# Analytical Models for Measuring the Mechanical Properties of Yeast

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Models for Measuring Mechanical Properties by Micromanipulation

_{1}being the index meridional direction and ε

_{2}the longitudinal direction), E is the modulus of elasticity, h

_{0}is the initial wall thickness, and v is Poisson’s ratio.

_{i})

_{i})

_{i}represents the functions of the principal tensions, and δ = λ

_{2}sinψ; and ω = dδ/dψ.

_{1}and λ

_{2}are principal stretches in the meridian and circumferential directions. The last modeling step is to use the strain energy functions from Equations (1)–(3) in Equations (4)–(7) to obtain the stresses.

_{0}and the final cell wall thickness h is considered negligible. The principal Cauchy stresses in the wall are expressed as follows:

_{1}and λ

_{2}are the principal stretch ratios, equal to the ratio of the length of the membrane section before and after deformation; indices 1 and 2 correspond to the meridional and circumferential directions, respectively; T

_{1}and T

_{2}are the tensions in the wall corresponding to λ

_{1}and λ

_{2}, respectively; E is the Young’s modulus of the cell wall; v is Poisson’s ratio; and h

_{0}is the initial thickness of the cell wall.

_{s}and k

_{φ}are differential equations with respect to s and φ, P is the internal turgor pressure of the cell, and σ

_{ss}(s,t) and σ

_{φφ}(s,t) are stresses along s and φ in the cell wall.

_{s}= δu/δs as well as to ὲ

_{φ}= (1/r)(dr/dt), where r is the local radius of the cell wall, and is related to stresses in the cell wall by Equations (20) and (21) [48]:

## 3. Mathematical Models for Measuring Mechanical Properties by the AFM Method

_{tip}, v

_{tip}, E

_{sample}, and v

_{sample}are the Young’s moduli and Poisson’s ratios for the tip and sample materials, respectively. In the case where the tip material is much harder than the sample material, the following equation is used [64]:

_{w}depends on the mechanical properties and dimensions of the cell wall but does not depend on the internal pressure of the cell. By transforming the correlation, they calculated the elastic modulus of the cell wall:

_{ind}/r = 0.01) in the case of a single-layer cell wall, Mercade’-Prieto et al. confirmed [38] that the Reissner equation (Equation (28)) is applicable only to thin shells (h/r < 0.02); in the case of thicker shells, in which the probe is pressed into the shell, the Hertz–Sneddon equations are valid [78]. The values of the point loading F and wall stiffness Eh (where h is the wall thickness) in this case are calculated according to Equations (28) and (29):

_{wd}is the cell wall elasticity constant.

_{out}with a Young’s modulus E

_{out}and is attached to an inner layer (core) of the cell wall which has a thickness h

_{in}with a Young’s modulus E

_{in}. For the micromanipulation method, in the case of compression of a two-layer core-shell sphere to large deformations, the Young’s modulus is calculated by Equation (30). The results of the FEM are consistent with the micromanipulation data, and the Young’s modulus E

_{in}is about 0.4–0.8 GPa.

_{in}/E

_{out}. When E

_{in}/E

_{out}= 1, the system behaves as a single-layer wall; when E

_{in}/E

_{out}= ∞, however, the system is analogous to the cell wall being pressed against a very rigid substrate [79]. In this regard, it is incorrect to use the Hertz–Sneddon analysis when E

_{in}/E

_{out}is low, while its use is optimal when E

_{in}/E

_{out}is high. FEM results obtained by Mercade’-Prieto et al. using the double layer model were determined using the pseudo-Hertz Equation (31) with pseudo-Hertz Young’s modulus (E

_{pH}) and Hencky strain ε = 0.004, in which the vertex displacement (d) is equal to the indentation depth (d

_{ind}) [38].

_{out}). At a higher value of E

_{in}/E

_{out}, the inner layer behaves as a rigid substrate, in which case d ≈ d

_{ind}and E

_{pH}~E

_{out}. It is worth considering that when a rigid inner layer is present the outer layer is highly deformed even with low indentation. Thus, the model of a two-layer cell wall suggests the possibility of estimating the elasticity modulus by the AFM method (with a sharp probe) of only the outer layer and using the micromanipulation method to estimate the total stiffness of the wall. It should be noted that the determination of the Young’s modulus is affected by the contribution of the inner layer and by the rigidity of the substrate.

