Bend or Twist? What Plectonemes Reveal about the Mysterious Motility of Spiroplasma

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Bend or Twist? What Plectonemes Reveal about the Mysterious Motility of Spiroplasma

Paul M. Ryan, Joshua W. Shaevitz, and Charles W. Wolgemuth

Phys. Rev. Lett. 131, 178401 – Published 24 October 2023

See synopsis: Two Experimental Observations of Helix Reversals

Abstract

Spiroplasma is a unique, helical bacterium that lacks a cell wall and swims using propagating helix hand inversions. These deformations are likely driven by a set of cytoskeletal filaments, but how remains perplexing. Here, we probe the underlying mechanism using a model where either twist or bend drive spiroplasma’s chirality inversions. We show that Spiroplasma should wrap into plectonemes at different values of the length and external viscosity, depending on the mechanism. Then, by experimentally measuring the bending modulus of Spiroplasma and if and when plectonemes form, we show that Spiroplasma’s helix hand inversions are likely driven by bending.

Received 22 February 2023
Accepted 15 September 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.178401

Physics Subject Headings (PhySH)

Cell locomotion Cell mechanics Elastic deformation Low Reynolds number flows Nonlinear dynamics in fluids Slender body theory Swimming

Physics of Living SystemsFluid DynamicsPolymers & Soft Matter

synopsis

Two Experimental Observations of Helix Reversals

Published 24 October 2023

Helical bacteria and corkscrew rods can both undergo handedness reversals that could be useful in future robotic systems.

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Authors & Affiliations

Paul M. Ryan ^1,*, Joshua W. Shaevitz², and Charles W. Wolgemuth ^1,3,†

¹Department of Physics, University of Arizona, Tucson, Arizona 85721, USA
²Joseph Henry Laboratories of Physics and Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey 08544, USA
³Department of Molecular and Cellular Biology, University of Arizona, Tucson, Arizona 85721, USA

^*paulryan@arizona.edu
^†wolg@arizona.edu

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Issue

Vol. 131, Iss. 17 — 27 October 2023

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Images

Figure 1
(a) To create a helix by twisting, a straight rod can first be bent into a planar curve with constant radius $κ_{0}^{- 1}$ and then twisted about its length with twist density $τ_{0}$ . (b) To create one by bending, first bend the rod into a planar, sinusoidal wave, and then bend it along the perpendicular direction into another sinusoidal wave, 90° out of phase from the first wave. In (a) and (b) the red arrows show the bend and twist directions, and the blue lines point along $e_{2}$ . Note that the resulting $e_{2}$ vectors do not point in the same direction in (a) and (b). (c) Cross section of Spiroplasma showing the internal cytoskeletal ribbon with (red) Fibril protein filaments and (blue) SpMreB filaments, as described in [1]. (d) Experiment (top) and bend-model simulation (bottom) of a swimming Spiroplasma at low viscosity (see Supplemental Material movie S1 [2]). The solid arrow indicates swimming direction, the $*$ represents the junction (“kink”) between RH and LH chirality, and time progresses left to right in the figure. The preferred torsion, $τ$ , and curvatures, $κ_{1, 2}$ as a function of arclength for a Spiroplasma with a chirality inversion versus length for the torsion-based (e) and curvature-based (f) models where $*$ indicates the location of the kink.
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Figure 2
(a) Schematic of the optical trapping setup. A Spiroplasma (green) is immobilized at one end by attachment to a glass cover slip (slide). A spherical bead (blue) trapped in an infrared laser (red) is attached to the Spiroplasma’s free end. The stage is then moved slowly so that the bacterium is stretched. The distance between laser axis and bead center, $Δ x$ , is measured to find the force. (b) Representative force vs extension curves with the best fit to the curve predicted for stretching an elastic helix, from which we find an average bending modulus $A = 0.15 \pm 0.04 pN μ m^{2}$ .
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Figure 3
Upper left: a Spiroplasma cell of length $L = 5.56 μ m$ swims through fluid with viscosity $η = 5.9 cP$ exhibiting a measured kink speed, $v_{k} = 10.0 μ m / s$ [movies S2(a), S2(b), and S2(c) in the Supplemental Material [2] ]. This bacterium never forms a plectoneme. Using the same parameters, the bend-based model’s dynamics (middle left panels) agree well with experiment, while the torsion-based model (lower left panels) forms a plectoneme. Upper right: Spiroplasma length $L = 4.76 μ m$ swimming in fluid with $η = 10$ cP [movies S3(a), S3(b), and S3(c) [2] ]. As a kink propagates along the cell ( $v_{k} = 11.5 μ m / s$ ), a plectoneme forms. Results of the bend (middle right panels) and torsion (lower left panels) model simulations with the same parameters. Plectonemic regions in the simulations are highlighted in red whereas the circled area is a plectoneme incorrectly predicted by the torsion-based model.
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Figure 4
Plectoneme formation as a function of length, viscosity, and kink speed. While the bend- and torsion-based models both predict that the length when plectonemes form depends on $1 / η v_{k}$ , the torsion-based model (black triangles) predicts plectonemes at shorter lengths than the bend-based model (black squares). Experimental observations found where in the $L^{2}$ vs $1 / η v_{k}$ phase-space plectonemes occurred (green triangles) or did not occur (red circles). This experimentally determined plectoneme-forming region agrees well with that predicted by the bend-based model (blue-colored region), while the torsion-based model substantially overestimates the area of this region, with the gray coloring denoting where only the torsion-based model predicted plectonemes. The dashed blue (gray) lines indicate the variability in the onset of plectonemes when varying the bend modulus from $A = 0.14 - 0.20 pN μ m^{2}$ for the bend (torsion) modalities while keeping the bend and twist moduli equal. The “*” indicates the experimental data points shown in Fig. 3.
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