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Deciphering two-dimensional calcium fractional diffusion of membrane flux in neuron

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Abstract

Calcium is a decisive messenger for neuronal vivid functions. The calcium intracellular sequestering major unit is the Endoplasmic Reticulum (ER). Brownian motion of calcium could be bound to different buffers like S100B, calmodulin, etc, and different organelles. Plasma membrane channels like voltage-gated calcium channels (VGCC) and Plasma Membrane Calcium ATPase (PMCA), Orai channel could perturb the calcium concentration. To investigate the calcium interplay for intracellular signaling we have developed the two-dimensional time fractional reaction–diffusion equation. To solve this model analytically, we have used the Laplace and Fourier cosine integral transform method. By using Green’s function we obtained the compact solution in closed form with Mainardi’s function and Wright’s function. Uniqueness and existence proved the more fundamental approach to the fractional reaction–diffusion problem. The fractional Caputo approach gives better insight into this real-life problem by its nonlocal nature. Significant effects of different parameters on free calcium ions were obtained and the results are interpreted with normal and Alzheimeric cells. Non-local property and dynamical aspects are graphically presented which might provide insight into the Stromal interaction molecule (STIM) and S100B parameters.

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References

  1. LaFerla, F.M.: Calcium dyshomeostasis and intracellular signaling in Alzheimer’s disease. Nat. Rev. Neurosci. 3, 862–872 (2002). https://doi.org/10.1038/nrn960

    Article  Google Scholar 

  2. Kraft, R.: STIM and ORAI proteins in the nervous system. Channels 9, 245–252 (2015). https://doi.org/10.1080/19336950.2015.1071747

    Article  Google Scholar 

  3. Bezprozvanny, I.B.: Calcium Signaling and Neurodegeneration. Acta Nat. 2, 72–80 (2010). https://doi.org/10.32607/20758251-2010-2-1-72-80

    Article  Google Scholar 

  4. Mattson, M.R.: Calcium and neurodegeneration. Aging Cell. 6, 337–350 (2007). https://doi.org/10.1111/j.1474-9726.2007.00275.x

    Article  Google Scholar 

  5. Smith, G.D.: Analytical steady-state solution to the rapid buffering approximation near an open Ca2+ channel. Biophys J. 71, 3064–3072 (1996). https://doi.org/10.1016/S0006-3495(96)79500-0

    Article  Google Scholar 

  6. Smith, G.D., Dai, L., Miura, R.M., Sherman, A.: Asymptotic analysis of buffered calcium diffusion near a point source. SIAM J. Appl. Math. 61, 1816–1838 (2001). https://doi.org/10.1137/S0036139900368996

    Article  MathSciNet  Google Scholar 

  7. González-Vélez, V., Piron, A., Dupont, G.: Calcium oscillations in pancreatic \(\alpha \)-cells Rely on noise and ATP-driven changes in membrane electrical activity. Front. Physiol. 11, 602844 (2020). https://doi.org/10.3389/fphys.2020.602844

    Article  Google Scholar 

  8. Dupont, G., Houart, G., De Koninck, P.: Sensitivity of CaM kinase II to the frequency of Ca2+ oscillations: a simple model. Cell Calcium 34, 485–497 (2003). https://doi.org/10.1016/S0143-4160(03)00152-0

    Article  Google Scholar 

  9. Friedhoff, V.N., Ramlow, L., Lindner, B., Falcke, M.: Models of stochastic \(\text{ Ca}^{2+}\) spiking. Eur. Phys. J. Spec. Top. 230, 2911–2928 (2021). https://doi.org/10.1140/epjs/s11734-021-00174-1

    Article  Google Scholar 

  10. Schmeitz, C., Hernandez-Vargas, E.A., Fliegert, R., Guse, A.H., Meyer-Hermann, M.: A mathematical model of T lymphocyte calcium dynamics derived from single transmembrane protein properties. Front. Immunol. 4, 277 (2013). https://doi.org/10.3389/fimmu.2013.00277

    Article  Google Scholar 

  11. Marhl, M., Haberichter, T., Brumen, M., Heinrich, R.: Complex calcium oscillations and the role of mitochondria and cytosolic proteins. BioSystems 57, 75–86 (2000). https://doi.org/10.1016/S0303-2647(00)00090-3

    Article  Google Scholar 

  12. Brusch, L., Lorenz, W., Or-Guil, M., Bär, M., Kummer, U.: Fold–Hopf bursting in a model for calcium signal transduction. Zeitschrift für Physikalische Chemie. 216, 487 (2002). https://doi.org/10.1524/zpch.2002.216.4.487

