Skip to main content
SpringerLink
Log in

SDE decomposition and A-type stochastic interpretation in nonequilibrium processes

  • Review Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

An innovative theoretical framework for stochastic dynamics based on the decomposition of a stochastic differential equation (SDE) into a dissipative component, a detailed-balance-breaking component, and a dual-role potential landscape has been developed, which has fruitful applications in physics, engineering, chemistry, and biology. It introduces the A-type stochastic interpretation of the SDE beyond the traditional Ito or Stratonovich interpretation or even the α-type interpretation for multidimensional systems. The potential landscape serves as a Hamiltonian-like function in nonequilibrium processes without detailed balance, which extends this important concept from equilibrium statistical physics to the nonequilibrium region. A question on the uniqueness of the SDE decomposition was recently raised. Our review of both the mathematical and physical aspects shows that uniqueness is guaranteed. The demonstration leads to a better understanding of the robustness of the novel framework. In addition, we discuss related issues including the limitations of an approach to obtaining the potential function from a steady-state distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Wright, The roles of mutation, inbreeding, crossbreeding, and selection in evolution, in: D. F. Jones (Ed.) Proceedings of the Sixth International Congress of Genetics, Vol. 1, 356–366 (1932)

    Google Scholar 

  2. C. H. Waddington, The strategy of the genes: A discussion of some aspects of theoretical biology, New York: MacMillan Company, 1957

    Google Scholar 

  3. S. A. Kauffman, The Origins of Order: Self Organization and Selection in Evolution, New York: Oxford University Press, 1993

    Google Scholar 

  4. X. M. Zhu, L. Yin, L. Hood, and P. Ao, Calculating biological behaviors of epigenetic states in the phage λ life cycle, Funct. Integr. Genomics 4(3), 188 (2004)

    Article  Google Scholar 

  5. P. Ao, Global view of bionetwork dynamics: Adaptive landscape, J. Genet. Genomics 36(2), 63 (2009)

    Article  Google Scholar 

  6. S. Huang, G. Eichler, Y. Bar-Yam, and D. E. Ingber, Cell fates as high-dimensional attractor states of a complex gene regulatory network, Phys. Rev. Lett. 94(12), 128701 (2005)

    Article  ADS  Google Scholar 

  7. P. Ao, Potential in stochastic differential equations: Novel construction, J. Phys. Math. Gen. 37(3), L25 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. R. Yuan and P. Ao, Beyond itô versus stratonovich, J. Stat. Mech. 2012(07), P07010 (2012)

    Article  Google Scholar 

  9. M. I. Freidlin and A. D. Wentzell, Random Perturbations of Dynamical Systems, 3rd Ed., Berlin: Springer-Verlag, 2012

    Book  MATH  Google Scholar 

  10. J. X. Zhou, M. D. S. Aliyu, E. Aurell, and S. Huang, Quasi-potential landscape in complex multi-stable systems, J. R. Soc. Interface Online 1–15 (2012)

    Google Scholar 

  11. R. Yuan, Y. A. Ma, B. Yuan, and P. Ao, Lyapunov function as potential function: A dynamical equivalence, Chin. Phys. B 23(1), 010505 (2014)

    Article  ADS  Google Scholar 

  12. C. Lv, X. Li, F. Li, and T. Li, Constructing the energy landscape for genetic switching system driven by intrinsic noise, PLoS One 9(2), e88167 (2014)

    Article  ADS  Google Scholar 

  13. D. K. Wells, W. L. Kath, and A. E. Motter, Control of stochastic and induced switching in biophysical networks, Phys. Rev. X 5(3), 031036 (2015)

    Google Scholar 

  14. P. Ao, D. Galas, L. Hood, and X. M. Zhu, Cancer as robust intrinsic state of endogenous molecular-cellular network shaped by evolution, Med. Hypotheses 70(3), 678 (2008)

    Article  Google Scholar 

  15. R. Yuan, X. Zhu, G. Wang, S. Li, and P. Ao, Cancer as robust intrinsic state shaped by evolution: A key issues review, Rep. Prog. Phys. 80(4), 042701 (2017)

    Article  ADS  Google Scholar 

  16. Y. Tang, R. Yuan, G. Wang, X. Zhu, and P. Ao, Potential landscape of high dimensional nonlinear stochastic dynamics and rare transitions with large noise, arXiv: 1611.07140 (2016)

    Google Scholar 

  17. G. W. Wang, X. M. Zhu, J. R. Gu, and P. Ao, Quantitative implementation of the endogenous molecularcellular network hypothesis in hepatocellular carcinoma, Interface Focus 4(3), 20130064 (2014)

    Article  Google Scholar 

  18. X. Zhu, R. Yuan, L. Hood, and P. Ao, Endogenous molecular-cellular hierarchical modeling of prostate carcinogenesis uncovers robust structure, Prog. Biophys. Mol. Biol. 117(1), 30 (2015)

    Article  Google Scholar 

  19. S. Li, X. Zhu, B. Liu, G. Wang, and P. Ao, Endogenous molecular network reveals two mechanisms of heterogeneity within gastric cancer, Oncotarget 6, 13607 (2015)

    Article  Google Scholar 

  20. R. Yuan, X. Zhu, J. P. Radich, and P. Ao, From molecular interaction to acute promyelo-cytic leukemia: Calculating leukemogenesis and remission from endogenous molecular-cellular network, Sci. Rep. 6(1), 24307 (2016)

    Article  ADS  Google Scholar 

  21. P. Zhou and T. Li, Construction of the landscape for multi-stable systems: Potential landscape, quasipotential, a-type integral and beyond, J. Chem. Phys. 144(9), 094109 (2016)

