**2. Methods of measuring cellular viscoelasticity**

124 Viscoelasticity – From Theory to Biological Applications

mechanical role in maintaining cellular shape and resisting turgor pressure, yet at the same time flexible enough to allow cell growth and division. Multi-cellular eukaryotes are arranged into specialized structures of varied composition (*e.g.* tissues) which protect them from harsh environments. Single-celled prokaryotes or eukaryotes can also form sophisticated structures, such as biofilms or mycelia, but even as single cells they are able to bear mechanical stress and maintain integrity which is quite astonishing. Cells range from soft mammalian tissue cells to those with very firm plant cellulosic fibers, so understanding how different cellular components provide diverse mechanical properties is of great interest. Further, how do mechanical properties support locomotion? Active movements depend on cell type and ultimate function within tissues or the greater environment, and their associated mechanical properties change according to environmental demands. Another question relates to how external factors, such as temperature, pH, aridity and xenobiotics affect cell mechanical properties. Further, what is the role of cell mechanics in processes such as active and passive diffusion, adhesion, community formation and evolution of complex tissues in higher eukaryotes, and how do the mechanics of isolated components

work in concert within dynamic live systems that are far from equilibrium?

new perspective of cellular mechanics is emerging [2].

**1.1. Viscoelasticity in biology** 

Several of these questions remain unanswered, however early success studying the rheological properties of non-living materials has provided a strong foundation for studying biological systems. Over the past decade, several obstacles have been overcome and thus a

If cellular metabolism could be frozen at any given moment, disrupting the production of metabolites and supra-molecular structures, the cell would simply become fluid-like with the associated mechanical properties. Living cells behave both as an elastic solid and as a viscous fluid, and so are considered viscoelastic. Such materials, including biological molecules and cells, cannot be fit using classical models of either elasticity or viscosity. Cellular viscoelasticity arises from the combination of high water content conflated with a polymerized structural matrix. On the one hand, the biopolymers which support cell shape provide strong enough mechanical properties to resist environmental pressures, but on the

other hand their organization is highly dynamic and linked to metabolic conditions.

Cellular mechanical properties can be characterized using viscosity, elasticity and creep compliance. Herein we focus on viscoelasticity studies measuring either biological sample deformation induced by an external force or the force resulting from sample deformation. The cell cytosol, which contains the majority of cell fluid, can be treated as the coexistence of liquid and solid phases. The latter is composed of proteins, DNA, RNA and cytoskeleton filaments as well as organelles suspended in the viscous buffered saline, saturated with metabolites and proteins. The liquid component has a high compression modulus, meaning that without the structural components and macromolecules it would be very fluid (less viscous). In contrast, the solid phase has a lower compression modulus, exhibiting more elastic properties. In cells as a whole, the viscosity of the cytoplasm dominates the transport In microrheological studies, several experimental techniques and theoretical models are combined to accurately quantify cellular mechanics. Several methods that are routinely used for cell rheology are briefly reviewed in the following section. Micropipette aspiration, microplate rheometry, and optical stretching are used to probe whole cell mechanics, while probe- or particle-based techniques such as magnetic probes, optical tweezers and particle tracking cell rheology only measure local mechanical properties. The probes or particles within a structure are subjected to an external force and the subsequent response measured, usually by tracking their displacement, to characterize associated mechanical properties. Atomic force microscopy uses a probe, but attached to the end of a cantilever with welldefined rigidity able to the mechanics of whole cells and their surface layers. A schematic diagram of various devices is shown in Figure 1.

The mechanical properties measured by a probe or particle technique are highly dependent on the strength of, and relationship between the particle/probe and the structure being studied. This relationship is highly complex, varies with cell type and can lead to significant misjudgement of the stiffness. Probes can be modified with different molecules such as antibodies, peptides and cadherins to target antigens, integrins and cytoskeletal components. However these measurements do not account for interactions between the probes and cells which can lead to confounding results [5]. Cells have a heterogeneous

composition and their various compartments have diverse mechanical properties. If only one cell position is probed, as is common, the mechanical properties of the whole cell will not be well represented. Comprehensive methods have been developed for measuring the mechanical properties of mammalian cells and precise methods have not yet been adequately developed for prokaryotes. A summary of the advantages and disadvantages are listed in the Table 1.