    Article  Google Scholar 

  13. Dave, D.D., Jha, B.K.: Modeling the alterations in calcium homeostasis in the presence of protein and VGCC for Alzheimer cell. In: Advances in Intelligent Systems and Computing, (pp. 181-189) (2018). https://doi.org/10.1007/978-981-10-5699-4_18

  14. Jha, A., Adlakha, N.: Two-dimensional finite element model to study unsteady state Ca2+ diffusion in neuron involving ER LEAK and SERCA. Int. J. Biomath. 8, 1550002 (2015). https://doi.org/10.1142/S1793524515500023

    Article  MathSciNet  Google Scholar 

  15. Tewari, S.G., Camara, A.K.S., Stowe, D.F., Dash, R.K.: Computational analysis of Ca2+ dynamics in isolated cardiac mitochondria predicts two distinct modes of Ca2+ uptake. J. Physiol. 592, 1917–1930 (2014). https://doi.org/10.1113/jphysiol.2013.268847

    Article  Google Scholar 

  16. Wei, N., Layton, A.T.: Theoretical assessment of the Ca 2+ oscillations in the afferent arteriole smooth muscle cell of the rat kidney. Int. J. Biomath. 11, 1850043 (2018). https://doi.org/10.1142/S1793524518500432

    Article  MathSciNet  Google Scholar 

  17. Pawar, A., Pardasani, K.R.: Fractional order interdependent nonlinear chaotic spatiotemporal calcium and \(A\beta \) dynamics in a neuron cell. Phys. Scr. 98, 085206 (2023). https://doi.org/10.1088/1402-4896/ace1b2

    Article  Google Scholar 

  18. Dave, D.D., Jha, B.K.: 2D finite element estimation of calcium diffusion in Alzheimer’s affected neuron. Netw. Model. Anal. Health Inform. Bioinform. 10, 43 (2021). https://doi.org/10.1007/s13721-021-00322-6

    Article  Google Scholar 

  19. Naik, P.A., Pardasani, K.R.: 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 16, 1850091 (2019). https://doi.org/10.1142/S0219876218500913

    Article  MathSciNet  Google Scholar 

  20. Naik, P.A., Eskandari, Z., Yavuz, M., Zu, J.: Complex dynamics of a discrete-time Bazykin–Berezovskaya prey-predator model with a strong Allee effect. J. Comput. Appl. Math. 413, 114401 (2022). https://doi.org/10.1016/j.cam.2022.114401

    Article  MathSciNet  Google Scholar 

  21. Pawar, A., Pardasani, K.R.: Effect of disturbances in neuronal calcium and IP3 dynamics on \(\beta \)-amyloid production and degradation. Cogn. Neurodyn. 17, 239–256 (2022). https://doi.org/10.1007/s11571-022-09815-0

    Article  Google Scholar 

  22. Jagtap, Y., Adlakha, N.: Numerical model of hepatic glycogen phosphorylase regulation by nonlinear interdependent dynamics of calcium and \(IP_{3}\). Eur. Phys. J. Plus 138, 399 (2023). https://doi.org/10.1140/epjp/s13360-023-03961-y

    Article  Google Scholar 

  23. Kothiya, A., Adlakha, N.: Simulation of biochemical dynamics of \(C{a}^{2+}\) and \(PLC\) in fibroblast cell. J. Bioenerg. Biomembr. 55, 267–287 (2023). https://doi.org/10.1007/s10863-023-09976-5

    Article  Google Scholar 

  24. Joshi, H., Jha, B.K.: Chaos of calcium diffusion in Parkinson’s infectious disease model and treatment mechanism via Hilfer fractional derivative. Math. Model. Numer. Simul. Appl. 1, 84–94 (2021). https://doi.org/10.53391/mmnsa.2021.01.008

    Article  Google Scholar 

  25. Jethanandanİ, H., Jha, B.K., Ubale, M.: The role of calcium dynamics with amyloid beta on neuron-astrocyte coupling. Math. Model. Numer. Simul. Appl. 3, 376–390 (2023). https://doi.org/10.53391/mmnsa.1398320

    Article  Google Scholar 

  26. Joshi, H., Jha, B.K.: On a reaction–diffusion model for calcium dynamics in neurons with Mittag–Leffler memory. Eur. Phys. J. Plus 136, 623 (2021). https://doi.org/10.1140/epjp/s13360-021-01610-w

