Modeling transcriptional co-regulation of mammalian circadian clock

  • Received: 30 May 2016 Accepted: 20 January 2017 Published: 01 October 2017
  • MSC : Primary: 34D99; Secondary: 34C23

  • The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.

    Citation: Yanqin Wang, Xin Ni, Jie Yan, Ling Yang. Modeling transcriptional co-regulation of mammalian circadian clock[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1447-1462. doi: 10.3934/mbe.2017075

    Related Papers:

    [1] Ying Li, Zhao Zhao, Yuan-yuan Tan, Xue Wang . Dynamical analysis of the effects of circadian clock on the neurotransmitter dopamine. Mathematical Biosciences and Engineering, 2023, 20(9): 16663-16677. doi: 10.3934/mbe.2023742
    [2] Jifa Jiang, Qiang Liu, Lei Niu . Theoretical investigation on models of circadian rhythms based on dimerization and proteolysis of PER and TIM. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1247-1259. doi: 10.3934/mbe.2017064
    [3] Changgui Gu, Ping Wang, Tongfeng Weng, Huijie Yang, Jos Rohling . Heterogeneity of neuronal properties determines the collective behavior of the neurons in the suprachiasmatic nucleus. Mathematical Biosciences and Engineering, 2019, 16(4): 1893-1913. doi: 10.3934/mbe.2019092
    [4] E.V. Presnov, Z. Agur . The Role Of Time Delays, Slow Processes And Chaos In Modulating The Cell-Cycle Clock. Mathematical Biosciences and Engineering, 2005, 2(3): 625-642. doi: 10.3934/mbe.2005.2.625
    [5] Shishi Wang, Yuting Ding, Hongfan Lu, Silin Gong . Stability and bifurcation analysis of $ SIQR $ for the COVID-19 epidemic model with time delay. Mathematical Biosciences and Engineering, 2021, 18(5): 5505-5524. doi: 10.3934/mbe.2021278
    [6] Zhichao Jiang, Xiaohua Bi, Tongqian Zhang, B.G. Sampath Aruna Pradeep . Global Hopf bifurcation of a delayed phytoplankton-zooplankton system considering toxin producing effect and delay dependent coefficient. Mathematical Biosciences and Engineering, 2019, 16(5): 3807-3829. doi: 10.3934/mbe.2019188
    [7] Xin-You Meng, Yu-Qian Wu . Bifurcation analysis in a singular Beddington-DeAngelis predator-prey model with two delays and nonlinear predator harvesting. Mathematical Biosciences and Engineering, 2019, 16(4): 2668-2696. doi: 10.3934/mbe.2019133
    [8] Hongying Shu, Wanxiao Xu, Zenghui Hao . Global dynamics of an immunosuppressive infection model with stage structure. Mathematical Biosciences and Engineering, 2020, 17(3): 2082-2102. doi: 10.3934/mbe.2020111
    [9] Hongfan Lu, Yuting Ding, Silin Gong, Shishi Wang . Mathematical modeling and dynamic analysis of SIQR model with delay for pandemic COVID-19. Mathematical Biosciences and Engineering, 2021, 18(4): 3197-3214. doi: 10.3934/mbe.2021159
    [10] Xiaomeng Ma, Zhanbing Bai, Sujing Sun . Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders. Mathematical Biosciences and Engineering, 2023, 20(1): 437-455. doi: 10.3934/mbe.2023020
  • The circadian clock is a self-sustaining oscillator that has a period of about 24 hours at the molecular level. The oscillator is a transcription-translation feedback loop system composed of several genes. In this paper, a scalar nonlinear differential equation with two delays, modeling the transcriptional co-regulation in mammalian circadian clock, is proposed and analyzed. Sufficient conditions are established for the asymptotic stability of the unique nontrivial positive equilibrium point of the model by studying an exponential polynomial characteristic equation with delay-dependent coefficients. The existence of the Hopf bifurcations can be also obtained. Numerical simulations of the model with proper parameter values coincide with the theoretical result.


    1. Introduction

    Circadian clocks are endogenous 24-h oscillations that regulate the temporal organization of physiology, metabolism and behavior[4,11]. Disruption of circadian rhythms leads to various diseases and may reduce lifespan in mice [2,8,9,14]. The central mechanism of the mammalian circadian oscillation is a negative feedback loop that involves the transcriptional activator genes: clock and bmal1, and five repressor genes: period (per1-per3) and cryptochrome (cry1 and cry2)[11,20]. CLOCK and BMAL1 are transcription factors that activate per and cry gene transcriptions. The resulting PER and CRY proteins accumulate, and then inhibit CLOCK/BMAL1-mediated transcription after a certain time delay. Various auxiliary loops also take part in the regulation of mammalian circadian clocks. Previous studies validated that cry1 expression is positively auto-regulated via the inhibition of its repressor rev-erbα, which forms a positive loop [16]. Some nuclear receptors, exhibiting circadian like patterns of expression [18], may also contribute to the circadian clock via other auxiliary loops. These auxiliary loops are believed to receive the environmental factors for adaption purpose, therefore provide multiple entry points to regulate the circadian oscillator. Therefore, an important question arises: How does the system with more than one feedback loop maintain the robustness of the oscillation?

