Supplemental Information
Methods
Media and Slide Preparation
LB (Luria-Bertani) medium1 was used for the routine cultivation of strains.
Experiments were conducted using M9 minimal medium1 supplemented with 100mM
leucine and 10% LB (vol:vol). E. coli TOP10 (Invitrogen) or DH5αPRO (BD
Biosciences) were used for all experiments. When appropriate, media were augmented
with kanamycin (50μg/ml, Sigma) and anhydrotetracylcine hydrochloride (Acros). Slides
were prepared using the modified M9 medium with 1% Sea Plaque low melt
electrophoresis-grade agarose (FMC). Antibiotics were added after partial cooling of the
melted agar medium and 1 ml was deposited onto glass microscope slides (Fisher
Scientific) and allowed to solidify. Ten µl of cell broth were then spread on the prepared
slides, cover slips were applied, and the slides were incubated at 28°C for ~1 hr prior to
imaging.
Synthetic Gene Circuits
The plasmid pGFPasv (pZE21-GFPasv2) was used directly and as the basis for the
generation of mutant forms of GFP with different half-lives. For the variants, pGFPasv
was digested overnight with StuI and HindIII (New England Biolabs), and the parent
vector was gel purified. Forward and reverse oligonucleotide sets for each GFP variant
were combined in a 1:1 molar ratio in ligation buffer and allowed to anneal at room
temperature for one hour. A 100:1 molar ratio of double-stranded oligonucleotide insert
DNA was added to a ligation mix containing 100 ng digested, gel-purified vector DNA.
Ligations were conducted in 25 μl reactions containing 4U T4 DNA ligase in LigaFast
rapid ligation buffer (Promega) at room temperature (RT) for 10 min. A 5-μl sample of
the ligation mix was then transformed into E. coli TOP10 cells per the manufacturer’s
instructions.
The tetR gene was amplified by PCR from the Repressilator plasmid2 and cloned
into pCR.2.1-TOPO (Invitrogen). The reverse primer was modified to include a stop
codon at the end of the normal coding sequence to eliminate the short half-life tag and
generate a mature protein with wild-type half-life, as well as to include a copy of the
untranslated region between the PLtet01 promoter and the KpnI site 5’ of the atg start
codon of the gfp-asv gene in pGFPasv. Plasmids were transformed, propagated,
screened, purified and sequenced. Plasmid DNA from the proper isolate was
subsequently digested overnight with KpnI (Promega), gel purified using the Qiaquick
gel extraction kit (Qiagen), and non-directionally cloned into the KpnI-digested,
dephosphorylated (using Antarctic Phosphatase per the manufacturer’s instructions, New
England Biolabs) pGFPasv plasmid. Ligation conditions were: 100 ng vector, 2:1 molar
ratio of insert:vector, 4U T4 DNA ligase (Promega), and 12 μl of Ligafast rapid ligation
buffer (Promega) in 24 μl total volume. Reactions were incubated for 10 min at RT and
transformed as described.
As a control, pGFPasv was transformed into E. coli DH5αPRO, which contains a
chromosomal insertion of the tetR gene behind the PN25 promoter. Chemically competent
cells of this strain were prepared using the RapidTransit transformation solution (Sigma)
and transformed with pGFPasv as per the manufacturer’s instructions. All other plasmids
were transformed into chemically competent E. coli TOP10 cells (Invitrogen) following
the manufacturer’s instructions. In all cases, plasmids were purified and the inserts were
verified by sequencing using standard methods.
Data acquisition and analysis
Confocal microscope settings were adjusted to collect light (500-550 nm) in slices
(thickness > cell height). Time series of noise in GFP concentrations (Xm(n·Ts)) for
individual cells (1, 2,..., M) were defined by their differences from the population mean,
and individual noise traces (trajectories) that spanned the entire growth time were
constructed by sequentially combining the noise traces of cells within a common line of
descent as shown in the figure below. Noise traces were extracted from the images for
nearly all possible trajectories in each experiment. Custom Matlab (Mathworks)
programs were used to find mean fluorescence of entire cell population and to estimate
population doubling time from an exponential growth curve.
Normalized autocorrelations functions (ACFs) for individual trajectories (m) were
found from the noise time series (Xm(n·Ts)) using a biased algorithm3
N j
 m  jTs  
 X m (nTs ) X m n  j Ts 
n 1
N

