2017, 14(5-6): 1247-1259. doi: 10.3934/mbe.2017064

Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM

1. 

Mathematics and Science College, Shanghai Normal University Shanghai 200234, China

2. 

Department of Mathematics, University of Science and Technology of China Hefei 230026, China

* Corresponding authorr: J. Jiang

Received  May 19, 2016 Revised  January 2017 Published  May 2017

Fund Project: This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No.11371252, Research and Innovation Project of Shanghai Education Committee under Grant No.14zz120, and Shanghai Gaofeng Project for University Academic Program Development

Circadian rhythms of physiology and behavior are widespread\break mechanisms in many organisms. The internal biological rhythms are driven by molecular clocks, which oscillate with a period nearly but not exactly $24$ hours. Many classic models of circadian rhythms are based on a time-delayed negative feedback, suggested by the protein products inhibiting transcription of their own genes. In 1999, based on stabilization of PER upon dimerization, Tyson et al. [J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, Biophys. J. 77 (1999) 2411-2417] proposed a crucial positive feedback to the circadian oscillator. This idea was mathematically expressed in a three-dimensional model. By imposing assumptions that the dimerization reactions were fast and dimeric proteins were in rapid equilibrium, they reduced the model to a pair of nonlinear ordinary differential equations of mRNA and total protein concentrations. Then they used phase plane analysis tools to investigate circadian rhythms. In this paper, the original three-dimensional model is studied. We explore the existence of oscillations and their periods. Much attention is paid to investigate how the periods depend on model parameters. The numerical simulations are in good agreement with their reduced work.

Citation: Jifa Jiang, Qiang Liu, Lei Niu. Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM. Mathematical Biosciences & Engineering, 2017, 14 (5-6) : 1247-1259. doi: 10.3934/mbe.2017064
References:
[1]

R. Allada, N. E. White, W. V. So, J. C. Hall, M. Rosbash, A mutant Drosophila homolog of mammalian clock disrupts circadian rhythms and transcription of period and timeless, Cell, 93 (1998), 791-804. doi: 10.1016/S0092-8674(00)81440-3.

[2]

K. Bae, C. Lee, D. Sidote, K-Y. Chuang, I. Edery, Circadian regulation of a Drosophila homolog of the mammalian clock gene: PER and TIM function as positive regulators, Mol. Cell. Biol., 18 (1998), 6142-6151. doi: 10.1128/MCB.18.10.6142.

[3]

T. K. Darlington, K. Wager-Smith, M. F. Ceriani, D. Staknis, N. Gekakis, T. D. L. Steeves, C. J. Weitz, J. S. Takahashi, S. A. Kay, Closing the circadian loop: CLOCK-induced transcription of its own inhibitors per and tim, Science, 280 (1998), 1599-1603.

[4]

A. Eskin, S. J. Yeung, M. R. Klass, Requirement for protein synthesis in the regulation of a circadian rhythm by serotonin, Proc. Natl. Acad. Sci. USA, 81 (1984), 7637-7641. doi: 10.1073/pnas.81.23.7637.

[5]

N. Gekakis, L. Saez, A.-M. Delahaye-Brown, M. P. Myers, A. Sehgal, M. W. Young, C. J. Weitz, Isolation of timeless by PER protein interaction: Defective interaction between timeless protein and long-period mutant $\mbox{PER}^{L}$, Science, 270 (1995), 811-815. doi: 10.1126/science.270.5237.811.

[6]

A. Goldbeter, A model for circadian oscillations in the Drosophila period protein (PER), Proc. R. Soc. Lond. B, 261 (1995), 319-324. doi: 10.1098/rspb.1995.0153.

[7]

P. E. Hardin, J. C. Hall, M. Rosbash, Feedback of the Drosophila Period gene product on circadian cycling of its messenger RNA levels, Nature, 343 (1990), 536-540. doi: 10.1038/343536a0.

[8]

M. W. Hirsch, Systems of differential equations which are competitive or cooperative. Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013.

[9]

M. W. Hirsch, Systems of differential equations that are competitive and cooperative. Ⅳ: Structural stability in three dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067.

[10]

Z. J. Huang, K. D. Curtin, M. Rosbash, PER protein interactions and temperature compensation of a circadian clock in Drosophila, Science, 267 (1995), 1169-1172. doi: 10.1126/science.7855598.

