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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 drugdelivery
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$m Briefly comment on the relation between the stationary solution of Eq. (2) and the ideal
gas law. Control of drug release: We next consider administration of a drug using these vesicle
capsules. Assuming that the vesicles have adhered to the proper target (eg, a tumor), we
apply the trigger that causes dissociation of the compound ABC (see Eq. (1)). The
resulting increase of interior osmotic pressure will cause the vesicles to swell. The
vesicles' membrane tension will remain negligible until the vesicle shape becomes
spherical. Further swelling will cause the membrane area to expand, which in turn
increases the membrane tension until it reaches the lysis tension, at which point the
vesicle will burst and release its content. > b. Estimate the time that elapses between trigger and release of the drug. Hints: The timing of vesicle swelling is determined by water transport across the
membrane, which is governed by Eq. (2). This differential equation does not have an
explicit solution in the form V0)  However, we can inspect the vesicle behavior for small volume changes AVU) where V=V0+AV and 3K«I. (3) Vs
In Eq. (2) we also neglect changes of the vesicle surface area, replacing A with A0.
However, to predict the membrane tension given by A—Au U=KA (4.) during swelling of the spherical vesicle, we will need to keep track of the actual surface
area at that stage. For spherical geometry, the membrane area can he obtained from the
vesicle volume. [Notatiom
o ... membrane tension
Kg .. . elastic modulus for area expansion] Answers for b.
Immediately after application of the trigger, What are the new values of the following
quantities to be used with Eq. (2)? A o = “L JJ‘013
V0 ? l.  : VI: 0
. \ my; wwl' : w 4RY 7 a W
A05 lo‘Of'LEm /6155W\55P % 3 0,392") ~B 3
. .. V“ 30.3:4/37H0 ZQQ,L163I$ 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
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arbx
”151% (at/W *‘ﬂﬁ
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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
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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(9q3315w0 2 "96 3’ m"
at 5
a; 92.26%le Lfri”1/5 b H [.11’2—‘63‘Ktﬁ‘? 5" ’ E
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£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} areaz /
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 6161(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
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he
— '51.
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—=_————T———————yg————————————T——————— ...
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 Spring '11
 Heinrich

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