Heterogeneous force network in 3D cellularized collagen networks

Strain-dependent dynamic re-alignment of collagen fibers in skeletal muscle extracellular matrix

Collagen fiber architecture within the skeletal muscle extracellular matrix (ECM) is significant to passive muscle mechanics. While it is thought that collagen fibers re-orient themselves in response to changes in muscle length, this has not been dynamically visualized and quantified within a muscle. The goal of this study was to measure changes in collagen alignment across a range of muscle lengths and compare the corresponding alignment to muscle mechanics. We hypothesized that collagen fibers dynamically increase alignment in response to muscle stretching, and this change in alignment is related to passive muscle stiffness. Further, we hypothesized that digesting collagen fibers with collagenase would reduce the re-alignment response to muscle stretching. Using DBA/2J and D2.mdx mice, we isolated extensor digitorum longus (EDL), soleus, and diaphragm muscles for collagenase or sham treatment and decellularization to isolate intact or collagenase-digested decellularized muscles (DCMs). These DCMs were mechanically tested and imaged using second harmonic generation microscopy to measure collagen alignment across a range of strains. We found that collagen alignment increased in a strain-dependent fashion, but collagenase did not significantly affect the strain-dependent change in alignment. We also saw that the collagen fibers in the diaphragm epimysium (surface ECM) and perimysium (deep ECM) started at different angles, but still re-oriented in the same direction in response to stretching. These robust changes in collagen alignment were weakly related to passive DCM stiffness. Overall, we demonstrated that the architecture of muscle ECM is dynamic in response to strain and is related to passive muscle mechanics. STATEMENT OF SIGNIFICANCE: Our study presents a unique visualization and quantification of strain-induced changes in muscle collagen fiber alignment as they relate to passive mechanics. Using dynamic imaging of collagen in skeletal muscle we demonstrate that as skeletal muscle is stretched, collagen fibers re-orient themselves along the axis of stretch and increase their alignment. The degree of alignment and the increase in alignment are each weakly related to passive muscle stiffness. Collagenase treatments further demonstrate that the basis for muscle Extracellular matrix stiffness is dependent on factors beyond collagen crosslinking and alignment. Together the study contributes to the knowledge of the structure-function relationships of muscle extracellular matrix to tissue stiffness relevant to conditions of fibrosis and aberrant stiffness.

Compressive instabilities enable cell-induced extreme densification patterns in the fibrous extracellular matrix: Discrete model predictions

We present a new model and extensive computations that explain the dramatic remodelling undergone by a fibrous collagen extracellular matrix (ECM), when subjected to contractile mechanical forces from embedded cells or cell clusters. This remodelling creates complex patterns, comprising multiple narrow localised bands of severe densification and fiber alignment, extending far into the ECM, often joining distant cells or cell clusters (such as tumours). Most previous models cannot capture this behaviour, as they assume stable mechanical fiber response with stress an increasing function of fiber stretch, and a restriction to small displacements. Our fully nonlinear network model distinguishes between two types of single-fiber nonlinearity: fibers that undergo stable (supercritical) buckling (as in previous work) versus fibers that suffer unstable (subcritical) buckling collapse. The model allows unrestricted, arbitrarily large displacements (geometric nonlinearity). Our assumptions on single-fiber instability are supported by recent simulations and experiments on buckling of individual beams with a hierarchical microstructure, such as collagen fibers. We use simple scenarios to illustrate, for the first time, two distinct compressive-instability mechanisms at work in our model: unstable buckling collapse of single fibers, and snap-through of multiple-fiber groups. The latter is possible even when single fibers are stable. Through simulations of large fiber networks, we show how these instabilities lead to spatially extended patterns of densification, fiber alignment and ECM remodelling induced by cell contraction. Our model is simple, but describes a very complex, multi-stable energy landscape, using sophisticated numerical optimisation methods that overcome the difficulties caused by instabilities in large systems. Our work opens up new ways of understanding the unique biomechanics of fibrous-network ECM, by fully accounting for nonlinearity and associated loss of stability in fiber networks. Our results provide new insights on tumour invasion and metastasis.

Altering Cell Junctional Tension in Spheroids through E-Cadherin Engagement Modulation

Cadherin-mediated tension at adherens junctions (AJs) is fundamental for cell-cell adhesion and maintaining epithelial integrity. Despite the importance of manipulating AJs to dissect cell-cell interactions, existing three-dimensional (3D) multicellular models have not adequately addressed the precise manipulation of these junctions. To fill this gap, we introduce E-cadherin-modified tension gauge tethers (TGTs) at the junctions within spheroids. The system enables both quantification and modulation of junctional tension with specific DNA triggers. Using rupture-induced fluorescence, we successfully measure mechanical forces in 3D spheroids. Furthermore, mechanically strong TGTs can maintain normal E-cadherin-mediated adhesion. Employing toehold-mediated strand displacement allowed us to disrupt E-cadherin-specific cell-cell adhesion, consequently altering intracellular tension within the spheroids. Our methodology offers a robust and precise way to manipulate cell-cell adhesion and intracellular mechanics in spheroid models.

Subcellular mechano-regulation of cell migration in confined extracellular microenvironment

Cell migration is a highly coordinated cellular event that determines diverse physiological and pathological processes in which the continuous interaction of a migrating cell with neighboring cells or the extracellular matrix is regulated by the physical setting of the extracellular microenvironment. In confined spaces, cell migration occurs differently compared to unconfined open spaces owing to the additional forces that limit cell motility, which create a driving bias for cells to invade the confined space, resulting in a distinct cell motility process compared to what is expected in open spaces. Moreover, cells in confined environments can be subjected to elevated mechanical compression, which causes physical stimuli and activates the damage repair cycle in the cell, including the DNA in the nucleus. Although cells have a self-restoring system to repair damage from the cell membrane to the genetic components of the nucleus, this process may result in genetic and/or epigenetic alterations that can increase the risk of the progression of diverse diseases, such as cancer and immune disorders. Furthermore, there has been a shift in the paradigm of bioengineering from the development of new biomaterials to controlling biophysical cues and fine-tuning cell behaviors to cure damaged/diseased tissues. The external physical cues perceived by cells are transduced along the mechanosensitive machinery, which is further channeled into the nucleus through subcellular molecular linkages of the nucleoskeleton and cytoskeleton or the biochemical translocation of transcription factors. Thus, external cues can directly or indirectly regulate genetic transcriptional processes and nuclear mechanics, ultimately determining cell fate. In this review, we discuss the importance of the biophysical cues, response mechanisms, and mechanical models of cell migration in confined environments. We also discuss the effect of force-dependent deformation of subcellular components, specifically focusing on subnuclear organelles, such as nuclear membranes and chromosomal organization. This review will provide a biophysical perspective on cancer progression and metastasis as well as abnormal cellular proliferation.

Inference of long-range cell-cell force transmission from ECM remodeling fluctuations

Cells sense, manipulate and respond to their mechanical microenvironment in a plethora of physiological processes, yet the understanding of how cells transmit, receive and interpret environmental cues to communicate with distant cells is severely limited due to lack of tools to quantitatively infer the complex tangle of dynamic cell-cell interactions in complicated environments. We present a computational method to systematically infer and quantify long-range cell-cell force transmission through the extracellular matrix (cell-ECM-cell communication) by correlating ECM remodeling fluctuations in between communicating cells and demonstrating that these fluctuations contain sufficient information to define unique signatures that robustly distinguish between different pairs of communicating cells. We demonstrate our method with finite element simulations and live 3D imaging of fibroblasts and cancer cells embedded in fibrin gels. While previous studies relied on the formation of a visible fibrous 'band' extending between cells to inform on mechanical communication, our method detected mechanical propagation even in cases where visible bands never formed. We revealed that while contractility is required, band formation is not necessary, for cell-ECM-cell communication, and that mechanical signals propagate from one cell to another even upon massive reduction in their contractility. Our method sets the stage to measure the fundamental aspects of intercellular long-range mechanical communication in physiological contexts and may provide a new functional readout for high content 3D image-based screening. The ability to infer cell-ECM-cell communication using standard confocal microscopy holds the promise for wide use and democratizing the method.

Hybrid cellular Potts and bead-spring modeling of cells in fibrous extracellular matrix

The mechanical interaction between cells and the extracellular matrix (ECM) is fundamental to coordinate collective cell behavior in tissues. Relating individual cell-level mechanics to tissue-scale collective behavior is a challenge that cell-based models such as the cellular Potts model (CPM) are well-positioned to address. These models generally represent the ECM with mean-field approaches, which assume substrate homogeneity. This assumption breaks down with fibrous ECM, which has nontrivial structure and mechanics. Here, we extend the CPM with a bead-spring model of ECM fiber networks modeled using molecular dynamics. We model a contractile cell pulling with discrete focal adhesion-like sites on the fiber network and demonstrate agreement with experimental spatiotemporal fiber densification and displacement. We show that at high network cross-linking, contractile cell forces propagate over at least eight cell diameters, decaying with distance with power law exponent n= 0.35 - 0.65 typical of viscoelastic ECMs. Further, we use in silico atomic force microscopy to measure local cell-induced network stiffening consistent with experiments. Our model lays the foundation for investigating how local and long-ranged cell-ECM mechanobiology contributes to multicellular morphogenesis.

Cross-linker Mobility Governs Fracture Behavior of Catch-Bonded Networks

While most chemical bonds weaken under the action of mechanical force (called slip bond behavior), nature has developed bonds that do the opposite: their lifetime increases as force is applied. While such catch bonds have been studied quite extensively at the single molecule level and in adhesive contacts, recent work has shown that they are also abundantly present as crosslinkers in the actin cytoskeleton. However, their role and the mechanism by which they operate in these networks have remained unclear. Here, we present computer simulations that show how polymer networks crosslinked with either slip or catch bonds respond to mechanical stress. Our results reveal that catch bonding may be required to protect dynamic networks against fracture, in particular for mobile linkers that can diffuse freely after unbinding. While mobile slip bonds lead to networks that are very weak at high stresses, mobile catch bonds accumulate in high stress regions and thereby stabilize cracks, leading to a more ductile fracture behavior. This allows cells to combine structural adaptivity at low stresses with mechanical stability at high stresses.

A cell-based framework for modeling cardiac mechanics

Cardiomyocytes are the functional building blocks of the heart-yet most models developed to simulate cardiac mechanics do not represent the individual cells and their surrounding matrix. Instead, they work on a homogenized tissue level, assuming that cellular and subcellular structures and processes scale uniformly. Here we present a mathematical and numerical framework for exploring tissue-level cardiac mechanics on a microscale given an explicit three-dimensional geometrical representation of cells embedded in a matrix. We defined a mathematical model over such a geometry and parametrized our model using publicly available data from tissue stretching and shearing experiments. We then used the model to explore mechanical differences between the extracellular and the intracellular space. Through sensitivity analysis, we found the stiffness in the extracellular matrix to be most important for the intracellular stress values under contraction. Strain and stress values were observed to follow a normal-tangential pattern concentrated along the membrane, with substantial spatial variations both under contraction and stretching. We also examined how it scales to larger size simulations, considering multicellular domains. Our work extends existing continuum models, providing a new geometrical-based framework for exploring complex cell-cell and cell-matrix interactions.