^{2}) + (y

^{2}))

^{1/2}, R is the radius of the probe, h is the height of the sample, δ is the indentation, F

_{0}is the applied force, and E

_{cell}is the Young’s modulus of the cell.

^{2}/R) and as a conical tip for large deformations:

^{1/2}/π and n = 2

^{3/2}/π for the top of the pyramid; and m = 1/2 and n = 1 for the cone.

_{m}being the duration of the rendezvous phase and t

_{ind}the duration of the full indentation cycle); t

_{1}is an auxiliary function defined by Equation (36); ξ is a dummy time variable required for integration; E(t) is Young’s modulus of relaxation; v is Poisson’s ratio; and R is the radius of the indenter.

_{0}is the instantaneous modulus, E

_{∞}is the long-term modulus, and τ is the relaxation time.

_{r}is the characteristic relaxation time.

_{ad}are represented by the following equations:

^{2}) is the sample elasticity constant, F is the normal loading force, and γ is Dupre’s work of adhesion [65].

## 4. Mathematical Models for Measuring Mechanical Properties by the SICM Method

_{0}applied to the upper end of the pipette and calculating the resulting deformation of an elastic sample as a function of z, the authors obtained IZ curves and their s between 98% and 99% of the current for various E/p

_{0}ratios. The empirically subject relationship between s and the Young’s modulus of the sample is described by the following equation:

_{∞}is the s for an infinitely rigid sample and A is a constant depending on the geometry of the pipette.

_{0}through a nanopipette with the pressure drop ΔP of the liquid in the capillary and the environment:

_{i}is the radius of the inner hole of the nanopipette tip, θ is the semi-cone angle of the inner wall of the nanopipette tip, and h is the viscosity of the liquid.

_{r}/dz tangential to the surface; it is maximum at r = r

_{i}and is equal to 12 nN/mm

^{2}[88].

_{0}is the ion current far from the sample surface, and ΔI is the drop in the ion current when approaching the sample.

_{0}is the cell height, and h is the height of the indented cell area.

_{c}is the height of the cell cortex, h

_{s}is the height of the soft area of the cell cytoskeleton, E

_{c}is the Young’s modulus of the cell cortex, and E

_{s}is the Young’s modulus of the cell cytoskeleton.

_{σ}is the surface tension force of the decan–salt layer.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**A**) Illustration of a sea urchin model, showing a yeast cell when compressed between two large rigid plates. (

**B**) Illustration of a model of a hollow sphere filled with gas; cell geometry: ψ

_{0}is the angular position of a point on the cell wall from the vertical axis of symmetry before compression and ψ determines the angle of a point on the edge of the contact area between the compression surface and the cell after compression. (

**C**) Illustration of the shell model showing the geometry of the system and the image of the increase in the viscosity of the cell wall. (

**D**) Illustration of the core–shell model; normalized compression force with total wall stiffness F/r(Eh)

_{tota}

_{l}under parallel compression for a two-layer model with an inner wall h

_{in}/r = 0.01 and an outer wall h

_{out}/r = 0.04 with different ratios of Young’s moduli.

**Figure 2.**(

**A**) Schematic representation of the elements of the AFM setup. The spherical tip interacts with the surface of the sample, which leads to the deflection of the microcantilever, which is recorded by the photodetector using a laser beam reflected from the microcantilever. In a typical measurement of mechanical curves, the base of the micro console is approached or retracted at a constant vertical speed and the force is recorded. (

**B**) An example of the resulting F-Z curves, in which the arrow indicates the point of contact of the probe with the sample. (

**C**) An example of converting an F-Z curve to a force–indentation curve, from which Young’s modulus is obtained.

**Figure 3.**(

**A**) Scheme of the Hertz model, where E is Young’s modulus, v is Poisson’s ratio, and X is indentation. Indexes 1 and 2 refer to two bodies, respectively. (

**B**) Scheme of the DMT model, where R is the radius of the elastic sphere, a is the contact radius, and α is the shift from the center of the elastic sphere. (

