# Mathematical Models of the Arabidopsis Circadian Oscillator

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## Abstract

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## 1. Introduction

## 2. Chronological Development of Arabidopsis Circadian Clock Models: Expansion Phase

_{mod}(see Figure 3) and indirectly via repression of PRR9; LHY then feeds back into TOC1. This network motif buffers noise in input stimuli, acting as a persistence detector [12].

_{mod}quickly elevates mRNA levels again through the aforementioned C1-FFL. However, the mechanism governing this switch from inhibition to activation is not explained.

_{mod}.

## 3. Recent Development of Arabidopsis Circadian Clock Models: Reduction Phase

## 4. Current Challenges and Overview

#### 4.1. Spatial Models

#### 4.2. Effects of Light on the Clock

#### 4.3. Effects of Temperature on the Clock

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bechtel, W.; Abrahamsen, A. Dynamic Mechanistic Explanation: Computational Modeling of Circadian Rhythms as an Exemplar for Cognitive Science. Stud. Hist. Philos. Sci. Part A
**2010**, 41, 321–333. [Google Scholar] [CrossRef] - Kauffman, S.A. Articulation of Parts Explanation in Biology and the Rational Search for Them. PSA Proc. Bienn. Meet. Philos. Sci. Assoc.
**1970**, 1970, 257–272. [Google Scholar] [CrossRef] - Alon, U. An Introduction to Systems Biology: Design Principles of Biological Circuits, 2nd ed.; CRC Press, Taylor & Francis Group: Boca Raton, FL, USA, 2020. [Google Scholar]
- Jaeger, J.; Monk, N. Dynamical Modules in Metabolism, Cell and Developmental Biology. Interface Focus
**2021**, 11, 20210011. [Google Scholar] [CrossRef] [PubMed] - Rodriguez-Maroto, G.; Catalán, P.; Nieto, C.; Prat, S.; Ares, S. Mathematical Modeling of Photo-and Thermomorphogenesis in Plants. In Thermomorphogenesis: Methods and Protocols; Springer: New York, NY, USA, 2024; pp. 247–261. [Google Scholar]
- Bujdoso, N.; Davis, S.J. Mathematical Modeling of an Oscillating Gene Circuit to Unravel the Circadian Clock Network of Arabidopsis Thaliana. Front. Plant Sci.
**2013**, 4, 3. [Google Scholar] [CrossRef] [PubMed] - Locke, J.C.W.; Millar, A.J.; Turner, M.S. Modelling Genetic Networks with Noisy and Varied Experimental Data: The Circadian Clock in Arabidopsis Thaliana. J. Theor. Biol.
**2005**, 234, 383–393. [Google Scholar] [CrossRef] - Locke, J.C.W.; Southern, M.M.; Kozma-Bognár, L.; Hibberd, V.; Brown, P.E.; Turner, M.S.; Millar, A.J. Extension of a Genetic Network Model by Iterative Experimentation and Mathematical Analysis. Mol. Syst. Biol.
**2005**, 1, 2005.0013. [Google Scholar] [CrossRef] - Zeilinger, M.N.; Farré, E.M.; Taylor, S.R.; Kay, S.A.; Doyle, F.J. A Novel Computational Model of the Circadian Clock in Arabidopsis That Incorporates PRR7 and PRR9. Mol. Syst. Biol.
**2006**, 2, 58. [Google Scholar] [CrossRef] - Locke, J.C.W.; Kozma-Bognár, L.; Gould, P.D.; Fehér, B.; Kevei, É.; Nagy, F.; Turner, M.S.; Hall, A.; Millar, A.J. Experimental Validation of a Predicted Feedback Loop in the Multi-oscillator Clock of Arabidopsis thaliana. Mol. Syst. Biol.
