# American Institute of Mathematical Sciences

October  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 15, 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 and 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.  Google Scholar [2] K. Bae, C. Lee, D. Sidote, K-Y. Chuang and 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.  Google Scholar [3] T. K. Darlington, K. Wager-Smith, M. F. Ceriani, D. Staknis, N. Gekakis, T. D. L. Steeves, C. J. Weitz, J. S. Takahashi and S. A. Kay, Closing the circadian loop: CLOCK-induced transcription of its own inhibitors per and tim, Science, 280 (1998), 1599-1603.   Google Scholar [4] A. Eskin, S. J. Yeung and 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.  Google Scholar [5] N. Gekakis, L. Saez, A.-M. Delahaye-Brown, M. P. Myers, A. Sehgal, M. W. Young and 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.  Google Scholar [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.  Google Scholar [7] P. E. Hardin, J. C. Hall and 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] Z. J. Huang, K. D. Curtin and M. Rosbash, PER protein interactions and temperature compensation of a circadian clock in Drosophila, Science, 267 (1995), 1169-1172.  doi: 10.1126/science.7855598.  Google Scholar [11] M. W. Karakashian and 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.  Google Scholar [12] S. B. S. Khalsa, D. Whitmore and 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.  Google Scholar [13] B. Kloss, J. L. Price, L. Saez, J. Blau, A. Rothenfluh, C. S. Wesley and M. W. Young, The Drosophila clock gene double-time encodes a protein closely related to human casein kinase l$\epsilon$, Cell, 94 (1998), 97-107.  doi: 10.1016/S0092-8674(00)81225-8.  Google Scholar [14] R. J. Konopka and S. Benzer, Clock mutants of Drosophila melanogaster, Proc. Natl. Acad. Sci. USA, 68 (1971), 2112-2116.  doi: 10.1073/pnas.68.9.2112.  Google Scholar [15] J.-C. Leloup and 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.  Google Scholar [16] J. L. Price, J. Blau, A. Rothenfluh, M. Abodeely, B. Kloss and 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.  Google Scholar [17] P. Ruoff and 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.  Google Scholar [18] J. E. Rutila, V. Suri, M. Le, M. V. So, M. Rosbash and 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.  Google Scholar [19] T. Scheper, D. Klinkenberg, C. Pennartz and J. van Pelt, A mathematical model for the intracellular circadian rhythm generator, J. Neurosci., 19 (1999), 40-47.   Google Scholar [20] A. Sehgal, J. L. Price, B. Man and 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc. Providence, Rhode Island, 1995.  Google Scholar [24] V. Suri, A. Lanjuin and 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.  Google Scholar [25] W. R. Taylor, J. C. Dunlap and J. W. Hastings, Inhibitors of protein synthesis on 80s ribosomes phase shift the Gonyaulax clock, J. Exp. Biol., 97 (1982), 121-136.   Google Scholar [26] J. J. Tyson, C. I. Hong, C. D. Thron and 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.  Google Scholar [27] L. B. Vosshall, J. L. Price, A. Sehgal, L. Saez and 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.  Google Scholar [28] Y. Wang and J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns., 176 (2001), 470-493.  doi: 10.1006/jdeq.2001.3989.  Google Scholar [29] H. Zeng, Z. Qian, M. P. Myers and M. Rosbash, A light-entrainment mechanism for the Drosophila circadian clock, Nature, 380 (1996), 129-135.  doi: 10.1038/380129a0.  Google Scholar [30] H.-R. Zhu and 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.  Google Scholar

show all references

##### References:
 [1] R. Allada, N. E. White, W. V. So, J. C. Hall and 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.  Google Scholar [2] K. Bae, C. Lee, D. Sidote, K-Y. Chuang and 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.  Google Scholar [3] T. K. Darlington, K. Wager-Smith, M. F. Ceriani, D. Staknis, N. Gekakis, T. D. L. Steeves, C. J. Weitz, J. S. Takahashi and S. A. Kay, Closing the circadian loop: CLOCK-induced transcription of its own inhibitors per and tim, Science, 280 (1998), 1599-1603.   Google Scholar [4] A. Eskin, S. J. Yeung and 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.  Google Scholar [5] N. Gekakis, L. Saez, A.-M. Delahaye-Brown, M. P. Myers, A. Sehgal, M. W. Young and 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.  