In this paper, we derive an electrostatic-electrodynamic model of the exchanges of ions between a cell and its exterior during its growth, as well as a model of exchange of ions within the cell. Observations show that, in the phase G1, the growth of the volume explains the variation of density of ions (by dilution), hence explains the change of electrostatic potential inside the cell. The potential encounters a threshold at the beginning of phase S, and the ion channels open (the conductance of the membrane increases). This afflux of ions leads to a change of potential, which will trigger the disappearance of the nucleus double membrane (through the calcium channels).

From these remarks on the electric phenomena in the cell, one deduces a simple mathematical model, which is a generalization of the Hodgkin-Huxley model for the axons, for the cell cycle.

@article{MSIA_2017__8_1_1_0, author = {Jorgelindo da Veiga and Olivier Lafitte and Laurent Schwartz}, title = {A simple mathematical model for the growth and division of cells}, journal = {MathematicS In Action}, publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles}, volume = {8}, number = {1}, year = {2017}, pages = {1-8}, doi = {10.5802/msia.10}, zbl = {1406.92146}, mrnumber = {3745382}, language = {en}, url={msia.centre-mersenne.org/item/MSIA_2017__8_1_1_0/} }

Jorgelindo da Veiga; Olivier Lafitte; Laurent Schwartz. A simple mathematical model for the growth and division of cells. MathematicS In Action, Volume 8 (2017) no. 1, pp. 1-8. doi : 10.5802/msia.10. https://msia.centre-mersenne.org/item/MSIA_2017__8_1_1_0/

[1] Cation transport and growth in neuroblastoma cells, modulation of ${K}^{+}$ transport and electrical membrane properties during the cell cycle, Journal of cellular physiology (1981)

[2] An anaphase calcium signal controls chromosome disjunction in early sea urchin embryos, Cell, Tome 92 (1998), pp. 193-204 | Article

[3] A quantitative description of membrane current and its application to conduction and excitation in nerves, Journal of Physiology, Tome 117 (1952), pp. 500-544 | Article

[4] Mathematical aspects of Hodgkin-Huxley neural theory, Cambridge University Press, 1987 | Article | Zbl 0627.92005

[5] Shaping the multiscale architecture of mitochondria, Current opinion in cell biology, Tome 38 (2016), pp. 45-51 | Article

[6] Mechanical stress related to brain atrophy in Alzheimer’s disease, Alzheimer’s and Demential, Tome 12 (2016) no. 1, pp. 11-20 | Article

[7] Classical electropermeabilization modelling at the cell scale, Journal of theoretical biology, Tome 68(1-2) (2014) no. 1-2, pp. 235-265 | Zbl 1300.92023

[8] Cancer, un traitement simple et non toxique, Thierry Souccar Collection ’Medecine’, 2016

[9] Diffusion Limited Aggregation from Shear Stress As a Simple Model of Vasculogenesis 1999, Fractals, Tome 7 (1999) no. 1, pp. 33-39 | Article

[10] A thermodynamic theoriy of microbial growth, The ISME Journal, Tome 8 (2014), pp. 1747-1751 | Article

[11] Membrane regulation of the $N{a}^{+}$, ${K}^{+}$-ATPase During the Neuroblastoma Cell Cycle: correlation with protein mobility, J. Cellular Biochemistry, Tome 21 (1983), pp. 77-91 | Article