Introduction
Indentation Experiments
The curvature of the cell wall can significantly affect the measured indentation forces due to friction [17]. To avoid this complicating factor, many choose to probe a relatively flat part of the cell or tissue [19].
Modeling Indentations
Discrete models idealize the elastic response of the cell as one or more spring components. 1.6), where 𝐸 is Young's modulus and 𝜈 is Poisson's ratio between the elastic half-space.
Outline of This Thesis
The relief of the turgor pressure enables isolation of the mechanical response of the cell wall [25, 70]. Therefore, in the subsequent inverse analysis, we focus exclusively on the turgor pressure and the elastic modulus of the cell envelope.
Structure and Biomechanics during Xylem Vessel Transdifferen-
Introduction
Plant cells have two key structural elements that collectively control their mechanical properties: the cell wall and the cytoskeleton. Therefore, quantifying the mechanical contributions of the cell wall(s) and cytoplasm during differentiation of xylem vessel elements is a complicated problem, and one that has not yet been resolved.
Results and Discussion
From our spring model, the overall stiffness of the cell in isotonic conditions is stage 0. Our analysis could not distinguish the stiffness of the PCW in water stage 1 (𝑘hypo*,PCW) from the stiffness of the PCW in growth medium at the same stage (𝑘iso, PCW) not.
Materials and Methods
The resulting indentation data were positionally corrected to account for system stiffness contributions. Microscope slides (AmScope, Irvine, CA, USA) were cleaned with isopropyl alcohol, surface activated with a high-frequency generator for 1 minute (BD-20A, Electro-Technic Products, Chicago, IL, USA), and a thin layer of 0.5 ml of poly- l -Lysine was coated by centrifugation on top of slides (SUSS MicroTec, Garching, Germany). Short-range nano-indentations to assess cell wall properties were performed with AFM (Asylum Research, MFP-3D-Bio, Goleta, CA, USA).
Most of the data processing for the microcompression tests follows that of Routier-Kierzkowska et al. The software was used to identify the contact point and extract a Young's modulus by applying the Hertz model.
Conclusions
In the first step, a constant temperature drop is applied over the entire thickness of the cell shell, Δ𝑇. The tension in the cell envelope causes the internal hydrostatic pressure of the fluid cavity, 𝑝, to increase. To represent the turgor pressure in the cytoplasm, the interior of the cell is filled with an incompressible fluid that is pressurized to 𝑝.
In the second step of the simulation, the AFM tip is lowered to insert the cell. The only difference between the MLE and the MAP is the inclusion of the prior distribution in the MAP [90, 91].
Cell Wall and Cytoskeletal Contributions in Single-Cell Biome-
Introduction
Cellulose, the carrier component of CW, plays a central role in the structure and mechanics of CW. The stiffness of CW allows the cell to survive in high osmolarity environments by sustaining huge turgor pressures (𝑝=0.3−1.0 MPa) in the vacuole without rupturing the cell [3]. It is clear that the internal structure of a cell changes as it prepares to divide, and this changing structure affects the apparent stiffness of the CW in mechanical measurements of growing cells.
To isolate the mechanical contribution of the cytoskeleton, drug treatments are used to depolymerize MTs and remove AFs. To validate the CW stiffness measurement from this model, we perform shallow nanoindentations on cells in the same osmotic conditions.
Results and Discussion
The stiffness contribution from the cytoplasm is further deconvoluted into the contributions from MTs, AFs and the rest of the cytoplasm (including the vacuole). An illustration of the spring model adapted to each test condition is shown in Figure 3.6. The stiffness analysis allows us to decouple the relative stiffness contributions of the MTs and AFs from the CW and the rest of the cytoplasm.
The stiffness of the MTs and AFs was similar to that of all other isolated components in PS. Cosgrove predicted that the high turgor pressure of plant cells would increase the stiffness of the CW by prestressing [79].
Materials & Methods
Contributions to the variance of the indentation force of each variable are presented in Table 4.1. Meshed parameter space with color contour representing the magnitude of the error in cubic coefficients objective function. The coefficient𝑑 is related to the change in curvature of the fitting polynomial through a series expansion.
Recently, there is increasing evidence that cytoskeletal filaments contribute to the mechanical response of the cell. Finally, we can write down the expression for the radial displacement of the sphere on the outer surface.
Inverse Analysis to Determine the Mechanical Properties of
Introduction
However, it is not clear whether the treatment reduced overall cell stiffness through weakening of the cell envelope, or through some other mechanism that affected the cells' ability to maintain turgor pressure. A mechanistic model that can reliably decouple turgor pressure and CW elasticity from their AFM power spectra will advance our understanding of the physical effect this antimicrobial peptide treatment has on the bacterial cells tested. Some have found evidence that for shallow indentations, AFM measurements only probe the mechanical response of the cell envelope or CW material.
