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

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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

References

  1. Augustine GJ, Santamaria F, Tanaka K (2003) Local calcium signaling in neurons. Neuron 40:331–346

    Article  Google Scholar 

  2. Bezprozvanny I (2011) Calcium signaling and neurodegenerative diseases. Trends Mol Med 15:89–100

    Article  Google Scholar 

  3. Brzyska M, Elbaum D (2003) Dysregulation of calcium in Alzheimer’s disease. Acta Neurobiol Exp 63:171–183

    Google Scholar 

  4. Carafoli E, Brini M (eds) (2007) Calcium signalling and disease. Springer, New York

    Google Scholar 

  5. Clapham DE (2007) Calcium signaling. Cell 131:1047–1058

    Article  Google Scholar 

  6. Coe H, Michalak M (2009) Calcium binding chaperones of the endoplasmic reticulum. Gen Physiol Biophys 28:96–103

    Google Scholar 

  7. Crank J (1975) The mathematics of diffusion, Second edn. Clarendon Press, Oxford

    Google Scholar 

  8. Dave DD, Jha BK (2018a) Analytically depicting the calcium diffusion for Alzheimer’s affected cell. Int J Biomath 11(6):1850088–1850101

    MathSciNet  Article  Google Scholar 

  9. Dave DD, Jha BK (2018b) Delineation of calcium diffusion in alzheimeric brain. J Mech Med Biol 18(2):1–15

    Google Scholar 

  10. Demuro A, Parker I, Stutzmann GE (2010) Calcium signaling and amyloid toxicity in Alzheimer’s disease. J Biol Chem 6:1–1

    Google Scholar 

  11. Fall C et al (2002) Computational cell biology. Springer, New York

    Google 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):1–14

    Article  Google 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):135–144

    MathSciNet  MATH  Google 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):1–11

    MathSciNet  Article  Google 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):1–22

    Google Scholar 

  16. Keener J, Sneyd J (2009) Mathematical physiology second. Springer, New York

    Google Scholar 

  17. Khachaturian ZS (1993) Calcium hypothesis of Alzheimer’s disease and brain aging. Ann N Y Acad Sci 1–11

  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):840–847

    Article  Google Scholar 

  19. Laferla FM (2002) Calcium dyshomeostasis and intracellular signalling in Alzheimer’s disease. Nat Rev Neurosci 3:862–872

    Article  Google Scholar 

  20. Magi S et al (2016) Intracellular calcium dysregulation: implications for Alzheimer’s disease. Biomed Res Int 2016:1–14

    Article  Google 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):1250065-01–1250065-15

    MathSciNet  Article  Google 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):1550074-01–1550074-30

    MathSciNet  Article  Google Scholar 

  23. Mattson MP et al (2000) Calcium signaling in the ER: its role in neuronal plasticity and neurodegenerative disorders. Trends Neurosci 23(5):222–229

    Article  Google Scholar 

  24. Morris G et al (2018) Could Alzheimer’s disease originate in the periphery and if so how so? Mol Neurobiol

  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):1–19

    MATH  Google 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 5:1–6

    Article  Google 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

    Article  Google 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

    Article  Google Scholar 

  30. Rajagopal S, Ponnusamy M (2017) Calcium signaling: from physiology to diseases. Springer, Singapore

    Google Scholar 

  31. Schmidt H (2012) Three functional facets of calbindin D-28k. Front Mol Neurosci 5:1–7

    Article  Google Scholar 

  32. Schwaller B (2010) Cytosolic \(Ca^{2+}\) Buffers. Cold Spring Harbor Perspect Biol 1–20

  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 34:1824–1829

    Article  Google Scholar 

  35. Smith GD (1996) Analytical steady-state solution to the rapid buffering approximation near an open \(Ca^{2+}\) channel. Biophys J 71:3064–3072

    Article  Google Scholar 

  36. Squire L et al (2008) Fundamental neuroscience, Third edn. Elsevier, Amsterdam

    Google Scholar 

  37. Supnet C, Bezprozvanny I (2010) Neuronal calcium signaling, mitochondrial dysfunction and Alzheimer’s disease. J Alzheimers Dis 20(2):S487–S498

    Article  Google Scholar 

  38. Tewari SG, Pardasani KR (2011) Finite element model to study two dimensional unsteady state cytosolic calcium diffusion. J Appl Math Inf 29:427–442

    MathSciNet  MATH  Google Scholar 

  39. Turkington C, Mitchell D (2010) The encyclopedia of Alzheimer’s disease second. Facts on file: an imprint. Infobase Publishing, New York

    Google Scholar 

  40. Verkhratsky A et al (2010) Astrocytes in Alzheimer’s Disease. Neurotherap J Am Soc Exp Neurotherap 7:399–412

    Article  Google 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):

  42. Yadav RR et al (2012) Three-dimensional temporally dependent dispersion through porous media: analytical solution. Environ Earth Sci 65:849–859

    Article  Google 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 1:4816–4827

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Devanshi D. Dave.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dave, D.D., Jha, B.K. 3D mathematical modeling of calcium signaling in Alzheimer’s disease. Netw Model Anal Health Inform Bioinforma 9, 1 (2020). https://doi.org/10.1007/s13721-019-0207-3

Download citation

Keywords

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

Mathematics Subject Classification

  • 92B05
  • 92C30
  • 35K57