Delineating the respective impacts of stochastic curl- and grad-forces in a family of idealized core genetic commitment circuits

Abstract

Stochastic dynamics pervades gene regulation. Despite being random, the dynamics displays a kind of innate structure. In fact, two stochastic forces combine driving efforts: one force originates from the gradient of the underlying stochastic potential, and the other originates from the mathematical curl of the probability flux. The curl force gives rise to rotation. The gradient force gives rise to drift. Together they give rise to helical behavior. Here, it is shown that around and about the vicinity of attractive fixed points, the gradient force naturally wanes but the curl force is found to remain high. This leads to a locally noticeably different type of stochastic track near and about attractive fixed points, compared to tracks in regions where drift dominates. The consistency of this observation with the experimental fact that, in biology, fate commitment appears to not be a-priory locked-in, but rather necessitating active maintenance, is discussed. Hence attractive fixed-points are not only fuzzy, but may effectively be, locally, “more free”.

Graphical abstract

Keywords

systems biology / theoretical biology / gene regulation / nonlinear dynamics / stochasticity

References

[1] Xi, H. and Turcotte, M. (2015) Parameter asymmetry and time-scale separation in core genetic commitment circuits. Quant. Biol., 3, 19–45

[2] Wang, J., Xu, L. and Wang, E. K. (2008) Potential landscape and flux framework of nonequilibrium networks: robustness, dissipation, and coherence of biochemical oscillations. Proc. Natl. Acad. Sci. USA, 105, 12271–12276

[3] Wang, J., Zhang, K., Xu, L. and Wang, E. (2011) Quantifying the Waddington landscape and biological paths for development and differentiation. Proc. Natl. Acad. Sci. USA, 108, 8257–8262

[4] Li, C., Wang, E. and Wang, J. (2011) Landscape and flux decomposition for exploring global natures of non-equilibrium dynamical systems under intrinsic statistical fluctuations. Chem. Phys. Lett., 505, 75–80

[5] Li, C., Wang, E. and Wang, J. (2011) Landscape, flux, correlation, resonance, coherence, stability, and key network wirings of stochastic circadian oscillation. Biophys. J., 101, 1335–1344

[6] Zhang, F., Xu, L., Zhang, K., Wang, E., Wang, J. (2012) The potential and flux landscape theory of evolution. J. Chem. Phys. 137, 065102

[7] Li, C., Wang, E. and Wang, J. (2012) Landscape topography determines global stability and robustness of a metabolic network. ACS Synth. Biol., 1, 229–239

[8] Li, C. and Wang, J. (2013) Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation. J. R. Soc. Interface, 10, 20130787

[9] Wu, W. and Wang, J. (2013) Potential and flux field landscape theory. I. Global stability and dynamics of spatially dependent non-equilibrium systems. J. Chem. Phys., 139, 121920

[10] Wu, W. and Wang, J. (2013) Landscape framework and global stability for stochastic reaction diffusion and general spatially extended systems with intrinsic fluctuations. J. Phys. Chem. B, 117, 12908–12934

[11] Li, C. and Wang, J. (2014) Quantifying the underlying landscape and paths of cancer. J. R. Soc. Interface, 11, 20140774

[12] Li, C. and Wang, J. (2014) Landscape and flux reveal a new global view and physical quantification of mammalian cell cycle. Proc. Natl. Acad. Sci. USA, 111, 14130–14135

[13] Xu, L., Zhang, F., Zhang, K., Wang, E. and Wang, J. (2014) The potential and flux landscape theory of ecology. PLoS One, 9, e86746

[14] Wang, J. (2015) Landscape and flux theory of non-equilibrium dynamical systems with application to biology. Adv. Phys., 64, 1–137

[15] Gilbert, D. R., Gilber, N. L. (1910) On Four-Dimensional Vector Analysis, and its application in electrical theory. Proc. Am. Acad. Arts Sci., 46, 165–181

[16] Wang, K. Zhang, L. Xu, E. Wang, E. (2011) Quantifying the Waddington landscape and biological paths for development and differentiation. Proc. Natl. Acad. Sci. USA, 108, 8257–8262

[17] Qian, H. (2011) Nonlinear stochastic dynamics of mesoscopic homogeneous biochemical reaction systems — an analytical theory. Nonlinearity, 24, R19

[18] Gillespie, D. T. (1991) Markov Processes, An Introduction for Physical Scientists. Boston : Academic Press

[19] Gillespie, D. T. (2007) Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58, 35–55

[20] Gillespie, D. T. (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22, 403–434

[21] Gillespie, D. T. (1977) Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81, 2340–2361

[22] Xi, H., Duan, L. and Turcotte, M. (2013) Point-cycle bistability and stochasticity in a regulatory circuit for Bacillus subtilis Competence. Math. Biosci., 244, 135–147

[23] Feng, H. and Wang, J. (2011) Potential and flux decomposition for dynamical systems and non-equilibrium thermodynamics: Curvature, gauge field, and generalized fluctuation-dissipation theorem. J. Chem. Phys., 135, 234511

[24] Feng, H., Zhang, K. and Wang, J. (2014) Non-equilibrium transition state rate theory. Chem. Sci., 5, 3761–3769

[25] Waddington, C. H. and Kacser, H. (1957) The Strategy of the Genes: A Discussion of Some Aspects of Theoretical Biology. London: George Allen & Unwin

[26] Xu, L., Zhang, K. and Wang, J. (2014) Exploring the mechanisms of differentiation, dedifferentiation, reprogramming and transdifferentiation. PLoS One, 9, e105216

[27] Ferrell, J. Jr. (2012) Bistability, bifurcations, and Waddington’s epigenetic landscape. Curr. Biol., 22, R458–R466

[28] Holmberg, J. and Perlmann, T. (2012) Maintaining differentiated cellular identity. Nat. Rev. Genet., 13, 429–439

[29] Li, C. and Wang, J. (2013) Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation. J. R. Soc. Interface, 10, 20130787

[30] Ao, P., Galas, D., Hood, L. and Zhu, X. (2008) Cancer as robust intrinsic state of endogenous molecular-cellular network shaped by evolution. Med. Hypotheses, 70, 678–684

[31] Huang, S. (2013) Genetic and non-genetic instability in tumor progression: link between the fitness landscape and the epigenetic landscape of cancer cells. Cancer Metastasis Rev., 32, 423–448

[32] Li, C. and Wang, J. (2015) Quantifying the landscape for development and cancer from a core cancer stem cell circuit. Cancer Res., 75, 2607–2618

[33] Lasorella, A., Benezra, R. and Iavarone, A. (2014) The ID proteins: master regulators of cancer stem cells and tumour aggressiveness. Nat. Rev. Cancer, 14, 77–91

[34] Strogatz, S. H. (1994) Nonlinear Dynamics and Chaos, Cambridge: Westview Press

SUPPLEMENTARY MATERIALS

The supplementary materials can be found online with this article at DOI 10.1007/s40484-016-0070-5.

ACKNOWLEDGEMENTS

Special thanks go to Hongguang Xi for his unfailing assistance in completing the extensive parallel numerical simulations over extended periods of time. Many thanks to Michael Zhang for his continued hospitality at UT Dallas where MT designed the research, wrote the code, performed the analysis, and wrote the paper. The author is indebted to anonymous reviewers for their help in improving the manuscript.

COMPLIANCE WITH ETHICS GUIDELINES

Marc Turcotte declares that he has no conflict of interest. This article does not contain any studies with human or animal subjects performed by the author.

Funding

RIGHTS & PERMISSIONS

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg