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ORIGINAL PAPER

Biophysical properties of Saccharomyces cerevisiae

and their relationship with HOG pathway activation

Jo ¨rg Schaber•Miquel A`ngel Adrover•Emma Eriksson•Serge Pelet•

Elzbieta Petelenz-Kurdziel•Dagmara Klein•Francesc Posas•

Mattias Gokso ¨r•Mathias Peter•Stefan Hohmann•Edda Klipp

Received: 11 January 2010/Revised: 18 March 2010/Accepted: 12 May 2010/Published online: 19 June 2010

? The Author(s) 2010. This article is published with open access at Springerlink.com

Abstract

mechanical cell properties are important for predictive

mathematical modeling of cellular processes. The concepts

of turgor, cell wall elasticity, osmotically active volume,

and intracellular osmolarity have been investigated for

decades, but a consistent rigorous parameterization of these

concepts is lacking. Here, we subjected several data sets

of minimum volume measurements in yeast obtained

after hyper-osmotic shock to a thermodynamic modeling

framework. We estimated parameters for several relevant

biophysical cell properties and tested alternative hypotheses

Parameterizedmodelsof biophysicaland

abouttheseconceptsusingamodeldiscriminationapproach.

In accordance with previous reports, we estimated an aver-

age initial turgor of 0.6 ± 0.2 MPa and found that turgor

becomes negligible at a relative volume of 93.3 ± 6.3%

corresponding to an osmotic shock of 0.4 ± 0.2 Osm/l. At

high stress levels (4 Osm/l), plasmolysis may occur. We

found that the volumetric elastic modulus, a measure of cell

wall elasticity, is 14.3 ± 10.4 MPa. Our model discrimina-

tion analysis suggests that other thermodynamic quantities

affecting the intracellular water potential, for example the

matrix potential, can be neglected under physiological con-

ditions. The parameterized turgor models showed that acti-

vation of the osmosensing high osmolarity glycerol (HOG)

signaling pathway correlates with turgor loss in a 1:1 rela-

tionship. This finding suggests that mechanical properties of

the membrane trigger HOG pathway activation, which can

be represented and quantitatively modeled by turgor.

M. A`. Adrover, E. Eriksson, S. Pelet, E. Petelenz-Kurdziel have

contributed equally to this work.

Electronic supplementary material

article (doi:10.1007/s00249-010-0612-0) contains supplementary

material, which is available to authorized users.

The online version of this

J. Schaber ? E. Klipp (&)

Theoretical Biophysics, Humboldt University,

Invaliden Str. 42, 10115 Berlin, Germany

e-mail: edda.klipp@rz.hu-berlin.de

M. A`. Adrover ? F. Posas

Cell Signaling Unit, Departament de Cie `ncies Experimentals i de

la Salut, Universitat Pompeu Fabra (UPF),

08003 Barcelona, Spain

E. Eriksson ? M. Gokso ¨r

Department of Physics, University of Gothenburg,

412 96 Gothenburg, Sweden

S. Pelet ? M. Peter

Institute of Biochemistry, ETH, Zurich, Switzerland

E. Petelenz-Kurdziel ? D. Klein ? S. Hohmann

Department of Cell and Molecular Biology/Microbiology,

University of Gothenburg, Box 462, 405 30 Go ¨teborg, Sweden

Present Address:

D. Klein

Clinical R&D Laboratory, Oxoid, Microbiology,

Basingstoke, Thermo Fisher Scientific, Perth, UK

J. Schaber (&)

Institute for Experimental Internal Medicine, Medical Faculty,

Otto von Guericke University, Leipziger Str. 44,

39120 Magdeburg, Germany

e-mail: schaber@med.ovgu.de

Present Address:

E. Eriksson

SP Technical Research Institute of Sweden,

Brinellgatan 4, Box 857, 501 15 Bora ˚s, Sweden

123

Eur Biophys J (2010) 39:1547–1556

DOI 10.1007/s00249-010-0612-0

Page 2

Keywords

Volumetric elastic modulus ? High osmolarity glycerol

(HOG) signaling ? Plasmolysis ? Model discrimination ?

Yeast

Turgor ? Cell wall elasticity ?

Introduction

All living cells maintain an osmotic pressure gradient

between their interior and the extracellular environment.

