MASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Electrical Engineering and Computer Science,Department of Mechanical Engineering,Division of Bioengineering and Environmental Health,Harvard-MIT Division of Health Sciences and TechnologyQuantitative Physiology: Cells and Tissues2.791J/2.794J/6.021J/6.521J/BE.370J/BE.470J/HST.541JHomework Assignment #3 Issued: September 22, 2005Due: September 29, 2005ReadingLecture 7 — Volume 1: 4.7-4.7.1.2Lecture 8 — Volume 1: 4.7.2-p.230 Fig.4.26 Fig.4.28 4.8.2-4.8.3Lecture 9 — Volume 1: 6.1-6.2.1.4AnnouncementsExam 1 will be held on Thursday, October 6, 2005 from 7:30 PM to 9:30 PM in room 1-190. Theexam is closed-book: notes on both sides of one 812× 11 sheet of paper may be used for reference.Calculators may be used, but computers and wireless devices may not be used.There will be no recitations on the day of the exam.Exercise 1. What is a colligative property of a solution? List 3 colligative properties of solutions.Exercise 2. Figure 4.21 in volume 1 of the text shows the volume of Arbacia eggs as a function ofextracellular osmolarity.a) What is the isotonic volume of these cells?b) What is the fraction of the isotonic volume of these cells that is not due to intracellular water?Exercise 3. A cell in a bath is subjected to changes in extracellular osmolarity. Assume that theflux of volume is given by ΦV= LVRT (CoΣ− CiΣ). A step change in the osmolarity of the bathproduces a change in the normalized volume of a cell as shown in the ν(t) waveform in Figure 1.Twelve possible normalized concentration waveforms ˜c(t) are shown in the lower panel of thefigure. Assume all solutes are impermeant and that the cell acts as an ideal osmometer so thatVc=NiΣCoΣ+ V′c,where V′cis the osmotically inactive portion of the total cell volume Vc, and V′c<< Vc.a) Which of the 12 ˜c(t) waveforms are possible causes of the ν(t) waveform? Explain yourchoices briefly.12 4 6 8 100.511.530.512.52iiiiiiivvviviiviiiixxxixiiν(t)˜c(t)Normalized time, ανtFigure 1: Change in normalized volume (upper panel) in response to a change in normalized exter-nal solute concentration (lower panel). The normalized volume is ν(t) = Vi(t)/Vinwhere Vi(t)and Vinare the cell water volume and its isotonic value. The normalized extracellular concentra-tion is ˜c(t) = CoΣ(t)/ConΣwhere CoΣ(t) and ConΣare the total extracellular solute concentration andits isotonic value.b) The temperature of all the components (cell and bath) is decreased by 10◦C. Determine ifthe initial slope of the normalized volume (dν(t)/dt at t = 0+) and final value (ν(∞))increased, decreased or remained unchanged? Explain your reasoning.Exercise 4. Describe what vasopressin does.Problem 1. A long, steel pipe filled with fresh water to a height hois lowered quickly into theocean until its bottom end is a distance hsbelow the surface of the ocean as shown in Figure 2.The bottom end of the pipe is closed with a semipermeable membrane permeable to water only.SaltFresh waterPipe ofMembraneArea AwaterΦVhshoFigure 2: Schematic diagram of a pipe terminated in a semipermeable membrane and immersedin the ocean. The fresh water in the pipe has mass density ρoand osmolarity 0; the salt water hasmass density ρsand osmolarity CΣ; the semipermeable membrane has hydraulic conductivity LV.2a) Derive an expression for the inward flux of water, ΦV, in terms of the variables shown inFigure 2, the acceleration of gravity g, and any other necessary constants.b) Determine the magnitude and sign of the initial derivative of ho(t), i.e., dho/dt evaluated att = 0, ifhs= 100 m ρs= 1.03 g/cm3ho(0) = 1 m ρo= 1.00 g/cm3A = 10 cm2LV= 3 × 10−12m/(Pa·s)CΣ= 1 osmol/L g = 980 cm/s2T = 300Kc) Show that, provided hsis greater than some critical depth, hc, that the final equilibrium valueof hois greater than hs. Find the value of hc.d) If hs> hc, it appears that both power and fresh water could be obtained free from the ocean.Is this reasonable? If not, which assumption(s) in the problem statement would you considerto be invalid?Problem 2. Each of two identical semipermeable membranes – (1) and (2) – is used to separatetwo solutions as shown in the following figure.Side 2Membrane 1Side 10 d x05Pressure (atmospheres)Side 2Membrane 2Side 10 d xp1p2RTC1ΣRTC2ΣRTCΣ(x)p1p2RTC1ΣRTC2ΣRTCΣ(x)In each figure, the solid lines indicate the osmotic pressures as a function of space, and the dashedlines indicate the hydraulic pressures in the baths. The scales for osmotic and hydraulic pressuresare the same, as indicated by the scale bar on the left of the figure.a) Determine the hydraulic pressure p(x) in the membrane as a function of distance x and plotp(x) for both membranes. Include the bath pressures p1and p2in your plots.b) Determine the profile of p(x) − RT CΣ(x) in the membrane for both (1) and (2).c) The concentration of solute on Side 1 is the same for both case (1) and (2). Find the totalconcentration of solute on Side 1.d) Is the volume flux through the membrane in Case 1 smaller, larger or equal to that in Case2? Explain.3Problem 3. The following figure shows the design of a miniature pump that can be implanted inthe body to deliver a drug. No batteries are required to run this pump!Chamber 1 Chamber 2Rigid,DrugFrictionless,3 cm0.7 cm(solute) (drug)orificemembranesemipermeablepistonimpermeableThe pump contains two cylindrical chambers filled with incompressible fluids: the two chamberstogether have a length of 3 cm and a diameter of 0.7 cm. Chamber 1 is filled with a solutionwhose concentration is 10 mol/L; the osmolarity of this solution greatly exceeds that of body flu-ids. 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 hy-draulic pressure between the chambers; the piston allows no transport of water, solute or drugbetween chambers. The pump walls are rigid, impermeable and cylindrical with an orifice at oneend for delivering the drug and a rigid, semipermeable membrane at the other end. The orificediameter is sufficiently large that the hydraulic pressure drop across this orifice is negligible andsufficiently small so that the diffusion of drug through the orifice is also negligible. The semi-permeable membrane is permeable to water only, and not permeable to the solute. Assume thatT = 300 K.a) Provide a discussion of 50 words or fewer for each of the