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2014, 11(4): 723-740. doi: 10.3934/mbe.2014.11.723

Critical transitions in a model of a genetic regulatory system

1. 

Institute for Mathematics and Its Applications, Minneapolis, MN 55455, United States

2. 

Yeshiva University, Department of Mathematical Sciences, New York, NY 10016, United States

Received  January 2013 Revised  November 2013 Published  March 2014

We consider a model for substrate-depletion oscillations in genetic systems, based on a stochastic differential equation with a slowly evolving external signal. We show the existence of critical transitions in the system. We apply two methods to numerically test the synthetic time series generated by the system for early indicators of critical transitions: a detrended fluctuation analysis method, and a novel method based on topological data analysis (persistence diagrams).
Citation: Jesse Berwald, Marian Gidea. Critical transitions in a model of a genetic regulatory system. Mathematical Biosciences & Engineering, 2014, 11 (4) : 723-740. doi: 10.3934/mbe.2014.11.723
References:
[1]

T. Bulter, S.-G. Lee, W.-W. Wong, E. Fung, M. R. Connor and J. C. Liao, Design of artificial cell-cell communication using gene and metabolic networks, Proc. Natl. Acad. Sci. USA, 101 (2004), 2299-2304. doi: 10.1073/pnas.0306484101.

[2]

R. M. Bryce and K. B. Sprague, Revisiting detrended fluctuation analysis, Scientific Reports 2, 2012. doi: 10.1038/srep00315.

[3]

F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Mémoli and S. Oudot, Gromov-hausdorff stable signatures for shapes using persistence, Computer Graphics Forum, 28 (2009), 1393-1403. doi: 10.1111/j.1467-8659.2009.01516.x.

[4]

F. Chazal, L.Guibas, S. Oudot and P. Skraba, Persistence-based clustering in riemannian manifolds, Proc. 27th Annual ACM Symposium on Computational Geometry, (2011), 97-106. doi: 10.1145/1998196.1998212.

[5]

F. Chazal, V. de Silva and S. Oudot, Persistence Stability for Geometric Complexes, Geometriae Dedicata, 2013. doi: 10.1007/s10711-013-9937-z.

[6]

D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz Functions Have $L_p$-Stable Persistence, Foundations of Computational Mathematics, 10 (2010), 127-139. doi: 10.1007/s10208-010-9060-6.

[7]

P. D. Ditlevsen and S. J. Johnsen, Tipping points: Early warning and wishful thinking, Geophysical Research Letters, 37 (2010), L19703. doi: 10.1029/2010GL044486.

[8]

H. Edelsbrunner and J. Harer, Persistent homology - a survey, in Surveys on Discrete and Computational Geometry. Twenty Years Later, 257-282, eds. J. E. Goodman, J. Pach and R. Pollack, Contemporary Mathematics 453, Amer. Math. Soc., Providence, Rhode Island, (2008). doi: 10.1090/conm/453/08802.

[9]

H. Edelsbrunner and M. Morozov, Persistent Homology: Theory and Applications, Proceedings of the European Congress of Mathematics, 2012.

[10]

M. Elowitz and S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature, 403 (2000), 335-338.

[11]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[12]

M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow and V. Nanda, A Topological Measurement of Protein Compressibility, preprint, 2013.

[13]

T. S. Gardner, C. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in Escherichia coli, Nature, 403 (2000), 339-342.

[14]

J. Hasty, J. Pradines, M. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression, PNAS, 97 (2000), 2075-2080. doi: 10.1073/pnas.040411297.

[15]

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

[16]

R. M. Jones, Matlab Code for the DFA Procedure, http://criticaltransitions.wikispot.org/ (2013).

[17]

C. Kuehn, A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics, Physica D: Nonlinear Phenomena, 240 (2011), 1020-1035. doi: 10.1016/j.physd.2011.02.012.

[18]

C. Kuehn, A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications, Journal of Nonlinear Science,, 23 (2013), 457-510. doi: 10.1007/s00332-012-9158-x.

[19]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points -fold and canard points in two dimensions, SIAM J. of Math. Anal., 33 (2001), 286- 314. doi: 10.1137/S0036141099360919.

[20]

L. Kondic, A. Goullet, C. S. O'Hern, M. Kramar, K. Mischaikow and R. P. Behringer, Topology of force networks in compressed granular media, Europhys. Lett., 97 (2012), 54001. doi: 10.1209/0295-5075/97/54001.