This gradient is counterbalanced by a hydrostatic pressure,

called turgor, which is especially prominent in plants and

unicellular organisms, most notably fungi. The concept of

turgor has been debated for many years (Burstro ¨m 1971). It

is recognized that turgor plays a vital role in growth

(Cosgrove 1981; Zimmermann 1978; Marechal and Gervais

1994), cell structure (Cosgrove 1993; Morris et al. 1986;

Zhongcan and Helfrich 1987; Munns et al. 1983), and

membrane transport processes (Zimmermann 1978; Lande

et al. 1995; Kamiya et al. 1963; Soveral et al. 2008) and

that it may be sensed by the cells as an indicator of external

osmotic changes (Tamas et al. 2000; Coster et al. 1976;

Schmalstig and Cosgrove 1988; Reiser et al. 2003).

Therefore, quantifying cell wall and membrane properties,

and thus the resulting turgor pressure, has been the aim of

numerous studies (Cosgrove 1981; Morris et al. 1986;

Munns et al. 1983; Kamiya et al. 1963; Coster et al. 1976;

Smith et al. 1998, 2000a, b, c; Marechal et al. 1995;

Gervais et al. 1996a; Cosgrove 2000). Parameterized

models of turgor pressure, based on cell wall and mem-

brane properties are especially important for quantitative

mathematical models describing the interdependence

between water transport and signaling processes (Schaber

and Klipp 2008; Klipp et al. 2005; Gennemark et al. 2006;

Kargol and Kargol 2003a, 2003).

At present, it is not possible to directly measure turgor

in yeast. However, the so-called linear elastic theory

states that the change in turgor pressure P is proportional

to a relative change in membrane enclosed cell volume

Vm, i.e.,

dP ¼ edVm

Vm;

ð1Þ

where e is a proportionality factor, called volumetric elastic

modulus or bulk modulus. This factor can be viewed as a

measure of the elasticity of the cell wall. Thus, the turgor

pressure can be derived from cell volume measurements

that have been performed repeatedly on yeast cells under

different experimental conditions (Munns et al. 1983;

Smith et al. 2000a, b, c; Touhami et al. 2003; Wei et al.

2001; Martinez de Maranon et al. 1996; Arnold and Lacy

1977). However, the results obtained for turgor pressure

were highly inconsistent, with variations spanning up to

three orders of magnitude from 0.05 (Martinez de Maranon

et al. 1996) to 2.9 (Arnold and Lacy 1977) MPa.

The concepts of turgor, cell wall elasticity, osmotically

active volume and intracellular osmolarity have never been

quantified and parameterized together in a consistent

framework, nor challenged by alternative concepts (Smith

et al. 2000c; Martinez de Maranon et al. 1996; Gervais

et al. 1996b). Moreover, the few reports in which turgor

and/or cell wall elasticity were quantitatively determined

usednon-physiologicalexperimental

example, one study used pre-stressed cells (Marechal et al.

1995) and in two others, the cells were suspended in dis-

tilled water before volume measurements (Smith et al.

2000b, c).

The objective of this study was to derive parameterized

models of turgor and other biophysical properties of

the cell, for example the volumetric elastic modulus, in

Saccharomyces cerevisiae. We developed a modeling frame-

work that includes the concepts of turgor pressure, intra and

extracellular osmotic pressure, and membrane and volume

properties and, for the first time, also other thermodynamic

quantities, for example the matrix potential, in a consistent

manner. By reformulating sub-modules while keeping

the overall framework consistent, we could test several

hypotheses describing the relationship between cell volume

and changes in the intracellular water potential. Verification

of these hypotheses was based on four independent data

sets. To ensure a wide applicability of the selected model,

the conditions for generating each data set were slightly

different. We also correlated cell volume measurements

with Hog1 phosphorylation levels and Hog1-GFP nuclear

localization to investigate the physiological consequences

of turgor loss under hyper-osmotic stress in a quantitative

fashion. Hog1 is the yeast p38 stress-activated protein

kinase whose phosphorylation level and subcellular locali-

zation are controlled by an osmo-sensing signaling pathway

(Hohmann 2002). We found a 1:1 relationship between

turgor loss and HOG pathway activation. A loss of turgor

relates to HOG pathway activation such that when turgor

becomes zero because of application of a hyperosmotic

solution the HOG pathway is maximally activated. This

finding suggests that the extent of HOG pathway activation

is directly linked to some mechanical property of the

membrane, which can be represented and quantitatively

modeled by the loss of turgor pressure.

conditions. For

Materials and methods

Descriptions and units of all variables and parameters are

listed in Table 1.

1548 Eur Biophys J (2010) 39:1547–1556

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Modeling framework

We assume that the change in osmotically active volume

Vosis equal to the water flux across the cell membrane,

which is driven by differences in chemical water potentials

inside and outside the cell. Based on these thermodynamic

principles (for details refer to the ‘‘Supplementary mate-

rial’’) we developed a model describing the minimal vol-

ume of yeast cells attained after osmotic shock, Vap

included several new aspects that are depicted in Fig. 1.

