Phys. Rev. Lett. 94, 028101. (doi:10.1103/Phys-RevLett.94.028101)
7. Qian, H. 2007 Phosphorylation energy hypothesis: open chemical systems and their biological functions, Annu. Rev. Phys. Chem. 58, 113-142. (doi:10.1146/annurev.physchem.58.032806.104550)
8. Samoilov, M. S. & Arkin, A. P. 2006 Deviant effects in molecular reaction pathways, Nat. Biotechnol. 24, 1235-1240. (doi:10.1038/nbt1253)
9. Gillespie, D. T. 2007 Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem. 58, 35-55. (doi:10.1146/annurev.physchem.58.032806.104637)
10. Beard, D. A. & Qian, H. 2008 Chemical biophysics: quantitative analysis of cellular systems. Cambridge, UK: Cambridge University Press.
11. Liang, J. & Qian, H. 2010 Computational cellular dynamics based on the chemical master equation: A challenge for understanding complexity. J. Comput. Sci. Tech. 25, 154. (doi:10.1007/s11390-010-9312-6)
12. Delbrück, M. 1940 Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120-124. (doi:10.1063/1.1750549)
13. McQuarrie, D. A. 1967 Stochastic approach to chemical kinetics, J. Appl. Probab. 4, 413. (doi:10.2307/3212214)
14. Nicolis, G. & Prigogine, I. 1977 Self-organization in nonequilibrium systems: from dissipative structures to order through fluctuations. New York, NY: Wiley.
15. Gaspard, P. 2004 Fluctuation theorem for nonequilibrium reactions, J. Chem. Phys. 120, 8898-8905. (doi:10.1063/1.1688758)
16. Lax, M. 1960 Fluctuations from the nonequilibrium steady state. Rev. Mod. Phys. 32, 25-64. (doi:10.1103/RevMod-Phys.32.25)
17. Klein, M. J. 1955 Principle of detailed balance, Phys. Rev. 97, 1446-1447. (doi:10.1103/PhysRev.97.1446)
18. Bergmann, P. G. & Lebowitz, J. L. 1955 New approach to nonequilibrium processes. Phys. Rev. 99, 578. (doi:10.1103/PhysRev.99.578)
19. Luo, J.-L., Van den Broeck, C. & Nicolis, G. 1984 Stability criteria and fluctuations around nonequilibrium states. Z. Phys. B 56, 165.
20. Jiang, D.-Q. & Qian, M.-P. 2004 Mathematical theory of nonequilibrium steady states. Lecture Notes in Mathematics, no. 1833. Berlin, Germany: Springer.
21. Kawai, R., Parrondo, J. M. R. & Van den Broeck, C. 2007 Dissipation: the phase-space perspective. Phys. Rev. Lett. 98, 080602.
22. Parrondo, J. M. R., Van den Broeck, C. & Kawai, R. 2009 Entropy production and the arrow of time. New J. Phys. 11, 073008. (doi:10.1088/1367- 2630/11/7/073008)
23. Kurtz, T. G. 1972 The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57, 2976-2978. (doi:10.1063/1.1678692)
24. Dembo, A. & Zeitouni, O. 1998 Large deviations techniques and applications, 2nd edn. New York, NY: Springer.
25. Ellis, R. S. 1985 Entropy, large deviations, and statistical mechanics. New York, NY: Springer.
26. Touchette, H. 2008 The large deviation approach to statistical mechanics. Phys. Rep. 478, 1-69.
27. Ge, H. & Qian, H. 2009 Thermodynamic limit of a nonequilibrium steady state: Maxwell-type construction for a bistable biochemical system. Phys. Rev. Lett. 103, 148103. (doi:10.1103/PhysRevLett.103.148103)
28. Vellela, M. & Qian, H. 2009 Stochastic dynamics and nonequilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited. J. R. Soc. Interface 6, 925.
29. Yang, C. N. & Lee, T. D. 1952 Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev. 87, 404.
30. Lee, T. D. & Yang, C. N. 1952 Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model. Phys. Rev. 87, 410.
31. Zimm, B. H. 1951 Contribution to the theory of critical phenomena. J. Chem. Phys. 19, 1019-1023. (doi:10.1063/1.1748444)
32. Blythe, R. A. & Evans, M. R. 2002 Lee-Yang zeros and phase transitions in nonequilibrium steady states. Phys. Rev. Lett. 89, 080601. (doi:10.1103/PhysRevLett.89.080601)
33. Chomaz, Ph. & Gulminelli, F. 2003 First-order phase transitions: equivalence between bimodalities and the Yang-Lee theorem. Physica A 330, 451-458. (doi:10.1016/j.physa.2003.01.001)
34. Touchette, H. 2005 Comment on 'First-order phase transitions: Equivalence between bimodalities and the Yang-Lee theorem'. Physica A 359, 375-379. (doi:10.1016/j.physa.2005.05.098)
35. Elf, J. & Ehrenberg, M. 2004 Spontaneous separation of bistable biochemical systems into spatial domains of opposite phases. Syst. Biol. 1, 230. (doi:10.1049/sb:20045021)
