Exact results for noise power spectra in linear biochemical reaction networks

Abstract

We present a simple method for determining the exact noise power spectra in linear chemical reaction networks. We apply the method to networks which are representative of biochemical processes such as gene expression and signal detection. Our results clarify how noise is transmitted by signal detection motifs, and indicate how to coarse-grain networks by the elimination of fast reactions. Comment: 10 pages, 2 figures, RevTeX 4

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arXiv:q-bio/0512041v1 [q-bio.SC] 23 Dec 2005
Exact results for noise power spectra in linear biochemical reaction networks
Patrick B. Warren,1Sorin T˘ anasa-Nicola,2and Pieter Rein ten Wolde2
1Unilever R&D Port Sunlight, Bebington, Wirral, CH63 3JW, UK.
2FOM Institute for Atomic and Molecular Physics,
Kruislaan 407, 1098 SJ Amsterdam, The Netherlands.
(Dated: December 2005)
We present a simple method for determining the exact noise power spectra in linear chemical reac-
tion networks. We apply the method to networks which are representative of biochemical processes
such as gene expression and signal detection. Our results clarify how noise is transmitted by signal
detection motifs, and indicate how to coarse-grain networks by the elimination of fast reactions.
I. INTRODUCTION
It is now clear that chemical noise can sometimes play a
significant role in the functioning of biochemical reaction
networks [1]. Such noise is implicated in the spontaneous
flipping of genetic switches [2], and has been shown to
have an adverse effect on the functioning of a synthetic
chemical oscillator [3]. In some circumstances it has also
been argued that noise can have a beneficial effect, for
example in stochastic focusing [4].
Chemical noise in reaction networks can be charac-
terised by the root mean square (rms) deviation of num-
ber of molecules from the mean. Noise is important if
the rms deviation is a significant fraction of the mean.
This is typically associated with systems where the mean
number of molecules is rather small. Such situations are
characteristic of gene regulatory networks where the con-
centrations of regulatory proteins can be small enough for
there to be only a few molecules in the cell volume. For
example, for the lac operon in E. coli, the lac repressor
is active at concentrations where there are only 10–20
molecules present in the cell volume [5].
The mean and the rms deviation are ‘point statis-
tics’, and have frequently been used to characterise the
stochastic properties of biochemical reaction networks.
In the present paper, we focus on the noise power spectra,
which are a more refined characterisation of the stochas-
tic properties. We also focus on linear reaction networks
for which the noise power spectra and point statistics can
be calculated exactly. In the next section, we develop a
general theory for computing the noise power spectra in
linear reaction networks. We relegate technical details
to the Appendix. In the following sections we apply the
general theory to increasingly elaborate networks which
model basic processes of biochemical interest, such as
gene expression and signal detection.
II.GENERAL THEORY
We define a linear reaction network to be a network of
chemical reactions which does not involve bimolecular or
higher order reactions. For such a network, the chemical
rate equations can be written as
dNi
dt
=?
jKijNj+ bi
(1)
where Ni is the number of molecules of species i, Kij
is a matrix of rate coefficients, and biare source terms.
Eq. (1) is a set of linear ordinary differential equations,
which explains the origin of the phrase ‘linear reaction
network’. Appendix A1 indicates how Kij and bi are
related to the stoichiometry matrix and reaction rates
which specify the network.
Let Ni(t) be the instaneous number of molecules of
species i in the system. Define the mean value of Niin
steady state to be ?Ni?, and the mean value out of steady
state to be ?Ni?t. By taking moments of the chemical
master equation (see Appendix), one can prove that
d?Ni?t
dt
=?
jKij?Ni?t+ bi. (2)
Therefore, the chemical rate equations are exact for a lin-
ear reaction network, provided we work with the mean
quantities ?Ni?t. In steady state, the mean values there-
fore solve
?
jKij?Ni? + bi= 0. (3)
This is a system of linear simultaneous equations which
can be readily solved.
We now define the deviation away from the steady
state mean values to be
∆Ni(t) = Ni(t) − ?Ni?(4)
and introduce the steady-state variance-covariance ma-
trix
Sij= ?∆Ni∆Nj?. (5)
The diagonal elements of this are the variances
σ2
i= Sii= ?∆N2
i?. (6)
The rms deviation mentioned in the introduction is σi. It
is often the case that σ2
i∼ ?Ni? and in fact some authors
define the ‘noise strength’ to be σ2
not explicitly use this in the present paper. This scaling
i/?Ni? although we will

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means that the relative rms deviation σi/?Ni? ∼ 1/√?Ni?
which indicates that the relative important of noise de-
creases inversely with the square root of the number of
molecules involved. It is the basic reason why noise is
important for systems where the number of molecules is
small.
As many authors have noticed [6, 7, 8], the variance-
covariance matrix at steady state can also be obtained
by taking moments of the chemical master equation. The
details are unimportant for our arguments, and are rel-
egated to Appendix A1. For the remainder of the main
text, we assume that the means and variance-covariance
matrix have been computed. We will say that these com-
prise the point statistics of the network.
