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1
Negative feedback through mRNA provides
the best control of gene-expression noise
Abhyudai Singh Member, IEEE
Abstract—Genetically identical cell populations exposed to the same environment can exhibit considerable cell-to-cell variation
in the levels of specific proteins. This variation or expression noise arises from the inherent stochastic nature of biochemical
reactions that constitute gene-expression. Negative feedback loops are common motifs in gene networks that reduce expression
noise and intercellular variability in protein levels. Using stochastic models of gene expression we here compare different
feedback architectures in their ability to reduce stochasticity in protein levels. A mathematically controlled comparison shows
that in physiologically relevant parameter regimes, feedback regulation through the mRNA provides the best suppression
of expression noise. Consistent with our theoretical results we find negative feedback loops though the mRNA in essential
eukaryotic genes, where feedback is mediated via intron-derived microRNAs. Finally, we find that contrary to previous results,
protein mediated translational regulation may not always provide significantly better noise suppression than protein mediated
transcriptional regulation.
Index Terms—Gene-expression noise, negative feedback, noise suppression, microRNAs, linear noise approximation
F
1
Protein
I NTRODUCTION
T
H e inherent probabilistic nature of biochemical reactions that constitute gene-expression together with
low copy numbers of mRNAs can lead to large stochastic
fluctuations in protein levels [1], [2], [3]. Intercellular
variability in protein levels generated by these stochastic
fluctuations is often referred to as gene-expression noise.
Increasing evidence suggests that gene-expression noise
can be detrimental for the functioning of essential and
housekeeping proteins whose levels have to be tightly
maintained within certain bounds for optimal performance
[4], [5], [6]. Moreover, many diseased states have been
attributed to an increase in expression noise in particular
genes [7], [8], [9]. Given that stochasticity in protein levels
can have significant effects on biological function and phenotype, cells actively use different regulatory mechanisms
to minimize expression noise [10], [11], [12], [13], [14],
[15], [16].
Negative feedback loops are key regulatory motifs within
cells that help reduce stochasticity in protein levels. A common and well characterized negative feedback mechanism
is protein mediated transcriptional regulation where the
protein expressed from a gene inhibits its own transcription
[17], [18], [19], [20]. For example, it is estimated that
over 40% of Escherichia coli transcription factors regulate
their own expression through this feedback mechanism
[21]. Both theoretical and experimental studies have shown
that such a negative feedback at the transcriptional level
reduces noise in protein numbers [22], [23], [24], [25], [26],
[27], [28]. Recent work has provided evidence of more
• A. Singh is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716.
E-mail: absingh@udel.edu
II
Translation
IV
I
III
Promoter
mRNA
Transcription
Gene
Fig. 1. The process of gene-expression where mRNAs
are transcribed from the gene and proteins are translated from individual mRNAs (red arrows). Different
feedback mechanisms in gene-expression where the
rate of transcription or translation is dependent on the
mRNA or protein count (dashed lines).
sophisticated negative feedback loops where the protein
inhibits the translation of its own mRNA [29], [30] or
mRNA inhibits the transcription of its gene [31], [32]. We
here compare and contrast the noise suppression ability of
these different feedback mechanisms in gene-expression.
Gene-expression is typically modeled by assuming that
mRNA transcription and protein translation from individual
mRNAs occurs at fixed constant rates. Feedback mechanisms can be incorporated in this model by assuming that
the transcriptional rate or translation rate is a monotonically
decreasing function of either the protein count or the mRNA
count. This procedure results in four different negative
feedback architectures, which are illustrated in Figure 1.
For example, feedback architecture I corresponds to protein
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TABLE 1
Frequency of different expression/degradation events and the corresponding reset maps.
Event
Reset in population count
Transcription
m(t) → m(t) + B
km dt
mRNA degradation
m(t) → m(t) − 1
γm m(t)dt
protein translation
p(t) → p(t) + 1
k p m(t)dt
protein degradation
p(t) → p(t) − 1
γ p p(t)dt
mediated transcriptional regulation where the transcription
rate is a decreasing function of the protein count. Similarly,
feedback architecture IV corresponds to a scenario where
the protein translation rate per mRNA is a decreasing
function of the mRNA count.
