## 2. The Michaelis-Menten model of substrate uptake – a simple rectangular hyperbola

Studies of the growth rates of microbes under controlled conditions commonly examine the relationship between some measure of growth rate, such as the incorporation of a radio-labelled substrate, and the substrate concentration. An example might be the uptake rate of bacteria grown at different concentrations of glucose, measured by the incorporation of 14C-labelled glucose. Such a relationship is very rarely linear, because microorganisms absorb dissolved materials through their cell membranes and either the cell surface area or the number of transport sites in the cell membrane place an upper limit on the amount of substrate that a cell can absorb over a given time. Because of this, the relationship typically looks like this:

At low substrate concentrations, there is a roughly linear relationship between the uptake rate and the substrate concentration. This means that if the concentration is doubled, the uptake rate will also double. But at higher concentrations, the cell cannot continue absorbing further substrate at the same rate. Here, the rate at which uptake rate increases declines gradually as the uptake rate approaches the maximum that the cell can sustain. The curve in the figure was produced using a Uptake kinetics modeled by a rectangular hyperbola Substrate concentration Uptake rate rectangular hyperbolic model, which is often called Michaelis-Menten kinetics in this context. The model takes this form:

v = v_{max} . (S/[S+K_{m}])

In this model:

- v is the uptake rate at a substrate concentration of S
- v
_{max}is the maximum uptake rate - K
_{m}is a constant, which is the substrate concentration at which the uptake rate is half of v_{max}.

If you look at the part of the rectangular hyperbola that is enclosed by brackets, (S/[S+K_{m}]), you can see that as the substrate concentration increases, S becomes much larger than K_{m}. This means that the value of (S/[S+K_{m}]) tends towards one, and v tends towards v_{max}.

We can show that K_{m} is the substrate concentration where v is half of v_{max} by considering the case where S is equal to K_{m}:

S = K_{m}

(S/[S+K_{m}]) = (K_{m}/[K_{m}+K_{m}]) = 0.5

v = v_{max} . 0.5

If the value of K_{m} is small, the initial slope of the curve is steep and the uptake rate soon approaches the value of v_{max}. Conversely, if K_{m} is large, the initial slope is small and uptake rates approaching v_{max} are only attained at very high substrate concentrations. In the model illustrated above, the value of v_{max} is 10, whilst K_{m} is 2.5. Note that even at the highest substrate concentration, which is ten times the value of K_{m}, _{v} is clearly less than v_{max}.