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Modeling the temporal interplay of molecularsignaling and gene expression by usingdynamic nested effects modelsBenedict Anchanga, Mohammad J. Sadeha, Juby Jacoba, Achim Treschb, Marcel O. Vlada,c,d, Peter J. Oefnera,and Rainer Spanga,1

aInstitute of Functional Genomics, University of Regensburg, Josef-Engert-Strasse 9, 93053 Regensburg, Germany; bGene Center Munich, Department of Chem-istry and Biochemistry, Ludwig-Maximilians-Universitaet Muenchen, Feodor-Lynen-Strasse 25, 81377 Munich, Germany; cInstitute of Mathematical Statisticsand Applied Mathematics, Casa Academiei Romane, Calea 13 Septembrie 13, Bucharest, 050711, Romania; and dDepartment of Chemistry, Stanford University,Stanford CA 94305-5080b;

Edited by John Ross, Stanford University, Stanford, CA, and approved February 12, 2009 (received for review October 1, 2008)

Cellular decision making in differentiation, proliferation, or celldeath is mediated by molecular signaling processes, which controlthe regulation and expression of genes. Vice versa, the expres-sion of genes can trigger the activity of signaling pathways. Weintroduce and describe a statistical method called Dynamic NestedEffects Model (D-NEM) for analyzing the temporal interplay of cellsignaling and gene expression. D-NEMs are Bayesian models ofsignal propagation in a network. They decompose observed timedelays of multiple step signaling processes into single steps. Timedelays are assumed to be exponentially distributed. Rate constantsof signal propagation are model parameters, whose joint posteriordistribution is assessed via Gibbs sampling. They hold informationon the interplay of different forms of biological signal propagation.Molecular signaling in the cytoplasm acts at high rates, direct signalpropagation via transcription and translation act at intermediaterates, while secondary effects operate at low rates. D-NEMs allowthe dissection of biological processes into signaling and expres-sion events, and analysis of cellular signal flow. An applicationof D-NEMs to embryonic stem cell development in mice revealsa feed-forward loop dominated network, which stabilizes the dif-ferentiated state of cells and points to Nanog as the key sensitizerof stem cells for differentiation stimuli.

perturbation data | network reconstruction

I ntracellular signaling processes control the activity of transcrip-tion factors and the expression of genes. Changes in gene expres-

sion can activate further signaling processes, leading to secondaryeffects, which themselves give rise to tertiary effects and so on. Theresult is an intricate interplay of cell signaling and gene regulation.Whereas protein modification in the cytoplasm can propagate sig-nals in seconds, transcription and translation processes last hours,and secondary effects often become visible only after days. Ourgoal is to model the temporal interplay of signaling and expres-sion in complex biological processes involving several signalingpathways and spanning multiple rounds of cell signaling, generegulation, and gene expression.

Numerous statistical methods have been suggested for theanalysis and reconstruction of regulatory networks. Among themost widely used are relevance networks (1), graphical Gauss-ian models (2, 3), methods from information theory (4), Bayesiannetworks (5), including dynamic Bayesian networks (6), and meth-ods based on ordinary differential equations (7, 8). All of thesemethods employ pure observational data, where the network wasnot perturbed experimentally. Simulation (9, 10) and experimen-tal studies (9, 11) show that perturbation experiments improveperformance in network reconstruction. Rung et al. (12) built adirected disruption graph by connecting two genes where per-turbation of the first gene resulted in expression changes in theother gene. However, disruption networks do not separate directfrom indirect effects. Wagner (13) uses transitive reductions to

find parsimonious subgraphs explaining a disruption network. Theframework of Bayesian networks was also extended to accountfor perturbation data (14, 15). Yeang et al. (16) searched fortopologies that are consistent with observed downstream effectsof interventions. Although this algorithm is not confined to thetranscriptional level of regulation, it requires that most signalinggenes show effects when perturbing others.

The method described here builds on Nested Effects Models(NEMs), which have been proposed by Markowetz et al. (15) forthe analysis of nontranscriptional signaling networks. NEMs inferthe graph of upstream/downstream relations for a set of signalinggenes from perturbation effects. Because nontranscriptional sig-naling is too fast to be analyzed by delays of downstream effects,time series have not been used in this approach. This changes whenanalyzing slow-going biological processes like cell differentiation.