    Goldbeter [5] proposed the first molecular model of circadian clock in Drosophila. This model takes account of the core negative feedback loop of per self-repression. Since then, a lot of mathematical models in different organisms based on the negative feedback loop have been developed and studied by many groups [6,10,15,19]. Recently, the circadian clock models with more than one feedback loops have been proposed, and the computational studies have revealed some interesting results [3,9,12,17]. However, the studies on the stability of the multi-loop regulation in circadian clock remain obscure.

    Here we consider a system of one differential equation modeling the transcriptional co-regulation in a mammalian circadian clock. In this system, there is a core feedback loop that CRY1 negatively regulates its own expression by inhibiting CLOCK/BMAL1-mediated transcription. A delay naturally appears in this process, representing the durations of post-translational regulations. There occurs another auxiliary loop in this model in which CRY1 positively mediates its own expression via the inhibition of its repressor rev-erbα (see Fig. 1). Similarly, this positive process also contains a time delay. Therefore, we consider a two-loop system with two time delays in this work. The theoretical approach is based on the work of Adimy et al.[1]. Our aim is to show that the auxiliary loops, which are linked to other pathways for adaption purpose, will not abandon the oscillation with reasonable parameters.

    Figure 1. The model of a mammalian circadian clock with two delays. Figure (a) is a schematic diagram of gene regulation in the mammalian circadian clock system, figure (b) is a schematic diagram of the simplified mathematical model of a mammalian circadian clock.

    This paper is organized as follows: Section 2 presents the model of a mammalian circadian clock, a scalar nonlinear delay differential equation with two delays, and investigates the existence of a nontrivial positive equilibrium point. Section 3 analyses the asymptotic stability of this equilibrium point. We first linearize the model about the equilibrium point and obtain a first degree exponential polynomial characteristic equation. Then we determine the conditions for the stability when only one delay is equal to zero, and, eventually, when both delays are nonzero. Besides, we also establish the existence of the Hopf bifurcations, which destabilizes the system and leads to the existence of periodic solutions. Section 4 numerically illustrates the theoretical results and Section 5 discusses the effect of time delays on the period of the model.


    2. The model

    In mammalian circadian clocks, there is a core negative feedback loop to drive the oscillation. CRY protein combines with PER to form the dimer CRY/PER, which translocates into nucleus to inhibit CLOCK/BMAL1-mediated transcription of cry1, see Fig. 1(a). Besides the core loop, there are also some auxiliary loops in the mammalian circadian system to co-regulate clock. In this work, we focus on an important auxiliary loop via rev-erbα. The transcription of rev-erbα is also mediated by CLOCK/BMAL1 complex, so that can be inhibited by CRY/PER complex. In return, REV-ERBα feeds back to repress the transcription of cry1. The double-inhibition process forms a positive feedback loop (see Fig. 1(a)).

    Our previous work about this system [17] illustrated some interesting numerical results. In order to obtain the theoretical insights of it, we first reduce the variables to form a simple one-dimensional system. We denote by x(t) the mRNA concentration of cry1 at time t. The transcription rate of cry1 (the synthesis rate of x(t)) is regulated by two items: one is the negative feedback through CLOCK/BMAL1 complex after a time delay τ1, and the other one is the positive feedback through REV-ERBα after a time delay τ2 (see Fig. 1(b)). The degradation rate of cry1 mRNA is simply assumed to be proportional to itself. Therefore, the equation modeling the co-regulation of multiple loops in mammalian circadian clocks is:

    dxdt=k1M+x2(tτ1)+k2x(tτ2)cx(t), (2.1)

    where k1 and M are positive constants describing the negative regulation, k2 is a positive constant characterizing the strength of positive regulation, and the constant c>0 reflects the linear degradation rate. The parameter τ1>0 denotes the average time needed by the negative feedback, which represents the durations of post-translational regulations. The parameter τ2>0 denotes the average time needed by the positive feedback.

    Obviously, system (2.1) has a unique continuous solution x(t) which is well-defined for all t0 and for a continuous initial condition. Furthermore, it is easy to see that, for nonnegative initial conditions, the solutions of (2.1) remain nonnegative for t0. In fact, if we let t0>0 be such that x(t)>0 for t<t0 and x(t0)=0, then, from (2.1), we have

    dx(t0)dt=k1M+x2(t0τ1)+k2x(t0τ2)>0,

    and the result follows.

