current and future time domain optical surveys , such as the sdss stripe 82 supernova survey , palomar transient factory ( ptf , * ? ? ?* ) , the catalina real - time transient survey ( crts , * ? ? ?* ) , pan - starrs , and the large synoptic survey telescope ( lsst , * ? ? ?* ) , are providing and will provide an unprecedented flood of variability data .such data sets will number in the millions to hundreds of billions of photometric data points , providing systematic multi - wavelength variability studies for thousands to eventually billions of objects .for example , lsst will have trillion total measurements on billion objects .for many classes of astronomical sources these will be the first systematic multi - passband variability studies including large numbers of objects with well - sampled multiwavelength light curves .moreover , these rich data sets will also enable the inclusion of multi - passband variability information for distinguishing different classes of objects .data sets generated by these surveys will present many exciting opportunities , providing astrophysical insight for known classes of objects , as well as the discovery of unknown variability classes , new subclasses of known variable classes , and anomalous outliers .the central data analysis problem for extracting science from these time - domain data sets is how to quantitatively characterize the variability . for periodic signals characterizing the variabilityis relatively straightforward , with the period being the obvious and most important feature to use . for transient phenomenon , such as supernovae or black hole tidal disruptions ,the light curve is often characterized by fitting a parameterized deterministic model to the data , either statistical ( e.g. , splines ) or astrophysical . for quasi - periodic and stochastic light curvesthe variability is often characterized through the power spectral density ( psd ) .the psd is the variability amplitude per frequency , so it describes the variability power contained within a frequency interval .a similar measure that is sometimes used is the structure function , which describes the variability amplitude as a function of time scale .the variability of quasi - periodic and stochastic light curves may then be characterized by summarizing the power spectrum through a parametric form , with power - laws and sums of lorentzian functions being common choices for agn and x - ray binaries , respectively .irregular sampling or sequences of regular sampling separated by gaps are often the source of the most problematic aspects of measuring variability features from a lightcurve .unfortunately , all ground - based astronomical data and many space - based data are subject to sampling which is not strictly uniform . for periodic signals , methods have been developed to estimate periods from irregularly sampled light curves , and assess their statistical significance ( e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) ; significance tests have also been developed for periodic signals against red noise for regularly sampled light curves . for deterministic models , fitting the light curveis a traditional regression problem so the sampling pattern does not bias the results .  deriving the psd features and their uncertainties from an irregularly sampled light curve is considerably more challenging for stochastic light curves .for a regularly sampled light curve the traditional way of estimating the psd is through the discrete fourier transform of the light curve , the modulus - squared of which is called the periodogram .the periodogram suffers from biases due to the fact that it is calculated from a time series that is a sample from a continuous - time stochastic process . as a result, the sampling pattern of the light curve distorts the periodogram relative to the true psd that generated the light curve ( e.g. , * ? ? ?* ; * ? ? ?these issues similarly cause the empirical structure function to be a distorted estimate of the true structure function .this is a concern because if one does not correct for this distortion , differences in variability properties caused by vagaries of the sampling pattern could mistakenly be interpreted to have an astrophysical origin .moreover , this distortion is also a problem for classification algorithms that utilize variability information , as objects may spuriously be interpreted to belong to different variability classes simply because they have different sampling patterns .clearly this issue must be dealt with in order to take advantage of the state of the art time domain data sets .there are two primary approaches used in astronomical studies to account for irregular sampling in stochastic light curves .the first approach is to use monte carlo simulations to forward model the periodogram as a function of a model power spectrum .the approach proceeds by first simulating a large number of light curves from an assumed power spectrum , sampling them with the same pattern as the measured light curve , computing the periodogram for each down sampled simulated light curve , and averaging the simulated periodograms to create a ` model ' periodogram .the best - fit power spectrum is found by minimizing the between the model periodogram and the measured periodogram .moreover , confidence intervals may be estimated from the monte carlo samples .this approach is extremely flexible , but can be computationally expensive .this is especially true if there are intervals of fine sampling separated by intervals of sparse sampling , as this requires either generating a very dense light curve at the finest sampling rate , or splitting the light curve into segments and computing their periodograms separately .unfortunately , because of the computational cost , it is difficult to see how this approach can be applied to massive time domain data sets .moreover , fitting complex multi - component for even a single data set can become computationally expensive in the monte carlo forward fitting approach .the second approach to accounting for irregular sampling in stochastic light curves is to fit the light curve in the time domain .this is almost always done by assuming the light curve is a realization of a gaussian process ( e.g. , * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) . in this case, the likelihood function is a multivariate normal distribution with unknown mean and covariance matrix .the covariance matrix is parameterized by the autocovariance function , which forms a fourier transform pair with the power spectrum . in this time - domain approach ,the autocovariance function is fit through maximum - likelihood or bayesian approaches , and the power spectrum is calculated from the inferred autocovariance function .the irregular sampling and measurement errors are automatically accounted for by the likelihood function .this approach is statistically powerful because of its reliance on the likelihood function , but in general is also computationally expensive . in order to calculate the multivariate gaussian likelihood functionit is necessary to invert the covariance matrix of the light curve , where is the number of data points in the light curve .in general matrix inversion scales as , which can be prohibitive for large samples of light curves that are sampled with even moderate density .there are special classes of gaussian processes for which the computational complexity only scales linearly with the length of the light curve . for linear processes which have a ` state - space 'representation ( discussed further in  [ s - statespace ] , see also * ? ? ?* ) the likelihood function can be computed using a computationally efficient ( scaling as ) algorithm known as the kalman filter .gaussian processes with an exponential autocorrelation fall into this class of stochastic models that have fast algorithms for computing their likelihood function .this particular process is known as both a first order continuous - time autoregressive process ( car(1 ) ) or an ornstein - uhlenbeck process , and was introduced by as a model for quasar optical light curves .subsequent work has confirmed that it provides a good description of quasar optical light curves on time scales of days to years at the level of data quality of the ogle and stripe 82 surveys .however , recent studies have found evidence for deviations from the car(1 ) model for optical light curves of agn . in addition , successful agn selection techniques have been developed based on the car(1 ) parameters .  