Chain and Multi-Spacial Markov Chains of Time Dependence of Allele Manifestation in Cancer: Metric, Algebras and Expectations of the Reward

<p><strong>Abstract</strong></p>\n<p>Time-continuous Chain and Multi-spacial Markov chains of time dependence of allele manifestation in cancer are newly written. The allele manifestation is associated with ’non-cancer states’ and ’cancer states’ which constitute nodes (and pathways). The allele manifestation is described as directed graphs (on the opportune manifold), which are modeled according to Markov Decision Processes. The measure from the algebra on the manifold of the directed graphs is newly proven to be matched with that of the Kantorovich metric on the Borel (sub-)sets of the Markov states space. The expectation of the reward is calculated in the case of ’non-cancer states’ and in that of ’cancer states’; the scheme is ready for a path-integral approach.</p>\n<p><strong>Keywords:</strong> Cancer; chains; first-order Markov processes; allele manifestation; allele loci; metric distances; Markov models; Kantorovich measure; directed graphs; grouppoids; free semi-grouppoids.</p>