Optimization of Ion Concentrations in an Animal Cell: A Mathematical Model Approach
Keywords:
Optimization, Homeostasis, Partial Differential Equations, Mathematical Modeling, Simulation, Membrane Potential, Optimal Ion Concentration, Ion Channel, Finite Difference Method
Abstract
The increasing world population has raised significant concerns about matching food production to demand, prompting extensive research in scientific fields aimed at enhancing plant and animal productivity. However, some of these advances have introduced unintended consequences, such as uncontrolled cell growth that leads to cancer. Effective regulation of cell size is essential to maintaining organismal health. This study focuses on the utilization of a numerical method to develop a mathematical model that optimizes ion transport within an animal cell as a mechanism to ensure a physiologically healthy cell. Through this model, optimal ion concentrations were identified using MATLAB-SIMULINK. The results were validated using experimental data to ensure that they promote healthy cell growth and stability. The results provide valuable insights for treating disorders associated with abnormal cell growth initiated by unregulated ions concentration. This research contributes to understanding cellular homeostasis and lays the groundwork for future bioengineering applications.
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[2] A.C. Vishnu and Harald S. (2011). Ion channels and transporters in cancer. American Journal of Physiology- Cell Physiology, 40(1):C541-C549.
[3] Allman E. S., Rhodes J. A. (2013). Mathematical models in biology: An introduction. Cambridge University Press.
[4] Atkins, Peter William and de Paula, Julio (2014). Physical chemistry. Oxford University Press., 10 edition.
[5] B., Hille (2001). Ionic Channels of Excitable Membranes. Sunderland, MA: Sinauer Associates, 3 edition.
[6] Bello J. F., Taiwo E. S. and I. Adinya (2024). Modified models for constrained mean absolute deviation portfolio optimization. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 10(1):12–24.
[7] Blaustein M. P., Hamlyn J. M. (2016). Sodium transport and hypertension. Current Opinion in Nephrology and Hypertension, 25(6):513–521.
[8] C., Skou J. (1957). The influence of some cations on an adenosine triphosphatase from peripheral nerves. Biochimica et Biophysica Acta, 23:394–401.
[9] Coalson, Rob D. and Kournikova, Maria G. (2005). Poisson–Nernst–Planck Theory Approach to the Calculation of Current Through Biological Ion Channels. Oxford University Press., 4 edition.
[10] Cohen, B. (2023). Concentrations of different ions in cells. Bionumbers.
[11] Cuevas E., Escobar H. (2024). Computational Methods with MATLAB. Springer Cham.
[12] Danjuma T., Onah E. S. and Aboiyar T. (2020). Asset optimization problem in a financial institution. International Journal of Mathematical Sciences and Optimization: Theory and Applications, (2):581–591.
[13] E., Goldman D. (1943). Potential, impedance, and rectification in membranes. The Journal of General Physiology, 27(1):37–60.
[14] F., Lang (2007). Mechanisms and significance of cell volume regulation. Journal of the American College of Nutrition, 26(5):613S–623S.
[15] Gadsby D. C., Nakao M. (2003). Ionic composition in mammalian organisms. The Journal of Physiology, 541(3):635–645.
[16] Gerisch A., Chaplain M. A. (2014). Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion. Mathematical Biosciences, 264:232–248.
[17] Hodgkin A. L., Huxley A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of Physiology, 117(4):500–544.
[18] Hodgkin A. L., Katz B. (1949). The effect of sodium ions on the electrical activity of the giant axon of the squid. The Journal of Physiology, 108(1):37–77.
[19] Hoffmann E. K., Dunham P. B. (1995). The effect of sodium ions on the electrical activity of the giant axon of the squid. International Review of Cytology, 161(1):173–262.
[20] House CD, Vaske CJ, Schwartz AM Obias V Frank B Luu T Sarvazyan N Irby R Strausberg RL Hales TG Stuart JM Lee NH (2010). Voltage-gated na+ channel scn5a is a key regulator of a gene transcriptional network that controls colon cancer invasion. Cancer Research, 70(1):6957–6967.
[21] J., Hu and X., Huang (2020). A fully discrete positivity-preserving and energy-dissipative finite difference scheme for poisson–nernst–planck equations. Numerische Mathematik, 145(1):77–135.
[22] Jeremy M B., John L.T. and Lubert S. (2002). Biochemistry. New York: W H Freeman, 5 edition.
[23] J.O., Hillesdon A.J.; Pedley T.J.; Kessler (1995). The development of concentration gradients in a suspension of chemotactic bacteria. Bulletin of Mathematical Biology, 57(2):299-344.
[24] Kandel E. R., Schwartz J. H., Jessell T. M. (2012). Principles of Neural Science. McGraw-Hill, 5 edition.
[25] Keener J., Sneyd J. (2009). Mathematical Physiology. Springer, 2 edition.
[26] Matejczyk, Bartlomiej (2019). Mathematical modelling and simulations of the ion transport through confined geometries. The University of Warwick, lib-publications.
[27] Morton, K.W. and Mayer, D.F. (2005). Numerical Solution of Partial Differential Equations; An introduction. Cambridge University Press, 2 edition.
[28] Murase, H. and Kitano, H. (2011). Mathematical modeling in systems biology. Humana Press., 2 edition.
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[30] Smith J., Brown R. (2020). A high-order finite difference method for solving the compressible euler equations: consistency, convergence, and stability analysis. Journal of Computational Physics, 402(10).
[31] Smith N. P., Crampin E. J., Niederer S. A. (2017). Computational modelling of the heart: from cell to organ. Mathematical, Physical and Engineering Sciences, 375(2102).
[32] V'yacheslav L., George S., Roman S. and Natalia P. (2011). Ion channels and transporters in cancer. 5. ion channels in control of cancer and cell apoptosis. American Journal of Physiology- Cell Physiology, 249(10).
[33] Wang C.X., Chertock A., Cui S.M. Kurganov A. and Zhang Z. (2022). A diffuse-domain based numerical method for a chemotaxis-fluid model. Numerical Analysis, 33(1):341-375.
[34] Wilfred, Stein (2012). Transport and diffusion across cell membranes. Oxford University Press., 3 edition.
[35] Yang N. J., Hinner M. J. (2015). Getting across the cell membrane: An overview for small molecules, peptides, and proteins. Methods and Protocols, 1266(1):29-53.
Published
2025-05-30
How to Cite
Mwangi, K. M., Karimi, K., Oke, T. O., & Kaneba, C. (2025). Optimization of Ion Concentrations in an Animal Cell: A Mathematical Model Approach. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 11(2), 62 73. Retrieved from https://ijmso.unilag.edu.ng/article/view/2894
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