    Now, we begin to consider the existence of equilibrium points for system (2.1). An equilibrium point of system (2.1) is a stationary solution x of (2.1), that is,

    (k2c)(x)3+M(k2c)x+k1=0. (2.2)

    Evidently, 0 is not an equilibrium point of system (2.1), so it is enough to search for its nonzero equilibrium points. For this purpose, we need to assume that c>k2 and recall the well-known Shengjin's formulas.

    Lemma 2.1. (Shengjin's formulas) Consider a variable cubic equationt

    ax3+bx2+cx+d=0, (2.3)

    where a,b,c,dR,and a0. Set repeated root discriminant A=b23ac, B=bc9ad, C=c23bd, and total discriminant Δ=B24AC. Then the following statements are true:

    ( i ) When A=B=0, (2.3) has a triple real root x1=x2=x3=b3a=cb=3dc;

    ( ii ) When Δ=B24AC>0, (2.3) has a real root x=b(3Y1+3Y2)3a and a pair of conjugate complex roots, where Y1,2=Ab+3a(B±B24AC2);

    ( iii ) When Δ=B24AC=0, (2.3) has three real roots x1=ba+K, x2=x3=K2, where K=BA,A0;

    ( iv ) When Δ=B24AC<0, (2.3) has three unequal real roots.

    Using Lemma 2.1, in Eq.(2.2), we calculate

    A=3M(k2c)2,B=9k1(k2c),C=M2(k2c)2,

    and obtain

    Δ=B24AC=(k2c)2(81k21+12M3(k2d)2)>0,

    then it follows from Lemma 2.1 (ii) that Eq.(2.2) has a real root x=x=3Y1+3Y23(ck2), where

    Y1=32(k2c)2(9k1+81k21+12M3(k2c)2),
    Y2=32(k2c)2(9k181k21+12M3(k2c)2).

    In addition, we can easily see that Y1+Y2=27k1(k2c)2>0, leading to Y1>Y2 and 3Y1>3Y2, which implies 3Y1+3Y2>0. Consequently, Eq. (2.2) has a positive solution, which is unique if and only if c>k2.

    This result is summarized in the following proposition.

    Proposition 2.1. Assume that c>k2. Then system (2.1) has a unique nontrivial positive equilibrium point x, a solution of Eq.(2.2).


    3. Local asymptotic stability

    In this section, we concentrate on the stability of the nontrivial equilibrium point x. Hence, we assume throughout this section that c>k2.

    We first start to linearize system (2.1) around x and deduce the characteristic equation.


    3.1. Linearization and characteristic equation

    Take y(t)=x(t)x, the linearized system of (2.1) about x is then given by

    dydt=2k1x(M+x2)2y(tτ1)+k2y(tτ2)cy(t), (3.1)

    Denote

    α=2k1x(M+x2)2,β=k2. (3.2)

    The characteristic equation associated with (3.1) is given by

    λ+αeλτ1βeλτ2+c=0. (3.3)

    Through studying the sign of the real parts of roots of (3.3), we can analyse the local asymptotic stability of the equilibrium point x. Here we recall that x is locally asymptotically stable if and only if all roots of (3.3) have negative real parts, and its stability can only be lost if roots cross the vertical axis, that is, if purely imaginary roots appear.

    Because of the presence of two different delays, τ1 and τ2, in Eq.(3.3), it is very complicated to analyse the sign of the real parts of eigenvalues, and there is no direct approaches to be considered. In the following, on the base of the analytic methods as in Ruan and Wei [13], we will discuss the stability of the equilibrium point when one delay is equal to zero, and deduce conditions for the stability of the equilibrium point when both time delays are nonzero.


    3.2. The case τ1=τ2=0

    Assume that τ1=τ2=0. Then, the characteristic Eq.(3.3) is written as a first degree polynomial equation

    λ+αβ+c=0. (3.4)

    Obviously, the only eigenvalue of (3.4) is λ=α+βc. It is seen that λ<0 because of the assumption c>β=k2. We can then conclude the asymptotic stability of x when τ1=τ2=0 in the next proposition.

    Proposition 3.1. Assume that c>β. Then all eigenvalues of (3.3) have negative real parts, and the equilibrium point x of system (2.1) is locally asymptotically stable.


    3.3. The case τ1>0, τ2=0

    We now consider the case τ1>0 and τ2=0. Setting τ2=0 in (3.3), the characteristic equation becomes

    λ+αeλτ1β+c=0. (3.5)

    When τ1 increases, the stability of the equilibrium point x can be lost only if pure imaginary roots appear. Hence we look for purely imaginary roots λ=±iω,ωR, of (3.5).

    If iω is a purely imaginary root of (3.5), then separating real and imaginary parts, ω satisfies

    αcos(ωτ1)=βc,αsin(ωτ1)=ω. (3.6)

    One can notice that if ω is a solution of (3.6), then so is ω. Thus, in the following, we only look for positive solutions ω of (3.6).