despite its recent success ,the car(1 ) model is very simple , as it assumes a psd that is a lorentzian centered at zero .thus , there are only two free parameters in the car(1 ) model : the bend frequency of the psd ( i.e. , the width of the lorentzian ) and the normalization .this makes the model inflexible , limiting its broader use .in this paper we overcome the inflexiblity of the car(1 ) model and present the general class of continuous time autoregressive moving average ( carma ) models .carma models are generated by adding higher order derivatives to the stochastic differential equation that defines the car(1 ) processes .the special case of a car( ) process is discussed by , who applied it to the light curves for two variable stars .the psd of a carma process is a sum of lorentzian functions , with the free parameters being the centroid , widths , and normalizations of the lorentzian functions .this provides a significant amount of flexibility in modeling the psd , making carma modeling applicable to many classes of astronomical variables .moreover , carma models have a state space representation , enabling the use of the kalman filter for calculating their likelihood function . because of this , the computational complexity of calculating the likelihood function for carma models still scales linearly with the number of data points in the light curve , making them scalable to massive time domain data sets .the format of this paper is as follows . in  [ s - carma_def ]we begin by defining the carma process via a stochastic differential equation , present the psd of a carma process , and present the autocovariance function of a carma process .then , in  [ s - statespace ] we express the carma process using a continuous - time state space representation and use this representation to derive the solution to the stochastic differential equation defining the process .this solution forms the basis for the statistical properties of a carma process sampled at a set of observational times , which are necessary for fitting the carma process parameters to a measured time series . in  [ s - inference ] we present the likelihood function for a sampled carma process , and in  [ s - likelihood ] present an efficient algorithm for computing the likelihood function based on the kalman filter . in order to derive the kalman filter for a carma processit is necessary to obtain a discrete - time state space representation of the sampled process , and we begin [ s - likelihood ] by using the results of  [ s - statespace ] to present this discrete time representation .we extend these results in [ s - interpolation ] and derive an algorithm for efficiently performing interpolation and extrapolation from a measured time series assuming a carma model . in [ s - bayes ] we describe bayesian inference for the carma model , including our adopted prior distribution , in [ s - assess_fit ] we describe how to assess the quality of fit based on the carma model , in  [ s - order ] we discuss how to choose the order of the carma model , and in [ s - computation ] we discuss computational aspects related to fitting the carma model to a measured time series . in [ s - simulations ] we illustrate statistical inference under a carma model on two simulated lightcurves , and in  [ s - applications ] we apply the carma model to astronomical lightcurves from an x - ray binary , agn , and periodic variable stars . in  [ s - discussion ]we discuss our results , and provide directions for future development .in this section we introduce the important mathematical properties of the carma(p , q ) process , including its definition , psd , autocovariance function , and solution .further details may be found in , e.g. , , , , , and .a zero - mean carma(p , q ) process is defined to be the solution to the stochastic differential equation here , is a continuous time white noise process with zero mean and variance .in addition , we define and = 1 .the parameters are the autoregressive coefficients , and the parameters are the moving average coefficients . for the process to be stationary ,it is necessary that and that the roots of the autoregressive polynomial have negative real parts .in addition , the process has ` minimum phase ' when the roots of the moving average polynomial have non - positive real parts .if the carma process is minimum phase , this basically means that we can uniquely determine the values of the input white noise process from the output carma process .  a stationary carma(p , q ) process has the psd and autocovariance function at lag  \left[\sum_{l=0}^q \beta_l ( -r_k)^l \right ] \exp(r_k \tau ) }     { -2 \operatorname{re}(r_k ) \prod_{l=1 , l \neq k}^p ( r_l - r_k)(r^*_l + r_k ) } .   \label{eq - carma_autocovar}\ ] ] here , denotes the real part and is the complex conjugate of . in this work we only deal with the case when the roots are unique , as thisis required in order to use the kalman filter to efficiently calculate the likelihood ( see  [ s - likelihood ] ) .  most previous work on using continuous - time autoregressive processes for characterizing astronomical time serieshas focused on the case when and , i.e. , a car(1 ) model .the car(1 ) model is also called an ornstein - uhlenbeck process , and has often been referred to as a ` damped random walk ' in the astronomical literature . using the notation above , the psd and autocovariance for the more familiar case of are , respectively , and as can be seen , the psd for the car(1 ) process is a lorentzian function centered at zero with a break frequency at , while the autocorrelations decay exponentially with an -folding time scale .  inspection of the autocovariance function and psd of a carma(p , q ) process provides some guidance on how to interpret the carma(p , q ) parameters . forthe carma(p , q ) process the autocorrelation function is a weighted sum of exponential functions , with the arguments of these exponential functions being the roots of the polynomial given by equation ( [ eq - characteristic ] ) and the weights being a function of the moving average coefficients .these roots may be complex - valued , although if is odd there is always at least one real root . as a result ,the autocorrelation function for the carma(p , q ) process is a sum of exponentially damped sinusoidal functions ( corresponding to the complex roots ) and exponential decays ( corresponding to the real roots ) .the -folding time scale of the decaying autocorrelations for each exponential function is , while the frequencies of the oscillations in the autocorrelations are .because the psd and the autocovariance function are a fourier transform pair , we can also connect the roots of to the psd . because the fourier transform of an exponentially damped sinusoidal function is a lorentzian function , the psd of a carma(p , q ) process can be expressed as a weighted sum of lorentzian functions .the widths of the lorentzian functions are proportional to while the centroids of the lorentzian functions are given by . as with the autocovariance function , the moving average coefficients help control the weights in the sum . incidentally ,a sum of lorentzian functions is a common model used to characterize the x - ray psds of x - ray binaries ( e.g. , * ? ? ?* ; * ? ? ?the solution to equation ( [ eq - carma ] ) may be obtained by introducing a state space representation of a carma(p , q ) process ( e.g. , * ? ? ?in addition , as discussed in section [ s - likelihood ] , representing a carma(p , q ) process in a state space representation enables efficient calculation of the likelihood function for a measured time series .a state space representation models a stochastic process as arising from an observation equation and a state equation .the observation equation relates the observed time series to an unknown latent state variable , and the state equation describes the evolution of the state variable .note that the state variable will in general not be scalar - valued . for the carma(p , q ) model ,the -element state vector is a vector containing the value of a latent process and its derivatives as a function of time : the state space representation of a carma(p , q ) process is where is a wiener process , is a white noise process with mean zero and unit variance ,  ] is a -element column vector , and is a matrix with elements   the solution to equation ( [ eq - state_equation ] ) with random initial condition is ( e.g. , * ? ? ?* ) the first term on the right hand side represents the deterministic contribution to the evolution of the state vector , given the initial condition , while the second term on the right hand side is a random variable representing the stochastic contribution to this evolution . note that because has an expectation value of zero , the stochastic integral in equation ( [ eq - carma_solution ] ) also has an expectation value given by the zero vectorthe process given by the solution to equation ( [ eq - carma_solution ] ) is stationary if and only if has an expectation value given by the zero vector and covariance matrix with elements  in this case , the stationary mean of is also the zero vector and the stationary covariance is also given by equation ( [ eq - state_covar ] ) . because is stationary , the carma process defined by equation ( [ eq - obs_equation ] ) is also stationary with mean zero and variance ; note that this is the same as that given by equation ( [ eq - carma_autocovar ] ) using a time lag of .if the white noise term in equation ( [ eq - carma ] ) is gaussian , the resulting carma(p , q ) process is also gaussian . in this case , the likelihood function for a carma(p , q ) model may be derived for a measured time series ^t ] .the kalman filter is used to calculate the means and variances for each of these normal distributions , efficiently calculating the likelihood in operations .because astronomical time series are measured with error , we introduce a measurement error term to the observation equation of the state space representation , modifying equation ( [ eq - obs_equation ] ) to become the measurement error for , , is assumed to be normally distributed with mean zero and variance . equations ( [ eq - obs_equation2 ] ) and ( [ eq - state_equation ] ) are a state space representation for a time series measured with error assuming a carma(p , q ) model .  in order to use the kalman filter for a sampled continuous time process, it is necessary to convert the continuous time state space representation to that of a discrete time process evaluated at the sampled time values .this requires integrating the state equation ( eq.[[eq - state_equation ] ] ) over the time intervals between subsequent observations .the discrete state space representation of a carma process sampled at times is derived using equation ( [ eq - carma_solution ] ) ( e.g. , * ? ? ?* ) to be where denotes the normally distributed measurement error at with mean zero and variance and is a random vector drawn from a multivariate normal distribution with mean given by the zero vector and covariance matrix ( e.g. , * ? ? ?* ) equations ( [ eq - discrete_obs_equation])([eq - discrete_error_covar ] ) provide everything we need to compute the kalman filter and the likelihood function for a carma model .calculation of the matrix exponentials needed in equations ( [ eq - discrete_state_equation ] ) and ( [ eq - discrete_error_covar ] ) is computationally expensive .however , for diagonal matrices the matrix exponential is trivial to calculate . in order to improve the efficiency of the kalman filter we use the diagonal form , where the columns of contain the right eigenvectors of and is a diagonal matrix : recall that are the roots of the autoregressive polynomial .we then transform equations ( [ eq - discrete_obs_equation ] ) and ( [ eq - discrete_state_equation ] ) to be in terms of the rotated state vectors : here is a diagonal matrix with diagonal elements , and . the stochastic term in the rotated state space formulation , , is a complex valued normally distributed random variable with mean given by the zero vector and hermitian covariance matrix .the elements of the covariance matrix of are not actually needed for the kalman filter ( e.g. , * ? ? ?* ) in our implementation , but they are given in .we note that the variables in the rotated state space notation are complex - valued , although they are all real - valued in the original representation . derived the kalman filter for a car(p ) model under the rotated state representation , and extended these results to a carma(p , q ) model . in  [ s - kfilter ] of the appendix we provide the kalman filter algorithm for a carma( ) model , and refer the reader to and for further details . after running the kalman filter, we will have the values of and needed for computing the likelihood function via equation ( [ eq - likhood ] ) .   in certain applicationsone may need to simulate an interpolated or extrapolated time series conditional on a measured time series .example of this include reverberation mapping of agn or studies of the time delay between images of gravitationally lensed quasars ( e.g. , * ? ? ? * ; * ? ? ? * ; * ? ? ? * ) .in addition , forecasting may also be useful for generating alerts .the probability distribution of a gaussian carma process at time given the measured time series is a normal distribution , and one can use the usual gaussian process machinery to calculate the conditional mean and variance of future data . however , as with the likelihood calculation this is expensive , also requiring an expensive matrix inversion .instead , for a carma process we can use the kalman filter to efficiently calculate the mean and variance of this normal distribution , and in this section we derive these quantities .denote the value of a carma process at time as , and define , i.e. , .assuming a gaussian carma process , we can write the probability distribution of given a carma model and a measured time series as as before , . the values of and can be obtained by running the kalman filter on the time series generated by inserting into the set of observation times .the sets of coefficients and can be obtained recursively through an algorithm similar to the kalman filter , which we describe in [ s - coefs ] of the appendix .from equation ( [ eq - linear_filter ] ) we can derive the mean and variance of as \label{eq - predicted_cmean } \\var(y_0|{\bf y},\theta ) & = \left [ \frac{1}{var(y_0|{\bf y}_{<j(t_0)},\theta ) } \right .\nonumber \\   & \left .+ \sum_{i = j(t_0)}^n \frac{\tilde{d}^2_i}{var(y_i|y_0,{\bf y}_{<i } , \theta ) } \right]^{-1 } \label{eq - predicted_cvar}.\end{aligned}\ ] ] equations ( [ eq - predicted_cmean ] ) and ( [ eq - predicted_cvar ] ) provide the interpolated or extrapolated value and its uncertainty , assuming the carma model .note that for forecasting and backcasting , the first and second terms appearing in the right hand sides of equations ( [ eq - predicted_cmean ] ) and ( [ eq - predicted_cvar ] ) are ignored , respectively .an interpolated or extrapolated time series may be simulated sequentially at time values by first using equations ( [ eq - predicted_cmean ] ) and ( [ eq - predicted_cvar ] ) to generate a value of from a normal distribution , inserting this value of into the measured time series array , and repeating for the remaining .   in this workwe focus on bayesian inference of the carma process model .this is primarily because in bayesian inference one derives the probability distribution of the carma process given the measured time series ( i.e. , the ` posterior ' distribution ) , providing a rigorous assessment of the uncertainties in the carma model , and consequently in the inferred power spectrum .the likelihood function of a carma model can exhibit multiple maxima for ( e.g. * ? ? ?* ) , and therefore traditional techniques based on the fisher information matrix and the asymptotic normality of the maximum likelihood estimate do not apply in general for carma models .bayesian inference is based on the posterior distribution , which is related to the likelihood function by the equation here , is the prior distribution on the model parameters . because we have already derived the likelihood function , the only additional thing we need to do to perform bayesian inference is to specify the prior distribution . in our modelwe assume a uniform prior on the standard deviation of the lightcurve subject to ^{1/2 } < r_0 ] .these constraints reduce the number of free parameters to five for this model , which can be a computational advantage .in addition , also showed that the solution to the stochastic linear diffusion equation is a mixture of car(1 ) processes , providing a physical interpretation of the variability model . investigated extensions to the gaussian car(1 ) process as a model for quasar variability .the processes investigated by included models with more flexible psds , such as a car(2 ) process and a carma(1,1 ) process .the former process provides the ability to capture quasi - periodic variations , while the latter process introduces additional smoothing of the stochastic driving noise .both processes are special cases of the general class of carma processes we discuss here ; however , we note that the carma(1,1 ) model is not stationary . also investigated car(1 ) models after relaxing the assumptions of a linear gaussian process .in particular , they also investigated non - gaussian car(1 ) models and non - linear processes where the light curve variance also stochastically varies in time .these models , while not as flexible in modeling the psd as a carma process , provide valuable alternatives to linear gaussian models and may provide a better description of the variability of some light curves . however , in the case of quasar light curves from stripe 82 concluded that the linear gaussian car(1 ) process provided the best model for most of the quasars , based on a bayesian model comparison.  while carma models are , in theory , computationally efficient , the likelihood space can be complex and exhibit multiple modes which can be problematic for numerical optimizers and samplers .this is especially true for higher order models , and seems to affect higher order models more strongly than higher order models .the likelihood space may also be very complex for light curves that have regular nearly deterministic variations , such as rr - lyrae stars .we consider the difficulty in optimizing or sampling from a multimodal complex posterior to be the primary short - coming of these models at this time , and dampens their computational efficiency .thus , researchers who utilize them must be careful to check that the primary posterior mode has been found , and that the dominant posterior modes have been sampled from .we deal with this in our mcmc sampler using a parallel tempering algorithm , which we have used with success .however , even this can fail to adequately sample the posterior if an insufficient number of chains are run , or if the algorithm is not run for a sufficiently long period of time .moreover , our maximum - likelihood estimation is rather simple , as we simply use 100 random starts and optimize by finding a local mode using a greedy gradient - based algorithm .future work should focus on improving the effectiveness and efficiency of the maximum - likelihood estimate and mcmc algorithm .  in order to illustrate the applicability of carma models for a variety of astronomical sources , as well as to provide a guide for interpreting their results , we applied these models to light curves for an x - ray binary , two agn , a long - period variable star , and a rr - lyrae star .in general we found that the carma models provide a good description of these light curves , suggesting that they can be applied to a broad range of astronomical sources that exhibit stochastic or quasi - periodic variations .the only exception was the rr - lyrae star light curve .this is to be expected , as rr - lyrae light curves exhibit regular non - sinusoidal variations and thus are unlikely to have a strong stochastic component .however , in spite of this the carma models still identified the period quoted by the catalogue that this light curve was taken from in of the mcmc samples .moreover , the deviation in the residuals from a normal distribution for the rr - lyrae star implies that the distribution of the residuals from a carma fit may provide an effective means of discriminating between different types of variables , even for those for which a carma model is not optimal .these results suggests that there may be value in using variability features derived from carma parameters even for non - stochastic light curves .further improvements to the carma modeling approach can be obtained by including a deterministic component , such as a periodic function , for modeling periodic light curves , such as those from rr - lyrae . in this casethe residuals from fitting a deterministic function are modeled as following a carma process .in fact , this is the motivation behind the periodic autoregressive moving average models ( parma , e.g. , * ? ? ?