Homework Assignment #3 Issued: Recitation 5 Due: Recitation 7

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science, Department of Mechanical Engineering, Division of Bioengineering and Environmental Health, Harvard-MIT Division of Health Sciences and Technology Quantitative Physiology: Cells and Tissues 2.791J/2.794J/6.21J/6.521J/BE37J/BE47J/HST541J Homework Assignment #3 Issued: Recitation 5 Due: Recitation 7 Reading Lecture 9 Volume 1: p.23 Fig.4.26 Fig Lecture 1 Volume 1: Exercise 1. What is a colligative property of a solution? List 3 colligative properties of solutions. Exercise 2. Figure 1 shows two chambers each containing a semipermeable membrane, permeable to water but not to the solute, separating two compartments. One of the compartments contains a W W Figure 1: Arrangement of weight and solute for two chambers with a semipermeable membrane separating two compartments. weight on a piston, the other does not. The only difference between the two arrangements is the location of the solute particles. Which of the two chambers could be in osmotic equilibrium? Explain. Exercise 3. Define the isotonic volume of a cell. Exercise 4. A cylindrical cell is bathed in solutions of a single impermeant substance that does not ionize. For each solution, the concentration of the impermeant substance and the equilibrium cell volume are measured and the results are shown in Figure 2, where represents the isotonic concentration. a) Estimate the isotonic volume of the cylindrical cell. b) Assume that the cell volume contains a portion that consists of water that is osmotically active and a portion that is osmotically inactive. Estimate and explain the basis of your estimate. c) Assume that mosm/l and that the cell membrane is impermeable to NaCl. Estimate the equilibrium cell volume that would result if the cell were bathed in a solution of NaCl with concentration 2 mmol/l. Explain your reasoning. 1

2 Cell volume (mm 3 ) CΣ on /CΣ o Figure 2: Relation between cell volume and extracellular solute concentration. d) Assume that the cell bath is changed from an isotonic solution to a 2 mmol/l NaCl solution at time. Sketch the time course for equilibration of the cell volume indicating numerical values where possible. List the physical parameters of the cell that influence the time course. Problem 1. A rigid, homogeneous membrane separates two well-stirred compartments with rigid walls which contain aqueous solutions of a solute. The cross-sectional area of the membrane and compartments is cm. The solute is transported through the membrane by diffusion only. The membrane:solution partition coefficient for the solute is 1., but the value of the diffusion coefficient for the solute in the membrane is unknown except that it is known that. The widths are: 1 cm for compartment 1, 1 cm for compartment 2, and.1 cm for the membrane. The geometry is shown in Figure 3. The concentration of the solute in compartments 1 and 2 are.1 cm Membrane Area A Compartment cm 1 cm Figure 3: Geometry for two-compartment diffusion between compartment 1 and 2 through a membrane. and, respectively. The concentration of the solute in the membrane depends on position in the membrane and is. At the concentration of the solute in compartment 1, mmol/cm, and the concentrations in the membrane and in compartment 2 are zero,. The resulting 2

3 initial spatial dependence of the concentration is shown in Figure 4. Compartment 1 Membrane Compartment 2 15 c 1 () c(x, ) (mmol/cm 3 ) 1 5 c(x, ).5.1 x (cm) c 2 () Figure 4: The initial concentration of solute in the two compartments and in the membrane are shown at. The figure illustrates the concentration in the membrane and in a portion of each compartment only. The concentrations in the compartments are uniform in space. a) Sketch the solute concentration distribution in the membrane and in the two compartments at equilibrium. Make a sketch similar to Figure 4 and label important values on the curve. b) For seconds, the concentration and flux in the membrane and the concentrations in the two baths, and, are shown in Figure 5. Compartment 1 Membrane Compartment c 1 (1) 1 c(x, 1) (mmol/cm 3 ) φ(x, 1) 5 5 c(x, 1).5.1 x (cm) 1 φ(x, 1) (µmol/(cm 2 s)) c 2 (1) Figure 5: The concentration of solute (solid line) in the two compartments and in the membrane are shown at seconds. The figure illustrates the concentration in the membrane and in a portion of each compartment only. The concentrations in the compartments are uniform in space. The flux of solute is shown (dashed line) in the membrane only. 3

4 i) Is the concentration in the membrane at its steady-state distribution? Explain. ii) Estimate the diffusion coefficient for the solute in the membrane,, from the data in Figure 5. Clearly indicate your method. c) For seconds, the concentration in the membrane,, and the concentrations in the bath are shown in Figure 6. In addition, Figure 6 also shows the dependence of concentration on time at two locations: one in the membrane, ; and the other in compartment 2,. Compartment 1 Membrane Compartment 2 15 c 1 (1) 1 c(x, 1) c(x, 1) (mmol/cm 3 ) x (cm) c 2 (1) 1 5 c(.4, t) (mmol/cm 3 ) 5 c(.4, t) c 2 (t) c 2 (t) (mmol/cm 3 ) 5 1 t (s) Figure 6: The concentration of solute in the two compartments and in the membrane are shown at seconds (upper panel) The figure illustrates the concentration in the membrane and in a portion of each compartment only. The concentrations in the compartments are uniform in space. Also shown (lower panel) are the dependence of concentration on time for the time interval for a location in the membrane, (indicated in the upper panel with a solid arrow) and in compartment 2, (indicated in the upper panel with a dashed arrow). i) Is the concentration in the membrane at its steady-state distribution? Explain. 4

