To investigate genomic responses that precede the response of sentinel 'primary NO₃ ^- response' genes (NIR, NRT2.1, NIA1, NIR1) to nitrate application, we first generated several time-series experiments (data not shown). These allowed us to identify the earliest time at which we were able to detect unambiguous NO₃ ^- induction of these sentinel response genes using real time quantitative PCR (RT-QPCR). Figure 1a shows the expression of selected sentinel genes over time (0, 3, 6, 9, 12, 15, 20, 25, 35, 45, 60 minutes) in response to treatment with 1 mM KNO₃ or controls of 1 mM KCl. These results (Figure 1a) demonstrate that a sentinel gene such as NRT1.1 is induced at 20 minutes (compared to KCl controls, and in comparison to gene expression at time 0 minutes). The timing of induction of other sentinel genes involved in the 'primary NO₃ ^- response' are NIR1 at 12 minutes and NRT2.1 and NIA1 at 15 minutes. Following these preliminary experiments, we next ran Affymetrix ATH1 chips on biological replicates corresponding to the beginning of sentinel gene induction and their preceding time points (0, 3, 6, 9, 12, 15, 20 minutes). Note that we kept the 20-minute time point as a reference, since it was the earliest time point that had previously been studied 6 .

The resulting nitrate-responsive transcriptome kinetic dataset corresponded to 26 ATH1 chips with 22,810 probes each. A sequential analysis involving linear modeling (detailed in Materials and methods) was carried out to identify genes regulated at each particular time point with highly stringent criteria (including control of the false discovery rate (FDR)). We detected 83, 192, 55, 149, 190, and 229 genes significantly regulated by nitrate treatment at the 3, 6, 9, 12, 15 and 20 minute time points, respectively (Additional file 1). The union of these gene lists corresponds to 550 distinct nitrate-responsive genes. We demonstrate that a large majority of the newly identified NO₃ ^--regulated genes are controlled at the earliest time points (3 and 6 minutes), which have never before been assayed (Figure 1b). In order to support these new findings, 15 genes have been validated by QPCR (Additional file 2) on three replicates (two were used for the microarray chips and one for QPCR only). The predicted behaviors of these genes were validated by the QPCR approach, as follows. One set of genes is shown to have a transient response to NO₃ ^- (for example, At1g55120, At3g50750, At1g64370, At4g16780, At1g27900, At1g22640, At1g52060, and At2g42200). While a second gene set is validated to be very early responsive genes (for example, At1g13300, At1g49000, At4g31910, At5g15830, At2g27830, At3g25790, and At5g65210). Quantitatively, the correlation between the NO₃ ^- induction (KNO₃/KCl ratio) detected by both approaches (ATH1 chip and QPCR) is R² > 0.5 for 8 genes, 0.5 > R² > 0.4 for 3 genes, R² < 0.4 for 4 genes. It is noteworthy that for the genes having a low correlation, their overall behavior is validated by QPCR (for example, constant versus transient induction by NO₃ ^-; Figure 2b; Additional file 2).

To probe the biological significance of these kinetic patterns of nitrate regulation of gene expression, we determined the functional categories that are over-represented in the lists of nitrate-regulated genes at each time point, separating the induced and repressed gene lists (Additional file 2). Interestingly, the biological functions induced earliest after nitrate addition do not concern nitrogen directly. Instead, within 3 minutes, the very first statistically significant over-represented functional category is ribosomal proteins (P-value 6.58e^-6). This finding generates the hypothesis that nitrogen could trigger a transient and very rapid reprogramming of key elements of the translation machinery needed to synthesize new proteins required for nitrogen acquisition. This idea might be further supported by the fact that many more genes are induced by the addition of nitrate than are repressed (see below). Moreover, later on in the time-course (as early as 9 minutes), the next biological function to be significantly induced is the oxidative pentose-phosphate-pathway, a function that is known to be a critical step providing reductants needed to assimilate NO₃ ^- 26 . The oxidative pentose-phosphate-pathway has also been shown to generate a signal controlling key effectors of the NO₃ ^- response, such as NRT2.1, NRT2.4, NRT1.1, NRT1.5, and AMT1.3 27 . Taken together, these observations suggest that the early nitrate response involves mechanisms needed to prepare the plant to respond to nitrate rather than mechanisms that relate directly to nitrogen. Such mechanisms - for example, nitrate transport and amino acid metabolism - are regulated later on in the time series (Additional file 3).

To begin to decipher the pattern of nitrate-regulated gene expression over the entire time series, we first clustered the gene expression ratio (Log₂(Signal KNO₃/Signal KCl) of the 550 significantly regulated genes) in order to gain insight into the genomic reprogramming during the first 20 minutes of KNO₃ treatment (Figure 1c). The vast majority of the reprogramming is an induction of gene expression by NO₃ ^-, rather than a repression. To quantify this observation, the numbers of genes that are detected as significantly induced by NO₃ ^- at 3, 6, 9, 12, 15, and 20 minutes are 63 (76% of regulated genes), 146 (76% of regulated genes), 54 (98% of regulated genes), 123 (82% of regulated genes), 164 (87% of regulated genes), and 209 (92% of regulated genes), respectively. One interpretation is that NO₃ ^- induces an adaptation program that is on 'stand-by' in NO₃ ^--free conditions, rather than a shut-down of a putative 'N-free-condition' program. Clustering analysis also allowed us to sort gene responses according to their overall behavior. This analysis demonstrated that rapid gene expression responses to nitrate could be classified into up to 20 clusters (according to figure of merit (FOM) analysis; see Materials and methods; Figure 2). Considering each cluster independently, we were able to identify over-represented biological functions for eight clusters, including chloroplast, the oxidative pentose-phosphate-pathway, and ribosomal proteins (Figure 2; see Additional file 4 for details).

Moreover, we identified and analyzed 146 genes that were consistently induced over the 20 minutes of nitrate treatment (corresponding to clusters 1, 9, 11, 13, and 14). This group of consistently nitrate-induced genes includes over-represented biological functions such as oxidoreduction coenzyme process (P-value = 0.00027), nicotinamide metabolic process (P-value = 6.50e-05), regulation of transcription (P-value = 0.00167), pentose phosphate shunt (P-value = 0.00073). We also identified 219 genes showing responses to nitrate that seem to represent a general pattern of transient regulation (clusters 2, 3, 4, 6, 7, 8, 10, 12, 16, 17, and 18). Interestingly, the oxygen and redox state of the cell seems to be a general function that is transiently adapted by KNO₃ treatment. Indeed, Munich Information Center for Protein Sequence (MIPS) functions such as oxygen radical detoxification (P-value = 0.00018), peroxidase reaction (P-value = 0.01479), and superoxide metabolism (P-value = 0.02472) are over-represented gene ontology terms in this group. This observation might indicate the effect of NO₃ ^- on the redox state of the cell. Finally, we show that 124 genes are repressed by NO₃ ^- treatment, transiently or otherwise (corresponding to clusters 5, 19, and 20). The common function overrepresented in this group is transcription (P-value = 0.00312). This could result from the extinction of the pre-existing transcriptome program preceding the NO₃ ^- treatment. Since the plants had been nitrogen starved for 24 hours before NO₃ ^- treatment, this might correspond to genes that are up-regulated by the pre-treatment (nitrogen starvation) and down-regulated by NO₃ ^- provision. To statistically test this hypothesis, we set up a randomization test (see Materials and methods) to quantify whether the genes that are down-regulated in our conditions correspond to genes that were up-regulated by nitrogen starvation in Peng et al. 28 ; this occurred with a P-value of 0.0089. Conversely, no significant overlap was detected for clusters induced bi NO₃ ^- (clusters 1, 2, 4, 9, 10, 11, 13, 14). This finding validates the idea that NO₃ ^--down-regulated clusters correspond to genes involved in the response of plants to the pre-treatment conditions. In summary, a large part of the NO₃ ^- gene expression reprogramming has been missed by previous genomic studies. The time-varying expression modulation newly identified here involves physiological functions that could be components of the nitrate signaling system itself.

