3D observations provide striking findings in rubber elasticity.

The mechanical response of rubbers has been ubiquitously assumed to be only a function of the imposed strain. Using innovative X-ray measurements capturing the three-dimensional spatial volumetric strain fields, we demonstrate that rubbers and indeed many common engineering polymers undergo significant local volume changes. But remarkably, the overall specimen volume remains constant regardless of the imposed loading. This strange behavior which also leads to apparent negative local bulk moduli is due to the presence of a mobile phase within these materials. Combining X-ray tomographic observations with high-speed radiography to track the motion of the mobile phase, we have revised classical thermodynamic frameworks of rubber elasticity. The work opens broad avenues to understand not only the mechanical behavior of rubbers but a large class of widely used engineering polymers.

Underexcitation prevents crystallization of granular assemblies subjected to high-frequency vibration.

Crystallization of dry particle assemblies via imposed vibrations is a scalable route to assemble micro/macro crystals. It is well understood that there exists an optimal frequency to maximize crystallization with broad acceptance that this optimal frequency emerges because high-frequency vibration results in overexcitation of the assembly. Using measurements that include interrupted X-ray computed tomography and high-speed photography combined with discrete-element simulations we show that, rather counterintuitively, high-frequency vibration underexcites the assembly. The large accelerations imposed by high-frequency vibrations create a fluidized boundary layer that prevents momentum transfer into the bulk of the granular assembly. This results in particle underexcitation which inhibits the rearrangements required for crystallization. This clear understanding of the mechanisms has allowed the development of a simple concept to inhibit fluidization which thereby allows crystallization under high-frequency vibrations.

The influence of entropic crowding in cell monolayers.

Cell-cell interaction dictates cell morphology and organization, which play a crucial role in the micro-architecture of tissues that guides their biological and mechanical functioning. Here, we investigate the effect of cell density on the responses of cells seeded on flat substrates using a novel statistical thermodynamics framework. The framework recognizes the existence of nonthermal fluctuations in cellular response and thereby naturally captures entropic interactions between cells in monolayers. In line with observations, the model predicts that cell area and elongation decrease with increasing cell seeding density-both are a direct outcome of the fluctuating nature of the cellular response that gives rise to enhanced cell-cell interactions with increasing cell crowding. The modeling framework also predicts the increase in cell alignment with increasing cell density: this cellular ordering is also due to enhanced entropic interactions and is akin to nematic ordering in liquid crystals. Our simulations provide physical insights that suggest that entropic cell-cell interactions play a crucial role in governing the responses of cell monolayers.

Role of boundary conditions in determining cell alignment in response to stretch.

The ability of cells to orient in response to mechanical stimuli is essential to embryonic development, cell migration, mechanotransduction, and other critical physiologic functions in a range of organs. Endothelial cells, fibroblasts, mesenchymal stem cells, and osteoblasts all orient perpendicular to an applied cyclic stretch when plated on stretchable elastic substrates, suggesting a common underlying mechanism. However, many of these same cells orient parallel to stretch in vivo and in 3D culture, and a compelling explanation for the different orientation responses in 2D and 3D has remained elusive. Here, we conducted a series of experiments designed specifically to test the hypothesis that differences in strains transverse to the primary loading direction give rise to the different alignment patterns observed in 2D and 3D cyclic stretch experiments ("strain avoidance"). We found that, in static or low-frequency stretch conditions, cell alignment in fibroblast-populated collagen gels correlated with the presence or absence of a restraining boundary condition rather than with compaction strains. Cyclic stretch could induce perpendicular alignment in 3D culture but only at frequencies an order of magnitude greater than reported to induce perpendicular alignment in 2D. We modified a published model of stress fiber dynamics and were able to reproduce our experimental findings across all conditions tested as well as published data from 2D cyclic stretch experiments. These experimental and model results suggest an explanation for the apparently contradictory alignment responses of cells subjected to cyclic stretch on 2D membranes and in 3D gels.

Entropic Forces Drive Cellular Contact Guidance.

