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it is now clear that chemical noise can sometimes play a significant role in the functioning of biochemical reaction networks @xcite . such noise is implicated in the spontaneous flipping of genetic switches @xcite , and has been shown to have an adverse effect on the functioning of a synthetic chemical oscillator @xcite . in some circumstances it has also been argued that noise can have a beneficial effect , for example in stochastic focusing @xcite . chemical noise in reaction networks can be characterised by the root mean square ( rms ) deviation of number 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 concentrations 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 1020 molecules present in the cell volume @xcite . the mean and the rms deviation are ` point statistics ' , 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 stochastic 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 . 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 @xmath0 where @xmath1 is the number of molecules of species @xmath2 , @xmath3 is a matrix of rate coefficients , and @xmath4 are source terms . is a set of linear ordinary differential equations , which explains the origin of the phrase ` linear reaction network ' . appendix [ sec : appkb ] indicates how @xmath3 and @xmath4 are related to the stoichiometry matrix and reaction rates which specify the network . let @xmath5 be the instaneous number of molecules of species @xmath2 in the system . define the mean value of @xmath1 in steady state to be @xmath6 , and the mean value out of steady state to be @xmath7 . by taking moments of the chemical master equation ( see appendix ) , one can prove that @xmath8 therefore , the chemical rate equations are _ exact _ for a linear reaction network , provided we work with the mean quantities @xmath7 . in steady state , the mean values therefore solve @xmath9 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 @xmath10 and introduce the steady - state variance - covariance matrix @xmath11 the diagonal elements of this are the variances @xmath12 the rms deviation mentioned in the introduction is @xmath13 . it is often the case that @xmath14 and in fact some authors define the ` noise strength ' to be @xmath15 although we will not explicitly use this in the present paper . this scaling means that the relative rms deviation @xmath16 which indicates that the relative important of noise decreases 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 @xcite , 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 relegated to appendix [ sec : appkb ] . for the remainder of the main text , we assume that the means and variance - covariance matrix have been computed . we will say that these comprise 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 @xmath17 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 @xmath18 c_{ij}(t ) \to 0\quad\text{as}\quad t\to\infty,\\[6pt ] c_{ij}(t ) = c_{ji}(-t ) \end{array}\ ] ] ( the first of these is not always true ; see section [ sec : fast ] ) . the last property shows that the autocorrelation functions are time - symmetric , @xmath19 , but it does not follow that the cross - correlation functions are time - symmetric , as is illustrated by the example in section [ sec : gem ] .    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 @xmath20 where @xmath21 is the fourier transform of @xmath22 . we shall use the second form in eq .   to compute the noise power spectra . usually the noise power spectra obey the sum rule ( for an exception , see section [ sec : fast ] ) , @xmath23    now we turn to the computation of the correlation functions and the noise power spectra . in appendix [ sec : appc ] we prove that the correlation functions satisfy @xmath24 if we subtract eq .   from eq .  , we find that the mean deviation out of steady state obeys @xmath25 this shows that the correlation functions decay in exactly the same way as deviations away from steady state , so that eq . is an example of a regression theorem @xcite . is to be solved with the initial conditions @xmath26 . 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 . shows that @xmath27 where @xmath28 eq . is a set of linear simultaneous equations which can readily be solved for @xmath29 . the power spectra now follow almost for free . comparing eq .   with eq . shows that @xmath30 in other words @xmath31 is twice the real part of the analytic continuation of @xmath32 to @xmath33 . eqs .   