Skip to main content
SpringerLink
Log in

Motile parameters of cell migration in anisotropic environment derived by speed power spectrum fitting with double exponential decay

  • Research Article
  • Published:
Frontiers of Physics Aims and scope Submit manuscript

Abstract

Cell migration through anisotropic microenvironment is critical to a wide variety of physiological and pathological processes. However, adequate analytical tools to derive motile parameters to characterize the anisotropic migration are lacking. Here, we proposed a method to obtain the four motile parameters of migration cells based on the anisotropic persistent random walk model which is described by two persistence times and two migration speeds at perpendicular directions. The key process is to calculate the velocity power spectra of cell migration along intrinsically perpendicular directions respectively, then to apply maximum likelihood estimation to derive the motile parameters from the power spectra fitting with double exponential decay. The simulation results show that the averaged persistence times and the corrected migration speeds can be good estimations for motile parameters of cell migration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Vicente-Manzanares and A. R. Horwitz, Cell migration: An overview, Methods Mol. Biol. 769, 1 (2011)

    Article  Google Scholar 

  2. D. S. Vasilev, N. M. Dubrovskaya, N. L. Tumanova, and I. A. Zhuravin, Prenatal hypoxia in different periods of embryogenesis differentially affects cell migration, neuronal plasticity, and rat behavior in postnatal ontogenesis, Front. Neurosci. 10, 126 (2016)

    Article  Google Scholar 

  3. M. Zalokar and I. Erk, J. Microsc. Biol. Cell. 25, 97 (1976)

    Google Scholar 

  4. P. Kulesa, D. L. Ellies, and P. A. Trainor, Comparative analysis of neural crest cell death, migration, and function during vertebrate embryogenesis, Dev. Dyn. 229(1), 14 (2004)

    Article  Google Scholar 

  5. W. S. Krawczyk, A pattern of epidermal cell migration during wound healing, J. Cell Biol. 49(2), 247 (1971)

    Article  Google Scholar 

  6. G. D. Sharma, J. He, and H. E. P. Bazan, p38 and ERK1/2 coordinate cellular migration and proliferation in epithelial wound healing: Evidence of cross-talk activation between MAP kinase cascades, J. Biol. Chem. 278(24), 21989 (2003)

    Article  Google Scholar 

  7. T. Tao, A. Robichaud, S. Nadeau, R. Savoie, B. Gallant, and R. Ouellette, Effect of cumulus cell removal on the fertilization and the day 3 embryo quality in human IVF, International Congress Series 1271, 135 (2004)

    Article  Google Scholar 

  8. J. Bowszyc, J. Bowszyc, and T. Machońko, 15-years of studies of the immunological cellular response in syphilis in humans using migration inhibition tests, Przegl. Dermatol. 72(2), 134 (1985)

    Google Scholar 

  9. H. Zhang, Y. Han, J. Tao, S. Liu, C. Yan, and S. Li, Cellular repressor of E1A-stimulated genes regulates vascular endothelial cell migration by the ILK/AKT/mTOR/VEGF(165) signaling pathway, Exp. Cell Res. 317(20), 2904 (2011)

    Article  Google Scholar 

  10. X. L. Fang Wei, M. Cai, Y. Liu, P. Jung, and J. W. Shuai, Regulation of 1, 4, 5-triphosphate receptor channel gating dynamics by mutant presenilin in Alzheimer’s disease cells, Front. Phys. 12(3), 128702 (2017)

    Article  ADS  Google Scholar 

  11. S. X. Liu, Y. Z. Geng, and S. W. Yan, Structural effects and competition mechanisms targeting the interactions between p53 and MDM2 for cancer therapy, Front. Phys. 12(3), 128908 (2017)

    Article  ADS  Google Scholar 

  12. H. C. Berg and F. Dyson, Random walks in biology, Phys. Today 40(3), 73 (1987)

    Article  Google Scholar 

  13. L. X. Li, Photon diffusion in a relativistically expanding sphere, Front. Phys. 8(5), 555 (2013)

    Article  ADS  Google Scholar 

  14. X. H. Li, G. Yang, and J. P. Huang, Chaotic-periodic transition in a two-sided minority game, Front. Phys. 11(4), 118901 (2016)