[11]

M. W. Karakashian, J. W. Hastings, The effects of inhibitors of macromolecular biosynthesis upon the persistent rhythm of luminescence in Gonyaulax, J. Gen. Physiol., 47 (1963), 1-12. doi: 10.1085/jgp.47.1.1.

[12]

S. B. S. Khalsa, D. Whitmore, G. D. Block, Stopping the circadian pacemaker with inhibitors of protein synthesis, Proc. Natl. Acad. Sci. USA, 89 (1992), 10862-10866. doi: 10.1073/pnas.89.22.10862.

[13]

B. Kloss, J. L. Price, L. Saez, J. Blau, A. Rothenfluh, C. S. Wesley, M. W. Young, The Drosophila clock gene double-time encodes a protein closely related to human casein kinase l$ε$, Cell, 94 (1998), 97-107. doi: 10.1016/S0092-8674(00)81225-8.

[14]

R. J. Konopka, S. Benzer, Clock mutants of Drosophila melanogaster, Proc. Natl. Acad. Sci. USA, 68 (1971), 2112-2116. doi: 10.1073/pnas.68.9.2112.

[15]

J.-C. Leloup, A. Goldbeter, A model for circadian rhythms in Drosophila incorporating the formation of a complex between PER and TIM proteins, J. Biol. Rhythms, 13 (1998), 70-87. doi: 10.1177/074873098128999934.

[16]

J. L. Price, J. Blau, A. Rothenfluh, M. Abodeely, B. Kloss, M. W. Young, double-time is a novel Drosophila clock gene that regulates PERIOD protein accumulation, Cell, 94 (1998), 83-95. doi: 10.1016/S0092-8674(00)81224-6.

[17]

P. Ruoff, L. Rensing, The temperature-compensated Goodwin model simulates many circadian clock properties, J. Theor. Biol., 179 (1996), 275-285. doi: 10.1006/jtbi.1996.0067.

[18]

J. E. Rutila, V. Suri, M. Le, M. V. So, M. Rosbash, J. C. Hall, CYCLE is a second bHLH-PAS clock protein essential for circadian rhythmicity and transcription of Drosophila period and timeless, Cell, 93 (1998), 805-814. doi: 10.1016/S0092-8674(00)81441-5.

[19]

T. Scheper, D. Klinkenberg, C. Pennartz, J. van Pelt, A mathematical model for the intracellular circadian rhythm generator, J. Neurosci., 19 (1999), 40-47.

[20]

A. Sehgal, J. L. Price, B. Man, M. W. Young, Loss of circadian behavioral rhythms and per RNA oscillations in the Drosophila mutant timeless, Science, 263 (1994), 1603-1606. doi: 10.1126/science.8128246.

[21]

J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Diff. Eqns., 38 (1980), 80-103. doi: 10.1016/0022-0396(80)90026-1.

[22]

H. L. Smith, Periodic orbits of competitive and cooperative systems, J. Diff. Eqns., 65 (1986), 361-373. doi: 10.1016/0022-0396(86)90024-0.

[23]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Amer. Math. Soc. Providence, Rhode Island, 1995.

[24]

V. Suri, A. Lanjuin, M. Rosbash, TIMELESS-dependent positive and negative autoregulation in the Drosophila circadian clock, EMBO J., 18 (1999), 501-791. doi: 10.1093/emboj/18.3.675.

[25]

W. R. Taylor, J. C. Dunlap, J. W. Hastings, Inhibitors of protein synthesis on 80s ribosomes phase shift the Gonyaulax clock, J. Exp. Biol., 97 (1982), 121-136.

[26]

J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417. doi: 10.1016/S0006-3495(99)77078-5.

[27]

L. B. Vosshall, J. L. Price, A. Sehgal, L. Saez, M. W. Young, Block in nuclear localization of period protein by a second clock mutation, timeless, Science, 263 (1994), 1606-1609. doi: 10.1126/science.8128247.

[28]

Y. Wang, J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns., 176 (2001), 470-493. doi: 10.1006/jdeq.2001.3989.

[29]

H. Zeng, Z. Qian, M. P. Myers, M. Rosbash, A light-entrainment mechanism for the Drosophila circadian clock, Nature, 380 (1996), 129-135. doi: 10.1038/380129a0.