and 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 [ sec : appl ] we prove that the same results can be obtained from the chemical langevin equations described by gillespie @xcite . it is interesting to note that the power spectra are simpler 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 polynomial be factorised , which is not always possible . in fact the route to the power spectra , via eqs  , and in the appendix , just involves the solution of linear simultaneous equations .    , @xmath34 , @xmath35 , and @xmath36 . these correspond to cro in a recent model of phage lambda @xcite . parameters for the implicit mrna model of eq . are @xmath37 , @xmath38 , @xmath39 . these are chosen so that the two models have matched point statistics , @xmath40 and @xmath41 , and matching values of @xmath42.[fig : fig1 ] ] we first consider a very simple model of gene expression @xmath43 which represents a birth - death process in which @xmath44 copies of a are generated each time the first ( birth ) reaction fires . this represents the ` burstiness ' of gene expression in prokaryotes @xcite . the point statistics are @xmath45 the chemical rate equation is @xmath46 so the autocorrelation function obeys @xmath47 . the solution , and its laplace transform , are @xmath48 the corresponding power spectrum is @xmath49    now we turn to a more realistic representation of gene expression which explicitly includes mrna , namely @xmath50 the first two reactions represent generation and degradation 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 @xcite , and we recover previously known results for the point statistics .    the mean values in steady state are @xmath51 in this , @xmath52 is the mean number of copies of a produced in the lifetime of one mrna and can be taken to represent the ` burstiness ' of gene expression . comparing the last of eq .   with the first of eq . , it appears that @xmath53 in the present model corresponds to @xmath44 in the previous 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 appendix [ sec : appkb ] ) @xmath54 \displaystyle s_{m\!a}=\frac{\rho{\langle{n_m}\rangle}}{\gamma+\lambda } = \frac{\gamma{\langle{n_a}\rangle}}{\gamma+\lambda}.\label{eq : mavcm } \end{array}\ ] ] in this case , comparing eq .   with eq . , we see that the correspondence @xmath55 fails for @xmath56 . now we turn to the computation of the correlation functions and the power spectra . the chemical rate equations are @xmath57 the correlation functions obey the homogeneous form of these equations . taking the laplace transform as indicated in the previous section , we find @xmath58 s\tilde c_{am } - s_{am } = - \lambda\tilde c_{am},\\[6pt ] s\tilde c_{m\!a } - s_{m\!a } = \rho \tilde c_{m\!m }   - \gamma \tilde c_{m\!a},\\[6pt ] s\tilde c_{aa } - \sigma_a^2 = \rho \tilde c_{am } - \gamma \tilde c_{aa}. \end{array}\ ] ] because @xmath59 , there are four different laplace - transformed correlation functions ( note though that @xmath60 ) . the solutions are @xmath61 \displaystyle \tilde c_{m\!a}=\frac{s_{m\!a}}{s+\gamma}+ \frac{\rho\sigma_m^2}{(s+\lambda)(s+\gamma)},\\[12pt ] \displaystyle \tilde c_{aa}=\frac{\sigma_a^2}{s+\gamma}+ \frac{\rho s_{am}}{(s+\lambda)(s+\gamma)}. \end{array}\ ] ] 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 , @xmath62 ( using @xmath63 for the second line ) . finally , the power spectra are obtained . the power spectrum for m is @xmath64 . this might have been anticipated as a special case of the previous model with @xmath65 , since as far as m is concerned it is undergoing a simple birth - death process . the power spectrum for a is @xmath66 with a suitable choice of @xmath44 and rate cofficients @xmath67 and @xmath68 , the point statistics of the previous model in eq . could be matched to the present model in eq .  . however , it is in general impossible to match the power spectra since the @xmath69 dependence is different . [ fig : fig1 ] 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 .      a very simple model of signal detection is the reaction @xmath70 where a is the input signal and b is the output signal . .   has a chemical rate equation @xmath71 we have included an additional noise term @xmath72 in this , so it is actually a chemical langevin equation . taking the fourier transform yields @xmath73 the mean square modulus is then @xmath74 we have assumed that @xmath72 is uncorrelated with @xmath75 . the theory of chemical langevin equations @xcite ( see also appendix [ sec : appl ] ) shows that @xmath76 , and for @xmath77 one has @xmath78 , thus finally @xmath79 this result has quite a striking interpretation . the total 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 circuit element @xcite . this result , in one form or another , was derived by paulsson @xcite , and by shibata and fujimoto @xcite . 