2804 Mol. BioSyst., 2011, 7, 2804–2812 This journal is c The Royal Society of Chemistry 2011

Cite this: Mol. BioSyst., 2011, 7, 2804–2812

A low number of SIC1 mRNA molecules ensures a low noise level in cell

cycle progression of budding yeastwzMatteo Barberis,y*a Claudia Beck,ya Aouefa Amoussouvi,

bcGabriele Schreiber,

a

Christian Diener,aAndreas Herrmann

band Edda Klipp*

a

Received 23rd February 2011, Accepted 7th June 2011

DOI: 10.1039/c1mb05073g

The budding yeast genome comprises roughly 6000 genes generating a number of about 10000

mRNA copies, which gives a general estimation of 1–2 mRNA copies generated per gene. What

does this observation implicate for cellular processes and their regulation? Whether the number

of mRNA molecules produced is important for setting the amount of proteins implicated in a

particular function is at present unknown. In this context, we studied cell cycle control as one of

the highly fine tuned processes that guarantee the precise timing of events essential for cell growth.

Here, we developed a stochastic model that addresses the effect of varying the mRNA amount of

Sic1, inhibitor of the Cdk1–Clb5 kinase activity, and the resulting noise on Sic1/Clb5 balance at the

G1/S transition. We considered a range of SIC1 transcripts number according to our experimental

data derived from the MS2 mRNA tagging system. Computational simulation revealed that an

increased amount of SIC1 mRNAs lead to an amplified dispersion of Sic1 protein levels,

suggesting mRNA control being critical to set timing of Sic1 downregulation and, therefore,

S phase onset. Moreover, Sic1/Clb5 balance is strongly influenced by Clb5 production in both

daughter and mother cells in order to maintain the characteristic time of S phase entry overall the

population. Furthermore, CLB5 mRNA molecules calculated to reproduce temporal dynamics of

Sic1 and Clb5 for daughter and mother cells agree with recent data obtained from more complex

networks. Thus, the results presented here provide novel insights into the influence that the mRNA

amount and, indirectly, the transcription process exploit on cell cycle progression.

Introduction

The budding yeast Saccharomyces cerevisiae was the first

eukaryotic genome completely sequenced, with a sequence of

12 068 kilobases which defines 5885 potential protein-encoding

genes.1 The translational frequency of this genome leads to an

approximate number of 10 000 mRNA sequences of average

size 1500 nucleotides,2 and few intron-containing genes

produce the majority of mRNAs.3 From these numbers, 1–2

mRNA copies actually present per gene can be estimated.

The majority of yeast proteins are produced from less than ten

copies of mRNA, which in turn are produced from 1–2 copies

of a gene per cell.4–6 Therefore, the expression of an individual

gene can vary considerably among genetically identical cells

because of stochastic fluctuations in transcription.7–9

Interestingly, recent evidence clearly showed that transcript

levels of temporally induced genes are highly correlated in

individual cells, whereas transcription of constitutive genes

encoding for essential regulators expressed throughout the

yeast cell cycle is not coordinated because of stochastic

fluctuations.10 This suggests that the transcriptional machinery

is able to control the protein level of cell cycle regulators to

guarantee the precise timing of events essential for cell growth

and viability. However, there is no direct correlation between

mRNA expression and protein expression. Usually, a high

mRNA amount leads to an increased protein level but

also other factors are involved, i.e. translational activity that

depends primarily on ribosome occupancy and ribosome

density,11 mRNA silencing and control of the protein stability.

Thus, the mechanisms at the basis of this control are not fully

understood.

Mathematical modelling has been demonstrated to be a

powerful tool to elucidate the dynamic behaviour of cellular

regulatory networks. Models of cell cycle regulation based on

systems of Ordinary Differential Equations (ODEs) have

a Institute for Biology, Theoretical Biophysics, Humboldt UniversityBerlin, Berlin, Germany. E-mail: matteo.barberis@biologie.hu-berlin.de,edda.klipp@rz.hu-berlin.de; Fax: +49 30-2093-8813;Tel: +49 30-2093-8383

bDepartment of Biology, Molecular Biophysics, Humboldt UniversityBerlin, Berlin, Germany

cMax Delbruck Center for Molecular Medicine, Berlin-Buch, Germanyw Published as part of a Molecular BioSystems themed issue onComputational Biology: Guest Editor Michael Blinov.z Electronic supplementary information (ESI) available: Details of thealgorithm for the GPU architecture, Fig. S1–S4 and Tables S1–S3. SeeDOI: 10.1039/c1mb05073gy These authors contributed equally to this work.

