Soft Elastic Substrates Research Articles

Substrate nesting guides cyst morphogenesis of human pluripotent stem cells without 3D extracellular matrix overlay

Understanding the principles underlying the self-organization of stem cells into tissues is fundamental for deciphering human embryo development. Here, we report that, without three-dimensional (3D) extracellular matrix (ECM) overlay, human pluripotent stem cells (hPSCs) cultured on two-dimensional soft elastic substrates can self-organize into 3D cysts resembling the human epiblast sac in a stiffness-dependent manner. Our theoretical modeling predicts that this cyst organization is facilitated and guided by the spontaneous nesting of the soft substrate, which results from the adhesion-dependent mechanical interaction between cells and substrate. Such substrate nesting is sufficient for the 3D assembly and polarization of hPSCs required for cyst organization, even without 3D ECM overlay. Furthermore, we identify that the reversible substrate nesting and cyst morphogenesis also require appropriate activation of ROCK-Myosin II pathway. This indicates a unique set of tissue morphomechanical signaling mechanisms that clearly differ from the canonical cystogenic mechanism previously reported in 3D ECM. Our findings highlight an unanticipated synergy between mechanical microenvironment and mechanotransduction in controlling tissue morphogenesis and suggest a mechanics-based strategy for generation of hPSCs-derived models for early human embryogenesis. Statement of significanceSoft substrates can induce the self-organization of human pluripotent stem cells (hPSCs) into cysts without three-dimensional (3D) extracellular matrix (ECM) overlay. However, the underlying mechanisms by which soft substrate guides cystogenesis are largely unknown. This study shows that substrate nesting, resulting from cell–substrate interaction, plays an important role in cyst organization, including 3D assembly and apical–basal polarization. Additionally, actomyosin contractility mediated by the ROCK-Myosin II pathway also contributes to the substrate deformation and cyst morphology. These findings demonstrate the interplay between the mechanical microenvironment and cells in tissue morphogenesis, suggesting a mechanics-based strategy in building hPSC-derived models for early human embryo development.

Static Wetting of a Liquid Droplet on a Soft Elastic Substrate

Abstract A refined spherical cap model, combined with an elastic foundation model for the elastic substrate, is proposed to study the static wetting of a liquid droplet on a soft elastic substrate. The strain energy of the substrate is evaluated by the Johnson–Kendall–Roberts (JKR) model, and the increase of the surface energy of the substrate outside the contact zone is calculated based on the elastic foundation model. The total potential energy of the droplet-substrate system is given in terms of four geometrical parameters: the contact radius, the contact angle of the droplet, the deflection angle inside the contact zone, and the maximum downward displacement of the substrate surface at the contact zone center. The equilibrium state is determined based on the stationary condition of total potential energy. The present model reduces to the Young’s equation for a rigid substrate and to the Neumann’s triangle for a liquid-like substrate. Three equations are given to determine the liquid droplet shape in terms of surface energies and substrate’s elastic modulus. Reasonable agreement with existing experimental data and simulation results shows that the present model with derived formulas has the potential to catch the role of substrate’s elastic deformation on static wetting and fill the gap between the Young’s equation and the Neumann’s triangle for a soft elastic substrate.

Cell Shape and Forces in Elastic and Structured Environments: From Single Cells to Organoids

AbstractWith the advent of mechanobiology, cell shape and forces have emerged as essential elements of cell behavior and fate, in addition to biochemical factors such as growth factors. Cell shape and forces are intrinsically linked to the physical properties of the environment. Extracellular stiffness guides migration of single cells and collectives as well as differentiation and developmental processes. In confined environments, cell division patterns are altered, cell death or extrusion might be initiated, and other modes of cell migration become possible. Tools from materials science such as adhesive micropatterning of soft elastic substrates or direct laser writing of 3D scaffolds have been established to control and quantify cell shape and forces in structured environments. Herein, a review is given on recent experimental and modeling advances in this field, which currently moves from single cells to cell collectives and tissue. A very exciting avenue is the combination of organoids with structured environments, because this will allow one to achieve organotypic function in a controlled setting well suited for long‐term and high‐throughput culture.