Contact guidance-the widely known phenomenon of cell alignment induced by anisotropic environmental features-is an essential step in the organization of adherent cells, but the mechanisms by which cells achieve this orientational ordering remain unclear. Here, we seeded myofibroblasts on substrates micropatterned with stripes of fibronectin and observed that contact guidance emerges at stripe widths much greater than the cell size. To understand the origins of this surprising observation, we combined morphometric analysis of cells and their subcellular components with a, to our knowledge, novel statistical framework for modeling nonthermal fluctuations of living cells. This modeling framework is shown to predict not only the trends but also the statistical variability of a wide range of biological observables, including cell (and nucleus) shapes, sizes, and orientations, as well as stress-fiber arrangements within the cells with remarkable fidelity with a single set of cell parameters. By comparing observations and theory, we identified two regimes of contact guidance: 1) guidance on stripe widths smaller than the cell size (w ≤ 160 µm), which is accompanied by biochemical changes within the cells, including increasing stress-fiber polarization and cell elongation; and 2) entropic guidance on larger stripe widths, which is governed by fluctuations in the cell morphology. Overall, our findings suggest an entropy-mediated mechanism for contact guidance associated with the tendency of cells to maximize their morphological entropy through shape fluctuations.

Thermodynamic Modeling of the Statistics of Cell Spreading on Ligand-Coated Elastic Substrates.

Biological spread cells exist in a perpetually fluctuating state and therefore cannot be described in terms of a unique deterministic system. For modeling approaches to provide novel insight and uncover new mechanisms that drive cell behavior, a framework is required that progresses from traditional deterministic methods (whereby simulation of an experiment predicts a single outcome). In this study, we implement a new, to our knowledge, modeling approach for the analysis of cell spreading on ligand-coated substrates, extending the framework for nonequilibrium thermodynamics of cells developed by Shishvan et al. to include active focal adhesion assembly. We demonstrate that the model correctly predicts the coupled influence of surface collagen density and substrate stiffness on cell spreading, as reported experimentally by Engler et al. Low surface collagen densities are shown to result in a high probability that cells will be restricted to low spread areas. Furthermore, elastic free energy induced by substrate deformation lowers the probability of observing a highly spread cell, and, consequentially, lower cell tractions affect the assembly of focal adhesions. Experimentally measurable observables such as cell spread area and aspect ratio can be directly postprocessed from the computed homeostatic ensemble of (several million) spread states. This allows for the prediction of mean and SDs of such experimental observables. This class of cell mechanics modeling presents a significant advance on conventional deterministic approaches.

Prediction of Cell Alignment on Cyclically Strained Grooved Substrates.

Cells respond to both mechanical and topographical stimuli by reorienting and reorganizing their cytoskeleton. Under certain conditions, such as for cells on cyclically stretched grooved substrates, the effects of these stimuli can be antagonistic. The biophysical processes that lead to the cellular reorientation resulting from such a competition are not clear yet. In this study, we hypothesized that mechanical cues and the diffusion of the intracellular signal produced by focal adhesions are determinants of the final cellular alignment. This hypothesis was investigated by means of a computational model, with the aim to simulate the (re)orientation of cells cultured on cyclically stretched grooved substrates. The computational results qualitatively agree with previous experimental studies, thereby supporting our hypothesis. Furthermore, cellular behavior resulting from experimental conditions different from the ones reported in the literature was simulated, which can contribute to the development of new experimental designs.

An Integrated Stochastic Model of Matrix-Stiffness-Dependent Filopodial Dynamics.

The ways that living cells regulate their behavior in response to their local mechanical environment underlie growth, development, and healing and are important to critical pathologies such as metastasis and fibrosis. Although extensive experimental evidence supports the hypothesis that this regulation is governed by the dependence of filopodial dynamics upon extracellular matrix stiffness, the pathways for this dependence are unclear. We therefore developed a model to relate filopodial focal adhesion dynamics to integrin-mediated Rho signaling kinetics. Results showed that focal adhesion maturation, i.e., focal adhesion links reinforcement and integrin clustering, dominates over filopodial dynamics. Downregulated focal adhesion maturation leads to the biphasic relationship between extracellular matrix stiffness and retrograde flow that has been observed in embryonic chick forebrain neurons, whereas upregulated maturation leads to the monotonically decreasing relationship that has been observed in mouse embryonic fibroblasts. When integrin-mediated Rho activation and stress-dependent focal adhesion maturation are combined, the model shows how filopodial dynamics endows cells with exquisite mechanosensing. Taken together, the results support the hypothesis that mechanical and structural factors combine with signaling kinetics to enable cells to probe their environments via filopodial dynamics.

Unsteady penetration of a target by a liquid jet.