Micromechanical remodeling of the extracellular matrix by invading tumors: anisotropy and heterogeneity

Altered tissue mechanics is an important signature of invasive solid tumors. While the phenomena have been extensively studied by measuring the bulk rheology of the extracellular matrix (ECM) surrounding tumors, micromechanical remodeling at the cellular scale remains poorly understood. By combining holographic optical tweezers and confocal microscopy on in vitro tumor models, we show that the micromechanics of collagen ECM surrounding an invading tumor demonstrate directional anisotropy, spatial heterogeneity and significant variations in time as tumors invade. To test the cellular mechanisms of ECM micromechanical remodeling, we construct a simple computational model and verify its predictions with experiments. We find that collective force generation of a tumor stiffens the ECM and leads to anisotropic local mechanics such that the extension direction is more rigid than the compression direction. ECM degradation by cell-secreted matrix metalloproteinase softens the ECM, and active traction forces from individual disseminated cells re-stiffen the matrix. Together, these results identify plausible biophysical mechanisms responsible for the remodeled ECM micromechanics surrounding an invading tumor.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

Abnormal Aggregation of Invasive Cancer Cells Induced by Collective Polarization and ECM-Mediated Mechanical Coupling in Coculture Systems

Studies on pattern formation in coculture cell systems can provide insights into many physiological and pathological processes. Here, we investigate how the extracellular matrix (ECM) may influence the patterning in coculture systems. The model coculture system we use is composed of highly motile invasive breast cancer cells, initially mixed with inert nonmetastatic cells on a 2D substrate and covered with a Matrigel layer introduced to mimic ECM. We observe that the invasive cells exhibit persistent centripetal motion and yield abnormal aggregation, rather than random spreading, due to a “collective pulling” effect resulting from ECM-mediated transmission of active contractile forces generated by the polarized migration of the invasive cells along the vertical direction. The mechanism we report may open a new window for the understanding of biological processes that involve multiple types of cells.

Weakening of resistance force by cell-ECM interactions regulate cell migration directionality and pattern formation

Collective migration of epithelial cells is a fundamental process in multicellular pattern formation. As they expand their territory, cells are exposed to various physical forces generated by cell-cell interactions and the surrounding microenvironment. While the physical stress applied by neighbouring cells has been well studied, little is known about how the niches that surround cells are spatio-temporally remodelled to regulate collective cell migration and pattern formation. Here, we analysed how the spatio-temporally remodelled extracellular matrix (ECM) alters the resistance force exerted on cells so that the cells can expand their territory. Multiple microfabrication techniques, optical tweezers, as well as mathematical models were employed to prove the simultaneous construction and breakage of ECM during cellular movement, and to show that this modification of the surrounding environment can guide cellular movement. Furthermore, by artificially remodelling the microenvironment, we showed that the directionality of collective cell migration, as well as the three-dimensional branch pattern formation of lung epithelial cells, can be controlled. Our results thus confirm that active remodelling of cellular microenvironment modulates the physical forces exerted on cells by the ECM, which contributes to the directionality of collective cell migration and consequently, pattern formation.

A computational framework for modeling cell-matrix interactions in soft biological tissues

Living soft tissues appear to promote the development and maintenance of a preferred mechanical state within a defined tolerance around a so-called set point. This phenomenon is often referred to as mechanical homeostasis. In contradiction to the prominent role of mechanical homeostasis in various (patho)physiological processes, its underlying micromechanical mechanisms acting on the level of individual cells and fibers remain poorly understood, especially how these mechanisms on the microscale lead to what we macroscopically call mechanical homeostasis. Here, we present a novel computational framework based on the finite element method that is constructed bottom up, that is, it models key mechanobiological mechanisms such as actin cytoskeleton contraction and molecular clutch behavior of individual cells interacting with a reconstructed three-dimensional extracellular fiber matrix. The framework reproduces many experimental observations regarding mechanical homeostasis on short time scales (hours), in which the deposition and degradation of extracellular matrix can largely be neglected. This model can serve as a systematic tool for future in silico studies of the origin of the numerous still unexplained experimental observations about mechanical homeostasis.

Deriving time-varying cellular motility parameters via wavelet analysis

Cell migration, which is regulated by intracellular signaling pathways (ICSP) and extracellular matrix (ECM), plays an indispensable role in many physiological and pathological process such as normal tissue development and cancer metastasis. However, there is a lack of rigorous and quantitative tools for analyzing the time-varying characteristics of cell migration in heterogeneous microenvironment, resulted from, e.g. the time-dependent local stiffness due to microstructural remodeling by migrating cells. Here, we develop a wavelet-analysis approach to derive the time-dependent motility parameters from cell migration trajectories, based on the time-varying persistent random walk model. In particular, the wavelet denoising and wavelet transform are employed to analyze migration velocities and obtain the wavelet power spectrum. Subsequently, the time-dependent motility parameters are derived via Lorentzian power spectrum. Our results based on synthetic data indicate the superiority of the method for estimating the intrinsic transient motility parameters, robust against a variety of stochastic noises. We also carry out a systematic parameter study and elaborate the effects of parameter selection on the performance of the method. Moreover, we demonstrate the utility of our approach via analyzing experimental data ofin vitrocell migration in distinct microenvironments, including the migration of MDA-MB-231 cells in confined micro-channel arrays and correlated migration of MCF-10A cells due to ECM-mediated mechanical coupling. Our analysis shows that our approach can be as a powerful tool to accurately derive the time-dependent motility parameters, and further analyze the time-dependent characteristics of cell migration regulated by complex microenvironment.

Shannon entropy for time-varying persistence of cell migration

Cell migration, which can be significantly affected by intracellular signaling pathways and extracellular matrix, plays a crucial role in many physiological and pathological processes. Cell migration is typically modeled as a persistent random walk, which depends on two critical motility parameters, i.e., migration speed and persistence time. It is generally very challenging to efficiently and accurately quantify the migration dynamics from noisy experimental data. Here, we introduce the normalized Shannon entropy (SE) based on the FPS of cellular velocity autocovariance function to quantify migration dynamics. The SE introduced here possesses a similar physical interpretation as the Gibbs entropy for thermal systems in that SE naturally reflects the degree of order or randomness of cellular migration, attaining the maximal value of unity for purely diffusive migration (i.e., SE = 1 for the most "random" dynamics) and the minimal value of 0 for purely ballistic dynamics (i.e., SE = 0 for the most "ordered" dynamics). We also find that SE is strongly correlated with the migration persistence but is less sensitive to the migration speed. Moreover, we introduce the time-varying SE based on the WPS of cellular dynamics and demonstrate its superior utility to characterize the time-dependent persistence of cell migration, which typically results from complex and time-varying intra- or extracellular mechanisms. We employ our approach to analyze experimental data of in vitro cell migration regulated by distinct intracellular and extracellular mechanisms, exhibiting a rich spectrum of dynamic characteristics. Our analysis indicates that the SE and wavelet transform (i.e., SE-based approach) offers a simple and efficient tool to quantify cell migration dynamics in complex microenvironment.

