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
[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 |
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-erb
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-erb
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.
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-erb
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
dxdt=k1M+x2(t−τ1)+k2x(t−τ2)−cx(t), | (2.1) |
where
Obviously, system (2.1) has a unique continuous solution
dx(t0)dt=k1M+x2(t0−τ1)+k2x(t0−τ2)>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
(k2−c)(x∗)3+M(k2−c)x∗+k1=0. | (2.2) |
Evidently,
Lemma 2.1. (Shengjin's formulas) Consider a variable cubic equationt
ax3+bx2+cx+d=0, | (2.3) |
where
(
(
(
(
Using Lemma 2.1, in Eq.(2.2), we calculate
A=−3M(k2−c)2,B=−9k1(k2−c),C=M2(k2−c)2, |
and obtain
Δ=B2−4AC=(k2−c)2(81k21+12M3(k2−d)2)>0, |
then it follows from Lemma 2.1 (ii) that Eq.(2.2) has a real root
Y1=32(k2−c)2(9k1+√81k21+12M3(k2−c)2), |
Y2=32(k2−c)2(9k1−√81k21+12M3(k2−c)2). |
In addition, we can easily see that
This result is summarized in the following proposition.
Proposition 2.1. Assume that
In this section, we concentrate on the stability of the nontrivial equilibrium point
We first start to linearize system (2.1) around
Take
dydt=−2k1x∗(M+x∗2)2y(t−τ1)+k2y(t−τ2)−cy(t), | (3.1) |
Denote
α=2k1x∗(M+x∗2)2,β=k2. | (3.2) |
The characteristic equation associated with (3.1) is given by
λ+αe−λτ1−βe−λτ2+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
Because of the presence of two different delays,
Assume that
λ+α−β+c=0. | (3.4) |
Obviously, the only eigenvalue of (3.4) is
Proposition 3.1. Assume that
We now consider the case
λ+αe−λτ1−β+c=0. | (3.5) |
When
If
αcos(ωτ1)=β−c,αsin(ωτ1)=ω. | (3.6) |
One can notice that if
Adding the squares of both hand sides of Eq.(3.6), we see that
ω2−α2+(c−β)2=0. | (3.7) |
Set
The function
τk1=1ω0(arctan(ω0β−c)+(2k+1)π), k=0,1,2,⋯. | (3.8) |
In addition, by differentiating (3.5) with respect to
(1−ατ1e−λτ1)dλdτ1−αλe−λτ1=0. | (3.9) |
From (3.9), we deduce that
(dλdτ1)−1=1−ατ1e−λτ1αλe−λτ1=eλτ1αλ−τ1λ. |
Then,
Re(dλdτ1)−1λ=iω0=Re(eλτ1αλ−τ1λ)λ=iω0=1ω0αsin(ω0τ1). |
Combining with (3.6), we get
Re(dλdτ1)−1λ=iω0=1α2>0, |
which means
dRe(λ)dτ1∣λ=iω0=Re(dλdτ1)λ=iω0>0. |
Based on the above analysis, we obtain the following Lemma 3.1.
