How Much Work to Displace a Microtubule's Free End?

[biophysgirl]

A microtubule is 100 microns long. Take the young modulus as 1 GPa and the microtubule as a
hollow cylinder with outer diameter of 12.5 nm and thickness 2.5nm.

a) If one end of the microtubule is clamped down, and is the other end is free to wobble.
Estimate the work that must be done to displace the free end of the microtubule by an amount
y in a direction perpendicular to the microtubule itself.

b) Thermal fluctuations will bend the microtubule spontaneously. Estimate the value of y under
thermal energy of room temp.

c) How much energy is needed to curve this microtubule by 60 degrees? Express your result in units of
kBT.

Any help is much appreciated!

 

[Bystander]

https://www.physicsforums.com/threads/guidelines-for-students-and-helpers.635513/
Pick a starting point, take things one step at a time, and we'll see where you get stuck.

 

[biophysgirl]

I'm not sure how to derive part c. I know that y=L*tan(theta/2) and that the bending energy=2*keffective*y^2/L^3. I'm not sure where to go from here

 

[Bystander]

How much energy is needed to curve this microtubule by 60 degrees?

That's pretty much open to any translation/interpretation you want, isn't it? I'd be inclined to call it a uniform curvature over length of the tube such that tangents to the fixed and free ends are 60 degrees apart.

 

[biophysgirl]

I am aware of that part. I'm not sure how to integrate that into the equations.

 

[Bystander]

bending energy=2*keffective*y^2/L^3

My inclination would be to differentiate this expression with respect to theta. I'll send up a flare to the mech. E's to come take a look and verify or correct that notion for you.

 

[biophysgirl]

Thanks!

 

[Bystander]

And while we're waiting for the experts, from Machinery's, 26th, p. 249, deflection of a round cantilevered beam is proportional to lW/d³, l the length, W the load, and d the diameter; I doubt the relation holds for the extreme deflection of this problem, but might be a test for form.

 

[haruspex]

I'm not sure how to derive part c. I know that y=L*tan(theta/2) and that the bending energy=2*keffective*y^2/L^3. I'm not sure where to go from here

I don't know where those equations come from or what their applicability is, but they look to me like approximations for small deflections. The deflection is not small here.
On the net I see equations like MR = EI (bending moment, radius of curvature, modulus, second moment of area respectively), $U = \frac{EI\theta}{2R}$, theta being the angular deflection. For a hollow tube $I=\pi({r_o}^4-{r_i}^4)$, the r's being the outer and inner radii.