Following Markowetz et al. (15), we call the perturbed genesS-genes for signaling genes and denote them by S = S1, . . . , Sn.The genes that change expression after perturbation are called E-genes and we denote them by E = E1, . . . , EN . We further denotethe set of E-genes displaying expression changes in response tothe perturbation of Si by Di. In a nutshell: NEMs infer that S1 actsupstream of S2:

S1 −→ S2 if and only if D2 ⊂ D1

All downstream effects of a perturbation in S2 can also be trig-gered by perturbing S1. This suggests that the perturbation of S1causes a perturbation of S2 and acts upstream of S2. The graphof upstream/downstream relations is estimated from the nestedstructure of downstream effects. Due to noise in the data, we donot expect strict super-/subset relations. Instead, NEMs recoverrough nesting.

In the Bayesian framework of Markowetz et al. (15), networksare scored by posterior probabilities. By enumerating all net-work topologies, the maximum posterior network is chosen. Theexhaustive search limits the method to small networks of up to 8 S-genes. Greedy search heuristics (17, 18) and divide-and-conquerapproaches (17, 19) enable the analysis of larger networks withhundreds of S-genes. The latter divide the graph into smaller units,use exhaustive enumeration for each subgraph, and then reassem-ble the complete network. The division into subgraphs can eitherbe into all pairs or triples of nodes (19) or data-dependent into

Author contributions: R.S. designed research; B.A., J.J., A.T., M.O.V., and R.S. performedresearch; B.A., M.J.S., and R.S. analyzed data; and M.J.S., P.J.O., and R.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: rainer.spang@klinik.uni-regensburg.de.

This article contains supporting information online at www.pnas.org/cgi/content/full/0809822106/DCSupplemental.

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coherent modules (17). For a review and software see articles byFroehlich et al. (20, 21).

Note that there is a difference between the upstream/downstreamrelations of a network and the actual signal flow. If S1 is upstreamof S2 and S2 is upstream of S3, consistency requires that S1 isalso upstream of S3. In fact, all proposed methods except ref. 18confine the model space to transitively closed graphs. Althoughthe consistency argument is valid for upstream/downstream rela-tions, it does not hold for signal flows. Assume we have a linearcascade of S-genes where the signal flows from S1 via S2 to S3.Whether there is an alternative signal flow from S1 directly to S3does not follow from upstream/downstream relations. However,evidence of the alternative signal flow comes from time delays ofdownstream effects. Assume that the time spent to propagate adownstream effect from S1 to S2 plus the time spent to propagateit from S2 to S3 is larger than the time to propagate the effect fromS1 to S3 directly, then there must exist an alternative shortcut path-way from S1 to S3. Thus, temporal expression measurements yieldadditional insight into the cellular signal flow.

Fig. 1 illustrates the idea of D-NEMs in an elementary exam-ple. Shown is the hierarchical structure of a network and discretetime series data for three E-genes. One indicates that a signal hasreached the E-gene, while zero indicates that the expression of thisgene has not yet changed. Note, that the graph topology is con-sistent with the nested structure of ones in the final time point t5,shown in red.

Signals starting in S1 reach E2 one time unit after they havearrived at E1 suggesting that signal propagation from S1 to S2takes one unit of time. The same argument using the data fromperturbation of S2 suggests that it takes two time units to propagatefrom S2 to S3. Consequently, going from S1 to S3 via S2 takes 3 timeunits. However, the time delay from perturbation of S1 to observ-ing effects in E3 is only 1 time unit (marked in blue). This suggeststhe existence of a direct signal flow from S1 to S3. Evidence comesfrom the two blue ones. In case they were zeros, the time delaybetween S1 and S3 would have been the sum of times spent whengoing via S2. In this case, there would be no evidence for a shortcutpathway and we would decide on the more parsimonious graph.A real world analysis is more difficult than the toy example. Signalpropagation is a stochastic process, measurements are prone to

Fig. 1. Elementary example of a D-NEM. Shown is a network of three S-genes together with binary time series tables for typical E-genes connectedto the S-genes. Each table holds three rows corresponding to the three possi-ble perturbation experiments of S-genes. A one in column ti , row Sj of tableEk represents the observation of a downstream effect in Ek , ti time units afterperturbation of Sj .

noise, and we do not know which E-genes are controlled by whichS-genes. These sources of uncertainty are addressed by D-NEMs.