Dynamically Re-Organized Collagen Fiber Bundles Transmit Mechanical Signals and Induce Strongly Correlated Cell Migration and Self-Organization

Correlated cell migration in fibrous extracellular matrix (ECM) is important in many biological processes. During migration, cells can remodel the ECM, leading to the formation of mesoscale structures such as fiber bundles. However, how such mesoscale structures regulate correlated single-cells migration remains to be elucidated. Here, using a quasi-3D in vitro model, we investigate how collagen fiber bundles are dynamically re-organized and guide cell migration. By combining laser ablation technique with 3D tracking and active-particle simulations, we definitively show that only the re-organized fiber bundles that carry significant tensile forces can guide strongly correlated cell migration, providing for the first time a direct experimental evidence supporting that matrix-transmitted long-range forces can regulate cell migration and self-organization. This force regulation mechanism can provide new insights for studies on cellular dynamics, fabrication or selection of biomedical materials in tissue repairing, and many other biomedical applications.

The role of temperature in the rigidity-controlled fracture of elastic networks

We study the influence of thermal fluctuations on the fracture of elastic networks, via simulations of the uniaxial extension of central-force spring networks with varying rigidity. Studying their failure response, both at the macroscopic and microscopic level, we find that an increase in temperature corresponds to a more homogeneous stress (re)distribution and induces thermally activated failure of springs. As a consequence, the material strength decreases upon increasing temperature, the microscopic damage spreads over a larger area and a more ductile fracture process is observed. These effects are modulated by network rigidity and can therefore be tuned via the network connectivity and the rupture threshold of the springs. Knowledge of the interplay between temperature and rigidity improves our understanding of the fracture of elastic network materials, such as (biological) polymer networks, and can help to refine design principles for tough soft materials.

Modeling multicellular dynamics regulated by extracellular-matrix-mediated mechanical communication via active particles with polarized effective attraction

Collective cell migration is crucial to many physiological and pathological processes such as embryo development, wound healing, and cancer invasion. Recent experimental studies have indicated that the active traction forces generated by migrating cells in a fibrous extracellular matrix (ECM) can mechanically remodel the ECM, giving rise to bundlelike mesostructures bridging individual cells. Such fiber bundles also enable long-range propagation of cellular forces, leading to correlated migration dynamics regulated by the mechanical communication among the cells. Motivated by these experimental discoveries, we develop an active-particle model with polarized effective attractions (APPA) to investigate emergent multicellular migration dynamics resulting from ECM-mediated mechanical communications. In particular, the APPA model generalizes the classic active-Brownian-particle (ABP) model by imposing a pairwise polarized attractive force between the particles, which depends on the instantaneous dynamic states of the particles and mimics the effective mutual pulling between the cells via the fiber bundle bridge. The APPA system exhibits enhanced aggregation behaviors compared to the classic ABP system, and the contrast is more apparent at lower particle densities and higher rotational diffusivities. Importantly, in contrast to the classic ABP system where the particle velocities are not correlated for all particle densities, the high-density phase of the APPA system exhibits strong dynamic correlations, which are characterized by the slowly decaying velocity correlation functions with a correlation length comparable to the linear size of the high-density phase domain (i.e., the cluster of particles). The strongly correlated multicellular dynamics predicted by the APPA model is subsequently verified in in vitro experiments using MCF-10A cells. Our studies indicate the importance of incorporating ECM-mediated mechanical coupling among the migrating cells for appropriately modeling emergent multicellular dynamics in complex microenvironments.

Sensing Traction Force on the Matrix Induces Cell-Cell Distant Mechanical Communications for Self-Assembly

The long-range biomechanical force propagating across a large scale may reserve the capability to trigger coordinative responses within cell population such as during angiogenesis, epithelial tubulogenesis, and cancer metastasis. How cells communicate in a distant manner within the group for self-assembly remains largely unknown. Here, we found that airway smooth muscle cells (ASMCs) rapidly self-assembled into a well-constructed network on 3D Matrigel containing type I collagen (COL), which relied on long-range biomechanical force across the matrix to direct cell-cell distant interactions. Similar results happened by HUVEC cells to mimic angiogenesis. Interestingly, single ASMCs initiated multiple extended protrusions precisely pointing to neighboring cells in distance (100-300 μm away or 5-10 folds of the diameter of a round single cell), depending on traction force sensing. Individual ASMCs mechanosensed each other to move directionally on both nonfibrous Matrigel only and Matrigel containing fibrous COL but lost mutual sensing on the cross-linked gel or coated glass due to no long-range force transmission. The bead tracking assay demonstrated distant transmission of traction force (up to 400 μm) during the matrix deformation, and finite element method modeling confirmed the consistency between maximum strain distribution on the matrix and cell directional movements in experiments. Furthermore, ASMCs recruited COL from the hydrogel to build a fibrous network to mechanically stabilize the cell network. Our results revealed principally that cells can sense traction force transmitted through the matrix to initiate cell-cell distant mechanical communications, resulting in cell directional migration and coordinated cell and COL self-assembly with active matrix remodeling. As an interesting phenomenon, cells seem to be able to "make a phone call" via long-range biomechanics, which implicates physiological importance such as for tissue pattern formation.