* ) , which allow for periodic variations in the mean and autocovariance function of a time series .in addition , it is possible to define multivariate carma models through a vector- and matrix - valued extension to equation ( [ eq - carma ] ) ( e.g. , * ? ? ? * ; * ? ? ? * ) .multivariate carma models hold considerable potential for characterizing the full multi - passband variability information obtained by time - domain surveys , and will be the subject of future work .  in summary , carma models provide an important addition to the astronomer s statistical toolbox in the era of massive time - domain surveys , and have the potential to play an important role in the analysis of variability as a probe of astrophysics , as well as in the use of variability as a means of identifying classes of astronomical sources .we would like to thank tommaso treu for helpful discussions and comments on our manuscript , lucy heil for graciously provided the light curve for xte 1550 - 564 and helpful comments on our manuscript , simon vaughan for helpful comments on our manuscript , and an anonymous referee for helping improve our discussion of carma models .bk acknowledges support from the southern california center for galaxy evolution , a multi - campus research program funded by the university of california office of research .acknowledges funding from nasa origins grant nnx09ab32 g .was partially supported by the eu fp7 grant no .312789 , and by the polish ncn grant no .2011/03/b / st9/03459 .this paper includes data collected by the kepler mission .funding for the kepler mission is provided by the nasa science mission directorate .funding for the sdss and sdss - ii has been provided by the alfred p. sloan foundation , the participating institutions , the national science foundation , the u.s .department of energy , the national aeronautics and space administration , the japanese monbukagakusho , the max planck society , and the higher education funding council for england .the sdss web site is http://www.sdss.org/.  the sdss is managed by the astrophysical research consortium for the participating institutions .the participating institutions are the american museum of natural history , astrophysical institute potsdam , university of basel , university of cambridge , case western reserve university , university of chicago , drexel university , fermilab , the institute for advanced study , the japan participation group , johns hopkins university , the joint institute for nuclear astrophysics , the kavli institute for particle astrophysics and cosmology , the korean scientist group , the chinese academy of sciences ( lamost ) , los alamos national laboratory , the max - 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mean time series , we will work with the centered values instead of .2 .  denote the covariance matrix of the predicted rotated state as .initialize the rotated state vector and its covariance at time to its stationary mean and covariance , and respectively : defining , the stationary covariance matrix for has elements  3 .calculate the mean and variance of the first measurement in the time series using the stationary values for a carma process : where is given by equation ( [ eq - carma_autocovar ] ) .4 .  initialize the kalman gain : here denotes the hermitian transpose of .update the estimate of the rotated state vector : 6 .  update the covariance matrix of the rotated state vector :   after the initializing the kalman filter as above , repeat the following steps for :  1 .predict the rotated state vector at the next observation time given the time series at the earlier observation times : 2 .calculate the covariance matrix of the predicted rotated state vector at time : 3 .calculate the mean and variance of the centered time series at time conditional on the earlier values : here we have used the notation $ ] .4 .  update the kalman gain : 5 .  update the estimated rotated state vector : 6 .  finally , update the covariance matrix of the estimated rotated state vector :   the values of and computed using the above algorithmcan then be used to efficiently calculate the likelihood function given by equation ( [ eq - likhood ] ) , noting that .the coefficients and needed to compute the expected value of as a function of for can be computed recursively using the following algorithm :  1 .first , run the kalman filter up to index . if then skip this step .2 .  compute and using equations ( [ eq - state_transition])([eq - ycvar ] ) .if , then nothing further needs to be calculated . otherwise , use these values to compute , and .3 .  initialize the rotated state vector coefficients and as \\   { \bf d}_{j(t_0 ) } & = \lambda_0 { \bf k}_0,\end{aligned}\ ] ] where is a diagonal matrix with . if then and the initial values are 4 .initialize the coefficients and as note that if then and .   1 .update the linear coefficients for the rotated state vector \\   { \bf d}_{i } & = \lambda_i [ { \bf d}_{i-1 } - \tilde{d}_{i-1 } { \bf k}_i ] .\end{aligned}\ ] ] 2 .update the linear coefficients :   because the kalman gains , , only depend on the observation times , and not on the measured time series , they are computed by performing the kalman filter using the observation times .

Abstract: we present the use of continuous - time autoregressive moving average ( carma ) models as a method for estimating the variability features of a light curve , and in particular its power spectral density ( psd ) . carma models fully account for irregular sampling and measurement errors , making them valuable for quantifying variability , forecasting and interpolating light curves , and for variability - based classification . we show that the psd of a carma model can be expressed as a sum of lorentzian functions , which makes them extremely flexible and able to model a broad range of psds . we present the likelihood function for light curves sampled from carma processes , placing them on a statistically rigorous foundation , and we present a bayesian method to infer the probability distribution of the psd given the measured lightcurve . because calculation of the likelihood function scales linearly with the number of data points , carma modeling scales to current and future massive time - domain data sets . we conclude by applying our carma modeling approach to light curves for an x - ray binary , two agn , a long - period variable star , and an rr - lyrae star , in order to illustrate their use , applicability , and interpretation .


high - quality genome - scale metabolic reconstructions are composed of thousands of reactions and metabolites . due to their complexity , the analysis of these metabolic reconstructions requires computational approaches like constraint - based optimization techniques and methodological frameworks like complex network science to elucidate features of their functional organization and pathway structure . some of the tools used with this purpose are elementary flux modes , extreme pathways , minimal metabolic behaviors , and minimal pathway structures , which are based on finding feasible subnetworks to relate them with definite functions .a different approach in this endeavor is provided by the concept of backbone .backbones maintain significant information while displaying a substantially decreased number of interconnections and , hence , can provide accurate but reduced versions of the whole system . in this direction , the work by almaas _ introduced a filtering technique that selects the reaction that dominates the production or consumption of each metabolite .this method is able to segregate classical pathways , but the selected high - flux subgraphs present a linear structure with very little interconnectivity and so they necessarily lack the characteristic complex features of real metabolic networks .filtering approaches have also interested researchers working on networks in a more general context .a filtering method for weighted networks based on the disparity measure was developed in ref .this approach exploits the heterogeneity present in the intensity of interactions ( weights ) in real networks , both at the global and local levels , to extract the dominant set of connections for each element .typically , the obtained disparity backbones preserve almost all nodes in the initial network and a large fraction of the total weight , while reducing considerably the number of links that pass the filter . at the same time, disparity backbones preserve the heterogeneity of the degree distribution , the level of clustering , and the bow - tie structure , and other complex features of the original networks .in this work , we use flux balance analysis ( fba ) maximizing the growth rate to determine reaction fluxes in the _ _ i__jo1366 version of the metabolic network of _ escherichia coli _k-12 mg1655 , and afterwards we use the disparity filter to extract its backbone .in contrast to the filtered linear structures from ref . , metabolic backbones obtained using the disparity filter conserve not only high - flux reactions but also many low - flux reactions provided that they are significant for the production or consumption of a certain metabolite , such that the complexity of the original networks is preserved in the backbone .we investigate the obtained backbone in glucose minimal medium for fingerprints of evolution and environmental adaptation finding that its central