Interfacial rheology of linearly growing polyelectrolyte multilayers at the water-air interface: from liquid to solid viscoelasticity.

Despite the long history of investigations of polyelectrolyte multilayer formation on solid or liquid surfaces, important questions remain open concerning the construction of the first set of layers. These are generally deposited on a first anchoring layer of different chemistry, influencing their construction and properties. We propose here an in-depth investigation of the formation of NaPSS/PAH multilayers at the air/water interface in the absence of a chemically different anchoring layer, profiting from the surface activity of NaPSS. To analyse the mechanical properties of the different layers, we combine recently established analysis techniques of an inflating/deflating bubble exploiting simultaneous shape and pressure measurement: bubble shape elastometry, general stress decomposition and capillary meniscus dynanometry. We complement these measurements by interfacial shear rheology. The obtained results allow us to confirm, first of all, the strength of the aforementioned techniques to characterize complex interfaces with non-linear viscoelastic properties. Furthermore, their sensitivity allows us to show that the multilayer properties are highly sensitive to the temporal and mechanical conditions under which they are constructed and manipulated. We nevertheless identify a robust trend showing a clear transition from a liquid-like viscoelastic membrane to a solid-like viscoelastic membrane after the deposition of 5 layers. We interpret this as the number of layers required to create a fully connected multilayer, which is consistent with previous results obtained on solid or liquid interfaces.

Enhancing robustness, precision, and speed of traction force microscopy with machine learning.

Traction patterns of adherent cells provide important information on their interaction with the environment, cell migration, or tissue patterns and morphogenesis. Traction force microscopy is a method aimed at revealing these traction patterns for adherent cells on engineered substrates with known constitutive elastic properties from deformation information obtained from substrate images. Conventionally, the substrate deformation information is processed by numerical algorithms of varying complexity to give the corresponding traction field via solution of an ill-posed inverse elastic problem. We explore the capabilities of a deep convolutional neural network as a computationally more efficient and robust approach to solve this inversion problem. We develop a general purpose training process based on collections of circular force patches as synthetic training data, which can be subjected to different noise levels for additional robustness. The performance and the robustness of our approach against noise is systematically characterized for synthetic data, artificial cell models, and real cell images, which are subjected to different noise levels. A comparison with state-of-the-art Bayesian Fourier transform traction cytometry reveals the precision, robustness, and speed improvements achieved by our approach, leading to an acceleration of traction force microscopy methods in practical applications.

Elastometry of Complex Fluid Pendant Capsules.

Oil/water interfaces are ubiquitous in nature. Opposing polarities at these interfaces attract surface-active molecules, which can seed complex viscoelastic or even solid interfacial structure. Biorelevant proteins such as hydrophobin, polymers such as PNIPAM, and the asphaltenes in crude oil (CRO) are examples of some systems where such layers can occur. When a pendant drop of CRO is aged in brine, it can form an interfacial elastic membrane of asphaltenes so stiff that it wrinkles and crumples upon retraction. Most of the work studying CRO/brine interfaces focuses on the viscoelastic liquid regime, leaving a wide range of fully solidified, elastic interfaces largely unexplored. In this work, we quantitatively measure elasticity in all phases of drop retraction. In early retraction, the interface shows a fluid viscoelasticity measurable using a Gibbs isotherm or dilatational rheology. Further retraction causes a phase transition to a 2D elastic solid with nonisotropic, nonhomogeneous surface stresses. In this regime, we use new techniques in the elastic membrane theory to fit for the elasticities of these solid capsules. These elastic measurements can help us develop a deeper understanding not only of CRO interfaces but also of the myriad fluid systems with solid interfacial layers.

Strain stiffening of Ndc80 complexes attached to microtubule plus ends.

