Advertisement

3D mathematical modeling of calcium signaling in Alzheimer’s disease

  • Devanshi D. DaveEmail author
  • Brajesh Kumar Jha
Original Article
  • 25 Downloads

Abstract

The present paper focuses on the solution of the three-dimensional calcium advection–diffusion equation in the presence of calcium-binding buffers. As buffers play an important role in maintaining cytosolic calcium concentration level, decrease in buffers leads to increase in cytoplasmic calcium which may further lead to toxicity of Alzheimer’s disease. The governing three-dimensional differential equation has been further converted into one-dimensional equation using similarity transforms. The solution is obtained analytically using Laplace transforms and suitable boundary conditions. The obtained solution is simulated in MATLAB. The graphs clearly show the impact of buffers on calcium concentration level for normal and Alzheimeric cells.

Keywords

Calcium Buffers 3D-advection–diffusion Alzheimer’s disease Analytical solution 

Mathematics Subject Classification

92B05 92C30 35K57 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. Augustine GJ, Santamaria F, Tanaka K (2003) Local calcium signaling in neurons. Neuron 40:331–346CrossRefGoogle Scholar
  2. Bezprozvanny I (2011) Calcium signaling and neurodegenerative diseases. Trends Mol Med 15:89–100CrossRefGoogle Scholar
  3. Brzyska M, Elbaum D (2003) Dysregulation of calcium in Alzheimer’s disease. Acta Neurobiol Exp 63:171–183Google Scholar
  4. Carafoli E, Brini M (eds) (2007) Calcium signalling and disease. Springer, New YorkGoogle Scholar
  5. Clapham DE (2007) Calcium signaling. Cell 131:1047–1058CrossRefGoogle Scholar
  6. Coe H, Michalak M (2009) Calcium binding chaperones of the endoplasmic reticulum. Gen Physiol Biophys 28:96–103Google Scholar
  7. Crank J (1975) The mathematics of diffusion, Second edn. Clarendon Press, OxfordzbMATHGoogle Scholar
  8. Dave DD, Jha BK (2018a) Analytically depicting the calcium diffusion for Alzheimer’s affected cell. Int J Biomath 11(6):1850088–1850101MathSciNetCrossRefGoogle Scholar
  9. Dave DD, Jha BK (2018b) Delineation of calcium diffusion in alzheimeric brain. J Mech Med Biol 18(2):1–15Google Scholar
  10. Demuro A, Parker I, Stutzmann GE (2010) Calcium signaling and amyloid toxicity in Alzheimer’s disease. J Biol Chem 6:1–1Google Scholar
  11. Fall C et al (2002) Computational cell biology. Springer, New YorkzbMATHGoogle Scholar
  12. Jha A, Adlakha N (2015) Two-dimensional finite element model to study unsteady state \(Ca^{2+}\) diffusion in neuron involving ER. LEAK and SERCA. Int J Biomath 8(1):1–14CrossRefGoogle Scholar
  13. Jha BK, Adlakha N, Mehta MN (2012) Analytic solution of two dimensional advection diffusion equation arising in cytosolic calcium concentration distribution. Int Math Forum 7(3):135–144MathSciNetzbMATHGoogle Scholar
  14. Jha BK, Adlakha N, Mehta MN (2014) Two-dimensional finite element model to study calcium distribution in astrocytes in presence of excess buffer. Int J Biomath 7(3):1–11MathSciNetCrossRefGoogle Scholar
  15. Jha A, Adlakha N, Jha BK (2015) Finite element model to study effect of \(Na^{+}-Ca^{2+}\) exchangers and source geometry on calcium dynamics in a neuron cell. J Mech Med Biol 16(2):1–22Google Scholar
  16. Keener J, Sneyd J (2009) Mathematical physiology second. Springer, New YorkCrossRefGoogle Scholar
  17. Khachaturian ZS (1993) Calcium hypothesis of Alzheimer’s disease and brain aging. Ann N Y Acad Sci 1–11CrossRefGoogle Scholar
  18. Kotwani M, Adlakha N, Mehta MN (2014) Finite element model to study the effect of buffers. Source amplitude and source geometry on spatio-temporal calcium distribution in fibroblast cell. J Med Imaging Health Inf 4(6):840–847CrossRefGoogle Scholar
  19. Laferla FM (2002) Calcium dyshomeostasis and intracellular signalling in Alzheimer’s disease. Nat Rev Neurosci 3:862–872CrossRefGoogle Scholar
  20. Magi S et al (2016) Intracellular calcium dysregulation: implications for Alzheimer’s disease. Biomed Res Int 2016:1–14CrossRefGoogle Scholar
