Structure preserving algorithms for mathematical model of auto-catalytic glycolysis chemical reaction and numerical simulations

Abstract

This paper aims to develop positivity preserving splitting techniques for glycolysis reaction–diffusion chemical model. The positivity of state variables in the glycolysis model is an essential property that must be preserved for all choices of parameters. We propose two splitting methods that remain dynamically consistent with the continuous glycolysis reaction–diffusion model. The proposed methods converge to a true steady-state or fixed point under the given condition. On contrary to the classical operator splitting finite difference methods, we use nonstandard finite difference theory to propose a new class of operator splitting techniques.

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References

  1. 1.

    P. Stoodley, K. Sauer, D.G. Davies, J.W. Costerton, Biofilms as complex differentiated communities. Ann. Rev. Microbiol. 56(1), 187–209 (2002)

    Article  Google Scholar 

  2. 2.

    J.D. Murray, Mathematical Biology, vol. 2 (Springer, Berlin, 2002)

    Google Scholar 

  3. 3.

    C. Liu, X. Fu, L. Liu, X. Ren, C.K.L. Chau, S. Li, L. Xiang, H. Zeng, G. Chen, L. Tang et al., Sequential establishment of stripe patterns in an expanding cell population. Science 334(6053), 238–241 (2011)

    ADS  Article  Google Scholar 

  4. 4.

    A. Jilkine, L. Edelstein-Keshet, A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues. PLoS Comput. Biol. 7(4), e1001121 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability. Chem. Eng. Sci. 38, 29–43 (1983)

    Article  Google Scholar 

  6. 6.

    P. Gray, S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and instabilities in the system A + 2B \(\rightarrow \) 3B, B\(\rightarrow \)C. Chem. Eng. Sci. 39, 1087–1097 (1984)

    Article  Google Scholar 

  7. 7.

    G. Nicolis, I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley-Interscience, New York, 1977)

    Google Scholar 

  8. 8.

    J. Schnakenberg, Simple chemical reaction systems with limit cycle behavior. Theor. Biol. 81, 389–400 (1979)

    MathSciNet  Article  Google Scholar 

  9. 9.

    A. Gierer, H. Meinhardt, A theory of biological pattern formation. Kybernetika 12, 30–39 (1972)

    Article  Google Scholar 

  10. 10.

    E.E. Sel’Kov, Self-oscillation in glycolysis. A simple model. Eur. J. Biochem. 4, 79–86 (1968)

    Article  Google Scholar 

  11. 11.

    A.H. Romano, T. Conway, Evolution of carbohydrate metabolic pathways. Res. Microbiol. 147, 448–55 (1996)

    Article  Google Scholar 

  12. 12.

    A.N. Lane, T.W.M. Fan, R.M. Higashi, Metabolic acidosis and the importance of balanced equation. Metabolomics 5(2), 163–165 (2009)

    Article  Google Scholar 

  13. 13.

    S.H. Strogatz, Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering (Adisson-Wesley, New York, 1994), pp. 205–209

    Google Scholar 

  14. 14.

    R.E. Mickens, Positivity preserving discrete model for the coupled ODE’s modeling glycolysis, in Proceeding of the Fourth International Conference on Dynamical Systems and Differential Equations, Wilmington, NC, USA (2002), pp. 623-629

  15. 15.

    J. Zhou, J. Shi, Pattern formation in a general glycolysis reaction–diffusion system. IMA J. Appl. Math. 80, 1703–1738 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    M. Wei, J. Wu, G. Guo, Steady state bifurcations for a glycolysis model in biochemical reaction. Nonlinear Anal. Real World Appl. 22, 155–175 (2015)

    MathSciNet  Article  Google Scholar 

  17. 17.

    A. Korkmaz, O. Ersoy, I. Dag, Motion of patterns modeled by the Gray-Scott autocatalysis system in one dimension. MATCH Commun. Math. Comput. Chem. 77, 507–526 (2017)

    MathSciNet  Google Scholar 

  18. 18.

    S. Dahiya, R.C. Mittal, A modified cubic B-spline differential quadrature method for three-dimensional non-linear diffusion equations. Open Phys. 15, 453–463 (2017)

    Article  Google Scholar 

  19. 19.

    R.I. Fernandes, G. Fairweather, An ADI extrapolated Crank-Nicolson orthogonal spline collocation method for nonlinear reaction–diffusion systems. J. Comput. Phys. 231, 6248–6267 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    E.H. Twizell, A.B. Gumel, Q. Cao, A second order scheme for the Brusselator reaction–diffusion system. J. Math. Chem. 26, 297–316 (1999)

    MathSciNet  Article  Google Scholar 

  21. 21.

    G. Adomian, The diffusion-Brusselator equation. Comput. Math. Appl. 59, 1–3 (1995)

    MathSciNet  Article  Google Scholar 

  22. 22.

