Here, we focus on the case where sRNAs negatively regulate a target mRNA. Positive regulation by sRNAs is discussed in the Supplementary information
. Post-transcriptional regulation through sRNAs is modeled using mass action equations with three molecular species: the number of sRNA molecules s
, the number of target mRNA molecules m
, and the number of regulated protein molecules p
(Elf et al
; Lenz et al, 2004
; Levine et al, 2007
; Mitarai et al, 2007
; Shimoni et al, 2007
). The effect of intrinsic noise is modeled by Langevin terms,
, that describe the statistical fluctuations in the underlying biochemical reactions (van Kampen, 1981
). The kinetics of the various species are described by the differential equations
The terms can be interpreted as follows. sRNAs (mRNAs) are transcribed at a rate αs (αm), and are degraded at a rate τs−1 (τm−1). Additionally, both sRNAs and mRNAs are stoichiometrically degraded by pairing through Hfq at a rate that depends on the sRNA–mRNA interaction strength μ. Proteins are translated from mRNAs at a rate αp and are degraded at a rate τp−1.
The Langevin terms,
, model intrinsic noise by treating the birth and death processes of the various species in equation (1)
as independent Poisson processes (van Kampen, 1981
model the noise in the creation and degradation of individual sRNAs, mRNAs, and the regulated protein, respectively.
models sRNA–mRNA mutual degradation noise. The Langevin terms are characterized within the linear noise approximation by two-point time correlation functions (j
, μ), which for steady states take the form
, and σμ2
denote the mean number of sRNA, mRNA, and protein molecules, respectively. It is noted that we have separated the noise due to RNA production and degradation,
, from the noise due to the binary reaction between mRNAs and sRNAs, eaμ
. This allows us to write equation (2)
in terms of four independent Langevin terms while still capturing the cross-correlation between sRNAs and mRNAs.
Recent evidence suggests that prokaryotic transcription may occur with RNA molecules being made in short intense bursts (Golding et al, 2005
). The effects of transcriptional bursting can be incorporated into our model by allowing two states of gene activation, as reviewed below (for a detailed discussion, see Paulsson, 2005
). Specifically, genes can be in a transcriptionally inactive ‘off' state or in a transcriptionally active ‘on' state. The average
transcription rate of RNA, αj
) in equation (1)
, is then related to the probability of the relevant gene being on, gjon
with αjon being the mean transcription rate of the relevant RNA when the gene is always on. We model the dynamics of a repressor-controlled gene using the equation
are the unbinding and binding rates of the repressor and
is a Langevin noise term. At steady state, it follows from the fluctuation dissipation theorem that
(Bialek and Setayeshgar, 2005
). Thus, a full model that includes transcriptional bursting is described by equation (1)
in conjunction with equations (3)
the number of mRNA molecules, p
the number of proteins, αm
the average rate of transcription, αp
the average rate of translation, and τm−1
the first-order degradation rates of mRNA molecules and proteins, respectively. The two Langevin terms,
, model noise in the synthesis and degradation of the mRNA and protein, respectively (see Supplementary information
) and obey the equations (j
The effects of transcriptional bursting can also be included in this model using equations (3)
Mean steady-state protein number
The mean steady-state protein number for regulation through sRNAs can be approximated by ignoring the Langevin terms and setting the time derivatives to zero in equation (1)
(see Supplementary information
; Paulsson, 2004
; Levine et al, 2007
). The mean as calculated within this mean-field approximation may differ from the actual mean especially where noise is large. Nonetheless, the qualitative steady-state behavior of the mean can be understood within this approximation.
As shown in Levine et al (2007)
and Elf et al (2005)
, the mean protein number exhibits a threshold linear behavior as a function of the mRNA transcription rate αm
, with the threshold at αs
(see ). This behavior should be contrasted with transcriptional regulation through TFs for which the mean protein number is a linear function of αm
(Thattai and van Oudenaarden, 2001
; Elowitz et al, 2002
; Swain et al, 2002
; Paulsson, 2004
). For sRNA-based regulation, the mean steady-state protein number depends on RNA transcription rates only through the difference αm
, and this dependence can be characterized by three distinct regimens: repressed αs
, expressing αs
, and a crossover regimen αs
. Increasing the sRNA–mRNA interaction strength μ results in a sharper crossover between the repressed and expressing regimens. The dashed line in depicts the μ → ∞ threshold linear behavior.