Controllable Wetting Transitions on Photoswitchable Physical Gels.

Softness plays a key role in the deformation of soft elastic substrates at the three-phase contact line, and the acting forces lead to the formation of a wetting ridge due to elastocapillarity. The change in wetting ridge and surface profiles at different softness has a great impact on the droplet behavior in different phenomena. Commonly used materials to study soft wetting are swollen polymeric gels or polymer brushes. These materials offer no possibility to change the softness on demand. Therefore, adjustable surfaces with tunable softness are highly sought-after to achieve on-demand transition between wetting states on soft surfaces. Here, we present a photorheological physical soft gel with adjustable stiffness based on the spiropyran photoswitch that shows the formation of wetting ridges upon droplet deposition. The presented photoswitchable gels allow the creation of reversibly switchable softness patterns with microscale resolution using UV light-switching of the spiropyran molecule. Gels with varying softness are analyzed, showing a decrease in the wetting ridge height at higher gel stiffness. Furthermore, wetting ridges before and after photoswitching are visualized using confocal microscopy, showing the transition in the wetting properties from soft wetting to liquid/liquid wetting.

Flat, wrinkled, or ridged: Relaxation of an elastic film on a viscous substrate undergoing continuous compression

We conduct a finite element computational study of the dynamics of a thin elastic film bonded to a much thicker viscous substrate undergoing compression at a fixed rate. The applied compression tends to continuously increase the strain, and hence the elastic energy, of the film. In contrast to the well-studied case of a soft elastic substrate, a viscous substrate cannot store elastic energy; instead it regulates the kinetics of the various mechanisms that dissipate elastic energy of the film. Sufficiently short films remain flat because shear flow in the liquid near the ends allows rapid relaxation of the strain over the entire film length. In longer films, end-relaxation cannot relax film strain in the mid-section, which therefore buckles. Buckles initially appear as packets of approximately-sinusoidal wrinkles. With increasing strain, these packets transform into tall localized ridges separated by nearly flat regions. In all cases, the buckles cause end-relaxation to become dynamically confined to a narrow region near the ends We construct a state map identifying regions of the parameter space of strain vs film length in which the film remains flat, develops wrinkle packets, or develops localized ridges. The evolution of the film energy during continuous compression shows that ridge localization appears due to a competition between two effects: a well-spaced ridges offer a lower energy state than uniform wrinkles, but wrinkles can develop faster because they require the viscous fluid to move over shorter distances.

The Arp2/3 complex enhances cell migration on elastic substrates.

Cell migration on soft surfaces occurs in both physiological and pathological processes such as corticogenesis during embryonic development and cancer invasion and metastasis. The Arp2/3 complex in neural progenitor cells was previously demonstrated to be necessary for cell migration on soft elastic substrate but not on stiff surfaces, but the underlying mechanism was unclear. Here, we integrate computational and experimental approaches to elucidate how the Arp2/3 complex enables cell migration on soft surfaces. We found that lamellipodia comprised of a branched actin network nucleated by the Arp2/3 complex distribute forces over a wider area, thus decreasing stress in the substrate. Additionally, we found that interactions between parallel focal adhesions within lamellipodia prolong cell-substrate interactions by compensating for the failure of neighboring adhesions. Together with decreased substrate stress, this leads to the observed improvements in migratory ability on soft substrates in cells utilizing lamellipodia-dependent mesenchymal migration when compared with filopodia-based migration. These results show that the Arp2/3 complex-dependent lamellipodia provide multiple distinct mechanical advantages to gliomas migrating on soft 2D substrates, which can contribute to their invasive potential.

