On the gene expression landscape of cancer.

Kauffman picture of normal and tumor states as attractors in an abstract state space is used in order to interpret gene expression data for 15 cancer localizations obtained from The Cancer Genome Atlas. A principal component analysis of this data unveils the following qualitative aspects about tumors: 1) The state of a tissue in gene expression space can be described by a few variables. In particular, there is a single variable describing the progression from a normal tissue to a tumor. 2) Each cancer localization is characterized by a gene expression profile, in which genes have specific weights in the definition of the cancer state. There are no less than 2500 differentially-expressed genes, which lead to power-like tails in the expression distribution functions. 3) Tumors in different localizations share hundreds or even thousands of differentially expressed genes. There are 6 genes common to the 15 studied tumor localizations. 4) The tumor region is a kind of attractor. Tumors in advanced stages converge to this region independently of patient age or genetic characteristics. 5) There is a landscape of cancer in gene expression space with an approximate border separating normal tissues from tumors.

Estimating the number of available states for normal and tumor tissues in gene expression space.

The topology of gene expression space for a set of 12 cancer types is studied by means of an entropy-like magnitude, which measures the volumes of the regions occupied by tumor and normal samples, i.e., the number of available states (genotypes) that can be classified as tumor-like or normal-like, respectively. Computations show that the number of available states is much greater for tumors than for normal tissues, suggesting the irreversibility of the progression to the tumor phase. The entropy is nearly constant for tumors, whereas it exhibits a higher variability in normal tissues, probably due to tissue differentiation. In addition, we show an interesting correlation between the fraction (tumor/normal) of available states and the overlap between the tumor and normal sample clouds, interpreted as a way of reducing the decay rate to the tumor phase in more ordered or structured tissues.

A one-dimensional parameter-free model for carcinogenesis in gene expression space.

A small portion of a tissue defines a microstate in gene expression space. Mutations, epigenetic events or external factors cause microstate displacements which are modeled by combining small independent gene expression variations and large Levy jumps, resulting from the collective variations of a set of genes. The risk of cancer in a tissue is estimated as the microstate probability to transit from the normal to the tumor region in gene expression space. The formula coming from the contribution of large Levy jumps seems to provide a qualitatively correct description of the lifetime risk of cancer in 8 tissues, and reveals an interesting connection between the risk and the way the tissue is protected against infections.

Mutations as Levy flights.

Data from a long time evolution experiment with Escherichia Coli and from a large study on copy number variations in subjects with European ancestry are analyzed in order to argue that mutations can be described as Levy flights in the mutation space. These Levy flights have at least two components: random single-base substitutions and large DNA rearrangements. From the data, we get estimations for the time rates of both events and the size distribution function of large rearrangements.

Gene expression rearrangements denoting changes in the biological state.

In many situations, the gene expression signature is a unique marker of the biological state. We study the modification of the gene expression distribution function when the biological state of a system experiences a change. This change may be the result of a selective pressure, as in the Long Term Evolution Experiment with E. Coli populations, or the progression to Alzheimer disease in aged brains, or the progression from a normal tissue to the cancer state. The first two cases seem to belong to a class of transitions, where the initial and final states are relatively close to each other, and the distribution function for the differential expressions is short ranged, with a tail of only a few dozens of strongly varying genes. In the latter case, cancer, the initial and final states are far apart and separated by a low-fitness barrier. The distribution function shows a very heavy tail, with thousands of silenced and over-expressed genes. We characterize the biological states by means of their principal component representations, and the expression distribution functions by their maximal and minimal differential expression values and the exponents of the Pareto laws describing the tails.