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doi: 10.3934/dcdsb.2020007

Transition between monostability and bistability of a genetic toggle switch in Escherichia coli

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Weinian Zhang

Received  January 2019 Revised  July 2019 Published  December 2019

Fund Project: Supported by NSFC grants #11771307, #11831012 and #11821001

In this paper, we investigate a genetic toggle switch in Escherichia Coli, which models an artificial double-negative feedback loop with two mutually repressors. This model is a planar differential system with three parameters, one of which is an integer power $ n\ge1 $, in the case that repressors 1 and 2 multimerize with $ n $ and 1 subunits respectively and its equilibria are decided by a polynomial of degree $ n+1 $. Since one hardly solves such a polynomial equation, a known result on bistability was given by omitting some small terms under the assumption that the promoters are strong and the expression ratio between the ON state and the OFF state is large. In this paper, determining distribution of zeros qualitatively for the polynomial of high degree, we analytically discuss on the system without the assumption and completely give qualitative properties for all equilibria, which corrects the known result of bistability. Furthermore, we prove that there may occur in the system a codimension 2 bifurcation, called cusp bifurcation, which is a collision of two saddle-node bifurcations and manifests the transition between bistability and monostability. We exhibit the global dynamics of repressors in various cases by analyzing equilibria at infinity and proving nonexistence of closed orbits.

Citation: Jie Li, Weinian Zhang. Transition between monostability and bistability of a genetic toggle switch in Escherichia coli. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020007
References:
[1]

M. R. AtkinsonM. A. SavageauJ. T. Myers and A. J. Ninfa, Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli, Cell, 113 (2003), 597-607.  doi: 10.1016/S0092-8674(03)00346-5.  Google Scholar

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M. Baron, N. Buchler, C. Cooper, M. Farnitano, C. Gersbach, C. Kim, H. S. Kim, A. Reid and J. Zhu, Designing Synthtic Gene Networks Using Artificial Trnscription Factors in Yeast, Available from: http://2013.igem.org/Team:Duke. Google Scholar

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J. Carr, Applications of Center Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981.  Google Scholar

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B. P. CormackR. H. Valdivia and S. Falkow, FACS-optimized mutants of the green fluorescent protein (GFP), Gene, 173 (1996), 33-38.  doi: 10.1016/0378-1119(95)00685-0.  Google Scholar

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J. E. Ferrell, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability, Curr. Opin. Cell Biol., 14 (2002), 140-148.  doi: 10.1016/S0955-0674(02)00314-9.  Google Scholar

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J. E. Ferrell and S. H. Ha, Ultrasensitivity part Ⅲ: Cascades, bistable switches, and oscillators, Trends Biochem. Sci., 12 (2014), 612-618.  doi: 10.1016/j.tibs.2014.10.002.  Google Scholar

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B. Gao and W. Zhang, Equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function, IEEE Trans. Neural Network, 19 (2008), 782-794.  doi: 10.1109/TNN.2007.912321.  Google Scholar

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T. S. GardnerC. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in Escherichia coli, Nature, 403 (2000), 339-342.  doi: 10.1038/35002131.  Google Scholar

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J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcation of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

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J. HastyJ. PradinessM. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression, Proc. Natl Acad. Sci. USA, 97 (2000), 2075-2080.  doi: 10.1073/pnas.040411297.  Google Scholar

[13]

X. R. HouR. Yan and W. N. Zhang, Bifurcations of a polynomial differentialsystem of degree $n$ in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.  doi: 10.1016/S0898-1221(02)00108-6.  Google Scholar

[14]

K. C. KeilerP. R. H. Waller and R. T. Sauer, Role of a peptide tagging system in degradation of proteins synthesized from damaged messenger RNA, Science, 271 (1996), 990-993.  doi: 10.1126/science.271.5251.990.  Google Scholar

[15]

L. Edelstein-Keshet, Mathematical Models in Biology, Random House, Inc., New York, 1988.  Google Scholar

[16]

A. S. Khalil and J. J. Collins, Synthetic biology: Applications come of age, Nat. Rev. Genet., 11 (2010), 367-379.  doi: 10.1038/nrg2775.  Google Scholar

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.  Google Scholar

[18]

J. E. Lisman, A mechanism for memory storage insensitive to molecular turnover: A bistable autophosphorylating kinase, Proc. Natl. Acad. Sci. USA, 82 (1985), 3055-3057.  doi: 10.1073/pnas.82.9.3055.  Google Scholar

