Question

A technique for radioactively labeling proteins is electrophilic radioiodination in which an aromatic substitution of \(^{131}\) I onto a tyrosine residue is performed as follows: Using the activity of \(^{131} \mathrm{I}\), one can measure protein lifetimes in a variety of biological processes. 131 I undergoes beta decay with a half- life of 8.02 days. Initially a protein labeled with \(^{131}\) I has a specific activity of \(1.0 \mu \mathrm{Ci}\), which corresponds to 37,000 decay events every second. The protein is suspended in aqueous solution and exposed to oxygen for 5 days. After isolating the protein from solution, the protein sample is found to have a specific activity of \(0.32 \mu \mathrm{Ci}\). Is oxygen reacting with the tyrosine residues of the protein, resulting in the loss of \(^{131}\) I?

Short answer

After calculating the expected specific activity (\(A_t\)) using the radioactive decay formula, we find that \(A_t = \left(1.0 \mu \text{Ci}\right)e^{\left(- \frac{ln(2)}{8.02}\right) \times 5}\). Comparing this with the given final specific activity (0.32 µCi), if these values are different, it indicates that oxygen is indeed reacting with the tyrosine residues of the protein, resulting in the loss of \(^{131}\text{I}\).

Step-by-step solution

Calculate the decay constant λ

To find the decay constant λ, we will use the formula for half-life, which is \(T_\frac{1}{2} = \frac{ln(2)}{λ}\). Rearranging this equation, we get \(λ = \frac{ln(2)}{T_\frac{1}{2}}\). Given half-life, \(T_\frac{1}{2}\) = 8.02 days. So, the decay constant λ can be calculated as follows: \(λ = \frac{ln(2)}{8.02}\) days\(^{-1}\).

Calculate the expected specific activity after 5 days

The radioactive decay formula, \(A_t = A_0e^{-λt}\), lets us calculate the remaining activity after a certain time. In our case, \(A_0\) is the initial specific activity (1.0 µCi), and \(t\) is the number of days for which the protein sample was exposed to oxygen (5 days). So, we can calculate the expected specific activity after 5 days (\(A_t\)) as: \(A_t = \left(1.0 \mu \text{Ci}\right)e^{-λ \times 5}\).

Compare the expected specific activity with the given final specific activity

Now, we will compare the calculated expected specific activity (\(A_t\)) with the final specific activity given in the problem (0.32 µCi). If the expected specific activity is equal to the given final specific activity, it means that the exposure to oxygen has not affected the protein, and the drop in specific activity is solely due to radioactive decay. If the expected specific activity is different from the given final specific activity, it means that oxygen has reacted with the protein, resulting in the loss of \(^{131}\text{I}\).

Conclusion

Based on the comparison made in Step 3, we can conclude whether or not oxygen is reacting with the tyrosine residues of the protein, causing the loss of \(^{131}\text{I}\).

Key concepts

Electrophilic Radioiodination

Electrophilic radioiodination is a method used to introduce radioactive iodine, specifically the isotope \(^{131}I\), into proteins. This process typically occurs through an aromatic substitution reaction on the tyrosine residues of proteins.
This technique is valuable because it allows scientists to track proteins within biological systems by measuring the radioactivity of the iodine attached to the tyrosine.

• **Aromatic Substitution:** The iodine (\(^{131}I\)) replaces a hydrogen atom in the aromatic ring of the tyrosine residue. This enables the tagged protein to undergo radioactive decay, which can be quantitatively measured.
• **Radioactivity Measurement:** After labeling, the radioactive iodine emits beta particles which can be detected. This emission helps in observing the protein's behavior in various biological processes.
• **Application:** This technique is widely used in research to study protein degradation, metabolism, and other dynamic processes within living organisms.

By using this method, researchers can gain insights into protein function and turnover by evaluating how long the labeled protein retains its radioactive marker as it interacts with its environment.

Beta Decay

Beta decay is a type of radioactive decay where an unstable atomic nucleus transforms into a more stable nucleus by emitting beta particles.
In the case of \(^{131}I\), beta decay involves the emission of an electron or a positron and an antineutrino or neutrino, respectively.

• **Process:** In \(^{131}I\), a neutron in the nucleus is converted into a proton with the emission of an electron (beta particle) and an antineutrino. This increases the atomic number by one, forming a new element, xenon \(^{131}Xe\).
• **Energy Release:** The beta particles carry kinetic energy, which is released in the process and can be detected by geiger counters or other radiation detectors.
• **Significance in Labeling:** The beta decay of radioactively labeled iodine is what allows researchers to measure the protein's presence as the emitted particles can be counted.

Understanding beta decay is crucial for employing radioiodination methods in scientific experiments, as it provides the means to quantify protein behavior over time.

Radioactive Decay Formula

The radioactive decay formula is a mathematical equation used to predict the activity of a radioactive substance over time. It is essential for calculating the change in specific activity of a sample due to radioactive decay.
This prediction is significant in experiments involving radioactively labeled compounds.

• **Formula:** The radioactive decay formula is expressed as:
  \(A_t = A_0e^{-\lambda t}\)
  where \(A_t\) is the remaining activity at time \(t\), \(A_0\) is the initial activity, \(\lambda\) is the decay constant, and \(e\) is the base of the natural logarithm.
• **Purpose:** This formula allows scientists to determine how much of a radioactive isotope remains at any given time after some has decayed. In the context of the exercise, it helps in predicting the expected specific activity of a protein sample after being exposed to a given condition for a period.
• **In Application:** By calculating the expected activity using this formula and comparing it with the observed activity, researchers can determine whether an external factor, such as reaction with oxygen, has caused additional loss of radioiodine.

Hence, the radioactive decay formula plays a central role in analyzing whether the observed activity changes are due to natural decay or induced by other factors.

Half-life Calculation

Half-life calculation is a fundamental concept that helps to determine how long it takes for half of a radioactive substance to decay. It is expressed mathematically and used to understand the stability and decay rate of radioactive isotopes like \(^{131}I\).

• **Definition:** The half-life is the time required for a quantity to reduce to one-half of its initial value. For \(^{131}I\), the half-life is 8.02 days.
• **Calculation:** The relationship between half-life \(T_{1/2}\) and the decay constant \(\lambda\) is given by the formula:
  \(\lambda = \frac{\ln(2)}{T_{1/2}}\)
  This relationship is vital in calculating the decay constant, which is further used in the radioactive decay formula to find remaining activity over time.
• **Practical Use:** By knowing the half-life, researchers can estimate the expected activity of a substance and compare it with experimental data to assess the progression of decay or investigate other potential reactions, like the interaction with oxygen in the given exercise.

The half-life thereby provides a standardized measure to discuss and predict the behavior of radioactive materials, making it crucial for studies involving radioactive labeling.