3.1 Stress, strain, elasticity, and viscoelasticity
Before embarking on a detailed discussion of how cellular mechanical properties can be measured, it is first necessary to define some terms. Mechanical stress is the force applied per unit area to an object (e.g., a cell), and strain is that object’s deformation normalized by its initial size. Thus, mechanical stress is expressed in units of force/area (e.g., N/m
2 or Pascals (Pa)), and strain is a dimensionless quantity. The Young’s Modulus (also known as the elastic modulus or elasticity (
E)), a measure of the deformability of the material, is stress divided by strain; the higher the Young’s Modulus, the stiffer the material. Because strain is a dimensionless quantity, the Young’s Modulus has the same units as stress, e.g., Pa. The Young’s Modulus offers a way to quantify mechanical differences between tissues, and indeed the measured bulk elasticities of human tissues span some five orders of magnitude, e.g., fat (17 Pa), mammary gland (160 Pa), brain (260–490 Pa), liver (640 Pa), kidney (2.5 kPa), skeletal muscle (50 kPa), cartilage (950 kPa) [
54]. Strictly speaking, elasticity describes the mechanical properties associated with the ability of a material to internally store mechanical energy and is therefore independent of the rate of deformation. However, many biological materials, including living cells, are capable of both storing and dissipating applied mechanical energy through internal frictional interactions, and do so in a way that depends strongly on the rate of deformation. For this reason, when measuring the mechanical properties of these materials, it is critical to capture both the elastic, or “storage” properties and the viscous, or “loss” properties. Such materials are referred to as viscoelastic materials, and the aggregate viscous and elastic response of a material to mechanical deformation is collectively referred to as its
rheology [
55].
3.2 Measuring cellular rheology in two-dimensional cell culture
Over the past decade, a sophisticated suite of technologies has been developed with the primary goal of quantifying the viscoelastic properties of cultured cells [
8,
56]. These include methods for measuring mean rheological properties of whole cells, such as optical stretching [
57‐
60], micropipette aspiration [
61‐
65], traction force microscopy (TFM) [
66‐
69], atomic force microscopy (AFM) [
70‐
73], and magnetic twisting microrheometry [
10,
74‐
77]; and microscale mechanics of portions of cells, such as subcellular laser ablation (SLA) [
78‐
82], micropost array detectors [
83‐
88], and particle tracking microrheometry [
89‐
93]. Some of these methods can be applied to both the subcellular and whole-cell scale; for example, AFM may be used both at low resolution to obtain mean indentational modulus of a population cells and at high resolution to spatially map mechanical properties across the surface of a single cell. All of these methods have been reviewed extensively elsewhere; to offer examples of how these techniques can be applied to cellular rheology in the context of tumor biology, we focus here on AFM and SLA.
1.
Atomic force microscopy (AFM)
In atomic force microscopy (AFM), one measures the interaction force between a sample surface, such as a living cell, and a microscale probe (“tip”) attached to a spring-like cantilever (Fig.
3A-C). The encounter between the tip and sample creates a force that deflects the cantilever, which in turn can be optically tracked and converted to an interaction force if the spring constant of the cantilever is known. Because contrast in AFM originates entirely from the interaction force between the tip and sample, it typically requires no fixation or staining and may readily be conducted in cell culture media. Thus, the method is perfectly suited to capture dynamic processes in living systems. One may acquire two types of information from the tip-sample interaction with the AFM: topographical images and force measurements. In the former measurement, the surface of a sample is scanned at constant force, and the compensatory motions of the stage needed to maintain force constant as the sample topography changes can be used to reconstruct an image. In the latter approach, the sample is vertically indented by the tip at a fixed position, and the resistance of the sample to that deformation may be analyzed to extract the material’s viscoelastic properties. AFM has been employed to image superficial cytoskeletal structures in living cells that may not be readily optically imaged, including cortical actin bundles [
94,
95]. Similarly, the force measurement capability of AFM has been used quite successfully to quantitatively measure properties relevant to cellular mechanics at length scales ranging from single molecules to whole cells. In the area of single molecule mechanics, AFM has been used to measure both the force-dependent unfolding of ECM proteins [
96] and cell-ECM adhesion proteins [
97] in an effort to understand how these systems accomplish mechanochemical conversions. AFM has also demonstrated tremendous value for quantifying the indentational rheology of living cells, including cellular elasticity [
98], spatial maps of elasticity across the cell surface [
99], and transduction of local compressive forces into biochemical signals [
100].
