1.  a.  One form of the therapy called "plasmapheresis" uses a membrane device to separate some of a patient's blood plasma from the rest of his/her blood.  Blood is taken from an arm vein and passed through a plasma separator.  A stream of blood, richer in cells and poorer in plasma, flows out of the separator and is pumped back into the patient's vein, downstream of where the blood was taken.  A stream of plasma also flows out of the separator.  This stream flows into a second membrane separator which removes only certain immunoglobulins from the plasma.  The purified plasma stream leaving this second device is blended with the returning blood stream before it reaches the patient.  All transport in this system, even that across the membranes, is convective and unidirectional.  Make a flow chart of the patient and the extracorporeal system just described, following the examples given in Figure 1 of the lecture notes.  Ignore any communication between the blood compartment and any other compartment of the body.

    b.  An anephric patient is being treated with an artificial kidney.  Consider the therapy to be primarily directed at the removal of urea -- a gross over-simplification.  The patient's body water is to be modeled as two compartments, blood and closely associated water, and less accessible water.  The artificial  kidney is fed with dialysate from a compartment to which the dialysate is returned after passing through the artificial kidney.  All transport in this system is bi-directional.  Make a flow chart of the patient and the artificial kidney system just described, following the examples given in Figure 1 of the lecture notes.

2.  Perl defines clearance in a manner that seems different from that used in class.  He says that clearance is equal to the rate of removal of a substance divided by the concentration of that substance in the inflow to the organ being considered.  Is this definition equivalent or different from that given in class?  Is your answer correct for all circumstances?

3. The concentration of albumin (MW = 69,000) is about 4 g/dl in normal plasma.  Plasma occupies about 55% of the total volume of normal blood.  Many relatively insoluble lipids are transported in blood by binding to lipophilic sites on albumin.  If an albumin molecule has, say, 6 lipophilic sites, how many grams of a lipid molecule  (MW = 200) can be carried by the plasma albumin in a liter of blood. when the albumin is fully loaded?  If it is determined that   10% of the albumin sites are loaded at a free lipid concentration of 71 mg/liter and 75% at a free lipid concentration of 245 mg/liter, how might the binding be modeled?   (Hint:  consider that half of the sites (3/molecule) have K_M1 and the other half of the sites, K_M2.)  What fraction of the lipid is free and what fraction is bound?  What is the free energy change associated with the binding of a free lipid molecule in blood whose free lipid content is 245 mg/liter to an albumin molecule 75% of whose sites are loaded with this lipid?

4.  a. Calculate the concentration of oxygen (g/cc) in a gas which is at 30 C and a total pressure of 1 ATM. if the gas is saturated with water, and is 95% O₂, 5% CO₂, dry basis.  Using the Henry's law constant stated in the notes, 5.15 10⁴ ATM, calculate the concentration of oxygen in water that is in equilibrium with this gas.  Normal blood contains about 15 g/dl of hemoglobin.  The molecular weight of hemoglobin is   68,000 and each molecule can bind 4 molecules of oxygen.  What is the concentration of oxygen bound to hemoglobin in normal blood, when the hemoglobin is fully saturated with oxygen?  Tabulate the three concentrations that you have calculated.  

   b.  Basal (resting) oxygen consumption in the adult human is about 200 STP ml/min of oxygen.  If the cardiac output is taken to be 5 liters/min and the arterial oxygen saturation in normal blood is 100%, what is the saturation of the venous return?  Using the Hill equation given in the notes for the fractional saturation,Y, calculate the pO₂ of the venous return, using, as in the notes n = 2.8 and P₅₀ equal to 21 torr.

    c.  In a blood oxygenator a microporous membrane of area 3 m² is observed to transfer 240 ml STP of oxygen per minute from the "95/5" gas described above to a blood simulant whose p_O2 is 20 torr.  Calculate the apparent permeability in cm/sec.  About 60% of the area of the membrane is believed to be made of microscopic pores, and the membrane is about 0.01 cm (0.1 mm) thick.  The diffusion coefficient of oxygen in air is about 0.2 cm²/sec and about 10 ^-5 cm²/sec in water.  Does this information allow you to decide whether the pores are filled with gas or with liquid water?

5.  An artificial kidney membrane has a permeability to urea of 3 10^-3 cm/sec.  The membrane is in contact with a compartment which is uniform except for a stagnant layer of water 60 micrometers thick adjacent to the membrane.  If the diffusion coefficient of urea in water is 2 10^-5 cm²/sec, what is the "apparent" permeability?  How much lower, in percent, is this permeability than the membrane permeability?

6.  At a certain pH ribonuclease caries a net negative charge of 3.  If 100 cc of a 3 mmolar solution of this enzyme (in the form of its sodium salt) is added to one side of a membrane to which ions but not ribonuclease is permeable, and 100 cc of 50 mmolar NaCl is added to the other side, what are the final (equilibrium) concentrations of ions on the two sides of the membrane?   If you wished to describe the kinetics of the approach to equilibrium, what driving force would you use?

7.  The concentrations of Na+ and K+ ions in tissues are about 11 and 92 mmolar respectively (intracellular) and 140 and 4 mmolar respectively (extracellular).  Calculate the free energy requirement for maintenance of each of these ion gradients at 310K.

8.  Two compartments (1 and 2) contain solutions of KCl.  The compartments are separated by a membrane that is permeable only to potassium.  The potential between compartment 1 and 2 is V_m.   The concentrations of KCl in compartments 1 and 2 are 100 mmolar and 10 mmolar, respectively.

a.  Determine the equilibrium value of V_m, and give a physical explanation of the sign of the potential.

b.  A battery is now connected to the solutions, so that V_m = - 30 mV.  In which direction will current flow through the membrane?  Explain.

c.  Draw an equivalent electrical network for the condition indicated in part b.  Label the nodes that represent compartments 1 and 2, V_m, and label I_m, defined as the current that flows through the membrane in the direction from compartment 1 to compartment 2.

