of 11
Quantitative Plant Biology
cambridge.org/qpb
Original Research Article
Cite this article:
L. Ginsberg et al. Cell wall and
cytoskeletal contributions in single cell
biomechanics of
Nicotiana tabacum
.
Quantitative Plant Biology
,
3
:
e1, 1–11
https://dx.doi.org/10.1017/qpb.2021.15
Received: 18 August 2021
Revised: 5 November 2021
Accepted: 26 November 2021
Keywords:
cell wall; cytoskeleton; micro-indentation;
nano-indentation;
Nicotiana tabacum
;
statistical modeling.
Author for correspondence:
E. Roumeli,
E-mail:
eroumeli@uw.edu
©TheAuthor(s),2022.PublishedbyCambridge
University Press in association with The John
Innes Centre. This is an Open Access article,
distributed under the terms of the Creative
Commons Attribution licence
(
https://creativecommons.org/licenses/by/4.0/
),
which permits unrestricted re-use, distribution,
and reproduction in any medium, provided the
original work is properly cited.
Cellwallandcytoskeletalcontributionsinsingle
cell biomechanics of
Nicotiana tabacum
Leah Ginsberg
1
, Robin McDonald
1
, Qinchen Lin
2
, Rodinde Hendrickx
1
,Giada
Spigolon
3
, Guruswami Ravichandran
1
, Chiara Daraio
1
and Eleftheria Roumeli
2
1
Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA;
2
Department of Materials Science and Engineering, University of Washington, Seattle, WA 98195, USA;
3
Biological
Imaging Facility, California Institute of Technology, Pasadena, CA 91125, USA
Abstract
Studiesonthemechanicsofplantcellsusuallyfocusonunderstandingtheeffectsofturgor
pressure and properties of the cell wall (CW). While the functional roles of the underlying
cytoskeleton have been studied, the extent to which it contributes to the mechanical properties
of cells is not elucidated. Here, we study the contributions of the CW, microtubules (MTs)
and actin filaments (AFs), in the mechanical properties of
Nicotiana tabacum
cells. We use a
multiscale biomechanical assay comprised of atomic force microscopy and micro-indentation
insolutionsthat(i)removeMTsandAFsand(ii)alterosmoticpressuresinthecells.Tocompare
measurements obtained by the two mechanical tests, we develop two generative statistical
models to describe the cell’s behaviour using one or both datasets. Our results illustrate that
MTs and AFs contribute significantly to cell stiffness and dissipated energy, while confirming
the dominant role of turgor pressure.
1. Introduction
The mechanical properties of plant cells are tightly related to their growth, function, adaptation
and survival (Cosgrove,
2016
;Milanietal.,
2013
;Szymanski&Cosgrove,
2009
). Enabled by
recent developments in mechanical testing and imaging capabilities, the mechanical properties
of plant cells have been increasingly studied (Bidhendi & Geitmann,
2019
; Burgert,
2006
;
Geitmann,
2006
;Vogleretal.,
2015
;Wuetal.,
2018
). The largest body of literature focuses on the
properties of the cell wall (CW) and turgor pressure to understand and model the mechanical
behaviour of the entire plant cell (Bidhendi & Geitmann,
2019
;Braybrook,
2015
;Tomos&Leigh,
1999
; Weber et al.,
2015
;Wuetal.,
2018
). This is in contrast to studies in (wall-less) animal
cells, in which the structural roles of the main cytoskeletal filaments, microtubules (MT) and
actin filaments (AF), have been established (Gardel et al.,
2008
; Huber et al.,
2015
;Janmey,
1991
;Pegoraroetal.,
2017
). In both plant and animal cells, the cytoskeletal filaments form an
interconnected network of polymer nanofibres that are responsible for providing structure, and
for transducing mechanical signals to assist cell growth, function and development (Durand-
Smet et al.,
2014
;Janmey,
1998
). Even though the contributions of cytoskeletal filaments in the
mechanical properties of plant cells are of high interest, their direct measurement is challenging
duetothepresenceofthestiffCWinadditiontothehighturgorpressureinsideplantcells.
Recent findings show that MTs have a leading role in guiding cellulose deposition in the CW,
which indirectly influences the mechanical properties of the CW (Durand-Smet et al.,
2014
;
Paredez et al.,
2006
). Additionally, rheological tests on plant cells treated to remove their CW
show that MTs, in particular, have non-negligible mechanical contributions compared to the
CW and turgor pressure (Durand-Smet et al.,
2014
). Nevertheless, the mechanical contributions
of the cytoskeleton in intact plant cells remain unexplored.