**2006**, 2, 59. [Google Scholar] [CrossRef] - Pokhilko, A.; Hodge, S.K.; Stratford, K.; Knox, K.; Edwards, K.D.; Thomson, A.W.; Mizuno, T.; Millar, A.J. Data Assimilation Constrains New Connections and Components in a Complex, Eukaryotic Circadian Clock Model. Mol. Syst. Biol.
**2010**, 6, 416. [Google Scholar] [CrossRef] - Mangan, S.; Zaslaver, A.; Alon, U. The Coherent Feedforward Loop Serves as a Sign-sensitive Delay Element in Transcription Networks. J. Mol. Biol.
**2003**, 334, 197–204. [Google Scholar] [CrossRef] - McWatters, H.G.; Kolmos, E.; Hall, A.; Doyle, M.R.; Amasino, R.M.; Gyula, P.; Nagy, F.; Millar, A.J.; Davis, S.J. ELF4 Is Required for Oscillatory Properties of the Circadian Clock. Plant Physiol.
**2007**, 144, 391–401. [Google Scholar] [CrossRef] [PubMed] - Kolmos, E.; Nowak, M.; Werner, M.; Fischer, K.; Schwarz, G.; Mathews, S.; Schoof, H.; Nagy, F.; Bujnicki, J.M.; Davis, S.J. Integrating ELF4 into the Circadian System through Combined Structural and Functional Studies. HFSP J.
**2009**, 3, 350–366. [Google Scholar] [CrossRef] [PubMed] - Nusinow, D.A.; Helfer, A.; Hamilton, E.E.; King, J.J.; Imaizumi, T.; Schultz, T.F.; Farré, E.M.; Kay, S.A. The ELF4–ELF3–LUX Complex Links the Circadian Clock to Diurnal Control of Hypocotyl Growth. Nature
**2011**, 475, 398–402. [Google Scholar] [CrossRef] - Herrero, E.; Kolmos, E.; Bujdoso, N.; Yuan, Y.; Wang, M.; Berns, M.C.; Uhlworm, H.; Coupland, G.; Saini, R.; Jaskolski, M.; et al. EARLY FLOWERING4 Recruitment of EARLY FLOWERING3 in the Nucleus Sustains the Arabidopsis Circadian Clock. Plant Cell
**2012**, 24, 428–443. [Google Scholar] [CrossRef] - Pokhilko, A.; Fernández, A.P.; Edwards, K.D.; Southern, M.M.; Halliday, K.J.; Millar, A.J. The Clock Gene Circuit in Arabidopsis Includes a Repressilator with Additional Feedback Loops. Mol. Syst. Biol.
**2012**, 8, 574. [Google Scholar] [CrossRef] - Nieto, C.; Catalán, P.; Luengo, L.M.; Legris, M.; López-Salmerón, V.; Davière, J.M.; Casal, J.J.; Ares, S.; Prat, S. COP1 Dynamics Integrate Conflicting Seasonal Light and Thermal Cues in the Control of Arabidopsis Elongation. Sci. Adv.
**2022**, 8, eabp8412. [Google Scholar] [CrossRef] [PubMed] - Elowitz, M.B.; Leibler, S. A synthetic oscillatory network of transcriptional regulators. Nature
**2000**, 403, 335–338. [Google Scholar] [CrossRef] [PubMed] - Pokhilko, A.; Mas, P.; Millar, A.J. Modelling the Widespread Effects of TOC1 Signalling on the Plant Circadian Clock and Its Outputs. BMC Syst. Biol.
**2013**, 7, 23. [Google Scholar] [CrossRef] - Fogelmark, K.; Troein, C. Rethinking Transcriptional Activation in the Arabidopsis Circadian Clock. PLoS Comput. Biol.
**2014**, 10, e1003705. [Google Scholar] [CrossRef] - Kim, Y.; Han, S.; Yeom, M.; Kim, H.; Lim, J.; Cha, J.Y.; Kim, W.Y.; Somers, D.E.; Putterill, J.; Nam, H.G.; et al. Balanced Nucleocytosolic Partitioning Defines a Spatial Network to Coordinate Circadian Physiology in Plants. Dev. Cell
**2013**, 26, 73–85. [Google Scholar] [CrossRef] - Mangan, S.; Alon, U. Structure and Function of the Feed-Forward Loop Network Motif. Proc. Natl. Acad. Sci. USA
**2003**, 100, 11980–11985. [Google Scholar] [CrossRef] [PubMed] - Gutenkunst, R.N.; Waterfall, J.J.; Casey, F.P.; Brown, K.S.; Myers, C.R.; Sethna, J.P. Universally sloppy parameter sensitivities in systems biology models. PLoS Comput. Biol.
**2007**, 3, e189. [Google Scholar] [CrossRef] [PubMed] - Transtrum, M.K.; Machta, B.B.; Brown, K.S.; Daniels, B.C.; Myers, C.R.; Sethna, J.P. Perspective: Sloppiness and emergent theories in physics, biology, and beyond. J. Chem. Phys.
**2015**, 143, 010901. [Google Scholar] [CrossRef] [PubMed] - Foo, M.; Somers, D.E.; Kim, P.J. Kernel Architecture of the Genetic Circuitry of the Arabidopsis Circadian System. PLoS Comput. Biol.
**2016**, 12, e1004748. [Google Scholar] [CrossRef] - Davidson, E.H.; Erwin, D.H. Gene Regulatory Networks and the Evolution of Animal Body Plans. Science
**2006**, 311, 796–800. [Google Scholar] [CrossRef] [PubMed] - Kim, J.; Park, S.M.; Cho, K.H. Discovery of a Kernel for Controlling Biomolecular Regulatory Networks. Sci. Rep.
**2013**, 3, 2223. [Google Scholar] [CrossRef] [PubMed] - Perez-Carrasco, R.; Barnes, C.P.; Schaerli, Y.; Isalan, M.; Briscoe, J.; Page, K.M. Combining a toggle switch and a repressilator within the AC-DC circuit generates distinct dynamical behaviors. Cell Syst.
**2018**, 6, 521–530. [Google Scholar] [CrossRef] [PubMed] - De Caluwé, J.; Xiao, Q.; Hermans, C.; Verbruggen, N.; Leloup, J.C.; Gonze, D. A Compact Model for the Complex Plant Circadian Clock. Front. Plant Sci.
**2016**, 7, 74. [Google Scholar] [CrossRef] [PubMed] - De Melo, J.R.F.; Gutsch, A.; Caluwé, T.D.; Leloup, J.C.; Gonze, D.; Hermans, C.; Webb, A.A.R.; Verbruggen, N. Magnesium Maintains the Length of the Circadian Period in Arabidopsis. Plant Physiol.
**2021**, 185, 519–532. [Google Scholar] [CrossRef] - Ohara, T.; Hearn, T.J.; Webb, A.A.; Satake, A. Gene Regulatory Network Models in Response to Sugars in the Plant Circadian System. J. Theor. Biol.
**2018**, 457, 137–151. [Google Scholar] [CrossRef] - Ohara, T.; Satake, A. Photosynthetic Entrainment of the Circadian Clock Facilitates Plant Growth under Environmental Fluctuations: Perspectives from an Integrated Model of Phase Oscillator and Phloem Transportation. Front. Plant Sci.
**2017**, 8, 1859. [Google Scholar] [CrossRef] [PubMed] - Zhang, R.; Gonze, D. Stochastic Simulation of a Model for Circadian Rhythms in Plants. J. Theor. Biol.
**2021**, 527, 110790. [Google Scholar] [CrossRef] [PubMed] - Joanito, I.; Chu, J.W.; Wu, S.H.; Hsu, C.P. An Incoherent Feed-Forward Loop Switches the Arabidopsis Clock Rapidly between Two Hysteretic States. Sci. Rep.
**2018**, 8, 13944. [Google Scholar] [CrossRef] [PubMed] - Tokuda, I.T.; Akman, O.E.; Locke, J.C.W. Reducing the Complexity of Mathematical Models for the Plant Circadian Clock by Distributed Delays. J. Theor. Biol.
**2019**, 463, 155–166. [Google Scholar] [CrossRef] [PubMed] - Davis, W.; Endo, M.; Locke, J.C.W. Spatially Specific Mechanisms and Functions of the Plant Circadian Clock. Plant Physiol.
**2022**, 190, 938–951. [Google Scholar] [CrossRef] [PubMed] - Nohales, M.A. Spatial Organization and Coordination of the Plant Circadian System. Genes
**2021**, 12, 442. [Google Scholar] [CrossRef] [PubMed] - Gould, P.D.; Domijan, M.; Greenwood, M.; Tokuda, I.T.; Rees, H.; Kozma-Bognar, L.; Hall, A.J.; Locke, J.C. Coordination of Robust Single Cell Rhythms in the Arabidopsis Circadian Clock via Spatial Waves of Gene Expression. eLife
**2018**, 7, e31700. [Google Scholar] [CrossRef] [PubMed] - Greenwood, M.; Tokuda, I.T.; Locke, J.C.W. A Spatial Model of the Plant Circadian Clock Reveals Design Principles for Coordinated Timing. Mol. Syst. Biol.
