**2.6. Particle-tracking microrheology**

In particle-tracking microrheology, fluorescent microbeads are injected into live cells and diffused randomly in their cytoplasm. These beads are so small (< 1μm) that their inertial forces are negligible and they move according to Brownian motion. The movement of the fluorescent beads can be observed by fluorescence microscopy, and route distance can be converted to bead displacement which is used to calculate frequency-dependent viscoelastic moduli and/or the creep compliance of the cytoplasm [14]. For particle-tracking microrheology of living cells, the applied deformation and resultant stress is not oscillatory and is used to probe the mechanical properties of adherent cells on planar substrates, showing strong elastic responses over short timescales but with dominant viscous responses over longer time periods [15].

Particle-tracking microrheology has been used to study the viscoelastic responses of live cells and their cytoplasm under pharmacological treatment, serum starvation and at the edge of tissue wounds, as well as the mechanical responses of their nuclei [35-37]. For these studies, target cells can be deeply embedded in a 3D matrix, a condition more similar to cells in their physiological environment and difficult to probe by other methods.

#### **2.7. Atomic force microscopy**

The advent of atomic force microscopy (AFM) provided a valuable tool to image cell surface structure at sub-nm resolution and to probe the global and local nano-mechanical properties of cells. Such a non-invasive method makes it possible to investigate live cells under physiological conditions. The key component of AFM is a sharp tip mounted on a cantilever (usually silicon or silicon nitride), which is raster-scanned over the sample surface by piezoelectric micropositioners (Figure 2). Lateral or vertical displacement of the cantilever is detected by a position sensitive photodiode, which signals the fast feedback loop to maintain a constant relationship (*e.g.* force or distance) between tip and sample and the computer which is used to generate an image of the sample surface. AFM can be operated in many different modes, including force spectroscopy (FS) which is used to probe the mechanical properties of the cell surface layer or whole cell [38].

**2.6. Particle-tracking microrheology** 

over longer time periods [15].

**2.7. Atomic force microscopy** 

rather than the whole cell.

viscoelasticity.

the cell, and thus are limited to probing the viscoelastic response of a microenvironment

Twisting magnetometry [32] and the more recently developed magnetic twisting cytometry [33] can also be used to measure the movement of magnetic beads, which usually consist of colloidal metal or polycrystalline iron oxide. The cell is deformed under a twisting magnetic field that is applied perpendicularly to the initial magnetic field once it has been turned off [34]. The change in the magnetic field direction causes reorientation of the magnetic bead towards the twisting field, and once both are turned off, the rate of magnetic bead rotation and the amount of recoil are measured to interpret local

In particle-tracking microrheology, fluorescent microbeads are injected into live cells and diffused randomly in their cytoplasm. These beads are so small (< 1μm) that their inertial forces are negligible and they move according to Brownian motion. The movement of the fluorescent beads can be observed by fluorescence microscopy, and route distance can be converted to bead displacement which is used to calculate frequency-dependent viscoelastic moduli and/or the creep compliance of the cytoplasm [14]. For particle-tracking microrheology of living cells, the applied deformation and resultant stress is not oscillatory and is used to probe the mechanical properties of adherent cells on planar substrates, showing strong elastic responses over short timescales but with dominant viscous responses

Particle-tracking microrheology has been used to study the viscoelastic responses of live cells and their cytoplasm under pharmacological treatment, serum starvation and at the edge of tissue wounds, as well as the mechanical responses of their nuclei [35-37]. For these studies, target cells can be deeply embedded in a 3D matrix, a condition more similar to cells

The advent of atomic force microscopy (AFM) provided a valuable tool to image cell surface structure at sub-nm resolution and to probe the global and local nano-mechanical properties of cells. Such a non-invasive method makes it possible to investigate live cells under physiological conditions. The key component of AFM is a sharp tip mounted on a cantilever (usually silicon or silicon nitride), which is raster-scanned over the sample surface by piezoelectric micropositioners (Figure 2). Lateral or vertical displacement of the cantilever is detected by a position sensitive photodiode, which signals the fast feedback loop to maintain a constant relationship (*e.g.* force or distance) between tip and sample and the computer which is used to generate an image of the sample surface. AFM can be operated in many different modes, including force spectroscopy (FS) which is

used to probe the mechanical properties of the cell surface layer or whole cell [38].

in their physiological environment and difficult to probe by other methods.

