Systems biology has recently re-emerged (Westerhoff and Palsson, 2004), after a long quiescent period since its firsts steps (Bertalanfy, 1928), due to the acknowledgement of the components' interactions importance for an enhanced understanding of living organisms. Analysis of these interactions can be achieved through the identification of biological networks. Typically, they are characterized by their connectivity distributions (Barabasi and Oltvai, 2004) or by the enrichment in small network motifs, as compared with randomly connected networks (Milo, et al., 2002). The present work studies transcription regulatory networks, in particular for the yeast Saccharomyces cerevisiae. The most abundant information available about these networks corresponds to a topological model or wiring diagram (Schlitt and Brazma, 2005). In other words, the network consists on a directed graph, where each node corresponds to a gene and respective gene product, and a directed interaction between A and B means that A gene product is a transcription factor (TF) that regulates the expression of gene B. For S. cerevisiae these interactions have been gathered from the literature (Guelzim, et al., 2002) and detected with high throughput experimental techniques like chromatin immunoprecipitation combined with DNA chip technology (ChIPchip) (Lee, et al., 2002). Additionally, this model organism is relatively rich in microarray gene expression studies. Soon after the availability of topological models of yeast's transcriptional regulatory network, several researchers evaluated the agreement between network structure information and its dynamical behaviour. Some studies only categorized pairs of genes into two or three classes: no common TFs, one or more common TF and two or more common TFs (Allocco, et al., 2004; Yeung, et al., 2004; Yu, et al., 2003). Others looked at blocks of target genes modulated by the same set of TFs, measuring the impact of the number of TFs on block co-expression (Herrgard, et al., 2003). These studies used the same general approach to detect co-expression (applying a threshold to an expression correlation measure), but relied on stringent criteria to recognize co-regulation. In the present report the association patterns that emerge under more generic definitions are explored. With this aim, a probabilistic framework was developed to predict when two genes respond simultaneously to a random perturbation.

Most of the common expression datasets are collections of expression values, relating two gene expression states, one before and the other after a specific environmental perturbation. The correlation coefficient between expression profiles is dependent on the number of times the two genes respond simultaneously and the number of times only one of the genes responds. In this section estimates are derived for the probabilities of observing each kind of response from available network data. This is done in the absence of information about strengths and signs of each regulatory interaction.

In the following derivation, a random perturbation is applied, where a limited number of TFs e are perturbed, and around them the perturbation propagates through all the possible pathways until a maximum depth d.

The response of a given pair of genes is in one of three classes: a) both genes respond (simultaneous response, SR); b) only one of the genes responds (differential response, DR); c) none of the genes respond (neutral response, NR).

NRs should only randomly affect the pair-wise correlation coefficients of expression profiles. SRs effect will depend on interaction signs and DRs should contribute to lower correlation coefficients.

For different d values and for every pair of genes A and B, it is possible to count the transcription factors (TFs) in each of the following four classes: a) number of TFs that regulate both genes - X_d; b) number of TFs that regulate only gene A - Y_d; c) number of TFs that regulate only gene B - W_d and d) number of TFs that do not regulate any of the genes - Z_d. Every transcription factor among the N present in the network can be classified in this way. In a random perturbation, e TFs are perturbed. For a given pair of genes, x_d, y_d, w_d and z_d will represent the numbers of perturbed TFs in each of the four possible classes.

Perturbing at least one common TF (x_d>0), irrespective of whatever other TFs are perturbed, is considered to be sufficient to elicit an SR. If x_d=0, it can still be possible to observe an SR if yd and wd are both greater than zero. This means that, by chance, the two target genes are regulated after a perturbation by TFs that exclusively regulate each one of the target genes separately. If none of the perturbed TFs regulates any of the target genes (z_d=e), then the pair of genes will show a NRs. All the other cases correspond to DRs.

In this deduction two main assumptions are made: a) when more than one TF acts on a given promoter, even if some are activating and others repressing, there is always a net response from the target gene, and b) every TF has an equal probability of being perturbed. The latter is not necessarily reasonable. It could be that TFs are involved in perturbation responses more frequently because their activity is modulated by a higher number of signal transduction cascades.However, in the absence of information about the activity frequency of different signalling pathways, the least biased hypothesis is the equal-probability one.

Knowing the six parameters (e, d, X_d, Y_d, W_d and Z_d) it is possible to compute the probability of observing any of the possible responses (SR, DR or NR) after a random network perturbation:

P(SR | e,d)=P(x_d > 0) + P((x_d = 0) ^ (y_d > 0) ^ (w_d > 0)) (1)

P(DR | e,d) = P((y_d > 0) ^ (w_d = 0) ^ (x_d = 0)) + p((w_d > 0)^(y_d = 0) ^ (x_d = 0)) (2)

P(NR | e,d) = P((z_d > 0) ^ (x_d = y_d = w_d = 0)) (3)

Applying probability calculus, expressions (1), (2) and (3) can be converted to algebraic functions. The simplest case to calculate is:

P(NR | e,d)= ^Z_dC_e / ^NC_e (4)

Where ⁿC_p is number of combinations of n elements taken p at a time.

P(DR | e,d)= ^Z_d+Y_dC_e + ^Z_d+W_dC_e - 2· ^Z_dC_e / ^NC_e (5)

P(SR | e,d)= ^NC_e - ^Z_d+Y_dC_e -^Z_d+W_dC_e + ^Z_dC_e / ^NC_e (6)

For e=1:

P(NR | e =1,d) = Z_d/N (7)

P(DR | e =1,d) = Y_d+W_d/N (8)

P(SR | e =1,d) = X_d/N (9)

As only one factor is perturbed, the factors in Y_d and W_d always contribute to DR and not to SR. It is also noticeable that P(DR) only depends on the sum (Y_d+W_d) and not on the relative proportion of each one alone.

The values of X_d, Y_d, W_d and Z_d at the various depths d, are completely defined by the regulatory network architecture. Consequently, for a perturbation with depth d, the network architecture also defines P(SR), P(DR) and P(NR) values for a given pair of genes. I surveyed the relations between the probability of observing strong correlations in gene expression and the values of P(SR) and P(DR) for different values of e (1 to 4) and d (1 to 9, the diameter of the regulatory network used in this study). A change in the parameters e and d induces smooth modifications in the strong correlation probability profiles for P(SR) and P(DR). Strong positive expression correlations are more clearly associated with high P(SR) for e=3 and d=3. The intensity of the strong co-expression enrichment grows from d=1 until d=3, and decreases for higher depths. P(DR) for those parameter values is also a characteristic example of the remaining P(DR) profiles. These particular results are shown in Figure 1. The two most remarkable observations made from Figure 1 are that the greatest enrichments of positive strong expression correlations happen near the maximum values of P(SR) and at the minimum possible values of P(DR). Additionally, strong positive correlations are less frequent for higher values of P(DR). Near a P(SR) of 0.5, half of the gene pairs have a correlation coefficient higher than the top 0.5% threshold, meaning that strong positive correlations are concentrated 100 times. Strong negative expression correlations only show some significant deviation from the random null model when they are stratified by P(DR) values. Although much less intense, they have an inverted profile relatively to the positive correlations. That is, they are less frequent for low values of P(DR) and slightly enriched for some intervals of greater P(DR) values.

Combining expressions (8) and (9):

X_d / X_d + Y_d + W_d = P(SR | e =1,d) / P(SR | e =1,d) + P(DR | e =1,d) = P(SR | z_e =0, e=1,d) (10)

According to expression (10), the proportion of common direct TFs is the probability of observing a simultaneous response knowing that it is not neutral - P(SR|z_d=0,e=1,d=1). If we recall that neutral responses should not affect correlation coefficients between expression profiles, the proportion in expression (10) should be more related with co-expression than the number of common TFs alone. After observing the gene expression profile of a sufficiently high number of perturbation experiments or distinct cellular states it would be expectable that genes sharing more common TFs are more frequently co-regulated. On the other hand, if two genes are regulated by very distinct TF sets, it would be expectable to observe perturbations where only one of the genes is regulated. Measuring the proportion of common TFs does effectively account for both the common TFs and the exclusive TFs of a given target gene pair.

In fact, Notebaart and colleagues (Notebaart, et al., 2008) successfully used a transcription factor similarity (TFS) measure to explain the coupling of metabolic flux between two enzyme-coding genes. This success was not attained when graph distance between the genes in the transcription regulatory network was used instead of the TFS. They defined TFS as the total number of shared TFs between two genes divided by the total number of unique TFs regulating the two genes, which is equivalent to expression (10).

Other well-known graph-theoretic measure used to analyse biological networks is the topological overlap measure (TOM) (Ravasz, et al., 2002). Interestingly, TOM is also closely related to the perturbation response probabilities derived in our model:

TOM = P(SR | e = 1,d = 1) / min(P(AR | e = 1,d = 1),P(BR | e = 1,d = 1)) = X₁ / min((X₁ + Y₁),(X₁ + W₁)) (11)

Where, for the pair of genes A and B, P(AR) and P(BR) are the probabilities that genes A and B, respectively, respond to the perturbation. Yip and Horvath expanded TOM to a generalized form (GTOM) that presents associations with gene function that are more robust to uncertainties in network topology data (Yip and Horvath, 2007). In this probabilistic model language, the generalization corresponds to the possibility to consider perturbations with higher depths:

GTOM(d) = P(SR | e =1,d) / min(P(AR | e =1,d),P(BR | e =1,d)) = X_d / min((X_d + Y_d),(X_d + W_d)) (12)

It is readily apparent from expression (12) that GTOM can be further generalized by allowing perturbations involving more than one TF:

GTOM(e,d) = P(SR | e,d) / min(P(AR | e,d),P(BR | e,d)) (13)

These relationships and equivalences between our probabilistic approach and common graph based measures further suggests that P(SR) and P(DR) estimates can provide valuable information about the correlation between regulatory network topology and function. On the other hand, measures like TSF, TOM or GTOM are also enriched with the use of the presented response probabilities. They gain a more concrete biological meaning, which can potentiate the interpretation of the associations between their values and network functional properties. TSF may be read as the probability of observing a simultaneous response to network perturbations, knowing that the response is not neutral. TOM and GTOM are also proportional to the probability of simultaneous response, but their values are normalized by the response probability of the gene that responds less frequently to network perturbations. Thus, when TOM or GTOM are 1, it does not mean that both genes respond exactly to the same network perturbations, but that one gene is sensible to a set of perturbations that is contained in the set of perturbations that have an effect in the other gene.

A main result of this work consisted in proposing a simple rational connection between the architecture of the regulatory network topology and its functional behaviour. Using a minimal probabilistic model it was possible to justify the differences between computing the proportion versus the absolute number of common TFs. In addition to the effect of common TFs, the use of the proportion has also accounts for the TFs exclusively regulating the expression of one gene in a target pair. As a consequence, it reflects both the probability of observing a simultaneous response, P(SR) and a differential response, P(DR), after a system perturbation. The absolute count is more directly related with the P(SR) and does not include the impact of P(DR) in the experimental correlation coefficients between expression profiles.

It is also shown that integrating information about indirect factors may be important for the analysis of gene expression data. The results obtained with the P(SR) indicate that including factor information until 3 regulatory levels up of the target genes provides the most strong association between the probability of simultaneous response and the probability of observing strong co-expression. This observation does not necessarily imply that on average, the perturbations associated with the used dataset had a depth of three connections. It could alternatively mean that the past history of network activations or inhibitions could be relevant for the response to newer perturbations.

Besides the associations found between response probabilities and strong correlation of expression profiles, this approach was further validated by the relationship with other methodologies (TFS, TOM, GTOM), which on their own have previously demonstrated their utility in providing valuable insights into the functional organization of biological regulatory networks. Perturbation response probabilities enhance the interpretation of graph-based measures by attributing them a biological meaning related to the network capacity to respond to and propagate random perturbations. In fact, the use of P(SR) and P(DR) allows an extra generalization of the GTOM measure. It is plausible that the latter can have an increased performance since the most strong association between P(SR) and expression profiles occurred for perturbations of 3 TFs propagating through 3 network connection levels (e=3 and d=3).

The presented approach uses minimal information about transcriptional regulatory network. I believe this approach can be useful as it can be easily validated with most common microarray experiments. More detailed models of transcriptional regulatory network dynamics, using Boolean, Bayesian, stochastic or differential equations frameworks would be optimally validated against expression profiles with long and well-sampled time series not so commonly produced. As the network information grows and becomes more complete, the probabilistic model shown here can provide better predictions. The simplicity of the input information and model assumptions may allow the analysis of integrated networks including simultaneously several mechanisms of gene transcription or protein activity regulation.

I finally propose that coarse level analysis like the one presented here can provide a useful bridge between large gene expression datasets and more fine-detailed models of biological network dynamics.