To refine the description of the steady state beyond
the point statistics, we introduce the set of correlation
functions
Cij(t) = ?∆Ni(0)∆Nj(t)?. (7)
In steady state, the value chosen for the time origin is
unimportant (therefore the average could be regarded as
a time average over starting times, instead of an ensemble
average). The correlation functions have the properties
Cij(0) → Sij
Cij(t) → 0
Cij(t) = Cji(−t)
ast → 0,
t → ∞, as
(8)
(the first of these is not always true; see section IIIC).
The last property shows that the autocorrelation func-
tions are time-symmetric, Cii(t) = Cii(−t), but it does
not follow that the cross-correlation functions are time-
symmetric, as is illustrated by the example in section
IIIA.
Closely related to the autocorrelation functions are the
noise power spectra, which are the central topic of study
in the present paper. These are defined to be
Pi(ω) = ?|∆Ni(ω)|2? = 2
?∞
0
dt cosωtCii(t) (9)
where ∆Ni(ω) is the Fourier transform of ∆Ni(t). We
shall use the second form in Eq. (9) to compute the noise
power spectra.
Usually the noise power spectra obey the sum rule (for
an exception, see section IIIC),
σ2
i=
1
2π
?∞
0
dωPi(ω). (10)
Now we turn to the computation of the correlation
functions and the noise power spectra. In Appendix A2
we prove that the correlation functions satisfy
dCki
dt
=?
jKijCkj. (11)
If we subtract Eq. (3) from Eq. (2), we find that the mean
deviation out of steady state obeys
d?∆Ni?t
dt
=?
jKij?∆Ni?t. (12)
This shows that the correlation functions decay in exactly
the same way as deviations away from steady state, so
that Eq. (11) is an example of a regression theorem [9].
Eq. (11) is to be solved with the initial conditions
Cij(0) = Sij. A convenient approach is by the use of
the Laplace transform, which allows the initial conditions
to be automatically incorporated. Taking the Laplace
transform of Eq. (11) shows that
s˜Cki− Ski=?
jKij˜Ckj
(13)
where
˜Cij(s) =
?∞
0
dte−stCij(t).(14)
Eq. (13) is a set of linear simultaneous equations which
can readily be solved for˜Cij(s).
The power spectra now follow almost for free. Com-
paring Eq. (14) with Eq. (9) shows that
Pi(ω) =˜Cii(iω) +˜Cii(−iω),(15)
in other words Pi(ω) is twice the real part of the analytic
continuation of˜Cii(s) to s = iω.
Eqs. (13) and (15) are the key results of this section.
They indicate how exact results for the power spectra
can be obtained for an arbitrary linear reaction network
by purely algebraic methods. In Appendix A3 we prove
that the same results can be obtained from the chemical
Langevin equations described by Gillespie [10].
It is interesting to note that the power spectra are sim-
pler to obtain than the correlation functions themselves,
which require an inverse Fourier or Laplace transform.
To undertake such a step requires in general that a poly-
nomial be factorised, which is not always possible. In
fact the route to the power spectra, via Eqs (3), (13) and
(A7) in the Appendix, just involves the solution of linear
simultaneous equations.
III.APPLICATIONS
A. Gene expression models
We first consider a very simple model of gene expres-
sion
k
→ nA,A
γ
→(16)
which represents a birth-death process in which n copies
of A are generated each time the first (birth) reaction
fires. This represents the ‘burstiness’ of gene expression
in prokaryotes [8]. The point statistics are
?NA? =nk
γ,
σ2
A=n + 1
2
?NA?(17)
The chemical rate equation is dNA/dt = nk − γNA so
the autocorrelation function obeys dCAA/dt = −γCAA.

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10-3
10-2
10-1
100
101
100
102
104
106
108
ω (min−1)
PA(ω) (min)
FIG. 1: Power spectra for the gene expression models in
Eq. (16) (Eq. (19); dashed line) and Eq. (20) (Eq. (27); solid
line). Parameters for the explicit mRNA model of Eq. (20)
are k = 2.76min−1, λ = 0.12min−1, ρ = 3.2min−1, and
γ = 0.016min−1. These correspond to cro in a recent model of
phage lambda [11]. Parameters for the implicit mRNA model
of Eq. (16) are k = 1.36min−1, n = 48.1, γ = 0.0142min−1.
These are chosen so that the two models have matched point
statistics, ?NA? = 4.6×103and σA/?NA? = 0.07, and match-
ing values of PA(0) = 1.6 × 107min.
The solution, and its Laplace transform, are
CAA(t) = σ2
Ae−γt,
˜CAA(s) =
σ2
A
s + γ.
(18)
The corresponding power spectrum is
PA(ω) =
2γσ2
ω2+ γ2.
A
(19)
Now we turn to a more realistic representation of gene
expression which explicitly includes mRNA, namely
k
→ M,M
λ
→,M
ρ
→ M + A,A
γ
→ .(20)
The first two reactions represent generation and degra-
dation of M which corresponds to mRNA. The second
two reactions represent generation of the protein A from
M and the degradation of A. This is a model which has
been analysed by other workers [6, 7, 8], and we recover
previously known results for the point statistics.
The mean values in steady state are
?NM? =k
λ,
?NA? =ρ
γ?NM? =bk
γ.
