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July  2019, 24(7): 3139-3155. doi: 10.3934/dcdsb.2018304

Interlocked multi-node positive and negative feedback loops facilitate oscillations

Qingqing Li and  Tianshou Zhou ,

Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China

* Corresponding author: Tianshou Zhou

Received  May 2018 Revised  July 2018 Published  July 2019 Early access  October 2018

Fund Project: This work was partially supported by the National Nature Science Foundation of China under Grant NO.91530320(T.Z.) and Grant NO.11775314(T.Z.) and 973 project of Science and Technology department of China under Grant NO.2014CB964703(T.Z.).

Positive and negative feedback loops in biological regulatory networks appear often in a multi-node manner since regulatory processes are in general multi-step. Although it is well known that interlocked positive and negative feedback loops (iPNFLs) can generate sustained oscillations, how the number of nodes in each loop affects the oscillations remains elusive. By analyzing a model of iPNFLs with multiple nodes, we find that the node number of the negative loop mainly plays a role of amplifying oscillation amplitudes whereas that of the positive loop mainly plays a role of reducing oscillatory regions, both depending on the (competitive or noncompetitive) way of interaction between the two loops. We also find that given an iPNFL network of the same structure, the noncompetitive model is more likely to produce large-amplitude oscillations than the competitive model. These results not only indicate that multi-node iPNFLs are an effective mechanism of promoting oscillations but also are helpful for the design of synthetic oscillators.

Keywords: Interlocked feedback loops, oscillation, combinatorial regulation, synthetic oscillator.
Mathematics Subject Classification: Primary: 92B05, 92C42; Secondary: 93C15.
Citation: Qingqing Li, Tianshou Zhou. Interlocked multi-node positive and negative feedback loops facilitate oscillations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3139-3155. doi: 10.3934/dcdsb.2018304
References:
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Y. An, G. S. Xu and Z. B. Yang, Calcium participates in feedback regulation of the oscillating ROP1 Rho GTPase in pollen tubes, Proc. Natl. Acad. Sci. U. S. A., 106 (2009), 22002-22007.  doi: 10.1073/pnas.0910811106.

[2]

B. Ananthasubramaniam and H. Herzel, Positive feedback promotes oscillations in negative feedback loops, PLoS One, 9 (2014), e104761. 

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D. Angeli, F. J. Jr and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad. Sci. U. S. A., 101 (2004), 1822-1827.  doi: 10.1073/pnas.0308265100.

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O. Brandman and T. Meyer, Feedback loops shape cellular signals in space and time, Science, 322 (2008), 390-395.  doi: 10.1126/science.1160617.

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O. Brandman, F. J. Jr, R. Li and T. Meyer, Interlinked fast and slow positive feedback loops drive reliable cell decisions, Science, 310 (2005), 496-498.  doi: 10.1126/science.1113834.

[6]

C. F. Calkhoven and G. Ab, Multiple steps in the regulation of transcription-factor level and activity, Biochemical Journal, 317 (1996), 329-342.  doi: 10.1042/bj3170329.

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S. M. Castillo-Hair, E. R. Villota and A. M. Coronado, Design principles for robust oscillatory behavior, Systems and Synthetic Biology, 9 (2015), 125-133.  doi: 10.1007/s11693-015-9178-6.

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O. Cinquin and J. Demongeot, Positive and Negative Feedback: Striking a balance between necessary antagonists, Journal of Theoretical Biology, 216 (2002), 229-241.  doi: 10.1006/jtbi.2002.2544.

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Z. Darieva, A. Clancy, R. Bulmer, E. Williams, A. Pic-Taylor, B. A. Morgan and A. D. Sharrocks, A competitive transcription factor binding mechanism determines the timing of late cell cycle-dependent gene expression, Molecular Cell, 38 (2010), 29-40.  doi: 10.1016/j.molcel.2010.02.030.

