Automated chemical synthesis of DNA has made it possible to custom-order moderate length DNA sequences from commercial suppliers. Assume that a single nucleotide weighs about 5.6 X 10^-22 grams and that there are four kinds of nucleotides. If you wish to order all possible DNA sequences of a fixed length, at what length will you order exceed
a) 100 kg and
b) the mass of the Earth (5.9736 X 10^24 kg)?
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To determine the length at which the order of all possible DNA sequences exceeds a certain mass, we need to calculate the total mass of all possible DNA sequences up to that length.
Let's start by determining the number of possible DNA sequences for a fixed length. Since DNA consists of four different nucleotides (A, T, G, and C), and each position in the DNA sequence can have any of these nucleotides, the number of possible sequences for a fixed length is given by 4^n, where n is the length of the sequence.
a) To find the length at which the mass exceeds 100 kg, we need to calculate the total mass of all possible sequences until that length.
Let's assume x represents the length of the DNA sequence. The total mass of all possible sequences up to length x is calculated using the formula:
Total mass = (number of possible sequences) * (mass of each nucleotide) * (length of each sequence)
In this case, the mass of each nucleotide is given as 5.6 x 10^-22 grams.
Total mass = (4^1 * mass of each nucleotide) + (4^2 * mass of each nucleotide) + ... + (4^x * mass of each nucleotide)
Now, we can solve for x by equating the total mass to 100 kg:
100 kg = (4^1 * 5.6 x 10^-22 g) + (4^2 * 5.6 x 10^-22 g) + ... + (4^x * 5.6 x 10^-22 g)
We can solve this equation using a numerical method or approximation techniques.
b) To find the length at which the mass exceeds the mass of the Earth, we follow the same approach as above. Instead of equating the total mass to 100 kg, we equate it to the mass of the Earth (5.9736 x 10^24 kg):
5.9736 x 10^24 kg = (4^1 * 5.6 x 10^-22 g) + (4^2 * 5.6 x 10^-22 g) + ... + (4^x * 5.6 x 10^-22 g)
Similarly, we can solve this equation using appropriate numerical methods or approximations.
Please note that due to the complexity of these calculations, it may be more practical to consult existing databases or scientific literature to find the required lengths.