It is widely acknowledged that ceramic armor experiences an unsteady penetration response: an impacting projectile may erode on the surface of a ceramic target without substantial penetration for a significant amount of time and then suddenly start to penetrate the target. Although known for more than four decades, this phenomenon, commonly referred to as dwell, remains largely unexplained. Here, we use scaled analog experiments with a low-speed water jet and a soft, translucent target material to investigate dwell. The transient target response, in terms of depth of penetration and impact force, is captured using a high-speed camera in combination with a piezoelectric force sensor. We observe the phenomenon of dwell using a soft (noncracking) target material. The results show that the penetration rate increases when the flow of the impacting water jet is reversed due to the deformation of the jet-target interface--this reversal is also associated with an increase in the force exerted by the jet on the target. Creep penetration experiments with a constant indentation force did not show an increase in the penetration rate, confirming that flow reversal is the cause of the unsteady penetration rate. Our results suggest that dwell can occur in a ductile noncracking target due to flow reversal. This phenomenon of flow reversal is rather widespread and present in a wide range of impact situations, including water-jet cutting, needleless injection, and deposit removal via a fluid jet.

Design and formulation of functional pluripotent stem cell-derived cardiac microtissues.

Access to robust and information-rich human cardiac tissue models would accelerate drug-based strategies for treating heart disease. Despite significant effort, the generation of high-fidelity adult-like human cardiac tissue analogs remains challenging. We used computational modeling of tissue contraction and assembly mechanics in conjunction with microfabricated constraints to guide the design of aligned and functional 3D human pluripotent stem cell (hPSC)-derived cardiac microtissues that we term cardiac microwires (CMWs). Miniaturization of the platform circumvented the need for tissue vascularization and enabled higher-throughput image-based analysis of CMW drug responsiveness. CMW tissue properties could be tuned using electromechanical stimuli and cell composition. Specifically, controlling self-assembly of 3D tissues in aligned collagen, and pacing with point stimulation electrodes, were found to promote cardiac maturation-associated gene expression and in vivo-like electrical signal propagation. Furthermore, screening a range of hPSC-derived cardiac cell ratios identified that 75% NKX2 Homeobox 5 (NKX2-5)+ cardiomyocytes and 25% Cluster of Differentiation 90 OR (CD90)+ nonmyocytes optimized tissue remodeling dynamics and yielded enhanced structural and functional properties. Finally, we demonstrate the utility of the optimized platform in a tachycardic model of arrhythmogenesis, an aspect of cardiac electrophysiology not previously recapitulated in 3D in vitro hPSC-derived cardiac microtissue models. The design criteria identified with our CMW platform should accelerate the development of predictive in vitro assays of human heart tissue function.

Simulation of the mechanical response of cells on micropost substrates.

Experimental studies where cells are seeded on micropost arrays in order to quantify their contractile behavior are becoming increasingly common. Interpretation of the data generated by this experimental technique is difficult, due to the complexity of the processes underlying cellular contractility and mechanotransduction. In the current study, a coupled framework that considers strain rate dependent contractility and remodeling of the cytoskeleton is used in tandem with a thermodynamic model of tension dependent focal adhesion formation to investigate the biomechanical response of cells adhered to micropost arrays. Computational investigations of the following experimental studies are presented: cell behavior on different sized arrays with a range of post stiffness; stress fiber and focal adhesion formation in irregularly shaped cells; the response of cells to deformations applied locally to individual posts; and the response of cells to equibiaxial stretching of micropost arrays. The predicted stress fiber and focal adhesion distributions; in addition to the predicted post tractions are quantitatively and qualitatively supported by previously published experimental data. The computational models presented in this study thus provide a framework for the design and interpretation of experimental micropost studies.

Contact guidance via heterogeneity of substrate elasticity.

