Universality of stable multi-cluster periodic solutions in a population model of the cell cycle with negative feedback. (English) Zbl 1484.92026
Summary: We study a population model where cells in one part of the cell cycle may affect the progress of cells in another part. If the influence, or feedback, from one part to another is negative, simulations of the model almost always result in multiple temporal clusters formed by groups of cells. We study regions in parameter space where periodic ‘\(k\)-cyclic’ solutions are stable. The regions of stability coincide with sub-triangles on which certain events occur in a fixed order. For boundary sub-triangles with order ‘\(\mathbf{rs1}\)’, we prove that the \(k\)-cyclic periodic solution is asymptotically stable if the index of the sub-triangle is relatively prime with respect to the number of clusters \(k\) and neutrally stable otherwise. For negative linear feedback, we prove that the interior of the parameter set is covered by stable sub-triangles, i.e. a stable \(k\)-cyclic solution always exists for some \(k\). We observe numerically that the result also holds for many forms of nonlinear feedback, but may break down in extreme cases.
References:
| [1] | Achuthan, S.; Canavier, C. C., Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators, J. Neurosci., 29, 16, 5218-5233 (2009) |
| [2] | Bàràny, B.; Moses, G.; Young, T. R., Instability of the steady state solution in cell cycle population structure models with feedback, J. Math. Biol., 78, 5, 1365-1387 (2019) · Zbl 1415.92143 |
| [3] | Boczko, E. M.; Gedeon, T.; Stowers, C. C.; Young, T. R., ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast, J. Biol. Dyn., 4, 328-345 (2010) · Zbl 1342.92119 |
| [4] | Bose, N., Tests for Hurwitz and Schur properties of convex combination of complex polynomials, IEEE Trans. Circuits Syst., 36, 9, 1245-1247 (1989) · Zbl 0683.30008 |
| [5] | Breitsch, N.; Moses, G.; Boczko, E.; Young, T. R., Cell cycle dynamics: Clustering is universal in negative feedback systems, J. Math. Biol., 70, 5, 1151-1175 (2014) · Zbl 1347.37137 |
| [6] | Buckalew, R.L., Mathematical models in cell cycle biology and pulmonary immunity, Ph.D. dissertation, Ohio University, 2014. Available at http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1395242276. |
| [7] | Cai, L.; Tu, B. P., Driving the cell cycle through metabolism, Annu. Rev. Cell Dev. Biol., 28, 59-87 (2012) |
| [8] | Chen, Z.; Odstrcil, E. A.; Tu, B. P.; McKnight, S. L., Restriction of DNA replication to the reductive phase of the metabolic cycle protects genome integrity, Science, 316, 1916-1919 (2007) |
| [9] | Duboc, P.; Marison, L.; von Stockar, U., Physiology of Saccharomyces cerevisiae during cell cycle oscillations, J. Biotechnol., 51, 57-72 (1996) |
| [10] | Fell, H. J., On the zeros of convex combinations of polynomials, Pac. J. Math., 89, 1, 43-50 (1980) · Zbl 0416.30005 |
| [11] | Fernandez, B.; Tsimring, L., Typical trajectories of coupled degrade-and-fire oscillators: From dispersed populations to massive clustering, J. Math. Biol., 68, 7, 1627-1652 (2014) · Zbl 1300.92013 |
| [12] | Finn, R. K.; Wilson, R. E., Population dynamic behavior of the chemostat system, J. Agric. Food Chem., 2, 66-69 (1954) |
| [13] | Futcher, B., Metabolic cycle, cell cycle and the finishing kick to start, Genome Biol., 7, 107-111 (2006) |
| [14] | Golomb, D.; Rinzel, J., Clustering in globally coupled inhibitory neurons, Physica D, 72, 3, 259-282 (1994) · Zbl 0809.92003 |
| [15] | Kilpatrick, Z. P.; Ermentrout, B., Sparse gamma rhythms arising through clustering in adapting neuronal networks, PLoS Comput. Biol., 7, 11, e10022810 (2011) |
| [16] | Kiss, I. Z.; Zhai, Y.; Hudson, J. L., Predicting mutual entrainment of oscillators with experiment-based phase models, Phys. Rev. Lett., 94, 24 (2005) |
| [17] | Kiss, I. Z.; Rusin, C. G.; Kori, H.; Hudson, J. L., Engineering complex dynamical structures: Sequential patterns and desynchronization, Science, 316, 5833, 1886-1889 (2007) · Zbl 1226.93068 |
| [18] | Klevecz, R. R.; Bolen, J.; Forrest, G.; Murray, D. B., A genome-wide oscillation in transcription gates DNA replication and cell cycle, Proc. Natl. Acad. Sci. USA, 101, 5, 1200-1205 (2004) |
| [19] | Kuenzi, M. T.; Fiechter, A., Changes in carbohydrate composition and trehalose activity during the budding cycle of Saccharomyces cerevisiae, Arch. Microbiol., 64, 396-407 (1969) |
