Supporting Information
MECHANICAL DOUBLE LAYER MODEL FOR SACCHAROMYCES CEREVISIAE
CELL WALL
Ruben Mercadé-Prieto*, Colin R. Thomas and Zhibing Zhang
School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham B15
2TT, UK.
*
corresponding author: [email protected]
(a) Single layer wall with sharp indenter (rind/r = 0.01)
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(b) Double layer wall with large indenter (rind/r = 1)
Fig. S1 Meshing of the single and double layer walls during FEM using (a) sharp or (b) large
spherical indenters. For both cases, the total thickness htotal = 0.05.
2
0.004
F/r(Eh)total
0.003
0.002
Empty Hencky
Liquid Hencky
Empty Green
0.001
Liquid Green
0
0
0.01
0.02
0.03
Fractional deformation 
0.04
Fig. S2 Normalized compression force with the total stiffness for AFM-like experiment
using a large spherical indenter (rind/r = 1), using the double layer model with hin/r = 0.01,
hout/r = 0.04 and Ein / Eout = 500. Simulations were performed using Green and Hencky
strains, for empty spheres and for spheres with a liquid core. Insets show the maximum
principal Hencky strain after a displacement of the indenter d = htotal = hin + hout (at  =
0.025, about 125 nm for a typical yeast cell).
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0.17
Green's strains
rind/r = 1
0.15
kwe / r(Eh)total
0.13
0.11
0.09
hout=0.044; hin=0.004
hout=0.044; hin=0.008
hout=0.044; hin=0.016
hout=0.036; hin=0.004
hout=0.036; hin=0.012
hout=0.036; hin=0.024
0.07
0.05
0.03
1
10
Ein / Eout
100
1000
Fig. S3 Effect of the elastic modulus ratio in a double layer wall on kw calculated at d < hout.
The compression was performed using a large spherical indenter (rind/r = 1) for several
thickness values of hout and hin using Green’s strains.
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Effect of turgor pressure on the elastic modulus
The effect of an inner or turgor pressure in a core-shell sphere using FEM for the simple case
of Green’s strains is explored briefly here. Firstly, for a linear-elastic, incompressible,
isotropic and homogeneous wall, membrane theory predicts that the increase of the cell
radius from r0 to r due to the existence of an internal pressure P is given by (Kraus 1967)
r
P E
 1  0.25
r0
h0 r0
(S1)
This equation was validated here using FEM, as shown in Fig. S4(a). It is important to notice
that the swelling of a cell is not determined solely by the turgor pressure, but by the ratio of
the pressure to the elastic modulus, i.e. P / E. Hence, it is impossible to study the effect of the
turgor pressure on compression if E is not known.
In practice, the reverse problem has been considered: r/r0 is determined experimentally at
different osmotic pressures, from which an “osmotic” elastic modulus is determined using
Eq. S1 or FEM. For yeast it has been shown that the value of E determined in this way is
~100 MPa (Smith et al. 2000). AFM studies commonly do not report r/r0. On the other hand,
studies using compression testing with micromanipulation have reported 1 < r/r0 < 1.1
(Chaudhari et al. 2012; Smith et al. 2000; Stenson et al. 2009; Stenson et al. 2011), with a
mean turgor pressure of 0.3-0.4 MPa when the cells where suspended in Isoton II solution,
i.e. with an external osmotic pressure of 0.8 MPa. Schaber et al. (2010) have reported a
turgor pressure of 0.6 ± 0.2 MPa. Considering only the most swollen case of r/r0 ~1.1 and
using h0/r0 = 0.05, eq. S1 suggests that P / E < 0.02.
The effects of different turgor pressures on sharp indentation (rind/r = 0.01) have been
evaluated by FEM using Green strains for simplicity, shown in Fig. S4(b). The spring cell
constant kw has been normalized using r and h from Fig. S4(a) or using the unswollen values
r0 and h0. When the latter were used, the spring cell constant does not depend greatly on P /
E, whereas when the swollen values are used kw/Erh increases with P / E due to the fast
decrease of h with P / E (Fig. S4(a)). Therefore, for the P / E values expected for yeast cells
(<0.02), the existence of an internal pressure has a negligible effect on the compression force
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for displacements common in AFM compression (d < h). This was also verified for large
spherical indenters (rind/r = 1, results not shown).
1.30
1.0
0.9
1.25
0.8
(a)
0.7
1.20
1.15
0.5
0.4
1.10
h / ho
r / ro
0.6
0.3
0.2
1.05
0.1
1.00
0
0.01
0.02
0.03
0.04
0.0
0.05
P/E
0.21
(b)
0.19
kw / Erh
0.17
0.15
kw /Erh
0.13
0.11
kw /Er0h0
0.09
0.07
0
0.01
0.02
0.03
0.04
0.05
P/ E
Fig. S4 (a) Increase of the cell radius after the swelling caused by an internal pressure,
continuous line is the prediction of eq. S1. As the cell expands, the wall gets thinner; the
dashed line is the best fit h/h0 = 1 – 12.25P/E. (b) Effect of the turgor pressure on the
calculated cell wall spring constant normalized using the swollen radius and wall thickness
(triangles), or using the unswollen values (squares). Simulation conditions h0/r0= 0.04 using
an indenter with rind/r0 = 0.01.
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Normalization factor for double-layer walls compressed at large deformations
In the compression of cells using micromanipulation the main mode of deformation is by
stretching. For a single layer core-shell sphere, the compression force causing stretching has
been reported to be proportional to the cube of the fractional deformation  (Lulevich et al.
2004), so that:
F  4Erh 3
This assumes a thin wall incapable of supporting bending moments and small deformations.
For a double layer system, if it is assumed that the total force is the addition of the stretching
resistance of both layers, and considering only the fractional deformation  of the external
layer for simplicity
Fdoublelayer  4r Ehout  3  4r Ehin  3
Then it is predicted the following normalization relationship regardless of the value of Ein
and Eout (and also of hin and hout, as bending stresses are assumed negligible)
Fdoublelayer
r Ehtotal
 3
where (Eh)total = (Eh)out + (Eh)in.
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Reference List
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