MolecularBioSystems

Dynamic Article Links

www.rsc.org/molecularbiosystems PAPER

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This journal is c The Royal Society of Chemistry 2011 Mol. BioSyst., 2011, 7, 2804–2812 2805

been established in budding yeast12–16 as well as logical,17–19

stochastic16,20,21 and numerical22 approaches. Focused models

of specific cell cycle phases have also been presented.23,24 In all

the models, stochastic fluctuation of the mRNA amount of cell

cycle regulators has not been considered in detail to explain

mechanisms of cell cycle control. However, previous studies

showed that a stochastic approach is favourable for species

present in low numbers, since it considers their discrete

behaviour corresponding to assumed probability distributions

and, therefore, reveals accurate temporal realisations.25,26 This is

valid especially for transcription and translation processes.25,26

To investigate the effect of mRNA fluctuations on cell cycle

progression in budding yeast, we developed here a stochastic

model centered on the mRNA amount of Sic1, inhibitor of the

kinase activity, and the resulting noise at the G1/S transition.

This cell cycle phase is characterized by an irreversible transi-

tion beyond which the cell is committed to divide.27,28 Sic1 is

largely accumulated in the newborn cell being synthesized at

anaphase/telophase and reaches maximal levels in the G1

phase.29,30 It plays a major role by preventing premature

DNA replication and provides timing for the G1/S transition

via inhibition of the Cdk1–Clb5,6 kinase activity.31 When the

Cdk1–Cln1,2 kinase activity targets Sic1 for degradation32

through multisite phosphorylation33 Sic1 is inactivated, thus

allowing activation of DNA replication by Cdk1–Clb5,6.34,35

Time course data of Sic1 and Clb5 protein levels measured

in elutriated cells are available,24 showing the competition

between Sic1 and Clb5, since Sic1 levels decrease at higher

Clb5 concentration. In cells grown on glucose, Clb5 reaches its

half-maximal level at about 80 min, which represents the

Sic1/Clb5 intersection point.24 The nuclear Cdk1–Clb5 initiates

then DNA replication.

In our stochastic model, we investigated the dynamics of the

Sic1/Clb5 intersection point, focusing on the effect of fluctua-

tion of the SIC1 mRNA amount. We considered a range of

SIC1 transcripts number according to our experimental data

derived from the MS2 mRNA tagging system. Our simulation

results reveal that an increased number of SIC1 mRNA

molecules lead to an amplified dispersion of Sic1 protein

levels, suggesting that the mRNA amount is critical to regulate

the G1/S transition. Moreover, Clb5,6 production and Sic1

degradation for setting the Sic1/Clb5 balance have been

calculated for the budding yeast population, by employing

cell cycle phase lengths and volumes for both mother and

daughter cells as reported.36,37 In fact, our simulations give

predictions of the optimal CLB5 mRNA molecules necessary

to reach the Sic1/Clb5 intersection point in both mother and

daughter cells, in agreement with recent data obtained from

more complex networks.16 All together, our findings provide

novel insights into the influence of the mRNA amount on cell

cycle progression in budding yeast.

Materials and methods

Cloning of tagged SIC1 strain

BY4741 strain of S. cerevisiae (EUROSCARF; MATa

his3D1 leu2D0 met15D0 ura3D03) has been used in the study.

Plasmid pMS2CPGFP (x3), which expresses MS2–CP fused to

a GPF triplet, and plasmid pLOXHIS5MS2L were also used.

Cloning of tagged SIC1 strain was performed following the

m-TAG gene-tagging procedure using primers A and B.38

Primer A: 50GCCAAAGGCATTGTTTCAATCTAGGGA

TCAAGAGCATTGAAACGCTGCAGGTCGACAACCC3 0

Primer B: 50TAAAATATAATCGTTCCAGAAACTTTTT

TTTTTCATTTCTGCATAGGCCACTAGTGGATC3 0

Yeast growth conditions

For fluorescence microscopy, cells were grown overnight at

30 1C in liquid synthetic dextrose medium containing 2% glucose

(w/v). For induction of MS2–CP fused to GFP (3�) complexes,

cells were shifted tomedium lackingmethionine for 1.5 h at 30 1C.

Yeast cells were fixed by adding 10% (w/v) formalin solution

(Sigma) to a final concentration of 4% (w/v) for 45 min at 25 1C.

Yeast samples were then washed from fixative, resuspended into

PBS and stored at 4 1C before microscopic measure acquisition.

Image acquisition and data analysis

Images were acquired on an Olympus IX81 epifluorescence

microscope with a UPlanApo 100�, 1.35 numerical aperture

oil-immersion objective (Olympus). An HBO 100 watts light

source was used for illumination with a U-MWNiba filter

(Olympus). The excitation and emission wavelengths were

480 nm and 530 nm, respectively. Vertical stacks of 21 images

with a z step size of 0.2 mm were acquired using a Clara E

Interline camera (Andor) with a 6.45 mm pixel size CCD.

