Determination of the average shape of flagellar bends: a gradient curvature model

Data obtained by manual digitization of photographs of flagellar bending waves have been analyzed by determining size parameters for the bends by least-squares fitting of a model waveform. These parameters were then used to normalize the data so that the average shape of the bends could be determined. Best fits were obtained with a model waveform derived from the constant curvature waveforms used previously but with provision for a linear change in curvature across the central region of the bend-the gradient curvature model (GCM). The central regions of the GCM bending waves are separated by transition regions with length determined by a parameter called the truncation factor (FT). Fitting the GCM to sine-generated bending waves give optimal fit when FT = 0.34. Fitting the GCM to four different samples of flagellar bending waves gave best fits with values of FT ranging from 0.17 for ATP-reactivated Lytechinus spermatozoa beating at approximately 10 Hz to 0.32 for live spermatozoa of Arbacia. The difference between the Arbacia waveforms and a sine-generated waveform is therefore very small, but a sine-generated waveform lacks the degree of freedom represented by FT that is required to fit other waveforms optimally. The residual differences between the waveform data and optimal GCM waveforms were averaged and found to be small. In most cases, the curvature in the central region of the optimal GCM decreased in magnitude towards the tip of the flagellum; however, this slope was highly variable and sometimes positive. Significant variations in both this slope and FT were found in individual bends as they propagated along a flagellum.