36. Fange, D. & Elf, J. 2006 Noise-induced Min phenotypes in E. coli. PLoS Comput. Biol. 2, 0637.
37. Kubo, R., Matsuo, K. & Kitahara, K. 1973 Fluctuation and relaxation of macrovariables. J. Stat. Phys. 9, 51-96. (doi:10.1007/BF01016797)
38. Graham, R. & T'el, T. 1984 Existence of a potential for dissipative dynamical systems. Phys. Rev. Lett. 52, 9-12. (doi:10.1103/PhysRevLett.52.9)
39. Hu, G. 1986 Stationary solution of master equations in the large-system-size limit. Z. Physik B 65, 103-106.
40. G'omez-Uribe, C. A. & Verghese, G. C. 2007 Mass fluctuation kinetics: capturing stochastic effects in systems of chemical reactions through coupled mean-variance computations. J. Chem. Phys. 126, 024109. (doi:10.1063/1.2408422)
41. Kim, K. H., Qian, H. & Sauro, H. M. 2008 Sensitivity regulation based on noise propagation in stochastic reaction networks. Preprint, University of Washington, USA (http://arxiv.org/abs/0805.4455).
42. Zeeman, E. C. 1976 In Structural stability, the theory of catastrophies, and applications in the sciences (eds A. Dold & B. Eckmann), Lecture notes in mathematics, no. 525, 328. Berlin, Germany: Springer.
43. Thom, R. 1977 Stabilite Structurelle et Morphogenese. Paris, France: Intereditions.
44. Gaite, J. A. 1990 Phase transitions as catastrophes: the tricritical point. Phys. Rev. A 41, 5320-5324. (doi:10.1103/PhysRevA.41.5320)
45. Laughlin, R. B., Pines, D., Schmalian, J., Stojkovi, B. P. & Wolynes, P. G. 2000 The middle way. Proc. Natl Acad. Sci. USA 97, 32-37. (doi:10.1073/pnas.97.1.32)
46. Qian, H., Shi, P. Z. & Xing, J. 2009 Stochastic bifurcation, slow fluctuations, and bistability as an origin of biochemical complexity. Phys. Chem. Chem. Phys. 11, 4861-4870. (doi:10.1039/b900335p)
47. Zeeman, E. C. 1988 Stability of dynamical systems. Nonlinearity 1, 115-155. (doi:10.1088/0951-7715/1/1/005)
48. Graham, R. & Haken, H. 1971 Generalized thermodynamic potential for Markoff systems in detailed balance and far from thermal equilibrium. Z. Physik A 243, 289-302.
49. Ross, J., Hunt, K. L. C. & Vlad, M. O. 2002 Determination of thermodynamic and stochastic potentials in nonequilibrium systems from macroscopic measurements. J. Phys. Chem. A 106, 10 951-10 960 (doi:10.1021/jp020736d)
50. Ao, P. 2004 Potential in stochastic differential equations: novel construction. J. Phys. A 37, L25-L30. (doi:10.1088/0305-4470/37/3/L01)
51. Nicolis, G. & Lefever, R. 1977 Comment on the kinetic potential and the maxwell construction in non-equilibrium chemical phase transitions. Phys. Lett. A 62, 469-471. (doi:10.1016/0375-9601(77)90069-X)
52. Nicolis, G. & Turner, J. W. 1977 Stochastic analysis of a nonequilibrium phase transition: Some exact results. Physica A 89, 245. (doi:10.1016/0378-4371(77)90107-8)
53. Van Kampen, N. G. 1983 Stochastic processes in physics and chemistry. Amsterdam, The Netherlands: North- Holland.
54. Gardiner, C. W. 1985 Handbook of stochastic methods. Berlin, Germany: Springer.
55. Baras, F., Pearson, J. E. & Malek Mansour, M. 1990 Microscopic simulation of chemical oscillations in homogeneous systems. J. Chem. Phys. 93, 5747-5750. (doi:10.1063/1.459568)
56. Geysermans, P. & Baras, F. 1996 Particle simulation of chemical chaos. J. Chem. Phys. 105, 1402-1408. (doi:10.1063/1.472032)
57. Ge, H. & Qian, H. 2010 Asymptotic limit of a singularly perturbed stationary diffusion equation: the case of a limit cycle. Preprint, Fudan University, China.
58. Kurtz, T. G. 1976 Limit theorems and diffusion approximations for density. Math. Progr. Stud. 5, 67.
59. Kurtz, T. G. 1978 Strong approximation theorems for density dependent Markov chains. Stochast. Proc. Appl. 6, 223-240. (doi:10.1016/0304-4149(78) 90020-0)
60. Baras, F. & Malek Mansour, M. 1997 Microscopic simulations of chemical instabilities. Adv. Chem Phys. 100, 393-474. (doi:10.1002/ 9780470141595.ch5)
61. Malek Mansour, M., Van den Broeck, C., Nicolis, G. & Turner, J. W. 1981 Asymptotic properties of markovian master equations. Ann. Phys. (USA) 131, 283-313. (doi:10.1016/0003-4916(81)90033-6)
62. Baras, F, Malek Mansour, M. & Pearson, J. E. 1996 Microscopic simulation of chemical bistability in homogeneous systems. J. Chem. Phys. 105, 8257-8261. (doi:10.1063/1.472679)
63. Horsthemke, W., Malek Mansour, M. & Brenig, L. 1977 Mean-field theory of critical behaviour and first order transition: a comparison between the master equation and the nonlinear Fokker-Planck equation. Z. Physik B 28, 135-139. (doi:10.1007/BF01325452)
64. Huang, K. 1963 Statistical mechanics. New York, NY: Wiley.