Protein-Level Fluctuation Correlation at the Microcolony Level and Its
Application to the Vibrio harveyi Quorum-Sensing Circuit
Yufang Wang,†* Kimberly C. Tu,‡N. P. Ong,†Bonnie L. Bassler,‡§and Ned S. Wingreen‡
†Department of Physics and‡Department of Molecular Biology, Princeton University, Princeton, New Jersey; and§Howard Hughes Medical
Institute, Chevy Chase, Maryland
gates through active regulatory links. Thus, correlations in gene-expression noise can provide information about regulatory links.
We present what to our knowledge is a new approach to measure and interpret such correlated fluctuations at the level of single
microcolonies, which derive from single cells. We demonstrated this approach mathematically using stochastic modeling, and
applied it to experimental time-lapse fluorescence microscopy data. Specifically, we investigated the relationships among LuxO,
LuxR, and the small regulatory RNA qrr4 in the model quorum-sensing bacterium Vibrio harveyi. Our results show that LuxR
positively regulates the qrr4 promoter. Under our conditions, we find that qrr regulation weakly depends on total LuxO levels
and that LuxO autorepression is saturated. We also find evidence that the fluctuations in LuxO levels are dominated by intrinsic
noise. We furthermore propose LuxO and LuxR interact at all autoinducer levels via an unknown mechanism. Of importance, our
new method of evaluating correlations at the microcolony level is unaffected by partition noise at cell division. Moreover, the
method is first-order accurate and requires less effort for data analysis than single-cell-based approaches. This new correlation
approach can be applied to other systems to aid analysis of gene regulatory circuits.
Gene expression is stochastic, and noise that arises from the stochastic nature of biochemical reactions propa-
a regulatory interaction between two protein molecules,
three questions arise: 1), What regulates what? 2), Is this
regulation positive or negative? 3), How tight is the regula-
tion? In some situations, knowledge about biochemical
properties suffices to answer these questions. However, at
it may not be operational. In such cases, changes in the
stream processes. This may occur because the concentration
of the upstream regulator lies beyond an effective range, or
a critical cofactor is lacking (1,2). A number of techniques
have been developed to address the above questions, such
as the deletion and overexpression methods that are widely
used in molecular biology (3). An alternate approach is to
improve our quantitative understanding of how a gene regu-
functional factors in the circuit.
Gene expression is inherently a stochastic process. As
a result, cells develop nongenetic individuality even under
uniform growth conditions, as has been confirmed by
single-cell microscopy (5–7). Herewe exploit thevariability
of protein levels and the temporal dynamics of this vari-
ability to address the three questions mentioned above. In
steady-state growth, cellular protein levels are assumed to
fluctuate around a mean value. With an active regulatory
link, upstream protein level fluctuations can propagate
downstream. For positive regulation, the downstream level
fluctuates in the same direction as the upstream regulator,
albeit with a time lag. For negative regulation, the down-
stream level fluctuates in the opposite direction to the
upstream regulator, also with a time lag. This time lag exists
because it takes time to dilute and degrade existing mole-
cules while new molecules are synthesized. One can infer
information about the activity level of the regulatory link
and the time lag for fluctuation propagation by investigating
certain temporal correlation functions of protein-level
Correlation techniques have previously been used to
quantify and analyze fluctuations at the molecular level in
biochemical reactions (8,9). The cross correlation reports
the time lag between similar patterns in upstream and down-
stream levels, and thus answers the first question (What
regulates what?). Most of the time, the sign of the correla-
tion where the amplitude is largest answers the second ques-
tion (Is regulation positive or negative?), because a positive
(negative) correlation indicates positive (negative) regula-
tion. The value of the cross correlation can be used to
address the third question (How tight is the regulation?).
Not all downstream level fluctuations can be correlated
with the upstream level fluctuation. For tight (loose) regula-
tion, the correlation between upstream and downstream is
high (low), and downstream components respond strongly
(weakly) to the upstream level changes. Autocorrelation
also carries information. It reveals how long it takes for cells
to eliminate the memory of previous protein levels. The
width of the autocorrelation peak is essentially the timescale
required for a cell with above- or below-average protein
Submitted November 23, 2010, and accepted for publication May 4, 2011.
Yufang Wang’s present address is Life Technologies, Foster City, CA.
Editor: Andre Levchenko.
? 2011 by the Biophysical Society
Biophysical Journal Volume 100 June 2011 3045–3053 3045
levels to relax to average levels. The process of memory
elimination is strongly affected by active regulation, espe-
cially active autoregulation (10).
Fluctuations also arise during cell division when mole-
cules are partitioned stochastically between two daughters.
