Active gyrotactic stability of microswimmers using hydromechanical signals

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Active gyrotactic stability of microswimmers using hydromechanical signals

Jingran Qiu, Navid Mousavi, Lihao Zhao, and Kristian Gustavsson

Phys. Rev. Fluids 7, 014311 – Published 27 January 2022

Abstract

Many plankton species undergo daily vertical migration to large depths in the turbulent ocean. To do this efficiently, the plankton can use a gyrotactic mechanism, aligning them with gravity to swim downwards or against gravity to swim upwards. Many species show passive mechanisms for gyrotactic stability. For example, bottom-heavy plankton tend to align upwards. This is efficient for upward migration in quiescent flows, but it is often sensitive to turbulence which upsets the alignment. Here we suggest a simple, robust active mechanism for gyrotactic stability, which is only lightly affected by turbulence and allows alignment both along and against gravity. We use a model for a plankton that swims with a constant speed and can actively steer in response to hydrodynamic signals encountered in simulations of a turbulent flow. Using reinforcement learning, we identify the optimal steering strategy. By using its setae to sense its settling velocity transversal to its swimming direction, the swimmer can deduce information about the direction of gravity, allowing it to actively align upwards. The mechanism leads to a rate of upward migration in a turbulent flow that is of the same order as in quiescent flows, unless the turbulence is very vigorous. In contrast, passive swimmers with typical parameters of copepods show much smaller upward velocity in turbulence. Settling may even cause them to migrate downwards in vigorous turbulence.

Received 25 May 2021
Accepted 21 December 2021

DOI:https://doi.org/10.1103/PhysRevFluids.7.014311

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Locomotion Machine learning Turbulence

Microswimmers Self-propelled particles

Fluid DynamicsPhysics of Living Systems

Authors & Affiliations

Jingran Qiu ¹, Navid Mousavi ², Lihao Zhao ^1,*, and Kristian Gustavsson ^2,†

¹AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
²Department of Physics, University of Gothenburg, SE-41296 Gothenburg, Sweden

^*zhaolihao@mail.tsinghua.edu.cn
^†kristian.gustafsson@physics.gu.se

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Vol. 7, Iss. 1 — January 2022

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Images

Figure 1
(a) Orientation axes $n, p$ , and $q$ for a spheroidal swimmer in a fixed Cartesian frame of reference ( $\hat{x}, \hat{y}, \hat{z}$ ) with direction of gravity $\hat{g} = - \hat{z}$ . (b),(c) Unit spheres illustrating the mechanism in Eq. (11) for active gyrotactic stability. Starting with orientation $n_{0}, p_{0}$ , and $q$ , the swimmer actively rotates around its $q$ axis until $| p_{z} | \leq p_{z}^{th}$ (shaded region) with positive rotation if (b) $p_{z, 0} > 0$ and negative rotation if (c) $p_{z, 0} < 0$ , leading to the final orientation $n_{f}, p_{f}$ , and $q$ with $n_{z, f} \geq n_{z, 0}$ in general.
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Figure 2
Summary of strategies obtained by reinforcement learning for the cases of (a), (b) the planar model with single signal $p_{z}$ and (c), (d) full 3D model with joint signal $p_{z}$ and $q_{z}$ . Results from (a), (c) the frozen DNS and (b), (d) the statistical model. Rows represent different states; the signs $-$ , 0, and $+$ represent the states $σ^{(-)}, σ^{(0)}$ , and $σ^{(+)}$ in the discretized signal in Eq. (7). Columns represent different actions, where the signs represent the values $- ω^{(s)}$ , 0, and $+ ω^{(s)}$ , respectively. The action selected in a given state for the best strategy found in our reinforcement learning is highlighted in red. In each panel, the resulting strategies from 60 completed training sessions are summarized. Each numerical value gives the nonzero percentage of strategies choosing a certain action in a given state. The background is color coded according to this percentage from white (0%) to blue (100%).
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Figure 3
Results of numerical simulations in the steady state for (a) average alignment and (b), (c) velocity against (a), (b) the threshold of the flow signal, $p_{z}^{th} = Δ u^{th} / v_{⊥}^{(g)}$ , and (c) the dimensionless angular swimming velocity $ω^{(s)} τ^{(η)}$ . Velocities are normalized using ${〈 v_{z} 〉}_{0}$ in Eq. (10). Symbols show results from simulations of Eqs. (1) with smart planar steering [ $ω^{(s)} = ω_{q}^{(s)} q$ using Eq. (11)] in the statistical model with $Ku = 10$ (colored symbols) and DNS [hollow symbols in (a) and (b)]. Numerical results for the naive strategy ( $ω^{(s)} = 0$ ) in the statistical model are shown as horizontal dashed green lines. The shaded region corresponds to threshold levels below the minimal resolution limit, $Δ u = 20 μ m / s$ . The dash-dotted line in (a) shows the theory for the simplified dynamics, given by Eq. (13). Flow parameters are $u^{rms} = 6.7 mm / s$ and $τ^{(η)} = 1 s$ . All swimmer parameters are according to the values used in Table 1, except for parameter values stated in the figure.
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Figure 4
(a), (d) Steady-state results from statistical model simulations for the average vertical velocity $〈 v_{z} 〉$ against the Kolmogorov time $τ^{(η)}$ (units of seconds) for a swimmer following either the planar strategy (11) (red $\circ$ ), the full 3D strategy using two signals (14) (blue $⋄$ ), the full 3D strategy using four signals (15) (magenta $▵$ ), or the naive passively gyrotactic strategy (green $□$ ). Parameters of the swimmer are given in Table 1. The flow velocity is $u^{rms} = 1 mm / s$ (top row) and $u^{rms} = 10 mm / s$ (bottom row) with a constant Kubo number, $Ku = 10$ . (b), (e) Same for the vertical velocity due to swimming, $v^{(s)} 〈 n_{z} 〉$ (the scale on the right axis shows the alignment $〈 n_{z} 〉$ ). (c), (f) Same for the vertical component of the flow velocity along swimmer trajectories, $〈 u_{z} 〉$ . The displayed velocities are normalized by ${〈 v_{z} 〉}_{0}$ in Eq. (10).
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Figure 5
Same as Fig. 3, but for the full 3D model. Symbols show numerical results following the strategy in Eq. (14). The dash-dotted line shows the theory: Eq. (B5) for $p_{z}^{th} < 1 / \sqrt{2}$ and two times the value in Eq. (13) if $1 / \sqrt{2} < p_{z}^{th} \leq 1$ .
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