# An Algebraic Approach to Modelling the Regulation of Gene Expression

Student thesis: Phd

### Abstract

Biotechnology is witnessing a remarkable growth evident both in the types of new products and in the innovative new processes developed. More efficient process design, optimisation and troubleshooting can be achieved through a better understanding of the underlying biological processes inside the cell; a key one of which is the regulation of gene expression. For engineers such understanding is attained through mathematical modelling, and the most commonly used models of gene expression regulation are those based on differential equations, as they give quantitative results. However, those results are undermined by several difficulties including the large number of parameters some of which, such as kinetic constants, are difficult to determine. This prompted the development of qualitative models, most notably Boolean models, based on the assumption that biological variables are binary in nature, e.g. a gene can be on or off and a chemical species present or absent. There are situations however, where different actions take place in the cell at different threshold values of the biological variables, and hence the binary assumption no longer holds.The purpose of this study was to develop a method for modelling gene regulatory functions where the variables can be thought of as taking more than two discrete values.A method was developed, where, with the appropriate assumptions the biological variables can be regarded as elements of an algebraic structure known as a finite field, in which case the regulatory function can be considered as a function on such a field.The formulation was adopted from electronic engineering, and leads to a polynomial known as the Reed-Muller expansion of the discrete function.The model was first developed for the more familiar binary case. It was given three different algebraic interpretations each enabling the study of a different biological problem, albeit related to gene regulation.The first interpretation is as a function on a Boolean algebra, but using the Exclusive OR (XOR) operation instead of the OR operation. The discriminating superiority of the XOR allows a more efficient determination of the gene regulatory function from the data, a problem known as reverse engineering.The second interpretation is as a polynomial on a finite field, where analogy with the Taylor series expansion of a real valued function allowed the coefficients of the expansion to be thought of as conveying sensitivity information. Furthermore a method was devised to detect mutation in the cell by regarding the problem as detecting a fault in a digital circuit.The third interpretation is as a transform on a discrete function space, which was demonstrated to be useful in synthetic biology design.The method was then extended to the multiple-valued case and demonstrated with modelling the gene regulation of a well known example system, the bacteriophage lambda.
Date of Award 1 Aug 2011 English The University of Manchester Ferda Mavituna (Supervisor)

### Keywords

• Gene Expression
• Finite Fields
• Modelling

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