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L. Giorgetti, T. Siggers, G. Tiana, G. Caprara, S. Notarbartolo, T. Corona, M. Pasparakis, P. Milani, M. L. Bulyk and G. Natoli, Noncooperative interactions between transcription factors and clustered DNA binding sites enable graded transcriptional responses to environmental inputs, Molecular Cell, 37 (2010), 418-428.  doi: 10.1016/j.molcel.2010.01.016.

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B. Huang, X. Tian, F. Liu and W. Wang, Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops, Phys. Rev. E., 94 (2016), 052413.  doi: 10.1103/PhysRevE.94.052413.

[12]

Z. Hui, Y. Chen and C. Yong, Noise propagation in gene regulation networks involving interlinked positive and negative feedback loops, PLoS One, 7 (2012), e51840.  doi: 10.1371/journal.pone.0051840.

[13]

F. J. Jr, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability, Current Opinion in Cell Biology, 14 (2002), 140-148.  doi: 10.1016/S0955-0674(02)00314-9.

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D. Kim, Y. K. Kwon and K. H. Cho, Coupled positive and negative feedback circuits form an essential building block of cellular signaling pathways, Bioessays, 29 (2007), 85-90.  doi: 10.1002/bies.20511.

[15]

J. R. Kim, Y. Yoon and K. H. Cho, Coupled feedback loops form dynamic motifs of cellular networks, Biophysical Journal, 94 (2008), 359-365.  doi: 10.1529/biophysj.107.105106.

[16]

K. N. Lan, Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops, Journal of the Royal Society Interface, 9 (2012), 1998-2010.  doi: 10.1098/rsif.2012.0028.

[17]

I. M. Lengyel, D. Soroldoni, A. C. Oates and L. G. Morelli, Nonlinearity arising from noncooperative transcription factor binding enhances negative feedback and promotes genetic oscillations, Papers in Physics, 6 (2014), 060012.  doi: 10.4279/PIP.060012.

[18]

W. A. Lim, C. M. Lee and C. Tang, Design principles of regulatory networks: Searching for the molecular algorithms of the cell, Molecular Cell, 49 (2013), 202-212.  doi: 10.1016/j.molcel.2012.12.020.

[19]

W. Ma, A. Trusina, H. El-Samad, W. A. Lim and C. Tang, Defining Network Topologies that Can Achieve Biochemical Adaptation, Cell, 138 (2009), 760-773.  doi: 10.1016/j.cell.2009.06.013.

[20]

K. Maeda and H. Kurata, Long negative feedback loop enhances period tunability of biological oscillators, Journal of Theoretical Biology, 440 (2018), 21-31.  doi: 10.1016/j.jtbi.2017.12.014.

[21]

W. Mather, M. R. Bennett, J. Hasty and L. S. Tsimring, Delay-induced degrade-and-fire oscillations in small genetic circuits, Physical Review Letters, 102 (2009), 068105.  doi: 10.1103/PhysRevLett.102.068105.

[22]

N. A. M. Monk, Oscillatory Expression of Hes1, p53, and NF-kappaB Driven by Transcriptional Time Delays, Current Biology, 13 (2003), 1409-1413.  doi: 10.1016/S0960-9822(03)00494-9.

[23]

K. Montagne, R. Plasson, Y. Sakai, T. Fujii and Y. Rondelez, Programming An In Vitro Dna Oscillator Using A Molecular Networking Strategy, Molecular Systems Biology, 7 (2011), 466-472.  doi: 10.1038/Msb.2010.120.

[24]

M. Monti and P. R. Wolde, The accuracy of telling time via oscillatory signals, Physical Biology, 13 (2016), 035005.  doi: 10.1088/1478-3975/13/3/035005.

[25]

A. Munteanu, M. Constante, M. Isalan and R. V. Solé, Avoiding transcription factor competition at promoter level increases the chances of obtaining oscillation, BMC Systems Biology, 4 (2010), p66. doi: 10.1186/1752-0509-4-66.

