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April  2022, 9(2): 133-158. doi: 10.3934/jcd.2021024

A mathematical analysis of an activator-inhibitor Rho GTPase model

1. 

University of Zaragoza, Department of Mechanical Engineering, Campus Rio Ebro, E-50018 Zaragoza, Spain

2. 

TU Dortmund University and Max-Planck-Institute of Molecular Physiology, Department of Chemistry and Chemical Biology and Department of Systemic Cell Biology, Otto-Hahn-Str. 4a, 44227, Dortmund, Germany

3. 

University of Manitoba, Department of Mathematics, 428 Machray Hall, Winnipeg, MB R3T 2N2 Canada

4. 

University of Sussex, Department of Mathematics, School of Mathematical and Physical Sciences, Pevensey 3 5C15, Brighton, BN1 9QH. UK

* Corresponding author: A.Madzvamuse@sussex.ac.uk

Received  April 2021 Revised  September 2021 Published  April 2022 Early access  December 2021

Recent experimental observations reveal that local cellular contraction pulses emerge via a combination of fast positive and slow negative feedbacks based on a signal network composed of Rho, GEF and Myosin interactions [22]. As an examplary, we propose to study a plausible, hypothetical temporal model that mirrors general principles of fast positive and slow negative feedback, a hallmark for activator-inhibitor models. The methodology involves (ⅰ) a qualitative analysis to unravel system switching between different states (stable, excitable, oscillatory and bistable) through model parameter variations; (ⅱ) a numerical bifurcation analysis using the positive feedback mediator concentration as a bifurcation parameter, (ⅲ) a sensitivity analysis to quantify the effect of parameter uncertainty on the model output for different dynamic regimes of the model system; and (ⅳ) numerical simulations of the model system for model predictions. Our methodological approach supports the role of mathematical and computational models in unravelling mechanisms for molecular and developmental processes and provides tools for analysis of temporal models of this nature.

Citation: Victor Ogesa Juma, Leif Dehmelt, Stéphanie Portet, Anotida Madzvamuse. A mathematical analysis of an activator-inhibitor Rho GTPase model. Journal of Computational Dynamics, 2022, 9 (2) : 133-158. doi: 10.3934/jcd.2021024
References:
[1]

E. BodineL. DeaettJ. McDonaldD. Olesky and P. van den Driessche, Sign patterns that require or allow particular refined inertias, Linear Algebra Appl., 437 (2012), 2228-2242.  doi: 10.1016/j.laa.2012.05.014.

[2]

A. Bolado-CarrancioO. S. RukhlenkoE. NikonovaM. A. TsyganovA. WheelerA. Garcia-MunozW. KolchA. von Kriegsheim and B. N. Kholodenko, Periodic propagating waves coordinate RhoGTPase network dynamics at the leading and trailing edges during cell migration, Elife, 9 (2020), e58165. 

[3]

A. BoureuxE. VignalS. Faure and P. Fort, Evolution of the Rho family of ras-like GTPases in eukaryotes, Molecular Biology and Evolution, 24 (2006), 203-216.  doi: 10.1093/molbev/msl145.

[4]

E. Campillo-FunolletC. Venkataraman and A. Madzvamuse, Bayesian parameter identification for Turing systems on stationary and evolving domains, Bull. Math. Biol., 81 (2019), 81-104.  doi: 10.1007/s11538-018-0518-z.

[5]

G. CulosD. Olesky and P. van den Driessche, Using sign patterns to detect the possibility of periodicity in biological systems, J. Math. Biol., 72 (2016), 1281-1300.  doi: 10.1007/s00285-015-0906-z.

[6]

C. DerMardirossian and G. M. Bokoch, GDIs: Central regulatory molecules in Rho GTPase activation, Trends in Cell Biology, 15 (2005), 356-363.  doi: 10.1016/j.tcb.2005.05.001.

[7]

A. DhoogeW. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362.

[8]

A. DhoogeW. GovaertsY. A. KuznetsovH. Meijer and B. Sautois, New features of the software Matcont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.  doi: 10.1080/13873950701742754.

[9]

J. DrewC. Johnson and D. Olesky, Spectrally arbitrary patterns, Linear Algebra Appl., 308 (2000), 121-137.  doi: 10.1016/S0024-3795(00)00026-4.

