Lecture 9b: Nutrient Uptake
The Michaelis-Menten Model in Detail
Nutrient uptake, Michaelis-Menten kinetics, Algal nutrition
1 Algal Nutrient Uptake Experiments
The animation below follows one nitrate pulse from the flask, through a depletion curve, to the Michaelis–Menten relationship. Pause at the prediction questions. The walk-through below explains the calculations and connects the moving points to the uptake process.
Download the animation · Illustrative measurements
The animation has on-screen explanations and no audio. Its numbers are illustrative, chosen to match the scale of the worked example in this lecture. They are not measurements from a particular species.
The uptake process of nitrogen by algae can be measured in situ or under controlled conditions in the laboratory. Nutrient uptake is often determined by measuring the disappearance of a nutrient from the culture medium over a time interval after the addition of the alga. These experiments allow the calculation of depletion curves, and from the depletion curve the uptake kinetics can be determined.
To do this, we use two types of experiments: multiple flask experiments and perturbation experiments. In both, a known amount of nutrient (the substrate) is introduced into the water. We measure the decrease in its concentration over a timed interval, convert that decrease into an amount using the water volume, and divide by the algal mass and elapsed time. This gives a positive uptake rate, \(V\), for a corresponding substrate concentration, \([S]\). The primary difference between the methods lies in the experimental setup and the data analysis.
The Blender film, Measuring a changing uptake rate, develops the same numerical example as the Manim animation above. It adds a comparison of the two experimental designs, a biomass intervention and a separate demonstration of sampling losses. Use its worked questions to check the distinction between concentration, amount and rate.
2 Multiple Flask Method
The multiple flask method is a widely used experimental technique for measuring the nutrient uptake kinetics of macroalgae. This method is used to determine how macroalgae absorb nutrients (such as nitrogen or phosphorus) from their environment under different substrate concentrations. The goal is to generate a relationship between the rate of nutrient uptake and the nutrient concentration available to the macroalgae. The experimental data can then be used to characterise the nutrient uptake kinetics, such as maximum uptake rate (\(V_{max}\)) and half-saturation constant (\(K_m\)). These model parameters tell us about the mechanisms that algae use to remove nitrogen or other nutrients from their environmental to be incorporated into biomass.
2.1 Experimental setup
In the multiple flask method, a series of flasks are prepared, each containing a different initial concentration of the nutrient (substrate) to span the range of nutrient levels typically encountered by the macroalgae in its natural habitat. This allows for measurements of nutrient uptake rates across a spectrum of substrate concentrations, from low to high.
2.2 Steps of the multiple flask experiment
- Substrate Preparation: Prepare several flasks, each with a known initial concentration of the nutrient (e.g., nitrogen) in solution. These concentrations should cover a range of interest, often from nutrient-limiting to saturating levels.
- Algal Introduction: Introduce a known biomass of macroalgae into each flask. The biomass should be standardised across all flasks (e.g., 4.5 g of fresh macroalgal tissue per flask).
- Incubation: The flasks are incubated for a defined time period, typically 20–30 minutes, under controlled environmental conditions such as light and temperature.
- Sampling: At the beginning of the incubation (\(t=0\)) and at the end of the incubation period (e.g., \(t=30\) minutes), water samples are taken from each flask to measure the concentration of the nutrient in the water.
- Nutrient Analysis: The concentration of the nutrient in each water sample is analysed using chemical methods (e.g., colorimetric analysis or ion chromatography).
The difference in nutrient concentration between the start and end of the incubation reflects the amount of nutrient taken up by the macroalgae during the experiment.
2.3 Data collected
- Initial substrate concentrations (\([S_{\text{initial}}]\)) in each flask.
- Final substrate concentrations (\([S_{\text{final}}]\)) after the incubation period.
- Time of incubation (\(\Delta_t\)).
- Algal biomass in each flask (usually standardised, e.g., 4.5 g fresh mass).
