Question

Common proteins are polymers of 20 different amino acids. How big a protein (how many amino acid residues) would be necessary to have an Avogadro's number of possible sequences?

Short answer

19 amino acid residues are necessary.

Step-by-step solution

Understand Avogadro's Number

Avogadro's number is the number of atoms, ions, or molecules in one mole of a substance. It is equal to approximately \(6.022 \times 10^{23}\).

Recognize Variables

Each protein sequence is made up of amino acid residues. There are 20 different types of amino acids that can be used to form these sequences.

Establish Relationship between Protein Length and Sequences

If a protein has n amino acid residues, the total number of possible sequences it can form is given by \(20^n\), because there are 20 possible choices for each position in the sequence.

Set Up Equation

We need to find the number of amino acid residues n such that the total number of different sequences is equal to Avogadro's number. The equation that must be solved is: $$20^n = 6.022 \times 10^{23}$$

Solve for n using Logarithms

Take the logarithm of both sides of the equation to solve for n: $$\text{log}(20^n) = \text{log}(6.022 \times 10^{23})$$ Applying the properties of logarithms, this becomes: $$n \times \text{log}(20) = \text{log}(6.022) + \text{log}(10^{23})$$ Because $$\text{log}(10^{23}) = 23$$ and $$\text{log}(6.022) \text{ is approximately equal to } 0.78,$$ Therefore, we can write: $$n \times \text{log}(20) \text{ is approximately equal to } 0.78 + 23$$ Calculate: $$n \times 1.301 \text{ is approximately equal to } 23.78$$ Finally, solve for n: $$n \text{ is approximately equal to } \frac{23.78}{1.301} \text{ is approximately equal to } 18.28$$ Since the number of residues must be a whole number, take the ceiling: $$n = 19$$

Key concepts

amino acid residues

Amino acids are the building blocks of proteins. Each amino acid has a central carbon atom (also called the alpha-carbon) bonded to an amino group, a carboxyl group, a hydrogen atom, and an R group or side chain. The unique properties of the side chain determine the characteristics and functions of each amino acid.
When amino acids link together to form proteins, they do so through peptide bonds, resulting in a chain known as a protein sequence. Each link in the chain is called an amino acid residue.
In biochemistry, understanding amino acid residues is crucial as they influence the structure and functionality of proteins. The variety and sequence of amino acid residues in a protein determine its unique properties and biological roles.

protein sequences

Proteins are composed of one or more long chains of amino acids, referred to as protein sequences. The sequence of amino acids in a protein is determined by the genetic code and is unique to each protein. This sequence defines the protein's three-dimensional structure and function.
In a protein sequence, each position is occupied by one of the 20 different amino acids, allowing for a vast diversity of possible sequences. For example, a protein with just two amino acid residues can have 20^2 (or 400) possible sequences. The complexity and diversity increase significantly as the length of the protein sequence grows.
Understanding protein sequences is essential for many applications in biochemistry, such as drug design, enzyme mechanism studies, and the development of therapeutic proteins.

logarithms in biochemistry

Logarithms are a mathematical tool that allows us to solve equations involving exponential relationships. In biochemistry, they are especially useful when dealing with large numbers, such as calculating the number of possible protein sequences.
For example, consider the equation determining the number of amino acid residues required to achieve Avogadro's number of possible sequences. We start with the equation: \(20^n = 6.022 \times 10^{23}\).
To solve for n, we take the logarithm of both sides: \(\text{log}(20^n) = \text{log}(6.022 \times 10^{23})\).
Using logarithmic properties, this becomes: \(n \times \text{log}(20) = \text{log}(6.022) + \text{log}(10^{23})\).
Logarithms simplify the multiplications and powers, making it easier to isolate and solve for n, demonstrating their practical application in biochemistry problem-solving.

Avogadro's number application

Avogadro's number, approximately \(6.022 \times 10^{23}\), is a fundamental constant in chemistry. It represents the number of atoms, ions, or molecules in one mole of a substance. In biochemistry, we use Avogadro's number to understand the scale of molecular and protein interactions.
For instance, when determining how many amino acid residues are needed to produce an Avogadro's number of possible protein sequences, we can set up the equation \(20^n = 6.022 \times 10^{23}\). This equation helps us relate the length of a protein sequence (n) to the total number of possible sequences, bridging molecular biology and macroscopic quantities.
Applying Avogadro's number in this context allows us to grasp the enormity and potential variability inherent in biological systems at the molecular level.

biochemistry problem-solving

Solving problems in biochemistry often involves applying concepts from chemistry, biology, and mathematics. Let's break down the steps used in the given protein sequence problem to understand the process:

• First, identify what needs to be calculated: the length of a protein (number of amino acid residues) to have Avogadro's number of possible sequences.
• Understand the key variables: 20 different amino acids and the total number of sequences equal to Avogadro's number.
• Set up the relationship between protein length and sequences using the equation \(20^n = 6.022 \times 10^{23}\).
• Solve the equation using logarithms to isolate the variable n.
• Calculate and interpret the result, rounding to the nearest whole number where necessary.

Approaching biochemistry problems in this structured way ensures thorough understanding and accurate solutions, enhancing your problem-solving skills.