## 3. Modeling uptake inhibition using an inverse hyperbola

In the previous example, the uptake rate started at zero at a substrate concentration of zero, and increased towards a maximum value as the substrate concentration increased. Models based on hyperbolic functions can also be used to describe a different situation, where the uptake rate starts at a maximum value and decreases as the substrate concentration increases.

An example is provided by marine microalgae (phytoplankton). At some times of the year, they have access to two main sources of nitrogen dissolved in the seawater. Nitrate is abundant, but the microalgae need to expend energy to transport nitrate into the cell. By contrast, dissolved ammonia is normally scarce (perhaps only 10% of the nitrate concentration) but is 'free' in that ammonia can diffuse through the cell membrane without any energy cost. If ammonia concentration in seawater increases, microalgae switch from using nitrate to using ammonia. A model of the uptake rate of nitrate (as a proportion of total nitrogen uptake) in relation to ammonia concentration would look like this:

Here, the uptake rate starts at its maximum value, which is 1 as all of the nitrogen uptake is in the form of nitrate. Nitrate uptake rate declines steeply initially, and then the curve gradually levels out. The curve in the figure was produced from an inverse rectangular hyperbolic model. The model takes this form:

v = v_{max} . 1 - (S/[S+K_{m}])

We have used the same notation as in the Michaelis-Menten model, so that:

- 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}.

Look at the expression 1 – (S/[S+K_{m}]) in the inverse hyperbola equation. You should see that for an ammonia concentration (S) of zero, (S/[S+K_{m}]) also has a value of zero. One minus zero is one, so v equals v_{max}. As the substrate concentration increases significantly above K_{m}, S becomes much larger than K_{m}, and the value of (S/[S+K_{m}]) tends towards one, and v tends towards zero. However, v will only come close zero when S is very much larger than K_{m}.

If the value of K_{m} is small, the initial (negative) slope of the curve is steep and the uptake rate soon decreases well below the value of v_{max}. Conversely, if K_{m} is large, the initial slope is small and uptake rate declines more slowly. In the model illustrated above, the value of v_{max} is 1, whilst K_{m} is 0.5.

## 4. Photosynthesis-light curves: another model in the hyperbola family

In previous examples, we have considered the use of a simple hyperbolic function to model the relationship between substrate uptake rate and substrate concentration. The examples in previous section used dissolved chemicals as substrates. Here, we are concerned with photosynthesis, where the substrate is light energy rather dissolved chemical nutrients, but the Inhibition of nitrate uptake modeled by an inverse rectangular hyperbola Ammonia concentration Nitrate uptake rate as a proportion of total N-uptake principles are identical and the shape of the curve is very similar. There is a wide variety of models of the relationship between photosynthetic rate and light (including variants of the rectangular hyperbola used earlier). Here, we are using a model that behaves in a similar way but is formulated in a different way.

All of the models described here are approximations – they are not necessarily based on realistic physiology although the parameters (constants) of the model can be interpreted in terms of physiological processes. This model of the relationship between photosynthetic rate and irradiance (light level) was selected by Trevor Platt and colleagues as being a good representation under a wide range of conditions, and also easy to fit to real data.

The model uses a hyperbolic trigonometric function, the hyperbolic tangent or 'tanh' (pronounced 'tansh'). This function is usually available in spreadsheets, so the model can be built easily in a spreadsheet. The model is written as:

P = P_{max} . tanh(α.E/P_{max})

- P is the rate of photosynthesis at an irradiance of E
- P
_{max}is the maximum photosynthetic rate - α is a constant corresponding to the initial slope of the curve, so has units of P/E (making the argument of the tanh dimensionless)

As E increases, the value of tanh(α.E/P_{max}) tends towards 1 and P tends towards P_{max}. The curve produced by the model looks like this:

Unlike the rectangular hyperbolae used in previous sections, the maximum value P_{max} is achieved within the range of irradiances modelled and so has some physiological relevance.

This model does not include the commonly observed inhibition of photosynthesis at very high irradiances. It can be modified to include an inhibition term, or other photosynthesis-irradiance models can be used.