The first dataset is the one from Figure one in Lapointe et al. [44]. This data is available from http://microarray-pubs.stanford.edu/prostateCA/images/fig1data.txt and supplemen-tary material is also available from http://microarray-pubs.stanford.edu/prostateCA/.

In the original study, the authors used a cDNA microarray technology that allowed them to measure gene expression of several thousand genes on 112 samples, including 41 normal prostate specimens, 62 primary prostate tumours and 9 lymph node metastases. From that set, a subset of 5,153 probes were selected as differentiating prostate cancer samples from normal and metastases (this is the set from figure one in Lapointe et al. [44] and available at the web address given above). After imputation of missing values, we first calculated the Normalized Shannon Entropy and the MPR-Statistical Complexity for the each sample.

The flowing section explains the context in which our results were generated (refer to the ‘Materials and Methods’ section for detail on how our quantities are computed). The Normalized Shannon Entropy measure is widely used in ecosystem modelling to quantify species diversity, where it is acknowledge as having great sensitivity to relative abundances of species in an ecosystem [45]. We utilise the same sensitivity to differentiate a samples in cancer datasets. Figure 1 shows that the Normalized Shannon Entropy of prostate cancer tumor samples do not differ much from normal samples. This is in contrast to lymph node metastasis samples that appear to have smaller values of Normalized Shannon Entropy.

A mathematical interpretation of this result is that the samples from lymph node metastases have cells that not only varied their transcriptomic profile, they have also “peaked” the distribution of expression values with significant fold increases on a smaller number of probes. This explains the reduction in Normalized Shannon Entropy. We note that there are several mechanisms that can explain a macroscopically observable global reduction of transcription. For instance, this may indicate that a relatively large number of genes have reduced their expression levels by genome damage, changes in gene regulation, or other silencing processes. It is reassuring to observe that the changes of the most prototypical quantitative measure we can draw from Information Theory, the Normalized Shannon Entropy correlate well with the transition between normal samples with to ones with metastases. However, it is also evident from that normal samples do not differentiate much from the tumor group (the Normalized Shannon Entropy values do not differ much). It is then not the number of genes with high expression values, but the change in the distribution of expression levels on the molecular profile, that can provide the other measure that could distinguish these other samples. This must be handled by the other statistical complexity measures to be discussed next.

Several statistical complexity measures can be defined which aim to clarify our argument. We will first discuss the results of computing the MPR-Statistical Complexity measure (in the previous figure the y-coordinates correspond to the MPR-Statistical Complexity values of each sample). The MPR-Statistical Complexity is proportional to both the Normalized Shannon Entropy associated to the transcription profile and the Jensen-Shannon's divergence between that probability density function and the uniform probability distribution. Again, we refer the reader to the ‘Materials and Methods’ section for an explanation of how these magnitudes are computed.

Although the results of using the MPR-Statistical Complexity might not seem particularly impressive, there are a few reasons why we introduce them at this stage. We want to illustrate a fact that can already be observed when we employ this measure on this dataset. In this dataset, for a given entropy value interval, normal tissue samples tend to have relatively lower MPR-Statistical Complexity values than tumor and lymph node metastasis. This means that both prostate cancer and metastases samples diverge from a “more uniform” distribution indicating that the distribution “peaks” in fewer active genes. It also means that, in terms of Jensen-Shannon's divergence, the transcriptional profile of a normal prostate cell sample is “closer” to a uniform distribution than to the one that is observed in a prostate cancer cell sample.

The reader will readily argue, and with reason, that the transcriptional profile of a normal cell is tissue-specific and that it hardly resembles that of a uniform distribution of expression values. That is correct and this observation motivates the introduction of two new statistical complexity measures. We generically call these two variants as ‘M-complexities’ (with ‘M’ standing for “modified”). They have the same functional form as the MPR-Statistical Complexity, but instead of computing the Jensen-Shannon's divergence from a uniform probability distribution we compute it against an ad hoc probability distribution functions derived from the data. In this sense, these measures are more supervised then the MPR-Statistical Complexity is. Another perspective is that the MPR-Statistical Complexity is a special case of this measure in which the ad hoc probability distribution function of reference is the equiprobability distribution. The relevance of this measure derives from being a general definition that allows accommodating several different reference states. We will use it to measure divergences to the “initial” and “final” transcriptomic states (two states of reference). Taken as computed averages over normal samples, and respectively metastatic ones, these measures will allow tracking the processes of differentiation of a cancer cell from a particular tissue type.

For example, using Lapointe et al.'s dataset, the M-Normal statistical complexity quantifier first requires the computation of the probability distribution function of the average gene expression profile of all normal prostate samples. Afterwards, the Normalized Shannon Entropy and the Jensen-Shannon's divergence of any sample profile will be computed using the divergence to that averaged normal distribution. Analogously, we compute the M-Metastases statistical complexity quantifier by first calculating the average profile of the metastases samples, and then generating the corresponding probability distribution function, finally computing the Jensen-Shannon's divergence with that profile. We refer to the ‘Materials and Methods’ section for details of the calculations.

The results can be observed in Figure 2. On the x-axis, the lymph node metastases have the largest values of M-Normal indicating a divergence from the normal profile. In addition, the M-metastases values of normal samples tend to be higher than most of the metastasis samples (with the exception of only one).

Figure 2 shows a gradual progression of the samples positions on this plane from a well-differentiated tissue type specific profile, first to a more heterogeneous primary tumor cluster, and finally to an even less differentiated metastatic profile.

The result presented in Figure 2 shows that the prostate cancer samples, which are not metastases and therefore could have been scattered anywhere on the plane, are clustered on a particular confined area between the two other groups. We understand that there are reasons to be sceptical about this result being not just a simple consequence of the gene selection process used by Lapointe et al. For example, if we assume that the 5,153 probes singled out by Lapointe et al. in their figure one of Ref. [44] (and that constitute our original data) have been selected with a supervised method that try to distinguish between normal and metastases, then the relative position of normal and metastases samples is perhaps something to be expected. However, even under that assumption, what is not expected is the position of all primary tumor prostate cancer samples, linking the normal cluster of samples with the metastases one. Note that the definition of both the M-Normal and M-Metastases measures do not use any information from the primary tumor prostate cancer samples, so the location of these samples between the normal cluster and the metastases, bridging them naturally is something to highlight. Together with Figure 1, it gives evidence that supports the working hypothesis that a gradual “progression” occurs, from the normal tissue specific profile to the metastasis one.

Indeed, following our line of argument, Figure 2 has even more relevance when we highlight the fact that the 5,153 probes have not been selected with a supervised method. The authors say that the only selection criteria was to single out the 5,153 cDNAs whose expression varied most across samples. In the supplementary notes of their paper the authors say: “We included for subsequent analysis only well measured genes whose expression varied, as determined by (1) signal intensity over background >1.5-fold in both test and reference channels in at least 75% of samples, and (2) 3-fold ratio variation from the mean in at least two samples; 5,153 genes met these criteria.” As a consequence, Figure 2 has been generated without class selection bias only using the genes that have the most varied expression pattern.

We now turn to another aspect of the statistical complexity and entropy analysis. We note that Figure 2 shows that the metastases samples have a clear reduction on Normalized Shannon Entropy in comparison with the values observed for the normal samples. At the same time, metastases samples, as expected, have higher M-normal complexity than the normal samples (Figure 2). It is then interesting to evaluate the value of the Jensen-Shannon divergence of these samples and to identify the genes that most correlate with the variations of Jensen-Shannon divergence to quantify one of the factors that is related to the statistical complexity changes.

We have computed the correlation of the gene expression profile corresponding to each of the 5,123 probes. For each of the 5,123 probes, we computed both the Pearson correlation (x-axis of Figure 3) and the Spearman correlation (y-axis of Figure 3) of each probe profile with the Jensen-Shannon divergence having as probability distribution of reference that of a metastasis profile (these values are called JSM2-Pearson and JSM2-Spearman in the accompanying Excel file provided). With this data, we have produced Figure 3, a scatter plot of the values associated to each probe. In this figure, there are two probes that are immediately recognizable by any cancer researcher, and in particular for those in prostate cancer: KLK3/PSA (Prostate Specific Antigen) and FOS.

The interpretation of these scatter plots is not immediate and needs an introductory explanation. Each dot corresponds to one probe of the array. For example, a dot that is very close to the origin of coordinates (0,0) indicates a probe such that its pattern of gene expression (across all samples) is not correlated with the Jensen-Shannon divergence to the average profile of a metastasis pattern. It is, in essence, a probe which is highly uninteresting in this regard. Probes that have a high correlation, across all samples, either positive or negative with the Jensen-Shannon divergence to the average profile of a metastasis pattern are highly informative. They “co-express” with this measure.

Although we provide in the supplementary material the information corresponding to all probes, we will discuss just a few of them. This will allow the reader to understand these plots and will put our results in the perspective with current research in prostate cancer. We particularly highlight the position of KLK3/PSA, FOS and CCL2. To our surprise, we have found which is perhaps the most famous biomarker in prostate cancer KLK3/PSA (Kallikrein-related peptidase 3), probe G_914588 (correlations of −0.9312 and −0.9000 respectively). FOS and KLK3/PSA are the second and the fourth most negatively correlated probes in this ranking of all the genes in the microarray. With opposite signs for correlations are CDKN2D, FOXM1, and BRCA2. The following is a discussion of a selection of probes (highlighted in Figure 3) in the context of prostate cancer.

The following sections present the results that we obtained with a melanoma dataset. Our aim is to observe if variations of the Normalized Shannon Entropy and the statistical complexity measures, MPR-complexity and the modified forms M-normal and M-metastases, provide interesting results in a different disease and experimental setting.

In this case we have selected a gene expression dataset from Haqq et al. [110] containing information of 14,772 cDNAs in 37 samples (Figure two from the [110]). The 37 samples include 3 normal skin, 9 nevi, 6 primary melanoma and 19 melanoma metastases. This datasets has more phenotypical characteristics for the group of samples.

After an initial process of data cleaning, we removed 35 probes which had an unsually high expression value on only a few samples, in some cases on a single one. The dataset we work with from original contributed by Haqq et al.consists of 14,737 probes. First, we computed the Normalized Shannon Entropy and the MPR-Statistical Complexity for each sample (refer to the ‘Materials and Methods’ section for a detailed presentation of these calculations). Figure 4 shows the values of these quantifiers for each sample.

We first observe an important difference between Figure 1 and Figure 4. In this melanoma dataset, neither the use of the Normalized Shannon Entropy nor the MPR-complexity helps to discriminate between normal skin, nevi, primary and metastastic melanomas. Nevertheless, we decided to present this figure for methodological reasons. We envision that some researchers will calculate the Normalized Shannon Entropy and MPR-complexity using all the probes. We note that in Figure one of Haqq et al's original paper, the whole probe set was previously filtered by selecting those which vary across samples, thus indicating that they may have information about disease subtypes (although the phenotypic types were not biasing the selection). In this case we want to illustrate both the Normalized Shannon Entropy and MPR-complexity calculated using all the probes does not give the expected benefits. We will now see the benefits of using the M-complexities.

As we did for prostate cancer (see Figure 2), we aim at identifying if the use of the modified forms of the statistical complexity (the M-complexities) could give some insight where the Normalized Shannon Entropy and MPR-complexity measures fail. To compute the M-normal measure, we need to define the average gene expression profile for a normal cell (which we call P_ave). We thus resort to the three normal skin profiles and we produce the average based on these profiles (details for computing the average profiles are given in the ‘Materials and Methods’ section). We call M-skin the resulting measure that relies on this profile. Analogously, we need to compute a pattern for M-metastasis, and we proceed to calculate the P_ave profile averaging over the 19 metastases samples. The result is encouraging, as samples plotted in the (M-skin, M-metastasis)-plane cluster in groups, showing an important M-skin complexity transition between normal skin cells and nevi. Most importantly, this method naturally shows that some of the metastatic samples have a large value of M-skin complexity, so we present the results of another experiment, aimed at clarifying this fact.

In their original publication, Haqq et al. classified the melanoma metastases in two groups due to their molecular profiles: five samples were classified as ‘Type I’ and fourteen as ‘Type 2’ based on a hierarchical clustering approach. Our result reinforced the view that the Type II melanomas metastasis is a pretty homogeneous group, we will present the results on the (M-skin, M-metastasis I)-plane. This means that now the P_ave profile will not be obtained by averaging over the 19 metastases samples, but instead using only the 14 samples which have been labelled as ‘Type II’. As such, we aim at revealing if Type I samples are indeed different in this plane, and if other clusters are also present.

Figure 5 presents the results. The first fact worth commenting is the pronounced gap between normal skin samples and the nevi, primary, and metastatic melanoma samples as revealed by the M-skin measure. Note also that the M-skin is based on the average profile that of the normal samples, which indicates that no information about the profiles of metastasis are used, yet M-skin reveals that increasing values of this measure may be linked with a ‘progression’ from nevi to primary and metastasis melanoma profiles.

We now introduce another useful technique to identify genes which correlate with the transitions. The challenge is to find genes which are related with the progression towards metastases profiles, even when we recognize that there the group of metastasis samples is heterogeneous (containing at least two groups). Since the final outcome of Figure 4 and Figure 5 is that the Normalized Shannon Entropy does not help much in this experimental scenario, we will concentrate only on one of the multiplicative factors of the M-complexities, the Jensen-Shannon divergence. We compute two P_ave profiles, one with the normal skin samples only, and the other with all the metastasis samples (regardless their type). We will call the two divergences JSM0 and JSM5 respectively. We then compute the Spearman correlation of the profile of all gene probes in the array across the 37 samples to both JSM0 and JSM5. We have listed all probes according to the absolute value of the difference of these correlations, i.e. Abs. Diff. (probe) = |JSM0(probe)−JSM5(probe)| in decreasing order. The results are provided as Haqq-PLoSONE-SupFile.xls, in the sheet labelled ‘Results-correlation’.

The rationale is to identify those probes which are highly correlated (both positively or negatively) with the Jensen-Shannon divergence of the normal tissue profile and that “reverse signs”. For instance, a probe for the TP63 gene (Tumor protein p63, keratinocyte transcription factor KET), AA455929, is ranked in the third position. Its correlation with the Jensen-Shannon divergence of the normal skin type is relatively high and negative (JSM0 = −0.63632) while at the same time is has a positive correlation with the Jensen-Shannon divergence of the metastasis profile (JSM5 = 0.62138). In the ranking, the first probe that presents the opposite behaviour is one for ADA (Adenosine deaminase), AA683578. Figure 6 helps to understand the relationship of these correlations with expression. Not only are these genes well correlated with the divergences, they also seem to be good markers of the progression from one tissue type profile to the metastasis profile.

We will now discuss three of these genes in the context of current biological knowledge on melanoma drivers and metastatic progression. We provide many references for one of them, SPP1 (Secreted phosphoprotein 1 or Osteopontin). The discussion on this gene will be left for later, when we will discuss specifc oncosystems related to cell proliferation, chemotaxis and responses to external simulus. Figure 7 shows the expression of ADA (Adenosine deaminase, AA683578) as a function of TP63 (keratinocyte transcription factor KET, AA455929). All normal skin samples, as well as nevi and a couple of primary melanomas have relatively low values of ADA but they express TP63. There is a change of roles in metastatic and some primary melanomas, which have reduced TP63 expression but increased values of expression of ADA. As we will later see, these events correlate with other major transcriptional modifications which involve dozens of genes and that we have been able to map thanks to functional genomics bioinformatics tools. The role of SPP1 will be discussed in that context after some references to TP63, ADA, and PLK1 which follow.

Another microarray dataset we have selected to evaluate for the relevance of transitions of Normalized Shannon Entropy and Statistical Complexity was contributed by True et al. [332] in 2006.

The original goal of True et al. was to identify a molecular correlate for Gleason patterns 3 and, if possible, the clinically most worrisome patterns 4 and 5. They partially succeeded by linking the expression of only 86 genes with Gleason pattern 3 [332] using a standard statistical analysis. In this study, we eliminated sample 02-209C since data was acquired using a different platform and would not be useful for our analysis. The remaining thirty one (31) samples were assayed with the GPL3834 (FHCRC Human Prostate PEDB cDNA Array v4) platform using 15,488 probes. We also eliminated all the probes with missing values, remaining 13,188 probes.

We have first plotted the samples on the (Normalized Shannon Entropy, MPR-Statistical Complexity) plane (Figure 12). It was interesting to observe that there exists a high correlation between the two measures. Samples that are entirely composed of Gleason pattern 3 tend to have a greater value of Normalized Shannon Entropy than 0.985. We can also identify a cluster of samples that present Gleason patterns which are either 4 or 5. Note that there seems to be two outliers (02_003E and 03_063) to the generic trend of the other 29 samples. The two outliers are samples that correspond to samples labelled as having Gleason 3 patterns and both have unusually low values of Normalized Shannon Entropy that are well below the values of the rest of the group.

This raised a suspicion about the true nature of this phenomenon. If the labelling is correct, this may indicate a subsampled group of prostate cancer that has Gleason 3 pattern characteristics but very low entropy. Alternatively, it may indicate an experimental bias for reasons we can not explain with the available clinical information. In order to clarify the situation, and see if we can declare these two samples as outliers of the other group, we performed another experiment. We have now computed two modified complexities, which we will call M-Gleason 3 and M-Gleason 5 (Figure 13). The names are probably self-explanatory, but a brief reminder follows. To calculate the MPR-Complexity, by definition, we have used the equiprobable distribution as our probability distribution of reference (for the computation of the Jensen-Shannon Divergence of the gene expression profile to this distribution). In the case of the M-Gleason 3, the probability distribution of the reference is obtained averaging all the probability distributions of the samples that have been labelled as Gleason 3 (analogously, we calculated M-Gleason 5). Samples that have Gleason pattern 3 and 5 appear as separate clusters in the (M-Gleason 3, M-Gleason 5) plane with the two putative outliers of the general trend far apart (even if they have been used to calculate the average probability distribution function of the Gleason 3 pattern). Even samples with Gleason 4 pattern are located closer to samples of Gleason patterns 3, and 5, indicating that, perhaps, there exists a subsampled subtype of prostate cancer or there might be another experimental bias or factor that at present we can not resolve with the information we have for these samples. Consequently, we have decided to eliminate both samples (02-003E and PNA_03-063A) from further calculations. With these considerations, we now have a dataset with 13,188 probes and 29 samples as our dataset for further analysis.

Figure 14 shows the distribution of the samples using the Normalized Shannon Entropy and the MPR-complexity. By definition, the positions of the 29 samples in the plane do not change (this figure is basically “zooming in” one region of Figure 12 that contains these samples). We note again, however, that the 29 samples seem to be separating in three different clusters. Whether we can argue about the existence or not of these gaps in Normalized Shannon Entropy, it is clear that there seems to be a progression as we have seen with Lapointe et al's dataset. There is a group of three samples with Gleason pattern 3 that seem to have the the largest Normalized Shannon Entropy values. There is also a cluster that only contains samples of either Gleason pattern 4 and 5, all with Normalized Shannon Entropy values smaller than 0.985.

There is also very little variation (see Figure 15) of the positions of the 29 samples on the (M-Gleason 3, M-Gleason 5)-plane, indicating a degree of robustness that the computation of these modified complexities have, even in the presence of some outliers.

After observing that Figure 14 shows a correlation of Gleason pattern score with Normalized Shannon Entropy, we asked ourselves: ‘which are the genes that most positively and negatively correlate with the transitions of Normalized Shannon Entropy?’ We have plotted Spearman versus Pearson correlation values of probe expressions to attempt to find those that best correlate, either positively or negatively, with the Normalized Shannon Entropy values of the samples. The results have revealed some of the most relevant biomarkers of progression, and some unexpected newcomers. Figure 16 shows the Pearson and Spearman correlations of all the 13,188 probes in the dataset with the Normalized Shannon Entropy values of the samples. We have highlighted some particular genes that are discussed below.

Another natural question can be asked: Which is the extra information that we can obtain the by analysing the correlations with the MPR-Statistical Complexity in this case? As we have discussed before, and can be appreciated from Figure 14, there is a strong correlation between the MPR-Statistical Complexity and the value of the Normalized Shannon Entropy. It appears in prostate cancer, as in this gene expression dataset, the reduction of Entropy is not the major factor responsible for the increase in MPR-Statistical Complexity. Again, it is perhaps better to now look at one of the multiplicative factors of the statistical complexity measure, the Jensen-Shannon divergence to the equiprobability distribution, as this is increasing the MPR-complexity.

From our results, we can not completely establish if the downregulation of CD40 and CD59 are enough to pinpoint an impaired or abnormal immune response. If we continue the inspection of the list, the first 20 probes give us more supporting evidence. The 20 probes correspond to 13 different genes. Five of these 13 genes have Genome Ontology information annotated as “defense response”, the above mentioned CD59 and CD40 as well as IL4R (interleukin 4 receptor, CR616481), XBP1 (X-box binding protein 1, AK093842) and HLA-A (major histocompatibility complex class I HLA-A29.1, BU075230). Takahashi et al. [544] report an inverse correlation between XBP1 expression and histological differentiation in a series of prostate cancers without hormonal therapy, the expression of XBP1 was localized in epithelial and adenocarcinoma cells of the prostate and the majority of refractory cancer cases exhibited weak XBP1 expression), MST1/STK4 (along with MST2/STK3) act as inhibitors of endogenous AKT1, a mediator of cell growth and survival [545].

We can not yet know what reason is behind their joint downregulation, but another interesting common denominator is that 12 out of 13 genes share a regulatory motif for NF-kappaB (according to TRANSFAC, V$NFKB_Q6_01). A putative role for NF-kappaB in prostate cancer has been reported based on the observation of the centrality of NFKB on two up- and down-regulated networks compairing prostate tumors and healthy tissue [546] and in a larger study by McDonnel et al. [547] (255 core prostate cancer tissue microarrays from 47 prostatectomy specimens). Several other researchers are currently investigating different roles of the NFKB family in prostate cancer [548], [549], [550], [551], [552], [553], [554], [555] and it could be a promising target for intervention [555], [556], [557], [558], [559], [560], [561], [562], [563], [564], [565], [566], [567], [568], [569], [570], [571]. If we include other genes following the ranking order, the first 38 genes in the ranking include 33 that have the regulatory motif V$NFKB_Q6_01 (GATHER reports for this list a p-value of 0.0006). Even when we double the list to the probes that correspond to the first 76 different genes recognized by GATHER, 58 of them have the regulatory motif V$NFKB_Q6_01, with p-value = 0.003 (ATP6AP2, BCAT1, BTG2 [572], [573], [574], [575], [576], [577], [578], C14orf123, C18orf45, CCL2, CD302, CD40 (already discussed), CD59 (already discussed), CHI3L1, COL16A1, COMMD6, CRABP2, CSRP1, CTBP2, CTGF (Connective tissue growth factor, [579], [580], [581], [582]), DES, DMN, DNAJB1, EGF, EMP1, FHL2 [583], [584], [585], [586], [587], [588], GRIPAP1, GSTM1 [589], [590], HBEGF, IL4R, ITGA3, ITGA7, JUNB [591], [592], KIAA0152, KIAA1191, KIAA1324, KLF6, LAMB2, LMNA (already discussed), NFATC1, NFKB2, NUDC [593], P4HB, PDK2, PIM1, PISD, PXN, RAP1B, RNF40, SARA1, SEC61A1, SGTA [594], SLC12A2, SRD5A2, STAT6 [595], [596], TACSTD2, TBX1, TMED3, VPS39, WDFY3, XBP1 [544], ZAK). This result indicates that our results support the importance of NFkappa-B and the huge amount of research effort to understand the role of the NFkappa-B activity and its potential as a target for intervention in prostate cancer (File S4).

The group of 58 biomarkers contains one of particular interest, STAT6. This gene is considered a survival factor in prostate cancer and a key regulator of the genetic transcriptional program responsible for progression [595]. STAT6 has been recently linked to HPN as one of the most robust pair of biomarkers for prostate cancer using an integrative approach that linked several microarray datasets [596].

Analysis using GATHER of this group reveals that six of these 58 genes are in KEGG pathway path:hsa04510, Focal adhesion (EGF, ITGA3, ITGA7, LAMB2, PXN, RAP1B, p-value<0.0007) and from these there are three in pathway:hsa045122, ECM-receptor interaction (ITGA3, ITGA7, LAMB2, p-value<0.005) while four of these six are also in path:hsa04810: Regulation of actin cytoskeleton, (EGF, ITGA3, ITGA7, PXN, p-value<0.01).

The changes of the Normalized Shannon Entropy and Statistical Complexity of the gene expression profile of a cancer cell are associated with the gradual deterioration of genome transcriptional information content due to the modification of its structural and functional integrity during disease progression. Our results clearly suggest that we can track the cancer cell's progression by following observable changes in the Shannon Entropy and, in particular, by employing the Jensen-Shannon Divergence of the gene expression profile of a sample to the normal expression profile. We have also shown if an average expression profile of some state of interest can be properly defined (i.e. distant metastasis) then the Jensen-Shannon Divergence can help us to identify which probes best correlate with these measures resulting in useful biomarkers.

Before any thermodynamical consideration could be discussed, we note that there is a clear and objective informational perspective that our study delivers. In this study we have chosen to position ourselves as the ‘receivers’ of a ‘transcriptional message’. In this experimental perspective the tumor tissue is the ‘sender’ (the source of information) and the high-throughput technology (gene expression microarrays in this case) can be regarded as the transmission medium (providing noise and distortion). As we explain in the ‘Materials and Methods’ section, the Shannon Entropy of a gene expression profile is the average expected surprisal of that profile understood as a message. The Normalized Shannon Entropy makes this surprisal an intensive measure and the correlation of the gene expression patterns across samples with this measure can deliver useful biomarkers to track the progression of transcriptional change. After normalization, we have a measure that does not depend of the number of probes of the high-throughput technology, although, it obviously does depend on the type of probes used.

We believe that the readers may have already noticed an apparent paradox. While some researchers understand cancer progression as a mechanism that increases entropy, we actually observe a reduction of Normalized Shannon Entropy in this work. This means that our normalized average expected surprisal, as receivers of the transcriptional message, is smaller. We must then discuss the physical meaning of thermodynamic entropy, its current use in systems biology and cancer research genetics and the informational measure we use in this paper to clarify these notions in this context.

In biomedical research there exists a certain consensus among cancer researchers that genetic instability or “mutability” is a major critical force of cancer progression, but it is not the only one to consider. It is clear that the mutational damage of key genes (like TP53, TERT, BRCA1, RB1, etc.), and the collective damage inflicted on key DNA repair mechanisms (like Nucleotide-excision repair and Base-excision repair) collaborate for an increasing acceleration of the number of genomic changes. Sub-microscopic alterations of the genome accumulate in cancer progression in an irreversible way and “are compounded by the widespread scrambling of the chromosome structure, and thus the karyotype, found in cells from the great majority of solid tumours” [599]. In Weinberg's own words [599]: “we learned that this chromosomal chaos also contributes this progression forward”.

This “chromosomal chaos” [600] or “cancer as a chromosomal disease” perspective is viewed by some researchers not as just a side consequence of mutational damage, but as the main core theme to understand a number of unexplained issues in cancer progression. “In sum, cancer is caused by chromosomal disorganization, which increases karyotypic entropy” [601]. Regarding the cancer types studied in this paper, one particular “measure of disorder of a system”, aneuploidy, has been observed in poorly-differentiated prostate cancer cells and it is often associated with a more agreessive phenotype [602], [603], increased PSA levels [604], [605], and correlate with Gleason score [606], [607], [608]. Gene fusions and chromosomal rearrangements are other source of increase in the “disorder” of the genome organization and they are increasingly being recognized as a major player in prostate cancer progression [609]. The increase in “karyotypic complexity” and “extended aneuploidy and heteroploidy” may be already enough to develop a malignant melanoma phenotype, as the report of Gagos et al. indicate [610]. The observed finding of aneuploidy in melanoma (also including uveal melanoma) is also increasingly important due to a number of different independent observations [247], [611], [612], [613], [614], [615], [616], [617], [618]. It is in this context that the word ‘entropy’ has been used.

The magnitude of the “chromosomal chaos” is also evident from comparative genomic hybridization (CGH) studies which show significant variations in the copy number of individual chromosomal segments. ‘Chaos’ is really a very appropriate word to describe what we observe from CGH data. The genomic changes are not distributed uniformly at random. ‘Chaos’ has been described by some researchers as “a kind of order without any periodicity”. Some common changes seem to consistently appear in several independently arising tumours of the same type, and sometimes the researchers suggest common links [619]. Our work has addressed, in part, this question: “Can we quantify the chaos observed in the genome from the increasingly available transcriptional data and relate it to tumour progression?” If no commonalities were observed, we would not have found interesting biomarkers that seem that strongly correlate with the divergences from normal tissue types. We know from our results that these commonalities do occur.

We need to go back to basics to explain these evolving concepts and resolve this apparent paradox. The phrase “karyotypic entropy” has been used in the past to define what is actually a divergence from the normal chromosome structure and it genomic organization. This denomination has also been employed by several authors, notably [601], but it has also been used in at least two other publications [620], [621]. These works have in common the use of this term to refer to a “disorder”, fuelled by the undergraduate textbooks indoctrination of associating increase of entropy in natural spontaneous processes with the increase of “observed disorder” in the system. We propose that the use of a natural measure of divergence, the Jensen-Shannon divergence, could not only be a more formal, but also more appropriate modelling approach. As such, we propose to introduce the term ‘karyotypic divergence’ or ‘karyotypic Jensen-Shannon divergence’ to replace this concept and to avoid a subjective approach.

Why is it the case that we observe the Normalized Shannon Entropy of the transcriptional profile decreasing with cancer progression when intuitively our average expected surprisal (Shannon Entropy) should increase with progression?

Arieh Ben-Naim in his recent book “A farewell to Entropy: Statistical Thermodynamics based on Information” [622] comments:“It is interesting to note that Landsberg (1978) not only contended that disorder is an ill-defined concept, but actually made the assertion that ‘it is reasonable to expect ‘disorder’ to be an intensive variable’”. Ben-Naim also states: “In my view, it does not make any difference if you refer to information or to disorder, as subjective or objective. What matters is that order and disorder are not well-defined scientific concepts. On the other hand, information is a well-defined scientific quantity, as much as a point or a line are scientific in geometry, or mass or charge of a particle are scientific in physics.” However, in a manuscript entitled “Can Entropy and ‘order’ increase together ?” Landberg defines (in an attempt to decouple the notions of order and entropy), for a thermodynamical system that can be on N states the ‘disorder’ D(N) to be the Normalized Entropy (which is a function of N) divided by Boltzmann's constant [623]. ‘Disorder’ then is an intensive magnitude bounded by 0 and 1, and ‘order’ is defined as 1-D(N).

While Landberg's decoupling argument between order and entropy [623] may still be controversial in Physics, the question is pertinent for our apparent paradox (the question that motivates this subsection). Borrowing from the title of his paper we could now state the central question as “Can Shannon Entropy increase while the Normalized Shannon Entropy decrease?” The solution of this apparent paradox is a trick of escapologism, perhaps also paralleled by what a cancer cell may be experiencing (or “reacting” in response to increased sources of stresses), and it is worth discussing in this context. Let H[X] be Shannon Entropy for an ensamble X with N different values. We will now assume, and here is the trick, that N is not a constant, but a function of time N(t). Let D(X(N(t))) be the Normalized Shannon Entropy. By definition D(X(N(t))) = H(X(N(t)))/ . Then, just by taking the time derivatives it can be shown that the time variation of D(X(N(t))) can be negative, although the time rate of H[X] can be positive.where k is a constant. The escape to our paradox is “achieved” via making explicit the time variability of N(t). Landberg explicitly mentions that biological systems are examples where growth processes increase N(t), and perhaps the increased diversity in the transcriptome of a cancer cell during progression is one of such examples.

This discussion somehow resolves the apparent disassociations due to language barriers that may exist between the different disciplines (physics, information theory, molecular biology and oncology). A biologist may regard a cancer cell as an entity that, during progression, may “spread” its transcriptomic profile, including the generation of a large number of novel molecular species (due to adquired characteristics during its “devolution” from the normal type). In our informational perspective, this would be analogous to a situation in which the sender of a message, after some time, decides to increase the size of the alphabet of transmitted symbols. Clearly, it is intuitive to think that the receiver would be in a situation of increased Shannon Entropy. However, if the receiver is not aware of the new symbols (or is not able to detect them) and some of the symbols of the previous alphabet are no longer used, the receiver would now perceive a reduction of Normalized Shannon Entropy, observing an increasing order.

We now borrow an illustrative example from Landberg [623], but we add a twist to this argument for the purpose of illustrating this discussion. Suppose we have a sender transmitting only two possible symbols (N = 2), and we will assume that we have the same probability, let's denote this as (1/2, 1/2). Then the average expected surprisal (Shannon Entropy), is H(X) = 1, and the Normalized Shannon Entropy is also equal to one. Assume now that now our sender starts to transmit using another symbol, so that we now have theoretical probabilities of (0.5, 0.25, 0.25). Then N = 3, and the average expected surprisal increases to H(X′) = 1.5 the Normalized Shannon Entropy is now 1.5/  = 0.946… (a reduction). This ‘third symbol’ could actually represent a new “molecular species” or a protein isoform that would not be normally expressed in that tissue type [624], or even something entirely new, product of a mutational/deletional event. If our hypothetical high-throughput technology can only be detecting the first two symbols, and following the conventions we established in the ‘Materials and Methods’ section, we would be “observing” frequencies of (2/3,1/3) since the other events would not be detected with our equipment. As a consequence, the both the  = 1, Shannon Entropy and the Normalized Shannon Entropy are both reduced to 0.918293. Obviously, we can not count what we can not observe. As a consequence, a degenerating transcriptional profile that produces novel molecular species, and at the same time reduces those which we can not measure with a particular technology, would look increasingly more ordered.

We envision that physicists may find here a fertile ground to explore new ideas and attempt novel mathematical formalisms for cancer progression from the realm of finite-state thermodynamics [625] and in particular endorevesible processes [626] and endoreversible thermodynamics [627]. Some molecular alterations would then be part of the set of revesible processes that could occur in a cancer cell, while other processes like aneuploidy or gene fusions could be truly “irreversible genetic switches” associated with cancer progression [628]. If we assume that the process is slow (i.e. the times required for significant variations of the transcriptome's profile is large in comparison with the cell's processes time scales), and follwing the results of Spirkl and Reis [626], it may be possible that we have a constant entropy production rate exists during cancer progression leading to Hauptmann's “entropic devolution” [629]. Hauptmann sees a malignant tumour as “a dissipative structure arising within the thermodynamical open system of the human body” that starts when “a localized surplus of energy exists and there is no possibility to export entropy. An energetic overload in most malignant cells is indicated by their abnormally high phosphorylation state.” His perspective, preceeded in part by Dimitrov [630], Klimek [631], [632] and Marinescu and Viculetz [633] might then fit well an endoreversible thermodynamic formalism. Hauptmann says in [629] “I believe that cancer is a special kind of adaptation to energetic overload, characterized by multiplication and mutation of genomic DNA (generation of new biomolecules which enhance the probability of survival under harmful conditions), and by chiral alterations (reduction of entropy by entrapping energy) leading to abnormal configurated biomolecules. In this regard the genetic alterations are probably secondary changes. Cancer serves to dissipate energy in a type of developmental process but one in which the results are harmful to the whole organism: an entropic devolution.”

This thermodynamical perspective is now worth exploring and we will discuss it in this context. Assuming that a cancer cell is in a state of “energy overload”, without “the possibility of exporting entropy”, could it lead to some type of “genetic alterations”? Which key mechanisms might be impaired? What consequences is this “system” delivering? Could this be another hallmark for oncosystems indentification?

In 1871, in this book called “Theory of Heat”, Maxwell speculated the idea of “a being, who can see the individual molecules” and who has enough reactive intelligence to open and close a unique small hole existing between two communicating vessels (called ‘A’ and ‘B’). An ideal gas filled both vessels, so that starting at uniform temperature the intelligent being could observe the molecules and close and open the hole accordingly to a mission: “to allow only the swifter molecules to pass from A to B, and only the slower ones pass from B to A.” The being, “without expenditure of work raise the temperature of B and lower that of A in contradiction to the second law of thermodynamics.” The ability of the “being” to use observable information about the system to lower the thermodynamical entropy has motivated many articles in physics and fuelled the imagination of many since it was originally introduced by Mawell, and named as “demon” by Thomson three years later [622]. An excellent collection of articles until 1990 [634], [635], [636], [637], [638], [639], [640], [641], [642], [643], [644] was edited by Leff and Rex [645]. The Maxwell “demon”, far from being “exorcised” from Physics, still inspires interesting new perspectives [634], [635], [636], [637], [638], [639], [640], [641], [642], [643], [644], [646], [647].

In a letter to Peter Guthrie Tait, Maxwell writes about the “demons”: “Is the production of an inequality of temperature their only occupation? No, for less intelligent demons can produce a difference in pressure as well as temperature by merely allowing all particles going in one direction while stopping all those going the other way. This reduces the demon to a valve. As such value him. Call him no more a demon but a valve like that of the hydraulic ram, suppose.” (from [645], p. 6). Maxwell gives again here a sign of his brilliant mind, “degrading” the demon to a valve, but also offering an inspiring perspective to oncosystems research. Which types of mechanisms exist in biological systems, and particularly in individual cells, to control these differential values in key parameters? Could changes of key physical parameters for metabolic processes of the cytoplasm and cell's organelles like temperature, volume, pH or electrochemical potentials be also implicated in cancer progression?

The influence of temperature may be giving an interesting working hypothesis for further research. What are the consequences if cancer cells are a different type of open system which also operates at a different temperature than a normal cell? Butler et al. have studied p53 and they argue that at temperatures above 37 degrees centigrades wild-type p53 spontaneously loses DNA binding activity. While folding kinetics do not show important changes in a range from 5 to 35 degrees C, the unfolding rates accelerate 10,000-fold. This leads to a somewhat unexpected mechanism of p53 inactivation. It could be the case that a fraction of p53 molecules become trapped in misfolded conformations with each folding-unfolding cycle due to the increased frequency of cycling. The occurrence of misfolded p53 proteins can lead to aggregation and subsequent ubiquitination in the cell, leading to p53 inactivation [648], [649]. If a key “guardian of the genome integrity” [650], [651] and its remarkable conformational flexibility [652] is challenged by an increase of temperature [653], its role in genotoxic damage and adaptive response (like that of the skin to UVB damage [654]) may be impaired. The same may occur for other members of the DNA damage response. An increment in temperature has already been linked to skin carcinogenesis. Boukamp et al. report in that [655] “exposure of immortal human HaCaT skin keratinocytes (possessing UV-type p53 mutations) to 40 degrees C reproducibly resulted in tumorigenic conversion and tumorigenicity was stably maintained after recultivation of the tumors.”

On the other hand, natural gradients on physical biochemical properties can also be challenged in a cancer cell. This in turn derives in metabolic processes running under abnormal parametric circumstances. It is well-known that compartimentalization, in biological systems, naturally require the existence of mechanisms that would keep some key state variables relatively constant, or within bounds, for normal operation of the metabolic processes. One example is very illustrative and a case in point. Instead of demons, holes, or valves, the cell requires pores in its membranes to allow osmotic regulatory processes, yet it should preclude the conduction of protons. This is a nanotechnological design problem not faced by Maxwell, but certainly solved by biological systems without the need of an “intelligent being” as Mawell cleverly pointed to Tait in his letter.

This discussion brings us to one of the gene families we have already discussed in this paper, the aquaporins [184], [656], [657], [658], [659], [660], [661]. They are considered the primary water channels of cell membranes [662], [663], [664], [665]. The specific functions of each member of this family are now being slowly mapped by several research labs around the world [666]. Their clinical role in cancer [667], [668], [669], [670], [671], [672], [673], [674], [675],obesity [676], malaria [677], [678] and other diseases is emerging [657], [679], [680], [681], [682], [683], [684], [685], [686], [687], [688], [689]. In [690], our group observed the dowregulation of AQP3 in all melanoma cell lines studied of the NCI-60 dataset of Ross et al.; this dowregulation was also observed for the CNS and Renal cell lines. AQP3 was relatively upregulated for Leukaemia and Colon cell-lines (we refer the reader to the Supplementary Material of [690] for details). Inhibition of AQP3 in prostate cancer cells was already proposed as a mechanism that increases the sensitivity to cryotherapy treatment [691].

The aquaporins are not “an intelligent being” in any real sense, yet they are so formidable selective that they could easily parallel Maxwell demon's efficiency in creating the right conditions for the cell. Wu et al. give us some clues on the role of point mutations in the AQP1 and how their effective electrostatic proton barrier can be impaired [692]. The elicitation of the detailed mechanistic explanation of this extraordinary selectivity is under intense investigation with a number of techniques, including sophisticated molecular dyanamics simulations, for an overview of this field see [665], [693], [694], [695], [696], [697], [698], [699], [700], [701], [702], [703], [704], [705], [706], [707], [708]. One less known feature of aquaporins is that they may not only channel water, but also carbon dioxide and ammonia [709], [710], [711], glycerol [712] and urea and other small solutes [713] and, very relevant for cancer research, hydrogen peroxide [188]. At least two of members of this family have been observed in the inner mitochondrial membrane in different tissues. This in turn may indicate mitochondrial roles for aquapotins in osmotic swelling induced by apoptotic stimuli [714].

Could it be possible that we can track cancer progression by looking at some of these “Maxwell demons”? We have seen in Figure 10, that AQP3 has a reduced expression with increased progression in our melanoma dataset. Cao et al., reported that ultraviolet radiation induced AQP3 down-regulation in human karatinocytes; thus AQP3 has become a strong and plausible link between UV radiation, skin dehydration [186], [715] and photoaging [189]. This may indicate an impared function on skin hydration [184], [185], [716], [717], [718], [719]. The expression of AQP3, as well as AQP1, AQP5, and AQP9 seem to be correlated with melanoma progression, indicating a common pattern of downregulation from the higher values in normal skin and benign nevi (see Figure 17).

Does a similar pattern of aquaporin downregulation exist in prostate cancer? Wang et al. have looked at the expression and localization of AQP3 in human prostate using cell lines as well as patient samples. They have observed AQP3 mRNA “in both normal and cancerous epithelia of human prostate tissues, but not in the mesenchyme. In the normal epithelia of the prostate, localization was limited to cell membranes, particularly the basolateral membranes. However, the expression of AQP3 protein in the cancer epithelia was not observed on the cell membranes.” This finding seems to implicate the subcellular localization of AQP3 as a possible indicator of a transition to a more malignant phenotype. Lapointe's dataset allows us to see the downregulation of AQP3 and AQP1. A large subgroup of primary prostate tumors has reduced levels of AQP3 and AQP1 as most of the lymph node metastasis samples [Figure 18].

One critique that we are aware we could receive is that the current manuscript presents a novel methodology and an underlying unifying theory based on retrodictions or postdictions. Indeed we have shown that the use of the Normalized Shannon Entropy and the Information Theory quantifiers (the M-complexities and the Jensen-Shannon divergence) allow to monitor cancer progression and to identify the best biomarkers that correlate with the transcriptomic changes. Our approach works in a retrodiction way in that it looks at data already obtained by other studies, but gives a unifying framework to track cancer progression. For instance, on True et al's dataset, our unifying hallmark of cancer gives not only MAOA, which was already identified in the original publication, but also AMACR, CD40, CDK4, etc. are very important biomarkers for prostate cancer. Analogously, the identification of KLK3/PSA in Lapointe's dataset is another important retrodiction which shows the power of the method.

In some sense our approach also works in a postdiction way, as it helps to evaluate the speculation that cancer cells have “an entropic devolution”. Our results show that the variations of Normalized Shannon Entropy and Jensen-Shannon divergences indeed give measurable changes, and that these changes are related to important biomarkers in the two types of cancer studied in this work.

In addition, we remark that we are literally making hundreds, or even thousands of predictions. The results in the ‘Supplementary Material’ provide this information for the detailed scrutiny of our peers. We believe that other probes with gene expression patterns in high correlation with the probes discussed in this paper, and perhaps less studied by immunohistochemistry and other methods in the two cancer types studied here, are worth exploring as a group of biomarkers. These predictions can be tested with further studies on staging and patient stratification.

A very recent study by Ballal et al. have linked BRCA1 to telomere length and maintenance and its loss from the telomere in response to DNA damage [720] (see also [721]). We have previously mentioned that BRCA1 is a conspiquous biomarker arising from the analysis of True et al.'s dataset using our methods. We found this to correlate with a preivous study that showed that BRCA1 has a reduced expression in immortalized prostate epithelial cells before and after their conversion to tumorigenicity [493]. We also mentioned that the knockdown of BRCA1 leads to anaccumulation of multinucleated cells [492], preserving chromosomal stability [490]. Ballal et al. telomeric ChIP assays to detect BRCA1 at the telomere and reported time-dependent loss of BRCA1 from the telomere following DNA damage. Due to the role of telomeres in maintaining chromosomal stability [722] and the inverse correlation of telomere length and divergent karyotypes in prostate cancer cell lines [723], [724] (as well as the recognized role of telomere dysfunction in the induction of apoptosis or senescence in vivo [725], [726], [727], [728], [729], [730], increase of mutation rates [731], DNA fragmentation [732], and their relation with DNA damage signalling [733]), we checked for other probes of genes involved in telomeric function.

From those which we were able to identify in True et al's dataset, we have found a strong high correlation of the expression of BRCA1 with TERF2/TRF2 (telomeric repeat binding factor 2) [734] and a negative correlation with the expression pattern of TERF2IP (telomeric repeat binding factor 2, interacting protein) [Figure 19].

Finally, one particular type of probes has also caught our attention, and we would like to refer to them before concluding this section.

With the denomination of ‘non-coding RNA’ we identify those RNA molecules which are functional but that are not translated into proteins. Many microarray chips contain probes that are annotated as ‘non-protein coding’, indicating that there might be some valuable expression data that we can also mine for information. We note that our method, although employing transcriptomic data, does not limit its application to protein-coding information, and that the combined use of protein-coding and non-coding protein probe expression would allow a more comprehensive view of the transcriptional state of the cell.

Among non-protein coding, microRNAs [735] are gaining acceptance as key players in several cancers [736], [737], [738] (including prostate cancer [739], [740]), but the so-called “long non-coding RNAs” [741] are also gaining a place in the scenario of cancer biomarkers (see [742], and [743], [744], [745]). We thus turned our attention to these probes that have been annotated as “non-protein coding” and we highlight some of them that have very high correlation values with the Normalized Shannon Entropy in True et al's prostate cancer dataset. In particular, the probes for MALAT1/MALAT-1 [742], [746], [747], [748], [749], [750], [751], [752], [753], [754], [755], [756], [757], [758] have a very conspiquous position (See Figure 20). They located very closely to other protein coding biomarkers that have also lost expression and have been discussed in this work like SFPQ, CD40, BRCA1, and TP53 (see Figure 16 ). MALAT1 has been recently pointed as a biomarker in primary human lobular breast cancer as a result of an analysis of over 132,000 Roche 454 high-confidence deep sequencing reads [749]. An international team, searching on thousands of novel non-coding transcripts of the breast cancer transcriptome, has been able to identify more than three hundred reads corresponding to MALAT1 [749]. This is a non-coding RNA which was identified in 2003 in non-small cell lung cancer, was shown to be highly expressed (relative to GAPDH) in lung, pancreas and prostate, but not in other tissues including muscle, skin, stomach, bone marrow, saliva, thyroid and adrenal glands, uterus and fetal liver [758]. MALAT-1, also known as NEAT2, is considered to be “extraordinarily conseved for a noncoding RNA, more so than even XIST” [754]. Our results indicate that the reduction of expression of some non-coding RNAs, in particular of MALAT-1, and SNORA60 with respect to their normal expression in prostate, as well as the upregulation of SNHG8 and SNHG1 should be monitored as useful biomarkers to track disease progression.

We will now address another non-coding RNA called NEAT1 which, like NEAT2, is also conserved in the mammalian lingeage. Before we move onto NEAT1, we will first recall a previous result. We have noted before the conspiquous position of SFPQ/PSF (Polypyrimidine tract-binding protein-associated splicing factor) in Figure 16. The expression of a probe for SPQF has the highest correlation with the values of the Normalized Shannon Entropy. We highlighted before that SFPQ/PSF is a putative regulator of growth factor-stimulated gene expression [498]. The loss of SFPQ expression during the progression of prostate cancer may be an important key to understand this disease or one of its subtypes. We have also mentioned that the AR/PSF complex interacts with the PSA gene (perhaps the most well-established prostate cancer biomarker) and that SFPQ/PSF inhibits AR transcriptional activity [499]. Kuwahara et al. showed that SFPQ together with NONO (Non-POU-domain-containing, octamer binding protein) and PSPC1 (Paraspeckle protein 1 alpha isoform, formerly known as PSP1) are expressed in mouse Sertoli cells of the testis and form complexes that function as coregulators of androgen receptor-mediated transcription [500]. While new research results [759] link SFPQ and NONO/P54NRB with the RAD51 family of proteins (largely regarded as another key protector of chromosome integrity as being involved in homologous recombination DNA repair), it is perhaps SFPQ and NONO's co-localization in paraspeckles that make this group also remarkable [760].

Paraspeckles [760], [761], [762], [763], [764], [765], [766], [767], [768], [769], [770], [771], [772], [773], [774], [775], [776], [777] are a novel nuclear compartment, of approximately 0.2–1 µm in size, discovered in 2002, by Fox et al. in Dundee Scotland, following the identification of the protein PSPC1 (AF448795) in the nucleolar proteomics project at Lamond's lab which is described well by Fox et al. [777]. Three years later, Fox, Bond and Lamond showed that NONO and PSPC1 form a heterodimer that localizes to paraspeckles in an RNA-dependent manner [773]. Paraspeckles are dynamic structures, observed in numbers that vary between 10 and 20, that seem to control gene expression via retention of RNA in the nucleus [772]. A long noncoding RNA called NEAT1/MEN epsilon/beta [754], [760], [762], [764], [778], that colocalizes with paraspeckles, seems to be integral to their structure. Depletion of NEAT1 erradicates paraspeckles and a biochemical analysis by Clemson et al indicates that the NEAT1 binds with paraspeckle proteins SFPQ/PSF, P54NRB/NONO and PSPC1. NEAT1 is also known as TncRNA (trophoblast-derived noncoding RNA) [754], [779], [780], [781], [782], [783], [784], [785], [786] and probes for TncRNA exist on this dataset, We have observed in True et al.'s dataset that there exists a high correlation between the Normalized Shannon Entropy with the expression of SFPQ/PSF, P54NRB/NONO, and TncRNA. Overall, this implies that the disruption of the function of the paraspeckles is correlated with the increasing signs of deterioration of normal transcriptomic state of the cells. While a causal relationship still needs to be proved, we admire the mathematical elegance of the Normalized Shannon Entropy of the samples, a global measure of the average expected surprisal of the transcriptome, which in turn has lead us to consider the dysfunction of the smallest nuclear body as a putative biomarker of disease progression. The role of SFPQ/PSF in the control of tumorigenesis is under investigation [787] and the information coming from these studies would need to be integrated with their role, together with P54NRB/NONO and TncRNA, in paraspeckles if we want to achieve a better understanding of these mechanisms.

In this contribution we have shown that for the melanoma and prostate cancer datasets studied, the quantitative changes of Information Theory measures, Normalized Shannon Entropy, Jensen–Shannon divergence and the novel Statistical Complexity quantifiers defined here are in high correlation with gene expression changes of well-established biomarkers associated to cancer progression. In addition, variations of the basic technique (i.e. a modified form of statistical complexity) which allows us to better understand the phenotypic changes observed in these samples which are associated with the progression and the transitions of the gene expression profiles. For instance, in a properly defined Statistical Complexity vs. Entropy plane, on a melanoma dataset first studied in Ref. [110], samples appear in well differentiated “clusters”. These clusters correlate well with the phonotypic characteristics of normal skin, nevi, primary and metastatic melanoma. In this “Complexity vs. Entropy” plane, primary melanomas samples appear “bridging” benign nevi and metastatic melanoma samples. Our results may also suggest that the evolution of metastatic melanoma leads to at least two different subtypes.

The Normalized Shannon Entropy of a transcriptional sample profile is calculated associating the measured expression values of a gene with the relatively probability of being expressed. We have observed that, in general, the transcriptomes of tumour progressing cells tend to have lower values of Normalized Shannon Entropy than normal ones. Given a population of normal cells of a given tissue type it is then possible to compute useful measure of divergence of cancer cell profiles from the normal expression average profile, in terms of Information Theory quantifiers, the Shannon Eveness normalized entropy and generalized statistical complexity [788], [789], [790].

In addition, our observation of the correlation of the statistical complexity of tumours with its natural progression allows an unprecedented way of finding biomarkers that links with the gradual deterioration of the genome integrity. The proposed methodology uncovered, for the first time, evidence of the putative role of impared centrosome cohesion in melanoma progression.

Statistical complexity has then been able to pinpoint otherwise unrecognized biomarkers in concert with existing ones, reinforcing the view that “chromosomal chaos” and “cancer as a chromosomal disease” can be a useful guiding principle to understand the molecular biology of cancer and uncover the timeline of its progression. This is a powerful method to uncover “oncosystems” instead of “oncogenes”. “Oncosystems” are a highly differentially disregulated set of genes that, if linked with the molecular “hallmarks of cancer” described in the introduction, and existing databases with putative common functional genomic annotations, can help to understand the biological progression pathways that drive the disease.

On one of the prostate cancer dataset studied (obtained from a previous published study, [44]), we observe a gradual pattern of reduction of Normalized Shannon Entropy from three well characterized tissue types: normal prostate, primary prostate tumours and lymph node metastases. On a different dataset on prostate cancer (from Ref [332]), we observe that a group of samples having Gleason patterns 4 and 5 (two patterns which are typically associated to an aggressive phenotype) have lower Normalized Shannon Entropy values than a subset of Gleason pattern 3 (a pattern which is normally associated to a less aggressive phenotype but which nevertheless is still of clinical concern). However, a group of samples having Gleason patterns 3, 4, and 5 is revealed; this mixed cluster has a mid-range entropy. This is an interesting fact which correlates with the limitations observed in Ref. [332]. We note the authors' comment: “We were unable to identify a cohort of genes that could distinguish between pattern 4 and 5 cancers with sufficiently high accuracy to be useful, suggesting a high degree of similarity between these cancer histologies or substantial molecular heterogeneity in one or both of these groups.” Our results provide a conciliatory middle ground that explains the perceived clinical usefulness of Gleason pattern classification, widely used around the world, while at the same time reveals the reason for the difficulties of obtaining a good transcriptional signature for the other two patterns [791].

We have seen, through a detailed discussion of several biomarkers in three different datasets, that the variation of the gene expression distributional profile can be characterized via Information Theory quantifiers. Our study also showed that current established biomarkers of the two diseases studied seem to correlate with those that best co-variate with these quantifiers. For instance, AMACR, in our second prostate cancer dataset studied, naturally appears as one of the most correlated genes (in both the Pearson and the Spearman sense) with the pattern of variation of Entropy of the samples. Together with MAOA, which is the highlighted gene in True et al.'s [332] original publication, AMACR is now being recognized as one of the best biomarkers in primary prostate cancer with approximately 180 publications dedicated to it in the past five years. We have also shown that many gene probes that best correlate with the divergence of the normal tissue profile have been identified as useful biomarkers (via other accepted validation methods). This said, the use of other sources of information, like pathway or gene ontology databases has lead as to the identification of other cell processes that may be altered.

We have presented a unifying hallmark of cancer, the cancer cell's transcriptome changes its Normalized Shannon Entropy (as measured by high-througput technologies), while it increments its physical Entropy (via creation of states we might not measure with our devices). This hallmark allows, via the use of the Jensen-Shannon divergence, to identify the arrow of time of the process, and helps to map the phenotypical and molecular hallmarks of cancer as major converging trends of the transcriptome. The methodology has produced remarkable postdictions and retrodictions that show that it can predictively guide biomarker discovery.

Samples were obtrained from radical prostatectomy surgical procedures. Samples are labelled as “tumors” if they contain at least 90% of cancerous epithelial cells, and they were considered as “non-tumor” if they contain no tumor epithelium and are from the noncancerous region of the prostate. The later samples were labelled “normals” although the authors alert that some may contain dysplasia. In this dataset, Lapointe et al. have performed a gene expression profiling by using cDNA microarrays containing 26,260 different human genes (UniGene clusters). Using 50 µg of total RNA from prostate samples Cy5-labeled cDNA was prepared and Cy3-labeled cDNA used 1.5 µg of mRNA common reference, pooled from 11 human cell lines (see Ref. [792]). The fluorescence ratios were subsequently normalized by mean centering genes for each array, a relatively standard procedure. In addition, to minimize potential print run specific bias, Lapointe et al. report that ratios were then mean centered for each gene across all arrays according to Ref. [793]. We have only used the genes that the authors report in their first figure, 5,153 genes that have been well measured and have significan variation in some of the samples. For the other details of their matrials and methods we refer the readers to the Supporting Notes and the Materials and Methods section of their original publication [44].

Samples were obtained from nevus volunteers and melanoma patients and only those samples that have more than 90% of tumor cells were profiled. The 20,862 cDNAs used (Research Genetics, Huntsville, AL) represent 19,740 independent loci. (Unigene build 166).median of ratio values from the experiment were subjected to linear normalization in nomad (which can be accessed at http://derisilab.ucsf.edu), log-transformed (base 2), and filtered for genes where data were present in 80% of experiments, and where the absolute value of at least one measurement was >1.

In this dataset, samples have information of 15,488 spots per array, with a total of 7,700 unique cDNAs represented. The samples were obtained from frozen tissue blocks from 29 radical prostatectomies accessioned and selected to represent Gleason grades 3, 4, and 5. The samples are “treatment naïve”, meaning that they were also selected such that their gene expression profile is also and the absence of any bias that the treatment before prostatectomy. The frozen sections (8 µm) were cut from optimal cutting temperature medium blocks and immediately fixed in cold 95% ethanol. Around 5,000 epithelial cells from both histologically benign glands and cancer glands were separately laser-capture microdissected (LCM). The authors of the study have also been very careful to include only one Gleason pattern in each laser-captured cancer sample, following a process in which the patterns were assessed independently by two investigators.The matched benign epithelium was captured for each cancer sample for a total of 121 samples.

An important characteristic of this dataset is the normalization procedure. For each spot and in each channel (Cy3 and Cy5), True et al. substracted the median background intensity from the median foreground intensity, and subsequently the log ratios of cancer expression to benign expression were computed. These ratios were obtained by first dividing the background-subtracted intensities (Prostate Cancer/Benign) and then taking the logarithm base 2. In the case that the median background intensity was greater than the median foreground intensity, the spot was considered missing. We refer to the original publication for the other aspects of imputation, spot quality and filtering, but, like in Lapointe et al's study, they also filter to keep informative (expression ratios of benign versus cancer should at least be 1.5-fold or greater in at least half of one of the Gleason groups as one of the selection criteria).

Given a probability distribution function over a discrete finite set, is then straightforward to calculate its Normalized Shannon Entropy if we have the theoretical frequencies. Several measures of “complexity” of a probability distribution function have been proposed. In this work we have used Statistical Complexity measures.

All the complexity measures used in this work are the product of a Normalized Shannon Entropy of the probability distribution function, and a divergence measure to a reference probability distribution function. We follow earlier proposals by López-Ruiz, Mancini and Calbet who first introduced a statistical complexity measure based on such a product in [797]. The LMC-Statistical Complexity is the product of the Normalized Shannon Entropy, H[P], times the disequilibrium, Q[P]; the latter given by the Euclidean distance from P to P _e, the uniform probability distribution over the ensemble. In this paper we used a later modification which we refer as the MPR-Statistical Complexity [43] which replaces the Euclidean distance between P to P _e by the Jensen-Shannon divergence [788], [798]. The Jensen-Shannon divergence is linked in physics to the thermodynamic length [799], [800], [801], [802].

We define the MPR-Statistical complexity [790] as:where , Q₀ is a normalization factor, and is the Jensen-Shannon's divergence between two probability density functions P⁽¹⁾ and P⁽²⁾, which in turn is defined asIn this work, in many cases we compute the Jensen-Shannon divergences of a probability with a probability of reference which is not the uniform probability distribution over the ensemble. In general, it is the average over a subset of probability distribution functions which are consider to be either the “initial” of “final” states of interest. Let be such an average, then the M-Statistical Complexity of a probability distribution function , given a of reference, is given by