Skye Dore-Hall

Topic

Comparisons of Ramp Functions and Michaelis-Menten Functions in Biochemical Dynamical Systems

Department of Mathematics and Statistics

Date & location

• Friday, April 12, 2024
• 9:00 A.M.
• Clearihue Building, Room B021

Examining Committee

Supervisory Committee

• Dr. Roderick Edwards, Department of Mathematics and Statistics, University of Victoria (Supervisor)
• Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Member)
• Dr. Stephanie Willerth, Department of Mechanical Engineering, UVic (Outside Member)

External Examiner

• Prof. Jean-Luc Gouzé, French National Institute for Research in Digital Science and Technology

Chair of Oral Examination

• Dr. Tao Wang, Department of Economics, UVic

Abstract

Analysis of nonlinear dynamical systems, such as those modeled using Michaelis-Menten kinetics, can be difficult. Thus, it is natural to consider whether such systems can be simplified in a way that facilitates analysis while preserving qualitative behaviour. Previously, we showed that when the Michaelis-Menten terms in a model of plant metabolism are replaced by piecewise linear approximations called ramp functions, the qualitative behaviour of the model is maintained. We then defined a limited class of systems containing ramp functions called biochemical ramp systems and studied their properties, including the existence and stability of equilibria and global flow.

Here, we expand on our previous work by reforming the definition of a biochemical ramp system to describe a wider class of systems. We study the properties of several types of biochemical ramp systems that were previously not covered by the definition, and show that their qualitative behaviour is similar to that of their Michaelis-Menten counterparts. We then introduce concepts from chemical reaction network theory, such as the Deficiency Zero and Deficiency One Theorems, and explain how they are applicable to the analysis of biochemical ramp functions, but cannot be applied to the corresponding Michaelis-Menten systems. In the last chapter, we show that when ramp functions are used in systems that do not fall under the expanded definition of a biochemical ramp system, there can be qualitative differences in behaviour between these ramp systems and their Michaelis-Menten counterparts. We end with a look at periodic behaviour in ramp systems by studying a version of the Lotka-Volterra predator-prey model containing ramp functions.