UA&P EnSci Summer 07-08 Project One Summary

Let me describe the approach we took in the algae experiments.

1. We set out to ask two questions regarding an important environmental issue. The BIG question was: "Are detergents toxic to plants?" We made an assumption that toxicity may be measured by its effects on birthrate. The more specific question, then, was "Do detergents in the water decrease the growth rate of algae?" Given that question, the best general strategy was to do a variation-type of experiment where we manipulate the concentration of detergent in water and then check for a co-variation in growthrate.

2. But, how do we measure growth rate? And then, what growth rate are we talking about: the intrinsic growth rate or the actual growth rate? In principle, measuring the intrinsic growth rate should be easy: Count the algae at some posterior day (say day 2), get the difference between that count and the previous day's count (day 1), and divide that difference by the previous day's count. The value we get should be the same for whatever pair of days we choose, if the growth of algae is EXPONENTIAL.

But are the actual algae cultures growing exponentially? Exponential growth is obtained under the very best conditions, i.e., practically unlimited space and resources, and no competition. A flask of algae, however, is a small world; so small that we may reasonably expect that as algae become many, their growth rate would decrease because of competition and rapid depletion of resources. Therefore, the growth rate would change rather quickly with time; that is, it will not be constant. The exponential model would be imprecise; it would be difficult for us to choose an appropriate anterior and posterior date to make the calculation.

So, it would seem that a CARRYING CAPACITY model is more appropriate to describe the real situation in the flask. But the model, like all models, requires some assumptions before it can be precisely defined in a form that can be tested. One assumption is that the actual growth rate is inversely proportional to the current population count.

Now, an inverse proportion is a linear relation that is easy to describe, but is not necessarily true. We had to test if the model containing that assumption was realistic.

3. So, we set up a growth experiment. We put various algae in flasks and counted their growth everyday for up to 12 days and plotted the population as a function of time. We then compared the actual growth curve to a theoretical growth curve predicted by a carrying capacity model, defined below:

dQ/dt = Q0 x birthrate x (1 - Q/carrying capacity), where Q0 is the algal population count at time 0 (the initial count) and Q the population at any particular time.

The Q for various times was calculated using a numerical method implemented on Excel. Thus, there was a theoretical Q and its corresponding actual Q at every point in time measured.

4. How do we know the model works? Check how close the theoretical and actual Q values are. This was done by first fixing the carrying capacity as the estimated maximum value of Q (which can be seen by the plot of real Q's as they approach some kind of stable value), then trying different birthrates, then getting the sum of the squares of the difference between actual and theoretical Q's for all time points tested, then choosing the birthrate that gives the lowest sum of squares.

5. What happened was that in most cases--if not all--the selected birthrate gave theoretical values that very closely matched the actual values. Graphically, this means that the plot of the actual and theoretical growth curves coincided rather well. This result suggested that we may use the carrying capacity model to calculate intrinsic birthrate even when the actual birthrate changes with time.

6. Thus equipped with a reliable way to calculate birthrate, we proceeded by repeating the algal growth curve experiment with various concentrations of a detergent called sodium dodecyl sulfate, or SDS, added to the culture medium. Using the carrying capacity model, we were able to estimate birthrates for every flask in the experiment.

7. A look at the all these growth rates showed that when the SDS concentration was higher, the growth rates were lower. This result suggested that SDS concentration does bring down growth rate. We deduce from this that SDS is toxic to algae.

8. In short

a. We wanted to know if detergent lowers growth rate.

b. We needed a way to estimate growth rate given realistic conditions. That way was to use a model, the carrying capacity model, which included growth rate as a factor. We defined the model mathematically so that it could be tested in Excel. An experiment comparing actual population with theoretical population (the number predicted by the Excel model) showed that we can estimate growth rate rather nicely, as seen by the beautifully fitted curves.

c. Equipped with a reliable and realistic (and rather beautiful) model, we proceeded to estimate growth rate in cultures to which detergent was added. The results showed that growth rate does go down as detergent concentration goes up.