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October  2021, 14(10): 3387-3399. doi: 10.3934/dcdss.2021017

## Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator

 1 Department of Mathematics, Malaviya National Institute of Technology, Jaipur-302017, INDIA 2 Department of HEAS(Mathematics), RTU, Kota-324010, INDIA 3 Department of Mathematics, University of Rajasthan, Jaipur 302004, Rajasthan, India

* Corresponding author: Devendra Kumar

Received  October 2019 Revised  December 2019 Published  October 2021 Early access  March 2021

The aim of this paper is to study the calcium profile governed by the advection diffusion equation. The mathematical and computational modeling has provided insights to understand the calcium signalling which depends upon cytosolic calcium concentration. Here the model includes the important physiological parameters like diffusion coefficient, flow velocity etc. The mathematical model is fractionalised using Hilfer derivative and appropriate boundary conditions have been framed. The use of fractional order derivative is more advantageous than the integer order because of the non-local property of the fractional order differentiation operator i.e. the next state of the system depends not only upon its current state but also upon all of its preceeding states. Analytic solution of the fractional advection diffusion equation arising in study of diffusion of cytosolic calcium in RBC is found using integral transform techniques. Since, the Hilfer derivative is generalisation of Riemann- Liouville and Caputo derivatives so, these two are also deduced as special cases. The numerical simulation has been done to observe the effects of the fractional order of the derivatives involved in the differential equation representing the model over the concentration of calcium which is function of time and distance. The concentration profile of calcium is significantly changed by the fractional order.

