Chemical Engineering Science 64 (2009) 4504 -- 4514

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Chemical Engineering Science

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A protein diffusionmodel of the sealing effect

Stewart Russella, Limary M. Cancela, John M. Tarbella, David S. Rumschitzkib,∗aDepartment of Biomedical Engineering, The City College of New York, The Graduate School, University Center of CUNY, New York, NY 10031, USAbDepartment of Chemical Engineering, The City College of New York, The Graduate School, University Center of CUNY, New York, NY 10031, USA

A R T I C L E I N F O A B S T R A C T

Article history:Received 22 October 2008Received in revised form 5 May 2009Accepted 7 May 2009Available online 20 May 2009

Keywords:Biomedical engineeringCellular biology and engineeringConvective transportMathematical modelingParameter identificationSealing effect

Water transport across the arterial endothelium is believed primarily to occur through breaks in thetight junction strands at the cell periphery between neighboring cells. Additional proteins arriving at thetight junction can close these breaks, thereby attenuating this water flux. Motivated by evidence that thediffusion of presynthesized protein from the interior of the cell to and incorporation into the cell borderis the mechanism of endothelial tight junctional sealing, we develop a diffusion-limited mathematicalmodel of intercellular gap sealing. A single endothelial cell is represented as a thin, axisymmetric disk,initially containing a uniform distribution of junctional protein that does not interact with the apical orbasal cell surfaces. Upon application of a transmural pressure gradient, water flows through the junctionalcleft, and tight junction remodeling begins. We assume that proteins at the junction are instantaneouslyincorporated into its strand, dropping the free protein concentration at the cell periphery to zero. This setsthe diffusion of intracellular proteins toward the junction in motion. The solution of this one-dimensionalinitial value problem provides excellent fits to current and previously published experimental data overa wide variety of conditions. It yields three physically meaningful parameters for each fit, including aprotein diffusivity in the cytoplasm that varies little within experimental treatments. Statistical variationof these parameters allows rational comparison of experimental runs and identification of outlier runs.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction: the sealing effect

The endothelium forms a selectively permeable barrier to thetransport of water and nutrients from the blood to other tissues ofthe body (Mehta and Malik, 2006; Tarbell, 2003). One interestingcharacteristic of this cell monolayer is that its permeability changesin response to environmental conditions. An increase in pressureacross the endothelium results in a transiently increased volume flowrate that gradually attenuates over a period of minutes to hours, evenwhen the pressure remains constant (DeMaio et al., 2004; Turner,1992). Although most observations of this effect have been madeon cell monolayers in vitro that are exposed to transmural flow forthe first time, there have been other observations of the sealingeffect in vivo where cells had been conditioned to transmural flow(Kim et al., 2005; Lever et al., 1992; Tedgui and Lever, 1984).

The tight junction is the principal modulator of the endothe-lium's barrier function. It takes the form of a band of proteins thatspan the cleft between adjacent cells, forming an impermeablejunctional strand along approximately 90% of the length of the

∗ Corresponding author.E-mail address: david@ccny.cuny.edu (D.S. Rumschitzki).

0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2009.05.016

intercellular junctions (Adamson et al., 2004) see Fig. 1. Tight junc-tion proteins form a nearly continuous strand that increases hy-draulic resistance—almost completely blocking the convective flowof water (Michel and Curry, 1999). It is primarily through gaps inthe junction strand that water flows. Endothelial hydraulic conduc-tivity (Lp), which quantifies its permeability to water, is the ratio oftransendothelial water flux to effective transendothelial pressure,

Lp = JV(�P − ���)

, (1)

where JV is the volumetric flux (not the flow rate, as in some otherworks) across the endothelium and�P(��) the hydrostatic (osmotic)pressure difference across the cell layer (Curry in Renkin et al., 1984).The reflection coefficient, �, which ranges from 0 to 1, accounts forthe difference between the transport through an ideal semiperme-able membrane and that through real experimental cell layers, whichin fact permit some solute to pass (Michel and Curry, 1999). The en-dothelial glycocalyx, a layer of long proteoglycan chains that coatsthe luminal surface of the endothelium, is believed to contributemainly to the exclusion of larger solutes, but also partially to thehydraulic resistance (1/Lp).

Two distinct protein complexes bind across the cleft betweenadjacent cells. The adherens protein complex keeps the width of

S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514 4505

Fig. 1. Schematic of the inter-endothelial cleft in a plane normal to the endothelial surface. A pressure gradient drives a fluid flow through the cleft from lumen (top)to sub-endothelial region. The tight junction (∼3nm wide) inhibits water flow. Adherens junctions (∼20nm wide) maintain the intercellular spacing. Occluding proteinstranslocate from the cell interior to its border. Adapted from Tarbell (2003).

the cleft at a fixed distance (order 20nm). But the tight junctioncomplex, typically found closer to the luminal surface, narrowsthe intracellular cleft to less than 3nm in width (Tarbell, 2003).Breaks in the tight junction strand allow the cleft to open to thefull width of the adherens junction, reducing local hydraulic re-sistance and increasing the convection of water and solutes there.Hu et al. (2000) observed that breaks in the tight junction, withan average length of 150nm, spaced approximately 4300nm apart,were consistent with an analytical description of flow through theendothelial cleft in frog mesentery (Hu et al., 2000) and later withbreaks of 315 at 3559nm spacing in rat mesentery (Adamson etal., 2004). In the case of an adaptive response to transmural pres-sure, it is the length and frequency of these breaks rather than thewidth of the cleft, that is thought to decrease (DeMaio et al., 2004).The response of the endothelium to flow by mechanotransductionhas been the subject of extensive study, and is reviewed in Davies(1995). It has been observed that, in response to an increase intransendothelial pressure, pre-existing stores of the tight junctionprotein zonula occludens (ZO)-1 and other proteins translocate tothe cell junction and reduce flow across the endothelium (DeMaioet al., 2004). DeMaio et al. (2004) found that disrupting cell cy-toplasm microtubules (MT) reduces the migration of ZO-1 and ofother proteins to the tight junction and consequent sealing. Shearstress introduced by transmural flow appears to initiate the dis-assembly and reassembly of adherens junctions, and results in anupregulation of diverse junctional proteins, indicating a continuousturnover (Noria et al., 1999). It is reasonable to assume that ZO-1is taking part in the closing of tight junction breaks, and that therate at which flow is attenuated may be proportional to the rate

at which this protein accumulates in the junctional interface andreduces total gap length; see Fig. 2.

Adamson et al. (2004) presented evidence that projections fromadjacent cells overlap each other in the area of the tight junction andthat this might also lower JV, but, as we shall see below, our model,which does not consider this effect that depends on an unknownjunction thickness profile, explains current data very well.

An early model to relate transendothelial flow to cell geometryis the slit-pore model, i.e., Poiseuille flow through a slit, which givesthe hydraulic conductivity in the cleft (Lpc) as

Lpc = Ltjw3

12�HfO, (2)

where Ltj is the total intercellular cleft length per unit vessel wallarea, w the width of the adherens junction, H the height of the cleftfrom the base of the cell to the lumen, � the viscosity of water, andfO the fraction of the cleft that is open: the break to total cleft length.Darcy's law describes the glycocalyx layer's hydraulic conductivity(LpEGL) as

LpEGL = Kp

h�, (3)

where Kp is the Darcy permeability and h the height of the glycocalyx.Since flow parallel to the endothelium in the glycocalyx is 2 orders ofmagnitude smaller than flow normal to it (Hu andWeinbaum, 1999),we neglect any flow through the glycocalyx other than above anunderlying tight junction break. Eqs. (2) and (3) combine as hydraulic

4506 S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514

Fig. 2. Top (en face) view of the endothelial cleft, showing a schematic of the theoretical arrangement of proteins forming the tight junction. Tight junctional proteinstransport from the cytoplasm to the junction. When the junction is formed, these proteins span the width of the cleft between cells, reducing the width w from 20nm toless than 3nm. The adherens junction (not visible) maintains a constant cleft width. As sealing progresses, the area available for flow changes as the length, not the width,of the break decreases.

resistances in series (L−1p = L−1

pc + L−1pEGL), or

Lp = 1�

(12HLtjw3 + h

Kp

)−1

fO. (4)

For additional details about the difference between the idealand experimentally-observed flow through the complex geom-etry and hemodyamic environment of the intercellular cleft see(Sugihara-Seki and Fu, 2005). Although there is general agreementon the basic organization of the junction (Fig. 1), the stoichiometryand kinetics of each step of junction formation are largely unknown.Moreover, the available data of water flux across the monolayervs. time is inadequate to model those steps. Instead, despite thenoted participation of microtubules in the transport of occludingprotein, we model this translocation as a one-dimensional (1-D),axisymmetric diffusive process from a disk-shaped cytoplasm (hav-ing an initially uniform protein concentration) to the cell periphery.We assume the kinetics of junction formation to be instantaneousrelative to the rate of protein translocation. The model's output isthe open slit fraction, fO as a function of time. Eq. (4) relates fO(t) toLp(t), which results in a decrease in water flux with time, as seenin Fig. 3. Finally, we integrate the results so as to compare directlywith the raw experimental data described below.

2. Experimental measurement of endothelial hydraulicconductivity

Endothelial monolayers are cultured to confluence on porouspolycarbonate filters, and placed in a fluid filled chamber with an up-per and lower compartment. The lower compartment is connectedto a precision glass capillary tube with an air bubble in it. The entireapparatus is maintained in at 37 ◦C, 5% CO2. Fig. 4 shows our sys-tem, an adaptation of Sill et al. (1995). A 10mmHg pressure differ-ence is introduced across the monolayer, causing fluid flow throughthe monolayer, into the lower compartment and pushing the bubblealong the tube. Bubble position is recorded for 20min before, dur-ing, and for 60min after the application of pressure. See AppendixA for a more detailed description of the setup.

Fig. 5 shows a typical bubble position vs. time curve. One convertsbubble displacement �x in time �t to water flux, JV, by multiplying

Fig. 3. Typical volume flow rate vs. time curves for bovine aortic endothelial cells(BAECs). Confluent monolayers grown on fibronectin-coated polyester filters with0.4�m pores are placed in a pressure-flow apparatus, and subjected to a 10 cm H2Otransmural pressure. A significant attenuation of the volume flow rate across thelayer occurs over 20–60min.

by the ratio, F = 0.02745, of bubble tracker tube to monolayer cross-sectional area:

JV = �x�t

F. (5)

To minimize error amplification via numerical differentiation, weshall integrate the model result below so as to directly compare withthe x(t) data.

3. Model development

Current studies of convective transport in cell culture do notaddress sealing; they only extract the long-time steady-state vol-ume flow rate (JV∞) from the data. By modeling the sealing process,we mimic the entire experimental data curve and, as a result, ex-tract two additional physical parameters related to monolayer in-tegrity and experimental preparation from the data that characterizesealing. These parameters provide statistical criteria for the evalu-ation of outlier experiments and a rational method for evaluating

S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514 4507

Fig. 4. Schematic of the pressure-flow apparatus. BAEC monolayers are culturedto confluence on porous polycarbonate filters and placed in a fluid filled chamberwith an upper and lower compartment. The lower compartment is connected toa glass capillary tube with an air bubble in it, followed by an albuminal reservoirthat is open to the atmosphere. Lowering the albuminal reservoir level below theluminal reservoir level applies a pressure across the monolayer; this drives waterthrough the monolayer and into the lower compartment, pushing the bubble alongthe tube. A photometer tracks bubble movement and records it in an electronicdata collection system. The entire apparatus is maintained in at 37 ◦C, 5% CO2.

Fig. 5. Typical raw data showing bubble position vs. time for BAEC monolayers.As sealing progresses, the bubble moves more slowly. After 30–60min, sealing iseffectively complete. Solid curve represents fit of model below.

different cell culture treatments. The latter evaluations are the sub-ject of a subsequent applications paper.