We made a distinction between the commonly measured

volume that is enclosed by the cell wall, which we refer to

as the apparent volume Vap, and the membrane-enclosed

cytoplasmic volume, Vm, where osmotic and hydrostatic

pressures are defined. Vmcomprises the osmotically active

volume Vosand the minimal solid cytosolic volume Vb, i.e.,

Vm= Vos? Vb. We assume that for osmotic shocks higher

than a critical value of cpl

value of Vap

wall (Vap), i.e., plasmolysis occurs, giving rise to the

min, which

eor volumes lower than a critical

pl, the membrane (Vm) detaches from the cell

periplasmic volume Vpl, such that Vap= Vm? Vpl. Fur-

thermore, we define Vm

becomes zero and Vm

as the volume where negative

hydrostatic effects and/or matrix potential effects become

important, corresponding to hyperosmotic shock levels of

cP=0

and cs

thresholds that affect the rate of water outflow, i.e.,

shrinkage. Between Vm

reciprocal function of external osmolarity, also called the

van’t Hoff relationship (Noble 1969) (Eq. S6 in ‘‘Supple-

mentary material’’). Based on these concepts, we derived a

formula for the apparent minimum volume after osmotic

shock Vap

larities and volumes:

?

PVmin

ap

aPCRTþ ce

where c0

‘‘Supplementary material’’), Vap

P=0as the volume at which the turgor

s

ee, respectively (Fig. 1). The latter two are

P=0and Vm

sapparent cell volume is a

minas a function of turgor pressure PVmin

ap, osmo-

Vmin

ap¼ Vbþ Vplþ

ci

0V0

ap? Vb? Vpl

?

0

stressþ ce

ð2Þ

iis the initial internal osmolarity (Eq. S4 in

0is the initial apparent cell

Table 1 Summary of quantities used in the main text

QuantityUnit Description

Vap

Vap

Vap

Vm

Vos

Vpl

% initial volume Apparent cell volume, enclosed by the cell wall

0

% initial volume Initial Vap

Minimal Vapattained after hyperosmotic shock

Cell volume, enclosed by the plasma membrane

min

% initial volume

% initial volume

% initial volume Osmotically active volume

% initial volume Periplasmic volume. We tested three alternative models of Vplas a function of cstress

‘‘Supplementary material’’)

e

(see

Vb

Vm

Vm

Vap

c0

c0

cstress

cP=0

cs

cpl

P0

PVmin

ap

% initial volumeSolid cytosolic volume

P=0

% initial volumeVolume, when turgor becomes zero

s

% initial volumeVolume below which negative hydrostatic or matrix potential effects are important

pl

% initial volumeApparent volume where plasmolysis occurs

i

Osm/lInitial internal osmolarity.

e

Osm/l Initial external osmolarity

e

Osm/l Applied osmotic stress

e

Osm/l

cstress

cstress

cstress

Initial turgor at the initial turgid cell volume of 100%

e

corresponding to Vm

corresponding to Vm

corresponding to Vap

P=0

e

Osm/l

e

s

e

Osm/l

e pl

MPa

MPa Turgor at Vap

‘‘Supplementary material’’) and the two-sided model (Eq. S8 in ‘‘Supplementary material’’)

(Fig. 2)

min. In general, there are two alternatives, i.e., the one-sided model (Eq. S7 in

e

MPaVolumetric elastic modulus, describing the elasticity of the cell wall, relating relative volume

change to hydrostatic pressure, i.e., turgor

es

MPaProportionality factor, relating relative volume change to the matrix potential or other effects

negatively influencing the intracellular water potential

R

J K-1mol-1

Gas constant

T

aPC

K Temperature

Conversion factor relating pressure to osmolarity, e.g., for relating MPa and Osm/l it is 10-3

Dimensionless

Eur Biophys J (2010) 39:1547–1556 1549

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volume, R is the gas constant, T is the temperature, aPCis a

dimensionless conversion factor, c0

osmolarity, and cstress

is the applied osmotic stress.

eis the initial external

e

Turgor pressure and matrix potential

Thermodynamic quantities other than turgor and osmolar-

ity, that can affect the cellular water potential, are usually

ignored. For example, a specific fraction of the intracellular

water might be bound to colloidal surfaces by electrostatic

interactions. This water is not freely available in solution,

which lowers the total intracellular water potential by a

value that is termed the matrix potential (Griffin 1981). In

addition, it is possible that the cytoskeleton mechanically

counteracts cell shrinkage such that the cell has an inherent

tendency to re-swell in a sponge-like fashion. This effect

would also reduce the total intracellular water potential.