[21]

M. Kramar, C++ code to compute the Wasserstein metric, Personal communication, http://www.math.rutgers.edu/ miroslav (2013).

[22]

A. Leier, P. D. Kuo, W. Banzhaf and K. Burrage, Evolving noisy oscillatory dynamics in genetic regulatory networks, In EuroGP'06 Proceedings of the 9th European conference on Genetic Programming, LNCS, 3905 (2006), 290-299. doi: 10.1007/11729976_26.

[23]

V. N. Livina and T. M. Lenton, A modified method for detecting incipient bifurcations in a dynamical system, Geophys. Res. Lett., 34 (2007), L03712. doi: 10.1029/2006GL028672.

[24]

V. Nanda, The Perseus Software Project for Rapid Computation of Persistent Homology, http://www.math.rutgers.edu/ vidit/perseus.html

[25]

M. Nicolau, A. J. Levine and G. Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, PNAS, 108 (2011), 7265-7270. doi: 10.1073/pnas.1102826108.

[26]

B. Novak and J. J. Tyson, Numerical analysis of a comprehensive model of M-phase control in Xenopus oocyte extracts and intact embryos, J. Cell. Sci., 106 (1993), 1153-1168.

[27]

A. A. Ptitsyn, S. Zvonic and J. M. Gimble, Digital signal processing reveals circadian baseline oscillation in majority of mammalian genes, PLoS Comput Biol, 3 (2007), 1108-1114. doi: 10.1371/journal.pcbi.0030120.

[28]

M. Scheffer et al., Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.

[29]

M. Scheffer et al, Anticipating critical transitions, Science, 338 (2012), 344, DOI: 10.1126/science.1225244

[30]

J. M. T. Thompson and J. Sieber, Climate tipping as a noisy bifurcation: A predictive technique, IMA Journal of Applied Mathematics, 76 (2011), 27-46. doi: 10.1093/imamat/hxq060.

[31]

J. J. Tyson, K. C. Chen and B. Novak, Sniffers, buzzers, toggles and blinkers: Dynamics of regulatory and signaling pathways in the cell, Curr. Opin. Cell. Biol., 15 (2003), 221-231. doi: 10.1016/S0955-0674(03)00017-6.

[32]

A. J. Zomorodian, Topology for Computing, Cambridge University, 2005. doi: 10.1017/CBO9780511546945.

show all references

References:
[1]

T. Bulter, S.-G. Lee, W.-W. Wong, E. Fung, M. R. Connor and J. C. Liao, Design of artificial cell-cell communication using gene and metabolic networks, Proc. Natl. Acad. Sci. USA, 101 (2004), 2299-2304. doi: 10.1073/pnas.0306484101.

[2]

R. M. Bryce and K. B. Sprague, Revisiting detrended fluctuation analysis, Scientific Reports 2, 2012. doi: 10.1038/srep00315.

[3]

F. Chazal, D. Cohen-Steiner, L. J. Guibas, F. Mémoli and S. Oudot, Gromov-hausdorff stable signatures for shapes using persistence, Computer Graphics Forum, 28 (2009), 1393-1403. doi: 10.1111/j.1467-8659.2009.01516.x.

[4]

F. Chazal, L.Guibas, S. Oudot and P. Skraba, Persistence-based clustering in riemannian manifolds, Proc. 27th Annual ACM Symposium on Computational Geometry, (2011), 97-106. doi: 10.1145/1998196.1998212.

[5]

F. Chazal, V. de Silva and S. Oudot, Persistence Stability for Geometric Complexes, Geometriae Dedicata, 2013. doi: 10.1007/s10711-013-9937-z.

[6]

D. Cohen-Steiner, H. Edelsbrunner, J. Harer and Y. Mileyko, Lipschitz Functions Have $L_p$-Stable Persistence, Foundations of Computational Mathematics, 10 (2010), 127-139. doi: 10.1007/s10208-010-9060-6.

[7]

P. D. Ditlevsen and S. J. Johnsen, Tipping points: Early warning and wishful thinking, Geophysical Research Letters, 37 (2010), L19703. doi: 10.1029/2010GL044486.

