We present a simple and \emph{non-invasive} experimental procedure to measure the linear viscoelastic properties of cells by passive video particle tracking microrheology. In order to do this, a generalised Langevin equation is adopted to relate the time-dependent thermal fluctuations of a bead, chemically bound to the cell's \emph{exterior}, to the frequency-dependent viscoelastic moduli of the cell. It is shown that these moduli are related to the cell's cytoskeletal structure, which in this work is changed by varying the solution osmolarity from iso- to hypo-osmotic conditions. At high frequencies, the viscoelastic moduli frequency dependence changes from $\propto \omega^{3/4}$ found in iso-osmotic solutions to $\propto \omega^{1/2}$ in hypo--osmotic solutions; the first situation is typical of bending modes in isotropic \textit{in vitro} reconstituted F--actin networks, and the second could indicate that the restructured cytoskeleton behaves as a gel with "\textit{dangling branches}". The insights gained from this form of rheological analysis could prove to be a valuable addition to studies that address cellular physiology and pathology.
As many authors have noted, the mechanical properties of a cell's cytoskeleton can influence factors such as growth, apoptosis, motility, signal transduction and gene expression [1]. Related to this, there is a desire to be able to provide a rheological interpretation of the cell's viscoelastic response that has the potential to yield quantitative information on the cell's cytoskeletal structure. Consequently, in this work, we have developed a means of using passive video particle tracking microrheology measurements to quantitatively measure changes in the viscoelastic properties of a cell as a consequence (in this case) of simple changes in its external environment, i.e. subjecting a cell to a hypo-osmotic shock.
Osmotic regulation and the transport of osmotically active molecules are fundamental to both metabolic processes and homeostasis of cells and require precise regulation and maintenance of intracellular water [2]. The ability of many cells to regulate their volume, via internal restructuring, in response to osmotic changes in their environment is an essential component of normal cellular function. This regulatory volume change is linked to a reorganisation of the cytoskeletal actin networks [3,4]. Exposure to a hypotonic solution typically induces an initial rapid swelling followed by a shrinkage of the cell that occurs over a slower timescale of several minutes. The modulation of the actin dynamics and polymerisation that accompanies these regulatory volume changes has been shown in various cell types by methods such as DNase I inhibition assay and fluorescence measurements of phalloidin-labelled actin [3].
In this work we have developed a video particle-tracking tool to study the microrheology of cells, using a Jurkat lymphocyte cell line as a model system. In general, lymphocytes have been shown to have a response to hypo-osmotic solutions that is influenced by tyrosine kinase activity [5] and cytoskeleton (F-actin) participation in ion channel activation [6]. Furthermore, the interest in T-cells stems from their important role in the regulation of immune responses. In particular, Jurkat cells, as used in this study, are CD4+ T-lymphoma cells that are often utilised as models to understand T-cell signalling [7] and HIV-1 dissemination in viral pathogenesis [8].
The linear viscoelastic properties of a material can be represented by the frequency-dependent dynamic complex modulus G * (ω), which provides information on both the viscous and the elastic nature of the material (i.e. on how the matter dissipates and stores the energy transferred to it) at different frequencies ω; it is defined as the ratio between the Fourier transforms (denoted by the symbol " ˆ") of the applied stress σ(t) (which is proportional to the applied force) and the resulting strain γ(t) (which is proportional to the material/cell deformation):
where i is the imaginary unit (i.e. i 2 = -1).
The standard method of measuring G * (ω) is based on the imposition of an oscillatory stress σ(ω, t)
given by a formula such as:
where σ 0 is the amplitude of the stress function, ω is the frequency and t the time. The resulting oscillatory strain γ(ω, t) then has the formula:
where γ 0 is the strain amplitude and δ(ω) is the frequency-dependent phase shift between the stress and the strain. The amplitudes of the complex modulus in-phase and out-of-phase components are proportional to the ratio between amplitudes of the stress and of the strain, with constants of proportionality defining the storage (elastic) G (ω) and the loss (viscous) G (ω) moduli, respectively:
For example, in the case of a purely elastic solid, the stress and the strain are in phase (i.e. Hooke’s law, the material deformation is directly proportional to the applied force) and δ(ω) = 0; whereas, for a purely viscous fluid, such as water or glycerol, δ(ω) = π/2. For complex solids (e.g. gels, cells) or viscoelastic fluids (e.g. blood, saliva) δ(ω) would take any value between these limits (i.e. 0 δ(ω) π/2) depending on the frequency at which the force (stress) is applied.
The aim of this article is to present a straightforward and non-invasive procedure for measuring the in vivo linear viscoelastic properties of single cells via passive video particle tracking microrheology of single beads attached to the cells’ exterior. This method has advantages over both complicated active microrheology techniques, where complex experimental set-ups are necessary to exert an external force for performing stress-controlled measurements; and invasive passive video particle tracking of submicronprobes embedded (either via endocytosis or micropipette injection) within the cell’s cytoskeleton (e.g. [9,10,11,12,13,14,15]. In particular, the procedure consists of measuring the thermal fluctuations of a bead chemically bound to the cell’s exterior (Figure 1), for a sufficiently long time. A generalised Langevin equation was adopted to relate the time-dependent bead trajectory, r(
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