For simplicity, we summarize these effects under the term

matrix potential (Ds in Eq. S1 of the ‘‘Supplementary

material’’). Quantitative descriptions of the matrix poten-

tial are not available and elaborated models of mechanical

forces induced by volume changes (Stamenovic and

Coughlin 1999) are beyond the scope of this study.

Therefore, we include the effects summarized under the

term of the matrix potential in the model of the turgor

pressure PVmin

ap

with a simple mathematical description. In

this way, we obtain an expression for PVmin

matrix potential (assuming Vpl= 0 for VmC Vm

?

eln

VP¼0

m

for

0 for

Vmin

ap?Vpl

Vs

m

for

ap

including the

P=0):

PVmin

ap

Vmin

8

>

ap;Vpl;VP¼0

Vmin

ap

m

;Vs

m;e;es

?

Vmin

Vs

¼

??

ap?VP¼0

m?Vmin

Vmin

m

ap? Vpl\VP¼0

ap? Vpl\Vs

m

esln

??

m

>

>

>

:

<

ð3Þ

of

InFig. 2

min- Vpl.

We tested Eq. 3 against the data, with and without the

term for matrix potential, which we refer to as the one-

sided and the two-sided models, respectively. We also

tested three hypotheses regarding the size of the periplas-

mic volume Vpl, (a) Vpl= 0, (b) Vpl[0 with a constant

apparent volume Vap

material’’), and (c) Vpl[0 with a variable apparent volume

Vap

Inserting Eq. 3 into Eq. 2 yields an implicit expression

for Vap

model for turgor, matrix effect, and periplasmic volume.

When a solution exists, this solution is unique (see ‘‘Sup-

plementary material’’).

PVmin

ap

isdisplayed asafunction

Vm= Vap

pl(Fig. 1; Eq. S10 in ‘‘Supplementary

pl(Eq. S11 in ‘‘Supplementary material’’).

minthat depends on various parameters of the specific

Fig. 1 Qualitative representation of cell volume concepts. Solid line:

Vap, the apparent cell wall enclosed volume. Dotted line: Vm, the

membrane enclosed volume. Vm

becomes zero and Vm

is the volume where negative hydrostatic

effects or matrix potential effects become important, corresponding to

hyperosmotic shock levels of cP=0

and cs

the stress and volume thresholds, respectively, where Vmdetaches

from Vap. Vplis the periplasmic volume. Vbis the minimum solid

cytosolic volume. Note, that it is not necessarily true that cs

Vm

membrane in different states after hyper-osmotic shock

P=0is the volume at which the turgor

s

ee, respectively. cpl

eand Vap

plare

e\cpl

eand

s[Vap

pl. The diagrams of the cells illustrate the cell wall and the

Fig. 2 Illustration of turgor and matrix potential. Turgor (black line),

which has an initial value of P0at the initial turgid cell volume of

100%, is displayed on the positive y-axis. When the cell shrinks, the

turgor decreases to zero at the volume Vm

‘‘one-sided turgor model’’. The matrix potential or other effects that

reduce the water potential are displayed on the negative y-axis. These

are assumed to become important below a specific threshold volume

Vm

the dashed line is referred to as the ‘‘two-sided turgor model’’

P=0. This is referred to as the

s. The model represented by the solid line up to Vm

stogether with

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Model discrimination

Using Eq. 2 together with a model for Vpl(Eqs. S10, S11 in

‘‘Supplementary material’’) and a turgor model (Eq. 3;

Eqs. S7, S8 in ‘‘Supplementary material’’), a solution for

Vap

sured, Eq. 2 can be fitted to the data by estimating the

respective parameters (Table S1 in ‘‘Supplementary mate-

rial’’), e.g., by minimizing the sum of squared residuals

(SSR)

X

where x = (x1, x2, …, xm) denote the parameters that are

to be estimated and that depend on the model (Table S1 in

‘‘Supplementary material’’), riis the standard deviation of

the measurements and n is the number of data points. For

the sake of simplicity, we assume that the measurement

errors are independently normally distributed. Thus, the

SSR becomes a maximum-likelihood estimator and we can

use the Akaike’s information criterion corrected for small

sample size (AICc) to rank and, thus, discriminate the

models according to their SSR and number of parameters

(Burnham and Anderson 2002):

?

where k is the number of parameters.

