• Previous Article
    Epidemic characteristics of two classic models and the dependence on the initial conditions
  • MBE Home
  • This Issue
  • Next Article
    Modeling the spread of bed bug infestation and optimal resource allocation for disinfestation
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, (2007). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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. doi: 10.1126/science.1070919. Google Scholar

[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. doi: 10.1007/s002200200605. Google Scholar

[10]

B. Ermentrout and J. Rinzel, XPPAUT: X-Windows PhasePlane plus Auto,, Computational Systems Neurobiology, (2012), 519. doi: 10.1007/978-94-007-3858-4_17. Google Scholar

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence,, Wiley, (1986). doi: 10.1002/9780470316658. Google Scholar

[12]

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

[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. doi: 10.1007/s10955-009-9819-9. Google Scholar

[14]

C. Godrèche, Rates for irreversible Gibbsian Ising models,, J. Stat. Mech-Theory E., (2013). Google Scholar

[15]

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

[16]

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

[17]

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

[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. doi: 10.1007/s00220-007-0201-y. Google Scholar

[19]

T. G. Kurtz, Approximation of Population Processes,, SIAM, (1981). Google Scholar

[20]

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

[21]

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

[22]

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

[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. doi: 10.1016/j.physa.2015.08.001. Google Scholar

[24]

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

[25]

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

[26]

E. Olivieri and M. E. Vares, Large Deviations and Metastability,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511543272. Google Scholar

[27]

L. Perko, Differential Equations and Dynamical Systems,, Third Edition. Springer, (2001). doi: 10.1007/978-1-4613-0003-8. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1038/nature04701. Google Scholar

[31]

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

[32]

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

[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. doi: 10.1073/pnas.162041399. Google Scholar

[34]

H. Thorisson, Coupling, Stationarity and Regeneration,, First Edition. Probability and its Applications, (2001). Google Scholar

[35]

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

[36]

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

[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. doi: 10.1016/j.bulm.2004.07.005. Google Scholar

[38]

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

show all references

References:
[1]

U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits,, CRC Press, (2007). Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[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. doi: 10.1126/science.1070919. Google Scholar

[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. doi: 10.1007/s002200200605. Google Scholar

[10]

B. Ermentrout and J. Rinzel, XPPAUT: X-Windows PhasePlane plus Auto,, Computational Systems Neurobiology, (2012), 519. doi: 10.1007/978-94-007-3858-4_17. Google Scholar

[11]

S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence,, Wiley, (1986). doi: 10.1002/9780470316658. Google Scholar

[12]

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

[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. doi: 10.1007/s10955-009-9819-9. Google Scholar

[14]

C. Godrèche, Rates for irreversible Gibbsian Ising models,, J. Stat. Mech-Theory E., (2013). Google Scholar

[15]

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

[16]

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

[17]

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

[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. doi: 10.1007/s00220-007-0201-y. Google Scholar

[19]

T. G. Kurtz, Approximation of Population Processes,, SIAM, (1981). Google Scholar

[20]

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

[21]

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

[22]

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

[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. doi: 10.1016/j.physa.2015.08.001. Google Scholar

[24]

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

[25]

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

[26]

E. Olivieri and M. E. Vares, Large Deviations and Metastability,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511543272. Google Scholar

[27]

L. Perko, Differential Equations and Dynamical Systems,, Third Edition. Springer, (2001). doi: 10.1007/978-1-4613-0003-8. Google Scholar

[28]

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

[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. Google Scholar

[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. doi: 10.1038/nature04701. Google Scholar

[31]

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

[32]

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

[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. doi: 10.1073/pnas.162041399. Google Scholar

[34]

H. Thorisson, Coupling, Stationarity and Regeneration,, First Edition. Probability and its Applications, (2001). Google Scholar

[35]

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

[36]

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

[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. doi: 10.1016/j.bulm.2004.07.005. Google Scholar

[38]

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

[1]

Hui Meng, Fei Lung Yuen, Tak Kuen Siu, Hailiang Yang. Optimal portfolio in a continuous-time self-exciting threshold model. Journal of Industrial & Management Optimization, 2013, 9 (2) : 487-504. doi: 10.3934/jimo.2013.9.487

[2]

Zhigang Zeng, Tingwen Huang. New passivity analysis of continuous-time recurrent neural networks with multiple discrete delays. Journal of Industrial & Management Optimization, 2011, 7 (2) : 283-289. doi: 10.3934/jimo.2011.7.283

[3]

Joon Kwon, Panayotis Mertikopoulos. A continuous-time approach to online optimization. Journal of Dynamics & Games, 2017, 4 (2) : 125-148. doi: 10.3934/jdg.2017008

[4]

Hanqing Jin, Xun Yu Zhou. Continuous-time portfolio selection under ambiguity. Mathematical Control & Related Fields, 2015, 5 (3) : 475-488. doi: 10.3934/mcrf.2015.5.475

[5]

Lakhdar Aggoun, Lakdere Benkherouf. A Markov modulated continuous-time capture-recapture population estimation model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1057-1075. doi: 10.3934/dcdsb.2005.5.1057

[6]

Willem Mélange, Herwig Bruneel, Bart Steyaert, Dieter Claeys, Joris Walraevens. A continuous-time queueing model with class clustering and global FCFS service discipline. Journal of Industrial & Management Optimization, 2014, 10 (1) : 193-206. doi: 10.3934/jimo.2014.10.193

[7]

Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha, Haomin Zhou. Influence prediction for continuous-time information propagation on networks. Networks & Heterogeneous Media, 2018, 13 (4) : 567-583. doi: 10.3934/nhm.2018026

[8]

Fritz Colonius, Guilherme Mazanti. Decay rates for stabilization of linear continuous-time systems with random switching. Mathematical Control & Related Fields, 2019, 9 (1) : 39-58. doi: 10.3934/mcrf.2019002

[9]

Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167

[10]

Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83

[11]

Huijuan Li, Junxia Wang. Input-to-state stability of continuous-time systems via finite-time Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019192

[12]

Qi Yang, Lei Wang, Enmin Feng, Hongchao Yin, Zhilong Xiu. Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018168

[13]

Haixiang Yao, Zhongfei Li, Xun Li, Yan Zeng. Optimal Sharpe ratio in continuous-time markets with and without a risk-free asset. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1273-1290. doi: 10.3934/jimo.2016072

[14]

Ping Chen, Haixiang Yao. Continuous-time mean-variance portfolio selection with no-shorting constraints and regime-switching. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018166

[15]

Huai-Nian Zhu, Cheng-Ke Zhang, Zhuo Jin. Continuous-time mean-variance asset-liability management with stochastic interest rates and inflation risks. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-22. doi: 10.3934/jimo.2018180

[16]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267

[17]

Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Analysis of a model coupling volume and surface processes in thermoviscoelasticity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2349-2403. doi: 10.3934/dcds.2015.35.2349

[18]

Rumi Ghosh, Kristina Lerman. Rethinking centrality: The role of dynamical processes in social network analysis. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1355-1372. doi: 10.3934/dcdsb.2014.19.1355

[19]

Shi Jin, Christof Sparber, Zhennan Zhou. On the classical limit of a time-dependent self-consistent field system: Analysis and computation. Kinetic & Related Models, 2017, 10 (1) : 263-298. doi: 10.3934/krm.2017011

[20]

Shubo Zhao, Ping Liu, Mingchao Jiang. Stability and bifurcation analysis in a chemotaxis bistable growth system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1165-1174. doi: 10.3934/dcdss.2017063

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]