**Figure 2.** A schematic representation of the atomic force microscope. Printed with permission (Springer, USA).

AFM offers the further advantage of being able to correlate sample topography with mechanical properties across the sample surface using indentation forces as small as 10 pN. With force spectroscopy (single point) or force mapping (multiple points), the tip approaches the sample, indents the sample and then retracts at each point, generating a force versus distance curve at a specific point on the cell surface (Figure 3). Cantilever deflection as a function of distance of the tip from the cell surface is initially represented by photodiode voltage as function of piezo displacement. This voltage is then converted to cantilever deflection and finally a force or indentation distance. The extent to which the sample is deformed depends on its viscoelastic properties.

Cantilever deflection can be converted to force using Hook's law:

$$\mathbf{F} = \mathbf{k} \times \mathbf{d} \tag{1}$$

where k is the cantilever spring constant, and d is cantilever deflection.

Force spectroscopy and mapping are used to quantify the mechanical behavior of the cell with the help of theoretical models. The indentation of the biological sample can be determined by subtracting the difference between cantilever deflection on hard surfaces and

on soft biological surfaces. Based on the Hertz model, Sneddon [40] developed a theory describing the relationship between loading force and indentation. Most commercially available AFM tips are either conical or parabolic, and hence these two types of AFM tips are considered during modeling. The relationship between loading force and indentation are given by following equations [41]:

$$F\_{parabolic} = \frac{{}^{4E\sqrt{R}}}{{}^{3(1-\nu^2)}}\delta^{3/2} \tag{2}$$

$$\mathbf{F}\_{\rm con} = \frac{\mathbf{z} \mathbf{E} \tan \alpha}{\pi (\mathbf{1} - \mathbf{v}^{\mathbf{z}})} \boldsymbol{\delta}^2 \tag{3}$$

Viscoelasticity in Biological Systems: A Special Focus on Microbes 133

���� (5)

����� (6)

� � (8)

<sup>2</sup> A=<sup>π</sup> p p 2R<sup>δ</sup> -δ (7)

cylindrical bacterial cells [42]. Models developed by Zhao et al. [20] can be used directly to calculate the Young's modulus of fungal hyphal walls. Fungal cell wall elasticity depends not only on the spring constant, but also hyphal radius (R), and cell wall thickness (h):

Since the slope of the approach portion of the force curve provides information on sample stiffness, the spring constant determined from the equation 4 can also be used to determine

� ��� �

� � �������

the elastic modulus of round-shaped fungal spores using the following equation [43]:

from the following equation [44]:

k� � �����

where E is the elastic modulus of the spore and A is the contact area between the AFM tip and sample. The contact area between an AFM tip and spore sample can be determined

where δ� is the indentation below the circle of contact calculated the from following equation:

δ� � ������

where δ� is the maximum indentation and δ� is the residual depth of indentation. These

In the above section we have outlined how to quantify the elastic behavior of microbial cells, and most of the available literature describes bacteria elastic properties with Young's modulus. However, biological samples are not purely elastic but viscoelastic. Therefore, the

values are determined experimentally from the force versus distance curve.

**Figure 4.** A schematic representation of standard solid model used to determine viscoelastic

k1 is the instantaneous elastic response, k2 is the delayed elastic response as a function of creep and η is

parameters. Adapted from [19] with permission.

the viscocity.

microbial cell can be modeled as a combination of both properties.

where R is radius of curvature for a parabolic AFM tip, α is the half opening angle of conical tip, δ is the indentation of the cell as a result of loading force F, 'E' is the Young's modulus of the sample, which describes the magnitude of elasticity and � is Poisson ratio, which is assumed to be 0.5 for soft biological materials. The Young's modulus of microbial cells is determined from the non-linear portion of the force indentation curve with equations 2 or 3 [41].

**Figure 3.** A representative force-distance curve taken on the surface of an *Aspergillus nidulans* cell wall. Solid and dashed lines represent approach and retract cycles respectively. Point b indicates jump into contact of the AFM tip to the sample. Section b-c represents the force required to indent the sample a given distance, and is used to measure cantilever deflection and to calculate sample indentation [39].