We derive analytical expressions for the protein noise
levels for each of these different feedback architectures.
Using these expressions we determine which feedback
provides the best noise suppression, and how does its
performance depend on gene-expression parameters such
as mRNA and protein half-life. It is important to point out
that comparisons between different feedback architectures
are done keeping the mean protein and mRNA count
fixed. Furthermore, we assume that different feedbacks
also have the same feedback strength, which is measured
by the sensitivity of the transcription/translation rate to
the mRNA/protein count. Such a form of comparison is
also referred to in literature as a mathematically controlled
comparison [33].
The paper is organized as follows: In Section 2 we
quantify the extent of stochasticity in protein levels in a
gene-expression model with no negative feedback. Protein
noise levels for feedback architectures I − IV are computed
in Section 3. In Section 4 we compare the noise suppression
abilities of the different feedback architectures. Finally, a
discussion of our results is provided in Section 5.
2
G ENE EXPRESSION MODEL WITH NO REG -
ULATION
We consider a gene-expression model where transcriptional
events take place at rate km with each event creating a burst
of B mRNA molecules, where B is an arbitrary discrete
random variable with probability distribution
Probability{B = z} = αz , z = {1, 2, 3, . . .}.
(1)
Typically B = 1 with probability one. However, many genes
encode promoters that allow for transcriptional bursting
where B > 1 and many mRNAs can be made per transcriptional event [34], [35], [36]. Protein molecules are
translated from each single mRNA at rate k p . We assume
that mRNAs and proteins degrade at constant rates γm and
γ p , respectively. In the stochastic formulation of this model,
transcription, translation and degradation are probabilistic
events that occur at exponentially distributed time intervals.
Probability event will
occur in (t,t + dt]
Moreover, whenever a particular event occurs, the mRNA
and protein population count is reset accordingly. Let m(t)
and p(t) denote the number of molecules of the mRNA and
protein at time t, respectively. Then, the reset in m(t) and
p(t) for different gene-expression and degradation events
is shown in the second column of Table 1. The frequency
with which different events occur is determined by the
third column of Table 1, which lists the probability that
a particular event will occur in the next infinitesimal time
interval (t,t + dt].
To quantify noise in protein levels we first write the
differential equations that describe the time evolution of
the different statistical moments of the mRNA and protein
count. The moment dynamics can be obtained using the
following result: For the above gene-expression model, the
time-derivative of the expected value of any differentiable
function ϕ(m, p) is given by equation (2) [37], [38]. Here,
and in the sequel we use the symbol h.i to denote the
expected value. Using (2) with appropriate choices for
ϕ(m, p) we obtain the following moment dynamics:
dhmi
dhpi
= km hBi − γm hmi,
= k p hmi − γ p hpi
(3a)
dt
dt
dhm2 i
= km hB2 i + γm hmi + 2km hBihmi − 2γm hm2 i (3b)
dt
dhp2 i
= k p hmi + γ p hpi + 2k p hmpi − 2γ p hp2 i
(3c)
dt
dhmpi
= k p hm2 i + km hBihpi − γ p hmpi − γm hmpi. (3d)
dt
As done in many studies we quantify noise in protein levels
through the coefficient of variation squared defined as
¯ 2,
CV 2 = σ̄ 2 /hpi
(4)
where σ̄ 2 is the steady-state variance in protein levels
¯ denotes the steady-state mean protein count [39],
and hpi
[40]. Quantifying the steady-state moments from (3) and
substituting in (4) we obtain
CV 2 =
(hB2 i + hBi)γ p
1
+ ¯
¯
2hBi(γ p + γm )hmi hpi
(5)
where
¯
¯ = hBikm , hpi
¯ = hmik p
hmi
γm
γp
(6)
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dhϕ(m, p)i
=
dt
*
3
+
∞
∑ km αz [ϕ(m + z, p) − ϕ(m, p)] + γm m[ϕ(m − 1, p) − ϕ(m, p)] + k p m[ϕ(m, p + 1) − ϕ(m, p)]
z=1
+ γ p p[ϕ(m, p − 1) − ϕ(m, p)] .