    Article  ADS  Google Scholar 

  15. S. Huang, C. P. Brangwynne, K. K. Parker, and D. E. Ingber, Symmetry-breaking in mammalian cell cohort migration during tissue pattern formation: Role of random-walk persistence, Cell Motil. Cytoskeleton 61(4), 201 (2005)

    Article  Google Scholar 

  16. Z. X. Niu, T. Hang, and Y. Chen, Molecular dynamics study of nanodroplet diffusion on smooth solid surfaces, Front. Phys. 13(5), 137804 (2018)

    Article  ADS  Google Scholar 

  17. Y. A. Yan and J. S. Shao, Stochastic description of quantum Brownian dynamics, Front. Phys. 11(4), 110309 (2016)

    Article  ADS  Google Scholar 

  18. M. Boguñá, J. M. Porrà, and J. Masoliver, Generalization of the persistent random walk to dimensions greater than 1, Phys. Rev. E 58(6), 6992 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  19. C. Peggy, A. L. Ny, B. D. Loynes, et al., Persistent random walks (I): Recurrence versus transience, J. Theor. Probab. 31(1), 1 (2016)

    MathSciNet  MATH  Google Scholar 

  20. H. I. Wu, B. L. Li, T. A. Springer, and W. H. Neill, Modelling animal movement as a persistent random walk in two dimensions: Expected magnitude of net displacement, Ecol. Modell. 132(1–2), 115 (2000)

    Article  Google Scholar 

  21. M. Schienbein and H. Gruler, Langevin equation, Fokker-Planck equation and cell migration, Bull. Math. Biol. 55(3), 585 (1993)

    Article  MATH  Google Scholar 

  22. D. S. Lemons and A. Gythiel, Paul Langevin’s 1908 paper “On the Theory of Brownian Motion” [Sur la théorie du mouvement brownien, C. R. Acad. Sci. (Paris) 146, 530–533 (1908)], Am. J. Phys. 65(11), 1079 (1997)

    Article  ADS  Google Scholar 

  23. J. L. Parada, G. Carrillo-Castañeda, and M. V. Ortega, Profile of the enzymes of the Krebs cycle in Salmonella typhimurium during the utilization of succinate, acetate, and citrate for growth, Rev. Latinoam. Microbiol. 15, 29 (1973)

    Google Scholar 

  24. C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys. 15(3), 311 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Wang, J. Kuo, S. C. Bae, and S. Granick, When Brownian diffusion is not Gaussian, Nat. Mater. 11(6), 481 (2012)

    Article  ADS  Google Scholar 

  26. T. H. Harris, E. J. Banigan, D. A. Christian, C. Konradt, E. D. Tait Wojno, K. Norose, E. H. Wilson, B. John, W. Weninger, A. D. Luster, A. J. Liu, and C. A. Hunter, Generalized Lévy walks and the role of chemokines in migration of effector CD8+ T cells, Nature 486(7404), 545 (2012)

    Article  ADS  Google Scholar 

  27. D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, C. J. A. Bradshaw, J. W. Pitchford, A. James, M. Z. Ahmed, A. S. Brierley, M. A. Hindell, D. Morritt, M. K. Musyl, D. Righton, E. L. C. Shepard, V. J. Wearmouth, R. P. Wilson, M. J. Witt, and J. D. Metcalfe, Scaling laws of marine predator search behaviour, Nature 451(7182), 1098 (2008)

    Article  ADS  Google Scholar 

  28. A. Czirók, K. Schlett, E. Madarasz, and T. Vicsek, Exponential distribution of locomotion activity in cell cultures, Phys. Rev. Lett. 81(14), 3038 (1998)

    Article  ADS  Google Scholar 

  29. A. Upadhyaya, J. P. Rieu, J. A. Glazier, and Y. Sawada, Anomalous diffusion and non-Gaussian velocity distribution of Hydra cells in cellular aggregates, Physica A 293(3–4), 549 (2001)

    Article  ADS  Google Scholar 

  30. D. Selmeczi, S. Mosler, P. H. Hagedorn, N. B. Larsen, and H. Flyvbjerg, Cell motility as persistent random motion: Theories from experiments, Biophys. J. 89(2), 912 (2005)