[19]

R. Lutz and H. Bujard, Independent and tight regulation of transcriptional units in Escherichia coli via the LacR/O, the TetR/O and AraC/I1-I2 regulatory elements, Nucleic Acids Res., 25 (1997), 1203-1210.  doi: 10.1093/nar/25.6.1203.  Google Scholar

[20]

H. Maamar and D. Dubnau, Bistability in the Bacillus subtilis K-state (competence) system requires a positive feedback loop, Mol. Microbiol., 56 (2005), 615-624.  doi: 10.1111/j.1365-2958.2005.04592.x.  Google Scholar

[21]

T. S. MoonE. J. ClarkeE. S. GrobanA. TamsirR. M. ClarkM. EamesT. Kortemme and C. A. Voigt, Construction of a genetic multiplexer to toggle between chemosensory pathways in Escherichia coli, J. Mol.Biol., 406 (2011), 215-227.  doi: 10.1016/j.jmb.2010.12.019.  Google Scholar

[22]

F. MosconiT. JulouN. DespratD. K. SinhaJ. F. AllemandC. Vincent and D. Bensimon, Some nonlinear challenges in biology, Nonlinearity, 21 (2008), T131-T147.  doi: 10.1088/0951-7715/21/8/T03.  Google Scholar

[23]

E. M. OzbudakM. ThattaiH. N. LimB. I. Shraiman and A. van Oudenaarden, Multistability in the lactose utilization networks of Escherichia coli, Nature, 427 (2004), 737-740.  doi: 10.1038/nature02298.  Google Scholar

[24]

L. Perko, Diffenrential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[25]

S. I. Rubinow, Introduction to Mathematical Biology, Dover Publications, Inc., Mineola, NY, 2002.  Google Scholar

[26]

S. SemseyL. JauffredZ. CsiszovszkiJ. ErdőssyV. StégerS. Hansen and S. Krishna, The effect of LacI autoregulation on the performance of the lactose utilization system in Escherichia coli, Nucleic Acids Res., 41 (2013), 6381-6390.  doi: 10.1093/nar/gkt351.  Google Scholar

[27]

P. SiutiJ. Yazbek and T. K. Lu, Synthetic circuits integrating logic and memory in living cells, Nat. Biotechnol., 31 (2013), 448-452.  doi: 10.1038/nbt.2510.  Google Scholar

[28]

Y. L. TangD. Q. Huang and W. N. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.  doi: 10.1093/imamat/hxr005.  Google Scholar

[29]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[30]

J. J. Y. TeoS. S. Woo and R. Sarpeshkar, Synthetic biology: A unifying view and review using analog circuits, IEEE Trans. Biomed. Circuits Syst., 9 (2015), 453-474.  doi: 10.1109/TBCAS.2015.2461446.  Google Scholar

[31]

A. van Oudenaarden, Systems Biology: Modeling Biological Networks, 7.32/7.81J/8.591J, 2009. Available from: http://web.mit.edu/biophysics/sbio/. Google Scholar

[32]

W. Xiong and J. E. Ferrell Jr., A positive-feedback-based bistable 'memory module' that governs a cell fate decision, Nature, 426 (2003), 460-465.  doi: 10.1038/nature02089.  Google Scholar

[33]

D. B. Xiu, Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), 293-309.   Google Scholar

[34]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992.  Google Scholar

show all references

References:
[1]

M. R. AtkinsonM. A. SavageauJ. T. Myers and A. J. Ninfa, Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in Escherichia coli, Cell, 113 (2003), 597-607.  doi: 10.1016/S0092-8674(03)00346-5.  Google Scholar

[2]

M. Baron, N. Buchler, C. Cooper, M. Farnitano, C. Gersbach, C. Kim, H. S. Kim, A. Reid and J. Zhu, Designing Synthtic Gene Networks Using Artificial Trnscription Factors in Yeast, Available from: http://2013.igem.org/Team:Duke. Google Scholar

[3]

I. Bendixson, Sur lés courbes définies par des équations différentielles, Acta Math., 24 (1901), 1-88.  doi: 10.1007/BF02403068.  Google Scholar

[4]

J. Carr, Applications of Center Manifold Theory, Applied Mathematical Sciences, 35. Springer-Verlag, New York-Berlin, 1981.  Google Scholar

[5] S. N. ChowC. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.  Google Scholar
[6]

B. P. CormackR. H. Valdivia and S. Falkow, FACS-optimized mutants of the green fluorescent protein (GFP), Gene, 173 (1996), 33-38.  doi: 10.1016/0378-1119(95)00685-0.  Google Scholar

