70
4 Motion of Microorganisms
Fig. 4.7 (a) Illustration of irreversibility of motion through a propagating wave: forces applied
during two half-cycles (green and red) are oppositely directed (Kaupp and Alvarez, 2016). (b)–(d)
Beating patterns according to their symmetry properties. In each figure the faint line shows the
filament half-a-period later. (b) A planar beating pattern. (c) A planar pattern that is symmetric with
respect to reversal of the horizontal coordinate and the time shift by half a period. (d) A pattern that
is symmetric with respect to rotation around the vertical axis and a simultaneous time shift (Vilfan,
2012)
vironment. The filament’s deformations obey the laws of elasticity of incompressible
slender bodies, reducible to a single longitudinal dimension (Landau and Lifshits,
1986) and dependent on the bending and torsional elastic moduli, but the drag force
by the surrounding fluid brings about additional complications (Cox, 1970). The
common way to compute flows generated by slim bodies is to approximate them as
a sequence of touching beads.
Various shapes and their time evolution can be analyzed by expanding them in
Fourier modes, both in time and along the filament’s length, most commonly, in the
form of traveling waves sin(ks − ωt), where k is a wavenumber along the arc length
s of the filament and ω is a frequency. Symmetric waveforms result in straightline net motion, as in the upper left-hand panel of Fig. 4.8. Asymmetry, leading to
Fig. 4.8 Left: Stroboscopic images of beating patterns (with the color changing from blue to yellow
in the course of an oscillation period) and the corresponding trajectories; straight or curved red
lines indicate, respectively, a zero or non-zero average flagellar curvature. Right: Trajectory with a
color-coded variable average flagellar curvature (Gong et al, 2019)
4 Motion of Microorganisms
Fig. 4.7 (a) Illustration of irreversibility of motion through a propagating wave: forces applied
during two half-cycles (green and red) are oppositely directed (Kaupp and Alvarez, 2016). (b)–(d)
Beating patterns according to their symmetry properties. In each figure the faint line shows the
filament half-a-period later. (b) A planar beating pattern. (c) A planar pattern that is symmetric with
respect to reversal of the horizontal coordinate and the time shift by half a period. (d) A pattern that
is symmetric with respect to rotation around the vertical axis and a simultaneous time shift (Vilfan,
2012)
vironment. The filament’s deformations obey the laws of elasticity of incompressible
slender bodies, reducible to a single longitudinal dimension (Landau and Lifshits,
1986) and dependent on the bending and torsional elastic moduli, but the drag force
by the surrounding fluid brings about additional complications (Cox, 1970). The
common way to compute flows generated by slim bodies is to approximate them as
a sequence of touching beads.
Various shapes and their time evolution can be analyzed by expanding them in
Fourier modes, both in time and along the filament’s length, most commonly, in the
form of traveling waves sin(ks − ωt), where k is a wavenumber along the arc length
s of the filament and ω is a frequency. Symmetric waveforms result in straightline net motion, as in the upper left-hand panel of Fig. 4.8. Asymmetry, leading to
Fig. 4.8 Left: Stroboscopic images of beating patterns (with the color changing from blue to yellow
in the course of an oscillation period) and the corresponding trajectories; straight or curved red
lines indicate, respectively, a zero or non-zero average flagellar curvature. Right: Trajectory with a
color-coded variable average flagellar curvature (Gong et al, 2019)