The resulting noise profiles are remarkably difficult to
separate from gene expression noise (11). Here, we intro-
duce a fluctuation-based approach that is applied at the
level of the microcolony, which is a collection all cells
derived from the same single cell. By taking averages over
a microcolony, we can eliminate partition noise. Compared
with correlations at the single-cell level, correlations at
the microcolony level therefore provide more-relevant
information about gene expression and active regulatory
In this work, we investigated the regulatory pathways in
the bioluminescent marine bacterium Vibrio haryeyi using
fluctuation correlation. V. harveyi communicates by synthe-
sizing, releasing, and detecting the population-dependent
accumulation of extracellular signal molecules called auto-
inducers (AIs) (12–15). This cell-to-cell communication
process, which coordinates collective behaviors, is called
quorum sensing (QS). At the heart of the V. harveyi QS
circuit (Fig. 1 A) are five small quorum regulatory RNAs
called Qrr1–5 (16,17). At low AI concentrations, phosphor-
ylated LuxO (LuxO~P) (3) activates transcription of the
qrr1–5 genes. At high AI concentrations, LuxO is not phos-
phorylated, and transcription of the qrr genes ceases. The
small RNAs Qrr1–5 block translation and destabilize several
mRNA targets, including that encoding luxR (18). LuxR is
the master transcriptional regulator that activates and
represses >70 genes (12,19–21). The V. harveyi QS circuit
produces highly uniform behavior in individual cells, sug-
gesting that the QS circuit has been optimized to synchro-
nize the response to AIs (22). The QS circuit apparently
differs from regulatory loops designed to generate diversity
among members of the population, e.g., in competence and
sporulation in Bacillus subtilis (23,24).
This study focuses on regulation among the luxO, luxR,
and qrr genes. In our experiments, LuxO protein was fused
to yellow fluorescent protein (YFP), and LuxR protein was
fused to mCherry. The fusion proteins preserve the function
of the native proteins. The strains also contain a transcrip-
tional fusion of cfp fused to the qrr4 to report the promotor
activity through the cyan fluorescence (Fig. 1 A). The
dynamics of fluorescence and fluorescent-tagged proteins
in individual cells were followed by time-lapse fluorescence
microscopy (Fig. 1, B–D). We calculated the cross- and
autocorrelations of different fluorescence fluctuation signals
to analyze this regulatory network. We showed that in
V. harveyi, LuxR tightly activates the qrr4 promoter at low
AI levels, which agrees with previous findings (18). We
observed a low correlation between the total YFP-LuxO
level and the qrr4 promoter activity, which helps to eluci-
date how LuxO~P regulates qrr4. We also found that
although LuxO represses its own expression, there was no
signature of this feedback in the measured autocorrelation
under our experimental conditions. Fluctuation of LuxO
concentration is apparently dominated by intrinsic noise.
time-lapse images of gene expression within microcolonies. (A) A portion
of the QS pathway in V. harveyi. The concentration of AIs regulates the
phosphorylation of the LuxO protein. LuxO~P activates production of the
Qrr sRNAs, which repress LuxR protein production by binding and
degrading its mRNA. Together with LuxO~P, LuxR activates the produc-
tion of Qrr sRNAs. LuxO represses its own transcription regardless of its
phosphorylation status. Qrr sRNAs repress LuxO protein production via
direct basepairing with luxO mRNA. Most lines denote regulation native
to V. harveyi. For imaging purposes, LuxO is fused to YFP, and LuxR is
fused to mCherry. CFP is produced together with Qrr4 (gray arrow). (B)
Typical time-lapse images of V. harveyi cells. This series shows the cyan
fluorescence channel (qrr4-cfp) overlaid with the white-light channel. (C)
Image partitioned at the microcolony level. The three cells at the beginning
of image acquisition grew into three microcolonies. (D) Image partitioned
at the single-cell level. Pixels denoting cells or microcolonies are shown in
white; background pixels are shown in black. (E) Typical time traces of
fluorescence per unit area within a microcolony in different channels:
qrr4-cfp (blue), YFP-LuxO (green), and mCherry-LuxR (red). The normal-
ized fluorescence values are very close to unity over time. The thick curves
are the microcolony averages, and the thin curves are from individual cells
within the microcolony.
A portion of the QS circuit in V. harveyi is shown along with
Biophysical Journal 100(12) 3045–3053
3046Wang et al.
At 0, 50 nM, and 1 mM AI, a positive correlation was
mCherry-LuxR level fluctuations. We therefore propose
that LuxO interacts with LuxR at all AI levels via a mecha-
nism other than the currently known small-RNA-based
MATERIALS AND METHODS
both have three fluorescent protein fusions: YFP-LuxO replaces the native
LuxO, mCherry-LuxR replaces the native LuxR, and qrr4-cfp is used to
was performed to disrupt LuxO autorepression. Cells were initially grown
in liquid culture and then continued to grow between a glass coverslip
and an agarose pad on an inverted microscope. Images were acquired in
multiple fluorescent channels as well as in transmitted light at 10-min inter-
vals (Fig. 1, B–D). We analyzed the images to obtain the size and total fluo-
rescence of individual cells and microcolonies at each time point. More
details about the cell strains, microscopy, and image analysis are provided
in the Supporting Material.
