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January  2017, 14(1): 249-262. doi: 10.3934/mbe.2017016

Spatio-temporal models of synthetic genetic oscillators

Cicely K. Macnamara and  Mark A. J. Chaplain ,

School of Mathematics and Statistics, Mathematical Institute, North Haugh, University of St Andrews, St Andrews KY16 9SS, Scotland

* Corresponding author: Mark Chaplain

Received  November 20, 2015 Accepted  April 15, 2016 Published  October 2016

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems.

Keywords: Gene regulatory networks, diffusion-driven oscillations, synthetic gene networks, repressilators, positive-negative feedback.
Mathematics Subject Classification: Primary: 92C40, 92C42; Secondary: 34C15, 34C23, 34C25, 35B05, 35K57, 37G15.
Citation: Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
References:
[1]

S. Busenberg and J. M. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol., 22 (1985), 313-333.  doi: 10.1007/BF00276489.

[2]

A. Cangiani and R. Natalini, A spatial model of cellular molecular trafficking including active transport along microtubules, J. Theor. Biol., 267 (2010), 614-625.  doi: 10.1016/j.jtbi.2010.08.017.

[3]

M. A. J. Chaplain, M. Ptashnyk and M. Sturrock, Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations, Math. Models Method Appl. Sci., 25 (2015), 1179-1215.  doi: 10.1142/S021820251550030X.

[4]

Y. Y. Chen, K. E. Galloway and C. D. Smolke, Synthetic biology: Advancing biological frontiers by building synthetic systems, Genome Biology, 13 (2012), p240.  doi: 10.1186/gb-2012-13-2-240.

[5]

L. Dimitrio, J. Clairambault and R. Natalini, A spatial physiological model for p53 intracellular dynamics, J. Theor. Biol., 316 (2013), 9-24.  doi: 10.1016/j.jtbi.2012.08.035.

[6]

J. Eliaš and J. Clairambault, Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples, Comp. Struct. Biotech. J., 10 (2014), 14-22. 

[7]

J. Eliaš, L. Dimitrio, J. Clairambault and R. Natalini, Modelling p53 dynamics in single cells: Physiologically based ODE and reaction-diffusion PDE models, Phys. Biol., 11 (2014), 045001. 

[8]

J. Eliaš, L. Dimitrio, J. Clairambault and R. Natalini, The p53 protein and its molecular network: Modelling a missing link between DNA damage and cell fate, Biochim. Biophys. Acta (BBA Proteins and Proteomics), 1844 (2014), 232-247. 

[9]

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

[10]

L. Glass and S. A. Kauffman, Co-operative components, spatial localization and oscillatory cellular dynamics, J. Theor. Biol., 34 (1972), 219-237.  doi: 10.1016/0022-5193(72)90157-9.

[11]

B. C. Goodwin, Oscillatory behaviour in enzymatic control processes, Adv. Enzyme Regul., 3 (1965), 425-428. 

[12]

K. E. Gordon, I. M. M. V. Leeuwen, S. Lain and M. A. J. Chaplain, Spatio-temporal modelling of the p53-{M}dm2 oscillatory system, Math. Model. Nat. Phenom., 4 (2009), 97-116.  doi: 10.1051/mmnp/20094304.

[13]

J. S. Griffith, Mathematics of cellular control processes. Ⅰ. negative feedback to one gene, J. Theor. Biol., 20 (1968), 202-208.  doi: 10.1016/0022-5193(68)90189-6.

[14]

H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa and R. Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science, 298 (2002), 840-843.  doi: 10.1126/science.1074560.

[15]

M. H. Jensen, J. Sneppen and G. Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett., 541 (2003), 176-177.  doi: 10.1016/S0014-5793(03)00279-5.

[16]

R. Kagemyama, T. Ohtsuka and T. Kobayashi, The Hes1 gene family: Repressors and oscillators that orchestrate embryogenesis, Development, 134 (2007), 1243-1251. 

[17]

T. Kobayashi and R. Kageyama, Hes1 regulates embryonic stem cell differentiation by suppressing notch signaling, Genes to Cells, 15 (2010), 689-698.  doi: 10.1111/j.1365-2443.2010.01413.x.