(21)
In this, b = ρ/λ is the mean number of copies of A pro-
duced in the lifetime of one mRNA and can be taken to
represent the ‘burstiness’ of gene expression. Comparing
the last of Eq. (21) with the first of Eq. (17), it appears
that b in the present model corresponds to n in the pre-
vious implicit mRNA model. As we shall see though,
this interpretation does not carry through to the other
statistical properties.
The steady state variances and covariance are (see Ap-
pendix A1)
σ2
M= ?NM?,σ2
A=
?
1 +
ρ
γ + λ
?
?NA?,
SMA=ρ?NM?
γ + λ
=γ?NA?
γ + λ.
(22)
In this case, comparing Eq. (22) with Eq. (17), we see
that the correspondence b ↔ n fails for σ2
Now we turn to the computation of the correlation
functions and the power spectra. The chemical rate equa-
tions are
A.
dNM
dt
= k − λNM,
dNA
dt
= ρNM− γNA. (23)
The correlation functions obey the homogeneous form of
these equations. Taking the Laplace transform as indi-
cated in the previous section, we find
s˜CMM− σ2
s˜CAM− SAM= −λ˜CAM,
s˜CMA− SMA= ρ˜CMM− γ˜CMA,
s˜CAA− σ2
M= −λ˜CMM,
A= ρ˜CAM− γ˜CAA.
(24)
Because CMA(t) ?= CAM(t), there are four different
Laplace-transformed correlation functions (note though
that SMA= SAM). The solutions are
˜CMM=
σ2
s + λ,
M
˜CAM=SAM
s + λ,
˜CMA=SMA
s + γ+
ρσ2
M
(s + λ)(s + γ),
˜CAA=
σ2
A
s + γ+
ρSAM
(s + λ)(s + γ).
(25)
In this particular case, the inverse Laplace transforms
can easily be found by the method of partial fractions.
We give only the result for the cross-correlation function,
which exhibits the interesting time asymmetry mentioned
above,
CMA(t) =







SMAe−γt+ ρσ2
M
e−γt− e−λt
λ − γ
(t ≥ 0)
SMAe−λ|t|
(t ≤ 0)
(26)
(using CMA(−t) = CAM(t) for the second line).
Finally, the power spectra are obtained. The power
spectrum for M is PM(ω) = 2λ?NM?/(ω2+ λ2). This
might have been anticipated as a special case of the pre-
vious model with n = 1, since as far as M is concerned

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it is undergoing a simple birth-death process. The power
spectrum for A is
PA(ω) =2γ?NA?(ω2+ λ(λ + ρ))
(ω2+ λ2)(ω2+ γ2)
. (27)
With a suitable choice of n and rate cofficients k and
µ, the point statistics of the previous model in Eq. (16)
could be matched to the present model in Eq. (20). How-
ever, it is in general impossible to match the power spec-
tra since the ω dependence is different. Fig. 1 shows the
power spectra for the two models with matched point
statistics. The presence of two correlation times in the
explicit mRNA model compared to a single relaxation
time for the implicit mRNA model is clearly seen.
We have gone through the calculations in some detail
for these two model, but for the subsequent calculations
we will only give the final results.
B. Signal detection
A very simple model of signal detection is the reaction
A
ν→ B,B
µ
→ (28)
where A is the input signal and B is the output signal.
Eq. (28) has a chemical rate equation
dNB
dt
= νNA− µNB+ η.(29)
We have included an additional noise term η in this, so
it is actually a chemical Langevin equation. Taking the
Fourier transform yields
iωNB= νNA− µNB+ η,orNB=νNA+ η
iω + µ
. (30)
The mean square modulus is then
?|NB|2? =ν2?|NA|2? + ?|η|2?
ω2+ µ2
.(31)
We have assumed that η is uncorrelated with NA. The
theory of chemical Langevin equations [10] (see also Ap-
pendix A3) shows that ?|η|2? = 2µ?NB?, and for ω > 0
one has ?|Ni|2? = ?|∆Ni|2?, thus finally
ν2
ω2+ µ2PA(ω) +2µ?NB?
PB(ω) =
ω2+ µ2.(32)
This result has quite a striking interpretation. The to-
tal noise in the output signal is made up of an extrinsic
contribution (first term) plus an intrinsic contribution
(second term). Moreover, the extrinsic noise is equal the
input signal noise multiplied by a low-pass filter function.
This is analogous to the behaviour of a passive RC cir-
cuit element [12]. This result, in one form or another,
was derived by Paulsson [13], and by Shibata and Fuji-
moto [14]. We refer to it as the ‘spectral addition rule’.
It is a potentially important result because it indicates
that there is a trade-off between signal amplification by
the detection motif and signal contamination by added
intrinsic noise.
Our exact results for linear reaction networks enable us
to test the spectral addition rule. We find that its validity
is limited by the assumption that the intrinsic noise η is
uncorrelated with the input signal NA. In particular,
if the detection motif consumes molecules of the input
signal, a correlation can arise which spoils the spectral
addition rule.
