Geometry of Wave Propagation on Active Deformable Surfaces

Pearson W. Miller, Norbert Stoop, and Jörn Dunkel

Phys. Rev. Lett. 120, 268001 – Published 29 June 2018

Abstract

Fundamental biological and biomimetic processes, from tissue morphogenesis to soft robotics, rely on the propagation of chemical and mechanical surface waves to signal and coordinate active force generation. The complex interplay between surface geometry and contraction wave dynamics remains poorly understood, but it will be essential for the future design of chemically driven soft robots and active materials. Here, we couple prototypical chemical wave and reaction-diffusion models to non-Euclidean shell mechanics to identify and characterize generic features of chemomechanical wave propagation on active deformable surfaces. Our theoretical framework is validated against recent data from contractile wave measurements on ascidian and starfish oocytes, producing good quantitative agreement in both cases. The theory is then applied to illustrate how geometry and preexisting discrete symmetries can be utilized to focus active elastic surface waves. We highlight the practical potential of chemomechanical coupling by demonstrating spontaneous wave-induced locomotion of elastic shells of various geometries. Altogether, our results show how geometry, elasticity, and chemical signaling can be harnessed to construct dynamically adaptable, autonomously moving mechanical surface waveguides.

Received 5 October 2017
Revised 5 March 2018

DOI:https://doi.org/10.1103/PhysRevLett.120.268001

Physics Subject Headings (PhySH)

Biomimetic & bio-inspired materials Waveguides

Living matter & active matter

Polymers & Soft Matter

Authors & Affiliations

Pearson W. Miller, Norbert Stoop, and Jörn Dunkel

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA

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Issue

Vol. 120, Iss. 26 — 29 June 2018

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Images

Figure 1
Traveling wave model reproduces in vivo mechanics of embryonic surface contraction waves. (a) For $A_{C} = 1.5$ and $A_{I} = 1.6$ , the contraction wave model reproduces sperm-triggered ascidian embryo shape dynamics. Top: Microscopy images of the deformed oocytes (adapted with permission from Ref. [19]) overlaid with cross sections of our simulation. Bottom: Corresponding 3D surfaces indicating the concentration field. Scale bar $50 μ m$ . See Video 1 [28] for full motion. (b) Simulation with $A_{C} = A_{I} = 0.5$ reproduces the surface contraction wave in starfish oocytes during anaphase in meiosis I. Experimental images adapted with permission from Ref. [21]. Scale bar $30 μ m$ . See Video 2 [28] for full motion. (c) Kymographs depicting the local radius of mean curvature for the experimental (top) and simulation (bottom) cases. Regions of maximal curvature indicate the center of the wave and travel at a constant speed. $μ_{V} = 24 Y h / R^{4}$ , $h / R = 0.05$ .
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Figure 2
For weak chemomechanical coupling, point wave propagation reflects the discrete symmetries of the elastic media. (a) Weakly coupled ( $A_{I} = 0$ , $A_{C} = 0.2$ , side length $L = 20 h$ ) waves propagating from a point on various geometries. On discrete surfaces, such as the cube and triangular prism shown here, self-interference increases concentration of the wave front near edges and vertices and breaks the initial radial symmetry of the wave; see Video 3 [28]. (b) Unfolded representations of the two triangular prism cases shown in (a). Through choice of starting point $^{*}$ , the wave front can be guided to exhibit a threefold or fivefold symmetry on the opposing face. Unique faces and vertices are color coded for clarity.
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Figure 3
Mechanically coupled chemical waves induce locomotion in elastic shells on a solid substrate under gravity; see Videos 4–7 [28]. (a) Locomotion in deformation-insensitive systems (hollow symbols, dashed lines) is slower than for deformation-coupled waves (full circles, solid lines) in spherical (blue) and prismatic shells (red). Time is normalized for each system in terms of oscillation period $T$ , and distance is measured in terms of shell radius $R$ . (b)–(e) Corresponding snapshots in time demonstrate that the wave shape is strongly affected by deformations (b),(d) as compared to deformation-insensitive wave propagation (c),(e). Spherical shells have radius $R = 25$ , while for prisms, all edges have length $L = 2.3 R$ . In all cases, thickness $h = 0.084 R$ and volume multiplier $μ_{V} = 0.4 Y h / R^{4} = 11.1 Y h / L^{4}$ .
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