Bingmei M. Fu1*
Bin Chen
Department of Mechanical Engineering,
Cancer Institute,1
University of Nevada, Las Vegas
A Model for the Modulation of
Microvessel Permeability by
Junction Strands
To investigate the effect of junction strands on microvessel permeability, we extend the
previous analytical model developed by Fu et al. (1994, J. Biomech. Eng., 116, pp.
502–513), for the interendothelial cleft to include multiple junction strands in the cleft
and an interface between the surface glycocalyx layer and the cleft entrance. Based on the
electron microscopic observations by Adamson et al. (1998, Am. J. Physiol., 274(43),
pp. H1885–H1894), that elevation of intracellular cAMP levels would increase number of
tight junction strands, this two-junction-strand and two-pore model can successfully account for the experimental data for the decreased permeability to water, small and
intermediate-sized solutes by cAMP. 关DOI: 10.1115/1.1611514兴
Introduction
Vascular endothelium is the principal barrier to, and regulator
of, material exchange between circulating blood and body tissues.
The ultrastructural pathways and mechanisms whereby endothelial cells and the cleft between the cells 共interendothelial cleft兲
modulate microvessel permeability to water and solutes have been
an unsolved subject in microvessel transport since early 1950’s
关3– 6兴. Microvessel permeability to water is hydraulic conductivity Lp and that to a solute, solute permeability P. In conjunction
with microperfusion techniques, electron microscopy, and quantitative imaging methods, Fu et al. 关1,7,8兴 developed a 3-D combined junction-orifice-fiber entrance layer model to investigate the
molecular structures of the interendothelial cleft, which determine
the normal permeability properties of the microvessel wall. These
structures include: 共1兲 endothelial cell surface glycocalyx which
determines selectivity to large molecules, and 共2兲 junction strands
in the cleft between adjacent endothelial cells which determine the
fraction of the cleft length that is effectively open to molecules of
various sizes and the geometry of the diffusion pathway within the
interendothelial cleft. These junction strands are believed to be
formed by occludin and cadherin-like proteins 关9,10兴. Figure 1
shows their 3-D model for the interendothelial cleft. To obtain an
analytical solution, this model has only one junction strand within
the cleft and uses an effective approximation, which converts the
surface glycocalyx layer into an equivalent of fiber layer inside
the cleft. It predicts that in order to account for the measured
hydraulic conductivity, the fiber 共glycocalyx兲 layer must be confined to a relatively narrow region at the entrance of the cleft. This
fiber layer also serves as the primary molecular filter. Furthermore, the model predicts that the junction strand in the cleft must
contain at least two types of pores: infrequent 150 nm⫻20 nm
large orifice breaks and a continuous ⬃1.5 nm narrow slit or
closely spaced 1.5 nm radius circular pores. For frog mesenteric
capillary, this one-junction strand and two-pore model provides an
excellent fit for the hydraulic conductivity Lp and the diffusive
permeability P data for solutes of size ranging from potassium to
albumin 关1兴.
Many previous studies have found that increased intracellular
levels of adenosine 3⬘,5⬘-cyclic monophosphate 共cAMP兲 can
block the inflammatory response in a variety of experimental
*Corresponding address: 4505 Maryland Parkway, Box 454027, Las Vegas, NV
89154. Telephone: 共702兲-895-3455; Fax: 共702兲-895-3936; e-mail: bmfu@nscee.edu.
Contributed by the Bioengineering Division for publication in the JOURNAL OF
BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division November 13, 2002; revision received February 18, 2003. Associate Editor: C.
Dong.
620 Õ Vol. 125, OCTOBER 2003
models 关11–17兴. Simultaneous stimulation of adenylate cyclase
共either by activating prostaglandin receptors or by stimulating
adenylate cyclase directly with forskolin兲 and phosphodiesterase
共PDE兲 inhibition with zardaverine 共a strong inhibitor of PDE type
III and type IV兲 was highly effective in inhibiting permeability
increases induced by H2 O2 in isolated rabbit lung 关15兴. A previous
study on intact microvessels also confirmed the blocking effect of
a cAMP analog on inflammatory stimulation 关13兴. Solutions containing ATP 共10 ␮M兲 stimulate a 5- to 10-fold increase in hydraulic conductivity Lp in both capillaries and postcapillary venules of
frog and hamster mesentery. This characteristic transient inflammatory response was nearly abolished by 15 min pretreatment
with 8-bromo adenosine 3⬘,5⬘-cyclic monophosphate 共cAMP, 2
mM兲 关13兴.