Google Scholar [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.  Google Scholar [7] P. E. Hardin, J. C. Hall and 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [10] Z. J. Huang, K. D. Curtin and M. Rosbash, PER protein interactions and temperature compensation of a circadian clock in Drosophila, Science, 267 (1995), 1169-1172.  doi: 10.1126/science.7855598.  Google Scholar [11] M. W. Karakashian and 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.  Google Scholar [12] S. B. S. Khalsa, D. Whitmore and 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.  Google Scholar [13] B. Kloss, J. L. Price, L. Saez, J. Blau, A. Rothenfluh, C. S. Wesley and M. W. Young, The Drosophila clock gene double-time encodes a protein closely related to human casein kinase l$\epsilon$, Cell, 94 (1998), 97-107.  doi: 10.1016/S0092-8674(00)81225-8.  Google Scholar [14] R. J. Konopka and S. Benzer, Clock mutants of Drosophila melanogaster, Proc. Natl. Acad. Sci. USA, 68 (1971), 2112-2116.  doi: 10.1073/pnas.68.9.2112.  Google Scholar [15] J.-C. Leloup and 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.  Google Scholar [16] J. L. Price, J. Blau, A. Rothenfluh, M. Abodeely, B. Kloss and 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.  Google Scholar [17] P. Ruoff and 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.  Google Scholar [18] J. E. Rutila, V. Suri, M. Le, M. V. So, M. Rosbash and 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.  Google Scholar [19] T. Scheper, D. Klinkenberg, C. Pennartz and J. van Pelt, A mathematical model for the intracellular circadian rhythm generator, J. Neurosci., 19 (1999), 40-47.   Google Scholar [20] A. Sehgal, J. L. Price, B. Man and 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc. Providence, Rhode Island, 1995.  Google Scholar [24] V. Suri, A. Lanjuin and 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.  Google Scholar [25] W. R. Taylor, J. C. Dunlap and J. W. Hastings, Inhibitors of protein synthesis on 80s ribosomes phase shift the Gonyaulax clock, J. Exp. Biol., 97 (1982), 121-136.   Google Scholar [26] J. J. Tyson, C. I. Hong, C. D. Thron and 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.  Google Scholar [27] L. B. Vosshall, J. L. Price, A. Sehgal, L. Saez and 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.  Google Scholar [28] Y. Wang and J. Jiang, The general properties of discrete-time competitive dynamical systems, J. Diff. Eqns., 176 (2001), 470-493.  doi: 10.1006/jdeq.2001.3989.  Google Scholar [29] H. Zeng, Z. Qian, M. P. Myers and M. Rosbash, A light-entrainment mechanism for the Drosophila circadian clock, Nature, 380 (1996), 129-135.  doi: 10.1038/380129a0.  Google Scholar [30] H.-R. Zhu and 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.  Google Scholar
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
Numerical solution of (1). Parameter values are chosen as in Table 1. We take $k_a=10^6$ and $k_d=k_a/K_{eq}$
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 $[\, c, d\, ] = [\, 50, 460\, ]$
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
The vector field for (1) on the boundary of $B(a, b, c)$
Parameter values suitable for circadian rhythm of wild-type fruit flies
 Name Value Units $E_{a}/RT$ Description $v_{m}$ 1 $\mathrm{\frac{C_m}{h}}$ 6 Maximum rate of synthesis of mRNA $k_{m}$ 0.1 $\mathrm{h^{-1}}$ 4 First-order rate constant for mRNA degradation $v_{p}$ 0.5 $\mathrm{\frac{C_{p}}{C_{m}h}}$ 6 Rate 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 $\mathrm{h^{-1}}$ 6 First-order rate constant for proteolysis $K_{eq}$ 200 $\mathrm{C_{p}^{-1}}$ -12 Equilibrium constant for dimerization $P_{crit}$ 0.1 $\mathrm{C_{p}}$ 6 Dimer concen at the half-maximum transcription rate $J_{P}$ 0.05 ${C_{p}}$ -16 Michaelis 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.
 Name Value Units $E_{a}/RT$ Description $v_{m}$ 1 $\mathrm{\frac{C_m}{h}}$ 6 Maximum rate of synthesis of mRNA $k_{m}$ 0.1 $\mathrm{h^{-1}}$ 4 First-order rate constant for mRNA degradation $v_{p}$ 0.5 $\mathrm{\frac{C_{p}}{C_{m}h}}$ 6 Rate 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 $\mathrm{h^{-1}}$ 6 First-order rate constant for proteolysis $K_{eq}$ 200 $\mathrm{C_{p}^{-1}}$ -12 Equilibrium constant for dimerization $P_{crit}$ 0.1 $\mathrm{C_{p}}$ 6 Dimer concen at the half-maximum transcription rate $J_{P}$ 0.05 ${C_{p}}$ -16 Michaelis 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.