[18]

T. Kobayashi and R. Kageyama, Hes1 oscillations contribute to heterogeneous differentiation responses in embryonic stem cells, Genes, 2 (2011), 219-228.  doi: 10.3390/genes2010219.

[19]

T. Kobayashi, H. Mizuno, I. Imayoshi, C. Furusawa, K. Shirahige and R. Kageyama, The cyclic gene Hes1 contributes to diverse differentiation responses of embryonic stem cells, Genes & Development, 23 (2009), 1870-1875.  doi: 10.1101/gad.1823109.

[20]

G. Lahav, N. Rosenfeld, A. Sigal, N. Geva-Zatorsky, A. J. Levine, M. B. Elowitz and U. Alon, Dynamics of the p53-Mdm2 feedback loop in individual cells, Nature Genet., 36 (2004), 147-150.  doi: 10.1038/ng1293.

[21]

E. Lieberman-Aiden, N. L. van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B. R. Lajoie, P. J. Sabo, M. O. Dorschner, R. Sandstrom, B. Bernstein, M. A. Bender, M. Groudine, A. Gnirke, J. Stamatoyannopoulos, L. A. Mirny, E. S. Lander and J. Dekker, Comprehensive mapping of long range interactions reveals folding principles of the human genome, Science, 326 (2009), 289-293.  doi: 10.1126/science.1181369.

[22]

R.-T. Liu, S.-S. Liaw and P. K. Maini, Oscillatory turing patterns in a simple reaction-diffusion system, J. Korean Phys. Soc., 50 (2007), 234-238. 

[23]

J. M. Mahaffy, Genetic control models with diffusion and delays, Math. Biosci., 90 (1988), 519-533.  doi: 10.1016/0025-5564(88)90081-8.

[24]

J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984), 39-57.  doi: 10.1007/BF00275860.

[25]

R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii and U. Alon, Network motifs: Simple building blocks of complex networks, Science, 298 (2002), 824-827.  doi: 10.1126/science.298.5594.824.

[26]

H. Momiji and N. A. M. Monk, Dissecting the dynamics of the Hes1 genetic oscillator, J. Theor. Biol., 254 (2008), 784-798.  doi: 10.1016/j.jtbi.2008.07.013.

[27]

N. A. M. Monk, Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. 

[28]

F. Naqib, T. Quail, L. Musa, H. Vulpe, J. Nadeau, J. Lei and L. Glass, Tunable oscillations and chaotic dynamics in systems with localized synthesis, Phys. Rev. E, 85 (2012), 046210.  doi: 10.1103/PhysRevE.85.046210.

[29]

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

[30]

O. Purcell, N. J. Savery, C. S. Grierson and M. di Bernardo, A comparative analysis of synthetic genetic oscillators, J. R. Soc. Interface, 7 (2010), 1503-1524.  doi: 10.1098/rsif.2010.0183.

[31]

L. Sang, H. A. Coller and M. J. Roberts, Control of the reversibility of cellular quiescence by the transcriptional repressor HES1, Science, 321 (2008), 1095-1100.  doi: 10.1126/science.1155998.

[32]

Y. Schaerli, A. Munteanu, M. Gili, J. Cotterell, J. Sharpe and M. Isalan, A unified design space of synthetic stripe-forming networks, Nat. Commun., 5 (2014), p4905.  doi: 10.1038/ncomms5905.

[33]

R. M. Shymko and L. Glass, Spatial switching in chemical reactions with heterogeneous catalysis, J. Chem. Phys., 60 (1974), 835-841.  doi: 10.1063/1.1681157.

[34]

R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM J. Sci. and Stat. Comput., 11 (1990), 1-32.  doi: 10.1137/0911001.

[35]

M. Sturrock, A. Hellander, A. Matzavinos and M. A. J. Chaplain, Spatial stochastic modelling of the Hes1 gene regulatory network: Intrinsic noise can explain heterogeneity in embryonic stem cell differentiation, J. R. Soc. Interface, 10 (2013), 20120988.  doi: 10.1098/rsif.2012.0988.

