So many problem sets by Jessie L. '07

Now that term has geared up a bit, most classes have given some sort of homework, usually in the form of a problem set. I thought the prospectives on here might be curious to see what problems on problem sets are actually like, so I’ve posted a few of the problems from my own problem sets (don’t give me the answers! I haven’t solved all of these yet!). 9.22 has presentations and readings, not psets, so I didn’t post anything from it. 6.004 has labs rather than problem sets, so I posted a problem from the first lab, which is due this week.

5.12 Organic Chemistry

3. Sphingosine, a component of sphingomyelin, is part of the cellular lipid bilayer. For sphingosine, a) draw in all of the hydrogen atoms. Label each sp3 carbon according to b) C-C connectivity and c) C-H connectivity (Y = methyl, E = methylene, I = methine). c) Label each amine, alcohol, and hydrogen by sp3 CC connectivity.

5. a) Draw Newman projects for the six energy maxima and minima for 2-
methylhexane, sighting along the C2-C3 bond.
b) Calculate the energy of each conformation. (Assume the same energy
values for –Pr that you would for –Me.)
c) Use these values to make a graph of potential energy versus dihedral angle.
d) What is the barrier to rotation around the C2-C3 bond?

7.03 Genetics

4. This problem deals with the following pedigree [not pictured in this entry], which shows the inheritance of a very rare trait.

(a) Assume that the inherited disorder is expressed with complete penetrance and that there are no new mutations. What mode(s) of inheritance is/are consistent with this pedigree? (Your choices are: X-linked dominant, X-linked recessive, autosomal recessive, autosomal dominant.)
(b) For each consistent mode of inheritance, what are the probabilities that Individuals 1 and 2 will have:
… an affected son?
… an affected daughter?
… an unaffected son?
… an unaffected daughter?
(c) Use Bayes’ theorem to calculate the probability that the next child of Individuals 1 and 2 will be affected with the disorder, given the new knowledge that the couple already has two
healthy sons. Do this calculation for each mode of inheritance consistent with the pedigree.

9.29/ Intro to Computational Neuroscience

1. Mathematics of the integrate-and-fire neuron
a) For a spherical neuron with a surface area of 0.03 mm2, a specific membrane capacitance of cm = 10 nF/mm2, a specific membrane resistance of rm = 1 MΩ-mm2, and a resting membrane potential EL = -70:
i) What is the total membrane capacitance Cm?
ii) What is the specific membrane conductance gm? What is the total membrane conductance Gm? What is the total membrane resistance Rm?
iii) What is the membrane time constant τm?

iv) How much external electrode current would be required to hold the neuron at a membrane potential of -60 mV?
v) If this amount of current is turned on at time t = 0 and held at this value, how long will the neuron take to reach a membrane potential of -65 mV if the cell is initially at a voltage V = E_L?
vi) Calculate the total charge required to polarize the neuron to the resting potential. Assume the predominant conductance is potassium and use [K]_in=140mM for the concentration of potassium in mammalian neurons. What fraction of the total number of K+ ions in the neuron is lost as the neuron is polarized?
b) Show that
V(t) = E_L + R_mI_e + (V(t_0) – E_L – R_mI_e)exp(-(t – t_0)/T_m)
where the constant t0 is any reference time, satisfies the equation
T_m(dV/dt) = E_L – V + R_m I_e

when Ie is a constant. Also verify that when t = t_0, the left and right sides of the equation agree and that when t → ∞, V(t) → V(∞) ≡ E_L + R_mI_e.

Setting t_0 equal to the current time in a computer simulation and t equal to the next time step a time ∆t later gives the one-time-step update rule we will use for simulating the integrate-and-fire neuron (and which we will use more generally for simulating any equation of the form of equation (0.2) above, e.g. in problem 3 for the spike rate adaptation conductance below if we substitute g_sra for V and set E_L=I_e=0):
V → E_L + R_mI_e + (V – E_L – R_mI_e)exp(-∆t/T_m),
where this shorthand notation means that, in the time step ∆t, the voltage gets updated from its old value to the value on the right of the arrow, i.e.
V(t-t_0) → E_L + R_mI_e + (V(t) – E_L – R_mI_e)exp(-∆t/T_m)

6.004 Computation Structures

(C) To maximize noise margins we want to have the transition in the voltage transfer characteristic (VTC) of the nand2 gate centered halfway between ground and the power supply voltage (3.3V). To determine the VTC for nand2, we’ll perform a dc analysis to plot the gate’s output voltage as a function of the input voltage using the following additional netlist statements:

* dc analysis to create VTC
Xtest vin vin vout nand2
Vin vin 0 0v
Vol vol 0 0.3v // make measurements easier!
Voh voh 0 3v // see part (D)
.dc Vin 0 3.3 .005
.plot vin vout voh vol

Combine this netlist fragment with the one given at the start of this section and run the simulation. To center the VTC transition, keep the size of the nfet in the nand2 definition as “SW=8 SL=1” and adjust the width of both pfets until the plots for vin and vout intersect at about 1.65 volts. Just try different integral widths (i.e, 9, 10, 11, …). Report the integral width that comes closest to having the curves intersect at 1.65V.