Lemma 3.1. Let
(
(
dRe(λ)dτ1∣λ=iω0>0. |
According to Lemma 3.1 and the Hopf bifurcation theorem for delay differential equations, we conclude, the stability of the equilibrium point
Theorem 3.1. Assume that
(
(
(
We now return to investigate Eq.(3.3) with
λ+αe−λτ∗1−βe−λτ2+c=0. | (3.10) |
We first verified a result concerning the sign of the real parts of characteristic roots of (3.10) with
Lemma 3.2. If all roots of Eq.(3.5) have negative real parts for
Proof. From Theorem 3.1 (ⅰ), we know that Eq.(3.5) has no root with nonnegative real part for
Since Eq.(3.10) with
Lemma 3.3. If
Proof. Let
λ=−αe−λτ∗1+βe−λτ2−c, |
and
|λ|≤αe−λτ∗1+βe−λτ2+c≤α+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
Since Eq.(3.5) has at least one root with strictly positive real parts when
From Lemmas 3.2 and 3.3, we have the following theorem concerning the stability of the equilibrium point
Theorem 3.2. Let
(
(
Proof. First we let
Secondly, we let
Next, we look for purely imaginary roots
iv+α(cos(vτ∗1)−isin(vτ∗1))−β(cos(vτ2)−isin(vτ2))+c=0, |
then separating real and imaginary parts in the above equality,
βcos(vτ2)=αcos(vτ∗1)+c,βsin(vτ2)=αsin(vτ∗1)−v. | (3.11) |
Adding the squares to both sides of each equation in (3.10), we see that
v2+2cαcos(vτ∗1)−2αvsin(vτ∗1)+α2+c2−β2=0. | (3.12) |
Let
f(v)=v2+α2+c2−β2,g(v)=−2cαcos(vτ∗1)+2αvsin(vτ∗1). | (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
f(˜v)<g(˜v), | (3.15) |
then there exists at least one positive number
Proof. Let
If the condition of Proposition 3.2 holds, then Eq.(3.12) has at least one positive solution
τj2={1v0(arctan(αsin(v0τ∗1)−v0αcos(v0τ∗1)+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(v0τ∗1)−v0αcos(v0τ∗1)+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(v0τ∗1)−v0αcos(v0τ∗1)+c)−π+2jπ), j=1,2,…, if v0τj2 lies in the third quadrant;(2j−12)πv0, j=1,2,…,if v0τj2 appears on the lower half imaginary axis;1v0(arctan(αsin(v0τ∗1)−v0αcos(v0τ∗1)+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
sign{dReλdτ2|λ=iv0}=sign(g(v0,τ∗1)), |
where
g(v0,τ∗1)=v0(1−ατ∗1cos(v0τ∗1))−(1+τ∗1c)αsin(v0τ∗1)≠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
(dλdτ2)−1=−eλτ2λβ+ατ∗1λβe−λ(τ∗1−τ2)−τ2λ. |
Then
Re(dλdτ2)−1λ=iv0=−ατ∗1v0βsin(v0τ∗1)cos(v0τ2)+(ατ∗1v0βcos(v0τ∗1)−1v0β)sin(v0τ2). |
Combining with (3.12), we simplify the above equality and obtain
Re(dλdτ2)−1λ=iv0=1v0β2[v0(1−ατ∗1cos(v0τ∗1))−(1+τ∗1c)αsin(v0τ∗1)]=1v0β2g(v0,τ∗1). |
Based on (3.17) and (3.18), we conclude that
sign{dReλdτ2|λ=iv0}=sign{Re(dλdτ2)−1λ=iv0}=sign(g(v0,τ∗1)). |
Remark 3.1. From Theorem 3.3, it is followed that
Remark 3.2. We can also make similar discussions if we let
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
Assuming
dxdt=4.50.5+x2(t−τ1)+0.15x(t−τ2)−x. | (4.1) |
Firstly, we illustrate the stability and Hopf bifurcation of the equilibrium of system (4.1) when
dxdt=4.50.5+x2(t−τ1)−0.85x. | (4.2) |
From (4.2), we get a positive equilibrium point
Secondly, we illustrate the stability and Hopf bifurcation of the equilibrium of system (4.1) according to Theorem 3.2 when
Next, we take
From Fig. 3 and Fig. 4, it is seen that we just fix
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
Furthermore, in order to verify the theoretical result of the bifurcation Fig. 5, we use numerical simulations and obtain all the values of
From Fig. 7, we can see that the equilibrium point
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,
When discussing a mammalian circadian clock model with two delays, we found delays can affect the period of the model. When we first fix
From Fig. 8(a), we can see the period is monotonously increasing with the increase of time delay
All in all, it is seen from Fig. 8 that
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. |
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 |