    Adding the squares of both hand sides of Eq.(3.6), we see that ω must be a root of the following equation:

    ω2α2+(cβ)2=0. (3.7)

    Set F(X)=X2α2+(cβ)2 and make the two assumptions as follows

    (H1) αcβ;

    (H2) α>cβ.

    The function F has no positive zeros when (H1) holds. When (H2) holds, the function F has a unique positive zero ω0=α2(cβ)2. Substituting ω0 into Eq.(3.6), we have

    τk1=1ω0(arctan(ω0βc)+(2k+1)π), k=0,1,2,. (3.8)

    In addition, by differentiating (3.5) with respect to τ1, we obtain

    (1ατ1eλτ1)dλdτ1αλeλτ1=0. (3.9)

    From (3.9), we deduce that

    (dλdτ1)1=1ατ1eλτ1αλeλτ1=eλτ1αλτ1λ.

    Then,

    Re(dλdτ1)1λ=iω0=Re(eλτ1αλτ1λ)λ=iω0=1ω0αsin(ω0τ1).

    Combining with (3.6), we get

    Re(dλdτ1)1λ=iω0=1α2>0,

    which means

    dRe(λ)dτ1λ=iω0=Re(dλdτ1)λ=iω0>0.

    Based on the above analysis, we obtain the following Lemma 3.1.

    Lemma 3.1. Let τk1 be defined by (3.8).

    ( i ) If (H1) holds, the characteristic Eq.(3.5) has no imaginary roots;

    ( ii ) If (H2) holds, for τ1=τk1, the characteristic Eq.(3.5) has a pair of simple conjugate pure imaginary roots ±iω0, satisfying

    dRe(λ)dτ1λ=iω0>0.

    According to Lemma 3.1 and the Hopf bifurcation theorem for delay differential equations, we conclude, the stability of the equilibrium point x of (2.1) when τ1>0, τ2=0, in the following theorem.

    Theorem 3.1. Assume that (H2) holds and τ2=0. Let τ01 be defined by (3.8). Then

    ( i ) when τ1(0,τ01), the positive equilibrium point x of system (2.1) is locally asymptotically stable;

    ( ii ) there exists an enough small number ϵ>0 such that the positive equilibrium point x of system (2.1) is unstable when τ1(τ01,τ01+ϵ);

    ( iii ) when τ1=τ01, a Hopf bifurcation occurs at the positive equilibrium point x of system (2.1).


    3.4. The case τ1>0, τ2>0

    We now return to investigate Eq.(3.3) with τ1>0, τ2>0. In order to study the local stability of the equilibrium point x of (2.1), we regard τ2 as a bifurcation parameter, take τ1=τ1(0,), and discuss the distribution of the roots of the following characteristic equation

    λ+αeλτ1βeλτ2+c=0. (3.10)

    We first verified a result concerning the sign of the real parts of characteristic roots of (3.10) with τ1(0,τ01) in the following lemma.

    Lemma 3.2. If all roots of Eq.(3.5) have negative real parts for τ1(0,τ01), then there exists a τ2(τ1)>0 such that all roots of Eq.(3.10) have negative real parts when τ2<τ2(τ1).

    Proof. From Theorem 3.1 (ⅰ), we know that Eq.(3.5) has no root with nonnegative real part for τ1(0,τ01). Obviously, the left hand side of Eq.(3.10) is analytic in λ and τ2. Following Theorem 2.1 of Ruan and Wei [13], as the parameter τ2 varies, the sum of the multiplicity of zeros of the left hand side of Eq.(3.3) in the open right half-plane can change only if a zero appears on or crosses the imaginary axis.

    Since Eq.(3.10) with τ2=0 has no root with nonnegative real part when τ1(0,τ01), there exists a τ2(τ1) such that all roots of Eq.(3.10) have negative real parts when τ2<τ2(τ1).

    Lemma 3.3. If τ1(τ01,τ01+ϵ), then there exists a positive number τ2(τ1) such that Eq.(3.10) has at least one root with strictly positive real parts when τ2<τ2(τ1).

    Proof. Let λ be a zero of the left hand side of Eq.(3.10) satisfying Reλ0. Then from (3.10), we can obtain

    λ=αeλτ1+βeλτ2c,

    and

    |λ|αeλτ1+βeλτ2+cα+β+c,

    which shows that the zeros of the left hand side of Eq.(3.10) in the open right half plane are uniformly bounded. Furthermore, the left hand side of Eq.(3.10) is an elementary analytic function with regard to λ and τ2. Therefore, according to Corollary 2.4 in Ruan and Wei [13], as the parameter τ2 varies, the sum of the multiplicity of zeros of the left hand side of Eq.(3.3) in the open right half-plane can only change if a zero appears on or crosses the imaginary axis.