[36]

M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways, J. Theor. Biol., 273 (2011), 15-31.  doi: 10.1016/j.jtbi.2010.12.016.

[37]

M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Influence of the nuclear membrane, active transport and cell shape on the Hes1 and p53--Mdm2 pathways: Insights from spatio-temporal modelling, Bull. Math. Biol., 74 (2012), 1531-1579.  doi: 10.1007/s11538-012-9725-1.

[38]

Z. Szymańska, M. Parisot and M. Lachowicz, Mathematical modeling of the intracellular protein dynamics: The importance of active transport along microtubules, J. Theor. Biol., 363 (2014), 118-128.  doi: 10.1016/j.jtbi.2014.07.022.

[39]

G. Tiana, M. H. Jensen and K. Sneppen, Time delay as a key to apoptosis induction in the p53 network, Eur. Phys. J., 29 (2002), 135-140.  doi: 10.1140/epjb/e2002-00271-1.

[40]

B. Yordanov, N. Dalchau, P. K. Grant, M. Pedersen, S. Emmott, J. Haseloff and A. Phillips, A computational method for automated characterization of genetic components, ACS Synthetic Biology, 3 (2014), 578-588.  doi: 10.1021/sb400152n.

show all references

References:
[1]

S. Busenberg and J. M. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol., 22 (1985), 313-333.  doi: 10.1007/BF00276489.

[2]

A. Cangiani and R. Natalini, A spatial model of cellular molecular trafficking including active transport along microtubules, J. Theor. Biol., 267 (2010), 614-625.  doi: 10.1016/j.jtbi.2010.08.017.

[3]

M. A. J. Chaplain, M. Ptashnyk and M. Sturrock, Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations, Math. Models Method Appl. Sci., 25 (2015), 1179-1215.  doi: 10.1142/S021820251550030X.

[4]

Y. Y. Chen, K. E. Galloway and C. D. Smolke, Synthetic biology: Advancing biological frontiers by building synthetic systems, Genome Biology, 13 (2012), p240.  doi: 10.1186/gb-2012-13-2-240.

[5]

L. Dimitrio, J. Clairambault and R. Natalini, A spatial physiological model for p53 intracellular dynamics, J. Theor. Biol., 316 (2013), 9-24.  doi: 10.1016/j.jtbi.2012.08.035.

[6]

J. Eliaš and J. Clairambault, Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples, Comp. Struct. Biotech. J., 10 (2014), 14-22. 

[7]

J. Eliaš, L. Dimitrio, J. Clairambault and R. Natalini, Modelling p53 dynamics in single cells: Physiologically based ODE and reaction-diffusion PDE models, Phys. Biol., 11 (2014), 045001. 

[8]

J. Eliaš, L. Dimitrio, J. Clairambault and R. Natalini, The p53 protein and its molecular network: Modelling a missing link between DNA damage and cell fate, Biochim. Biophys. Acta (BBA Proteins and Proteomics), 1844 (2014), 232-247. 

[9]

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

[10]

L. Glass and S. A. Kauffman, Co-operative components, spatial localization and oscillatory cellular dynamics, J. Theor. Biol., 34 (1972), 219-237.  doi: 10.1016/0022-5193(72)90157-9.

[11]

B. C. Goodwin, Oscillatory behaviour in enzymatic control processes, Adv. Enzyme Regul., 3 (1965), 425-428. 

[12]

K. E. Gordon, I. M. M. V. Leeuwen, S. Lain and M. A. J. Chaplain, Spatio-temporal modelling of the p53-{M}dm2 oscillatory system, Math. Model. Nat. Phenom., 4 (2009), 97-116.  doi: 10.1051/mmnp/20094304.

[13]

J. S. Griffith, Mathematics of cellular control processes. Ⅰ. negative feedback to one gene, J. Theor. Biol., 20 (1968), 202-208.  doi: 10.1016/0022-5193(68)90189-6.

[14]

H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa and R. Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science, 298 (2002), 840-843.  doi: 10.1126/science.1074560.