n 1
,
X m2 (nTs )
where Ts was the five-minute sampling interval, n was the sample number (1, 2,…, N),
and j had integer values from 0 to N-1. As high copy number plasmids were used, it was
not necessary to correct for cell-cycle variations due to chromosome replication. The
composite autocorrelation function (c) for M cell trajectories was found using
M
 c  jTs  
N j
 X
m1 n1
M
m
(nTs ) X m (n  j Ts )
N
 X
m1 n1
2
m
(nTs )
The solid curve shows a single trajectory through five generations of cell growth and
the dashed lines show alternate routes that produce other trajectories. A representative
noise trace is shown next to each cell in the trajectory. The noise trace of the complete
trajectory (shown at the bottom) is constructed by sequentially combining the noise
traces of each cell in the trajectory.
The normalized biased (B) and unbiased (UB) autocorrelation functions (ACFs)
differ in their scaling factors as follows:
M
 B   
N
 X
m1 n0
M
m
(nTs ) X m (nTs   )
N
 X m (nTs )
m1 n0
2
M
UB   
N

m1 n 0
X m (nTs ) X m (nTs   )
N  Ts   
M
N
2
X m (nTs )

m1 n 0  N  Ts 
Generally, the unbiased algorithm is more accurate for the average ACF at particular time
lags () but can be less accurate at larger lag values for limited data3. Exact stochastic
simulations showed that calculated biased ACFs for 8-hour data sets yielded better
agreement to the true ACFs for the simulated processes (data not shown).
Variability of Individual Trajectory Autocorrelation Functions
The ACFs of individual cell trajectories shown in figure 1d display large variability.
The ACFs of some trajectories decay smoothly in a manner similar to the composite
ACF, while many others were characterized by a decaying sinusoidal-type behavior. To
verify that the latter behavior is an artifact related to the small sample size of a single
trajectory, we conducted simulations of individual cell trajectories under conditions
similar to the experiments (see figure below). The simulations are known to have
underlying stochastic behavior similar to the composite ACFs. The simulated single-cell
trajectories show similar behavior as exhibited in the experimental single-cell trajectories,
demonstrating that the unexpected shape of some of the curves is most likely an artifact
of the limited sample size. By combining measurements from many cells to obtain a
larger sample size, the composite ACF (dark line in figure below) provides a better
estimate of the true ACF of the random process.
1
Normalized Autocorrelaton
0.75
0.5
0.25
0
-0.25
-0.5
0
100
200
300
400
 (min)
Error Bars
Noise frequency range, FN, was found from the biased ACF using
FN 
1
1 / 2
,
where τ1/2 was the value of τ where the normalized ACF reached a value of ½. At a
minimum there is error in the estimation of FN due to the finite duration of the noise
traces. Instrumental noise in the fluorescence measurement would include drift and noise
of the confocal microscope laser and photomultiplier tube. Instrumental drift showed up
as trends in the measurements that were at least partially compensated by the subtraction
of the population mean described above, and background measurements showed
instrumental noise to be very small compared to the magnitude of the total measured
noise. Additionally, the electronic instrumental noise was wideband (i.e. extended to
high frequencies) while the gene circuit noise was subject to severe band limiting by the
underlying gene circuits. Our measurements showed only negligible amounts of wide
band noise, leading us to conclude that instrumental noise was negligible. Accordingly,
the error bars in figure 4 of the main text were estimated assuming that most of the error
was due to the finite duration of the noise traces using a series of simulated experiments
containing approximately the same number of total trajectories of the same duration as
the actual experiments. The error bars were set at + one standard deviation as found from
these simulations. For pTetR-GFPasv experiments, error bars were set using simulations
of the pGFPasv + 100 ng/ml ATc experiments.
This error bar assumption was tested by measuring the noise frequency range in five
different pGFPasv experiments. Although the same experimental temperature (32ºC) was
used in three of these runs, there was some variation in the growth rate (figure 4a).
However, regardless of temperature the measured noise frequency ranges were within the
error bars of the noise frequency range predicted using the measured cell growth rate and
literature values of the protein decay time (red curve in figure 4a).
TetR Control Distribution
Figure 4a demonstrates that the TetR control circuit does not exhibit the increase in
noise frequency range associated with autoregulation. The figure below shows that this
control circuit also does not exhibit the characteristic shift in the shape of the noise
frequency range distribution. These distributions are for ~60 minute cell doubling times
and 100 ng/ml ATc.
Normalized Probability (%)
100
pTetR-GFPasv+ATc
75
TetR ctrl
50
25
0
0
3
6
9
12
-4
Noise Frequency Range (x10 sec-1)
Analysis, Modeling, and Simulation
Frequency Domain Analysis
The frequency domain (FD) approach is equivalent to the Langevin approach 4 and
shares the same limitations and caveats. In particular, this is a linear systems analysis
technique that has been adapted to non-linear systems analysis, and may not be valid for
highly non-linear circuits. In FD analysis the noise is dealt with using its power spectral
density (PSD; the frequency distribution of the noise) and the signal processing elements
(i.e. the components of the gene circuit) are dealt with in terms of their transfer functions,
H  f   o( f ) i( f ) , where o and i are output (e.g. GFP fluorescence) and input (e.g.
induction level) signals respectively, and f is the frequency in Hz 5. For a circuit with l
nodes (chemical species) with g noise sources, the noise PSDs at all nodes within the
gene circuit are given by
2
[Sout ( f )]  [ H ( f ) ][ Ssource ( f )]
,
where Sout  f  is an l x 1 matrix of the PSDs of the total noise in the concentration of
each chemical species; Ssource f  is an g x 1 matrix of the PSDs of the noise sources;
and [ H  f  ] is an l x g matrix of the power gains between node k (source) to node j
2
(output). The elements of the noise power gain matrix are given by
n j ,k 
2
H j , k ( f ) H *j ,k ( f )  H const
2
 