Cell size and actin architecture determine force generation in optogenetically activated cells

Adherent cells use actomyosin contractility to generate mechanical force and to sense the physical properties of their environment, with dramatic consequences for migration, division, differentiation, and fate. However, the organization of the actomyosin system within cells is highly variable, with its assembly and function being controlled by small GTPases from the Rho family. To understand better how activation of these regulators translates into cell-scale force generation in the context of different physical environments, here we combine recent advances in non-neuronal optogenetics with micropatterning and traction force microscopy on soft elastic substrates. We find that, after whole-cell RhoA activation by the CRY2/CIBN optogenetic system with a short pulse of 100 ms, single cells contract on a minute timescale in proportion to their original traction force, before returning to their original tension setpoint with near perfect precision, on a longer timescale of several minutes. To decouple the biochemical and mechanical elements of this response, we introduce a mathematical model that is parametrized by fits to the dynamics of the substrate deformation energy. We find that the RhoA response builds up quickly on a timescale of 20 s, but decays slowly on a timescale of 50 s. The larger the cells and the more polarized their actin cytoskeleton, the more substrate deformation energy is generated. RhoA activation starts to saturate if optogenetic pulse length exceeds 50 ms, revealing the intrinsic limits of biochemical activation. Together our results suggest that adherent cells establish tensional homeostasis by the RhoA system, but that the setpoint and the dynamics around it are strongly determined by cell size and the architecture of the actin cytoskeleton, which both are controlled by the extracellular environment.

Solving transient problems in soft Elasto-Hydrodynamic lubrication

Many important problems in elasto-hydrodynamic lubrication arise when one of the contacting bodies is elastically soft. An example is the squeezing of a thin liquid film between a rigid sphere/cylinder and a soft elastic substrate, in which the sphere may not be sliding and be subjected only to normal loads (stationary contact), or it can be sheared at the same time and undergo steady or non-steady sliding. These problems are notoriously difficult to solve for soft solids under large loads and thin lubricant films. For this reason, indentation and sliding problems have usually been analyzed separately, and solutions for transient sliding problems are few. Here we develop a numerical method that avoids difficult-to-converge iterative methods and is able to solve both normal indentation and/or transient sliding lubrication problems. The scheme is fully automatic, stable, and efficient and requires only a linear matrix equation to be solved at every time step. We present solutions of two transient sliding problems: in the first of which, a rigid sphere undergoes transient normal lubricated contact followed by transient sliding on an elastic half space. In the second, a rigid cylinder undergoes transient lubricated sliding on an elastic foundation.

An Analytic Model of Transient Heat Conduction for Bi-Layered Flexible Electronic Heaters by Symplectic Superposition.

In a flexible electronic heater (FEH), periodic metal wires are often encapsulated into the soft elastic substrate as heat sources. It is of great significance to develop analytic models on transient heat conduction of such an FEH in order to provide a rapid analysis and preliminary designs based on a rapid parameter analysis. In this study, an analytic model of transient heat conduction for bi-layered FEHs is proposed, which is solved by a novel symplectic superposition method (SSM). In the Laplace transform domain, the Hamiltonian system-based governing equation for transient heat conduction is introduced, and the mathematical techniques incorporating the separation of variables and symplectic eigen expansion are manipulated to yield the temperature solutions of two subproblems, which is followed by superposition for the temperature solution of the general problem. The Laplace inversion gives the eventual temperature solution in the time domain. Comprehensive time-dependent temperatures by the SSM are presented in tables and figures for benchmark use, which agree well with their counterparts by the finite element method. A parameter analysis on the influence of the thermal conductivity ratio is also studied. The exceptional merit of the SSM is on a direct rigorous derivation without any assumption/predetermination of solution forms, and thus, the method may be extended to more heat conduction problems of FEHs with more complex structures.

Substrate elasticity and surface tension mediate the spontaneous rotation of active chiral droplet on soft substrates

