# Self-organized shape dynamics of active surfaces

^{a}Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany;^{b}Chair of Scientific Computing for Systems Biology, Faculty of Computer Science, Technische Universität, Dresden, 01187 Dresden, Germany;^{c}Center for Systems Biology Dresden, 01307 Dresden, Germany;^{d}MOSAIC Group, Max Planck Institute of Molecular Cell Biology and Genetics, 01307 Dresden, Germany

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Edited by Erik Luijten, Northwestern University, Evanston, IL, and accepted by Editorial Board Member John A. Rogers November 12, 2018 (received for review July 5, 2018)

## Significance

Morphogenesis, the emergence of shape and form in biological systems, is a process that is fundamentally mechanochemical: Shape changes of material are driven by active mechanical forces that are generated by chemical processes, which in turn can be affected by the deformations and flows that occur. We provide a framework that integrates these interactions between the geometry of deforming materials and active processes in them by introducing the shape dynamics of self-organized active surfaces. We show that the tight coupling between surface mechanics and active processes gives rise to the spontaneous formation of nontrivial shapes, shape oscillations, and directed peristaltic motion. Our simple yet general description lays the foundation to explore the regulatory role of shape in morphogenetic processes.

## Abstract

Mechanochemical processes in thin biological structures, such as the cellular cortex or epithelial sheets, play a key role during the morphogenesis of cells and tissues. In particular, they are responsible for the dynamical organization of active stresses that lead to flows and deformations of the material. Consequently, advective transport redistributes force-generating molecules and thereby contributes to a complex mechanochemical feedback loop. It has been shown in fixed geometries that this mechanism enables patterning, but the interplay of these processes with shape changes of the material remains to be explored. In this work, we study the fully self-organized shape dynamics using the theory of active fluids on deforming surfaces and develop a numerical approach to solve the corresponding force and torque balance equations. We describe the spontaneous generation of nontrivial surface shapes, shape oscillations, and directed surface flows that resemble peristaltic waves from self-organized, mechanochemical processes on the deforming surface. Our approach provides opportunities to explore the dynamics of self-organized active surfaces and can help to understand the role of shape as an integral element of the mechanochemical organization of morphogenetic processes.

Morphogenesis is the generation of patterns and shapes by dynamic processes. In the biological context, cells and tissues are shaped during developmental processes that give rise to organisms starting from a fertilized egg. Biological morphogenesis is fundamentally mechanochemical; i.e., it relies on an interplay between chemical signals and active mechanics (1⇓–3). Active mechanical processes in cells are generated in the cytoskeleton, a gel-like network of protein filaments. Motor proteins, such as myosin, interact with cytoskeletal filaments and generate movement and forces driven by the hydrolysis of ATP (4), thus making the cytoskeleton an active material. The capacity of living matter to generate spontaneous motion, flows, and material deformations as a result of molecular force-generating processes can be captured by the theory of active matter (5, 6).

The cellular actomyosin cortex is a thin layer of an active material near the cell membrane, which governs cell mechanics and cell shape changes. For example, during cell division a contractile ring forms in the cell cortex that drives cortical flows and constricts the cell to create two daughter cells (7⇓⇓–10). Active cortical contractions also play a role in tissue shape changes (11, 12). For example, during gastrulation it has been suggested that contraction of the apical cell cortex drives the invagination of a 2D cell layer, called the epithelium (13). These examples show that active mechanical processes on surfaces play a key role in driving morphogenesis.

The emergent character of morphogenetic processes naturally arises from a feedback loop in which chemically regulated active stresses induce material flows and deformations that in turn affect the chemical regulators (14). This raises the fundamental question of how active stresses are dynamically organized during shape changes of cells and tissues. In the absence of shape changes, it has already been demonstrated that advection of regulators of active stress gives rise to mechanochemical self-organization and patterning (14⇓–16). The general process of morphogenesis introduces additional feedback via changes of the underlying geometry that enables the generation of shapes by self-organizing processes. Here, we present a general framework to describe the dynamics of shapes that arise from the self-organization of mechanochemical processes on an active surface and develop a numerical technique to capture the corresponding shape changes of surfaces with axial symmetry.

Our formulation can be seen as a generalization of previous work, in which shapes are determined from a minimization of energy functionals (17⇓–19) or from a deformations dynamic that is guided by phenomenological rules (20⇓–22). Here, we derive the deformation dynamics and the steady states from the general force and torque balance on curved surfaces. This allows including constitutive relations that are associated with energy functionals of the surface shape (23), as well as constitutive relations that describe active elastic or viscous materials (24, 25). In particular, we consider in this work an active thin film description, which has been extensively applied as a generic model to account for the dynamic behavior of the cellular cortex and tissue layers (7, 16, 26, 27). In our description, active stresses in the material can depend on the concentration of a stress-regulating molecular species that is dynamically changing in response to flows on and deformations of the surface. We study this system on surfaces with spherical and tubular geometries, which represent a broad class of biological structures including the cellular cortex (7, 9, 17, 24, 26, 27) and various types of biological tissue (12, 16, 19, 28).