To calculate the fluctuation correlation, we define Xi(t) and Yi(t) to repre-
sent any two protein concentrations at time t, and use i to index the single
microcolony or single cell. We also define xi(t) and yi(t) to be any of the
concentration deviations from the expected values. The expected values
and the concentration deviations (fluctuations) are expressed by
XiðtÞ;xiðtÞ ¼ XiðtÞ ? hXðtÞi:
The correlation S and normalized correlation R between the fluctuations x
and y are given by
The normalized correlation matrix Rx,y(t1, t2) defined in Eq. 3 plays a major
role in our subsequent analysis. The time coordinates (t1, t2) serve as the
indices specifying a matrix element of Rx,y(t1, t2). In practice, Sx,x(t1, t1)
and Sy,y(t2, t2) in the denominator are not directly calculated but are extrap-
olated from neighboring elements to exclude imaging noise (Supporting
Ideally, when the regulation process is time-invariant, the values of the
Rx,y(t1, t2) elements depend only on the time difference t1– t2. Hence,
elements that lie on lines parallel to the diagonal have identical values
(these lines are defined by t1– t2¼ t, where t is a constant). In a heat
map of Rx,y(t1, t2) values, all of the contour lines should be parallel to the
diagonal. The two-dimensional correlation heat map Rx,y(t1, t2) can then
be reduced to a one-dimensional correlation curve:
However, when the correlation matrix is constructed from experimental
data, deviations from this simple pattern indicate that the regulation param-
eters have changed over the course of the experiment. Our experiments
were designed to minimize such changes. Data sets in which large regions
of the correlation heat map display horizontal or vertical swaths with nomi-
nally uniform values might be caused by individual cells that failed to
follow the population behavior. Such individual cells and their associated
microcolonies were carefully excluded from the analysis.
Correlation at the microcolony level
Analysis of fluorescence microscopy at the single-cell level
usually requires integration over each whole cell volume.
However, cell volume usually grows during the cell cycle
concentration or fluorescence per unit volume is at steady
cell, the amplitude changes with cell volume. One approach
is to limit the sampling volume to a constant value so it is
not affected by cell growth. Here we introduce an alternative
approach that still utilizes data from the whole cell volume.
The sources of fluctuation in protein levels inside the cell
may be complicated. Themostfundamentalsourceoffluctu-
ation is the stochastic nature of molecule synthesis, degrada-
tion, and dilution. Given fixed synthesis, degradation, and
dilution rates, the steady-state molecule-number fluctuation
inside a fixed volume is Poissonian. Hence, the molecule-
molecule number in that volume. Now consider an exponen-
tially growing volume with the concentration at steady state.
The number SD will be proportional to the square root of
the total volume. Therefore, for the measured concentration
tude is proportional to V?1=2. As a result, in the dynamics of
halved due to normalization (Eqs. S66–S68), whereas the
synthesis and degradation rates are unchanged. Therefore,
for an exponentially growing volume, as in a microcolony,
the fixed-volume correlation results can still be used only
relation of an unregulated stable protein is Rx;xðtÞ ¼ e?b0jtj
at the microcolony level, whereas if it is measured in a fixed
volume, Rx,x(t) ¼ e–bjtj, where b ¼ 2b0¼ log 2 generation–1
(for details, see Supporting Material).
Many proteins are regulated or affected by extrinsic
noise. Their fluctuations are not Poissonian; however, even
a complicated regulation process can be dissected into small
steps. Given fixed rates, the fluctuation introduced in each
step will be Poissonian and can be normalized as above.
The final fluctuation is an integration of the Poissonian
fluctuations of all of the intermediate steps. Thus, in many
cases, even with complicated regulation, protein-level
fluctuations can still be normalized and analyzed as in
the fixed-volume case. Eq. 3 automatically yields the fluc-
tuation correlation normalized to V?1=2as Sx;yðt1;t2Þf
Cross correlation between mCherry-LuxR
We used mCherry-LuxR (red fluorescence) and qrr4-cfp
(cyan fluorescence) to investigate regulation between
Biophysical Journal 100(12) 3045–3053
Protein Level Fluctuation Correlation3047
LuxR and Qrr4. The cross correlation between red and cyan
fluorescence fluctuations was compared only at 0 and 50 nM
AI because at 1 mM AI, the expression of qrr4 is minimal
and the cyan fluorescence is too weak to be resolved from
the cell’s autofluorescence background.
According to Eq. 3, with the assignment X / red fluores-
cence from mCherry-LuxR and Y / cyan fluorescence
from qrr4-cfp, we computed the cross-correlation matrix
RmCherry, CFPbetween fluctuations in the red and cyan fluo-
rescence. Heat maps of the correlation matrixes are dis-
played at 0 nM (Fig. 2 A) and 50 nM AI (Fig. 2 B). In
Fig. 2 A, the diagonal clearly separates two regions with
strikingly different cross-correlationvalues. The high values
in the region above the diagonal contrasts sharply with the
low values below. The enhanced correlation in the former
implies that the qrr4-cfp fluctuation positively correlates
with mCherry-LuxR fluctuations with a time lag. The corre-
lation curve shown in Fig. 2 C (blue solid curve) displays
a broad peak centered near ?1.4 generations. This agrees
latory link (Supporting Material).
