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Model on cell movement, growth, differentiation and de-differentiation: Reaction-diffusion equation and wave propagation

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Abstract.

We construct a model for cell proliferation with differentiation into different cell types, allowing backward de-differentiation and cell movement. With different cell types labeled by state variables, the model can be formulated in terms of the associated transition probabilities between various states. The cell population densities can be described by coupled reaction-diffusion partial differential equations, allowing steady wavefront propagation solutions. The wavefront profile is calculated analytically for the simple pure growth case (2-states), and analytic expressions for the steady wavefront propagating speeds and population growth rates are obtained for the simpler cases of 2-, 3- and 4-states systems. These analytic results are verified by direct numerical solutions of the reaction-diffusion PDEs. Furthermore, in the absence of de-differentiation, it is found that, as the mobility and/or self-proliferation rate of the down-lineage descendant cells become sufficiently large, the propagation dynamics can switch from a steady propagating wavefront to the interesting situation of propagation of a faster wavefront with a slower waveback. For the case of a non-vanishing de-differentiation probability, the cell growth rate and wavefront propagation speed are both enhanced, and the wavefront speeds can be obtained analytically and confirmed by numerical solution of the reaction-diffusion equations.

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References

  1. M. Poujade et al., Proc. Natl. Acad. Sci. U.S.A. 104, 15988 (2007)

    Article  ADS  Google Scholar 

  2. P. Rosen, D.S. Misfeldt, Proc. Natl. Acad. Sci. U.S.A. 77, 4760 (1980)

    Article  ADS  Google Scholar 

  3. A.D. Lander, Biophys. J. 99, 3145 (2010)

    Article  Google Scholar 

  4. S. Sell, Environ Health Perspect. 101, 15 (1993)

    Google Scholar 

  5. G.Q. Daley, Cold Spring Harb. Symp. Quant. Biol. 73, 171 (2008)

    Article  Google Scholar 

  6. T. Kawamura et al., Nature 460, 1140 (2009)

    Article  ADS  Google Scholar 

  7. C.N. Shen et al., Mech. Dev. 120, 107 (2003)

    Article  Google Scholar 

  8. M.C. Harrishingh et al., EMBO J. 23, 3061 (2004)

    Article  Google Scholar 

  9. D.J. Pearson, Y. Yang, D. Dhouailly, Proc. Natl. Acad. Sci. U.S.A. 102, 3714 (2005)

    Article  ADS  Google Scholar 

  10. S. Zhuang et al., J. Biol. Chem. 280, 21036 (2005)

    Article  Google Scholar 

  11. X.S. Zhang et al., Endocrinology 147, 1237 (2006)

    Article  Google Scholar 

  12. A. Duckmanton et al., Chem. Biol. 12, 1058 (2005)

    Article  Google Scholar 

  13. S. Cai et al., J. Health Sci. 55, 709 (2009)

    Article  Google Scholar 

  14. M. Ben Amar, C. Chatelain, P. Ciarletta, Phys. Rev. Lett. 106, 148101 (2011)

    Article  ADS  Google Scholar 

  15. S. Fedotov, A. Iomin, Phys. Rev. Lett. 98, 118101 (2007)

    Article  ADS  Google Scholar 

  16. S. Fedotov, A. Iomin, Phys. Rev. E 77, 031911 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  17. C. Deroulers, M. Aubert, M. Badoual, B. Grammaticos, Phys. Rev. E 79, 031917 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  18. S.A. Menchon, C.A. Condat, Phys. Rev. E 78, 022901 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  19. L. Liu et al., Proc. Natl. Acad. Sci. U.S.A. 108, 6853 (2011)

