Gene Networks

Abstract

It might surprise you that networks play a role in biology, but networks are ubiquitous. All living things have DNA within their cells. This DNA contains the building blocks of an organism. The process of creating such a building block requires mRNA and proteins. The concentration of a certain mRNA molecule plays a part in the making of these proteins. In biology these processes are called transcription and translation. These interacting processes can be transformed into a mathematical model, a network. This network is a collection of nodes and edges, which represent the interactions between different mRNA concentrations. Since these interactions are unknown, random matrix theory is used to model these interactions. It is interesting how the concentrations of mRNA molecules evolve over time. A system of differential equations can be used to model these changes of concentrations over time. This report aims to discover the properties of the model of interacting organisms. For this a linear model is found, which is a system of differential equations linearised around a given equilibrium. A system can be written as a matrix, to model multiple organisms this results in a block matrix. Each block can then be envisioned to model a certain gene pool that corresponds to an organism. Interactions between these gene pools can be modelled by adding an interaction block to the off diagonal blocks of the block matrix. Later on this linear model is improved with a non-­negative constraint, concentrations are after all non-­negative, which results in a new nonlinear model. Properties of both models are found by studying the distribution of the eigenvalues. Girko’s law and Wigner’s law are two important laws from random matrix theory, that help with the determination of the eigenvalues of a random interaction matrix. For the linear model it was found that the distribution of the eigenvalues is influenced by the entries of the block matrix and by the strength of the connection between the block matrices. Once the eigenvalues and the corresponding eigenvectors are found, the solution is deterministic. For the nonlinear model it was found that the distribution of the eigenvalues are influenced in a similar way as the linear model. But due the non­negative constraint, the stability of the system is not deterministic. The system can be partly asymptotically stable, partly stable and partly unstable for different time windows. It is living on the ‘edge of chaos’.