Modeling Markovian biological systems via optimization operators

摘要

A general, computer-oriented method permitting to derive Markovian models with required (desired) properties is suggested and illustrated by examples. The method is based on the concept of a transition matrices generating (tmg) optimization operator, which is defined as a pair involving a (linear) transformationTand the associate optimization problemLT. When the latter one is solved a set of transition matrices with required properties (ergodicity, regularity etc.) is get by starting from a sequence of probability vectors {Pk} which expresses the test data. Since the corresponding measurements are inevitably subjected to errors, it is not required that {Pk} be “reached” in the step-wise evolution of the process. Instead, it is required to minimize the so-calledv-distance with respect to the probability vectors {Pk}. The optimization is performed by taking into account some constraints expressing the prior-known properties of the chain. This enables to solve the following problem: Given a sequence of (measured) probability vectors {Pk}, find a sequence of transition matrices {Pk} leading to the smallestv-distance with respect to {Pk} subject to given constraints. Some fundamental properties of the resulting Markov chains are emphasized, which are useful in modeling concrete biological systems. Thus, more realistic Markovian models are obtained starting from test data, as compared with the methods using conventional me