During the past decade, advances in mechanical testing instrumentation have enabled
remarkable new insights on the importance of plant cell mechanics in plant development.
Atomic-force microscopy (AFM), has been used to quantify the elastic modulus of the CW
(Braybrook,
2015
;Peaucelleetal.,
2011
), as well as to estimate the turgor pressure (Beauzamy
et al.,
2015
;Vellaetal.,
2012
). AFM results combined with finite element modelling (FEM)
https://doi.org/10.1017/qpb.2021.15
Published online by Cambridge University Press
2
L. Ginsberg et al.
provided evidence that in
Arabidopsis thaliana
(Arabidopsis) pave-
ment cells, the orientation of MTs to mechanical stresses plays a
dominant role in guiding cell shape (Sampathkumar et al.,
2014
).
Moreover, AFM has been used to reveal the different elastic proper-
ties of the CW in turgid versus plasmolysing solutions in Arabidop-
sis epidermal cells, highlighting the effects of different stress states
in the CW modulus (Braybrook,
2015
). Overall, the AFM nano-
indentation method allows for the simultaneous acquisition of
highly resolved topographical information and mechanical prop-
erty mapping (Peaucelle et al.,
2011
;Yilmazetal.,
2020
). The
applied forces are typically in the pico- to nano-Newton range, and
the indenter sizes are a few nanometres wide, which makes the
method suitable for highly localised cell properties. When global
cell properties are of interest, micron-sized indenters and higher
force load cells are required.
Cellular-force microscopy, a method coupling a micro-
indentation device with a light microscope, has been applied
for such global, cell-level measurements (Nelson,
2011
;Routier-
Kierzkowska et al.,
2012
; Vogler et al.,
2015
; Weber et al.,
2015
). This
apparatus allows for acquisition of micro-indentation data on iso-
lated plant cells, with applied forces in the micro-Newton range. It
hasbeenusedtoobtaindirectstiffnessmeasurementsofonionepi-
dermal cells which revealed turgor pressure-induced stiffening of
theCWandspatialstiffnessvariationsacrossthetissuesurface.In
particular,inturgidcells,thesurfaceabovethecross-walljunction
wassoftercomparedtothemiddlepartofthecellswhichwasstiffer
(Routier-Kierzkowska et al.,
2012
). When used in combination
with a computational mechanics model, cellular-force microscopy
can be used to extract material properties of subcellular compo-
nents, such as the elastic modulus of the CW material (Weber et al.,
2015
).
As more experimental methods to characterise the mechanical
properties of plant cells have been adopted, discrepancies arising
from comparing results from separate studies have emerged (Bid-
hendi & Geitmann,
2019
;Braybrook,
2015
; Vogler et al.,
2015
).
Differences in the sample preparation, loading rate and orienta-
tion, indenter shape and size, extent of deformation, models and
assumptions for data analysis, on top of variations between samples,
justify the literature discrepancies even when the same experimen-
tal method is applied (Bidhendi & Geitmann,
2019
).
Here, we present a method to compare extracted mechanical
properties of plant cells using two techniques: AFM and micro-
indentation. This method provides insights into the mechanical
contributions of CW, turgor pressure and cytoskeletal filaments
in intact
Nicotiana tabacum
Bright Yellow-2 (BY-2) cells, without
requiring a complex computational mechanics model of the system.
Our multiscale biomechanical assay allows us to probe mechan-
ical properties across multiple length scales which is essential to
evaluate the contributions of cytoskeletal fibres, that are a few
nanometres in diameter, the CW, which when hydrated is around
a micrometre thick, and the bulk protoplasm which is tens of
micrometres in diameter and length. To evaluate the effects of tur-
gor pressure, we test cells in solutions of two different osmolarities.
In order to isolate the mechanical contribution of the cytoskeleton,
we test cells after short exposure to drug treatments that depoly-
merise MTs and AFs, respectively. We propose a combination of
a generative statistical model and a simplified mechanical spring
model to analyse the mechanical testing results. This approach
allows us to determine the relative stiffness contributions from the
CW, MTs, AFs and the rest of the protoplasm, from two indepen-
dent experimental methods and without the need to create a FEM.