To solve the resulting equations and the shape dynamics numerically, we use an integral representation of the deforming surface and develop a dynamic coordinate transformation. We show that a mechanochemical instability of a spherical active surface leads to spontaneous polarization and the formation of deformed stationary shapes. We point out that surface deformations, even in the absence of this instability, lead to an accumulation of stress regulators in regions of high curvature due to geometric effects. Furthermore, we show that the self-organization of active stress regulators can spontaneously constrict tubular surfaces and can lead to mechanochemical shape oscillations. Finally, we describe the emergent formation of directed flows on tubular surfaces with bending rigidity that resemble a peristaltic wave resulting from the coupling between self-organized active stresses and surface geometry.

## Geometry and Mechanics of Self-Organized Active Surfaces

We consider a time-dependent surface *A*), where

### Force and Torque Balance Equations of Curved Surfaces.

To describe deformations of Γ as a result of mechanical stresses within and onto the surface, surface configurations should obey force and torque balance equations that we briefly introduce in the following. Consider a line element of length *A*). The forces acting on this line element can be written as *SI Appendix*). Eqs. **1** and **2** describe the force balance in the directions tangential and normal to the surface, respectively, and we have omitted inertial effects. Eq. **3** determines how bending moments in the surface are balanced by out-of-plane shear tension **4** determines their coupling to the antisymmetric part of the tension tensor *Constitutive Relations of Active Fluid Surfaces*.

### Constitutive Relations of Active Fluid Surfaces.

To define the active thin film description studied in this work, we first introduce the equilibrium properties of the surface. We consider here a surface with constant surface tension γ and bending rigidity κ, described by the Helfrich free energy (29)**5** read (23)**5**–**7**, which we do not consider here for simplicity (25). Note that the normal force balance Eq. **2** yields for

To characterize flows and deformations of the thin film, we use the symmetric part of the in-plane strain rate tensor **6**, **7**, and **9**. Additionally, we consider **3**. Furthermore, we assume that the volume enclosed by the surface is conserved, which defines a pressure p that enters the normal force balance Eq. **2** as *SI Appendix*).

### Chemical Regulation of Active Tension.

We consider an active tension amplitude of the form

## Dynamic Representation of Deforming Axisymmetric Surfaces

For a given deformation velocity **14** for a given deformation velocity

An arbitrary axisymmetric surface can be represented explicitly using an arc-length parameterization*B*). In the following, we use ϕ and s explicitly as tensor indexes. The nonvanishing components of the curvature tensor are the azimuthal curvature **16** as**15**, via**14**, can be found by determining the time evolution of the meridional curvature

An arc-length parameterization **17**–**19**. However, the time dependence of the domain S makes it difficult to evaluate the total time derivative in the equation of the shape dynamics, Eq. **14**, and renders the arc-length parameterization impractical for the numerical treatment of differential equations on a deforming surface. We therefore introduce additionally a Eulerian parameterization of Γ given by**15**. Note that X and **14** can be written as**21** and the definition of the metric tensor that for an Eulerian parameterization (*SI Appendix*) **23** with initial condition

To reconstruct the deforming surface via Eqs. **17**–**19** as a function of time, we need the meridional curvature *SI Appendix*)**17**–**19** and **22**–**24** provide a framework to solve Eq. **14** for a given deformation velocity

## Self-Organization of Active Surface Deformations

We now combine the dynamics of surfaces with the physics of an active fluid film. For a given surface shape and distribution of active tension, the instantaneous deformation velocity **1**–**4** with the constitutive relations Eqs. **10** and **11** (*SI Appendix*). Note that these equations determine the velocities up to a constant velocity vector because the system is Galilei invariant. The dynamics of the shape together with dynamics of the stress regulator are then obtained by solving Eqs. **13** and **14**, using at all times the instantaneously determined velocity fields. This combination of geometry and active hydrodynamics captures the mechanochemical feedback mediated by the chemical regulator, where the latter organizes active stress patterns and depends itself on material flows and on changes of the geometry.

We now study self-organized surface deformations of axisymmetric spherical and tubular surfaces. To characterize the strength of the mechanochemical feedback, we introduce a dimensionless contractility parameter*SI Appendix*.

### Self-Organized Shape Dynamics of Spherical Surfaces.

We first consider axisymmetric surfaces of spherical topology and analyze the linear stability of the homogeneous steady state with **1**–**4** and the dynamic Eq. **13** for the concentration field to linear order (*SI Appendix*), we find that for increasing contractility parameter α the mode

The growing mode *B* and Movie S1). In the final steady state the advective influx of stress regulator into the contractile region is balanced by a diffusive outflux away from it. The resulting inhomogeneous tension across the surface leads to an oblate shape with broken mirror symmetry with respect to the z axis and thus spatially varying curvature (Fig. 2*B*, *Right*).