Metamorph (Molecular Devices) software platform was used

for instrument control, image acquisition and to reduce three-

dimensional fluorescence image stacks to two-dimensional images

by maximum intensity projection along the z-axis.

Computational model formulation

The stochastic model of the G1/S transition has been imple-

mented with Cain, a software built to simulate chemical model

systems (http://cain.sourceforge.net/), by using mass action

kinetics for all reactions. Rate constants for reactions re3 to

re6 were taken from the G1/S network reported in the literature24

and translated into stochastic rate constants according to the

procedure described by Gillespie.39 Consequently, k3 and k5were taken directly having already the right dimension, k4 was

divided by the volume of the cell and the Avogadro constant

and k6 was multiplied by 1 (k6a), 10 (k6b) or 100 (k6c) CLB5

mRNAmolecules assuming a first-order reaction. Besides, para-

meters k1 and k2 were used to generate a specific SIC1 mRNA

amount. A volume of about 65 fl has been considered and divided

into 25 fl and 40 fl for daughter and mother cells, respectively

(http://bionumbers.hms.harvard.edu/bionumber.aspx?s=y&id=

100452&ver=1&lnsh=1; http://bionumbers.hms.harvard.edu/bio

number.aspx?s=y&id=101794&ver=7&lnsh=1).40 Initial mole-

cule numbers for Sic1 protein and SIC1mRNAwere taken from

the literature37 and divided between daughter and mother cells

according to 25/65 and 40/65 ratios. To produce specific realisa-

tions of the stochastic model, we considered production of

Clb5,6 after a characteristic time for G1 phase length (tG1).

We implemented this assumption in Cain by setting events and

the first reaction method.41 Moreover, we set k6b = 3 min�1 and

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2806 Mol. BioSyst., 2011, 7, 2804–2812 This journal is c The Royal Society of Chemistry 2011

determined 10 trajectories for a time of 250 min. The number of

frames was set to 125.

Calculation of coefficients of variation

The coefficient of variation is a statistical measurement for

dispersion of data points around the mean. It is defined for a

mean m (m a 0) and a standard deviation s by the ratio

CV ¼ sm, also called the weighted noise to confront two data

series with drastically different means. We compared the

coefficient of variation of SIC1 mRNA (CV1) with the

coefficient of variation of Sic1 protein (CV2). Furthermore,

we defined a ratio Q ¼ CV2CV1

as the percentage of how much

larger is the CV of Sic1 protein compared to the CV of

SIC1 mRNA.

In order to simulate a large number of trajectories for

varying parameters (initial number of SIC1 mRNAs, k2) we

ported a fast implementation of the Gillespie algorithm41 to a

Graphics Processing Unit (GPU) (for details of the algorithm

for the GPU architecture see Fig. S1 and S2 in ESIz).Modifying some aspects of the algorithm in order to fully

utilize the hardware of the GPU together with efficient

memory management we ended up with an algorithm which

enables us to simulate more than 1.8 billion reactions

per second. 1 00 000 trajectories were calculated for each

simulation for a daughter cell, and mean and standard devia-

tion were obtained at each time point. To estimate the value of

Q and CV2, the value corresponding to the first observed peak

was always selected in the case of multiple local maxima.

Implementation of cell growth

The cell growth for both daughter and mother cells was

implemented in the software Cain. Because the volume is not

constant but time dependent, the second-order reaction re4was transformed manually, since it results in the probability of

the molecules to be in a reaction distance and in a volume-

independent probability of interaction. For two molecules

with independent and uniformly distributed positions in a

sphere volume, the probability of both being in reaction

distance is inversely proportional to the volume.42 The other

reactions were not affected by the cell volume and were

retained. Considering that the volume increases roughly

linearly40 and doubles during the G1 phase, the following

equation was derived for the volume V(t) at the time point t1:

Vðt1Þ ¼ V0 þ t1 �V0

tG1

where tG1 is the duration of the G1 phase and V0 the initial

volume. Hence, the volume at tG1, V(tG1), is two times V0. On

this basis, the volume of the daughter cell at a time point t1 is

equal to:

VD(t1) = 25 fl + t10.68 fl min�1

and the volume of the mother cell is equal to:

VM(t1) = 40 fl + t12.5 fl min�1

Thus, the probability results in: Pðre4; t1Þ ¼ Sic1 � Clb5 � k4Vðt1Þ

Nevertheless, the probability for reaction re4 is very high

because of the order of magnitude of volumes (denominator),

hence the Sic1–Clb5,6 complex forms instantaneously.