Equilibrium of (1) and corresponding eigenvalues of its Jacobian matrix vary with $K_{eq}$ and $k_a$
 $K_{eq}$ $k_{a}$ Equilibrium1 Eigenvalues $200$ $10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$ $10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$ $1$ $(8.62, 0.10, 0.04)$ $\{-25.96, 0.01\pm0.11\mathrm{i} \}$ $10^{3}$ $(1.38, 0.04, 0.24)$ $\{-164.97, 0.11\pm0.41\mathrm{i} \}$ $10^{6}$ $(1.36, 0.04, 0.25)$ $\{-1.47\times10^5, 0.12\pm0.42\mathrm{i} \}$ $15$ $10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$ $10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$ $1$ $(9.60, 0.08, 0.10)$ $\{-30.77, -0.03\pm0.08\mathrm{i} \}$ $10^{2}$ $(5.09, 0.08, 0.10)$ $\{-63.98, 0.66\pm0.28\mathrm{i} \}$ $10^{3}$ $(5.03, 0.08, 0.10)$ $\{-417.94, 1.43, 0.54 \}$ $10^{6}$ $(5.02, 0.08, 0.10)$ $\{-3.9\times10^5, 1.57, 0.52\}$ $1$ $10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$ $10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$ $1$ $(10.00, 0.05, 2\times10^{-3})$ $\{-46.81, -1.13, -0.10 \}$ $10^{3}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1240, -28.12, -0.10 \}$ $10^{6}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1.2\times10^6, -28.49, -0.10 \}$ 1 Those zeros in equilibrium terms are actually very small positive numbers. Other parameter values are as given in Table 1.
 $K_{eq}$ $k_{a}$ Equilibrium1 Eigenvalues $200$ $10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$ $10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$ $1$ $(8.62, 0.10, 0.04)$ $\{-25.96, 0.01\pm0.11\mathrm{i} \}$ $10^{3}$ $(1.38, 0.04, 0.24)$ $\{-164.97, 0.11\pm0.41\mathrm{i} \}$ $10^{6}$ $(1.36, 0.04, 0.25)$ $\{-1.47\times10^5, 0.12\pm0.42\mathrm{i} \}$ $15$ $10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$ $10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$ $1$ $(9.60, 0.08, 0.10)$ $\{-30.77, -0.03\pm0.08\mathrm{i} \}$ $10^{2}$ $(5.09, 0.08, 0.10)$ $\{-63.98, 0.66\pm0.28\mathrm{i} \}$ $10^{3}$ $(5.03, 0.08, 0.10)$ $\{-417.94, 1.43, 0.54 \}$ $10^{6}$ $(5.02, 0.08, 0.10)$ $\{-3.9\times10^5, 1.57, 0.52\}$ $1$ $10^{-6}$ $(10.00, 0.05, 0)$ $\{-50.20, -0.40, -0.1\}$ $10^{-3}$ $(10.00, 0.05, 6\times10^{-6})$ $\{-50.19, -0.40, -0.1\}$ $1$ $(10.00, 0.05, 2\times10^{-3})$ $\{-46.81, -1.13, -0.10 \}$ $10^{3}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1240, -28.12, -0.10 \}$ $10^{6}$ $(10.00, 0.05, 3\times10^{-3})$ $\{-1.2\times10^6, -28.49, -0.10 \}$ 1 Those zeros in equilibrium terms are actually very small positive numbers. Other parameter values are as given in Table 1.
Period of endogenous rhythms of wild-type flies varies as $k_{a}$ ($K_{eq}=200$) varies
 $k_{a}$ 0.001 0.1 0.8 0.9 1 10 100 Period none none none 72.44 63.10 50.89 32.51 $k_{a}$ 500 1000 5000 $10^4$ $5\times10^4$ $10^{5}$ $5\times10^{5}$ Period 28.61 26.90 24.86 24.54 24.27 24.24 24.21 $k_{a}$ $10^{6}$ $2\times10^{6}$ $2.5\times10^{6}$ $2.9\times10^{6}$ $3\times10^{6}$ Period 24.21 24.21 24.21 24.30 24.44 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.