Figure 2 Steady-state behavior for gene regulation through sRNAs. For the regulated protein, the steady-state mean number exhibits an approximately threshold linear behavior as a function of the mRNA transcription rate αm. The threshold (more ...)
In the repressed regimen, on average, there are many more sRNAs transcribed than mRNAs. Consequently, almost all free mRNAs are quickly bound by sRNAs and degraded. This results in low levels of expression of the regulated protein. By contrast, in the expressing regimen, the average number of mRNAs greatly exceeds the number of sRNAs. The sRNAs degrade only a small fraction of the total mRNA population so mRNAs accumulate and are translated into proteins.
To compare the signal-transduction properties of sRNA-based regulation with TF-based regulation, we consider the two regulation schemes as signal processing systems. depicts how sRNA-based regulation, e.g. in quorum sensing, can be viewed as a signal processing system (see also Supplementary information
; ). In the context of quorum sensing, the input signal is the time-averaged number of phosphorylated LuxO (LuxO~P) molecules in the cell, which, after a series of intermediate biochemical reactions, is converted into the output signal, the average number of LuxR molecules. Fluctuations in LuxO~P and LuxR about their averages can be thought of as the input and output noise, respectively. The noise in the output is a combination of input noise
(fluctuations in the input signal), intrinsic noise
(stochasticity inherent in gene regulation), and extrinsic noise
(other sources of noise impinging on the signal processing system not explicitly considered in the model, such as ribosome and RNA polymerase fluctuations).
Figure 3 Schematic drawing showing our comparison of transcriptional and post-transcriptional sRNA-mediated regulation. We take as the input signal to both systems a protein regulator (blue discs) that either directly transcriptionally regulates the relevant gene (more ...)
The fidelity of a signaling system is ultimately limited by the output noise of the system. The output noise, defined as the ratio of the variance in the output protein number to the square of the mean output protein number, can be thought of as the square of the ‘percentage error' in the output. The higher the output noise, the poorer the signaling fidelity of a gene regulation scheme. Thus, examining the noise properties of sRNA-based and transcription factor gene regulation is important for comparing these two forms of gene regulation.
Gene regulation takes place as part of a larger genetic and biomolecular network, the purpose of which is to convert a measured signal into a concentration of the regulated protein. A simple but important observation is that sRNA-based regulation also requires protein regulators to couple to external signals. In particular, a protein regulator is necessary to vary the transcription rate of the sRNAs in response to an input. For this reason, we take as the input signal to both systems a protein that either transcriptionally regulates the relevant protein directly or else transcriptionally regulates the sRNAs. In the case of direct transcriptional regulation, the protein regulator acts as a repressor, whereas for post-transcriptional, sRNA-based regulation, it acts as an activator (see ). Furthermore, the kinetics of the protein regulator are chosen to be identical in both cases. The upstream components of the network that controls the level of the relevant protein regulator are also assumed to be identical. This allows for a principled comparison of the two regulatory schemes.
Gene regulation is intrinsically noisy. In this paper, we define intrinsic noise as the fluctuations in the output protein number, given a fixed steady-state input, due to the stochastic nature of the underlying biochemical reactions. When calculating intrinsic noise, we neglect the contributions to output noise from fluctuations in the input and from extrinsic noise sources such as variations in the number of ribosomes and RNA polymerase molecules (see ).
is the protein burst size (the average number of proteins made from an mRNA molecule) and pmax
is the mean protein level in the absence of repressor. The first term in equation (7)
captures the noise due to translational bursting (the protein burst from each mRNA due to the translation of multiple proteins from each mRNA molecule) and the second captures the noise due to transcriptional bursting (the RNA burst while no repressor is bound). The transcriptional bursting contribution is typically much smaller than that of translational bursting as the unbinding rate of the repressor is generally much faster than the protein degradation rate, k−
1. Consequently, the intrinsic noise for protein-based regulation is often approximated as σp2
The intrinsic noise of an sRNA-regulated protein differs significantly from that of a transcriptionally regulated protein. Noise in stoichiometrically coupled systems such as sRNA-based gene regulation has been studied earlier (Paulsson and Ehrenberg, 2001
; Elf and Ehrenberg, 2003
; Elf et al, 2003
). It was found by Elf et al (2005)
that the ultrasensitivity of stoichiometric systems in the crossover regimen necessarily gives rise to enhanced stochastic fluctuations. This ‘near-critical' behavior was related to the behavior at phase transitions where fluctuations also diverge (McNeil and Walls, 1974
). We have extended these previous analyses to the context of gene regulation by sRNAs, and have calculated the intrinsic protein noise within the linear noise approximation (see Supplementary information
; van Kampen, 1981
; Elf and Ehrenberg, 2003
), including the effects of transcriptional and translational bursting. We have checked our results using exact stochastic simulations (see Supplementary information
; Supplementary Figures 2 and 3
). The simulations confirm the existence of three regimens and verify that noise is enhanced in the crossover region due to critical fluctuations.
