Applications of second order differential equations to bioreactors

Timothy Nicholls

MATH345 Mathematics Advanced Project

Spring 2006


Biochemical and bioprocess engineering is swiftly becoming one of the most important areas of applied mathematics as the use of living cells to produce marketable chemical products is becoming increasingly important. There have been many powerful advances that are transforming the field, including such applications to genetic sequencing, the production of pharmaceuticals, biologics, commodities and medical applications such as tissue engineering and gene therapy. It is a broad field drawing its fundamentals from biochemistry, microbiology, and molecular biology.

If particular interest in this area is the concept of bioreactors where enzymatic reactions are involved in the growth of micro-organisms. In fact, by 2000, products produced in this way way have seen the biotechnology industry soar in value to over $17 billion. Bioreactors also play a significant role in the wastewater industry where such processes are used to produce clean drinking water.

There are various different classifications of bioreactors including batch, fed batch or continuous, which generally refer to when and how much substrate is fed into the bioreactor over time. Bioreactor design is quite a complex engineering task. Under optimum conditions the micro-organisms or cells are able to perform their desired function with great efficiency. The bioreactor's environmental conditions like gas (i.e. air, oxygen, nitrogen, carbon dioxide), flow rates, temperature, pH and dissolved oxygen levels, and agitation speed/circulation rate need to be closely monitored and controlled to ensure that the process continues as desired.

Mathematics has had a strong history in this area, where various models have been produced to model the complex bioprocesses that have been observed. Some well known (though can be simplified_ models such as the flow reactor model and the membrane reactor model have been applied to such processes. The most notable models in this area are those proposed by Monod and later Andrews (1968) drawing on research by Haldane (1930). These models have been popular due to their ability to explain empirical data, and have thus had wide application in many industries.

As the importance of biochemical and bioprocess engineering, the amount of work done in this area is growing exponentially with significant advances being made. The literature review analyses and compiles some of the work that others have done in this area. Of particular importance in this literature is the introduction of growth inhibition into the model, as was proposed by Andrews (1968). Much of the work done in recent decades builds on his model.

This report seeks to build on previous research by analysing a generalised version of the Andrews growth inhibition model that introduces properties such as maintenance and the death of micro-organisms into the model. As with previous research, both the flow reactor and membrane reactor have been analysed in detail.

Following the analysis of these model, this report includes a discussion of future work to be done in the area. This report demonstrates the usefulness of mathematics in modelling real-life processes such as the complex chemical processes so often associated with biotechnology and highlights the importance of mathematics to future advances in this area.

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