To determine the minimum size of a codon for the hypothetical Martian life-forms, we first need to calculate how many unique combinations of codons can be generated with the given number of bases and then match that to the number of amino acids they use.
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Number of Bases: The Martian organisms use only 2 bases (A and T).
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Number of Amino Acids: There are 17 amino acids.
A codon is defined as a sequence of bases that corresponds to an amino acid. The number of unique codons that can be formed is calculated as \( n^k \), where \( n \) is the number of bases and \( k \) is the length of the codon.
Given:
- \( n = 2 \) (A and T)
- \( k \) is the length of the codon we need to determine.
To find the minimum \( k \) such that the number of distinct codons is at least 17, we start calculating:
- For \( k = 1 \): \( 2^1 = 2 \)
- For \( k = 2 \): \( 2^2 = 4 \)
- For \( k = 3 \): \( 2^3 = 8 \)
- For \( k = 4 \): \( 2^4 = 16 \)
- For \( k = 5 \): \( 2^5 = 32 \)
From the calculations:
- At \( k = 4 \) (codon length of 4), there are 16 unique combinations, which is not enough to code for 17 different amino acids.
- At \( k = 5 \), there are 32 unique combinations, which is more than sufficient to represent the 17 amino acids.
Therefore, the minimum size of a codon for these Martian life-forms is 5 bases.
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