The spring constant of a fungal cell wall can be determined using the following equation:

$$\mathbf{k}\_{\mathbf{w}} = \frac{\mathbf{K}\_{\mathbf{c}} \mathbf{m}}{\mathbf{1} - \mathbf{m}} \tag{4}$$

where k� is the spring constant of the hyphal cell wall, also called relative rigidity, K� is the spring constant of the cantilever and m is the slope of the approach curve, corrected for that of a hard surface. This equation can also be used to determine the spring constant of cylindrical bacterial cells [42]. Models developed by Zhao et al. [20] can be used directly to calculate the Young's modulus of fungal hyphal walls. Fungal cell wall elasticity depends not only on the spring constant, but also hyphal radius (R), and cell wall thickness (h):

132 Viscoelasticity – From Theory to Biological Applications

are given by following equations [41]:

on soft biological surfaces. Based on the Hertz model, Sneddon [40] developed a theory describing the relationship between loading force and indentation. Most commercially available AFM tips are either conical or parabolic, and hence these two types of AFM tips are considered during modeling. The relationship between loading force and indentation

���������� � ��√�

F��� � �� ���� �������

the non-linear portion of the force indentation curve with equations 2 or 3 [41].

where R is radius of curvature for a parabolic AFM tip, α is the half opening angle of conical tip, δ is the indentation of the cell as a result of loading force F, 'E' is the Young's modulus of the sample, which describes the magnitude of elasticity and � is Poisson ratio, which is assumed to be 0.5 for soft biological materials. The Young's modulus of microbial cells is determined from

**Figure 3.** A representative force-distance curve taken on the surface of an *Aspergillus nidulans* cell wall. Solid and dashed lines represent approach and retract cycles respectively. Point b indicates jump into contact of the AFM tip to the sample. Section b-c represents the force required to indent the sample a given distance, and is used to measure cantilever deflection and to calculate sample indentation [39].

The spring constant of a fungal cell wall can be determined using the following equation:

k� � ���

where k� is the spring constant of the hyphal cell wall, also called relative rigidity, K� is the spring constant of the cantilever and m is the slope of the approach curve, corrected for that of a hard surface. This equation can also be used to determine the spring constant of

�������

���� (2)

δ� (3)

��� (4)

$$E = 0.8 \, (\frac{\text{k}\_{\text{w}}}{\text{h}}) (\frac{\text{R}}{\text{h}})^{1.5} \tag{5}$$

Since the slope of the approach portion of the force curve provides information on sample stiffness, the spring constant determined from the equation 4 can also be used to determine the elastic modulus of round-shaped fungal spores using the following equation [43]:

$$\mathbf{k}\_{\rm w} = \mathrm{ZE}(\frac{\mathbf{A}}{\pi})^{1/2} \tag{6}$$

where E is the elastic modulus of the spore and A is the contact area between the AFM tip and sample. The contact area between an AFM tip and spore sample can be determined from the following equation [44]:

$$\mathbf{A} \triangleq \pi \left( \text{2R} \boldsymbol{\eth}\_{\text{p}} \text{-} \boldsymbol{\eth}\_{\text{p}} \text{ }^{2} \right) \tag{7}$$

where δ� is the indentation below the circle of contact calculated the from following equation:

$$
\delta\_\mathrm{P} = \left(\frac{\delta\_\mathrm{t} - \delta\_\mathrm{r}}{2}\right) \tag{8}
$$

where δ� is the maximum indentation and δ� is the residual depth of indentation. These values are determined experimentally from the force versus distance curve.

In the above section we have outlined how to quantify the elastic behavior of microbial cells, and most of the available literature describes bacteria elastic properties with Young's modulus. However, biological samples are not purely elastic but viscoelastic. Therefore, the microbial cell can be modeled as a combination of both properties.

**Figure 4.** A schematic representation of standard solid model used to determine viscoelastic parameters. Adapted from [19] with permission.

k1 is the instantaneous elastic response, k2 is the delayed elastic response as a function of creep and η is the viscocity.

Vadillo-Rodrigue et al. (2009) [19] explained the viscoelastic properties of bacterial cell walls using a standard solid model which describes both an instantaneous and a delayed elastic deformation. Based on this model they have derived the following equation that describes the experimentally obtained creep response data:

$$\mathbf{Z(t)} = \frac{\mathbf{F\_0}}{\mathbf{K\_1}} + \frac{\mathbf{F\_0}}{\mathbf{K\_2}} \left[ 1 - \exp\left(-\mathbf{t} \frac{\mathbf{K\_2}}{\eta\_2}\right) \right] \tag{9}$$