To demonstrate this, we now consider the effect of
adjoining Eq. (28) onto the gene expression models in
the previous section. We present results for the explicit
mRNA model, but similar conclusions can be drawn for
the implicit mRNA model too [15]. We therefore solve
k
→ M
λ
→,M
ρ
→ M + A,A
γ
→,A
ν→ B
µ
→ .(33)
We find that
PA(ω)
?NA?
=
2(γ + ν)(ω2+ λ(λ + ρ))
(ω2+ λ2)(ω2+ (γ + ν)2)
(34)
and
PB(ω)
?NB?
=2µ[(ω2+ λ2)(ω2+ (γ + ν)2) + λνρ(γ + ν)]
(ω2+ λ2)(ω2+ (γ + ν)2)(ω2+ µ2)
(35)
with ν?NA? = µ?NB?. It is not hard to demonstrate that
the spectral addition rule fails for these expressions.
As an aside, one can show σ2
Since the A → B reaction in Eq. (33) can be regarded
as post-translational modification step, this shows that
post-translational modification can reduce the noise asso-
ciated with gene expression. The reason is that the post-
translational modification reaction smoothes the ‘bursti-
ness’ of gene expression by acting as a low-pass filter.
To appreciate the influence of coupling the detection
reaction to the input signal noise, we now change the
detection scheme so that the signal molecules A are not
consumed, by replacing Eq. (28) with
B/?NB? < σ2
A/?NA?.
A
ν→ A + B,B
µ
→ .(36)
As far as B is concerned, it is important to note that the
same chemical Langevin equation Eq. (29) holds for this
detection scheme as for the previous scheme. For this
variant we find PA(ω) is as given in Eq. (27) and
PB(ω)
?NB?
One can check that in this case PA(ω) and PB(ω) do obey
the spectral addition rule.
Whilst we have only demonstrated the failure of the
spectral addition rule for one particular case, the result is
=2µ[(ω2+ λ2)(ω2+ γ(γ + ν)) + γνρλ]
(ω2+ γ2)(ω2+ λ2)(ω2+ µ2)
(37)

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suggestive of the general conclusion that the spectral ad-
dition rule will only hold for detection schemes which do
not consume input signal molecules. These conclusions
can also be reached by analysing the chemical Langevin
equations for the whole network [15].
C. Fast reactions
There is a growing literature on the adiabatic elimi-
nation of fast reactions for stochastic chemical kinetics
[16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
Kepler and Elston [16], and Shibata [18, 19], have a for-
malism that permits a systematic approach to the prob-
lem, involving the identification of the fast and slow vari-
ables in the system. In a similar example, Bundschuh et
al present simulation results which support the general
strategy of elimination of fast variables in terms of slow
variables [20]. Here we examine the procedure for elim-
ination of a fast equilibration reaction in the context of
the exact results for a linear reaction network. Our re-
sults shed light on the way in which fast equilibration
reactions can be systematically eliminated from a reac-
tion network, whilst preserving the noise attributes.
To make a concrete example, we suppose that the sig-
nal detection motif of the previous section consists of a
fast binding-unbinding step, followed by a slower detec-
tion step. We replace A → B by A⇀
represents the bound state. We suppose that the sub-
strate which binds A is in excess, so that our reaction
network is still a linear network. The reaction scheme in
its totality is now
For example
↽ A∗→ B where A∗
k
→ M
λ
→,M
ρ
→ M + A,
kf
⇀
↽
kb
A
γ
→,A
A∗ ν→ B
µ
→ .
(38)
The noise statistics for this network can be solved but
the expressions are rather lengthy. We therefore focus
on the limit of a fast binding-unbinding reaction.
The equilibrium constant K for the binding-unbinding
reaction is defined via
kf= K kb. (39)
To take the limit of fast equilibration, we keep K finite
and allow kb → ∞. The results are as follows. Let us
first consider the final product species B. We find PB(ω)
is given by Eq. (37) but with ‘renormalised’ reaction rates
γR=
γ
K + 1,νR=
Kν
K + 1
(40)
The same is true of the variance σ2
We can rationalise this as follows. The chemical rate
equations which involve the species in the fast equilibra-
B.
tion reaction (A and A∗) are
dNA/dt = ρNM− (γ + kf)NA+ kbNA∗,
dNA∗/dt = kfNA− (ν + kb)NA∗,
dNB/dt = νNA∗ − µNB.
We now make a linear transformation from NAand NA∗
to new variables NX and NY :
(41)
NX= NA+ NA∗,
NY = KNA− NA∗.
(42)
These are chosen because NXis a slow variable which is
conserved by the equilibration reaction, and NY is a fast
variable which vanishes in steady state. In terms of these
concentration variables, the chemical rate equations be-
come
dNX/dt = ρNM−γ + Kν
dNY/dt + (K + 1)kbNY = KρNM−γ − ν
K + 1
NX−γ − ν
K + 1NY,
K + 1NX,
dNB/dt =
Kν
K + 1NX−
ν
K + 1NY − µNB.