Viscoelasticity in Biological Systems: A Special Focus on Microbes 127

**Figure 1.** Schematics for the common methods used to measure cell rheology.

Micropipette aspiration is a widely used method and offers a versatile way of measuring the mechanical properties of living cells. A cell is aspired onto the end of a micropipette by a negative pressure gradient, and the aspiration length of the cell inside the micropipette is recorded as a function of time [21]. Cells experience large non-linear deformations in response to aspiration suction pressure (0.1 pN/μm2), giving rise to forces (10 - 104 pN) that allow tracking of edges with high accuracy (± 25 nm). This method can measure the elastic and viscous properties of very soft materials like red and white blood cells, and stiffer cells such as endothelial (skin) cells and chondrocytes (cartilage) [7]. Measurements are interpreted using basic continuum models to solve for elasticity and viscosity parameters.

In most cases, cells suspended in buffer or saline solution that are aspirated appear round, but shape and degree of deformation depends on cell type. Many mammalian cells are naturally spherical (*e.g.* white blood cells) or spontaneously adopt this shape when detached from a surface. Recall that cells can behave as liquids and/or solids [7], the response of which is similar until a hemispherical projection is formed in the pipette. Beyond that point, a further increase in the suction pressure causes constant cortical tension, creating a liquidlike cell able to flow completely into the pipette [22]. On the other hand, the surface of a solid cell will extend into the pipette to a new equilibrium position. Under constant pressure, the mechanical properties of the cell determine how far it is pulled inside the

**2.1. Micropipette aspiration** 


**Table 1.** Comparison summary of different methods to measure cell rheology.

Viscoelasticity in Biological Systems: A Special Focus on Microbes 127

**Figure 1.** Schematics for the common methods used to measure cell rheology.

## **2.1. Micropipette aspiration**

126 Viscoelasticity – From Theory to Biological Applications

listed in the Table 1.

**Micropipette aspiration** 

**Microplate rheometer** 

**Optical stretcher** 

**Magnetic probes** 

**Particle tracking microrheology** 

**Optical tweezers** 

**Atomic force microscopy** 

composition and their various compartments have diverse mechanical properties. If only one cell position is probed, as is common, the mechanical properties of the whole cell will not be well represented. Comprehensive methods have been developed for measuring the mechanical properties of mammalian cells and precise methods have not yet been adequately developed for prokaryotes. A summary of the advantages and disadvantages are

Methods Applications Drawbacks References

models

profiles

cell

Can only probe

Local heating and prototoxicity can result Can only be used at low forces in the linear regime

measurements

Quantitative measurements rely heavily on theoretical

Pipette geometry can limit

Causes heating of cells Extensive modeling is required to obtain force

microenvironment inside the

Used only for soft tissue cells [14-16]

Slow scan rate [19, 20]

No subcellular resolution [8,9]

[6,7]

[10,11]

[12,13]

[17,18]

Measurements of non-linear deformations with high

Both soft and rigid cells can

Several manipulations in the same instrument are possible A large range of forces are measurable (1nN - 1μN) Control of cellular pre-stress

No physical contact required

Large range of frequencies

Parallel measurements of large number of cells possible High timescale resolution

Quantitative measurements of shear modulus possible

High timescale resolution Force and position can be controlled more accurately

High spatial resolution Measures large range of

More precise and easy to control cantilever position

**Table 1.** Comparison summary of different methods to measure cell rheology.

forces

Can be used under physiological conditions

and non-destructive Less time consuming with

simple setup

[0.01–1,000 Hz]

accuracy

be used

Micropipette aspiration is a widely used method and offers a versatile way of measuring the mechanical properties of living cells. A cell is aspired onto the end of a micropipette by a negative pressure gradient, and the aspiration length of the cell inside the micropipette is recorded as a function of time [21]. Cells experience large non-linear deformations in response to aspiration suction pressure (0.1 pN/μm2), giving rise to forces (10 - 104 pN) that allow tracking of edges with high accuracy (± 25 nm). This method can measure the elastic and viscous properties of very soft materials like red and white blood cells, and stiffer cells such as endothelial (skin) cells and chondrocytes (cartilage) [7]. Measurements are interpreted using basic continuum models to solve for elasticity and viscosity parameters.

In most cases, cells suspended in buffer or saline solution that are aspirated appear round, but shape and degree of deformation depends on cell type. Many mammalian cells are naturally spherical (*e.g.* white blood cells) or spontaneously adopt this shape when detached from a surface. Recall that cells can behave as liquids and/or solids [7], the response of which is similar until a hemispherical projection is formed in the pipette. Beyond that point, a further increase in the suction pressure causes constant cortical tension, creating a liquidlike cell able to flow completely into the pipette [22]. On the other hand, the surface of a solid cell will extend into the pipette to a new equilibrium position. Under constant pressure, the mechanical properties of the cell determine how far it is pulled inside the micropipette. The measurement of suction pressure, location of the cell and movement of its edge in the micropipette can be used to calculate the viscous and elastic parameters [7].