By contrast, the correlation matrix at 50 nM AI is mark-
edly lower in value over the entire heat map (Fig. 2 B). The
corresponding correlation curve is roughly symmetric about
tmCherry–LuxR– tqrr4–cfp¼ 0 (Fig. 2 C, green dashed curve).
We attribute this weak, symmetric profile to extrinsic noise.
Thus, at 0 nM AI, mCherry-LuxR tightly and positively
regulates qrr4, whereas this regulation is significantly weak-
ened at intermediate AI concentration (50 nM).
LuxR positive regulation of the qrr promoters was previ-
ously investigated in bulk assays. A co-activator, LuxO~P, is
required for such regulation (16). It was reported that LuxR
directly activates the expression of qrr2, qrr3, and qrr4
genes by binding to their promoters (18,25). Similarly, our
correlation results show that mCherry-LuxR positively
regulates the qrr4 promoter at low AI concentrations,
when YFP-LuxO~P is abundant.
There is yet another regulatory link between LuxR and
Qrr4: specifically, Qrr sRNAs destabilize luxR mRNA. If
Qrr4 strongly represses luxR expression, the cross-correla-
tionvalues might be negative for tmCherry–LuxR– tqrr4–cfp> 0.
one fluorescent protein regulates another, such as mCherry-
LuxR upregulating qrr4-cfp, the correlation time lag is the
time required for relaxation of the upstream protein-level
fluctuation and expression of the downstream protein.
That is the time for protein to be degraded or diluted. When
it comestoRNAinteractions, assuming thatthe time periods
and luxR-mCherry mRNA are similar, the time lag of the
cyan-red correlation due to this regulation is the lifetime of
the RNA molecules. The kinetics of Qrr-mRNA binding
and degradation are much faster than the timescales for
changes in protein levels (26,27). Therefore, the correlation
due to this regulation should have a negative value and
almost zero time lag. However, cells usually also have
extrinsic noise that affects the synthesis of all proteins,
resulting in a positive correlation between almost any two
proteins. Generally, the correlation due to extrinsic noise
has almost zero time lag. Because these two effects act in
opposite directions, they may cancel, resulting in the
apparent lack of correlation between qrr4-cfp and mCherry-
may be the result of this negative regulation.
At the transition between low cell density (LCD) and high
cell density (HCD), LuxO~P and LuxR are simultaneously
present. A previous study of V. cholerae showed that
HapR (a homolog of V. harveyi’s LuxR) dramatically accel-
erates the transition from HCD to LCD (26). In a recent
study, the fluorescence from mCherry-LuxR revealed that
the concentration of LuxR is significant in the LCD state
(28) (Table 1). A strong positive correlation between early
[AI] = 0 nM
[AI] = 50 nM
−3 −2 −1
−3 −2 −10123
cfp and concentrations of two upstream proteins
(mCherry-LuxR and YFP-LuxO) in the WT strain.
(A and B) Heat maps of the correlation matrix
RmCherry, CFPbetween concentration fluctuations of
mCherry-LuxR and qrr4-cfp, at 0 nM AI (A) and
50 nM (B). (A) The qrr4-cfp fluctuation correlates
strongly with that of the earlier mCherry-LuxR,
with a time lag of ~1.5 generation. (B) At 50 nM
AI, such an asymmetric correlation is essentially
absent. (C) Correlation curves at 0 nM of AI (blue
solid) and at 50 nM (green dashed). (D and E) Heat
maps of the correlation matrix RYFP, CFPmeasuring
correlations between fluctuations in qrr4-cfp and
total YFP-LuxO concentrations. The relatively
uniform values for both 0 nM AI (D) and 50 nM
(E) implies a very weak correlation. (F) Correlation
curves at 0 nM AI (blue solid) and 50 nM (green
produced from four independent movies.
Fluctuation correlation between qrr4-
Biophysical Journal 100(12) 3045–3053
3048Wang et al.
mCherry-LuxR levels and late qrr4-cfp levels at 0 nM AI
indicates that LuxR is active and positively regulates the
qrr4 gene at LCD. This finding suggests that in addition to
accelerating the HCD-LCD transition, LuxR activation of
the qrr expression also reduces LuxR concentration at LCD.
Because the qrr promoters are predominantly regulated
by LuxO~P, and not dramatically by LuxR (18), the weaker
regulation between LuxR and qrr at intermediate AI
concentrations (50 nM) may be a consequence of two possi-
bilities: 1), LuxR saturates the binding site in the qrr4
promoter; or 2), the LuxO~P concentration is too low for
the LuxO~P binding site to be occupied. At 50 nM AI, the
significant level of cyan fluorescent protein (CFP; Table 1)
suggests that LuxO~P remains abundant. The CFP level
becomes close to zero only in the limit of a very high AI
concentration (1 mM), when almost all LuxO molecules
are unphosphorylated. Our observation that the regulatory
link between LuxR and the qrr4 promoter weakens consid-
erably at 50 nM AI may therefore reflect saturation of the
LuxR binding sites at the qrr4 promoter. A previous bio-
informatics analysis showed that LuxR binding sites at the
qrr promoters more closely resemble the LuxR consensus
binding site than do LuxR binding sites at other known
LuxR-regulated genes (25). Therefore, the LuxR binding
sites at the qrr promoters are easier to saturate than other
LuxR binding sites.