In order to further document the potential of this dynamic transcriptome response to mediate cross-talk between nitrate signaling and other well-studied signaling pathways in plants, we evaluated if the gene sets regulated by NO₃ ^- at the different time points in our analysis overlap more than expected by chance with genes regulated by hormones using data generated by the Chory lab 29 . To do this, we compared the nitrate-regulated gene lists (over six time points) with the lists of hormone-regulated genes 29 and generated a matrix that assembled the randomization test P-values (see Materials and methods) between each pair of gene lists. The lists included genes regulated by NO₃ ^- across each of the six time points (our study), and lists of genes regulated by seven different hormones by the Chory lab (abscisic acid, cytokinins, auxin (IAA), methyl jasmonate, brassinolides, gibberellic acid, ethylene)] 29 . These results (Figure 3) lead to three main conclusions supporting the existence of gene modules responding to nitrate and hormone signaling.

First, we considered only the overlap between the NO₃ ^--responsive gene lists at different time points. We found evidence for two linked 'modules' of nitrate-regulated gene expression (modules 1 and 2 in Figure 3b). The first nitrate-regulated module consists of the nitrate-regulated genes in the union of the 3- and 6-minute gene lists. The overlap between these two lists is far beyond what we would expect by chance (P-value < 0.001). However, the 3-minute gene list overlaps very little with the rest of the nitrate-regulated genes in the time-course study. As such, the second nitrate-regulated module is made up of the union of the 6-, 9-, 12-, 15-, and 20-minute gene lists (these gene lists overlap significantly more than random). The 6-minute gene list acts as the link between the very early nitrate-response genes (before 6 minutes) and the more delayed ones (after 6 minutes).

Second, the overlap of the nitrate-regulated genes with the hormone-regulated genes (modules 3 and 4 in Figure 3b) is significantly higher than expected at the 9-minute nitrate time point for abscisic acid-, indole acetic acid-, brassinolide- and methyl jasmonate-regulated genes, while the 12-minute nitrate time point overlaps significantly with cytokinin-regulated genes. This suggests that the interaction of nitrate signaling with other hormone signals is likely to involve the genes regulated by nitrate after 9 minutes. This leads to the hypothesis that, from 0 to 6 minutes, the genomic reprogramming concerns a pure NO₃ ^- signaling pathway, and thereafter (for example, 9 minutes after nitrate treatment) interactions with developmental signals such as hormones occur (Figure 3). This enables us to derive the hypothesis that the early nitrate controllers (for example, transcription factors, kinases, and so on) regulated at 3, 6, and 9 minutes are involved in the control of the nitrate signaling itself, rather than in the interaction between NO₃ ^- and other signals such as hormones.

Third, this analysis shows that the different hormonal treatments control largely overlapping gene modules, as has been described previously 29 .

In conclusion, connections between NO₃ ^- and hormone-related signaling are common features of plant molecular networks at several layers of integration (for a review, see 30 ). For instance, transcriptional connections have been identified where genes involved in a NO₃ ^--responsive 'biomodule' have been shown to be more responsive to NO₃ ^- if they are also strongly regulated by hormones 13 . More recently, we provided a mechanistic hypothesis to explain the role of NRT1.1 as a NO₃ ^- sensor controlling lateral root development. Indeed, NRT1.1 is a transceptor able to transport both auxin and nitrate. The sensing mechanism results from the ability of nitrate to inhibit auxin transport by NRT1.1, leading to low lateral root development at low nitrate concentrations 16 . To determine whether this mechanism is also involved in the transcriptional induction studied in the present work will require further investigation. However, the fact that hormones can be involved at the beginning of NO₃ ^- sensing mechanisms 13 16 and downstream of NO₃ ^- transcriptional activation (this analysis) is an intriguing observation that deserves further investigation to understand what is the purpose of such signal entanglement.