f
 

 1   f
 
zn _ j , k  
n 1



2
m j ,k 
 
f
 

1

   f
 
pm _ j , k  
m 1



,
where H j ,k  f   o j ( f ) ik ( f ) , Hconst is a frequency independent term (usually Hconst =
Hj,k(0)), H *j ,k  f  is the complex conjugate of Hj,k(f), and fz1…..n and fp1….m are the critical
frequencies associated with the zeros and poles of the transfer functions from a source at
location k to an output at location j. This assumes no poles or zeros at the origin.
The FD analysis of a genetic circuit then consists of (1) selecting a convenient output
signal, which here is GFP fluorescence; (2) determining the location and PSDs of the
noise sources; and (3) finding the power gains between each noise source and the selected
output.
The transfer function (and hence the power gains) are found from the differential
equation describing the relevant molecular populations. We consider GFP existing in an
initial immature (i.e. non-fluorescent; GFP) and a mature (fluorescent; GFPm) state such
that
dmRNA
  t    R mRNA
dt
dGFP
 k p mRNA   pGFP  kmGFP
dt
dGFPm
 kmGFP   pGFPm
dt
,
where
α(t) = transcription rate
mRNA/GFP/GFPm = population of mRNA/GFP molecules
γR, γp = decay rate constants for mRNA and GFP
kp, km = rate constant for translation and maturation of GFP.
We assume that both mature and immature GFP decay at the same rate.
In our experiments the average GFP fluorescence was measured, which corresponds
to molecular concentration (not population), and we rewrite the differential equations in
terms of concentration:
d mRNA  t    R mRNA mRNA dV (t )



dt
V (t )
V (t )
dt
 t    R mRNA  mRNA dV (t )
V (t )
dt
,
dV (t )
1 dV (t )
t
t
V (t )  Vo 1  e
 Vo e

dt
V (t ) dt
d mRNA
  t    R mRNA   mRNA
dt


where
[mRNA] = mRNA concentration
[α(t)] = rate of transcription per unit volume
V(t) = cell volume
Vo = initial cell volume
(S-1)
δ = cell volume growth rate constant.
By similar analysis
d GFP
 k p mRNA   p GFP  k m GFP   GFP
dt
.
d GFPm 
 k m GFP   p GFPm    GFPm 
dt
(S-2)
The intrinsic noise sources are due to fluctuations in the rates of transcription,
translation, GFP maturation, and the decay of mRNA and GFP (mature and immature),
which can be collected into three noise sources located at the point of transcription,
translation, and maturation, respectively. The transfer functions and power gains from
each of these sources to the output (mature GFP concentration) are found by Fourier
transformation and solution of equations (S-1) and (S-2) to yield
H  p m  f 
2

