Continuum Theory of Active Phase Separation in Cellular Aggregates

Open Access

Continuum Theory of Active Phase Separation in Cellular Aggregates

Hui-Shun Kuan, Wolfram Pönisch, Frank Jülicher, and Vasily Zaburdaev

Phys. Rev. Lett. 126, 018102 – Published 4 January 2021

Abstract

Dense cellular aggregates are common in biology, ranging from bacterial biofilms to organoids, cell spheroids, and tumors. Their dynamics, driven by intercellular forces, is intrinsically out of equilibrium. Motivated by bacterial colonies as a model system, we present a continuum theory to study dense, active, cellular aggregates. We describe the process of aggregate formation as an active phase separation phenomenon, while the merging of aggregates is rationalized as a coalescence of viscoelastic droplets where the key timescales are linked to the turnover of the active force. Our theory provides a general framework for studying the rheology and nonequilibrium dynamics of dense cellular aggregates.

Received 31 March 2020
Accepted 29 November 2020

DOI:https://doi.org/10.1103/PhysRevLett.126.018102

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. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Cell aggregation Cell mechanics Collective behavior Phase separation

Statistical Physics & ThermodynamicsPhysics of Living Systems

Authors & Affiliations

Hui-Shun Kuan ^1,2,3, Wolfram Pönisch ^2,4,5, Frank Jülicher ^2,6,7, and Vasily Zaburdaev ^1,2,3,*

¹Department of Biology, Friedrich-Alexander-Universität Erlangen-Nürnberg, 91058 Erlangen, Germany
²Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany
³Max Planck Zentrum für Physik und Medizin, 91058 Erlangen, Germany
⁴MRC Laboratory for Molecular Cell Biology, University College London, WC1E 6BT London, United Kingdom
⁵Department of Physiology, Development and Neuroscience, University of Cambridge, CB2 3DY Cambridge, United Kingdom
⁶Center for Systems Biology Dresden, 01307 Dresden, Germany
⁷Cluster of Excellence Physics of Life, Technische Universität Dresden, 01307 Dresden, Germany

^*vasily.zaburdaev@fau.de

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Issue

Vol. 126, Iss. 1 — 8 January 2021

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Images

Figure 1
Bacterial colonies and interactions. (a) N. gonorrhoeae aggregates formed on a solid substrate. (b) Two merging aggregates and pili network. (c) A single cell and its pili. The images are from (a) optical, (b) electron, and (c) transmission electron microscopy. (d) Microscopic forces acting on the cells. Cells are represented as disks of radius $R$ . Solid and dashed lines denote pairs of bound and free pili, respectively. Pili length $l$ is exponentially distributed with the mean $l_{0}$ . Cells $i$ and $j$ have $n_{p}^{i j}$ bound pili pairs between them. Each pair generates a force dipole of strength $f_{p}$ . Thus, $f_{p} n_{p}^{i j}$ is the magnitude of the force acting from cell $j$ on cell $i$ due to the pili, while the green arrow shows its direction. $f_{i k}$ and red arrows show the magnitude and the direction of the steric repulsive force.
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Figure 2
Aggregate formation as a phase separation phenomenon. (a) Difference of high and low cell number density phases as a function of time. The critical initial density of the onset of the phase separation is at $n_{c} R^{2} ≃ 0.049$ for $k_{on} / k_{off} R^{2} = 1.78$ , the initial density fluctuation level of 0.001, and pili length is set $l_{0} / R = 2.67$ [for all panels except (c)]. (b) The snapshot of the density after the onset of the phase separation [initial density is $n_{0} R^{2} = 0.079$ as in the second line from top of panel (a), and $f_{p} / π R E = 6$ ]. (c) The cell number density (solid curve) and total number of bound pili density (dashed curve) of stable aggregates for two different pili lengths. The bulk elastic moduli are taken as $f_{p} / π R E = 6$ (smaller plateau) and $f_{p} / π R E = 7.5$ (larger plateau) to provide the same degree of phase separation. The pili length controls the width of the aggregate’s boundary ( $n_{0} R^{2} = 0.072$ ). (d) The nematic tensor $\underline{\underline{N}} - \frac{1}{2} Tr [\underline{\underline{N}}]$ inside of a stabilized aggregate ( $n_{0} R^{2} = 0.072$ ). Lines and their length indicate the nematic ordering direction and the order parameter, respectively.
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Figure 3
Coalescence of aggregates. (a) The snapshots of two aggregates merging at different times ( $k_{on} ξ / f_{p} R = 0.083$ , $k_{off} ξ R / f_{p} = 0.047$ , and $f_{p} / π R E = 6$ , Movie S2). Lines indicate the magnitude and direction of the nematic ordering as in Fig. 2. (b) The bridge height $h / R$ as the function of the rescaled time $t f_{p} / R ξ$ . Top: inset shows definition of $h (t)$ . The boundary of the aggregate is defined as the level set of midpoint between the low and high density. Changing $f_{p}$ , while $k_{on}$ and $k_{off}$ are kept constant, is shown with gradient of colors, with darker color corresponding to higher force. Middle: changing $k_{off}$ , where $f_{p} / π R E = 6$ and $k_{on} / R^{2} k_{off} = 1.78$ are kept constant to guarantee the same degree of phase separation. The gradient of colors indicates the values of $k_{off}$ , with the darker color corresponding to higher value of $k_{off}$ . The inset shows the asymptotic merging time $τ_{m}$ (corresponding to the dashed lines in the main plot) which linearly depends on $k_{off}^{- 1}$ due to the effective viscosity. Bottom: changing both $k_{off}$ and $f_{p}$ . The inset shows the master curve after rescaling as discussed in the text. The top curve is close to the parameter regime considered in the previous study [21].
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