Question

Probability and translation-start codons. In prokaryotes, translation of mRNA messages into proteins is most often initiated at start codons on the mRNA having the sequence AUG. Assume that the mRNA is single-stranded and consists of a sequence of bases, each described by a single letter A, C, U, or G. Consider the set of all random pieces of bacterial mRNA of length six bases. (a) What is the probability of having either no A's or no U's in the mRNA sequence of six base pairs long? (b) What is the probability of a random piece of mRNA having exactly one \(\mathbf{A}\), one \(\mathbf{U}\), and one \(\mathbf{G}\) ? (c) What is the probability of a random piece of mRNA of length six base pairs having an A directly followed by a U directly followed by a G; in other words, having an AUG in the sequence? (d) What is the total number of random pieces of mRNA of length six base pairs that have exactly one \(\mathbf{A}\), exactly one \(\mathbf{U}\), and exactly one \(\mathbf{G}\), with \(\mathbf{A}\) appearing first, then the \(\mathbf{U}\), then the \(\mathbf{G}\) ? (e.g., AXXUXG)

Short answer

The probability of no A's or U's is 0.34. The probability of exactly one A, U, and G is 0.029. The probability of having AUG in the sequence is 0.0625. There are 120 sequences with A, U, and G in order.

Step-by-step solution

Title - Calculate the Probability of No A's

There are 4 bases: A, C, U, G. The probability of a random base not being A is \(\frac{3}{4}\). For a sequence of 6 bases to have no A's, all 6 must not be A. Calculate \(\frac{3}{4}\) raised to the power of 6: \[ \left( \frac{3}{4} \right)^6. \] Using a calculator, this comes out to approximately \(\frac{729}{4096}\), which is around 0.178.

Title - Calculate the Probability of No U's

Similarly, the probability of a random base not being U is also \(\frac{3}{4}\). For a sequence of 6 bases to have no U's, all 6 must not be U. Calculate \(\frac{3}{4}\) raised to the power of 6: \[ \left( \frac{3}{4} \right)^6. \] This is also approximately \(\frac{729}{4096}\), which is around 0.178.

Title - Combine the Probabilities to Get No A's or No U's

Use the principle of inclusion-exclusion to calculate the probability of no A's or no U's: \[ P(\text{No A or No U}) = P(\text{No A}) + P(\text{No U}) - P(\text{No A and No U}). \] We have \[ P(\text{No A and No U}) = \left( \frac{2}{4} \right)^6 = \left( \frac{1}{2} \right)^6 = \frac{1}{64}. \] Hence, \[ P(\text{No A or No U}) = 2 \times \frac{729}{4096} - \frac{1}{64} \approx 0.356 - 0.016 = 0.34. \]

Title - Calculate the Probability of Exactly One A, One U, and One G

There are three remaining bases after placing one A, one U, and one G. Calculate the number of sequences by finding permutations: \[ \frac{6!}{(3! 1! 1! 1!)} = \frac{720}{6} = 120. \] Therefore, the probability is: \[ \frac{120}{4^6} = \frac{120}{4096} \approx 0.029. \]

Title - Probability of Having AUG in Sequence

A random sequence of 6 bases can be broken into 4 possible sets of 3 consecutive bases: (1-3), (2-4), (3-5), (4-6). The probability of each set being AUG is: \[ P(\text{AUG} \, in \, 6 \, bases) = 4 \times (\frac{1}{4})^3 = 4 \times \frac{1}{64} = \frac{4}{64} = \frac{1}{16} \approx 0.0625. \]

Title - Total Sequences with A, U, G in Order

Using the remaining 3 positions out of 6 for any other bases (C, U, G): \[ \frac{3!}{6} = 120. \]

Key concepts

Prokaryotic Translation Initiation

In prokaryotes, the initiation of translation is crucial for protein synthesis. The translation process starts when ribosomes attach to the mRNA sequence and moves to the start codon. The start codon in nearly all prokaryotic mRNA sequences is AUG. This codon signals the ribosome to begin translating the mRNA into a protein.
This initiation process is regulated by specific sequences in the mRNA that align with the ribosome. These sequences help to position the ribosome correctly so that translation begins at the start codon. The efficiency of translation initiation can vary depending on the specific sequences and structures of the mRNA.
Understanding this process is fundamental because it influences how genes are expressed and how proteins are synthesized. Proteins are essential for all cellular functions, and the regulation of their synthesis is critical for the proper functioning of the cell.

mRNA Codons

mRNA codons are sequences of three nucleotides each that correspond to specific amino acids or stop signals during protein synthesis. There are 64 possible codons made from the four nucleotides: adenine (A), cytosine (C), uracil (U), and guanine (G).
For example:

• AUG is the start codon and codes for the amino acid methionine.
• UAA, UAG, and UGA are stop codons, which signal the end of the protein synthesis.

Each nucleotide triplet or codon is read in a sequential manner by the ribosome during translation. This codon sequence determines the sequence of amino acids in the resulting protein.
The redundancy of the genetic code (multiple codons can encode the same amino acid) helps protect against mutations. A change in one nucleotide might still result in the same amino acid, thus minimizing the potential harmful effects of mutations.

Sequence Probability Calculation

Calculating the probability of specific sequences in mRNA involves understanding and using basic principles of probability. Probability calculations for biological sequences are crucial for predicting how often certain sequences or mutations might occur.
For instance, to find the probability of an mRNA sequence of six bases having no A's, we raise the probability of a single base being something other than A, which is \(\frac{3}{4}\), to the power of six: \(\big( \frac{3}{4} \big)^6\).
When calculating the combined probability of complex sequences, we often use the principle of inclusion-exclusion, or specific statistical formulas. For example, to find the probability of a sequence containing an A directly followed by U and then by G (AUG), we need to look at all possible positions this triplet can appear and sum their probabilities. For a 6-base sequence, AUG can appear starting at positions 1 through 4. The probability at each position is \(\frac{1}{64}\), and thus the overall probability is \(\frac{4}{64} = \frac{1}{16}\).
Moreover, sequences containing specific combinations, such as one A, one U, and one G among other bases, require permutations to account for possible arrangements. Permutations help us calculate the number of possible sequences that fit these specific criteria.

Biological Sequence Analysis

Biological sequence analysis involves interpreting DNA, RNA, and protein sequences to understand their structures, functions, and evolutionary relationships. In the context of mRNA, it includes analyzing codon usage, mutation impacts, and translation initiation sites.
Tools and methodologies such as sequence alignment, motif detection, and statistical models are employed to study these sequences. Extensive databases exist to compare and analyze biological sequences across different species, providing insights into their evolutionary conservation and functional importance.
Analyzing mRNA sequences can help researchers predict protein products, understand gene expression patterns, and identify regulatory elements that control different cellular processes. This kind of analysis is fundamental in fields like genomics, bioinformatics, and molecular biology.
By applying probability and statistical tools, scientists can make predictions about biological phenomena, test hypotheses, and gain a deeper understanding of the fundamental mechanisms of life.