This paper presents methods that quantify the structure of statistical interactions within a given data set, and was first used in [58]. It establishes new results on the k-multivariate mutual-informations (I k ) inspired by the topological formulation of Information introduced in [4,63]. In particular we show that the vanishing of all I k for 2 ≤ k ≤ n of n random variables is equivalent to their statistical independence. Pursuing the work of Hu Kuo Ting and Te Sun Han [23,21,22], we show that information functions provide co-ordinates for binary variables, and that they are analytically independent on the probability simplex for any set of finite variables. The maximal positive I k identifies the variables that co-vary the most in the population, whereas the minimal negative I k identifies synergistic clusters and the variables that differentiate-segregate the most the population. Finite data size effects and estimation biases severely constrain the effective computation of the information topology on data, and we provide simple statistical tests for the undersampling bias and the k-dependences following [43]. We give an example of application of these methods to genetic expression and unsupervised cell-type classification. The methods unravel biologically relevant subtypes, with a sample size of 41 genes and with few errors. It establishes generic basic methods to quantify the epigenetic information storage and a unified epigenetic unsupervised learning formalism. We propose that higher-order statistical interactions and non identically distributed variables are constitutive characteristics of biological systems that should be estimated in order to unravel their significant statistical structure and diversity. The topological information data analysis presented here allows to precisely estimate this higher-order structure characteristic of biological systems."When you use the word information, you should rather use the word form"René Thom
The development of morphological neuronal polarity starts by the formation and elongation of an axon. At the same time the axon initial segment (AIS) is generated and creates a diffusion barrier which differentiate axon and somatodendritic compartment. Different structural and functional proteins that contribute to the generation of neuronal action potential are concentrated at the axon initial segment. While axonal elongation is controlled by signalling pathways that regulate cytoskeleton through microtubule associated proteins and tubulin modifications, the microtubule cytoskeleton under the AIS is mostly unknown. Thus, understanding which proteins modify tubulin, where in the neuron and at which developmental stage is crucial to understanding how morphological and functional neuronal polarity is achieved. In this study performed in mice and using a well established model of murine cultured hippocampal neurons, we report that the tubulin deacetylase HDAC6 is localized at the distal region of the axon, and its inhibition with TSA or tubacin slows down axonal growth. Suppression of HDAC6 expression with HDAC6 shRNAs or expression of a non-active mutant of HDAC6 also reduces axonal length. Furthermore, HDAC6 inhibition or suppression avoids the concentration of ankyrinG and sodium channels at the axon initial segment (AIS). Moreover, treatment of mouse cultured hippocampal neurons with detergents to eliminate the soluble pool of microtubules identified a pool of detergent resistant acetylated microtubules at the AIS, not present at the rest of the axon. Inhibition or suppression of HDAC6 increases acetylation all along the axon and disrupts the specificity of AIS cytoskeleton, modifying the axonal distal gradient localization of KIF5C to a somatodendritic and axonal localization. In conclusion, our results reveal a new role of HDAC6 tubulin deacetylase as a regulator of microtubule characteristics in the axon distal region where axonal elongation takes place, and allowing the development of acetylated microtubules microdomains where HDAC6 is not concentrated, such as the axon initial segment.
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