[26]

R. Murugan, Theory on the dynamics of oscillatory loops in the transcription factor networks, PLoS One, 7 (2014), 3736-3739. 

[27]

M. Namiko, J. M. Hogh and S. Szabolcs, Coupled positive and negative feedbacks produce diverse gene expression patterns in colonies, MBio, 6 (2015), e00059-15.  doi: 10.1128/mBio.00059-15.

[28]

B. Novák and J. J. Tyson, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell. Biol, 9 (2008), 981-991. 

[29]

E. L. O'Brien, E. V. Itallie and M. R. Bennett, Modeling synthetic gene oscillators, Mathematical Biosciences, 236 (2012), 1-15.  doi: 10.1016/j.mbs.2012.01.001.

[30]

S. Pigolotti, S. Krishna and M. H. Jensen, Oscillation patterns in negative feedback loops, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 6533-6537.  doi: 10.1073/pnas.0610759104.

[31]

J. R. Pomerening, S. Y. Kim and F. J. Jr, Systems-level dissection of the cell-cycle oscillator: bypassing positive feedback produces damped oscillations, Cell, 122 (2005), 565-578.  doi: 10.1016/j.cell.2005.06.016.

[32]

T. Shopera, W. R. Henson, A. Ng, Y. J. Lee, K. Ng and T. S. Moon, Robust, tunable genetic memory from protein sequestration combined with positive feedback, Nucleic Acids Research, 43 (2015), 9086-9094.  doi: 10.1093/nar/gkv936.

[33]

H. Song, P. Smolen, E. Avron, D. A. Baxter and J. H. Byrne, Dynamics of a minimal model of interlocked positive and negative feedback loops of transcriptional regulation by cAMP-response element binding proteins, Biophysical Journal, 92 (2007), 3407-3424.  doi: 10.1529/biophysj.106.096891.

[34]

J. Stricker, S. Cookson, M. R. Bennett, W. H. Mather, L. S. Tsimring and J. Hasty, A fast, robust and tunable synthetic gene oscillator, Nature, 456 (2008), 516-519.  doi: 10.1038/nature07389.

[35]

P. K. Tapaswi, P. Bhattacharya, An extended mathematical-model of transcription and translation during embryogenesis, Cybernetica, 24 (1981), 61-84. Available from: http://library.isical.ac.in:8080/jspui/bitstream/10263/938/1/CYB-24-1-1981-P61-84.pdf.

[36]

X. J. Tian, X. P. Zhang, F. Liu and W. Wang, Interlinking positive and negative feedback loops creates a tunable motif in gene regulatory networks, Physical Review E Statistical Nonlinear and Soft Matter Physics, 80 (2009), 011926.  doi: 10.1103/PhysRevE.80.011926.

[37]

T. Y. Tsai, Y. S. Choi, W. Ma, J. R. Pomerening, C. Tang and J. E. Ferrell, Robust, tunable biological oscillations from interlinked positive and negative feedback loops, Science, 321 (2008), 126-129.  doi: 10.1126/science.1156951.

[38]

K. Uriu and H. Tei, Feedback loops interlocked at competitive binding sites amplify and facilitate genetic oscillations, Journal of Theoretical Biology, 428 (2017), 56-64.  doi: 10.1016/j.jtbi.2017.06.005.

[39]

A. Verdugo and R. Rand, Hopf bifurcation in a DDE model of gene expression, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 235-242.  doi: 10.1016/j.cnsns.2006.05.001.

[40]

Y. C. Wang, S. E. Peterson and J. F. Loring, Protein post-translational modifications and regulation of pluripotency in human stem cells, Cell Research, 24 (2014), 143-160.  doi: 10.1038/cr.2013.151.

[41]

J. J. Wei and C. B. Yu, Hopf bifurcation analysis in a model of oscillatory gene expression with delay, Proceedings of the Royal Society of Edinburgh, 139 (2009), 879-895.  doi: 10.1017/S0308210507000091.