(2)
denote the steady-state mean mRNA and protein count,
respectively. The first term on the right-hand-side of (5)
corresponds to noise in protein levels that arises from
stochastic production and degradation of mRNA molecules,
¯
and is inversely proportional to the mean mRNA count hmi.
The second term in (5) represents Poissonian noise arising
from random birth-death of individual protein molecules.
Given that mRNA population counts are typically orders of magnitude smaller than protein population counts
¯ hpi
¯ ≈ 10−3 from [2] , we ignore the second term in
hmi/
(5) and approximate CV 2 as
CV 2 ≈
(hB2 i + hBi)γ p
¯ .
2hBi(γ p + γm )hmi
(7)
This approximation implies that gene-expression noise primarily arises from fluctuations in mRNA counts that are
transmitted downstream to the protein level. In summary,
(7) represents the steady-state noise in protein level when
there is no feedback in gene-expression. Next, we quantify
protein noise levels for different feedback mechanisms.
3
I NTRODUCING REGULATORY
NISMS IN GENE - EXPRESSION
MECHA -
We first consider protein mediated transcriptional regulation
which corresponds to feedback architecture I in Figure 1.
3.1 Protein mediated transcriptional regulation
Transcriptional regulation is incorporated in the model by
assuming that the transcription rate is dependent on the protein levels. More specifically, transcriptional events occur
at rate km (p), which is a monotonically decreasing function
of the protein count p(t). This corresponds to a negative
feedback mechanism where any increase (decrease) in
protein numbers is compensated by a decrease (increase)
in the transcription rate. To quantify the protein noise
levels we use the linear noise approximation [41], which
involves linearizing the transcription rate km (p) about the
¯ This
steady-state average number of protein molecules hpi.
approximation is valid as long as the stochastic fluctuations
in protein counts are small, which is likely to be true for
tightly regulated essential proteins. Towards this end, we
assume
¯ p(t) − hpi
¯
(8)
km (p) ≈ km (hpi)
1−κ
¯
hpi
¯ is the average transcription rate. The dimenwhere km (hpi)
sionless constant
¯
hpi
dkm (p)
κ =−
| ¯ >0
(9)
¯
km (hpi) d p p=hpi
determines the sensitivity of the transcription rate to the
protein count and can be interpreted as the strength of the
negative feedback.
To obtain the time evolution of the statistical moments
we use (2), with km now replaced by (8). This results in
the following moment dynamics:
dhmi
= hkm (p)ihBi − γm hmi
(10a)
dt
dhpi
= k p hmi − γ p hpi
(10b)
dt
dhm2 i
= hkm (p)ihB2 i + γm hmi + 2hkm (p)mihBi − 2γm hm2 i
dt
(10c)
dhp2 i
= k p hmi + γ p hpi + 2k p hmpi − 2γ p hp2 i
(10d)
dt
dhmpi
= k p hm2 i + hkm (p)pihBi − γ p hmpi − γm hmpi.
dt
(10e)
Quantifying the steady-state moments from (10) and substituting in (4) gives the following protein noise level for
feedback architecture I:
CVI2 =
γ p (hBi + hB2 i)
¯ ,
2hBi(γ p + γm )(1 + κ)hmi
(11)
where the steady-state mean protein count is the unique
solution to the equation
¯
hBik p km (hpi)
¯
= hpi
γm γ p
(12)
and the steady-state mean mRNA count is given by
¯
¯ = hpiγ p .
hmi
kp
(13)
As done in the previous section, to obtain the noise level
(11) we assumed that the protein population count is
much larger than the mRNA population count, and hence
ignored expression noise arising from random birth and
death of individual protein molecules. Throughout the paper
we use CVX2 , X ∈ {I, II, III, IV} to denote the steady-state
protein noise level corresponding to feedback architecture
X. Moreover, CV 2 , given by equation (7) represents the
noise level when there is no feedback. Comparing the
analytical expressions in (7) and (11) one can see that for
κ > 0, the noise level with protein mediated transcriptional
regulation is always smaller than the noise level with no
feedback. As expected, when κ = 0 (i.e., the transcription
rate is independent of the protein count and there is no
feedback), CVI2 = CV 2 .