Fig. 6 shows a schematic of themodel. By abuse of geometry (sincecircles do not tile the plane) we describe the endothelial cell as a thin,axisymmetric disk adjoining neighboring cells along its perimeterin a two-dimensional periodic array. We do not consider gaps atthe 3-cell contact points. Although endothelial cells in vivo oftenhave an elongated shape, aligned with the direction of flow, cellscultured at zero pressure in vitro are close-packed in a characteristic,approximately hexagonal cobblestone pattern, with an aspect ratio(out of plane height, to in plane radius) ranging from 1:15 to 1:30.Since tight junctions form only at the cell periphery, we assume theoccluding proteins have no interaction with the apical and basal cellsurfaces. Although the cell has finite (z) thickness and the junction

Fig. 6. Idealized cell and junction geometry showing (A) an oblique projection, (B)an en face view, and (C) a lateral view. The break in the junction strand betweencells is presumed to be the pathway for water flow and solute transport. The tightjunction is modeled as a single strand of unit height (in the z direction). Radialdiffusion of protein from the cell interior to, and incorporation into, its border allowfor tight junction remodeling, resulting in flow attenuation.

strand is a thin protein band, we ignore any z-dependence and ourdependent variables depend only on radial position r and time t.

Tight junction assembly incorporates various pre-existing pro-teins, of which ZO-1 appears to be controlling (Anderson et al., 1989).Let C(r, t) be the concentration of free ZO-1 (or of a lumped speciesrepresenting all relevant diffusing junctional proteins) in the cell andD be the diffusivity of this species in the cytoplasm. The axisymmet-ric diffusion equation is

�C�t

= D

(�2

C

�r2+ 1

r�C�r

). (6)

We assume the protein instantaneously and irreversibly binds to thejunction at r = R upon arrival; C(r, 0) is bounded; and the occludingprotein concentration C0 is initially uniform (C0 unknown):

C(r = R, t) = 0, |C(r = 0, t)|<∞, C(r, 0) = C0. (7)

Nondimensionalizing as in (8) gives the nondimensional problem (9)and (10):

C̄ : = CC0

, R̄ : = rR, t̄ : = tD

R2, (8)

�C̄�t̄

= �2C̄

�r̄2+ 1

r̄�C̄�r̄

, (9)

C̄(r̄ = 1, t̄) = 0, C̄(r̄ = 0, t̄)<∞, C̄(r̄, 0) = 1, (10)

where nondimensional variables have ¯ . The flux, JP , of free proteininto the tight junction is

JP = −D�C�r

∣∣∣∣∣r=R

or J̄P : = RDC0

JP = − �C̄�r̄

∣∣∣∣∣r̄=1

. (11)

Let A∞ =�R2C0 be the total amount of intracellular protein availablefor sealing at the beginning of the experiment, and A(t̄) be the totalamount of protein entering the junction from time 0 to t. The fraction

4508 S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514

A(t̄)/A∞ (�1) incorporated into the junction at time t is

A(t̄)A∞

= A(t̄)�R2C0

= −2∫ t̄

0

�C̄(r̄, t̄′)�r̄

∣∣∣∣∣r̄=1

dt̄′. (12)

It contains no parameters, and we shall calculate it once and for all.In addition, we allow that some portion of the junction is already

sealed at t=0. This amount of protein, defined as B0, depends on theability of the confluent monolayer to form tight junctions withouttransmural flow, and is unknown. Therefore, the sealing effect startswith an amount B0 in the junction, and ends when the total amountof protein in the junction is B0 +A∞. The monolayer's instantaneousfraction of its maximum sealability per cell as a function of time isthen

fS(t̄) =(B0 + A(t̄)B0 + A∞

), (13)

and the unsealed fraction is

fO(t̄) =(A∞ − A(t̄)B0 + A∞

)= A∞

B0 + A∞

(1 − A(t̄)

A∞

). (14)

The experimental data is bubble position, x, vs. time, t. With theapplication of pressure, the bubble begins to move, and the bubbletracker picks up the moving bubble at position x0 at time t0. Mono-layer JV (t) is now a function of t. By analogy with Eq. (5)

x(t) − x0 = 1F

∫ t

t0JV (t′)dt′. (15)

JV(t) has both a dynamic sealing part, and a contribution JV∞ thatpersists at infinite time due to gaps in the junctions and other holesthat do not seal. Traditionally, it is only JV∞ that one extracts fromthese experiments. It would be useful to separate JV∞ into a mono-layer defect that is not associated with the junction strand and anunsealed tight junction part, but x(t) data alone cannot make thisdistinction; use of tracers too large to pass through unsealed tightjunctions might help achieve it. Eqs. (1) and (4) give JV (t)− JV∞, rep-resenting junctional sealing, as

JV (t) = �fO(t) + JV∞, (16)

� =(

�P − ����

)(12HLtjw3 + h

Kp

)−1

. (17)

Recall that �P is kept constant during the experiment and ��� isalso presumed time-invariant. The recent Michel–Weinbaum modelof a revised Starling principle (Hu and Weinbaum, 1999) predictsthat a transendothelial flow will, in general, create a time-dependentgradient in protein (albumin) concentration across the endothelialglycocalyx that can generate a significant osmotic force opposingtransmural flow, i.e.,�� in Eq. (17) may be time-dependent. A simpleorder-of-magnitude analysis (see Appendix B) shows that the timescale of its development is ∼5–50 s, far shorter than the 20–60minfor sealing. This estimate is consistent with (Michel, 1987)'s experi-ments showing that these osmotic forces reach steady state in intactmesentery within 2min. With Eq. (16), Eq. (15) for x(t) becomes

x(t) − x0 = 1F

∫ t

0(�fO(t) + JV∞)dt. (18)