Results

Detection of SIC1 mRNA molecules in single cells

Results frommicroarrays studies on quantification of transcripts

in yeast cells revealed a common tendency for most of mRNAs

to be present less than 2 copies per cell.4 Moreover, many genes

seem to be transcribed less than once during an entire cell

cycle.43 In addition, a recent work that used the single cell

detection Fixed In situ Hybridization (FISH) method showed

that cells contain a number of mRNA molecules in the range

0–10 for many genes involved in cell cycle regulation.10

To obtain an estimation of SIC1 transcript abundance in single

yeast cells the in vivoMS2 mRNA tagging system was used. This

method integrates a series of hairpin loops from MS2 bacterio-

phage in the 30 untranslated region (30UTR) of the SIC1 gene by

homologous recombination.38,44 The hairpins are binding sites

for a MS2 coat protein (MS2–CP) fused to a triplet of the GFP

(MS2–CP–GFP(3�)) complex that are expressed in yeast cells.

To confirm that the genomic manipulation of the 30UTR does

not interfere with Sic1 expression and cell cycle progression,

growth rates of the MS2 loop containing strain in comparison

with wild type strain were measured and equal growth rates were

revealed (data not shown). The binding of MS2–CP–GFP(3�)complexes on the mRNA results in the accumulation of fluores-

cence that allows visualization and quantification of transcript

abundance in single cells. The population was followed in a

logarithmic growth phase and contained yeast cells at different

stages of the cell cycle. This population of unsynchronized

cells—carrying SIC1 modified with the MS2 mRNA tagging

system—showed only a low number of fluorescent granules in

each cell. Fig. 1A shows that the majority of the unsynchronized

cells contained 0 or 1 mRNA fluorescent granule. However, some

of the cells contain several mRNA fluorescent granules, up to 7

(not shown).

Implementation of the stochastic model

The stochastic model of the G1/S transition consists of six

basic reactions (Fig. 2, re1 to re6) and an additional degrada-

tion rate of Sic1 (re7) that is included and analysed during

later simulations. The reaction re1 describes the transcription

process to produce SIC1 mRNA and degradation of SIC1

mRNA occurs in reaction re2. The reaction re3 describes the

translation from SIC1 mRNA to Sic1 protein. Furthermore,

in reaction re4, Sic1 forms a protein complex with cytoplasmic

Cdk1–Clb5,6 (here indicated as Clb5,6 for simplicity since

Cdk1 is present during the whole cell cycle at a non-limiting

level and, thus, is supposed to be always available). Sic1 binds

to Cdk1–Clb5,6 and transports it from cytosol to nucleus.45

The distinction between cytosolic and nuclear Clb5,6 is retained

in this model, but transport over the nuclear membrane is

surmised through the complex formation. Upon transport into

the nucleus, the Sic1–Clb5,6 complex generates free nuclear

Clb5,6 (reaction re5). The free nuclear Clb5,6 is able to initiate

DNA replication. The cytoplasmic Clb5,6 is produced by

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reaction re6. The reaction re7 accounts for Cdk1-mediated

degradation of Sic1.33,46 In the model, Sic1 is not recycled in

the cytoplasm after titration into the Cdk1–Clb5,6 complex,

hence we always refer to Sic1 as its cytoplasmic form. The best

values for the parameters of reaction rates re6 and re7 to fit the

Sic1/Clb5 intersection point are calculated in later simulations.

The equations describing the stochastic model are listed in

Table S1 in ESI.z

Temporal dynamics of the stochastic model

With the purpose to address the dynamics at the G1/S transition,

we examined the stochastic behaviour of Sic1 and Clb5 for a

single yeast daughter cell. As mentioned above, Sic1 is present

at its maximum level at the beginning of the G1 phase, therefore

in the model an initial value for Sic1 has been set, whereas

the other species involved were set to 0. Sic1 was initialised

with 284 molecules, a value derived from its total number of

molecules found in budding yeast,47 and recalculated for the

size of a daughter cell (about 25 fl). Furthermore, the initial

number of SIC1 mRNA molecules was varied between 0 and

10 during later simulations, according to our experimental

observations (Fig. 1).

The rate constants from k3 to k6 were taken from the

published G1/S network24 and then recalculated to generate

stochastic rate constants, as described in Materials and

methods (see Table S2 in ESIz). Clb5,6 production is included

after 37 min, which represents the length of the G1 phase (tG1)

determined experimentally for newborn daughters.37 We fixed

the rate constant of reaction re6 to the parameter z, which is set

to 0 at the beginning of the simulation and changed to 3 min�1

at tG1 as reported.16

An advantage of stochastic modelling is to observe potential

behaviours of systems that can result in prominent differences.

In order to investigate the variability in the system perfor-

mance, it is useful to compare directly single trajectories.

To represent the possible individual realisations of the

stochastic model, Fig. 3 shows 10 possible trajectories of each

species in the model during a simulation time of 250 min.