In the mitotic spindle, microtubules attach to chromosomes via kinetochores. The microtubule-binding Ndc80 complex is an integral part of kinetochores, and is essential for kinetochores to attach to microtubules and to transmit forces from dynamic microtubule ends to the chromosomes. The Ndc80 complex has a rod-like appearance with globular domains at its ends that are separated by a long coiled coil. Its mechanical properties are considered important for the dynamic interaction between kinetochores and microtubules. Here, we present a novel method that allows us to time trace the effective stiffness of Ndc80 complexes following shortening microtubule ends against applied force in optical trap experiments. Applying this method to wild-type Ndc80 and three variants (calponin homology (CH) domains mutated or Hec1 tail unphosphorylated, phosphorylated, or truncated), we reveal that each variant exhibits strain stiffening; i.e., the effective stiffness increases under tension that is built up by a depolymerizing microtubule. The strain stiffening relation is roughly linear and independent of the state of the microtubule. We introduce structure-based models that show that the strain stiffening can be traced back to the specific architecture of the Ndc80 complex with a characteristic flexible kink, to thermal fluctuations of the microtubule, and to the bending elasticity of flaring protofilaments, which exert force to move the Ndc80 complexes. Our model accounts for changes in the amount of load-bearing attachments at various force levels and reproduces the roughly linear strain stiffening behavior, highlighting the importance of force-dependent binding affinity.

Pressure-deformation relations of elasto-capillary drops (droploons) on capillaries.

An increasing number of multi-phase systems exploit complex interfaces in which capillary stresses are coupled with solid-like elastic stresses. Despite growing efforts, simple and reliable experimental characterisation of these interfaces remains a challenge, especially of their dilational properties. Pendant drop techniques are convenient, but suffer from complex shape changes and associated fitting procedures with multiple parameters. Here we show that simple analytical relationships can be derived to describe reliably the pressure-deformation relations of nearly spherical elasto-capillary droplets ("droploons") attached to a capillary. We consider a model interface in which stresses arising from a constant interfacial tension are superimposed with mechanical extra-stresses arising from the deformation of a solid-like, incompressible interfacial layer of finite thickness described by a neo-Hookean material law. We compare some standard models of liquid-like (Gibbs) and solid-like (Hookean and neo-Hookean elasticity) elastic interfaces which may be used to describe the pressure-deformation relations when the presence of the capillary can be considered negligible. Combining Surface Evolver simulations and direct numerical integration of the drop shape equations, we analyse in depth the influence of the anisotropic deformation imposed by the capillary on the pressure-deformation relation and show that in many experimentally relevant circumstances either the analytical relations of the perfect sphere may be used or a slightly modified relation which takes into account the geometrical change imposed by the capillary. Using the analogy with the stress concentration around a rigid inclusion in an elastic membrane, we provide simple non-dimensional criteria to predict under which conditions the simple analytical expressions can be used to fit pressure-deformation relations to analyse the elastic properties of the interfaces via "Capillary Pressure Elastometry". We show that these criteria depend essentially on the drop geometry and deformation, but not on the interfacial elasticity. Moreover, this benchmark case shows for the first time that Surface Evolver is a reliable tool for predictive simulations of elastocapillary interfaces. This opens doors to the treatment of more complex geometries/conditions, where theory is not available for comparison. Our Surface Evolver code is available for download in the ESI.

From diffusive mass transfer in Stokes flow to low Reynolds number Marangoni boats.

We present a theory for the self-propulsion of symmetric, half-spherical Marangoni boats (soap or camphor boats) at low Reynolds numbers. Propulsion is generated by release (diffusive emission or dissolution) of water-soluble surfactant molecules, which modulate the air-water interfacial tension. Propulsion either requires asymmetric release or spontaneous symmetry breaking by coupling to advection for a perfectly symmetrical swimmer. We study the diffusion-advection problem for a sphere in Stokes flow analytically and numerically both for constant concentration and constant flux boundary conditions. We derive novel results for concentration profiles under constant flux boundary conditions and for the Nusselt number (the dimensionless ratio of total emitted flux and diffusive flux). Based on these results, we analyze the Marangoni boat for small Marangoni propulsion (low Peclet number) and show that two swimming regimes exist, a diffusive regime at low velocities and an advection-dominated regime at high swimmer velocities. We describe both the limit of large Marangoni propulsion (high Peclet number) and the effects from evaporation by approximative analytical theories. The swimming velocity is determined by force balance, and we obtain a general expression for the Marangoni forces, which comprises both direct Marangoni forces from the surface tension gradient along the air-water-swimmer contact line and Marangoni flow forces. We unravel whether the Marangoni flow contribution is exerting a forward or backward force during propulsion. Our main result is the relation between Peclet number and swimming velocity. Spontaneous symmetry breaking and, thus, swimming occur for a perfectly symmetrical swimmer above a critical Peclet number, which becomes small for large system sizes. We find a supercritical swimming bifurcation for a symmetric swimmer and an avoided bifurcation in the presence of an asymmetry.

