Local Mechanical Modulation-Driven Evagination in Invaginated Epithelia.

Local cells can actively create reverse bending (evagination) in invaginated epithelia, which plays a crucial role in the formation of elaborate organisms. However, the precise physical mechanism driving the evagination remains elusive. Here, we present a three-dimensional vertex model, incorporating the intrinsic cell polarity, to explore the complex morphogenesis induced by local mechanical modulations. We find that invaginated tissues can spontaneously generate local reverse bending due to the shift of the apicobasal polarity. Their exact shapes can be analytically determined by the local apicobasal differential tension and the internal stress. Our continuum theory exhibits three regions in a phase diagram controlled by these two parameters, showing curvature transitions from ordered to disordered states. Additionally, we delve into epithelial curvature transition induced by the nucleus repositioning, revealing its active contribution to the apicobasal force generation. The uncovered mechanical principles could potentially guide more studies on epithelial folding in diverse systems.

Emergence, Pattern, and Frequency of Spontaneous Waves in Spreading Epithelial Monolayers.

A striking phenomenon of collective cell motion is that they can exhibit a spontaneously emerging wave during epithelia expansions. However, the fundamental mechanism, governing the emergence and its crucial characteristics (e.g., the eigenfrequency and the pattern), remains an enigma. By introducing a mechanochemical feedback loop, we develop a highly efficient discrete vertex model to investigate the spatiotemporal evolution of spreading epithelia. We find both numerically and analytically that expanding cell monolayers display a power-law dependence of wave frequency on the local heterogeneities (i.e., cell density) with a scaling exponent of -1/2. Moreover, our study demonstrates the quantitative capability of the proposed model in capturing distinct X-, W-, and V-mode wave patterns. We unveil that the phase transition between these modes is governed by the distribution of active self-propulsion forces. Our work provides an avenue for rigorous quantitative investigations into the collective motion and pattern formation of cell groups.

Beyond stiffness: Multiscale viscoelastic features as biomechanical markers for assessing cell types and states.

Cell mechanics are pivotal in regulating cellular activities, diseases progression, and cancer development. However, the understanding of how cellular viscoelastic properties vary in physiological and pathological stimuli remains scarce. Here, we develop a hybrid self-similar hierarchical theory-microrheology approach to accurately and efficiently characterize cellular viscoelasticity. Focusing on two key cell types associated with livers fibrosis-the capillarized liver sinusoidal endothelial cells and activated hepatic stellate cells-we uncover a universal two-stage power-law rheology characterized by two distinct exponents, αshort and αlong. The mechanical profiles derived from both exponents exhibit significant potential for discriminating among diverse cells. This finding suggests a potential common dynamic creep characteristic across biological systems, extending our earlier observations in soft tissues. Using a tailored hierarchical model for cellular mechanical structures, we discern significant variations in the viscoelastic properties and their distribution profiles across different cell types and states from the cytoplasm (elastic stiffness E1 and viscosity η), to a single cytoskeleton fiber (elastic stiffness E2), and then to the cell level (transverse expansion stiffness E3). Importantly, we construct a logistic-regression-based machine-learning model using the dynamic parameters that outperforms conventional cell-stiffness-based classifiers in assessing cell states, achieving an area under the curve of 97% vs. 78%. Our findings not only advance a robust framework for monitoring intricate cell dynamics but also highlight the crucial role of cellular viscoelasticity in discerning cell states across a spectrum of liver diseases and prognosis, offering new avenues for developing diagnostic and therapeutic strategies based on cellular viscoelasticity.

Mechanical guidance to self-organization and pattern formation of stem cells.

