- Problems[[Student version, January 17, 2003]] 215
Repeat Problem 6.5 for a slightly different situation: Instead of having just two discrete allowed
values, our system has a continuous, unit-vector variableˆnthat can point in any direction in space.
Its energy is a constant plus−anˆ·ˆz,or−aˆnz=−acosθ.Hereais a positive constant with units
of energy andθis the polar angle ofnˆ.
a. Find the probability distributionP(θ, φ)dθdφfor the directions thatˆnmay point.
b. Compute the partition functionZ(a)and the free energyF(a)for this system. Then compute the
quantity〈nˆz〉.Your answer is sometimes called theLangevin function.Find the limiting behavior
at high temperature and make sure your answer is reasonable.
6.10 T 2 Gating compliance
(Continuation of Problem 6.7.) We can model the system in Figure 6.13 quantitatively as follows.
Wethink of the bundle of stereocilia as a collection ofNelastic units in parallel. Each element
has two springs: One, with spring constantkaand equilibrium positionxa,represents the elasticity
of the tip link filament. The other spring, characterized bykbandxb,represents the stiffness of
the stereocilium’s attachment point (provided by a bundle of actin filaments). See panel (d) of the
figure.
The first spring attaches via a hinged element (the “trap door”). When the hinge is in its open
state, the attachment point is a distanceδto the right of its closed state relative to the body of
the stereocilium. The trap door is a 2-state system with a free energy change ∆F 0 to jump to its
open state.
a. Derive the formulafclosed(x)=ka(x−xa)+kb(x−xb)for the net force on the stereocilium in
the closed state. Rewrite this in the more compact formfclosed=k(x−x 1 ), and find the effective
parameterskandx 1 in terms of the earlier quantities. Then find the analogous formula for the
state where the trap door has opened by a distancey,where 0<y<δ.That is, find the net force
f(x, y)onthe stereocilium in this state.
b. The total forceftotis the sum ofNcopies of the formula you just found. InPopenNof these
terms the trap door is open; in the remaining (1−Popen)Nit is closed. To find the open probability
using Equation 6.34 on page 199, we need the free energy difference. Get a formula for ∆F(x)by
working out ∆F 0 +
∫δ
0 dyf(x, y), wheref(x, y)isyour answer to part (a).
c. Assemble the pieces of your answer to get the forceftot(x)interms of the five unknown parameters
N,k,x 1 ,δ,and ∆F 1 ,where ∆F 1 =∆F 0 +^12 kaδ^2 .Graph your solution using the illustrative values
N=65,k=0. 017 pN nm−^1 ,x 1 =− 22. 7 nm,∆F 1 =14. 8 pN nm,and various values forδstarting
from zero and moving upward. What value ofδ gives a curve resembling the data? Is this a
reasonable value?