Find the average speed of a ratchet moving by steps of

Question

Question 1: Backward motion of the ratchet:

a) Find the average speed of a ratchet moving by steps of length L through a medium where its diffusion constant is D with no backward motion, as was done in the lectures.

b) Find the average forward speed if at each step a fraction of the ratchets step backward. Check the two limits ε = 0 and ε → ∞. Comment.

Question 2: Contributions to the capacitance of a lipid bilayer:

a) Estimate the capacitance of a lipid bilayer. Consider only the electrical insulating part of the bilayer, the lipid tails, as a layer of oil about 2 mu thick. The dielectric constant of oil is ε_oil / ε_o = 2.

b) The charge screening layers in the water (ε_water = 80) on either side of the lipid bilayer also contribute to its capacitance. In physiological salt concentrations, these layers are each roughly 0.7 nm thick. Use the formula for capacitors in series from first year physics and your result in a) to estimate the contribution to the total capacitance from these layers.

Question 3: Discharging the battery: Imagine the resting membrane of the axon as a capacitor with specific capacitance C = 1 µF/cm².

a) How much charge per unit area must pass through the membrane to discharge the capacitor (that is bring V from V_rest = -60mV to zero)?

b) Re-express your answer in a) by giving the surface area per excess proton charge needed to maintain V_rest = -60 mV. Then express it a third time, as the charge per unit length of the axon, taking the squid giant axon to be a cylinder of radius 0.5 nm.

c) As we shall see Wed. April 6, depolarization of the membrane (e.g. raising V_mem above 0 mV) is largely the result of the inflow of Na⁺ ions. Estimate the effect on the interior ion concentration of a charge transfer of the sort just described in a) and b) as follows. Again imagine the giant axon as a cylinder with salt solution, with ion concentrations given in the Table below:

II  Ion	Valence z	Interior c2,i (mM)	Relation	Exterior c1,i (mM)	Nernst potential ViNernst (mV)
KK+ Na+	+1 +1	400 50	> <	20 440	-75 +59
Cl-	-1	52	<	560	-59

Find the total number of interior Na⁺ ions per unit length. Find the corresponding number if the interior Na⁺ concentration matched the exterior value. Subtract these two numbers and compare with the total number of sodium ions passing through the membrane as estimated in b).

d) Explain in the light of those numbers why an axon can continue to transmit many action potentials after the its ion pumps have been shut down.

Question 4: Vacuole equilibrium:

Here are some data for the marine alga Chaetomorpha. The extracellular fluid is seawater: the "plasmalemma" (outer cell membrane) separates the outside from the cytoplasm; a second membrane ("tonoplast membrane') separates the cytoplasm from an interior organelle, the vacuole.

Ion	Vacuole (mM)	Cytoplasm (mM)	Extracellular (mM)	r*"' (plasmalemma) (my)	remat (tonoplast) (my)
K+ Na+	530 56	425 50	10 490	? +57	-5.5 ?
Cl-	620	30	573	-74	+76

a. The table gives some of the Nernst potentials across the two membranes. Fill in the missing ones.

b. The table does not list the charge density ρ_q,macro arising from impermeant macroions in the cytoplasm. What is - ρ_q,macro /e in mM?

c. The actual measured membrane potential difference across the tonoplast membrane is +76 mV. Suppose all the quoted numbers are accurate to about 2%. Which ion(s) must be actively pumped across the tonoplast membrane, and in which direction(s)?

Question 5: We analyzed the polymerization ratchet in the two limiting cases of diffusion-limited and reaction-limited polymerization. By comparing the time for a load, polystyrene sphere 1 µm in diameter, to diffuse a distance given by the actin monomer size, and the average time for an actin monomer to be added to the growing end of the filament, find the condition for the free actin monomer concentration that is necessary for the polymerization in the presence of the load to be reaction-limited. Compare this concentration with the critical concentration for actin filament growth.