**C**) Diagram of a cylindrical shell model, where the AFM cantilever tip applies a normal force F, indenting (δ) a hypha with internal pressure P, radius R, and thickness h. (

**D**) Illustration of the elastic shell model. A spherical shell with thickness h and undeformed radius R experiences internal pressure P and is loaded with a vertical point force F at the pole. This causes a vertical deflection w(r) and, in particular, a displacement w(0) = −w

_{0}at the point of application of the force. (

**E**) Illustration of the Non-Hertz Model. (

**F**) Schematic of the FEM for sharp tip (1) or spherical tip (2). (

**G**) Viscoelastic models can be used to relate the stress and strain by using elastic (springs, denoted E) and viscous (dashpots, denoted η) parts. (

**H**) Scheme of the Johnson–Kendall–Roberts model, where a is the contact radius and F

_{ad}is the maximum adhesive force.

**Figure 4.**(

**A**) Schematic representation of obtaining topography via the SICM method, where the probe stops when the ion current drops by 0.5% while being approximately at the nanocapillary radius from the sample surface. (

**B**) Schematic representation of the measurement of sample stiffness by indenting it due to the internal colloidal pressure of the nanocapillary; the ion current drop is 2%.

Source | Model | Function | An Object | Description |
---|---|---|---|---|

J.D. Stenson et al. [31,35] | Sea urchin egg model | Infinitely small deformation in Equations (10)–(12), final deformation in Equations (13)–(15), Hankey’s deformation in Equations (16)–(18) | yeast cell wall | In the model, cells are thin-walled, liquid-filled spheres; the desired characteristics depend on the Poisson’s ratio and the thickness of the cell wall. It is possible to neglect cell wall permeability at high strain rates. Fixing the initial stretch factor leads to an inaccurate estimate of the elastic modulus. |

Feng and Yang [36] | Model of compression of hollow spheres filled with gas | Equations (4)–(7) constitutive equations for contact and non-contact regions | cell wall of tomato cells | The cell wall in this model is divided into areas in contact and areas not in contact with compressive forces. |

Banavar et al. [37] | Shell theory | Local normal balance of forces of the cell wall in Equation (19) Stresses in the cell wall according to Equations (20) and (21) | growing cell wall dynamics | The growing cell wall behaves like an inhomogeneous viscous liquid with a spatially changing viscosity that increases with distance from the growth apex |

Mercade’-Prieto et al. [38] | Core-shell model | wall stiffness F/r(Eh) _{out} | cell wall | The model gives an estimate of the overall stiffness of the cell wall (Figure 1D). |

Source | Model | Function | An Object | Description |
---|---|---|---|---|

H. Hertz [52] | Hertz Model | Cantilever Force Equation (22), effective Young’s modulus Equation (23) (When the material of the tip is significantly harder than the material of the sample, Equation (24)) | homogeneous smooth bodies | The model is used under the assumptions that the indenter shape is parabolic, and the sample thickness is much greater than the indentation depth. The model does not allow the probe to stick to the sample. |

B. Derjaguin [53] | DTM model | cell wall | The model is applicable in the presence of long-range surface forces outside the area of contact between the probe and the sample and is valid in the event of weak adhesion between the nanoindenter and the outer surface of the sample. Its use is a priority for objects with low cohesion and a small radius of curvature. | |

Zhao et al. [46] | Cylindrical shell model | The modulus of elasticity of the cell wall in Equation (28) | cell wall | In the technique, F and δ are linearly dependent on each other, while the cell wall elasticity constant kw depends on the mechanical properties and dimensions of the cell wall but does not depend on the internal pressure of the cell. |

Vella et al. [54] | Elastic shell model | internal pressure in yeast cells | Young’s modulus is an order of magnitude higher than the values obtained using the Hertz model. | |

Mercade’-Prieto et al. [38] | Single layer sphere | The values of F and Eh are calculated from Equations (28) and (29) | cell wall | Corrected values of the Young’s modulus are higher than using Hertz–Sneddon analysis but lower than using micromanipulation compression. |

