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Pattern formation in a one-dimensional MARCKS protein cyclic model with spatially inhomogeneous diffusion coefficients

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Abstract

We analytically investigate the conditions for the wave instability in a reaction-diffusion system describing the nonlinear dynamics of the myristoylated alanine-rich C kinase substrate (MARCKS) between cytosol and cytoplasmic membrane. Taking into account the effect of spatial inhomogeneous diffusion coefficients, and by applying the discrete multiple scale expansion method, we show that the nonlinear generic model can be transformed into a one-dimensional discrete nonlinear Schrödinger equation. We perform a linear stability analysis on the plane wave solutions to derive the criterion of the modulational instability (MI) phenomenon. This analysis reveals that the critical amplitude of the plane wave is highly influenced by the phosphorylation rate and weakly influenced by the inhomogeneous diffusion coefficients. The exact analytical solutions show that the system exhibits traveling waves and periodic array of patterns. The results seem to indicate the features of synchronization in the collective dynamics. In homogenous state, we obtained a spatial pattern of horizontal stripes. By considering the spatial inhomogeneity effect, we obtain a spatial pattern of oblique stripes. We also notice that an increase in wavenumber induces the increase in the number of stripes in the model.

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References

  1. M. El Amri, U. Fitzgerald, G. Schlosser, MARCKS and MARCKS-like proteins in development and regeneration. J. Biomed. Sci. 25, 43 (2018)

    Article  Google Scholar 

  2. A. Arbuzova, J. Wang, D. Murray, J. Jacob, D.S. Cafiso, S. Mc Laughlin, Kinetics of interaction of the Myristoylated alanine-rich C kinase substrate, membranes, and calmodulin. J. Biol. Chem. 272, 27167–27177 (1997)

    Article  Google Scholar 

  3. J. Wang, A. Gambhir, G. Mihalyne, D. Murray, U. Golebiewska, S. McLaughlin, Lateral sequestration of phosphatidylinositol4,5- bisphosphate by the basic effector domain of Myristoylated alanine-rich C kinase substrate is due to non specific electrostatic interactions. J. Biol. Chem. 277, 34401–34412 (2002)

    Article  Google Scholar 

  4. L. Rusu, A. Gambhir, S. McLaughlin, J. Rdler, Fluorescence correlation spectroscopy studies of peptide and protein binding to phospholipid vesicles. Biophys. J. 87, 1044–1053 (2004)

    Article  ADS  Google Scholar 

  5. G.M. Verghese, J.D. Johnson, C. Vasulka, D.M. Haupt, D.J. Stumpo, P.J. Blackshear, protein kinase C-mediated phosphorylation and calmodulin binding of recombinant Myristoylated alanine-rich C kinase substrate (MARCKS) and MARCKS-related protein. J. Biol. Chem. 269(12), 9361–9367 (1994)

    Article  Google Scholar 

  6. P.J. Blackshear, G.M. Verghese, J.D. Johnson, D.M. Haupt, D.J. Stumpo, Characteristics of the F52 protein, a MARCKS homologue. J. Biol. Chem. 267, 13540–13546 (1992)

    Article  Google Scholar 

  7. M. Glaser, S. Wanaski, A. BuserC, V. Boguslavsky, W. Rashidzada, A. Morris, M. Rebecchi, S.F. Scarlata, L.W. Runnels, G.D. Prestwich, J. Chen, A. Aderemi, J. Ahni, S. McLaughlin, Myristoylated alanine-rich C kinase substrate(MARCKS) produces reversible inhibition of phospholipase C by sequestering phosphatidylinositol4, 5-Bisphosphate inlateraldomains. J. Biol. Chem. 271, 26187–26193 (1996)

    Article  Google Scholar 

  8. S. Alonso, M. Bär, Phase separation and bistability in a three-dimensional model for protein domain formation at biomembranes. Phys. Biol. 7, 046012 (2010)

    Article  ADS  Google Scholar 

  9. A. Mvogo, Germain Hubert Ben-Bolie, and Timoléon Crépin Kofané, Solitary waves of a-helix propagation in media with arbitrary inhomogeneities. Eur. Phys. J. B 86, 217 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  10. A. Mvogo, G.H. Ben-Bolie, T.C. Kofané, Solitary waves in an inhomogeneous chain of a-helical proteins. Int. J. Modern Phys. B 28, 17 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Gizzi, A. Loppini, R. Ruiz-Baier, A. Ippolito, A. Camassa, A. La Camera, E. Emmi, L. Di Perna, V. Garofalo, C. Cherubini, S. Filippi, Nonlinear diffusion and thermo-electric coupling in a two-variable model of cardiac action potential. Chaos 27, 093919 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  12. K.M. Pagea, P.K. Mainib, N.A.M. Monk, Complex pattern formation in reaction-diffusion systems with spatially varying parameters. Phys. D 202(1–2), 95–115 (2005)