    Since Eq.(3.5) has at least one root with strictly positive real parts when τ1(τ01,τ01+ϵ) with ϵ>0, the left hand side of Eq.(3.10) with τ2=0 has no zeros appearing on the imaginary axis by Lemma 3.1 (ii). Thus, there must exist a positive number τ2(τ1) such that the sum of the orders of the zeros of the left hand side of Eq.(3.10) in the open right half plane is a fixed number when τ2<τ2(τ1), which means that Eq.(3.10) has at least one root with strictly positive real parts when τ2<τ2(τ1).

    From Lemmas 3.2 and 3.3, we have the following theorem concerning the stability of the equilibrium point x of (2.1) when τ1>0, τ2>0.

    Theorem 3.2. Let τ1=τ1(0,), τ2(0,).

    ( i ) If τ1(0,τ01), then there exists a number τ2(τ1)>0 such that the positive equilibrium point x of system (2.1) is locally asymptotically stable when τ2<τ2(τ1).

    ( ii ) If τ1(τ01,τ01+ϵ) with ϵ>0, then there exists a number τ2(τ1)>0 such that the positive equilibrium point x of system (2.1) is unstable when τ2<τ2(τ1).

    Proof. First we let τ1(0,τ01). Then from Lemma 3.2, there exists a τ2(τ1)>0 such that all roots of Eq.(3.10) have negative real parts when τ2<τ2(τ1), which shows that (ⅰ) follows.

    Secondly, we let τ1(τ01,τ01+ϵ) with ϵ>0. Then according to Lemma 3.3, that there exists a positive number τ2(τ1) such that Eq.(3.10) has at least one root with strictly positive real parts when τ2<τ2(τ1), which means that statement (ⅱ) is true.

    Next, we look for purely imaginary roots λ=±iv of Eq.(3.10) for τ1(0,), where v=v(τ2)>0. Substituting λ=iv into (3.10), we have

    iv+α(cos(vτ1)isin(vτ1))β(cos(vτ2)isin(vτ2))+c=0,

    then separating real and imaginary parts in the above equality, v satisfies

    βcos(vτ2)=αcos(vτ1)+c,βsin(vτ2)=αsin(vτ1)v. (3.11)

    Adding the squares to both sides of each equation in (3.10), we see that v must be a root of the following equation:

    v2+2cαcos(vτ1)2αvsin(vτ1)+α2+c2β2=0. (3.12)

    Let

    f(v)=v2+α2+c2β2,g(v)=2cαcos(vτ1)+2αvsin(vτ1). (3.13)

    Then Eq.(3.12) is equivalent to the following equation

    f(v)=g(v). (3.14)

    Now, we establish the sufficient conditions for the existence of positive solutions of Eq.(3.14) as follows.

    Proposition 3.2. If there exists a positive number ˜v satisfying

    f(˜v)<g(˜v), (3.15)

    then there exists at least one positive number v0(0,˜v) such that f(v0)=g(v0).

    Proof. Let F(v)=f(v)g(v). Obviously, F(v) is continuous on the interval [0, ˜v]. From (3.13) and (3.15), we see F(0)>0, F(˜v)<0. Then the conclusion is true according to intermediate value theorem.

    If the condition of Proposition 3.2 holds, then Eq.(3.12) has at least one positive solution v0. Thus we obtain from (3.11)

    τj2={1v0(arctan(αsin(v0τ1)v0αcos(v0τ1)+c)+2jπ), j=0,1,, if v0τj2 lies in the first quadrant;(2j+12)πv0, j=0,1,,if v0τj2 appears on the upper half imaginary axis;1v0(arctan(αsin(v0τ1)v0αcos(v0τ1)+c)+π+2jπ), j=0,1,, if v0τj2 lies in the second quadrant;(2j+1)πv0, j=0,1,,if v0τj2 appears on the negative real axis;1v0(arctan(αsin(v0τ1)v0αcos(v0τ1)+c)π+2jπ), j=1,2,, if v0τj2 lies in the third quadrant;(2j12)πv0, j=1,2,,if v0τj2 appears on the lower half imaginary axis;1v0(arctan(αsin(v0τ1)v0αcos(v0τ1)+c)+2jπ), j=1,2,,if v0τj2 lies in the fourth quadrant;2jπv0, j=1,2,,if v0τj2 appears on the positive real axis. (3.16)

    Based on the above analysis, we obtain the following theorem.