[42]

X. P. Zhang, Z. Cheng, F. Liu and W. Wang, Linking fast and slow positive feedback loops creates an optimal bistable switch in cell signaling, Physical Review E Statistical Nonlinear and Soft Matter Physics, 76 (2007), 031924.  doi: 10.1103/PhysRevE.76.031924.

[43]

X. Zhao, T. Hirota, X. M. Han, H. Cho, L. W. Chong, K. Lamia, S. Liu, A. R. Atkins, E. Banayo, C. Liddle, R. T. Yu, J.R. Yates, S. A. Kay, M. Downes and R. M. Evans, Circadian amplitude regulation via FBXW7-targeted REV-ERB $α$ degradation, Cell, 165 (2016), 1644-1657.  doi: 10.1016/j.cell.2016.05.012.

show all references

References:
[1]

Y. An, G. S. Xu and Z. B. Yang, Calcium participates in feedback regulation of the oscillating ROP1 Rho GTPase in pollen tubes, Proc. Natl. Acad. Sci. U. S. A., 106 (2009), 22002-22007.  doi: 10.1073/pnas.0910811106.

[2]

B. Ananthasubramaniam and H. Herzel, Positive feedback promotes oscillations in negative feedback loops, PLoS One, 9 (2014), e104761. 

[3]

D. Angeli, F. J. Jr and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Natl. Acad. Sci. U. S. A., 101 (2004), 1822-1827.  doi: 10.1073/pnas.0308265100.

[4]

O. Brandman and T. Meyer, Feedback loops shape cellular signals in space and time, Science, 322 (2008), 390-395.  doi: 10.1126/science.1160617.

[5]

O. Brandman, F. J. Jr, R. Li and T. Meyer, Interlinked fast and slow positive feedback loops drive reliable cell decisions, Science, 310 (2005), 496-498.  doi: 10.1126/science.1113834.

[6]

C. F. Calkhoven and G. Ab, Multiple steps in the regulation of transcription-factor level and activity, Biochemical Journal, 317 (1996), 329-342.  doi: 10.1042/bj3170329.

[7]

S. M. Castillo-Hair, E. R. Villota and A. M. Coronado, Design principles for robust oscillatory behavior, Systems and Synthetic Biology, 9 (2015), 125-133.  doi: 10.1007/s11693-015-9178-6.

[8]

O. Cinquin and J. Demongeot, Positive and Negative Feedback: Striking a balance between necessary antagonists, Journal of Theoretical Biology, 216 (2002), 229-241.  doi: 10.1006/jtbi.2002.2544.

[9]

Z. Darieva, A. Clancy, R. Bulmer, E. Williams, A. Pic-Taylor, B. A. Morgan and A. D. Sharrocks, A competitive transcription factor binding mechanism determines the timing of late cell cycle-dependent gene expression, Molecular Cell, 38 (2010), 29-40.  doi: 10.1016/j.molcel.2010.02.030.

[10]

L. Giorgetti, T. Siggers, G. Tiana, G. Caprara, S. Notarbartolo, T. Corona, M. Pasparakis, P. Milani, M. L. Bulyk and G. Natoli, Noncooperative interactions between transcription factors and clustered DNA binding sites enable graded transcriptional responses to environmental inputs, Molecular Cell, 37 (2010), 418-428.  doi: 10.1016/j.molcel.2010.01.016.

[11]

B. Huang, X. Tian, F. Liu and W. Wang, Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops, Phys. Rev. E., 94 (2016), 052413.  doi: 10.1103/PhysRevE.94.052413.

[12]

Z. Hui, Y. Chen and C. Yong, Noise propagation in gene regulation networks involving interlinked positive and negative feedback loops, PLoS One, 7 (2012), e51840.  doi: 10.1371/journal.pone.0051840.

[13]

F. J. Jr, Self-perpetuating states in signal transduction: Positive feedback, double-negative feedback and bistability, Current Opinion in Cell Biology, 14 (2002), 140-148.  doi: 10.1016/S0955-0674(02)00314-9.