    Article  MathSciNet  Google Scholar 

  13. T. Alarcon, H.M. Byrneb, P.K. Maini, A cellular automaton model for tumour growth in inhomogeneous environment. J. Theor. Biol. 225(2), 257–274 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. P.K. Maini, D.L. Benson, J.A. Sherratt, Pattern formation in reaction-diffusion models with spatially inhomogeneoos diffusion coefficients. IMA J. Math. Appl. Med. Biol. 9, 197–213 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. A.S. Etémé, C.B. Tabi, A. Mohamadou, T.C. Kofané, Long-range memory effects in a magnetized Hindmarsh-Rose neural network. Commun. Nonlinear Sci. Numer. Simul. 84, 105208 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  16. A.S. Etémé, C.B. Tabi, A. Mohamadou, T.C. Kofané, Unstable discrete modes in HindmarshRose neural networks under magnetic flow effect. Chaos Solitons Fractals 123, 116123 (2019)

    MATH  Google Scholar 

  17. C.N. Takembo, H.P.E. Fouda, Effect of temperature fuctuation on the localized pattern of action potential in cardiac tissue. Sci. Rep. 10, 15087 (2020)

    Article  Google Scholar 

  18. A.S. Etémé, Consistency between modulational instability and energy localization in time-delay -memristive neural network. Europhys. Lett. 143, 42002 (2023)

    Article  ADS  Google Scholar 

  19. C.B. Tabi, I. Maïna, A. Mohamadou, H.P.F. Ekobena, T.C. Kofané, Fractional unstable patterns of energy in alpha- helix proteins with long-range interactions. Chaos Solitons Fractal 116, 386391 (2018)

    Article  Google Scholar 

  20. A.D. Koko, C.B. Tabi, H.P.F. Ekobena, A. Mohamadou, T.C. Kofané, Nonlinear charge transport in the helicoidal DNA molecule. Chaos 22, 043110 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. C.B. Tabi, S. Veni, E. Wamba, T.C. Kofané, Modulational instability and droplet formation in Bose-Bose mixtures with Lee-Huang-Yang correction and polaron-like impurity. Phys. Lett. A 485, 129087 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  22. C.B. Tabi, P. Otlaadisa, T.C. Kofané, Modulational instability in two-dimensional dissipative open Bose-Einstein condensates with mixed helicoidal and Rashba-Dresselhaus spin-orbit couplings. Phys. Lett. A 481, 129004 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.J. Brudvig, J.M. Weimer, X MARCKS the spot: myristoylated alanine-rich C kinase substrate in neuronal function and disease. Front. Cell. Neurosci. 9, 407 (2015)

    Article  Google Scholar 

  24. J. Herrera, M.A. Maza, A.A. Minzoni, N.F. Smyth, A.L. Worthy, Davydov soliton evolution in temperature gradients driven by hyperbolic waves. Phys. D 191, 156–177 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. G.R.Y. Mefire, C.B. Tabi, A. Mohamadou, H.P.F. Ekobena, T.C. Kofané, Modulated pressure waves in large elastic tubes. Chaos 23, 033128 (2013)

    Article  ADS  MATH  Google Scholar 

  26. A.S. Etémé, C.B. Tabi, A. Mohamadou, Long-range patterns in Hindmarsh-Rose networks. Commun. Nonlinear Sci. Numer. Simul. 43, 211–219 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. C.B. Tabi, A.S. Etémé, A. Mohamadou, T.C. Kofané, Unstable discrete modes in Hindmarsh-Rose neural networks under magnetic flow effect. Chaos Solitons Fractals 123, 116–123 (2019)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. J. Leon, M. Manna, Multiscale analysis of discrete nonlinear evolution equations, physique mathematique et theorique. CNRS-UMR.5825, (1999)

  29. N. Akhmediev, J.M. Soto-Crespo, A. Ankiewicz, Extreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A 373, 2137–2145 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. A.N. Zaikin, A.M. Zhabotinsky, Concentration wave propagation in two-dimensional liquid-phase self-oscillating system. Nature 225, 535–537 (1970)

