(662d) Double Layer CELL Wall MODEL for Yeast CELLS

In the last decade there has been
extensive research on the mechanical properties of biological walls, using in
particular atomic force microscope (AFM) and compression testing by
micromanipulation. Small indentation experiments using AFM and sharp indenters
consistently give an elastic modulus (E) of the yeast cell (Saccharomyces
cerevisiae) wall of 0.2-1.6 MPa [1], whereas micromanipulation and other large
deformation techniques generate values of 100-200 MPa [2]. No explanation has
yet been reported to account for this difference of two orders of magnitude. It
will be shown here using finite element modeling (FEM) that the Hertzian
equation used to calculate E from AFM data is inappropriate for core-shell
spheres like yeast cells. Secondly, a double-layer cell wall model is presented
to explain the difference in E values obtained with the AFM and
micromanipulation  techniques.

Finite element modelling (FEM)
was performed using ABAQUS/Standard 6.5. Half core-shell spheres were simulated
in 2D under axis-symmetrical conditions, as performed previously for the
compression of microcapsules with a core-shell structure [3]. The AFM indenter
and the bottom substrate were modeled as rigid materials, while the cell wall
was modeled using 7100 deformable CAX4RH elements; the mesh size was more
refined close to the indentation area (see Fig. 1). For a double -layer wall, a
second shell was introduced under the previous external layer using 1500 CAX4RH
elements. The liquid core was modelled using FAX2 elements with a density of 1
kg L-1. The wall material was modeled as isotropic and linearly elastic; a
Hookean approach has been considered as it has been shown to be a good
mechanical model for the yeast wall up to very large deformations [1]. Different
inner pressure values common for yeast cells were also modelled.

Single-layer cell wall model

AFM compression studies commonly
calculate the mechanical properties of the cell wall using the Hertzian
equation (1), assuming a spherical shape of the indenter:

                                  (1)

where r_ind is the
indenter radius and d_ind is the indentation depth. However, this
equation cannot be used for core-shell spheres, a possible model for yeast
cells, because of the significant inward bending of the wall, as shown in Fig.
1, regardless of the inner pressure. The displacement of the indenter tip (d)
is much larger than d_ind. In practice, what has been calculated is a
pseudo Hertzian E assuming d_ind = d. Considering this, the E values
reported with AFM for yeast cell are underestimated by 5-10 times.

Figure 1: FEM results showing
compression with a sharp indenter, for a displacement twice the wall thickness
(d = 2h), yet the penetration indentation is d_ind ~0.18h.

Double-layer cell wall model

The common assumption of all
previous biomechanical studies is that the yeast cell wall is homogeneous,
although possibly having chitin-rich bud scars. However, there are two very
different layers in the yeast cell wall encapsulating the plasma membrane. The
outer layer is comprised of mannoproteins, reported to control the cell
porosity, while the inner layer is composed of b1,3-glucan
and chitin, which is believed to impart mechanical strength to the cell wall
[4].

A FEM model was constructed for a
double-layer wall with different thickness and elastic modulus values, assuming
a soft external layer and a stiff inner layer. Micromanipulation experiments
were conducted to compress single yeast cells until rupture, which usually
occurred at fractional deformations e of
0.6-0.7, defined by the ratio of displacement to initial cell diameter (2r)
[1]. Figure 2 shows that using (Eh)_total, defined as the sum of (Eh)_out
and (Eh)_in, a unique relationship for the normalized  compression
force versus deformation is obtained regardless of the actual elastic modulus
values of the two layers within the chosen ranges. Therefore, assuming a
double-layer model of the yeast cell wall, micromanipulation results provide
(Eh)_total, found to be in the range of 11-15 N m^-1.

Figure 2: Normalized compression
force with the total wall stiffness using parallel compression for a
double-layer model with an inner wall of h_in/r = 1% and an outer
wall of h_out/r = 4%.

Under a double-layer model, the
associated errors in using a Hertzian analysis, such as eq. (1), for the
determination of E are smaller than those in the single-layer wall model due to
the existence of the stiff inner layer that limits the inward bending of the
cell wall. Figure 3 shows that Hertz-like curve fitting can still be performed
at small deformations, as also observed for single layers, despite eq. (1) not
being fully valid. In addition, the estimated E is close to that of the outer
layer. Hence, AFM results using a sharp tip are reasonably correct only if a stiff
inner layer is assumed.

AFM and micromanipulation results
can be combined to determine the biomechanical properties of the two layers in
the yeast cell wall. The external later is characterized with sharp tip
indentation, E_out ~1 MPa. For a total wall thickness h of ~130 nm,
the thickness of the b1,3-glucan fiber
layer (h_in) is estimated to be around 10-40 nm. Finally, the elastic
modulus of the stiff inner layer was calculated to be E_in ~ 0.3-1.3
GPa depending on chosen value of h_in.

Figure 3: Normalized compression
force with the outer wall stiffness using a sharp indenter for a double-layer
model with h_in/r = 0.8%, E_in = 100 MPa and h_out/r
= 4.4%. The FEM data has been fitted by Hertzian analysis.

Conclusions

A biomechanical double-layer cell
wall model has been developed following the layered structure of the S.
cerevisiae wall. Assuming an external soft layer plus an internal stiff layer,
it is shown that the reported AFM elastic modulus corresponds to that of the
external layer, while micromanipulation results provide the total wall
stiffness. The results can be combined, using the thickness of the different
layers, to estimate the elastic modulus of the stiff b1,3-glucan layer for the first time.

REFERENCES

1.            Dague E, et al.
Yeast 27, 673-684, 2010.

2.            Smith AE, et al.
PNAS USA 97, 9871-9874, 2000.

3.            Mercadé-Prieto R,
et al. Chem Eng Sci 66, 1835-1843, 2011.

4.            Klis FM, et al.
FEMS Microbiol Rev 26, 239-256, 2002.