Active chirality is an universal property of macromolecules as well as cellular systems and processes, such as nucleic acid, proteins, bacterial flagella and its motions, actomyosin filaments and network, and the left–right (LR) asymmetric morphogenesis, etc. Despite of attempts to elucidate the relations among cellular chirality, spontaneous cellular rotation and extracellular matrix properties, many features of the chiral active matter that are in contact with elastic substrates are still not clear in terms of fundamental physical principles. For instance, when an active chiral droplet in contact with an elastic substrate, the underline interplay among surface elasticity, chirality, and surface tension is largely unknown. In this work, based on the Gurtin–Murdoch surface elasticity we developed an adhesive surface hydro-elasticity theory of moving line to model and simulate the spontaneous rotation of a chiral active soft matter droplet on soft elastic substrates to study the rich morphological, conformational, and dynamic behaviors of the droplet/substrate interaction. In this study, we have found that both surface tension and substrate elasticity significantly affect the spontaneous rotation of the active chiral droplet on the substrate from several crucial aspects: (1) the geometry and morphology of the rotating droplet; (2) the angular velocity and helical vertices of the rotating droplet; (3) the order parameter distribution inside the rotating droplet, (4) the dynamic contact area of the rotating droplet, and (5) the contact stress distribution in both the droplet as well as the elastic substrate.

Rotation-translation coupling of soft objects in lubricated contact.

We study the coupling between rotation and translation of a submerged cylinder in lubricated contact with a soft elastic substrate. Using numerical solutions and asymptotic theory, we analyze the elastohydrodynamic problem over the entire range of substrate deformations relative to the thickness of the intervening fluid film. We find a strong coupling between the rotation and translation of the cylinder when the surface deformation of the substrate is comparable to the thickness of the lubricating fluid layer. In the limit of large deformations, we show that the bodies are in near-Hertzian contact and cylinder rolls without slip, reminiscent of dry frictional contact. When the surface deformation is small relative to the separation between the surfaces, the coupling persists but is weaker, and the rotation rate scales with the translation speed to the one-third power. We then show how the external application of a torque modifies these behaviors by generating different combinations of rotational and translational motions, including back-spinning and top-spinning states. We demonstrate that these behaviors are robust regardless of whether the elastic substrate is thick or thin relative to the length scales of the flow.

Gradient-dynamics model for liquid drops on elastic substrates.

The wetting of soft elastic substrates exhibits many features that have no counterpart on rigid surfaces. Modelling the detailed elastocapillary interactions is challenging, and has so far been limited to single contact lines or single drops. Here we propose a reduced long-wave model that captures the main qualitative features of statics and dynamics of soft wetting, but which can be applied to ensembles of droplets. The model has the form of a gradient dynamics on an underlying free energy that reflects capillarity, wettability and compressional elasticity. With the model we first recover the double transition in the equilibrium contact angles that occurs when increasing substrate softness from ideally rigid towards very soft (i.e., liquid). Second, the spreading of single drops of partially and completely wetting liquids is considered showing that known dependencies of the dynamic contact angle on contact line velocity are well reproduced. Finally, we go beyond the single droplet picture and consider the coarsening for a two-drop system as well as for a large ensemble of drops. It is shown that the dominant coarsening mode changes with substrate softness in a nontrivial way.

Splashing on Soft Elastic Substrates.

Drop impact onto soft substrates is important in applications such as bioprinting, spray coating, and aerosol drug delivery. Experiments are conducted to determine the effect of elasticity on the splash morphology, as defined by the splashing threshold, spine number, spreading factor, and retraction factor. PDMS silicone gel and gelatin hydrogel are used as the substrates because they have different wetting properties and a large range of elasticities. The splash threshold, as defined by the Weber number We, increases as the substrate elasticity decreases indicating that it is harder to splash on soft substrates. After impact, the drop spreads to a maximum diameter that decreases for soft substrates, irrespective of wetting properties, illustrating the role of substrate deformation in the energy balance during splashing. The number of spines that form at the leading edge of the drop depends upon the elasticity and the wetting properties of the liquid/substrate system. Following spreading, the drop retracts to an equilibrium diameter which does not show a strong correlation with any material properties. The reported results agree well with the existing literature for most cases and also provide new insights into gels with small elasticity.

Magnetically Superamphiphobic Nanoparticles for Magnetic Response Surface Preparation.

