S15PrismAutoCollimation
Yeast Metabolic Rate Characterization Using Refractive Index Measurement by Prism Autocollimation
Kyle Crocker & Henry Duran
Methods of Experimental Physics
Introduction
Yeast metabolizes sugar to create water and carbon dioxide. The decreasing sugar concentration causes a proportional decrease in the index of refraction. Measurement of the change of the index of refraction over time can be used to examine the rate at which yeast cells metabolize sugar. This experiment studies this rate, and may be useful in baking, beer brewing, and winemaking industries--industries which use yeast.
Theory
Index of refraction is proportional to mass concentration of a solute, in this case sugar. Specifically
where n is the index of refraction and nw is the index of refraction of water, C is the mass concentration of the sugar, and A is constant.
The yeast consumes sugar at a rate given by the mass action law:
where R is a constant that governs the rate of reaction. We predict that R is proportional to the number of yeast cells and solve to obtain
with r the rate constant per yeast cell per volume, N the number of yeast cells per unit volume, and t is the time for which the yeast has been metabolizing the sugar in the solution.
Refractive Index Measurement
Figure 1: Image of apparatus
Following reference [1], Snell's law is used to determine the index of refraction of the fluid inside of a prism when light reflects out of the prism along the same path that it entered. In particular,
where theta is the incident angle of light onto the prism, and α is the refracted angle.
The refracted angle is known when autocollimation is satisfied, so the prism is rotated and the incident angle recorded when autocollimation is detected.
Figure 2: Experimental Setup. Adapted from reference [1].
Specifically, the incident angle is the difference between the peak detected at reflection off of the front face of the prism and the peak detected at reflection off of the mirror at the back of the prism. Fig. 3 shows these peaks, which are plotted as a function of step as a stepper motor rotates the prism.
Figure 3: Light intensity (in terms of voltage from photodiode) as a function of stepper motor step. Original figure.
Calibration & Testing
The refracted angle in the prism is determined by finding the average incident angle with distilled water in the prism, where,
The index of refraction of isopropyl alcohol was then measured at 1.375, which is within 0.2% of the listed value, 1.377.
Data Collection
Solutions with different mass concentrations of sugar are prepared, and Fleischman's yeast is put into each of these solutions at 24 degrees Celsius. At given time intervals, the yeast is strained out of a sample, and the index of refraction is measured.
After we thus obtain the index of refraction, a mixture is made with the same initial conditions as sample 5 and 6, and a spectrum analyzer is used to measure the optical density of the mixture at given time intervals. This measurement is used to find the yeast density. This is only done for sample 5, since this sample had initial conditions most suitable for the precision of our experiment.
Figs. 4 and 5 show the measured indices of refraction and yeast density over time.
Figure 4: Measured yeast cell density for sample 5 at times measured in minutes. Average value with standard deviation from this value on plot. Original figure.
Figure 5: Index of refraction vs time for sample 5. The trend is roughly exponential, as expected, although the measurement is affected by fluctuations in the yeast cell density, and doesn’t appear to asymptotically approach the index of refraction of water used as a solute. Original figure.
Analysis
The yeast concentration seems to fluctuate around a relatively constant value, so the average yeast density is assumed to be N, and the standard deviation from the average is the uncertainty in N.
-ln(n-nw )/N vs t is plotted using a least squares linear fit. The uncertainty is propagated in the usual way, where the uncertainty in N and the 0.2% uncertainty in refractive index measurement as the dominant uncertaties.
The slope of this plot is r, which is the rate constant per yeast density that governs the rate at which yeast metabolizes sugar.
Results & Discussion
Fig. 6 shows the result of the least squares linear fit described.
Figure 6: Logarithmic plot of the measured indices of refraction versus the time, for sample 5. The line is a least squares linear fit of the data points. The slope of the fit line for each of these fit equations corresponds to the measured rate constant per yeast cell per unit volume. Original Figure.
From this fit we obtain,
with reduced chi-squared
There appears to be a non-linear trend in the data. This is likely because the relationship is only linear to a first order approximation. In particular, if the yeast increases initially due to the availability of sugar, but decreases as time goes on due to a lack of nutrients, we would expect to see the index change more rapidly at early times and less rapidly at later times than predicted by a linear fit, which is observed. This demonstrates that our assumption that the number of yeast cells remains constant is only approximately true, but that this was not unexpected. Similarly, our assumption that the index is proportional to the concentration is also an approximation.
Additionally, the measured refractive index of the solution doesn't seem to assymptotically approach the index of refraction of water, as predicted. This is something that future experiments should examine, but may be due to preservatives in the yeast, minerals in the tap water which was used to create the solution, or even poor calibration data.
Conclusion
These results seem to confirm (to a first order approximation) our theoretical prediction regarding yeast metabolization, and it seems to provide a method by which index refraction measurement by prism autocollimation may be used to study the rate at which yeast metabolizes sugar. Such a study may be done with many types of yeast to determine how the metabolic rate of each depends on its density.
More precise index of refraction measurements were expected, so future experiments may try to discover the source of error in these measurements and minimize them. One way that this may be done is through the use of a spatial filter and a more powerful laser than what was used in this experiment. The lack of an assymptotic approach of the index of refraction of the solution to that of water should also be studied, since this is unexpected.
Additionally, more frequent and precise measurements of yeast density may make possible a better fit than the first order approximation that was used in this experiment.
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