We assume exponentially distributed time delays for individualsignal propagation steps. The rate constants of the exponentialdistributions differ from case to case and are the main parame-ters of the model. All edges of a transitively closed network areassociated with an individual rate constant, whose posterior distri-bution is inferred by using Gibbs sampling. As explained before,molecular signaling in the cytoplasm occurs at high rates, directsignal propagation via transcription and translation at interme-diate rates, and secondary effects at low rates. The joint poste-rior of the rate constants will be used to analyze the interplayof signaling networks and gene expression in complex biologi-cal processes. It is also used to unravel molecular signal flow incells.

ModelThe input of a D-NEM consists of (i) a set of microarray time seriesthat measure the response of cells to molecular perturbations, and(ii) a transitively closed directed graph on vertex set S represent-ing a hypothetical hierarchical structure of upstream/downstreamrelations. The output consists of (i) the joint posterior distributionof rate constants describing the dynamics of signal propagation,and (ii) a not necessarily transitive subgraph of the input graphthat describes signal flow rather than hierarchical structure. LetD(i, k, l, s) denote the expression measurement of Ek in time pointts of the lth replication of a time series recorded after perturbationof Si. Following Markowetz et al. (19), we assume that the dataare binary, where zero encodes the wild-type expression level of agene, and one encodes that the expression of this E-gene changedbecause of perturbation-induced signal propagation.

We assume that the time spent for propagating a signal fromnode Si to node Sj is exponentially distributed with a rate constantkij. Note that the expected time spent in this step of signal transduc-tion is 1/kij. Fast processes are associated with high rate constants,but slow processes are associated with small rate constants. Expo-nential distributions are widely used to model temporal processesin complex systems (22, 23).

We do not observe the time spent for signal propagationbetween S-genes directly. Instead, we observe the time delaybetween a perturbation of an S-gene and the occurrence of down-stream effects in E-genes. Following Markowetz et al. (19) weintroduce parameters � = (θ1, . . . , θN ) to link E- to S-genes. Ifθk = i, then Ek is linked to Si. Moreover, we assume that everyE-gene is linked to a single S-gene. The set of E-genes attached tothe same S-gene is a regulatory module under the common regu-latory control of the S-gene. The module of E-genes attached toSi is denoted by Ei. Finally, we introduce additional rate constantskiE that represent the time delay between activation of Si and reg-ulation of its target module Ei. A single common rate is used forall E-genes in the module. Note that the exponential distributionsof time delays nevertheless yield model flexibility to accommo-date potential variability across absolute time delays for S-gene toE-genes signal propagation within the same target module. Notealso, that for exponentially distributed time delays, the recipro-cal rate constant of the distribution is equal to the average timedelay.

Following ideas in Tresch and Markowetz (18), we add an addi-tional node denoted by +, which is not connected to any of theS-genes. However, E-genes can be linked to this node, if they do notfit in any of the Ei. The +-node implicitly selects E-genes. Geneslinked to + are excluded from the model. We denote the com-plete set of rate constants including rates between S-genes andrates between S- and E-genes by K.

Although the θk are discrete parameters by nature, rate con-stants are usually modeled as continuous parameters. However,for the sake of computational efficiency, we confine the rates to adiscrete set of values denoted by (κ0, . . . , κT+1). If the data include

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time points (t1, . . . , tT ), we choose (κ0, 1/t1, . . . , 1/tT , κT+1), whereκ0 is set to a high value (i.e., 1,000) that represents the very fast sig-nal transduction through posttranslational protein modificationlike phosphorylation. Moreover, κT+1 is set to a value close tozero, indicating that no signal at all flows through this edge, suchedges can be excluded from the network. Overall, we have a set ofdiscrete parameters (K, �).

Prior Distributions. Assuming independent prior distributions forK and �, Bayes’s theorem yields P(�, K |D) = P(D|K , �)P(K)P(�)/P(D). The prior distribution P(�) can be chosen to incor-porate prior knowledge on the interactions of S- with E-genes.Such information might be derived from ChIP data or regulatorymotif analysis. The prior provides an interface through which themodel can be linked to different biological data types in integra-tive modeling approaches. Here we use the prior for calibratingE-gene selection. We set P(θk = +) to �, while distributing theremaining weight of 1 − � uniformly on the values 1, . . . , n.