Note that fO(t) refers only to the open fraction of the sealable cleft,which is zero at infinite time. Let x̄ : =x/R, J̄V : =JVR/D, and �̄ : =�R/D.Eqs. (14) and (18) yield

x̄(t) − x̄0 = 1F

[�̄(

A∞B0 + A∞

) (t̄ −

∫ t

0

A(t̄′)A∞

dt̄′)

+ J̄V∞ t̄]. (19)

The total amount of sealing protein is B0+A ∞ . Let � = B0/A ∞ bethe ratio of the initial amount of available sealing protein in the

junction to that inside the cell. If �̄−1?1, then the sealing effectwill be dramatic; if �̄−1>1, then the sealing effect will be small.The sealing potential (1+�)−1 = A ∞ /(B0+A ∞) measures how muchsealing occurs during the experiment. From (14), one can relate fO(t̄)to A(t̄)/A∞ and rewrite (19) as

fO(t̄) =(1 − A(t̄)/A∞

1 + �

), (20)

x̄(t) − x̄0 = 1F

[�̄

1 + �

(t̄ −

∫ t

0

A(t̄′)A∞

dt̄′)

+ J̄V∞ t̄]. (21)

4. Model parameters

Table 1 lists known physical and experimental model parame-ters. The transmural pressure �P is the same for all experimentsexcept where noted. With these numbers, our model predictsan unconstrained flow (junctions fully unsealed, i.e., fO = 1) of22.0×10−6 cms−1. This JV∞ value, about half that observed in ourexperiments, should correspond to the in vivo percent sealable gap,typically ∼90% of the total flux (Adamson et al., 2004). Only values ofw � 25nm fit experimental data well; all fits below use w = 25nm,corresponding to flows ranging from 5.4 to 54.0×10−6 cms−1.

5. Solution of the diffusion model equations (9), (12), and (19)

The separation of variables solution to Eq. (9) is

C̄(r̄, t̄) =∞∑n=1

2�n

e−�2nt̄J0(�nr̄)J1(�n)

, (22)

where �n is the nth eigenvalue (there is no zero eigenvalue) of theBessel function J0, satisfying

J0(�n) = 0; n = 1, 2, . . . . (23)

Eq. (12) for A(t)/A∞ and its integral become

A(t)A∞

=∞∑n=1

4

�2n

(1 − e−�2nt̄), (24)

∫ t̄

0

A(t′)A∞

dt̄′ =∞∑n=1

4

�4n

{�2nt̄ − (1 − e−�2

nt̄)}, (25)

allowing the solution of the bubble position Eq. (19) (Appendixworksout its limits):

x̄(t) − x̄0 = 1F

[�̄

1 + �

(t̄ −

∞∑n=1

4

�4n

{�2nt̄ − (1 − e−�2

nt̄)})

+ J̄V∞ t̄

]. (26)

The dimensional form of (26) is convenient for data fitting. For x̄0 ≡0, it is

x(t) = 1F

[{�

1 + �+JV∞

}t − �

1 + �

∞∑n=1

4

�4n

{�2nt−

R2

D(1 − e−�2

n(tD/R2))

}].

(27)

6. Algorithm development

The model was implemented using MATLAB� using its built-infunction fmincon, a steepest-descent approach to the optimizationof a constrained nonlinear multivariable function. It minimized theweighted residual sum of squares (SSE) of position data vs. modelprediction by fitting parameters D, �, and JV∞. A weighting func-tion (1 + e−t̄/0.3) was used to force a closer fit to the early, curvedportion of the data but give weights of approximately 1 in the non-curved region. A full description of fmincon is found in (MATLAB�

S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514 4509

Table 1Structural parameters of the intercellular junction from literature and experiment.

Parameter Source Value

R Cell radius BAEC in vivo (data not shown) 15�m�P Transmural pressure Known from experiment 9.80×103 dyn cm−2 (10 cm H2O)�� Osmotic pressure Calculated from experimental values 3.87×103 dyn cm−2 (10 cm H2O)Ltj Length of tight junction per unit area Calculated from model 2/RKd Darcy permeability of the glycocalyx Sugihara-Seki and Fu (2005) 6.10nm2

H Depth of tight junction Adamson et al. (2004) 411nmh Thickness of glycocalyx Zhang et al. (2006) 150nmw Width of adherens junction Adamson et al. (2004) for rat mesentery 18nmw* Width of adherens junction Predicted by curve fitting of the model 25nmwc Width of tight junction Tarbell (2003) 3nm� Viscosity of water at 37 ◦C Massey (1971) 7.544×10−3 dyn s cm−2

Fig. 7. Raw bubble position vs. time data, with fitted curve overlaid. Data(+ symbol) show a fairly consistent noise signal throughout. The best values foundare D = 2.261×10−8 cm2 s−1 and JV ∞ = 3.674×10−6 cms−1. D represents the early(t < 20min) curvature and JV ∞ the steady (t > 40min) slope. The sum of thesquares of the residual of fitted values against data (SSE) is 0.835 cms−1 for 410data points.

Optimization Toolbox technical reference). The (RMS) criterion fortruncation of the infinite sum in (27) was that the square root of theintegral of the square of the difference between a solution involvingthe first N terms and a solution involving the first 2N terms, dividedby the integral of the 2N term solution was less than 10−3 for a de-pendent variable on the order of 101. 100 terms fulfilled this crite-rion; we used Halley's method (von Winckel, 2005) to find the first100 zeros of the Bessel function J0(r). As a check, we implementeda Runge–Kutta solution to Eq. (9) with a finite difference mesh of100 points in r, and 1000 in t and a Simpson's rule integration ofEqs. (12) and (19). Solutions and the sum of squares error each agreedto 4 significant figures, but calculation time was an order of magni-tude longer.

7. Model validation

7.1. Data fitting

Figs 7–9 show typical examples of best-fit model curves (smoothlines) overlaid upon experimental data (points). The data curvesshow uniform noise throughout. Note the close agreement betweenmodel curves and data for both high and low curvatures, indepen-dent of the noise level. The sealing potential (1+�)−1 determines theinitial slope of the curve, D its rate of relaxation to a steady slope,JV∞, of the linear portion. The standard error is on the order of 10−6

on dependent variables of order 10−1.