The fitting was performed by differential evolution, a

global optimization method (Storn and Price 1997), and

the models were ranked according to the AICc (Eq. 4).

Additionally, for the best model, we performed a Monte–

Carlo analysis by re-sampling the data a hundred times

assuming a normal distribution with the respective stan-

dard deviations for each data point. In this way, means

and confidence intervals for the estimated parameters

could be obtained.

mincan be numerically determined. When Vap

minis mea-

SSRðxÞ ¼

n

i¼1

1

ri

Vmin

ap;iðxÞ ? Vmin

measured;i

??2;

AICc ¼ 2k þ n ln

2pSSR

n

?

þ 1

??

þ2kðk þ 1Þ

n ? k ? 1;

ð4Þ

Data sets

We collected three independent data sets (data sets 1–3 in

Fig. 3; Fig. S1 in ‘‘Supplementary material’’) and took one

data set from the literature (Marechal et al. 1995) (data set

4 in Fig. 3; Fig. S2 in ‘‘Supplementary material’’). The cell

volume was quantified immediately before and after the

hyperosmotic stress, ensuring that the minimum volume

after osmotic shock was measured. For our experiments we

used NaCl as an osmoticum, whereas in the data set from

literature yeast cells were subjected to a wide range

of glycerol concentrations (refer to Marechal et al. 1995

for details). Table 2 summarizes the different data sets.

Experimental details for data sets 1–3 are described in the

‘‘Supplementary material’’.

Results

Parameter fitting and model selection predicts turgor

pressure and other biophysical cell properties

We fitted Vap

alternative models for PVmin

different models (see Table S1 in ‘‘Supplementary mate-

rial’’ for a complete listing). For each data set the best

model according to the AICc is described in Table 3. The

fitted and measured Vap

corresponding predicted turgor pressure.

minfrom Eq. 2 to four different data sets using

and Vpl, resulting in seven

m

minare displayed in Fig. 3, with the

One-sided turgor model is most appropriate

for physiological stress conditions

The three data sets produced for this study were all fitted

best by the one-sided turgor model based on Eqs. 1 and 3,

where only the positive turgor pressure plays a role

(Fig. 2). Even though this is the generally accepted model

(Cosgrove 1981; Marechal and Gervais 1994; Klipp et al.

2005; Gennemark et al. 2006; Meikle et al. 1988), we

tested it with six other models, because to our knowledge it

has never been challenged by other models, nor rigorously

fitted to data. Thus, we show for the first time that effects

resisting cell shrinkage, for example the matrix potential or

mechanical forces, can be neglected under physiological

conditions. Consequently, when the turgor becomes zero

after an external osmotic shock of cP=0

the measured apparent volume Vapis well predicted by the

membrane enclosed volume Vm according to the van’t

Hoff’s equation (Eq. S6 in ‘‘Supplementary material’’).

This means that the cell wall tightly follows the shrinkage

of the membrane enclosed volume. Data set 4, which was

collected by applying extreme stress levels, also favored

the one-sided turgor model. However, at 4 Osm/l plas-

molysis was predicted, with a constant minimal apparent

volume Vap

e

(Table 1; Fig. 3),

plof 65%.

Estimated biophysical cell properties are similar

between experiments

Parameters estimated on the basis of the datasets give a

consistent picture of important biophysical cell properties.

For data sets 2 and 3, where single cell information was

available, we distinguished between small and large cells

and repeated the analysis to test whether cell size had an

effect on the results. We found no significant differences

between small and large cells with regard to the estimated

parameters, which is in agreement with earlier reports

(Martinez de Maranon et al. 1996). The solid volume Vb

was estimated to be between 33 and 49% of the initial

cell volume, which also corresponds to earlier reports

Eur Biophys J (2010) 39:1547–1556 1551

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(Marechal et al. 1995; Meikle et al. 1988). The initial

turgor estimated on the basis of the three data sets with

similar experimental conditions (data sets 1–3) was

0.6 ± 0.2 MPa on average, which confirmed results from

earlier studies (Smith et al. 2000c; Gervais et al. 1996a;

Meikle et al. 1988). The estimated membrane rigidity e was

the parameter that varied the most between experiments

and it turned out to be a sensitive, poorly determined

parameter, which is reflected by the relatively high stan-

dard deviations. However, at least for the three directly

comparable data sets 1–3, the results were similar, giving

an average volumetric elastic modulus of 14.3 ± 10.4 MPa

ranging from 5.7 to 25.8 MPa.

