Question

A protein has a diffusion coefficient in water of D = 10-6 cm2/s. How long does it take for the protein to diffuse 10 μm (typical radius of a cell)? Consider 3D diffusion.

Answer 1

Using the formula Xrms = √(2Dt), we can calculate the time it takes for the protein to diffuse 10 µm by rearranging it to t = (Xrms2) / (2D) and substituting the given values.

Step-by-step explanation:

The question asks how long it takes for a protein with a diffusion coefficient (D) of 10-6 cm2/s to diffuse 10 µm, considering 3D diffusion. The protein's movement through water can be described using the root mean square distance (Xrms) formula for diffusion: Xrms = √(2Dt), where D is the diffusion constant and t is the time.

Given that the typical radius of a cell is 10 µm, which equals 0.001 cm, we can rearrange the formula to solve for t (time): t = (Xrms2) / (2D). Using the given value of D, we plug in the values to calculate the time it would take for such a diffusion process.

By substituting Xrms with 0.001 cm and D with 10-6 cm2/s, we will find the value for t, which is the average time required for the protein to diffuse the specified distance. Remember that diffusion rates can be affected by molecule mass, temperature, as well as cohesive and adhesive forces.

Answer 2

0.5s

Step-by-step explanation:

Relevant Equation:

Einstein's Approximation

`t=(d^2)/(2D)`

Where

• t is time used for diffusion
• d is the mean value of distance diffused
• D is diffusion coefficient

Typically, Einstein's approximation equation is accurate enough to be used in 3D biological diffusion while still being simple to use, thus this will be our go-to equation.

When plugging in the given parameters (Don't forget to convert units)

`t=(d^2)/(2D)((10*10^(-6))^2)/(2*(10^-6)^(-4))`

`t= 0.5\ s`

Which is considered slow for usual chemical reaction in cells, which is why transport proteins are usually used to move protein across the cell