To understand the underlying mechanisms of the regulation of
microvessel permeability by elevated intracellular cAMP levels,
Adamson et al. 关2兴 tested the cAMP effect on microvessel hydraulic conductivity Lp from the control state in an in vivo study. They
found that increase cAMP levels by simultaneous adenylate cyclase activation 共by forskolin兲 and phosphodiesterase inhibition
共by rolipram兲 reduced frog mesenteric microvessel hydraulic conductivity Lp to 37% of its baseline value in ⬃20 min. A parallel
study by Fu et al. 关18兴 showed that in 20 min after exposure to
rolipram and forskolin, permeability of small solute sodium fluorescein 共MW⫽376, Stokes radius⫽0.45 nm兲 Psf and that of
intermediate-sized solute ␣-lactalbumin 共MW⫽14,176, Stokes
radius⫽2.01 nm兲 P␣ -lactalbumin was decreased to 67% and 64% of
their baseline values, respectively. Furthermore, Adamson et al.
关2兴 found in their electron microscopy study that cAMP effect
would induce an increase of the number of junction stands from a
mean value of 1.7 to a mean value of 2.2 per cleft in ⬃20 min.
More junction strands will induce more tortuosity in the interendothelial cleft, thus will decrease the microvessel permeability to
water and solutes.
To quantitatively understand the ultrastructural mechanisms of
altered microvessel permeability by the enhancement of intracellular cAMP, we develop in this study a new model for the prediction of the structural changes in the interendothelial cleft induced
by cAMP, based on aforementioned experimental observations.
This new model is shown in Fig. 2. One new feature of our model
is that there is an interface between the surface glycocalyx layer
and the cleft entrance. Another new feature is that there are two
junction strands in the interendothelial cleft instead of one in previous models.
Copyright © 2003 by ASME
Transactions of the ASME
Fig. 1 Previous model for the interendothelial cleft structure
†1,7,8, with permission‡. „a… 3-D sketch; „b… plane view. Ljun is
the junction strand thickness. L1 and L3 are the depths between
the junction strand and the luminal and abluminal fronts of the
cleft. L is the total length of the cleft. The distance between two
adjacent breaks in the junction strand is 2-D. At the entrance of
the cleft on the luminal side, surface glycocalyx is represented
by a periodic square array of cylindrical fibers. Lf is the fiber
layer thickness. The radius of these fibers is a and the gap
spacing between fibers is ⌬. In the junction strand, there are
periodically distributed large pores 2dÃ2B and a continuous
small slit of height 2bs. This model can successfully explain
the microvessel permeability under normal conditions †1‡.
Model Description
Model Geometry. The top view of our new model for
explaining the decrease in microvessel permeability is shown in
Fig. 2共b兲. There is a surface fiber layer of thickness Lf at the
entrance of the cleft and there are two junction strands in the cleft.
Figure 2共a兲 shows the 3-D schematic of a periodic unit. In the
junction strand, large breaks observed by Adamson and Michel
关19兴 are represented as orifice openings of dimensions 2d⫻2B in a
zero thickness barrier. The spacing between orifices is 2-D. In
addition to the large orifices, there is a small continuous slit of
height 2bs along the junction strand. This small slit is needed to
provide an optimal fit for small ion permeability 关1,7兴. For the
junction strand with this small slit of height 2bs⬃1.5 nm, the
thickness of the junction strand Ljun is taken as 10 nm 关11兴. L1 is
the distance between the first junction strand and the luminal front
of the cleft, L2 , the distance between the second junction strand
and the luminal front. yc is the distance between the center of the
periodic junction strand unit in the second strand and the centerline across the junction break in the first strand.