In order to probe the role of a transcription factor/hub in the predicted regulatory network presented in Figure 4b, transgenic plants (pSLP9:rSPL9) expressing an altered version of the mRNA for the SPL9 transcription factor were compared to wild-type plants for their response to NO₃ ^- provision. This gene has been selected for several reasons: (i) it is induced at very early time points (3 and 6 minutes); (ii) the inferred network predicts that SPL9 potentially controls at least six genes, including two nitrogen assimilation sentinel genes - this places it as the third-most influential transcription factor on the nitrogen assimilation gene sentinels; (iii) it is also the most strongly influenced gene in both its number of 'in' connections, and by the magnitude of the regulations controlling it (Figure 4b); and (iv) SPL9 displays some strong correlations with key regulators across the NASC (Nottingham Arabidopsis Stock Centre) array dataset (see below). As such, SPL9 constitutes a potential crucial bottleneck in the flux of information mediated by the proposed nitrogen regulatory network (Figure 4). We first considered SPL9 mutants and monitored sentinel expression in this genetic background. However, even if some defects have been observed, no consistent phenotype can be reported (data not shown). This may be readily explained by the topological redundancy of the network (Figure 4). Thus, one could expect that over-expression of SPL9 mRNA would trigger a detectable effect on the predicted sentinel targets and on the network behavior. SPL9 is a miR156-targeted SBP-box transcription factor identified to control shoot development 45 and flowering transition 46 47 , and it also appears as a potential central regulator in the network derived from the SSM (Figure 4). SBP-box transcription factors are potentially redundant, but the use of miR156-resistant SPL9 transgenic plants (resulting in over-expression/gain of function plants) has been extensively used to decipher their role. In our experimental set-up, transgenic SPL9 mRNA is over-expressed 4- to 20-fold in the miR156-resistant plants compared to wild type (Figure 5). Moreover, although the rSPL9 mRNA resulting from the modified gene is resistant to degradation by miR156 (explaining the over-expression) the transgene is still under the control of its native promoter. Thus, this over-expression reflects, at least in part, the promoter activity or the effect of other post-transcriptional controls (independent of miR156). This demonstrates that the SPL9 promoter activity is potentially also under the effect of NO₃ ^-, since rSPL9 mRNA exhibits a transitory depression in the transgenic plants (Figure 5).

mRNA transcription levels of several sentinel genes have been followed in this pSPL9:rSPL9 transgenic line. The most dramatic effect recorded is for the NIR target sentinel gene. Interestingly, the NIR gene has previously been demonstrated to be one of the most robustly NO₃ ^--regulated genes based on a meta-analysis of microarray data from nitrogen treated plants 14 . In support of this, over-expression of the SPL9 gene leads to a significant advance in the NIR NO₃ ^- response by about 10 minutes, and attenuates its magnitude of regulation at later time points (60 minutes). Less dramatic but still significant effects (over three independent experiments) have been recorded for NRT1.1/CIPK23 genes, belonging to the NO₃ ^- sensing module 20 , and for the NIA2 gene. These results demonstrate a role of the SPL9 transcription factor in the control of genes involved in the NO₃ ^- primary response. We also further investigated the role of SPL9 over-expression on the transcription levels of genes in the network over time (Figure 4b). Interestingly, SPL9 seems to have an effect on the vast majority of the genes in the regulatory network that we have tested (Additional file 5). The diversity of the misregulation of gene expression is high. For instance, 4 out of the 14 genes tested display an early effect (between 0 and 20 minutes) after the SPL9 over-expression. However, 11 genes display modified gene expression at later time points (40 and 60 minutes).

An observation needs to be made at this point. We compared the predicted effect of SPL9 on its putative targets (inferred by ML), with the actual effect of over-expression of SPL9 on these genes measured by Q-PCR. The only systematic effect that we found is that genes predicted to be negatively regulated by SPL9 display an early induction in the pSPL9:rSPL9 line and are indeed less induced at later time points (this is an interesting feature found for four genes, including NIR, At1g13300, At5g10030, and At5g65210). However, the opposite is not true, since the genes predicted to be up-regulated by SPL9 also display a 'down-regulation' at later time points (by 60 minutes), but no 'up-regulation' at early time points (10 to 20 minutes), in response to rSPL9 over-expression. This relative absence of logic can be very easily explained by the predicted functional redundancy found in the network (also discussed below). The question about predicting the over-expression of a network hub is intriguing and will need further investigation, including the generation of transcriptome data to probe and learn how the whole system is perturbed by gene over-expression.

