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Modeling mechanochemical pattern formation in elastic sheets of biological matter

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Abstract

Inspired by active shape morphing in developing tissues and biomaterials, we investigate two generic mechanochemical models where the deformations of a thin elastic sheet are driven by, and in turn affect, the concentration gradients of a chemical signal. We develop numerical methods to study the coupled elastic deformations and chemical concentration kinetics, and illustrate with computations the formation of different patterns depending on shell thickness, strength of mechanochemical coupling and diffusivity. In the first model, the sheet curvature governs the production of a contractility inhibitor and depending on the threshold in the coupling, qualitatively different patterns occur. The second model is based on the stress-dependent activity of myosin motors and demonstrates how the concentration distribution patterns of molecular motors are affected by the long-range deformations generated by them. Since the propagation of mechanical deformations is typically faster than chemical kinetics (of molecular motors or signaling agents that affect motors), we describe in detail and implement a numerical method based on separation of timescales to effectively simulate such systems. We show that mechanochemical coupling leads to long-range propagation of patterns in disparate systems through elastic instabilities even without the diffusive or advective transport of the chemicals.

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References

  1. T. Lecuit, P.F. Lenne, Nature Rev. Molecular Cell Biol. 8, 633 EP (2007)

    Article  Google Scholar 

  2. J.M. Sawyer, J.R. Harrell, G. Shemer, J. Sullivan-Brown, M. Roh-Johnson, B. Goldstein, Developmental Biol. 341(1), 5 (2010)

    Article  Google Scholar 

  3. M. Labouesse (ed.), Forces and Tension in Development, vol. 95, 1st edn. (Academic Press, 2011)

  4. J. Howard, S.W. Grill, J.S. Bois, Nat. Rev. Mol. Cell Biol. 12(6), 392 (2011)

    Article  Google Scholar 

  5. J. Howard, Mechanics of motor proteins and the cytoskeleton (Sinauer Associates, Sunderland, Mass, 2001)

    Google Scholar 

  6. M. Murrell, P.W. Oakes, M. Lenz, M.L. Gardel, Nature reviews molecular cell biology 16, 486 EP (2015)

    Article  Google Scholar 

  7. T. Lecuit, P.F. Lenne, E. Munro, Ann. Rev. Cell Dev. Biol. 27(1), 157 (2011)

    Article  Google Scholar 

  8. M. Greenberg, G. Arpağ, E. Tüzel, E. Ostap, Biophys. J. 110(12), 2568 (2016)

    Article  Google Scholar 

  9. B. Guo, W.H. Guilford, Proc. Nat. Acad. Sci. 103(26), 9844 (2006)

    Article  ADS  Google Scholar 

  10. E. Hannezo, C.P. Heisenberg, Cell 178(1), 12 (2019)

    Article  Google Scholar 

  11. K.A. Jansen, D.M. Donato, H.E. Balcioglu, T. Schmidt, E.H. Danen, G.H. Koenderink, Biochimica et Biophysica Acta (BBA) - molecular cell research. Mechanobiology 1853(11 Part B), 3043 (2015)

    Google Scholar 

  12. K. Dasbiswas, E. Alster, S. Safran, Sci. Rep. 6, 27692 (2016)

    Article  ADS  Google Scholar 

  13. P. Recho, A. Hallou, E. Hannezo, Proc. Nat. Acad. Sci. 116(12), 5344 (2019)

    Article  Google Scholar 

  14. D. Gilmour, M. Rembold, M. Leptin, Nature 541(7637), 311 (2017)

    Article  ADS  Google Scholar 

  15. Y. Ideses, V. Erukhimovitch, R. Brand, D. Jourdain, J.S. Hernandez, U.R. Gabinet, S.A. Safran, K. Kruse, A. Bernheim-Groswasser, Nature Commun. 9(1), 2461 (2018)

    Article  ADS  Google Scholar 

  16. D.P. Holmes, Current Opin. Colloid Inter. Sci. 40, 118 (2019)

    Article  Google Scholar 

  17. A. Zakharov, L. Pismen, Soft Matter 13(15), 2886 (2017)

    Article  ADS  Google Scholar 

  18. A. Senoussi, S. Kashida, R. Voituriez, J.C. Galas, A. Maitra, A. Estevez-Torres 116(45), 22464 (2019)

    Google Scholar 

  19. B. Hobmayer, F. Rentzsch, K. Kuhn, C.M. Happel, C.C. von Laue, P. Snyder, U. Rothbächer, T.W. Holstein, Nature 407(6801), 186 (2000)

