We grew and irradiated a human leukemia cell line (MOLT4) containing functional p53 and harvested protein and RNA at regular intervals after irradiation. The time course was performed in triplicate, and Affymetrix U133A microarrays (Affymetrix Inc., Santa Clara, CA, USA) were run to measure the global transcriptional response. Before irradiation, we assumed the p53 network to be in equilibrium (that is, that the rate of change in its constituents is zero). Irradiating the cells disrupts the equilibrium and activates transcription of numerous p53 target genes. The rate at which p53-dependent mRNA transcripts accumulate depends on the basal transcription rate of a target gene, the sensitivity of the gene to p53, the level of activity of p53, and the transcript degradation rate. We can connect these factors to represent the overall behavior of the system. The time evolution of each gene transcript is described by the following non-autonomous linear differential equation for the rate of change in transcript concentration x_j(t) of gene j at time t:

Where B_j is the constant basal transcription rate of j; S_jf(t) is the transcription induced by p53, composed of a constant S_j, which is the sensitivity of gene j to p53, and f(t), which is the activity of p53 at time t; and D_jx_j(t) is a degradation term, with D_j being a constant degradation rate. For a full description of the model, see Mathematical methodology (below).

In order to predict whether a gene is likely to be a p53 target, it is necessary to estimate its sensitivity (S_j) to p53 and to ensure that parameter values can be found that, when combined in the model equation, result in an expression profile similar to the experimentally determined profile. However, the p53 activity f(t) is not experimentally available and is the key 'hidden variable' in the system. To estimate this profile we used prior biologic knowledge rather than adopting a 'black box' approach. We selected a small training set of five known p53 targets (DDB2, p21^WAF1/CIP1, SESN1/hPA26, BIK, and TNFRSF10b/TRAILreceptor 2) 171819202122 and used the microarray time series observations for this set to derive the p53 activity profile f(t), and the parameter values of basal transcription rate, sensitivity to p53, and degradation rate. These values and their confidence intervals were obtained by applying Markov Chain Monte Carlo (MCMC) with a Metropolis-Gibbs sampler 23 (see Mathematical methodology, below). Normally, the calculations involved in these estimations are very demanding on computer time. In terms of systems biology, in which many such calculations are likely to be linked, this poses a major barrier to network analysis. We therefore discretized the model equation and devised a fast matrix-based algorithm to solve it efficiently (see Mathematical methodology, below).

Initial estimates of the parameters and the hidden profile f(t) exhibited a very high degree of variance. Repeated modeling of artificial data indicated that this was a general characteristic of the model and not peculiar to the particular experimental data set. We noticed that the estimates were highly correlated with each other (Figure 1a). This suggested that experimentally determining the value of one additional parameter might constrain the others and so reduce the overall variance. We therefore measured the rate of degradation of one transcript (p21^WAF1/CIP1) using quantitative polymerase chain reaction (PCR; Figure 1b). We found that this single measurement was sufficient to reduce dramatically the variance and greatly improve the final estimates (Figure 1c). We term this process 'data anchoring'. We found that obtaining the degradation rate of any element in the training set was equally sufficient to anchor the model, provided that the same gene was also used as the reference point for estimating sensitivity (see Mathematical methodology, below). The inclusion of the degradation rates of more genes did not significantly improve parameter estimation.

Incorporation of the degradation data allowed efficient estimation of the parameters B_j, S_j and D_j, and the p53 activity profile f(t) for the training set of known targets (Figure 2). This process was performed simultaneously on three replicate time series to improve the robustness of the outcome (Figure 2b). We found that the model-estimated profile approximated the experimentally determined activity profile based on measuring p53 phosphorylation at serine 15 24 (Figure 3). The profiles show a close match early in the response, but the model predicts a more rapid decline in activity. This discrepancy can be explained by the operation of other regulatory mechanisms that affect p53 activity but not concentration, for example relocation of phosphorylated p53 to the cytoplasm 25.

The use of a training set of known targets takes advantage of the fact that prior biologic knowledge exists for many TFs. Because the p53 response is well studied, we were able to examine the optimum model requirements. We found that three training genes are sufficient for the model to make accurate parameter estimates (Figure 4a). The inclusion of more (up to ten) genes narrowed the confidence intervals but the improvement was small beyond five genes. We also found that inclusion of genes not regulated by p53 (for example TNFSF10) led to a poor gene-specific model score, enabling these genes to be excluded from the training set. We found the method to be very robust, and the exact choice of target genes does not appear to affect estimation greatly, providing that the measurement error is not excessive (namely, the detection P value should be below 0.001 for Affymetrix data) and that the anchoring gene is clearly differentially regulated (Figure 4b; also see Mathematical methodology, below).