5 Concentration (mmol/l) Ion Giant sequoia Red mangrove soil sap seawater sap K Na 2 45 Cl NO Ca Mg 8 5 Table 1: Composition of sap and extracellular solutions for the Giant sequoia and the Red mangrove trees. ii) Estimate the equilibrium time constant for equilibration of the solute in the two compartments,, from the data in Figure 6. Clearly indicate your method. iii) Estimate the steady-state time constant for the solute in the membrane,, from the data in Figure 6. Clearly indicate your method. iv) Estimate the diffusion coefficient of the solute in the membrane from the data in Figure 6. Clearly indicate your method. v) Estimate the quantity from the data in Figure 6. Clearly indicate your method. vi) Sketch and versus from to. Problem 2. The water content of plants is very high (up to 9% by weight) but this water is in flux; water is absorbed through the roots, rises as sap, and evaporates from the leaves. The total water content of a plant can be replaced many times per hour. The mechanisms that determine water flow include: gravity, osmosis, and capillarity. In this problem we will consider the effect of the gravity and osmosis. Assume that water flow is steady and is due to gravitational, osmotic, and other forces. Let the pressure due to gravity be and the osmotic pressure be. Let the pressure due to other sources in the trees be ; assume these sources are not appreciable in the medium that is in contact with the roots. Consider two trees: 1. The General Sherman giant sequoia (Sequoia gigantea) which is located in Sequoia National Park in California stands 272 feet high and has a diameter of 36.5 feet at its base; you can drive your car through a tunnel that has been cut through the base. The tree was estimated to be 3,8 years old. 2. The red mangrove (Rhizophora mangle) grows in the tropics to a height of as much as 8 feet. It is found in tidal creeks and estuaries and it can grow with its roots in seawater. Yet the sap has the composition of fresh water. The compositions of the relevant media are given in Table 1. a) Find a numerical bound on tree. b) Which tree requires the larger value of (expressed in atmospheres) such that the sap will rise in each? Explain. 5

6 Problem 3. Figure 7 shows the design of a miniature pump that can be implanted in the body to deliver a drug. No batteries are required to run this pump! The pump contains two cylindrical Rigid, semipermeable membrane.7 cm Frictionless, impermeable piston Chamber 1 Chamber 2 (solute) (drug) Drug orifice 3 cm Figure 7: Design of a miniature implantable pump. chambers filled with incompressible fluids: the two chambers together have a length of 3 cm and a diameter of.7 cm. Chamber 1 is filled with a solution whose concentration is 1 mol/l; the osmolarity of this solution greatly exceeds that of body fluids. Chamber 2 is filled with the drug solution. The two chambers are separated by a frictionless, massless, and impermeable piston. The piston moves freely and supports no difference in hydraulic pressure between the chambers; the piston allows no transport of water, solute or drug between chambers. The pump walls are rigid, impermeable and cylindrical with an orifice at one end for delivering the drug and a rigid, semipermeable membrane at the other end. The orifice diameter is sufficiently large that the hydraulic pressure drop across this orifice is negligible and sufficiently small so that the diffusion of drug through the orifice is also negligible. The semipermeable membrane is permeable to water only, and not permeable to the solute. Assume that K. a) Provide a discussion of 5 words or fewer for each of the following: i) What is the physical mechanism of drug delivery implied by the pump design? ii) What is(are) the source(s) of energy for pumping the drug? iii) Assume there is an adequate supply of drug in the pump for the lifetime of the implanted subject and that it is necessary to provide a constant rate of drug delivery. Which fundamental factors limit the useful lifetime of this pump in the body? b) When implanted in the body, the pump delivers the drug at a rate of 1 L/h. Find the value of the hydraulic conductivity,, of the semipermeable membrane. 6

7 Problem 4. A volume element with constant cross-sectional area has rigid walls and is divided into two parts by a rigid, semipermeable membrane that is mounted on frictionless bearings so that the membrane is free to move in the -direction as shown in the following figure. c 1 c 2 x 1 (cm) The semipermeable membrane is permeable to water but not to the solutes (glucose or NaCl or CaCl ). At, solute 1 is added to side 1 to give an initial concentration of and solute 2 is added to side 2 to give an initial concentration of. Concentrations are specified as the number of millimoles of glucose or NaCl or CaCl per liter of solution. The initial position of the membrane is. For each of the following parts, find the final (equilibrium) values of the membrane position, and the concentrations, and. (a.) (b.) (c.) (d.) ; ; ; ;. ;. ;. ; ;. (e.) ; ;. 7