Fig. 8. Typical data set with fitted curve for D = 2.498×10−8 cm2 s−1, an typical valuefor these curves, and a high JV ∞ = 6.1×10−6 cms−1 value. Data (+ symbol) show afairly consistent noise signal throughout. At t∼27min, a discontinuity in the datashows that the bubble tracker lost the position of the bubble for approximately1.5min; nevertheless, the model fit is unimpaired. SSE is 0.835 cms−1 for 398 datapoints.

Fig. 9. Data (+) set with very little curvature with fitted curve overlayed. Lowdata curvature yields a D = 0.4195×10−8 cm2 s−1 and sharp slope yields a highJV ∞ = 8.519×10−6 cms−1. SSE is 0.689 cms−1 for 415 data points.

7.2. Data conditioning

The model, and therefore its optimal fit to the data, requires pre-cise determination of the bubble position x0 at the time t0 of pressure

4510 S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514

Fig. 10. Typical figures for optimization of time t0 of bubble capture. Residual sumof squares (SSE) is calculated for a series of starting points before and after theapplication of pressure: (A) (SSE) (open points) drops sharply after the early high--noise region of data acquisition. Data point 0 is known from the raw experimentaldata to be the point at which pressure was applied. Fitting SSE to a numericalsloped step function (line) determines the first data point beyond the noisy region,in this example at approximately 2min, and SSE stabilizes by 5min and (B) for thetime adjustment found in A, the corresponding D is plotted against time. D dropsand stabilizes within 2min of the application of pressure. A drop in D is alwaysstrongly correlated with the drop in SSE.

application, where the sealing process is believed to begin. However,(x0, t0) is often not known to sufficient precision, since the trackermust first search before capturing the moving bubble, thereby intro-ducing a time lag between pressure application and accurate bub-ble position data recording. Erroneous values of D and � will resultboth from applying the initial condition at the wrong t0 and if datafrom before bubble capture is passed to the fitting algorithm. Theexperimental apparatus' difficulty in tracking the bubble positionat early times where flow is high has been noted before, and hasmotivated improved bubble-tracking methods (Hubert et al., 2006).Even if (x0, t0) were known exactly and the tracker instantaneouslycaptured the bubble at t0, there is early noise associated with, e.g.,the initial formation of a lubricating layer, upon initiation of bubblemotion.

To minimize errors associated with the unknown (x0, t0), we in-clude the optimum starting point (x0, t0) as two data conditioningparameters, and determine them by least squares fitting. To iden-tify the point (after t0) at which the bubble tracker has most likelycaptured the bubble, Fig. 10 A (open circles) plots the residual sumof squares of the best fit for a series of starting points before (nega-tive values) and after (positive values) the known experimental timeof pressure application (data point 0). When the tracker locks ontothe bubble, the error drops precipitously. The drop in SSE coincides

Table 2Fitted parameters for 7 experiments.

Sample number Diffusivity(×10−8 cm2 s−1)

JV ∞(×10−6 cms−1)

Sealing potential(1+�)−1

1 0.4195a 8.519b 0.10502 0.5089a 2.851 0.25973 2.261 3.674 0.21394 2.984a 5.518b 0.15345 2.521 6.084b 0.14126 2.498 6.080b 0.14137 2.240 1.380 0.4202

Mean 1.855 5.141 0.2049Std. dev. 0.9956 1.888 0.100195% confidence interval ± 0.6901 ± 1.308 ± 0.06938

BAEC cultured in identical growth conditions, to either confluence plus 4 days,(samples 1 and 2), or confluence plus one day (samples 3–7), on polycarbonatefilters and subjected to transmural flow in the pressure flow apparatus describedabove and, in more detail, in Sill et al. (1995).

aOutliers greater or less than 1.5 times the interquartile range in diffusivity.bOutliers with Lp less than 5×10−7 cms−1.

with a stabilization of D shown in Fig. 10B (open circles). Note that,after their drops, both curves remain flat in a neighborhood of thedrop, indicating that the fits are insensitive to the precise start ofdata collection after the tracker has locked onto the bubble. We fitthese results to a numerical step function (Fig. 10, lines) to find thedata index that corresponds to the first point of data retained.

8. Results and discussion

Table 2 lists fitted parameters for 7 experiments for BAECs platedon fibronectin-coated polycarbonate filters and tested as describedabove.

Our model is based on the assumption that sealing is the resultof the diffusion of sealing protein. Therefore, the diffusivity found byfitting the data should be essentially the same for all cells of simi-lar type and culture conditions. It is not easy to tell by eye whethertwo curves have similar shapes when they vary in amplitude andtime scale. Samples 1 and 2 were grown for four additional days postconfluence; samples 3–7 were grown to only one day post conflu-ence. All other culture conditions were identical. Data for samples1, 3, 5 and 6 were collected during one experimental session overa 4-h period. Data from samples 2, 4, and 7 were collected duringanother 4-h experimental session on a different day. Note the simi-larity in our fitted D between samples 1 and 2, and between samples3–7. The data show a low standard error within cell treatments, andbetween experimental runs. The nondimensional data for all sevencurves collapse to a single curve in Fig. 11, showing that our modelprediction of the sealing behavior adequately describes all of theseexperiments, even for varying conditions.

9. Comparison with previous results

Since this is the first model to describe sealing from first prin-ciples, Table 3 compares our model's fitted parameter Lp ∞ (by (1),equivalent to JV ∞ ) to previously published results.

Our Lp ∞ values are in line with previous results. Our D's arealso in general agreement with diffusion velocities found by otherexperiments. Movement of protein in the cell is reported in a varietyof ways: diffusivity, speed of translocation, and speed of cell elonga-tion. To compare our D values with published results for various pro-teins in cells, Table 4 converts reported measurements to equivalentD's. Its values range from that of free diffusion, order 10−7 cm2 s−1,to motor protein transport, order 10−10 cm2 s−1. Our values are

S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514 4511

Fig. 11. Experimental flow rate vs. time curves for 7 experiments. BAEC monolayerscultured in normal 5mM glucose. Low outliers in diffusivity are shown with whiteboxes. High outlier in diffusivity is shown with shaded boxes. Other samples arewith open circles.