To summarize, our cross-correlation results show that at
very low AI concentrations, LuxR proteins bind to the
qrr4 promoter to activate the qrr4 gene expression, which
agrees with previous findings. These results raise the possi-
bility that the LuxR binding site at the qrr4 promoter is
readily saturated by LuxR, and thus this regulatory link
becomes less sensitive to LuxR concentration at interme-
diate or higher AI concentrations (50 nM).
Cross correlation between total YFP-LuxO
We now consider cross correlation between the fluorescence
with the assignment X / YFP-LuxO and Y / qrr4-cfp,
we obtained heat maps of the cross-correlation matrix
tively). At 0 nM AI, there is a weak trend of matrix values
being higher above the diagonal than below. At 50 nM AI,
there is no discernible asymmetrical feature about the diag-
onal that would be suggestive of regulation. These heat
maps suggest that the regulatory links between LuxO
concentration and qrr4 expression are weak.
It is known that LuxO~P activates the qrr genes, whereas
unphosphorylated LuxO does not (3,16). LuxO molecules
are phosphorylated through LuxU by AI receptors acting as
kinases in the absence of AI ligands. We expect AI binding
and unbinding and phosphate transfer to be rapid processes
relative to protein turnover. We assume a fixed probability
r that any particular LuxO molecule is phosphorylated.
Withagiventotal LuxO number NLuxO,the LuxO~P number
NLuxO~Pfollows a binomial distribution with hNLuxO~Pi ¼
mean is a measure of how NLuxO~Ptracks NLuxO, because
loose tracking of NLuxO:
rð1 ? rÞNLuxO
. The SD over the
rð1 ? rÞNLuxO
1 ? r
Either a low phosphorylation level or a low NLuxOnumber
would result in NLuxO~Pnot closely tracking NLuxO, which
could explain the low correlation between YFP-LuxO and
qrr4-cfp fluctuations. In the WT strain, the YFP-LuxO level
is estimated to be as low as 8.3 partition units per cell
(Table 1). However, because LuxO is likely to oligomerize
(3), the true copy number per cell is unknown.
Alternatively, if unphosphorylated LuxO also binds to the
qrr promoters but does not activate transcription, then only
the ratio between LuxO~P and LuxO concentrations
controls the qrr transcription. This concentration ratio is
independent of total LuxO levels, and thus the cross corre-
lation between YFP-LuxO and qrr4-cfp is also expected to
be low in this case.
In WT V. harveyi, luxO mRNA is one of the targets of Qrr
regulation. Our chromosomal luxO-yfp fusion has the WT
luxO promoter and ribosomal binding site, and is therefore
lation is barely visible in the correlation matrix, likely for the
same reason that the negative regulation between qrr4-cfp
and mCherry-LuxR is not visible in the correlation matrix.
To summarize, although the qrr promoters are predomi-
nantly regulated by LuxO~P, the total YFP-LuxO amount
correlates poorly with qrr4-cfp expressed.
Autocorrelation of YFP-LuxO
LuxO is known to repress its own expression (27). In
general, a protein can repress its own expression to
concentrations of 0 nM, 50 nM, and 1 mM
Steady-state fluorescence in V. harveyi at AI
0 nM50 nM1 mM
163 5 61
113 5 40
8.2 5 5.3
41.3 5 15.5
101 5 23
90 5 25
92 5 49*
135 5 63*
17.6 5 10.1
69.5 5 16.9
577 5 97
439 5 89
7.6 5 22*
2.8 5 21*
21.6 5 9.2
76.6 5 18.6
670 5 116
634 5 107
The fluorescence units are partition units per cell, which are calculated ac-
cording to Teng et al. (28).
Partition unit can be a dimer or an oligomer if the protein dimerizes or
oligomerizes; 260 cell division events were used for the estimation.
*SD of cell autofluorescence is comparable to that of signal fluorescence,
making the error bars large.