Smart surfaces prepared using superhydrophobic coatings have been used to control the movement of tiny aqueous droplets for many years. However, the control of both aqueous droplets and oily droplets is still a challenge. Herein, a novel smart superamphiphobic composite film is developed with a superamphiphobic and magnetic surface as well as a soft elastic substrate for liquid droplets manipulation. The raspberry-like nanoparticles on the surface are synthesized by co-hydrolyzation of fluoroalkyl silane and tetraethoxysilane on iron oxide nanoparticles. The resulting composite nanoparticles (F/SiO2@Fe3O4 NPs) exhibit excellent superhydrophobicity (WCA of about 170°) and superoleophobicity (OCA of about 160°) as well as magnetism (saturated magnetization value of 12.0 emug-1). The morphologies of the F/SiO2@Fe3O4 NPs were investigated by transmission electron microscopy and scanning electron microscopy. Chemical composition and magnetization value of these magnetic nanoparticles as well as the magnetic field induced droplets manipulation behavior of the smart surface were also evaluated. The smart surface can realize the manipulation of both water droplets and oil droplets, which demonstrates potential applications in microfluidic technologies.

Disclination regulated depression and elevation of glassy nematic coatings on soft elastic substrates

We study photo-activated nonlinear deformation of periodic polydomain glassy nematic coatings on soft elastic substrates. The director fields are piecewise constant with rank-1 connection at the domain boundaries and form regular arrays of disclinations of different charges at the junctions. Based on numerical simulations, we show that conical depression and elevation appear at the positive and negative disclinations respectively. The phenomenon is attributed to bend of the coating controlled by interfacial shear force, and the mechanism can be utilized to create various novel surface topographies which have not been reported previously. Although our exploration is concerned with the theoretical aspect, the results may inspire future experimentation and provide insights for designing photo-switchable surfaces.

Ordered ring-shaped cracks induced by indentation in metal films on soft elastic substrates.

Ordered crack patterns contain plentiful physical mechanisms and are useful for technological applications such as lithography, template, and biomimicry. Here we report on ordered multiple ring-shaped cracks induced by indentation in metal films on soft elastic polydimethylsiloxane (PDMS) substrates. It is shown that the indentation triggers the deformation of PDMS substrate and generates a radial tensile stress in the film, leading to the formation of ring-shaped cracks with a nearly uniform spacing. The morphological characteristics and evolution behaviors of the multiple ring-shaped cracks are revealed by optical microscopy, atomic force microscopy, and scanning electron microscopy. Their formation mechanisms are discussed by theoretical analysis based on the fracture mechanics. The report in this work can promote better understanding of the indentation-induced stress anisotropy and mode competition in rigid-film-soft-substrate systems and provide a facile strategy to control the crack patterns by simple mechanical loading.

The emergence of local wrinkling or global buckling in thin freestanding bilayer films

Periodic wrinkling of a rigid capping layer on a deformable substrate provides a useful method for templating surface topography for a variety of novel applications. Many experiments have studied wrinkle formation during the compression of a rigid film on a relatively soft pre-strained elastic substrate, and most have focused on the regime where the substrate thickness can be considered semi-infinite relative to that of the film. As the relative thickness of the substrate is decreased, the bending stiffness of the film dominates, causing the bilayer to transition to either local wrinkling or a global buckling instability. In this work optical microscopy was used to study the critical parameters that determine the emergence of local wrinkling or global buckling of freestanding bilayer films consisting of a thin rigid polymer capping layer on a pre-strained elastomeric substrate. The thickness ratio of the film and substrate as well as the pre-strain were controlled and used to create a buckling phase diagram which describes the behaviour of the system as the ratio of the thickness of the substrate is decreased. A simple force balance model was developed to understand the thickness and strain dependences of the wrinkling and buckling modes, with excellent quantitative agreement being obtained with experiments using only independently measured material parameters.

Lubricated steady sliding of a rigid sphere on a soft elastic substrate: hydrodynamic friction in the Hertz limit.