The full expressions for the intrinsic noise are lengthy and in the main text we present only our major findings. and show typical intrinsic noise profiles as functions of the transcription rate ratio, αs
, and of the average protein level of the regulated protein, for various magnitudes of transcriptional bursting. For a given sRNA–mRNA interaction strength μ, the intrinsic noise increases with larger transcriptional bursts (smaller k−
). Furthermore, for a fixed k−
, the intrinsic noise increases with increasing sRNA–mRNA interaction strength μ, (see Supplementary Figure 1
; Elf and Ehrenberg, 2003
). The intrinsic noise is small in the repressed regimen αs
, and shows a pronounced peak in the crossover region, αs
(see ) as expected for a stoichiometric system. We have also obtained simplified, asymptotic expressions for the noise in the repressing and expressing regimens when τm
, and there is no transcriptional bursting (see Supplementary information
). The expressions for the intrinsic noise in the repressing and expressing regimens are given by, respectively:
Figure 4 Protein noise with or without transcriptional bursting. Noise in protein expression σp2/2 (variance divided by mean squared) as a function of the ratio of the sRNA and mRNA transcription rates, αs/αm, for different (more ...)
Figure 5 Comparison of analytic expressions for the intrinsic protein noise for TF- and sRNA-based regulation. The intrinsic noise for sRNA-based regulation as a function of normalized average protein concentration, /pmax, with and without transcriptional (more ...)
We have written these expressions so that the contribution of sRNA–mRNA mutual degradation noise is contained entirely in the second term of equations (8)
Comparing the intrinsic noise of protein- and sRNA-based regulators in , we observe that sRNA regulators are significantly less noisy than TFs in the repressed regimen. The dominant source of intrinsic noise for a TF-regulated protein, in the limit τm
, is that proteins are made in bursts of average size b
1. For an sRNA-regulated protein, the average size of a protein burst, beff
, is much smaller (see equation (8)
). This can be understood by noting that there are many more sRNAs than mRNAs in the repressed regimen, and therefore any free mRNA is quickly bound by an sRNA and degraded. This leads to a reduction in the effective mRNA lifetime and consequently a reduced beff
(Levine et al, 2007
). The reduction in effective mRNA lifetimes and intrinsic noise takes place even when mRNAs and sRNAs are produced in bursts.
The fidelity of a signaling system can be characterized by the output noise (σptotal
. In general, high-fidelity signaling requires (σptotal
1. Thus, from it is clear that over a large range of output protein levels, the large intrinsic noise due to transcriptional bursting makes it difficult for sRNAs to perform high-fidelity signaling.
One of the most striking features of is that sRNA-based regulation is much more sensitive to transcriptional bursting than protein-based regulation. For sRNAs, transcriptional bursting greatly enhances the near-critical fluctuations because the production of RNAs in bursts increases the anticorrelated sRNA–mRNA fluctuations in the crossover regimen (see Elf et al, 2003
; Elf and Ehrenberg, 2003
for more details on the near-critical fluctuations). In contrast, for transcriptional regulation directly by a TF, the contribution of transcriptional bursting to the intrinsic noise is relatively small for most choices of parameters (see ). As recent experiments suggest that prokaryotic transcription may generically produce RNAs in bursts (Golding et al, 2005
), this is likely to be a physiologically relevant effect for sRNA-based gene regulation.