Viscoelasticity in Biological Systems: A Special Focus on Microbes 129

structure, proliferation, motility, or viability [25]. Another potential limitation is the complexity of the force profiles, which require extensive theoretical modeling. Nonetheless, the optical stretcher has proven to be an effective way of measuring the contribution of cytoskeletal filaments to cell viscoelasticity since single suspended cells are probed as well-

Optical tweezers can be used to capture tiny dielectric particles with a highly focused laser beam [17]. The movement of dielectric particles can be controlled with two main optical forces. The scattering force acts along the direction of beam propagation, while the electric field induces a dipole in the dielectric particles, producing an electric field gradient that pulls particles towards the focus. When the gradient force dominates, the dielectric particles can be confined in a stable three-dimensional optical trap. The trap is then moved to manipulate the bead, so the applied force and resultant particle displacement are interpreted in terms of mechanical response [26]. To obtain viscoelastic information, an oscillatory force is applied to the dielectric bead by oscillating the laser position with a movable external mirror. The resultant amplitude of the bead motion and the phase shift are interpreted in terms of viscoelastic response. The experimental data from optical tweezers can only be used to study the viscoelastic responses at low force in the linear regime [18]. Local heating and phototoxicity ("opticution" as coined by Block) can result from the intense laser powers required to trap

Another way to explore the viscoelasticity of cells involves manipulation of their movement through an externally applied magnetic field. The earliest application of this technique was pioneered by Freundlich and other researchers [12,27], in which magnetic particles were embedded in the material of interest (*e.g.* cells). The viscoelasticity can then be probed using magnetic tweezers or magnetic twisting. The ability to functionalize magnetic colloidal

The magnetic tweezer technique involves the manipulation of a super paramagnetic bead with an applied magnetic field generated by four pairs of soft ferromagnetic cores, each wound with a separate field coil arranged at special angles [30]. The movement of the bead is monitored by its induced magnetic dipole as it interacts with the field gradient of the strong magnet to which it is exposed. The corresponding displacement of the magnetic bead is used to measure cell properties. The movement of magnetic beads can also be controlled, albeit only in one direction, by a strong magnetic field gradient arising from electromagnets generated by axis-symmetrically arranged magnetic coils on a sharp iron tip [31]. Multiple pairs of electromagnetic tips are required for more complex movements of the magnetic bead, which for these experiments are smaller than the size of

beads allows for their specific localization within the cell [28, 29].

defined viscoelastic objects [25].

**2.4. Optical tweezers** 

biological samples.

**2.5. Magnetic probes** 

There are several draw-backs associated with this method since quantitative measurement of cell stiffness relies heavily on theoretical models [6]. The most recent application of micropipette aspiration is to capture and hold cells for manipulation [7].

### **2.2. Microplate rheometry**

Over the past decade there has been much progress in the study of rheological behaviour for single cells, including the development of several models which explain the mechanical properties of cytoskeletal networks. A microplate rheometer, invented by Thoumine and Ott in 1997 [8], consists of two parallel microplates that support cell adhesion and spreading. The plates can be made flexible and in some cases are coated with an adhesive material. A whole cell is stretched or compressed between the two plates while the flexible microplate is used as a nN-scale force sensor. Integration with an inverted microscope allows cells that are being manipulated to be directly observed. Several adaptations to the instrumentation allow measurement of creep function, area of expansion modulus, contraction forces of single cells, adhesive interactions and stress-strain responses. The latter can be used to measure viscoelasticity.

The force applied to the cell is directly proportional to the relative stiffness of the microplate, hence measurement of cell deformation offers valuable information on cell mechanical properties. Measureable forces range between 1 nN – 1 μN, and several manipulation modes are possible, including compression, traction, aspiration and adhesive rupture. Following their invention, microplate rheometers have been improved by a number of scientists. The behaviour of cells in response to compression, traction, aspiration or adhesion, for which elastic, viscous and contractile regimes can be distinguished based on time scale, can be used to calculate the viscoelastic modulus of living cells. The instrument has been modified for probing single layers of cells at once using the novel cell monolayer rheology (CMR) technique, making possible harmonic oscillation experiments and step shear or step stress experiments which reveal different viscoelastic regimes [9].