2.4 Calculations for determining nutrient uptake rate (\(V\))
The following steps outline how to calculate the nutrient uptake rate (\(V\)) from the experimental data obtained using the multiple flask method (apply to data obtained from each flask). These steps convert changes in nutrient concentration into actual uptake rates, adjusted for algal biomass and incubation time.
Step 1: calculate the change in nutrient concentration (\(\Delta[S]\))
To determine how much nutrient was taken up during the incubation, subtract the final nutrient concentration from the initial nutrient concentration:
\[ \Delta [S] = [S_{\text{initial}}] - [S_{\text{final}}] \]
For example: \[ \Delta [S] = 25 \, \mu M - 9.9 \, \mu M = 15.1 \, \mu M \]
This gives the reduction in nutrient concentration over the time period but does not yet account for the volume of the flask or the biomass of algae.
Step 2: convert concentrations to mass of nutrient present per flask
Convert the concentration of the nutrient (in μmol.L⁻¹) into the actual mass of nutrient (in μg) present in the flask. To do this, use the molecular mass (MM) of the nutrient (e.g., nitrogen), which is 14.0067 g.mol⁻¹ for N.
For example: \[ 25 \, \mu M = 25 \, \mu mol/L \times 14.0067 \, \frac{g}{mol} = 350.17 \, \mu g \, N \, \text{(per liter)} \] \[ 9.9 \, \mu M = 9.9 \, \mu mol/L \times 14.0067 \, \frac{g}{mol} = 138.67 \, \mu g \, N \, \text{(per liter)} \]
Next, account for the volume of the flask (e.g., 500 mL). Since the above values are for 1 liter, divide by 2 to find the mass in 500 mL:
\[ \text{Mass of N at the start} = 350.17 \, \mu g / 2 = 175.09 \, \mu g \] \[ \text{Mass of N at the end} = 138.67 \, \mu g / 2 = 69.34 \, \mu g \]
Step 3: calculate the amount of nutrient taken up by the alga
Now, calculate how much nutrient was taken up by the algae during the incubation:
\[ \Delta \text{Mass of N} = 175.09 \, \mu g - 69.34 \, \mu g = 105.75 \, \mu g \, N \]
This represents the total amount of nitrogen removed from the water by the algal biomass during the 20-minute incubation.
Step 4: normalise nutrient uptake by algal biomass
To determine how much nutrient was taken up per unit mass of algae, divide the total nutrient uptake by the biomass of algae in the flask (e.g., 4.5 g):
\[ \text{N taken up per gram} = \frac{105.75 \, \mu g \, N}{4.5 \, g} = 23.5 \, \mu g \, N/g \]
This is the amount taken up per gram during the 20-minute incubation. Divide by elapsed time to obtain a rate.
Step 5: calculate the nutrient uptake rate per hour
If the experiment lasted 20 minutes, but the uptake rate needs to be expressed on an hourly basis, multiply this amount per gram by 3 h⁻¹ (because 20 minutes is one third of an hour):
\[ \text{Nutrient uptake rate per hour} = 23.5 \, \mu g \, N/g \times 3\,\mathrm{h^{-1}} = 70.50 \, \mu g \, N/g/hr \]
2.5 Final workflow for calculating nutrient uptake rate
- Determine the change in nutrient concentration between the start and end of the experiment for each flask: \[ \Delta [S] = [S_{\text{initial}}] - [S_{\text{final}}] \]
- Convert concentrations to mass of nutrient (e.g., μg N) using the molecular mass and flask volume: \[ \text{Mass of nutrient} = [S] \times \Omega \times \text{MM of nutrient} \]
- Calculate the amount of nutrient taken up by the algae: \[ \Delta \text{Mass of nutrient} = \text{Mass of nutrient (initial)} - \text{Mass of nutrient (final)} \]
- Normalise the nutrient uptake by the algal biomass: \[ \text{N taken up per gram} = \frac{\Delta \text{Mass of nutrient}}{\text{Algal biomass}} \]
- Divide the amount per gram by elapsed time, using \(\Delta t\) in minutes to obtain an hourly rate: \[ \text{Nutrient uptake rate (hourly)} = \text{N taken up per gram} \times \frac{60}{\Delta t} \]
3 Perturbation Method
The perturbation method is an alternative approach for measuring nutrient uptake kinetics in macroalgae. This method involves a single flask where a relatively high initial concentration of a substrate nutrient is introduced, typically set at a level that is ecologically relevant to the system being studied. Rather than using multiple flasks with different initial nutrient concentrations, the perturbation method measures the decrease in nutrient concentration over time within the same flask. By sampling the substrate concentration at regular time intervals (e.g., every 10 or 20 minutes), we can calculate nutrient uptake rates for each time period.
