bim162hw8 - BIM 162 Homework 08 Due date Thursday 03110111...

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Unformatted text preview: BIM 162 Homework 08 Due date: Thursday, 03110111 Problem 1 (Fill in the blanks. Use additional paper if needed.) Aim: We‘d like to design, and understand the behavior of, a vesicular drug-delivery capsule that can be activated osmotically. The design is based on a biologically inert compound "ABC" that can be split into three individual parts in response to an external trigger: -ABCM>A+B+C. (I) For simplicity, we assume that the effective concentration of the drug itself will be so small that it can be neglected in all osmotic balances. Design: Assuming that we can prepare a uniform suspension of vesicles, we choose to produce capsules with a surface area of 100 pm). (For comparison, the typical surface area of a human red blood cell is 140 umz.) Our vesicle preparation initially gives spherical vesicles that are filled with, and surrounded by, the ABC solution. The spherical geometry is not ideal because such vesicles lack the freedom to change their shape and may lyse prematurely. Therefore, we will gently shrink the vesicles until their volume is 80% of the volume of a sphere (at the given, constant surface area). This volume reduction is achieved by leaving the container open to the air (in a clean environment) and letting water evaporate. Since the vesicle membrane is permeable to water (but not to the ABC compound), osmotic balance causes water to be removed from the vesicles at about the same slow rate at which it evaporates from the container. The .goal is to obtain vesiclesthat have the desired volume when the osmolarity of the suspension reaches the physiological value of ~30!) mOsm. (We continually monitor the osmolarity using an osmometer. "Osm" here denotes "osmolar concentration", i.e., the number of moles of osmotically active particles per volume.) —) a. In order to achieve this goal: What osmolarity must the solution have that initially is used to prepare the (spherical) vesicles? Include in your answer the value of the desired, final vesicle volume. (see next page; explain steps) Hints: The answer to the first question can be found from the stationary solution of the differential equation of water transport across the vesicle membrane: d3? = 'm,H20AP[ dgmfi % — Sims]- (2) (The derivation of this equation is given in the Appendix.) The symbols denote: t time V vesicle volume A vesicle surface area (membrane area) V0, A0 initial vesicle volume and surface area -1- mL Vm,H20 molar volume of water; Vm,H20 :18— mol P permeability of membrane to water 1 . . C35,“ : —c“ osmolanty in compartment "u." NA NA Avogadro's number 0" # of (dissociated) molecules/volume in compartment "a" (:1st initial osmolarity in compartment "a" Recall what "stationary solution" of a differential equation means. The considered process is analogous (though more gentle) to placing the initially spherical vesicles (which have been prepared at the sought osmolarity dime) into a suspension with ex physiological osmolarity osm,0 , and then waiting until the vesicle volume has completely adjusted to its final value V = V00 . Answers for a. 140:? 3": ‘2 0'8 Va ~_—? V00: ngvg lg (Jinn—I? p *1" Va 6)“ Alf C r —- C \ Elf fl vawA P “mo V 05%”? {In- _ . V~o g Cm V0 1 ca H 300m05m' Cma " 30—0 3: 05%? x; ”We” :: R40mm0$m Briefly comment on the relation between the stationary solution of Eq. (2) and the ideal gas law. Control of drug release: We next consider administration of a drug using these vesicle capsules. Assuming that the vesicles have adhered to the proper target (e-g-, a tumor), we apply the trigger that causes dissociation of the compound ABC (see Eq. (1)). The resulting increase of interior osmotic pressure will cause the vesicles to swell. The vesicles' membrane tension will remain negligible until the vesicle shape becomes spherical. Further swelling will cause the membrane area to expand, which in turn increases the membrane tension until it reaches the lysis tension, at which point the vesicle will burst and release its content. -> b. Estimate the time that elapses between trigger and release of the drug. Hints: The timing of vesicle swelling is determined by water transport across the membrane, which is governed by Eq. (2). This differential equation does not have an explicit solution