XXIV ICTAM, 21-26 August 2016, Montreal, Canada
STOCHASTIC GAIT-SWITCHING IN MOTILE MICROALGAE
Kirsty Y. Wan1 and Raymond E. Goldstein ∗2
1
DAMTP, University of Cambridge, UK
Summary Self-propulsion by flagella is a significant selective advantage for microorganisms which possess such capabilities. There is great
diversity in the number of flagella, their beating modes, and greater still in the ultrastructure of the basal apparatus into which the flagella are
inserted. For free-living unicellular eukaryotes with few flagella the question of their coordination has been receiving increasing attention
from theorists and experimentalists alike. We propose that this coordination is driven intracellularly, and use select algal species to perform
a comparative study of their swimming gaits. Over longer timescales, we can visualize at high spatiotemporal resolution the stochastic
transitions between different gaits which produce striking manifestations of a bacteria-like run-and-tumble locomotion.
GAITS OF FREE-LIVING FLAGELLATES
Uniflagellates including many species of plant and animal sperms navigate fluidic environments by undulating a posteriorlyoriented flagellum. This single appendage can be used to effectuate directional changes, such as turning towards a source of
chemoattractant. In contrast, ciliated microorganisms including the freshwater protozoan Paramecium and the large colonial
alga Volvox [3] rely on a large-scale coordinated motion of large numbers of surface-attached cilia for swimming and feeding.
In this case coordination is thought to be largely due to hydrodynamic interactions between the neighbouring cilia, which tend
to drive pairs of cilia into in-phase synchrony, and arrays of cilia into metachrony. The generality of this phenomenon is challenged when one examines the robust in-phase breaststroke swimming of the unicellular alga Chlamydomonas, the physical
mechanism for synchrony between its two front-mounted flagella is the subject of extensive study and some contention. The
current view is that some form of intracellular coupling must be present, most likely through the contractility of connecting
fibers between the basal bodies whence emanate the flagella [1].
If this were true, then a correlation must be expected between observed gaits or modes of flagellar actuation and the
symmetries of the basal apparatus which not only define the configuration of flagella but also the strength and directionality
of the coupling between flagella. Here we explore the morphospace of distinct modes of flagellar synchrony maintained
by flagella across species of naturally-occurring quadriflagellate algae (for free-swimming or pipette-held individuals) in the
context of their differing basal architectures. Unicellular flagellates possessing more than four flagella are rare, with only a
few species that have either eight or sixteen flagella [1]– the selective advantage of having ever greater numbers of flagella
must be eventually be outweighed by the energetic and activity costs of flagellar synthesis and coordination. By analogy with
well-documented forms of locomotion in quadrupeds, we find that quadriflagellate gaits include the trot, the pronk, the gallop,
and even a stand gait in which all flagella are non-moving. Species-specific symmetries existing within the basal apparatus
are therefore implicated in constraining the possible phase relations between flagella.
GAIT SYMMETRIES
As in many biological systems exhibiting complex periodic behaviour, it has proved useful to consider a dimensionreduced analysis of oscillations in terms of phase. We were able to measure the phases of individual flagella in several species
of quadriflagellates through a combination of micromanipulation, cell tracking, and high-speed imaging. For flagellar phases
ψj (flagellum index j) we compute the matrix ∆ij = ψi − ψj (i, j = 1, . . . , 4), where ∆ij = ∆ji , ∆ii = 0, and ∆ik =
∆ij + ∆jk . Each gait is then associated with a 3-tuple of phase differences: [∆12 ∆13 ∆14 ]. The primitive Prasinophyte alga
Figure 1: A) In free-living quadriflagellates, two possible configurations are possible (types 1,2). The flagella emerge from basal bodies
comprising a species-specific network whose symmetries may constrain observed swimming gaits. B) In T. suecica (type 2) the relative
phase between the remaining flagella is unchanged when beating is purposefully stalled in one flagellum.
∗ Corresponding
author. Email: [email protected]
Figure 2: Flagellated eukaryote performs biased random walk by actively modulating the orientation and coordination of anteriorlypositioned flagellla.
P. parkeae swims with two pairs of precisely alternating breaststrokes akin to the ‘trot’ of a horse, for which [∆12 ∆13 ∆14 ] =
[π 0 π]. On the other hand the advanced green alga C. crucifera exhibits a rotary gallop gait ([∆12 ∆13 ∆14 ] = [π/2 π 3π/2]).
Inspired by studies of central pattern generators (CPGs) in the control of multilegged locomotion in vertebrates, we sought
features of the basal apparatus in these species that may correlated with the spatiotemporal symmetries of observed gaits
(where a spatiotemporal symmetry comprises a permutation of flagella and a temporal phase shift that fixes the dynamics).
For instance the rotary gallop of four oscillators (Figure 1A) is fixed by the identity transformation, but also by interchange
of diagonals with a phase shift of π, and can emerge from a square network in which the coupling between all oscillators are
identical. This square symmetry is shown in electronmicrographs of sections through C. crucifera, in which basal bodies are
inserted radially at corners of a square.
GAITS SWITCHING DURING FREE-SWIMMING
The flagellate stand gait has the most symmetries (reflection in the horizontal, in the vertical, interchange of diagonals and
arbitrary phase shift) and has the symmetry group D2 × S 1 , where D is a dihedral group and S the circle group. Intriguingly,
bifurcation from the stand to an alternate swimming gait is observed in many species. Forward swimming can further bifurcate
to a shock response in which the front-mounted flagella change from a puller-type to a pusher-type motility, and the swimming
direction is dramatically reversed or altered. Such gait transitions can occur either stochastically or be induced by mechanical
perturbations, but the backward-swimming phase is transient and a steady-state beating pattern is re-established within a
few beats [2]. Over time (Figure 2), these episodic runs (forward-swimming) and tumbles (gait-switching) allow the cell to
navigate a much larger region (effective diffusion constant of D ∝ v 2 τ ≈ 10−4 cm2 /s for an alga swimming at v = 200
µm/s and mean run duration τ of 0.5 s, compared to passive Brownian diffusion Db = 10−9 cm2 /s for a 20 µm cell body).
CONCLUSIONS
Somewhat surprisingly, we find that motile algae can i) maintain precise gaits of swimming defined by specific phase
relations between flagella, yet ii) also rapidly modify these gaits in response to mechanical or biochemical cues to overcome
microscale diffusion barriers. Thus the propensity for a run-and-tumble like dynamics appears to be a general strategy adopted
not only by peritrichously flagellated bacteria [4, 5], but many species of eukaryotic flagellates alike. In light of these results
the control of flagellar coordination and synchrony must now be re-evaluated in the context of intracellular signalling.
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