[14]

D. Kim, Y. K. Kwon and K. H. Cho, Coupled positive and negative feedback circuits form an essential building block of cellular signaling pathways, Bioessays, 29 (2007), 85-90.  doi: 10.1002/bies.20511.

[15]

J. R. Kim, Y. Yoon and K. H. Cho, Coupled feedback loops form dynamic motifs of cellular networks, Biophysical Journal, 94 (2008), 359-365.  doi: 10.1529/biophysj.107.105106.

[16]

K. N. Lan, Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops, Journal of the Royal Society Interface, 9 (2012), 1998-2010.  doi: 10.1098/rsif.2012.0028.

[17]

I. M. Lengyel, D. Soroldoni, A. C. Oates and L. G. Morelli, Nonlinearity arising from noncooperative transcription factor binding enhances negative feedback and promotes genetic oscillations, Papers in Physics, 6 (2014), 060012.  doi: 10.4279/PIP.060012.

[18]

W. A. Lim, C. M. Lee and C. Tang, Design principles of regulatory networks: Searching for the molecular algorithms of the cell, Molecular Cell, 49 (2013), 202-212.  doi: 10.1016/j.molcel.2012.12.020.

[19]

W. Ma, A. Trusina, H. El-Samad, W. A. Lim and C. Tang, Defining Network Topologies that Can Achieve Biochemical Adaptation, Cell, 138 (2009), 760-773.  doi: 10.1016/j.cell.2009.06.013.

[20]

K. Maeda and H. Kurata, Long negative feedback loop enhances period tunability of biological oscillators, Journal of Theoretical Biology, 440 (2018), 21-31.  doi: 10.1016/j.jtbi.2017.12.014.

[21]

W. Mather, M. R. Bennett, J. Hasty and L. S. Tsimring, Delay-induced degrade-and-fire oscillations in small genetic circuits, Physical Review Letters, 102 (2009), 068105.  doi: 10.1103/PhysRevLett.102.068105.

[22]

N. A. M. Monk, Oscillatory Expression of Hes1, p53, and NF-kappaB Driven by Transcriptional Time Delays, Current Biology, 13 (2003), 1409-1413.  doi: 10.1016/S0960-9822(03)00494-9.

[23]

K. Montagne, R. Plasson, Y. Sakai, T. Fujii and Y. Rondelez, Programming An In Vitro Dna Oscillator Using A Molecular Networking Strategy, Molecular Systems Biology, 7 (2011), 466-472.  doi: 10.1038/Msb.2010.120.

[24]

M. Monti and P. R. Wolde, The accuracy of telling time via oscillatory signals, Physical Biology, 13 (2016), 035005.  doi: 10.1088/1478-3975/13/3/035005.

[25]

A. Munteanu, M. Constante, M. Isalan and R. V. Solé, Avoiding transcription factor competition at promoter level increases the chances of obtaining oscillation, BMC Systems Biology, 4 (2010), p66. doi: 10.1186/1752-0509-4-66.

[26]

R. Murugan, Theory on the dynamics of oscillatory loops in the transcription factor networks, PLoS One, 7 (2014), 3736-3739. 

[27]

M. Namiko, J. M. Hogh and S. Szabolcs, Coupled positive and negative feedbacks produce diverse gene expression patterns in colonies, MBio, 6 (2015), e00059-15.  doi: 10.1128/mBio.00059-15.

[28]

B. Novák and J. J. Tyson, Design principles of biochemical oscillators, Nat. Rev. Mol. Cell. Biol, 9 (2008), 981-991. 

[29]

E. L. O'Brien, E. V. Itallie and M. R. Bennett, Modeling synthetic gene oscillators, Mathematical Biosciences, 236 (2012), 1-15.  doi: 10.1016/j.mbs.2012.01.001.

[30]

S. Pigolotti, S. Krishna and M. H. Jensen, Oscillation patterns in negative feedback loops, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 6533-6537.  doi: 10.1073/pnas.0610759104.