One of the more innovative recent applications of AFM to cellular mechanics is the measurement of protrusive forces generated by growing actin networks, such as those found in invadopodia and pseudopodia. For example, Fletcher and coworkers recently nucleated a dendritic actin network from an AFM cantilever and allowed the network to polymerize against a solid support and deflect the cantilever [
101]. With this system, they measured network protrusive forces under various applied loads, analogous to a pseudopodium squeezing its way through an endothelial barrier. Surprisingly, these studies show that the growth velocity depends on the loading history of the network and not merely the instantaneous load. These data therefore suggest that these cytoskeletal networks likely remodel to adapt to applied loads (e.g., by recruiting additional actin filaments), and that these remodeling events are progressively recorded in the evolving structure of the network. These investigators later used a similar approach to measure the oscillatory viscoelastic properties of these growing networks and were able to observe predictable and reversible stress-softening phenomena [
102]. These results are particularly exciting in light of the parallel efforts of Radmacher and colleagues to measure forces associated with cell migration in living cells [
103]. By orienting the AFM cantilever perpendicularly to a glass coverslip containing a culture of migrating keratocytes, these authors could directly measure cellular propulsive forces as individual cells encountered the cantilever during migration and attempted to push the cantilever by extending a lamellipodium against it.
AFM has also recently been employed as a diagnostic tool for measuring stiffness differences in leukemia cells, and for tracking changes in stiffness in response to chemotherapy [
104‐
106]. In these studies, myeloblastic cell lines were found to be more than an order of magnitude stiffer than corresponding lymphoblastic cell lines. Taken together with the clinical observation that acute myeologenous leukemia produces leukostasis much more frequently than acute lymphocytic leukiemia, these observations serve as a conceptual basis for a model in which low cell deformability likely contributes directly to cellular occlusion of blood vessels. This model has been further supported by the observation that when these cells are treated with chemotherapeutic agents and undergo apoptosis, they stiffen further, consistent with the clinical observation that leukostatic episodes often correlate with the induction of chemotherapy.
2.
Subcellular laser ablation
Although AFM has yielded much insight into cellular rheological properties relevant to tumor cell invasion and metastasis, it suffers from two important limitations. First, it can only probe the exterior surface of a living cell, thereby offering limited access to the mechanical properties of internal structures. Second, AFM measurements represent the collective contribution of many cytoskeletal filaments and motor proteins and do not permit dissection of the contribution of individual structural elements in localized microscale regions within the cell. As described earlier, the elucidation of specific cytoskeletal structures in specific places and times in the cell (e.g., stress fibers, filopodial actin bundles) are likely to be critical as the cell journeys towards invasion and metastasis.
Subcellular laser ablation (SLA) has emerged as a complementary method that is capable of overcoming both limitations (Fig.
3D-E). First applied towards cell biology by Michael Berns and coworkers [
78,
107‐
111], SLA uses a tightly focused laser beam to irradiate and vaporize nano- to microscale structures in living cells. Upon irradiation, material at the laser focus undergoes nonlinear multiphoton absorption, leading to optical breakdown and material destruction. Importantly, if the pulse energy, pulse width, and repetition rate are chosen correctly, structures in living cells may be selectively incised with sub-micrometer precision without compromising the plasma membrane or killing the cell. For example, it was recently demonstrated that delivery of femtosecond laser pulses at kilohertz repetition rates and at pulse energies ranging from 1.4 nJ—2.3 nJ can produce zones of photodamage as small as ~150 nm [
79].