To test the stability of our generative statistical model, we perform
an analysis solely based on the micro-indentation data, and then,
perform a combined AFM and micro-indentation data analysis.
The combined AFM and micro-indentation data analysis more
accurately captures the difference in stiffness between the MTs and
AFs by taking into account the observed connection between the
cytoskeletal filaments and the CW using AFM in conjunction with
the micro-indentation data.
2. Results and discussion
2.1. Cell morphology
We observe the morphology of the unstained BY-2 cells using light
microscopy, and upon staining with calcofluor white, we image
the cells with confocal laser scanning microscopy (CLSM) (Figure
S1a,b). The hydrated CW thickness as visualised in a near-native
state from CLSM images is measured to be 0
.
79
±
0
.
02
μ
m(mean
±
standard error), which is similar to values reported for other
thin-walled cells in the literature (Moghaddam & Wilman,
1998
;
Radoti
́
cetal.,
2012
; Yakubov et al.,
2016
). The observed BY-2
cells are elongated, approximately cylindrical, with cell length and
diameter values presented in Figure S1c,d, as measured from light
microscopy. The mean observed cell length is 105
.
43
±
3
.
45
μ
m,
and the mean observed cell diameter is 39
.
12
±
0
.
55
μ
m, in agree-
ment with prior literature (
ˇ
Covanová et al.,
2013
;Siebereretal.,
2009
).
BY-2 marker cell lines expressing GFP-tubulin
α
that visualises
MTs (GFP-BY2-
α
), and GFP-AtFim1 to visualise AFs (GFP-BY2-F)
were used to study the cytoskeletal changes in response to the
selected chemical treatments, which disrupt each of the two net-
works so that their mechanical property contributions can be iso-
lated. By visualising the cells and their cytoskeleton through CLSM
in normal growth media and after short exposures (2 min) to
250
μ
MlatrunculinB(LatB)or50
μ
M oryzalin solutions, fluo-
rescent and transmitted light images demonstrate that the short
treatment was enough to disrupt the AF and MT networks, without
causing plasmolysis or other observable microscopic defects in the
cells. Short-term exposures to the drug treatments are chosen to
avoid secondary effects of removing components of the cytoskele-
ton. For example, MTs are known to be linked to the orientation
of cellulose microfibrils in the CW, so long-term disruption of MTs
could alter the alignment of the cellulose microfibrils, which would
in turn inhibit the biological function of the CW (Cosgrove,
2014
).
Example images of the marker lines before and after short expo-
sures to drug treatments and plasmolysing solution are presented in
Figure 1
. Additional Z-stacked images of BY-2 marker cells in GM
illustratingthe transversely oriented(with respect tothe cellgrowth
axis) MTs and the more isotropically oriented AFs are provided in
Figure S2.
2.2. AFM analysis
We subject the wild-type BY-2 cells to AFM tests in GM (growth
media) and PS (plasmolysing solution) to evaluate indentation
moduli of the CW in solutions of different osmotic pressures. To
determine any effects of MT and AF removal on the elastic proper-
ties of the CW, we subject the cells to short treatments of oryzalin
(Durand-Smet et al.,
2014
), or LatB (Durst et al.,
2014
;Maischetal.,
2009
), which are added to the GM or PS (see Section
2
). There are
six testing conditions: GM, GM–MT, GM–AF, PS, PS–MT, PS–AF,
where -MT indicates the oryzalin treatment which depolymerises
MTs, and -AF indicates the LatB treatment which removes AFs.
https://doi.org/10.1017/qpb.2021.15
Published online by Cambridge University Press
Quantitative Plant Biology
3
Fig. 1.