On the basis of our general modeling framework and the

estimated parameters, we could derive several other inter-

esting cell properties. The initial osmolarity within the cell

c0

of the outside medium. Thus, apparently, yeast can main-

tain substantial osmotic gradients between the inside and

outside of the cell, which are equilibrated by turgor. The

volume at which turgor becomes zero was estimated to be

of 93.3 ± 6.3% on average for data sets 1–3. The corre-

sponding external shock for the salt experiments was

0.4 ± 0.2 Osm/l, corresponding to a salt concentration of

roughly 0.2 M NaCl. As this is within the range of con-

centrations in which saturation of HOG pathway activation

iwas estimated to be up to more than twice the osmolarity

Fig. 3 Best fit models and data. The data and the best fit model (solid

line) refer to the left y-axis. The dark gray region depicts the 95%

confidence region of the fitted model. The right y-axis refers to

the estimated turgor pressure (dashed line). The light gray region

depicts the 95% confidence region of the estimated turgor pressure.

The x-axis refers to concentrations of the respective stress agent. The

confidence regions were calculated on the basis of the confidence

intervals of the estimated parameters (Table 3). The four data sets

are described in the ‘‘Methods’’ sections and the ‘‘Supplementary

material’’

Table 2 Data sets used for model discrimination

No. MethodStrainMedium Agent

?CGrowth

c0

e(Osm/l)# n

Refs.

1Particle sizerW303 YPD NaCl30Early log0.26 16 This study

2 Single cells bright fieldBY4741 YNBNaCl20 Log 0.2711 This study

3 Single cells fluorescenceW303 SDNaCl 30Log 0.25 13 This study

4Single cells bright field CBS1171 WickerhamGlycerol 25 Stationary0.86 15Marechal et al. (1995)

c0

ewas measured for data sets 1–3 with an osmometer and calculated for data set 4 from Marechal et al. (1995), # is the number of data points

1552Eur Biophys J (2010) 39:1547–1556

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is observed (Maeda et al. 1995), we hypothesized that the

initial HOG pathway response may be correlated with

turgor.

Loss of turgor correlates with HOG pathway activation

The exact biophysical mechanisms that trigger the activa-

tion of the HOG pathway in yeast have not yet been

determined (Hohmann 2002), but there is evidence that the

turgor is one of the factors involved (Tamas et al. 2000;

Reiser et al. 2003). As our computational analysis provided

us with parameterized models of turgor pressure (Fig. 3;

Table 3), we tested whether there is a correlation between

the loss of turgor for different degrees of osmotic shock

and the extent of HOG pathway activation. Thus, in addi-

tion to cell volume measurements, we collected two inde-

pendent data sets reflecting the degree of HOG pathway

activation depending on the intensity of the hyperosmotic

stress. In the first data set, the extent of Hog1 nuclear

accumulation was measured, whereas the second data set

represents the extent of Hog1 phosphorylation in response

to different NaCl concentrations (Fig. 4, for details about

the measurements refer to the ‘‘Supplementary material’’).

By considering both Hog1 activation (Fig. 4) and the

predicted turgor (dotted line in Fig. 3) as a function of

applied osmotic stress, we could investigate Hog1 activa-

tion as a function of predicted turgor (Fig. 5). With three

different parameterizations for the turgor model (from data

sets 1–3, data set 4 was disregarded, because it used a

different stress agent) and two different Hog1 activation

data sets, we obtained six turgor-Hog1 activation correla-

tions, of which we display three in Fig. 5. The others are

displayed in the ‘‘Supplementary material’’ (Fig. S3).

We fitted a linear relationship between HOG pathway

activation and relative turgor for the three different

parameterized models, corresponding to data sets 1–3, and

the two different HOG pathway activation data sets (black

lines in Fig. 5) by a weighted orthogonal regression. Our

null hypothesis claimed a direct 1:1 linear relationship,

i.e., H0: a ? bx, with (a, b) = (100, -1) (gray lines in

Fig. 5). To test H0we computed confidence regions for

the estimated parameter pairs (a, b) by a Monte–Carlo

analysis. As can be seen from the insets in Fig. 5, in all

cases the hypothesized relationship (gray points) was

within the 95% confidence region of the actual estimated

parameter pair. Thus, the null hypothesis could not be

Table 3 Results of parameter fits of Eq. 2 for the best selected model according to the AICc