The self-organization of stem cells (SCs) constitutes the fundamental basis of the development of biological organs and structures. SC-driven patterns are essential for tissue engineering, yet unguided SCs tend to form chaotic patterns, impeding progress in biomedical engineering. Here, we show that simple geometric constraints can be used as an effective mechanical modulation approach that promotes the development of controlled self-organization and pattern formation of SCs. Using the applied SC guidance with geometric constraints, we experimentally uncover a remarkable deviation in cell aggregate orientation from a random direction to a specific orientation. Subsequently, we propose a dynamic mechanical framework, including cells, the extracellular matrix (ECM), and the culture environment, to characterize the specific orientation deflection of guided cell aggregates relative to initial geometric constraints, which agrees well with experimental observation. Based on this framework, we further devise various theoretical strategies to realize complex biological patterns, such as radial and concentric structures. Our study highlights the key role of mechanical factors and geometric constraints in governing SCs' self-organization. These findings yield critical insights into the regulation of SC-driven pattern formation and hold great promise for advancements in tissue engineering and bioactive material design for regenerative application.

New Mechanical Markers for Tracking the Progression of Myocardial Infarction.

The mechanical properties of soft tissues can often be strongly correlated with the progression of various diseases, such as myocardial infarction (MI). However, the dynamic mechanical properties of cardiac tissues during MI progression remain poorly understood. Herein, we investigate the rheological responses of cardiac tissues at different stages of MI (i.e., early-stage, mid-stage, and late-stage) with atomic force microscopy-based microrheology. Surprisingly, we discover that all cardiac tissues exhibit a universal two-stage power-law rheological behavior at different time scales. The experimentally found power-law exponents can capture an inconspicuous initial rheological change, making them particularly suitable as markers for early-stage MI diagnosis. We further develop a self-similar hierarchical model to characterize the progressive mechanical changes from subcellular to tissue scales. The theoretically calculated mechanical indexes are found to markedly vary among different stages of MI. These new mechanical markers are applicable for tracking the subtle changes of cardiac tissues during MI progression.

Viscoelastic Multiscale Mechanical Indexes for Assessing Liver Fibrosis and Treatment Outcomes.

Understanding liver tissue mechanics, particularly in the context of liver pathologies like fibrosis, cirrhosis, and carcinoma, holds pivotal significance for assessing disease severity and prognosis. Although the static mechanical properties of livers have been gradually studied, the intricacies of their dynamic mechanics remain enigmatic. Here, we characterize the dynamic creep responses of healthy, fibrotic, and mesenchymal stem cells (MSCs)-treated fibrotic lives. Strikingly, we unearth a ubiquitous two-stage power-law rheology of livers across different time scales with the exponents and their distribution profiles highly correlated to liver status. Moreover, our self-similar hierarchical theory effectively captures the delicate changes in the dynamical mechanics of livers. Notably, the viscoelastic multiscale mechanical indexes (i.e., power-law exponents and elastic stiffnesses of different hierarchies) and their distribution characteristics prominently vary with liver fibrosis and MSCs therapy. This study unveils the viscoelastic characteristics of livers and underscores the potential of proposed mechanical criteria for assessing disease evolution and prognosis.

Size-induced motion mode transitions in collective cell invasion toward free spaces.

Collective cell migration plays a vital role in various physiological and pathological processes, such as embryonic development and tumor metastasis. Recent experiments have shown that different from isolated cells, the moving cell groups exhibit rich emerging motion modes in response to external geometrical constraints. By considering the interactions between neighboring cells and internal biomechanical processes of each cell (i.e., cell sociality and cell individuality), we develop an active vertex model to investigate the emerging modes of collective cell migration in microchannels. Single-cell polarization is propelled by continuous protrusion of its leading edge and retraction of the rear. We here introduce the contribution of continuous protrusions and retractions of lamellipodia, named the protrusion alignment mechanism, to the cell individuality. Using the present model, it is found that altering the width of channels can trigger the motion mode transitions of cell groups. When cells move in narrow channels, the protrusion alignment mechanism brings neighboring groups of coordinated cells into conflicts and in turn induces the caterpillar-like motion mode. As the channel width increases, local swirls spanning the channel in width first appear as long as the channel width is smaller than the intrinsic correlation length of cell groups. Then, only local swirls with a maximum diameter of the intrinsic correlation length are formed, when the channel is sufficiently wider. These rich dynamical modes of collective cells originate from the competition between cell individuality and sociality. In addition, the velocity of the cell sheet invading free spaces varies with the channel size-induced transitions of migration modes. Our predictions are in broad agreement with many experiments and may shed light on the spatiotemporal dynamics of active matter.