    Article  ADS  Google Scholar 

  31. L. Liu, G. Duclos, B. Sun, J. Lee, A. Wu, Y. Kam, E. D. Sontag, H. A. Stone, J. C. Sturm, R. A. Gatenby, and R. H. Austin, Minimization of thermodynamic costs in cancer cell invasion, Proc. Natl. Acad. Sci. USA 110(5), 1686 (2013)

    Article  ADS  Google Scholar 

  32. J. Zhu, L. Liang, Y. Jiao, and L. Liu, Enhanced invasion of metastatic cancer cells via extracellular matrix interface, PLOS One 10(2), e0118058 (2015)

    Article  Google Scholar 

  33. P. H. Wu, A. Giri, S. X. Sun, and D. Wirtz, Three-dimensional cell migration does not follow a random walk, Proc. Natl. Acad. Sci. USA 111(11), 3949 (2014)

    Article  ADS  Google Scholar 

  34. P. H. Wu, A. Giri, and D. Wirtz, Statistical analysis of cell migration in 3D using the anisotropic persistent random walk model, Nat. Protoc. 10(3), 517 (2015)

    Article  Google Scholar 

  35. B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev. 10(4), 422 (1968)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  36. R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339(1), 1 (2000)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  37. B. Liu and J. Goree, Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma, Phys. Rev. Lett. 100(5), 055003 (2008)

    Article  ADS  Google Scholar 

  38. M. Weiss, M. Elsner, F. Kartberg, and T. Nilsson, Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells, Biophys. J. 87(5), 3518 (2004)

    Article  ADS  Google Scholar 

  39. C. L. Stokes, D. A. Lauffenburger, and S. K. Williams, Migration of individual microvessel endothelial cells: Stochastic model and parameter measurement, J. Cell. Sci. 99(Pt 2), 419 (1991)

    Article  Google Scholar 

  40. L. Li, E. C. Cox, and H. Flyvbjerg, “Dicty dynamics”: Dictyostelium motility as persistent random motion, Phys. Biol. 8(4), 046006 (2011)

    Article  ADS  Google Scholar 

  41. F. Höfling and T. Franosch, Anomalous transport in the crowded world of biological cells, Rep. Prog. Phys. 76(4), 046602 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  42. V. Zaburdaev, S. Denisov, and J. Klafter, Lévy walks, Rev. Mod. Phys. 87(2), 483 (2015)

    Article  ADS  Google Scholar 

  43. R. T. Tranquillo, D. A. Lauffenburger, and S. H. Zigmond, A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations, J. Cell Biol. 106(2), 303 (1988)

    Article  Google Scholar 

  44. G. A. Dunn, Characterising a kinesis response: Time averaged measures of cell speed and directional persistence, Agents Actions Suppl. 12, 14 (1983)

    Google Scholar 

  45. C. L. Stokes and D. A. Lauffenburger, Analysis of the roles of microvessel endothelial cell random motility and chemotaxis in angiogenesis, J. Theor. Biol. 152(3), 377 (1991)

    Article  ADS  Google Scholar 

  46. M. R. Parkhurst and W. M. Saltzman, Quantification of human neutrophil motility in three-dimensional collagen gels: Effect of collagen concentration, Biophys. J. 61(2), 306 (1992)

    Article  ADS  Google Scholar 

  47. R. Gorelik and A. Gautreau, Quantitative and unbiased analysis of directional persistence in cell migration, Nat. Protoc. 9(8), 1931 (2014)

    Article  Google Scholar 

  48. J. N. Pedersen, L. Li, C. Grǎdinaru, R. H. Austin, E. C. Cox, and H. Flyvbjerg, How to connect time-lapse recorded trajectories of motile microorganisms with dynamical models in continuous time, Phys. Rev. E 94(6–1), 062401 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  49. A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. R. Stat. Soc. B 39(1), 1 (1977)

    MathSciNet  MATH  Google Scholar 

  50. C. L. Vestergaard, J. N. Pedersen, K. I. Mortensen, and H. Flyvbjerg, Estimation of motility parameters from trajectory data, Eur. Phys. J. Spec. Top. 224(7), 1151 (2015)

    Article  Google Scholar 

  51. C. L. Vestergaard, P. C. Blainey, and H. Flyvbjerg, Optimal estimation of diffusion coefficients from single-particle trajectories, Phys. Rev. E 89(2), 022726 (2014)