Table 3Comparison of our fitted Lp ∞ values to previous studies of flow across BAECmonolayers.

Current work Cell type Lp ∞ (cms−1)

Theoretical prediction (10% tight junction break) 5.4×10−7

Experiment (average for normal cells) BAEC 5.3×10−7

Previous workCancel et al. (2007) BAEC 5.4×10−7

DeMaio et al. (2004) BAEC 8×10−7

Dull et al. (1991) BAEC 5×10−7

Turner (1992) BAEC 36.7×10−7

Results reported as JV ∞ , values were converted to Lp ∞ by Eq. (1).

in the physiological range, typically about 3–4 times smaller thanactin diffusion in the cytoplasm, close to the elongation rate of thepseudopodia in a migrating cell and much larger than motor pro-tein transport. As noted, we do not consider some of the morecomplex geometries found in nature, e.g., Adamson et al's. (2004)evidence that projections from adjacent cells overlap each othernear, and thus lengthening, the tight junction. The rate of sealing isvery close to (3–4 times slower than) that of actin polymerizationin an elongating cell. As such, cell-cell overlap may be relevant.If elongation of these projections, like growing pseudopodia, areinvolved in the sealing process, then the parameter Ltj(�, t), � theazimuthal angle, in Eq. (4) would be time dependent rather than auniform constant Ltj. Since the present model explains the availablesealing data adequately (with only three physically meaningful pa-rameters), the development of a model with an unknown and, asyet unmeasurable time-dependent function to determine the dis-tribution and relative state of overlaying projections between cells(even if azimuthally averaged) must await the development of moresophisticated methods of sealing data acquisition.

10. Statistical issues and the rational identification of outliersin the data

Two statistical problems plague this type of experiment andimpact the differentiation of outlier data due to some unknownexperimental artifact from a valid representation of a biologicaleffect. They are the problem of heteroskedascity (when the sample

Table 4Comparison of our fitted values of D against D values of various protein systems inthe cell.

Current predictions Diffusivity

Model results 2.50×10−8 cm2 s−1

Free diffusion in mediaZhang et al. (2006) 9.29×10−7 cm2 s−1 (albumin in lumen)

Speed of microtubule elongationWalker (1991) 6.25×10−7 cm2 s−1 (microtubules in vivo)

Speed of cell migrationPickering (1997) 13.5 ± 3.6×10−8 cm2 s−1 (fibroblast)Yao (2007) 6.9×10−8 cm2 s−1 (BAEC)Osborne (2006) 3.75 ± 1.6×10−8 cm2 s−1 (BAEC under shear)

Free diffusion in the cytoplasmMcGrath et al. (1998) 5.8 ± 1.2×10−8 cm2 s−1 (actin)Waharte et al. (2005) 15×10−8 cm2 s−1 (actin)Waharte et al. (2005) 4×10−8 cm2 s−1 (myosin-1)

Diffusion by motor protein transportKalchishkova and Bohm (2008) 1.94×10−9 cm2 s−1 (no load)Shtridelman et al. (2008) 1.67×10−9 cm2 s−1 (under load)

Free diffusion in the plasmamembrane

Wang et al. (1999) 2.16×10−10 cm2 s−1

The speed of cell migration is typically presumed to be proportional to actin cy-toskeletal elongation. Where published results are reported as a velocity (cell mi-gration, motor protein transport), or a rate of elongation (microtubules), the Tablehas converted them to diffusivity D using the relation 〈x2〉/t = 4 ∗ D, where 〈x2〉is the mean square displacement (distance) and t is time. Four is the constant ofproportionality for two-dimensional diffusion.

variance does not decrease with increasing number of observations),and skewness, since the lower bound for JV ∞ is zero, and the stan-dard error is often of the same order as the experimental variable.Suttorp et al. (1988) and Dull et al. (1991) used a threshold cri-terion to eliminate samples with a hydraulic conductivity abovea pre-determined limit. This upper hyperpermeability (for JV ∞ )limit is arbitrary and may dismiss valid data that is high due toexperimental treatment.

An alternativemethod is to label as outliers those samples that are1.5 times the interquartile range greater or lesser than the mean ofone of the fitted parameters (Crawley, 2005). The interquartile rangeis the difference between the first and third quartile, which, for D, is0.974×10−8 cm2 s−1, and the mean is 1.84×10−8 cm2 s−1. So, outliersare those with D> 2.83×10−8 cm2 s−1 or D< 0.881×10−8 cm2 s−1. Ifwe consider only JV�, by this method, then there are no outliers, butthe JV vs. t curves in Fig. 12 clearly show differences in diffusivity(as shown by curvature).

Table 5 compares the two different methods of evaluating out-liers among the seven data sets presented in Table 2. Applying the in-terquartile rule, one sees that samples 1 and 2, are low outliers and 4is a high outlier for diffusivity. Applying the thresholding rule to iden-tify outliers as monolayers with an Lp∞ higher than 5.0×10−7 cms−1,we see that samples 1, 4, 5, and 6 are outliers in JV ∞ . Removing Doutliers from the data set reduces the standard deviation for JV ∞and D by 48% and 73%, respectively, while increasing their means by3.7% and 37%, respectively. Removing outliers by thresholding hasalmost no effect on the standard deviation for D but lowers that ofJV∞ by 38% . The thresholding method lowers the mean for D by 10%and for JV∞ by 49%. Whether samples are comparable as part of abiologically similar group, or should be excluded, can only be deter-mined by additional experimental considerations, but application ofstatistical methods to the fitted parameters D (and �) other than JV∞provide simple criteria to characterize similar and dissimilar exper-imental treatments.