Hydraulic Conductivity Lp. Similar to that in Fu et al. 关1兴,
Lp is obtained by solving pressure p共x,y兲 and velocity V共x,y,z兲
fields in the cleft region. Since the height of the cleft 2B is small,
compared to both the distance between the junction pores 2D and
Journal of Biomechanical Engineering
Fig. 2 New model for explaining cAMP effect showing a surface glycocalyx layer and two junction strands in the cleft. „a…
3-D sketch; „b… plane view. L1 and L2 are the distances between
the first and the second junction strands and the luminal front
of the cleft, respectively. There is an interface between the surface glycocalyx and the cleft entrance at xÄ0. yc is the distance
between the center of the periodic junction strand unit in the
second strand and the centerline across the large junction pore
in the first strand.
the depths L1 , L2 of the cleft, the water flow in the cleft can be
approximated by a Hele-Shaw channel flow 关1兴. The velocity in
the cleft can be expressed as:
冉 冊
V共 x,y,z兲 ⫽V0 共 x,y兲 1⫺
Z2
B2
V0 共 x,y兲 ⫽u0 共 x,y兲 i⫹v0 共 x,y兲 j
(1)
(2)
which satisfies the non-slip condition at z⫽⫾B. V0 (x,y), the velocity in the center plane z⫽0, is given by:
V0 共 x,y兲 ⫽⫺
B2
ⵜp
2␮
(3)
Here ␮ is the fluid viscosity. For Hele-Shaw flow, the pressure in
the cleft satisfies:
⳵ 2p
⳵x
2
⫹
⳵ 2p
⳵ y2
⫽0
(4)
Integrating Eq. 1 over the height of the cleft gives the average
velocity V̄共x,y兲:
V̄共 x,y兲 ⫽
2
V 共 x,y兲
3 0
(5)
A linear 1-D Darcy flow approximation is applied to the unbounded surface glycocalyx layer of thickness Lf 关20兴. The local
average velocity along the length of the cleft in the x direction is
OCTOBER 2003, Vol. 125 Õ 621
ū共y)⫽
Kp pL⫺p共 1 兲 共 0,y兲
␮
Lf
(6)
Here, Kp is the Darcy permeability. pL and p( 1 ) (0,y) are pressures
in the lumen and at the entrance of the cleft behind the surface
glycocalyx, respectively. Continuity in water flux at the interface
of the fiber layer and the cleft entrance 共combining Eqs. 3, 5, 6兲
gives,
x⫽0
⫺
B2 Lf ⳵ p共 1 兲
3Kp ⳵ x
冏
⫽pL⫺p共 1 兲 共 0,y兲
(7a)
and cleft regions. A 1-D diffusion was approximated for the surface fiber matrix region. The concentration in the fiber region was
obtained as:
x⫽0
⳵ p共 1 兲 ⳵ p共 2 兲
⫽
⳵x
⳵x
p共 1 兲 ⫽p共 2 兲
兩 y兩 ⭐d
x⫽0
⫺
共i兲
共1兲
x⫽L2
p共 2 兲 ⫽p共 3 兲
pore region
strand region
i⫽1,2
共2兲
(7d)
(7e)
⳵ p共i)
⫽0
⳵y
i⫽1,2,3
Q2D Ljt
pL⫺pA 2D
(7g)
(8)
where Q2D is the volume flow rate through one period of the
junction strand including one large 2d⫻2B break and 2共D-d兲 long
2bs wide small slit. A numerical method similar to that in Hu and
Weinbaum 关20兴 was applied to solve Eq. 4 with corresponding
boundary conditions for pressure field p共i兲共x,y) in regions 1, 2 and
3. The convergence condition is that the relative error between the
nth and 共n⫹1兲th iteration values for p at each point, 兩 p ( n⫹1 )
⫺p ( n ) /p ( n⫹1 ) 兩 , is less than 10⫺6 . Equations 1 and 3 determine
the velocity V共x,y,z兲 from p共i兲共x,y). Integration of u共L1 ,y,z)
across the cross-sectional area of one period of the junction strand
共⫺D⬍y⬍D,⫺B⬍z⬍B兲 gives the value of Q2D . pL and pA, which
are constants here, are pressures in the lumen and in the tissue
space, respectively. 2D is the spacing between adjacent junction
breaks. Ljt is the total length of the cleft per unit surface area of
microvessel wall.
