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2016, 13(5): 981-998. doi: 10.3934/mbe.2016026

Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module

1. 

Institute of Mathematics and Statistics, Universidade de São Paulo, Rua do Matão, 1010, CEP 05508-090, São Paulo, Brazil

Received  April 2015 Revised  March 2016 Published  July 2016

We study an alternative approach to model the dynamical behaviors of biological feedback loop, that is, a type-dependent spin system, this class of stochastic models was introduced by Fernández et. al [13], and are useful since take account to inherent variability of gene expression. We analyze a non-symmetric feedback module being an extension for the repressilator, the first synthetic biological oscillator, invented by Elowitz and Leibler [7]. We consider a mean-field dynamics for a type-dependent Ising model, and then study the empirical-magnetization vector representing concentration of molecules. We apply a convergence result from stochastic jump processes to deterministic trajectories and present a bifurcation analysis for the associated dynamical system. We show that non-symmetric module under study can exhibit very rich behaviours, including the empirical oscillations described by repressilator.
Citation: Manuel González-Navarrete. Type-dependent stochastic Ising model describing the dynamics of a non-symmetric feedback module. Mathematical Biosciences & Engineering, 2016, 13 (5) : 981-998. doi: 10.3934/mbe.2016026
References:
[1]

U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, CRC Press, Boca Raton, 2007.

[2]

U. Alon, Network motifs: Theory and experimental approaches, Nat. Rev. Genet., 8 (2007), 450-461. doi: 10.1038/nrg2102.

[3]

F. Blanchini, E. Franco and G. Giordano, A Structural Classification of Candidate Oscillatory and Multistationary Biochemical Systems, Bull. Math. Bio., 76 (2014), 2542-2569. doi: 10.1007/s11538-014-0023-y.

[4]

L. Chen and K. Aihara, Stability of genetic regulatory networks with time delay, IEEE T. Circuits-I, 49 (2002), 602-608. doi: 10.1109/TCSI.2002.1001949.

[5]

L. Chen, R. Wang, T. Kobayashi and K. Aihara, Dynamics of gene regulatory networks with cell division cycles, Phys. Rev. E, 70 (2004), 011909, 13 pp. doi: 10.1103/PhysRevE.70.011909.

[6]

X. Descombes and E. Zhizhina, The Gibbs fields approach and related dynamics in image processing, Condensed Matter Physics, 11 (2008), 293-312. doi: 10.5488/CMP.11.2.293.

[7]

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

[8]

M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919.

[9]

A. C. D. van Enter, R. Fernández, F. den Hollander and F. Redig, Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures, Commun. Math. Phys., 226 (2002), 101-130. doi: 10.1007/s002200200605.

[10]

B. Ermentrout and J. Rinzel, XPPAUT: X-Windows PhasePlane plus Auto, Computational Systems Neurobiology, (2012), 519-531, http://www.math.pitt.edu/ bard/xpp/xpp.html doi: 10.1007/978-94-007-3858-4_17.

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.

[12]

T. d'Eysmond and F. Naef, Systems biology and modeling of circadian rhythms, in The Circadian Clock (ed. U. Albrecht), Springer-Verlag, New York, 12 (2009), 283-293. doi: 10.1007/978-1-4419-1262-6_11.

[13]

R. Fernández, L. R. Fontes and E. J. Neves, Density-profile processes describing biological signaling networks: Almost sure convergence to deterministic trajectories, J. Stat. Phys., 136 (2009), 875-901. doi: 10.1007/s10955-009-9819-9.

[14]

C. Godrèche, Rates for irreversible Gibbsian Ising models, J. Stat. Mech-Theory E., (2013), P05011, 30pp.

[15]

C. Godrèche and A. J. Bray, Nonequilibrium stationary states and phase transitions in directed Ising models, J. Stat. Mech-Theory E., (2009), P12016.

[16]

E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik, 31 (1925), 253-258. doi: 10.1007/BF02980577.

[17]

R. Kotecký and E. Olivieri, Droplet dynamics for asymmetric Ising model, J. Stat. Phys., 70 (1993), 1121-1148. doi: 10.1007/BF01049425.

[18]

C. Kulske and A. Le Ny, Spin-flip dynamics of the Curie-Weiss model: Loss of Gibbsianness with possibly broken symmetry, Commun. Math. Phys., 271 (2007), 431-454. doi: 10.1007/s00220-007-0201-y.

[19]

T. G. Kurtz, Approximation of Population Processes, SIAM, Philadelphia, 1981.

[20]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third Edition. Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[21]

M. C. A. Leite and Y. Wang, Multistability, oscillations and bifurcations in feedback loops, Math. Biosc. and Eng., 7 (2010), 83-97. doi: 10.3934/mbe.2010.7.83.

[22]

T. M. Liggett, Interacting Particle Systems, Springer, Berlin, 1985. doi: 10.1007/978-1-4613-8542-4.

[23]

J. R. G Mendonça and M. J. de Oliveira, Type-dependent irreversible stochastic spin models for genetic regulatory networks at the level of promotion-inhibition circuitry, Physica. A, 440 (2015), 33-41. doi: 10.1016/j.physa.2015.08.001.