    Article  ADS  Google Scholar 

  22. R. Yuan, Y. Tang, and P. Ao, Comment on “construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, a-type integral and beyond”, J. Chem. Phys. 145(14), 147104 (2016) [J. Chem. Phys. 144, 094109 (2016)]

    Article  ADS  Google Scholar 

  23. C. Kwon, P. Ao, and D. J. Thouless, Structure of stochastic dynamics near fixed points, Proc. Natl Acad. Sci. USA 102, 13029 (2005)

    Article  ADS  Google Scholar 

  24. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44, New York: Springer-Verlag, 2012

  25. R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Physics, 2nd Ed., Heidelberg: Springer-Verlag, 1995

    MATH  Google Scholar 

  26. W. M. Haddad and V. S. Chellaboina, Nonlinear Dynamical Systems and Control: A Lyapunov-based Approach, Princeton: Princeton University Press, 2008

    MATH  Google Scholar 

  27. P. Ao, Emerging of stochastic dynamical equalities and steady state thermodynamics from Darwinian dynamics, Commum. Theor. Phys. 49(5), 1073 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  28. L. Yin and P. Ao, Existence and construction of dynamical potential in nonequilibrium processes without detailed balance, J. Phys. A: Math. Gen. 39, 8593 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. H. Ge and H. Qian, Landscapes of non-gradient dynamics without detailed balance: Stable limit cycles and multiple attractors, Chaos 22(2), 023140 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. R. Yuan, X. Wang, Y. Ma, B. Yuan, and P. Ao, Exploring a noisy van der Pol type oscillator with a stochastic approach, Phys. Rev. E 87(6), 062109 (2013)

    Article  ADS  Google Scholar 

  31. Y. Tang, R. Yuan, and Y. Ma, Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems, Phys. Rev. E 87(1), 012708 (2013)

    Article  ADS  Google Scholar 

  32. Y. A. Ma, R. Yuan, Y. Li, B. Yuan, and P. Ao, Lyapunov functions in piecewise linear systems: From fixed point to limit cycle, arXiv: 1306.6880 (2013)

    Google Scholar 

  33. Y. Ma, Q. Tan, R. Yuan, B. Yuan, and P. Ao, Potential function in a continuous dissipative chaotic system: decomposition scheme and role of strange attractor, Int. J. Bifurcat. Chaos 24(02), 1450015 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  34. S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, 2nd Ed., Boulder: Westview Press, 2015

    MATH  Google Scholar 

  35. P. Ao and J. Rammer, Influence of an environment on equilibrium properties of a charged quantum bead constrained to a ring, Superlattices Microstruct. 11(3), 265 (1992)

    Article  ADS  Google Scholar 

  36. Y. C. Chen, M. P. A. Fisher, and A. J. Leggett, The return of a hysteretic Josephson junction to the zerovoltage state: I–V characteristic and quantum retrapping, J. Appl. Phys. 64(6), 3119 (1988)

    Article  ADS  Google Scholar 

  37. Y. Tang, R. Yuan, and P. Ao, Anomalous free energy changes induced by topology, Phys. Rev. E 92(6), 062129 (2015)

    Article  ADS  Google Scholar 

  38. S. Xu, S. Jiao, P. Jiang, and P. Ao, Two-time-scale population evolution on a singular land-scape, Phys. Rev. E 89(1), 012724 (2014)

    Article  ADS  Google Scholar 

  39. P. Ao, C. Kwon, and H. Qian, On the existence of potential landscape in the evolution of complex systems, Complexity 12(4), 19 (2007)

    Article  MathSciNet  Google Scholar 

  40. H. Qian, P. Ao, Y. Tu, and J. Wang, A framework towards understanding mesoscopic phenomena: Emergent unpredictability, symmetry breaking and dynamics across scales, Chem. Phys. Lett. 665(16), 153 (2016)

    Article  ADS  Google Scholar 

  41. Y. Cao, H. M. Lu, and J. Liang, Probability landscape of heritable and robust epigenetic state of lysogeny in phage lambda, Proc. Natl. Acad. Sci. USA 107(43), 18445 (2010)

    Article  ADS  Google Scholar 

  42. M. Lu, J. Onuchic, and E. Ben-Jacob, Construction of an effective landscape for multistate genetic switches, Phys. Rev. Lett. 113(7), 078102 (2014)

    Article  ADS  Google Scholar 

  43. H. Qian, The zeroth law of thermodynamics and volume-preserving conservative system in equilibrium with stochastic damping, Phys. Lett. A 378(7–8), 609 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Y. Tang, R. Yuan, and P. Ao, Summing over trajectories of stochastic dynamics with multi-plicative noise, J. Chem. Phys. 141(4), 044125 (2014)

    Article  ADS  Google Scholar 

  45. P. Ao, T. Q. Chen, and J. H. Shi, Dynamical decomposition of Markov processes without detailed balance, Chin. Phys. Lett. 30(7), 070201 (2013)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. NSFC91329301 and NSFC9152930016) and grants from the State Key Laboratory of Oncogenes and Related Genes (Grant No. 90-10-11).

Author information

Authors and Affiliations

  1. Key Laboratory of Systems Biomedicine, Ministry of Education, Shanghai Center for Systems Biomedicine, Shanghai Jiao Tong University, Shanghai, 200240, China

    Ruoshi Yuan & Ping Ao

  2. Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China

    Ying Tang

Authors
  1. Ruoshi Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ying Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ping Ao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ping Ao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, R., Tang, Y. & Ao, P. SDE decomposition and A-type stochastic interpretation in nonequilibrium processes. Front. Phys. 12, 120201 (2017). https://doi.org/10.1007/s11467-017-0718-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-017-0718-2

Keywords

Search

Navigation