Biophysical Journal 100(12) 3045–3053
Protein Level Fluctuation Correlation3049
accelerate the process of damping large concentration fluc-
tuations and returning the concentration to the basal value
(29–31). A too-low steady-state protein level or lack of an
essential cofactor can result in ineffective repression. At
the other extreme, the steady-state protein level can be so
high that it saturates autorepression, in which case the
mean protein level is repressed compared with a strain lack-
ing autorepression. For both of these cases, the fluctuation
autocorrelation timescales are simply those required for
dilution and degradation of the protein, as if there were no
A LuxO binding site is located upstream from the luxO
gene. Point mutations can impair this binding site and elim-
inate autorepression (the luxO-ar strain) (27). Compared
with the WT strain, the luxO-ar strain produces more
LuxO protein at all AI concentrations. Specifically, in our
strains, when LuxO is substituted with YFP-LuxO, higher
YFP fluorescence is observed in the luxO-ar strain than in
the WT strain (27) (Table 1). We found that LuxO autore-
pression was significant at all AI levels in our experiments,
even in the absence of AI, when LuxO levels are at their
To investigate whether autorepression reduces the corre-
lation time of fluctuations in the LuxO level, we produced
autocorrelation heat maps for fluctuations in the YFP fluo-
rescence in both WT and luxO-ar strains, as shown in
Fig. 3, A and B, respectively. The autocorrelations are com-
pared at the 1 mM AI concentration to minimize the influ-
ence from sRNAs (Fig. 3 C). In fact, the autocorrelation
curves are very similar at all three AI concentrations (data
not shown). The fluctuation timescale is inferred from the
width of the autocorrelation curve. The correlation curve
is most reliable in the central region (t near 0) where the
data density is highest. In this region, the two autocorrela-
tion curves for the WT and the luxO-ar strain overlap well
within the error ranges. This overlap indicates that LuxO au-
torepression is saturated, because active autorepression
would shorten the correlation time of fluctuations.
The autocorrelation curves also overlap well with theoret-
ical prediction for a stable protein without an upstream regu-
lator RðtÞ ¼ e?b0t, where b0 ¼ (log 2)/2 generation
(Fig. 3 C, gray curve; also see Supporting Material), sug-
gesting no active upstream protein regulators. On the
contrary, it was previously reported that another protein
LuxT represses the luxO promoter by binding between
117 and 149 bases upstream of the luxO initiation codon
(32). Any upstream protein regulation noise or extrinsic
noise should result in a broader autocorrelation peak, with
tight regulation producing a wide peak with a round top.
In principle, if an upstream regulator or extrinsic noise
coexists with autorepression, it is possible to obtain a similar
autocorrelation peak as for an unregulated protein. If
wt, 1μM AI
luxO−ar, 1μM AI
wt, 0nM AI
wt, 1μM AI
−3 −2 −10123
τ = t1 − t2
−3 −2 −10123
τ = t1 − t2
YFP-LuxO, qrr4-cfp, and mCherry-LuxR concen-
fluctuation autocorrelation matrixes for YFP-LuxO
in the WT strain (A), YFP-LuxO in the luxO-ar
strain (B), qrr4-cfp in the WT strain (D), and
mCherry-LuxR in the WT strain (E). AI concentra-
curves inferred from A (green dashed) and B (black
autocorrelation for a stable protein with noisy
production (RðtÞ ¼ e?b0jtj, where b0¼ (log 2)/2,
curves inferred from D (blue dashed) and E (red
solid). The lower gray curve is the same as the
retical autocorrelation for proteins whose produc-
tion rate depends on an upstream stable protein
with noisy production, RðtÞ ¼ ð1 þ b0jtjÞe?b0jtj
10 (also Eq. S47). (G–I) Single-molecule fluores-
cence squared SD (s2) versus mean (hNi) at the
beginning of image acquisition for qrr4-cfp (G),
YFP-LuxO (H), and mCherry-LuxR (I). s2is
proportional to hNi if intrinsic noise dominates.
Such a relation is observed for YFP-LuxO but not
for qrr4-cfp or mCheery-LuxR, which are known
to be regulated by multiple upstream factors.
WT 1 mM AI (square), luxO-ar 0 nM AI (cross),
luxO-ar 50 nM AI (triangle), and luxO-ar 0 nM AI
(pentagram). The insets show SD (s) versus mean
(hNi) on a linear scale.
Autocorrelation of fluctuations for
Biophysical Journal 100(12) 3045–3053
3050Wang et al.
fluctuations in the copy number of YFP-LuxO proteins are
dominated by intrinsic noise, one would expect the square
of the fluorescence SD to scale linearly with the total fluo-
rescence. Such scaling should not apply if there is a compa-
rable amount of extrinsic noise. The SD squared sI
total fluorescence intensity I per cell is plotted for qrr4-
cfp, YFP-LuxO, and mCherry-LuxR in Fig. 3, G–I, at the
single-cell level. A clear linear scaling is observed for
YFP-LuxO, consistent with the case that intrinsic noise
dominates. For the other two proteins, which are extrinsi-
cally regulated, simple linear scling does not apply. The
linear fitting of sI
y-intercept, suggesting that the extrinsic noise is almost
zero for this protein. Because intrinsic noise appears to
dominate the fluctuation of YFP-LuxO level,noise upstream
of YFP-LuxO cannot propagate downstream to factors regu-
lated by YFP-LuxO.
We found I2/sI2for YFP-LuxO per cell at 0 nM, 50 nM,
and 1 mM AIs to be 1.89, 4.04 and 4.97 for the WT strain;
and 9.49, 16.0, and 17.6 for the luxO-ar strain, respectively.
The total protein number N ¼ (1 þ b)I2/sI
size b is the average number of protein produced per mRNA
molecule (33) (Eq. S34). YFP-LuxO amount is estimated
based on the partition at cell division (Table 1). If a partition
unit is a YFP-LuxO monomer, we get b ¼ 3.35. However,
because LuxO is likely to oligomerize (3), an analysis based
on partition noise would underestimate the real copy
number and b would be larger.
