Question

Nucleic acids are polymers of just four different monomers in a linear arrangement. How many different sequences are available if one makes a polymer with only 40 monomers? How does this number compare with Avogadro's number?

Short answer

The number of possible sequences is \[ 2^{80} \], about \[ 1.2 \times 10^{24} \], comparable to Avogadro's number \[ 6.022 \times 10^{23} \].

Step-by-step solution

- Identify the Number of Monomers

Recognize that the polymer consists of 40 monomers. In this exercise, we are considering nucleic acids.

- Number of Possible Monomer Sequences

Given there are 4 different types of nucleotides (monomers), for each position in the 40-monomer polymer, a choice can be made from these 4 nucleotides.

- Calculate the Total Number of Sequences

Use the formula for the number of sequences possible with a fixed length of polymer: \[ 4^{40} \]. Calculate the value of \[ 4^{40} \].

- Simplify the Exponential Calculation

4 can be written as \[ 2^2 \], so: \[ 4^{40} = (2^2)^{40} = 2^{80} \]. Calculate \[ 2^{80} \] to find the total number of sequences.

- Compare with Avogadro's Number

Avogadro's number is approximately \[ 6.022 \times 10^{23} \]. Compare the calculated number of sequences, \[ 2^{80}\], which is approximately \[ 1.2 \times 10^{24} \], with Avogadro's number to find it's of the same order of magnitude.

Key concepts

Monomers

In the world of nucleic acids, monomers refer to the building blocks that create polymers like DNA and RNA. These monomers are essentially nucleotides. When we say a polymer has 40 monomers, we mean that it is made up of a chain of 40 nucleotide units.
Each nucleotide comprises three components:

• A phosphate group
• A sugar molecule
• A nitrogenous base (adenine, thymine, cytosine, or guanine in DNA)

Understanding the structure of these nucleotides helps in grasping how they come together to form nucleic acids. The sequence in which these nucleotides are arranged determines the genetic information carried by the polymer.

Nucleotides

Nucleotides are the actual monomers that form nucleic acids. A nucleic acid sequence is a linear arrangement of these nucleotides. When we talk about the different sequences, each position in the sequence can be occupied by one of four types of nucleotides.
These four types include:

• Adenine (A)
• Thymine (T) - replaced by Uracil (U) in RNA
• Cytosine (C)
• Guanine (G)

Given this diversity, the number of possible sequences for any length of nucleic acid can be calculated. For a polymer of 40 monomers, we are concerned with arranging these 40 slots with any of the four nucleotides, leading to numerous possible sequences.

Exponential Calculation

The number of different sequences possible in a nucleic acid polymer can be calculated using exponential notation. For a polymer of 40 monomers, where each position could be one of 4 nucleotides, we use the formula \( 4^{40} \).
This can be simplified using powers of 2, since 4 is essentially \( 2^2 \left( 4^{40} = (2^2)^{40} = 2^{80} \).
Understanding these exponential calculations helps break down seemingly complex numbers into more manageable terms. Calculating \( 2^{80} \) in simpler steps reveals the large number of possible sequences, demonstrating the vast diversity of potential nucleic acid configurations. Specifically, \( 2^{80} \) results in approximately \( 1.2 \times 10^{24} \) sequences.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that represents the number of atoms, ions, or molecules in one mole of a substance. Its value is approximately \( 6.022 \times 10^{23} \).
Comparing this to the number of possible 40-monomer nucleic acid sequences (approximately \( 1.2 \times 10^{24} \)), we find that the number of sequences is on the same order of magnitude as Avogadro's number.
This comparison highlights the immense variety in potential genetic combinations, showcasing how the seemingly simple components of nucleic acids can lead to a staggering number of possibilities, far exceeding the number of molecules in a mole.