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Feedback type
4
Description
Expression noise (CV2)
Ref.
No feedback, feedback strength !=0
Fixed transcription and
translation rates
Transcription rate
Feedback architecture I, feedback strength !>0
decreasing function of
[19,20]
protein count
Translation rate
Feedback architecture II, feedback strength !>0
decreasing function of
[29,30]
protein count
Transcription rate
Feedback architecture III, feedback strength !>0
decreasing function of
[31,32]
mRNA count
Translation rate
Feedback architecture IV, feedback strength 0<!<1
decreasing function of
[45,46]
mRNA count
Fig. 2. Gene expression noise measured by the coefficient of variation squared (CV 2 ) of the protein count
corresponding to different feedback circuits. Here the random variable B denotes the transcriptional burst size,
¯ denotes the steady-state mean mRNA count, γ p and γm represent the protein and mRNA degradation rate,
hmi
respectively. The dimensionless constant κ represents the strength of the negative feedback. The last column
provides references of genes that encode the corresponding feedback architecture.
3.2
Protein mediated translational regulation
We now consider a more sophisticated form of negative
feedback where proteins regulate gene-expression at the
translational level (feedback architecture II in Figure 1).
We model protein mediated translational regulation by
assuming that the protein translation rate per mRNA is
a monotonically decreasing function k p (p) of the protein
count p(t). Thus, the total protein production rate k p (p)m
is dependent on both the mRNA and protein population
count.
As before, we assume that the stochastic fluctuations
¯ and
in p(t) and m(t) around their respective means hpi
¯
hmi are sufficiently small and approximate the total protein
production rate as
¯ p(t) − hpi
¯
¯
k p (p)m ≈ k p (hpi) m(t) − κ hmi
, (14)
¯
hpi
¯ is the average protein translation rate per
where k p (hpi)
mRNA and the dimensionless constant
¯
dk p (p)
hpi
κ =−
| ¯ >0
(15)
¯
k p (hpi) d p p=hpi
is the strength of the negative feedback architecture II. The
moment dynamics corresponding to this feedback can again
be obtained from (2), with k p m replaced by the right-handside of (14). Following steps similar to those in the previous
section, we obtain the following protein noise level for
feedback architecture II:
CVII2 =
γ p (hBi + hB2 i)
¯ .
2hBi(γm + γ p (1 + κ))(1 + κ)hmi
(16)
Finally, we consider feedback architectures III and IV.
The procedure for quantifying protein noise levels for III
and IV is very similar to that used for feedback architectures I and II, except that the transcription and translation
rates are now monotonically decreasing functions of the
mRNA count rather the protein count. Due to space considerations we only present the final result in Figure 2,
which lists the protein noise levels for different feedback
architectures.
4
C OMPARISONS BETWEEN DIFFERENT
GENE REGULATORY ARCHITECTURES
Analytical expressions in Figure 2 show two general trends
across different feedback architectures. Firstly, the protein
noise levels are inversely proportional to the steady-state
mean mRNA count. Secondly, increasing the feedback
strength κ, i.e., making the transcriptional/translational
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5
rates more sensitive to the protein/mRNA count, results in
lower protein noise levels. To assess the noise suppression
abilities of different feedback architectures we perform a
mathematically controlled comparison where all circuits are
assumed to have the same feedback strength and steadystate mean mRNA count.
We first compare the noise suppression abilities of protein mediated transcriptional and translational regulation
(i.e., feedback architectures I and II). Towards that end,
we compute the ratio
We here analyzed the noise suppression properties of four
different negative feedback loops within gene-expression
(Figure 1). Assuming that stochastic fluctuations in the populations of the protein and the mRNA are sufficiently small,
we derived explicit analytical formulas for the protein noise
level for each of the four feedback mechanisms. These
formulas reveal that some feedback architectures are inherently better at noise suppression, while the performance of
others is dependent on the parameters of gene-expression.