To summarize, we conclude from the autocorrelation
curves that LuxO saturates the binding site in the promoter
that represses its own expression. Our results also suggest
2versus I for YFP-LuxO has a near-zero
2, where the burst
LuxO fluctuations correlate positively with
subsequent LuxR fluctuations
Broadened autocorrelation curves reflect regulation by
slowly varying upstream components, such as an upstream
regulatory protein. The heat maps of the autocorrelation of
qrr4-cfp and mCherry-LuxR are shown in Fig. 3, D and E,
respectively. The profiles of the autocorrelation curves for
both qrr4-cfp and mCherry-LuxR (Fig. 3 F) differ consider-
ably from those expected for stable proteins undergoing
noisy production (Fig. 3 F, lower gray curve). This may
result from the propagation of noise from upstream compo-
nents or from extrinsic noise. In the case of qrr4-cfp, the
qrr4 promoter is regulated by several upstream long-lived
proteins (LuxR as shown in Fig. 2, and s54). Therefore,
the greater width of the autocorrelation curve of qrr4-cfp
relative to unregulated proteins is expected. (The autocorre-
lation curve of qrr4-cfp fluctuations will differ from that of
Qrr4 fluctuations, if measurable, because the sRNA is
subject to rapid degradations.) Likewise, LuxR may have
an active upstream regulator or LuxR may be affected by
The cross-correlation matrixes between YFP-LuxO and
mCherry-LuxR show that YFP-LuxO fluctuations positively
correlatewith time-lagged mCherry-LuxR fluctuations at all
three AI concentrations investigated (Fig. 4). The asym-
metric structure of the cross-correlation matrix is significant
enough to show that noise propagates positively from YFP-
LuxO to mCherry-LuxR levels.
It is true that the presence of a correlation with a time
lag can, in principle, occur without causal regulation in
complex gene networks. Certain network architectures can
produce qualitatively similar cross-correlation functions.
An examplewould be A regulates both B and C, and B regu-
lates D. In this case, the correlation between C and D is
similar to the correlation between B and D, because part
of B and D’s fluctuation comes from their common regu-
lator, A (9). Our data suggest that YFP-LuxO level fluctua-
tion is mostly due to intrinsic noise. In this case, YFP-LuxO
fluctuations do not reflect an upstream noise source, which
disfavors the above scenario for noncausal correlation. A
cross correlation similar to that observed for LuxO and
LuxR can also occur if YFP-LuxO regulates some other
factor that regulates mCherry-LuxR in this system. In this
case, the time lag between YFP-LuxO and mCherry-LuxR
fluctuation is expected to be longer than direct regulation.
The theoretical peak position is ?1.4 generations for direct
regulation. Adding another layer of intermediate regulatory
protein would move the correlation peak to between ?1.4
and ?2.0 generations (see Supporting Material). Protein
degradation and feedback can also affect the peak position.
[AI] = 0 nM
[AI] = 50 nM
[AI] = 1 μM
−3 −2 −1
measuring cross correlations between fluctuations in YFP-LuxO and
mCherry-LuxR concentrations. (A–C) Heat maps of RYFP, mCherryat three
AI concentrations: 0 nM (A), 50 nM (B), and 1 mM (C). The correlation
curves are plotted in D. In panels A–C, the higher values above the diagonal
correspond to the finding that YFP-LuxO actively and positively regulates
mCherry-LuxR. Regulation does not depend on LuxO phosphorylation
status, because it is observed at all three AI concentrations. Each matrix
was averaged over the matrixes produced from four independent movies.
Heat maps of the cross-correlation matrix RYFP, mCherry
Biophysical Journal 100(12) 3045–3053
Protein Level Fluctuation Correlation3051
In the QS circuit, the main regulator of LuxR is the Qrr
sRNAs. The lifetime of sRNA molecules can be enhanced
upon binding to the sRNA chaperone Hfq (16), which
protects the sRNA molecules from degradation (34). In
this case, slow fluctuations of Qrr levels could also lead to
a broad autocorrelation peak for mCherry-LuxR. In the
QS circuit as it is currently known, LuxO and LuxR interact
exclusively through the Qrr sRNAs. As a result, no interac-
tion between LuxO and LuxR is expected at high AI levels
when very few Qrr sRNAs are produced (Table 1).
Although the correlation analysis suggests that LuxO
upregulates LuxR at all AI concentrations, the mCherry-
LuxR level is not higher in the luxO-ar strain, where there
are more YFP-LuxO proteins per cell. Because LuxR
and LuxO also interact through sRNAs, if LuxO upregulates
LuxR via an sRNA independent pathway, one would expect
LuxR levels to be higher without sRNAs, e.g., at saturating
AI concentrations. A previous bulk experiment that
unphosphorylated state) showed no difference in relative
light units (3), which are indicators of the LuxR level. It is
therefore very puzzling that LuxR fluctuation correlates
may be additional links within the regulatory circuit that are
We studied protein concentration fluctuations in live
V. harveyi cells. Time-lapse microscopy was used to monitor
fluorescent fusion proteins and a fluorescent reporter. Single
cells were traced as they grew into microcolonies. Protein-
level fluctuations were also measured at the microcolony
level, which treats the microcolony as if it were a single
large cell. Autocorrelation and cross-correlation matrixes
were calculated between different protein concentration
fluctuations at the microcolony level.
