Bulletin of Mathematical Biology

, Volume 80, Issue 3, pp 583–597 | Cite as

Mathematical Model of Contractile Ring-Driven Cytokinesis in a Three-Dimensional Domain

Original Article


In this paper, a mathematical model of contractile ring-driven cytokinesis is presented by using both phase-field and immersed-boundary methods in a three-dimensional domain. It is one of the powerful hypotheses that cytokinesis happens driven by the contractile ring; however, there are only few mathematical models following the hypothesis, to the author’s knowledge. I consider a hybrid method to model the phenomenon. First, a cell membrane is represented by a zero-contour of a phase-field implicitly because of its topological change. Otherwise, immersed-boundary particles represent a contractile ring explicitly based on the author’s previous work. Here, the multi-component (or vector-valued) phase-field equation is considered to avoid the emerging of each cell membrane right after their divisions. Using a convex splitting scheme, the governing equation of the phase-field method has unique solvability. The numerical convergence of contractile ring to cell membrane is proved. Several numerical simulations are performed to validate the proposed model.


Cytokinesis Contractile ring Phase-field Immersed-boundary 



The author was supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korean government (No. A21300000) and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2017R1C1B1001937).


  1. Alberts B, Johnson A, Lewis J, Raff M, Roberts P (2002) Molecular biology of the cell, 4th edn. Garland Science, New YorkGoogle Scholar
  2. Bathe M, Chang F (2010) Cytokinesis and the contractile ring in fission yeast: towards a systems-level understanding. Trends Microbiol 18:38–45CrossRefGoogle Scholar
  3. Bertozzi A, Esedoglu S, Gilette A (2007) Inpainting of binary images using the Cahn–Hilliard equation. IEEE Trans Image Process 16:285–291MathSciNetCrossRefMATHGoogle Scholar
  4. Bi E, Maddox P, Lew D, Salmon E, McMilland E, Yeh E, Prihngle J (1998) Involvement of an actomyosin contractile ring in Saccharomyces cerevisiae cytokinesis. J Cell Biol 142:1301–1312CrossRefGoogle Scholar
  5. Botella O, Ait-Messaoud M, Pertat A, Cheny Y, Rigal C (2015) The LS-STAG immersed boundary method for non-Newtonian flows in irregular geometries: flow of shear-thinning liquids between eccentric rotating cylinders. Theor Comput Fluid Dyn 29:93–110CrossRefGoogle Scholar
  6. Britton N (2003) Essential mathematical biology. Springer, BerlinCrossRefMATHGoogle Scholar
  7. Cahn J, Hilliard J (1958) Free energy of a nonuniform system. I. Interfacial free energy. J Chem Phys 28(2):258–267CrossRefGoogle Scholar
  8. Calvert M, Wright G, Lenong F, Chiam K, Chen Y, Jedd G, Balasubramanian M (2011) Myosin concentration underlies cell size-dependent scalability of actomyosin ring constriction. J Cell Biol 195:799–813CrossRefGoogle Scholar
  9. Carvalgo A, Oegema ADK (2009) Structural memory in the contractile ring makes the duration of cytokinesis independent of cell size. Cell 137:926–937CrossRefGoogle Scholar
  10. Celton-Morizur S, Dordes N, Fraisier V, Tran P, Paoletti A (2004) C-terminal anchoring of mid1p to membranes stabilized cytokinetic ring position in early mitosis in fission yeast. Mol Cellul Biol 24:10621–10635CrossRefGoogle Scholar
  11. Chang F, Drubin D, Nurse P (1997) cdc12p, a protein required for cytokineses in fission yeast, is a component of the cell division ring and interacts with profilin. J Cell Biol 137:169–182CrossRefGoogle Scholar
