Probing Local Force Propagation in Tensed Fibrous Gels

Micro-tensile rheology of fibrous gels quantifies strain-dependent anisotropy

Semiflexible fiber gels such as collagen and fibrin have unique nonlinear mechanical properties that play an important role in tissue morphogenesis, wound healing, and cancer metastasis. Optical tweezers microrheology has greatly contributed to the understanding of the mechanics of fibrous gels at the microscale, including its heterogeneity and anisotropy. However, the explicit relationship between micromechanical properties and gel deformation has been largely overlooked. We introduce a unique gel-stretching apparatus and employ it to study the relationship between microscale strain and stiffening in fibrin and collagen gels, focusing on the development of anisotropy in the gel. We find that gels stretched by as much as 15 % stiffen significantly both in parallel and perpendicular to the stretching axis, and that the parallel axis is 2-3 times stiffer than the transverse axis. We also measure the stiffening and anisotropy along bands of aligned fibers created by aggregates of cancer cells, and find similar effects as in gels stretched with the tensile apparatus. Our results illustrate that the extracellular microenvironment is highly sensitive to deformation, with implications for tissue homeostasis and pathology. STATEMENT OF SIGNIFICANCE: The inherent fibrous architecture of the extracellular matrix (ECM) gives rise to unique strain-stiffening mechanics. The micromechanics of fibrous networks has been studied extensively, but the deformations involved in its stiffening at the microscale were not quantified. Here we introduce an apparatus that enables measuring the deformations in the gel as it is being stretched while simultaneously using optical tweezers to measure its microscale anisotropic stiffness. We reveal that fibrin and collagen both stiffen dramatically already at ∼10 % deformation, accompanied by the emergence of significant, yet moderate anisotropy. We measure similar stiffening and anisotropy in the matrix remodeled by the tensile apparatus to those found between cancer cell aggregates. Our results emphasize that small strains are enough to introduce substantial stiffening and anisotropy. These have been shown to result in directional cell migration and enhanced force propagation, and possibly control processes like morphogenesis and cancer metastasis.

Micromechanics of fibrous scaffolds and their stiffness sensing by cells

Mechanical properties of the tissue engineering scaffolds are known to play a crucial role in cell response. Therefore, an understanding of the cell-scaffold interactions is of high importance. Here, we have utilized discrete fiber network model to quantitatively study the micromechanics of fibrous scaffolds with different fiber arrangements and cross-linking densities. We observe that localized forces on the scaffold result in its anisotropic deformation even for isotropic fiber arrangements. We also see an exponential decay of the displacement field with distance from the location of applied force. This nature of the decay allows us to estimate the characteristic length for force transmission in fibrous scaffolds. Furthermore, we also looked at the stiffness sensing of fibrous scaffolds by individual cells and its dependence on the cellular sensing mechanism. For this, we considered two conditions- stress-controlled, and strain-controlled application of forces by a cell. With fixed strain, we find that the stiffness sensed by a cell is proportional to the scaffold's 'macroscopic' elastic modulus. However, under fixed stress application by the cell, the stiffness sensed by the cell also depends on the cell's own stiffness. In fact, the stiffness values for the same scaffold sensed by the stiff and soft cells can differ from each other by an order of magnitude. The insights from this work will help in designing tissue engineering scaffolds for applications where mechanical stimuli are a critical factor.