core is mainly composed of evolutionary conserved reactions in energy metabolism whose fluxes still retain at present a key role in the evolved organisms .this feature is also observed in the central core reactions of the backbone of the __ i__jw145 metabolic network of _ mycoplasma pneumoniae _ , a much simpler organism . in _e. coli _ , the analysis of the backbone reveals that the synthesis of nucleotides and the metabolism of lipids form smaller cores which rely critically on energy metabolism ; but not conversely. we also study how the structure of the backbone in _e. coli _ depends on the composition of the minimal media , which allows us to identify pathways that are more sensitive to environmental changes and nutrient availability .we use fba to compute the fluxes of the reactions in a metabolic network which maximize the growth rate of the organism ( see methods , section [ sa ] ) .the fluxes are treated as weights by the disparity filter ( see methods , section [ sb ] ) .the large - scale connectivity structure of the obtained backbone is analyzed in terms of connected components ( see methods , section [ sc ] ) . additional media are considered to analyze the environmental sensitivity of the backbone composition ( see methods , section [ sd ] ) .an important feature of the flux solutions obtained using fba is that they capture the heterogeneity of the flux distribution within a single flux state .the probability distribution function of the obtained fba fluxes , insets of figs .[ fig:1]a and [ fig:1]b , is characterized by a broad scale distribution of values .we disregarded zero - flux reactions , such that the set of active reactions and metabolites is markedly reduced as compared to their number when all reactions in the genome - scale reconstruction are considered , from 411 active reactions and 445 metabolites to 404 reactions and 445 metabolites in _ e. coli _ , and from 163 active reactions and 227 metabolites to 162 reactions and 227 metabolites in _m. pneumoniae_. notice that all metabolites and nearly all reactions are conserved for both organisms , meaning that we do not lose information about the constituents of both metabolic networks .heterogeneity is also present at the local level in the fluxes that produce and consume a given metabolite .we calculate the disparity measure for every metabolite participating in producing or consuming reactions as , where is the flux of reaction normalized by the total flux consuming or producing metabolite , , ( see methods , section [ sb ] ) .we treat separately producing and consuming reactions . figs .[ fig:1]a and [ fig:1]b display the disparity values for all metabolites as a function of their number of producing and consuming reactions ( incoming and outgoing degree ) in _ e. coli _ and in _ m. pneumoniae _ , respectively .the shadowed areas correspond to disparity values compatible with the null hypothesis that the total incoming and outgoing flux is uniformly distributed at random among the incoming and outgoing reactions in which the metabolite participates .the null hypothesis helps to discount local heterogeneities produced by random fluctuations ( see caption of fig .[ fig:1 ] ) .as shown , most metabolites , both in _ e. coli _ and _ m. pneumoniae _ , present flux disparity values that can not be explained by the null hypothesis , meaning that the local distribution of the fluxes of reactions associated to metabolites is significantly heterogeneous .we conclude then that the disparity filter will be able to efficiently extract the backbone with the most significant connections for both organisms , while preserving the characteristic features of metabolism as a complex network .notice that significant fluxes are those with values much above the average expectation given by the null hypothesis and that , in absolute terms , they can be high or low ( see methods , section [ sb ] ) .the filter preserves a reaction in the backbone if the probability that its normalized flux is compatible with the null hypothesis ( _ p_-value ) is smaller than a chosen significance threshold , which determines the filtering intensity . for each metabolite , we compute the _p_-value for each producing and consuming reaction and compare the obtained _p_-value with the significance level . in this way , all the reaction with fluxes which are significant for the production or consumption of a metabolite can be selected , in contrast to the approach in where only a single most significant flux was selected so that the obtained subgraphs presented an obvious linear structure .the disparity filter can be adjusted by tuning the critical threshold to observe how the metabolic networks of both _e. coli _ and _ m. pneumoniae _ are reduced as we decrease from 1 to 0 , meaning the complete genome - scale reconstruction . noticethat , after applying the filter with a specific value of , we recover a bipartite representation of the metabolic backbone . to avoid working with stoichiometrically non - balanced reactions , we transform the filtered bipartite representation into a one - mode projection of metabolites placing a directed link between two metabolites if there is a reaction whose flux is simultaneously significant for the consumption of one metabolite and for the production of the other ( see fig .[ fig:1]c ) .next , we compute the fractions of links , metabolites and total flux remaining in the one - mode filtered networks as a function of the significance level .these magnitudes are normalized by dividing them by the corresponding values in the original network , , , and . in figs .[ fig:1]d and [ fig:1]e , we show vs , and vs in the associated insets , for the one - mode projections of the filtered networks for both _e. coli _ and _ m. pneumoniae_. while the filter can reduce considerably the fraction of links , the corresponding fraction of metabolites is maintained at almost the original value .in addition , the total flux in the backbone only starts to drop appreciably after more than 50% of the links are removed .we take the critical value as the point where the fraction of metabolites starts to decay ( see figs . [ fig:1]d and e ) .this critical value can be seen as an optimal point which greatly reduces the number of links in the network preserving , at the same time , most nodes and therefore much biochemical and structural information as possible .the values are for _ e. coli _ and for _ m. pneumoniae_. notice that the value of is lower for _ e. coli_ due to its stronger heterogeneity in the local distribution of fluxes , which allows to reduce further the fraction of links while preserving all the nodes .the backbone of a metabolic network corresponds to the filtered one - mode projection graph using the critical value as the _ p_-value threshold .the backbone unveils the pivotal pathways of fluxes in metabolism and at the same time preserves the connectivity structure , the heterogeneity of the degree distribution , and the high level of clustering typical of complex networks .we filter the metabolic backbones of both _e. coli _ and _ m. pneumoniae _ using the identified critical values for the significance level .the backbone of each organism retains all metabolites , 445 and 227 respectively .next , we analyze their structure in terms of connectedness .metabolic networks have been found to display typical large - scale connectivity patterns of directed complex networks , characterized by a bow - tie structure ( see methods , section [ sc ] ) , with most reactions in a interconnected core , named the strongly connected component ( scc ) , together with in ( in ) and out ( out ) components formed mainly by nodes directly connected to the scc component .this is the case also for the genome - scale reconstruction of both _e. coli _ and _ m. pneumoniae _ used in this work , whose sccs contain the largest part of metabolites and whose in and out components are formed , respectively , by nutrients and waste metabolites .metabolites in the backbone of _e. coli _ are arranged in a large connected component of 178 metabolites and 51 disconnected small components .three different sccs can be identified in the largest connected component of the backbone , each one with 25% , 10% , and 6% of the metabolites ( see fig . [fig:2]a ) .the two smallest sccs are in the out component of the largest scc .central compounds of metabolism are identified in these sccs : protons , water , adenosine triphosphate ( atp ) , l - glutamate , phosphate , nicotinamide adenine dinucleotide ( nad ) , diphosphate , adenosine diphosphate ( adp ) and flavin adenine dinucleotide ( fad ) .these metabolites are highly - connected even in the backbone , highlighting the ability of the disparity filter to preserve the same structural features of the complete metabolic network while in a reduced version . however , a comparison of the hubs in the backbone with the hubs in the non - filtered metabolic network , _i.e. _ , considering , shows some deviations .the most important difference happens for l - glutamate .it occupies the 4th position in the ranking of the most connected metabolites , whereas it is located in the 12th position in the non - filtered network .besides , the fraction of reactions producing l - glutamate in relation to the total number of reactions in which this metabolite participates is 0.27 in the backbone vs 0.65 in the non - filtered network .regarding other metabolites , which in this case can not be considered as a hub due to its smaller degree , we find that fad ranks 12th in the backbone , while its position for the non - filtered case is 22nd .recall that the number of metabolites in the backbone and in the non - filtered network is the same .another interesting case concerns the metabolite glucose 6-phosphate .it contains no outgoing connections in the backbone , whereas its outgoing degree in the non - filtered network is four .since links in the metabolic backbone denote reactions , it is interesting to assess the composition of the backbone of _ e. coli _ in terms of pathways .first , we start by computing the percentages of pathways in the original metabolic network .we find that the five pathways with more reactions are transport , inner membrane ( 17% ) , cofactor and prosthetic group biosynthesis ( 13% ) , glycerophospholipid metabolism ( 12% ) , alternate carbon metabolism ( 8% ) , and cell envelope biosynthesis ( 8% ) .the same analysis in the metabolic backbone gives a different composition , with pathways such as oxidative phosphorylation and the citric acid cycle gaining weight , as shown in fig .