4512 S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514

Fig. 12. Dimensionless JV vs. time curves. Dimensionless flux is found by subtractingJV ∞ then normalizing by �/(1+�). Dimensionless time is tD/R2. Collapse of all curvesto the theoretical curve demonstrates that the simple diffusion with fast junctionincorporation model adequately models data from monolayers subject to varyingpreparations. Statistical outliers in D are shown with square markers (white boxes,low outliers, gray boxes high outlier). Other series are shown with round markers.

Table 5Comparison of methods of identifying outliers.

Diffusivity(×10−8 cm2 s−1)

JV ∞(×10−6 cms−1)

Sealing potential(1+�)−1

Entire data set (7 experiments)Mean 1.86 5.14 0.205Std. dev. 1.00 1.89 0.10095% confidence interval ± 0.69 ± 1.3 ± 0.069

After removal of outliers by JV ∞ threshold < 5.0×10−7 cms−1

Mean 1.67 2.64 0.275Std. dev. 1.01 1.16 0.1095% confidence interval ± 1.1 ± 1.3 ± 0.098

After removal of outliers in diffusivityMean 2.54 5.33 0.224Std. dev. 0.268 0.988 0.11595% confidenceinterval

± 0.23 ± 0.87 ± 0.10

Data are mean, standard deviation, and 95% confidence interval of fitted parametersfor the 7 experiments in Table 2.3. Using the statistical criterion, we identify outliersin D as those that are above 2.83×10−8 cm2 s−1 or below 0.881×10−8 cm2 s−1. Forthe threshold method, we defined outliers in JV ∞ as those with JV ∞ greater than5.0×10−6 cms−1. After removal of outliers in D samples 1, 2, and 4, the 95% confidenceinterval of the true value of the mean was reduced by 33.7% for JV ∞ and 66.0% forD. Removal of outliers by thresholding JV ∞ , samples 1, 4, and 5, increased the 95%confidence interval for D by 64.8%, and increased the 95% confidence interval forJV ∞ by 40.5%. The mean and standard deviation of the sealing potential increasedby both methods, but were not statistically different from one method to the other(p > 0.14).

11. Conclusions

We have presented the first attempt at a biologically relevantmodel of the sealing effect. Our model's hypothesis is that the pro-tein diffusion to and rapid incorporation into, the inter cellular junc-tion cause the attenuation of pressure-driven water flow across BAECmonolayers. The model derives an expression for the bubble motionvs. time that one can directly compare with the data, eliminating thenoise-amplification of numerical differentiation used in previous ex-tractions of JV∞ from similar data. The model extracts two physically

meaningful parameters, in addition to JV∞, from data. One �, charac-terizes the initial state of the junction, and thus JV at the start of theexperiment and the other, the protein diffusivity, D, derives from thecurvature of the raw data of bubble position vs. time, characterizesJ′Vs its change over the course of the experiment. The fitted parame-ters are user-independent, reducing the need for subjective tests onthe part of the experimentalist. Statistical use of these parametersreduces the sample variance of the most important derived param-eter JV∞. Model results accurately reproduce literature JV∞ values.

Given that our mathematical model is drawn from a simpletheory of intracellular mechanics, the extracted parameters directlyrelate to physical elements of the cell or of the experimental en-vironment. Motor protein transport, which we represent by aneffective diffusion, depends upon the availability of motor proteinsand the presence of intact cytoskeletal elements appropriate fortransport. The sealing potential depends upon the concentration ofavailable sealing proteins in the cytoplasm and in the junction at thestart of the experiment. One can change other physical parameters,such as the cell radius R, fluid viscosity �, protein reflection coeffi-cient �, and the osmotic pressure ��, by conducting experimentswith different cell types and media conditions. Media conditionsmay also be changed as a function of time, providing yet anotheravenue for investigation.

Clearly cells with pathological effective diffusivity will exhibitpathological sealing. What constitutes pathological diffusivity withinthe cytoplasm is a field ripe for speculation. The activity of motorproteins in concert with microtubules can now be manipulated withsome precision (Drake and Pack, 2008), allowing quantitative inves-tigation of the role of this system in maintaining appropriate mono-layer sealing. Wu et al. (2007) report that altering of MT stability bynocodozole (depolymerization) and taxol (hyperstabilization) bothcause an increase in endothelial permeability. However, their obser-vation of the state of the cytoskeleton cannot distinguish whetherthis effect is from the inhibition of protein diffusion or from grossmorphological disorder. Our subsequent application paper will usethis model to characterize D of junctional proteins under these var-ious conditions.

Recent studies by our group have shown that a fraction of thetransmural water flow appears to be transcellular, occurring throughthe aquaporin-1 (AQP-1) membrane water channel, a route that isnot associated with the tight junction. Our application paper willinvestigate whether chemical or genetic reduction of flow throughAQP-1 reduces JV ∞ without changing D or the sealing potential. Ina second test, we will conduct experiments to see if fixation of theplasma membrane changes D and/or the sealing potential withoutchanging JV ∞ .

Notation

C concentration, mmolmL−1

D diffusivity of protein, cm2 s−1

fO fraction of gap length in total tight junction lengthh height of endothelial glycocalyx, nmH depth of intercellular junction, nmJ0 Bessel function of the first kind, order 0J1 Bessel function of the first kind, order 1JP protein flux, mmol s−1 cm2