Cell chirality regulates coherent angular motion on small circular substrates.

Collective cell migration occurs in a wide range of physiological and pathological processes, such as wound healing and tumor metastasis. Experiments showed that many types of cells confined in circular islands can perform coherent angular rotation, yet the underlying mechanisms remain unclear. Here we propose a biomechanical model, including the membrane, microtubules, and nucleus, to study the spatiotemporal evolutions of small cell clusters in confined space. We show that cells can spontaneously transfer from "radial pattern" to "chiral pattern" due to fluctuations. For a pair of cells with identical chiral orientation, the cluster rotates in the opposite direction of the chiral orientation, and the fluctuations can reverse the cluster's rotational direction. Interestingly, during the persistent rotation, each cell rotates around its own centroid while it is revolving around the island center and shows a constant side to the island center, as tidal locking in astronomy. Furthermore, for a few more cells, coherent angular rotation also appears, and the emergence of a central cell can accelerate the cluster rotation. These findings shed light on collective cell migration in life processes and help to understand the spatiotemporal dynamics of active matter.

Network dynamics of the nonlinear power-law relaxation of cell cortex.

Living cells are known to exhibit universal power-law rheological behaviors, but their underlying biomechanical principles are still not fully understood. Here, we present a network dynamics picture to decipher the nonlinear power-law relaxation of cortical cytoskeleton. Under step strains, we present a scaling relation between instantaneous differential stiffness and external stress as a result of chain reorientation. Then, during the relaxation, we show how the scaling law theoretically originates from an exponential form of cortical disorder, with the scaling exponent decreased by the imposed strain or crosslinker density in the nonlinear regime. We attribute this exponent variation to the molecular realignment along the stretch direction or the transition of network structure from in-series to in-parallel modes, both solidifying the network toward our one-dimensional theoretical limit. In addition, the rebinding of crosslinkers is found to be crucial for moderating the relaxation speed under small strains. Together with the disorder nature, we demonstrate that the structural effects of networks provide a unified interpretation for the nonlinear power-law relaxation of cell cortex, and may help to understand cell mechanics from the molecular scale.

Stress-driven cell extrusion can maintain homeostatic cell density in response to overcrowding.

Epithelial tissues can become overcrowded during proliferation and growth, in which case excessive cells need to be constantly removed. Here, we employed a vertex dynamics model to explore the microscopic mechanisms that govern homeostasis in an overcrowded monolayer of epithelial cells. It is demonstrated that the monolayer under mechanical compression can maintain a constant cell density and an optimal stress level through cell extrusion. Interestingly, cells are always extruded at sites of stress singularity in the monolayer, which may be spontaneously generated through random movements of cell groups near the extruding cells, and the fluctuation of protein molecules aggregating along the cell surface facilitates the restoration of the monolayer to its equilibrium state. Our results provide a foundation to interpret recent experiments as well as shed light on the mechanisms that underlie epithelial development and maintenance.

Orientations of Cells on Compliant Substrates under Biaxial Stretches: A Theoretical Study.

Mechanical cues from the microenvironments play a regulating role in many physiological and pathological processes, such as stem cell differentiation and cancer cell metastasis. Experiments showed that cells adhered on a compliant substrate may change orientation with an externally applied strain in the substrate. By accounting for actin polymerization, actin retrograde flow, and integrin binding dynamics, here we develop a mechanism-based tensegrity model to study the orientations of polarized cells on a compliant substrate under biaxial stretches. We show that the cell can actively regulate its mechanical state by generating different traction force levels along its polarized direction. Under static or ultralow-frequency cyclic stretches, stretching a softer substrate leads to a higher increase in the traction force and induces a narrower distribution of cell alignment. Compared to static loadings, high-frequency cyclic loadings have a more significant influence on cell reorientation on a stiff substrate. In addition, the width of the cellular angular distribution scales inversely with the stretch amplitude under both static and cyclic stretches. Our results are in agreement with a wide range of experimental observations, and provide fundamental insights into the functioning of cellular mechanosensing systems.