The large intrinsic noise in the crossover regimen, αs≈αm can be understood by considering the special case αs=αm for very strong sRNA–mRNA binding, μ → ∞. In this limit, sRNAs and mRNAs, transcribed at the same average rate, quickly bind to each other and degrade and almost no protein is made. However, once in a while there is a fluctuation that produces more mRNAs than average. In this case, unless there is a corresponding fluctuation in sRNAs, the mRNAs cannot be degraded by sRNA–mRNA binding. The mRNAs produced in such a fluctuation will degrade by the usual slow degradation rate τm−1 resulting in a large burst of protein production, contributing to the large intrinsic noise. Transcriptional bursting further increases the magnitude of the aforementioned sRNA and mRNA fluctuations and consequently further increases the intrinsic noise in the crossover regimen.
Gain and filtering
We now consider, in the absence of noise, the change in output protein number about some steady state or ‘operating point' in response to a small, time-varying input signal. A small time-varying change from the steady-state value of the number of proteins controlling the sRNA transcription rate, δc
, results in a corresponding time-varying change of the output protein number from its steady-state value, δp
. For small enough signals, the dynamics are captured by linearized versions of the mass action equations (equation (1)
) (see Supplementary information
). In the frequency domain, the relationship between the output protein response at frequency ω and the input signal at frequency ω takes the simple form
where the frequency-dependent gain is given by
the characteristic time the sRNA gene is ‘on' and τ±
two times related to—and of the same order of magnitude as––the mRNA and sRNA lifetimes (see Supplementary information
for exact definition of τ±
). Each term of the form (i
can be interpreted as a low-pass filter with a cutoff frequency τ−1
. The four low-pass filters in the frequency-dependent gain come from different intermediate steps: I
from the binding–unbinding of the protein regulator (activator), II
from the transcription of RNAs and the sRNA–mRNA interaction, and III
from the translation of mRNAs into proteins. The amplitude of the frequency-dependent gain decreases rapidly
at high frequencies. This can be compared with the gain in TF-based regulation, which has only three low-pass filters and falls of at high frequencies
(see ; Supplementary information
). Thus, we conclude that sRNA-based regulation is less sensitive to high-frequency input noise than TF-based regulation.
Figure 6 Normalized frequency-dependent gain, g(ω)/g(0), as a function of the frequency, ω, for a small input signal for TF- and sRNA-based regulation in the repressed and expressing regimens. The amplitude of the frequency-dependent gain decreases (more ...)
The underlying reason for the enhanced noise filtering properties of sRNAs is that sRNA-based regulation involves an additional step when compared with transcriptional regulation. Namely, the input signal from upstream components in the genetic network is transmitted to the mRNAs encoding the output protein through sRNAs, which corresponds to an additional noise filter. This extra filtering could also be achieved by introducing an additional layer of transcriptional regulation in the genetic network. However, adding an extra layer of transcriptional regulation also leads to a slower kinetic response of the signaling network to changes in the input signal because an additional protein regulator must be synthesized or degraded to transmit signals. This kinetic cost is much smaller for sRNA-based regulation (see below). Consequently, sRNA-based regulation allows for an extra layer of noise filtering without sacrificing the ability to respond quickly to changes in input.
The above results hold only when the input signal is coupled to the sRNAs. Small input signals can also modulate the transcription of the protein-coding mRNAs instead of the sRNAs. In this case, at high frequencies, the gain falls off as
similar to TF-based regulation, as the input signal does not pass through the sRNAs (see Supplementary information
). Thus, coupling the input signal to sRNAs instead of mRNAs is necessary to achieve the advantageous high-frequency filtering properties of sRNA-based gene regulation. This may explain why input signals are often found coupled to the sRNAs rather than to the mRNAs in sRNA-based regulatory circuits.