    Article  ADS  Google Scholar 

  31. J.J. Tyson, P.C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction. J. Chem. Phys 73, 2224–2237 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  32. S. Alonso, U. Dietrich, C. Händel, J.A. Käs, M. Bär, Oscillations in the lateral pressure of lipid monolayers induced by nonlinear chemical dynamics of the second messengers MARCKS and Protein Kinase C. Biophys. J. 100, 939–947 (2011)

    Article  ADS  Google Scholar 

  33. K. John, M. Bär, Travelling lipid domains in a dynamic model for protein induced pattern formation in biomembranes. Phys. Biol. B2, 23–32 (2005)

    Google Scholar 

  34. I. Sali, C.B. Tabi, H.P. Ekobena, T.C. Kofané, Modulational instability in a biexciton molecular chain with saturable nonlinearity effects. Int. J. Modern Phys. B 30, 1550244 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  35. J. Lippoldt, C. Händel, U. Dietrich, J.A. Käs, Dynamic membrane structure induces temporal pattern formation. Biochim. Biophys. Acta 1838, 2380–2390 (2014)

    Article  Google Scholar 

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Acknowledgements

The corresponding authors (Chenceline Fouedji) is very grateful to Blaise Tabi Dzou for stimulating discussions.

Author information

Authors and Affiliations

  1. Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon

    Chenceline Fouedji, Armand Sylvin Etémé & Henri Paul Ekobena Fouda

  2. Botswana International University of Science and Technology, P/Bag 16, Palapye, Botswana

    Conrad Bertrand Tabi

Authors
  1. Chenceline Fouedji
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  2. Armand Sylvin Etémé
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  3. Conrad Bertrand Tabi
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  4. Henri Paul Ekobena Fouda
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Correspondence to Chenceline Fouedji.

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Appendices

Appendix A

Coefficients at other order (1, l)

$$\begin{aligned} a=\frac{\alpha _0}{i\omega +\gamma _2-4D\sin ^2\frac{q}{2}}, b=\frac{a\alpha _0}{i\omega +\gamma _3-4D\sin ^2\frac{q}{2}}. \end{aligned}$$

Appendix B

Coefficients at other order (2, 2)

$$\begin{aligned}{} & {} A=\bigg (\alpha _1-\gamma _3b+4D_0\sin ^2\frac{q}{2}\bigg )\bigg (2i\omega +\gamma _3)(2i\omega +\gamma _2\bigg )\\{} & {} \quad \quad +\gamma _3\bigg (2i\omega +\gamma _2)(\gamma _3b+4D\sin ^2\frac{q}{2}\bigg )+\gamma _2\gamma _3\bigg (-\alpha _1+4D\sin ^2\frac{q}{2}\bigg )\\{} & {} A_1=(2i\omega +\alpha _0+4D_1\sin ^2q)(2i\omega +\gamma _3)(2i\omega +\gamma _2)-\gamma _2\gamma _3\alpha _0,\,\, a_2=\frac{A}{A1}\\{} & {} B=\bigg (\gamma _3b+4D\sin ^2\frac{q}{2}\bigg )(2i\omega +\gamma _2)(2i\omega -\alpha _0-4D_1\sin ^2q)+\gamma _2((2i\omega -\alpha _0-4D_1\sin ^2q)\\{} & {} \quad \quad (4D\sin ^2\frac{q}{2}-\alpha _1)+\gamma _2\alpha _0(\alpha _1-\gamma _3b+4D_0\sin ^2\frac{q}{2}),\,\, a_3=\frac{B}{A_1}\\{} & {} C=\bigg (2i\omega -\alpha _0-4D_1\sin ^2q)(2i\omega +\gamma _3\bigg )\bigg (-\alpha _1+4D\sin ^2\frac{q}{2}\bigg )+\gamma _3\alpha _0(\gamma _3b+4D\sin ^2\frac{q}{2})\\{} & {} \quad \quad +\alpha _0(2i\omega +\gamma _3)\bigg (\alpha _1-\gamma _3b+4D_0\sin ^2\frac{q}{2}\bigg ),\,\, a_4=\frac{C}{A_1}. \end{aligned}$$

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Fouedji, C., Sylvin Etémé, A., Tabi, C.B. et al. Pattern formation in a one-dimensional MARCKS protein cyclic model with spatially inhomogeneous diffusion coefficients. Eur. Phys. J. Plus 138, 987 (2023). https://doi.org/10.1140/epjp/s13360-023-04606-w

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