Surfactant-loaded capsules as Marangoni microswimmers at the air-water interface: Symmetry breaking and spontaneous propulsion by surfactant diffusion and advection.

We present a realization of a fast interfacial Marangoni microswimmer by a half-spherical alginate capsule at the air-water interface, which diffusively releases water-soluble spreading molecules (weak surfactants such as polyethylene glycol (PEG)), which act as "fuel" by modulating the air-water interfacial tension. For a number of different fuels, we can observe symmetry breaking and spontaneous propulsion although the alginate particle and emission are isotropic. The propulsion mechanism is similar to soap or camphor boats, which are, however, typically asymmetric in shape or emission to select a swimming direction. We develop a theory of Marangoni boat propulsion starting from low Reynolds numbers by analyzing the coupled problems of surfactant diffusion and advection and fluid flow, which includes surfactant-induced fluid Marangoni flow, and surfactant adsorption at the air-water interface; we also include a possible evaporation of surfactant. The swimming velocity is determined by the balance of drag and Marangoni forces. We show that spontaneous symmetry breaking resulting in propulsion is possible above a critical dimensionless surfactant emission rate (Peclet number). We derive the relation between Peclet number and swimming speed and generalize to higher Reynolds numbers utilizing the concept of the Nusselt number. The theory explains the observed swimming speeds for PEG-alginate capsules, and we unravel the differences to other Marangoni boat systems based on camphor, which are mainly caused by surfactant evaporation from the liquid-air interface. The capsule Marangoni microswimmers also exhibit surfactant-mediated repulsive interactions with walls, which can be qualitatively explained by surfactant accumulation at the wall.

Snapping elastic disks as microswimmers: swimming at low Reynolds numbers by shape hysteresis.

We illustrate a concept for shape-changing microswimmers, which exploits the hysteresis of a shape transition of an elastic object, by an elastic disk undergoing cyclic localized swelling. Driving the control parameter of a hysteretic shape transition in a completely time-reversible manner gives rise to a non-time-reversible shape sequence and a net swimming motion if the elastic object is immersed into a viscous fluid. We prove this concept with a microswimmer which is a flat circular elastic disk that undergoes a transition into a dome-like shape by localized swelling of an inner disk. The control parameter of this shape transition is a scalar swelling factor of the disk material. With a fixed outer frame with an additional attractive interaction in the central region, the shape transition between flat and dome-like shape becomes hysteretic and resembles a hysteretic opening and closing of a scallop. Employing Stokesian dynamics simulations of a discretized version of the disk we show that the swimmer is effectively moving into the direction of the opening of the dome in a viscous fluid if the swelling parameter is changed in a time-reversible manner. The swimming mechanism can be qualitatively reproduced by a simple 9-bead model.

Pendant drop tensiometry: A machine learning approach.

Modern pendant drop tensiometry relies on the numerical solution of the Young-Laplace equation and allows us to determine the surface tension from a single picture of a pendant drop with high precision. Most of these techniques solve the Young-Laplace equation many times over to find the material parameters that provide a fit to a supplied image of a real droplet. Here, we introduce a machine learning approach to solve this problem in a computationally more efficient way. We train a deep neural network to determine the surface tension of a given droplet shape using a large training set of numerically generated droplet shapes. We show that the deep learning approach is superior to the current state of the art shape fitting approach in speed and precision, in particular if shapes in the training set reflect the sensitivity of the droplet shape with respect to surface tension. In order to derive such an optimized training set, we clarify the role of the Worthington number as a quality indicator in conventional shape fitting and in the machine learning approach. Our approach demonstrates the capabilities of deep neural networks in the material parameter determination from rheological deformation experiments, in general.

Strong Deformation of Ferrofluid-Filled Elastic Alginate Capsules in Inhomogenous Magnetic Fields.