km

 km   p  





2




b2
2
 p      2f

 1  
  r


2 
km
H k p  p m  f  
 km   p  


2







1
2 
2 
2 
  
   2f   
2f

 1 
1       k   






m  
   p
   p






1
2 
 p      2f
 1  
  p 







1
2
2 
  
  
2

f
 1 

    p    k m   
  
 
H k m  pm  f 
2




1
1

2 
 p      2f
 1  
  p 






,
2 
  
 
  
 
where
H   p m  f  = the transfer function from the point of transcriptional noise to mature
GFP concentration
H k p  pm  f  = the transfer function from the point of translational noise to mature
GFP concentration
H km  pm  f  = the transfer function from the point of GFP maturation to mature
GFP concentration
b = kp/γr.
The intrinsic noise sources are white (i.e. equal noise power at all frequencies) and can be
found using the shot noise equation5 to yield
S  f   4 
S k p  f   4k p mRNA  4b 
S k m  f   4k m GFP 
4k mb 
 p    km

,

where
S  f  = the power spectral density of the noise in the rate of transcription and
mRNA decay
Sk p  f  = the power spectral density of the noise in the rate of translation and
immature GFP decay
S km  f  = the power spectral density of the noise in the rate of GFP maturation
and mature GFP decay
b = the burst rate (average number of GFP molecules translated from each mRNA
molecule).


The PSD of the total intrinsic noise in the mature GFP concentration S pI m  f  is the
sum of each of the noise sources weighted by their noise power gains:






2

 4b 2   

k
1
I
m

S p m  f  

2
2 
 km   p       2 
   2f  2   2f   
  

 p
2

f
 1 
 
 1  
 1  


    r       p       p    k m   











2




k


4
b

1
m


2 
2
 km   p       2  
   2f   
 

 p
2f
 1 
 
 1  


    p       p    k m   











4k mb 
1

2 
 p    k m  p      2f
 1  
  p 








2 
  
 
  
 
,
where the first term on the RHS of the equation is the noise due to transcription, the
second term is the noise due to translation, and the third term is noise due to maturation.
For most cases this equation can be greatly simplified by considering usual values of the
rate constants. The mRNA typically decays much faster than even destabilized protein in
prokaryotes, so the high frequency pole associated with mRNA decay can be neglected
with little error. Additionally, the burst rate is typically large enough that translational
and maturation noise can be neglected6,7, and S pI m  f  can be approximated as
S pI m


km

 f  

k




m
p


2



4b 2   

 p   2    2f
 1  
  p 






.
1
2 
2
  
 
2f
 1 



    p    k m   
  
 
Following translation, the GFP protein must undergo several maturation steps to
develop the chromophore. These include folding, cyclization, and oxidation, which we
have lumped into one step characterized by the rate constant km. Folding and cyclization
occur relatively rapidly with respective half times of 10 and 3 min8, while oxidation is the
rate limiting step with a half time ranging from 19-83 min8,9. At the lower end of the
range of oxidation half time the pole associated with km is at a high enough frequency that
it can be neglected and the gain factor
km
 1. For this case
km   p  



4b 2   
1
I
S pm  f  
2 
 p       2f

 1     
p
 



.
2 

 

  
 
If the oxidation half time were on the upper end of the cited range, the pole associated
with km would noticeably lower the noise frequency range. However, our measurements
had no indication of this pole, suggesting that the oxidation half time was on the lower
end of the cited range, and we adopt this assumption for our modeling and analysis.
The second significant noise term is extrinsic noise, which is the intrinsic noise of
other gene circuits (e.g. those producing RNAP and ribosomes) that couple in through
transcription, translation, and active protein decay pathways. In this case the noise
sources are not white, but are band limited by the filtering of their gene circuits. As all
the sources, locations, and magnitudes of extrinsic noise have not been elucidated, we
approximate extrinsic noise as a single source located at the point of translation with a
PSD given by
SE  f  
S E 0
  2f  2 
1  

    