Diffusive Permeability P. The diffusive permeability or solute permeability P is obtained by solving the concentration field in
the cleft. Under the experimental conditions of Fu et al. 关18兴 at
low perfusion pressures of 5 to 8 cmH2 O, which induced effective
transmural pressure drop in the range of 1 to 4 cmH2 O, the solute
transport can be simplified as a pure diffusion process in the fiber
622 Õ Vol. 125, OCTOBER 2003
⳵ 2C
⳵ y2
⫽0
(11)
C共 1 兲 ⫽C共 2 兲
兩 y兩 ⭐d
x⫽L1
⳵ C共 1 兲 ⳵ C共 2 兲
⫽
⳵x
⳵x
(10b)
d⬍ 兩 y 兩 ⭐D
x⫽L 1
D ssl b s C 共 1 兲 共 L 1 ,y 兲 ⫺C 共 2 兲 共 L 1 ,y 兲
⳵C共i兲
⫽⫺ c
⳵x
L jun
DsB
i⫽1,2
(10c)
(7f)
Boundary conditions 共7b兲, 共7d兲 require that pressure and velocity
be continuous across the large junction break while 共7c兲, 共7e兲
require that the volume flow be continuous across the junction
strand where there is a continuous small slit. Boundary condition
共7f兲 indicates that the pressure at the cleft exit equals the pressure
in tissue space pA, which is a constant. Boundary condition 共7g兲 is
the periodicity condition. Ljun is the thickness of the junction
strand.
The hydraulic conductivity is defined as,
Lp⫽
(10a)
Other boundary conditions in each region of the cleft 共Fig. 2兲 are,
bs3 p共 2 兲 共 L2 ,y兲 ⫺p共 3 兲 共 L2 ,y兲
⳵ p共 i 兲
⫽⫺ 3
⳵x
Ljun
B
y⫽⫾D
⳵ x2
(7c)
⫹
共3兲
⳵p
⳵p
⫽
⳵x
⳵x
x⫽L 兩y兩 ⭐D p共 3 兲 ⫽pA
⫽C L ⫺C 共 0,y 兲
x⫽0
Here Dsf and Dsc are diffusion coefficients of a solute in the fiber
layer and cleft region, respectively. The governing equation for
the cleft region is a 2-D diffusion equation,
⳵ 2C
i⫽2,3
0⭐x⭐L
⳵x
D sf
共2兲
⳵p
p 共 L1 ,y兲 ⫺p 共 L1 ,y兲
⫽⫺ 3
⳵x
Ljun
B
bs3
冏
D sc L f ⳵ C 共 1 兲
d⬍ 兩 y兩 ⭐D
x⫽L1
x⫽L2
(7b)
(9)
Here, Lf is the fiber layer thickness, CL is the solute concentration
in the lumen, and C( 1 ) (0,y) is the concentration at the cleft entrance behind the surface fiber layer. Continuity in mass flow at
the interface of the fiber layer and the cleft entrance gave,
The other boundary and matching conditions for Eq. 4 in each
region of Fig. 2 are:
x⫽L1
CL⫺C共 1 兲 共 0,y兲
x⫹C共 1 兲 共 0,y兲
Lf
C共f) ⫽⫺
x⫽L2
x⫽L 2
共i兲
C共 2 兲 ⫽C共 3 兲
pore region
⳵C
⫽⫺
⳵x
D ssl b s
D sc B
C
共2兲
共2兲
共3兲
⳵C
⳵C
⫽
⳵x
⳵x
strand region
共 L 2 ,y 兲 ⫺C 共 3 兲 共 L 2 ,y 兲
L jun
i⫽2,3 (10e)
x⫽L 兩y兩 ⭐D C共 3 兲 ⫽CA
0⭐x⭐L
(10d)
y⫽⫾D
⳵ C共 i 兲
⫽0
⳵y
i⫽1,2,3
(10f)
(10g)
Boundary conditions 共10b–g兲 for the concentration field mean the
same as boundary conditions 共Eqs. 7b–g兲 for the pressure field.
CA here is the concentration in the tissue space.
The diffusive permeability P is defined as,
P⫽
s
Q2D
Ljt
CL⫺CA 2D
(12)
S
where Q2D
is the solute mass flow rate through one period of the
junction strand 共⫺D⬍y⬍D兲. The same numerical technique used
to solve p共i) was applied to solve Eq. 11 with corresponding
boundary conditions for C共i兲共x,y) in regions 1, 2 and 3. Integration
of Dsc⳵ C共x,y)/ ⳵ x at x⫽L1 across the cross-sectional area of one
period of the junction strand 共⫺D⬍y⬍D,⫺B⬍z⬍B兲 gave the
S
value of Q2D
. Constants CL and CA are solute concentrations in
the lumen and in the tissue space, respectively. Other parameters
are the same as in Eqs. 7b–g.