We present a new system based on alginate gels for the encapsulation of a ferrofluid drop, which allows us to create millimeter-sized elastic capsules that are highly deformable by inhomogeneous magnetic fields. We use a combination of experimental and theoretical work in order to characterize and quantify the deformation behavior of these ferrofluid-filled capsules. We introduce a novel method for the direct encapsulation of unpolar liquids by sodium alginate. By adding 1-hexanol to the unpolar liquid, we can dissolve sufficient amounts of CaCl2 in the resulting mixture for ionotropic gelation of sodium alginate. The addition of polar alcohol molecules allows us to encapsulate a ferrofluid as a single phase rather than an emulsion without impairing ferrofluid stability. This encapsulation method increases the amount of encapsulated magnetic nanoparticles resulting in high deformations of approximately 30% (in height-to-width ratio) in inhomogeneous magnetic field with magnetic field variations of 50 mT over the size of the capsule. This offers possible applications of capsules as actuators, switches, or valves in confined spaces like microfluidic devices. We determine both elastic moduli of the capsule shell, Young's modulus and Poisson's ratio, by employing two independent mechanical methods, spinning capsule measurements and capsule compression between parallel plates. We then show that the observed magnetic deformation can be fully understood from magnetic forces exerted by the ferrofluid on the capsule shell if the magnetic field distribution and magnetization properties of the ferrofluid are known. We perform a detailed analysis of the magnetic deformation by employing a theoretical model based on nonlinear elasticity theory. Using an iterative solution scheme that couples a finite element/boundary element method for the magnetic field calculation to the solution of the elastic shape equations, we achieve quantitative agreement between theory and experiment for deformed capsule shapes using the Young modulus from mechanical characterization and the surface Poisson ratio as a fit parameter. This detailed analysis confirms the results from mechanical characterization that the surface Poisson ratio of the alginate shell is close to unity, that is, deformations of the alginate shell are almost area conserving.

Elastic capsules at liquid-liquid interfaces.

We investigate the deformation of elastic microcapsules adsorbed at liquid-liquid interfaces. An initially spherical elastic capsule at a liquid-liquid interface undergoes circumferential stretching due to the liquid-liquid surface tension and becomes lens- or discus-shaped, depending on its bending rigidity. The resulting elastic capsule deformation is qualitatively similar, but distinct from the deformation of a liquid droplet into a liquid lens at a liquid-liquid interface. We discuss the deformed shapes of droplets and capsules adsorbed at liquid-liquid interfaces for a whole range of different surface elasticities: from droplets (only surface tension) deforming into liquid lenses, droplets with a Hookean membrane (finite stretching modulus, zero bending modulus) deforming into elastic lenses, to microcapsules (finite stretching and bending modulus) deforming into rounded elastic lenses. We calculate capsule shapes at liquid-liquid interfaces numerically using shape equations from nonlinear elastic shell theory. Finally, we present theoretical results for the contact angle (or the capsule height) and the maximal capsule curvature at the three phase contact line. These results can be used to infer information about the elastic moduli from optical measurements. During capsule deformation into a lens-like shape, surface energy of the liquid-liquid interface is converted into elastic energy of the capsule shell giving rise to an overall adsorption energy gain by deformation. Soft hollow capsules exhibit a pronounced increase of the adsorption energy as compared to filled soft particles and, thus, are attractive candidates as foam and emulsion stabilizers.

Adsorption of finite semiflexible polymers and their loop and tail distributions.

We discuss the adsorption of semiflexible polymers to a planar attractive wall and focus on the questions of the adsorption threshold for polymers of finite length and their loop and tail distributions using both Monte Carlo simulations and analytical arguments. For the adsorption threshold, we find three regimes: (i) a flexible or Gaussian regime if the persistence length is smaller than the adsorption potential range, (ii) a semiflexible regime if the persistence length is larger than the potential range, and (iii) for finite polymers, a novel crossover to a rigid rod regime if the deflection length exceeds the contour length. In the flexible and semiflexible regimes, finite size corrections arise because the correlation length exceeds the contour length. In the rigid rod regime, however, it is essential how the global orientational or translational degrees of freedom are restricted by grafting or confinement. We discuss finite size corrections for polymers grafted to the adsorbing surface and for polymers confined by a second (parallel) hard wall. Based on these results, we obtain a method to analyze adsorption data for finite semiflexible polymers such as filamentous actin. For the loop and tail distributions, we find power laws with an exponential decay on length scales exceeding the correlation length. We derive and confirm the loop and tail power law exponents for flexible and semiflexible polymers. This allows us to explain that, close to the transition, semiflexible polymers have significantly smaller loops and both flexible and semiflexible polymers desorb by expanding their tail length. The tail distribution allows us to extract the free energy per length of adsorption for actin filaments from experimental data [D. Welch et al., Soft Matter 11, 7507 (2015)].