    Article  ADS  Google Scholar 

  52. J. H. Chen and H. Y. Fan, On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations, Front. Phys. 10(1), 100301 (2015)

    Google Scholar 

  53. Z. K. Wu, P. Li, and Y. Z. Gu, Propagation dynamics of finite-energy Airy beams in nonlocal nonlinear media, Front. Phys. 12(5), 124203 (2017)

    Article  ADS  Google Scholar 

  54. K. Katoh, K. Misawa, K. Kuma, and T. Miyata, MAFFT: A novel method for rapid multiple sequence alignment based on fast Fourier transform, Nucleic Acids Res. 30(14), 3059 (2002)

    Article  Google Scholar 

  55. S. B. Weinstein and P. M. Ebert, Data transmission by frequency division multiplexing using the discrete Fourier transform, IEEE Trans. Commun. Tech. 19, 628 (1971)

    Article  Google Scholar 

  56. S. F. Nørrelykke and H. Flyvbjerg, Power spectrum analysis with least-squares fitting: Amplitude bias and its elimination, with application to optical tweezers and atomic force microscope cantilevers, Rev. Sci. Instrum. 81(7), 075103 (2010)

    Article  ADS  Google Scholar 

  57. Y. Liu, X. Zhang, Y. Wu, W. Liu, X. Li, R. Liu, L. Liu, and J. Shuai, Derivation of persistent time for anisotropic migration of cells, Chin. Phys. B 26(12), 128707 (2017)

    Article  ADS  Google Scholar 

  58. N. Le Bihan and J. Mars, Singular value decomposition of quaternion matrices: A new tool for vector-sensor signal processing, Signal Processing 84(7), 1177 (2004)

    Article  MATH  Google Scholar 

  59. M. E. Wall, A. Rechtsteiner, and L. M. Rocha, Singular value decomposition and principal component analysis, arXiv: physics/0208101v4 (2002)

  60. K. Berg-Sørensen and H. Flyvbjerg, Power spectrum analysis for optical tweezers, Rev. Sci. Instrum. 75(3), 594 (2004)

    Article  ADS  Google Scholar 

  61. J. R. Xie and B. H. Wang, Modularity-like objective function in annotated networks, Front. Phys. 12(6), 128903 (2017)

    Article  ADS  Google Scholar 

  62. R. Gorelik and A. Gautreau, The Arp2/3 inhibitory protein arpin induces cell turning by pausing cell migration, Cytoskeleton (Hoboken) 72(7), 362 (2015)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Grant Nos. 11675134, 11874310, 11474345, and 11604030), the 111 Project (Grant No. B160229), and the China Postdoctoral Science Foundation (Grant No. 2016M602071).

Author information

Authors and Affiliations

  1. Department of Physics, Xiamen University, Xiamen, 361005, China

    Yan-Ping Liu  (刘艳平), Xiang Li  (李翔), Jing Qu  (屈静), Xue-Juan Gao  (高学娟), Qing-Zu He  (何情祖) & Jian-Wei Shuai  (帅建伟)

  2. State Key Laboratory of Cellular Stress Biology, Innovation Center for Cell Signaling Network, Xiamen University, Xiamen, 361102, China

    Xiang Li  (李翔) & Jian-Wei Shuai  (帅建伟)

  3. College of Physics, Chongqing University, Chongqing, 401331, China

    Li-Yu Liu  (刘雳宇) & Ru-Chuan Liu  (刘如川)

  4. National Institute for Big Data in Healthcare at Xiamen University, Xiamen, 361102, China

    Jian-Wei Shuai  (帅建伟)

Authors
  1. Yan-Ping Liu  (刘艳平)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiang Li  (李翔)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Qu  (屈静)
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xue-Juan Gao  (高学娟)
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Qing-Zu He  (何情祖)
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Li-Yu Liu  (刘雳宇)
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Ru-Chuan Liu  (刘如川)
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Jian-Wei Shuai  (帅建伟)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Ru-Chuan Liu  (刘如川) or Jian-Wei Shuai  (帅建伟).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, YP., Li, X., Qu, J. et al. Motile parameters of cell migration in anisotropic environment derived by speed power spectrum fitting with double exponential decay. Front. Phys. 15, 13602 (2020). https://doi.org/10.1007/s11467-019-0929-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11467-019-0929-9

Keywords

Search

Navigation