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

  • * Corresponding authorr: J. Jiang

    * Corresponding authorr: J. Jiang 
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.
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  • 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.

    Mathematics Subject Classification: Primary: 34Cxx; Secondary: 92Bxx.


    \begin{equation} \\ \end{equation}
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  • 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.
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    Table 2.  Equilibrium of (1) and corresponding eigenvalues of its Jacobian matrix vary with $K_{eq}$ and $k_a$

    $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.
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    Table 3.  Period of endogenous rhythms of wild-type flies varies as $k_{a}$ ($K_{eq}=200$) varies

    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.
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    Table 4.  Period of endogenous rhythms of $per^{L}$ mutant varies as $k_{a}$ ($K_{eq}=15$) varies

    Periodic oscillations occur when $k_a$ is beyond the bifurcation value $k_a^*=1.2$. Other parameter values are as in Table 1.
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    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.
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    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.
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