How Many Ribosomes Can Translate a Single Bacterial mRNA During Its Lifetime?

Understanding the Given Information

This question combines three important concepts of bacterial protein synthesis: the rate of translation, the length of the mRNA and the average lifetime of the mRNA. To calculate how many ribosomes can translate the given mRNA during its lifetime, we must first determine the speed at which one ribosome synthesizes amino acids.

The question states that a bacterial ribosome synthesizes a protein containing 300 amino acids in 20 seconds. This information allows us to calculate the translation rate in amino acids per second.

The mRNA under consideration contains 1350 nucleotides. Since each amino acid is specified by a codon consisting of three nucleotides, the number of codons and therefore the approximate number of amino acid positions can be determined by dividing the mRNA length by three.

Finally, the mRNA survives for an average of 2 minutes. This lifetime must be converted into seconds so that it can be directly compared with the time required for translation.

Step-by-Step Calculation of the Number of Ribosomes

Step 1: Calculate the Translation Rate of One Ribosome

A ribosome synthesizes a protein containing 300 amino acids in 20 seconds. Therefore, the translation rate can be calculated by dividing the number of amino acids synthesized by the time required.

Translation rate = Number of amino acids synthesized ÷ Time

Translation rate = 300 ÷ 20

Translation rate = 15 amino acids per second

Therefore, under the conditions described in the question, one bacterial ribosome adds amino acids to the growing polypeptide chain at an average rate of 15 amino acids per second.

Step 2: Calculate the Number of Codons in the 1350-Nucleotide mRNA

The given mRNA contains 1350 nucleotides. A codon consists of three consecutive nucleotides. Therefore, the total number of codons can be calculated by dividing the number of nucleotides by three.

Number of codons = Total number of nucleotides ÷ 3

Number of codons = 1350 ÷ 3

Number of codons = 450

Thus, the 1350-nucleotide mRNA corresponds to 450 codon positions. For the calculation intended in this question, the ribosome therefore has to translate a sequence corresponding to 450 amino acid positions.

Step 3: Calculate the Time Required to Translate the 1350-Nucleotide mRNA

We have already calculated that the ribosome synthesizes protein at a rate of 15 amino acids per second. The given mRNA corresponds to 450 amino acid positions.

Therefore, the time required for one ribosome to translate the complete mRNA is:

Translation time = Number of amino acids ÷ Translation rate

Translation time = 450 ÷ 15

Translation time = 30 seconds

Therefore, one ribosome requires 30 seconds to translate the complete 1350-nucleotide mRNA under the conditions given in the question.

Step 4: Convert the mRNA Lifetime into Seconds

The average lifetime of the mRNA is given as 2 minutes. Since the translation time has been calculated in seconds, the mRNA lifetime must also be expressed in seconds.

1 minute = 60 seconds

Therefore:

2 minutes = 2 × 60 seconds

2 minutes = 120 seconds

Thus, the mRNA remains available for translation for an average of 120 seconds.

Step 5: Calculate the Number of Ribosomes That Can Translate the mRNA

One ribosome requires 30 seconds to translate the complete mRNA, while the mRNA remains functional for 120 seconds.

Therefore, the number of complete translation events possible during the lifetime of the mRNA is calculated as:

Number of ribosomes = mRNA lifetime ÷ Time required for one complete translation

Number of ribosomes = 120 ÷ 30

Number of ribosomes = 4

Therefore, according to the calculation intended in this question, 4 ribosomes can translate the single mRNA during its average lifetime.

Complete Calculation in a Single Expression

The complete solution can also be written in a compact mathematical form.

First, calculate the translation rate:

300 amino acids ÷ 20 seconds = 15 amino acids per second

Next, calculate the number of codons in the mRNA:

1350 nucleotides ÷ 3 nucleotides per codon = 450 codons

Now calculate the time required to translate the complete mRNA:

450 ÷ 15 = 30 seconds

Convert the mRNA lifetime into seconds:

2 minutes × 60 = 120 seconds

Finally:

Number of ribosomes = 120 ÷ 30 = 4

Final Answer = 4 ribosomes

Why Is the Translation Rate 15 Amino Acids per Second?

The translation rate describes how rapidly a ribosome moves along an mRNA while adding amino acids to a growing polypeptide chain. The question directly provides enough information to calculate this rate.

