Imagine you are purifying a protein which accounts for 0.001% of the total protein within a cell. How many steps of purification are required to yield pure protein, assuming a 10 fold purification with each step of purification?

To calculate the number of purification steps required, we need to determine how many times we need to purify the protein to reach a purity level of 100%.

Given that each step of purification provides a 10-fold purification, we can calculate the fold increase in purity for each step using the formula:

Fold Increase = 1 / (Percentage Remaining/100)

First, let's determine the initial purity of the protein before any purification steps are taken.

Initial purity = 0.001% = 0.00001 (as a decimal)

Next, we'll calculate the fold increase in purity for each purification step:

Fold Increase = 1 / (0.00001) = 100,000

To find the number of purification steps required, we need to keep multiplying the fold increase until we reach a purity level of 100%.

Let's say the number of purification steps required is 'n'.

Fold Increase^ n = 100,000^n = 1 (since we want a purity level of 100%)

Solving this equation for 'n', we find:

n = log(1)/log(100,000)

Using a calculator, the logarithm of 1 to any base is zero, so:

n = 0 / log(100,000)

Taking the logarithm with base 10, we can calculate:

n ≈ 0 / 5

n = 0

Therefore, we require 0 purification steps to reach a purity level of 100%.

However, it's important to note that this result assumes perfect purification efficiency, which is unlikely to be achievable in practice. Various factors such as losses during the purification process, sample losses, and impurities can affect the actual number of purification steps required.