Bio-Microrheology: A Frontier in Microrheology

In BMR, nanometer- or micrometer-sized particles, synthetic and endogenous, as well as fluorescently tagged molecules are used as local probes for live-cell measurements. The motions of appropriately sized probes embedded within cells provide an evaluation of the local, nonbulk, viscoelastic properties of heterogeneous cellular regions. The use of local probes for measuring rheological properties has motivated several kinds of experiments and theoretical interpretations. Whereas in macroscopic rheology stress-strain relationships are measured through mechanical deformation of bulk materials, in BMR embedded probe motion is tracked and its relationship to the local microenvironment inferred. Hence, distinct analytical concepts are employed in BMR, with the trajectories or time-dependent displacements of probe particles transformed into measurements of regional deformation 1., 11.. A recent review provides an introduction to cellular sensing and response to forces and gives several examples of linked mechanical and biochemical effects in cells (23).

In a system at thermodynamic equilibrium where no external forces are applied, particles suspended in a liquid undergo translational and rotational diffusion due to forces exerted by molecules in the surrounding medium (24). In other words, particles diffusing freely through a liquid typically exhibit a “random walk” that is nondirectional and characterized by a Gaussian distribution in step size. This random walk is propelled by thermal fluctuations as the probe particles interact with cellular structures and the liquid surrounding them.

Thermal, or passive, microrheology for viscoelastic materials is based on an extension of the concepts of Brownian (25) motion of particles in simple liquids. The motion of particles within a viscous liquid can be quantified with the diffusion coefficient, D, which is a measure of how rapidly particles execute a thermally driven random walk. Given the particle size, temperature, and viscosity, η, the diffusion coefficient in a viscous liquid can be determined by the Stokes-Einstein relation: $D=k_{\text{B}}T/6\pi a\eta$24., 26., 27., 28.. This relation is valid for thermal fluctuation-induced particle motion where no energy consumption, such as ATP-driven motion or convective flows of liquids, is present (1). In addition, it is assumed in the equation that particles are spherical and rigid and no heterogeneities exist. Therefore, in microrheological experiments, particle dynamics are related to the medium and probe properties and provide quantitative information about the local microenvironment.

The dynamics of particle motion are usually described by the time-dependent mean-square displacement (MSD), $〈\Delta r^2(t)〉$. An equivalent statistical representation is the position autocorrelation function, but since the MSD is a more physically based property we adhere to it. When particles diffuse through viscoelastic media or are transported in a nondiffusive manner the $〈\Delta r^2(t)〉$ becomes nonlinear with time and can be described with a time-dependent power law, $〈\Delta r^2(t)〉\sim t^{\alpha }$. The slope of the log-log plot of the $〈\Delta r^2(t)〉$, α, which is also referred to as the diffusive exponent (29), describes the mode of motion a particle is undergoing and is defined for physical processes between 0 ≤ α ≤ 2.

The MSD can be used to obtain rheological properties of a complex fluid microenvironment (1). The generalized Stokes-Einstein relation (GSER) correlates the particle radius, a, and the $〈\Delta r^2(t)〉$to provide the creep compliance: $J(t)=\pi a〈\Delta r^2(t)〉/k_{\text{B}}T$, where k_B is Boltzmann’s constant, and T is the absolute temperature 30., 31.. The time-dependent creep compliance 30., 31., or material deformation under a step-increase in stress, can be directly obtained from the MSD. This provides a measure of the viscosity or the elastic modulus in viscous or elastic samples, respectively. In addition, a method was developed to estimate the elastic, storage modulus and the viscous, loss modulus algebraically based on the logarithmic slope of the MSD (30). This approach was later extended to provide more accurate estimates of the moduli, including rapidly changing MSDs (32). The material moduli provide equivalent rheological information to that obtained by the creep compliance and are typically used when comparisons to bulk macrorheology measurements are needed.

Convective motion revealed by an α > 1 at long lag times is attributed to cell motility. The particles with broader distribution of step sizes (particles 5 and 6 in Fig. 1) exhibited near-Brownian motion as indicated by the proximity of the diffusive exponent to 1 (Fig. 1 C). The MSD remained constant at ∼0.8 at all lag times, as the larger random steps masked the convective motion due to cell crawling. We divide the six particles in Fig. 1 into three categories: assuming that particles with similar diffusive exponents are undergoing comparable motion; spatially confined at short timescales with diffusive exponent 0.5 and 0.25 corresponding to particles 1,2 and 3,4, respectively; and nearly diffusive transport for particles 5 and 6. Each pair of particles is ensemble averaged, an operation typically performed on multiple trajectories, and the time-dependent creep compliance, J(t) given in Fig. 1 D, is found by using the GSER. The $〈\Delta r^2(t)〉$ data are only statistically reliable up to the last decade of time (dashed line in Fig. 1 C) as there are few long lag-time measurements to be averaged; hence, those data are not presented in Fig. 1 D. In addition, the compliance is only physical when its log slope ≤1, thus any instances where the slope is >1 are marked with dashed lines. Similar values have been obtained for microrheological measurements performed in vitro with reconstituted actin polymers (34). It is not currently possible to fully interpret the meaning of these data, as models explaining this complex behavior are lacking. This example also demonstrates that the concepts of microrheology cannot be directly applied to studies in cells.