[10]

D. G. Drubin and W. J. Nelson, Origins of cell polarity, Cell, 84 (1996), 335-344.  doi: 10.1016/S0092-8674(00)81278-7.

[11]

A. J. EnglerS. SenH. L. Sweeney and D. E. Discher, Matrix elasticity directs stem cell lineage specification, Cell, 126 (2006), 677-689.  doi: 10.1016/j.cell.2006.06.044.

[12]

P. Friedl and S. Alexander, Cancer invasion and the microenvironment: Plasticity and reciprocity, Cell, 147 (2011), 992-1009.  doi: 10.1016/j.cell.2011.11.016.

[13]

C. GarnettD. Olesky and P. van den Driessche, A note on sign patterns of order 3 that require particular refined inertias, Linear Algebra Appl., 450 (2014), 293-300.  doi: 10.1016/j.laa.2014.03.007.

[14]

A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, Proc. Nat. Acad. Sci., 78 (1981), 6840-6844.  doi: 10.1073/pnas.78.11.6840.

[15]

M. GraesslJ. KochA. CalderonD. KampsS. BanerjeeT. MazelN. SchulzeJ. K. JungkurthR. Patwardhan and D. Solouk, An excitable Rho GTPase signaling network generates dynamic subcellular contraction patterns, JCB, 216 (2017), 4271-4285.  doi: 10.1083/jcb.201706052.

[16]

C. GuilluyR. Garcia-Mata and K. Burridge, Rho protein crosstalk: Another social network?, Trends in Cell Biology, 21 (2011), 718-726.  doi: 10.1016/j.tcb.2011.08.002.

[17]

R. G. Hodge and A. J. Ridley, Regulating Rho GTPases and their regulators, Nature Reviews Molecular Cell Biology, 17 (2016), 496-510.  doi: 10.1038/nrm.2016.67.

[18]

W. R. HolmesM. A. Mata and L. Edelstein-Keshet, Local perturbation analysis: A computational tool for biophysical reaction-diffusion models, Biophysical Journal, 108 (2015), 230-236.  doi: 10.1016/j.bpj.2014.11.3457.

[19]

M. JacquierS. KuriakoseA. BhardwajY. ZhangA. ShrivastavS. Portet and S. V. Shrivastav, Investigation of novel regulation of N-myristoyltransferase by mammalian target of rapamycin in breast cancer cells, Scientific reports, 8 (2018), 1-11.  doi: 10.1038/s41598-018-30447-0.

[20] D. W. Jordan and P. Smith, Nonlinear ordinary differential equations: An introduction to dynamical systems, 3$^rd$ edition, Oxford University Press, USA, 1999. 
[21]

V. O. Juma, Data-Driven Mathematical Modelling and Simulation of Rho-Myosin Dynamics, PhD thesis, University of Sussex, 2019.

[22]

D. KampsJ. KochV. O. JumaE. Campillo-FunolletM. GraesslS. BanerjeeT. MazelX. ChenY.-W. Wu and S. Portet, Optogenetic tuning reveals rho amplification-dependent dynamics of a cell contraction signal network, Cell Reports, 33 (2020), 108467.  doi: 10.1016/j.celrep.2020.108467.

[23]

E. J. Y. KimE. Korotkevich and T. Hiiragi, Coordination of cell polarity, mechanics and fate in tissue self-organization, Trends in Cell Biology, 28 (2018), 541-550.  doi: 10.1016/j.tcb.2018.02.008.

[24]

I.-J. KimD. D. OleskyB. L. ShaderP. van den DriesscheH. van der Holst and K. N. Vander Meulen, Generating potentially nilpotent full sign patterns, Electron. J. Linear Algebra, 18 (2009), 162-175.  doi: 10.13001/1081-3810.1302.

[25]

R. Larter, Sensitivity analysis of autonomous oscillators. separation of secular terms and determination of structural stability, J. Phys. Chem., 87 (1983), 3114-3121.  doi: 10.1021/j100239a032.

[26]

D. LeeA. KoulN. LubnaS. A. McKenna and S. Portet, Mathematical modelling of OAS2 activation by dsRNA and effects of dsRNA lengths, AIMS Math., 6 (2021), 5924-5941.  doi: 10.3934/math.2021351.

[27]

B. Lu and H. Yue, Sensitivity analysis of oscillatory biological systems with a SVD-based algorithm, Systemics and Informatics World Network, 10 (2010), 85-92. 

[28]

Y. Lu and H. Yue, Objective sensitivity analysis of biological oscillatory systems, IFAC Proceedings Volumes, 44 (2011), 10466-10471.  doi: 10.3182/20110828-6-IT-1002.02967.