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 @xmath72 is uncorrelated with the input signal @xmath75 . 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 .   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 @xcite . we therefore solve @xmath80 we find that @xmath81 and @xmath82 } { ( \omega^2+\lambda^2)(\omega^2+(\gamma+\nu)^2)(\omega^2+\mu^2)}\ ] ] with @xmath83 . it is not hard to demonstrate that the spectral addition rule _ fails _ for these expressions .    as an aside , one can show @xmath84 . since the @xmath85 reaction in eq . can be regarded as post - translational modification step , this shows that post - translational modification can reduce the noise associated with gene expression . the reason is that the post - translational modification reaction smoothes the ` burstiness ' 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 .   with @xmath86 as far as b is concerned , it is important to note that the _ same _ chemical langevin equation eq .   holds for this detection scheme as for the previous scheme . for this variant we find @xmath87 is as given in eq .   and @xmath88 } { ( \omega^2+\gamma^2)(\omega^2+\lambda^2)(\omega^2+\mu^2 ) } \label{eq : pbmab}\ ] ] one can check that in this case @xmath87 and @xmath89 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 suggestive of the general conclusion that the spectral addition 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 @xcite . there is a growing literature on the adiabatic elimination of fast reactions for stochastic chemical kinetics @xcite . for example kepler and elston @xcite , and shibata @xcite , have a formalism that permits a systematic approach to the problem , involving the identification of the fast and slow variables 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 @xcite . here we examine the procedure for elimination of a fast equilibration reaction in the context of the _ exact _ results for a linear reaction network . our results shed light on the way in which fast equilibration reactions can be systematically eliminated from a reaction network , whilst preserving the noise attributes .    to make a concrete example , we suppose that the signal detection motif of the previous section consists of a fast binding - unbinding step , followed by a slower detection step . we replace @xmath85 by @xmath90 where a@xmath91 represents the bound state . we suppose that the substrate which binds a is in excess , so that our reaction network is still a linear network . the reaction scheme in its totality is now @xmath92 \text{a}{\overset{\gamma}{\to } } , & { } \quad \text{a}{\underset{k_b}{\overset{k_{\!f}}{\rightleftharpoons}}}\text{a}^*{\overset{\nu}{\to}}\text{b}{\overset{\mu}{\to}}. \end{array } \label{eq : maasb}\ ] ] 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 @xmath93 for the binding - unbinding reaction is defined via @xmath94 to take the limit of fast equilibration , we keep @xmath93 finite and allow @xmath95 . the results are as follows . let us first consider the final product species b. we find @xmath89 is given by eq . but with ` renormalised ' reaction rates @xmath96 the same is true of the variance @xmath97 . we can rationalise this as follows . the chemical rate equations which involve the species in the fast equilibration reaction ( a and a@xmath91 ) are @xmath98 dn_{a^*}/dt = k_{\!f}n_a-(\nu+k_b)n_{a^*},\\[6pt ] dn_b / dt = \nu n_{a^*}-\mu n_b . \end{array}\ ] ] we now make a linear transformation from @xmath75 and @xmath99 to new variables @xmath100 and @xmath101 : @xmath102 n_y = k n_a - n_{a^*}. \end{array } \label{eq : lintran}\ ] ] these are chosen because @xmath100 is a _ variable which is conserved by the equilibration reaction , and @xmath101 is a _ fast _ variable which vanishes in steady state . in terms of these concentration variables , the chemical rate equations become @xmath103 \displaystyle dn_y / dt + ( k+1 ) k_b n_y = k\rho n_m-\frac{\gamma-\nu}{k+1}\,n_x,\\[12pt ] \displaystyle dn_b / dt = \frac{k\nu}{k+1}\,n_x-\frac{\nu}{k+1}\,n_y-\mu n_b . \end{array } \label{eq : cxyb}\ ] ] the second of these shows that @xmath101 relaxes at a rate @xmath104 to a constant @xmath105 , for large @xmath106 . this formalises the separation of timescales between @xmath101 and all the other concentration variables . in the large @xmath106 limit therefore , @xmath107 and eqs . become @xmath108 dn_b / dt = \nu_r n_x - \mu n_b . \end{array}\ ] ] with the reaction rates given by eq .  . these are ( a subset of ) the chemical rate equations for the following reaction network @xmath109 this is the same as the previously discussed scheme in eq .  , with the replacement @xmath110 . the power spectra follow from eqs .   