Similarly, the prior distribution P(K) yields an interface forincorporating biological knowledge. If one knows that S1 andS2 fall into the same molecular signaling pathway, one can setP(k12 = κ0) to one, because signaling will operate on a highrate. In this article we exploit the fact that transcription takeshours and set P(kiE = κ0) to zero while assuming a uniformprior for the remaining values. Moreover, we set the prior prob-ability for the assumption that a given transitive edge exists toP(kij = κT+1) = 0.5, while again assuming a uniform prior for theremaining values.

Likelihood. Let us first consider a fixed linear path g in �, whichconnects the S-gene Si with the E-gene Ek:

Sik1−→ Sj1 · · · kq−1−−→ Sjq−1

kq−→ Ek,

where for simplicity of notation we reduce the double indices ofrate constants to single indices and write k1, k2, . . . , kq to denotethe rate constants. We are interested in the time needed for prop-agating a signal from Si down the path to Ek. More precisely, wewant to calculate the probability that the signal has reached Ekbefore some fixed time point t∗. If Zg is the sum of q indepen-dent, and exponentially distributed random variables with rateconstants k1, . . . , kq, then this probability equals P(Zg < t∗). Thedensity function of Zg is given by the convolution of independentexponential distributions

�(t)g =∫ ∞

0· · ·

∫ ∞

0δ

(t −

q∑u=1

τu

) q∏u=1

ψu(τu)dτ1 . . . dτq,

where ψu(τ ) = ku exp(−kuτ ) is the density of an exponentialwith rate ku. Laplace transformation yields a closed form for thecumulative distribution function of Zg

Fg(t) =q∑

b=1

∏a�=b

{ka

ka − kb

}[1 − exp(−tkb)] [1]

Note that the right-hand side is not defined if two or more of theku are identical. However, as right and left limits exist and areidentical, we can evaluate the probability by adding tiny distinctjitter values to the ku.

In the general case a signal can be propagated from Si to Ek viamultiple alternative paths. In this case we assume that the fastestpath determines the time delay for downstream effects to be seen.

We enumerate all linear paths connecting Si to Ek. For each pathwe construct a random variable Zu as described above. If the alter-native paths do not share edges, the probability that the signal hasarrived at Ek before time t∗ via at least one of the paths is given by

PSi→Ek (t∗) = 1 −∏

u

(1 − Fu(t∗)) [2]

In the general case, paths share edges, which lead to dependenciesof signal propagation times. Nevertheless, simulations show thatEq. 2 is a good approximation of the distribution of time delays,except maybe in some very unfortunate topological constellations.It is an approximation based on the assumption that the interac-tions among merging pathways can be neglected similar to themean-field approximation from many body theories in statisticalphysics. In the examples discussed below the approximation erroris not affecting any of the conclusions (simulation data not shown).

Eqs. 1 and 2 describe the stochastic nature of signal propaga-tion in the cell. Note that by Eq. 2 the average overall time delaybetween Si and Ek is smaller than the average time delay associatedwith the fastest path connecting them because, with some positiveprobability, the in average slower process will be the actually fasterone. This speedup by stochasticity effect is a consequence of thestochastic nature of time delays.

Before calculating the likelihood, we need to consider a sec-ond source of stochasticity, namely measurement error. FollowingMarkowetz et al. (19), we denote the probabilities for false posi-tive and false negative signals by α and β, respectively. Assumingconditional independence, the likelihood factorizes into

P(D|K , �) =∏D=1

PSi→Ek (ts)(1 − β) + (1 − PSi→Ek (ts))α

×∏D=0

PSi→Ek (ts)β + (1 − PSi→Ek (ts))(1 − α),

where the first product is over all data points, for which we observea downstream effect, and the second product is over those forwhich we do not.

Gibbs Sampling. With N E-genes, n S-genes, and L edges in theinput graph, the model comprises N + n + L discrete parameters.For simplicity of notation, we reduce the double indices of rateconstants to single indices such that the joint posterior is written

P(k1, . . . , kL+n, θ1, . . . , θN |D).

We initialize the parameters with random values from theirdomains. Then we iteratively cycle through all rate constantsupdating them by sampling from the conditional posterior dis-tributions

p(ki|K − {ki}, �, D).