Dynamic Migration Modes of Collective Cells.

Collective cell migration occurs in a diversity of physiological processes such as wound healing, cancer metastasis, and embryonic morphogenesis. In the collective context, cohesive cells may move as a translational solid, swirl as a fluid, or even rotate like a disk, with scales ranging from several to dozens of cells. In this work, an active vertex model is presented to explore the regulatory roles of social interactions of neighboring cells and environmental confinements in collective cell migration in a confluent monolayer. It is found that the competition between two kinds of intercellular social interactions-local alignment and contact inhibition of locomotion-drives the cells to self-organize into various dynamic coherent structures with a spatial correlation scale. The interplay between this intrinsic length scale and the external confinement dictates the migration modes of collective cells confined in a finite space. We also show that the local alignment-contact inhibition of locomotion coordination can induce giant density fluctuations in a confluent cell monolayer without gaps, which triggers the spontaneous breaking of orientational symmetry and leads to phase separation.

Are elastic moduli of biological cells depth dependent or not? Another explanation using a contact mechanics model with surface tension.

Atomic force microscopy (AFM) has become the most commonly used tool to measure the mechanical properties of biological cells. In AFM indentation experiments, the Hertz and Sneddon models of contact mechanics are usually adopted to extract the elastic modulus by analyzing the load-indent depth curves for spherical and conical tips, respectively. However, the effects of surface tension, neglected in existing contact models, become more significant in indentation responses due to the lower elastic moduli of living cells. Here, we present two simple yet robust relations between load and indent depth considering surface tension effects for spherical and conical indentations, through dimensional analysis and finite element simulations. When the indent depth is smaller than the intrinsic length defined as the ratio of surface tension to elastic modulus, the elastic modulus obtained by classical contact mechanics theories would be overestimated. Contrary to the majority of reported results, we find that the elastic modulus of a cell could be independent of indent depths if surface tension is taken into account. Our model seems to be in agreement with experimental data available. A comprehensive comparison will be done in the future.

A Dynamic Biochemomechanical Model of Geometry-Confined Cell Spreading.

Cell spreading is involved in many physiological and pathological processes. The spreading behavior of a cell significantly depends on its microenvironment, but the biochemomechanical mechanisms of geometry-confined cell spreading remain unclear. A dynamic model is here established to investigate the spreading of cells confined in a finite region with different geometries, e.g., rectangle, ellipse, triangle, and L-shape. This model incorporates both biophysical and biochemical mechanisms, including actin polymerization, integrin-mediated binding, plasma viscoelasticity, and the elasticity of membranes and microtubules. We simulate the dynamic configurational evolution of a cell under different geometric microenvironments, including the angular distribution of microtubule forces and the deformation of the nucleus. The results indicate that the positioning of the cell-division plane is affected by its boundary confinement: a cell divides in a plane perpendicular to its minimal principal axis of inertia of area. In addition, the effects of such physical factors as the adhesive bond density, membrane tension, and microtubule number are examined on the cell spreading dynamics. The theoretical predictions show a good agreement with relevant experimental results. This work sheds light on the geometry-confined spreading dynamics of cells and holds potential applications in regulating cell division and designing cell-based sensors.

A Tensegrity Model of Cell Reorientation on Cyclically Stretched Substrates.

