Spectrophotometer
Background
Spectrophotometers have been, and continue to be, the workhorses of molecular and cell biology. The basic idea is simple. A beam of light is shone through a sample and the intensity of the light coming out of the other side of the sample is measured. The applications are numerous, but two have been particularly prominent in biology. First, the concentration of unicellular organisms can be ascertained. Due to their size, each cell of a microbe scatter light of specific wavelengths (about 600 nanometers for E. coli). As a result, less transmitted light is detected from samples with higher concentration of microbes. Second, some molecular species absorb light at a given wavelength due to their electronic structure. A sample of higher concentration will have lower detected intensity. Measurements of concentrations of cells and molecules over time enables study of the kinetics, e.g., of microbial growth and rate of chemical reactions.
Your task in this project is to construct a spectrophotometer. Though it would certainly be fun to apply the device to bacterial growth curves, such curves take a long time to acquire (since you have to wait for the bugs to grow!). Instead, we will perform an assay of the kinetics of an enzyme catalyzed reaction. Optical assays have long been the workhorses of kinetics assays. As a classic example, Leonor Michaelis and Maud Menten studied the hydrolysis of sucrose into fructose and glucose by the enzyme invertase. Sucrose rotates plane polarized light to the right, while glucose and fructose rotate it to the left. Michaelis and Menten could therefore make measurements of the rotation of polarized light to determine the relative concentrations of the sugars in their reaction mixture and could thus work out the kinetcs of the enzyme-catalyzed hydrolysis. You will instead measure the rate at which the enzyme amylase catalyzed the breakdown of starch into sugars.
Basics of spectrophotometry in microbiology
We will introduce the concepts behind spectrophotometry as applied to measurement of transmitted light through microbial solutions, since this is a “classic” application of spectrophotometry in biology. In fact, the molecular revolution of biology which transformed the way we think about the living world was largely pushed forward by spectroscopic measurements of microbes.
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. (It is a very fortunate coincidence that the wavelength of light absorbed by the starch-iodine complexes we will use in our enzyme kinetics assay is very close to 600 nm, so you are coincidentally building a dual-purpose spectrophotometer.)
Cells, having a different index of refraction as the surrounding medium, randomly reflect and scatter light out of the incident light path. The passage of a photon through a medium full of cells without getting scattered can be readily understood through the math.
To set up the problem, let us consider a medium with area \(A\) and thickness \(x\). Suppose the number concentration of cells is \(n\) and each cell has a scattering cross-section of \(\sigma\). Let’s now cut the medium into slices of thickness \(\delta_x\). Now let us consider one of these slices. The number of cells \(N\) in the slice is given by:
This means that the cells will put up a total area of \(nA\delta_x \sigma\) for scattering. If a photon passes through this area sum, it will be scattered. If a photon passes through the rest of the area \(A\), it will not be scattered. As such, we can express the probability that the photon is scattered by the slice as:
The probability that the photon is not scattered is given by the following. We simply perform a reverse Fourier series simplification to arrive at the exponential form. (When \(\delta_x\) is vanishingly small, the approximation increases in accuracy.)
The probability that a photon is not scattered going through a series of these slices is basically given by multiplying this probability by itself over and over again. To calculate the probability of a photon going through unscathed through the entire medium of thickness \(x\), the product should have a total of \(\frac{x}{\delta_x}\) terms. This gives us:
We can now derive the transmission intensity through the medium by noting that the total incident and total transmission is simply an aggregation of individual photons. As such, the probability equation translate directly to an intensity relationship:
Thus, if the light entering the sample had intensity \(I_0\), that leaving has intensity \(I_0 e^{-n\sigma x}\). This equation is known as the Beer-Lambert Law.
The transmission efficiency of the sample is the ratio of the light intensity leaving the sample to that impinging on the sample, \(T=I/I_0\), and the reported quantity is called the optical density, \(\text{OD} = -\log_{10} T = n\sigma x \cdot \log_{10} e\). The general rule of thumb is that the accurate OD readings lie in the range of \(0.01 < \text{OD} < 1\). 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 \(\text{OD} = \text{OD}_\text{measured} - \text{OD}_\text{blank}\).