Fig. 3A presents realisations exemplarily for 1 initial SIC1

mRNA molecule and an average mRNA level equal to 3,

generated by the ratio between mRNA production (k1) and

degradation (k2) rate constants. A variance is denoted between

trajectories, however the 10 curves show a similar trend, which

is described by mean and standard deviation (Fig. 3B). The

initial Sic1 amount (blue curve) decreases quickly after the

production of nuclear Clb5,6 (pink curve), due to the increase

of the Sic1–Clb5,6 complex formation. After about 170 min

Sic1 converges to 0, because in the model Sic1 is not recycled

in the cytoplasm after titration into the Cdk1–Clb5,6 complex,

and cytoplasmic Clb5,6 (green curve) begins to accumulate,

while the slope of nuclear Clb5,6 gradually diminishes, indi-

cating that at this point Sic1 is the limiting factor for the

appearance of nuclear Clb5,6. Fig. 3C provides a detailed view

of the SIC1 mRNA performance. During the simulation time

of 100 min, the trajectories split up from 1 initial molecule to a

distribution between 0 and 3 molecules. From 100 to 250 min,

the distribution reveals a tendency to a mean value of about

3 molecules, which is derived by the ratio k1/k2 = 3 (Fig. 3C).

Fig. 1 Visualization of the number of endogenous SIC1 mRNA

molecules in single yeast cells. Fluorescence microscopic image (A)

and Differential Interference Contrast (DIC) (B) of cells genetically

modified to integrate MS2 loop sequences between the coding

sequence and 30UTR of SIC1 gene and transformed with a plasmid

expressing MS2–CP–GFP(3�). (A) Fluorescent granules represent

MS2–CP–GFP(3�) enriched at the MS2 loop sequences of SIC1

mRNA which provide binding sites for the vector encoded MS2–CP

proteins. Maximum intensity projection of the fluorescent image

stacks is shown. (B) Corresponding DIC image of the focused plane.

Scale bar represents 5 mm.

Fig. 2 Schematic representation showing the biochemical reactions

of individual molecular species in the stochastic model.

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Effect of initial SIC1 mRNA molecules on S phase onset

With the purpose to examine how fluctuations of the mRNA

affect protein production during the translation process—

which finally influences the onset of DNA replication—

the values for mean (m) and standard deviation (s) of SIC1

mRNA and Sic1 protein were used to calculate two coefficients

of variation. The coefficient of variation (CV), also called

weighted noise, can be considered a measure to compare the

degree of variation of two data sets with drastically different

means. In our case, CV1 represents the coefficient of variation

of SIC1 mRNA whereas CV2 the coefficient of variation of

Sic1 protein. The simulations were run each time by generating

1 00 000 trajectories. The ratio Q = CV2/CV1 was calculated

to compare the CVs, i.e. weighted noise of the protein com-

pared to weighted noise of the mRNA.

Fig. 4 shows the simulated dynamics of Sic1 (blue curve),

CV2 (pink curve) and Q (red curve) exemplarily for a k1/k2ratio of SIC1 mRNA equal to 3 and selected initial SIC1

mRNA molecule numbers (simulations of the complete range

of initial mRNAmolecules from 0 to 10 are reported in Fig. S3

in ESIz). It is observed that Sic1 (blue curve) decreases linearly

after 37 min (tG1), when nuclear Clb5,6 is produced, and it

drops exponentially when becoming smaller than the molecule

number at the Sic1/Clb5 intersection point. Then, Sic1 is the

limiting factor for complex formation (reaction re4), since the

probability that a second-order reaction takes place consists

of: (i) the number of molecules of Sic1, (ii) the number of

molecules of Clb5,6 and (iii) probability that they interact

expressed by k4. Hence, there is a saturation effect for re4 after

which Sic1 drops below the saturation point. Therefore, the

probability that re4 takes place decreases together with the

decrease of Sic1, hence together with (i) and (iii), and Sic1

converges to 0. This leads to a decrease of its standard

deviation (green curve), too. Consequently, the curve of

CV2 and, thus, Q rises exponentially during this time and

the peak of CV2 occurs when Sic1 is roughly 0. Moreover,

CV2 and Q increase, and the peak later, with increasing initial

SIC1 mRNA molecules. The profiles of CV2 and Q are

similar, whereas CV1 remains constant during the simulations

(not shown) due to the fact that SIC1 mRNA is only affected

by production (k1) and degradation (k2) rate constants.

Fig. 3 Simulated dynamics of the system generating 10 trajectories.

(A) Sic1 (blue) decreases after production of nuclear Clb5,6 (pink)

and, when its levels are close to 0, cytoplasmic Clb5,6 (green)

accumulates. The crossing between blue and pink curves is the Sic1/

Clb5 intersection point. (B) Mean and standard deviation calculated

from the 10 trajectories shown in panel A. (C) Fluctuations of SIC1

mRNA molecules and mean (dark red).

