 ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Instability of $k$-Cluster Solutions in a Cell Cycle Population Model when $k$ is Prime

Journal of Applied Nonlinear Dynamics 11(1) (2022) 87--138 | DOI:10.5890/JAND.2022.03.007

Rabi K.C., Abdalnaser Algoud, Todd R.Young

Department of Mathematics, Ohio University, Athens, OH 45701, USA

Abstract

We study periodic cyclic' solutions of a population model of the cell cycle. In this model, clusters of cells in one fixed phase of the cell cycle may exert a negative influence on the progress of clusters in another phase. Previous results showed that stability of cyclic solutions is determined by the values of model parameters $s$ and $r$, and by which of two possible orderings of certain events the cyclic solution follows. The parameter triangle $\triangle = \{(s,r): 0 \le s \le r \le 1 \}$ is subdivided into sub-triangles on which the stability of all cyclic solutions are the same. The stability for sub-triangles on the boundary of $\triangle$ was fully characterized in terms of number theoretic relationships between the number of clusters $k$ and certain indices of the sub-triangles. Interior sub-triangles with the order of events called $\mathbf{sr1}$, were shown to have unstable solutions. In the present work, we focus on interior sub-triangles with the other order of events $\mathbf{rs1}$. We show that when $k$ is prime, then the cyclic solutions are unstable for all interior sub-triangles. When $k$ is even, we show that there always exist a small number of sub-triangles on which cyclic solutions are at least neutrally stable. For $k$ odd and composite, we show that there are stable sub-triangles when $k = 9$ and $k=15$ and no others.

Acknowledgments

The authors thank Saad Aldosari, Herath Mudiyanselage Indupama Herath and Daniel Ntiamoah for many long, tedious and helpful discussions.

References

1.   B\{a}r\`{a}ny, B., Moses, G., and Young, T.R. (2019), Instability of the Steady State Solution in Cell Cycle Population Structure Models with Feedback, Journal of Mathematical Biology, 78(5), 1365-1387. DOI: 10.1007/s00285-018-1312-0.
2.   Boczko, E.M., Gedeon, T., Stowers, C.C., and Young, T.R. (2010), ODE, RDE and SDE models of cell cycle dynamics and clustering in yeast, Journal of Biological Dynamics, 4, 328-345. DOI: 10.1080/17513750903288003.
3.   Bose, N. (1989), Tests for Hurwitz and Schur properties of convex combination of complex polynomials, IEEE Transactions on Circuits and Systems, 36(9), 1245-1247. DOI: 10.1109/31.34672.
4.   Breitsch, N., Moses, G., Boczko, E.M., and Young, T.R. (2014), Cell cycle dynamics: clustering is universal in negative feedback systems, Journal of Mathematical Biology, 70(5), 1151-1175. DOI: 10.1007/s00285-014-0786-7.
5.   Buckalew, R. (2014), Cell Cycle Clustering in a Nonlinear Mediated Feedback Model, DCDS B, 19(4), 867-881.
6.   Diekmann, O., Heijmans, H., and Thieme, H. (1984), On the stability of the cell size distribution, Journal of Mathematical Biology, 19, 227-248. DOI:10.1007/BF00277748.
7.   Diekmann, O., Heijmans, H., and Thieme, H. (1993), Perturbing semigroups by solving Stieltjes renewal equations, Journal of Differential and Integral Equations, 6, 155-181.
8.   Fell, H.J. (1980), On the zeros of convex combinations of polynomials, Pacific Journal of Mathematics, 89(1), 43-50. https://projecteuclid.org/euclid.pjm/1102779366.
9.   Gong, X., Buckalew, R., Young, T.R., and Boczko, E. (2014), Cell cycle dynamics in a response/signaling feedback system with a gap, Journal of Biological Dynamics, 8(1), 79-98. DOI:10.1080/17513758.2014.904526.
10.   Gong, X., Moses, G., Neiman, A., and Young, T.R. (2014), Noise-induced dispersion and breakup of clusters in cell cycle dynamics, Journal of Theoretical Biology, 335, 160-169. DOI: j.jtbi.2014.03.034.
11.   Hannsgen, K.B., Tyson, J.J., and Watson, L.T. (1985), Steady-state size distributions in probabilistic models of the cell division cycle, SIAM Journal of Applied Mathematics, 45(4), 523-540. DOI:10.1137/0145031.
12.   Heijmans, H.J.A.M. (1984), On the stable size distribution of populations reproducing by fission into two unequal parts, Mathematical Biosciences, 72, 19-50. DOI:10.1016/0025-5564(84)90059-2.
13.   Lang, S. (2002), Algebra, Springer-Verlag.
14.   Morgan, L., Moses, G., and Young, T.R. (2018), Coupling of the cell cycle and metabolism in yeast cell-cycle-related oscillations via resource criticality and checkpoint gating, Letters in Biomathematics, 5, 113-128. DOI: 10.1080/23737867.2018.1456366.
15.   Moses, G. (2015), Dynamical systems in biological modeling: clustering in the cell division cycle of yeast, Dissertation, Ohio University, http://rave.ohiolink.edu/etdc/.
16.   Moses, G. and Scalfano, D. (2016), Cell cycle dynamics in a response/signaling feedback system with overlap, Journal of Applied Nonlinear Dynamics, 5(3), 243-267.
17.   Rabi, K.C. (2020), 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 Dissertation, Ohio University.
18.   Rabi, K.C., and Abdalnaser, A. Roots of staircase palindromic polynomials. Preprint, arxiv:2012.15663.
19.   Perko, L. (2000), Differential Equations and Dynamical Systems, Springer-Verlag, 3rd edition.
20.   Prathom, K. (2019), Stability Regions of Cyclic Solutions under Negative Feedback and Uniqueness of Periodic Solutions for Uneven Cluster Systems, Dissertation, Ohio University, http://rave.ohiolink.edu/etdc/.
21.   Rombouts, J., Prathom, K. and Young, T.R. (2020), Clusters tend to be of equal size in a negative feedback population model of cell cycle dynamics, SIAM Journal of Applied Dynamical Systems, 19(2), 1540-1573. DOI: 10.1137/19M129070X.
22.   Rotman, J. and Cuoco, A. (2013), Learning Modern Algebra, Mathematical Association of America.
23.   Young, T.R., Fernandez, B., Buckalew, R., Moses, G., and Boczko, E.M. (2012), Clustering in cell cycle dynamics with general response/signaling feedback, Journal of Theoretical Biology, 292, 103-115. DOI: 10.1016/j.jtbi.2011.10.002.
24.   Young, T.R., Prathom, K., and Rombouts, J. (2019), Temporal clustering in cell cycle dynamics, Dynamical Systems Magazine, https://dsweb.siam.org/.