Contact guidance, the widely-known phenomenon of cell alignment, is an essential step in the organization of adherent cells. This guidance is known to occur by, amongst other things, anisotropic features in the environment including elastic heterogeneity. To understand the origins of this guidance we employed a novel statistical thermodynamics framework, which recognises the non-thermal fluctuations in the cellular response, for modelling the response of the cells seeded on substrates with alternating soft and stiff stripes. Consistent with observations, the modelling framework predicts the existence of three regimes of cell guidance: (i) in regime I for stripe widths much larger than the cell size guidance is primarily entropic; (ii) for stripe widths on the order of the cell size in regime II guidance is biochemically mediated and accompanied by changes to the cell morphology while (iii) in regime III for stripe widths much less than the cell size there is no guidance as cells cannot sense the substrate heterogeneity. Guidance in regimes I and II is due to "molli-avoidance" with cells primarily residing on the stiff stripes. While the molli-avoidance tendency is not lost with decreasing density of collagen coating the substrate, the reduced focal adhesion formation with decreasing collagen density tends to inhibit contact guidance. Our results provide clear physical insights into the interplay between cell mechano-sensitivity and substrate elastic heterogeneity that ultimately leads to the contact guidance of cells in heterogeneous tissues. STATEMENT OF SIGNIFICANCE: Cellular morphology and organization play a crucial role in the micro-architecture of tissues and dictates their biological and mechanical functioning. Despite the importance of cellular organization in all facets of tissue biology, the fundamental question of how a cell organizes itself in an anisotropic environment is still poorly understood. We employ a novel statistical thermodynamics framework which recognises the non-thermal fluctuations in the cellular response to investigate cell guidance on substrates with alternating soft and stiff stripes. The propensity of cells to primarily reside on stiff stripes results in strong guidance when the period of the stripes is larger than the cell size. For smaller stripe periods, cells sense a homogeneous substrate and guidance is lost.

Microfabricated tissue gauges to measure and manipulate forces from 3D microtissues.

Physical forces generated by cells drive morphologic changes during development and can feedback to regulate cellular phenotypes. Because these phenomena typically occur within a 3-dimensional (3D) matrix in vivo, we used microelectromechanical systems (MEMS) technology to generate arrays of microtissues consisting of cells encapsulated within 3D micropatterned matrices. Microcantilevers were used to simultaneously constrain the remodeling of a collagen gel and to report forces generated during this process. By concurrently measuring forces and observing matrix remodeling at cellular length scales, we report an initial correlation and later decoupling between cellular contractile forces and changes in tissue morphology. Independently varying the mechanical stiffness of the cantilevers and collagen matrix revealed that cellular forces increased with boundary or matrix rigidity whereas levels of cytoskeletal and extracellular matrix (ECM) proteins correlated with levels of mechanical stress. By mapping these relationships between cellular and matrix mechanics, cellular forces, and protein expression onto a bio-chemo-mechanical model of microtissue contractility, we demonstrate how intratissue gradients of mechanical stress can emerge from collective cellular contractility and finally, how such gradients can be used to engineer protein composition and organization within a 3D tissue. Together, these findings highlight a complex and dynamic relationship between cellular forces, ECM remodeling, and cellular phenotype and describe a system to study and apply this relationship within engineered 3D microtissues.

Contact guidance as a consequence of coupled morphological evolution and motility of adherent cells.

Adherent cells seeded on substrates spread and evolve their morphology while simultaneously displaying motility. Phenomena such as contact guidance, viz. the alignment of cells on patterned substrates, are strongly linked to the coupling of morphological evolution with motility. Here, we employ a recently developed statistical thermodynamics framework for modelling the non-thermal fluctuating response of cells to probe this coupling. This thermodynamic framework is first extended via a Langevin style model to predict temporal responses of cells to unpatterned and patterned substrates. The Langevin model is then shown to not only predict the different experimentally observed temporal scales for morphological observables such as cell area and elongation but also the interplay of morphology with motility that ultimately leads to contact guidance.

Cell reorientation on a cyclically strained substrate.

Cyclic strain avoidance, the phenomenon of cell and cytoskeleton alignment perpendicular to the direction of cyclic strain of the underlying 2D substrate, is an important characteristic of the adherent cell organization. This alignment has typically been attributed to the stress-fiber reorganization although observations clearly show that stress-fiber reorganization under cyclic loading is closely coupled to cell morphology and reorientation of the cells. Here, we develop a statistical mechanics framework that couples the cytoskeletal stress-fiber organization with cell morphology under imposed cyclic straining and make quantitative comparisons with observations. The framework accurately predicts that cyclic strain avoidance stems primarily from cell reorientation away from the cyclic straining rather than cytoskeletal reorganization within the cell. The reorientation of the cell is a consequence of the cell lowering its free energy by largely avoiding the imposed cyclic straining. Furthermore, we investigate the kinetics of the cyclic strain avoidance mechanism and demonstrate that it emerges primarily due to the rigid body rotation of the cell rather than via a trajectory involving cell straining. Our results provide clear physical insights into the coupled dynamics of cell morphology and stress-fibers, which ultimately leads to cellular organization in cyclically strained tissues.

Nanofibril-mediated fracture resistance of bone.

