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.


A set of nonlinear differential equations describing flagellar motion in an external viscous medium is derived. Because of the local nature of these equations and the use of a Crank-Nicolson-type forward time step, which is stable for large deltat, numerical solution of these equations on a digital computer is relatively fast. Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding. The elastic shear resistance is derived from a plausible structural model of the radial link system. The active shear force for the dynein system is specified by a history-dependent functional of curvature characterized by the parameters m0, a proportionality constant between the maximum active shear moment and curvature, and tau, a relaxation time which essentially determines the delay between curvature and active moment.

MeSH terms

  • Cell Movement
  • Flagella*
  • Mathematics
  • Models, Biological
  • Viscosity