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@xmath91 in the original network . we find that @xmath111 again these results are easy to rationalise , since inverting eqs .   and setting @xmath112 shows that @xmath113 ( the proportionality factors become squared when computing the noise power spectra ) .    for the variances , @xmath56 and @xmath114 , we find a different story though . in fact , in the limit @xmath95 , we find a _ breakdown _ of the sum rule in eq .  . the reason is that @xmath115 for @xmath116 and @xmath117 . the sum rule always holds for the left hand side of eq .  , but the limiting power spectra are defined in terms of the right hand side of eq .  . ( as an aside , we find that the sum rule is satisfied for @xmath118 , which is equal to @xmath119 , 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 ' @xmath120 for the present situation , we find that @xmath121 interestingly , the same result is found for _ all _ situations involving a fast equilibration reaction which we have examined .    the origin of the sum rule deficit lies in the noise generated by the fast equilibration reaction . formally , at any finite @xmath106 we can define the contribution of the equilibration reaction to the total noise of species @xmath2 to be @xmath122 for @xmath116 and a@xmath91 , one has that @xmath123 as @xmath95 , at any finite value of @xmath69 , but that @xmath124 . the reason is that , whilst @xmath125 vanishes as @xmath126 , it extends over a frequency range @xmath127 , so the total integrated contribution does not vanish in the limit @xmath95 . 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 @xmath128 solving for the point statistics in this system one has @xmath129 and , from eqs .   in the appendix , @xmath130 -k s_{aa^*}+\sigma_{a^*}^2 + k\sigma_a^2-s_{aa^*}-2{\langle{n_{a^*}}\rangle}=0\\[6pt ] k s_{aa^*}-\sigma_{a^*}^2+{\langle{n_{a^*}}\rangle}=0 \end{array } \label{eq : aasvcm}\ ] ] these equations are linearly dependent and can not by themselves be solved for the elements of the variance - covariance matrix . the reason is that the slow variable @xmath131 is exactly conserved by the reaction scheme in eq .  . however this implies that @xmath132 and so @xmath133 . these are consistent with eqs .  , which can now be solved to get @xmath134 this explains the universal value found for the sum rule deficit above .     and @xmath135 . the lines are for @xmath136 , 1.0 , @xmath137 , and @xmath95 ( dashed line).[fig : fig2 ] ]    to complete the discussion , we turn finally to the correlation functions where the analogue of eq . is @xmath138 for @xmath116 and @xmath117 . the left hand side is always equal to @xmath139 . one can show that the difference between the two sides gives an alternative expression for the sum rule deficit , @xmath140 this can be made clear by another example . consider @xmath141 the steady state point statistics , which hold for arbitrary @xmath106 , are @xmath142 , @xmath143 , @xmath144 , @xmath145 . we solve for the correlation functions using the methods described in section [ sec : gen ] . similar results are obtained for both a and @xmath117 so we report results only for species a. the laplace - transformed autocorrelation function is @xmath146 writing @xmath147 allows us to perform the inverse laplace transform by the method of partial fractions . the decay rates @xmath148 are given by @xmath149 the net result is @xmath150 with @xmath151 for any finite value of @xmath106 , it is clear from this that @xmath152 and there is no sum rule deficit . now let us take the limit @xmath95 with @xmath153 being held constant . to leading order we have @xmath154 \displaystyle a\to\frac{k}{k+1 } , & \displaystyle b\to\frac{1}{k+1 } , \end{array}\ ] ] and thus @xmath155.\ ] ] apart from the amplitude , this correlation function is characteristic of @xmath156 with @xmath157 . this is entirely in accord with rewriting the original scheme in terms of the relevant slow variable . the analysis shows that in the limit @xmath95 , we have @xmath158 . thus there is a non - zero sum rule deficit @xmath159 where we have used @xmath160 . again , the 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 @xmath161 ) . at any finite @xmath106 there are two decay rates in the correlation function . in the limit @xmath95 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 . [ fig : fig2 ] which shows typical results for @xmath162 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 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@xmath91 ) . our study has provided examples where one can rigorously prove that this strategy is successful . firstly , let us make some general remarks about the use of noise power spectra to characterise the stochastic properties 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 experiments and simulations . both face similar difficulties in measuring the power spectra . a signal of sufficiently long duration needs to be captured at sufficiently fine resolution in order to be able to estimate the power spectrum . event - driven simulations though , such as those based on the gillespie algorithm @xcite , 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 results . our results indicate that the way noise is transmitted through a signal detection motif may be more complicated than previously thought @xcite . in particular , if the detection motif consumes the input signal molecules , a correlation can be set up between the intrinsic 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 reaction 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 network 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 fundamenteel onderzoek der materie ( fom ) '' , which is financially supported by the `` nederlandse organisatie voor wetenschappelijk onderzoek ( nwo ) '' . pbw acknowledges the hospitality of trey ideker and the california institute for telecommunications and information technology ( calit2 ) at the university of california , san diego , where part of this work was written up . technical details and proofs are relegated to this appendix . for further reading we recommend gardiner @xcite , van kampen @xcite , gillespie @xcite , and swain @xcite . 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 stoichiometry matrix is a non - square matrix @xmath163 equal to the change in species @xmath2 due to the firing of reaction @xmath164 in the network . we shall use roman indices to label chemical species and the greek symbol @xmath164 to label reactions . in terms of @xmath163 , the chemical rate equations can be written @xmath165 where @xmath166 is the flux through reaction channel @xmath164 . the quantity @xmath166 is usually dependent on the current values of @xmath1 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 @xmath167 of being in the state characterised by @xmath1 molecules of species @xmath2 obeys @xmath168 if a reaction channel is impossible , for instance where @xmath169 , we set @xmath170 . eq .   is the chemical master equation . 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 ` @xmath173 ' . 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 @xmath174 and the set of first - order reactions by @xmath175 . the propensity function for a zeroth - order reaction is @xmath176 where @xmath177 is the reaction rate . for the first order reactions , we introduce an indicator matrix @xmath178 , such that @xmath179 if the reaction @xmath164 involves species @xmath2 as the reactant , and @xmath180 otherwise . armed with this , the propensity function for a first - order reaction is @xmath181 inserting eqs .   and into eq .   obtains eq .   in the main text , with @xmath182 the first moment of the chemical master equation is found by setting @xmath183 in eq .  . it is easy to show that this gives eq .   in the main text , with the above definitions of @xmath4 and @xmath3 . the second moment of the chemical master equation is found by setting @xmath184 in eq .  . this gives a closed set of equations for the elements of the variance - covariance matrix . in steady state , these equations can be reduced to @xmath185 where @xmath186 eq . is a set of linear simultaneous equations for the elements @xmath187 of the variance - covariance matrix .    as an example of the application of this machinery , consider the model for gene expression in eq . in the main text . there are two species , m and a , and four reactions . the stoichiometry matrix is @xmath188 the reaction rates are @xmath189 the indicator matrix is @xmath190 in these , the reactions to the left of the vertical line correspond to @xmath191 and the reactions to the right of the line correspond to @xmath192 ( the indicator matrix is only defined for the latter reactions ) .    from these one computes @xmath193 obviously this could have been written down by inspection . less trivially , one also finds @xmath194 \displaystyle & = \bigl(\begin{array}{cc}2\lambda{\langle{n_m}\rangle } & 0\\ 0 & 2\gamma{\langle{n_a}\rangle}\end{array}\bigr ) . \end{array}\ ] ] eqs . simplify to @xmath195 \rho\sigma_m^2-\gamma s_{m\!a}-\lambda s_{m\!a } = 0,\\[6pt ] 2\rho s_{m\!a}-2\gamma\sigma_a^2 + 2\gamma{\langle{n_a}\rangle}=0 . \end{array}\ ] ] there are only three equations since eq . is symmetric in @xmath2 and @xmath196 . these are solved to get eqs .   