Microcapsule Buckling Triggered by Compression-Induced Interfacial Phase Change.

There is an emerging trend toward the fabrication of microcapsules at liquid interfaces. In order to control the parameters of such capsules, the interfacial processes governing their formation must be understood. Here, poly(vinyl alcohol) films are assembled at the interface of water-in-oil microfluidic droplets. The polymer is cross-linked using cucurbit[8]uril ternary supramolecular complexes. It is shown that compression-induced phase change causes the onset of buckling in the interfacial film. On evaporative compression, the interfacial film both increases in density and thickens, until it reaches a critical density and a phase change occurs. We show that this increase in density can be simply related to the film Poisson ratio and area compression. This description captures fundamentals of many compressive interfacial phase changes and can also explain the observation of a fixed thickness-to-radius ratio at buckling, [Formula: see text].

Elasticity of interfacial rafts of hard particles with soft shells.

We study an elasticity model for compressed protein monolayers or particle rafts at a liquid interface. Based on the microscopic view of hard-core particles with soft shells, a bead-spring model is formulated and analyzed in terms of continuum elasticity theory. The theory can be applied, for example, to hydrophobin-coated air-water interfaces or, more generally, to liquid interfaces coated with an adsorbed monolayer of interacting hard-core particles. We derive constitutive relations for such particle rafts and describe the buckling of compressed planar liquid interfaces as well as their apparent Poisson ratio. We also use the constitutive relations to obtain shape equations for pendant or buoyant capsules attached to a capillary, and to compute deflated shapes of such capsules. A comparison with capsules obeying the usual Hookean elasticity (without hard cores) reveals that the hard cores trigger capsule wrinkling. Furthermore, it is shown that a shape analysis of deflated capsules with hard-core/soft-shell elasticity gives apparent elastic moduli which can be much higher than the original values if Hookean elasticity is assumed.

Monte Carlo simulation of dense polymer melts using event chain algorithms.

We propose an efficient Monte Carlo algorithm for the off-lattice simulation of dense hard sphere polymer melts using cluster moves, called event chains, which allow for a rejection-free treatment of the excluded volume. Event chains also allow for an efficient preparation of initial configurations in polymer melts. We parallelize the event chain Monte Carlo algorithm to further increase simulation speeds and suggest additional local topology-changing moves ("swap" moves) to accelerate equilibration. By comparison with other Monte Carlo and molecular dynamics simulations, we verify that the event chain algorithm reproduces the correct equilibrium behavior of polymer chains in the melt. By comparing intrapolymer diffusion time scales, we show that event chain Monte Carlo algorithms can achieve simulation speeds comparable to optimized molecular dynamics simulations. The event chain Monte Carlo algorithm exhibits Rouse dynamics on short time scales. In the absence of swap moves, we find reptation dynamics on intermediate time scales for long chains.

Feedback mechanism for microtubule length regulation by stathmin gradients.

We formulate and analyze a theoretical model for the regulation of microtubule (MT) polymerization dynamics by the signaling proteins Rac1 and stathmin. In cells, the MT growth rate is inhibited by cytosolic stathmin, which, in turn, is inactivated by Rac1. Growing MTs activate Rac1 at the cell edge, which closes a positive feedback loop. We investigate both tubulin sequestering and catastrophe promotion as mechanisms for MT growth inhibition by stathmin. For a homogeneous stathmin concentration in the absence of Rac1, we find a switchlike regulation of the MT mean length by stathmin. For constitutively active Rac1 at the cell edge, stathmin is deactivated locally, which establishes a spatial gradient of active stathmin. In this gradient, we find a stationary bimodal MT-length distribution for both mechanisms of MT growth inhibition by stathmin. One subpopulation of the bimodal length distribution can be identified with fast-growing and long pioneering MTs in the region near the cell edge, which have been observed experimentally. The feedback loop is closed through Rac1 activation by MTs. For tubulin sequestering by stathmin, this establishes a bistable switch with two stable states: one stable state corresponds to upregulated MT mean length and bimodal MT length distributions, i.e., pioneering MTs; the other stable state corresponds to an interrupted feedback with short MTs. Stochastic effects as well as external perturbations can trigger switching events. For catastrophe-promoting stathmin, we do not find bistability.