  12. Chen Y, Wise S, Shenoy V, Lowengrub J (2014a) A stable scheme for a nonlinear multiphase tumor growth model with an elastic membrane. Int J Numer Methods Biomed Eng 30(7):726–754MathSciNetCrossRefGoogle Scholar
  13. Chen Z, Hickel S, Devesa A, Berland J, Adams N (2014b) Wall modeling for implicit large-eddy simulation and immersed-interface methods. Theor Comput Fluid Dyn 28(1):1–21CrossRefGoogle Scholar
  14. Chorin A (1968) Numerical solution of the Navier–Stokes equation. Math Comput 22:745–762MathSciNetCrossRefMATHGoogle Scholar
  15. Daniels M, Wang Y, Lee M, Venkitaraman A (2004) Abnormal cytokinesis in cells deficient in the breast cancer susceptibility protein brca2. Science 306(5697):876–879CrossRefGoogle Scholar
  16. de Fontaine D (1967) A computer simulation of the evolution of coherent composition variations in solid solutions. Ph.D. thesis, Northwestern UniversityGoogle Scholar
  17. Eyer D (1998) Unconditionally gradient stable scheme marching the Cahn–Hilliard equation. MRS Proc 529:39–46CrossRefGoogle Scholar
  18. Gisselsson D, Jin Y, Lindgren D, Persson J, Gisselsson L, Hanks S, Sehic D, Mengelbier L, Øra I, Rahman N et al (2010) Generation of trisomies in cancer cells by multipolar mitosis and incomplete cytokinesis. Proc Natl Acad Sci 107(47):20489–20493CrossRefGoogle Scholar
  19. Gompper G, Zschoke S (1991) Elastic properties of interfaces in a Ginzburg–Landau theory of swollen micelles, droplet crystals and lamellar phases. Europhys Lett 16:731–736CrossRefGoogle Scholar
  20. Harlow E, Welch J (1965) Numerical calculation of time dependent viscous incompressible flow with free surface. Phys Fluid 8:2182–2189MathSciNetCrossRefMATHGoogle Scholar
  21. Helfrich W (1973) Elastic properties of lipid bilayers: theory and possible experiments. Z Naturforschung C 28:693–703Google Scholar
  22. Jochova J, Rupes I, Streiblova E (1991) F-actin contractile rings in protoplasts of the yeast schizosaccharomyces. Cell Biol Int Rep 15:607–610CrossRefGoogle Scholar
  23. Kamasaki T, Osumi M, Mabuchi I (2007) Three-dimensional arrangement of f-actin in the contractile ring of fission yeast. J Cell Biol 178:765–771CrossRefGoogle Scholar
  24. Kang B, Mackey M, El-Sayed M (2010) Nuclear targeting of gold nanoparticles in cancer cells induces dna damage, causing cytokinesis arrest and apoptosis. J Am Chem Soc 132(5):1517–1519CrossRefGoogle Scholar
  25. Kim J (2005) A continuous surface tension force formulation for diffuse-interface models. J Comput Phys 204(2):784–804MathSciNetCrossRefMATHGoogle Scholar
  26. Koudehi M, Tang H, Vavylonis D (2016) Simulation of the effect of confinement in actin ring formation. Biophys J 110(3):126aCrossRefGoogle Scholar
  27. Lee H, Kim J (2008) A second-order accurate non-linear difference scheme for the n-component Cahn–Hilliard system. Physica A 387:4787–4799MathSciNetCrossRefGoogle Scholar
  28. Lee H, Choi J, Kim J (2012) A practically unconditionally gradient stable scheme for the n-component Cahn–Hilliard system. Physica A 391:1009–1019CrossRefGoogle Scholar
  29. Lee S, Jeong D, Choi Y, Kim J (2016a) Comparison of numerical methods for ternary fluid flows: immersed boundary, level-set, and phase-field methods. J KSIAM 20(1):83–106MathSciNetMATHGoogle Scholar
  30. Lee S, Jeong D, Lee W, Kim J (2016b) An immersed boundary method for a contractile elastic ring in a three-dimensional newtonian fluid. J Sci Comput 67(3):909–925MathSciNetCrossRefMATHGoogle Scholar