[7]

J. E. Ferrell, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability, Curr. Opin. Cell Biol., 14 (2002), 140-148.  doi: 10.1016/S0955-0674(02)00314-9.  Google Scholar

[8]

J. E. Ferrell and S. H. Ha, Ultrasensitivity part Ⅲ: Cascades, bistable switches, and oscillators, Trends Biochem. Sci., 12 (2014), 612-618.  doi: 10.1016/j.tibs.2014.10.002.  Google Scholar

[9]

B. Gao and W. Zhang, Equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function, IEEE Trans. Neural Network, 19 (2008), 782-794.  doi: 10.1109/TNN.2007.912321.  Google Scholar

[10]

T. S. GardnerC. R. Cantor and J. J. Collins, Construction of a genetic toggle switch in Escherichia coli, Nature, 403 (2000), 339-342.  doi: 10.1038/35002131.  Google Scholar

[11]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations Dynamical Systems and Bifurcation of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[12]

J. HastyJ. PradinessM. Dolnik and J. J. Collins, Noise-based switches and amplifiers for gene expression, Proc. Natl Acad. Sci. USA, 97 (2000), 2075-2080.  doi: 10.1073/pnas.040411297.  Google Scholar

[13]

X. R. HouR. Yan and W. N. Zhang, Bifurcations of a polynomial differentialsystem of degree $n$ in biochemical reactions, Comput. Math. Appl., 43 (2002), 1407-1423.  doi: 10.1016/S0898-1221(02)00108-6.  Google Scholar

[14]

K. C. KeilerP. R. H. Waller and R. T. Sauer, Role of a peptide tagging system in degradation of proteins synthesized from damaged messenger RNA, Science, 271 (1996), 990-993.  doi: 10.1126/science.271.5251.990.  Google Scholar

[15]

L. Edelstein-Keshet, Mathematical Models in Biology, Random House, Inc., New York, 1988.  Google Scholar

[16]

A. S. Khalil and J. J. Collins, Synthetic biology: Applications come of age, Nat. Rev. Genet., 11 (2010), 367-379.  doi: 10.1038/nrg2775.  Google Scholar

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.  Google Scholar

[18]

J. E. Lisman, A mechanism for memory storage insensitive to molecular turnover: A bistable autophosphorylating kinase, Proc. Natl. Acad. Sci. USA, 82 (1985), 3055-3057.  doi: 10.1073/pnas.82.9.3055.  Google Scholar

[19]

R. Lutz and H. Bujard, Independent and tight regulation of transcriptional units in Escherichia coli via the LacR/O, the TetR/O and AraC/I1-I2 regulatory elements, Nucleic Acids Res., 25 (1997), 1203-1210.  doi: 10.1093/nar/25.6.1203.  Google Scholar

[20]

H. Maamar and D. Dubnau, Bistability in the Bacillus subtilis K-state (competence) system requires a positive feedback loop, Mol. Microbiol., 56 (2005), 615-624.  doi: 10.1111/j.1365-2958.2005.04592.x.  Google Scholar

[21]

T. S. MoonE. J. ClarkeE. S. GrobanA. TamsirR. M. ClarkM. EamesT. Kortemme and C. A. Voigt, Construction of a genetic multiplexer to toggle between chemosensory pathways in Escherichia coli, J. Mol.Biol., 406 (2011), 215-227.  doi: 10.1016/j.jmb.2010.12.019.  Google Scholar

[22]

F. MosconiT. JulouN. DespratD. K. SinhaJ. F. AllemandC. Vincent and D. Bensimon, Some nonlinear challenges in biology, Nonlinearity, 21 (2008), T131-T147.  doi: 10.1088/0951-7715/21/8/T03.  Google Scholar

[23]

E. M. OzbudakM. ThattaiH. N. LimB. I. Shraiman and A. van Oudenaarden, Multistability in the lactose utilization networks of Escherichia coli, Nature, 427 (2004), 737-740.  doi: 10.1038/nature02298.  Google Scholar

[24]

L. Perko, Diffenrential Equations and Dynamical Systems, Texts in Applied Mathematics, 7. Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3.  Google Scholar

[25]

S. I. Rubinow, Introduction to Mathematical Biology, Dover Publications, Inc., Mineola, NY, 2002.  Google Scholar

[26]