Data set 1Data set 2 Data set 3 Data set 4

#2224 Selected model

PVmin

Vpl

k

m

One-sided One-sided One-sidedOne-sided

000 Constant Vap

4

pl

333

Vb

P0

e

Vap

c0

Vm

cP=0

cpl

49.0 ± 3.4 40.2 ± 5.533.0 ± 4.8 40.1 ± 11.1 Estimated parameters

0.5 ± 0.1 0.9 ± 0.2 0.5 ± 0.13.4 ± 1.0

25.8 ± 7.0 5.7 ± 2.011.4 ± 2.128.6 ± 11.3

pl

65.0 ± 5.9

i

0.4 ± 0.10.6 ± 0.10.5 ± 0.0 2.2 ± 0.4 Derived parameters

P=0

98.2 ± 3.1 86.1 ± 6.395.5 ± 6.8 88.9 ± 10.8

e

0.2 ± 0.10.5 ± 0.2 0.2 ± 0.11.9 ± 1.9

e

4.4 ± 2.2

For a complete description of parameters and units refer to Table 1

# Number of best selected model (Table S1 in ‘‘Supplementary material’’), PVmin

(Fig. 1; Eq. S10 in ‘‘Supplementary material’’), k: number of fitted parameters, Vb: solid base volume, P0: initial turgor, e: volumetric elastic

modulus, Vap

to equilibrate turgor, cpl

the data points were sampled from their respective standard deviations indicated in Figs. 3 and 4. The mean of parameters in the lower part are

derived from the mean parameters in the upper part

m: selected turgor model (Fig. 2; Eq. 3), Vpl: selected Vplmodel

plapparent volume where plasmolysis occurs, c0

e: applied external stress to reach Vap

i: initial internal osmolarity, Vm

pl. Estimated parameters are given as mean ± standard deviation from 100 fits where

P=0: volume at zero turgor, cP=0

e

: applied external stress

Fig. 4 Hog1 activation data. Circles: maximum Hog1 nuclear

concentration after osmotic shock. Squares: Hog1 phosphorylation

after 2 min of shock treatment. Data are scaled to the respective

measured maximum

Eur Biophys J (2010) 39:1547–15561553

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rejected. Moreover, the Kendall-rank correlation coeffi-

cient (Kendall and Gibbons 1990) was highly significant

in all cases (P\0.01).

Discussion

Parameterized models of biophysical and mechanical cell

properties become increasingly important for quantitative

and predictive descriptions of cellular behavior. Our

modeling framework, which included the concepts of tur-

gor, membrane elasticity, intracellular osmolarity, osmoti-

cally active volume, non-turgid volume, and solid cytosolic

volume in a coherent and thermodynamically accurate

way, enabled us to parameterize all these concepts con-

sistently, by fitting the relevant parameters to experimental

data. In addition, it enabled us to discriminate between

alternative hypotheses for turgor and volume changes

by including alternative sub-models within the general

framework. Even though the data sets were based on very

different techniques, the overall conclusions are strikingly

similar indicatingthatthe

description is both accurate and universal.

We confirmed previous reports, which suggested an

initial turgor pressure of around 0.6 MPa (Smith et al.

2000b; Meikle et al. 1988). Turgor pressures reported for

yeast differ substantially. Part of this variation might be

because they have never been rigorously fitted to data. In

our study, we obtained similar results for data sets 1–3.

However, the initial turgor estimated from data set 4 was

sixfold higher. We assume that this discrepancy is caused

by different growth phases of yeast cells employed in the

experiments. This explanation is supported by earlier

reports, in which differences in turgor between cells in

different physiological states were observed. Specifically,

a higher turgor pressure was reported for cells in sta-

tionary phase (Smith et al. 2000c; Martinez de Maranon

et al. 1996). After examining different possible explana-

tions (Eq. 3), we confirmed the hypothesis that turgor

pressure is significant only above a specific volume

threshold, below which it can be neglected, as previously

proposed (Cosgrove 1981; Marechal and Gervais 1994;

Klipp et al. 2005; Gennemark et al. 2006; Meikle et al.

1988). Other forces potentially affecting the water

potential, and therefore water flux within the cell, can be

neglected under the studied conditions of osmotic shock

up to 1.5 M NaCl.