Basics of spectrophotometry in enzyme kinetics assays
We can use optical methods to measure the concentration of a given chemical species, known as absorption spectroscopy. When a photon of an wavelength whose energy matches that of an electronic transition of a molecule of interest impinges upon the molecule, the electronic transition is made and the photon is absorbed. These photons do not make it through the sample.
The reasoning with which we derived the Beer-Lambert Law for scattering is exactly the same as for scattering by microbes. The only difference is that the photons that fail to pass through the sample are absorbed rather than scattered. The reported quantity has exactly the same form as the optical density above, but is referred to as absorbance, \(A = -\log_{10} T = n\sigma x \cdot \log_{10} e\).
Basics of microbial growth
Because microbial growth is such an important application of spectroscopy, we give a brief description of how the growth rate of a microbe under given conditions can be obtained by spectroscopic measurement. Obtaining a growth curve is not required for your project, but you should be able to use your spectrophotometer to obtain one.
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
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
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.
Growth media is inoculated with the microbes.
The microbes are allowed to grow for some time (usually about two hours). This ensures they are in exponential growth phase.
A sample is taken and diluted, usually such that the OD is about 0.05 or so.
From this diluted sample, OD measurements are obtained at regular time intervals.
A plot of OD versus time is generated and a regression is performed to get an estimate of the parameter \(k\).
Basics of enzyme kinetics
This section was written in collaboration with Victoria Chen.
The Michaelis-Menten model is commonly and effectively used to model enzyme kinetics. It models enzyme kinetics as a set of chemical reactions between an enzyme, E, and a substrate, S, to give product P.
The enzyme reversibly binds the substrate with binding rate constant \(k_+\). The substrate may also unbind with unbinding rate constant \(k_-\). When bound, the enzyme can serve to convert the substrate to product with rate constant \(v\).
We can explore how the concentration of each species changes over time by using mass action kinetics, which says that the rate of a reaction is proportional to the product of the concentrations of the reacting species. Applying this rule, we can write the dynamics as a system of ordinary differential equations.
where \(c_{i}\) denotes the concentration of species \(i\). Note that
which is a statement of conservation of enzyme. This means that we need to specify a total enzyme amount to fully specify the problem. We define this to be \(c_\mathrm{e}^{0}\) such that \(c_\mathrm{e}^{0}=c_\mathrm{e} + c_\mathrm{es}\). With this conservation law, we can write the ODEs as
These equations describe the full dynamics of the enzyme-catalyzed system. To simplify the analysis, we often make the quasi-steady state approximation that the bound substrate intermediate \(ES\) does not see appreciable change in its concentration on the time scale of the production of the product \(P\). That is,
This enables us to solve for the quasi-steady state fraction of enzyme that is bound to substrate.
where we have defined the Michaelis constant
It has dimension of concentration and is analogous to a dissociation constant in that it is the ratio of the total rate constant for dissociation of the enzyme from the catalyst to that of binding the enzyme to the catalyst.
Finally, substitution of this expression gives the rate of product formation,
If \(\mathrm{d}c_\mathrm{es} / \mathrm{d} t \approx 0\), then the rate of substrate consumption is
Inspection of this expression identifies two constants. The first is \(V_0 \equiv v\,c_\mathrm{e}^0\), commonly referred to as the maximum velocity, which is the fastest that the enzyme-catalyzed reaction can proceed. It makes sense that it is the product of the rate constant for conversion of enzyme-substrate complex to product, \(v\), and the total enzyme concentration \(c_\mathrm{e}^0\). The second parameter is the Michaelis constant \(K\).
There is no closed-form solution for the Michaelis-Menten equation, but it can be written in terms of the Lambert-W special function. The result is
where \(c_\mathrm{s}(t=0) = c_\mathrm{s}^0\) and \(W_0(x)\) denotes the primary branch of the Lambert-W function.
In an enzyme kinetics experiment, the concentration of substrate or product is measured over time by taking optical density measurements.