Thermal fluctuations induce particle motion, but are not the only forces acting within cells. Molecules or particles within a cell may diffuse to accumulation sites 18., 35. or may be actively transported by molecular motors. Molecular motor transport processes, which expend ATP-stored energy, result in non-Brownian dynamics. This directed motion is typically saltatory (36), suggesting the simultaneous operation of several motors, and is a superposition of directed transport and Brownian motion (37). Molecular motors, in the presence of ATP, have been shown to transport organelles and particles at velocities as high as a few micrometers per second. A summary of forces and motion resulting from the three main molecular motors is given in Table 1. Molecular-motor driven probe displacement does not typically yield rheological data because particle motion is nonrandom, directional, and mediated by an ATP-driven force.

In addition to thermal fluctuation-driven and ATP-driven internal cell forces, probes may be externally manipulated using so-called “active” microrheology techniques. Externally applied forces acting on particles in cells result in local physical stress and movement of the particles through more elastic regions. Thermal fluctuation cannot excite soft materials out of equilibrium, necessitating active techniques to extend measurements to larger strains or deformations, often into the nonlinear range. However, active stress application may result in disruption of internal microstructures. Hence, BMR studies usually begin with passive experiments and proceed to active measurements if small deformations produced by thermal fluctuations are not enough to probe the material properties of stiff cellular regions. Similarly, experiments on stiff cells, or direct measurements of stiff structures such as the cytoskeleton, benefit from active force application techniques.

Forces applied to probes within cells and within ex vivo reconstituted bio-macromolecules must be highly localized so that they affect only the region of interest. For this purpose, magnetic or laser tweezers have mainly been used. Typical ranges of applicability for magnetic and laser tweezers are given in Table 1. A more detailed summary of the measurement applications for active methods is given elsewhere (38).

Active forces can be applied to particles on the surface and inside cells through magnetic and laser tweezers. Magnetic BMR has mainly been used for measurements of membrane elasticity 4., 39., 40.. Magnetic particles have been functionalized to bind to a particular receptor type or region on the cell surface 4., 39.. Studies with paramagnetic or ferromagnetic microspheres have shown that displacement fields, or affected areas after force application on the cell surface (4) or in the cytoplasm (41), decay rapidly with distance from the magnetic microsphere. Hence, microspheres in one region can be manipulated (42) without mechanically affecting adjacent regions, and measured responses are therefore local. Similarly, laser tweezers have been used to manipulate particles, cells, and bacteria 43., 44., 45. by applying small forces to them and then measuring their displacements with high precision and accuracy. Trapped particles can be restricted to a specific region and passively monitored (46), or an active force can be locally applied and its effects on internal structure measured. For example, the force required to bend actin filaments 22., 47. can be quantified. Laser tweezers have been used to trap spherical, polymeric particles 22., 47. or naturally occurring granules within cells (46). They have also been used to stably trap thin, coin-like, wax microdisks (48) and examine the nonlinear rheological properties of liquids (49).

Probes used to study cell interiors have been magnetic 41., 50., 51. or polymeric 35., 41., 47., 50., 52., 53., 54., 55., 56. particles, fluorescently tagged molecules 57., 58., and endogenous lipid granules and organelles 46., 59., 60., 61., 62., 63.. Synthetic particles are introduced into cells by cellular uptake, microinjection, or electroporation (see Table 2) depending on particle size, particle surface chemistry, cell size, and cell type. Cellular uptake mechanisms have been identified, but the process is not quantified or well understood at the molecular level. Particles introduced by natural cellular uptake processes, e.g., endocytosis and phagocytosis, may become trapped within membrane-bound vesicles. Particle-containing vesicles will be transported differently than “free” cytoplasmic particles within the cell interior, and their resulting environmental interactions will be unique. Thus, BMR of particles introduced by cellular uptake processes may provide a biased picture of internal cell structure. Also, exogenous particles can disrupt the local cell structure as they can bind to or push-and-pull internal structures, such as the cytoskeleton (64). Conversely, endogenous particles are naturally present and do not perturb the cell structure. However, since endogenous particle size and surface chemistry cannot be controlled their interactions also cannot be predicted, so that certain granules may, for example, unknowingly become transported by molecular motors. Thus, a compromise between control of particle and transport parameters and perturbation of the natural cell state is typically encountered in BMR studies, and a mixture of information about the rheological environment and about molecular motor activity is obtained.

