The kinetics of cell growth and generation of products is one of the important features of monitoring a fermentation and helps to explain the functioning of microbial cell models. Much of microbial cell growth is about measuring the increase in cell mass which is related to substrate use and product formation.
Each of these can be defined by a rate equation. That means there is a measure that can be made over a time period. It functions in the same way as in food kinetics where a characteristic of a product is monitored such as a rate of browning or the rate of loss of a nutrient. So, the same principles are applied to cell growth etc.
From the outset there are two types of rate equation to be thought about. The first is a volumetric rate which relates to a change in a parameter per unit volume and time. With cells, this may be a change in cell mass per unit volume of a fermentation per unit time.
Cell growth is modelled in a variety of methods. It involves unstructured segregated models, substrate inhibited models and product inhibited models.
The Monod Model
The most common model used for cell growth is that devised by Monod from 1940. he proposed this equation to relate microbial growth rate μ in an aqueous environment to the concentration of the limiting nutrient [S]. It has the same form as the Michaelis-Menten equation for enzyme catalysed reactions.
μ = μm [S] / (Ks + [S] )
Here, [S] is the concentration of the limiting substrate. The μ is the growth rate for a defined microorganism whilst μm is the maximum growth rate. The constant Ks is the ‘half-velocity constant’ or saturation constant and is the value of [S] when the growth rate is half that of the maximum growth rate, i.e. μ/μm .= 1/2.
The maximum microbial growth rate and the half-velocity constant Ks will be different for each microorganism and also the conditions of the fermentation.
The Monod model is very simple because cell growth is extremely complex. It works however when the concentrations of any inhibitors to cell growth are low. It is worth checking the alternative models.
Approaches To Cell Growth Modelling
There are four basic types of model which apply to other model systems. The unstructured models are ones where the cell population is treated as a single component. The structured model is more complex because the cell population is treated as a multi-component system.
The nonsegregated models are ones where cells are treated as homogenous whilst segregated models treat cells as heterogenous.
The unstructured nonsegregated models are the simplest and apply to many different types of fermentations. Here the cell population is viewed as a single entity and the cells are regarded as homogenous. The structured segregated models are ones where the cell population is regarded as a multi-component system and the cells are treated as heterogenous. This particular model is most realistic but complex to analyse and mathematically compute.
The Monod model is a great example of an unstructured, nonsegregated model. The main assumptions are that there is a single limiting substrate. There is a semi-empirical relatonship to be understood here. It means there is a simngle enzyme sytem that follows Michaelis-Menten kinetics which is responsible for uptake of the substrate. The amoun tof enzyme is sufficiently low to be growth limiting. It implies when the cell population density and low and the cell growth is slow. There are at least 4 other unstructured, nonsehgrated models where we assume one limiting substrate.
The Blackman equation has two forms:
μ = μm when [S] is greater than or equal to 2Ks
When [S] is less than 2Ks:-
μ = μm [S] / 2Ks
This model fits the data better but there is a discontinuity here which presents issues.
The Tessier equation:
μ = μm [1 – exp (-K[S]]
The Tessier equation with its exponent also provides a good fit.
The Moser equation is:-
μ = μm [S]n / (Ks + [S]n )
The Moser equation only obeys the Monod equation when n = 1. The value of n is the key figure when it comes to fitting this equation.
The final equation is the Contois equation:-
μ = μm [S] / (Ksx*[S]x + [S] )
This equation has a saturation constant Ksx*[S]x
The saturation constant is proportional to the cell concentration.
The next type of model is a development of the Monod equation which is called the Extended Monod model.
μ = μm {[S] – [S]min} / (Ks + [S] + [S]min )
The Extended Monod model uses a term denoting the minimal substrate concentration needed for cell growth, [S]min
The Monod model has also been applied to two limiting substrates:-
μ = μm *[S1] / (Ks1 + [S1] )*[S2] / (Ks2 + [S2] )
The Monod model has been modified for rapidly-growing and dense cultures. Some new models have been developed:-
μ = μm [S] / (Ks0[S]0 + [S] )
μ = μm [S] / (Ks0[S]0 +Ks1 + [S] )
The [S]0*is the initial substrate concentration and Ks0* is a dimensionless constant.
The Monod model cannot model any forms of substrate inhibition. So, substrate inhibition means increasing the substrate concentration beyond particular values reduces the cell growth rate.
The Luedeking-Piret Model
The Luedeking-Piret model is based on an unstructured equation that was first developed in 1959 for lactic acid production. It applies to both batch and continuous culture. It has been used with some modifications at times for a variety of fermentations be they prokaryotic or eukaryotic.
The basic equation is:
dP/dt = α (dX/dt) + βX
The equation relates the rate of product formation as a sum of two terms.
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