[30]

H.-R. Zhu, H. L. Smith, Stable periodic orbits for a class of three dimensional competitive systems, J. Diff. Eqns., 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.

show all references

References:
[1]

R. Allada, N. E. White, W. V. So, J. C. Hall, M. Rosbash, A mutant Drosophila homolog of mammalian clock disrupts circadian rhythms and transcription of period and timeless, Cell, 93 (1998), 791-804. doi: 10.1016/S0092-8674(00)81440-3.

[2]

K. Bae, C. Lee, D. Sidote, K-Y. Chuang, I. Edery, Circadian regulation of a Drosophila homolog of the mammalian clock gene: PER and TIM function as positive regulators, Mol. Cell. Biol., 18 (1998), 6142-6151. doi: 10.1128/MCB.18.10.6142.

[3]

T. K. Darlington, K. Wager-Smith, M. F. Ceriani, D. Staknis, N. Gekakis, T. D. L. Steeves, C. J. Weitz, J. S. Takahashi, S. A. Kay, Closing the circadian loop: CLOCK-induced transcription of its own inhibitors per and tim, Science, 280 (1998), 1599-1603.

[4]

A. Eskin, S. J. Yeung, M. R. Klass, Requirement for protein synthesis in the regulation of a circadian rhythm by serotonin, Proc. Natl. Acad. Sci. USA, 81 (1984), 7637-7641. doi: 10.1073/pnas.81.23.7637.

[5]

N. Gekakis, L. Saez, A.-M. Delahaye-Brown, M. P. Myers, A. Sehgal, M. W. Young, C. J. Weitz, Isolation of timeless by PER protein interaction: Defective interaction between timeless protein and long-period mutant $\mbox{PER}^{L}$, Science, 270 (1995), 811-815. doi: 10.1126/science.270.5237.811.

[6]

A. Goldbeter, A model for circadian oscillations in the Drosophila period protein (PER), Proc. R. Soc. Lond. B, 261 (1995), 319-324. doi: 10.1098/rspb.1995.0153.

[7]

P. E. Hardin, J. C. Hall, M. Rosbash, Feedback of the Drosophila Period gene product on circadian cycling of its messenger RNA levels, Nature, 343 (1990), 536-540. doi: 10.1038/343536a0.

[8]

M. W. Hirsch, Systems of differential equations which are competitive or cooperative. Ⅰ: Limit sets, SIAM J. Math. Anal., 13 (1982), 167-179. doi: 10.1137/0513013.

[9]

M. W. Hirsch, Systems of differential equations that are competitive and cooperative. Ⅳ: Structural stability in three dimensional systems, SIAM J. Math. Anal., 21 (1990), 1225-1234. doi: 10.1137/0521067.

[10]

Z. J. Huang, K. D. Curtin, M. Rosbash, PER protein interactions and temperature compensation of a circadian clock in Drosophila, Science, 267 (1995), 1169-1172. doi: 10.1126/science.7855598.

[11]

M. W. Karakashian, J. W. Hastings, The effects of inhibitors of macromolecular biosynthesis upon the persistent rhythm of luminescence in Gonyaulax, J. Gen. Physiol., 47 (1963), 1-12. doi: 10.1085/jgp.47.1.1.

[12]

S. B. S. Khalsa, D. Whitmore, G. D. Block, Stopping the circadian pacemaker with inhibitors of protein synthesis, Proc. Natl. Acad. Sci. USA, 89 (1992), 10862-10866. doi: 10.1073/pnas.89.22.10862.

[13]

B. Kloss, J. L. Price, L. Saez, J. Blau, A. Rothenfluh, C. S. Wesley, M. W. Young, The Drosophila clock gene double-time encodes a protein closely related to human casein kinase l$ε$, Cell, 94 (1998), 97-107. doi: 10.1016/S0092-8674(00)81225-8.

[14]

R. J. Konopka, S. Benzer, Clock mutants of Drosophila melanogaster, Proc. Natl. Acad. Sci. USA, 68 (1971), 2112-2116. doi: 10.1073/pnas.68.9.2112.

[15]

J.-C. Leloup, A. Goldbeter, A model for circadian rhythms in Drosophila incorporating the formation of a complex between PER and TIM proteins, J. Biol. Rhythms, 13 (1998), 70-87. doi: 10.1177/074873098128999934.

[16]

J. L. Price, J. Blau, A. Rothenfluh, M. Abodeely, B. Kloss, M. W. Young, double-time is a novel Drosophila clock gene that regulates PERIOD protein accumulation, Cell, 94 (1998), 83-95. doi: 10.1016/S0092-8674(00)81224-6.