    Article  ADS  Google Scholar 

  20. J.V. Bonventre, J. Am. Soc. Nephrol. 14, S55 (2003)

    Article  Google Scholar 

  21. X. Fu et al., Lancet 358, 1067 (2001)

    Article  Google Scholar 

  22. C. Zhang et al., J. Cell. Mol. Med. 14, 1135 (2010)

    Google Scholar 

  23. X. Sun et al., Biol. Pharm. Bull. 34, 1037 (2011)

    Article  Google Scholar 

  24. C. Zhang et al., Aging Cell 11, 14 (2012)

    Article  Google Scholar 

  25. J.Y. Chang, P.Y. Lai, Phys. Rev. E 85, 041926 (2012)

    Article  ADS  Google Scholar 

  26. F. Graner, J.A. Glazier, Phys. Rev. Lett. 69, 2013 (1992)

    Article  ADS  Google Scholar 

  27. R.M.H. Merks, J.A. Glazier, Physica A 352, 113 (2005)

    Article  ADS  Google Scholar 

  28. N. Chen et al., Comput. Phys. Commun. 176, 670 (2007)

    Article  ADS  Google Scholar 

  29. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Math. Models Methods Appl. Sci. 20, 1179 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. N. Bellomo, A. Bellouquid, J. Nieto, J. Soler, Math. Models Methods Appl. Sci. 22, 1130001 (2012)

    Article  MathSciNet  Google Scholar 

  31. P. Hogeweg, Biosystems 64, 97 (2002)

    Article  Google Scholar 

  32. E.L. Bearer et al., Cancer Res. 69, 4493 (2009)

    Article  Google Scholar 

  33. N. Bellomo, M. Delitala, Phys. Life Rev. 5, 183 (2008)

    Article  ADS  Google Scholar 

  34. J.T. Oden, A. Hawkins, S. Prudhomme, Math. Models Methods Appl. Sci. 20, 477 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. C.H. Waddington, in The Strategy of the Genes. A Discussion of some Aspects of Theoretical Biology (Alen & Unwin, 1957)

  36. K. Takahashi, S. Yamanaka, Cell 126, 663 (2006)

    Article  Google Scholar 

  37. K. Takahashi et al., Cell 131, 861 (2007)

    Article  Google Scholar 

  38. S. Yamanaka, Nature 460, 08180 (2009)

    Article  Google Scholar 

  39. J.-P. Capp, BioEssays 34, 170 (2012)

    Article  Google Scholar 

  40. A. Bellouquid, M. Delitala, Mathematical Modeling of Complex Biological Systems (Birkhäuser, Inc., Boston, 2006)

  41. A. Bagorda, C.A. Parent, J. Cell Sci. 121, 2621 (2008)

    Article  Google Scholar 

  42. K.F. Swaney, C.H. Huang, P.N. Devreotes, Annu. Rev. Biophys. 39, 265 (2010)

    Article  Google Scholar 

  43. E.T. Roussos, J.S. Condeelis, A. Patsialou, Nature Rev. Cancer 11, 573 (2011)

    Article  Google Scholar 

  44. W.Y. Chiang, Y.X. Li, P.Y. Lai, Phys. Rev. E 84, 041921 (2011)

    Article  ADS  Google Scholar 

  45. R.A. Fisher, Ann. Eugen. 7, 353 (1937) A. Kolmogorov, I. Petrovskii, N. Piscounov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem

    Google Scholar 

  46. J.D. Murray, Mathematical Biology, 3rd edition (Springer, New York, 2002)

  47. M.X. Wang, P.Y. Lai, in preparation

  48. M.X. Wang, P.Y. Lai, Phys. Rev. E 86, 051908 (2012)

    Article  ADS  Google Scholar 

  49. G.F. Oster, J.D. Murray, J. Exp. Zool. 251, 186 (1989)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Graduate Institute of Biophysics, and Center for Complex Systems, National Central University, Chungli, Taiwan, 320, R.O.C.

    Mao-Xiang Wang, Yu-Jung Li, Pik-Yin Lai & C. K. Chan

  2. Institute of Physics, Academia Sinica, Nankang, Taipei, Taiwan, 115, R.O.C.

    C. K. Chan

  3. School of Science, Nanjing University of Science and Technology, Nanjing, 210094, China

    Mao-Xiang Wang

Authors
  1. Mao-Xiang Wang
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  2. Yu-Jung Li
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  3. Pik-Yin Lai
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  4. C. K. Chan
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Correspondence to Pik-Yin Lai.

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Wang, MX., Li, YJ., Lai, PY. et al. Model on cell movement, growth, differentiation and de-differentiation: Reaction-diffusion equation and wave propagation. Eur. Phys. J. E 36, 65 (2013). https://doi.org/10.1140/epje/i2013-13065-4

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  • DOI: https://doi.org/10.1140/epje/i2013-13065-4

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