On the other hand, it is noteworthy that with its intrinsically highly connected topology, the network is able to amplify effects (such as SPL9 over-expression) across steps/time. Indeed, consider a simple positive feedback loop where A→B and also B→A; if the coefficient of A→B and B→A is only 10%, this is enough that when the expression of A and B is 100 at time 0, it will reach 160 after 5 steps. Here, we hypothesize that this is what happens for SPL9; that it takes time (more than 20 minutes) to amplify the differences recorded for the 11 genes with modified gene expression at later time points.

This analysis defined SPL9 as a potential hub for short time scale regulatory behavior, with small-amplitude regulatory effects (36% maximum). Those effects are going to be amplified across consecutive time steps and have longer-term (beyond 40 minutes) effects on additional genes. Thus, the interesting part of machine learning approach is that it can be used to predict the eventual behavior at time points that were not used in the ML process.

Interestingly, in our experimental setup, pSPL9:rSPL9 does not display any obvious developmental phenotype, contrary to what was described by Wang et al. 48 . These diverging results may be explained by the different plant growth conditions used in the two studies. In particular, in our pre-treatment conditions, plants were grown for 14 days in ammonium succinate without any NO₃ ^-. By contrast, in the Wang et al. studies the phenotypes were observed in plants grown on nitrate as a nitrogen source. The hypothesis that the phenotypes are nitrate-dependent is supported by the fact that the majority of the pSPL9:rSPL9 gene regulation phenotypes (Figure 5; Additional file 5) are triggered by NO₃ ^- provision. Out of the 18 genes displaying a mis-regulation in the transgenic plants, only the CIPK23 gene displays a phenotype before nitrate treatment. For the 17 other genes, NO₃ ^- is necessary to promote the rSPL9 over-expression effect.

In order to test the hypothesis that nitrate induces the phenotypes in transgenic plants, plants (wild type and pSPL9:rSPL9) were plated on the same background media used for the kinetic experiments (see Materials and methods) and complemented with 0.5 mM (NH₄)₂-succinate or 1 mM KNO₃. Following 8 days after germination, different root traits were scored. These results show that indeed the presence of NO₃ ^- enhances pSPL9:rSPL9 phenotypes (Additional file 6). Overall, only minor developmental differences were recorded on the (NH₄)₂-succinate media that cannot explain the distinct molecular phenotypes shown in Figure 5 and Additional file 5.

From a biological point of view, the modification of the nitrogen status of the plant (induced by nitrogen deprivation) has been shown to increase pri-miR156 accumulation 49 and mature miR156 in Hsieh et al. 50 . However, it is noteworthy that miR156 induction is not restricted to nitrogen deprivation, as phosphorus deprivation can also induce miR156 accumulation 50 . On the SPL9 side, we found that its expression across the Affymetrix NASC dataset is either positively or negatively correlated with key molecular actors in NO₃ ^- sensing, metabolism and development (NRT1.1, -0.68; NRT1.2, -0.55; ARF8, 0.73). These results provide additional support for a role for the mir156/SPL9 partners in the control of the nitrogen response in plants and potentially in its coordination with other signals, such as phosphorus status.

Our machine learning approach (state-space modeling) proposes a regulatory network learned from a high-resolution dynamic transcriptome analysis made in response to KNO₃ provision. A first interesting feature of this regulatory network is that it predicts a high level of connectivity (Figure 4b). Indeed, for the 76 studied genes, 60 have 500 significant connections (P-value < 0.001; see Materials and methods). This high level of connectivity (favoring functional redundancy) may explain why, to date, experimental analyses have uncovered only few molecular actors specifically involved in the control of NO₃ ^--induced gene expression (NRT1.1, CIPK8, CIPK23, NLP7, LBD37/LBD38/LBD39) 2 20 21 22 23 . Moreover, this level of potential transcription factor redundancy can be a cause of the variable and conditional NO₃ ^- genomic responses registered over different laboratories and discussed in Gutierrez et al. 14 and Krouk et al. 2 .

