Algebraic Expressions of Conditional Expectations in Gene Regulatory Networks

Please always quote using this URN: urn:nbn:de:0297-zib-70526
  • Gene Regulatory Networks are powerful models for describing the mechanisms and dynamics inside a cell. These networks are generally large in dimension and seldom yield analytical formulations. It was shown that studying the conditional expectations between dimensions (vertices or species) of a network could lead to drastic dimension reduction. These conditional expectations were classically given by solving equations of motions derived from the Chemical Master Equation. In this paper we deviate from this convention and take an Algebraic approach instead. That is, we explore the consequences of conditional expectations being described by a polynomial function. There are two main results in this work. Firstly: if the conditional expectation can be described by a polynomial function, then coefficients of this polynomial function can be reconstructed using the classical moments. And secondly: there are dimensions in Gene Regulatory Networks which inherently have conditional expectations with algebraic forms. We demonstrate through examples, that the theory derived in this work can be used to develop new and effective numerical schemes for forward simulation and parameter inference. The algebraic line of investigation of conditional expectations has considerable scope to be applied to many different aspects of Gene Regulatory Networks; this paper serves as a preliminary commentary in this direction.

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Author:Vikram Sunkara
Document Type:ZIB-Report
Tag:Chemical Master Equations; Dimension Reduction; Markov Chains
MSC-Classification:60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Gxx Stochastic processes / 60G20 Generalized stochastic processes
65-XX NUMERICAL ANALYSIS / 65Cxx Probabilistic methods, simulation and stochastic differential equations (For theoretical aspects, see 68U20 and 60H35) / 65C40 Computational Markov chains
92-XX BIOLOGY AND OTHER NATURAL SCIENCES / 92Bxx Mathematical biology in general / 92B05 General biology and biomathematics
Date of first Publication:2018/09/24
Series (Serial Number):ZIB-Report (18-52)
ISSN:1438-0064