γ p + γm
CVII2
=
< 1,
2
γ
+
CVI
m γ p (1 + κ)
5.1 Which feedback provides the best noise suppression?
(17)
which shows that feedback architecture II always provides
better noise suppression than I. For a fixed feedback gain
κ, the above ratio monotonically decreases with increasing
γ p /γm and
CVII2
= 1,
γ p /γm →0 CVI2
lim
CVII2
1
=
< 1.
2
1+κ
γ p /γm →∞ CVI
lim
(18a)
Thus, when the protein half-life is much smaller than the
mRNA half-life (γ p /γm 1) then the noise suppression
ability of feedback architecture II is far superior to that
of I. However, when the mRNA half-life is much smaller
than the protein half-life (γ p /γm 1), the difference in the
noise suppression abilities of the two feedback mechanisms
is small.
Comparing the noise suppression abilities of protein
mediated translation regulation and mRNA mediated transcriptional regulation (i.e., feedback architectures II and III)
we find
γm (1 + κ) + γ p
CVII2
.
=
2
γm + γ p (1 + κ)
CVIII
(19)
The above ratio shows that in genetic circuits where protein
molecules are more stable than the mRNA (γ p < γm ),
mRNA mediated transcriptional regulation provides better
noise suppression than protein mediated translation regulation. On the other hand, if a gene encodes a very unstable
protein such that γ p > γm , then protein mediated translation
regulation outperforms mRNA mediated transcriptional regulation.
Finally, our results indicate that feedback architecture
IV provides the best noise suppression, irrespective of the
mRNA and protein degradation rates. This is illustrated
in Figure 3 which plots CVX2 , X ∈ {I, II, III, IV} as a
function of the negative feedback strength κ. As can be
seen from the figure, feedback architecture IV provides
the best noise suppression while feedback architecture I
is the least effective in reducing gene-expression noise.
Moreover, depending on the mRNA and protein half-life,
feedback architecture II or III will provide the second best
suppression of expression noise.
5
D ISCUSSION
What regulatory mechanisms control stochasticity in protein levels such that cellular process can occur with sufficient high fidelity is a fundamental question in biology.
Our results indicate that in a mathematically controlled
comparison, feedback architecture IV provides the best
suppression of gene-expression noise. More specifically, for
a fixed feedback strength, a feedback mechanism where
the protein translation rate per mRNA is a monotonically
decreasing function of the mRNA count, provides the least
amount of statistical fluctuations in the protein count. This
result raises an interesting question of how this feedback
architecture is implemented within genes.
In recent years microRNAs, which represent a class
of short non-coding RNAs, have been recognized as key
regulators of gene expression [42]. These microRNAs typically regulate gene-expression at the translational level
by directly binding to a mRNA transcript and inhibiting
protein translation [43]. In eukaryotes, genes contain noncoding segments called introns, which are removed from the
transcribed mRNAs through splicing. Removal of introns
is a necessary step before mRNAs can start translating
proteins and microRNAs can be derived from the excised
introns [44], [42]. Recent studies have shown that many
microRNAs are contained within the intronic regions of
the same gene, whose translational activity is regulated by
that microRNA [45], [46]. In other words, both the microRNA and its target mRNA originate from the same gene,
and hence, are coexpressed. This structural relationship
between them creates a negative feedback circuit, where
any random increase in mRNA counts automatically increases microRNA levels, which in turn reduces the mRNA
translation rate. Given that disruption of these microRNA
mediated feedback loops have been associated with cancer
progression and neurodegenerative diseases [45], feedback
IV type architectures play a critical role in these genes to
minimize protein level fluctuations about desired set points.