The identification of regulatory networks is a major aim of systems biology. Relatively few studies have determined regulatory networks precisely enough so that the model can predict behavior in untested conditions; the successful studies concern unicellular organisms such as Halobacterium salinarum 32 33 . Our regulatory network model is far more simple; it includes less than a hundred genes and is much less predictive than the one developed for Halobacterium since our model is fed only with transcriptomic data and vastly fewer experiments. Interestingly, however, both approaches predict rather low transcription factor influences. Indeed, the maximum predicted influence of one transcription factor on one gene of our model ranges between 10% and 30% (positive or negative influences). Low influence rate might reinforce the notion that functional redundancy is a built-in feature of regulatory networks that helps organisms adapt themselves to the context of interacting environmental and/or evolutionary forces. Also, the fact that Arabidopsis is a multicellular organism suggests a potential for some 'hidden' regulatory networks, as cell-specific studies have suggested 10 . This level of complexity, combined with the fact that genes in plants are not organized in functional clusters (as they are in bacteria), are likely to be some of the numerous reasons why our model is less predictive compared to that for Halobacterium. Thus, the next challenge of the plant systems approach will be to reach a level of understanding able to infer regulatory networks at several levels of integration 3 .

In order to measure the improvement of our SSM over the non-state-space methods, given our Arabidopsis dataset, we also ran our SSM with the state-space coefficient set to γ = 0 for the following methods: 1) kinetic ODE with LARS optimization, as in Bonneau et al. 32 33 ; 2) first-order vector-autoregressive model with Elastic Nets optimization, as in Shimamura et al. 35 ; and 3) 'Brownian motion' ODE, without mRNA degradation, similar in principle to the method used in Wang et al. 34 .

As Table 2 shows, the quality of fit (signal-to-noise ratio) on the training data was slightly worse (32.1 dB) in all non-SSM methods than in our SSM (32.4 dB). The predictive performance on the training data set was 90% of correct signs for all methods. The leave-out-last performance of 3) was worse (below 65% correct signs on the test set), while it was the same for 1) and 2) as in the LARS-based SSM, that is, 74% correct signs on the test set.

The non-SSM approaches that we considered in Table 2 would each give a unique solution, while our SSM would give slightly different solutions over consecutive runs, thus enabling a bootstrapping procedure. For this reason, our SSM offers a more principled way to deal with uncertainty and avoid over-fitting in microarray measurements than non-SSM methods. Second, our SSM is flexible because it enables adding unobserved variables as additional transcription factors.

Arabidopsis plants, ecotype Columbia or transgenic plants (pSPL9:rSPL9) 46 , were grown in sterile hydroponics as adapted from 10 . In detail, sterilized seeds were sown on Nitex 03-250/47 mesh (Sefar America, Bricarcliff Manor, NY, USA) supported by a plastic platform to allow roots to grow in hydroponics inside a sterile Phytatray (Sigma-Aldrich, St Louis, MO, USA). Hydroponic media consisted of 1X Murashige and Skoog basal medium containing no nitrogen or sucrose (custom-ordered, GibcoBRL, Gaithersburg, MD, USA) supplemented with 3 mM sucrose, 0.5 mM ammonium succinate, MES buffered at pH 5.7 (0.5 g.l^-1). Plants were grown for 14 to 16 days in day/night cycles (16/8 h; 150 μmol photons m^-2.s^-1; Percival Scientific Inc., Perry, IA, USA) at 22°C. Twenty-four hours before the treatments, plants were transferred toward an equivalent fresh nitrogen-free medium in continuous light to avoid gene regulation induced by light. KNO₃ was added to the media at 1 mM, or otherwise as stated in the figures. Control plants were mock-treated by adding the same concentration of KCl. The samples corresponding to time 0 were first harvested and then the treatment (KNO₃ or KCl was applied by transferring the plants onto the refreshed nitrogen-free complemented media (with KNO₃ and KCL). Then, the stopwatch was started. Roots were then harvested every 3 minutes and immediately frozen in liquid nitrogen. It takes an average of 20 to 30 seconds to harvest a sample. For growth experiments, plants were directly plated on petri dishes containing agar (1%) solidified media identical to the hydroponic one described above.