[15]

M. H. Jensen, J. Sneppen and G. Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett., 541 (2003), 176-177.  doi: 10.1016/S0014-5793(03)00279-5.

[16]

R. Kagemyama, T. Ohtsuka and T. Kobayashi, The Hes1 gene family: Repressors and oscillators that orchestrate embryogenesis, Development, 134 (2007), 1243-1251. 

[17]

T. Kobayashi and R. Kageyama, Hes1 regulates embryonic stem cell differentiation by suppressing notch signaling, Genes to Cells, 15 (2010), 689-698.  doi: 10.1111/j.1365-2443.2010.01413.x.

[18]

T. Kobayashi and R. Kageyama, Hes1 oscillations contribute to heterogeneous differentiation responses in embryonic stem cells, Genes, 2 (2011), 219-228.  doi: 10.3390/genes2010219.

[19]

T. Kobayashi, H. Mizuno, I. Imayoshi, C. Furusawa, K. Shirahige and R. Kageyama, The cyclic gene Hes1 contributes to diverse differentiation responses of embryonic stem cells, Genes & Development, 23 (2009), 1870-1875.  doi: 10.1101/gad.1823109.

[20]

G. Lahav, N. Rosenfeld, A. Sigal, N. Geva-Zatorsky, A. J. Levine, M. B. Elowitz and U. Alon, Dynamics of the p53-Mdm2 feedback loop in individual cells, Nature Genet., 36 (2004), 147-150.  doi: 10.1038/ng1293.

[21]

E. Lieberman-Aiden, N. L. van Berkum, L. Williams, M. Imakaev, T. Ragoczy, A. Telling, I. Amit, B. R. Lajoie, P. J. Sabo, M. O. Dorschner, R. Sandstrom, B. Bernstein, M. A. Bender, M. Groudine, A. Gnirke, J. Stamatoyannopoulos, L. A. Mirny, E. S. Lander and J. Dekker, Comprehensive mapping of long range interactions reveals folding principles of the human genome, Science, 326 (2009), 289-293.  doi: 10.1126/science.1181369.

[22]

R.-T. Liu, S.-S. Liaw and P. K. Maini, Oscillatory turing patterns in a simple reaction-diffusion system, J. Korean Phys. Soc., 50 (2007), 234-238. 

[23]

J. M. Mahaffy, Genetic control models with diffusion and delays, Math. Biosci., 90 (1988), 519-533.  doi: 10.1016/0025-5564(88)90081-8.

[24]

J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984), 39-57.  doi: 10.1007/BF00275860.

[25]

R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii and U. Alon, Network motifs: Simple building blocks of complex networks, Science, 298 (2002), 824-827.  doi: 10.1126/science.298.5594.824.

[26]

H. Momiji and N. A. M. Monk, Dissecting the dynamics of the Hes1 genetic oscillator, J. Theor. Biol., 254 (2008), 784-798.  doi: 10.1016/j.jtbi.2008.07.013.

[27]

N. A. M. Monk, Oscillatory expression of Hes1, p53, and NF-κB driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. 

[28]

F. Naqib, T. Quail, L. Musa, H. Vulpe, J. Nadeau, J. Lei and L. Glass, Tunable oscillations and chaotic dynamics in systems with localized synthesis, Phys. Rev. E, 85 (2012), 046210.  doi: 10.1103/PhysRevE.85.046210.

[29]

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

[30]

O. Purcell, N. J. Savery, C. S. Grierson and M. di Bernardo, A comparative analysis of synthetic genetic oscillators, J. R. Soc. Interface, 7 (2010), 1503-1524.  doi: 10.1098/rsif.2010.0183.

[31]

L. Sang, H. A. Coller and M. J. Roberts, Control of the reversibility of cellular quiescence by the transcriptional repressor HES1, Science, 321 (2008), 1095-1100.  doi: 10.1126/science.1155998.

[32]

Y. Schaerli, A. Munteanu, M. Gili, J. Cotterell, J. Sharpe and M. Isalan, A unified design space of synthetic stripe-forming networks, Nat. Commun., 5 (2014), p4905.  doi: 10.1038/ncomms5905.

