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 #2 Issued: September 15, 2005Due: September 22, 2005ReadingLecture 5 — Volume 1: 3.7-3.7.2 3.8-3.8.5Lecture 6 — Volume 1: 4.1-4.3.2.3 4.4-4.5.1.2Exercise 1. Cell a and cell b have identical dimensions but different permeabilities for solute n, Panand Pbn, respectively. The cells are placed in identical solutions that contain the permeant solute n.The intracellular concentrations for cell a and for cell b are shown in Figure 1. Is Pan> Pbnor istcan(t)cbn(t)Figure 1: Concentration of solute n as a function of time in two cells, can(t) and cbn(t) that havedifferent permeabilities for solute n.Pan< Pbn? Explain.Exercise 2. Two solutes a and b diffuse in the steady state through a membrane with the concentra-tion profiles shown in Figure 2. The membrane:solution partition coefficients for the two solutes,MembraneBath1Bath2ca(x)cb(x)Figure 2: Concentration of solutes a and b as a function of position.kaand kbdiffer. Is ka> kbor is ka< kb? Explain.1Exercise 3. The concentration of a solute at some time t0is shown as a function of x in Figure 3.Determine and sketch the flux of solute at time t0.(mmol/L)5 mmx01020c(x, t0)Figure 3: Concentration of solute as a function of position at time t0.Problem 1. Figure 4 shows the concentration of a solute as a function of position 200 seconds−3 −2 −1 0 1 2 3100200300400500x (mm)c(x, t) (mol/m3)Figure 4: Concentration versus distance.after a point source of strength 1 mol/m2was applied at t = 0 and x = 0. The temperature is300 K. Assume that the solute particles are conserved (none are created or destroyed) and move bydiffusion alone.a. Find a general expression for c(x, t).b. Determine the numerical value of the diffusion coefficient D.c. Determine the flux of solute through x = 1 mm at the time shown in Figure 4.Problem 2. As your first assignment at Tinyfluidics Inc., you are asked to design a microfluidicdevice that will remove small molecules from a sample of fluid that contains both large moleculesand small molecules. After some thinking, you design the laminar flow device shown below.LWbuffer inbuffer insample inwaste outwaste outfiltrate outdδThe sample is injected in a port with width δ. The sample flow is surrounded by buffers injectedon both sides of the sample. The combined flow then passes through a channel that has width Wand length L after which the fluids are separated into a desired filtrate output (in a channel of width2d) and two waste outputs. Assume that the fluid moves with the same speed v in all parts of themicrofluidic device (although this is not generally true, it is a convenient starting point). Assumethat δ << d, and that W is sufficiently large that it can be taken to be infinity.To test the device, you mix a solution that contains equal concentrations of 2 proteins, A and B.The diffusivity of solute B is four times that of solute A.Part a. Briefly explain how this device takes advantage of differences in diffusivities to achieveseparation.Part b. Let fArepresent the ratio of the amount of solute A found in the filtrate divided by theamount of solute A in the sample. Determine an expression for fA. Determine an expression forthe analogous ratio fBfor solute B.The following figure shows a plot of the dependence of fAand fBon L, when δ = 1µm, d = 20µm,v = 1 mm/s, and DA= 10−7cm2/s.0 5L (mm)10 00.51 fA and fBPart c. Which curve (dashed or solid) represents fA? Explain.Part d. Determine L0, the value of L that maximizes the difference between fAand fB. Brieflyexplain why this difference is smaller when L << L0and when L >> L0.Problem 3. All cells are surrounded by a cell membrane. The cytoplasm of most cells contains avariety of organelles that are also enclosed within membranes. Assume that a spherical cell withradius R = 50µm contains a spherical organelle called a vesicle, with radius r = 1µm, as shownin the following figure.spherical vesicleradius r = 1 µmspherical cellradius R = 50 µmbathAssume that the membranes surrounding the cell and vesicle are uniform lipid bilayers with iden-tical compositions and the same thickness d = 10 nm. Assume that solute X is transported across3both the cell and vesicle membrane via the dissolve and diffuse mechanism. Assume that X dis-solves equally well in the bath and in the aqueous interiors of the vesicle and cell. Assume thatthe solute X dissolves 100 times less readily in the membrane (i.e., the partitioning coefficient is0.01). Assume the diffusivity of X in the membranes is 10−7cm2/s.Initially, the concentration of X is zero inside the cell and inside the vesicle. At time t = 0, thecell is plunged into a bath that contains X with concentration 1 mmol/L.a) Estimate the time that is required for the concentration of X in the cell to reach 0.5 mmol/L.Find a numerical value or explain why it is not possible to obtain a numerical value with theinformation that is given.b) Estimate the time that is required for the concentration of X in the vesicle to reach0.5 mmol/L. Find a numerical value or explain why it is not possible to obtain a numeri-cal value with the information that is given.Problem 4. A rigid, homogeneous membrane separates two well-stirred compartments with rigidwalls which contain aqueous solutions of a solute. The cross-sectional area of the membrane andcompartments is A = 64 cm2. The solute is transported through the membrane by diffusion only.The membrane:solution partition coefficient for the solute is 1.0, but the value of the diffusioncoefficient for the solute in the membrane is unknown except that it is known that D ≥ 0. Thewidths are: 10 cm for compartment 1, 1 cm for compartment 2, and 0.1 cm for the membrane. Thegeometry is shown in Figure 5. The concentration of the solute in compartments 1 and 2 are c1(t)10 cm 1 cm0.1 cmCompartment 1 2MembraneArea AFigure 5: Geometry for two-compartment diffusion between compartment 1 and 2 through a mem-brane.and c2(t), respectively. The concentration of the solute in the membrane depends on position inthe membrane and is c(x, t).At t = 0 the concentration of the solute in compartment 1, c1(0) = 100 mmol/cm3, and