Fig. 4 Simulated dynamics of mean (blue) and standard deviation

(green) of Sic1, of CV2 (pink) and Q (red) for 2 (A), 6 (B) and 10 (C)

initial SIC1mRNAmolecules at a ratio of k1/k2 equal to 3. The curves

of CV2 and Q were multiplied by 50 for purpose of visualization.

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Fig. 5 shows the relation between Q, the initial SIC1 mRNA

molecule number and different k1/k2 ratios of SIC1 mRNA

(the graph for CV2 is reported in Fig. S4 in ESIz). As observed

in Fig. 4, the profiles of CV2 and Q reveal a similar trend,

showing an increase of Q (Fig. 5A and B) and CV2 (Fig. S4A

and B, ESIz) together with the initial SIC1 mRNA molecule

number. Moreover, there is also a clear increase for higher k1/k2ratios of SIC1 mRNA. One possible explanation of this result

is that with a higher mRNA level, the probability of the

translation reaction for the SIC1 gene to occur is increased.

It is known that the translational activity depends generally on

ribosome occupancy and ribosome density.11 Nevertheless, if

the translation reaction is considered from a stochastical point

of view, the probability to produce Sic1 depends above all on

the SIC1 mRNA amount and on the rate constant of this

reaction. The results reveal that high SIC1mRNA levels lead to

an amplified dispersion in the amount of the Sic1 protein. Since

the behaviour of the system is time dependent, peaks of CV2

and Q are shifted in time according to the initial conditions (see

Fig. 4), hence Fig. 5A does not refer to a specific time instant.

Temporal profiles of CV2 andQ peaks are reported in Fig. S5A

and B (ESIz). From Fig. 5B, it is observed that the weighted

noise of Sic1 protein ranges between 200 and 2400% of the

SIC1 mRNAs weighted noise, which agrees with the indication

of an amplified dispersion of the protein. Therefore, a low

number of SIC1 mRNA molecules ensure a low noise level,

which is apparently more dependent on the k1/k2 ratio of SIC1

mRNA than on the initial SIC1 mRNA molecule number.

Interplay between Clb5 production and Sic1 degradation in

setting S phase onset

In the second part of our analysis, cell growth was introduced

into the stochastic model for both a single daughter cell and a

single mother cell (see Materials and methods). Volumes of

25 fl and 40 fl were used as initial values for daughter and

mother cells, respectively, as reported.40 Proportionally to the

volume, the mother cell has a larger initial amount of Sic1 than

the daughter cell. Moreover, we considered duration of the G1

phase equal to tG1D = 37 min for the daughter cell and to

tG1M = 15.6 min for the mother cells, as previously reported.37

For this reason, in our simulation Clb5,6 production was

initiated after tG1M and tG1D. Initial conditions and rate

constants for both cells are listed in Table S3 in ESI.zTo investigate to which extent the timing of initiation of

DNA replication is influenced by Clb5,6 production, we tested

different rate constant values: 0.3 min�1 (k6a), 3 min�1 (k6b)

and 30 min�1 (k6c) for both daughter and mother cells that

double their volume during the G1 phase.48 In our model, the

total amount of Sic1 is the sum of free Sic1 and Sic1 in the

Sic1–Clb5,6 complex, and no additional degradation is con-

sidered for Sic1. For a low Clb5,6 production (k6a), the

molecule number of Sic1 protein (blue curve) increases for

both daughter (Fig. 6A) and mother (Fig. 6B) cells. This is

Fig. 5 Relation between Q, the initial SIC1mRNAmolecule number

and different k1/k2 ratios of SIC1 mRNA shown with a three-

dimensional representation (A) and a bi-dimensional representation

(B). In panel B, dash-dotted, dashed and solid lines represent the ratios

k1/k2 from 1 to 6.

Fig. 6 Simulated dynamics of a daughter cell (left column) and a

mother cell (right column) for different Clb5,6 production rate

constants: k6a (A, B), k6b (C, D) and k6c (E, F). Sic1 (blue), cytoplasmic

Clb5,6 (green), Sic1–Clb5 complex (light blue) and nuclear Clb5,6

(pink) are shown. Note the different scales of the y-axis for daughter

and mother cells.