Probing information content of hierarchical n-point polytope functions for quantifying and reconstructing disordered systems

Disordered systems are ubiquitous in physical, biological, and material sciences. Examples include liquid and glassy states of condensed matter, colloids, granular materials, porous media, composites, alloys, packings of cells in avian retina, and tumor spheroids, to name but a few. A comprehensive understanding of such disordered systems requires, as the first step, systematic quantification, modeling, and representation of the underlying complex configurations and microstructure, which is generally very challenging to achieve. Recently, we introduced a set of hierarchical statistical microstructural descriptors, i.e., the "n-point polytope functions" P_{n}, which are derived from the standard n-point correlation functions S_{n}, and successively included higher-order n-point statistics of the morphological features of interest in a concise, explainable, and expressive manner. Here we investigate the information content of the P_{n} functions via optimization-based realization rendering. This is achieved by successively incorporating higher-order P_{n} functions up to n=8 and quantitatively assessing the accuracy of the reconstructed systems via unconstrained statistical morphological descriptors (e.g., the lineal-path function). We examine a wide spectrum of representative random systems with distinct geometrical and topological features. We find that, generally, successively incorporating higher-order P_{n} functions and, thus, the higher-order morphological information encoded in these descriptors leads to superior accuracy of the reconstructions. However, incorporating more P_{n} functions into the reconstruction also significantly increases the complexity and roughness of the associated energy landscape for the underlying stochastic optimization, making it difficult to convergence numerically.

Compression stiffening of fibrous networks with stiff inclusions

Tissues commonly consist of cells embedded within a fibrous biopolymer network. Whereas cell-free reconstituted biopolymer networks typically soften under applied uniaxial compression, various tissues, including liver, brain, and fat, have been observed to instead stiffen when compressed. The mechanism for this compression-stiffening effect is not yet clear. Here, we demonstrate that when a material composed of stiff inclusions embedded in a fibrous network is compressed, heterogeneous rearrangement of the inclusions can induce tension within the interstitial network, leading to a macroscopic crossover from an initial bending-dominated softening regime to a stretching-dominated stiffening regime, which occurs before and independently of jamming of the inclusions. Using a coarse-grained particle-network model, we first establish a phase diagram for compression-driven, stretching-dominated stress propagation and jamming in uniaxially compressed two- and three-dimensional systems. Then, we demonstrate that a more detailed computational model of stiff inclusions in a subisostatic semiflexible fiber network exhibits quantitative agreement with the predictions of our coarse-grained model as well as qualitative agreement with experiments.

Development of a 3-D Physical Dynamics Monitoring System Using OCM with DVC for Quantification of Sprouting Endothelial Cells Interacting with a Collagen Matrix

The extracellular matrix (ECM) plays a key role during cell migration, proliferation, and differentiation by providing adhesion sites and serving as a physical scaffold. Elucidating the interaction between the cell and ECM can reveal the underlying mechanisms of cellular behavior that are currently unclear. Analysis of the deformation of the ECM due to cell-matrix interactions requires microscopic, three-dimensional (3-D) imaging methods, such as confocal microscopy and second-harmonic generation microscopy, which are currently limited by phototoxicity and bleaching as a result of the point-scanning approach. In this study, we suggest the use of optical coherence microscopy (OCM) as a live-cell, volumetric, fast imaging tool for analyzing the deformation of fibrous ECM. We optimized such OCM parameters as the sampling rate to obtain images of the best quality that meet the requirements for robust digital volume correlation (DVC) analysis. Visualization and analysis of the mechanical interaction between collagen ECM and human umbilical vein endothelial cells (HUVECs) show that cellular adhesion during protrusion can be analyzed and quantified. The advantages of OCM, such as fine isotropic spatial resolution, fast time resolution, and low phototoxicity, make it the ideal optic tool for 3-D traction force microscopy.

Modeling cell migration regulated by cell extracellular-matrix micromechanical coupling

Cell migration in fibrous extracellular matrix (ECM) is crucial to many physiological and pathological processes such as tissue regeneration, immune response, and cancer progression. During migration, individual cells can generate active pulling forces via actomyosin contraction, which are transmitted to the ECM fibers through focal adhesion complexes, remodel the ECM, and eventually propagate to and can be sensed by other cells in the system. The microstructure and physical properties of the ECM can also significantly influence cell migration, e.g., via durotaxis and contact guidance. Here, we develop a computational model for two-dimensional cell migration regulated by cell-ECM micromechanical coupling. Our model explicitly takes into account a variety of cellular-level processes, including focal adhesion formation and disassembly, active traction force generation and cell locomotion due to actin filament contraction, transmission and propagation of tensile forces in the ECM, as well as the resulting ECM remodeling. We validate our model by accurately reproducing single-cell dynamics of MCF-10A breast cancer cells migrating on collagen gels and show that the durotaxis and contact guidance effects naturally arise as a consequence of the cell-ECM micromechanical interactions considered in the model. Moreover, our model predicts strongly correlated multicellular migration dynamics, which are resulted from the ECM-mediated mechanical coupling among the migrating cell and are subsequently verified in in vitro experiments using MCF-10A cells. Our computational model provides a robust tool to investigate emergent collective dynamics of multicellular systems in complex in vivo microenvironment and can be utilized to design in vitro microenvironments to guide collective behaviors and self-organization of cells.

Geometric Dependence of 3D Collective Cancer Invasion

Metastasis of mesenchymal tumor cells is traditionally considered as a single-cell process. Here, we report an emergent collective phenomenon in which the dissemination rate of mesenchymal breast cancer cells from three-dimensional tumors depends on the tumor geometry. Combining experimental measurements and computational modeling, we demonstrate that the collective dynamics is coordinated by the mechanical feedback between individual cells and their surrounding extracellular matrix (ECM). We find the tissue-like fibrous ECM supports long-range physical interactions between cells, which turn geometric cues into regulated cell dissemination dynamics. Our results suggest that migrating cells in three-dimensional ECM represent a distinct class of an active particle system in which the collective dynamics is governed by the remodeling of the environment rather than direct particle-particle interactions.