Once we had constructed the estimate for the key 'hidden variable', namely the p53 activity profile f(t), we were able to apply the model to the remaining expression data to predict p53 targets. Data was filtered to identify upregulated and detected genes (754 in total). These were then tested to determine how well they fitted the model of activation by p53. We derived a score M (> 0) based on the closeness of experimental data to model predictions (in which lower scores are better). Because nonchanging genes with a flat profile would also fit the model, another score was computed that captures the predicted sensitivity to p53. This sensitivity score is a measure of how significantly S_j differs from zero, represented by a Z score. Z scores are the distance between the observed value and the population mean in units of standard deviation, and are therefore a measure of estimation robustness. Z scores are inversely related to P values (see Materials and methods, below).

We ranked the model scores, first in terms of model fit and then on predicted sensitivity to p53. Three broad classes of upregulated genes could be discerned, the composition of which depending on the stringency of the M score and sensitivity Z score threshold applied. At thresholds of M < 100 and sensitivity Z > 2 (and degradation estimates limited to 0.1/hour < D_j < 5/hour), class 1 consisted of 237 genes that fitted the model well and exhibited high probability of p53 sensitivity, exemplified by LRMP and p53TG1 (Figure 5a,b). Class 1 genes were therefore most likely to include genes regulated by p53, with the probability of sensitivity being the key indicator. As expected, the five known targets composing the training set were found among the 20 highest scoring genes (ranked by decreasing sensitivity Z score), alongside other established p53 targets and genes not previously known to be p53 regulated (Table 1).

Under the same thresholds, in a second class of 105 genes a relatively high sensitivity score was achieved despite a poor model fit, as in the case of TNFSF10 (TRAIL) and IER3 (Figure 5c,d). The model attempts to accommodate genes strongly regulated by factors other than p53 by varying degradation and sensitivity scores, which often results in apparently high sensitivity predictions. However, the poor overall model fit suggests that class 2 genes are either completely independent of p53 or exhibit more complex co-regulation.

Genes in class 3 have either poor sensitivity or poor model score (M > 100, sensitivity Z < 2), or both. The majority of the 412 genes in this group are likely to be regulated independently from p53 in a manner that exhibits no similarity to the p53 activity profile. However, class 3 will also include genes that are p53 dependent but that are not distinguishable by the model.

To validate the predictions made by the model, we transfected MOLT4 cells with small interfering (si)RNA to p53 to deplete p53 protein to below control levels (Figure 6a) 26. siRNAp53 substantially reduced ionizing irradiation-induced increases in the transcripts of three p53 target genes, namely HDM2, P21, and GADD45α (Figure 6b). We then ran microarrays to measure the effect of siRNAp53 on the transcriptional response to irradiation at the whole genome level. Validation was carried out at 4 hours to maximize the number of p53 targets and to minimize the inclusion of secondary targets. Data were filtered to identify those genes that were upregulated in both the time course and in the pSuper transfected control at 4 hours (see Materials and methods, below). This identified a total of 162 genes that were upregulated significantly by irradiation at 4 hours.

To quantify sensitivity to siRNAp53 at the individual gene level, we computed new Z scores that measured the difference between genes upregulated by irradiation in control cells and those upregulated in siRNAp53 treated cells. For clarity, these are referred to as validation scores. The higher the validation score, the more effectively siRNAp53 eliminates change in transcript concentration, and so the more likely the gene is to be dependent on p53. Seventy-four of the 162 4-hour-upregulated genes were predicted by the model to be p53 targets because they fell into class 1 (M < 100 and sensitivity Z score > 2). Of these 74, 66 (90%) exhibited high (Z > 1) validation scores (namely sensitivity to siRNAp53), confirming that they are p53 targets (Figure 7a). This figure rises to 73 out of 74 (98%) if a lower sensitivity Z score threshold (> 0.5) is applied or falls to 39 out of 74 (53%) if the sensitivity Z score threshold is set at 3. Higher sensitivity Z score thresholds therefore result in greater accuracy but at the expense of identifying a lower proportion of the targets (Figure 7b). Sensitivity Z score correlated well with validation score, indicating that predicted rank of p53 targets reflected the strength of p53 regulation (Figure 7c).