Model Parameters Under Normal Conditions
We use the experimental data for frog mesenteric capillaries in
our model. All the values for the cleft parameters under normal
共control兲 conditions are the same as those in 关1兴. They are as the
following: the total cleft depth L⫽400 nm, the thickness of the
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junction strand Ljun⫽10 nm, the total cleft length per unit vascular
surface area Ljt⫽2000 cm/cm2 , the cleft height 2B⫽20 nm, the
junction break width 2d⫽150 nm, and the average spacing between adjacent breaks is 2D⫽2640 nm. The distance between the
junction strand and the front of the cleft L1 ⫽200 nm 共Fig. 1兲. For
the surface fiber layer, we use fiber radius a⫽0.6 nm and the fiber
density Sf⫽0.039 for a periodic fiber array or Sf⫽0.11 for a random array. These Sf values are larger than that in Fu et al. 关1兴 in
order to account for the measured permeability data in Adamson
et al. 关21兴 and Fu et al. 关18兴. For these fiber arrangements, Dsf/Dsfree
for intermediate-sized solute ␣-lactalbumin 共Stokes radius⫽2.01
nm兲 is 0.025 and for small solute sodium fluorescein 共Stokes
radius⫽0.45 nm兲, 0.2. Dsf is the effective diffusion coefficient of a
solute in the fiber region and Dsfree is the free diffusion coefficient
of a solute in aqueous solutions. The diffusion coefficient for a
solute in the cleft region Dsc is calculated using the same theory as
in Weinbaum et al. 关6兴 and Fu et al. 关1兴. The Darcy permeability
Kp in Eq. 6 is also determined in the same way as in 关1,6兴. K p
⫽2⫻10⫺14 cm2 . The fiber entrance thickness Lf is chosen as 100
nm, based on the observation of Adamson and Clough 关1兴, although there may be thicker layer in other preparation 关22兴. Lf of
100 nm can account for the experimental result for Lp under normal conditions, Lp normal⫽2.4⫻10⫺7 cm/s/cm H2 O 关2兴. L f of 100
nm also fits the experimental data for P of various sized solutes
sf
ranged from 2.5 to 12.1⫻10⫺5 cm/s with a justified
关1兴. Pexperiment
average value of 6.9⫻10⫺5 cm/s 关18兴. In our model, we chose
sf
Pnormal
⫽8.5⫻10⫺5 cm/s 共two pores in junction strands when 2bs
␣ -lactalbumin
⫽1.1 nm) and Pnormal
⫽2.4⫻10⫺6 cm/s 关18,23兴.
In Adamson et al. 关2兴, an average of 1.7 junction strands per
cleft under normal conditions came from a distribution of the
number of junction strands from 0 to 5 with 41% clefts having 1
strand, ⬃39% having 2 strands, and ⬃14% having 3 strands.
Fig. 3 Lp and P as a function of the strand location L1 for the
single junction strand case. „a… Large pore „2dÃ2BÄ150
nmÃ20 nm… only in the strand; „b… Two pores in the strand
„small slit height 2bsÄ1.1 nm…. Lp, P values at L1 ÕLÄ0.5 are
taken as the normal control values for nondimensionalization
in Figs. 5, 6, and 7 in each case.
Fig. 4 Concentration distributions of sodium fluorescein in the cleft region when there are two junction strands. The first strand
is located at xÄ200 nm and the second at xÄ300 nm. Upper row: Large pore only in the strands; Bottom row: Two pores in the
strand. „a… ycÄ0, when the center of the second strand unit lines with the center of the large pore in the first strand, „b… yc
Ä75 nm, when the large pore in the second strand is located on one side the periodic unit and „c… ycÄ1320 nm, when the large
pore in the second strand lines exactly with the large pore in the first strand. The solute flux for the case shown in upper „a… is
9.3Ã10À6 cmÕs, and in lower „a…, 44.5Ã10À6 cmÕs; in upper „b…, 5.8Ã10À6 cmÕs, and in lower „b…, 41.9Ã10À6 cmÕs; in upper „c…,
35.3Ã10À6 cmÕs, and in lower „c…, 57.7Ã10À6 cmÕs.
Journal of Biomechanical Engineering
OCTOBER 2003, Vol. 125 Õ 623
Fig. 5 Dimensionless Lp and P as a function of the locations of the junction strands L1 and
L2 and the alignment of large junction pores in the strands yc „see Fig. 2…. „a… L1 Ä25 nm,
L2 Ä200 nm; „b… L1 Ä100 nm, L2 Ä200 nm; „c… L1 Ä200 nm, L2 Ä300 nm. yc ÕDÄ0 corresponds
to the case when the center of the second strand unit lines with the center of the large pore
in the first strand „see Fig. 4a…; yc ÕDÄ75Õ1320 is the case when the large pore in the second
strand is located on one side the periodic unit „see Fig. 4b…; yc ÕDÄ1 is when the large pore
in the second strand lines exactly with the large pore in the first strand „see Fig. 4c…. The left
column is for the case when there are only large pores in both strands and the right column
for the case when two pores in both strands.
