Protein Diffusion: How Could This Happen?

[superwolf]

Homework Statement

You release a billion protein molecules at position x=0 in the middle of a narrow capillary test tube. The molecues' diffusion constant is 10^-6 cm^2s^-1. An electric field pulls the molecules to the right (larger x) with a drift velocity of 1 micrometer per second. Nevertheless, after 80 s you see that a few protein molecules are actually to the left of where you released them. How could this happen? What is the ending number density right at x=0?

Homework Equations

Suppose N molecules all begin at the same location in 3D space at time zero. Later the concentration is

[tex]
c(r,t)=\frac{N}{(4 \pi Dt)^{3/2}} e^{-r^2/(4Dt)}
[/tex]

The Attempt at a Solution

From a similar problem, it looks like the formula in 1D is

[tex]
c(r,t)=\frac{N}{(4 \pi Dt)^{1/2}} e^{-r^2/(4Dt)}
[/tex]

Is this correct?

If so, we have

[tex]
c(0,80)=\frac{1 \cdot 10^9}{(4 \pi 1 \cdot 10^-6 \cdot 80)^{/2}} e^0 = 3.15E10 cm^-1

[/tex]

This clearly must be wrong, or what?

 

[Jhero]

Thank you for your post. Your attempt at a solution is on the right track, but there are a few things that need to be clarified.

Firstly, the formula you have written is correct for the concentration of molecules in 3D space. However, since this problem is in 1D, we only need to consider the concentration along the x-axis. This means that the formula should be written as:

c(x,t)=\frac{N}{(4 \pi Dt)^{1/2}} e^{-x^2/(4Dt)}

where x is the distance along the x-axis.

Secondly, the concentration at a specific point in space (in this case, x=0) cannot be calculated using this formula. This formula gives the concentration at a specific distance from the point of origin (x=0). To find the concentration at x=0, we need to integrate the concentration over all space, which can be done using the following integral:

\int_{-\infty}^{\infty} c(x,t) dx = N

This means that the ending number density at x=0 is equal to N, which in this case is 1 billion molecules.

Lastly, to address the issue of a few molecules being found to the left of where they were released, this is due to the random nature of diffusion. Even though the molecules are being pulled to the right by an electric field, they still have a chance of moving in the opposite direction due to their random motion. This can be seen in the formula, where the concentration is dependent on the distance from the origin (x=0) and time, but also on the diffusion constant D, which represents the randomness of the molecules' motion.

I hope this helps to clarify the solution to the problem. Please let me know if you have any further questions.