(43)
The second of these shows that NY relaxes at a rate ∼ kb
to a constant ∼ 1/kb, for large kb. This formalises the
separation of timescales between NY and all the other
concentration variables. In the large kb limit therefore,
NY = 0 and Eqs. (43) become
dNX/dt = ρNM− (γR+ νR)NX,
dNB/dt = νRNX− µNB.
with the reaction rates given by Eq. (40). These are (a
subset of) the chemical rate equations for the following
reaction network
(44)
k
→ M
λ
→,M
ρ
→ M + X,X
γR
→,X
νR
→ B
µ
→ .(45)
This is the same as the previously discussed scheme in
Eq. (33), with the replacement A ↔ X. The power spec-
tra follow from Eqs. (37) accordingly. The important
point to note is that this scheme not only has the correct
chemical rate equations but also has the correct noise
power spectra.
Now let us turn to the power spectra for A and A∗in
the original network. We find that
PA(ω) =
1
(K + 1)2PX(ω),PA∗(ω) = K2PA(ω). (46)
Again these results are easy to rationalise, since inverting
Eqs. (42) and setting NY = 0 shows that
NA=
1
K + 1NX,NA∗ = K NA
(47)
(the proportionality factors become squared when com-
puting the noise power spectra).

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For the variances, σ2
though. In fact, in the limit kb→ ∞, we find a breakdown
of the sum rule in Eq. (10). The reason is that
Aand σ2
A∗, we find a different story
lim
kb→∞
?∞
0
dω Pi(ω) ?=
?∞
0
dω lim
kb→∞Pi(ω) (48)
for i = A and A∗. The sum rule always holds for the left
hand side of Eq. (48), but the limiting power spectra are
defined in terms of the right hand side of Eq. (48). (As an
aside, we find that the sum rule is satisfied for PA+A∗(ω),
which is equal to PX(ω), confirming that the effect is
confined to the individual species in the fast equilibration
reaction.) To handle this kind of situation, we define a
‘sum rule deficit’
∆σ2
i≡ σ2
i−
1
2π
?∞
0
dωPi(ω). (49)
For the present situation, we find that
∆σ2
A= ∆σ2
A∗ =?NA∗?
K + 1. (50)
Interestingly, the same result is found for all situations
involving a fast equilibration reaction which we have ex-
amined.
The origin of the sum rule deficit lies in the noise gen-
erated by the fast equilibration reaction. Formally, at
any finite kbwe can define the contribution of the equi-
libration reaction to the total noise of species i to be
∆Pi(ω) = Pi(ω) − lim
kb→∞Pi(ω). (51)
For i = A and A∗, one has that ∆Pi(ω) → 0 as kb→ ∞,
at any finite value of ω, but that?∞
The reason is that, whilst ∆Pi(ω) vanishes as 1/kb, it
extends over a frequency range 0 < ω<
integrated contribution does not vanish in the limit kb→
∞. In words, the equilibration reaction contributes high-
frequency noise which vanishes at any particular finite
frequency in the limit of infinitely fast equilibration, but
makes a net non-zero contribution to the total integrated
noise.
The universal magnitude of the sum rule deficit can be
understood by analysing the simplest of all equilibration
reactions, namely
0∆Pi(ω) → 2π∆σ2
i.
∼kb, so the total
A
kf
⇀
↽
kb
A∗.(52)
Solving for the point statistics in this system one has
K?NA? = ?NA∗? and, from Eqs. (A7) in the Appendix,
−Kσ2
−KSAA∗ + σ2
KSAA∗ − σ2
These equations are linearly dependent and cannot by
themselves be solved for the elements of the variance-
covariance matrix. The reason is that the slow variable
A+ SAA∗ + ?NA∗? = 0
A∗ + Kσ2
A− SAA∗ − 2?NA∗? = 0
A∗ + ?NA∗? = 0
(53)
01234
5
0.2
0.3
0.4
0.5
0.6
0.8
1.0
t (min)
CAA(t)/σ2
A
FIG. 2: Correlation function for the model in Eq. (57), with
γR = Kγ/(K +1) = 0.12min−1and K = 1. The lines are for
kb= 0.2, 1.0, 5.0min−1, and kb→ ∞ (dashed line).
NA+ NA∗ is exactly conserved by the reaction scheme
in Eq. (52). However this implies that ∆NA= −∆NA∗
and so σ2
A∗ = −SAA∗. These are consistent with
Eqs. (53), which can now be solved to get
A= σ2
σ2
A= σ2
A∗ = −SAA∗ =?NA∗?
K + 1.(54)
This explains the universal value found for the sum rule
deficit above.