JV water flux, cms−1

Kd Darcy permeability of the glycocalyx, nm2

Lp hydraulic conductivity, cms−1 · cmH2O−1

Ltj length of tight junction per unit area, �m−1

P hydrostatic pressure, dyn cm−2

R cell radius, �mSA surface area, cm2

W width of intercellular cleft, nm

S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514 4513

Greek letters

�n the nth zero of the Bessel function J0� fluid viscosity, dyn s cm−2

� osmotic pressure, dyn cm−2

� reflection coefficient

Subscripts and superscripts

c endothelial cleftEGL endothelial glycocalyx layertj tight junction

Acknowledgments

JMT would like to acknowledge Professor Morton Denn's sup-port throughout his career and to point out that Professor Denn firstintroduced him to mathematical modeling in a graduate course atthe University of Delaware. DR would like to acknowledge Profes-sor Morton Denn's vast contributions to Chemical Engineering andthank him for his guidance as a Professor at Berkeley and as a seniorcolleague at CCNY. During his graduate research, SR was a memberof the IGERT program at The Levich Institute at City College underDr. Denn, where they discussed an early formulation of the seal-ing effect problem. This work was partially funded by the NationalInstitutes of Health Grants #1RO1HL67383 to DR (for SR and DSR)and #R01HL57093 to JMT (for LMC and JMT), and by the BenjaminLevich Institute of Physiochemical Hydrodynamics NSF IGERT grantin Soft Materials and Multi-Scale Phenomena #DGE-0221589, Pro-fessor Morton Denn, PI. Additional funding to SR was provided bythe CUNY Office of Student Research and the Barbara Slifka CUNYDissertation Year Fellowship.

Appendix A. Detailed description of experimental apparatus andconsiderations

The first experiments to determine the hydraulic conductivityof the endothelium were those carried out by Landis (1927) on mi-crovessels, and later improved upon by Smaje et al. (1970), Zweifach(1971), and Michel et al. (1974), by tracking the movement of redblood cells in occluded vessels to measure transendothelial out-flow. Although simple in concept, system complexity necessitatedmathematical modeling to account for capillary distensibility andother storage and resistive effects in order to extract meaningful pa-rameters from the data. Cell culture models for the same purpose(Baetscher and Brune, 1983; Tarbell et al., 1988; Suttorp et al., 1988;Turner, 1992; Dull et al., 1991) greatly simplified such measure-ments. They introduced a flow marker (an air bubble) into a separateimpermeable chamber (a calibrated capillary tube) in-stream withthe experimental cell monolayer; this ensured that marker move-ment resulted only from volume flow across the monolayer. Sill et al.(1995) added automated bubble-tracking components and comput-erized data acquisition; that paper contains a more comprehensivedescription of our complete apparatus and method. Nguyen (2008)adapted this automatic bubble tracker system to ex vivo measure-ments of Lp of whole vessel explants. More recent improvements(Hubert et al., 2006) incorporate a live video camera to record bubbleposition and integrate a proprietary-platform-based data processingalgorithm to reduce errors in data acquisition and processing.

Ideally, bubble movement is continuous and smooth, and bubbleposition x increases monotonically with t. However, slight changesin ambient temperature, environmental vibrations, etc., can all havea small effect on the progress of the bubble, resulting in noisydata. At low bubble velocities, rupture of the lubricating liquid film

surrounding the bubble can create a three-phase contact line onthe tube wall. The resulting contact line forces can lead to bubblesticking and lurching once a sufficient pressure to induce rewettingbuilds up behind the bubble. In addition, as the flow attenuates,the movement of the bubble slows, causing the signal to noise ratioto decrease. The numerical approximation of the volume flow ratefrom noisy marker position data is the largest source of error.

Appendix B. Theoretical considerations

B.1. Limiting cases of the solution, Eq. (23)

The limit as t→ ∞ of the time derivative of Eq. (27) is

limt→∞

dx(t)dt

= 1F

[JV∞ + �

1 + �

{1 −

∞∑n=1

4

�4n

(�2n − �2

ne−�2

n(tD/R2))

}]

= 1FJV∞. (28)

A(t̄)/A∞ → 1 as t→ ∞ and (12) imply the sum∑∞

n=14/�2n = 1. Its

value at t = 0 is

dxdt

∣∣∣∣t=0

= 1F

[JV∞ + �

1 + �

{1 −

∞∑n=1

4

�4n

(�2n − �2

n)

}]

= 1F

[JV∞ + �

1 + �

]. (29)

B.2. Time dependent changes in pressure due to osmotic forcesopposing filtration

To take into account the Michel–Weinbaum model, writeEq. (16) as

JV (t) − JV∞ = �′ (�P − ���(t))fO(t); �′ = �/(�P − ��n). (30)

The initial concentrations are CJ0 = CL in the junction and CL in thelumen. We now estimate the time �t until the junction concentra-tion reaches its steady state value CJSS. At experimental pressure inour in vitro device, we assume that the primary method of albumintransport is by convection, which is fast relative to diffusion, andthat the final concentration in the junction will be the convectivelimit CJSS =CL(1−�EGL), as defined in Zhang et al. (2006), where �EGLis the reflection coefficient of albumin in the glycocalyx. The solutetransport flux, JS, is related to JV by

JS = JV (1 − �)CL. (31)

An integral (from t = 0 to�t) solute mass balance over the glycocalyxlayer gives

(CJSS − CJ0)HA =∫ �t

0(JS(t′)|IN − JS(t′)|OUT )dt′A, (32)

where H is the junction height and A = 1 is a unit area. Inserting(31), CJ0 and CJSS into (32) gives

�EGLCLH =∫ �t

0JV (t′)[(1 − �TJ)CJ − (1 − �EGL)CL]dt′. (33)

By the integral mean value theorem (〈•〉 represents the mean value),

�EGLCLH = [−CL〈JV (t)〉(1 − �EGL) + 〈CJJV 〉(1 − �TJ)]�t. (34)

4514 S. Russell et al. / Chemical Engineering Science 64 (2009) 4504 -- 4514

We approximate 〈JVCJ〉<JV > [(CJ0 + CJSS)/2] and 〈JV 〉 ≈ JV-min. Thisgives the �t estimate

�t <H�EGL

[(1 − �TJ)(1 − �EGL/2) − (1 − �EGL)]JV-min. (35)

Substituting known parameters from the 1-D model, and our exper-imental value for JV-min gives

�t <4.11 × 10−5 cm · 0.95

[(1 − 0.197)(1 − 0.95/2) − (1 − 0.95)]10−6 cm/s≈ 105 s (36)

much shorter than the time for sealing (20–120min). Thus ourmodeltreats �� as a constant.

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