[33]

R. M. Shymko and L. Glass, Spatial switching in chemical reactions with heterogeneous catalysis, J. Chem. Phys., 60 (1974), 835-841.  doi: 10.1063/1.1681157.

[34]

R. D. Skeel and M. Berzins, A method for the spatial discretization of parabolic equations in one space variable, SIAM J. Sci. and Stat. Comput., 11 (1990), 1-32.  doi: 10.1137/0911001.

[35]

M. Sturrock, A. Hellander, A. Matzavinos and M. A. J. Chaplain, Spatial stochastic modelling of the Hes1 gene regulatory network: Intrinsic noise can explain heterogeneity in embryonic stem cell differentiation, J. R. Soc. Interface, 10 (2013), 20120988.  doi: 10.1098/rsif.2012.0988.

[36]

M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways, J. Theor. Biol., 273 (2011), 15-31.  doi: 10.1016/j.jtbi.2010.12.016.

[37]

M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. J. Chaplain, Influence of the nuclear membrane, active transport and cell shape on the Hes1 and p53--Mdm2 pathways: Insights from spatio-temporal modelling, Bull. Math. Biol., 74 (2012), 1531-1579.  doi: 10.1007/s11538-012-9725-1.

[38]

Z. Szymańska, M. Parisot and M. Lachowicz, Mathematical modeling of the intracellular protein dynamics: The importance of active transport along microtubules, J. Theor. Biol., 363 (2014), 118-128.  doi: 10.1016/j.jtbi.2014.07.022.

[39]

G. Tiana, M. H. Jensen and K. Sneppen, Time delay as a key to apoptosis induction in the p53 network, Eur. Phys. J., 29 (2002), 135-140.  doi: 10.1140/epjb/e2002-00271-1.

[40]

B. Yordanov, N. Dalchau, P. K. Grant, M. Pedersen, S. Emmott, J. Haseloff and A. Phillips, A computational method for automated characterization of genetic components, ACS Synthetic Biology, 3 (2014), 578-588.  doi: 10.1021/sb400152n.