[24]

A. Y. Mitrophanov and E. A. Groisman, Positive feedback in cellular control systems, Bioessays, 30 (2008), 542-555. doi: 10.1002/bies.20769.

[25]

M. J. de Oliveira, Irreversible models with Boltzmann-Gibbs probability distribution and entropy production, J. Stat. Mech-Theory E., (2011), P12012. Erratum J. Stat. Mech-Theory E. (2013), E04001.

[26]

E. Olivieri and M. E. Vares, Large Deviations and Metastability, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511543272.

[27]

L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[28]

N. Radde, The role of feedback mechanisms in biological network models - A tutorial, Asian J. Control, 13 (2011), 597-610. doi: 10.1002/asjc.376.

[29]

D. A. Rand, B. V. Shulgin, D. Salazar and A. J. Millar, Design principles underlying circadian clocks, J. Roy. Soc. Interface, 1 (2004), 119-130.

[30]

E. Schneidman, M. J. II Berry, R. Segev and W. Bialek, Weak pairwise correlations imply strongly correlated network states in a neural population, Nature, 440 (2006), 1007-1012. doi: 10.1038/nature04701.

[31]

V. Shahrezaei and P. S. Swain, The stochastic nature of biochemical networks, Curr. Opin. Biotech., 19 (2008), 369-374. doi: 10.1016/j.copbio.2008.06.011.

[32]

E. D. Sontag, Some new directions in control theory inspired by systems biology, Systems Biol., 1 (2004), 9-18. doi: 10.1049/sb:20045006.

[33]

P. S. Swain, M. B. Elowitz and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, P. Natl. Acad. Sci. USA, 99 (2002), 12795-12800. doi: 10.1073/pnas.162041399.

[34]

H. Thorisson, Coupling, Stationarity and Regeneration, First Edition. Probability and its Applications, Springer, New York, 2001.

[35]

R. Toussaint and S. R. Pride, Interacting damage models mapped onto Ising and percolation models, Phys. Rev. E, 71 (2005), 046127. doi: 10.1103/PhysRevE.71.046127.

[36]

J. Tyson and B. Novak, Functional motifs in biochemical reaction networks, Annu. Rev. Phys. Chem., 61 (2010), 219-240. doi: 10.1146/annurev.physchem.012809.103457.

[37]

R. Wang, Z. Jing and L. Chen, Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems, B. Math. Biol., 67 (2005), 339-367. doi: 10.1016/j.bulm.2004.07.005.

[38]

R. Wang, X.-M. Zhao and Z. Liu, Modeling and dynamical analysis of molecular networks, in Complex Sciences (ed. J. Zhou), Springer Berlin Heidelberg, 5 (2009), 2139-2148. doi: 10.1007/978-3-642-02469-6_90.

show all references

References:
[1]

U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, CRC Press, Boca Raton, 2007.

[2]

U. Alon, Network motifs: Theory and experimental approaches, Nat. Rev. Genet., 8 (2007), 450-461. doi: 10.1038/nrg2102.

[3]

F. Blanchini, E. Franco and G. Giordano, A Structural Classification of Candidate Oscillatory and Multistationary Biochemical Systems, Bull. Math. Bio., 76 (2014), 2542-2569. doi: 10.1007/s11538-014-0023-y.

[4]

L. Chen and K. Aihara, Stability of genetic regulatory networks with time delay, IEEE T. Circuits-I, 49 (2002), 602-608. doi: 10.1109/TCSI.2002.1001949.

[5]

L. Chen, R. Wang, T. Kobayashi and K. Aihara, Dynamics of gene regulatory networks with cell division cycles, Phys. Rev. E, 70 (2004), 011909, 13 pp. doi: 10.1103/PhysRevE.70.011909.

[6]

X. Descombes and E. Zhizhina, The Gibbs fields approach and related dynamics in image processing, Condensed Matter Physics, 11 (2008), 293-312. doi: 10.5488/CMP.11.2.293.

[7]

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

[8]

M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919.

[9]

A. C. D. van Enter, R. Fernández, F. den Hollander and F. Redig, Possible loss and recovery of Gibbsianness during the stochastic evolution of Gibbs measures, Commun. Math. Phys., 226 (2002), 101-130. doi: 10.1007/s002200200605.

[10]

B. Ermentrout and J. Rinzel, XPPAUT: X-Windows PhasePlane plus Auto, Computational Systems Neurobiology, (2012), 519-531, http://www.math.pitt.edu/ bard/xpp/xpp.html doi: 10.1007/978-94-007-3858-4_17.

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.

[12]

T. d'Eysmond and F. Naef, Systems biology and modeling of circadian rhythms, in The Circadian Clock (ed. U. Albrecht), Springer-Verlag, New York, 12 (2009), 283-293. doi: 10.1007/978-1-4419-1262-6_11.

