Gene expression data for a set of 12 localizations from The Cancer Genome Atlas are processed in order to evaluate an entropy-like magnitude allowing the characterization of tumors and comparison with the corresponding normal tissues. The comparison
indicates that the number of available states in gene expression space is much greater for tumors than for normal tissues and points out to a scaling relation between the fraction of available states and the overlapping between the tumor and normal sample clouds.
A principal component analysis of the TCGA data for 15 cancer localizations unveils the following qualitative facts about tumors: 1) The state of a tissue in gene expression space may 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 variability. 5) There is a landscape of cancer in gene expression space with an approximate border separating normal tissues from tumors.
Taking into account an evolutionary model of mutations in term of Levy Fights that was previously constructed, we designed an algorithm to reproduce the evolutionary dynamics of the Long-Term Evolution Experiment (LTEE) with E. Coli bacteria. The alg
orithm enables us to simulate mutations under natural selection conditions. The results of simulations on competition of clones, mean fitness, etc., are compared with experimental data. We attained to reproduce the behavior of the mean fitness of the bacteria cultures, get our own interpretations and more tuned descriptions of some phenomena taking part within the experiment, such as fixation and drift processes, clonal interference and epistasis.
Compiled data for the stem cell numbers, Ns, and division rates, ms, is reanalized in order to show that we can distinguish two groups of human tissues. In the first one, there is a relatively high fraction of maintenance (stem and transit) cells in
the tissue, but the division rates are low. The second group, on the other hand, is characterized by very high transit cell division rates, of around one division per day. These groups do not have an embrionary origin. We argue that their properties arise from a combination of the needs of tissue homeostasis (in particular turnover rate) and a bound on cancer risk, which is roughly a linear function of the product Ns ms. The bound on cancer risk leads to a threshold at ms = 8/year, where the fraction of stem cells falls down two orders of magnitude.
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 re
sult 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.
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.
I provide a simple estimation for the number of macrophages in a tissue, arising from the hypothesis that they should keep infections below a certain threshold, above which neutrophils are recruited from blood circulation. The estimation reads Nm=a N
cel^{alpha}/Nmax, where a is a numerical coefficient, the exponent {alpha} is near 2/3, and Nmax is the maximal number of pathogens a macrophage may engulf in the time interval, tr, between pathogen replications.
Simple ideas, endowed from the mathematical theory of control, are used in order to analyze in general grounds the human immune system. The general principles are minimization of the pathogen load and economy of resources. They should constrain the p
arameters describing the immune system. In the simplest linear model, for example, where the response is proportional to the load, the annihilation rate of pathogens in any tissue should be greater than the pathogens average rate of growth. When nonlinearities are added, a reference value for the number of pathogens is set, and a stability condition emerges, which relates strength of regular threats, barrier height and annihilation rate. The stability condition allows a qualitative comparison between tissues. On the other hand, in cancer immunity, the linear model leads to an expression for the lifetime risk, which accounts for both the effects of carcinogens (endogenous or external) and the immune response.
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 jump
s, 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, and reveals an interesting connection between the risk and the way the tissue is protected against infections.
Levy flights in the space of mutations model time evolution of bacterial DNA. Parameters in the model are adjusted in order to fit observations coming from the Long Time Evolution Experiment with E. Coli.