Fidelity of small signal response
Intrinsic noise limits the ability of a signaling system to faithfully respond to small signals. Typically, the ability of a system to transduce small signals is quantified by its gain (amplification factor) (Detwiler et al, 2000
; Elf and Ehrenberg, 2003
; Elf et al, 2003
). A large gain is interpreted to mean the system can differentiate small changes in the input signal. However, even if the gain is large, if there is also high intrinsic noise—as is the case in sRNA-based regulation—it may be impossible to distinguish the output signal from the output noise (see Detwiler et al, 2000
; Supplementary information
). Furthermore, the gain often depends on how input and output signals are defined (e.g. logarithmic gain versus linear gain). For this reason, we consider an alternative measure to compare the small signal responses of sRNA- and protein-based regulators, namely the minimal signal that can be faithfully transmitted by the system (Detwiler et al, 2000
As discussed above, the noise in the output protein limits the detection of small input signals. For an input signal to be detectable, the corresponding output signal must be greater than the output noise (Detwiler et al, 2000
). In particular, the power of the output signal must be greater than the power of the output noise. Consider a periodic input signal at a frequency ω0
and amplitude δc
. For small input signals, the output signal is related to the input signal by the frequency-dependent gain g
(ω). Thus, the output signal is O(t)
and the power of the output signal is by definition
On the other hand, the power of the output noise is calculated by integrating fluctuations over all frequencies, and is given within the linear noise approximation by the expression
(ω′) is just the fluctuation in the output protein level at a frequency ω due to intrinsic noise as calculated in the Supplementary information
. For a signal to be detectable, we must have
For a step input signal with amplitude δco
→ 0 in the above expressions), the requirement that the output signal is larger than the noise sets a lower bound on the detectable input signal δcomin
(Detwiler et al, 2000
). Of course, by time-averaging the output, one can reduce the output noise and hence detect smaller signals, but this does not affect our comparison. Therefore, we computed the minimum input signal without time-averaging for both sRNA- and TF-based regulation and found that, for even moderate amounts of transcriptional bursting, protein regulators are better than sRNAs at responding to small signals across the whole range of output protein levels. At low protein levels (repressed regimen), the minimum detectable signal for sRNA-based regulation is larger due to the lower gain for sRNA-based regulation than for TF-based regulation. At intermediate to high levels of output protein (crossover and expressing regimens), the minimum detectable signal for sRNAs is also larger due to the large protein noise σp2
arising from transcriptional bursting for sRNA-based regulation.
Consequently, contrary to previous speculations (Levine et al, 2007
), results indicate that sRNA-based regulation is unlikely to be useful for amplifying small signals despite the large gain of sRNA-based regulation in the crossover region. Our results also imply that it is more advantageous to use TF-based regulation than sRNA-based regulation in genetic networks designed to respond to small changes in upstream components.
Large signal response
In nature, an organism may benefit from switching quickly between two different gene expression states in response to a large persistent input signal. We have compared here the rates at which a regulated protein can switch between ‘off' and ‘on' states in response to an input signal when its mRNA is directly regulated by a TF or indirectly regulated by an sRNA.
shows the time evolution of the average mRNA level for both sRNA- and TF-based regulation in response to a step change in the input. The response for sRNA-based regulation depends on the initial conditions, and can be tuned by changing the location in which the system is initially located in the repressed regimen. In particular, the effective mRNA degradation (and dilution) rate depends on the sRNA pool size and on the sRNA–mRNA interaction strength μ. However, our conclusions do not strongly depend on the choice of parameters (see Supplementary information
Figure 7 Large signal switching. Normalized mRNA level m/mmax, as a function of time, in response to step changes in the input, for both the sRNA- and TF-based regulation. Switching from high mRNA level (on state) to low mRNA level (off state) and vice versa. (more ...)
We find that using sRNAs to switch protein expression on, i.e. going from low output protein number to high output protein number, is slower than direct TF regulation. This slower response is due to the sRNA pool that needs to be depleted before target mRNAs can accumulate. On the other hand, sRNA-based regulation can be faster than TF-based regulation when switching off expression of a protein—the large input signal rapidly increases the concentration of sRNAs resulting in fast degradation of target mRNAs (see ; Shimoni et al, 2007
). The slower response of the sRNA-based regulation at turning on protein expression stems from the delay introduced by having an additional layer of sRNA regulation in the signal-transduction pathway when compared with protein-based regulation (see ). However, this delay is much smaller than that which would be introduced by having an additional layer of transcriptional regulation as the synthesis and degradation rates of proteins are much slower than those of RNAs (see Supplementary information
for a discussion comparing our results with Shimoni et al, 2007
Thus far, we have considered the case where a protein is negatively regulated by sRNAs. However, a protein can also be positively regulated by sRNAs (see Storz et al, 2004
; Hammer and Bassler, 2007
; Supplementary information
), and in this case switching protein expression on using sRNAs can be faster than TF-based regulation. Typically, sRNAs positively regulate protein expression by preventing the formation of inhibitory secondary structures that occlude the ribosome-binding sites of the regulated mRNA. As there is generally a background pool of translationally inactive target mRNAs, a large input signal that produces sRNAs allows the target mRNAs to be quickly converted into the translationally active form.