    Article  Google Scholar 

  27. Luchko, Y., Suzuki, A., Yamamoto, M.: On the maximum principle for the multi-term fractional transport equation. J. Math. Anal. Appl. 505, 125579 (2022). https://doi.org/10.1016/j.jmaa.2021.125579

    Article  MathSciNet  Google Scholar 

  28. Vatsal, V.H., Jha, B.K., Singh, T.P.: To study the effect of ER flux with buffer on the neuronal calcium. Eur. Phys. J. Plus 138(494), 1–14 (2023). https://doi.org/10.1140/epjp/s13360-023-04077-z

    Article  Google Scholar 

  29. Naik, P.A., Pardasani, K.R.: Finite element model to study calcium distribution in oocytes involving voltage gated Ca2+ channel, ryanodine receptor and buffers. Alex. J. Med. 52, 43–49 (2016). https://doi.org/10.1016/j.ajme.2015.02.002

    Article  Google Scholar 

  30. Naik, P.A., Pardasani, K.R.: Three-dimensional finite element model to study calcium distribution in oocytes. Netw. Model. Anal. Health Inform. Bioinform. 6(16), 1–11 (2017). https://doi.org/10.1007/s13721-017-0158-5

    Article  Google Scholar 

  31. Naik, P.A., Farman, M., Zehra, A., Nisar, K.S., Hincal, E.: Analysis and modeling with fractal-fractional operator for an epidemic model with reference to COVID-19 modeling. Partial Differ. Equ. Appl. Math. 10, 100663 (2024). https://doi.org/10.1016/j.padiff.2024.100663

    Article  Google Scholar 

  32. Singh, T., Adlakha, N.: Numerical investigations and simulation of calcium distribution in the alpha-cell. Bull. Biomath. 1, 40–57 (2023). https://doi.org/10.59292/bulletinbiomath.2023003

    Article  Google Scholar 

  33. Kumar, H., Naik, P.A., Pardasani, K.R.: Finite element model to study calcium distribution in T lymphocyte involving buffers and ryanodine receptors. Proc. Natl. Acad. Sci. India Sect. A Phys. Sci. 88, 585–590 (2018). https://doi.org/10.1007/s40010-017-0380-7

    Article  Google Scholar 

  34. Naik, P.A.: Modeling the mechanics of calcium regulation in T lymphocyte: a finite element method approach. Int. J. Biomath. 13, 2050038 (2020). https://doi.org/10.1142/S1793524520500382

    Article  MathSciNet  Google Scholar 

  35. Jha, B.K., Joshi, H., Dave, D.D.: Portraying the effect of calcium-binding proteins on cytosolic calcium concentration distribution fractionally in nerve cells. Interdiscip. Sci. 10, 674–685 (2018). https://doi.org/10.1007/s12539-016-0202-7

    Article  Google Scholar 

  36. Joshi, H., Yavuz, M.: Numerical analysis of compound biochemical calcium oscillations process in hepatocyte cells. Adv. Biol. (2024). https://doi.org/10.1002/adbi.202300647

    Article  Google Scholar 

  37. Joshi, H., Yavuz, M., Stamova, I.: Analysis of the disturbance effect in intracellular calcium dynamic on fibroblast cells with an exponential kernel law. Bull. Biomath. 1, 24–39 (2023). https://doi.org/10.59292/bulletinbiomath.2023002

    Article  Google Scholar 

  38. Jethanandani, H., Jha, B.K., Ubale, M.: Bifurcation analysis of calcium dynamics in nerve cell. Eur. Phys. J. Plus 138, 1159 (2023). https://doi.org/10.1140/epjp/s13360-023-04699-3

    Article  Google Scholar 

  39. Bhattacharyya, R., Jha, B.K.: Analyzing fuzzy boundary value problems: a study on the influence of mitochondria and ER fluxes on calcium ions in neuron cells. J. Bioenerg. Biomembr. (2024). https://doi.org/10.1007/s10863-023-09994-3

    Article  Google Scholar 

  40. Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press (1998)

  41. Mainardi, F., Pagnini, G.: The wright functions as solutions of the time-fractional diffusion equation. Appl. Math. Comput. 141, 51–62 (2003)

    MathSciNet  Google Scholar 

  42. Keener, J., Sneyd, J. (eds.): Mathematical Physiology. Springer, New York, New York, NY (2009)

    Google Scholar 

  43. Zhang, H., Sun, S., Wu, L., Pchitskaya, E., Zakharova, O., Tacer, K.F., Bezprozvanny, I.: Store-operated calcium channel complex in postsynaptic spines: a new therapeutic target for Alzheimer’s disease treatment. J. Neurosci. 36, 11837–11850 (2016). https://doi.org/10.1523/JNEUROSCI.1188-16.2016

    Article  Google Scholar 

  44. Gil, D., Guse, A.H., Dupont, G.: Three-dimensional model of sub-plasmalemmal Ca2+ microdomains evoked by the interplay between ORAI1 and InsP3 receptors. Front. Immunol. 12, 659790 (2021). https://doi.org/10.3389/fimmu.2021.659790

    Article  Google Scholar 

  45. Manhas, N., Sneyd, J., Pardasani, K.R.: Modelling the transition from simple to complex Ca2+ oscillations in pancreatic acinar cells. J. Biosci. 39, 463–484 (2014). https://doi.org/10.1007/s12038-014-9430-3

    Article  Google Scholar 

  46. Sneyd, J., Tsaneva-Atanasova, K., Bruce, J.I.E., Straub, S.V., Giovannucci, D.R., Yule, D.I.: A model of calcium waves in pancreatic and parotid acinar cells. Biophys. J. 85, 1392–1405 (2003). https://doi.org/10.1016/S0006-3495(03)74572-X

    Article  Google Scholar 

  47. Dave, D.D., Jha, B.K.: Analytically depicting the calcium diffusion for Alzheimer’s affected cell. Int. J. Biomath. 11, 1850088 (2018). https://doi.org/10.1142/S1793524518500882

    Article  MathSciNet  Google Scholar 

  48. Berrocal, M., Mata, A.M.: The plasma membrane Ca2+-ATPase, a molecular target for Tau-induced cytosolic calcium dysregulation. Neuroscience 518, 112–118 (2022). https://doi.org/10.1016/j.neuroscience.2022.04.016

    Article  Google Scholar 

  49. Marambaud, P., Dreses-Werringloer, U., Vingtdeux, V.: Calcium signaling in neurodegeneration. Mol. Neurodegener. 4, 1–15 (2009). https://doi.org/10.1186/1750-1326-4-20

    Article  Google Scholar 

  50. Yagami, T., Kohma, H., Yamamoto, Y.: L-type voltage-dependent calcium channels as therapeutic targets for neurodegenerative diseases. Curr. Med. Chem. 19, 4816–4827 (2012). https://doi.org/10.2174/092986712803341430

    Article  Google Scholar 

  51. Jha, B.K., Adlakha, N., Mehta, M.N.: Finite volume model to study the effect of ER flux on cytosolic calcium distribution in astrocytes. J. Comput. 3, 74–80 (2011)

    Google Scholar 

  52. Miller, K.S., Ross, B.: An Introduction to The Fractional Calculus and Fractional Differential Equations, (1993)

  53. Diethelm, K.: The analysis of fractional differential equations: an application-oriented exposition using differential operators of caputo type. Lecture Notes in Mathematics. 2004 (2010)

  54. Du, A.T., Schuff, N., Amend, D., Laakso, M.P., Hsu, Y.Y., Jagust, W.J., Yaffe, K., Kramer, J.H., Reed, B., Norman, D., Chui, H.C., Weiner, M.W.: Magnetic resonance imaging of the entorhinal cortex and hippocampus in mild cognitive impairment and Alzheimer’s disease. J. Neurol. Neurosurg. Psychiatry 71, 441–447 (2001). https://doi.org/10.1136/jnnp.71.4.441

    Article  Google Scholar 

  55. Dave, D.D., Jha, B.K.: 3D mathematical modeling of calcium signaling in Alzheimer’s disease. Netw. Model. Anal. Health Inform. Bioinform. 9, 1–10 (2020). https://doi.org/10.1007/s13721-019-0207-3

    Article  Google Scholar 

  56. Cristóvaõ, J.S., Gomes, C.M.: S100 proteins in Alzheimer’s disease. Front. Neurosci. 13, 446874 (2019). https://doi.org/10.3389/fnins.2019.00463

    Article  Google Scholar 

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Acknowledgements

The authors are highly thankful to the SHODH scheme, the education department, Government of Gujarat, India for financial support for carrying out this research work.

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  1. Department of Mathematics, Pandit Deendayal Energy University, Gandhinagar, Gujarat, India

    Vora Hardagna Vatsal, Brajesh Kumar Jha & Tajinder Pal Singh

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Vatsal, V.H., Jha, B.K. & Singh, T.P. Deciphering two-dimensional calcium fractional diffusion of membrane flux in neuron. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02115-2

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