[ fig:2]c .this is in fact due to the pathway participation in the largest scc , where the major contribution comes from these two pathways along with glycolysis / gluconeogenesis and pentose phosphate pathway .it has been found that glycolysis and pentose phosphate pathway can take place without the need of enzymes . concerning the citric acid cycle, it is an ancient pathway that has evolved in order to achieve maximum atp efficiency by being coupled to oxidative phosporylation and glycolysis .in addition , this coupling helps the organism to decrease their quantity of reactive oxygen species by modulation of their participating metabolites , conforming in this way one of the central pillars of carbon metabolism and energy production .another pathway significantly present in the largest scc is glutamate metabolism .glutamate has been reported to be one of the oldest amino acids used in the earliest stages of life .the second largest scc contains links that belong mainly to membrane lipid metabolism ( 97% ) and cofactor and prosthetic group biosynthesis ( 3% ) .membrane lipid metabolism supplies the necessary lipids to generate the cell membrane needing the participation of the cofactor fad/fadh .it has been shown that the pathways involved in lipid metabolism exhibit differences between different lineages in organisms , whereas pathways related to central metabolism are more conserved . finally , pathways related to the smallest scc are purine and pyrimidine biosynthesis ( 91% ) .purines and pyrimidines serve as activated precursors of rna and dna , glycogen , etc .it has been found that the synthesis of purines and pyrimidines was the first pathway involving enzyme - based metabolism .interestingly , the other contribution to this scc is glycine and serine metabolism .glycine is a precursor of purines and pyrimidines .when considering values smaller than the critical one , implying that the filter is more restrictive , we observe that the smallest sccs disappears from the more stringent sub - backbone .more precisely , it happens for a value of .decreasing even more the significance level to , the second largest scc containing reactions in the purine and pyrimidine biosynthesis pathway still retains 30% of the nodes in the backbone , whereas the largest scc still contains a 86% . at a value of , the second scc finally disappears and the sub - backbone only remains a single scc , still preserving 82% of the nodes in the backbone .hence , the scc containing links belonging to pathways related to energy metabolism shows a large resistance to get fragmented , even though the filter becomes progressively more and more restrictive , which points to increased levels of local flux heterogeneities and so of flux dependencies for the consumption and production of metabolites .we perform the same analysis in _ m. pneumoniae _ , a simpler organism which has been proposed as a bacterial model due to its simplicity and its reduced genome .the non - filtered metabolic network is dominated by reactions in amino acid and nucleotide metabolism , consistent with the composition of the backbone .the connected component of the backbone is shown in fig .[ fig:2]b .it contains two sccs , one of them with only two metabolites ( see fig . [ fig:2]b ) .the relevant scc contains 21% of the metabolites in the connected component , and most of its links are related also with energy metabolism , as in _ e. coli_. the dominant pathways in the scc are glycolysis and pyruvate metabolism , see fig .[ fig:2]d where this composition is compared with that of the backbone as a whole .both glycolysis and pyruvate metabolism , which dominate the composition of the core of the backbone , are pathways that were present in the earliest stages of life , when no oxygen was present in the early atmosphere .+   in the previous section , we analyzed the pathway composition of the metabolic backbone of _e. coli _ in glucose minimal medium and found that pathways in the central core are related to energy metabolism .the analysis of the backbone also revealed that the synthesis of nucleotides and the metabolism of lipids form smaller cores which rely critically on energy metabolism ; but not conversely . in this section ,we study how changes in the environment modify the backbones of _ e. coli _ in relation to the one obtained in glucose minimal medium , which exposes adaptation capabilities .first , we calculate the fba fluxes that maximize the growth rate of _e. coli _ in the rich medium luria - bertani ( lb ) broth ( see methods , section [ sd ] ) .afterwards , we apply the disparity filter to extract the metabolic backbone in this new environment , that is obtained for a significance critical threshold .this value is noticeably larger than identified for the glucose minimal medium .interestingly , this rich medium activates 400 reactions , 11 less than in glucose minimal medium . of them , 279 are active in both media , of which 247 have a larger flux in lb broth .an analysis of the connected components in the metabolic backbone of _ e. coli _ in rich medium is also performed .we find that it contains a large connected component with 188 metabolites and 60 small disconnected components .the connected component contains also three sccs .however , two of them are tiny with only two metabolites , whereas the largest one encloses 34% of the metabolites in the connected component .interestingly , the pathway contributing more reactions to this large scc is membrane lipid metabolism ( see fig . [ fig:3]a ) .this fact is in accordance with ref . , where the authors found that the expression of the genes which synthesize fatty acids was generally elevated in rich medium .another important difference is the relative loss of prominence of oxidative phosphorylation and the pentose phosphate pathway .this might seem surprising since the pentose phosphate pathway is typically the main source of nicotinamide adenine dinucleotide phosphate ( nadph ) .however , in rich medium the functionally significant production of this metabolite takes place in the citric acid cycle pathway .this also evidences the importance of the citric acid cycle to produce nadph , and so its importance in the synthesis of membrane lipids .nevertheless , links associated to both oxidative phosphorylation and pentose phosphate pathway are also present in the backbone , located outside the sccs .next , we consider the set of minimal media given in ref . ( see methods , section [ sd ] ) where different carbon , nitrogen , phosphorus and sulfur sources are alternated . for each minimal medium, we scan for as in figs .[ fig:1]d and [ fig:1]e . in fig .[ fig:3]b and [ fig:3]c we plot , respectively , the probability distribution function of the collection of values and of the fraction of links remaining in the metabolic backbone for all media .we find that these magnitudes present a characteristic value , meaning that the flux structure is very similar across media in spite of the difference in the composition of nutrients .the presence of these characteristic value of and the stability of the retained fraction of links in the metabolic backbone in the different media motivate us to combine all of them into a single merged metabolic backbone , obtaining a network which we call superbackbone .the links in this superbackbone correspond to reactions that passed the filter in any of the external media considered , and are annotated with a weight that corresponds to the number of media in which the corresponding metabolic backbone contains the link .the histogram of the distribution of these weights is shown in fig .[ fig:3]d , characterized by a clear bimodal behavior .one peak corresponds to links being common to all media , and the other corresponds to the most common situation of links specific to a few media .an analysis of connectedness shows that this superbackbone contains a large connected component and 11 disconnected components . the connected component with 1090 metabolitesis composed by a large scc containing 43% of its metabolites , in addition to three small sccs containing only two nodes each .a pathway composition analysis in the large scc indicates that , again , we obtain significantly different results from the glucose minimal medium ( see fig .[ fig:2]c and fig .