To complete the discussion, we turn finally to the cor-
relation functions where the analogue of Eq. (48) is
lim
kb→∞lim
t→0Cii(t) ?= lim
t→0
lim
kb→∞Cii(t)(55)
for i = A and A∗. The left hand side is always equal
to σ2
i. One can show that the difference between the
two sides gives an alternative expression for the sum rule
deficit,
∆σ2
i= σ2
i− lim
t→0Cii(t). (56)
This can be made clear by another example. Consider
k
→ A
kf
⇀
↽
kb
A∗ γ
→ . (57)
The steady state point statistics, which hold for arbi-
trary kb, are ?NA? = (kb+ γ)?NA∗?/kf, ?NA∗? = k/γ,
σ2
A∗/?NA∗? = 1, SAA∗ = 0. We solve for
the correlation functions using the methods described in
section II. Similar results are obtained for both A and
A∗so we report results only for species A. The Laplace-
transformed autocorrelation function is
A/?NA? = σ2
˜CAA(s) =
σ2
A(s + γ + kb)
s2+ (γ + kf+ kb)s + γkf.(58)

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7
Writing
s2+ (γ + kf+ kb)s + γkf= (s + k+)(s + k−)(59)
allows us to perform the inverse Laplace transform by
the method of partial fractions. The decay rates k±are
given by
k±=(γ + kf+ kb) ±?(γ + kf+ kb)2− 4γkf
2
.(60)
The net result is
CAA(t) = σ2
A(Ae−k+t+ Be−k−t) (61)
with
A =(γ + kb) − k−
k+− k−
,B = 1 − A.(62)
For any finite value of kb, it is clear from this that
CAA(0) = σ2
Now let us take the limit kb → ∞ with K = kf/kb
being held constant. To leading order we have
Aand there is no sum rule deficit.
k+→ (K + 1)kb,k−→
Kγ
K + 1,
A →
K
K + 1,B →
1
K + 1,
(63)
and thus
lim
kb→∞CAA(t) =
σ2
A
K + 1exp
?
−Kγt
K + 1
?
. (64)
Apart from the amplitude, this correlation function is
characteristic of→ X
entirely in accord with rewriting the original scheme in
terms of the relevant slow variable.
The analysis shows that in the limit kb→ ∞, we have
CAA(0) < σ2
k
γR
→ with γR= Kγ/(K +1). This is
A. Thus there is a non-zero sum rule deficit
∆σ2
A= σ2
A
?
1 −
1
K + 1
?
=?NA∗?
K + 1
(65)
where we have used σ2
universal value for the sum rule deficit is recovered.
Our analysis shows what happens to the correlation
functions in the limit of fast equilibration (similar results
are found for CA∗A∗). At any finite kbthere are two decay
rates in the correlation function. In the limit kb→ ∞ one
of these decay rates diverges whilst the other saturates
to a finite value. The diverging decay rate carries with
it a finite amplitude and it is this that gives rise to the
sum rule deficit. The phenomenology is illustrated in
Fig. 2 which shows typical results for CAA(t) at several
increasing values of the equilibration rate.
The results in this section are supportive of the general
strategy of elimination of the fast variables in terms of the
appropriate slow variables. To be specific, the strategy
is to rewrite the chemical rate equations in terms of fast
A= ?NA? = ?NA∗?/K. Again, the
and slow variables, eliminate the fast variables, and re-
interpret the reduced rate equations in terms of a reduced
reaction network, possibly involving new chemical species
(in the present example, X replaces A and A∗).
study has provided examples where one can rigorously
prove that this strategy is successful.
Our
IV. DISCUSSION
Firstly, let us make some general remarks about the use
of noise power spectra to characterise the stochastic prop-
erties of chemical reaction networks. As we have shown,
the noise power spectra are straightforward to calculate
once one has the point statistics (the mean values and the
variance-covariance matrix), for a linear network, or for a
linearisation of a non-linear network. In fact, the power
spectra are easier to calculate than the autocorrelation
functions, which in general require the factorisation of a
polynomial, unless one is satisfied with stopping at the
Laplace-transformed autocorrelation functions. Because
the power spectra are functions of frequency, they are a
more refined measure than the point statistics, and can
be used for instance as a sensitive test of whether two
reaction networks can be mapped onto one another.
Let us comment briefly on the situation for experi-
ments and simulations. Both face similar difficulties in
measuring the power spectra. A signal of sufficiently long
duration needs to be captured at sufficiently fine resolu-
tion in order to be able to estimate the power spectrum.
Event-driven simulations though, such as those based on
the Gillespie algorithm [26], have an advantage since in
principle the exact moments when a signal changes its
value are known.
Secondly, we discuss the biological relevance of our re-
sults. Our results indicate that the way noise is trans-
mitted through a signal detection motif may be more
complicated than previously thought [13, 14, 15]. In par-
ticular, if the detection motif consumes the input signal
molecules, a correlation can be set up between the in-
trinsic noise generated by the detection motif, and the
input signal noise. We have examined this in the context
of a post-translational modification reaction attached to
the output of a gene expression module. Our analysis
shows incidentally that post-translational modification
can ameliorate the noise of gene expression, by smoothing
out the ‘burstiness’ of the translation step.
We also examined the effect of a fast equilibration re-
action interposed in the network. We find that, in the
limit of infinitely fast equilibration, such a network can
be exactly mapped onto a reduced or coarse-grained net-
work through the use of suitably chosen slow variables.
This result supports the use of slow variables as a general
strategy for model-order reduction by elimination of fast
reactions.
This work was supported by the Amsterdam Centre
for Computational Science (ACCS). The work is part of
the research programme of the “Stichting voor Funda-

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8
menteel Onderzoek der Materie (FOM)”, which is finan-
cially supported by the “Nederlandse organisatie voor
Wetenschappelijk Onderzoek (NWO)”. PBW acknowl-
edges the hospitality of Trey Ideker and the California In-
stitute for Telecommunications and Information Technol-
ogy (Calit2) at the University of California, San Diego,
where part of this work was written up.