[17]

P. Ruoff, L. Rensing, The temperature-compensated Goodwin model simulates many circadian clock properties, J. Theor. Biol., 179 (1996), 275-285. doi: 10.1006/jtbi.1996.0067.

[18]

J. E. Rutila, V. Suri, M. Le, M. V. So, M. Rosbash, J. C. Hall, CYCLE is a second bHLH-PAS clock protein essential for circadian rhythmicity and transcription of Drosophila period and timeless, Cell, 93 (1998), 805-814. doi: 10.1016/S0092-8674(00)81441-5.

[19]

T. Scheper, D. Klinkenberg, C. Pennartz, J. van Pelt, A mathematical model for the intracellular circadian rhythm generator, J. Neurosci., 19 (1999), 40-47.

[20]

A. Sehgal, J. L. Price, B. Man, M. W. Young, Loss of circadian behavioral rhythms and per RNA oscillations in the Drosophila mutant timeless, Science, 263 (1994), 1603-1606. doi: 10.1126/science.8128246.

[21]

J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Diff. Eqns., 38 (1980), 80-103. doi: 10.1016/0022-0396(80)90026-1.

[22]

H. L. Smith, Periodic orbits of competitive and cooperative systems, J. Diff. Eqns., 65 (1986), 361-373. doi: 10.1016/0022-0396(86)90024-0.

[23]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems Amer. Math. Soc. Providence, Rhode Island, 1995.

[24]

V. Suri, A. Lanjuin, M. Rosbash, TIMELESS-dependent positive and negative autoregulation in the Drosophila circadian clock, EMBO J., 18 (1999), 501-791. doi: 10.1093/emboj/18.3.675.

[25]

W. R. Taylor, J. C. Dunlap, J. W. Hastings, Inhibitors of protein synthesis on 80s ribosomes phase shift the Gonyaulax clock, J. Exp. Biol., 97 (1982), 121-136.

[26]

J. J. Tyson, C. I. Hong, C. D. Thron, B. Novak, A simple model of circadian rhythms based on dimerization and proteolysis of PER and TIM, Biophys. J., 77 (1999), 2411-2417. doi: 10.1016/S0006-3495(99)77078-5.

[27]

L. B. Vosshall, J. L. Price, A. Sehgal, L. Saez, M. W. Young, Block in nuclear localization of period protein by a second clock mutation, timeless, Science, 263 (1994), 1606-1609. doi: 10.1126/science.8128247.

[28]

Y. Wang, J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns., 176 (2001), 470-493. doi: 10.1006/jdeq.2001.3989.

[29]

H. Zeng, Z. Qian, M. P. Myers, M. Rosbash, A light-entrainment mechanism for the Drosophila circadian clock, Nature, 380 (1996), 129-135. doi: 10.1038/380129a0.

[30]

H.-R. Zhu, H. L. Smith, Stable periodic orbits for a class of three dimensional competitive systems, J. Diff. Eqns., 110 (1994), 143-156. doi: 10.1006/jdeq.1994.1063.