Viscoelasticity in Biological Systems: A Special Focus on Microbes 135

AFM has been employed to measure the elasticity of a wide variety of cells ranging from bacteria, fungi, cancer cells, stem cells, osteoblasts, fibroblasts, leukocytes, cardiocytes developing embryos, cells at different cell cycle stages, and those treated with drugs. A broad spectrum of new measurements is possible by exploiting and manipulating the interaction between tip and sample in a quantitative way. Elasticity is most often measured with conical AFM tips. Spherical tips give rise to elasticity measurements 2-3 times that of conical tips, likely based on the large contact surface area. In comparison with other methods, AFM is more advantageous based on its ability to image the sample surface at high resolution while measuring an indentation map of the sample. The combination of imaging and force spectroscopy provides information about how cell surface structure affects elasticity and viscoelasticity. However, measurements depend on tip shape, which cannot be determined during sample scanning. Despite some limitations, AFM applications are rapidly developing. New instrumental designs and modification of the associated theoretical models will ensure an effective way to measure the elasticity and viscoelasticity for a wide variety of biological

Although several methods have been developed to quantify cellular responses to deformation during locomotion, adhesion and mitosis, reliable tools are not available to quantify the distribution of mechanical forces between the various sub-cellular components [26]. Biological cells range in size between 1-100 μm and are comprised of constituents that provide mechanical strength, such as the cell envelope composed of multiple complex and distinct structures, cell walls composed primarily of polysaccharides interspersed with proteins, the cell membrane composed of phospholipid bilayers and membrane proteins, complex cell organelles of different sizes and shapes made of a variety of macromolecules, the cytoskeleton composed of microtubule networks, actin and intermediate filaments, other proteins and macromolecules such as DNA and RNA. The structure and function of each of these constituents may vary depending upon cell type. For instance, fungi are encased in cell walls, whereas bacteria have more elaborate cell envelopes with a peptidoglycan (polysaccharides cross-linked with peptides) layer and one or more cell membranes. Human cells, generally by virtue of being part of more elaborate structures, have only a cell membrane. It is not well understood how cells and their associated components sense mechanical forces or deformation, and convert such signals into

The small size of prokaryotes, in comparison with larger eukaryotic cells, was a considerable obstacle in the development of methods for directly measuring their mechanical properties [47], solved largely by FS methods now routinely used. Cellular mechanical strength mainly relies on the outermost layers, such as the cell wall, envelope, or membrane, in addition to internal structural components such as the cytoskeleton. Extracellular components, such as those used to help form elaborate community structures (*e.g*. biofilms) also contribute to viscoelasticity and mechanical strength. There has been a major focus on the viscoelastic properties imbued to the cell by its cytoskeleton, which has been highly conserved

**3. Viscoelastic cellular components and super structures** 

samples.

biological responses [46].

where, ሺሻ is the position of the z piezoelectric transducer as a function of time t, K1 is the spring constant that represents initial deformation, K2 is the spring constant after creep response, and η2 is viscosity.

The contribution of elastic and viscous components can be determined from the force-time curve taken at the center of cells when applying a constant force, F0, for at least a 10 second period. Cantilever deflection is determined and using equation 1 is converted to force and then to an indentation-time curve, which is also called creep response. The indention of the cell over time at a constant force can be theoretically determined from equation 4 and fitted to the indentation-time curve shown in Figure 5. The experimentally determined data fit very well with the theoretical data obtained from the model. Microbial cells in particular exhibit two types of responses when a force is exerted on their surface. The first is the instantaneous linear relationship of the force versus distance curve, attributed to whole cell turgor pressure, while the non-linear region is thought to correspond to the response of the cell envelope.

**Figure 5.** A typical creep deformation of an *Escherichia coli* cell at a constant force as a function of time. Adapted from [45] with permission.

AFM has been employed to measure the elasticity of a wide variety of cells ranging from bacteria, fungi, cancer cells, stem cells, osteoblasts, fibroblasts, leukocytes, cardiocytes developing embryos, cells at different cell cycle stages, and those treated with drugs. A broad spectrum of new measurements is possible by exploiting and manipulating the interaction between tip and sample in a quantitative way. Elasticity is most often measured with conical AFM tips. Spherical tips give rise to elasticity measurements 2-3 times that of conical tips, likely based on the large contact surface area. In comparison with other methods, AFM is more advantageous based on its ability to image the sample surface at high resolution while measuring an indentation map of the sample. The combination of imaging and force spectroscopy provides information about how cell surface structure affects elasticity and viscoelasticity. However, measurements depend on tip shape, which cannot be determined during sample scanning. Despite some limitations, AFM applications are rapidly developing. New instrumental designs and modification of the associated theoretical models will ensure an effective way to measure the elasticity and viscoelasticity for a wide variety of biological samples.