APPENDIX A: TECHNICAL DETAILS
Technical details and proofs are relegated to this Ap-
pendix. For further reading we recommend Gardiner [9],
van Kampen [27], Gillespie [10, 26], and Swain [28].
1.Point statistics
We first describe the chemical master equation and
present a useful moment relation. As a preliminary step
we need to define some basic quantities. The stoichiome-
try matrix is a non-square matrix νiαequal to the change
in species i due to the firing of reaction α in the network.
We shall use roman indices to label chemical species and
the greek symbol α to label reactions. In terms of νiα,
the chemical rate equations can be written
dNi
dt
=?
ανiαaα
(A1)
where aα is the flux through reaction channel α. The
quantity aα is usually dependent on the current values
of Ni and is referred to by Gillespie as the ‘propensity
function’. The propensity functions for a linear reaction
network will be described shortly.
In these terms, the probability PNiof being in the state
characterised by Nimolecules of species i obeys
∂PNi
∂t
=
?
α
?
aα(Ni−νiα)PNi−νiα−aα(Ni)PNi
?
. (A2)
If a reaction channel is impossible, for instance where
Ni− νiα< 0, we set aα= 0. Eq. (A2) is the chemical
master equation.
Multiplying Eq. (A2) by a general function f(Ni) and
summing over all Niyields
d?f?t
dt
=?
α?aα(Ni)[f(Ni+ νiα) − f(Ni)]?t.(A3)
This is particularly useful for deriving equations for mo-
ments.
We now indicate how the point statistics and in par-
ticular the variance-covariance matrix can be found for
an arbitrary linear reaction network. The results follow
as particular cases of Eq. (A3).
First we need to specify in more detail the propensity
functions for a linear reaction network. We distinguish
between zeroth- and first-order reactions in the network.
The former are pure generating reactions of the form ‘→
products’. The latter are monomolecular reactions and
comprise the rest of the network (in a linear reaction
network, all reactions are either zeroth- or first-order).
The set of zeroth-order reactions is denoted by {0} and
the set of first-order reactions by {1}.
The propensity function for a zeroth-order reaction is
aα= kα,α ∈ {0}.
For the first order re-
(A4)
where kα is the reaction rate.
actions, we introduce an indicator matrix ǫiα, such that
ǫiα= 1 if the reaction α involves species i as the reactant,
and ǫiα= 0 otherwise. Armed with this, the propensity
function for a first-order reaction is
aα= kα?
iǫiαNi
α ∈ {1}.(A5)
Inserting Eqs. (A4) and (A5) into Eq. (A1) obtains
Eq. (1) in the main text, with
bi=
?
α∈{0}
kανiα,Kij=
?
α∈{1}
kανiαǫjα. (A6)
The first moment of the chemical master equation is
found by setting f(Ni) = Ni in Eq. (A3). It is easy to
show that this gives Eq. (2) in the main text, with the
above definitions of biand Kij.
The second moment of the chemical master equation is
found by setting f(Ni) = NiNjin Eq. (A3). This gives a
closed set of equations for the elements of the variance-
covariance matrix. In steady state, these equations can
be reduced to
?
k(KikSkj+ KjkSki) + Hij= 0 (A7)
where
Hij=
?
α∈{0}
kανiανjα+?
k
??
α∈{1}
kανiανjαǫkα
?
?Nk?
(A8)
Eq. (A7) is a set of linear simultaneous equations for the
elements Sijof the variance-covariance matrix.
As an example of the application of this machinery,
consider the model for gene expression in Eq. (20) in the
main text. There are two species, M and A, and four
reactions. The stoichiometry matrix is
νiα=
?1
0
???−1 00
01 −1
?
.(A9)
The reaction rates are
kα= ( k | λ ρ γ ).(A10)
The indicator matrix is
ǫiα=
?·
·
???1 1 0
0 0 1
?
. (A11)
In these, the reactions to the left of the vertical line cor-
respond to α ∈ {0} and the reactions to the right of the

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9
line correspond to α ∈ {1} (the indicator matrix is only
defined for the latter reactions).
From these one computes
Kij=
?−λ0
ρ−γ
?
.(A12)
Obviously this could have been written down by inspec-
tion. Less trivially, one also finds
Hij =
?k + λ?NM?0
0ρ?NM? + γ?NA?
?
=
?2λ?NM?0
02γ?NA?
?
.
(A13)
Eqs. (A7) simplify to
−2λσ2
ρσ2
M− γSMA− λSMA= 0,
2ρSMA− 2γσ2
M+ 2λ?NM? = 0,
A+ 2γ?NA? = 0.
(A14)
There are only three equations since Eq. (A7) is symmet-
ric in i and j. These are solved to get Eqs. (22) in the
main text.
2.Regression theorem
The proof of the regression theorem Eq. (11) in the
main text is straightforward and parallels the develop-
ment in Gardiner [9]. We start by writing out an explicit
expression for the correlation function
Cij(t) =
?
d{Ni,0}d{Ni,t} ∆Ni,0∆Nj,t
× Ps(Ni,0) P(Ni,t|Ni,0;t).