Citation: Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3387-3399. doi: 10.3934/dcdss.2021017
##### References:
 [1] R. Agarwal, S. Jain and R. P. Agarwal, Mathematical modeling and analysis of dynamics cytosolic calcium in astrocytes using fractional calculus, Journal of Fractional Calculus and Application, 9 (2018), 1-12. [2] R. Agarwal and S. D. Purohit, A mathematical fractional model with non-singular kernel for thrombin receptor activation in calcium signalling, Math Meth Appl Sci., 42 (2019), 7160-7171.  doi: 10.1002/mma.5822. [3] B. S. T. Alkahtani and A. Atangana, Analysis of non-homogenous heat model with new trend of derivative with fractional order, Chaos, Soltons and Fractals, 89 (2016), 566-571.  doi: 10.1016/j.chaos.2016.03.027. [4] K. S. Al-Ghafri and H. Rezazadeh, Solitons and other solutions of (3+ 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 289-304. [5] N. A. Asif, Z. Hammouch, M. B. Riaz and H. Bulut, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, The European Physical Journal Plus, 133 (2018), 272. doi: 10.1140/epjp/i2018-12098-6. [6] I. Area, H. Batarfi, J. Losada, J. Nieto, W. Shammakh and A. Torres, On a fractional order Ebola epidemic model, Advances in Difference Equations, 2015 (2015), 278-300.  doi: 10.1186/s13662-015-0613-5. [7] A. Atangana and B. Alkahtani, Analysis of the Keller–Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453.  doi: 10.3390/e17064439. [8] A. Atangana and Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors, The European Physical Journal Plus, 134 (2019), 429. doi: 10.1140/epjp/i2019-12777-8. [9] D. Baleanu, Z. B. Güvenç and J. T. Machado, New trends in nanotechnology and fractional calculus applications, Springer, (2010). [10] D. Baleanu, D. Kumar and S. D. Purohit, Generalized fractional integrals of product of two H-functions and a general class of polynomials, International Journal of Computer Mathematics, 93 (2016), 1320-1329.  doi: 10.1080/00207160.2015.1045886. [11] W. D. Brzeziński, Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus, Applied Mathematics and Nonlinear Sciences, 3 (2018), 487-502. [12] M. Caputo, Linear models of dissipation whos Q is almost frequency independent-II, Geophysical Journal International, 13 (1967), 529-539. [13] M. Caputo, Elasticitò e Dissipazione, Zanichelli, Bologna, 1969. [14] G. Dupont and A. Goldbeter, Cam kinase II as frequency decoder of Ca2+ oscillations, Bioessays, 20 (1998), 607-610.  doi: 10.1002/(SICI)1521-1878(199808)20:8<607::AID-BIES2>3.0.CO;2-F. [15] H. Haubold, A. Mathai and R. Saxena, Solution of fractional reaction-diffusion equations in terms of H-function, J. Comput. Appl. Math., 235 (2011), 1311-1316.  doi: 10.1016/j.cam.2010.08.016. [16] R. Hilfer, Fractional time evolution, Applications of Fractional Calculus in Physics, (2000), 87–130. doi: 10.1142/9789812817747_0002. [17] M. A. Imran, M. Aleem, M. B. Riaz, R. Ali and I. Khan., A comprehensive report on convective flow of fractional (ABC) and (CF) MHD viscous fluid subject to generalized boundary conditions, Chaos, Solitons & Fractals, 118 (2019), 274-289.  doi: 10.1016/j.chaos.2018.12.001. [18] B. K. Jha, N. Adlakha and M. Mehta, Finite element model to study calcium diffusion in astrocytes, Int. J. of Pure and Appl. Math, 78 (2012), 945-955. [19] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006. [20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, (2006). [21] D. Kumar, J. Singh and D. Baleanu, A fractional model of convective radial fins with temperature-dependent thermal conductivity, Rom. Rep. Phys, 69 (2017), 103. [22] D. Kumar, J. Singh and D. Baleanu, Numerical computation of a fractional model of differential-difference equation, Journal of Computational and Nonlinear Dynamics, 11 (2016), 061004, 6 pp. doi: 10.1115/1.4033899. [23] D. Kumar, J. Singh and D. Baleanu, A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel, The European Physical Journal Plus, 133 (2018), 70. [24] D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, International Journal of Heat and Mass Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094. [25] D. Kumar, J. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences, 43 (2020), 443-457.  doi: 10.1002/mma.5903. [26] D. Kumar, J. Singh and D. Baleanu, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A., 492 (2018), 155-167.  doi: 10.1016/j.physa.2017.10.002. [27] D. Kumar, F. Tchier, J. Singh and D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. doi: 10.3390/e20040259. [28] D. Kumar, J. Singh and D. Baleanu, A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 2791-2802.  doi: 10.2298/TSCI170129096K. [29] K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley Interscience, 1993. [30] G. Mittag-Lffler, Sur la nouvelle function E(x), CR Acad. Sci. Paris, 137 (1903), 554-558. [31] M. B. Riaz and A. A. Zafar, Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives, Mathematical Modelling of Natural Phenomena, 13 (2018), 8-20.  doi: 10.1051/mmnp/2018005. [32] J. Singh, D. Kumar and A. Kilçman, Numerical solution of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abstract and Applied Analysis, 2014 (2014), 535793, 12 pp. doi: 10.1155/2014/535793. [33] J. Singh, A. Kilicman, D. Kumar, R. Swroop and F. M. Ali, Numerical study for fractional model of nonlinear predator-prey biological population dynamical system, Thermal Science, 23 (2019), 2017-2025.  doi: 10.2298/TSCI190725366S. [34] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068. [35] J. Singh, D. Kumar and D. Baleanu, On the analysis of fractional diabetes model with exponential law, Advances in Difference Equations, 2018 (2018), 231-246.  doi: 10.1186/s13662-018-1680-1. [36] J. Singh, A new analysis for fractional rumor spreading dynamical model in a social network with Mittag-Leffler law, Chaos, 29 (2019), 013137, 7 pp. doi: 10.1063/1.5080691. [37] I. N. Sneddon, Fourier Transforms, Inc., New York, 1951. [38] Z. Tomovski, Generalised cauchy type problems for nonlinear fractional differential equation with composite fractional derivative operator, Nonlinear Anal-Theor., 75 (2012), 3364-3384.  doi: 10.1016/j.na.2011.12.034. [39] T. F. Wiesner, B. C. Berk and R. M. Nerem, A mathematical model of cytosolic calcium dynamics in human umbilical vein endothelial cells, American Journal of Physiology, 270 (1996), C1556–C1569. doi: 10.1152/ajpcell.1996.270.5.C1556. [40] A. Wiman, On the fundamental theorem in the theory of functions, 29 (1905), 191–201. [41] F. K. Winston, L. E. Thibault and E. J. Macarak, An analysis of the time-dependent changes in intracellular calcium concentration in endothelial cells in culture induced by mechanical stimulation, J. Biomech. Eng., 115 (1993), 160-168.  doi: 10.1115/1.2894116. [42] M. P. Yadav and R. Agarwal, Numerical investigation of fractional-fractal boussinesq equation, Chaos, 29 (2019), 013109, 7 pp. doi: 10.1063/1.5080139. [43] I. K. Youssef and M. H. Dewaik, Solving Poisson's Equations with fractional order using Haarwavelet, Applied Mathematics and Nonlinear Sciences, 2 (2017), 271-284.  doi: 10.21042/AMNS.2017.1.00023. [44] A. Yokuş, Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics B, 32 (2018), 1850365, 12pp. doi: 10.1142/S0217979218503654. [45] A. Yokuş and S. Gülbahar, Numerical solutions with linearization techniques of the fractional harry dym equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 35-41. [46] Y. Zhang, C. Cattani and X. Yang, Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains, Entropy, 17 (2015), 6753-6764.  doi: 10.3390/e17106753.