Figure 1.  A simple molecular mechanism for the circadian clock in Drosophila. Redrawn from [26]. PER and TIM proteins are synthesized in the cytoplasm, where they may be destroyed by proteolysis or they may combine to form relatively stable heterodimers. Heteromeric complexes are transported into the nucleus, where they inhibit transcription of per and tim mRNA. Here it is assumed that PER monomers are rapidly phosphorylated by DBT and then degraded. Dimers are assumed to be poorer substrates for DBT
Figure 2.  Numerical solution of (1). Parameter values are chosen as in Table 1. We take $k_a=10^6$ and $k_d=k_a/K_{eq}$
Figure 3.  Relation between the oscillator period of (1) and some parameter values. In each diagram, other parameter values are chosen as in Table 1 and $k_a=10^6$, and periodic oscillations occur only when the correlate parameter is in the interval $[\, a, b\, ]$. In case A, $a=0.2$ and $b=1.4$; in case B, $a=0.02$ and $b=0.44$; in case C, $a=7$ and $b=46$; in case D, $a=0$ and $b=0.4$; in case E, $a=0.9$ and $b=\infty$; in case F, $a_1=a_2=4$, $b_1=570$ and $b_2=588$. For the convenience of numerical integration, curve $(1)$ is shown only with $K_{eq}\geq 40$ in case F. As for $4\leq K_{eq}\leq40$, a decreasing period is suggested by curve (2) with increasing $K_{eq}$. Particularly, on curve $(1)$ the period maintains $24.2$-$25.2$ when the parameter $K_{eq}$ varies in the interval $[\, c, d\, ] = [\, 50, 460\, ]$
Figure 4.  Two-parameter ($K_{eq}$ and $k_{p_1}$) bifurcation diagram for system \eqref{Eqs. 2.1}. Here $K_{eq}$ and $k_{p_1}$ are allowed to vary, and other parameter values are fixed as in Table 1. We take $k_a=10^6$. Periodic oscillations happen only within the U-shape region bounded by the two curves. Outside this region the system evolves toward a stable steady state. We note that for any $K_{eq}$ one can find a $k_{p_1}$ such that oscillations happen, which differs from the boundedness requirement of $K_{eq}$ as in Figure 3F
Figure 5.  The vector field for (1) on the boundary of $B(a, b, c)$
Table 1.  Parameter values suitable for circadian rhythm of wild-type fruit flies
NameValueUnits $E_{a}/RT$Description
$v_{m}$1 $\mathrm{\frac{C_m}{h}}$6Maximum rate of synthesis of mRNA
$k_{m}$0.1 $\mathrm{h^{-1}}$4First-order rate constant for mRNA degradation
$v_{p}$0.5 $\mathrm{\frac{C_{p}}{C_{m}h}}$6Rate constant for translation of mRNA
$k_{p_1}$10 $\mathrm{\frac{C_p}{h}}$6 $V_{max}$ for monomer phosphorylation
$k_{p_2}$0.03 $\mathrm{\frac{C_p}{h}}$6 $V_{max}$ for dimer phosphorylation
$k_{p_3}$0.1 $\mathrm{h^{-1}}$6First-order rate constant for proteolysis
$K_{eq}$200 $\mathrm{C_{p}^{-1}}$-12Equilibrium constant for dimerization
$P_{crit}$0.1 $\mathrm{C_{p}}$6Dimer concen at the half-maximum transcription rate
$J_{P}$0.05 ${C_{p}}$-16Michaelis constant for protein kinase (DBT)
This table is adapted from Tyson et al. [26]. Parameters $\mathrm{C_{m}}$ and $\mathrm{C_{p}}$ represent characteristic concentrations for mRNA and protein, respectively. $E_{a}$ is the activation energy of each rate constant (necessarily positive) or the standard enthalpy change for each equilibrium binding constant (may be positive or negative). The parameter values are chosen to ensure temperature compensation of the wild-type oscillator.
NameValueUnits $E_{a}/RT$Description
$v_{m}$1 $\mathrm{\frac{C_m}{h}}$6Maximum rate of synthesis of mRNA
$k_{m}$0.1 $\mathrm{h^{-1}}$4First-order rate constant for mRNA degradation
$v_{p}$0.5 $\mathrm{\frac{C_{p}}{C_{m}h}}$6Rate constant for translation of mRNA
$k_{p_1}$10 $\mathrm{\frac{C_p}{h}}$6 $V_{max}$ for monomer phosphorylation
$k_{p_2}$0.03 $\mathrm{\frac{C_p}{h}}$6 $V_{max}$ for dimer phosphorylation