Total RNA extraction was performed using Trizol™ Reagent according to the manufacturer's recommendations (Invitrogen, San Diego, CA, USA). Total RNA (1 to 2 μg) was digested by DNase I (Sigma). RNA was then reverse transcribed to one-strand cDNA using Thermo™ script RT (Invitrogen) according to the manufacturer's protocol. Gene expression was determined by RT-QPCR (LightCycler; Roche Diagnostics, Basel, Switzerland) using gene-specific primers (Additional file 7) and LightCycler FastStart DNA Master SYBR Green (Roche Diagnostics). Expression levels of tested genes were normalized to expression levels of the ACT2/8 and clathrin genes as described in 17 .

cDNA was synthesized from 2 μg total RNA using T7- Oligo(dT) promoter primer and reagents recommended by Affymetrix (Santa Clara, CA, USA). Biotin-labeled cRNA was synthesized according to the manufacturer's protocol. Labeled cRNA (8 μg) was hybridized to Arabidopsis ATH1 Affymetrix gene chip for 16 h at 45°C. Washing, staining and scanning were performed as recommended by Affymetrix. Image analysis and normalization to a target median intensity of 150 was performed with the Affymetrix MAS v5.0 set at default values. We analyzed the reproducibility of replicates using the correlation coefficient and visual inspection of scatter plots of pairs of replicates.

The microarray data reported in this paper have been deposited in the Gene Expression Omnibus (GEO) database [GEO:GSE20044]. Data manipulations were performed in R 51 . All filtering steps were carried out by controlling the FDR. The data set, corresponding to 26 ATH1 chips times 22,810 probes, were analyzed in two successive steps in order to increase the stringency and precision of the analysis. The intent of the first step was to narrow the focus to genes regulated by nitrate over the entire data set or in interaction with time. Thus, we ran an ANOVA (aov() function) over the data set where the signal of a probe i is Pi ~ μ + αN + βT + γT*N + ε, where N is the effect of the nitrate treatment, T is the effect of time, and T*N is the effect of their interaction, μ is the mean signal over the data set, and ε the unexplained variance. We sorted out probes having a significant call (P-value < 0.01; corresponding to 10% < FDR < 15%), for the effect of N and T*N. This first pass identified 2,159 probes corresponding to nitrogen-regulated genes. On this narrowed data set (26 ATH1 chips times 2,159 probes), we modeled the gene expression using a linear modeling approach (lm()) in order to determine for each gene the particular time point that showed the most marked effect of nitrate. Thus, the expression of a gene (signal of a probe) was modeled such as Pi = α₀ + α₁T + α₂N + α₃T*N + ε, where α₀ represents the signal at 'time 0' (before the treatment). This modeling approach forces the model to take 'time 0' as a baseline for the possible effect of NO₃ ^- or KCl, allowing us to eventually discriminate between the KNO₃ and KCl effects. Thus, we called a probe regulated if it has a positive call (FDR < 5%) for interaction with nitrate at a particular time point.

The clustering shown in Figures 1 and 2 was performed through MeV software 52 . The number of clusters was determined using the figure of merit (FOM) method 53 , followed by a clustering using Pearson correlation as the distance, and the average as method of aggregation. Additional file 8 provides the KNO₃/KCl ratio along the kinetic for the entire 550 regulated genes.

Data manipulations were performed in R 51 . We set up a test to monitor whether the overlap between two gene lists is higher than expected by chance. The test consists of randomly selecting 10,000 gene lists of Arabidopsis thaliana Gene Index (AGI) numbers, out of the genome, having the same size as the observed lists. Thus, we counted the number (n) of times that the intersection size of the random lists is equal or higher than the intersection observed for the two tested gene lists. A P-value is thus generated equal to n/10,000. For the generation of the matrix in Figure 3, the randomization test was limited to 1,000 gene lists in order to speed up the analysis.

