Analytical Models for Measuring the Mechanical Properties of Yeast
Abstract
:1. Introduction
2. Mathematical Models for Measuring Mechanical Properties by Micromanipulation
3. Mathematical Models for Measuring Mechanical Properties by the AFM Method
4. Mathematical Models for Measuring Mechanical Properties by the SICM Method
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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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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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
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 StyleSavin, 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