Mercade’-Prieto et al. [38] | Double layer model | Force profile at small deformations in Equation (31). | cell wall | The model of a two-layer cell wall suggests the possibility of estimating the elastic modulus by AFM only for the outer layer. |

E. A-Hassan, S.P. Timoshenko [55,56] | Theory of elastic shells | Young’s modulus is estimated from the ratio between the effective Young’s modulus, shell thickness and bending modulus | cells | Cells in the model are represented as shells filled with liquid. |

P. Garcia & R. Garcia [57] | Non-Hertz model | In the case of a paraboloid probe, the force is expressed by Equation (33). | mammalian cells attached to a solid support | The cell’s Young’s modulus depends on the solid substrate, and the bottom effect artifact is determined by the ratio between the contact radius and cell thickness. The model is applicable when the indentation is less than or equal to the tip radius. |

R. Vargas-Pinto et al. [58] | Hertz Model and Contact Model | The force, in the case of a spherical tip, is expressed by Equation (29) In the case of a sharp tip, the model is used Rico et al. [59] and Briscoe et al. [60], where the force is expressed in Equation (34) | mammalian cells with cortex | Combining the models resolved the issue of inaccuracy in determining the rigidity of the cage. Sharp probes examine the cortical layer, and spherical probes record the rigidity of the cortical layer together with the cytoskeleton. In the model presented, the elastic component and the active stress component are combined into an effective elastic response for ease of calculation. |

Y. Efremov et al. [61] | Elastic-Viscoelastic Compliance | Ting’s solution for indentation of a viscoelastic sample with a rigid spherical tip Equations (35) and (36). | living cells and hydrogels | It reflects the approach-retraction hysteresis well but requires an appropriate choice of the viscoelastic function. |

Y. Efremov et al. [61] P. Cai et al. [62] | Standard Linear Solid-State Rheology and Power Rheology | Relaxation time Equations (35) and (36), Kohlrausch–Williams–Watts function Equation (37). | living cells | The standard linear rigid body model is a combination of a spring and damper, in which the spring is parallel to the Maxwell element. |

Y. Efremov et al. [63] | Johnson-Kendall-Roberts model | The indentation depth, contact radius, and maximum adhesive force are presented in Equations (38)–(40), respectively) | living cells and hydrogels | The model fits the retraction part well with force-distance curves. |

Source | Model | Function | An Object | Description |
---|---|---|---|---|

D. Sanchez et al. [88], Rheinlaender, J., & Schäffer [89] | Hydrodynamic model | The force exerted on a flat surface in Equation (43). Young’s modulus of the sample in Equation (41). | cell membrane | To obtain the mechanical properties of the cell, hydrostatic pressure is applied through a nanopipette, which can lead to a mechanical response of the cell. |

R. Clarke et al. [90] | Internal colloidal pressure model | The modulus of elasticity of the cell wall in Equation (28) | cells with glycocalyx and cells without glycocalyx | Indentation is performed by means of internal colloidal pressure between the cell surface and the surface of the nanopipette tip, which significantly reduces the invasiveness of the method. |

Kolmogorov et al. [61], Savin N. et al. [8] | Hertz Model | The internal force is presented in Equation (51). | Mammalian cells [61], yeast cells [8] | The technique is based on the deformation of a double electric layer of decan-saline solution with a nanopipette. The displacement from the tip surface to the cell surface is minimized. However, there is no method for obtaining viscoelastic properties in all presented SICM models. |

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**MDPI and ACS Style**

Savin, N.; Erofeev, A.; Gorelkin, P.
Analytical Models for Measuring the Mechanical Properties of Yeast. *Cells* **2023**, *12*, 1946.
https://doi.org/10.3390/cells12151946

**AMA Style**

Savin N, Erofeev A, Gorelkin P.
Analytical Models for Measuring the Mechanical Properties of Yeast. *Cells*. 2023; 12(15):1946.
https://doi.org/10.3390/cells12151946

**Chicago/Turabian Style**

Savin, Nikita, Alexander Erofeev, and Petr Gorelkin.
2023. "Analytical Models for Measuring the Mechanical Properties of Yeast" *Cells* 12, no. 15: 1946.
https://doi.org/10.3390/cells12151946