Electric Forces and Electric Fields in DNA

[Dita]

I need help with the following, I don’t even know how to start

A molecule of DNA ( deoxyribonucleic acid) is 2.17μm long. The ends of the molecule become singly ionized – negative on one end, positive on the other. The helical molecule acts as a spring and compresses 1.00 % on becoming charged. Determine the effective spring constant of the molecule.

 

[Chen]

Maybe you're having problems with "singly ionized"? It means that each side of the molecule is charged with by one elementary charge. That is obviously the charge of the electron/proton, so you can just say that one side gains an electron while the other side loses one. Because of this there is an electric force between the two sides of the molecule, which is equal to:

$$F_{ele} = K\frac{q_e^2}{d^2}$$

Where d is the distance between the charges, in our case 2.17μm. That is the force that causes the spring-like molecule to compress. Now you can use Hooke's law to find the spring's constant, since you know the magnitude of the force as well as the compression length (1% of 2.17μm). The answer should be 2.45x10^{-9 N}/_m.

 

[M98Ranger]

First, let's break down the given information. We know that a molecule of DNA is 2.17μm long and that it becomes singly ionized, with one end becoming negative and the other becoming positive. We also know that when this happens, the helical molecule compresses by 1.00%. This means that the distance between the two charged ends decreases by 1.00% of 2.17μm, or 0.0217μm.

Now, let's consider the concept of electric forces and electric fields in DNA. When a molecule becomes charged, it creates an electric field around it. This electric field exerts a force on any other charged particles nearby. In the case of DNA, the charged ends of the molecule will create an electric field that will interact with other charged particles, such as ions or other molecules.

In this scenario, the negatively charged end of the DNA molecule will repel other negative charges and attract positive charges, while the positively charged end will repel positive charges and attract negative charges. This creates a net force that compresses the molecule, causing it to act like a spring.

To determine the effective spring constant of the DNA molecule, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. In this case, the force exerted by the electric field is equivalent to the force exerted by the spring. Therefore, we can write the following equation:

F = -kx

Where:
F is the force exerted by the electric field
k is the spring constant
x is the displacement of the spring (in this case, 0.0217μm)

We also know that the magnitude of the force exerted by the electric field is equal to the product of the charge of the molecule (q) and the strength of the electric field (E). Therefore, we can rewrite the equation as:

qE = -kx

Now, we need to determine the value of q and E. We know that the molecule becomes singly ionized, meaning that one end has a charge of +q and the other has a charge of -q. We also know that the distance between the two charged ends is 0.0217μm. Therefore, we can use Coulomb's Law to calculate the strength of the electric field:

E = kq/r^2

Where:
k is

 

[M98Ranger]

To begin, we can use the given information to calculate the change in length of the DNA molecule when it becomes charged. We know that the original length of the molecule is 2.17μm and it compresses 1.00% when charged, so the change in length is 0.01(2.17μm) = 0.0217μm.

Next, we can use the equation for spring constant (k) to find the effective spring constant of the DNA molecule. The equation is k = F/x, where F is the force applied and x is the change in length of the spring. In this case, the force applied is the electric force between the two charged ends of the DNA molecule.

To find the electric force, we can use Coulomb's law, which states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In this case, the charges are equal in magnitude (one negative and one positive) and the distance between them is the change in length of the DNA molecule, which we calculated earlier to be 0.0217μm.

So, the electric force between the two charged ends of the DNA molecule is given by F = kQ^2/d^2, where Q is the magnitude of the charge and d is the distance between the charges. Plugging in the values, we get F = k(-Q)^2/(0.0217μm)^2.

Now, we can substitute this value for force into the equation for spring constant and solve for k. We get k = F/x = [k(-Q)^2/(0.0217μm)^2]/(0.0217μm) = k(-Q)^2/(0.0217μm)^3.

Finally, we can plug in the given values for Q (the magnitude of the charge on each end of the DNA molecule) and solve for k. The magnitude of the charge is not given, so we can use a variable, Q, to represent it. We get k = kQ^2/(0.0217μm)^3.

So, the effective spring constant of the DNA molecule is given by k = kQ^2/(0.0217μm)^3. This equation shows that the spring constant is directly proportional to the square of the charge and inversely