5.2
Feedback architecture II versus III
A mathematically controlled comparison between protein
mediated translation regulation (feedback architecture II)
and mRNA mediated transcriptional regulation (feedback
architecture III) showed that their relative noise suppression
abilities is dependent on the protein and mRNA half-life
(Figure 3). Reference [47] does a survey of about 2000
genes in budding yeast and shows that for most genes the
ratio γ p /γm is much smaller than one, i.e., genes encode
short-lived mRNAs but long-lived proteins. In this physiologically relevant parameter regime, feedback architecture
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I
6
II
III
mRNA half-life > Protein half-life
mRNA half-life < Protein half-life
No feedback
Gene expression noise
IV
No feedback
I
I
III
II
II
III
IV
IV
Negative feedback strength
II
Negative feedback strength
Fig. 3. Decrease in gene expression noise with increasing negative feedback strength (κ) for different circuit
architectures when mRNA half-life is smaller than protein half-life (right) and when mRNA half-life is larger than
protein half-life (left). The y-axis is normalized by the gene-expression noise when there is no feedback. These
plots correspond to a 5-hour protein half-life with a 1-hour (right) and 10-hour (left) mRNA half-life.
III is predicted to provide better noise suppression than
architecture II, and second best noise suppression among
all the different feedback architectures.
In addition to inhibiting translation, increasing evidence
suggests that microRNAs can also modulate the transcriptional activity of genes [48]. Thus, like IV, feedback architecture III can also be implemented using intron-derived
microRNAs. An example of feedback architecture III is
present in the endothelial nitric-oxide synthase (eNOS)
gene. This gene encodes an essential protein required for
generating nitric oxide and its levels need to be tightly
regulated as its over-expression and under-expression have
been related to diseased states [32]. Data suggests that small
RNAs derived from introns within the eNOS gene repress
the transcriptional activity of the eNOS gene [31], [32].
Thus, any increase in eNOS mRNA transcripts would inhibit further transcriptional events from the gene creating an
effective negative feedback circuit for eNOS homeostasis.
5.3 Protein mediated feedback circuits
Above analysis suggests that in physiologically relevant
parameter regime, mRNA mediated feedback circuits (feedback architecture III & IV) provide better noise suppression
than protein mediated feedback circuits (feedback architecture I & II). However, these comparisons where done at a
fixed feedback strength. Typically, protein molecules do not
function as monomers but bind together to form dimers,
tetramers, etc. Such protein multimerization can induce
cooperativity in a feedback circuit that makes transcription/translation rates ultra sensitive to protein levels, and
hence, substantially increase the negative feedback strength
[26], [27]. Thus protein mediated feedback loops can also
provide effective noise suppression by operating at much
higher values of κ compared to mRNA mediated feedback
loops.
Consistent with the observations of [16], we find that
feedback architecture II always provides better noise suppression than architecture I. However, the difference in the
protein noise level with feedback architectures I and II critically depends on γ p /γm . In particular, increasing γ p /γm , i.e.,
making the protein half-life much shorter than the mRNA
half-life, increases this difference, and enhances the noise
suppression ability of feedback architecture II compared
to I. On the other hand, decreasing γ p /γm , i.e., making
the protein half-life much longer than the mRNA half-life,
decreases this difference, and diminishes the advantage of
using feedback architecture II over I for noise reduction.
In physiologically relevant parameter regime where γ p /γm
is much smaller than one [47], our analysis predicts that
protein mediated translational regulation may not provide
significantly better noise suppression than protein mediated
transcriptional regulation.
In summary, we have developed analytical formulas
that connect stochasticity in protein levels to the negative
feedback architecture. These formulas reveal that mRNA
mediated feedback circuits are superior to protein mediated
feedback circuits for noise reduction under physiologically
relevant parameter regimes. Consistent with this prediction
we find many essential genes encoding mRNA dependent
feedback circuit through intro-derived microRNAs. Our
analysis not only make experimentally testable predictions
but also will be helpful in designing precise synthetic gene
networks with minimal fluctuations in protein levels.
ACKNOWLEDGMENTS
The author would like to thank Joao Hespanha, Mustafa
Khammash, Leor Weinberger and members of the Weinberger lab for many discussion on this topic.
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7
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