3.1 Experimental setup
In the perturbation method, a single flask is prepared with a high initial concentration of the nutrient (e.g., nitrogen), and a known amount of macroalgal biomass is introduced. The flask is incubated, and water samples are taken at regular intervals to track the decrease in nutrient concentration. The resultant data allow us to calculate the nutrient uptake rate for each time interval.
Steps of the perturbation experiment
- Substrate Preparation: Add a known and high concentration of the nutrient (e.g., 25 μM nitrogen) to the flask. The concentration should be high enough to ensure measurable changes over the module of the experiment but ecologically relevant.
- Algal Introduction: Introduce a known biomass of a macroalga into the flask (e.g., 4.5 g of fresh macroalgal tissue).
- Incubation and Sampling: The flask is incubated, and water samples are taken at regular intervals (e.g., every 10 or 20 minutes) to measure the nutrient concentration at each time point.
- Nutrient Analysis: The concentration of the nutrient in each water sample is analysed to determine how much nutrient remains at each time point.
- Data Collection: The change in nutrient concentration between each successive time point is used to calculate the nutrient uptake rate over the interval.
The resulting data from the perturbation method consist of a time series of substrate concentrations paired with calculated nutrient uptake rates over specific time intervals.
3.2 Data collected
- Initial substrate concentration (\([S_{initial}]\)) and substrate concentrations at subsequent time points (\([S_{t1}]\), \([S_{t2}]\), …).
- Time intervals (\(\Delta_t\), e.g., every 5 or 10 minutes).
- Algal biomass in the flask (e.g., 4.5 g of fresh mass).
By plotting the remaining substrate concentration against each time point at which we sampled the water for nutrient measurement, we can construct a nutrient depletion curve. From this, we can observe how the nutrient is taken up by the macroalga and calculate the nutrient uptake rate at different stages of the experiment.
3.3 Calculations for determining nutrient uptake rate (\(V\))
Once the experimental data have been collected, the next step is to calculate the nutrient uptake rate for each time interval based on the reduction in nutrient concentration between successive time points. The following steps outline how to perform these calculations.
Step 1: calculate the change in nutrient concentration (\(\delta [S]\))
To determine how much nutrient has been taken up during a specific time interval (e.g., the first 5 minutes, for which \(\Delta t = 5\;\mathrm{min}\)), subtract the nutrient concentration at the end of the interval from the concentration at the start:
\[ \Delta [S] = [S_{\text{start}}] - [S_{\text{end}}] \]
For example: \[ \Delta [S] = 25 \, \mu M - 21.3 \, \mu M = 3.7 \, \mu M \]
This gives the reduction in nutrient concentration over the 5-minute interval.
Step 2: convert concentrations to mass of nutrient present per flask
Convert the nutrient concentration (in μmol.L⁻¹) into the mass of nutrient (in μg) present in the flask. To do this, use the molecular mass (MM) of the nutrient, which is 14.0067 g.mol⁻¹ for nitrogen.