  31. Li Y, Kim J (2016) Three-dimensional simulations of the cell growth and cytokinesis using the immersed boundary method. Math Biosci 271:118–127MathSciNetCrossRefMATHGoogle Scholar
  32. Li Y, Yun A, Kim J (2012) An immersed boundary method for simulating a single axisymmetric cell growth and division. J Math Biol 65:653–675MathSciNetCrossRefMATHGoogle Scholar
  33. Li Y, Jeong D, Choi J, Lee S, Kim J (2015) Fast local image inpainting based on the local Allen–Cahn model. Digital Signal Process 37:65–74CrossRefGoogle Scholar
  34. Lim S (2010) Dynamics of an open elastic rod with intrinsic curvature and twist in a viscous fluid. Phys Fluids 22(2):024104CrossRefMATHGoogle Scholar
  35. Lim S, Ferent A, Wang X, Peskin C (2008) Dynamics of a closed rod with twist and bend in fluid. SIAM J Sci Comput 31(1):273–302MathSciNetCrossRefMATHGoogle Scholar
  36. Mandato C, Berment W (2001) Contraction and polymerization cooperate to assemble and close actomyosin rings round xenopus oocyte wounds. J Cell Biol 154:785–797CrossRefGoogle Scholar
  37. Miller A (2011) The contractile ring. Curr Biol 21:R976–R978CrossRefGoogle Scholar
  38. Pelham R, Chang F (2002) Actin dynamics in the contractile ring during cytokinesis in fission yeast. Nature 419:82–86CrossRefGoogle Scholar
  39. Peskin C (1972) Flow patterns around heart valves: a numerical method. J Comput Phys 10(2):252–271MathSciNetCrossRefMATHGoogle Scholar
  40. Pollard T, Cooper J (2008) Actin, a central player in cell shape and movement. Science 326:1208–1212CrossRefGoogle Scholar
  41. Posa A, Balaras E (2014) Model-based near-wall reconstructions for immersed-boundary methods. Theor Comput Fluid Dyn 28(4):473–483CrossRefGoogle Scholar
  42. Shlomovitz R, Gov N (2008) Physical model of contractile ring initiation in dividing cells. Biophys J 94:1155–1168CrossRefGoogle Scholar
  43. Trottenberg U, Oosterlee C, Schüller A (2001) Multigrid. Academic Press, LondonMATHGoogle Scholar
  44. Vahidkhah K, Abdollahi V (2012) Numerical simulation of a flexible fiber deformation in a viscous flow by the immersed boundary-lattice Boltzmann method. Commun Nonlinear Sci Numer Simul 17(3):1475–1484MathSciNetCrossRefMATHGoogle Scholar
  45. Vavylonis D, Wu J, Hao S, O’Shaughnessy B, Pollard T (2008) Assembly mechanism of the contractile ring for cytokinesis by fission yeast. Science 319:97–100CrossRefGoogle Scholar
  46. Wang MZY (2008) Distinct pathways for the early recruitment of myosin ii and actin to the cytokinetic furrow. Mol Biol Cell 19(1):318–326CrossRefGoogle Scholar
  47. Wheeler A, Boettinger W, McFadden G (1992) Phase-field model for isothermal phase transitions in binary alloys. Phys Rev A 45(10):7424–7439CrossRefGoogle Scholar
  48. Zang J, Spudich J (1998) Myosin ii localization during cytokinesis occurs by a mechanism that does not require its motor domain. Proc Natl Acad Sci 95(23):13652–13657CrossRefGoogle Scholar
  49. Zhao J, Wang Q (2016a) A 3d multi-phase hydrodynamic model for cytokinesis of eukaryotic cells. Commun Comput Phys 19(03):663–681MathSciNetCrossRefMATHGoogle Scholar
  50. Zhao J, Wang Q (2016b) Modeling cytokinesis of eukaryotic cells driven by the actomyosin contractile ring. Int J Numer Methods Biomed Eng 32(12):e027774Google Scholar
  51. Zhou Z, Munteanu E, He J, Ursell T, Bathe M, Huang K, Chang F (2015) The contractile ring coordinates curvature-dependent septum assembly during fission yeast cytokinesis. Mol Biol Cell 26(1):78–90CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.National Institute for Mathematical SciencesDaejeonRepublic of Korea

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