.
We have assumed that extrinsic noise is dominantly band limited by dilution10. The value
of the constant term (SE(0)) is set by other rates (transcription, translation, etc. of RNAP,
ribosomes, proteases). For further analysis and simulation it is convenient to also collect
all the intrinsic noise terms at the point of translation, resulting in a single noise source
with a PSD given by
S source f  
S E 0
  2f  2 
1  

    


 SI .
This noise is processed by the GFP gene circuit, so that
S Tp m _ N
f 
S pEm _ N 0 
2
  2f  2   2f  
1  

  1 
         p  
 

 
S pEm _ N 0   S pI m _ N 0   1
where

S pI m _ N 0 
2
 
  2f  
1        ,
 
p 




S Tpm _ N  f  = the normalized S Tpm _ N 0  1 PSD of total noise (extrinsic +
intrinsic) in the mature GFP concentration
S pEm _ N 0 = the normalized PSD of extrinsic noise in the mature GFP
concentration at f = 0
S pI m _ N 0 = PSD of intrinsic noise in the mature GFP concentration at f = 0.
The normalized autocorrelation function, Φ(τ), is given by the inverse Fourier
transformation of the S T
pm
_N
f 
to obtain
          
   WE 
e
 e
  WI e    

 

S pEm _ N 0  S pI m _ N 0
WE 
  
1 

  




E
I




S
0
S
0
pm _ N
1  p m _ N



  
1 



  


,
WI  1  WE
.
ATc-Mediated Shift of Noise Frequency Range
The limited set of mechanisms proposed to increase noise frequency suggests that
ATc increased the GFP decay rate; produced negative autoregulation of the gene circuit;
or changed the nature of the extrinsic noise. However, autoregulation seems unlikely as
there was no TetR repressor in this control experiment to mediate an interaction between
ATc and the pGFPasv circuit. Furthermore, there is no experimental evidence to suggest
that ATc increases the rate of protein degradation. A change in the nature of the extrinsic
noise would seem to be a more plausible explanation for the ATc-mediated noise
frequency range modification.
Tetracycline and many of its derivatives inhibit translation by reversible binding
with the 30S ribosomal subunit11-13. The addition of ATc produced a cell growth rate that
was approximately 50%-70% that of ATc-free cultures at the same temperature (data not
shown), suggesting that a large fraction of 30S ribosomal subunits were bound in nontranslating heterodimers with ATc. We approached the analysis of the
anhydrotetracycline (ATc) effect on the constitutively expressed GFP circuit by
considering extrinsic noise effects of the reversible formation of a non-translating ATcribosome heterodimer (see figure in simulation section). Previous theoretical analysis
predicts that an additional white noise component is generated in the ribosome
concentration by the creation and dissolution of the heterodimers14 such that
SE  f  
S E1 0
  2f  2 
1  

    


 SE 2 ,
and
S Tp m _ N
f 
S pEm1 _ N 0 
2
  2f  2   2f  
1  

  1 
         p  
 

 

S pI m _ N 0   S pEm2 _ N 0 
2
 
  2f  
1       
 
p 


,
(S-3)
S pEm _ N 0   S pI m _ N 0   S pEm2 _ N 0   1
where
S pEm1 _ N 0 = the normalized PSD of the band limited extrinsic noise in the mature
GFP concentration at f = 0
S pEm2 _ N 0 = the normalized PSD of the heterodimer produced white extrinsic
noise in the mature GFP concentration at f = 0
As the equation above shows, the dimerization noise behaves like intrinsic noise and has
the effect of decreasing WE and increasing WI. However, as it is difficult to determine
new values of WE and WI analytically, simulation was used to analyze the pGFPasv +
ATc case.
Modeling and Simulation
We constructed a stochastic simulation model (shown below) that included intrinsic
and extrinsic noise sources and allowed investigation of the mechanism of ATc-mediated
noise frequency range remodeling. In this model all extrinsic noise was collected in the
ribosome concentration, and was limited in frequency range by dilution. All intrinsic
noise (transcription and translation) was represented by a single source at the point of
translation. The composite ATc curve in figure 4 of the main text was generated using
stochastic simulation software (BioSpreadsheet; available for download at
http://biocomp.ece.utk.edu). Custom software was used to automate the stochastic
simulations required for the simulated noise frequency distributions in figures 2 and 3 of
the main text. All stochastic simulations were based on variations of the Gillespie
stochastic simulation algorithm15-17.
The stochastic model of GFP expression for an individual cell was as follows:
Reaction
Rate
1. R  R + ribo
k1
2. ribo  ribo + GFP
bnoise*
3. ribo  *

