Team:Manchester/Model/Continuous Culture
Continuous Culture of Bacteria
Introduction
In an attempt to envision scaling up our project in an industrial scale, we are modelling the mathematical and theoretical background of continuous culture. Our aim is to determine the maximum output of bacteria production per unit of time which would allow us to estimate production cost and determine the profitability cost of our project.
We envision our bacteria to be cultured in a chemostat for production. A chemostat is a bioreactor in which fresh medium is continually added while culture liquid containing leftover nutrients, microorganisms, and metabolic products are continuously removed at the same rate. This technique is called continuous culture and allows microbial growth to take place under steady-state conditions - growth that occurs at a constant rate and in a constant environment. Unlike a batch culture method where bacterial cells undergo the full bacterial cell cycle, a continuous culture keeps the bacteria growing on its exponential phase of the bacterial cell cycle, thus a continuous supply of bacteria can be produced.
Background Theory
Variables - Kinetics of Bacterial Growth
The growth of bacteria in its exponential phase can be represented in the following exponential growth equation:
\begin{equation} \frac{1}{x} \frac{dx}{dt} = \mu = \frac{log_e 2}{t_d} \end{equation} where:
\(x\) is the bacteria concentration (dry weight mass/unit volume) at time \(t\)
\(μ\) is the specific growth rate
\(t_d\) is the doubling time (time required for the concentration of organism to double)
In 1942, Jacques Monod showed that there is a relationship between the specific growth rate and the concentration of a limiting growth substrate that can be represented in this equation:
\begin{equation} \mu = \mu_{max} \bigg(\frac{s}{K_s + s}\bigg) \end{equation} where:
\(s\) the concentration of a limiting growth substrate
\(μ_{max}\) is the maximum growth rate (growth rate when organism is placed in excess nutrients without any limiting factors)
\(K_s\) is the saturation constant – the value of \(s\) when: \begin{equation*} \frac{\mu}{\mu_{max}} = \frac{1}{2} \end{equation*}
A relationship between growth and utilization of substrate has also been shown by Monod by the equation::
\begin{equation} \frac{dx}{dt} = −Y \frac{ds}{dt} \end{equation} \begin{equation*} Y = \frac{\textrm{weight of bacteria formed}}{\textrm{substrate utilized}} \end{equation*}where \(Y\) is known as the yield constant
If the values of the three growth constants: \(μ_{max}\), \(K_s\) and \(Y\) are known, equation (1) to (3) provides a complete quantitative description of the ‘growth cycle’ of a batch culture.
A chemostat is a continuous flow system in which fresh growth medium is added into the vessel at a steady flow-rate (\(F\)) and culture liquid exits at the same rate. Contents within the vessel are stirred so that the growth medium is uniformly dispersed. The rate in which nutrient is exchanged in the vessel is expressed as the dilution rate (\(D\)):
\begin{equation} D = \frac{\textrm{medium flow rate}}{\textrm{culture volume}} = \frac{F}{V} \end{equation}If we assume that the bacteria within the vessel stops growing and dividing, with equal stirring and continuous flow of medium, every organism will have an equal probability of leaving the vessel within a given time. The wash-out rate (rate in which organism initially present in the vessel will be washed out) can be expressed as:
\begin{equation} - \frac{dx}{dt} = Dx \end{equation}where: \(x\) is the concentration of organisms in the vessel