Athermal Fracture of Elastic Networks: How Rigidity Challenges the Unavoidable Size-Induced Brittleness

By performing extensive simulations with unprecedentedly large system sizes, we unveil how rigidity influences the fracture of disordered materials. We observe the largest damage in networks with connectivity close to the isostatic point and when the rupture thresholds are small. However, irrespective of network and spring properties, a more brittle fracture is observed upon increasing system size. Differently from most of the fracture descriptors, the maximum stress drop, a proxy for brittleness, displays a universal nonmonotonic dependence on system size. Based on this uncommon trend it is possible to identify the characteristic system size L^{*} at which brittleness kicks in. The more the disorder in network connectivity or in spring thresholds, the larger L^{*}. Finally, we speculate how this size-induced brittleness is influenced by thermal fluctuations.

Elastic Anisotropy Governs the Range of Cell-Induced Displacements

The unique nonlinear mechanics of the fibrous extracellular matrix (ECM) facilitates long-range cell-cell mechanical communications that would be impossible for linear elastic substrates. Past research has described the contribution of two separated effects on the range of force transmission, including ECM elastic nonlinearity and fiber alignment. However, the relation between these different effects is unclear, and how they combine to dictate force transmission range is still elusive. Here, we combine discrete fiber simulations with continuum modeling to study the decay of displacements induced by a contractile cell in fibrous networks. We demonstrate that fiber nonlinearity and fiber reorientation both contribute to the strain-induced elastic anisotropy of the cell's local environment. This elastic anisotropy is a "lumped" parameter that governs the slow decay of displacements, and it depends on the magnitude of applied strain, either an external tension or an internal contraction, as a model of the cell. Furthermore, we show that accounting for artificially prescribed elastic anisotropy dictates the decay of displacements induced by a contracting cell. Our findings unify previous single effects into a mechanical theory that explains force transmission in fibrous networks. This work may provide insights into biological processes that involve communication of distant cells mediated by the ECM, such as those occurring in morphogenesis, wound healing, angiogenesis, and cancer metastasis. It may also provide design parameters for biomaterials to control force transmission between cells as a way to guide morphogenesis in tissue engineering.

Force chains in cell-cell mechanical communication

Force chains (FCs) are a key determinant of the micromechanical properties and behaviour of heterogeneous materials, such as granular systems. However, less is known about FCs in fibrous materials, such as the networks composing the extracellular matrix (ECM) of biological systems. Using a finite-element computational model, we simulated the contraction of a single cell and two nearby cells embedded in two-dimensional fibrous elastic networks and analysed the tensile FCs that developed in the ECM. The role of ECM nonlinear elasticity on FC formation was evaluated by considering linear and nonlinear, i.e. exhibiting 'buckling' and/or 'strain-stiffening', stress-strain curves. The effect of the degree of cell contraction and network coordination value was assessed. We found that nonlinear elasticity of the ECM fibres influenced the structure of the FCs, facilitating the transition towards more distinct chains that were less branched and more radially oriented than the chains formed in linear elastic networks. When two neighbouring cells contract, a larger number of FCs bridged between the cells in nonlinear networks, and these chains had a larger effective rigidity than the chains that did not reach a neighbouring cell. These results suggest that FCs function as a route for mechanical communication between distant cells and highlight the contribution of ECM fibre nonlinear elasticity to the formation of FCs.

Absorbing-active transition in a multi-cellular system regulated by a dynamic force network

Collective cell migration in 3D extracellular matrix (ECM) is crucial to many physiological and pathological processes. Migrating cells can generate active pulling forces via actin filament contraction, which are transmitted to the ECM fibers and lead to a dynamically evolving force network in the system. Here, we elucidate the role of this force network in regulating collective cell behaviors using a minimal active-particle-on-network (APN) model, in which active particles can pull the fibers and hop between neighboring nodes of the network following local durotaxis. Our model reveals a dynamic transition as the particle number density approaches a critical value, from an "absorbing" state containing isolated stationary small particle clusters, to an "active" state containing a single large cluster undergoing constant dynamic reorganization. This reorganization is dominated by a subset of highly dynamic "radical" particles in the cluster, whose number also exhibits a transition at the same critical density. The transition is underlaid by the percolation of "influence spheres" due to the particle pulling forces. Our results suggest a robust mechanism based on ECM-mediated mechanical coupling for collective cell behaviors in 3D ECM.

Biomechanics of Collective Cell Migration in Cancer Progression: Experimental and Computational Methods

Cell migration is essential for regulating many biological processes in physiological or pathological conditions, including embryonic development and cancer invasion. In vitro and in silico studies suggest that collective cell migration is associated with some biomechanical particularities such as restructuring of extracellular matrix (ECM), stress and force distribution profiles, and reorganization of the cytoskeleton. Therefore, the phenomenon could be understood by an in-depth study of cells' behavior determinants, including but not limited to mechanical cues from the environment and from fellow "travelers". This review article aims to cover the recent development of experimental and computational methods for studying the biomechanics of collective cell migration during cancer progression and invasion. We also summarized the tested hypotheses regarding the mechanism underlying collective cell migration enabled by these methods. Together, the paper enables a broad overview on the methods and tools currently available to unravel the biophysical mechanisms pertinent to cell collective migration as well as providing perspectives on future development toward eventually deciphering the key mechanisms behind the most lethal feature of cancer.

Capturing 3D large-strain Euler-bending filament dynamics in fibrous media simulations; sample case of compression collapse in dendritic actin network

Cytoskeletal networks to transmission towers are comprised of slender elements. Slender filaments bend and buckle more easily than stretch. Therefore a deforming network is expected to exhaust all possible bending-based modes before engaging filament stretch. While the large-strain bending critically determines fibrous-media response, simulations use small-strain and jointed approximations. At low resolution, these approximations inflate bending resistance and delay buckling onset. The proposed string-of-continuous-beams (SOCB) approach captures 3D nonlinear Euler bending of filaments with high fidelity at low cost. Bending geometry (i.e. angles and its differentials) is solved as primary variables, to fit a 5th order polynomial of the contour angle. Displacement, solved simultaneously as length conservation, is predicted with C3 and C6 smoothness between and within segments, using only 2 nodes. In the chosen analysis frame, in-plane and out-plane moments can be decoupled for arbitrarily-curved segments. Complex crosslink force-transfers can be specified. Simulations show that when a daughter branch is appended, the buckling resistance of a filament changes from linear to nonlinear before reversible collapse. An actin outcrop with 8 generations of mother-daughter branching produced the linear, nonlinear, and collapse regimes observed in compression experiments. 'Collapse' was a redistribution of outcrop forces following the buckling of few strands.

