# Cell cycle dynamics: clustering is universal in negative feedback systems

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## Abstract

We study a model of cell cycle ensemble dynamics with cell–cell feedback in which cells in one fixed phase of the cycle \(S\) (Signaling) produce chemical agents that affect the growth and development rate of cells that are in another phase \(R\) (Responsive). For this type of system there are special periodic solutions that we call \(k\)-cyclic or clustered. Biologically, a \(k\)-cyclic solution represents \(k\) cohorts of synchronized cells spaced nearly evenly around the cell cycle. We show, under very general nonlinear feedback, that for a fixed \(k\) the stability of the \(k\)-cyclic solutions can be characterized completely in parameter space, a 2 dimensional triangle \(T\). We show that \(T\) is naturally partitioned into \(k^2\) sub-triangles on each of which the \(k\)-cyclic solutions all have the same stability type. For negative feedback we observe that while the synchronous solution (\(k=1\)) is unstable, regions of stability of \(k \ge 2\) clustered solutions seem to occupy all of \(T\). We also observe bi-stability or multi-stability for many parameter values in negative feedback systems. Thus in systems with negative feedback we should expect to observe cyclic solutions for some \(k\). This is in contrast to the case of positive feedback, where we observe that the only asymptotically stable periodic orbit is the synchronous solution.

## Mathematics Subject Classification

Primary 37N25 92C37 34B60## Notes

### Acknowledgments

Richard Buckalew (Ohio U.) provided essential help with some of the images. We thank Kara Finley (Ohio U.) for helpful comments. E.B., T.Y. and this work were partially supported by the NIH-NIGMS grant R01GM090207.

## References

- Bier M, Bakker B, Westerhoff H (2000) How yeast cells synchronize their glycolytic oscillations: a perturbation analytic treatment. Biophys J 78:1087–1093CrossRefGoogle Scholar
- Boczko E, Cooper T, Gedeon T, Mischaikow K (2005) Structure theorems and the dynamics of nitrogen catabolite repression in yeast. PNAS 102:5647–5652CrossRefGoogle Scholar
- Boczko E, Stowers C, Gedeon T, Young T (2010) Ode, rde and sde models of cell cycle dynamics and clustering in yeast. J Biol Dyn 4:328–345CrossRefMathSciNetGoogle Scholar
- Chen Z, Odstrcil E, Tu B, McKnight S (2007) Restriction of DNA replication to the reductive phase of the metabolic cycle protects genome integrity. Science 316:1916–1919CrossRefGoogle Scholar
- Diekmann O, Gyllenberg M, Verduyn Lunel SM (1993) A cell-cycle model revisited. CWI. Department of Analysis, Algebra and Geometry [AM] Report No. 9305, pp 1–18. http://www.helsinki.fi/~mgyllenb/publications.html
- Diekmann O, Heijmans H, Thieme H (1984) On the stability of the cell size distribution. J Math Biol 19:227–248CrossRefzbMATHMathSciNetGoogle Scholar
- Finn R, Wilson R (1954) Population dynamic behavior of the chemostat system. Agric Food Chem 2:66–69CrossRefGoogle Scholar
- Hannsgen K, Tyson J (1985) Stability of the steady-state size distribution in a model of cell growth and division. J Math Biol 22:293–301CrossRefzbMATHMathSciNetGoogle Scholar
- Henson M (2005) Cell ensemble modeling of metabolic oscillations in continuous yeast cultures. Comput Chem Eng 29:645–661CrossRefMathSciNetGoogle Scholar
- Keulers M, Satroutdinov A, Sazuki T, Kuriyama H (1996) Synchronization affector of autonomous short period sustained oscillation of Saccharomyces cerevisiae. Yeast 12:673–682CrossRefGoogle Scholar
- Kilpatrick Z, Ermentrout B (2011) Sparse gamma rhythms arising through clustering in adapting neuronal networks. PLoS Comput Biol 7(11):e1002281.Google Scholar
- Kjeldsen T, Ludvigsen S, Diers I, Balshmidt P, Sorensen A, Kaarshold N (2002) Engineering-enhanced protein secretory expression in yeast with applications to insulin. J Biol Chem 277:18245–18248CrossRefGoogle Scholar
- Klevecz R (1976) Quantized generation time in mammalian cells as an expression of the cellular clock. PNAS 73:4012–4016CrossRefGoogle Scholar
- Klevecz R, Murray D (2001) Genome wide oscillations in expression. Mol Biol Rep 28:73–82CrossRefGoogle Scholar
- Kopmann A, Diekmann H, Thoma M (1998) Oxygen, ph value, and carbon source induced changes of the mode of oscillation in synchronous continuous culture of Saccharomyces cervisiae. Biotechnol Bioeng 63:410–417Google Scholar
- Kuenzi M, Fiechter A (1969) Changes in carbohydrate composition and trehalose activity during the budding cycle of Saccharomyces cerevisiae. Arch Microbiol 64:396–407Google Scholar
- Mauroy A, Sepulchre R (2008) Clustering behaviors in networks of integrate-and-fire oscillators. Chaos 18(3):037122Google Scholar
- Meyenburg HV (1969) Energetics of the budding cycle of Saccharomyces cerevisiae during glucose limited aerobic growth. Arch Microbiol 66:289–303Google Scholar
- Monte SD, d’Ovidio F, Danø S, Sørensen P (2007) Dynamical quorum sensing: population density encoded in cellular dynamics. PNAS 104:18377–18381Google Scholar
- Pr PP (2003) Oscillatory metabolism of Saccharomyces cerevisiae: an overview of mechanisms and models. Biotechnol Adv 21:183–192CrossRefGoogle Scholar
- Robertson J, Stowers C, Boczko E, Johnson C (2008) Real-time luminescence monitoring of cell-cycle and respiratory oscillations in yeast. PNAS 105:17988–93CrossRefGoogle Scholar
- Rotenberg M (1977) Selective synchrony of cells of differing cycle times. J Theor Biol 66:389–398CrossRefMathSciNetGoogle Scholar
- Slavov N, Botstein D (2011) Coupling among growth rate response, metabolic cycle, and cell division cycle in yeast. Mol Biol Cell 22:1999–2009CrossRefGoogle Scholar
- Stowers C, Young T, Boczko E (2011) The structure of populations of budding yeast in response to feedback. Hypotheses Life Sci 1:71–84Google Scholar
- Tu B, Kudlicki A, Rowicka M, McKnight S (2005) Logic of the yeast metabolic cycle: temporal compartmentation of cellular processes. Science 310:1152–1158CrossRefGoogle Scholar
- Uchiyama K, Morimoto M, Yokoyama Y, Shioya S (1996) Cell cycle dependency of rice \(\alpha \)-amylase production in a recombinant yeast. Biotechnol Bioeng 54:262–271CrossRefGoogle Scholar
- Von Meyenburg K (1973) Stable synchronous oscillations in continuous cultures of S. cerevisiae under glucose limitation. In: Biological and Biochemical Oscillators. Academic Press, NYGoogle Scholar
- Young T, Fernandez B, Buckalew R, Moses G, Boczko E (2012) Clustering in cell cycle dynamics with general responsive/signaling feedback. J Theor Biol 292:103–115CrossRefMathSciNetGoogle Scholar
- Zhu G, Zamamiri A, Henson M, Hjortsø M (2000) Model predictive control of continuous yeast bioreactors using cell population balance models. Chem Eng Sci 55:6155–6167Google Scholar