9.  The ionic concentrations of an isolated cell are given in mmolar as in Problem 7, above.  An electrode is inserted into the cell and connected to a current source so that the current through the cell membrane is I_m.   The Steady-state voltage across the cell membrane, V_m, is determined as a function of the current and is found to vary linearly between -40 mV at zero current to 0 mV at a current of 0.4 nanoamperes.  Assume that (1) the cell membrane is permeable to only K+ and Na+ ions, (2) the Nernst equilibrium potentials are V_n = (60/z_n) log₁₀(c^o_n/cⁱ_n) (mV), (3) the ion concentrations are constant, (4)  active transport processes make no contribution to these measurements.

a.  Determine the equilibrium potentials for sodium and potassium ions, V_Na and V_K.

b.  What is the resting potential of the cell and with these ionic concentrations?

c.  With the current I_m adjusted so that V_m = V_K, what is the ratio of the sodium current to the total membrane current, I_Na/I_m?

d.  What is the total conductance of the cell membrane G_m = G_Na + G_K?

e.  Determine G_Na and G_K .

10.  A uniform isolated small cell has a membrane that is permeable only to sodium and potassium ions and contains an active transport mechanism that transports three sodium ions outward and two potassium ions inward for every molecule of ATP split into ADP and phosphate.  Summed over the entire membrane of this cell, the active transport system splits 10^-17 moles of ATP per second.  Assume that the cell is at quasi-equilibrium, so that the concentrations of all ions are constant.  The cell has a total membrane conductance of 10^-10 A/V, (i.e. siemens).  The temperature is 24 C.  The ionic concentrations of sodium and potassium across the the membrane are respectively 15 and 150 mmolar inside and 106 and 3 mmolar outside.  The potassium conductance exceeds the sodium conductance of this cell.

a.  Determine the value of the component of the resting membrane potential, V^o_m, that is directly attributable to active transport.

b.  Determine the value of the resting membrane potential, V^o_m.

c.  Determine the values of the sodium, G_Na, and potassium, G_K, conductances of the membrane.

11.  Dr. Tropsnart Evitca has found a cell with a resting potential of -70 mV and a sodium ion concentration c^o_Na/cⁱ_Na = 8.  On the basis of measurements using radioactive tracers, the good doctor has proposed that Na+ is transported actively across the membrane of this cell and that four molecules of Na+ are extruded from the cell for each molecule of ATP hydrolyzed.  At the intracellular concentration of ATP, you may assume that hydrolyzing one mole of ATP liberates 30 KJ of energy.  Is the doctor's model energetically feasible?

12.  The following three formulae for the trisaccharide, raffinose, (found in sugar beets) have been proposed:  C₁₂H₂₂O₁₁ + 3H₂O; C₁₈H₃₂O₁₆ + 5H₂O; and C₃₆H₆₄O₃₂ + 10H₂O .  In 1888, de Vries used a plan cell to determine that plasmolysis occurred with a solution containing 59.4 grams of raffinose per liter of water whereas plasmolysis occurred in a solution of sucrose at a concentration of 0.1 molar.   Based on these measurements, de Vries determined the correct formula for raffinose.   Which formula would you choose and why would you choose it?

13.  A method for measuring the hydraulic conductivity, L_P, of an artificial membrane consists of placing a vertical tube of solution (solute and solvent) in a large container of water.  The tube has a cross-sectional area of 0.5 cm².  The tube is filled with 2 cm³ of a sucrose solution of concentration 0.1 molar.  The membrane is impermeable to sucrose but is permeable to water.  At t = 0 the glass tube is inserted into the water.  The height, h(t), of the solution in the tube is measured as a function of time.  Assume that the effects of hydraulic pressure are negligible.   Assume T = 300 K.

a.  Derive a differential equation for h(t).

b.  Find and sketch h(t), the solution of the differential equation you derived in part a.

c.  Measurements show that for a brief time interval after the tube is placed in the water, the height of the solution in the tube rises linearly with time, i.e., lim_t-->0 h(t) = h(0) + 0.02t, cm.   Find the numerical value of L_P.  Note that you do not need to solve the differential equation of part a to do this part of the problem.

14. Weiss reports a study of Brahm (1983) in which he used a rapid-mixing apparatus to inject cells pre-loaded with a solute into a large volume of solution free of that solute.  Brahm was able to measure the concentration of the solute in the cells as a function of time over the short time range 0 --> 30 ms.  The data were reported as y = log₁₀ (1 - c(t)/c(0)).  The slope of the line for the solute propanol was -16 ms^-1.    Given Brahm's estimates that the surface area of the erythrocytes was 136.9 micrometer² and the volume was 104.2 micrometer³, estimate the permeability of erythrocytes to propanol.  Compare your result with the information given in your notes for glycerol.

15.  Two adjoining cells have membranes that nearly touch each other.  The concentrations of uncharged solute n are c¹_n and c²_n inside cells 1 and 2 respectively and c⁰_n in the small intercellular gap.  The membrane permeabilities for this solute are P₁ and P₂ for the membranes of cells 1 and 2, respectively.  Find the overall permeability, P, between the inside of cell 1 and the inside of cell 2 in terms of P₁ and P₂.

Return to course main page.   Return to notes (Transport among Compartments).