    Theorem 3.3. Let τj2, j=0,1,2, be defined by (3.16). For τ1=τ1(0,),τ2=τj2, j=0,1,2,, the characteristic Eq.(3.3) has a pair of simple conjugate purely imaginary roots ±iv0 satisfying

    sign{dReλdτ2|λ=iv0}=sign(g(v0,τ1)),

    where

    g(v0,τ1)=v0(1ατ1cos(v0τ1))(1+τ1c)αsin(v0τ1)0. (3.17)

    Proof. It is easy to see from (3.11)-(3.14) that the characteristic Eq.(3.7) has a pair of simple conjugate pure imaginary roots ±iv0. Further, by differentiating (3.10) with respect to τ2, we deduce that

    (dλdτ2)1=eλτ2λβ+ατ1λβeλ(τ1τ2)τ2λ.

    Then

    Re(dλdτ2)1λ=iv0=ατ1v0βsin(v0τ1)cos(v0τ2)+(ατ1v0βcos(v0τ1)1v0β)sin(v0τ2).

    Combining with (3.12), we simplify the above equality and obtain

    Re(dλdτ2)1λ=iv0=1v0β2[v0(1ατ1cos(v0τ1))(1+τ1c)αsin(v0τ1)]=1v0β2g(v0,τ1).

    Based on (3.17) and (3.18), we conclude that

    sign{dReλdτ2|λ=iv0}=sign{Re(dλdτ2)1λ=iv0}=sign(g(v0,τ1)).

    Remark 3.1. From Theorem 3.3, it is followed that dReλdτ2λ=iv0>0 if g(v0,τ1)>0, and dReλdτ2λ=iv0<0 if g(v0,τ1)<0.

    Remark 3.2. We can also make similar discussions if we let τ1=0, and regard τ2 as a parameter.


    4. Numerical simulations

    In this section, we illustrate the different stability results obtained in the previous sections, mainly in Theorems 3.1 and 3.2. We also focus on periodic solutions appearing through a Hopf bifurcation. Without loss of generality, we take time unit as an hour, and let initial condition be x0=0.1.

    Assuming k1=4.5, k2=0.15, c=1, M=0.5, then system (2.1) becomes

    dxdt=4.50.5+x2(tτ1)+0.15x(tτ2)x. (4.1)

    Firstly, we illustrate the stability and Hopf bifurcation of the equilibrium of system (4.1) when τ1>0, τ2=0. In this case, system (4.1) is reduced to

    dxdt=4.50.5+x2(tτ1)0.85x. (4.2)

    From (4.2), we get a positive equilibrium point x=1.6473. From (3.2), we calculate α=1.4355, cβ=0.85, satisfying (H2). Based on (3.8), we have ω0=α2(cβ)2=1.1568, τ01=1.9057. Then according to Theorem 3.1 we see that the equilibrium point x=1.6473 of system (4.2) is locally asymptotically stable when τ1(0,τ01). When τ1 is gradually increasing and exceeds the critical value τ1=τ01=1.9057, system (4.2) changes from stability to instability at the positive equilibrium point and thus a Hopf bifurcation occurs. Then we stimulate the change of system (4.2) when τ1=0.5, τ1=1.5 and τ1=2.0, τ1=4.0 respectively, and find the simulation results are in accordance with Theorem 3.1 (see Fig. 2).

    Figure 2. Stability and Hopf bifurcation of system (4.1) for different τ1[0,) when τ2=0. The equilibrium point x of (4.2) is locally asymptotically stable when τ1=0.5 in figure (a) and τ1=1.5 in figure (b), respectively. The equilibrium point x of (4.2) losts its stability and stable bifurcation periodic solutions appear when τ1=2.0 in figure (c) and τ1=4.0 in figure (d), respectively.

    Secondly, we illustrate the stability and Hopf bifurcation of the equilibrium of system (4.1) according to Theorem 3.2 when τ1>0, τ2>0. Now we take τ1=1.85(0,τ01). Then according to Theorem 3.2 (i) and (3.16), we can find a critical value τ2(τ1)=4.05 such that the equilibrium point x of (4.1) is locally asymptotically stable when 0<τ2<4.05. We simulate the change of system (4.1) when 0<τ2<4.05 and find the simulation results are in accordance with the theoretical analysis results when τ2=0.5, τ2=1.5, τ2=3.5, τ2=4, respectively (see Fig. 3).

    Figure 3. Stability of system (4.1) with different τ2 when τ1=1.85(0,τ01) and 0<τ2<4.05. The equilibrium point x of (4.1) is locally asymptotically stable when τ2=0.5 in figure (a), τ2=1.5 in figure (b), τ2=3.5 in figure (c), τ2=4 in figure (d), respectively.

    Next, we take τ1=2.8(τ01,). According to Theorem 3.2 (ii) and (3.16), we can find a critical value τ2(τ1)=2.1 such that the equilibrium point x of (4.1) is unstable when 0<τ2<2.1. Then we simulate the change of system (4.1) when 0<τ2<2.1 and find the simulation results correspond to the theoretical analysis results when τ2=0.5, τ2=1, τ2=1.5, τ2=2, respectively (see Fig. 4).