For example: \[ 25 \, \mu M = 25 \, \mu mol/L \times 14.0067 \, \frac{g}{mol} = 350.17 \, \mu g \, N \, \text{(per liter)} \] \[ 21.3 \, \mu M = 21.3 \, \mu mol/L \times 14.0067 \, \frac{g}{mol} = 298.34 \, \mu g \, N \, \text{(per liter)} \]
Since the flask contains 500 mL (0.5 L) of solution, divide the values by 2 to get the mass of nitrogen in the 500 mL flask:
\[ \text{Mass of N at the start} = 350.17 \, \mu g / 2 = 175.09 \, \mu g \] \[ \text{Mass of N at the end} = 298.34 \, \mu g / 2 = 149.17 \, \mu g \]
Step 3: calculate the amount of nutrient taken up by the alga
Next, calculate the amount of nitrogen taken up by the algae during the 5-minute interval:
\[ \Delta \text{Mass of N} = 175.09 \, \mu g - 149.17 \, \mu g = 25.92 \, \mu g \, N \]
This represents the total amount of nitrogen removed from the water by the algal biomass in the 5-minute period.
Step 4: normalise nutrient uptake by algal biomass
To determine how much nitrogen was taken up per unit mass of algae, divide the total nitrogen uptake by the algal biomass (e.g., 4.5 g):
\[ \text{N taken up per gram} = \frac{25.92 \, \mu g \, N}{4.5 \, g} = 5.76 \, \mu g \, N/g \]
This is the amount taken up per gram during the five-minute interval. Divide by elapsed time to obtain a rate.
Step 5: calculate the nutrient uptake rate per hour
Divide the amount taken up per gram by \(5/60\;\mathrm{h}\) to obtain an hourly rate. This is equivalent to multiplying by \(12\;\mathrm{h^{-1}}\):
\[ \text{Nutrient uptake rate per hour} = 5.76 \, \mu g \, N/g \times 12\,\mathrm{h^{-1}} = 69.12 \, \mu g \, N/g/hr \]
This uptake rate relates to the specific time interval and can be used to track changes in \([V]\) over time. In this example, this uptake rate relates to the first 5 minutes of the experiment. The average \([S]\) during this intervals was \((25 \, \mu M + 21.3 \, \mu M)/2 = 23.15 \, \mu M\).
Repeat these steps for each remaining intervals and express \([V]\) relative the the mean \([S]\) for each interval (some authors use the \([S]\) at the start of the interval instead of the mean for the interval).
3.4 Final workflow for calculating nutrient uptake rate
- Determine the change in nutrient concentration between successive time points: \[ \Delta [S] = [S_{\text{start}}] - [S_{\text{end}}] \]
- Convert concentrations to mass of nutrient using the molecular mass and flask volume: \[ \text{Mass of nutrient} = [S] \times \Omega \times \text{MM of nutrient} \]
- Calculate the amount of nutrient taken up by the algae during the time interval: \[ \Delta \text{Mass of nutrient} = \text{Mass of nutrient (start)} - \text{Mass of nutrient (end)} \]
- Normalise the nutrient uptake by the algal biomass: \[ \text{N taken up per gram} = \frac{\Delta \text{Mass of nutrient}}{\text{Algal biomass}} \]
- Divide the amount per gram by elapsed time, using \(\Delta t\) in minutes to obtain an hourly rate: \[ \text{Nutrient uptake rate (hourly)} = \text{N taken up per gram} \times \frac{60}{\Delta t} \]
The important differences between the multiple flask and perturbation experiments are summarised in Table 1.
| Feature | Multiple Flask Experiments | Perturbation Experiments |
|---|---|---|
| Experimental Setup | Multiple flasks, each with different \([S]\) | Single flask with initial high \([S]\) |
| Data Independence | Data points are independent | Data points are correlated (repeated measures) |
| Analysis | Nonlinear least squares regression (NLS) | Nonlinear mixed model (NLMM) |
| R Function | nls() |
nlme::nlme() |
Our choice between multiple flask and perturbation experiments depends on our research questions and experimental constraints. In both methods, we must consider all sources of error and variability, such as measurement error, the type of nutrient, the physiological state of the alga, the light intensity, the experimental temperature, and other variables that might affect the uptake response.
4 Follow the Nitrogen: Reading the Animation
4.1 First, keep track of where the nitrogen is
Imagine a piece of seaweed in \(0.50\;\mathrm{L}\) of well-mixed seawater. Its fresh mass is \(4.5\;\mathrm{g}\). Add a small volume of concentrated nitrate solution to bring the external concentration to \(25\;\mu\mathrm{mol\,N\,L^{-1}}\). This is the perturbation: a sudden change in the supply of nitrogen.
The gold dots represent parcels of nitrate-N in the water. Follow them into the seaweed. Their disappearance from the water is accompanied by an increase in the nitrogen held inside the tissue, where it may be stored or assimilated. Uptake does not require an immediately visible increase in the size of the seaweed.
There are initially \(25\times0.50=12.5\;\mu\mathrm{mol}\) of added N in the flask. We keep a balance of this added nitrogen:
\[ N_{\mathrm{water}}(t)+U(t)=12.5\;\mu\mathrm{mol\,N}, \qquad U(t)=\Omega\,[S_0-S(t)]. \]
Here, \(U(t)\) is the amount taken up since the pulse, and \(\Omega\) is the water volume in litres. The seaweed already contained nitrogen before the experiment. That pre-existing nitrogen is outside this balance of the newly added pulse.
4.2 Then read the depletion curve
The horizontal axis is time. The vertical axis is the concentration still in the water, \(S\). Each new measurement extends the depletion curve. A steep decline means that a large amount of nitrogen has left the water during that interval. A shallow decline means that less has left over the same amount of time.
In the animation, the no-seaweed control remains at \(25\;\mu\mathrm{mol\,L^{-1}}\). This is what we would expect if the experimental conditions cause no other removal of dissolved N. In a real experiment, compare that control with the seaweed flask before attributing all disappearance to the seaweed. Microbial uptake, adsorption, nutrient release and sampling losses can complicate the balance.
4.3 Turn a slope into an uptake rate
The depletion slope is negative because \(S\) falls with time. We report uptake as a positive quantity by using \(S_{\mathrm{start}}-S_{\mathrm{end}}\). For a five-minute interval, with concentrations in \(\mu\mathrm{mol\,N\,L^{-1}}\):
\[ \overline V =\frac{(S_{\mathrm{start}}-S_{\mathrm{end}})\,\Omega}{M\,(5/60)} \quad\left[\mu\mathrm{mol\,N\,g^{-1}\,h^{-1}}\right]. \]
The volume converts concentration into an amount, \(M\) standardises that amount per gram of seaweed, and \(5/60\) expresses the interval in hours. Keep those three operations in mind. They tell you what each part of the calculation is doing.
This expression assumes that the water volume is effectively constant over the interval and that the concentration change has been attributed appropriately to uptake. If sample withdrawals appreciably reduce volume, use the remaining volume for the following interval and account for the exported nutrient in the overall inventory. The sampling example shows why loss of nitrogen from the flask is not always uptake.
| Interval | Start \(S\) | End \(S\) | Mean of endpoint concentrations | Mean uptake rate \(\overline V\) |
|---|---|---|---|---|
| 0–5 min | 25.00 | 21.30 | 23.15 | 4.93 |
| 30–35 min | 5.54 | 3.43 | 4.48 | 2.81 |
Concentrations are in \(\mu\mathrm{mol\,N\,L^{-1}}\). Rates are in \(\mu\mathrm{mol\,N\,g^{-1}\,h^{-1}}\), using fresh mass. These are rounded values from the animation’s model. The first rate is about \(69.1\;\mu\mathrm{g\,N\,g^{-1}\,h^{-1}}\) after multiplying by \(14.0067\;\mu\mathrm{g\,N}\) per \(\mu\mathrm{mol\,N}\), consistent with the worked example above.
4.4 Move each interval onto the \(V\)–\(S\) graph
Take the first pair, approximately \((23.15,\;4.93)\). Plot the concentration horizontally and the uptake rate vertically. Repeat for the next interval. You are now constructing a different graph from the same experiment.
Watch the order in which the points appear. The early intervals lie towards the upper right, where both concentration and rate are high. Later intervals appear further left and lower down. Time carries us from right to left along this curve during depletion. Although we conventionally read an increasing concentration axis from left to right, the flask is losing nitrate as the experiment proceeds.
The midpoint of the two endpoint concentrations is a convenient approximation to the concentration experienced over a short interval. The calculated rate is also an interval average. It therefore sits close to, rather than necessarily exactly on, the instantaneous Michaelis–Menten curve. Very long intervals can hide substantial changes in rate.
4.5 Separate an amount from a rate
During the replay, the green curve shows accumulated uptake, \(U(t)\). It rises as nitrogen enters the seaweed. Its slope becomes shallower as uptake slows. Near the end of the experiment, the seaweed has acquired almost all of the added nitrogen, while its current uptake rate is very low.
The moving tangents connect these observations. With time expressed in hours,
\[ V(t)=-\frac{\Omega}{M}\frac{\mathrm dS}{\mathrm dt} =\frac{1}{M}\frac{\mathrm dU}{\mathrm dt}. \]
If time is expressed in minutes, multiply the right-hand expressions by 60 to report an hourly rate. A high point on the accumulated-uptake curve means that much N has already been taken up. A steep tangent means that uptake is currently fast.
4.6 Explain the bend, then test your explanation
The animation uses \(V_{\max}=6\;\mu\mathrm{mol\,N\,g^{-1}\,h^{-1}}\) and \(K_s=5\;\mu\mathrm{mol\,N\,L^{-1}}\). At \(S=K_s\), the uptake rate is \(3\), half of \(V_{\max}\). This occurs about \(31.2\) minutes into this particular experiment. \(K_s\) is a concentration, so it does not itself tell us how many minutes an experiment will take.
At low external concentration, nutrient arrival restricts uptake. The membrane sketch shows sites spending time waiting. When nitrate is abundant, the finite capacity of transport and subsequent processing becomes increasingly important. Doubling \(S\) from \(25\) to \(50\) raises the modelled rate from \(5.00\) to only \(5.45\). The rate approaches \(V_{\max}\) asymptotically. A saturating curve alone does not identify the detailed transport mechanism.
At very low \(S\), \(V\approx\alpha S\), where \(\alpha=V_{\max}/K_s\). This initial slope describes uptake responsiveness when nitrate is scarce. Compare \(V_{\max}\) as well as \(K_s\) when comparing algae: a smaller \(K_s\) alone does not guarantee a larger absolute uptake rate if their capacities differ.
Now predict the second pulse. If we restore the nitrate concentration to \(25\), while keeping the uptake system unchanged, the point returns up and right along the same \(V\)–\(S\) curve. The next depletion segment becomes steep again. The accumulated amount already taken up remains in the seaweed. This intervention tests the explanation that the first slowdown resulted from declining external supply.
This illustration assumes fixed water volume, algal fresh mass, mixing, temperature, light, \(V_{\max}\) and \(K_s\), with no other N inputs or losses between pulses. Sampling removes a negligible volume. In real experiments, account for the water and nutrients removed in samples, and measure controls and replicate flasks. Internal storage, nutritional history and regulation can also change uptake capacity during a run. A failure to recover after the second pulse would give us a reason to investigate those processes.
The nitrate example is consistent with the rate-saturating response described for Gracilaria gracilis in Smit (2002), which also shows why water movement and the seaweed’s nutritional history matter. The animation’s parameter values are illustrative.
- A depletion curve becomes almost horizontal. What has happened to the external concentration and to the uptake rate? What additional evidence would you need before concluding that the seaweed had stopped functioning?
- Late in the experiment, accumulated uptake is high but uptake rate is low. Explain how both statements can be true.
- Double the seaweed mass while holding water volume, initial \(S\), and uptake parameters per gram constant. Predict the initial depletion slope, the initial rate per gram, and the total amount of pulse N eventually available for uptake.
- After a second nitrate pulse, uptake remains slow. Suggest an explanation and an experiment that would distinguish it from low external nutrient supply.
Check your reasoning
- In this model, external N is nearly depleted and uptake is slow. A no-seaweed control helps identify other losses. A second pulse tests whether uptake can recover when supply is restored.
- The accumulated amount records the history of uptake. Its slope gives the current rate, which can be small after most available N has already entered the seaweed.
- The initial concentration decline doubles in magnitude because twice as much seaweed removes N. The initial rate per gram remains the same at the same \(S\). The flask still contains only \(12.5\;\mu\mathrm{mol}\) of added N, so doubling biomass does not double this available amount.
- For example, internal N stores may now suppress further uptake, or the incubation conditions may have changed. Compare the response with fresh tissue under matched light, temperature and mixing, and measure internal N if storage is your explanation. State the result you would expect under each explanation.
5 The Michaelis-Menten Model
We apply the Michaelis–Menten model (Equation 1) to data from multiple flask and perturbation experiments when a saturating concentration response describes nutrient uptake under the experimental conditions. It estimates limiting capacity and half-saturation. The fitted curve does not, by itself, establish irreversibility or identify the membrane’s energy-coupling mechanism.
The Michaelis-Menten equation is given by:
\[V_i = \frac{V_{max} \cdot [S_i]}{K_m + [S_i]} + \epsilon_i \tag{1}\]
Here \(K_m\) denotes the half-saturation concentration, written \(K_s\) in Lecture 9a and the animation. We use \(K_s\) for the whole-alga uptake response. It is an operational half-saturation parameter, and need not be a direct measure of molecular binding affinity.
The Blender film on capacity and concentration shows how uptake-site occupancy can produce this relationship. That film maintains concentration at the membrane; here we measure changing concentration in the flask. Surface and bulk concentrations need not be equal when external delivery is restrictive.
Where:
- \(V_i\) is the uptake rate at the \(i\)-th observation,
- \(V_{max}\) is the limiting uptake rate approached at high substrate concentration,
- \([S_i]\) is the substrate concentration at the \(i\)-th observation,
- \(K_m\) is the Michaelis constant, which represents the substrate concentration at which the uptake rate is half of \(V_{max}\), and
- \(\epsilon_i\) is the error term at the \(i\)-th observation.
Both parameters have ecophysiological interpretations. \(V_{max}\) represents the uptake capacity per unit algal mass under the experimental conditions. The half-saturation concentration tells us the external concentration needed to attain half of that capacity. For whole-alga uptake, this response can reflect nutrient delivery to the surface as well as transport and processing within the tissue. The initial slope, \(\alpha=V_{max}/K_s\), is the useful comparison of absolute uptake responsiveness at low nutrient concentration.
6 Data Analysis: Fitting the Michaelis-Menten Model with Nonlinear Regression
I provide this information only as a matter of interest to BDC223 students, as this is an advanced topic that will only be covered in your BSc (Hons) degrees. It is intended to provide a glimpse into the type of analysis that can be performed on nutrient uptake data. For the purpose of this module, you will not be required to perform this analysis and can instead rely on fitting the \(V\) vs \([S]\) curve by hand.
To formally model the Michaelis-Menten relationship, we use the nls() function in R to apply a non-linear regression the data from multiple flask experiments. For the perturbation experiment, things are a bit more complicated. This method includes dependent data points because the measurements are taken from the same flask at different times, introducing a correlation between observations. This violates the independence assumption required for standard regression models. To accurately analyse these data, I recommend a nonlinear mixed-effects model implemented in the nlme() function. Mixed-effects models account for fixed effects (overall trends across all observations) and random effects (variations specific to individual experimental units, in this case, time points within the same flask). This helps handle the correlation between repeated measures and produces reliable estimates of the uptake dynamics within the flask.
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Citation
@online{smit2026,
author = {Smit, A. J. and J. Smit, A.},
title = {Lecture 9b: {Nutrient} {Uptake}},
date = {2026-10-07},
url = {https://tangledbank.netlify.app/BDC223/L09b-nutrients_michaelis_menten.html},
langid = {en}
}