In native states, animal cells of many types are supported by a fibrous network that forms the main structural component of the ECM. Mechanical interactions between cells and the 3D ECM critically regulate cell function, including growth and migration. However, the physical mechanism that governs the cell interaction with fibrous 3D ECM is still not known. In this article, we present single-cell traction force measurements using breast tumor cells embedded within 3D collagen matrices. We recreate the breast tumor mechanical environment by controlling the microstructure and density of type I collagen matrices. Our results reveal a positive mechanical feedback loop: cells pulling on collagen locally align and stiffen the matrix, and stiffer matrices, in return, promote greater cell force generation and a stiffer cell body. Furthermore, cell force transmission distance increases with the degree of strain-induced fiber alignment and stiffening of the collagen matrices. These findings highlight the importance of the nonlinear elasticity of fibrous matrices in regulating cell-ECM interactions within a 3D context, and the cell force regulation principle that we uncover may contribute to the rapid mechanical tissue stiffening occurring in many diseases, including cancer and fibrosis.cell traction force | 3D cell traction force microscopy | fibrous nonlinear elasticity | cell-ECM interaction | collagen A nimal cells of most cell types, including breast tumor cells, are supported structurally by a fibrous ECM within a 3D context (1, 2). Cells adhere to the fibers via the linkages between integrin receptors on the membrane surface and the adhesion molecules within the ECM. To migrate, cells pull/push along the fibers or squeeze through the pore structure of the network (3). The tensional balance between the cell and the ECM critically regulates many physiological and pathological processes, including immune response, tissue formation, and tumor progression (4-7). In the breast tumor, stiffening of the mechanical environment disrupts force balance between epithelial cells and the ECM, promoting a malignant phenotype (5, 8). Tumors stiffen as cells deposit more collagen than they digest (9-11), increasingly express cross-linking enzymes (8, 12), and exert traction forces to reorganize the ECM (13).The main structural component of the ECM is a network of cross-linked protein fibers. The fiber network aligns, stiffens, and sometimes, undergoes permanent changes when subjected to strain (14, 15). These adaptive mechanical properties of the fiber network provide cells entry points to modify their local microenvironment (16-18) and as such, perform physiologically realistic functions (1,3,(19)(20)(21)(22)(23). It has been reported that the nonlinear elasticity of fibrous matrices enables cells to transmit forces over distances of hundreds of micrometers, facilitating long-range communication between individual cells (24-26) and between tumor spheroids (27). Recent work has shown that individual cells are capable of stiffening the...

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In vitro, animal cells are mostly cultured on a gel substrate. It was recently shown that substrate stiffness affects cellular behaviors in a significant way, including adhesion, differentiation, and migration. Therefore, an accurate method is needed to characterize the modulus of the substrate. In situ microscopic measurements of the gel substrate modulus are based on Hertz contact mechanics, where Young's modulus is derived from the indentation force and displacement measurements. In Hertz theory, the substrate is modeled as a linear elastic half-space with an infinite depth, whereas in practice, the thickness of the substrate, h, can be comparable to the contact radius and other relevant dimensions such as the radius of the indenter or steel ball, R. As a result, measurements based on Hertz theory overestimate the Young's modulus. In this work, we discuss the limitations of Hertz theory and then modify it, taking into consideration the nonlinearity of the material and large deformation using a finite-element method. We present our results in a simple correction factor, ψ, the ratio of the corrected Young's modulus and the Hertz modulus in the parameter regime of δ/h ≤ min (0.6, R/h) and 0.3 ≤R/h ≤ 12.7. The ψ factor depends on two dimensionless parameters, R/h and δ/h (where δ is the indentation depth), both of which are easily accessible to experiments. This correction factor agrees with experimental observations obtained with the use of polyacrylamide gel and a microsphere indentation method in the parameter range of 0.1 ≤δ/h ≤ 0.4 and 0.3 ≤R/h ≤ 6.2. The effect of adhesion on the use of Hertz theory for small indentation depth is also discussed.

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