CLSM images on BY-2 marker lines to visualise the effects of short duration drug treatments on MTs and AFs. The model representation of MTs and AFs are inc
luded as
insets in each image panel. (a)–(c) CLSM images of GFP-BY2-
α
cells in growth media-based solutions. White arrows point to larger bundles of MTs that are visible near the CW. (a)
Fluorescence image in pure growth media. (b) Fluorescence image after exposure to growth media-based oryzalin solution. (c) Corresponding transmis
sion light image for (b),
showing no visual morphological change in the cell as a result of the short-term exposure to the drug treatment. (d)–(f) CLSM images of GFP-BY2-F cells
in growth media-based
solutions. Red arrows point to visible larger bundles of polymerised AFs. (d) Fluorescence image in pure growth media. (e) Fluorescence image after ex
posure to growth
media-based LatB solution. (f) Corresponding transmission light image for (e), showing no evident morphological change in the cell as a result of the
treatment. (g)–(i) CLSM
images of GFP-BY2-
α
cells in sorbitol. (g) Fluorescence image. (h) Combined fluorescence and transmission light image of (g and i). (i) Transmission light image. (j)–(l
)CLSM
images of GFP-BY2-F cells in sorbitol. (j) Fluorescence image. Red arrows point to visible larger bundles of polymerised AFs. (k) Combined fluoresce
nce and transmission light
image of(j and l). (l) Transmission light image. In panels (h, i, k and l) white arrowspointto CWsand black arrowspoint to plasmamembranes, which have
retracted from the CW.
All scale bars are 20
μ
m.
Considering that cellulose fibrils, which comprise the structural
backbone of the CW, are known to be organised in 5–50 nm-thick
bundles (Moon et al.,
2011
), and are immersed in a continuous,
heterogeneous matrix of hemicellulose, pectin and proteins, we
useanAFMtipwithasphericalbeadof1
μ
mdiametertoprobe
the bulk behaviour of the CW (Braybrook,
2015
). The average
indentation depth for turgid cells is 84
.
7
±
4
.
7 nm, which is shallow
enough (with respect to the hydrated total CW thickness) to assume
thattheobservedmechanicalresponseissolelyfromtheCW(Bray-
brook,
2015
;Milanietal.,
2013
;Radoti
́
cetal.,
2012
;Sampathkumar
et al.,
2014
). Plasmolysis of cells removes turgor pressure and
enables deeper nano-indentations of the CW, without probing the
protoplasm. The average indentation depth of plasmolysed cells is
217
.
0
±
45
.
8 nm, which is approximately 20% of the hydrated CW
thickness. This indentation depth i
s shallower, with respect to cell
size, than other literature-reported indentations aiming to isolate
theresponseoftheCWalone(Peaucelleetal.,
2011
). In all cases, the
extracted CW modulus is reflective of the mechanical properties of
the top layers of the CW material since that is the area of the CW
which we are stressing with a shallow indentation force. Typical
force-indentation and retraction data with an overlaid Hertz model
fit is presented in
Figure 2
a,withexampleimagesfromcellsineach
solution in the insets. The Young’s modulus results, separated by
treatment, are presented in
Figure 2
b,c.
Our data show that cells in all growth media-based solutions
have a CW Young’s modulus ranging from 0.65 to 15.2 MPa, while
in all plasmolysing solutions cells have a modulus ranging between
0.03 and 0.49 MPa. The non-parametric Kolmogorov–Smirnov test
reveals that there is a significant difference between the CW moduli
ofcellstestedinGMversusthoseinPS,with
p
=
4
.
4
×
10
16
.
Furthermore, the removal of AFs results in the largest reduction
of Young’s modulus, in cells in both GM and PS. This observation
suggests that there must be a connection between AFs and the
CW that is detectable from the conducted AFM tests which probe
local, exterior layers of the CW responses. The depolymerisation
of MTs also reduces the Young’s modulus in GM, but does not
make a significant difference in the cells tested in PS. Specifically, in
absence of drug treatments in the GM solution, we observe a CW
modulus of
E
GM
=
6
.
3
±
1
.
1 MPa, which is significantly different
from the moduli in GM–MT (
E
GM-MT
=
4
.
2
±
0
.
6MPa)andGM–
AF (
E
GM-AF
=
2
.
2
±
0
.
1MPa)treatments,with
p
-values of 0.049
and 5
.
8
×
10
5
, respectively. The GM–MT and GM–AF treatments
also lead to significantly different CW moduli, with a
p
-value of
1
.
1
×
10
4
. The CW modulus in pure PS treatment is
E
PS
=
270
±
60
kPa and is significantly different from the PS–AF (
E
PS-AF
=
130
±
30
kPa) treatment, with a
p
-value of 0.0015. The CW moduli in PS–MT
(
E
PS-MT
=
300
±
30 kPa) and PS–AF conditions are also significantly
different, with a
p
-value of 1
.