Our SSM involves noisy but observed gene expression data y(t) (black circles in Figure 4a), as well as idealized but unobserved ('latent') gene expression values z(t) (red circles in Figure 4a). As shown in Figure 4a, the relationship between consecutive latent variables z(t _k ) and z(t _{k +1}) is a Markov chain: each latent gene's expression value at time t _{k +1} is assumed to depend only on the state of potentially all the latent gene expressions at the previous time point t. For each gene i, this relationship stems from the kinetic ODE involving the rate of mRNA change (with a kinetic time constant τ), mRNA degradation, and a linear function fi of transcription factor concentrations for that specific gene. So-called 'Brownian motion' dynamics correspond to kinetic dynamics without the mRNA degradation term. In linearized (discretized) form, the overall dynamical model f can be represented by an n × m matrix F where n is the total number of genes and m the number of transcription factors (m<n, and transcription factors are given indexes from 1 to m), plus a bias term b and a Gaussian error term with zero mean and fixed covariance:

τ d z i ( t ) d t + z i ( t ) = f i ( z ( t ) ) + η i ( t ) τ t k + 1 − t k ( z i ( t k + 1 ) − z i ( t k ) ) + z i ( t k ) = ∑ j = 1 N i F i , j z j ( t k ) + b i , 0 + η i ( t k )

The observation model h is essentially an n × n identity matrix with a Gaussian error term:

y i ( t ) = h ( z i ( t ) ) + ϵ i ( t ) y i ( t k ) = z i ( t k ) + ϵ i ( t k )

An iterative procedure tries to learn the dynamical relationship between latent gene expression variables z(t) while maintaining the latent variables z(t) as close as possible to the observed Affimetrix measures y(t). The algorithm consists of a) minimizing the sum of quadratic errors of the dynamical and the observation models with respect to the latent variables Z by using gradient descent on the latent variables 25 (this is the inference step), and b) minimizing the sum of quadratic errors of the dynamical model using conjugate gradient, LARS 42 or Elastic Nets 43 optimization on the parameters of F (this is the learning step). On a sequence Y of T microarray measurements (including replicate sequences) over n genes, corresponding latent variables Z, under a dynamic model parameterized by transcription factor-gene influence matrix F and bias term b, and for a given hyperparameter γ (which controls the weight of the dynamical and observation errors), the total error term for the inference step is:

e γ ( Y , Z ; F , b ) = ∑ i = 1 n ∑ k = 1 T ( γ | | η i ( t k ) | | 2 2 + | | ϵ i ( t k ) | | 2 2 )

The solution of a non-SSM can be recovered by setting γ = 0, which amounts to having exactly Y = Z.

During the learning step, sparse gene regulation networks are obtained by penalizing dense solutions using L1-norm regularization, which amounts to adding a λ-weighted penalty to the dynamical error term, as in the LASSO initially described by Tibshirani 44 . Employing regularization on parameters of F also helps to avoid local optima in the solutions.

The learning algorithm is run for 100 consecutive epochs over all the replicate sequences (four replicate sequences here). In order to retain the optimal set of parameters of f, one selects the epoch where the dynamic error on the training dataset is minimal. The cross-validation data are unseen during the optimization procedure. One run of the learning procedure yields a matrix F of signed (positive (excitatory) or negative (inhibitory)) interactions between transcription factors and genes. Each element F_i,j represents the action of the j-th transcription factor on the i-th gene.

The algorithm described above for learning SSMs starts with random initial values for both the dynamical model (in other words, matrix F) and for the latent variables Z. We repeat the whole procedure 20 times in order to perform the following bootstrapping evaluation. Each run k of the algorithm might converge to a slightly different solution F ^*(k) . We then take the average transcription factor-gene interaction weights obtained from all solutions F ^*(k) and call it F*. Table 1 reports comparative results on the average solutions.

In parallel, we also generate 1,000 random permutations P ^*(k,1) , P ^*(k,2) , ..., P ^*(k,1000) of each matrix F*^(k), and then compute 1,000 average matrices P ^*(1) , P ^*(2) , ..., P ^*(1000) of those 'scrambled' matrices (we take the averages over the 20 runs). We compare each average element F _i,j * to the empirical distribution of the 1,000 permuted averages and thus obtain an empirical P-value. The final genetic regulation network consists of elements F _i,j * that have a P-value < 0.001.

We provide online 54 our own SSM software, which is based on publication 25 and implements the algorithm described above. The software runs under Matlab (The Mathworks Inc., Natick, MA, USA) and requires the LARS library, designed by Karl Skoglund (IMM, DTU), which is available as part of the BoLASSO Matlab package.