The studies in Table 2 have focused on evaluations of probe transport in the cell cytoplasm. An interesting and important extension would be evaluations within the nucleus, particularly during mechanically intensive processes such as nuclear membrane assembly and disassembly, DNA replication, and nuclear division. A potential obstacle for nuclear BMR measurements is that introducing particles into the nucleus of small animal cells may be technically challenging. Fluorescently tagged molecules, such as proteins (65), transfected into the cell and targeted to the nucleus have typically been used to examine nuclear dynamics. However, tracking the precise motion of molecules directly is difficult because of their small size and resulting fast diffusion. In addition, several molecules may aggregate together, making accurate size determination impossible. The diffusion coefficients and MSD of molecules can be estimated by examining the concentration changes due to molecule motion through a stationary volume (66), as indicated by changes in fluorescent intensity; this approach is termed “fluorescence correlation spectroscopy” and may be a useful initial approach for nuclear BMR.

Probe displacement in cells can be tracked by video particle tracking, such as Nomarski optics (differential interference contrast, DIC) (33), fluorescence microscopy (67), laser particle tracking, or three-dimensional (3-D) confocal microscopy. In contrast, probe motion in biomaterials, where large sample volumes are available, is tracked by dynamic light scattering (1), diffusing-wave spectroscopy (11), interferometric microscopy (68), or x-ray photon correlation spectroscopy (69). The simplest and most common method in cells is video tracking by DIC or fluorescent microscopy. Each of these techniques has advantages and disadvantages, and selection of a method is usually based upon a compromise between spatial and temporal resolution and the desired number of simultaneously tracked particles. Digital video tracking is often sufficient to track several particles simultaneously with a reasonably high signal/noise ratio. Beads with a diameter larger than the cytoskeletal network mesh size push through surrounding filaments to move (47) and can provide an accurate measure of the viscoelasticity in the entire region (64). Particles smaller than the cell filament mesh size are sensitive to the viscosity of the solvent and hydrodynamic interactions within the network but do not reflect the bulk viscoelasticity of that region (64).

The internal microenvironments of cells are a complex, heterogeneous combination of flexible cytoskeletal macromolecules and viscous liquid containing small molecules and ions, molecular complexes, and organelles (70). Studies in Table 2 have shown that cell interiors are heterogeneous and can be viscous, elastic, or viscoelastic depending on the scale and on cell state. BMR can be used to mechanically “map” distinct regions in the cell under various conditions. In the following paragraphs, we briefly discuss two BMR studies, one within and one on the surface of cells, as examples of the utility of BMR approaches.

As discussed above, in microrheological studies the motion of probes embedded in the sample are tracked and used to infer the rheological properties of the sample. In addition to examining each particle separately, pairs of particles can be examined (71). In two-particle microrheology the correlated movements of pairs of neighboring particles are used to measure the relative viscoelastic response on the timescale of a single probe particle 72., 73.. This approach has been shown by Lau et al. (62) to be of particular interest for cell BMR studies, since they provide a more accurate, simplified view of the transport in the complex microenvironments of cells. In that study, the motion of endogenous lipid granules and mitochondria in cells were tracked using DIC microscopy. Probe motion was mostly random, but a few convective molecular motor induced trajectories were observed. Single particle trajectories suggested diffusive motion, whereas two-particle measurements showed convective motion. The reasoning for the apparent discrepancy between one- and two-particle trajectories is ascribed to the inability of single particle measurements to distinguish between convection of the network surrounding the probe, e.g., cell crawling on the surface as in Fig. 1, and active transport of a probe relative to a stationary network. The two-point MSD results unambiguously indicate that the cytoskeleton itself is actively fluctuating due to nonthermal forces, effectively causing the entire axis system to be moving. Hence, employing two-point microrheology in heterogeneous samples, such as cells, can aid in interpretation of complex trajectories and provide more accurate information about the motion of the probes and their origins.

A locally applied stress may link biochemical and mechanical cell properties by triggering or blocking specific signaling pathways 23., 74. and can lead to receptor or protein clustering (75) and cytoskeleton-based mechanical deformation. A highly deformed but intact cell will likely not return to its original shape, since energy has been expended on rearranging the cytoskeleton and other molecules. The extent of elastic energy storage and the viscous energy dissipation define the elastic and viscous components of the cell response, respectively. In a recent study, Bursac et al. (76) examined cytoskeletal remodeling after application of a mechanical load to cells in the physiologic range of stress (e.g., from <1 Pa to >100 Pa) (77). Mechanical stress is felt by cells when, for example, lymphocytes squeeze through the endothelium during extravasation. Magnetic beads are used as a tool to apply local stress by coating with an Arg-Gly-Asp (RGD) peptide sequence that facilitated binding to the surface of human airway smooth muscle cells. In doing so, the beads are linked indirectly to the actin cytoskeleton through integrin receptors (39). This local approach was validated, as a magnetic pulse applied to the beads did not generate long-distance effects and responses were regional (41). Shear forces were used to disrupt the cell’s cytoskeleton as local torque was applied through integrins. The cell compliance increased with the time of applied step stress, but its functional behavior was independent of the aging time. Thus, no distinct molecular relaxation time or time constant could characterize the process (77), implying that relaxation within a cell includes several simultaneous processes (6). Hence, mechanical aging and rejuvenation processes in live cells can be systematically studied by surface microrheology, providing an indication of the rate of aging and rejuvenation and the complexity of the involved processes.