1
×
10
4
. Thus, from the nano-scale
measurements, we draw two main conclusions: (a) the biggest
changes in the CW modulus are caused by the changes in turgor
pressure, and we confirm experimentally that the higher internal
pressure stiffens the CW through stress, as predicted in (Cosgrove,
2016
) and (b) there is an evident interconnection between the
cytoskeleton and CW, which is manifested through CW softening
in response to the cell being subjected to drug treatments targeting
the cytoskeleton.
2.3. Micro-indentationexperimentsandgenerativespringmodel
For the micro-indentation experiments, BY-2 cells are tested in
the same testing conditions as in the AFM experiments (see Sec-
tion
2
). Representative force curves and images from the micro-
indentation test are provided in
Figure 3
.Theimagingcapabilities
during mechanical testing allow us to clearly observe plasmolysis
effects (
Figure 3
c) where a cell in PS has the plasma membrane
peeled away from the outer CW and the protoplasm retracted.
We calculate the initial effective stiffness by a linear fit to the
first 1
μ
m of indentation data after contact is initiated. This inden-
tation depth is close to the thickness of the CW as measured from
CLSM. Hence, the recorded mechanical response of the cell can
https://doi.org/10.1017/qpb.2021.15
Published online by Cambridge University Press
4
L. Ginsberg et al.
Fig. 2.
(a) Typical AFM force-indentation and retraction data from a cell in GM and in PS with Hertz fit to indentation data overlaid. Insets show correspondin
g images of cells in
the AFM test in GM (left) and PS (right). Arrows point to CW (white) and retracted plasma membrane (black). Scale bars are 40
μ
m. (b) Plot of indentation moduli for cells in all
drug treatments in GM. (c) Plot of indentation moduli for cells in all drug treatments in PS. Note the difference in scales between (b) and (c). Each point
in the plot represents an
indentation test. In each test condition, there are
n
9
tests from five biological replicates. Stars indicate significant differences in distributions according to the nonparametric
Kolmogorov–Smirnov test.
∗∗
p
<
0
.
05
,
∗∗∗
p
<
0
.
01
.
Fig. 3.
(a) Representative force-indentation and retraction data obtained in micro-indentation experiments on cells in GM (growth media) and in PS (plasmo
lysing solution). (b)
Image of BY-2 cells in GM taken from optical microscope of the micro-indentation testing apparatus. (c) Image of BY-2 cells in PS taken from optical mic
roscope in
micro-indentation testing apparatus with arrows pointing to the CW (white) and retracted plasma membrane (black). Scale bars are 100
μ
m. (d) Box and whiskers plot overlayed
on initial cell stiffness data in each test condition. Each point in the plot represents an indentation test on a different cell (
n
6
).
be attributed to a combination of the CW and the underlying
protoplasmic materials. In the plots of
Figure 3
d, the quantiles of
each dataset are overlaid as a boxplot on the stiffness data. In Figure
S3a, the empirical cumulative distribution functions (ECDFs) are
shown, which enable the visualisation of the distribution of cell
stiffness measurements across treatments.
The evident increase in stiffness observed in cells tested in a
solution of lower osmotic pressure illustrates the dominant effects
https://doi.org/10.1017/qpb.2021.15
Published online by Cambridge University Press
Quantitative Plant Biology
5
of turgor pressure, in comparison to all other effects under con-
sideration. Specifically, we note two distinct groupings in the mea-
sured distributions: cells in GM suspensions (
k
GM,all
=
8
.
95
±
0
.
86
N/m), and cells in PS suspensions (
k
PS,all
=
1
.
99
±
0
.
18 N/m). The
p
-value which separates the stiffness of cells in GM and PS condi-
tions is
p
=
7
.
22
×
10
17
.BY-2cellstiffnessmeasurementspreviously
reported in literature are in good agreement with the average cell
stiffnesses shown in
Figure 3
d (Felekis et al.,
2012
; Weber et al.,
2015
). Specifically, stiffness ranges of 10–33 N/m were reported
from (Felekis et al.,
2012
) for turgid cells, while back-calculated
values of 10 and 5 N/m can be extracted from (Weber et al.,
2015
)
for BY-2 cells in water and 0.2 M mannitol, respectively. The trend
of turgor pressure increasing the stiffness of the cell is also reflected
in measurements by (Weber et al.,
2015
). There was no statistically
significant difference between any of the groups within the GM or
PS categories.
