Skip to main content
Log in

A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells

Bulletin of Mathematical Biology Aims and scope Submit manuscript

Cite this article


We introduce two time-delay models of metabolic oscillations in yeast cells. Our model tests a hypothesis that the oscillations occur as multiple pathways share a limited resource which we equate to the number of available ribosomes. We initially explore a single-protein model with a constraint equation governing the total resource available to the cell. The model is then extended to include three proteins that share a resource pool. Three approaches are considered at constant delay to numerically detect oscillations. First, we use a spectral element method to approximate the system as a discrete map and evaluate the stability of the linearized system about its equilibria by examining its eigenvalues. For the second method, we plot amplitudes of the simulation trajectories in 2D projections of the parameter space. We use a history function that is consistent with published experimental results to obtain metabolic oscillations. Finally, the spectral element method is used to convert the system to a boundary value problem whose solutions correspond to approximate periodic solutions of the system. Our results show that certain combinations of total resource available and the time delay, lead to oscillations. We observe that an oscillation region in the parameter space is between regions admitting steady states that correspond to zero and constant production. Similar behavior is found with the three-protein model where all proteins require the same production time. However, a shift in the protein production rates peaks occurs for low available resource suggesting that our model captures the shared resource pool dynamics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.


  • Adams CA, Kuriyama H, Lloyd D, Murray DB (2003) The Gts1 protein stabilizes the autonomous oscillator in yeast. Yeast 20(6):463–470

    Article  Google Scholar 

  • Beyn W-j, Champneys A, Doedel E, Govaerts W, Kuznetsov Y, Sandstede B (1999) Numerical continuation, and computation of normal forms 2

  • Boczko EM, Gedeon T, Stowers CC, Young TR (2010) ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast. J Biol Dyn 4(4):328–345

    Article  MathSciNet  MATH  Google Scholar 

  • Brauer M et al (2008) Coordination of growth rate, cell cycle, stress response, and metabolic activity in yeast. Mol Biol Cell 19:352–367

    Article  Google Scholar 

  • Breda D, Maset S, Vermiglio R (2005) Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM J Sci Comput 27(2):482–495

    Article  MathSciNet  MATH  Google Scholar 

  • Datseris G (2018) Dynamicalsystems.jl: A julia software library for chaos and nonlinear dynamics. J Open Source Softw 3(23):598.

    Article  Google Scholar 

  • Fowler A (1981) Approximate solution of a model of biological immune responses incorporating delay. J Math Biol 13:23–45

    Article  MathSciNet  MATH  Google Scholar 

  • Gedeon T, Humphries AR, Mackey MC, Walther H-O, Wang Z (2022) Operon dynamics with state dependent transcription and/or translation delays. J Math Biol 84(1–2):2

    Article  MathSciNet  MATH  Google Scholar 

  • Gopalsamy K, Aggarwala B (1981) The logistic equation with a diffusionally coupled delay. Bull Math Biol 43(2):125–140

    Article  MathSciNet  MATH  Google Scholar 

  • Gulbudak H, Salceanu PL, Wolkowicz GS (2021) A delay model for persistent viral infections in replicating cells. J Math Biol 82(7):59

    Article  MathSciNet  MATH  Google Scholar 

  • Henson MA (2004) Modeling the sychronization of yeast respiratory oscillations. J Theor Biol 231:443–458

    Article  MATH  Google Scholar 

  • Huang G, Takeuchi Y, Ma W, Wei D (2010) Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull Math Biol 72:1192–1207

    Article  MathSciNet  MATH  Google Scholar 

  • Jules M, Francois J, Parrou J (2005) Autonomous oscillations in Saccharomyces cerevisiae during batch cultures on trehalose. FEBS J 272:1490–1500

    Article  Google Scholar 

  • Khasawneh FA, Mann BP (2013) A spectral element approach for the stability analysis of time-periodic delay equations with multiple delays. Commun Nonlinear Sci Numer Simul 18(8):2129–2141.

    Article  MathSciNet  MATH  Google Scholar 

  • Khasawneh FA, Barton DA, Mann BP (2012) Periodic solutions of nonlinear delay differential equations using spectral element method. Nonlinear Dyn 67:641–658

    Article  MathSciNet  MATH  Google Scholar 

  • Klevecz R, Bolen J, Forrest G, Murray D (2004) A genomewide oscillation in transcription gates DNA replication and cell cycle. PNAS 101(5):1200–1205

    Article  Google Scholar 

  • Kuenzi M, Fiechter A (1969) Changes in carbohydrate composition and trehalase activity during the budding cycle of Saccharomyces cerevisiae. Arch Mikrobiol 64:396–407

    Article  Google Scholar 

  • Kuznetsov YA (2004) Numerical analysis of bifurcations. Springer, New York, pp 505–585.