**2022**, 18, e10140. [Google Scholar] [CrossRef] [PubMed] - Juarrero, A. Context Changes Everything: How Constraints Create Coherence; The MIT Press: Cambridge, MA, USA, 2023. [Google Scholar] [CrossRef]
- Galvão, V.C.; Fankhauser, C. Sensing the light environment in plants: Photoreceptors and early signaling steps. Curr. Opin. Neurobiol.
**2015**, 34, 46–53. [Google Scholar] [CrossRef] - Ohara, T.; Fukuda, H.; Tokuda, I.T. An Extended Mathematical Model for Reproducing the Phase Response of Arabidopsis Thaliana under Various Light Conditions. J. Theor. Biol.
**2015**, 382, 337–344. [Google Scholar] [CrossRef] - Pay, M.L.; Kim, D.W.; Somers, D.E.; Kim, J.K.; Foo, M. Modelling of Plant Circadian Clock for Characterizing Hypocotyl Growth under Different Light Quality Conditions. In Silico Plants
**2022**, 4, diac001. [Google Scholar] [CrossRef] [PubMed] - Pay, M.L.; Christensen, J.; He, F.; Roden, L.; Ahmed, H.; Foo, M. An Extended Plant Circadian Clock Model for Characterising Flowering Time under Different Light Quality Conditions. In Proceedings of the 2022 22nd International Conference on Control, Automation and Systems (ICCAS), Jeju, Republic of Korea, 27 November–1 December 2022; pp. 1848–1853. [Google Scholar] [CrossRef]
- Huang, T.; Shui, Y.; Wu, Y.; Hou, X.; You, X. Red Light Resets the Expression Pattern, Phase, and Period of the Circadian Clock in Plants: A Computational Approach. Biology
**2022**, 11, 1479. [Google Scholar] [CrossRef] [PubMed] - De Caluwé, J.; De Melo, J.R.F.; Tosenberger, A.; Hermans, C.; Verbruggen, N.; Leloup, J.C.; Gonze, D. Modeling the Photoperiodic Entrainment of the Plant Circadian Clock. J. Theor. Biol.
**2017**, 420, 220–231. [Google Scholar] [CrossRef] [PubMed] - Schmal, C.; Myung, J.; Herzel, H.; Bordyugov, G. A Theoretical Study on Seasonality. Front. Neurol.
**2015**, 6, 94. [Google Scholar] [CrossRef] [PubMed] - Gould, P.D.; Ugarte, N.; Domijan, M.; Costa, M.; Foreman, J.; MacGregor, D.; Rose, K.; Griffiths, J.; Millar, A.J.; Finkenstädt, B.; et al. Network Balance via CRY Signalling Controls the Arabidopsis Circadian Clock over Ambient Temperatures. Mol. Syst. Biol.
**2013**, 9, 650. [Google Scholar] [CrossRef] - Avello, P.A.; Davis, S.J.; Ronald, J.; Pitchford, J.W. Heat the Clock: Entrainment and Compensation in Arabidopsis Circadian Rhythms. J. Circadian Rhythm.
**2019**, 17, 5. [Google Scholar] [CrossRef] [PubMed] - Avello, P.; Davis, S.J.; Pitchford, J.W. Temperature Robustness in Arabidopsis Circadian Clock Models Is Facilitated by Repressive Interactions, Autoregulation, and Three-Node Feedbacks. J. Theor. Biol.
**2021**, 509, 110495. [Google Scholar] [CrossRef] - Yuan, L.; Avello, P.; Zhu, Z.; Lock, S.C.; McCarthy, K.; Redmond, E.J.; Davis, A.M.; Song, Y.; Ezer, D.; Pitchford, J.W.; et al. Complex epistatic interactions between ELF3, PRR9, and PRR7 regulate the circadian clock and plant physiology. Genetics
**2024**, 226, iyad217. [Google Scholar] [CrossRef]