Lubricated sliding on soft elastic substrates occurs in a variety of natural and technological settings. It very often occurs in the iso-viscous elasto-hydrodynamic lubrication (EHL) regime (e.g., soft solid, low pressure). In this regime, for sliding of a smooth sphere on a soft solid, a "Hertz-like" effective contact region forms. Much of the fluid is squeezed out of the contact region although enough is retained to keep the solid surfaces fully separated. This is accompanied by complex deformation of the soft solid. The behavior of such soft lubricated contacts is controlled by a single dimensionless parameter 1/β that can be interpreted as a normalized sliding velocity. Solving this fundamental soft-lubrication problem poses significant computational difficulty for large β, which is the limit relevant for soft solids. As a consequence, little is known about the structure of the flow field under soft lubrication in the intake and outlet regions. Here we present a new solution of this soft lubrication problem focusing on the "Hertz" limit. We develop a formulation in polar coordinates that handles difficult computational issues much better than previous methods. We study how hydrodynamic pressure, film thickness and hydrodynamic friction vary with β. Scaling laws for these relationships are given in closed form for a range of β not previously accessible theoretically but that is typical in applications. The computational method presented here can be used to study other soft lubrication problems.

Elastocapillarity and rolling dynamics of solid nanoparticles on soft elastic substrates.

The motion of nanoparticles on soft surfaces is the result of interplay between capillary, elastic and friction forces. To elucidate the importance of the different contributions controlling nanoparticle rolling dynamics on soft surfaces, we performed molecular dynamics simulations of solid nanoparticles in contact with soft elastic substrates. The nanoparticle motion is initiated by applying a constant force resulting in stationary, steady rolling, and accelerating states, depending on the nanoparticle-substrate work of adhesion, W, the magnitude of the net applied force, F, and the substrate shear modulus G. In the stationary state, the restoring torque produced in the contact area balances the torque due to the external force. The rolling force Fr, determining the crossover to the rolling state, is proportional to the product of the work of adhesion W and nanoparticle size Rp, Fr ∼ WRp. In the steady rolling state, F > Fr, the nanoparticle maintains a constant rolling velocity which is a manifestation of the balance between the rolling friction force and the applied force. The observed scaling relationships between the applied force and nanoparticle velocity reflect a viscoelastic nature of the substrate deformation dynamics. A nanoparticle begins to accelerate when the energy supplied to the nanoparticle exceeds the energy dissipated in the contact area due to viscoelastic substrate deformation. Using these simulation results, we have constructed a diagram of states in terms of the dimensionless parameters F/WRp and W/GRp.

Polyacrylamide Hydrogels with Rigidity-Independent Surface Chemistry Show Limited Long-Term Maintenance of Pluripotency of Human Induced Pluripotent Stem Cells on Soft Substrates

In general, cells are cultured and adapted to the in vitro rigidities of plastic or glass ranging between 1 and 10 GPa, which is very far from physiological values that are mostly in the kilopascal range. Stem cells however show a high sensitivity to the rigidity of their culture environment, which impacts their differentiation program. Here, we address the impact of rigidity on the long-term maintenance of pluripotency in human induced pluripotent stem cells (hiPSCs) to determine whether soft substrates could provide a new standard for hiPSC expansion and maintenance. To do this, we set up a fabrication process of polyacrylamide-based culture supports with a rigidity-decoupled surface chemistry. Soft elastic substrates with uniform and reproducible physicochemical properties were designed. The maintenance of pluripotency of two hiPSCs lines on substrates with stiffnesses ranging from 3 to 25 kPa was studied with an identical chemical coating consisting of a truncated recombinant vitronectin with defined surface density. Based on the analysis of cellular adhesion, survival, growth kinetics, three-dimensional distribution, and gene and protein expressions, we demonstrate that below 25 kPa hiPSCs do not maintain pluripotency on long-term culture, while pluripotency and self-renewal capacities are maintained above 25 kPa. In contrast to previous studies, no drift toward a specific germ line lineage was revealed. On soft substrates, cell colonies started to grow in three-dimensional (3D), suggesting that softness allows cells to limit contact with the synthetic matrix and to build their own microenvironment. These observations drastically limit the benefit of using standardized soft substrates to expand hiPSCs, at least with the current culture conditions. The development of a robust technology for the design of soft substrates nevertheless opens up perspectives to fine-tune physicochemical properties of the culture environment in addition to or in replacement of soluble growth factors to finely direct cell fate.