Thirty upregulated (4 hours) genes fell into class 2 (M > 100 and sensitivity Z score > 2). As expected, the response of class 2 genes to siRNAp53 was divided. Fourteen genes, including TNFSF10 (TRAIL), remained unaffected by siRNAp53, showing them to be p53 independent/irradiation dependent. Sixteen class 2 genes were affected to some degree by the treatment, confirming predictions that this group included co-activated or co-repressed genes such as IER3, which is known to be synergistically regulated by nuclear factor-κB and p53 27. The remaining 58 upregulated (4 hours) genes fell into class 3, 34 of which were affected by siRNAp53.

Overall the Z score for S_j (sensitivity to p53) was a good discriminator for identifying p53 targets. The model was able to predict with confidence, and at high accuracy, 66 out of 115 (57%) genes verified as p53 targets at 4 hours, based on a sensitivity Z score threshold of 2. A further 16 class 2 genes exhibited evidence of co-regulation, suggesting an explanation for 71% of the interpretable data. Many of the remaining class 3 targets were expressed at low levels, or exhibited low (> 1.5-fold) levels of differential expression. This raises questions about their biologic significance, and suggests that the true success rate of hidden variable dynamic modeling (HVDM) is actually higher than reported above. A larger number of replicates would be required to be confident of the status of class 3 genes.

As seen for the validation data set, tightening thresholds (by choosing a higher sensitivity Z score) results in more confidence that the targets are regulated by p53 but at the cost of explaining a lower percentage of the data (Figure 7). When applied to the entire upregulated data set, HVDM can accurately predict a large number of p53 targets from a short time course without any further experimental input (Figure 8). These predictions included a number of genes not previously known to be p53 targets, including CD38, DENN-domain protein FLJ22457, CROT, GLS2, HERC5, ASCC3, LRMP, and CIKS/ACT1/TRAF3IP2 (Table 1). siRNA validation at an early time point (4 hours) indicates that these genes are most likely to be direct targets. CD38 is best known as a prognostic marker in the leukemia B cell lymphocytic leukemia (B-CLL associated) with poor outcome. It functions as a powerful regulator of calcium dependent signaling via the generation of cyclic ADP ribose and NAADP+ (nicotinic acid adenine dinucleotide phosphate) Its regulation by p53 suggests a possible role for calcium-dependent signaling in the DNA damage response.

Since both HVDM and clustering approaches aim to identify TF targets, we compared our results with a typical clustering approach, namely K means clustering. From the 754 genes identified as upregulated by irradiation, HVDM generated a ranked list of predicted p53 targets based on model score and best sensitivity Z scores (Table 1). Forty-eight of the 50 highest ranked targets (96%) predicted by HVDM were confirmed by siRNA to be p53 targets. These 50 HVDM predicted target genes were divided by K means clustering between six of eight clusters, each with a distinct response profile (Figure 9). For example, the HVDM predicted target TP53TG1 has a late expression profile (cluster 1, Figure 9), along with seven other top 50 targets. This profile is quite different from the activation profile of p53 or its 'typical' correlated targets (Figure 5b). Only two genes were probably false positives.

K means clustering of the 754 detected and upregulated genes based on expression levels alone grouped the genes into eight clusters based on transcript time profile (Figure 9). Visual examination of the profiles suggested that one of these classes (cluster 7, Figure 9) was most similar to the p53 activity profile determined by Western blot (Figure 5b), and indeed this cluster contained many of the well known p53 targets (including GADD45α, p21, and DDB2). However, because clustering approaches typically do not provide confidence intervals, it is impossible to identify which genes within the cluster are most or least likely to be real targets. We found that 25 out of 79 genes in cluster 7 were verified as p53 targets in the siRNA experiment (32%; data not shown). Verified genes also occurred in cluster 1 (11 out of 135 (8%)), cluster 3 (35 out of 102 (34%)), cluster 4 (21 out of 120 (17.5%)), cluster 5 (3 out of 90 (3.3%)), and cluster 6 (20 out of 51 (39%)).

In summary, HVDM can generate an accurate list of p53 targets with different expression profiles, ranked by probability of sensitivity to p53. In contrast, although K means clustering generates clusters enriched or depleted in p53 targets, it fails to identify targets with different profiles, is unable to quantify the level of sensitivity of a gene to p53, and cannot distinguish between true and false p53 targets. We also assessed the performance of self-organizing maps (SOM) clustering, with a similar outcome. This is expected, given that all processes that cluster on expression profile alone are bound to suffer similar deficiencies. Predictions made by HVDM are therefore accurate, explain a significant proportion of true targets, give indications about potential for co-regulation, and provide an excellent basis for prioritization of downstream bioinformatics and experimental analysis.