**Figure 1.**Structure of the circadian clock in L2005a and L2005b. (

**A**) The circadian clock in L2005a consists of a single feedback loop, where LHY/CCA1 inhibits the expression of TOC1, which activates it. (

**B**) Two new unknown components, X and Y, are included in L2005b to account for experimental data. Reproduced with permission from [7] (

**A**), and with modifications from [8] (

**B**), Creative Commons license.

**Figure 2.**Structure of the circadian clock in L2006. The clock now consists of three loops: a morning loop (yellow), an evening loop (blue), and an overarching loop that connects both, including LHY/CCA1, TOC1, and the unknown component X. Only genes are shown, and arrows represent regulatory interactions. Reproduced from [10], Creative Commons license.

**Figure 3.**Structure of the circadian clock in P2010. New components enter the model, both in the morning (NI) and evening (ZTL) loops. Post-transcriptional modification of proteins is included for the first time. Unknown component X is substituted by a TOC1-dependent complex. Modified from [11], Creative Commons license.

**Figure 4.**Structure of the circadian clock in P2011. The main novelty is the introduction of the Evening Complex (EC) into the model, substituting the unknown component Y and changing the relationship between the evening and morning loops. Modified from [17], Creative Commons license.

**Figure 6.**Structure of the circadian clock in F2014. Two new components are included in the model, RVE8 and NOX. Blue arrows indicate new interactions included in the model. Reproduced from [21], Creative Commons license.

**Figure 7.**Structure of the kernel of the circadian clock in MF2015. The kernel consists of four loops, all containing at least one PRR and sharing LHY/CCA1. (

**A**) Loop A, a repressilator involving TOC1, PRR5, and LHY/CCA1. (

**B**) Loop B, a repressilator involving TOC1, PRR7, and LHY/CCA1. (

**C**) Loop C, a repressilator involving the EC, PRR7, and LHY/CCA1. And (

**D**) Loop D, a negative feedback loop between PRR9 and LHY/CCA1. Reproduced from [26], Creative Commons license.

**Figure 8.**Structure of the circadian clock in DC2016. Several pairs of genes are combined, thus considerably reducing the complexity of the model. To the previous merging of CCA1/LHY (CL) and PRR9/7 (P97), they add PRR5/TOC1 (P51) and ELF4/LUX (EL). Reproduced from [30], Creative Commons license.

**Figure 9.**Structure of the core of the circadian clock in J2018. (

**Left**) Model A is proposed as a core of the circadian clock, including the merged genes CCA1/LHY, PRR5/TOC1, and PRR9/7. As a novelty, the model includes activation by LWD 1/2. (

**Right**) Model B adds the Evening Complex and its components with their respective interactions. This second model better captures the overall behavior of the clock. Modified from [35], Creative Commons license.

**Figure 10.**Structure of the circadian clock in G2022. The model presents an updated version of DC2016. The novelty here is the model of plant structure in hundreds of compartments, each of which has its own set of ODEs. Modified from [40], Creative Commons license.

**Figure 11.**Structure of the circadian clock in Pa2022a. The model includes the perception of light by different photoreceptors (

**A**) as well as their repression of COP1. The model also incorporates competitive binding between COP1 and phyA, phyB, and cry (

**B**). Reproduced from [44], Creative Commons license.

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**MDPI and ACS Style**

Henao, L.; Ares, S.; Catalán, P.
Mathematical Models of the *Arabidopsis* Circadian Oscillator. *Biophysica* **2024**, *4*, 267-282.
https://doi.org/10.3390/biophysica4020019

**AMA Style**

Henao L, Ares S, Catalán P.
Mathematical Models of the *Arabidopsis* Circadian Oscillator. *Biophysica*. 2024; 4(2):267-282.
https://doi.org/10.3390/biophysica4020019

**Chicago/Turabian Style**

Henao, Lucas, Saúl Ares, and Pablo Catalán.
2024. "Mathematical Models of the *Arabidopsis* Circadian Oscillator" *Biophysica* 4, no. 2: 267-282.
https://doi.org/10.3390/biophysica4020019