Problem 10 - Biochemistry: Concepts and Connections

Question

Suppose that the probability of making a mistake in translation at each translational step is a small number, \(\delta\). Show that the probability, \(p\), that a given protein molecule, containing \(n\) residues, will be completely crror-free is \((1-\delta)^{n}\).

Step-by-Step Solution

Answer

The probability is \((1 - \delta)^{n}\).

Step 1: Define the Event

We need to find the probability that a protein with \(n\) residues is error-free. This means that at every translation step, which corresponds to each residue, no mistake occurs.

Step 2: Determine the Probability of No Error at a Step

The probability of making an error in translation at each step is given as \(\delta\). Therefore, the probability of not making an error (error-free) at each step is \(1 - \delta\).

Step 3: Apply the Multiplication Rule for Independent Events

Since each step is independent, the probability that all \(n\) steps are error-free is the product of the probabilities of each step being error-free: \((1 - \delta) \times (1 - \delta) \times \cdots \times (1 - \delta)\) a total of \(n\) times.

Step 4: Derive the Probability Expression

Using the multiplication rule, the probability that all \(n\) steps are error-free is \((1 - \delta)^{n}\). This shows that the protein is completely error-free for all \(n\) residues.

Key Concepts

Translation Error

In biochemistry, protein synthesis is a critical process that converts genetic information from nucleic acids into functional proteins. However, errors can occur during this process, known as translation errors. These errors happen when an incorrect amino acid is incorporated into a growing protein chain. Although errors are generally rare, they can impact protein function, stability, and efficiency. Understanding translation errors is essential as they can have broad implications, affecting everything from cell function to disease development.
To model the likelihood of translation errors, we often assume that there is a small probability, denoted by \( \delta \), that an error occurs at each step of the translation process. This allows for a probabilistic analysis of how these translation errors might accumulate during the synthesis of a protein chain, leading to potential defects in the finished protein.

Independent Events

When discussing the probability of translation errors, it's crucial to understand the concept of independent events. In probability theory, independent events are those whose outcomes do not affect one another. This means the occurrence of one event does not change the probability of another.
In the context of protein synthesis, each translational step where an amino acid is added to the chain can be considered independent. The chance of making a mistake—where an incorrect amino acid is added—is not influenced by whether previous steps have been error-free or not. This independence simplifies the calculation of the overall probability of a translation error occurring within a protein.

Multiplication Rule

The multiplication rule is a fundamental concept in probability, used for calculating the probability of multiple independent events all occurring. When applying this rule, the overall probability is the product of the individual probabilities of each event taking place.
For the synthesis of a protein, the probability that each step is error-free is \( 1 - \delta \). Since each step is an independent event, we use the multiplication rule to find the probability that all \(n\) steps are error-free by multiplying \((1 - \delta)\) by itself \( n \) times. This results in the expression \((1 - \delta)^{n}\), which gives the probability that the entire protein molecule is free of translation errors.

Protein Synthesis

Protein synthesis is the biological process by which cells generate new proteins. This process involves two main stages: transcription of DNA to mRNA and translation of mRNA to a protein. The final stage, translation, occurs in ribosomes, where the mRNA sequence is read in sets of three nucleotides, known as codons, each specifying a particular amino acid.
During translation, tRNA molecules bring amino acids to the ribosome, where they are added to the growing protein chain. The fidelity of this process is vital for ensuring that proteins are synthesized correctly, without errors. Any mistake in translation can lead to a protein that doesn't function properly, which is why understanding and predicting the probability of errors, such as through the formula \((1 - \delta)^{n}\), is important. This ensures scientists and researchers can model the likelihood of errors throughout this critical biological process.