 $k_{a}$ 0.001 0.1 0.8 0.9 1 10 100 Period none none none 72.44 63.10 50.89 32.51 $k_{a}$ 500 1000 5000 $10^4$ $5\times10^4$ $10^{5}$ $5\times10^{5}$ Period 28.61 26.90 24.86 24.54 24.27 24.24 24.21 $k_{a}$ $10^{6}$ $2\times10^{6}$ $2.5\times10^{6}$ $2.9\times10^{6}$ $3\times10^{6}$ Period 24.21 24.21 24.21 24.30 24.44 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.
Period of endogenous rhythms of $per^{L}$ mutant varies as $k_{a}$ ($K_{eq}=15$) varies
 $k_{a}$ 0.001 0.1 1.1 1.2 2 10 100 500 Period none none none 57.19 55.67 41.34 30.98 29.21 $k_{a}$ 1000 2000 5000 $10^4$ $10^5$ $7\times10^5$ $7\times10^5$ $7\times10^5$ Period 28.94 28.80 28.71 28.67 28.65 28.65 29.20 30.37 Periodic oscillations occur when $k_a$ is beyond the bifurcation value $k_a^*=1.2$. Other parameter values are as in Table 1.
 $k_{a}$ 0.001 0.1 1.1 1.2 2 10 100 500 Period none none none 57.19 55.67 41.34 30.98 29.21 $k_{a}$ 1000 2000 5000 $10^4$ $10^5$ $7\times10^5$ $7\times10^5$ $7\times10^5$ Period 28.94 28.80 28.71 28.67 28.65 28.65 29.20 30.37 Periodic oscillations occur when $k_a$ is beyond the bifurcation value $k_a^*=1.2$. Other parameter values are as in Table 1.
Period of the endogenous rhythms of wild-type and mutant flies based on (1)
 Genotype $K_{eq}$ Temp Period Genotype $k_{p_{1}}$ $k_{p_{2}}$ Period Wild type 245 20 24.2 $dbt^{+}(1\times)$ 10 0.03 24.2 200 25 24.2 $dbt^{+}(2\times)$ 15 0.06 24.3 164 30 24.2 $dbt^{+}(3\times)$ 20 0.09 25.7 $per^{L}$ 18.4 20 26.5 $dbt^{S}$ 10 0.3 17.6 15.0 25 28.7 $dbt^{+}$ 10 0.03 24.2 12.3 30 30.4 $dbt^{L}$ 10 0.003 25.1 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 $K_{eq}$ Temp Period Genotype $k_{p_{1}}$ $k_{p_{2}}$ Period Wild type 245 20 24.2 $dbt^{+}(1\times)$ 10 0.03 24.2 200 25 24.2 $dbt^{+}(2\times)$ 15 0.06 24.3 164 30 24.2 $dbt^{+}(3\times)$ 20 0.09 25.7 $per^{L}$ 18.4 20 26.5 $dbt^{S}$ 10 0.3 17.6 15.0 25 28.7 $dbt^{+}$ 10 0.03 24.2 12.3 30 30.4 $dbt^{L}$ 10 0.003 25.1 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.
Period of the endogenous rhythms of wild-type and mutant flies based on (2)
 Genotype $K_{eq}$ Temp Period Genotype $k_{p_{1}}$ $k_{p_{2}}$ Period Wild type 245 20 24.2 $dbt^{+}(1\times)$ 10 0.03 24.2 200 25 24.2 $dbt^{+}(2\times)$ 15 0.06 24.4 164 30 24.2 $dbt^{+}(3\times)$ 20 0.09 25.7 $per^{L}$ 18.4 20 26.5 $dbt^{S}$ 10 0.3 17.6 15.0 25 28.7 $dbt^{+}$ 10 0.03 24.2 12.3 30 30.5 $dbt^{L}$ 10 0.003 25.2 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 $K_{eq}$ Temp Period Genotype $k_{p_{1}}$ $k_{p_{2}}$ Period Wild type 245 20 24.2 $dbt^{+}(1\times)$ 10 0.03 24.2 200 25 24.2 $dbt^{+}(2\times)$ 15 0.06 24.4 164 30 24.2 $dbt^{+}(3\times)$ 20 0.09 25.7 $per^{L}$ 18.4 20 26.5 $dbt^{S}$ 10 0.3 17.6 15.0 25 28.7 $dbt^{+}$ 10 0.03 24.2 12.3 30 30.5 $dbt^{L}$ 10 0.003 25.2 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.

2018 Impact Factor: 1.313