The stiffness results support the dominance of turgor pressure
on the cell stiffness, which is in agreement with our aforemen-
tioned AFM analysis, and literature (Routier-Kierzkowska et al.,
2012
). Directly from the experimental results, we conclude that
an increase in turgor pressure results in a dramatic increase in
cell stiffness at multiple measurement scales. Beyond this conclu-
sion, we aim to extract insights for the mechanical properties of
other subcellular structures, especially the cytoskeletal filaments.
Although there was no statistically significant difference between
the measured stiffnesses of the cells in all the different treatments,
we propose using a mechanical model to elucidate trends and
effects caused by the different treatments on the mechanical con-
tributions of sub-cellular components.
We apply a generalised two-spring model, which was introduced
in our prior work (Roumeli et al.,
2020
), to separate the stiffness
contributions from the CW and the protoplasm. In the two-spring
model (Figure S4), the mechanical response of a cell to micro-
indentation experiments is modeled as two springs acting in series.
Previous literature reports modeled the mechanical response of a
cell as a single spring by reporting apparent cell stiffness (Beauzamy
et al.,
2015
). In our model, the apparent cell stiffness is separated
into contributions from the CW and protoplasm, using spring
constants
k
CW
and
k
prot
,respectively.
k
total
=
k
CW
k
prot
k
CW
+
k
prot
.
(1)
This simplified model of the cell response relies upon assump-
tions about the structure and materials that constitute the cell. The
CW and protoplasmic materials are assumed to behave as homo-
geneous, isotropic, linear elastic materials at shallow indentation
depths,andanynonlinearbehaviours,suchasviscosity,adhesion,
or plasticity are not captured by the model. The stress in the cell
away from the indenter is assumed to be negligible for shallow
indentations (Boussinesq,
1885
). The interpretation of stiffness
with respect to subcellular structures is somewhat controversial
due to the heterogeneity, directionality and variability inherent to
biological systems.
2.4. Analysis of micro-indentation data
As a first iteration on the micro-indentation results, we assume
that the CW stiffness remains constant across drug treatments,
but not across osmotic solutions. This assumption is in accordance
with the observation from our AFM data that changes in the CW
elastic modulus caused by depolymerising MTs and removing AFs
were much less significant than the change caused by different
osmotic pressures. In the succeeding analysis section, which com-
bines results from the AFM and micro-indentation experiments,
we will remove this assumption, and analyse results from both
experiments simultaneously.
An illustration of the spring model adapted to each test condi-
tion is presented in Figure S5. To extract the stiffness contributions
from the MTs and AFs, we model them as springs in parallel to the
rest of the protoplasmic response, with coefficients
k
MT
and
k
AF
.
To account for the change in the protoplasm in different osmotic
conditions, the protoplasmic response is differentiated between
cells in GM and cells in PS. The GM is a hypotonic solution that
allows the cell to maintain turgor pressure, nutrients to flow into
the cell, and the cell to expand. The PS is a hypertonic solution since
the osmotic pressure of a solution that causes plasmolysis (instant
response visible through microscopic views of both mechanical
testing methods, see
Figure 1
g–l, inset in
Figures 2
aand
3
c) must be
higher than the osmotic pressure of the cell. The spring constants
k
hypo
and
k
hyper
represent the stiffness contribution from all pro-
toplasmic components in GM and PS, respectively, excluding the
MTs and AFs, which are already represented by
k
AF
and
k
MT
in the
spring model.
In total, we have six spring stiffnesses that are calculated though
our analysis:
k
CW,hypo
,
k
CW,hyper
,
k
hypo
,
k
hyper
,
k
MT
and
k
AF
.Wealso
have six measurements of the effective stiffnesses from the six
testing conditions: GM, GM–MT, GM–AF, PS, PS–MT and PS–
AF. Since we have an equal number of variables and datasets,
a unique solution to the system of effective stiffness equations
is possible. However, the equations are nonlinear and cannot be
solved analytically. To tackle this, we develop a generative statistical
model (Betancourt,
2019
;Bois,
2018
).