[13]

R. Fernández, L. R. Fontes and E. J. Neves, Density-profile processes describing biological signaling networks: Almost sure convergence to deterministic trajectories, J. Stat. Phys., 136 (2009), 875-901. doi: 10.1007/s10955-009-9819-9.

[14]

C. Godrèche, Rates for irreversible Gibbsian Ising models, J. Stat. Mech-Theory E., (2013), P05011, 30pp.

[15]

C. Godrèche and A. J. Bray, Nonequilibrium stationary states and phase transitions in directed Ising models, J. Stat. Mech-Theory E., (2009), P12016.

[16]

E. Ising, Beitrag zur theorie des ferromagnetismus, Z. Physik, 31 (1925), 253-258. doi: 10.1007/BF02980577.

[17]

R. Kotecký and E. Olivieri, Droplet dynamics for asymmetric Ising model, J. Stat. Phys., 70 (1993), 1121-1148. doi: 10.1007/BF01049425.

[18]

C. Kulske and A. Le Ny, Spin-flip dynamics of the Curie-Weiss model: Loss of Gibbsianness with possibly broken symmetry, Commun. Math. Phys., 271 (2007), 431-454. doi: 10.1007/s00220-007-0201-y.

[19]

T. G. Kurtz, Approximation of Population Processes, SIAM, Philadelphia, 1981.

[20]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Third Edition. Springer, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[21]

M. C. A. Leite and Y. Wang, Multistability, oscillations and bifurcations in feedback loops, Math. Biosc. and Eng., 7 (2010), 83-97. doi: 10.3934/mbe.2010.7.83.

[22]

T. M. Liggett, Interacting Particle Systems, Springer, Berlin, 1985. doi: 10.1007/978-1-4613-8542-4.

[23]

J. R. G Mendonça and M. J. de Oliveira, Type-dependent irreversible stochastic spin models for genetic regulatory networks at the level of promotion-inhibition circuitry, Physica. A, 440 (2015), 33-41. doi: 10.1016/j.physa.2015.08.001.

[24]

A. Y. Mitrophanov and E. A. Groisman, Positive feedback in cellular control systems, Bioessays, 30 (2008), 542-555. doi: 10.1002/bies.20769.

[25]

M. J. de Oliveira, Irreversible models with Boltzmann-Gibbs probability distribution and entropy production, J. Stat. Mech-Theory E., (2011), P12012. Erratum J. Stat. Mech-Theory E. (2013), E04001.

[26]

E. Olivieri and M. E. Vares, Large Deviations and Metastability, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511543272.

[27]

L. Perko, Differential Equations and Dynamical Systems, Third Edition. Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[28]

N. Radde, The role of feedback mechanisms in biological network models - A tutorial, Asian J. Control, 13 (2011), 597-610. doi: 10.1002/asjc.376.

[29]

D. A. Rand, B. V. Shulgin, D. Salazar and A. J. Millar, Design principles underlying circadian clocks, J. Roy. Soc. Interface, 1 (2004), 119-130.

[30]

E. Schneidman, M. J. II Berry, R. Segev and W. Bialek, Weak pairwise correlations imply strongly correlated network states in a neural population, Nature, 440 (2006), 1007-1012. doi: 10.1038/nature04701.

[31]

V. Shahrezaei and P. S. Swain, The stochastic nature of biochemical networks, Curr. Opin. Biotech., 19 (2008), 369-374. doi: 10.1016/j.copbio.2008.06.011.

[32]

E. D. Sontag, Some new directions in control theory inspired by systems biology, Systems Biol., 1 (2004), 9-18. doi: 10.1049/sb:20045006.

[33]

P. S. Swain, M. B. Elowitz and E. D. Siggia, Intrinsic and extrinsic contributions to stochasticity in gene expression, P. Natl. Acad. Sci. USA, 99 (2002), 12795-12800. doi: 10.1073/pnas.162041399.

[34]

H. Thorisson, Coupling, Stationarity and Regeneration, First Edition. Probability and its Applications, Springer, New York, 2001.

[35]

R. Toussaint and S. R. Pride, Interacting damage models mapped onto Ising and percolation models, Phys. Rev. E, 71 (2005), 046127. doi: 10.1103/PhysRevE.71.046127.

[36]

J. Tyson and B. Novak, Functional motifs in biochemical reaction networks, Annu. Rev. Phys. Chem., 61 (2010), 219-240. doi: 10.1146/annurev.physchem.012809.103457.

[37]

R. Wang, Z. Jing and L. Chen, Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems, B. Math. Biol., 67 (2005), 339-367. doi: 10.1016/j.bulm.2004.07.005.

[38]

R. Wang, X.-M. Zhao and Z. Liu, Modeling and dynamical analysis of molecular networks, in Complex Sciences (ed. J. Zhou), Springer Berlin Heidelberg, 5 (2009), 2139-2148. doi: 10.1007/978-3-642-02469-6_90.

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