How Much Work Does Motor φ29 Need to Overcome Entropy Loss in DNA Packing?

[emoboya3]

Homework Statement

Estimate total work that motor $$\phi$$29 needs to perform to overcome entropy loss of packing DNA

Homework Equations

P(R,N)=(3/2$$\pi$$Na$$^{2}$$)$$^{3/2}$$e$$^{(-3R^{2})/(2Na^{2}}$$)
This equation gives the end to end distribution or a 3D random walk with N segments length a.

The Attempt at a Solution

I'm not 100% sure where to start on this. I've been trying to figure it out all day, honestly. I know that DNA has a persistence length a$$\approx$$100nm because it's fairly rigid over about 300 base pairs. It is therefore modeled as N=65 segments.
I'm not sure where to go from here though. How do I come up with the entropy change by putting the DNA in the capsule?

I planned to model the capsule as a sphere of radius 20nm. Any help on where to go from here?

Thanks

 

[Jhero]

for your post and for sharing your thoughts on this problem. It's an interesting and challenging one! Let's see if we can work through it together.

First, let's define some variables to make things a bit clearer. Let's say that R is the radius of the capsule, N is the number of segments in the DNA, and a is the persistence length of the DNA. We can also define the volume of the capsule as V=4/3πR^3. Now, let's break down the problem into smaller steps.

Step 1: Calculate the end-to-end distribution
As you mentioned, the end-to-end distribution of a 3D random walk with N segments of length a is given by the equation P(R,N)=(3/2πNa^2)^3/2e^(-3R^2/2Na^2). This equation gives us the probability of finding the end of the DNA chain at a distance R from the starting point. We can use this equation to calculate the probability of finding the end of the DNA chain at the surface of the capsule, which we can call P(R=Rc,N). This will give us an idea of how much DNA is packed into the capsule.

Step 2: Calculate the volume of the DNA
Now, we need to calculate the volume of the DNA that is packed into the capsule. We can do this by multiplying the number of segments N by the length of each segment a. This will give us the total length of DNA in the capsule, which we can then multiply by the cross-sectional area of the DNA (πa^2) to get the volume. So the volume of DNA in the capsule is VDNA=Nπa^3.

Step 3: Calculate the change in entropy
Next, we need to calculate the change in entropy when we pack the DNA into the capsule. Entropy is a measure of disorder or randomness, and in this case, it is a measure of how much the DNA is confined or restricted in its movement. To calculate the change in entropy, we can use the equation ΔS=kln(W), where k is the Boltzmann constant and W is the number of possible configurations of the DNA. For a 3D random walk, the number of possible configurations is given by W=(2N)!/N!(N+1)!. So we can calculate the change in entropy as ΔS=kln[(2N)!/N!(N+