In the context of understanding biophysical signaling between capillary endothelial cells and the ECM in tumor angiogenesis [
112], SLA has been employed to probe the micromechanical properties of actomyosin stress fiber bundles (stress fibers), which are the contractile structures that anchor and enable endothelial cells to exert tractional forces against the ECM [
80]. These tractional forces play central roles in endothelial and epithelial cell shape, polarity, and motility both
in vitro [
113‐
116] and
in vivo [
117,
118]. The actin cytoskeletons of living endothelial cells were visualized using yellow fluorescent protein (YFP)-tagged actin, and selected stress fibers at the cell base were irradiated and severed with femtosecond laser pulses. These studies show that severed stress fibers retract in parallel with the axis of the fiber, providing
prima facie evidence that these structures bear tensile loads; and that the quantitative retraction kinetics are consistent with that of a viscoelastic cable. Perhaps the most surprising result to emerge from this study is that the coupling between one fiber and the cytoskeletal architecture and shape of the rest of the cell depend strongly on the stiffness of the ECM onto which cells are cultured. For cells cultured on rigid substrates with an elasticity on the order of 1 MPa—1 GPa (e.g., glass), severing a single stress fiber, or even multiple parallel fibers, does not appreciably alter cell shape. Conversely, severing a stress fiber in cells cultured on relatively soft (~4 kPa) polyacrylamide-based substrates produces a 4–5% elongation of the cell along the axis of the stress fiber, as well as a thinning and extension of cytoskeletal structures tens of microns from the site of incision. Parallel studies with TFM revealed that a single stress fiber contributes to ECM strain across nearly the entire cell-ECM interface and strains the ECM most strongly near the points at which the cytoskeletal element inserts into the focal adhesion. Thus, these studies illustrate how SLA can be used to show direct connections between individual micron-scale cellular contractile structures and tractional forces exerted by cells that are distributed over hundreds of square microns.
3.3 Measuring cellular mechanics in three dimensions and in vivo
The application of AFM and SLA to the measurement of cellular mechanics has largely been limited to cells in two-dimensional culture formats. Recently, however, both of these methods have been extended to more physiologically relevant systems. For example, AFM has been used to measure the regional elasticity of cultured brain slices [
119] and excised mammary tissue (VMW, unpublished observations). And as described earlier, laser ablation has been used to disrupt mechanical interactions between groups of cells in the developing three-dimensional embryo [
13]. Recently, in an effort to understand biophysical mechanisms regulating cadherin-mediated cell-cell adhesion in living epithelia, Cavey and colleagues successfully used SLA to sever junctional actin networks in
Drosophila embryonic epithelia in the presence of actin-severing agents and Rho kinase inhibitors, and in the context of siRNA-mediated knockdown of α-catenin [
120]. Similar efforts have been used to extend other cellular mechanics methods to living, three-dimensional organisms, including particle-tracking microrheology [
121].
Additional new methods are emerging that enable real-time tracking of cell-directed ECM dynamics during various stages of tumorigenesis. In many cases, this has involved creative extensions of two-dimensional mechanics approaches to three-dimensional cultures. For example, three-dimensional particle tracking microrheology has recently been used to quantify both cellular mechanics [
122] and matrix remodeling during migration of cells within hydrogels [
123]. Similarly, modified versions of TFM have been used to track ECM stresses and strains in three dimensions [
124]. These methods have also been correlated with molecular-scale events during cell migration, such as the formation and disassembly of focal adhesions [
125] and generation of contractile forces [
126]. An important challenge for the future will be to develop mechanical methodologies that are as quantitatively sophisticated as current two-dimensional approaches but that also allow access to more complex and physiologically relevant ECM environments.