Natural hard composites like human bone possess a combination of strength and toughness that exceeds that of their constituents and of many engineered composites. This augmentation is attributed to their complex hierarchical structure, spanning multiple length scales; in bone, characteristic dimensions range from nanoscale fibrils to microscale lamellae to mesoscale osteons and macroscale organs. The mechanical properties of bone have been studied, with the understanding that the isolated microstructure at micro- and nano-scales gives rise to superior strength compared to that of whole tissue, and the tissue possesses an amplified toughness relative to that of its nanoscale constituents. Nanoscale toughening mechanisms of bone are not adequately understood at sample dimensions that allow for isolating salient microstructural features, because of the challenge of performing fracture experiments on small-sized samples. We developed anin situthree-point bend experimental methodology that probes site-specific fracture behavior of micron-sized specimens of hard material. Using this, we quantify crack initiation and growth toughness of human trabecular bone with sharp fatigue pre-cracks and blunt notches. Our findings indicate that bone with fatigue cracks is two times tougher than that with blunt cracks.In situdata-correlated electron microscopy videos reveal this behavior arises from crack-bridging by nanoscale fibril structure. The results reveal a transition between fibril-bridging (∼1µm) and crack deflection/twist (∼500µm) as a function of length-scale, and quantitatively demonstrate hierarchy-induced toughening in a complex material. This versatile approach enables quantifying the relationship between toughness and microstructure in various complex material systems and provides direct insight for designing biomimetic composites.

Cellular Contact Guidance Emerges from Gap Avoidance.

In the presence of anisotropic biochemical or topographical patterns, cells tend to align in the direction of these cues-a widely reported phenomenon known as "contact guidance." To investigate the origins of contact guidance, here, we created substrates micropatterned with parallel lines of fibronectin with dimensions spanning multiple orders of magnitude. Quantitative morphometric analysis of our experimental data reveals two regimes of contact guidance governed by the length scale of the cues that cannot be explained by enforced alignment of focal adhesions. Adopting computational simulations of cell remodeling on inhomogeneous substrates based on a statistical mechanics framework for living cells, we show that contact guidance emerges from anisotropic cell shape fluctuation and "gap avoidance," i.e., the energetic penalty of cell adhesions on non-adhesive gaps. Our findings therefore point to general biophysical mechanisms underlying cellular contact guidance, without the necessity of invoking specific molecular pathways.

Transient active force generation and stress fibre remodelling in cells under cyclic loading.

The active cytoskeleton is known to play an important mechanistic role in cellular structure, spreading, and contractility. Contractility is actively generated by stress fibres (SF), which continuously remodel in response to physiological dynamic loading conditions. The influence of actin-myosin cross-bridge cycling on SF remodelling under dynamic loading conditions has not previously been uncovered. In this study, a novel SF cross-bridge cycling model is developed to predict transient active force generation in cells subjected to dynamic loading. Rates of formation of cross-bridges within SFs are governed by the chemical potentials of attached and unattached myosin heads. This transient cross-bridge cycling model is coupled with a thermodynamically motivated framework for SF remodelling to analyse the influence of transient force generation on cytoskeletal evolution. A 1D implementation of the model is shown to correctly predict complex patterns of active cell force generation under a range of dynamic loading conditions, as reported in previous experimental studies.

A microscopically motivated model for the remodeling of cardiomyocytes.

We present a thermodynamically based model that captures the remodeling effects in cardiac muscle cells. This work begins with the formulation of the kinematics of a cardiomyocyte resulting from a prescribed macroscopic deformation and the reorganization of the internal structure. Specifically, relations between the macroscopic deformation and the number of sarcomeres, the sarcomere stretch, and the number of myofibrils in the cell are determined. The remodeling process is split into two separate phases-(1) elongation/shortening of the existing myofibrils by addition/detachment of sarcomeres and (2) formation of new myofibrils. The remodeling associated with each phase is modeled through a dissipation postulate. We show that remodeling is based on a competition between the internal energy, the entropy, the energy supplied to the system by ATP and other sources to drive the remodeling process, and dissipation mechanisms. While the variations in entropy associated with phase (1) are neglected, the substantial entropy loss associated with the formation of new myofibrils is determined. To illustrate the merit of the proposed framework, we compute the response of cardiomyocytes subjected to isometric axial stretch that are either free to deform or fixed in the transverse direction. We also examine the predictions of this model for cardiomyocytes subjected to various cyclic loadings. The proposed framework is capable of capturing a wide range of remodeling effects and agrees with experimental observations.