[31]

J. R. Pomerening, S. Y. Kim and F. J. Jr, Systems-level dissection of the cell-cycle oscillator: bypassing positive feedback produces damped oscillations, Cell, 122 (2005), 565-578.  doi: 10.1016/j.cell.2005.06.016.

[32]

T. Shopera, W. R. Henson, A. Ng, Y. J. Lee, K. Ng and T. S. Moon, Robust, tunable genetic memory from protein sequestration combined with positive feedback, Nucleic Acids Research, 43 (2015), 9086-9094.  doi: 10.1093/nar/gkv936.

[33]

H. Song, P. Smolen, E. Avron, D. A. Baxter and J. H. Byrne, Dynamics of a minimal model of interlocked positive and negative feedback loops of transcriptional regulation by cAMP-response element binding proteins, Biophysical Journal, 92 (2007), 3407-3424.  doi: 10.1529/biophysj.106.096891.

[34]

J. Stricker, S. Cookson, M. R. Bennett, W. H. Mather, L. S. Tsimring and J. Hasty, A fast, robust and tunable synthetic gene oscillator, Nature, 456 (2008), 516-519.  doi: 10.1038/nature07389.

[35]

P. K. Tapaswi, P. Bhattacharya, An extended mathematical-model of transcription and translation during embryogenesis, Cybernetica, 24 (1981), 61-84. Available from: http://library.isical.ac.in:8080/jspui/bitstream/10263/938/1/CYB-24-1-1981-P61-84.pdf.

[36]

X. J. Tian, X. P. Zhang, F. Liu and W. Wang, Interlinking positive and negative feedback loops creates a tunable motif in gene regulatory networks, Physical Review E Statistical Nonlinear and Soft Matter Physics, 80 (2009), 011926.  doi: 10.1103/PhysRevE.80.011926.

[37]

T. Y. Tsai, Y. S. Choi, W. Ma, J. R. Pomerening, C. Tang and J. E. Ferrell, Robust, tunable biological oscillations from interlinked positive and negative feedback loops, Science, 321 (2008), 126-129.  doi: 10.1126/science.1156951.

[38]

K. Uriu and H. Tei, Feedback loops interlocked at competitive binding sites amplify and facilitate genetic oscillations, Journal of Theoretical Biology, 428 (2017), 56-64.  doi: 10.1016/j.jtbi.2017.06.005.

[39]

A. Verdugo and R. Rand, Hopf bifurcation in a DDE model of gene expression, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 235-242.  doi: 10.1016/j.cnsns.2006.05.001.

[40]

Y. C. Wang, S. E. Peterson and J. F. Loring, Protein post-translational modifications and regulation of pluripotency in human stem cells, Cell Research, 24 (2014), 143-160.  doi: 10.1038/cr.2013.151.

[41]

J. J. Wei and C. B. Yu, Hopf bifurcation analysis in a model of oscillatory gene expression with delay, Proceedings of the Royal Society of Edinburgh, 139 (2009), 879-895.  doi: 10.1017/S0308210507000091.

[42]

X. P. Zhang, Z. Cheng, F. Liu and W. Wang, Linking fast and slow positive feedback loops creates an optimal bistable switch in cell signaling, Physical Review E Statistical Nonlinear and Soft Matter Physics, 76 (2007), 031924.  doi: 10.1103/PhysRevE.76.031924.

[43]

X. Zhao, T. Hirota, X. M. Han, H. Cho, L. W. Chong, K. Lamia, S. Liu, A. R. Atkins, E. Banayo, C. Liddle, R. T. Yu, J.R. Yates, S. A. Kay, M. Downes and R. M. Evans, Circadian amplitude regulation via FBXW7-targeted REV-ERB $α$ degradation, Cell, 165 (2016), 1644-1657.  doi: 10.1016/j.cell.2016.05.012.