Normal stress anisotropy and marginal stability in athermal elastic networks

Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component σzz on long (compressible) timescales, crossing over to the first normal stress difference, N1 = σxx - σzz at short (incompressible) times. While the behavior of N1 is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain.

Realizations of highly heterogeneous collagen networks via stochastic reconstruction for micromechanical analysis of tumor cell invasion

Three-dimensional (3D) collective cell migration in a collagen-based extracellular matrix (ECM) is among one of the most significant topics in developmental biology, cancer progression, tissue regeneration, and immune response. Recent studies have suggested that collagen-fiber mediated force transmission in cellularized ECM plays an important role in stress homeostasis and regulation of collective cellular behaviors. Motivated by the recent in vitro observation that oriented collagen can significantly enhance the penetration of migrating breast cancer cells into dense Matrigel which mimics the intravasation process in vivo [Han et al. Proc. Natl. Acad. Sci. USA 113, 11208 (2016)PNASA60027-842410.1073/pnas.1610347113], we devise a procedure for generating realizations of highly heterogeneous 3D collagen networks with prescribed microstructural statistics via stochastic optimization. Specifically, a collagen network is represented via the graph (node-bond) model and the microstructural statistics considered include the cross-link (node) density, valence distribution, fiber (bond) length distribution, as well as fiber orientation distribution. An optimization problem is formulated in which the objective function is defined as the squared difference between a set of target microstructural statistics and the corresponding statistics for the simulated network. Simulated annealing is employed to solve the optimization problem by evolving an initial network via random perturbations to generate realizations of homogeneous networks with randomly oriented fibers, homogeneous networks with aligned fibers, heterogeneous networks with a continuous variation of fiber orientation along a prescribed direction, as well as a binary system containing a collagen region with aligned fibers and a dense Matrigel region with randomly oriented fibers. The generation and propagation of active forces in the simulated networks due to polarized contraction of an embedded ellipsoidal cell and a small group of cells are analyzed by considering a nonlinear fiber model incorporating strain hardening upon large stretching and buckling upon compression. Our analysis shows that oriented fibers can significantly enhance long-range force transmission in the network. Moreover, in the oriented-collagen-Matrigel system, the forces generated by a polarized cell in collagen can penetrate deeply into the Matrigel region. The stressed Matrigel fibers could provide contact guidance for the migrating cell cells, and thus enhance their penetration into Matrigel. This suggests a possible mechanism for the observed enhanced intravasation by oriented collagen.

Nonlinear Elasticity of the ECM Fibers Facilitates Efficient Intercellular Communication

Biological cells embedded in fibrous matrices have been observed to form intercellular bands of dense and aligned fibers through which they mechanically interact over long distances. Such matrix-mediated cellular interactions have been shown to regulate various biological processes. This study aimed to explore the effects of elastic nonlinearity of the fibers contained in the extracellular matrix (ECM) on the transmission of mechanical loads between contracting cells. Based on our biological experiments, we developed a finite-element model of two contracting cells embedded within a fibrous network. The individual fibers were modeled as showing linear elasticity, compression microbuckling, tension stiffening, or both of the latter two. Fiber compression buckling resulted in smaller loads in the ECM, which were primarily directed toward the neighboring cell. These loads decreased with increasing cell-to-cell distance; when cells were >9 cell diameters apart, no such intercellular interaction was observed. Tension stiffening further contributed to directing the loads toward the neighboring cell, though to a smaller extent. The contraction of two neighboring cells resulted in mutual attraction forces, which were considerably increased by tension stiffening and decayed with increasing cell-to-cell distances. Nonlinear elasticity contributed also to the onset of force polarity on the cell boundaries, manifested by larger contractile forces pointing toward the neighboring cell. The density and alignment of the fibers within the intercellular band were greater when fibers buckled under compression, with tension stiffening further contributing to this structural remodeling. Although previous studies have established the role of the ECM nonlinear mechanical behavior in increasing the range of force transmission, our model demonstrates the contribution of nonlinear elasticity of biological gels to directional and efficient mechanical signal transfer between distant cells, and rehighlights the importance of using fibrous gels in experimental settings for facilitating intercellular communication. VIDEO ABSTRACT.

The Role of Network Architecture in Collagen Mechanics

Collagen forms fibrous networks that reinforce tissues and provide an extracellular matrix for cells. These networks exhibit remarkable strain-stiffening properties that tailor the mechanical functions of tissues and regulate cell behavior. Recent models explain this nonlinear behavior as an intrinsic feature of disordered networks of stiff fibers. Here, we experimentally validate this theoretical framework by measuring the elastic properties of collagen networks over a wide range of self-assembly conditions. We show that the model allows us to quantitatively relate both the linear and nonlinear elastic behavior of collagen networks to their underlying architecture. Specifically, we identify the local coordination number (or connectivity) 〈z〉 as a key architectural parameter that governs the elastic response of collagen. The network elastic response reveals that 〈z〉 decreases from 3.5 to 3 as the polymerization temperature is raised from 26 to 37°C while being weakly dependent on concentration. We furthermore infer a Young's modulus of 1.1 MPa for the collagen fibrils from the linear modulus. Scanning electron microscopy confirms that 〈z〉 is between three and four but is unable to detect the subtle changes in 〈z〉 with polymerization conditions that rheology is sensitive to. Finally, we show that, consistent with the model, the initial stress-stiffening response of collagen networks is controlled by the negative normal stress that builds up under shear. Our work provides a predictive framework to facilitate future studies of the regulatory effect of extracellular matrix molecules on collagen mechanics. Moreover, our findings can aid mechanobiological studies of wound healing, fibrosis, and cancer metastasis, which require collagen matrices with tunable mechanical properties.