4. GFP  *

Reactions 1 and 3 represent extrinsic noise that is filtered by the dilution rate. It is
convenient to consider the extrinsic noise to be the stochastic variation of ribosome
concentrations when ATc experiments are modeled. FN is independent of the value of k1
and the dilution rate  was calculated for each cell as described below. Reaction 2
represents the translation of mRNA whose stochastic variation is an intrinsic noise
component that was modeled in the translation noise component. The mRNA decay rate
was neglected as it is usually short compared to the dilution rate.
The weighting of extrinsic and intrinsic noise was set by bnoise according to
S E 0
 bnoise ,
SI
and we achieved the best fit to our measurements for SE(0)/SI ≈ 4, which is consistent
with previous reports10,18. Note that the bnoise term used here does not represent the true
burst rate of the system, but rather is a modeling device used only to achieve the correct
ratio between extrinsic and intrinsic noise. Reaction 4 represents dilution and decay of
GFP. GFP half-lives of 60 and 110 min for aav- and asv- tagged GFP were obtained from
Andersen et al.19
Simulated histograms were generated from sets of 500 simulated experiments, each
comprised of the time histories of about 100 cells descending from a single ancestor.
Simulated time histories were processed identically to experimental time histories to
obtain histograms. Since dilution plays an important role in both noise filtering processes
considered, variability in cell doubling time was a major source of the variability in
experimental measurements of FN. The distribution of doubling times of individual cells
during a given experiment was characterized by a rightward-skewed histogram. A multistage growth model was used to characterize this variability during simulation. Cells
progressed from stage M0 to stage Mn according to the following stochastic process:
/  DT 
/  DT 
/  DT 
M 0 n

 M 1 n

 M 2 ...M n 1 n

 M n
where <DT> was the experimental mean doubling time and n was adjusted to match the
simulated and experimental standard deviations of DT. Cells immediately divided upon
reaching stage Mn producing two daughter cells at M0 having the same species
concentrations as the parent cell. GFP concentration was assumed high enough that the
noise of uneven division of molecules could be neglected. The dilution rate for an
individual cell was calculated as ln(2)/ t0n where t0n is the time required for the cell
to transition from M0 to Mn.
Noise range histograms were observed to shift to higher frequencies with an increase
in cell growth rate (figure 2a) or protein decay rate (figure 2b), consistent with
histograms determined by simulation of 500 experiments. Some noise range histograms
suggested bimodal distributions not predicted by the simulations. To test the possibility
that the bimodal distributions were artifacts caused by the limited number of cell
trajectories traced, we compared the histograms of several individual simulated
experiments to the experimental histogram based on similar numbers of pGFPaav cells.
The simulated histograms were entirely consistent with the characteristics of the
experimental histogram, including an apparent bimodal distribution in some cases. This
suggests that bimodal distributions in the histograms could be an artifact of the limited
sample size available and that the simulated histograms based on 500 experiments were
likely a reasonable approximation of the system behavior.
100%
Noramlized Probability
Experimental
Sim 1
80%
Sim 2
Sim 3
60%
Sim 4
Average 500 Sims
40%
20%
0%
0
3
6
9
-4
12
-1
Noise Frequency Range (x10 sec )
Temperature variations were used to manipulate cell doubling times, which also
affects the rates of other biochemical reactions in the cell. There are four parameters in
the GFP model (k1, bnoise, and ) for which the effect of temperature on FN needs to be
considered. Since the effect of temperature on was incorporated into the analysis
presented herein and normalized FN is independent of the scaling factor k1, only bnoise and
require detailed consideration. No information on the effect of temperature on the in
vivo decay rate of destabilized GFP could be found. To investigate the magnitude of
effect possible, we assumed an exponential increase in GFPasv half-life from 110 min at
-1
Noise Frequency Range (sec )
7.E-04
6.E-04
b=4
b=4, gamma=f(T)
5.E-04
b=10
b=1
4.E-04
GFPasv data
3.E-04
2.E-04
1.E-04
0.E+00
20
40
60
80
100
120
cell doubling time (min)
37oC (reported by Andersen et al.19) to 200 min at 22oC. This change in activity over the
temperature range of interest was somewhat larger than the temperature dependence
reported for activity of the high temperature requirement protein A2 (HtrA2)20, a protease
that binds using the same PDZ C-terminal domain as the tail-specific protease responsible
for -asv tagged GFP. Consideration of the temperature dependence of resulted in
changes in simulated FN that were significantly smaller than the experimental error bars
(see figure above). A mechanism by which changes in temperature would affect the
relative weighting of extrinsic and intrinsic noise, as reflected in the value of bnoise, has
not been described. To investigate this possibility, we determined the sensitivity of the
model to changes in bnoise as shown in the figure below. Based on a comparison of the
simulations with the data, a decrease in bnoise with decreasing temperature (increasing
doubling time) can be ruled out. The assumption that bnoise increases with decreasing
temperature (increasing doubling time), slightly improves the fit of the data, although the
effect on FN is modest. Based on the relatively small effect of changes in  and bnoise and
lack of direct evidence of the magnitude or existence of the temperature dependence, we
elected to consider only the effect of temperature on cell cycle in simulations of the base
GFP model.
Inhibition of translation in ATc experiments was modeled by assuming that the
fraction of bound ribosomes was proportional to the fractional reduction in growth rate
resulting from the addition of ATc at constant temperature. The effect of ribosome
inactivation by ATc at constant concentration can be simulated with the following
additional reactions (see figure below):
Reaction
Rate
5. ribo  ribo-ATc
kf
6. ribo-ATc  ribo
kr
The ratio of the forward and reverse kinetic constants is equal to the conditional
equilibrium complexation constant K. The van’t Hoff equation was used to estimate the
effect of temperature on the equilibrium complexation constant:
K1 H o
1 T2  1 T1 
ln