Generative statistical models are used to build a posterior prob-
ability distribution
g
(
θ
y
)
, which is the probability that a set of
parameters
θ
describes the given experimental data
y
.Here,weare
interested in the posterior probability distribution for the param-
eters
θ
=
{
k
CW,hypo
,
k
CW,hyper
,
k
hypo
,
k
hyper
,
k
MT
,
k
AF
}
given the dataset
y
={
k
GM
,
k
GM-MT
,
k
GM-AF
,
k
PS
,
k
PS-MT
,
k
PS-AF
}
,whereeachvariablein
y
represents a set of measurements of the stiffness from the selected
treatment. Thus, the posterior probability distribution details the
probability that a set of deconvoluted subcellular stiffness constants
describe the observed experiments. We use six separate poste-
rior probability distributions, all of which are dependent on each
other through the subcellular stiffness constants
θ
.UsingBayes’
theorem, we solve for
g
(
θ
y
)
, using the likelihood of observing
our experimental data given a selected set of parameters,
f
(
y
θ
)
,
and prior information about our parameters of interest,
g
(
θ
)
.The
likelihood is defined separately for each treatment using a Gaussian
distribution, and the prior distribution is defined empirically (see
Supplementary Materials).
Wemodeltheoverallstiffnessofeachcellmeasuredineachtest
condition using the two-spring model in Figure S4. Adaptations
of equation (
1
) for each testing condition gives the relationship
between the mean overall stiffness in each treatment (
μ
)andthe
stiffness of each sub-cellular component (
k
). The equivalent equa-
tions for the spring stiffness are:
μ
GM
=
k
CW,hypo
(
k
hypo
+
k
AF
+
k
MT
)
k
CW,hypo
+
k
hypo
+
k
AF
+
k
MT
,
(2)
μ
GM-MT
=
R
GM-MT
k
CW,hypo
(
k
hypo
+
k
AF
)
R
GM-MT
k
CW,hypo
+
k
hypo
+
k
AF
,
(3)
https://doi.org/10.1017/qpb.2021.15
Published online by Cambridge University Press
6
L. Ginsberg et al.
μ
GM-AF
=
R
GM-AF
k
CW,hypo
(
k
hypo
+
k
MT
)
R
GM-AF
k
CW,hypo
+
k
hypo
+
k
MT
,
(4)
μ
PS
=
k
CW,hyper
(
k
hyper
+
k
AF
+
k
MT
)
k
CW,hyper
+
k
hyper
+
k
AF
+
k
MT
,
(5)
μ
PS-MT
=
R
PS-MT
k
CW,hyper
(
k
hyper
+
k
AF
)
R
PS-MT
k
CW,hyper
+
k
hyper
+
k
AF
,
(6)
μ
PS-AF
=
R
PS-AF
k
CW,hyper
(
k
hyper
+
k
MT
)
R
PS-AF
k
CW,hyper
+
k
hyper
+
k
MT
,
(7)
where the ratios
R
are all equal to 1 for this initial analysis, since we
assume that the removal of MTs or AFs has no effect on the stiffness
of the CW.
With these six equations, we transform the means of each of
the six treatments to identify the six parameters of interest,
θ
=
{
k
CW,hypo
,
k
CW,hyper
,
k
hypo
,
k
hyper
,
k
MT
,
k
AF
}
. To optimise the posterior
distributions for all six parameters of interest simultaneously, we
combine the six separate posterior distributions into one objective
function and add six coefficients (
a
,
b
,
c
,
d
,
e
and
f
)thatwillbe
optimised concurrently. These six coefficients are used to balance
the final objective function, in absence of finding the true solution
to the system of six equations. Mathematically, we maximise
F
over
a
,
b
,
c
,
d
,
e
,
f
,and
θ
:
F
=
a
g
GM
(
θ
y
)+
b
g
GM-MT
(
θ
y
)+
c
g
GM-AF
(
θ
y
)+
d
g
PS
(
θ
y
)
+
e
g
PS-MT
(
θ
y
)+
f
g
PS-AF
(
θ
y
)
.
(8)
Projections of the objective function into two-dimensional space
are presented in Figure S6, illustrating the correlations between
each pair of the six stiffness parameters.
Coefficients
a
,
b
,
c
,
d
,
e
and
f
are weights multiplied in front of
the posterior distributions for each of the six parameters of interest.
Theweightsshouldallsumuptounity.Wealsoaddconstraintson
the size of the coefficients and the size of the spring constants to
ensure that all components are included, and none dominate the
optimisation function
a
+
b
+
c
+
d
+
e
+
f
=
1
,
(9)
0
.
05
a
,
b
,
c
,
d
,
e
,
f
0
.
5
,
(10)
0
.
01
θ
100
.