[29]

A. F. MaréeA. JilkineA. DawesV. A. Grieneisen and L. Edelstein-Keshet, Polarization and movement of keratocytes: A multiscale modelling approach, Bulletin of Mathematical Biology, 68 (2006), 1169-1211. 

[30]

MATLAB, 9.7.0.1190202 (R2019b), The MathWorks Inc., Natick, Massachusetts, 2018.

[31]

J. D. Murray, Mathematical Biology I: An Introduction, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[32]

P. NalbantY.-C. ChangJ. BirkenfeldZ.-F. Chang and G. M. Bokoch, Guanine nucleotide exchange factor-H1 regulates cell migration via localized activation of Rhoa at the leading edge, Molecular Biology of the Cell, 20 (2009), 3985-4129.  doi: 10.1091/mbc.e09-01-0041.

[33]

S. Portet, Dynamics of in vitro intermediate filament length distributions, Journal of Theoretical Biology, 332 (2013), 20-29.  doi: 10.1016/j.jtbi.2013.04.004.

[34]

S. PortetA. MadzvamuseA. ChungR. E. Leube and R. Windoffer, Keratin dynamics: Modeling the interplay between turnover and transport, PloS One, 10 (2015), 1-29.  doi: 10.1371/journal.pone.0121090.

[35]

H. Rabitz and D. Edelson, Numerical techniques for modelling and analysis of oscillating chemical reactions, Oscillations and Traveling Waves in Chemical Systems, (1985), 193-222. 

[36]

M. Raftopoulou and A. Hall, Cell migration: Rho GTPases lead the way, Developmental Biology, 265 (2004), 23-32.  doi: 10.1016/j.ydbio.2003.06.003.

[37]

A. J. Ridley, Rho GTPase signalling in cell migration, Current Opinion in Cell Biology, 36 (2015), 103-112.  doi: 10.1016/j.ceb.2015.08.005.

[38]

A. J. RidleyM. A. SchwartzK. BurridgeR. A. FirtelM. H. GinsbergG. BorisyJ. T. Parsons and A. R. Horwitz, Cell migration: Integrating signals from front to back, Science, 302 (2003), 1704-1709.  doi: 10.1126/science.1092053.

[39]

S. SahaT. L. Nagy and O. D. Weiner, Joining forces: Crosstalk between biochemical signalling and physical forces orchestrates cellular polarity and dynamics, Philosophical Transactions of the Royal Society B: Biological Sciences, 373 (2018), 20170145.  doi: 10.1098/rstb.2017.0145.

[40]

L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), 1-22.  doi: 10.1137/S1064827594276424.

[41]

C. M. SimonE. M. VaughanW. M. Bement and L. Edelstein-Keshet, Pattern formation of Rho GTPases in single cell wound healing, Molecular Biology of the Cell, 24 (2013), 421-432.  doi: 10.1091/mbc.e12-08-0634.

[42]

R. Tomovic and M. Vukobratovic, General Sensitivity Theory, vol. 1 of Mod. Analytic Comput. Methods Sci. Math., 35, New York, NY: North-Holland, 1972.

[43]

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

[44]

J. J. TysonK. Chen and B. Novak, Network dynamics and cell physiology, Nature Reviews Molecular Cell Biology, 2 (2001), 908-916.  doi: 10.1038/35103078.

[45]

J. J. Tyson and B. Novak, Temporal organization of the cell cycle, Current Biology, 18 (2008), 759-768.  doi: 10.1016/j.cub.2008.07.001.

[46]

K. WolfM. Te LindertM. KrauseS. AlexanderJ. Te RietA. L. WillisR. M. HoffmanC. G. FigdorS. J. Weiss and P. Friedl, Physical limits of cell migration: Control by ECM space and nuclear deformation and tuning by proteolysis and traction force, Journal of Cell Biology, 201 (2013), 1069-1084.  doi: 10.1083/jcb.201210152.

[47]

D. E. ZakJ. Stelling and F. J. Doyle, Sensitivity analysis of oscillatory (bio) chemical systems, Computers & Chemical Engineering, 29 (2005), 663-673.  doi: 10.1016/j.compchemeng.2004.08.021.

show all references

References:
[1]

E. BodineL. DeaettJ. McDonaldD. Olesky and P. van den Driessche, Sign patterns that require or allow particular refined inertias, Linear Algebra Appl., 437 (2012), 2228-2242.  doi: 10.1016/j.laa.2012.05.014.

