Continuous Description of Discrete Biological Data: Algorithms Based on a Stochastic Flow Model

Continuous Description of Discrete Biological Data: Algorithms Based on a Stochastic Flow Model

Serge V. Chernyshenko (Moscow Region State University, Moscow, Russia)
Copyright: © 2019 |Pages: 14
DOI: 10.4018/IJARB.2019010103

Abstract

The applicability of differential equations to description of integer values dynamics in bio-informatics is investigated. It is shown that a differential model may be interpreted as a continuous analogue of a stochastic flow. The method of construction of a quasi-Poisson flow on the base of multi-dimension differential equations is proposed. Mathematical correctness of the algorithm is proven. The system has been studied by a computer simulation and a discrete nature of processes has been taken into account. The proposed schema has been applied to the classical Volterra's models, which are widely used for description of biological systems. It has been demonstrated that although behaviour of discrete and continuous models is similar, some essential qualitative and quantitative differences in their dynamics take place.
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Background

The bioinformatics, as a field, includes a number of mathematical approaches to description of various aspects of biological systems’ functioning (Bioinformatics…, 2013; Shen, Tuszynski, 2008). At the present time, two ways of modelling – by differential equations and by stochastic processes – are developing, mainly, independently. An observation of the approaches can be finding, particularly, in (Brown, Rothery, 1993; Erciyes, 2015; Kunita, 1986). In some meaning they are opposite ways: differential equations are continuous and, usually, deterministic models; and stochastic processes are discrete and, naturally, stochastic ones. In the same time they are used in many cases for modelling of the same real processes (as city traffic, epidermis and many others). They can reflect different aspects of modelling systems and can be good supplements each other; but their harmonisation is a separate complex problem. It would be very important to integrate the both approaches in one general approach. It is especially actual for researches on bioinformatics, when one deals with discrete data (like genome), but directs to describe and predict continuous effects on a macro level.

Within studies, devoted to combination of the two approaches, there are a number of articles, especially oriented to bioinformatics problems (Anderson, Chaplain,1998). The work (Smilde,.et al, 2015) shows a way to understand some potential stochastic features of biological systems on the base of their differential models. The authors used well-known covariance analysis to estimate variance in some biological indices. In the article (Mei et al, 2015), stochastic flows are directly built into the bioinformatics differential equations and can be used for processing web-based biological data through a web-interface of the model. Also an Internet-based tools for biological data processing, combined different modelling technique, including differential equations and stochastic processes, are proposed in (Yu et al, 2013).

To describe stochastic effects in biological systems, there are a number of approaches based on different computation methods (Tavassoly et al., 2018). Constructed computational algorithms can be distributed (Boczkowski et al., 2018) or realized as a special modeling tool (Azeloglu, Iyengar, 2015).

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