[8]

H. Edelsbrunner and J. Harer, Persistent homology - a survey, in Surveys on Discrete and Computational Geometry. Twenty Years Later, 257-282, eds. J. E. Goodman, J. Pach and R. Pollack, Contemporary Mathematics 453, Amer. Math. Soc., Providence, Rhode Island, (2008). doi: 10.1090/conm/453/08802.

[9]

H. Edelsbrunner and M. Morozov, Persistent Homology: Theory and Applications, Proceedings of the European Congress of Mathematics, 2012.

[10]

M. Elowitz and S. Leibler, A synthetic oscillatory network of transcriptional regulators, Nature, 403 (2000), 335-338.

[11]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, Journal of Differential Equations, 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.

[12]

M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow and V. Nanda, A Topological Measurement of Protein Compressibility, preprint, 2013.

[13]

T. S. Gardner, C. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in Escherichia coli, Nature, 403 (2000), 339-342.

[14]

J. Hasty, J. Pradines, M. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression, PNAS, 97 (2000), 2075-2080. doi: 10.1073/pnas.040411297.

[15]

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

[16]

R. M. Jones, Matlab Code for the DFA Procedure, http://criticaltransitions.wikispot.org/ (2013).

[17]

C. Kuehn, A mathematical framework for critical transitions: Bifurcations, fast-slow systems and stochastic dynamics, Physica D: Nonlinear Phenomena, 240 (2011), 1020-1035. doi: 10.1016/j.physd.2011.02.012.

[18]

C. Kuehn, A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications, Journal of Nonlinear Science,, 23 (2013), 457-510. doi: 10.1007/s00332-012-9158-x.

[19]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points -fold and canard points in two dimensions, SIAM J. of Math. Anal., 33 (2001), 286- 314. doi: 10.1137/S0036141099360919.

[20]

L. Kondic, A. Goullet, C. S. O'Hern, M. Kramar, K. Mischaikow and R. P. Behringer, Topology of force networks in compressed granular media, Europhys. Lett., 97 (2012), 54001. doi: 10.1209/0295-5075/97/54001.

[21]

M. Kramar, C++ code to compute the Wasserstein metric, Personal communication, http://www.math.rutgers.edu/ miroslav (2013).

[22]

A. Leier, P. D. Kuo, W. Banzhaf and K. Burrage, Evolving noisy oscillatory dynamics in genetic regulatory networks, In EuroGP'06 Proceedings of the 9th European conference on Genetic Programming, LNCS, 3905 (2006), 290-299. doi: 10.1007/11729976_26.

[23]

V. N. Livina and T. M. Lenton, A modified method for detecting incipient bifurcations in a dynamical system, Geophys. Res. Lett., 34 (2007), L03712. doi: 10.1029/2006GL028672.

[24]

V. Nanda, The Perseus Software Project for Rapid Computation of Persistent Homology, http://www.math.rutgers.edu/ vidit/perseus.html

[25]

M. Nicolau, A. J. Levine and G. Carlsson, Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival, PNAS, 108 (2011), 7265-7270. doi: 10.1073/pnas.1102826108.

[26]

B. Novak and J. J. Tyson, Numerical analysis of a comprehensive model of M-phase control in Xenopus oocyte extracts and intact embryos, J. Cell. Sci., 106 (1993), 1153-1168.

[27]

A. A. Ptitsyn, S. Zvonic and J. M. Gimble, Digital signal processing reveals circadian baseline oscillation in majority of mammalian genes, PLoS Comput Biol, 3 (2007), 1108-1114. doi: 10.1371/journal.pcbi.0030120.

[28]

M. Scheffer et al., Early-warning signals for critical transitions, Nature, 461 (2009), 53-59.

[29]

M. Scheffer et al, Anticipating critical transitions, Science, 338 (2012), 344, DOI: 10.1126/science.1225244

[30]

J. M. T. Thompson and J. Sieber, Climate tipping as a noisy bifurcation: A predictive technique, IMA Journal of Applied Mathematics, 76 (2011), 27-46. doi: 10.1093/imamat/hxq060.

[31]

J. J. Tyson, K. C. Chen and B. Novak, Sniffers, buzzers, toggles and blinkers: Dynamics of regulatory and signaling pathways in the cell, Curr. Opin. Cell. Biol., 15 (2003), 221-231. doi: 10.1016/S0955-0674(03)00017-6.

[32]

A. J. Zomorodian, Topology for Computing, Cambridge University, 2005. doi: 10.1017/CBO9780511546945.

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