ROBUST DEPENDENCIES AND STRUCTURES IN STEM CELL DIFFERENTIATION

Cell differentiation is a complex process governed by the timely activation of genes resulting in a specific phenotype or observable physical change. Recent reports have indicated heterogeneity in gene expression even amongst identical colonies (clones). While some genes are always expressed, others are expressed with a finite probability. In this report, a mathematical framework is provided to understand the mechanism of osteoblast (bone forming cell) differentiation. A systematic approach using a combination of entropy, pair-wise dependency and Bayesian approach is used to gain insight into the dependencies and underlying network structure. Pairwise dependencies are estimated using linear correlation and mutual information. An algorithm is proposed to identify statistically significant mutual information estimates. The robustness of the dependencies and the network structure to decreasing number of colonies (colony size) and perturbation is investigated. Perturbation is achieved by generating bootstrap samples. The methods discussed are generic in nature and can be extended to similar experimental paradigms.

Entropy pairs for a measure

We define entropy pairs for an invariant measure µ on a topological dynamical system (X, T), and show they allow one to construct the maximal topological factorwith entropy 0 for µ. Then we prove that for any µ, a µ-entropy pair is always topologically so, and the reverse is true when (X, T) is uniquely ergodic.