Deciphering the mechanisms underlying the high sensitivity of cells to mechanical microenvironments is crucial for understanding many physiological and pathological processes, e.g., stem cell differentiation and cancer cell metastasis. Here, a cytoskeletal tensegrity model is proposed to study the reorientation of polarized cells on a substrate under biaxial cyclic deformation. The model consists of four bars, representing the longitudinal stress fibers and lateral actin network, and eight strings, denoting the microfilaments. It is found that the lateral bars in the tensegrity, which have been neglected in most of the existing models, can play a vital role in regulating the cellular orientation. The steady orientation of cells can be quantitatively determined by the geometric dimensions and elastic properties of the tensegrity elements, as well as the frequency and biaxial ratio of the cyclic stretches. It is shown that this tensegrity model can reproduce all available experimental observations. For example, the dynamics of cell reorientation is captured by an exponential scaling law with a characteristic time that is independent of the loading frequency at high frequencies and scales inversely with the square of the strain amplitude. This study suggests that tensegrity type models may be further developed to understand cellular responses to mechanical microenvironments and provide guidance for engineering delicate cellular mechanosensing systems.

The glycocalyx promotes cooperative binding and clustering of adhesion receptors.

Cell adhesion plays a pivotal role in various biological processes, e.g., immune responses, cancer metastasis, and stem cell differentiation. The adhesion behaviors depend subtly on the binding kinetics of receptors and ligands restricted at the cell-substrate interfaces. Although much effort has been directed toward investigating the kinetics of adhesion molecules, the role of the glycocalyx, anchored on cell surfaces as an exterior layer, is still unclear. In this paper, we propose a theoretical approach to study the collective binding kinetics of a few and a large number of binders in the presence of the glycocalyx, representing the cases of initial and mature adhesions of cells, respectively. The analytical results are validated by finding good agreement with our Monte Carlo simulations. In the force loading case, the on-rate and affinity increase as more bonds form, whereas this cooperative effect is not observed in the displacement loading case. The increased thickness and stiffness of the glycocalyx tend to decrease the affinity for a few bonds, while they have less influence on the affinity for a large number of bonds. Moreover, for a flexible membrane with thermally-excited shape fluctuations, the glycocalyx is exhibited to promote the formation of bond clusters, mainly due to the cooperative binding of binders. This study helps to understand the cooperative kinetics of adhesion receptors under physiologically relevant loading conditions and sheds light on the novel role of the glycocalyx in cell adhesion.

How do changes at the cell level affect the mechanical properties of epithelial monolayers?

Epithelial monolayers play a vital role in gastrulation, tumor metastasis and wound healing, and protect the tissue from pathogens. During these processes, the monolayers sense, generate, and exert mechanical forces to perform their biological functions, but their mechanical properties are rarely known. Here, we use the vertex dynamics models to investigate the mechanical behaviors of an epithelial monolayer and the configurations of the cells within the monolayer during stretch. It was found that the epithelial monolayer exhibited elastic and plastic properties, due to the geometric extension of cells and cell division, respectively. Moreover, the elasticity of monolayers was increased by enhancing the cell adhesion or by reducing the active contractility of actin-myosin rings. This study furthers our understanding of the relationship between the mechanical properties of individual cells and of their monolayers, and may shed light on linking cell behavior to the patterning and morphogenesis of tissues.

Binding constants of membrane-anchored receptors and ligands: A general theory corroborated by Monte Carlo simulations.

Adhesion processes of biological membranes that enclose cells and cellular organelles are essential for immune responses, tissue formation, and signaling. These processes depend sensitively on the binding constant K2D of the membrane-anchored receptor and ligand proteins that mediate adhesion, which is difficult to measure in the "two-dimensional" (2D) membrane environment of the proteins. An important problem therefore is to relate K2D to the binding constant K3D of soluble variants of the receptors and ligands that lack the membrane anchors and are free to diffuse in three dimensions (3D). In this article, we present a general theory for the binding constants K2D and K3D of rather stiff proteins whose main degrees of freedom are translation and rotation, along membranes and around anchor points "in 2D," or unconstrained "in 3D." The theory generalizes previous results by describing how K2D depends both on the average separation and thermal nanoscale roughness of the apposing membranes, and on the length and anchoring flexibility of the receptors and ligands. Our theoretical results for the ratio K2D/K3D of the binding constants agree with detailed results from Monte Carlo simulations without any data fitting, which indicates that the theory captures the essential features of the "dimensionality reduction" due to membrane anchoring. In our Monte Carlo simulations, we consider a novel coarse-grained model of biomembrane adhesion in which the membranes are represented as discretized elastic surfaces, and the receptors and ligands as anchored molecules that diffuse continuously along the membranes and rotate at their anchor points.