We found values for the volumetric elastic modulus e

between 6 and 29 MPa. We could not find any published

values of e for yeast. In a recent study, where yeast cells

were subjected to mechanical compression, values of

proposedmathematical

Fig. 5 Turgor-HOG pathway activation relationship. The x-axes

are relative turgor (%) as predicted by the parameterized models

for different shocks of NaCl (dotted line in Fig. 3). The y-axes are

relative HOG pathway activation (%) according to different shocks

of NaCl (Fig. 4). The black lines are a fitted linear relationship

(y = a ? bx) based on a weighted orthogonal regression. The gray

lines represent the null hypothesis H0: y = 100 - x, i.e., a direct

1:1 linear relationship between relative loss of turgor and relative

HOG pathway activation. The insets are plots of the (25, 50, 75,

90, 95%)-confidence regions of the respective estimated parameter

pair (a, b) of a ? bx with the outermost line being the 95%

confidence region. The black points correspond to the black lines

in the plots, the gray points correspond to the gray lines. The

confidence regions are obtained by a Monte–Carlo analysis with

1,000 runs

1554Eur Biophys J (2010) 39:1547–1556

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Young’s modulus E for the cell wall of around 110 MPa

were reported (Smith et al. 2000a, c). Unfortunately, we

could not relate these values to our volumetric elastic

modulus, because those values for Young’s modulus E

were derived assuming a Poisson’s ratio m of 0.5 and,

hence, the formula E = 3e (1 - 2m) was not applicable.

We noted that cells in stationary phase (data set 4) are

estimated to be more rigid, i.e., have a higher volumetric

elastic modulus, but to a much lesser extent than in Smith

et al. (2000c). The cells used in data set 1 were grown to

saturation, i.e., beyond proliferation, before they were re-

suspended in fresh medium for 1 h before measurements.

It is possible that they were still recovering from satura-

tion phase and therefore had higher cell wall rigidity than

cells from data sets 2 and 3, which were measured at

logarithmic growth phase. This reflects that e is a function of

the physiological state of the cell. Thus, when turgor is being

modeled over longer time intervals than in this study, the

assumption of e being constant might be compromised.

Our modeling results support the notion of a rather

elastic cell wall, because the cell wall tightly follows the

cell membrane on shrinkage, and plasmolysis only occurs

under extremely high stresses. There is experimental evi-

dence from mammalian cells that our model of an elastic

membrane/cell wall surrounding a viscous medium is an

over-simplification (Wang et al. 2001). However, our

model proved sufficient to explain our data. More com-

plicated models were not supported by the data in respect

of the AICc. It is known that upon osmotic shock the actin

cytoskeleton in yeast de-polymerizes (Brewster and Gustin

1994). This is in line with our results in which effects of the

skeleton, which we included into the model of the matrix

potential, were refuted by the model discrimination. A

recent compression study of mammalian cells also ruled

out the cytoskeleton as being responsible for the increase in

cell stiffness as the volume decreases (Zhou et al. 2009).

Other authors identified the concentration at which

plasmolysis occurs as approximately 1 Osm/l (Arnold and

Lacy 1977), but under different experimental conditions.

The predicted plasmolysis point of 65% in data set 4 was

higher than the lowest volumes measured in the other data

sets (44–54%, Fig. 3). Therefore, cells might have been at

the threshold of plasmolysis in those experiments. Because

no data were collected at higher stress levels this effect was

not distinguishable by model selection.

For the first time, we provide evidence that there is

direct link between loss of turgor upon osmotic shock and

HOG pathway activation, not only qualitatively but also

quantitatively. Moreover, the hypothesis of a direct 1:1

relationship between turgor loss and HOG pathway acti-

vation could not be rejected. The physical basis of such a

direct relationship remains elusive, but could for example

be a change in Sln1 and Sho1 membrane protein

conformation as a function of membrane stretch or other

mechanical forces that relate to turgor.

Acknowledgments

ital version of data set 4, Rosie Perkins, Javier Macia, and Karlheinz

Schaber for useful suggestions on the manuscript, and Martina

Fro ¨hlich for measurement support. This work was supported via

several projects funded by the European Commission: QUASI

(Contract No. 503230 to SH, EK, FP and MP), CELLCOMPUT

(Contract No. 043310 to SH, EK and FP), UNICELLSYS (Contract

No. 201142 to SH, EK, FP, MP and MG), SYSTEMSBIOLOGY

(Contract No. 514169 to SH and EK), and AMPKIN (Contract No.

518181 to SH and MG). In addition work was funded by grants from

the Swedish Foundation for Strategic Research SSF (Bio-X to MG),

the Swedish Research Council (project grants to SH and MG), the

Carl Trygger Foundation (to MG), the Science Faculty, University of

Gothenburg (to SH and MG), and the Swiss systemsX.ch (to MP).

We thank Patrick Gervais for providing a dig-

Open Access

Creative Commons Attribution Noncommercial License which per-

mits any noncommercial use, distribution, and reproduction in any

medium, provided the original author(s) and source are credited.

This article is distributed under the terms of the

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