[ fig:3]a ) .the most prominent pathway is alternate carbon metabolism , in agreement with ref . , where the authors found that alternate carbon metabolism is related to genes whose expression depends on external stimuli , particularly on alteration of carbon sources .it is also in agreement with results in ref . , where the authors hypothesize that alternate carbon metabolism can adapt to different nutritional environments , and also with results in ref . , where alternate carbon metabolism is found to be an important intermediate pathway in the network of pathways .the second most abundant pathway corresponds to transport , inner membrane , which again is in agreement with ref . and ref .this pathway is in charge of the transport of metabolites between periplasm and cytosol . finally , if we retain links present at least in 25% of the minimal media , the network fragments into 40 components with the largest one containing five sccs , which indicates that links with small weight , _i.e. _ links specific for a few media , have an important role in order to prevent the fragmentation of the superbackbone .identifying high - flux routes in metabolic networks has been useful in order to , for example , identify principal chains of metabolic transformations . in this work ,we go beyond the mere identification of high - flux routes with metabolic pathways .we are able to identify pivotal pathways containing both high and low fluxes which are central for the maximization of the growth rate and , on the other hand , to identify pathways more responsive to changes in the environment . to perform the analysis, we used fba and a filtering tool which needs no _ a priori _ assumptions for the connectivity of the filtered network , but which produces a backbone as a reduced version of the metabolic network .the backbone is globally connected and retains the characteristic complex features of the metabolic network as a whole .it contains all functionally significant connections given a set of external nutrients , recovering both intra- and inter - pathway connections which mark the most significant routes in metabolism formed by the most important interdependencies for the production and consumption of metabolites . as an explanation for the the strong dependence between between both intra- and inter - pathway connections present in the backbone, one could say that the overall performance of a cell is not only improved due to a better and more efficient utilization of the available resources but depends critically on cross interactions .this fact reinforces the idea that pathways are not isolated identities performing their tasks independently of one another .  as stated in ref . , properties that originate from evolutionary pressure should not be observed in random networks . in our investigations ,the effect of evolutionary pressure is understood to favor the maximization of the growth of the organism , in accordance to fba and its assumptions .it is important to note that the assumed objective function used to compute fba can have a direct effect on the diversity of fluxes in the metabolic network and so in the proportions of pathways in the filtered backbone and its cores .the analysis of the metabolic backbone of _e. coli _ and _ m. pneumoniae _ in glucose minimal medium shows that , for both organisms , the sccs is mainly composed by reactions that belong to ancient pathways that were already present at the first stages of life .a very different composition is obtained when considering the backbone as a whole ( see fig .[ fig:2]c and fig .[ fig:2]d ) and even more when comparing with the non - filtered metabolic network . in the backbone of _ e. coli _ , each of the three scc has a different and definite metabolic function .the largest scc contains pathways related to energy metabolism , as for _ m. pneumoniae _ , and its composition is dominated by the citric acid cycle , oxidative phosphorylation , and glucolysis / gluconegenesis , in accordance with the fact that the citric acid cycle has evolved towards an optimal chemical design by coupling to oxidative phosporylation and glycolysis .a second sccs corresponds to the metabolism of lipids , the most important constituents that compose the cell membrane .the smaller scc is responsible for the synthesis of purines and pyrimidines , vital for dna / rna synthesis .two findings relating the two small sccs deserve also special attention .firstly , the two small sccs are located in the out component of the large scc . secondly , as the filter becomes more restrictive , the small sccs fragment , while the large scc still maintains a large part of links and nodes .these features could be explained in terms of functional requirements . on the one side ,the smaller cores need chemical energy to perform their tasks and , on the other side , they need also basic building blocks .these needs are covered by the large scc , which suggests that the smaller sccs were added to the out component of the largest scc in later steps of evolution .an analysis of the connectivity of the hubs reveals that l - glutamate acquires in the backbone a functional relevance as a reactant which emphasizes its importance as the amino group donor for nearly all nitrogen - containing metabolites of the cell , in contrast to its role in the non - filtered network , where it could be classified rather as a product since in minimal medium it must be synthesized .the case of glucose 6-phosphate is also very interesting in the sense that it presents no outgoing connections in the backbone , but it is known that this metabolite has many metabolic fates .hence , we conclude that , in glucose minimal medium , no route consuming this metabolite can be considered more significant than the others , being thus all of them equally significant . therefore , glucose 6-phosphate can not be considered as a functional precursor but a functional product , a fact confirmed due to its presence in the out component of the backbone .a simpler organism , _ m. pneumoniae _ , has no other relevant sccs apart from energy metabolism , as a result of its parasitism , which has led to the loss of many metabolic functions .more precisely , in _m. pneumoniae _ the citric acid cycle and oxidative phosphorylation do not take place , meaning that it must rely on organic acid fermentation to obtain energy .moreover , changes in the growth rate greatly affect the fluxes through glycolysis and pyruvate metabolism .the study of the dependence on the environment of the _ e. coli _ metabolic backbone allows us to identify adaptation capabilities .regarding rich medium , we observe that the critical value of is substantially different than the one in glucose minimal medium , suggesting that this enriched medium modifies significantly the flux structure compared to the glucose minimal medium .the bacterium in rich medium displays less active reactions than in glucose minimal medium since , in minimal medium , many reactions must be active in order to synthesize biosynthetic precursors that in the rich medium can be obtained from the environment , in agreement with ref .the pathway called membrane lipid metabolism achieves a high relevance , being the most abundant pathway in the largest core of the rich medium metabolic backbone .this may happen because the proliferation response of _e. coli _ to the rich medium increases lipid biosynthesis , since lipids are not present as nutrients even in rich media due to its water insolubility .in contrast , biosynthesis of other components , specially amino acids , is obviously reduced . to satisfy the high lipid demand to generate new cells , fast - growing cells must synthesize membrane lipid components more rapidly .this leads to an over - expression of the genes related to membrane lipids which , in terms of metabolism , is observed in terms of a high relevancy of the membrane lipid metabolism pathway .another feature that we find is the loss of prominence of the pentose phosphate pathway .although this may suggest to a deficiency of the metabolite nadph , we find that , in the rich medium backbone , the functionally significant production of this metabolite takes place in the citric acid cycle pathway .this evidences the importance of citric acid cycle to produce nadph , which is tightly related to the synthesis of membrane lipids .an alternative explanation for the increase of the appearance of the citric acid cycle in the backbone is that more metabolites need to be catabolized in rich medium , and the tca cycle is the primary site for catabolism of many non - sugar substrates through anaplerotic reactions .the analysis of the adaptation of _e. coli _ to 333 different minimal media shows that the fraction of links in the backbone is practically independent on the composition of the nutrients present in these environments ( see fig .[ fig:3]c ) .this permits the construction of a merged backbone that comprises all the links in the backbone in each different media and in which each contributing backbone has uniform representation in fraction of links .this leads to the identification of pathways whose associated reactions are more sensitive to changes in the environment , unveiling alternate carbon metabolism as the pathway with more capabilities to respond to external stimuli , in accordance with existing works .this finding could also be seen as a successful positive control for the methodology applied here .the use of filtering methods usually implies a drastic reduction of the complexity of metabolic maps , which weakens the validity of potentially inferred conclusions .in contrast , the application of the disparity filter , based on a flux significance analysis to produce metabolic backbones , enables to reduce the system while maintaining all significant interactions according to and so it becomes a useful tool to unveil sound biological information .notice that these reduced versions must be seen as a map of the most significant connections , even though other metabolic reactions must be present to achieve viability and to satisfy the physico - chemical laws governing metabolic networks depending on nutritional conditions and other stresses applied to the cell .our investigations of _ e. coli _ and _ m. pneumoniae _ revealed metabolic backbones mainly composed of a core of reactions belonging to ancient pathways that still retain at present a central role in the