    N. Ahmed, M. Rafiq, M.A. Rehman, M.S. Iqbal, M. Ali, Numerical modelling of three dimensional Brusselator reaction diffusion system. AIP Adv. 9, 015205 (2019)

    ADS  Article  Google Scholar 

  23. 23.

    N. Ahmed, S.S. Tahira, M. Imran, M. Rafiq, M.A. Rehman, M. Younis, Numerical analysis of auto-catalytic glycolysis model. AIP Adv. 9, 085213 (2019)

    ADS  Article  Google Scholar 

  24. 24.

    N. Tahira, S.S. Ahmed, M. Rafiq, M.A. Rehman, M. Ali, M.O. Ahmad, Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model. Open Math. 17, 313–330 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    A. Chakrabrty, M. Singh, B. Lucy, P. Ridland, Predator–prey model with prey-taxis and diffusion. Math. Comput. Model. 46, 482–498 (2007)

    MathSciNet  Article  Google Scholar 

  26. 26.

    R.C. Harwood, Operator splitting method and applications for semilinear parabolic partial differential equations. Ph.D. dissertation (Dept. Math., Washington State Univ., Pullman, 2011)

  27. 27.

    R.C. Harwood, V.S. Manoranjan, D.B. Edwards, Lead-acid battery model under discharge with a fast splitting method. IEEE Trans. Energy Convers. 26(4), 1109–1117 (2011)

    ADS  Article  Google Scholar 

  28. 28.

    N.N. Yanenko, The Method of Fractional Steps (Springer, Berlin, 1971)

    Google Scholar 

  29. 29.

    V. Zharnitsky, Averaging for split-step scheme. Nonlinearity 16, 1359–1366 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    F. Ansarizadeh, M. Singh, D. Richards, Modelling of tumor cells regression in response to chemotherapeutic treatment. Appl. Math. Model. 48, 96–112 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    H.Q. Wang, Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations. Appl. Math. Comput. 170, 17–35 (2005)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations (World Scientific, Singapore, 1994)

    Google Scholar 

  33. 33.

    U. Fatima, M. Ali, N. Ahmed, M. Rafiq, Numerical modeling of susceptible latent breaking-out quarantine computer virus epidemic dynamics. Heliyon 4, e00631 (2018)

    Article  Google Scholar 

  34. 34.

    N. Ahmed, N. Shahid, Z. Iqbal, M. Jawaz, M. Rafiq, S.S. Tahira, M.O. Ahmad, Numerical modeling of SEIQV epidemic model with saturated incidence rate. J. Appl. Environ. Biol. Sci. 8(4), 67–82 (2018)

    Google Scholar 

  35. 35.

    J.E. Macias-Diaz, J. Ruiz-Ramirez, A non-standard symmetry-preserving method to compute bounded solutions of a generalized Newell–Whitehead–Segel equation. Appl. Numer. Math. 61, 630–640 (2011)

    MathSciNet  Article  Google Scholar 

  36. 36.

    R.E. Mickens, A nonstandard finite difference scheme for an advection–reaction equation. J. Differ. Equ. Appl. 10, 1307–1312 (2004)

    MathSciNet  Article  Google Scholar 

  37. 37.

    R.E. Mickens, A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion. Comput. Math. Appl. 45, 429–436 (2003)

    MathSciNet  Article  Google Scholar 

  38. 38.

    R.E. Mickens, Relation between the time and space step-sizes in nonstandard finite-difference schemes for the Fisher equation. Numer. Methods Partial Differ. Equ. 13, 51–55 (1997)

    MathSciNet  Article  Google Scholar 

  39. 39.

    T. Fujimoto, R. Ranade, Two characterizations of inverse-positive matrices: the Hawkins-Simon condition and the Le Chatelier-Braun principle. Electron. J. Linear Algebra 11, 59–65 (2004)

    MathSciNet  Article  Google Scholar 

  40. 40.

    K. Manna, S.P. Chakrabarty, Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids. J. Differ. Equ. Appl. 21, 918–933 (2015)

    MathSciNet  Article  Google Scholar 

  41. 41.

    W. Qin, L. Wang, X. Ding, A non-standard finite difference method for a hepatitis B virus infection model with spatial diffusion. J. Differ. Equ. Appl. 20(12), 1641–1651 (2014)

    MathSciNet  Article  Google Scholar 

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Correspondence to Ilyas Khan.

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Ahmed, N., Rafiq, M., Baleanu, D. et al. Structure preserving algorithms for mathematical model of auto-catalytic glycolysis chemical reaction and numerical simulations. Eur. Phys. J. Plus 135, 522 (2020). https://doi.org/10.1140/epjp/s13360-020-00539-w

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