Figure 1.  An example of interlocked positive and negative feedback loops. (a) Network topology, where the homogenous negative feedback loop contains $N$ node with $N$ being an odd number, the homogeneous positive feedback loop contains $M+1$ node with $M$ being a positive integer, and node $X_N$ is common. (b, c) Two representative modes for the combinatorial regulation of two transcription factors, where (b) corresponds to noncompetitive binding whereas (c) to competitive binding
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Figure 2.  Influence of parameter $\gamma$ on the real and imaginary parts of the root of the characteristic equation $f(\lambda)=0$:non-competition model. (a)$N=3$, $M=8$(corresponding to the case of $N<M+1$); (b)$N=3$, $M=2$ (corresponding to the case of$N=M+1$; (C)$N=5$, $M=2$) (corresponding to the case of $N>M+1$); (d) Bifurcation diagram of $x_3$ versus $\gamma$ for $N=3$, $M=2$.Here green solid and red dashed lines represent stable and unstable steady states respectively, and the symbol 'HB' represents the Hopf bifurcation point. Other parameter values are set as $\alpha_n=\alpha_p=3 $, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\alpha_n=\alpha_p=\alpha$, and $K_1=K_2=1$
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Figure 3.  The occurrence of oscillations in the system of interlocked multi-node positive and negative loops. (a, c) correspond to the noncompetitive model whereas (b, d) to the competitive model. (a, b) show time series of component $x_5$, where the inset is a phase trajectory in the $(x_4,x_5)$ plane. (c, d) show both a stable region for the fixed point (corresponding to 'nonoscillation' indicated in the diagram) and an oscillatory region (corresponding to 'oscillation' indicated in the diagram) in the $(N,M)$ plane, where the green das line is the border of the two regions. Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\alpha_n=\alpha_p=1$, and $K_1=0.5$, and $K_2=4$
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Figure 4.  The effect of the node number of the negative feedback loop on the amplitude and period of oscillations, where $M=2$.(a, c) correspond to the noncompetitive model, (b, d) correspond to the competitive model. (a, b) for oscillation amplitude, (c, d) for oscillation period. Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\alpha_n=\alpha_p=1$, and $K_2=0.5$
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Figure 5.  The effect of the node number of the positive feedback loop on the amplitude and period of oscillations, where $N=5$.(a, c) correspond to the noncompetitive model, (b, d) correspond to the competitive model. (a, b) for oscillation amplitude, (c, d) is related to period. Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$, $\gamma_n=\gamma_p=1$, and $K_2=0.5$
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Figure 6.  Three dimensional pseudo-diagram in the $(\gamma,K)$ plane, where the color bar represents oscillation amplitude, where parameter $N$ is fixed at $N=3$.(a, b, c, d) correspond to non-competitive binding with $M=1,2,6,8$ from left to right (corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely); (e, f, g, h) correspond to competitive binding with $M=1,2,6,8$ from left to right(corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely). Parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$
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Figure 7.  Three dimensional pseudo-diagram in the $(\gamma,K)$ plane, where the color bar represents oscillation peroid, where parameter $N$ is fixed at $N=3$. (a, b, c, d) correspond to non-competitive binding with $M=1,2,6,8$ from left to right (corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely); (e, f, g, h) correspond to competitive binding with $M=1,2,6,8$ from left to right(corresponding to the cases of $N>M+1$, $N=M+1$ and $N<M+1$, respecitvely). Other parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$
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Figure 8.  Comparison between influences of NFL node number and time delay on oscillating region, where the number of PFL nodes, $M$ is fixed at $M=2$. (a) displays the dependence of the maximum oscillation amplitude on the ratio of $K_1/K_2$; (b) shows the oscillation region of in the $(K_1,K_2)$ plane; (c) shows the oscillation region of $\tau=1.04$ also in the $(K_1,K_2)$ plane. Other parameter values are set as $\alpha_n=\alpha_p=3$, $c_n=c_p=0.1$, $h_n=3$, $h_p=1$
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