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. 1998 Apr;74(4):1658–1676. doi: 10.1016/S0006-3495(98)77879-8

Computation of the internal forces in cilia: application to ciliary motion, the effects of viscosity, and cilia interactions.

S Gueron 1, K Levit-Gurevich 1
PMCID: PMC1299513  PMID: 9545031

Abstract

This paper presents a simple and reasonable method for generating a phenomenological model of the internal mechanism of cilia. The model uses a relatively small number of parameters whose values can be obtained by fitting to ciliary beat shapes. Here, we use beat patterns observed in Paramecium. The forces that generate these beats are computed and fit to a simple functional form called the "engine." This engine is incorporated into a recently developed hydrodynamic model that accounts for interactions between neighboring cilia and between the cilia and the surface from which they emerge. The model results are compared to data on ciliary beat patterns of Paramecium obtained under conditions where the beats are two-dimensional. Many essential features of the motion, including several properties that are not built in explicitly, are shown to be captured. In particular, the model displays a realistic change in beat pattern and frequency in response to increased viscosity and to the presence of neighboring cilia in configurations such as rows of cilia and two-dimensional arrays of cilia. We found that when two adjacent model cilia start beating at different phases they become synchronized within several beat periods, as observed in experiments where two flagella are brought into close proximity. Furthermore, examination of various multiciliary configurations shows that an approximately antiplectic wave pattern evolves autonomously. This modeling evidence supports earlier conjectures that metachronism may occur, at least partially, as a self-organized phenomenon due to hydrodynamic interactions between neighboring cilia.

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Selected References

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  1. Blum J. J., Hines M. Biophysics of flagellar motility. Q Rev Biophys. 1979 May;12(2):103–180. doi: 10.1017/s0033583500002742. [DOI] [PubMed] [Google Scholar]
  2. Brokaw C. J. Computer simulation of flagellar movement. I. Demonstration of stable bend propagation and bend initiation by the sliding filament model. Biophys J. 1972 May;12(5):564–586. doi: 10.1016/S0006-3495(72)86104-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  3. Brokaw C. J. Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified. Biophys J. 1985 Oct;48(4):633–642. doi: 10.1016/S0006-3495(85)83819-4. [DOI] [PMC free article] [PubMed] [Google Scholar]
  4. Ermentrout G. B. The behavior of rings of coupled oscillators. J Math Biol. 1985;23(1):55–74. doi: 10.1007/BF00276558. [DOI] [PubMed] [Google Scholar]
  5. Eshel D., Gibbons I. R. External mechanical control of the timing of bend initiation in sea urchin sperm flagella. Cell Motil Cytoskeleton. 1989;14(3):416–423. doi: 10.1002/cm.970140311. [DOI] [PubMed] [Google Scholar]
  6. Gheber L., Priel Z. Synchronization between beating cilia. Biophys J. 1989 Jan;55(1):183–191. doi: 10.1016/S0006-3495(89)82790-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. Gueron S., Levit-Gurevich K., Liron N., Blum J. J. Cilia internal mechanism and metachronal coordination as the result of hydrodynamical coupling. Proc Natl Acad Sci U S A. 1997 Jun 10;94(12):6001–6006. doi: 10.1073/pnas.94.12.6001. [DOI] [PMC free article] [PubMed] [Google Scholar]
  8. Gueron S., Liron N. Ciliary motion modeling, and dynamic multicilia interactions. Biophys J. 1992 Oct;63(4):1045–1058. doi: 10.1016/S0006-3495(92)81683-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  9. Gueron S., Liron N. Simulations of three-dimensional ciliary beats and cilia interactions. Biophys J. 1993 Jul;65(1):499–507. doi: 10.1016/S0006-3495(93)81062-2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  10. Hines M., Blum J. J. Bend propagation in flagella. I. Derivation of equations of motion and their simulation. Biophys J. 1978 Jul;23(1):41–57. doi: 10.1016/S0006-3495(78)85431-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
  11. Hines M., Blum J. J. Bend propagation in flagella. II. Incorporation of dynein cross-bridge kinetics into the equations of motion. Biophys J. 1979 Mar;25(3):421–441. doi: 10.1016/S0006-3495(79)85313-8. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. Johnson R. E., Brokaw C. J. Flagellar hydrodynamics. A comparison between resistive-force theory and slender-body theory. Biophys J. 1979 Jan;25(1):113–127. doi: 10.1016/S0006-3495(79)85281-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
  13. Lubliner J. An analysis of interfilament shear in flagella. J Theor Biol. 1973 Sep 14;41(1):119–125. doi: 10.1016/0022-5193(73)90192-6. [DOI] [PubMed] [Google Scholar]
  14. Machemer H. Ciliary activity and the origin of metachrony in Paramecium: effects of increased viscosity. J Exp Biol. 1972 Aug;57(1):239–259. doi: 10.1242/jeb.57.1.239. [DOI] [PubMed] [Google Scholar]
  15. Murase M. Excitable dynein model with multiple active sites for large-amplitude oscillations and bend propagation in flagella. J Theor Biol. 1991 Mar 21;149(2):181–202. doi: 10.1016/s0022-5193(05)80276-0. [DOI] [PubMed] [Google Scholar]
  16. Murase M., Hines M., Blum J. J. Properties of an excitable dynein model for bend propagation in cilia and flagella. J Theor Biol. 1989 Aug 9;139(3):413–430. doi: 10.1016/s0022-5193(89)80219-x. [DOI] [PubMed] [Google Scholar]
  17. Murase M., Shimizu H. A model of flagellar movement based on cooperative dynamics of dynein-tubulin cross-bridges. J Theor Biol. 1986 Apr 21;119(4):409–433. doi: 10.1016/s0022-5193(86)80192-8. [DOI] [PubMed] [Google Scholar]
  18. Murase M. Simulation of ciliary beating by an excitable dynein model: oscillations, quiescence and mechano-sensitivity. J Theor Biol. 1990 Sep 21;146(2):209–231. doi: 10.1016/s0022-5193(05)80136-5. [DOI] [PubMed] [Google Scholar]
  19. Myerscough M. R., Swan M. A. A model for swimming unipolar spirilla. J Theor Biol. 1989 Jul 21;139(2):201–218. doi: 10.1016/s0022-5193(89)80100-6. [DOI] [PubMed] [Google Scholar]
  20. Okuno M., Hiramoto Y. Mechanical stimulation of starfish sperm flagella. J Exp Biol. 1976 Oct;65(2):401–413. doi: 10.1242/jeb.65.2.401. [DOI] [PubMed] [Google Scholar]
  21. Ramia M. Numerical model for the locomotion of spirilla. Biophys J. 1991 Nov;60(5):1057–1078. doi: 10.1016/S0006-3495(91)82143-9. [DOI] [PMC free article] [PubMed] [Google Scholar]
  22. Sleigh M. A., Barlow D. I. How are different ciliary beat patterns produced? Symp Soc Exp Biol. 1982;35:139–157. [PubMed] [Google Scholar]
  23. Sleigh M. A. Patterns of ciliary beating. Symp Soc Exp Biol. 1968;22:131–150. [PubMed] [Google Scholar]
  24. Tamm S. L. Cillary motion in Paramecium. A scanning electron microscope study. J Cell Biol. 1972 Oct;55(1):250–255. doi: 10.1083/jcb.55.1.250. [DOI] [PMC free article] [PubMed] [Google Scholar]

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