    Article  ADS  Google Scholar 

  20. F. Brinkmann, M. Mercker, T. Richter, A. Marciniak-Czochra, PLoS Comput. Biol. 14(7), e1006259 (2018)

    Article  ADS  Google Scholar 

  21. A.R. Harris, L. Peter, J. Bellis, B. Baum, A.J. Kabla, G.T. Charras, Proc. Nat. Acad. Sci. 109(41), 16449 (2012)

  22. L.D. Landau, E.M. Lifshits, Theory of Elasticity (Pergamon Press, New York, 1986)

    Google Scholar 

  23. B. Li, Y.P. Cao, X.Q. Feng, H. Gao, Soft Matter 8(21), 5728 (2012)

    Article  ADS  Google Scholar 

  24. D.A. Matoz-Fernandez, F.A. Davidson, N.R. Stanley-Wall, R. Sknepnek, Phys. Rev. Res. 2, 013165 (2020)

    Article  Google Scholar 

  25. E. Cerda, L. Mahadevan, Phys. Rev. Lett. 90, 074302 (2003)

    Article  ADS  Google Scholar 

  26. L. Wolpert, J. Theor. Biol. 25, 1 (1969)

    Article  ADS  Google Scholar 

  27. S. Okuda, T. Miura, Y. Inoue, T. Adachi, M. Eiraku, Sci. Rep. 8(1), 1 (2018)

    Google Scholar 

  28. M. Mercker, D. Hartmann, A. Marciniak-Czochra, PLoS ONE 8(12), e82617 (2013)

    Article  ADS  Google Scholar 

  29. A. Mietke, F. Jülicher, I.F. Sbalzarini, Proc. Nat. Acad. Sci. 116(1), 29 (2019)

    Article  ADS  Google Scholar 

  30. R. Yoshida, Adv. Mater. 22(31), 3463 (2010)

    Article  Google Scholar 

  31. A.P. Zakharov, K. Dasbiswas, Soft Matter 17, 4738 (2021)

    Article  ADS  Google Scholar 

  32. S. Li, D.A. Matoz-Fernandez, A. Aggarwal, M.O. de la Cruz, Proc. Nat. Acad. Sci. 118, 10 (2021)

    Google Scholar 

  33. K. Dasbiswas, E. Hannezo, N.S. Gov, Biophys. J. 114(4), 968 (2018)

    Article  ADS  Google Scholar 

  34. T.J. Harris, M. Peifer, J. Cell Biol. 170(5), 813 (2005)

    Article  Google Scholar 

  35. A.E. Shyer, T.R. Huycke, C. Lee, L. Mahadevan, C.J. Tabin, Cell 161(3), 569 (2015)

    Article  Google Scholar 

  36. C.L.E. Helm, M.E. Fleury, A.H. Zisch, F. Boschetti, M.A. Swartz, Proc. Nat. Acad. Sci. 102(44), 15779 (2005)

    Article  ADS  Google Scholar 

  37. S. Banerjee, M.L. Gardel, U.S. Schwarz, Ann. Rev. Condens. Matter Phys. 11(1), 421 (2020)

    Article  Google Scholar 

  38. K. Dasbiswas, H. Shiqiong, F. Schnorrer, S.A. Safran, A.D. Bershadsky, Philos. Trans. Royal Soc. B: Biol. Sci. 373(1747), 20170114 (2018)

    Article  Google Scholar 

  39. F. Burla, Y. Mulla, B.E. Vos, A. Aufderhorst-Roberts, G.H. Koenderink, Nature Rev. Phys. 1(4), 249 (2019)

    Article  ADS  Google Scholar 

  40. J. Alvarado, M. Sheinman, A. Sharma, F.C. MacKintosh, G.H. Koenderink, Soft Matter 13, 5624 (2017)

    Article  ADS  Google Scholar 

  41. B.T. Marshall, M. Long, J.W. Piper, T. Yago, R.P. McEver, C. Zhu, Nature 423(6936), 190 (2003)

    Article  ADS  Google Scholar 

  42. G. Bell, Science 200(4342), 618 (1978)

    Article  ADS  Google Scholar 

  43. G. Gompper, D. Kroll, J. Phys. I 6(10), 1305 (1996)

    Google Scholar 

  44. M. Meyer, M. Desbrun, P. Schröder, A.H. Barr, Visualization and mathematics III (Springer, Berlin, 2003), pp. 35–57

    Book  Google Scholar 

  45. X. Bian, S. Litvinov, P. Koumoutsakos, Comput. Methods Appl. Mech. Eng. 359, 112758 (2020)

    Article  ADS  Google Scholar 

  46. P.O. Persson, G. Strang, SIAM Rev. 46(2), 329 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  47. The OpenMP API specification for parallel programming http://www.openmp.org