[2]

A. Bolado-CarrancioO. S. RukhlenkoE. NikonovaM. A. TsyganovA. WheelerA. Garcia-MunozW. KolchA. von Kriegsheim and B. N. Kholodenko, Periodic propagating waves coordinate RhoGTPase network dynamics at the leading and trailing edges during cell migration, Elife, 9 (2020), e58165. 

[3]

A. BoureuxE. VignalS. Faure and P. Fort, Evolution of the Rho family of ras-like GTPases in eukaryotes, Molecular Biology and Evolution, 24 (2006), 203-216.  doi: 10.1093/molbev/msl145.

[4]

E. Campillo-FunolletC. Venkataraman and A. Madzvamuse, Bayesian parameter identification for Turing systems on stationary and evolving domains, Bull. Math. Biol., 81 (2019), 81-104.  doi: 10.1007/s11538-018-0518-z.

[5]

G. CulosD. Olesky and P. van den Driessche, Using sign patterns to detect the possibility of periodicity in biological systems, J. Math. Biol., 72 (2016), 1281-1300.  doi: 10.1007/s00285-015-0906-z.

[6]

C. DerMardirossian and G. M. Bokoch, GDIs: Central regulatory molecules in Rho GTPase activation, Trends in Cell Biology, 15 (2005), 356-363.  doi: 10.1016/j.tcb.2005.05.001.

[7]

A. DhoogeW. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362.

[8]

A. DhoogeW. GovaertsY. A. KuznetsovH. Meijer and B. Sautois, New features of the software Matcont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), 147-175.  doi: 10.1080/13873950701742754.

[9]

J. DrewC. Johnson and D. Olesky, Spectrally arbitrary patterns, Linear Algebra Appl., 308 (2000), 121-137.  doi: 10.1016/S0024-3795(00)00026-4.

[10]

D. G. Drubin and W. J. Nelson, Origins of cell polarity, Cell, 84 (1996), 335-344.  doi: 10.1016/S0092-8674(00)81278-7.

[11]

A. J. EnglerS. SenH. L. Sweeney and D. E. Discher, Matrix elasticity directs stem cell lineage specification, Cell, 126 (2006), 677-689.  doi: 10.1016/j.cell.2006.06.044.

[12]

P. Friedl and S. Alexander, Cancer invasion and the microenvironment: Plasticity and reciprocity, Cell, 147 (2011), 992-1009.  doi: 10.1016/j.cell.2011.11.016.

[13]

C. GarnettD. Olesky and P. van den Driessche, A note on sign patterns of order 3 that require particular refined inertias, Linear Algebra Appl., 450 (2014), 293-300.  doi: 10.1016/j.laa.2014.03.007.

[14]

A. Goldbeter and D. E. Koshland, An amplified sensitivity arising from covalent modification in biological systems, Proc. Nat. Acad. Sci., 78 (1981), 6840-6844.  doi: 10.1073/pnas.78.11.6840.

[15]

M. GraesslJ. KochA. CalderonD. KampsS. BanerjeeT. MazelN. SchulzeJ. K. JungkurthR. Patwardhan and D. Solouk, An excitable Rho GTPase signaling network generates dynamic subcellular contraction patterns, JCB, 216 (2017), 4271-4285.  doi: 10.1083/jcb.201706052.

[16]

C. GuilluyR. Garcia-Mata and K. Burridge, Rho protein crosstalk: Another social network?, Trends in Cell Biology, 21 (2011), 718-726.  doi: 10.1016/j.tcb.2011.08.002.

[17]

R. G. Hodge and A. J. Ridley, Regulating Rho GTPases and their regulators, Nature Reviews Molecular Cell Biology, 17 (2016), 496-510.  doi: 10.1038/nrm.2016.67.

[18]

W. R. HolmesM. A. Mata and L. Edelstein-Keshet, Local perturbation analysis: A computational tool for biophysical reaction-diffusion models, Biophysical Journal, 108 (2015), 230-236.  doi: 10.1016/j.bpj.2014.11.3457.

[19]

M. JacquierS. KuriakoseA. BhardwajY. ZhangA. ShrivastavS. Portet and S. V. Shrivastav, Investigation of novel regulation of N-myristoyltransferase by mammalian target of rapamycin in breast cancer cells, Scientific reports, 8 (2018), 1-11.  doi: 10.1038/s41598-018-30447-0.