When the cAMP was increased, the increased average number of
junction strands per cleft, 2.2, was from a distribution of ⬃20%
clefts having 1 strand, ⬃48% having 2 strands, ⬃20% having 3
strands, and the rest having 0, 4 or 5 strands. The largest shift was
from 1 to 2 strands. To simplify the problem, we used 1 strand in
the normal conditions as an average of 0 to 5 strands as in Fu
et al. 关1兴 共Fig. 1兲, and used 2 strands when cAMP was increased
共Fig. 2兲.
Results
Effect of Junction Strand Location on Lp, P under Normal
Conditions. Figure 3 shows the results for Lp and P as a function of the strand location L1 under normal conditions when there
is a single junction strand in the cleft. Figure 3共a兲 is for largepore-only cases and 共b兲 for two-pore cases. The effect of the
strand location L1 on Lp and P is similar in both cases. In general,
Lp and P are not sensitive to the change in L1 , only slightly
increase when the strand moves towards the abluminal front of the
624 Õ Vol. 125, OCTOBER 2003
cleft. Therefore, we choose the values when the strand is in the
middle of the cleft, L1 /L⫽0.5, as the normal control values for Lp
and P.
Effect of Locations of Junction Strands and Junction Pores
on Concentration Distributions. Figure 4 shows the spread
patterns of sodium fluorescein when there are two junction strands
in the cleft. The upper row in Fig. 4 is for large-pore-only cases
and the lower row for two-pore cases. The first strand is located in
the middle of the cleft (L1 ⫽200 nm) and the second is at L2
⫽300 nm. Figure 4共a兲 shows the concentration distributions when
the middle of the second junction strand unit is lined with the
center of the large pore in the first strand; half of the large pore in
the second strand is on each side of the strand unit. When the large
pore in the second strand is on one side of the strand unit, the
concentration distributions are shown in Fig. 4共b兲. Figure 4共c兲
shows the concentration distributions when these two large pores
are exactly lined up with each other. From the spread patterns of
sodium fluorescein shown in Fig. 4, we can see that different
arrangements of the strand and large pore locations would provide
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Fig. 6 The mean dimensionless Lp and P as a function of the second strand
location L2 when L1 Ä100 nm averaged over all possible yc „see Fig. 2b…
different resistance to the transport of water and solutes. For example, the solute flux for the case shown in upper Fig. 4共a兲 is
9.3⫻10⫺6 cm/s, and lower, 44.5⫻10⫺6 cm/s; in upper Fig. 4共b兲,
5.8⫻10⫺6 cm/s, and lower, 41.9⫻10⫺6 cm/s; in upper Fig. 4共c兲,
35.3⫻10⫺6 cm/s, and lower, 57.7⫻10⫺6 cm/s. The changes in the
water volume flux 共in the unit of cm/s兲 have similar patterns as for
the solute flux. These observations are summarized in Fig. 5.
Effect of Locations of Junction Strands and Junction Pores
on Lp and P Under Increased cAMP. Figure 5 shows representative cases for Lp, P␣ -lactalbumin, and Psf as a function of junction strand locations L1 and L2 and the alignment of the large
junction pores in the strands yc. Results are expressed as the ratio
to normal control values when there is a single strand, with both
large pores and a small slit, in the middle of the cleft 共see Fig. 3兲.
These normal values for the single strand with two pores are the
same as those measured in the experiments 关2,18,23兴. We assume
that there are no other changes in the cleft except the formation of
a new junction strand 共see Discussion兲. Figure 5共a兲 is for cases
when the first strand is located 25 nm and the second strand 200
nm away from the luminal front (L1 ⫽25 nm, L2 ⫽200 nm). Figure 5共b兲 is for cases when L1 ⫽100 nm, L2 ⫽200 nm, and Fig. 5共c兲
is for cases when L1 ⫽200 nm, L2 ⫽300 nm. The left column in
Fig. 5 shows cases when there are only large pores in both junction strands; the right column shows cases when there are two
pores in both strands. Figure 5 shows representative results of all
other cases when we change L1 , L2 , and yc /D. In all cases, the
Fig. 7 The mean dimensionless Lp and P as a function of the first strand location L1
averaged over all possible L2 „L2 ÌL1 … and yc „see Fig. 2b…. L2 ÌL1 means that the
location of the second junction strand L2 is always further than that of the first one L1
to the entrance of the cleft.