    Figure 4. Instability of system (4.1) with different τ2 when τ1=2.8(τ01,) and 0<τ2<2.1. The equilibrium point x of (4.1) is unstable when τ2=0.5 in figure (a), τ2=1 in figure (b), τ2=1.5 in figure (c), τ2=2 in figure (d), respectively.

    From Fig. 3 and Fig. 4, it is seen that we just fix τ1=1.85 or τ1=2.8, and consider τ2 impacting on the stability of system (4.1). In fact, based on (3.16), we can obtain a series of corresponding critical value of τ2 for every given τ1. Thus, we have the bifurcation diagram of the parameters τ1 and τ2 (see Fig. 5).

    Figure 5. Bifurcation diagram of (τ1,τ2) for system (4.1). S denotes stable regions, US denotes oscillating regions. The black solid line is made up of critical bifurcation points for (τ1,τ2), the rest solid lines with different colours are lines consisting of critical bifurcation points when τ2 pluses different period respectively, and the marked six different points represent different values of (τ1,τ2).

    From Fig. 5, we can see that system (4.1) turns its instability into stability, or turns its stability into instability on both sides of the bifurcation line. Especially, when we fix τ1=2.4 and take τ2=0.5, 2, 5, 9, 12, 15.5 which correspond to the six different points in Fig. 5, we simulate the stability of system 4.1 (see Fig. 6).

    Figure 6. Stability of system (4.1) with different τ2 when τ1=2.4(τ01,) and τ2>0. The equilibrium point x of (4.1) is locally asymptotically stable when τ2=2 in figure (b), τ2=9 in figure (d), τ2=15.5 in figure (f), respectively, it is unstable when τ2=0.5 in figure (a), τ2=5 in figure (c), τ2=12 in figure (e), respectively.

    Furthermore, in order to verify the theoretical result of the bifurcation Fig. 5, we use numerical simulations and obtain all the values of τ1,τ2 which make system (4.1) generate oscillations (see Fig. 7).

    Figure 7. Oscillating range of (τ1,τ2) for system (4.1). Black regions represent oscillating solutions with periods for system (4.1) when (τ1,τ2) locates in the black region.

    From Fig. 7, we can see that the equilibrium point x of system (4.1) is unstable when (τ1,τ2) locates in the black region, that is to say, system (4.1) generates oscillating solutions. The oscillating range of (τ1,τ2) is in accordance with the result of theoretical calculations in Fig. 5.


    5. Discussion

    In the previous researched circadian clock models with a time delay, it is found that the period of the model monotonously increases with the increase of the delay. In this paper, there are two different delays in our discussing biological clock model. To analyse the characteristic equation with two delays, we first concentrated on the case when one of the delays, τ2 equals zero and obtained a critical value for the delay τ1 : when τ1<τ01 all roots of the characteristic equation have negative parts and when τ1=τ01 purely imaginary roots appear. Then we assumed that τ1=τ1<τ01 and considered the delay τ2 as a parameter. It showed that there exists a τ2(τ1) such that all roots of the characteristic equation have negative real parts when τ2(0,τ2(τ1)). Finally, we assumed that τ1=τ1(τ01,τ01+ϵ) with ϵ>0 and considered the delay τ2 as a parameter. We concluded that there exists a τ2(τ1) such that the characteristic equation has at least one root with strictly positive real parts when τ2(0,τ2(τ1)). Consequently, we obtained stability and instability results for the mammalian circadian clock model with two independent delays respectively. The numerical results in Fig. 5 also indicate the existence of a Hopf bifurcation that leads to the emergence of periodic solutions.

    When discussing a mammalian circadian clock model with two delays, we found delays can affect the period of the model. When we first fix τ2=29, assuming that τ1 ranges from 0 to 12, then fix τ1=10, assuming that τ2 ranges from 0 to 35, we numerically simulated the influence of delays on the period of model (4.1)(see Fig. 8).

    Figure 8. The effect of time delays on the period of system (4.1). In figure (a), we fix τ2=29, the black solid line represents the relation between τ1 and the period. In figure (b), we fix τ1=10, the black solid line represents τ2 and the period.

    From Fig. 8(a), we can see the period is monotonously increasing with the increase of time delay τ1 when τ2=29, which is in accord with that of a biological clock model with a single delay. However, from Fig. 8(b), we find that the period emerges non-monotonic regular variety with the increase of time delay τ2 when τ1=10.

    All in all, it is seen from Fig. 8 that τ1 determines the length of time. When τ1 is fixed, τ2 is changed, the period can only change in a certain range, which shows that the positive feedback between CRY1 and REV-ERBα in mammalian circadian clocks can benefit the robustness of the period length.


    Acknowledgments

    We would like to thank the reviewers and the editor for their valuable suggestions.