[20] D. W. Jordan and P. Smith, Nonlinear ordinary differential equations: An introduction to dynamical systems, 3$^rd$ edition, Oxford University Press, USA, 1999. 
[21]

V. O. Juma, Data-Driven Mathematical Modelling and Simulation of Rho-Myosin Dynamics, PhD thesis, University of Sussex, 2019.

[22]

D. KampsJ. KochV. O. JumaE. Campillo-FunolletM. GraesslS. BanerjeeT. MazelX. ChenY.-W. Wu and S. Portet, Optogenetic tuning reveals rho amplification-dependent dynamics of a cell contraction signal network, Cell Reports, 33 (2020), 108467.  doi: 10.1016/j.celrep.2020.108467.

[23]

E. J. Y. KimE. Korotkevich and T. Hiiragi, Coordination of cell polarity, mechanics and fate in tissue self-organization, Trends in Cell Biology, 28 (2018), 541-550.  doi: 10.1016/j.tcb.2018.02.008.

[24]

I.-J. KimD. D. OleskyB. L. ShaderP. van den DriesscheH. van der Holst and K. N. Vander Meulen, Generating potentially nilpotent full sign patterns, Electron. J. Linear Algebra, 18 (2009), 162-175.  doi: 10.13001/1081-3810.1302.

[25]

R. Larter, Sensitivity analysis of autonomous oscillators. separation of secular terms and determination of structural stability, J. Phys. Chem., 87 (1983), 3114-3121.  doi: 10.1021/j100239a032.

[26]

D. LeeA. KoulN. LubnaS. A. McKenna and S. Portet, Mathematical modelling of OAS2 activation by dsRNA and effects of dsRNA lengths, AIMS Math., 6 (2021), 5924-5941.  doi: 10.3934/math.2021351.

[27]

B. Lu and H. Yue, Sensitivity analysis of oscillatory biological systems with a SVD-based algorithm, Systemics and Informatics World Network, 10 (2010), 85-92. 

[28]

Y. Lu and H. Yue, Objective sensitivity analysis of biological oscillatory systems, IFAC Proceedings Volumes, 44 (2011), 10466-10471.  doi: 10.3182/20110828-6-IT-1002.02967.

[29]

A. F. MaréeA. JilkineA. DawesV. A. Grieneisen and L. Edelstein-Keshet, Polarization and movement of keratocytes: A multiscale modelling approach, Bulletin of Mathematical Biology, 68 (2006), 1169-1211. 

[30]

MATLAB, 9.7.0.1190202 (R2019b), The MathWorks Inc., Natick, Massachusetts, 2018.

[31]

J. D. Murray, Mathematical Biology I: An Introduction, 3$^{rd}$ edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.

[32]

P. NalbantY.-C. ChangJ. BirkenfeldZ.-F. Chang and G. M. Bokoch, Guanine nucleotide exchange factor-H1 regulates cell migration via localized activation of Rhoa at the leading edge, Molecular Biology of the Cell, 20 (2009), 3985-4129.  doi: 10.1091/mbc.e09-01-0041.

[33]

S. Portet, Dynamics of in vitro intermediate filament length distributions, Journal of Theoretical Biology, 332 (2013), 20-29.  doi: 10.1016/j.jtbi.2013.04.004.

[34]

S. PortetA. MadzvamuseA. ChungR. E. Leube and R. Windoffer, Keratin dynamics: Modeling the interplay between turnover and transport, PloS One, 10 (2015), 1-29.  doi: 10.1371/journal.pone.0121090.

[35]

H. Rabitz and D. Edelson, Numerical techniques for modelling and analysis of oscillating chemical reactions, Oscillations and Traveling Waves in Chemical Systems, (1985), 193-222. 

[36]

M. Raftopoulou and A. Hall, Cell migration: Rho GTPases lead the way, Developmental Biology, 265 (2004), 23-32.  doi: 10.1016/j.ydbio.2003.06.003.

[37]

A. J. Ridley, Rho GTPase signalling in cell migration, Current Opinion in Cell Biology, 36 (2015), 103-112.  doi: 10.1016/j.ceb.2015.08.005.

[38]

A. J. RidleyM. A. SchwartzK. BurridgeR. A. FirtelM. H. GinsbergG. BorisyJ. T. Parsons and A. R. Horwitz, Cell migration: Integrating signals from front to back, Science, 302 (2003), 1704-1709.  doi: 10.1126/science.1092053.

[39]

S. SahaT. L. Nagy and O. D. Weiner, Joining forces: Crosstalk between biochemical signalling and physical forces orchestrates cellular polarity and dynamics, Philosophical Transactions of the Royal Society B: Biological Sciences, 373 (2018), 20170145.  doi: 10.1098/rstb.2017.0145.

[40]

L. F. Shampine and M. W. Reichelt, The MATLAB ODE suite, SIAM J. Sci. Comput., 18 (1997), 1-22.  doi: 10.1137/S1064827594276424.

[41]

C. M. SimonE. M. VaughanW. M. Bement and L. Edelstein-Keshet, Pattern formation of Rho GTPases in single cell wound healing, Molecular Biology of the Cell, 24 (2013), 421-432.  doi: 10.1091/mbc.e12-08-0634.

[42]

R. Tomovic and M. Vukobratovic, General Sensitivity Theory, vol. 1 of Mod. Analytic Comput. Methods Sci. Math., 35, New York, NY: North-Holland, 1972.

[43]

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

[44]

J. J. TysonK. Chen and B. Novak, Network dynamics and cell physiology, Nature Reviews Molecular Cell Biology, 2 (2001), 908-916.  doi: 10.1038/35103078.

[45]

J. J. Tyson and B. Novak, Temporal organization of the cell cycle, Current Biology, 18 (2008), 759-768.  doi: 10.1016/j.cub.2008.07.001.

[46]

K. WolfM. Te LindertM. KrauseS. AlexanderJ. Te RietA. L. WillisR. M. HoffmanC. G. FigdorS. J. Weiss and P. Friedl, Physical limits of cell migration: Control by ECM space and nuclear deformation and tuning by proteolysis and traction force, Journal of Cell Biology, 201 (2013), 1069-1084.  doi: 10.1083/jcb.201210152.

[47]

D. E. ZakJ. Stelling and F. J. Doyle, Sensitivity analysis of oscillatory (bio) chemical systems, Computers & Chemical Engineering, 29 (2005), 663-673.  doi: 10.1016/j.compchemeng.2004.08.021.

Figure 1.  Schematic representation of an activator-inhibitor system. A: An illustrative activator-inhibitor network model. In this set-up $ R $ activates $ M $; which in turn inhibits both $ R $ and $ G $. $ R $ and $ G $ form a positive feedback loop [15,22]. B: Flow diagram representing the interactions between active and inactive forms of species. The active species are denoted respectively by the variables; $ R(t) $, $ M(t) $ and $ G(t) $. Their total concentrations are respectively denoted, $ R_T $, $ M_T $ and $ G_T $ and these are conserved. Dotted arrows represent the catalytic activity while full arrows represent chemical reactions
Figure 2.  Typical shapes for nullclines corresponding to System (8) using different values of $ G_T $. As the parameter $ G_T $ varies, we obtain up to three different nullcline intersections. The blue curve indicates the M-nullcline, while the brownish to redish curves indicate R-nullclines at different values of $ G_T $
Figure 3.  Qualitative forms of nullcline intersections corresponding to System (8) as the parameter $ G_T $ varies. (a) For $ G_T = 0.5 $, the steady state $ E_1 $ is globally asymptotic stable (G.A.S.). (b) With $ G_T = 1.2 $, there is possibility of $ \mathbb{H}_2 $ and hence periodic solutions might occur around $ E_2 $, or $ E_2 $ is asymptotically stable. (c) For $ G_T = 7 $, the steady state in the form of $ E_3 $ is G.A.S. (d) For $ G_T = 20 $, there exists a bistable behaviour ($ E_4 $ and $ E_6 $ are locally asymptotically stable (L.A.S.) and $ E_5 $ is a saddle point)
Figure 4.  Numerical simulations illustrating time-series dynamics of $ R(t) $ and $ M(t) $ for System (8) corresponding to different dynamic regimes. As the value of $ G_T $ increases, the system transitions from stable ((a), $ G_T = 0.5 $), oscillatory ((b), $ G_T = 2 $), excitable ((c), $ G_T = 7 $) and then to bistable ((d), $ G_T = 15 $). A minimum perturbation of $ 0.09 $ from the steady state has to be taken to exhibit excitable dynamics. In the bistable regime, we used initial conditions $ (0.4,0.7) $ and $ (0.6,0.3) $ to approach both L.A.S. equilibria. Base values for parameters are shown Table 1
Figure 5.  Bifurcation diagrams for System (8). (a) Equilibrium values of the $ R- $component are plotted as a function of the parameter $ G_T $. Colors of plain and dashed lines indicate the nature of the equilibria: a blue plain line represents a stable equilibrium, a red dashed line represents an unstable equilibrium and a yellow dashed line indicates an unstable saddle point. The green doted lines show the minimum and maximum amplitudes of limit cycles. Hopf bifurcations are labelled (HB) while fold bifurcations are labelled (LP). The values of $ G_T $ at Hopf bifurcations are $ 1.002672, $ $ 2.55130 $ and $ 7.576458 $, while at the fold bifurcation, $ G_T = 6.841074 $. (b) Two-parameters bifurcation diagram: the oscillatory region is represented in red, the bistable region in yellow and the stable region is in white
Figure 6.  Local sensitivity profiles for System (8) with $ G_T = 0.25 $ and $ G_T = 0.4 $ selected to represent stable and oscillatory regimes respectively. Other parameters are fixed as shown in Table 1. The sensitivity matrix is bounded in the stable regime, but unbounded in the oscillatory regime. The colour codes used in (d) and (e) are the same as those used in (a) and (b)
Figure 7.  Period sensitivity results corresponding to System (8). (a) shows convergent time series of period sensitivity calculated from Equation (22) while (b) shows a bar graph of normalised period sensitivities extracted from (a) after convergence
Figure 8.  Cleaned-out and amplitude sensitivity results for System (8). (a) The blue curve is the cleaned-out sensitivity obtained from Equation (23). The red curve is the corresponding unbounded state sensitivity before SVD was applied as shown in Figures 6 (d) and (e). (b) $ R(t) $ and $ M(t) $ amplitude sensitivities to parameters calculated with Equation (25)
Table 1.  Parameters, their descriptions and values used for simulations and bifurcation analysis. Base line parameter values are taken from [43]. Some of them were adjusted to illustrate the qualitative dynamics hypothesised. These values could be estimated when considering experimental data through a parameter inference approach, this forms part of our current work (see for example [4]). The parameters $ k_i $, $ i = 0,\cdots, 7 $, represent the reaction constants, while $ K_{i} $, $ i = r0 $, $ r2 $, $ m5 $, $ g3 $, $ g4 $, represent Michaelis-Menten constants
Parameter Description Base value
$ k_0 $ R activation by G per unit time 4
$ k_1 $ Rate of R baseline activation 0.6
$ k_2 $ R baseline inhibition per unit time 1
$ k_2' $ R inhibition by M per unit time 1
$ k_3 $ Rate of G activation by R 1
$ k_4 $ Rate of G inhibition by M 1
$ k_5 $ M activation rate 0.035
$ k_6 $ M decay rate 0.01
$ k_7 $ M baseline recruitment rate 0.001
$ K_{r0} $ Michaelis-Menten constant for R activation 1
$ K_{r2} $ Michaelis-Menten constant for R self inhibition 1
$ K_{m5} $ Michaelis-Menten constant for M activation 1
$ K_{g3} $ Michaelis-Menten constant for G activation 0.12
$ K_{g4} $ Michaelis-Menten constant for G inhibition 0.075
$ R_T $ R total concentration 1
$ M_T $ M total concentration 1
$ G_T $ G total concentration variable
Parameter Description Base value
$ k_0 $ R activation by G per unit time 4
$ k_1 $ Rate of R baseline activation 0.6
$ k_2 $ R baseline inhibition per unit time 1
$ k_2' $ R inhibition by M per unit time 1
$ k_3 $ Rate of G activation by R 1
$ k_4 $ Rate of G inhibition by M 1
$ k_5 $ M activation rate 0.035
$ k_6 $ M decay rate 0.01
$ k_7 $ M baseline recruitment rate 0.001
$ K_{r0} $ Michaelis-Menten constant for R activation 1
$ K_{r2} $ Michaelis-Menten constant for R self inhibition 1
$ K_{m5} $ Michaelis-Menten constant for M activation 1
$ K_{g3} $ Michaelis-Menten constant for G activation 0.12
$ K_{g4} $ Michaelis-Menten constant for G inhibition 0.075
$ R_T $ R total concentration 1
$ M_T $ M total concentration 1
$ G_T $ G total concentration variable
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