    Book  Google Scholar 

  • Longtin A, Milton JG (1989) Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations. Bull Math Biol 51(5):605–624

    Article  MATH  Google Scholar 

  • Mier-y-Terán-Romero L, Silber M, Hatzimanikatis V (2010) The origins of time-delay in template biopolymerization processes. PLoS Comput Biol 6(4):1000726.

    Article  MathSciNet  Google Scholar 

  • Morgan L, Moses G, Young TR (2018) Coupling of the cell cycle and metabolism in yeast cell-cycle-related oscillations via resource criticality and checkpoint gating. Lett Biomath 5(1):113–128

    Article  MathSciNet  Google Scholar 

  • Muller D, Exler S, Aguilera-Vazquez L, Guerrero-Martin E, Reuss M (2003) Cyclic amp mediates the cell cycle dynamics of energy metabolism in Saccharomyces cerevisiae. Yeast 20:351–367

    Article  Google Scholar 

  • Murray D, Klevecz R, Lloyd D (2003) Generation and maintenance of synchrony in Saccharomyces cerevisiae continuous culture. Exp Cell Res 287:10–15

    Article  Google Scholar 

  • Pell B, Brozak S, Phan T, Wu F, Kuang Y (2023) The emergence of a virus variant: dynamics of a competition model with cross-immunity time-delay validated by wastewater surveillance data for COVID-19. J Math Biol 86(5):63

    Article  MathSciNet  MATH  Google Scholar 

  • Rihan FA, Tunc C, Saker S, Lakshmanan S, Rakkiyappan R (2018) Applications of delay differential equations in biological systems. Complexity 2018

  • Robertson J, Stowers C, Boczko E, Johnson C (2008) Real-time luminescence monitoring of cell-cycle and respiratory oscillations in yeast. PNAS 105(46):17988–17993

    Article  Google Scholar 

  • Rosen G (1987) Time delays produced by essential nonlinearity in population growth models. Bull Math Biol 49(2):253–255

    Article  MathSciNet  MATH  Google Scholar 

  • Scott M, Klumpp S, Mateescu EM, Hwa T (2014) Emergence of robust growth laws from optimal regulation of ribosome synthesis. Mol Syst Biol 10(8):747

    Article  Google Scholar 

  • Seydel R (2010) Practical bifurcation and stability analysis. Springer, New York

    Book  MATH  Google Scholar 

  • Silverman S et al (2010) Metabolic cycling in single yeast cells from unsynchronized steady-state populations limited on glucose or phosphate. PNAS 107:6946–6951

    Article  Google Scholar 

  • Slavov N, Botstein D (2011) Coupling among growth rate response, metabolic cycle, and cell division cycle in yeast. Mol Biol Cell 22:1997–2009

    Article  Google Scholar 

  • Slavov N, Macinskas J, Caudy A, Botstein D (2011) Metabolic cycling without cell division cycling in respiring yeast. PNAS 108(47):19090–19095

    Article  Google Scholar 

  • Sohn HY, Kuriyama H (2001) Ultradian metabolic oscillation of Saccharomyces cerevisiae during aerobic continuous culture: hydrogen sulphide, a population synchronizer, is produced by sulphite reductase. Yeast 18(2):125–135

    Article  Google Scholar 

  • Stépán G (1989) Retarded dynamical systems: stability and characteristic functions. Longman, Co-published with Wiley, London, New York

    MATH  Google Scholar 

  • Stowers CC, Young TR, Boczko EM (2011) The structure of populations of budding yeast in response to feedback. Hypotheses Life Sci 1:71–84

    Google Scholar 

  • Tu T, Kudlicki A, Rowicka M, McKnight S (2005) Logic of the yeast metabolic cycle: temporal compartmentalization of cellular processes. Science 310:1152–1158

    Article  Google Scholar 

  • vpasolve (Mathworks) Solve symbolic equations numerically. Accessed 20 June 2023

  • Woolford JJ, Baserga S (2013) Ribosome biogenesis in the yeast Saccharomyces cerevisiae. Genetics 195(3):643–681

    Article  Google Scholar 

Download references


This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-22-1-0007. Work of T.G. was partially supported by National Science Foundation grant DMS-1951510.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Firas A. Khasawneh.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chumley, M.M., Khasawneh, F.A., Otto, A. et al. A Nonlinear Delay Model for Metabolic Oscillations in Yeast Cells. Bull Math Biol 85, 122 (2023).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: