There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principa
l curves and principal trees, and so on. For each type of approximators the measure of the approximator complexity was developed too. These measures are necessary to find the balance between accuracy and complexity and to define the optimal approximations of a given type. We propose a measure of complexity (geometrical complexity) which is applicable to approximators of several types and which allows comparing data approximations of different types.
MicroRNAs can affect the protein translation using nine mechanistically different mechanisms, including repression of initiation and degradation of the transcript. There is a hot debate in the current literature about which mechanism and in which sit
uations has a dominant role in living cells. The worst, same experimental systems dealing with the same pairs of mRNA and miRNA can provide ambiguous evidences about which is the actual mechanism of translation repression observed in the experiment. We start with reviewing the current knowledge of various mechanisms of miRNA action and suggest that mathematical modeling can help resolving some of the controversial interpretations. We describe three simple mathematical models of miRNA translation that can be used as tools in interpreting the experimental data on the dynamics of protein synthesis. The most complex model developed by us includes all known mechanisms of miRNA action. It allowed us to study possible dynamical patterns corresponding to different miRNA-mediated mechanisms of translation repression and to suggest concrete recipes on determining the dominant mechanism of miRNA action in the form of kinetic signatures. Using computational experiments and systematizing existing evidences from the literature, we justify a hypothesis about co-existence of distinct miRNA-mediated mechanisms of translation repression. The actually observed mechanism will be that acting on or changing the limiting place of the translation process. The limiting place can vary from one experimental setting to another. This model explains the majority of existing controversies reported.
We present details of the analysis of the nonlinear quality of life index for 171 countries. This index is based on four indicators: GDP per capita by Purchasing Power Parities, Life expectancy at birth, Infant mortality rate, and Tuberculosis incide
nce. We analyze the structure of the data in order to find the optimal and independent on experts opinion way to map several numerical indicators from a multidimensional space onto the one-dimensional space of the quality of life. In the 4D space we found a principal curve that goes through the middle of the dataset and project the data points on this curve. The order along this principal curve gives us the ranking of countries. Projection onto the principal curve provides a solution to the classical problem of unsupervised ranking of objects. It allows us to find the independent on experts opinion way to project several numerical indicators from a multidimensional space onto the one-dimensional space of the index values. This projection is, in some sense, optimal and preserves as much information as possible. For computation we used ViDaExpert, a tool for visualization and analysis of multidimensional vectorial data (arXiv:1406.5550).
