Transactions in Theoretical and Mathematical Physics

Transactions in Theoretical and Mathematical Physics

A Dynamical Physics Equation for Cell Proliferation

Document Type : Original Article

Author
Javad Sharifi
Electrical and Computer Engineering Department, Qom University of Technology
10.30511/ttmp.2026.2083409.1070
Abstract
A dynamical model for mitosis cell proliferation from a geometric viewpoint is derived. The derived kinetics are nonlinear partial differential equations of cell volume and mass density. We consider the process of inherent cell growth due to biological processes as the mass increase in the cell which exerts force inside of the cell. Likewise, the process of segregation is viewed as a mass decrease. We consider the physical aspects of inherent biological processes such as anabolism of structural proteins and lipids swelling due to mass influx and shrinking due to mass outflow. To this aim, by first and second equations of motion together with Fick's Laws of diffusion for nutrition, we model this force and obtain the cell radius function and mass density evolution equations. The control action will appear in our dynamic equation.
Keywords
Dynamic Physics Equation
Cell Proliferation
Fick Laws
Subjects
1. M. B. Elowitz and S. Leibler, Nature, vol. 403, no.
20, pp. 335–338, 2000. DOI: https://doi.org/10.
1038/35002125#

2. L. Chen and K. Aihara, IEEE Transactions on Cir
cuits and Systems, vol. 49, no. 10, 2002. DOI: https:
//doi.org/10.1109/TCSI.2002.803354#

3. B. S. Chen and Y. C. Wang, BMC Bioinformatics, 7,
52, doi:10.1186/1471-2105-7-52, 2006. DOI: https:
//doi.org/10.1186/1471-2105-7-52#

4. R. Jothi et al., Molecular Systems Biology, vol. 5,
doi:10.1038/msb.2009.52, 2009. DOI: https://doi.
org/10.1038/msb.2009.52#

5. R. Siammas et al., Molecular Systems Biology, vol.
7, no. 25, 2011. DOI: https://doi.org/10.1038/
msb.2011.25#

6. T. S. Gardner, C. R. Cantor and J. J. Collins, Nature,
vol. 403, no. 20, pp. 339–342, 2000. DOI: https://
doi.org/10.1038/35002131#

7. V. A. Rhodius et al., Molecular Systems Biology, vol.
9, doi:10.138/msb.2013.58, 2013. DOI: https://doi.
org/10.1038/msb.2013.58#

8. L. M. Adleman, Science, 266, 5187, pp. 1021–1024,
1994. DOI: https://doi.org/10.1126/science.
7973651#

9. Y. Benenson et al., Nature, 414, 6862, pp. 430–434,
2001. DOI: https://doi.org/10.1038/35106533#

10. M. N. Stojanovic, T. E. Mitchell and D. Stefanovic,
Journal of the American Chemical Society, 124, 14,
pp. 3555–3561, 2002. DOI: https://doi.org/10.
1021/ja016756v#

11. J. I. Farren, J. Hasty, C. R. Cantor and J. J. Collins,
PNAS, vol. 100, no. 13, pp. 7714–7719, 2003. DOI:
https://doi.org/10.1073/pnas.1332628100#

12. J. MacDonald et al., Nano Letters, vol. 6, no. 11,
pp. 2598–2603, 2006. DOI: https://doi.org/10.
1021/nl0620684#

13. H. Lodish et al., Molecular Cell Biology, sixth edition,
W. H. Freeman, 2007.#

14. M. Gthocleous and W. Harris, Trends in Cell Biol
ogy, vol. 23, no. 10, pp. 484–491, 2013. DOI: https:
//doi.org/10.1016/j.tcb.2013.05.004#

15. Y. E. Korchev et al., Biophysical Journal, 78, 1, pp.
451–457, 2000. DOI: https://doi.org/10.1016/
S0006-3495(00)76605-7#

16. W. B. Bonnor, Bulletin of Mathematical Biology, 29,
3, pp. 429–436, 1967. DOI: https://doi.org/10.
1007/BF02476582#

17. E. C. Anderson et al., Biophysical Journal, 9, 2, pp.
246–263, 1969. DOI: https://doi.org/10.1016/
S0006-3495(69)86383-6#

18. G. I. Bell, Biophysical Journal, 8, 4, pp. 431
444, 1968. DOI: https://doi.org/10.1016/
S0006-3495(68)86498-7#

19. U. C. Selhuber et al., Physical Biology, 6, 2,
025015, 2009. DOI: https://doi.org/10.1088/
1478-3975/6/2/025015#

20. J. A. Hernandez, Bulletin of Mathematical Biology, 69,
5, pp. 1631–1648, 2007. DOI: https://doi.org/10.
1007/s11538-006-9181-5#

21. M. Terasaka et al., Cell, vol. 154, no. 2, pp. 285–296,
2013. DOI: https://doi.org/10.1016/j.cell.
2013.06.043#

22. F. M. Wallas, Cell, vol. 154, issue 2, pp. 265–266,
2013. DOI: https://doi.org/10.1016/j.cell.
2013.06.044#

23. M. Potter and D. C. Wiggart, Fluid Mechanics,
McGraw-Hill, 2008.#

24. T. Chow, Mathematical Methods for Physicists: A Con
cise Introduction, Cambridge University Press, 2003.#

25. D. H. Worthington, M. Salamone and D. S. Nachtwey,
Cell Proliferation, vol. 9, 2, pp. 119–130, 1976. DOI:
https://doi.org/10.1111/j.1365-2184.1976.
tb01260.x#
 
Transactions in Theoretical and Mathematical Physics
Volume 3, Issue 1
Winter 2026
Pages 25-29

  • Receive Date 07 January 2026
  • Accept Date 20 February 2026
  • First Publish Date 20 February 2026
  • Publish Date 01 February 2026