Quantification of focal adhesion dynamics of cell movement based on cell-induced collagen matrix deformation using second-harmonic generation microscopy

Mechanical interactions of living cells with the surrounding environment via focal adhesion (FA) in three dimensions (3-D) play a key role in dynamic biological events, such as tissue regeneration, wound healing, and cancer invasion. Recently, several methods for observing 3-D cell-extracellular matrix (ECM) interactions have been reported, lacking solid and quantitative analysis on the dynamics of the physical interaction between the cell and the ECM. We measured the submicron displacements of ECM deformation in 3-D due to protrusion-retraction dynamics during cell migration, using second-harmonic generation without labeling the matrix structures. We then quantitatively analyzed the mechanical deformation between the ECM and the cells based on spatiotemporal volumetric correlations. The greatest deformations within the collagen matrix were found to occur at sites of colocalization of the FA site-related proteins vinculin and actin, which confirms that FA sites play a critical role in living cells within the ECM as a point for adhesion, traction, and migration. We believe that this modality can be used in studies of cell-ECM interaction during angiogenesis, wound healing, and metastasis.

Displacement Propagation in Fibrous Networks Due to Local Contraction

The extracellular matrix provides macroscale structure to tissues and microscale guidance for cell contraction, adhesion, and migration. The matrix is composed of a network of fibers, which each deform by stretching, bending, and buckling. Whereas the mechanics has been well characterized in uniform shear and extension, the response to more general loading conditions remains less clear, because the associated displacement fields cannot be predicted a priori. Studies simulating contraction, such as due to a cell, have observed displacements that propagate over a long range, suggesting mechanisms such as reorientation of fibers toward directions of tensile force and nonlinearity due to buckling of fibers under compression. It remains unclear which of these two mechanisms produces the long-range displacements and how properties like fiber bending stiffness and fiber length affect the displacement field. Here, we simulate contraction of an inclusion within a fibrous network and fit the resulting radial displacements to ur ∼ r−n where the power n quantifies the decay of displacements over distance, and a value of n less than that predicted by classical linear elasticity indicates displacements that propagate over a long range. We observed displacements to propagate over a longer range for greater contraction of the inclusion, for networks having longer fibers, and for networks with lower fiber bending stiffness. Contraction of the inclusion also caused fibers to reorient into the radial direction, but, surprisingly, the reorientation was minimally affected by bending stiffness. We conclude that both reorientation and nonlinearity are responsible for the long-range displacements.

Probing three-dimensional collective cancer invasion with DIGME

BACKGROUND

Multicellular pattern formation plays an important role in developmental biology, cancer metastasis and wound healing. While many physical factors have been shown to regulate these multicellular processes, the role of ECM micro-to-meso scale geometry has been poorly understood in 3D collective cancer invasion.

RESULTS

We have developed a mechanical-based strategy, Diskoid In Geometrically Micropatterned ECM (DIGME). DIGME allows easy engineering of the shape of 3D tissue organoid, the mesoscale ECM heterogeneity, and the fiber alignment of collagen-based ECM all at the same time. We have employed DIGME to study the 3D invasion of MDA-MB-231 diskoids in engineered collagen matrix. We find that the collective cancer invasion is closely regulated by the micro-to-meso scale geometry of the ECM.

CONCLUSIONS

We conclude that DIGME provides a simple yet powerful tool to probe 3D dynamics of tissue organoids in physically patterned microenvironments.

Mechanical response of collagen networks to nonuniform microscale loads

As force is applied to fibrous proteins such as collagen or fibrin, the fibers respond by bending, stretching, or buckling, which together bring about a nonlinear relationship between force and displacement. The nonlinearity is typically understood in terms of strain stiffening in uniform extension or shear, but there remains a critical lack of data on how fibrous materials respond to other more complicated loadings. Here we study the mechanics of collagen networks in response to nonuniform loads applied on the local scale of the fibers. For this, we use particles made of an active hydrogel that undergoes a temperature-induced phase transition causing a large decrease in volume. We embed these particles in networks of fibrous collagen and use them as microactuators to apply controlled microscale loading. The resulting fiber displacements propagate over a long range with radial displacements u scaling as r^-n with n ≈ 1. By contrast, we find linear homogeneous materials have n ≈ 2, in agreement with classical linear elastic theory. Our experimental data supports the notion that the long range displacements result from buckling of fibers in compression and local straightening of fibers in tension, in agreement with previous studies. Surprisingly, global network anisotropy appears to have only a modest effect on the displacement propagation. These insights into the microscale mechanics demonstrate that the decay power n provides a useful metric to quantify the mechanics of fibrous materials. We therefore suggest it is a means to compare new theories with experimental data.

Probing cooperative force generation in collective cancer invasion

Collective cellular dynamics in the three-dimensional extracellular matrix (ECM) plays a crucial role in many physiological processes such as cancer invasion. Both chemical and mechanical signaling support cell-cell communications on a variety of length scales, leading to collective migratory behaviors. Here we conduct experiments using 3D in vitro tumor models and develop a phenomenological model in order to probe the cooperativity of force generation in the collective invasion of breast cancer cells. In our model, cell-cell communication is characterized by a single parameter that quantifies the correlation length of cellular migration cycles. We devise a stochastic reconstruction method to generate realizations of cell colonies with specific contraction phase correlation functions and correlation length a. We find that as a increases, the characteristic size of regions containing cells with similar contraction phases grows. For small a values, the large fluctuations in individual cell contraction phases smooth out the temporal fluctuations in the time-dependent deformation field in the ECM. For large a values, the periodicity of an individual cell contraction cycle is clearly manifested in the temporal variation of the overall deformation field in the ECM. Through quantitative comparisons of the simulated and experimentally measured deformation fields, we find that the correlation length for collective force generation in the breast cancer diskoid in geometrically micropatterned ECM (DIGME) system is [Formula: see text], which is roughly twice the linear size of a single cell. One possible mechanism for this intermediate cell correlation length is the fiber-mediated stress propagation in the 3D ECM network in the DIGME system.