For *C*, where we use a spheroidal surface with a homogeneous concentration as initial condition. During the relaxation process, we observe the transient formation of concentration maxima at the poles (Movie S2). These maxima appear as a consequence of the large mean curvature H at those locations, which leads to a locally increased surface compression during deformations (Eq. **13**).

### Self-Organized Shape Dynamics of Tubular Surfaces.

We now study the self-organization of an active surface with a tubular geometry. We analyze the linear stability of the homogeneous steady state with **1**–**4** and the dynamic Eq. **13** for the concentration field to linear order, we derive the stability diagram as a function of the contractility parameter α and the aspect ratio *A* and *SI Appendix*). For *A*), which indicates oscillatory behavior at the instability. These characteristics of the stability diagram remain qualitatively unchanged when the bending rigidity κ is finite. The instability line for *A*. The red-shaded region indicates complex eigenvalues. Note that the value of *SI Appendix*).

We also study the surface dynamics beyond the linear regime, using our numerical approach. For *A*), the cylinder surface constricts and generates a thin cylindrical neck region with decreasing radius (Fig. 3*B* and Movie S3). The numerical analysis indicates that this radius vanishes at finite time. The concentration of the stress regulator increases along the tubular neck. For parameters that correspond to complex eigenvalues in the linear stability diagram (blue cross in Fig. 3*A*), the cylinder constricts and expands periodically (Fig. 3*C* and Movie S4) with increasing amplitude until the neck radius vanishes. For

For *D* and Movie S5). As a consequence, average flows directed along the z axis occur in a reference frame, where the constriction does not move. This corresponds to a peristaltic contraction wave that propagates in a reference frame where the average flow vanishes. Our numerical results reveal that such propagating solutions emerge in all parameter regimes for which the cylinder surface is linearly unstable and

## Discussion

We have introduced a simple but general model for the mechanochemical self-organization of surface geometry. Active stresses in the surface are regulated by a diffusible and advected molecular species. Gradients of active stress induce surface flows and shape changes, which in turn influence the distribution of the stress regulator. As a consequence, shape changes, shape oscillations, and spontaneous surface flows can be generated via mechanochemical instabilities. In contrast to mechanochemical instabilities that have been discussed in fixed geometries (14⇓–16), the phenomena described in the present paper give rise to shape changes and depend themselves crucially on the shape changes that occur.

To solve the dynamic equations for the shape, the flows, and the concentration fields, we have developed a numerical approach based on an integral representation of axisymmetric surfaces. We use a time-dependent coordinate transformation, which allows us to obtain the shape dynamics in an implicit surface representation. The explicit coordinates of surface points can be calculated independently.

Using our approach, we have identified a mechanochemical shape instability of a sphere that leads to concentration and flow patterns with a polar asymmetry and an accordingly asymmetric oblate shape. For periodic cylinder surfaces, we find contractility-induced instabilities beyond a critical value

Our framework provides a basis to explore a large variety of systems that involve the mechanochemical self-organization of deforming active surfaces. Here, we have focused on simple cases, where the active material is embedded in an environment with a homogeneous pressure. The latter exerts a force per area normal to the surface with position-independent magnitude. In many real situations, the surrounding medium has material properties that can give rise to more complex patterns of external forces, including tangential shear forces acting on the surface. Furthermore, it will be interesting to consider active surface material properties with different constitutive relations, such as viscoelastic systems and materials that are anisotropic or chiral. Biological examples of such materials are given by epithelial tissues with planar polarity (1) and by anisotropic cytoskeletal systems (30). Furthermore, the minimal model of a single stress regulator chosen here could be extended to more complex chemical schemes that involve several species and interactions between them. These extended models could represent, for example, the actin dynamics present in the cell cortex and its biochemical regulators (3) or the behavior of sets of morphogens spreading in epithelial tissues (11). Finally, we note that many biological processes, such as the migration of cells or invagination events in tissues (13, 27), lead to shapes that are not axisymmetric. A generalization of our approach to nonaxisymmetric surfaces therefore provides an important challenge.

## Acknowledgments

The authors thank Mads Hejlesen, Rajesh Ramaswamy, Bevan Cheeseman, Suryanarayana Maddu, and Guillaume Salbreux for helpful discussions. A.M. acknowledges funding from an ELBE PhD fellowship. This work was financially supported by the German Federal Ministry of Research and Education, Grant 031L0044.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: ivos{at}mpi-cbg.de or julicher{at}pks.mpg.de.

Author contributions: A.M., F.J., and I.F.S. designed research; A.M. performed research; and A.M., F.J., and I.F.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. E.L. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1810896115/-/DCSupplemental.

- Copyright © 2019 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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