Mathematical Models of Facilitated Diffusion Processes in Physiology

Author: Dalal Zaben Alshammari

Alshammari, Dalal Zaben, 2018 Mathematical Models of Facilitated Diffusion Processes in Physiology, Flinders University, College of Science and Engineering

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Abstract

Combining diffusion and chemical reactions introduces many interesting effects in nature, particularly in physiology. Diffusion by itself, called simple diffusion, can occur in gases, liquids and solids. A simple example of diffusion in gases is when we spray a perfume and after a few minutes its smell spreads throughout the room. The diffusion process can be observed in liquids as well: for instance, when we put a small drop of food colouring into a cup of water, after a while, the colour of the water changes. Simple diffusion also occurs continuously in the human body while we breathe, since gas exchange occurs between our lungs and the air that we breathe. The reason behind the diffusion process is the movement of molecules from areas of high concentration to areas of lower concentration to achieve an equilibrium situation of uniform concentration between those two areas. The more complex case where diffusion occurs simultaneously as a chemical or other reaction between the component substances is called reaction diffusion.

The facilitated diffusion process is a special case of reaction diffusion and it affects many aspects of life, making it one of the most important processes in all of biology. During facilitated diffusion, a molecule (the ligand) joins another molecule (the carrier), which is typically a large protein, to form a ligand{carrier complex. This complex then provides an alternative way for the ligand to move or diffuse. Facilitated diffusion and simple diffusion are similar in that both involve the movement of molecules from a high-concentration region to a low-concentration region. They are also both passive processes since the molecules transport is performed without any energy input. However, in facilitated diffusion, the transport of molecules will only occur if it is facilitated or assisted by an appropriate carrier. Facilitated diffusion has two transport pathways, the direct and the facilitated. Further, facilitated diffusion requires both a forward reaction and a backward reaction to occur easily, the first one to associate the ligand with the carrier and the second to dissociate it from the carrier. The most common example of facilitated diffusion in the human body is that of oxygen by haemoglobin in red blood cells or by myoglobin in muscle tissues. In this thesis, a mathematical model of the facilitated diffusion of oxygen within muscle cells is studied. In this case, the carrier molecule is myoglobin. This dissertation has many goals, and one is to solve the second-order partial differential equation that expresses the mathematical model in two ways. This first is to solve the equation by using an equilibrium approximation. The second way is to solve the full partial differentiation equations numerically using the FEniCS program. We explore the extent to which various system parameters aect the enhancement of transport in some simple situations and compare the results of the two different solution methods. We show that a balance between the forward reaction and backward reaction rates is important to ensure maximum enhanced transport.

Keywords: Diffusion, Facilitated diffusion, Reaction diffusion, Oxygen transport, Carrier molecule

Subject: Mathematics thesis

Thesis type: Masters
Completed: 2018
School: College of Science and Engineering
Supervisor: Murk Bottema