Binding kinetics of membrane-anchored receptors and ligands: Molecular dynamics simulations and theory.

The adhesion of biological membranes is mediated by the binding of membrane-anchored receptor and ligand proteins. Central questions are how the binding kinetics of these proteins is affected by the membranes and by the membrane anchoring of the proteins. In this article, we (i) present detailed data for the binding of membrane-anchored proteins from coarse-grained molecular dynamics simulations and (ii) provide a theory that describes how the binding kinetics depends on the average separation and thermal roughness of the adhering membranes and on the anchoring, lengths, and length variations of the proteins. An important element of our theory is the tilt of bound receptor-ligand complexes and transition-state complexes relative to the membrane normals. This tilt results from an interplay of the anchoring energy and rotational entropy of the complexes and facilitates the formation of receptor-ligand bonds at membrane separations smaller than the preferred separation for binding. In our simulations, we have considered both lipid-anchored and transmembrane receptor and ligand proteins. We find that the binding equilibrium constant and binding on-rate constant of lipid-anchored proteins are considerably smaller than the binding constant and on-rate constant of rigid transmembrane proteins with identical binding domains.

Anisotropic power-law viscoelasticity of living cells is dominated by cytoskeletal network structure.

During physiological and pathological processes, cells experience significant morphological alterations with the re-arrangement of cytoskeletal filaments, resulting in anisotropic viscoelasticity. Here, a structure-based cell model is proposed to study the anisotropic viscoelastic mechanical behaviors of living cells. We investigate how cell shape affects its creep responses in longitudinal and perpendicular directions. It is shown that cells exhibit power-law rheological behavior in both longitudinal and perpendicular directions under step stress, with a more solid-like behavior along the longitudinal direction. We reveal that the cell volume and cytoskeletal filament orientation, which have been neglected in most existing models, play a critical role in regulating cellular anisotropic viscoelasticity. The stiffness of the cell in both directions increases linearly with increasing its aspect ratio, due to the decrease of cell volume. Moreover, the increase in the cell's aspect ratio produces the aggregation of cytoskeletal filaments along the longitudinal direction, resulting in higher stiffness in this direction. It is also shown that the increase in cell's aspect ratio corresponds to a process of cellular ordering, which can be quantitatively characterized by the orientational entropy of cytoskeletal filaments. In addition, we present a simple yet robust method to establish the relationship between cell's aspect ratio and cell volume, thus providing a theoretical framework to capture the anisotropic viscoelastic behavior of cells. This study suggests that the structure-based cell models may be further developed to investigate cellular rheological responses to external mechanical stimuli and may be extended to the tissue scale. STATEMENT OF SIGNIFICANCE: The viscoelastic behaviors of cells hold significant importance in comprehending the roles of mechanical forces in embryo development, invasion, and metastasis of cancer cells. Here, a structure-based cell model is proposed to study the anisotropic viscoelastic mechanical behaviors of living cells. Our study highlights the crucial role of previously neglected factors, such as cell volume and cytoskeletal filament orientation, in regulating cellular anisotropic viscoelasticity. We further propose an orientational entropy of cytoskeletal filaments to quantitatively characterize the ordering process of cells with increasing aspect ratios. Moreover, we derived the analytical interrelationships between cell aspect ratio, cell stiffness, cell volume, and cytoskeletal fiber orientation. This study provides a theoretical framework to describe the anisotropic viscoelastic mechanical behavior of cells.