With only discrete parameters, updating is straightforward. Wecalculate all values

p(ki = κj)p(D|K − ki, �, ki = κj),

normalize them to sum up to one, and draw a new value for ki fromthis distribution. The iteration is completed by similarly updatingall θk. In the supporting information (SI) Appendix, we analyze theconvergence and mixing properties of the Gibbs sampler. In gen-eral, convergence is fast and scale reduction factors between 1 and1.1 are reached after a burn in of 500 iterations. We typically start 2independent runs of the Gibbs sampler with random start points,discard the first 500 iterations in each trajectory, and combinethe remaining samples for further inference of signal propaga-tion. Choosing positive values for the tuning parameters α and

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β protects the conditional posterior distributions from singular-ities, and ensures the good convergence properties of the Gibbssampler.

Inference of Signal Flow. Under the natural assumption that per-turbation effects propagate down the signaling network to alldescendants of a perturbed gene, the nested structure of down-stream effects resolves the network only up to its transitivity class.Network topologies with identical transitive closures produce thesame nesting of downstream effects and, hence, can not be distin-guished. As explained above, temporal data hold the potential offurther resolving these transitivity classes. D-NEMs start from atransitively closed network. Posterior distributions are calculatedacross a discrete set of rate constants including a very small rateconstant κT+1. As explained above, kij=κT+1 reflects network con-stellation, in which no signal is flowing through the edge from Sito Sj. Note that if a rate constant is set to κT+1, the correspondingedge is not contributing to the likelihood according to Eq. 2. Theedge is effectively excluded from the model. Hence, in additionto estimating average time delays the Gibbs sampling procedurefacilitates network refinement. If the posterior probability of theedge from Si to Sj is P[kij=κT+1|D] > 0.6, we exclude the edgefrom the network.

Because of the long running times of the Gibbs samplerit is not possible to reconstruct the network topology fromscratch as was done for standard NEMs in refs. 18–20. Never-theless, we use our method to discriminate between small num-bers of candidate topologies. Model selection in the Bayesiancontext is based on Bayes factors and requires the compu-tation of marginal likelihoods. This is known to be a hardproblem, and approximative methods are therefore adopted.Here, we use the deviance information criterion (DIC) ofSpiegelhalter et al. (24).

A first test of a complex data model is to validate its perfor-mance in simulation scenarios where data are artificially gen-erated according to the model assumption. In SI Appendix weshow that our model recovers average time delays in noisydata and detects transitive shortcut edges even in situationswhere noise is high and average time delay differences aresubtle.

Application to Murine Stem Cell Development. We apply the D-NEM approach to a dataset on molecular mechanisms of self-renewal in murine embryonic stem cells. Ivanova et al. (25) usedRNA interference techniques to down-regulate six gene productsassociated with self-renewal regulatory function, namely Nanog,Oct4, Sox2, Esrrb, Tbx3, and Tcl1. They combined perturbation ofthese gene products with time series of microarray gene expressionmeasurements. Mouse embryonic stem cells (ESCs) were grownin the presence of the leukemia inhibitory factor LIF, thus retain-ing their undifferentiated self-renewing state (positive controls).Cell differentiation associated changes in gene expression weredetected by inducing differentiation of stem cells through remov-ing LIF and adding retinoic acid (RA) (negative controls). Finally,RNAi-based silencing of the 6 regulatory genes was used in (LIF+,RA−) cell cultures to investigate, whether silencing of these genespartially activates cell differentiation mechanisms. Time seriesat 6-7 time points in one-day intervals were taken for the posi-tive control culture (LIF+, RA−), the negative control culture(LIF−, RA+), and the six RNAi assays. In the context of theD-NEM framework the 6 regulatory gene products Nanog, Oct4,Sox2, Esrrb, Tbx3, and Tcl1 are S-genes, whereas all genes showingsignificant expression changes in response to LIF depletion areused as E-genes. Downstream effects of interest are those wherethe expression of an E-gene is pushed from its level in self-renewingcells to its level in differentiated cells. Our goal is to modelthe temporal occurrence of these effects across all time seriessimultaneously.

In a comparison of the (LIF+, RA−) to the (LIF−, RA+)cell cultures 137 genes showed a >2-fold up- or down-regulationacross all time points. These were used as E-genes in our analy-sis. The two time series without RNAi were used to discretizethe time series of perturbation experiments following a sim-ple discretization method detailed in SI Appendix, thereby set-ting an E-gene state to 1 in an RNAi experiment, if its expres-sion value is far from the positive controls, and 0 otherwise.Genes that did not show any 1 after discretization across allexperiments were removed, leaving 122 E-genes for furtheranalysis.