The microcolony method of calculating fluctuation corre-
lations is accurate when protein-level fluctuations can be
modeled as linear perturbations around a steady state,
because the microcolony is a linear combination of all
descendent cells. Moreover, the microcolony method can
improve the accuracy of single-cell-level analysis. Usually,
the volume of a single cell doubles every cell cycle, which
for Poissonian fluctuations implies that the concentration
fluctuation amplitude decreases by a factor of
previous studies, the protein concentration fluctuation
amplitude at the single-cell levelwas assumed tobe constant
over time. This approximation was regarded as adequate
when fluctuation data were obtained from a large number
of unsynchronized cells (9) or when relative protein concen-
tration ranks instead of the true protein concentrations
were used for correlation calculations (10). We were able
correlations of microcolony-level fluctuations. Our theoret-
ical predictions agree well with Monte Carlo simulations
(see Supporting Material).
The microcolony method eliminates some noise sources.
Because it averages over the offspring cells, differences
between sister cells are averaged out. Such differences are
at the single-celllevelispassedon tothe descendant cells. In
autocorrelation curves obtained at the single-cell level, the
peaks appear wider due to partition noise. This can be seen,
for example, in the YFP-LuxO autocorrelation (Fig. S2 e).
There are fewer copies of YFP-LuxO in the WT strain than
in the luxO-ar strain, and thus there is a higher fraction of
correlation curve is wider for the WT stain than for the
luxO-ar strain. In contrast, both curves have the same width
at the microcolony level (Fig. 3 C). In this sense, correlation
at the microcolonylevel provides more-relevant information
about gene expression and protein regulation, with less
disturbance from cell division and molecule partitioning.
Measuring protein-level fluctuation at the microcolony
level can also reduce the complexity of image processing.
Fluorescence microscopy image resolution is limited by
optical resolution and camera pixel size. In practice, accu-
rately partitioning each image from the microcolony into
single-cell areas is the most error-prone and time-
consuming part of the data analysis. Because most regula-
tory links have a timescale of ~1 generation, to accurately
study a regulatory link, one must follow the cells for
n R 3 generations. With the same set of time-lapse images,
an ~2n– 1 greater effort is required to partition the images
into the single-cell level compared with the microcolony
level. Moreover, in a microscopy image, the fluorescent
intensity reading of a single cell is affected by the neigh-
boring cells. Therefore, even if the image can be accurately
divided into regions of single cells, the fluorescence reading
may not truly reflect the protein level in each cell. The
microcolonies usually have smaller ratios of border length
to area than do single cells, so they are less affected by
neighboring microcolonies compared with single cells. It
was recently reported that a microfluidic device enables
multiple cell lineages to be tracked in parallel (35). Such
a device could reduce the image-processing complexity to
a minimum for microcolony fluctuation correlation analysis.
Furthermore, if we can accurately follow the fluorescence
of individual microcolonies, we will no longer have to
perform microscopy when using correlation to study regula-
tory links in cell circuits. Single cells can be distributed
to multi-well plates, and the total fluorescence can be
accurately followed in each well to generate correlation
matrixes. With current wavelength demixing technology,
we can use multiple fluorescent tags in the same cells to
study the interaction between multiple regulators.
We applied the improved correlation method to analyze
V. harveyi strains that were engineered to harbor YFP-LuxO,
Biophysical Journal 100(12) 3045–3053
3052 Wang et al.
regulatory interactions are active all times, and not all genes
are equally affected by extrinsic noise. We have demon-
strated that fluctuation correlation can lead to a better under-
standing of a regulatory circuit as a whole. Fluctuation
correlation is increasingly being used as a tool to study
genetic regulation and noise (36); however, the process of
calculating fluctuation correlation at the single-cell level
can be cumbersome and time-consuming. Our approach for
studying fluctuation correlation at the microcolony level
can simplify this process and facilitate its use as a tool for
investigating regulatory circuits.
Additional text, equations, references, and figures are available at http://
We thank Tao Long, Pankaj Mehta, Shu-wen Teng, and Sine L. Svenning-
sen of Princeton University for insightful discussions. Y.W. thanks Hongye
Sun and Joe Beechem at Life Technologies for encouragement.
This work was funded by the Howard Hughes Medical Institute, National
Institutes of Health (grant 5R01GM065859), and National Science Founda-
tion (grant MEB-0343821), and was partially supported by the Defense
Advanced Research Projects Agency (grant HR0011-05-1-0055). Y.W.
and N.P.O. received support from Princeton University.
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Protein Level Fluctuation Correlation 3053