Abstract
Colonies of the green alga Volvox are spheres that swim through the beating of pairs of flagella on their surface somatic cells. The somatic cells themselves are mounted rigidly in a polymeric extracellular matrix, fixing the orientation of the flagella so that they beat approximately in a meridional plane, with axis of symmetry in the swimming direction, but with a roughly azimuthal offset which results in the eponymous rotation of the colonies about a body-fixed axis. Experiments on colonies of Volvox carteri held stationary on a micropipette show that the beating pattern takes the form of a symplectic metachronal wave (Brumley et al. Phys. Rev. Lett., vol. 109, 2012, 268102). Here we extend the Lighthill/Blake axisymmetric, Stokes-flow model of a free-swimming spherical squirmer (Lighthill Commun. Pure Appl. Maths, vol. 5, 1952, pp. 109–118; Blake J. Fluid Mech., vol. 46, 1971b, pp. 199–208) to include azimuthal swirl. The measured kinematics of the metachronal wave for 60 different colonies are used to calculate the coefficients in the eigenfunction expansions and hence predict the mean swimming speeds and rotation rates, proportional to the square of the beating amplitude, as functions of colony radius. As a test of the squirmer model, the results are compared with measurements (Drescher et al. Phys. Rev. Lett., vol. 102, 2009, 168101) of the mean swimming speeds and angular velocities of a different set of 220 colonies, also given as functions of colony radius. The predicted variation with radius is qualitatively correct, but the model underestimates both the mean swimming speed and the mean angular velocity unless the amplitude of the flagellar beat is taken to be larger than previously thought. The reasons for this discrepancy are discussed.
Highlights
Volvox is a genus of algae with spherical, free-swimming colonies consisting of up to 50 000 surface somatic cells embedded in an extracellular matrix and a smallT
The colonies are about 0.3 % denser than water, and swim upwards in still water; this is because the relatively dense interior cells are clustered towards the rear, so when the anterior–posterior axis is deflected from vertical, the colony experiences a restoring gravitational torque that competes with a viscous torque to right the colony on a timescale of ∼10 s
This is further demonstrated in figure 5 which shows kymographs of ur and uθ measured at a distance r = 1.3 × a0 from the colony surface: the propagating wave is clearly seen in figure 5(a), which includes evidence of an interesting phase defect, while figure 5(b) suggests that the tangential velocity behaves more like a standing wave, dominated by the power stroke near the equator
Summary
Volvox is a genus of algae with spherical, free-swimming colonies consisting of up to 50 000 surface somatic cells embedded in an extracellular matrix and a small. The colony swims in a direction parallel to its anterior–posterior axis thanks to the beating of a pair of flagella on each somatic cell. Drescher et al (2009) measured the swimming speeds, sedimentation speeds, and angular velocities of 78, 81 and 61 colonies of V. carteri, respectively, ranging in radius from about 100 μm to about 500 μm. The (projected) velocity field was measured using particle image velocimetry (PIV); a total of 60 different colonies were investigated, ranging in radius from 48 μm to 251 μm (mean 144 ± 43 μm), the distribution of which is shown in figure 3. One example of the time-averaged magnitude of the velocity distribution is shown in figure 4(a) This is a maximum near the equator because the flagellar beating drives a non-zero mean flow past the colony, parallel to the axis of symmetry and directed
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