K2
R
Experimental estimates of the conditional equilibrium constant were obtained for ATc
experiments at 26 and 30 oC. The resulting value of Ho was –32 kcal/mol. The van’t
Hoff equation was then used to estimate K for other temperatures of interest.
ribo-ATc
kf
k
R
k1
r
ribo
δ
b·δ
GFP
*
In this stochastic model of the gene circuit, all
extrinsic noise is modeled by the stochastic
production of the ribo species, while all intrinsic
noise is modeled by the stochastic production of
GFP. The proper weighting between extrinsic
and intrinsic noise is achieved by varying b.
The effect of ATc on the extrinsic noise term is
modeled by the reversible reaction denoted with
the dashed arrows.
Connell et al. (2003) studied the binding kinetics of ribosomes by [H3]tetracycline
using a nitrocellulose binding assay21. They determined that the apparent forward
reaction rate constant had an activation energy, Ea, of about 14 kcal/mol, allowing us to
predict the change in kf with temperature using the Arrhenius law:

k f ,T  k f ,ref exp ( Ea / R)  (1 / Tref  1 / T )

where kf,ref is the forward rate constant at reference temperature Tref. We further assumed
that the kinetics of binding between ATc and ribosome were rapid. Simulations were
relatively insensitive to alternate assumptions for the relationship between kf and kr as
long as the binding kinetics were rapid.
The relationship between FN and doubling time for ATc experiments was developed
using stochastic simulation. Values of FN were obtained from the biased autocorrelation
of a long time series (9x106 seconds) generated by stochastic simulation using the defined
rate constants.
In addition to the reversible binding of 30S ribosomal subunits, we considered
several alternative mechanisms to explain the increase in FN upon addition of ATc to
pGFPasv cells. However, only a limited number of mechanisms by which an increase in
the noise range can be achieved have been identified, which we will consider in turn: 1)
an increase in the protein decay/dilution rate5; 2) negative autoregulation5; 3) a reduction
in the relative contribution of extrinsic noise to the total noise10; 4) and whitening of the
noise spectrum caused by reversible binding14. Increases in the cell growth rate or the
protein decay rate will shift the FN to higher values. The effect of adding ATc on cell
growth has already been considered in the analysis. In order to explain the observed shift
in FN the addition of ATc would need to decrease the half-life of GFPasv from 110
minutes to 35 minutes (see figure below). No mechanism by which ATc could so
dramatically change the half life is known. Similarly, a mechanism by which ATc
addition introduces negative autoregulation in the expression of GFP has not been
identified.
Addition of ATc could increase FN by reducing of the relative contribution of
extrinsic noise. This would be reflected in a decrease in bnoise in the base model. In fact,
the 30S ribosomal binding model above accounts for this mechanism. The fractional
decrease in free ribosomes, if assumed to be proportional to the fractional decrease in
growth rate upon addition of ATc, corresponds to effective values of bnoise of 2.92 and
2.28 at cell doubling times of 55 min and 104 min, respectively. However, to fully
