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. 1978 Aug;23(2):177–206. doi: 10.1016/S0006-3495(78)85442-3

The equation of motion for sperm flagella.

R Rikmenspoel
PMCID: PMC1473518  PMID: 687760

Abstract

The equation of motion for sperm flagella, in which the elastic bending moment and the active contractile moment are balanced by the moment from the viscous resistance of the surrounding fluid, is solved for a wave solution that superimposes partial solutions. Substitution of the expression for the wave solution into the equation leads to an expression for the active contractile moment. This active moment can be decomposed into two parts. The first part describes an active moment that travels over the flagellum with the mechanical flagellar wave, the second part represents a moment in phase over the entire length of the flagellum, which decreases linearly towards the distal tip. The linear synchronous moment, to which an amount of traveling moment has been added as a perturbation, leads to wave solutions that closely resemble flagellar waves. Properties such as wavelength and wave amplitudes and also the shape of the waves in sea urchin sperm flagella at different frequencies are accurately described by the theory. The change in wave shape in sea urchin sperm flagella at raised viscosity is predicted well by the theory. The different wave properties caused in bull sperm flagella by different boundary conditions at the proximal junction are explained. When only a traveling active moment is present in a flagellum, the wave solutions describe waves of a small wave length in a long flagellum. Some properties of the wave motion of sperm flagella are derived from the theory and verified experimentally.

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Selected References

These references are in PubMed. This may not be the complete list of references from this article.

  1. 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]
  2. Brokaw C. J. Effects of increased viscosity on the movements of some invertebrate spermatozoa. J Exp Biol. 1966 Aug;45(1):113–139. doi: 10.1242/jeb.45.1.113. [DOI] [PubMed] [Google Scholar]
  3. Brokaw C. J., Rintala D. R. Computer simulation of flagellar movement. III. Models incorporating cross-bridge kinetics. J Mechanochem Cell Motil. 1975;3(2):77–86. [PubMed] [Google Scholar]
  4. KAYE J. S. THE FINE STRUCTURE OF FLAGELLA IN SPERMATIDS OF THE HOUSE CRICKET. J Cell Biol. 1964 Sep;22:710–714. doi: 10.1083/jcb.22.3.710. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Lindemann C. B., Gibbons I. R. Adenosine triphosphate-induced motility and sliding of filaments in mammalian sperm extracted with Triton X-100. J Cell Biol. 1975 Apr;65(1):147–162. doi: 10.1083/jcb.65.1.147. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Lindemann C. B., Rudd W. G., Rikmenspoel R. The stiffness of the flagella of impaled bull sperm. Biophys J. 1973 May;13(5):437–448. doi: 10.1016/S0006-3495(73)85997-1. [DOI] [PMC free article] [PubMed] [Google Scholar]
  7. Lindemann C., Rikmenspoel R. Intracellular potentials in bull spermatozoa. J Physiol. 1971 Dec;219(1):127–138. doi: 10.1113/jphysiol.1971.sp009653. [DOI] [PMC free article] [PubMed] [Google Scholar]
  8. Lubliner J., Blum J. J. Analysis of form and speed of flagellar waves according to a sliding filament model. J Mechanochem Cell Motil. 1972 Aug;1(3):157–167. [PubMed] [Google Scholar]
  9. Lubliner J., Blum J. J. Model for bend propagation in flagella. J Theor Biol. 1971 Apr;31(1):1–24. doi: 10.1016/0022-5193(71)90117-2. [DOI] [PubMed] [Google Scholar]
  10. Phillips D. M. Comparative analysis of mammalian sperm motility. J Cell Biol. 1972 May;53(2):561–573. doi: 10.1083/jcb.53.2.561. [DOI] [PMC free article] [PubMed] [Google Scholar]
  11. Phillips D. M. Insect sperm: their structure and morphogenesis. J Cell Biol. 1970 Feb;44(2):243–277. doi: 10.1083/jcb.44.2.243. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. Rikmenspoel R. Contractile events in the cilia of Paramecium, Opalina, Mytilus, and Phragmatopoma. Biophys J. 1976 May;16(5):445–470. doi: 10.1016/S0006-3495(76)85701-3. [DOI] [PMC free article] [PubMed] [Google Scholar]
  13. Rikmenspoel R. Contractile mechanisms in flagella. Biophys J. 1971 May;11(5):446–463. doi: 10.1016/S0006-3495(71)86227-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  14. Rikmenspoel R. Elastic properties of the sea urchin sperm flagellum. Biophys J. 2008 Dec 31;6(4):471–479. doi: 10.1016/S0006-3495(66)86670-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
  15. Rikmenspoel R., Jacklet A. C., Orris S. E., Lindemann C. B. Control of bull sperm motility. Effects of viscosity, KCN and thiourea. J Mechanochem Cell Motil. 1973 May;2(1):7–24. [PubMed] [Google Scholar]
  16. Rikmenspoel R. Movement of sea urchin sperm flagella. J Cell Biol. 1978 Feb;76(2):310–322. doi: 10.1083/jcb.76.2.310. [DOI] [PMC free article] [PubMed] [Google Scholar]
  17. Rikmenspoel R., Rudd W. G. The contractile mechanism in cilia. Biophys J. 1973 Sep;13(9):955–993. doi: 10.1016/S0006-3495(73)86037-0. [DOI] [PMC free article] [PubMed] [Google Scholar]
  18. Rikmenspoel R. The tail movement of bull spermatozoa. Observations and model calculations. Biophys J. 1965 Jul;5(4):365–392. doi: 10.1016/S0006-3495(65)86723-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  19. Satir P. Studies on cilia. 3. Further studies on the cilium tip and a "sliding filament" model of ciliary motility. J Cell Biol. 1968 Oct;39(1):77–94. doi: 10.1083/jcb.39.1.77. [DOI] [PMC free article] [PubMed] [Google Scholar]
  20. Summers K. E., Gibbons I. R. Adenosine triphosphate-induced sliding of tubules in trypsin-treated flagella of sea-urchin sperm. Proc Natl Acad Sci U S A. 1971 Dec;68(12):3092–3096. doi: 10.1073/pnas.68.12.3092. [DOI] [PMC free article] [PubMed] [Google Scholar]
  21. Warner F. D. New observations on flagellar fine structure. The relationship between matrix structure and the microtubule component of the axoneme. J Cell Biol. 1970 Oct;47(1):159–182. doi: 10.1083/jcb.47.1.159. [DOI] [PMC free article] [PubMed] [Google Scholar]

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