evolved metabolism . besides , in _e. coli _ , the analysis of the core reveals a dominant direction with the synthesis of purines and pyrimidines and the metabolism of lipids ensuing after energy metabolism . at the same time, our approach can be appropriate in applications exploiting its capability to recognize pathways and particular reactions more sensitive to environmental changes , which makes it potentially useful in biotechnology and biomedicine .flux balance analysis ( fba ) is a technique which allows to compute metabolic fluxes without the need of kinetic parameters , just by using constrained - optimization .fba proceeds by writing the stoichiometric matrix of the whole network and multiplying it by the vector of fluxes .this stoichiometric matrix contains the stoichiometric coefficients of each metabolite in each reaction of the network .this product is then equal to the vector of the time variation of the concentrations .steady - state is assumed , thus . since in general metabolic networkscontain more reactions than metabolites , we have an underdetermined system of equations .hence , a biological objective function must be defined in order to have a biologically meaningful solution . in this work , the chosen objective function is the growth rate of the organism , which means that fba finds the solution that optimizes the growth of the organism , which is equivalent to maximize biomass formation .reversibility of reactions are also added in order to constrain the solutions . since we have a linear system of equations with linear constraints , linear programming is used in order to compute a flux solution in a small amount of time ( of the order of 1 s ) , which implies a computationally cheap method .we implement fba using gnu linear programming kit ( glpk ) .fba calculations are performed on the _ _ i__jo1366 version of _ e. coli _k-12 mg1655 , which contains 1805 metabolites and 2583 reactions , and on the _ _ i__jw145 version of _ m. pneumoniae _ , which contains 266 metabolites and 306 reactions . in both organisms ,the versions include the growth reaction , auxiliary reactions such as exchange or sink reactions , and all cellular compartments are taken into account .we model these metabolic networks as bipartite graphs with two kind of nodes , metabolites and reactions , and with links containing directionality , which leads to incoming , outgoing , and bidirectional links . for _, fba calculations are performed in a glucose minimal medium with a maximum uptake of glucose limited to 10 mmol gdw h , whereas for _ m. pneumoniae _ , fba computations are carried in a defined medium with a maximum glucose uptake of 7.37 mmol gdw h and a supply of d - ribose to simulate the availability of ribosylated bases .the disparity filter takes advantage of the local heterogeneity present in the fluxes of reactions associated to a given metabolite .the filter is able to retain those fluxes which are significant with respect to a null hypothesis .notice that , since we work with directed metabolic networks , we have three kinds of links : incoming , outgoing and bidirectional links .the latter are decoupled into incoming and outgoing links , leading to a network containing only incoming and outgoing links .we treat incoming and outgoing connections separately .the filtering method starts by normalizing the fluxes producing or consuming a metabolite , .the key point is that a few links carry a significant value of .we characterize the disparity in the local distribution of fluxes around with the measure     under perfect homogeneity , when all the links share the same amount of the flux producing or consuming the metabolite , independently of , whereas for perfect heterogeneity , when one of the links carries all the flux , .usually , an intermediate behavior is observed in real systems .  to assess the relevance of the difference fluxes associated to a metabolite , a null hypothesis is used which provides the expectation of the disparity measure .the null hypothesis consists on assuming that the intensity of the different fluxes associated to a metabolite is consistent with a random assignment coming from a uniform distribution .incoming and outgoing links are treated independently .the filter then proceeds by identifying which fluxes must be preserved . to do this, we compute the probability that a normalized flux is non - compatible with the null hypothesis .this probability is a _p_-value which is compared with a significance level , and thus links that carry fluxes with a probability can be considered non - consistent with the null model and so significant for the metabolite .the probability is computed with the expression ( see ref . ) .  once the significant fluxes have been selected , we construct the backbone by placing a directed link between two metabolites if there is a reaction whose flux is simultaneously significant for the consumption of one metabolite and for the production of the other .the filter can not decide on nodes with just one connection . according to our maximization of nodes minimization of links principle ,we use the prescription to preserve the reaction associated to metabolites with only one incoming or outgoing connection .a connected component of an undirected network is a subset of the network in which any two nodes are connected by at least one path .nodes in a connected component do not share connections with nodes belonging to a different connected component .directedness in network connections introduces a rich substructure in the connected components . inside the connected component of a directed network ,the so - called bow - tie structure emerges .bow - tie structures are formed by a strongly connected component ( scc ) , in and out components , tubes and tendrils .a scc is a subset of the connected component where nodes are reachable from any other by a directed path .the in component contains nodes that can access the scc but not vice versa . the out component is formed by nodes that can be reached from the scc but that can not return there .a tube is a sequence of nodes that connect the in and the out component without going through the scc .tendrils are composed by nodes that have no access to the scc and are not reachable from it .we consider a rich medium called luria - bertani broth .this nutritionally rich medium contains the set of compounds defining the minimal medium , _ i.e. _ , a set of minerals salts and four metabolites representing carbon , nitrogen , phosphorus and sulfur sources , in addition to the following compounds : amino acids , purines and pyrimidines , biotin , pyridoxine , thiamin , and the nucleotide nicotinamide mononucleotide ( see ref . for specific details ) .we use the different minimal media defined in ref . .more precisely , these media contain a set of minerals salts and four extra metabolites representing carbon , nitrogen , phosphorus and sulfur sources .to determine fba solutions in specific media we change the carbon source while we fix the sources of nitrogen , phosphorus and sulfur to the standard metabolites in each class , which are ammonia , phosphate , and sulfate respectively .in this way , each carbon source determines a different minimal medium . in the same way, other minimal media are constructed using the same procedure of changing the nitrogen , phosphorus or sulfur source while keeping the standard metabolite unchanged for the rest of categories ; note that , in these cases , the standard carbon source is fixed to glucose .555 media can be constructed using this procedure , 333 of them allowing growth after computing the fba solution in each media .we acknowledge support from a james s. mcdonnell foundation 21st century science initiative in studying complex systems scholar award ; mineco projects no . fis2013 - 47282-c02 - 01 and no .fis2013 - 41144-p ; and the _ generalitat de catalunya _ grants no .2014sgr608 and no .o. g. acknowledges support from a fpu grant funded by mineco .47ifxundefined [ 1 ] ifx#1 ifnum [ 1 ] # 1firstoftwo secondoftwo ifx [ 1 ] # 1firstoftwo secondoftwo `` `` # 1'''' [ 0]secondoftwosanitize [ 0 ] + 12$12 & 12#1212_12%12[1][0] * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) _ _ ,  ed .( , , ) _ _ ( , , ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) _ _ ( , , ) pp .  * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * , ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( ) * * ,  ( )

Abstract: the heterogeneity of reaction fluxes present in a metabolic network within a single flux state can be exploited to construct the so - called backbone as a reduced version of metabolism . the backbone maintains all significant fluxes producing or consuming metabolites while displaying a substantially decreased number of interconnections and , hence , it becomes a useful tool to extract primary metabolic routes . here , we disclose the metabolic backbone of _ escherichia coli _ using the computationally predicted fluxes which maximize the growth rate in glucose minimal medium , and we compare it with the backbone of _ mycoplasma pneumoniae _ , a much simpler organism . we find that the central core in both backbones is mainly composed of reactions in ancient pathways , still playing at present a key role in energy metabolism . in _ e. coli _ , the analysis of the backbone reveals that the synthesis of nucleotides and the metabolism of lipids form smaller cores which rely critically on energy metabolism ; but not conversely . at the same time , an analysis of the dependence of this backbone on media composition leads to the identification of pathways sensitive to environmental changes . the metabolic backbone of an organism is thus useful to trace simultaneously both its evolution and adaptation fingerprints .