(11)
Using a trust-region constrained optimisation method, we can
find the parameters
θ
that maximise the posterior distributions in
alltreatments,undertheaboveconstraints.Wecanalsoconstruct
a credible region by calculating the Hessian at the optimised point
in parameter space. The optimum parameter values (also known as
the maximum a posteriori or MAP parameter values) and credible
region (which contains approximately 68% of the total probability)
arereportedinthefollowingdiscussionas
k
MAP
±
σ
MAP
(mean
±
standard deviation), and the results are visualised in Figure S6
as a red ‘x’ overlaying the projections of the combined posterior
distributions.
By optimising the objective function from the combined pos-
terior distributions, we can decouple the relative stiffness contri-
butions from the six identified subcellular components of interest,
and the results are in line with previous reports and predictions
in literature (Cosgrove,
2016
; Durand-Smet et al.,
2014
;Routier-
Kierzkowska et al.,
2012
). The contribution from the protoplasm
without MTs and AFs in hypotonic conditions is the highest com-
ponent evaluated (
k
hypo
=
42
.
03
±
2
.
01 N/m), and about four times
greater than that of the protoplasm without MTs and AFs in hyper-
tonic conditions (
k
hyper
=
9
.
68
±
1
.
50 N/m). This is in agreement
with literature which shows that turgor pressure supplies most of
the stiffness for turgid cells in compression (Routier-Kierzkowska
et al.,
2012
).
High turgor pressure in hypotonic conditions stresses the CW,
making its response to compression appear stiffer. The stiffness of
the CW from AFM indentations in GM was 5.5 times greater than
the stiffness of the CW in PS. Without the inclusion of these results
in the current analysis, the model predicts that the CW stiffness in
hypotonic conditions (
k
CW,hypo
=
12
.
43
±
0
.
68 N/m) is about twice
as high as the CW stiffness in hypertonic conditions (
k
CW,hyper
=
6
.
95
±
0
.
32 N/m). The results from this analysis represent both
material and structure of the CW, while the AFM results probe only
the CW material. The fact that both quantities are higher in GM
than in PS could be merely a result of CW stiffening under high tur-
gorpressure,oritcouldbearesultofbothCWstrain-stiffeningand
an increase in the bending rigidity of the CW under a higher turgor
pressure (or another unknown geometric or structural change in
the cell under pressure).
The credible regions for the relative stiffness contribution from
AFs (
k
AF
=
11
.
81
±
4
.
69 N/m) and MTs (
k
MT
=
6
.
82
±
2
.
48 N/m)
overlap,andareonthesameorderofmagnitudeastheCWstiff-
ness. This result clearly demonstrates that the cytoskeleton is an
important structural component for the cell.
2.5. Combined analysis of AFM and micro-indentation data
Literature results confirm that MTs and AFs are physically con-
nectedtotheCW,andthustheremovalofthesefilamentsshould
affect the mechanical behaviour of the CW (Szymanski & Cosgrove,
2009
). Our AFM experiments support this fact, as we measure that
theCWstiffnessisindeedaffectedbytheremovalofcytoskeletalfil-
aments with drug treatments, albeit appreciably less than the effect
of altering the osmotic pressure of the solution. In this part of our
analysis, we introduce the observed effect of the drug treatments
on the stiffness of the CW in our generative statistical model using
ratios of the mean measured CW stiffnesses from the AFM tests.
We calculate the CW stiffness values from the AFM data using a
linear interpolation of the first 10% of the maximum force data
after contact is detected. The use of ratios instead of absolute values
of stiffness is selected to overcome discrepancies in measuring
the same properties with different experimental techniques, which
have also been reported in literature (Bidhendi & Geitmann,
2019
;
Wu et al.,
2018
).
The ratios used are:
R
GM-MT
=
k
AFM, GM-MT
k
AFM, GM
,
(12)
R
GM-AF
=
k
AFM, GM-AF
k
AFM, GM
,
(13)
R
PS-MT
=
k
AFM, PS-MT
k
AFM, PS
,
(14)
R
PS-AF
=
k
AFM, PS-AF
k
AFM, PS
,
(15)
where
k
AFM, treatment
is the mean indentation stiffness measured in
the specified treatment in AFM experiments. The use of these ratios
allows us to introduce the change in CW stiffness, as observed
in the AFM experiments, in the micro-indentation analysis,
https://doi.org/10.1017/qpb.2021.15
Published online by Cambridge University Press