Wrinkling of random and regular semiflexible polymer networks.

We investigate wrinkling of two-dimensional random and triangular semiflexible polymer networks under shear. Both types of semiflexible networks exhibit wrinkling above a small critical shear angle, which scales with an exponent of the bending modulus between 1.9 and 2.0. Random networks exhibit hysteresis at the wrinkling threshold. Wrinkling lowers the total elastic energy by up to 20% and strongly affects the elastic properties of all semiflexible networks such as the crossover between bending and stretching dominated behavior. In random networks, we also find evidence for metastable wrinkled configurations. While the disordered microstructure of random networks affects the scaling behavior of wrinkle amplitudes, it has little effect on wrinkle wavelength. Therefore, wrinkles represent a robust, microstructure-independent assay of shear strain or elastic properties.

Osmotic buckling of spherical capsules.

We study the buckling of elastic spherical shells under osmotic pressure with the osmolyte concentration of the exterior solution as a control parameter. We compare our results for the bifurcation behavior with results for buckling under mechanical pressure control, that is, with an empty capsule interior. We find striking differences for the buckling states between osmotic and mechanical buckling. Mechanical pressure control always leads to fully collapsed states with opposite sides in contact, whereas uncollapsed states with a single finite dimple are generic for osmotic pressure control. For sufficiently large interior osmolyte concentrations, osmotic pressure control is qualitatively similar to buckling under volume control with the volume prescribed by the osmolyte concentrations inside and outside the shell. We present a quantitative theory which also captures the influence of shell elasticity on the relationship between osmotic pressure and volume. These findings are relevant for the control of buckled shapes in applications. We show how the osmolyte concentration can be used to control the volume of buckled shells. An accurate analytical formula is derived for the relationship between the osmotic pressure, the elastic moduli and the volume of buckled capsules. This also allows use of elastic capsules as osmotic pressure sensors or deduction of elastic properties and the internal osmolyte concentration from shape changes in response to osmotic pressure changes. We apply our findings to published experimental data on polyelectrolyte capsules.

The secondary buckling transition: wrinkling of buckled spherical shells.

We theoretically explain the complete sequence of shapes of deflated spherical shells. Decreasing the volume, the shell remains spherical initially, then undergoes the classical buckling instability, where an axisymmetric dimple appears, and, finally, loses its axisymmetry by wrinkles developing in the vicinity of the dimple edge in a secondary buckling transition. We describe the first axisymmetric buckling transition by numerical integration of the complete set of shape equations and an approximate analytic model due to Pogorelov. In the buckled shape, both approaches exhibit a locally compressive hoop stress in a region where experiments and simulations show the development of polygonal wrinkles, along the dimple edge. In a simplified model based on the stability equations of shallow shells, a critical value for the compressive hoop stress is derived, for which the compressed circumferential fibres will buckle out of their circular shape in order to release the compression. By applying this wrinkling criterion to the solutions of the axisymmetric models, we can calculate the critical volume for the secondary buckling transition. Using the Pogorelov approach, we also obtain an analytical expression for the critical volume at the secondary buckling transition: The critical volume difference scales linearly with the bending stiffness, whereas the critical volume reduction at the classical axisymmetric buckling transition scales with the square root of the bending stiffness. These results are confirmed by another stability analysis in the framework of Donnel, Mushtari and Vlasov (DMV) shell theory, and by numerical simulations available in the literature.

Triggered contraction of self-assembled micron-scale DNA nanotube rings.

Contractile rings are formed from cytoskeletal filaments during cell division. Ring formation is induced by specific crosslinkers, while contraction is typically associated with motor protein activity. Here, we engineer DNA nanotubes and peptide-functionalized starPEG constructs as synthetic crosslinkers to mimic this process. The crosslinker induces bundling of ten to hundred DNA nanotubes into closed micron-scale rings in a one-pot self-assembly process yielding several thousand rings per microliter. Molecular dynamics simulations reproduce the detailed architectural properties of the DNA rings observed in electron microscopy. Theory and simulations predict DNA ring contraction - without motor proteins - providing mechanistic insights into the parameter space relevant for efficient nanotube sliding. In agreement between simulation and experiment, we obtain ring contraction to less than half of the initial ring diameter. DNA-based contractile rings hold promise for an artificial division machinery or contractile muscle-like materials.