D-NEMs assume that once a perturbation effect has reachedan E-gene, it persists until the end of the time series. In otherwords, a one at time point t indicates that a downstream effect hasreached the E-gene prior to t and not that it is still observable atthis time. Hence, a typical discretized time series starts with zeros,eventually switches to ones, and then stays one until the end ofthe series. We refer to these patterns as admissible patterns. Forthe vast majority of E-genes, the discretized data roughly followedadmissible patterns. Nevertheless, exceptions were observed mostlikely due to measurement noise. We replaced the time seriesfor each gene by the closest admissible pattern, based on editdistances. In the case where several admissible patterns had thesame edit distance to the time series, we chose the pattern hold-ing the most ones. These curated data were used in furtheranalysis.

Since long computation times for Gibbs sampling prohibit thereconstruction of the network’s topology from scratch by usingD-NEMs, we used the triplet search approach for the standardnested effect approach (19) applied to the final time point todetermine a topology for the network. Note that the final timepoint of an admissible pattern accumulates information along thetime series, because it reports a one whenever a downstream sig-nal has reached the E-gene at any time. The binary data of thelast time point across all S-gene perturbations is shown in Fig.2A, while Fig. 2B shows the reconstructed network. A nestedstructure is visible. For example, the top 4 rows in Fig. 2A showa staircase-like pattern of nested sets consistent with the linearcascade Nanog −→ Sox2 −→ Oct4 −→ Tcl1. We refer to thiscascade as the inner backbone of the network. The discretizationstep involves a cutoff parameter, whose value has some influenceon the derived network. Although some edges can vary for differ-ent parameter settings, key features of the network like the innerbackbone are not affected. All inference on stem cell developmentin the rest of this article is exclusively based on stable substruc-tures of the network. SI Appendix gives the details of our networkstability analysis.

The topology is based exclusively on the nesting of down-stream effects. Time delays of signal propagation can now beused for fine tuning the topology. Originally, the NEM analy-sis suggested a bidirectional arrow between Oct4 and Tcl1 sug-gesting that the nesting of downstream effects in the final timepoint can not resolve the direction of interaction between thesegenes. Time delays in contrast strongly favor a model, whichplaces Oct4 upstream of Tcl1. To show this, we fitted indepen-dent D-NEM models for the two networks, which place Oct4 up-or downstream of Tcl1. We used the deviance information cri-terion DIC (24) to decide which hypothesis is better supportedby the observed time delays. The DIC strongly favors the model,which places Oct4 upstream of Tcl1 (DIC of 5491.1 compared with5581.7).

Next, we exploit the D-NEM Gibbs sampler trajectories asso-ciated with the network topology from Fig. 2B to infer averagetime delays and regulatory control of E-genes. Fig. 2C shows thehistogram of average time delays (reciprocal rate constants) alongthe Gibbs sampling trajectory for the edge between Oct4 and itstarget E-genes. It is equivalent to the top-most gray-scale inten-sity profile of the heat map in Fig. 2D. The histogram reflects the

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Fig. 2. Stem cell data analysis. (A) Discretized data of the last time point across E-genes (rows) and S-gene perturbations (columns), with black representingdownstream effects and white no effects. (B) The transitively closed nested effects model estimated from the data shown in A using static NEM. (C) A histogramof the posterior probabilities for the average time delay associated with the edge from Oct4 to its target E-genes. (D) Heat map of the posterior distributionof average time delays. Rows correspond to edges of the network including those between S- and E-genes, whereas columns refer to average time delays.Marginal posterior probabilities are gray-scale coded. The top row corresponds to the histogram shown above. (E) The final network structure estimated bytime delay analysis using D-NEM. Edge colors correspond to estimated average time delays: fast signal propagation (green), intermediate signal propagation(blue), and slow signal propagation (red).

marginal posterior probability of this parameter. The posteriorheat map for all edges is shown in Fig. 2D. Light gray indicateshigh marginal posterior probability and dark-gray tones stand forlow marginal posterior probabilities. The posterior mass eitherconcentrates around zero indicating no time delay for this step ofsignal propagation, or intermediate values explaining secondaryand tertiary effects, or high values with most of the posterior masson κT+1 (shown as x) suggesting that no signal is flowing throughthis edge. We exclude an edge if the posterior mass on κT+1 is>0.6. The resulting network is shown in Fig. 2E. Strikingly, thetime delay data provide evidence that all but three of the edgesfrom Fig. 2B actually transport signal. Note that the time delaydata have also overruled the static NEM in one instance, in thatit has removed the nontransitive edge between Nanog and Tbx3.

