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Homework #5:  Assigned 7/18/01  Due: 7/25/01 at the start of class. 

Reading (one of the following)
Aidley  p 19-21, 26-29, 149-165
or KSJ (3rd edition) p 81-89, 135-144, 153-158, 173-186

Useful formulae:
charge(Coulombs) = capacitance(Farads) * voltage(Volts)
1 Coulomb = 1 Amp * 1 second
charge of an electron = -1.6 e-19 Coulombs
charge of a K+ or Na+ ion = 1.6 e-19 Coulombs
Number of particles in a mole = 6.02 e23
Energy (Joules) = charge (coulombs) * voltage change (volts)
1 Liter = 1000 cubic centimeter

Please use SI units…anyone using cgs units on this homework will be whipped with a soldering iron.

1. Recall from lecture #4 that cells have a capacitance due to their outer membrane being a good insulator.  The specific capacitance of cellular membranes is 1 uF/cm^2.  Consider the case of a spherical cell with a capacitance of 20 pF with an internal K+ and Cl- concentration of 140 mM (millimolar, or millimoles per liter) and an external concentration of 2 mM K+ and Cl-.  The cell's membrane is permeable to K+ only.  Initially (time = 0) the cell is at 0 mV (same voltage inside & outside of the cell).  However, diffusion of K+ across the cell membrane causes the cell very quickly to become negative (because K+ diffuses out of the cell). 

A) What is the steady state voltage of the cell?
B) What is the net number of K+ ions that moved out of the cell? 
C) How much has the K+ concentration changed inside of the cell?

2. Boltzmann's constant, k = 1.38 e-23 Joules/degree Kelvin.  What is kT in SI units at room temperature (T=300 K)?  What is kT in terms of milli-electron volts (energy required to move an electron through a voltage change of 1 millivolt)?  I'm asking this question not to have you do a trivial unit conversion, but because this number appears EVERYWHERE you're dealing with electronics--the Nerdst equation, the voltage dependence of ion channels and semiconductor junctions, Johnson noise, etc.  Why?  Because this is the order of magnitude of the thermal energy of charge carriers in any medium, and if we're in the realm of electronics, we want that energy in terms of volts (times charge).

3. A typical photoreceptor has synapses with many cells.  Retinal cells have a capacitance of about 20 pF.  Suppose that rather than having chemical synapses, photoreceptors were directly electrically coupled to other cells--i.e. all cells change their voltage together.  This scheme has many problems.  First is that there's no way to make an inverter (an ON bipolar cell).  Another problem comes from the capacitance.  Recall that a single photon produces a voltage change of about 2 mV in 200 ms.  Suppose a rod is coupled to 50 other cells.  How much current must flow through to change the voltage by 2 mV in 200 ms (assume current is constant in time)?  How does this compare with the magnitude of the single photon current change in the rod?

4. In honor of Erika's ethanol research, let's look at the (hypothetical) effects of high ethanol concentration on vision.  Turns out that ethanol acts on neurotransmitter receptors:  specifically, it reduces the current that flows in response to glutamate.

Imagine you were monitoring the current flowing into an ON and OFF bipolar cell.  Now imagine that we dump ethanol onto the cells.  Suppose this causes the current flowing through ionotropic receptors in bipolar cells to be reduced.  What effect would this have on the magnitude of current flowing into the two types of bipolar cells in total darkness?  In bright (saturating) light?   Would the single-photon response be affected?  How?

5. Suppose I build a cell that has a membrane permeable only to sodium ions (Na+).  The Na+ channels (that let Na pass through the membrane) are always open.  I can control the voltage E_M of the cell & measure the current I_M flowing across the membrane.  This is plotted in the graph below:

What is the relative ratio of sodium ions outside vs. inside of the cell?  Also, give a hand-wavy explanation for why the slopes of the current-voltage relation above are different for extremely positive and negative potentials.  Hint:  think about diffusion.  What do you think the above graph should look like if the internal & external sodium were equal?