| [20] | Mauroy, A.; Sepulchre, R., Clustering behaviors in networks of integrate-and-fire oscillators, Chaos, 18, 3 (2008) · Zbl 1309.34056 |
| [21] | von Meyenburg, H. K., Energetics of the budding cycle of Saccharomyces cerevisiae during glucose limited aerobic growth, Arch. Mikrobiol., 66, 289-303 (1969) |
| [22] | Morgan, L.; Moses, G.; Young, T. R., Coupling of the cell cycle and metabolism in yeast cell-cycle-related oscillations via resource criticality and checkpoint gating, Lett. Biomath., 5, 113-128 (2018) · doi:10.1080/23737867.2018.1456366 |
| [23] | Moses, G., Dynamical systems in biological modeling: Clustering in the cell division cycle of yeast, Ph.D. dissertation, Ohio University, 2015. Available at http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1438170442. |
| [24] | Muller, D.; Exler, S.; Aguilera-Vazquez, L.; Guerrero-Martin, E.; Reuss, M., Cyclic AMP mediates the cell cycle dynamics of energy metabolism in Saccharomyces cervisiae, Yeast, 20, 351-367 (2003) |
| [25] | Munch, T.; Sonnleitner, B.; Fiechter, A., The decisive role of the Saccharomyces cervisiae cell cycle behavior for dynamic growth characterization, J. Biotechnol., 22, 329-352 (1992) |
| [26] | Newcomb, L. L.; Diderich, J. A.; Slattery, M. G.; Heideman, W., Glucose regulation of Saccharomyces cerevisiae cell cycle genes, Eukaryot. Cell, 2, 143-149 (2003) |
| [27] | Patnaik, P. R., Oscillatory metabolism of Saccharomyces cerevisiae: An overview of mechanisms and models, Biotech. Adv., 21, 183-192 (2003) |
| [28] | Prathom, K., Stability regions of cyclic solutions under negative feedback and uniqueness of periodic solutions for uneven cluster systems, Ph.D. dissertation, Ohio University, 2019. Available at https://etd.ohiolink.edu/etdc/view?acc_num=ohiou1562778288743268. |
| [29] | Rabi, K.C., Study of some biologically relevant dynamical system models: (In)stability regions of cyclic solutions in cell cycle population structure model under negative feedback and random connectivities in multi-type neuronal network models, Ph.D. dissertation, Ohio University, 2020. Available at http://rave.ohiolink.edu/etdc/view?acc_num=ohiou16049254273607. |
| [30] | Rabi, K. C.; Algoud, A.; Young, T. R., Instability of k-cluster solutions in a cell cycle population model when k is prime, J. Appl. Nonlinear Dyn., 11, 1, 87-138 (2022) · Zbl 1486.37053 |
| [31] | Robertson, J. B.; Stowers, C. C.; Boczko, E. M.; Johnson, C. H., Real-time luminescence monitoring of cell-cycle and respiratory oscillations in yeast, Proc. Natl. Acad. Sci. USA, 105, 17988-17993 (2008) |
| [32] | Rombouts, J.; Prathom, K.; Young, T., Clusters tend to be of equal size in a negative feedback population model of cell cycle dynamics, SIAM J. Appl. Dyn. Syst., 19, 2, 1540-1573 (2020) · Zbl 1447.37083 · doi:10.1137/19M129070X |
| [33] | Stowers, C.; Young, T. R.; Boczko, E., The structure of populations of budding yeast in response to feedback, Hypoth. Life Sci., 1, 3, 71-84 (2011) |
| [34] | Taylor, A. F.; Kapetanopoulos, P.; Whitaker, B. J.; Toth, R.; Bull, L.; Tinsley, M. R., Clusters and switchers in globally coupled photochemical oscillators, Phys. Rev. Lett., 100, 21 (2008) |
| [35] | Taylor, A.; Tinsley, M.; Showalter, K., Insights into collective cell behaviour from populations of coupled chemical oscillators, Phys. Chem. Chem. Phys., 17, 31, 20047-20055 (2015) |
| [36] | Uchiyama, K.; Morimoto, M.; Yokoyama, Y.; Shioya, S., Cell cycle dependency of rice α-amylase production in a recombinant yeast, Biotechnol. Bioeng., 54, 262-271 (1996) |
| [37] | Wickramasinghe, M.; Kiss, I. Z., Spatially organized dynamical states in chemical oscillator networks: Synchronization, dynamical differentiation, and chimera patterns, PLoS ONE, 8, 11, e80586 (2013) |
| [38] | Wilson, D.; Moehlis, J., Clustered desynchronization, from high-frequency deep brain stimulation, PLoS Comput. Biol., 11, e1004673 (2015) |
| [39] | Young, T. R.; Fernandez, B.; Buckalew, R.; Moses, G.; Boczko, E. M., Clustering in cell cycle dynamics with general response/signaling feedback, J. Theor. Biol., 292, 103-115 (2012) · Zbl 1307.92101 |
| [40] | Zhang, J.; Yuan, Z.; Zhou, T., Synchronization and clustering of synthetic genetic networks: A role for CIS-regulatory modules, Phys. Rev. E, 79, e041903 (2009) |
| [41] | Zhao, G.; Chen, Y.; Carey, L.; Futcher, B., Cyclin-dependent kinase co-ordinates carbohydrate metabolism and cell cycle in S. cerevisiae, Mol. Cell, 62, 546-557 (2016) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