show all references

##### References:
 [1] R. Agarwal, S. Jain and R. P. Agarwal, Mathematical modeling and analysis of dynamics cytosolic calcium in astrocytes using fractional calculus, Journal of Fractional Calculus and Application, 9 (2018), 1-12. [2] R. Agarwal and S. D. Purohit, A mathematical fractional model with non-singular kernel for thrombin receptor activation in calcium signalling, Math Meth Appl Sci., 42 (2019), 7160-7171.  doi: 10.1002/mma.5822. [3] B. S. T. Alkahtani and A. Atangana, Analysis of non-homogenous heat model with new trend of derivative with fractional order, Chaos, Soltons and Fractals, 89 (2016), 566-571.  doi: 10.1016/j.chaos.2016.03.027. [4] K. S. Al-Ghafri and H. Rezazadeh, Solitons and other solutions of (3+ 1)-dimensional space–time fractional modified KdV–Zakharov–Kuznetsov equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 289-304. [5] N. A. Asif, Z. Hammouch, M. B. Riaz and H. Bulut, Analytical solution of a Maxwell fluid with slip effects in view of the Caputo-Fabrizio derivative, The European Physical Journal Plus, 133 (2018), 272. doi: 10.1140/epjp/i2018-12098-6. [6] I. Area, H. Batarfi, J. Losada, J. Nieto, W. Shammakh and A. Torres, On a fractional order Ebola epidemic model, Advances in Difference Equations, 2015 (2015), 278-300.  doi: 10.1186/s13662-015-0613-5. [7] A. Atangana and B. Alkahtani, Analysis of the Keller–Segel model with a fractional derivative without singular kernel, Entropy, 17 (2015), 4439-4453.  doi: 10.3390/e17064439. [8] A. Atangana and Z. Hammouch, Fractional calculus with power law: The cradle of our ancestors, The European Physical Journal Plus, 134 (2019), 429. doi: 10.1140/epjp/i2019-12777-8. [9] D. Baleanu, Z. B. Güvenç and J. T. Machado, New trends in nanotechnology and fractional calculus applications, Springer, (2010). [10] D. Baleanu, D. Kumar and S. D. Purohit, Generalized fractional integrals of product of two H-functions and a general class of polynomials, International Journal of Computer Mathematics, 93 (2016), 1320-1329.  doi: 10.1080/00207160.2015.1045886. [11] W. D. Brzeziński, Review of numerical methods for NumILPT with computational accuracy assessment for fractional calculus, Applied Mathematics and Nonlinear Sciences, 3 (2018), 487-502. [12] M. Caputo, Linear models of dissipation whos Q is almost frequency independent-II, Geophysical Journal International, 13 (1967), 529-539. [13] M. Caputo, Elasticitò e Dissipazione, Zanichelli, Bologna, 1969. [14] G. Dupont and A. Goldbeter, Cam kinase II as frequency decoder of Ca2+ oscillations, Bioessays, 20 (1998), 607-610.  doi: 10.1002/(SICI)1521-1878(199808)20:8<607::AID-BIES2>3.0.CO;2-F. [15] H. Haubold, A. Mathai and R. Saxena, Solution of fractional reaction-diffusion equations in terms of H-function, J. Comput. Appl. Math., 235 (2011), 1311-1316.  doi: 10.1016/j.cam.2010.08.016. [16] R. Hilfer, Fractional time evolution, Applications of Fractional Calculus in Physics, (2000), 87–130. doi: 10.1142/9789812817747_0002. [17] M. A. Imran, M. Aleem, M. B. Riaz, R. Ali and I. Khan., A comprehensive report on convective flow of fractional (ABC) and (CF) MHD viscous fluid subject to generalized boundary conditions, Chaos, Solitons & Fractals, 118 (2019), 274-289.  doi: 10.1016/j.chaos.2018.12.001. [18] B. K. Jha, N. Adlakha and M. Mehta, Finite element model to study calcium diffusion in astrocytes, Int. J. of Pure and Appl. Math, 78 (2012), 945-955. [19] F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana–Baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006. [20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science Limited, (2006). [21] D. Kumar, J. Singh and D. Baleanu, A fractional model of convective radial fins with temperature-dependent thermal conductivity, Rom. Rep. Phys, 69 (2017), 103. [22] D. Kumar, J. Singh and D. Baleanu, Numerical computation of a fractional model of differential-difference equation, Journal of Computational and Nonlinear Dynamics, 11 (2016), 061004, 6 pp. doi: 10.1115/1.4033899. [23] D. Kumar, J. Singh and D. Baleanu, A new analysis of the Fornberg-Whitham equation pertaining to a fractional derivative with Mittag-Leffler-type kernel, The European Physical Journal Plus, 133 (2018), 70. [24] D. Kumar, J. Singh, K. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, International Journal of Heat and Mass Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094. [25] D. Kumar, J. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences, 43 (2020), 443-457.  doi: 10.1002/mma.5903. [26] D. Kumar, J. Singh and D. Baleanu, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A., 492 (2018), 155-167.  doi: 10.1016/j.physa.2017.10.002. [27] D. Kumar, F. Tchier, J. Singh and D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. doi: 10.3390/e20040259. [28] D. Kumar, J. Singh and D. Baleanu, A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 2791-2802.  doi: 10.2298/TSCI170129096K. [29] K. S. Miller and B. Ross, An Introduction to The Fractional Calculus and Fractional Differential Equations, Wiley Interscience, 1993. [30] G. Mittag-Lffler, Sur la nouvelle function E(x), CR Acad. Sci. Paris, 137 (1903), 554-558. [31] M. B. Riaz and A. A. Zafar, Exact solutions for the blood flow through a circular tube under the influence of a magnetic field using fractional Caputo-Fabrizio derivatives, Mathematical Modelling of Natural Phenomena, 13 (2018), 8-20.  doi: 10.1051/mmnp/2018005. [32] J. Singh, D. Kumar and A. Kilçman, Numerical solution of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abstract and Applied Analysis, 2014 (2014), 535793, 12 pp. doi: 10.1155/2014/535793. [33] J. Singh, A. Kilicman, D. Kumar, R. Swroop and F. M. Ali, Numerical study for fractional model of nonlinear predator-prey biological population dynamical system, Thermal Science, 23 (2019), 2017-2025.  doi: 10.2298/TSCI190725366S. [34] J. Singh, D. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068. [35] J. Singh, D. Kumar and D. Baleanu, On the analysis of fractional diabetes model with exponential law, Advances in Difference Equations, 2018 (2018), 231-246.  doi: 10.1186/s13662-018-1680-1. [36] J. Singh, A new analysis for fractional rumor spreading dynamical model in a social network with Mittag-Leffler law, Chaos, 29 (2019), 013137, 7 pp. doi: 10.1063/1.5080691. [37] I. N. Sneddon, Fourier Transforms, Inc., New York, 1951. [38] Z. Tomovski, Generalised cauchy type problems for nonlinear fractional differential equation with composite fractional derivative operator, Nonlinear Anal-Theor., 75 (2012), 3364-3384.  doi: 10.1016/j.na.2011.12.034. [39] T. F. Wiesner, B. C. Berk and R. M. Nerem, A mathematical model of cytosolic calcium dynamics in human umbilical vein endothelial cells, American Journal of Physiology, 270 (1996), C1556–C1569. doi: 10.1152/ajpcell.1996.270.5.C1556. [40] A. Wiman, On the fundamental theorem in the theory of functions, 29 (1905), 191–201. [41] F. K. Winston, L. E. Thibault and E. J. Macarak, An analysis of the time-dependent changes in intracellular calcium concentration in endothelial cells in culture induced by mechanical stimulation, J. Biomech. Eng., 115 (1993), 160-168.  doi: 10.1115/1.2894116. [42] M. P. Yadav and R. Agarwal, Numerical investigation of fractional-fractal boussinesq equation, Chaos, 29 (2019), 013109, 7 pp. doi: 10.1063/1.5080139. [43] I. K. Youssef and M. H. Dewaik, Solving Poisson's Equations with fractional order using Haarwavelet, Applied Mathematics and Nonlinear Sciences, 2 (2017), 271-284.  doi: 10.21042/AMNS.2017.1.00023. [44] A. Yokuş, Comparison of Caputo and conformable derivatives for time-fractional Korteweg–de Vries equation via the finite difference method, International Journal of Modern Physics B, 32 (2018), 1850365, 12pp. doi: 10.1142/S0217979218503654. [45] A. Yokuş and S. Gülbahar, Numerical solutions with linearization techniques of the fractional harry dym equation, Applied Mathematics and Nonlinear Sciences, 4 (2019), 35-41. [46] Y. Zhang, C. Cattani and X. Yang, Local fractional homotopy perturbation method for solving non-homogeneous heat conduction equations in fractal domains, Entropy, 17 (2015), 6753-6764.  doi: 10.3390/e17106753.
Graph between $C(x, t)$ and $t$ for various values of $\xi$ for $\eta = 0$ which corresponds to the Riemann- Liouville derivative
Graph between $C(x, t)$ and t for various values of $\xi$ for $\eta = 1$ which corresponds to Caputo derivative
Graph between $C(x, t)$ and t for various values of $\xi$ for $\eta = 0.90$
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