DiscussionThe most striking feature of our early stem cell differentiationmodel is the high frequency of transitive edges. The circuitry isnonparsimonious, raising the question of why evolution has cho-sen this complex network topology. Note that a transitive edge isconsistent with the concept of feed-forward loops first introducedin ref. 28 and summarized in ref. 29. The authors have shownthat feed-forward loops are the most frequent network motif intranscriptional networks (28). In this light, the high density ofthe early stem cell differentiation model with its transcriptionalcomponents is not surprising.

To understand the regulatory dynamics mediated by transitiveedges, consider the subnetwork consisting of Nanog, Oct4, andTcl1. The E-genes controlled by Tcl1 change from self-renewalexpression levels to levels typical for differentiated cells bothin response to blocking the signal from Nanog via Oct4 to Tcl1and in response to blocking the transitive edge from Nanog toTcl1. Signals from both branches are jointly needed to activateTcl1. Their inputs are integrated by an AND-gate. AND-gates infeed-forward loops are known to facilitate relative accelerationof OFF-Step signaling compared with ON-Step signaling (29). Inour example, down-regulation of Nanog causes a fast response inthe Tcl1-E-genes, whereas the model suggests that the responseof Tcl1-E-genes to up-regulation of Nanog is delayed, if there isany at all. These E-genes can be shifted from stem cell levels todifferentiated levels simply by blocking Tcl1’s input from the short-cut path. Down-regulation of Nanog alone achieves this swiftly.Shifting E-genes from differentiated levels back to stem cell levelrequires reestablishing both pathways in the feed-forward loop.

Nanog needs to reestablish the expression of Oct4, which not onlydelays responses but also requires stimuli for activating Oct4 otherthan those included in the model.

In the early stem cell differentiation model, all transitive edgescorrespond to feed-forward loops. The top ranking gene in thehierarchy Nanog has fast control over most E-genes via direct edgesconnecting it with all other S-genes. Moreover, down-regulationof Nanog alone shifts the expression of all E-genes to levels of dif-ferentiated cells. Because signal propagation along the transitiveedges is fast, short fluctuations of Nanog trigger differentiation.The position on top of the regulatory hierarchy puts Nanog into therole of a key sensitizer for cell differentiation. The other S-genescontrol only parts of the E-genes and in many cases signal propaga-tion is considerably slower. We hypothesize that one possible roleof the other S-genes is to ensure that the differentiation processis virtually unidirectional. Support for this hypothesis comes fromthe frequent AND-gates within the network. To shift the expres-sion values of E-genes from the differentiated state back to thestem cell state the expression of Nanog needs to be raised again.However, because of the AND-gates a raise of Nanog alone doesnot trigger a fast cellular response. The transitive outer edges con-trol differentiation but not the reverse process. This model-basedprediction is in line with the observation that down-regulation ofNanog alone triggers stem cell differentiation (26), whereas a con-stitutive overexpression of several genes is needed to revert the dif-ferentiation process; like in the generation of induced pluripotentstem cells (27).

Taken together, the feed-forward loop dominated circuitry ofthe early stem cell development network stabilizes the differenti-ated state of cells relative to the self-renewal state. The transitiveedges guard against redifferentiation events. They filter noisy fluc-tuations in the activity of key regulatory genes like Nanog, Oct4,and Sox2. Evolution has developed this non-parsimonious cir-cuitry to make cell differentiation a predominantly unidirectionalprocess and thus to maintain the integrity of differentiated tissues.At the same time the circuitry destabilizes the self-renewal state,which can only be maintained through a joint and tight controlof all S-genes in concert. Fluctuations in the activity of individ-ual S-genes can trigger differentiation, with Nanog being the keysensitizer for differentiation stimuli.

ACKNOWLEDGMENTS. This work was supported by the Bavarian GenomeNetwork BayGene and the ReForM-M program of the Regensburg Schoolof Medicine. M.O.V. was supported by the National Science Foundation andProgramul Cercetare de Excelenta Grant M1-C2-3004/2006-Response of theRomanian Ministry of Research and Education.

Anchang et al. PNAS April 21, 2009 vol. 106 no. 16 6451

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