$k_{p_3}$0.1 $\mathrm{h^{-1}}$6First-order rate constant for proteolysis
$K_{eq}$200 $\mathrm{C_{p}^{-1}}$-12Equilibrium constant for dimerization
$P_{crit}$0.1 $\mathrm{C_{p}}$6Dimer concen at the half-maximum transcription rate
$J_{P}$0.05 ${C_{p}}$-16Michaelis constant for protein kinase (DBT)
This table is adapted from Tyson et al. [26]. Parameters $\mathrm{C_{m}}$ and $\mathrm{C_{p}}$ represent characteristic concentrations for mRNA and protein, respectively. $E_{a}$ is the activation energy of each rate constant (necessarily positive) or the standard enthalpy change for each equilibrium binding constant (may be positive or negative). The parameter values are chosen to ensure temperature compensation of the wild-type oscillator.
Table 2.  Equilibrium of (1) and corresponding eigenvalues of its Jacobian matrix vary with $K_{eq}$ and $k_a$
$K_{eq}$$k_{a}$Equilibrium1Eigenvalues
$200$$10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$
$10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$
$1$ $(8.62, 0.10, 0.04)$$\{-25.96, 0.01\pm0.11\mathrm{i} \}$
$10^{3}$ $(1.38, 0.04, 0.24)$ $\{-164.97, 0.11\pm0.41\mathrm{i} \}$
$10^{6}$ $(1.36, 0.04, 0.25)$ $\{-1.47\times10^5, 0.12\pm0.42\mathrm{i} \}$
$15$$10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$
$10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$
$1$ $(9.60, 0.08, 0.10)$ $\{-30.77, -0.03\pm0.08\mathrm{i} \}$
$10^{2}$ $(5.09, 0.08, 0.10)$ $\{-63.98, 0.66\pm0.28\mathrm{i} \}$
$10^{3}$ $(5.03, 0.08, 0.10)$ $\{-417.94, 1.43, 0.54 \}$
$10^{6}$ $(5.02, 0.08, 0.10)$ $\{-3.9\times10^5, 1.57, 0.52\}$
$1$$10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$
$10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$
$1$ $(10.00, 0.05, 2\times10^{-3})$ $\{-46.81, -1.13, -0.10 \}$
$10^{3}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1240, -28.12, -0.10 \}$
$10^{6}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1.2\times10^6, -28.49, -0.10 \}$
1 Those zeros in equilibrium terms are actually very small positive numbers. Other parameter values are as given in Table 1.
$K_{eq}$$k_{a}$Equilibrium1Eigenvalues
$200$$10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$
$10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$
$1$ $(8.62, 0.10, 0.04)$$\{-25.96, 0.01\pm0.11\mathrm{i} \}$
$10^{3}$ $(1.38, 0.04, 0.24)$ $\{-164.97, 0.11\pm0.41\mathrm{i} \}$
$10^{6}$ $(1.36, 0.04, 0.25)$ $\{-1.47\times10^5, 0.12\pm0.42\mathrm{i} \}$
$15$$10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$
$10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$
$1$ $(9.60, 0.08, 0.10)$ $\{-30.77, -0.03\pm0.08\mathrm{i} \}$
$10^{2}$ $(5.09, 0.08, 0.10)$ $\{-63.98, 0.66\pm0.28\mathrm{i} \}$
$10^{3}$ $(5.03, 0.08, 0.10)$ $\{-417.94, 1.43, 0.54 \}$
$10^{6}$ $(5.02, 0.08, 0.10)$ $\{-3.9\times10^5, 1.57, 0.52\}$
$1$$10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$
$10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$
$1$ $(10.00, 0.05, 2\times10^{-3})$ $\{-46.81, -1.13, -0.10 \}$
$10^{3}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1240, -28.12, -0.10 \}$
$10^{6}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1.2\times10^6, -28.49, -0.10 \}$
1 Those zeros in equilibrium terms are actually very small positive numbers. Other parameter values are as given in Table 1.
Table 3.  Period of endogenous rhythms of wild-type flies varies as $k_{a}$ ($K_{eq}=200$) varies
$k_{a}$0.0010.10.80.9110100
Periodnonenonenone72.4463.1050.8932.51
$k_{a}$50010005000$10^4$$5\times10^4$$10^{5}$$5\times10^{5}$
Period28.6126.9024.8624.5424.2724.2424.21
$k_{a}$$10^{6}$$2\times10^{6}$$2.5\times10^{6}$$2.9\times10^{6}$$3\times10^{6}$