These seemingly contradictory findings highlight the importance of developing a model of the whole cell as a fluid-filled structure with an internal hydrostatic pressure to investigate the intricate mechanics of the CW and turgor pressure. A new objective function, which accounts for two different facets of the force-indentation data, is then used to determine optimal CW modulus and turgor pressure.
Model
The displacement of the outer cell surface due to thermal shrinkage in this step of the simulation can be calculated analytically. The final result for the radial displacement (𝑢) of the outer surface of the fluid-filled thermoelastic sphere is The final result for the displacement of the outer surface of a fluid-filled thermoelastic cylinder is.
Due to the non-linear variation of the shell thickness 𝑡 with each simulation increment, this equation is an approximation that systematically overestimates the turgor pressure. Therefore, reducing uncertainty in aspect ratio, cell envelope thickness, or cell envelope compressibility would have modest or negligible impact on model accuracy.
Results and Discussion
A filled circle represents an analysis that converged within 0–10% of the true values provided in the simulated experiment. The second objective function we tested is a measure of the change in curvature of fitting polynomials to the force-indentation depth curves. The color contour in Figure 4.3B represents the size of. evaluated this objective function for a simulated experimental data set with the same known parameters as in Figure 4.3A. The final objective function we tested is a weighted average of the other two objective functions, with the error in third-order adjusted coefficients weighted the highest, 𝐽ave =0.75∗𝐽d,fout+0.25∗𝐽SOLS. 4.14).
Each point in the graphs of Figures 4.3C-F represents such a simulation that converged to within 25% of the true parameter values. Simulations that have not converged within 25% of the actual parameter values are shown by blank spaces.
Conclusions
We can solve the above constitutive equations for the two stress components in terms of the strains. By applying this boundary condition to the above equation, we can solve for one of the constants of integration. To apply the outer boundary condition, we first need to find𝜎𝑟 in terms of the displacement𝑢.
For example, a spring could be added sequentially in the cytoplasmic layer of a two-spring model to represent the nucleus of a cell. Therefore, the only difference between MLE and MAP optimizations is the inclusion of prior information.
Conclusions and Outlook
Future Work in Models of Walled Cells
To be able to decouple the contributions of each of the subcellular components in the model, the only experimental requirement is that the component can be removed from the cell in the micro- and nano-indentation experiments. Assuming that the core can be removed from the cell, and the cell subsequently tested in indentation experiments, the contribution of the core to the overall stiffness of the cell in compression can be determined. If the cell in stage 2 does have both a primary cell wall (PCW) and a secondary cell wall (SCW) acting as a combined cell wall, then the stiffness of the combined cell wall should be close to the addition of the stiffness measured from the isolated PCW and SCW.
Instead, the combined PCW and SCW stage 2 transdifferentiation is more than double the added isolated PCW and SCW stiffnesses. Combining the finite element modeling methodology presented in Chapter 4 with the experimental techniques presented in Chapters 2 and 3 would help to calculate the material properties of the subcellular components of the plant cell separated from the cellular structure.
Future Work in Parameter Fitting for Walled Cell Models
Modeling individual filaments can be very time intensive, but a surrogate feature is still possible to model. For example, introducing compressibility in the cytoplasm can be used to represent the bending behavior of cytoskeletal filaments in compression. Another aspect that may be of interest is the time-dependent behavior of walled cells, namely their viscoelastic or poroviscoelastic behavior.
In principle, such effects can be included in mechanical models including damping in the form of droplets.
Extensions to Biocomposites and Living Materials
Indentation methods have been successfully used to measure the mechanics of the constituent cells, matrix materials and the composite living material or biocomposite. In the future, a multi-scale model that uses information from indentation experiments at these vastly different length scales could be used to advance our knowledge of the mechanical behavior of living materials and biocomposites, and see widespread use in practice . Subsequently, the distribution of the measured indentation force is also normally distributed, 𝐹𝑖 |Δ𝑖 ∼ N (𝑎+𝑘Δ𝑖, 𝜎2). A.4) The likelihood function is the joint probability of observing our collected force-indentation depth data given regression parameters𝑎and𝑏and the variance𝜎2.
Atomic force microscopy stiffness tomography on living Arabidopsis thaliana cells reveals the mechanical properties of surface and deep cell wall layers during growth. Water uptake by growing cells: An assessment of the controlling roles of wall relaxation, solute uptake and hydraulic conductance.