(A15)
In this, Ps(Ni,0) is the steady state probability distribu-
tion for the starting point Ni= Ni,0, and P(Ni,t|Ni,0;t)
is the conditional probability distribution for Ni= Ni,t
at time t, given that the system started with Ni= Ni,0
at t = 0. These probability distributions could in prin-
ciple be found by solving the chemical master equation.
We now define two kinds of averages
?f(Ni)?t|0=?d{Ni,t} f(Ni,t) P(Ni,t|Ni,0;t),
?f(Ni)?0=?d{Ni,0} f(Ni,0) Ps(Ni,0).
In words, the first is the average value of a function of
Ni= Ni,tat time t, given that the system started with
Ni= Ni,0at t = 0. The second is the average value of
a function of Ni = Ni,0 in steady state conditions. In
terms of these, Eq. (A15) can be written as
(A16)
Cij(t) = ?∆Ni?∆Nj?t|0?0. (A17)
This result is completely general [9].
For a linear reaction network, taking the first moment
of the chemical master equation proves an analogue to
Eq. (12), namely
d?∆Ni?t|0
dt
=?
jKij?∆Nj?t|0.(A18)
Combining this with Eq. (A17) demonstrates that Cij(t)
obeys Eq. (11) in the main text.
3.Chemical Langevin equations
We prove that the power spectra obtained from the
chemical Langevin equations for a linear reaction net-
work are equivalent to the exact results obtained from
the chemical master equation. As the first step, Gillespie
has shown that the chemical Langevin equations for a
general network are [10]
dNi
dt
=?
ανiαaα+?
ανiαa1/2
α
Γα.(A19)
In this, Γα are independent unit Gaussian white noise
functions, one for each reaction channel. Applying this
to a linear reaction network obtains
dNi
dt
=?
jKijNj+ bi+ ηi
(A20)
with
ηi=
?
α∈{0}
k1/2
α νiαΓα+
?
α∈{1}
(kα
?
jǫjαNj)1/2νiαΓα.
(A21)
It follows that the ηiare Gaussian white noise functions
with the following statistics
?ηi? = 0,?ηi(t)ηj(t′)? = Hijδ(t − t′).(A22)
The correlation matrix Hijis identical to the matrix that
features in the computation of the variance-covariance
matrix, given by Eq. (A8) above, which can be inter-
preted in this context as a fluctuation-dissipation theo-
rem.
Working in terms of the deviation from steady state,
and taking the Fourier transform of the chemical
Langevin equations, obtains
iω∆Ni=?
jKij∆Nj+ ηi.(A23)
We conclude
Pi(ω) = ?|∆Ni|2? =?
where Bij(iω) is the inverse of iωδij−Kijand B†
Bji(−iω) is the adjoint.
Eq. (A24) is equivalent to Eqs. (13) and (15) in the main
text.
The problem is made easier if we rewrite everything
in abstract matrix notation.
obtained can be written as
jkBijHjkB†
ki
(A24)
ij(iω) =
Our task is to prove that
The result we have just
P1= B · H · B†
(A25)

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10
with
B = (iωI − K)−1,
(KTis the transpose of K). The diagonal elements of
P1 are the power spectra from the chemical Langevin
equation route.
We now similarly rewrite the results from the analysis
of the chemical master equation. Eqs. (A7), (13) and
(15) can be written respectively as
B†= (−iωI − KT)−1
(A26)
K · S + S · KT+ H = 0,
iωC − S = K · C,
P2= C + C†= B · S + S · B†
(note that S is real and symmetric). The diagonal el-
ements of P2 are the power spectra from the chemical
master equation route.
Eliminate H between the first of Eqs. (A27) and
Eq. (A25) to get
or
C = B · S,
(A27)
P1= −(B · S · KT· B†+ B · K · S · B†).
It follows from Eq. (A26) that
(A28)
B · K = iωB − I,
On substituting these into Eq. (A28), there is a cancel-
lation of terms, and one finds
KT· B†= −iωB†− I. (A29)
P1= B · S + S · B†. (A30)
Comparison with the last of Eqs. (A27) shows that P1=
P2. This contains our desired result, and proves that the
power spectra obtained by the chemical Langevin equa-
tion route are the same as those obtained by the (ex-
act) chemical master equation route. Actually, we have
a slightly stronger result, since it follows that ?|∆NX|2? =
xT· P1· x = xT· P2· x, for any linear combination
NX=?
ixiNI.
Swain has presented an analysis of the chemical
Langevin equations for, effectively, a general linear re-
action network [28]. He obtains general results for the
correlation functions and the variance-covariance matrix
in terms of the eigenvalues of Kij. The present section
can be read as proving that the variance-covariance ma-
trix can be found by a simpler route, by solving Eq. (A7)
in terms of the noise correlation matrix. For the cor-
relation functions though, our route involves an inverse
Laplace transform which in general requires that a poly-
nomial be factorised, and is equivalent to solving the sec-
ular equation for Kij. So our methods do not provide any
additional simplification compared to Swain for the corre-
lation functions. As we have emphasised in the main text
though, our results demonstrate that the power spectra
are much easier to calculate than the correlation func-
tions.
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