  48. J.H. Ferziger, M. Perić, R.L. Street, Computational Methods for Fluid Dynamics, vol. 3 (Springer, Berlin, 2002)

    Book  MATH  Google Scholar 

  49. M. Deka, S. Brahmachary, R. Thirumalaisamy, A. Dalal, G. Natarajan, J. Comput. Phys. 422, 108325 (2020)

  50. A. Kicheva, T. Bollenbach, O. Wartlick, F. Jülicher, M. Gonzalez-Gaitan, Current Opinion Genet. Develop. 22(6), 527 (2012)

    Article  Google Scholar 

  51. P.F. Egan, J.R. Moore, A.J. Ehrlicher, D.A. Weitz, C. Schunn, J. Cagan, P. LeDuc, Proc. Nat. Acad. Sci. 114(39), E8147 (2017)

    Article  ADS  Google Scholar 

  52. J. Yuval, S.A. Safran, Phys. Rev. E 87, 042703 (2013)

    Article  ADS  Google Scholar 

  53. G. Salbreux, G. Charras, E. Paluch, Trends Cell Biol. 22(10), 536 (2012)

    Article  Google Scholar 

  54. J.C. Butcher, N. Goodwin, Numerical methods for ordinary differential equations, vol. 2 (Wiley, New York, 2008)

    Book  Google Scholar 

  55. E. Moeendarbary, L. Valon, M. Fritzsche, A.R. Harris, D.A. Moulding, A.J. Thrasher, E. Stride, L. Mahadevan, G.T. Charras, Nat. Mater. 12, 253 (2013)

    Article  ADS  Google Scholar 

  56. M. Ortiz, G. Gioia, J. Mech. Phys. Solids 42(3), 531 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  57. D.P. Holmes, M. Ursiny, A.J. Crosby, Soft Matter 4(1), 82 (2008)

    Article  ADS  Google Scholar 

  58. J.D. Paulsen, E. Hohlfeld, H. King, J. Huang, Z. Qiu, T.P. Russell, N. Menon, D. Vella, B. Davidovitch, Proc. Nat. Acad. Sci. 113(5), 1144 (2016)

    Article  ADS  Google Scholar 

  59. M. Schuppler, F.C. Keber, M. Kröger, A.R. Bausch, Nature Commun. 7(1), 1 (2016)

    Article  Google Scholar 

  60. B. Davidovitch, R.D. Schroll, D. Vella, M. Adda-Bedia, E.A. Cerda, Proc. Nat. Acad. Sci. 108(45), 18227 (2011)

  61. J. Paulose, D.R. Nelson, Soft Matter 9(34), 8227 (2013)

    Article  ADS  Google Scholar 

  62. M.O. Lavrentovich, E.M. Horsley, A. Radja, A.M. Sweeney, R.D. Kamien, Proc. Nat. Acad. Sci. 113(19), 5189 (2016)

    Article  ADS  Google Scholar 

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Acknowledgements

This work was supported by funding from the National Science Foundation: NSF-CREST: Center for Cellular and Biomolecular Machines (CCBM) at the University of California, Merced: NSF-HRD-1547848. We gratefully acknowledge computing time on the Multi-Environment Computer for Exploration and Discovery (MERCED) cluster at UC Merced, which was funded by National Science Foundation Grant No. ACI-1429783.

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Authors and Affiliations

  1. Department of Physics, University of California, Merced, CA, 95343, USA

    Andrei Zakharov & Kinjal Dasbiswas

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  1. Andrei Zakharov
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  2. Kinjal Dasbiswas
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KD and AZ contributed to designing the research, interpretation of the results, analyzing the data and writing the manuscript. AZ performed the computer simulations, collected the data and prepared figures.

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Correspondence to Kinjal Dasbiswas.

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Advances in Computational Methods for Biological Physics—edited by Moumita Das, Alexander G. Fletcher, Karsten Kruse and Rastko Sknepnek.

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Zakharov, A., Dasbiswas, K. Modeling mechanochemical pattern formation in elastic sheets of biological matter. Eur. Phys. J. E 44, 82 (2021). https://doi.org/10.1140/epje/s10189-021-00086-x

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