A non local model for cell migration in response to mechanical stimuli

Cell migration is one of the most studied phenomena in biology since it plays a fundamental role in many physiological and pathological processes such as morphogenesis, wound healing and tumorigenesis. In recent years, researchers have performed experiments showing that cells can migrate in response to mechanical stimuli of the substrate they adhere to. Motion towards regions of the substrate with higher stiffness is called durotaxis, while motion guided by the stress or the deformation of the substrate itself is called tensotaxis. Unlike chemotaxis (i.e. the motion in response to a chemical stimulus), these migratory processes are not yet fully understood from a biological point of view. In this respect, we present a mathematical model of single-cell migration in response to mechanical stimuli, in order to simulate these two processes. Specifically, the cell moves by changing its direction of polarization and its motility according to material properties of the substrate (e.g., stiffness) or in response to proper scalar measures of the substrate strain or stress. The equations of motion of the cell are non-local integro-differential equations, with the addition of a stochastic term to account for random Brownian motion. The mechanical stimulus to be integrated in the equations of motion is defined according to experimental measurements found in literature, in the case of durotaxis. Conversely, in the case of tensotaxis, substrate strain and stress are given by the solution of the mechanical problem, assuming that the extracellular matrix behaves as a hyperelastic Yeoh's solid. In both cases, the proposed model is validated through numerical simulations that qualitatively reproduce different experimental scenarios.

Micropatterning the organization of multicellular structures in 3D biological hydrogels; insights into collective cellular mechanical interactions

Control over the organization of cells at the microscale level within supporting biomaterials can push forward the construction of complex tissue architectures for tissue engineering applications and enable fundamental studies of how tissue structure relates to its function. While cells patterning on 2D substrates is a relatively established and available procedure, micropatterning cells in biomimetic 3D hydrogels has been more challenging, especially with micro-scale resolution, and currently relies on sophisticated tools and protocols. We present a robust and accessible 'peel-off' method to micropattern large arrays of individual cells or cell-clusters of precise sizes in biological 3D hydrogels, such as fibrin and collagen gels, with control over cell-cell separation distance and neighboring cells position. We further demonstrate partial control over cell position in thez-dimension by stacking two layers in varying distances between the layers. To demonstrate the potential of the micropatterning gel platform, we study the matrix-mediated mechanical interaction between array of cells that are accurately separated in defined distances. A collective process of intense cell-generated densified bands emerging in the gel between near neighbors was identified, along which cells preferentially migrate, a process relevant to tissue morphogenesis. The presented 3D gel micropatterning method can be used to reveal fundamental morphogenetic processes, and to reconstruct any tissue geometry with micrometer resolution in 3D biomimetic gel environments, leveraging the engineering of tissues in complex architectures.

Local response and emerging nonlinear elastic length scale in biopolymer matrices

Nonlinear stiffening is a ubiquitous property of major types of biopolymers that make up the extracellular matrices (ECM) including collagen, fibrin, and basement membrane. Within the ECM, many types of cells such as fibroblasts and cancer cells have a spindle-like shape that acts like two equal and opposite force monopoles, which anisotropically stretch their surroundings and locally stiffen the matrix. Here, we first use optical tweezers to study the nonlinear force-displacement response to localized monopole forces. We then propose an effective-probe scaling argument that a local point force application can induce a stiffened region in the matrix, which can be characterized by a nonlinear length scale R* that increases with the increasing force magnitude; the local nonlinear force-displacement response is a result of the nonlinear growth of this effective probe that linearly deforms an increasing portion of the surrounding matrix. Furthermore, we show that this emerging nonlinear length scale R* can be observed around living cells and can be perturbed by varying matrix concentration or inhibiting cell contractility.

Anomalous linear elasticity of disordered networks

Continuum elasticity is a powerful tool applicable in a broad range of physical systems and phenomena. Yet, understanding how and on what scales material disorder may lead to the breakdown of continuum elasticity is not fully understood. We show, based on recent theoretical developments and extensive numerical computations, that disordered elastic networks near a critical rigidity transition, such as strain-stiffened fibrous biopolymer networks that are abundant in living systems, reveal an anomalous long-range linear elastic response below a correlation length. This emergent anomalous elasticity, which is non-affine in nature, is shown to feature a qualitatively different multipole expansion structure compared to ordinary continuum elasticity, and a slower spatial decay of perturbations. The potential degree of universality of these results, their implications (e.g. for cell-cell communication through biological extracellular matrices) and open questions are briefly discussed.