Consider the following model for the electrostatic stretching of DNA. A spring with spring constant kDNA = 250 pN/m is tethered to an immobile surface on one end and a charge of qDNA = −350e on the other. A uniform electric field of magnitude |E| = 2.5 N/C is applied pointing perpendicular to the surface.

a) How far does the spring stretch in response to the field?

b) Now an transcription factor (TF), with charge qRNA = +15e, is introduced at the location specified in the figure. Calculate the electric field due to the TF at the location of the DNA. Remember, biophysics experiments are done in water, which has a dielectric constant K = 80. (hint: the TF is very far from both the DNA and the surface, so you can ignore the string stretching distance in this calculation)

c) What is the angle θ that the spring makes with the x-axis in the presence of both the TF and the external field?

I would love any help with this question, as I do not really know where to start.

Sure! I can help you with that. Let's go through each part of the question one by one.

a) To find how far the spring stretches in response to the electric field, we can use Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position. In this case, the force is caused by the electric field, and the displacement is the stretching of the spring.

The equation for Hooke's Law is:

F = -k * x

where F is the force, k is the spring constant, and x is the displacement.

In this case, the force is caused by the electric field, which is calculated as the product of the charge and the electric field strength:

F = q * E

where q is the charge and E is the electric field strength.

Combining these equations and solving for x, we get:

x = -F / k = -(q * E) / k

Substituting the given values, we have:

q = -350e (charge of DNA)
E = 2.5 N/C (electric field strength)
k = 250 pN/m = 250 * 10^(-12) N/m (spring constant)

x = -(-350e * 2.5 N/C) / (250 * 10^(-12) N/m)

You can calculate the value of x using these numbers. Remember to convert the charge and spring constant to SI units if necessary.

b) To calculate the electric field due to the transcription factor (TF) at the location of the DNA, we can use Coulomb's Law. Coulomb's Law states that the electric field created by a point charge is given by:

E = k * |q| / r^2

where E is the electric field strength, k is the Coulomb's constant (k = 8.99 * 10^9 N*m^2/C^2), |q| is the magnitude of the charge, and r is the distance between the charges.

In this case, the charge of the TF is given as qRNA = +15e, and we need to find the electric field strength at the location of the DNA. Since the TF is very far from both the DNA and the surface, we can ignore the stretching of the spring in this calculation.

The distance between the charges (TF and DNA) is not given in the question. You need to refer to the figure or any additional information provided to determine the distance.

Using the equation and the given charge, you can calculate the electric field due to the TF at the location of the DNA.

c) To find the angle θ that the spring makes with the x-axis in the presence of both the TF and the external field, we need to consider the forces acting on the spring.

The electrostatic force due to the electric field will exert a force on the DNA charge, pulling it in the direction of the field. The spring will counteract this force, pulling the DNA back towards the surface.

The angle θ can be determined by considering the balance of forces. The force due to the electric field and the force due to the spring will be perpendicular to each other, so they can be treated as components of a right triangle.

Using trigonometry, we can relate the angle θ to the forces acting on the spring. The tangent of the angle θ is given by:

tan(θ) = F_spring / F_electric

where F_spring is the force due to the spring and F_electric is the force due to the electric field.

You can calculate the forces using the given information and then find the angle θ using this equation.

Remember to use SI units in all the calculations and convert any given values to SI units if necessary.