in the main text . the proof of the regression theorem eq .   in the main text is straightforward and parallels the development in gardiner @xcite . we start by writing out an explicit expression for the correlation function @xmath197 { } \hspace{8em}{}\times p_s(n_{i,0})\;p(n_{i , t}|n_{i,0 } ; t ) . \end{array } \label{eq : app : ceq}\ ] ] in this , @xmath198 is the steady state probability distribution for the starting point @xmath199 , and @xmath200 is the conditional probability distribution for @xmath201 at time @xmath202 , given that the system started with @xmath199 at @xmath203 . these probability distributions could in principle be found by solving the chemical master equation . we now define two kinds of averages @xmath204 { \langle{f(n_i)}\rangle}_0= \int d\{n_{i,0}\}\;f(n_{i,0})\;p_s(n_{i,0 } ) . \end{array}\ ] ] in words , the first is the average value of a function of @xmath205 at time @xmath202 , given that the system started with @xmath206 at @xmath203 . the second is the average value of a function of @xmath199 in steady state conditions . in terms of these , eq . can be written as @xmath207 this result is completely general @xcite .    for a linear reaction network , taking the first moment of the chemical master equation proves an analogue to eq . , namely @xmath208 combining this with eq .   demonstrates that @xmath209 obeys eq .   in the main text . we prove that the power spectra obtained from the chemical langevin equations for a linear reaction network 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 @xcite @xmath210 in this , @xmath211 are independent unit gaussian white noise functions , one for each reaction channel . applying this to a linear reaction network obtains @xmath212 with @xmath213 it follows that the @xmath214 are gaussian white noise functions with the following statistics @xmath215 the correlation matrix @xmath216 is _ identical _ to the matrix that features in the computation of the variance - covariance matrix , given by eq .   above , which can be interpreted in this context as a fluctuation - dissipation theorem . working in terms of the deviation from steady state , and taking the fourier transform of the chemical langevin equations , obtains @xmath217 we conclude @xmath218 where @xmath219 is the inverse of @xmath220 and @xmath221 is the adjoint . our task is to prove that eq . is equivalent to eqs .   and in the main text . the problem is made easier if we rewrite everything in abstract matrix notation . the result we have just obtained can be written as @xmath222 with @xmath223 ( @xmath224 is the transpose of @xmath225 ) . the diagonal elements of @xmath226 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 .  , and can be written respectively as @xmath227 i\omega { { \mathbf c}}-{{\mathbf s}}={{\mathbf k}}\cdot{{\mathbf c } } , \quad\text{or}\quad{{\mathbf c } } = { { \mathbf b}}\cdot{{\mathbf s}},\\[6pt ] { { \mathbf p}}_2={{\mathbf c}}+{{\mathbf c}}^\dag = { { \mathbf b}}\cdot{{\mathbf s } } + { { \mathbf s}}\cdot{{\mathbf b}}^\dag \end{array } \label{eq : app : p2}\ ] ] ( note that @xmath228 is real and symmetric ) . the diagonal elements of @xmath229 are the power spectra from the chemical master equation route . eliminate @xmath230 between the first of eqs .   and eq .   to get @xmath231 it follows from eq . that @xmath232 on substituting these into eq . , there is a cancellation of terms , and one finds @xmath233 comparison with the last of eqs . shows that @xmath234 . this contains our desired result , and proves that the power spectra obtained by the chemical langevin equation route are the same as those obtained by the ( exact ) chemical master equation route . actually , we have a slightly stronger result , since it follows that @xmath235 , for any linear combination @xmath236 . swain has presented an analysis of the chemical langevin equations for , effectively , a general linear reaction network @xcite . he obtains general results for the correlation functions and the variance - covariance matrix in terms of the eigenvalues of @xmath3 . the present section can be read as proving that the variance - covariance matrix can be found by a simpler route , by solving eq .   in terms of the noise correlation matrix . for the correlation functions though , our route involves an inverse laplace transform which in general requires that a polynomial be factorised , and is equivalent to solving the secular equation for @xmath3 . so our methods do not provide any additional simplification compared to swain for the correlation 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 functions .
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 .