Journal of Biomechanical Engineering
OCTOBER 2003, Vol. 125 Õ 625
Table 1 Comparison of the experimental data with the model
predictions for the mean values of the decreased permeability
over all possible locations of junction strands and the large
pores
P/Pnormal
Experimental Data
Large pore only
Two pore
Lp /Lpnormal
␣-lactalbumin
Sodium fluorescein
0.37
0.32
0.37
0.64
0.47
0.47
0.67
0.16
0.56
further the large pore in the second strand is away from the centerline across the large pore in the first strand, the larger resistance
the second strand induces. The largest resistance occurs when the
center of the large pore in the second strand is located at 共D-d兲
from the centerline across the large pore in the first strand (yc /D
⫽d/D⫽0.0568). The resistance in this arrangement can be up to 6
folds of that when the second large pore lines exactly with the first
large pore (yc /D⫽1), where the lowest resistance appears.
We can see from Fig. 5 that when the large pores in two strands
are close to each other, yc /D⬎0.6, especially when they are lined
up, the effect of junction strand locations is significant. The closer
the junction strands are to the cleft entrance, the larger the resistance they induce. When the large pores in two strands are far way
from each other, yc /D⬍0.6, the effect of the first strand location
L1 can be neglected 共Fig. 5兲, while the effect of relative locations
of two strands matters 共Fig. 6兲.
Figure 6 shows the mean effect of the second strand location L2
on permeability for a representative case when L1 ⫽100 nm over
all possible yc. It indicates that the closer the second strand L2 is
to the cleft entrance or to the first strand, the higher the resistance
is.
Figure 7 shows the mean effect of the first junction strand location L1 on permeability over all possible L2 (L2 ⬎L1 ) and yc.
L2 ⬎L1 means that the location of the second junction strand L2 is
always further than that of the first one L1 to the entrance of the
cleft. It seems that the first strand location L1 共from 25 nm to 300
nm兲 does have some effect on ␣-lactalbumin permeability
P␣ -lactalbumin, but not much on Lp and sodium fluorescein permeability Psf. When the first strand is located in the middle of the
cleft, L1 ⫽200 nm, the resistance to water and solute transport is
minimum.
Table 1 shows a comparison of the experimental data with the
model predictions for the mean values of decreased permeability
over all the cases when changing L1 , L2 , and yc. The mean value
for Lp /Lpnormal is obtained through three steps: 共1兲 finding the
mean over all possible yc by 兰 10 Lp /Lpnormald共yc /D) for various
combinations of L1 and L2 , such as cases shown in Fig. 5; 共2兲
finding
the
mean
over
all
possible
L2
by
L
兰 L2bLp /LpnormaldL2 /(L2b⫺L2a) for various L1 , such as cases
2a
shown in Fig. 6 共In Fig. 6, L2a⫽150 nm and L2b⫽375 nm); 共3兲
finally finding the mean over all possible L1 by
L
兰 L1bLp /LpnormaldL1 /(L1b⫺L1a), such as cases shown in Fig. 7 共In
1a
Fig. 7, L1a⫽25 nm and L1b⫽300 nm). The mean values for P
and P␣ -lactalbumin are obtained through the same process. The first
row in Table 1 shows the experimental data, the second row
shows the model predictions for the case when there are only large
pores in the strands and the third one shows the predictions for the
case when there are two pores in the strands.
The experimental data for the decrease in Lp, P␣ -lactalbumin and
sf
P by the elevation of intracellular cAMP levels are 37%, 64%,
and 67% of their control values, respectively in ⬃20 min 关2,18兴.
We can see from Table 1 that the mean values for Lp, P in the
cases when both strands have two pores fit the experimental results much better than those in cases when there are only large
pores in the strands.