Figure 1.  Plots showing (a) the hes1 mRNA concentration and (b) the Hes1 protein concentration varying in space and time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$ and the Hes1 protein production sites where translation occurs are located at $x_{p}=\pm0.5$. The plots show sustained oscillations of both hes1 mRNA and Hes1 protein. Baseline parameter set $\mathbb{P}$
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Figure 2.  Plots showing (a) the total hes1 mRNA concentration and (b) the total Hes1 protein concentration over time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$ and the Hes1 protein production sites where translation occurs are located at $x_{p}=\{\pm0.1, \pm0.3, \pm0.5, \pm0.7, \pm0.9\}$ (see legend). The plots show that if the Hes1 protein production sites are either too close or too far away from the hes1 mRNA gene-site, then oscillations are lost. Baseline parameter set $\mathbb{P}$
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Figure 3.  Plots showing (a) the total hes1 mRNA concentration and (b) the total Hes1 protein concentration over time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$ and the Hes1 protein production sites where translation occurs are located at $x_{p}=\{\pm0.1, \pm0.3, \pm0.5, \pm0.7, \pm0.9\}$ (see legend). The plots show that if the Hes1 protein production sites are either too close or too far away from the hes1 mRNA gene-site, then oscillations are lost. Parameters are as for the baseline parameter set $\mathbb{P}$, except $\epsilon$ which is taken to be $0.25$
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Figure 4.  Plots showing (a) the total hes1 mRNA concentration and (b) the total Hes1 protein concentration over time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$. Translation occurs uniformly in the cytoplasm as per equations (3) and (4), with the boundaries beyond which protein production occurs located at $x_{p}=\{0.1, 0.3, 0.5, 0.7, 0.9\}$ (see legend). The plots show that if Hes1 protein production is either too close or too far away from the hes1 mRNA gene-site, then oscillations are lost. Parameters are as for the baseline parameter set $\mathbb{P}$
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Figure 5.  Plots showing (a) the mRNA concentration and (b) the protein concentration varying in space and time for the three-gene repressilator system. $m_1, p_1$ top row; $m_2, p_2$ middle row; $m_3, p_3$ bottom row. The mRNA gene-sites are in different locations, i.e. $x_{m1}=0.0$, $x_{m2}=\pm0.2$, $x_{m3}=\pm0.4$, and the protein production sites are in different locations, i.e. $x_{p1}=\pm0.5$, $x_{p2}=\pm0.7$ and $x_{p3}=\pm0.9$. The plots show sustained oscillations of each of the three mRNA and protein species. Baseline parameter set $\mathbb{P}$
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Figure 6.  Plots showing (a) the total mRNA concentration for species 1 and (b) the total protein concentration for species 1 over time for the three-gene repressilator. Case (a) -solid black curve: the gene sites are $x_{m1}=x_{m2}=x_{m3}=0.0$ and protein production sites are $x_{p1}=x_{p2}=x_{p3}=\pm0.5$. Case (b) -blue dotted curve: the gene sites are $x_{m1}=0.0$, $x_{m2}=\pm0.2$, $x_{m2}=\pm0.4$, and protein production sites are $x_{p1}=\pm0.5$, $x_{p2}=\pm0.7$ and $x_{p3}=\pm0.9$. Case (c) -red dashed curve: the gene sites are $x_{m1}=x_{m2}=x_{m3}=0.0$ and protein production sites are $x_{p1}=x_{p2}=x_{p3}=\pm0.9$. Case (d) -cyan dot-dashed curve: the gene sites are $x_{m1}=0.0$, $x_{m2}=\pm0.05$, $x_{m3}=\pm0.1$, and protein production sites are $x_{p1}=\pm0.9$, $x_{m2}=\pm0.95$ and $x_{p3}=\pm1.0$. Baseline parameter set $\mathbb{P}$
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Figure 7.  Plots showing (a) the total mRNA concentration for species 1 and (b) the total protein concentration for species 1 over time for the three-gene repressilator. For all cases protein production (translation) is constant throughout the cytoplasm $|x|>0.5$. Case (a) -solid black curve: the gene sites are $x_{m1}=x_{m2}=x_{m3}=0.0$. Case (b) -blue dotted curve: the gene sites are $x_{m1}=x_{m2}=x_{m2}=\pm0.2$. Case (c) -red dashed curve: the gene sites are $x_{m1}=0.0$, $x_{m2}=\pm0.2$ and $x_{m3}=\pm0.4$. Case (d) -cyan dot-dashed curve: the gene sites are $x_{m1}=\pm0.1$, $x_{m2}=\pm0.3$ and $x_{m3}=\pm0.5$. Baseline parameter set $\mathbb{P}$
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Figure 8.  Plots showing (a) the mRNA concentration and (b) the protein concentration varying in space and time for a two-gene activator-repressor system. $m_1, p_1$ top row; $m_2, p_2$ bottom row. The mRNA gene-sites where transcription occurs are located at $x_{m_i}=0.0, i=1, 2$ and the protein production sites where translation occurs are located at $x_{p_i}= \pm0.9, i=1, 2$. The plots show sustained oscillations of mRNA and protein concentrations for both species. Baseline parameter set $\mathbb{P}$, $\beta_m=10.0$
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Figure 9.  Plots showing (a) the total mRNA concentration for species 1 and (b) the total protein concentration for species 1 over time for the two gene activator-repressor system. The mRNA gene-sites where transcription occurs are located at $x_{m_i}=0.0, i=1, 2$ and the protein production sites where translation occurs are located at $x_{p_i}= \pm0.8, \pm0.88, \pm0.94, \pm1.0, i=1, 2$ [(a) solid line, black, (b) dotted line, blue, (c) dashed line, red, (d) dot-dashed line, cyan, respectively]. The plots show that if the protein production sites are either too close or too far away from the mRNA gene-sites, then sustained oscillations are lost. Baseline parameter set $\mathbb{P}$, $\beta_m=10.0$
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