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2810 Mol. BioSyst., 2011, 7, 2804–2812 This journal is c The Royal Society of Chemistry 2011

reasonably due to the fact that formation of the Sic1–Clb5,6

complex is slower compared to Clb5,6 production and, there-

fore, there is no titration of Sic1 into the Cdk1–Clb5,6

complex in order to reduce its levels. Sic1 is consumed solely

by complex titration and without additional Sic1 degradation,

thus resulting in the increase of its levels which is normally not

observed in living cells. Since only a small part of Sic1 is used

to form the Sic1–Clb5,6 complex, and as well nuclear Clb5,6,

there is no intersection point between Sic1 and Clb5 during

this simulation time. It has to be emphasised that a value of

0.3 min�1 was also used by Tyson’s group because it generated

the best fit to their experimental data.16 By increasing the

Clb5,6 production rate (k6b, Fig. 6C and D; k6c, Fig. 6E and F),

Sic1 is titrated into the Cdk1–Clb5,6 complex and pro-

gressively downregulated whereas cytosolic Clb5,6 cannot be

completely consumed anymore. Sic1 becomes a limiting

factor of Sic1–Clb5,6 complex formation (reaction re4) and

cytosolic Clb5,6 accumulates when Sic1 converges to 0. While

converging to 0, Sic1 shows an exponential behaviour, which is

also observed by semi-log plots reported in Fig. S6C (ESIz).Hence, when Sic1 decreases, the production rate of nuclear

Clb5,6 diminishes too (Fig. 6C–F). Therefore, to reach the

Sic1/Clb5 intersection point for precise timing of the G1/S

transition, the Clb5,6 production rate has to have a value

between k6b and k6c for both daughter and mother cells, in the

case that Clb5 would be the only factor regulating Sic1 levels.

However, other processes influence the decrease of Sic1

during cell cycle progression.32,33 For this reason, an addi-

tional degradation rate for Sic1 was included (reaction re7).

As a consequence, values for the Sic1 degradation rate (k7) and

Clb5 production rate (k6) were found such that the temporal

behaviour of Sic1 and Clb5 matched previous experimental

results,24 yielding a Sic1/Clb5 intersection point at around

80 min. Therefore, we estimated the best parameters for k6 and

k7 which produce a Sic1/Clb5 intersection point in a specific

time window. The simulation time we considered started 20 min

after tG1 (tG1D = 37 min for the daughter cell, tG1M = 15.6 min

for the mother cell), which represents the middle point of the

S phase where half of the maximal Clb5 level is reached,24 and

ended after 20 min, to consider the duration of the S phase

(typically of about 40 min). Thus, we searched for Sic1/Clb5

intersection points within a time window of 20 min for both

mother and daughter cells. The result of multiple simulations is

shown in Fig. 7. The red dots show values of k6 and k7 for both

daughter (Fig. 7A) and mother (Fig. 7B) cells that permit to

match the intersection point. From this analysis, we derived

that the Sic1/Clb5 intersection point is reached for initial

mRNA molecule numbers of CLB5,6 equal to about 9 mole-

cules for the daughter cell and about 15 molecules for the

mother cell. These values match with the ones computed from

asynchronous populations of mother and daughter cells as

recently published.16 Values of k6 = 2.9 min�1 and k7 =

0.0065 min�1 for the daughter cell and k6 = 4.6 min�1 and

k7 = 0.007 min�1 for the mother cell ensure optimal inter-

section points. Therefore, we found that a Sic1 degradation rate of

about 0.007 min�1 leads to the best parameter choice matching

the timing of the Sic1/Clb5 intersection point for both daughter

and mother cells. The similarity of the plots confirms the

available experimental data,36 which show that daughter and

mother cells are characterized by the same timing during the

cell cycle, except for different growth rates and lengths of the

G1 phase. In our case, differences in the G1 phase were balanced

in the model by using distinct initial conditions (growth rates,

volumes) as well as times of Clb5,6 initialization.

Discussion and conclusions

Stochastic approaches are favourable to investigate the behaviour

of cellular species present in low numbers, especially for those

involved in transcription and translation processes.25,26 In fact,

it has been shown that expression of a gene can vary among

genetically identical cells because of stochastic fluctuations in

transcription.7–9 Recent evidence showed that transcription of

constitutive genes encoding for essential regulators expressed

throughout the yeast cell cycle is not coordinated because of

stochastic fluctuations.10 Moreover, experimental observations

on single cells allowed us to measure the noisiness of the G1/S

transition in a population of budding yeast cells.37,49 Despite

the importance of transcriptional events, existing mathematical

models of cell cycle regulation do not consider stochastic fluctua-

tion of the mRNA amount.

In the present work, our aim was to investigate the G1/S

transition in the cell cycle reproducing its essential dynamics

by generating a stochastic model that considers only the

balance between two key components: the cyclin-dependent

inhibitor, Sic1, and the activator of DNA replication,

Fig. 7 Representation of combinations of k6 and k7 for daughter

(A) and mother (B) cells. The red dots show possible combinations

yielding the Sic1/Clb5 intersection point within the time window

considered for the analysis and for possible Sic1 molecule numbers.

The blue dots match with the time window and the green dots match

with the molecule numbers.