    [1] [ M. Adimy,F. Crauste,S. G. Ruan, Periodic oscillations in leukopoiesis models with two delays, Journal of Theoretical Biology, 242 (2006): 288-299.
    [2] [ M. P. Antoch,V. Y. Gorbacheva,O. Vykhovanets,A. Y. Nikitin, Disruption of the circadian clock due to the Clock mutation has discrete effects on aging and carcinogenesis, Cell Cycle, 7 (2008): 1197-1204.
    [3] [ D. B. Forger,C. S. Peskin, A detailed predictive model of the mammalian circadian clock, Proceedings of the National Academy of Sciences of the United States of America, 100 (2003): 14806-14811.
    [4] [ A. Goldbeter, A model for circadian oscillations in the Drosophila period protein (PER), Proceedings. Biological sciences / The Royal Society, 261 (1995): 319-324.
    [5] [ A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, 1996.
    [6] [ C. I. Hong,J. J. Tyson, A proposal for temperature compensation of the circadian rhythm in Drosophila based on dimerization of the per protein, Chronobiology International, 14 (1997): 521-529.
    [7] [ J. K. Kim and D. B. Forger, A mechanism for robust circadian timekeeping via stoichiometric balance, Molecular Systems Biology, 8 (2012), 630.
    [8] [ R. V. Kondratov,A. A. Kondratova,V. Y. Gorbacheva, Early aging and age-related pathologies in mice deficient in BMAL1, the core component of the circadian clock, Genes & Developoment, 20 (2006): 1868-1873.
    [9] [ C. C. Lee, Tumor suppression by the mammalian Period genes, Cancer Causes Control, 17 (2006): 525-530 [PubMed: 16596306].
    [10] [ J. C. Leloup,A. Goldbeter, Toward a detailed computational model for the mammalian circadian clock: Sensitivity analysis and multiplicity of oscillatory mechanisms, J. Theoret. Biol., 230 (2004): 541-562.
    [11] [ P. L. Lowrey,J. S. Takahashi, Mammalian circadian biology: elucidating genome-wide levels of temporal organization, Annual Review of Genomics and Human Genetics, 5 (2004): 407-441.
    [12] [ H. P. Mirsky,A. C. Liu,D. K. Welsh,S. A. Kay,F. J. Doyle, A model of the cell-autonomous mammalian circadian clock, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009): 11107-11112.
    [13] [ S. G. Ruan,J. J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 10 (2003): 863-874.
    [14] [ F. A. Scheer,M. F. Hilton,C. S. Mantzoros,S. A. Shea, Adverse metabolic and cardiovascular consequences of circadian misalignment, Proceedings of the National Academy of Sciences of the United States of America, 106 (2009): 4453-4458.
    [15] [ 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 Journal, 77 (1999): 2411-2417.
    [16] [ M. Ukai-Tadenuma,R. G. Yamada,H. Xu,J. A. Ripperger,A. C. Liu,H. R. Ueda, Delay in feedback repression by cryptochrome 1 is required for circadian clock function, Cell, 144 (2011): 268-281.
    [17] [ J. Yan,G. Shi,Z. Zhang,X. Wu,Z. Liu,L. Xing,Z. Qu,Z. Dong,L. Yang,Y. Xu, An intensity ratio of interlocking loops determines circadian period length, Nucleic Acids Research, 42 (2014): 10278-10287.
    [18] [ X. Yang,M. Downes,R. T. Yu,A. L. Bookout,W. He,M. Straume,D. J. Mangelsdorf,R. M. Evans, Nuclear receptor expression links the circadian clock to metabolism, Cell, 126 (2006): 801-810.
    [19] [ W. Yu,M. Nomura,M. Ikeda, Interactivating feedback loops within the mammalian clock: BMAL1 is negatively autoregulated and upregulated by CRY1, CRY2, and PER2, Biochemical and Biophysical Research Communications, 290 (2002): 933-941.
    [20] [ E. E. Zhang,S. A. Kay, Clocks not winding down: Unravelling circadian networks, Nature Reviews Molecular Cell Biology, 11 (2010): 764-776.
  • This article has been cited by:

    1. X. D. Chen, J. Wang, M. Z. Zhao, F. Zhao, Characterization and expression analysis of circadian clock genes in the diploid woodland strawberry Fragaria vesca, 2018, 62, 00063134, 451, 10.1007/s10535-018-0793-4
    2. Seok Joo Chae, Dae Wook Kim, Seunggyu Lee, Jae Kyoung Kim, Spatially coordinated collective phosphorylation filters spatiotemporal noises for precise circadian timekeeping, 2023, 26, 25890042, 106554, 10.1016/j.isci.2023.106554
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3393) PDF downloads(554) Cited by(2)

Article outline

Figures and Tables

Figures(8)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog