# Spectrophotometer¶

## Background¶

At the center of the molecular revolution of biology that took place in the middle of the 20th century were measurement of bacterial growth curves. In its most basic form, a solution of growth media is inoculated with bacteria. As the bacteria convert the sugars and salts in the media into more bacteria (literally making life from inanimate matter!), they multiply. Measuring the rate of multiplication of bacteria under different media and growth conditions ultimately enabled researchers to make great leaps in the understanding of bacterial physiology and biochemistry, including how regulation of gene expression works.

Jacques Monod was a leader, if not the leader, in this field. In a beautiful review article about the subject (Monod, Ann. Rev. Microbiol., 1949) Monod stated in the strongest of terms the importance of studying bacterial growth: “The study of the growth of bacterial cultures does not constitute a specialized subject or branch of research: it is the basic method of Microbiology. It would be a foolish enterprise, and doomed to failure, to attempt reviewing briefly a ‘subject’ which covers actually our whole discipline.”

Monod had comments on methods for measuring growth curves as well, and they are true today as they were more than 70 years ago. “The most widely used methods, by far, are based on determinations of transmitted or scattered light. (Actually, the introduction around 1935 of instruments fitted with photoelectric cells has contributed to a very large degree to the development of quantitative studies of bacterial growth.)”

Owing to their central importance in microbial research, acquisition and analysis of growth curves has also become a central part of undergraduate study of biology. In the ensuing decades, the optical instruments used to acquire growth curves, called spectrophotometers, have played a central role in education as well as research.

In the spring, dozens of freshmen will take our freshman biology lab course. Because the pandemic shows little signs of relenting, we almost certainly will be offering that course remotely. Your task is to build a spectrophotometer out of the components in your kit, in addition to materials you can find around your dwelling. Newton developed a powerful new way to calculate pi() while quarantining at home during a bubonic plague outbreak (see this YouTube video. Perhaps you can come up with a better spectrophotometer design during your COVID quarantine! We will take one of the designs from this class and ship it to our spring enrollees so that they can study the ever-important microbial growth curve.

### Basics of spectrophotometry in microbiology¶

The block diagram of a typical spectrophotometer is shown below.

Image by Yassine Mrabet, CC-BY-4.0 License.

To determine the density (concentration) of microbes in liquid media, we load a cuvette with the suspension of cells. Some of the light is absorbed or scattered, and some is transmitted. The amount of transmitted light is measured. As will be described momentarily, the intensity of transmitted light is related to the concentration of microbes.

Most spectrophotometers use a prism to allow measurement of transmitted light at multiple wavelengths. The best wavelength of light to measure microbial growth, i.e., that which has strong scattering, is about 600 nm. For your design, you will not include a prism, since we do not need measurements at multiple wavelengths (and we are taking a minimalist approach). Rather, you will use an LED that emits light near 600 nm.

Cells, having a different index of refraction as the surrounding medium, randomly reflect and scatter light out of the incident light path. Imagine a photon traversing the media. If the cell concentration is low, then collisions of a photon with a cell may be modeled as a Poisson process. The time of arrival of a Poisson process is Exponentially distributed with cumulative distribution function

\begin{aligned} F(t) = 1 - \mathrm{e}^{-\beta t}. \end{aligned}

The probability that no Poisson process arrives within time $$t$$, meaning that the photon manages to pass through the sample without being scattered, is $$1 - F(t) = \mathrm{e}^{-\beta t}$$. Thus, if the light entering the sample had intensity $$I_0$$, that leaving has intensity $$I_0\,\mathrm{e}^{-\beta t}$$.

The time it takes to traverse the sample is proportional to the length of the cuvette, $$l$$. The parameter $$\beta$$, which parametrizes the distribution and describes how likely it is to hit a cell, is a function of the concentration $$c$$ of the cells in the media. If the media is dilute, we can write $$\beta$$ as a Taylor series in $$c$$ to first order such that $$\beta \propto c$$. We then arrive at

\begin{aligned} I = I_0 \mathrm{e}^{-\epsilon l c}, \end{aligned}

where the wavelength-dependent parameter $$\epsilon$$ that contains the constants of proportionality is referred to as the molar attenuation coefficient. This equation is known as the Beer-Lambert Law.

The transmission efficiency of the sample is the ratio of the final and initial light intensity, $$T = I/I_0$$, and the reported quantity is called the optical density, $$\mathrm{OD} = -\log_{10} T = \epsilon l c / \log_{10} \mathrm{e}$$. The general rule of thumb is that the accurate OD readings lie in the range of $$0.01 < \mathrm{OD} < 1$$. Serial dilutions of a sample may be necessary to bring the OD into this range. For microbial growth, the OD is often referred to as the OD600, since it is a measure of the optical density of light at 600 nm.

As a final note, it is important to measure a blank sample as well. The media itself can scatter light, and you need to correct for this. To do this, measure the OD of a cuvette containing only media. Then, your adjusted OD is $$\mathrm{OD} = \mathrm{OD}_\mathrm{measured} - \mathrm{OD}_\mathrm{blank}$$.

### Basics of microbial growth¶

Say we have a collection of cells in media that is in ideal conditions for growth. They are expressing all of the necessary genes for growth in the media and are consuming media and doing the “stuff of life” at a constant rate. Now say we pluck out one of these cells, the daughter of a very recent cell division, and immediately put it in fresh media. It will grow and divide, and we will have two cells after a division time $$t_\mathrm{div}$$. Those two cells will grow and divide, and after time $$2 t_\mathrm{div}$$, we will have four cells. After a time $$3t_\mathrm{div}$$ we will have eight cells, and so on. So, the rate at which new cells are produced is proportional to the number of cells in solution, or

\begin{aligned} \frac{\mathrm{d}N}{\mathrm{d}t} = k N, \end{aligned}

where $$N$$ is the number of cells and $$k$$ is the so called growth rate. If we initially have $$N_0$$ cells, this differential equation is solved to give

\begin{aligned} N(t) = N_0\,\mathrm{e}^{kt}. \end{aligned}

This is so-called exponential growth, and is typically the phase of growth we study in growth experiments, at least in freshmen labs. The important parameter here is $$k$$, referred to as the growth rate. This is the parameter we seek to measure in a growth experiment.

The mechanics of the experiment goes as follows.

1. Growth media is inoculated with the microbes.

2. The microbes are allowed to grow for some time (usually about two hours). This ensures they are in exponential growth phase.

3. A sample is taken and diluted, usually such that the OD is about 0.05 or so.

4. From this diluted sample, OD measurements are obtained at regular time intervals.

5. A plot of OD versus time is generated and a regression is performed to get an estimate of the parameter $$k$$.

## Specification¶

For this project, you are to build a spectrophotometer and obtain an example growth curve. The spectrophotometer must have the following characteristics.

1. It must be built from the components in your kit, plus any household items you may wish to use. (These household items should be common and/or low cost.)

2. It must operate in stand-alone mode. In this mode, the device does not need to be connected to a computer (only a power source). The user pushes a button and the resulting OD is displayed on the LCD display.

3. It must also operate while controlled by a computer. It is up to the user to choose the mode of operation. In this mode, the user has an on-screen control panel and can click buttons to obtain and display OD measurements.

You must demonstrate the instrument by acquiring a growth curve and determining the growth rate $$k$$.

### Documentation¶

You must include documentation that we will send along to the freshmen in the spring term. The documentation can be written in Jupyter notebook(s) (which can also be rendered as HTML) or as PDFs. It must contain:

1. Instructions to build the device from the items we will ship them and from household items.

2. Instructions for operating the instrument.

3. Instructions for calibrating the instrument.

### Suggestions¶

Here are some things to consider in your design and the operation of your instrument.

1. There are two types of ≈600 nm emitting LEDs, one with 10 candela luminosity, and one with 1 candela luminosity. You can try either in your design. For both LEDs, the light is primarily forward directed, coming from the top of the LED.

2. It is important to block stray light. Think about how to accomplish that with household items.

3. You may want to attempt to collimate the light with the glass beads (actually from children’s jewelry) provided in your kit or any other lens-like object you can find. You may also want to set up a pinhole or slit for the light to pass through the cuvette. If you do so, think about whether you need to account for diffraction or not.

4. Think about ways of dealing with noise and amplification, if necessary.

5. Think carefully about how to calibrate relationship between measured voltage across the photoresistor to OD.

6. In stand-alone mode, the user will use physical pushbuttons or toggles to control the instrument. You may want to read about debouncing.

### Instructions for obtaining growth curves¶

Because of difficulties associated with shipping and storing microbial agents, you will test your device by growing yeast, Saccharomyces cerevisiae. The measurement and analysis are the same as for bacteria, but we are instead using a common baking ingredient that is safe to have in the home.

The media for S. cerevisiae is yeast peptone dextrose (YPD) broth. The broth contains yeast extract that has all of the necessary vitamins, peptone (source of carbon, nitrogen, and some vitamins), and dextrose (sugar). You have a Falcon tube containing YPD powder) that you can use to make broth. The broth is 5% powder. E.g., to make 100 mL of broth, add 5 g of YPD powder. If you do not have a scale in your house, as a rule of thumb, a tablespoon of the YPD power is about 9 g. You should add it to water as pure as you can find (for example, if you have a Britta filter in your house, you should use filtered water). Normally, you would autoclave the broth to sterilize it, but that is not an option for home use. Instead, boil the broth for one minute (this will also ensure everything is dissolved). The broth can be stored at room temperature, and you can store it in the provided falcon tubes.

You can grow the yeast in the falcon tubes. Try to keep them in a place that does not have temperature fluctuations. You should occasionally shake the tube to be sure the oxygen in the solution is well mixed. (In a lab, these would be grown in a water bath shaker.) You can use the transfer pipettes in your kit to extract samples for measurement in the spectrophotometer. The cuvette volume is 4.5 mL, but do you not need to fill it up all the way, just high enough that the light from the LED passes through the media (with a bit of a buffer).

In a lab, you would dispose of the pipette tip used in the transfer and throw away the cuvette after a measurement. In our at-home version, you will likely need to re-use the Falcon tubes, pipettes, and cuvettes. You can wash them with water between uses. You may even want to re-use the solution you measured, replacing it in the main growth mixture after using it if you are running low on supplies. (You may wish to do this anyway in practice runs before you get your “final” growth curve.) The inevitable contamination may affect the results, but this is a necessary consequence of of operating in outside-of-laboratory conditions.

Bear in mind that you cannot calibrate the OD measurements to known microbe concentrations, which you should absolutely do in a lab (as Monod vehemently argues, “What should be noted is that in spite of the widespread use of the optical techniques, not enough efforts have been made to check them against direct estimations of cell concentrations or bacterial densities.”) This is not accessible to us because we do not have equipment or storage necessary to do those measurements in our living spaces, though any biology lab would be equipped with them.