626 Õ Vol. 125, OCTOBER 2003
sf
Discussion
Tight and adherens junctions create a regulated paracellular barrier to the movement of water, solutes, and immune cells between
endothelial cells 关9,14,24兴. Very little is known about the assembly of these junctions, but several kinds of evidence suggest that
they are very dynamic structures 关10,25兴. Many previous studies
have found in a variety of experimental models that increased
intracellular cAMP levels can decrease the water and solute permeability by possibly increasing the number of junction strands or
complexity in the paracellular cleft 关2兴. However, the quantitative
relationship between the permeability and the numbers of junction
strands, especially in an in vivo model, has never been investigated.
Based on the experimental results of Adamson et al. 关2兴 and Fu
et al. 关18兴 for frog mesenteric microvessel permeability, we test in
this paper the hypothesis that the decrease in hydraulic conductivity Lp and solute permeability P␣ -lactalbumin and Psf by elevation of
intracellular cAMP levels is due to the formation of new junction
strands in the interendothelial cleft.
Figure 5 shows how locations of two strands and large pores in
the strands affect Lp and P. Table 1 gives an averaged value over
all the possible locations. It clearly shows that in order to explain
the experimental results for water and solute permeability measured by Adamson et al. 关2兴 and Fu et al. 关18兴, the mean number
of the junction strand in the cleft would be increased from one to
two, and there must be two types of pores in the junction strands.
The model predicted values in Table 1 appear to be smaller than
the measured values. The explanation for this is that there is an
overestimation in the model predictions because the number of
junction strands was in fact only increased by 29% 共from 1.7 to
2.2, 关2兴兲 instead of 100% 共from 1 to 2兲 in the model.
The model presented in this manuscript is based on the assumption that there are no other structural changes in the cleft such as
the number of large pores, the large pore size, the height and the
length of the cleft, and the thickness and arrangement of the surface glycocalyx. This assumption has been validated by direct and
indirect evidences using electron microscopy and reflection coefficient measurements. After examining and surveying their
samples in the electron microscopy study, Adamson et al. 关2兴 concluded that cAMP did not close preexisting gaps 共junction pores兲
and in contrast to cell culture studies, in vivo the action of cAMP
did not increase cell to cell overlap or endothelial cell spreading.
Although there have been no direct investigations to test the effect
of cAMP on the structure of cell surface glycocalyx, the reflection
coefficient of ␣-lactalbumin was not changed by cAMP 关18兴.
In summary, we have developed a new model that predicts the
effect of the junction strands on microvessel permeability. In conjunction with the experimental observations, this model can successfully explain the effect of enhancement of intracellular cAMP
levels on microvessel permeability to water, small- and
intermediate-sized solutes in in vivo measurements on frog mesenteric microvessels.
Acknowledgments
This work was supported by National Institutes of Health 共NCI
R15CA86847兲, National Science Foundation CAREER award and
University of Nevada New Investigator Award.
Nomenclature
a ⫽ fiber radius
B ⫽ half cleft width
bs ⫽ half width of the continuous small slit along the junction strand
C共i) ⫽ concentration in fiber layer 共i⫽f兲 and in the cleft
共i⫽1,2,3兲
CL ⫽ concentration in the lumen
CA ⫽ concentration in the tissue
D ⫽ half spacing between adjacent large breaks
Transactions of the ASME
Dsfree
Dsc
Dsf
Dssl
⫽
⫽
⫽
⫽
Kp ⫽
L ⫽
L1 ⫽
L2 ⫽
L3 ⫽
Lf
Ljun
Lp
P
p共i)
⫽
⫽
⫽
⫽
⫽
pL
pA
Q2D
s
Q2D
rs
Sf
V
V0
V̄
⌬
␮
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
free diffusion coefficient in aqueous solution
effective diffusion coefficient in the cleft
effective diffusion coefficient in the fiber layer
effective diffusion coefficient in the small slit of the
junction strand
Darcy permeability
total length of the cleft region
the distance between the first junction strand and the
luminal front of the cleft in the new model
the distance between the second junction strand and
the luminal front of the cleft in the new model
the distance between the junction strand and the abluminal front of the cleft in the previous model
thickness of fiber layer
thickness of junction strand
hydraulic conductivity
solute permeability
pressure in the cleft 共i⫽1, 2, 3 for regions 1, 2
and 3兲
pressure in the lumen
pressure in the tissue
volume flow rate through a period 2D
solute flow rate through a period 2D
solute radius
fiber density
共u,v,w兲 velocity
(u0 ,v0 ) velocity at the center plane
共ū,v̄兲 average velocity over cleft height
spacing between adjacent fibers
viscosity
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