Theoretical Study of the One Self-Regulating Gene in the Modified Wagner Model

Christophe Guyeux,Luigi Marangio,Jean-François Couchot,Arnaud Rouzic,Jacques Bahi

doi:10.3390/math6040058

Abstract

Predicting how a genetic change affects a given character is a major challenge in biology, and being able to tackle this problem relies on our ability to develop realistic models of gene networks. However, such models are rarely tractable mathematically. In this paper, we propose a mathematical analysis of the sigmoid variant of the Wagner gene-network model. By considering the simplest case, that is, one unique self-regulating gene, we show that numerical simulations are not the only tool available to study such models: theoretical studies can be done too, by mathematical analysis of discrete dynamical systems. It is first shown that the particular sigmoid function can be theoretically investigated. Secondly, we provide an illustration of how to apply such investigations in the case of the dynamical system representing the one self-regulating gene. In this context, we focused on the composite function f a ( m . x ) where f a is the parametric sigmoid function and m is a scalar not in { 0 , 1 } and we have proven that the number of fixed-point can be deduced theoretically, according to the values of a and m.

Highlights

Predicting the effect of a genetic change on a character of interest remains a major challenge in biology [1]
[8] provides a complete review of biological systems inspired by network science
An interesting subset of gene network models does not aim at predicting the behavior of a specific group of identified genes in an organism, but are rather used as a general abstraction of a gene network, in order to study their evolutionary properties in individual-based simulations [9,10]

Summary

Introduction

Predicting the effect of a genetic change (differences in the DNA molecule) on a character of interest (which can be related to, e.g., human health, plant and animal production, or evolutionary differences between species) remains a major challenge in biology [1]. The main purpose of such models is to ensure that any combination M, S0 can be solved computationally (e.g., as the state of the network ST after T time-steps) within a predictable (most of the time, constant) amount of time This is of major importance in individual-based computer simulations or other numerical studies in which the network structure M can mutate and evolve over time. The lack of mathematical tractability remains, problematic, as it makes it difficult to compare simulation results with classical predictions from population and quantitative genetics models [17,18] This is why, in this article, a mathematical analysis of the sigmoid variant of the Wagner gene-network model is presented, focusing on the simplest case, that is, one unique self-regulating gene. This article ends by a conclusion section, in which the contribution is summarized and intended future work is outlined

Studying the Sigmoid Function

Introducing the Considered Sigmoid

About λ and μ Parameters

Fixed Point of f a

The Discrete Dynamical System under Consideration

Conclusions and Future Work