S. SemseyL. JauffredZ. CsiszovszkiJ. ErdőssyV. StégerS. Hansen and S. Krishna, The effect of LacI autoregulation on the performance of the lactose utilization system in Escherichia coli, Nucleic Acids Res., 41 (2013), 6381-6390.  doi: 10.1093/nar/gkt351.  Google Scholar

[27]

P. SiutiJ. Yazbek and T. K. Lu, Synthetic circuits integrating logic and memory in living cells, Nat. Biotechnol., 31 (2013), 448-452.  doi: 10.1038/nbt.2510.  Google Scholar

[28]

Y. L. TangD. Q. Huang and W. N. Zhang, Direct parametric analysis of an enzyme-catalyzed reaction model, IMA J. Appl. Math., 76 (2011), 876-898.  doi: 10.1093/imamat/hxr005.  Google Scholar

[29]

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[30]

J. J. Y. TeoS. S. Woo and R. Sarpeshkar, Synthetic biology: A unifying view and review using analog circuits, IEEE Trans. Biomed. Circuits Syst., 9 (2015), 453-474.  doi: 10.1109/TBCAS.2015.2461446.  Google Scholar

[31]

A. van Oudenaarden, Systems Biology: Modeling Biological Networks, 7.32/7.81J/8.591J, 2009. Available from: http://web.mit.edu/biophysics/sbio/. Google Scholar

[32]

W. Xiong and J. E. Ferrell Jr., A positive-feedback-based bistable 'memory module' that governs a cell fate decision, Nature, 426 (2003), 460-465.  doi: 10.1038/nature02089.  Google Scholar

[33]

D. B. Xiu, Efficient collocational approach for parametric uncertainty analysis, Commun. Comput. Phys., 2 (2007), 293-309.   Google Scholar

[34]

Z. F. Zhang, T. R. Ding, W. Z. Huang and Z. X. Dong, Qualitative Theory of Differential Equations, Translations of Mathematical Monographs, 101. American Mathematical Society, Providence, RI, 1992.  Google Scholar

Figure 1.  Genetic toggle switch
Figure 3.  One intersection
Figure 4.  Three intersections
Figure 2.  One intersection
Figure 15.  Critical conditions
Figure 16.  a = 2.8, b = 3
Figure 17.  a = 1.8, b = 3
Figure 5.  $ \Phi'(u_2)>0 $
Figure 6.  $ \Phi'(u_2) = 0 $
Figure 7.  $ \Phi'(u_2)<0 $
Figure 8.  (S1)
Figure 9.  (S2)
Figure 10.  (S3)
Figure 11.  (S4)
Figure 12.  (S5)
Figure 14.  Global phase portraits
Figure 13.  Invariant manifolds for systems (45) and (47)
Table 1.  Distribution of zeros $ u_{01} $ and $ u_{02} $ of $ \Phi' $
(C3-1-1) $ 0<u_{01}<\zeta_-<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-1-2) $ 0<u_{01}<\zeta_-<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-1-3) $ 0<u_{01}<\zeta_-<\tilde{u}_2<u_{02}<\zeta_+<a $,
(C3-2-1) $ 0<u_{01}=\zeta_-<\tilde{u}_2<u_{02}<\zeta_+<a $,
(C3-2-2) $ 0<u_{01}=\zeta_-<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-2-3) $ 0<u_{01}=\zeta_-<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-3-1) $ 0<\zeta_-<u_{01}<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-3-2) $ 0<\zeta_-<u_{01}<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-3-3) $ 0<\zeta_-<u_{01}<\tilde{u}_2<u_{02}<\zeta_+<a $.
(C3-1-1) $ 0<u_{01}<\zeta_-<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-1-2) $ 0<u_{01}<\zeta_-<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-1-3) $ 0<u_{01}<\zeta_-<\tilde{u}_2<u_{02}<\zeta_+<a $,
(C3-2-1) $ 0<u_{01}=\zeta_-<\tilde{u}_2<u_{02}<\zeta_+<a $,
(C3-2-2) $ 0<u_{01}=\zeta_-<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-2-3) $ 0<u_{01}=\zeta_-<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-3-1) $ 0<\zeta_-<u_{01}<\tilde{u}_2<\zeta_+<u_{02}<a $,
(C3-3-2) $ 0<\zeta_-<u_{01}<\tilde{u}_2<u_{02}=\zeta_+<a $,
(C3-3-3) $ 0<\zeta_-<u_{01}<\tilde{u}_2<u_{02}<\zeta_+<a $.
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