in the form V0) - However, we can inspect the vesicle behavior for small volume changes AVU) where V=V0+AV and 3K«I. (3) Vs In Eq. (2) we also neglect changes of the vesicle surface area, replacing A with A0. However, to predict the membrane tension given by A—Au U=KA (4.) during swelling of the spherical vesicle, we will need to keep track of the actual surface area at that stage. For spherical geometry, the membrane area can he obtained from the vesicle volume. [Notatiom o ... membrane tension Kg .. . elastic modulus for area expansion] Answers for b. Immediately after application of the trigger, What are the new values of the following quantities to be used with Eq. (2)? A o = “L JJ‘013 V0 ? l. - : VI:- 0 -. \ my; wwl' : w 4RY 7 a W A05 lo‘Of'LEm /6155W\55P % 3 0,392") ~B 3 . -.. V“ 30.3:4/37H0 ZQQ,L163I$ W0 W3 (flaw? £7.3’2ma165nm 6 I') was“, " 6M. 24L. 1 «r5 _ 72m- ggm,0=? 30s m 09"" E -3- Show how the assumptions of Eq. (3) lead to a linearized differential equation of the form dAV —Ea—5AV S and find the expressions for the constants a and b. (Don’t insert any values yet.) “0 Bonus (not required; use separate paper): Solve the differential equation Eq. (5) for the initial condition Eq. (6) given on the next page. Explain what you are doing. flag) MD V;Vo+bv 4 {3" Aél Va divfib‘v) m V U .— V 4, A: C ‘J" '— ”AT _ who 6’? (”’10 wv CW) GUN _ U. 'm ”7‘ -. Va mv ' \‘n _ q I V'mna A9EP[VD ((OSMJDH Cosmo __..._____.__——-—-——— VOHW gun ca Av .44 U0 VOHW X r- 1:“ \fYnJ’HLoa ASP (Cdshr; DJWID : 0V , m1 * {on We (1 — m i. , UWQLOASMCW Como) &&*@J fl.v:x ML ,.--_ I; 51be a” pm ,9 2 A», arbx ”151% (at/W *‘flfi 4&7th an)” army? 3 With initial condition AV(0) : 0 . (.6) Eq- (5) has the solution AV afloat”? (7') Assuming that this approximation is valid, and using a membrane permeability to water of P = 30 mills, how long will it take until the vesicle shape: becomes spherical? \ L? _ . _ l m .. C 0. - VMHvLo FEE)? (thfi 055ml?) '0 *3 : 18:: '00,; 4 33310 W; gums -—360¥l::): M233 E F; , . , l5 We / 02- 193w: WV ,9 fl 7'3 l0 : VWIHU}. A5,? (#:03er E \f D 96% b —6 550 “53 ‘ffiifi/ grfineoeloie 30W 705 j / / 6'3“? QflBIb $1087; -: mazes n5 5-” ”/hlow that the “membrane slack” has been used up, any further swelling will increase the size of the sphere, and thus also expand the membrane area and increase the membrane tension. We assume that the vesicle will rupture at a lysis tension of 0* = 10 mN/m. Use Eq. (4) (where KA = 240 mNz’m) to determine at which value of the surface area the vesicle will lyse. Then, find the volume that the sphere will have at that point. Eventually, calculate the total time elapsed between trigger and drugrelease. E .: 1 \“x *3 0C ate—E 3 ‘\ Av: 0.124(9-q3315w0 2 "96 3’ m" at 5 a; 92.26%le Lfri”1/5 b H [.11’2—‘63‘Ktfi‘? 5" ’ E v ; 530%,“) :7 ”bl: : M Q‘ 135%") .- i’, c; 39W”? SeeS ‘— 0‘:KA{:_E° fiesta/ME -Iogw a, m“ “'"" io‘o £290 APPENDIX=__ '\ Definition of e We start from Fie 's 1St law: J(.x)=—Dd:E:c) /"X (8) /. J . .. concentrati (net) flux, i.e., number of molecules thyd/ve across an area A where per unit time units: #farea/time) c concentration; ere: number of molecules per volum (units: #lvolume) Applied to diffusion through membrane, we rewrite Eq. (8) _ Ac(x) J(x) — —D Ax ./ (9) K x" A0 = 1::I — c“ difference b een the concentrations on both sides of the membrane (both solutio s are assumedio be ideally mixed) where E? (10) Thus, for the flux through the membrane, /’2 J=—PA JOE—c"). (11') Osmoticall driven volume chan of a mem rane ca sale We consider a vesicle that has Vplaced into a on—isotonic solution. Notation: V vesicle volume A vesicle surface. a a (membrane area) Vi], A0 initial ves' 1e volume and area Ni number of lecules of kind "i" The osmotic pressur}? erence causes the vesicle vol to change. Only water crosses the membrane. Ther ore, the change in vesicle volume i iven by the flux of water across the membrm} area-z / dV dNuzo —— — v - — v AJ 12 dr H10 d: H10 H0 ( ) where VHZ'O ... volume per water molecule [H20 flux of water across the membrane surface \. Using q. (11), dV - E = —vH20AP(cfi’ZO — c3120). (13) 35M @ _ ,8 Ah 61-61(15qu «n5 - q.qa3lé»<l58m5 " 5&3??? K1530“; ': ”#Lr—a 1..— ____,__,—r—7—' -Lf -7 L— Inlafiyfilo g. 51%177amocr' *T [.7ZZijxo (2'1“? We n71 . he — '51. a . - I —=-_-——-—-—T———-————yg—-———————————T——————— ...
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