If 300 amino acids are incorporated into a protein in 20 seconds, then the number of amino acids incorporated in one second is:

300 ÷ 20 = 15 amino acids per second

This translation rate is then applied to the second mRNA because the question assumes that the ribosome translates at the same average speed.

The longer mRNA contains 1350 nucleotides, corresponding to 450 codons. At a speed of 15 amino acid positions per second, the ribosome requires 30 seconds to move through the complete coding sequence.

Why Do We Divide the mRNA Length by Three?

The genetic information in mRNA is read in groups of three nucleotides called codons. Each codon generally specifies one amino acid during protein synthesis, except for termination codons that signal the end of translation.

Therefore, to convert the nucleotide length of a coding sequence into the corresponding number of codon positions, the total number of nucleotides is divided by three.

For the given mRNA:

1350 ÷ 3 = 450 codons

This conversion is essential because the translation rate provided in the question is based on the number of amino acids synthesized rather than the number of nucleotides traversed.

Why Does the 1350-Nucleotide mRNA Require 30 Seconds for Translation?

The ribosome synthesizes protein at a rate of 15 amino acids per second. The 1350-nucleotide mRNA contains 450 codon positions.

Therefore:

Time required = 450 ÷ 15

Time required = 30 seconds

This means that one complete passage of a ribosome along the given mRNA requires 30 seconds. Since the mRNA remains functional for 120 seconds, four such complete translation intervals fit within its average lifetime.

Why Is the mRNA Lifetime Important?

Bacterial mRNAs are generally short-lived compared with many eukaryotic mRNAs. An mRNA molecule can be translated only while it remains sufficiently intact and functional.

In this question, the mRNA has an average lifetime of 2 minutes, which is equivalent to 120 seconds. During this period, the translation machinery can use the mRNA as a template for protein synthesis.

Since each complete translation requires 30 seconds, the number of complete translation intervals available during the 120-second lifetime is:

120 ÷ 30 = 4

Therefore, the lifetime of the mRNA directly determines the total number of complete translation events considered in the calculation.

Understanding the Relationship Between mRNA Length and Translation Time

The time required for a ribosome to translate an mRNA depends primarily on the length of the coding sequence and the rate at which amino acids are incorporated into the growing polypeptide chain.

A shorter coding sequence requires less time for complete translation, whereas a longer coding sequence requires more time if the ribosome moves at the same rate.

In the first part of the question, a 300-amino-acid protein is synthesized in 20 seconds. The second mRNA corresponds to 450 codon positions, which is 1.5 times longer.

Therefore, the translation time is also 1.5 times longer:

20 seconds × 1.5 = 30 seconds

This provides another direct way to determine the translation time of the 1350-nucleotide mRNA.

Alternative Method Using Proportional Calculation

The problem can also be solved without separately calculating the translation rate. A ribosome synthesizes 300 amino acids in 20 seconds. The given 1350-nucleotide mRNA corresponds to 450 codon positions.

Therefore:

300 amino acids → 20 seconds

450 amino acid positions → x seconds

Using direct proportion:

x = (450 × 20) ÷ 300

x = 30 seconds

The mRNA lifetime is 120 seconds. Therefore:

Number of ribosomes = 120 ÷ 30 = 4

This alternative method gives the same final answer.

Biological Significance of Multiple Ribosomes Translating an mRNA

In bacterial cells, a single mRNA molecule can be used repeatedly for protein synthesis during its functional lifetime. This allows the cell to produce multiple protein molecules from one transcript and greatly increases the efficiency of gene expression.

In real cells, several ribosomes can also occupy and translate the same mRNA simultaneously, producing a structure known as a polysome or polyribosome. The numerical calculation in this question, however, uses the given translation time and mRNA lifetime to determine the number of complete translation events represented by the provided data.

The key mathematical relationship is therefore the ratio between the total functional lifetime of the mRNA and the time required for one complete translation.

Final Answer

Correct Answer: 4

A bacterial ribosome synthesizes 300 amino acids in 20 seconds, giving a translation rate of 15 amino acids per second. An mRNA containing 1350 nucleotides corresponds to 450 codon positions.

At a translation rate of 15 amino acids per second, one ribosome requires:

450 ÷ 15 = 30 seconds

The average lifetime of the mRNA is 2 minutes:

2 × 60 = 120 seconds

Therefore:

Number of ribosomes = 120 ÷ 30 = 4

Final Answer: 4 ribosomes