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This journal is c The Royal Society of Chemistry 2011 Mol. BioSyst., 2011, 7, 2804–2812 2811

Cdk1–Clb5,6 (referred to as Clb5,6 for simplicity). Our simu-

lations reveal a critical effect of the SIC1 mRNA molecule

number on the Sic1/Clb5 intersection point and, therefore, on

timing of the G1/S transition. In fact, high SIC1mRNA levels

shift the Sic1/Clb5 intersection point as well as complete

extinction of Sic1 to a later time. In detail, high initial SIC1

mRNAmolecules as well as a high k1/k2 ratio of SIC1mRNA

lead to an amplified dispersion of the different realisations

(trajectories) for Sic1 (increasing weighted noise CV2 and Q),

producing more Sic1 protein and Sic1–Clb5,6 complexes until

the total amount of Sic1 becomes 0. Consequently, higher

initial SIC1 mRNA molecules influence the onset into the

S phase by delaying Sic1 production for the following cell cycle.

Therefore, a low number of SIC1 mRNA molecules ensure a

low noise level providing a robust timing of cell cycle progres-

sion. This result is supported by our experimental data, which

indicate a range of SIC1 transcripts between 0 and 7. Due to

these findings, additional considerations can be made. The

timing of Sic1 extinction might be important to regulate its

level during the next cell cycle, i.e. an earlier decrease of Sic1

might result in an additional time to produce more Sic1

protein. Consequently, the SIC1 mRNA amount could influ-

ence the production of the next Sic1 wave. The potential

regulatory mechanism through which Sic1 exploits this regula-

tion could be by affecting its own transcription factor Swi5.

Swi5 exploits its role at the exit from mitosis, where it

promotes SIC1 production in anaphase/telophase.30,50,51 This

could be an interesting mechanism of a possible regulatory

property of Sic1 that was not expected before.

Importantly, our simulation results reveal novel insights of

the Sic1/Clb5 balance on the timing of S phase onset. In order

to address this point, simulations of a daughter and a mother

yeast cell were considered by including cell growth, appro-

priate volumes and cell cycle phase lengths as reported.36,37

Our findings indicate that if Sic1 was only downregulated by

the available Clb5,6, the production rate of Clb5,6 would

range between 3 and 30 min�1 for both daughter and mother

cells. A value of 0.3 min�1 was used in the stochastic simula-

tions of a recent cell cycle model from Tyson’s group,16

whereas a value of 0.32 min�1 was introduced in a kinetic

model of the G1/S network.24 In the present work, the value of

0.3 min�1 was considered, which represents the production

rate for 1 molecule of CLB5 mRNA. Nevertheless, a value

between 3 and 30 min�1 would imply that there must be

between 10 and 100 CLB5 mRNA molecules, which appears

to be not realistic. Indeed, as known, Clb5 is not the only

regulator of Sic1 downregulation. Considering this fact,

including an additional Sic1 degradation rate (a value adjusted

to 0.007 min�1) yielded best results by shifting the Sic1/Clb5

intersection point to a reasonable time together with Clb5,6

production rates of 2.9 min�1 and 4.6 min�1 for daughter and

mother cells, respectively. It has to be emphasised that the

optimal parameter for Sic1 degradation is similar for both

daughter and mother cells, despite the value for Clb5,6

production in the mother cell is larger compared to the one

in the daughter cell. This finding agrees with recent sensitivity

analyses showing that the rate of Sic1 degradation is a critical

parameter that influences the setting of the critical cell size

required at the G1/S transition and, therefore, starting of

DNA replication.52,53 In addition, with these parameter sets,

we have been able to derive values for optimal CLB5 mRNA

molecule numbers necessary to reach the Sic1/Clb5 inter-

section point in both mother and daughter cells, which agree

with recent data obtained from more complex network of cell

cycle regulation developed by Tyson’s group.16

Although the stochastic model reproduces correctly the timing

of S phase onset, it cannot provide an explanation for dynamics

of late cell cycle events. In fact, accumulation of Clb5,6 after

decrease of Sic1 levels is not observed in yeast cells due to Clb5,6

downregulation after the S phase by Cdk1–Clb complexes

involved in G2/M regulation, which are not considered in the

present model. This feature as well as further details, i.e. the

mRNA amount for Clb5 and Cln2, main regulators of Sic1

degradation,32,36,46 will be introduced in the future to describe

precise timing of the G1/S transition in a more detailed manner.

Acknowledgements

We thank J. Gerst at the Weizmann Institute of Science,

Rehovot, Israel for providing plasmids used in this study. This

work was supported by grants from ENFIN, a Network of

Excellence funded by the European Commission (contract

number LSHG-CT-2005-518254) and UNICELLSYS (contract

number HEALTH-2007-201142) to E.K. A.H. and E.K.

acknowledge funding by German Research Council (SFB740).

A.A. is funded by the PhD Program of the Max Delbruck

Center for Molecular Medicine, Berlin-Buch.

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