  21. Makrariya A, Adlakha N (2013) Two-dimensional finite element model of temperature distribution in dermal tissues of extended spherical organs of a human body. Int J Biomath 6(1):1250065-01–1250065-15MathSciNetCrossRefGoogle Scholar
  22. Makrariya A, Adlakha N (2015) Two-dimensional finite element model to study temperature distribution in peripheral regions of extended spherical human organs involving uniformly perfused tumors. Int J Biomath 8(6):1550074-01–1550074-30MathSciNetCrossRefGoogle Scholar
  23. Mattson MP et al (2000) Calcium signaling in the ER: its role in neuronal plasticity and neurodegenerative disorders. Trends Neurosci 23(5):222–229CrossRefGoogle Scholar
  24. Morris G et al (2018) Could Alzheimer’s disease originate in the periphery and if so how so? Mol NeurobiolGoogle Scholar
  25. Naik PA, Pardasani KR (2018a) Three-dimensional finite element model to study effect of RyR calcium channel. ER leak and SERCA pump on calcium distribution in oocyte cell. Int J Comput Methods 15(3):1–19zbMATHGoogle Scholar
  26. Naik PA, Pardasani KR (2018b) 2D finite-element analysis of calcium distribution in oocytes. Netw Model Anal Health Inf Bioinf.  https://doi.org/10.1007/s13721-018-0172-2
  27. Pathak K, Adlakha N (2015a) Finite element model to study calcium signalling in cardiac myocytes involving pump. Leak and excess buffer. J Med Imaging Health Inf 5:1–6CrossRefGoogle Scholar
  28. Pathak K, Adlakha N (2015b) Finite element model to study two dimensional unsteady state calcium distribution in cardiac myocytes. Alexandria J Med.  https://doi.org/10.1016/j.ajme.2015.09.007 CrossRefGoogle Scholar
  29. Pchitskaya E, Popugaeva E, Bezprozvanny I (2017) Calcium signaling and molecular mechanisms underlying neurodegenerative diseases. Cell Calcium.  https://doi.org/10.1016/j.ceca.2017.06.008 CrossRefGoogle Scholar
  30. Rajagopal S, Ponnusamy M (2017) Calcium signaling: from physiology to diseases. Springer, SingaporeCrossRefGoogle Scholar
  31. Schmidt H (2012) Three functional facets of calbindin D-28k. Front Mol Neurosci 5:1–7CrossRefGoogle Scholar
  32. Schwaller B (2010) Cytosolic \(Ca^{2+}\) Buffers. Cold Spring Harbor Perspect Biol 1–20Google Scholar
  33. Singh N, Adlakha N (2019) A mathematical model for interdependent calcium and inositol 1,4,5trisphosphate in cardiac myocyte. Netw Model Anal Health Inf Bioinf.  https://doi.org/10.1007/s13721-019-0198-0
  34. Small DH (2009) Dysregulation of calcium homeostasis in Alzheimer’s disease. Neurochem Res 34:1824–1829CrossRefGoogle Scholar
  35. Smith GD (1996) Analytical steady-state solution to the rapid buffering approximation near an open \(Ca^{2+}\) channel. Biophys J 71:3064–3072CrossRefGoogle Scholar
  36. Squire L et al (2008) Fundamental neuroscience, Third edn. Elsevier, AmsterdamGoogle Scholar
  37. Supnet C, Bezprozvanny I (2010) Neuronal calcium signaling, mitochondrial dysfunction and Alzheimer’s disease. J Alzheimers Dis 20(2):S487–S498CrossRefGoogle Scholar
  38. Tewari SG, Pardasani KR (2011) Finite element model to study two dimensional unsteady state cytosolic calcium diffusion. J Appl Math Inf 29:427–442MathSciNetzbMATHGoogle Scholar
  39. Turkington C, Mitchell D (2010) The encyclopedia of Alzheimer’s disease second. Facts on file: an imprint. Infobase Publishing, New YorkGoogle Scholar
  40. Verkhratsky A et al (2010) Astrocytes in Alzheimer’s Disease. Neurotherap J Am Soc Exp Neurotherap 7:399–412CrossRefGoogle Scholar
  41. Wang Y, Shi Y, Wei H (2017) Calcium dysregulation in Alzheimer’s disease: a target for new drug development. J Alzheimer’s Dis Parkinsinism 7(5):Google Scholar
  42. Yadav RR et al (2012) Three-dimensional temporally dependent dispersion through porous media: analytical solution. Environ Earth Sci 65:849–859CrossRefGoogle Scholar
  43. Yagami T, Kohma H, Yamamoto Y (2012) L-type voltage-dependent calcium channels as therapeutic targets for neuro- degenerative diseases. Curr Med Chem 1:4816–4827CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of TechnologyPDPUGandhinagarIndia

Personalised recommendations