The above expression may be used for curve-fitting applications to ascertain the parameters of interest. Alternatively, the differential equation for \(c_\mathrm{s}(t)\) may be solved numerically.
Specification
For this project, you are to build a spectrophotometer and obtain an example curve from an optically monitored enzyme catalyzed reaction. The spectrophotometer must have the following characteristics.
It must be built from the components we have in lab, plus any household items you may wish to use. (These household items should be common and/or low cost.) You may also 3D print objects if you like.
For this prototype, you may calibrate the relationship between voltage and absorbance using a commercial spectrophotometer provided in lab. The calibration may be hard-coded into the software of the device.
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/absorbance is displayed on the LCD display.
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/absorbance measurements. The user may also select continuous acquisition with a time between acquisitions to automate acquisition, e.g., for kinetics experiments. The output must be savable in CSV format.
You must demonstrate use of the instrument by acquiring a measurement of the concentration of starch-iodine complex (or something proportional to it) and performing a curve fit. You must minimally ascertain the time constant given by the ratio \(V_0/K\), but you may try to obtain individual values for \(V_0\) and \(K\) if you wish. This will require careful calibration of your spectrophotometer to obtain the molar attenuation coefficient for the starch-iodine complex.
Documentation
You must include documentation for your device. The documentation can be written in Jupyter notebook(s) (which can also be rendered as HTML) or as PDFs. It must contain:
Instructions for calibrating the instrument.
Instructions for operating the instrument.
Suggestions
Here are some things to consider in your design and the operation of your instrument.
There are two types of ≈600 nm emitting LEDs in the lab, 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.
It is important to block stray light. Think about how to accomplish that.
Think about ways of dealing with noise and amplification, if necessary.
You may wish to use a photodiode instead of a photoresistor to measuring light intensity. Indeed, many modern spectrophotometers use photodiodes. If you do that, bear in mind that a photodiode has a nice linear relationship between intensity of light and current through the photodiode. To convert the current to a voltage that you can measure with Arduino, you will need a transimpedance amplifier.
In stand-alone mode, the user will use physical pushbuttons or toggles to control the instrument. You may want to read about debouncing.
You may also want to think about the following considerations when demonstrating your spectrophotometer with the amylase kinetics experiment.
Think carefully about how to calibrate relationship between measured voltage across the photoresistor or current through a photodiode to OD/absorbance.
As you can read in the instructions below, you will use your own saliva as a source for amylase. The concentration of amylase will vary from experiment to experiment as a result. Therefore, \(V_0\) will be different from experiment to experiment. The Michaelis constant, however, should not vary from experiment to experiment, provided you use the same starch source for each experiment.
Instructions for obtaining a curve for enzymatic catalysis
The reaction we will monitor is starch-iodine complex ⟶ sugars + free iodine, catalyzed by amylase in your saliva. The starch-iodine complex absorbs near 600 nm and is the substrate in the Michaelis-Menten mechanism. We will monitor its concentration over time.
Obtaining a saliva sample
Do not eat, brush your teeth, or rinse your mouth for about 30 minutes.
Take a sip of water, and try to secrete saliva for 2-5 min. You should collect the saliva in a small cup or other container.
Filter the saliva through a coffee filter to obtain a clear saliva solution.
Measuring the chemical kinetics
Because you want to capture as much of the reaction dynamics as possible, you will need to work quickly to get the reaction mixture assembled and into your spectrophotometer.
Using a pipette, dispense a small amount of the provided iodine solution into an empty cuvette. You will not need much; around 20-50 µL should be sufficient. You may need to experiment with the amount, adjusting so the absorbance readings fall into the accurate range below A ≈ 1.
In a separate container, add 2 mL of the provided starch solution. Then add a small amount of the filtered saliva. You should use a volume that goes up to the first gradation in the provided Beral pipettes. This is about 30 µL. Quickly mix the solution by pipetting up-and-down several times.
Working quickly, pipette approximately 1 mL of starch-saliva solution into each cuvette. Pipette the mixture up and down a couple times to mix. Take measurements until the absorbance approaches zero, or until there is no more observable decrease in absorbance.
Submission
Submit your write-up as a ZIP file via Canvas.