account for the magnitude in shift in FN caused by the addition of ATc, a much smaller
value (ca. bnoise= 1) is required. Mechanisms that could cause a shift in the importance of
extrinsic noise of this extent cannot be further investigated using the simplified model
presented here. Furthermore, both extrinsic noise whitening by heterodimer formation
and the lowering of the extrinsic noise weighting factor have the same effect on the noise
frequency range (see equation (S-3)). That is, both mechanisms increase the proportion
of the noise that is only filtered by one significant pole, and it is not possible to
distinguish between the two effects with the noise spectral measurements reported here.
The 30S ribosomal binding model we discuss in this work combines the following
mechanisms to describe the shift in FN upon addition of ATc: 1) a change in the dilution
(cell growth) rate, 2) a change in the effective weighting of extrinsic and intrinsic noise,
-1
Noise Frequency Range (sec )
and 3) whitening of the extrinsic noise source by ribosome-ATc heterodimer formation.
7.E-04
GFP model b=4
6.E-04
GFP model b=1
data pGFPasv
5.E-04
GFP model, half life 35 min
ATc-ribosomal binding model
4.E-04
data pGFPasv + ATc
3.E-04
2.E-04
1.E-04
0.E+00
20
40
60
80
100
120
cell doubling time (min)
Autoregulation
Theoretical analysis predicts that negative autoregulation extends the noise
frequency range by a factor of (1+|T|), where T is measurement of the incremental
strength of regulation5 and is calculated as
T
d d TetR2 
d d TetRtot  d TetR2 
1
d
d TetR2 


,
d TetR2  d
d TetR2  d
d TetRtot   ATc d TetR2  d TetRtot 
where
α = transcription rate per unit volume
[TetR2] = concentration of free TetR dimer
[TetRtot]= total concentration of TetR (free and bound with ATc)
Transcriptional repression is accomplished by TetR dimer binding to an operator region
in the promoter, a process previously analyzed22,23 to show that

   B  pGFP1 

K o TetR2  

1  K o TetR2  
 K o TetR2 

d
1

 K o B  pGFP

 1  K TetR 2 1  K o TetR2  
d TetR2 
o
2


,
where
αB = maximum transcription rate per unit volume
Ko= ratio of forward and reverse rate constants for TetR dimer – operator binding
[pGFP] = plasmid concentration
At low values of [TetR2]
d
  K o B  pGFP ,
d TetR2 
while at high dimer concentration


d
1
1
  0
 K o B  pGFP

d TetR2 
 K o TetR2  K o TetR2  
Assuming that the total concentration is large enough that almost all TetR exist in the
dimer form,
k f TetR2   kr
d TetR2 
.

d TetRtot  k f 2ATctot   TetRtot   4k f TetR2   2kr

At low TetR concentrations this reduces to

d TetR2 
kr

d TetRtot  2k f ATctot   2kr
which has a maximum value of ½ ([ATctot]=0) and approaches 0 for [ATctot] >> [TetRtot].
At the other extreme where [TetRtot] >> [ATctot]
k f TetR2   k r
d TetR2 
1


d TetRtot  2k f TetR2   2k r  2
Combining terms at low TetR concentrations
T 
K o B  pGFP

kr
ATctot   TetRtot 

 0 ,
2k f ATctot   2kr
and at higher TetR concentrations
T 

K o B  pGFP 
1
1

  0 .







2
K
TetR
K
TetR
2
o
2 
 o
|T| reaches a peak at intermediate levels of [TetR2] (i.e. at intermediate cell growth rates)
where the concentration of TetR is large enough that not all TetR is bound with ATc, but
not so large that the repression curve has saturated.
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