Period24.2124.2124.2124.3024.44
Periodic oscillations happen when $k_a$ is larger than the bifurcation value $k_a^*=0.9$. Other parameter values are as given in Table 1.
$k_{a}$0.0010.10.80.9110100
Periodnonenonenone72.4463.1050.8932.51
$k_{a}$50010005000$10^4$$5\times10^4$$10^{5}$$5\times10^{5}$
Period28.6126.9024.8624.5424.2724.2424.21
$k_{a}$$10^{6}$$2\times10^{6}$$2.5\times10^{6}$$2.9\times10^{6}$$3\times10^{6}$
Period24.2124.2124.2124.3024.44
Periodic oscillations happen when $k_a$ is larger than the bifurcation value $k_a^*=0.9$. Other parameter values are as given in Table 1.
Table 4.  Period of endogenous rhythms of $per^{L}$ mutant varies as $k_{a}$ ($K_{eq}=15$) varies
$k_{a}$0.0010.11.11.2210100500
Periodnonenonenone57.1955.6741.3430.9829.21
$k_{a}$100020005000$10^4$$10^5$$7\times10^5$$7\times10^5$$7\times10^5$
Period28.9428.8028.7128.6728.6528.6529.2030.37
Periodic oscillations occur when $k_a$ is beyond the bifurcation value $k_a^*=1.2$. Other parameter values are as in Table 1.
$k_{a}$0.0010.11.11.2210100500
Periodnonenonenone57.1955.6741.3430.9829.21
$k_{a}$100020005000$10^4$$10^5$$7\times10^5$$7\times10^5$$7\times10^5$
Period28.9428.8028.7128.6728.6528.6529.2030.37
Periodic oscillations occur when $k_a$ is beyond the bifurcation value $k_a^*=1.2$. Other parameter values are as in Table 1.
Table 5.  Period of the endogenous rhythms of wild-type and mutant flies based on (1)
Genotype $K_{eq}$TempPeriodGenotype $k_{p_{1}}$ $k_{p_{2}}$Period
Wild type2452024.2 $dbt^{+}(1\times)$100.0324.2
2002524.2 $dbt^{+}(2\times)$150.0624.3
1643024.2 $dbt^{+}(3\times)$200.0925.7
$per^{L}$18.42026.5 $dbt^{S}$100.317.6
15.02528.7 $dbt^{+}$100.0324.2
12.33030.4 $dbt^{L}$100.00325.1
To simplify the integration, we take $k_{a}=10^{6}$ for wild-type flies and $k_{a}=5000$ for mutant flies. Other conditions are as in Table 6.
Genotype $K_{eq}$TempPeriodGenotype $k_{p_{1}}$ $k_{p_{2}}$Period
Wild type2452024.2 $dbt^{+}(1\times)$100.0324.2
2002524.2 $dbt^{+}(2\times)$150.0624.3
1643024.2 $dbt^{+}(3\times)$200.0925.7
$per^{L}$18.42026.5 $dbt^{S}$100.317.6
15.02528.7 $dbt^{+}$100.0324.2
12.33030.4 $dbt^{L}$100.00325.1
To simplify the integration, we take $k_{a}=10^{6}$ for wild-type flies and $k_{a}=5000$ for mutant flies. Other conditions are as in Table 6.
Table 6.  Period of the endogenous rhythms of wild-type and mutant flies based on (2)
Genotype $K_{eq}$TempPeriodGenotype $k_{p_{1}}$ $k_{p_{2}}$Period
Wild type2452024.2 $dbt^{+}(1\times)$100.0324.2
2002524.2 $dbt^{+}(2\times)$150.0624.4
1643024.2 $dbt^{+}(3\times)$200.0925.7
$per^{L}$18.42026.5 $dbt^{S}$100.317.6
15.02528.7 $dbt^{+}$100.0324.2
12.33030.5 $dbt^{L}$100.00325.2
This table is copied out of Tyson et al. [26]. It is assumed that each parameter $k$ varies with temperature according to $k(T)=k(298)\exp\{\varepsilon_{a}(1-298/T)\}$, with values for $k(298)$ and $\varepsilon_{a}=E_{a}/(0.592 \mathrm{kcal} \mathrm{mol}^{-1})$ given in Table 1. The $dbt^{+}(n\times)$ means $n$ copies of the wild-type allele.
Genotype $K_{eq}$TempPeriodGenotype $k_{p_{1}}$ $k_{p_{2}}$Period
Wild type2452024.2 $dbt^{+}(1\times)$100.0324.2
2002524.2 $dbt^{+}(2\times)$150.0624.4
1643024.2 $dbt^{+}(3\times)$200.0925.7
$per^{L}$18.42026.5 $dbt^{S}$100.317.6
15.02528.7 $dbt^{+}$100.0324.2
12.33030.5 $dbt^{L}$100.00325.2
This table is copied out of Tyson et al. [26]. It is assumed that each parameter $k$ varies with temperature according to $k(T)=k(298)\exp\{\varepsilon_{a}(1-298/T)\}$, with values for $k(298)$ and $\varepsilon_{a}=E_{a}/(0.592 \mathrm{kcal} \mathrm{mol}^{-1})$ given in Table 1. The $dbt^{+}(n\times)$ means $n$ copies of the wild-type allele.
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