A hydrodynamic mechanosensory hypothesis for brush border
microvilli
P.
Guo1,
A. M.
Weinstein2, and
S.
Weinbaum1
1 CUNY Graduate School and Center for Biomedical
Engineering, the City College of the City University of New York, New
York 10031; and 2 Department of Physiology, Weill Medical
College of Cornell University, New York, New York 10021
 |
ABSTRACT |
In the proximal tubule of the kidney, Na+ and
HCO3
reabsorption vary proportionally with changes in
axial flow rate. This feature is a critical component of
glomerulotubular balance, but the basic mechanism by which the tubule
epithelial cells sense axial flow remains unexplained. We propose that
the microvilli, which constitute the brush border, are physically
suitable to act as a mechanosensor of fluid flow. To examine this
hypothesis quantitatively, we have developed an elastohydrodynamic
model to predict the forces and torques along each microvillus and its resulting elastic bending deformation. This model indicates that: 1) the spacing of the microvilli is so dense that there is
virtually no axial velocity within the brush border and that drag
forces on the microvilli are at least 200 times greater than the shear force on the cell's apical membrane at the base of the microvilli; 2) of the total drag on a 2.5-µm microvillus, 74% appears
within 0.2 µm from the tip; and 3) assuming that the
structural strength of the microvillus derives from its axial actin
filaments, then a luminal fluid flow of 30 nl/min produces a deflection
of the microvillus tip which varies from about 1 to 5% of its 90-nm
diameter, depending on the microvilli length. The microvilli thus
appear as a set of stiff bristles, in a configuration in which changes in drag will produce maximal torque.
glomerulotubular balance; mechanosensory mechanism; actin
cytoskeleton; microvilli force and torque
 |
INTRODUCTION |
PERHAPS THE MOST
IMPORTANT characteristic of solute and water transport in the
proximal tubule is the observation that reabsorption varies
proportionally with delivered load, i.e., with the glomerular filtration rate (GFR) (14). In large measure this
"glomerulotubular balance" derives from a "perfusion-absorption
balance," that is, the capability of the proximal tubule epithelial
cells to sense changes in luminal flow rate and translate this signal
into changes in volume reabsorption (53). This system is
remarkably precise, since the fraction of filtered fluid reabsorbed by
the proximal tubule is nearly constant over the entire physiological
range of flow (48). In this report, we shall propose a new
role for the microvilli as a mechanosensory system, which not
only senses fluid shear and drag forces, but has the capability of
greatly amplifying the mechanical stresses that are felt on the
intracellular cytoskeleton. To quantitatively explore this hypothesis,
we shall develop a mathematical model to predict the hydrodynamic
forces and torques on the microvilli, their distribution along the
length of the microvillus, and the bending deformation of the F-actin microfilaments in the microvilli due to this hydrodynamic loading. These hydrodynamic forces and torques will also be related to the
flow-dependent spacing of the microvilli that has been measured (36).
The efferent limb of the control of proximal tubule fluid reabsorption
appears to be reasonably well established. Volume reabsorption is
driven by Na+ reabsorption, and this reabsorption rate is
determined by Na+ entry across the luminal membrane in
which the Na+/H+ antiporter is the most
important entry step (50). The afferent limb, namely the
mechanism by which the proximal tubule senses axial flow rate, is
unknown. In this regard attention has been focused on the brush
border of proximal tubule cells, since this is the interface with the
tubule lumen. These 2- to 3-µm, densely packed luminal projections
not only greatly amplify luminal membrane surface area
(52), they define a region near the luminal cell surface,
whose geometry can change with changes in perfusion conditions (36). A number of workers have considered the possibility
that changes in axial flow might produce changes in solute ion
concentration within the brush border region and that such ion
concentration changes could somehow be sensed by the cell
(3). However, all model calculations attempting to
estimate such solute gradients have indicated that this diffusion is so
rapid as to preclude any significant concentration difference within
the brush border (27).
One possibility, which has not been previously proposed or
quantitatively explored, is that the microvilli can serve a
mechanosensory role in the afferent limb of proximal tubule
perfusion-absorption balance. This possibility is suggested by the
axial cytoskeletal structure of the microvilli reported by Hassen and
Hermann (20). Subsequent studies (44, 37)
showed that each microvillus contains a distributed array of 6-10
long axial microfilaments of 7 nm diameter, which immunocytochemical
analysis showed were F-actin microfilaments (5). These
filaments, we hypothesize, would provide a structural rigidity to the
microvilli that could enable them to resist bending when subject to
hydrodynamic forces and thus be capable of serving as a
mechanotransducer, much like the hair cells in the inner ear.
A logical mechanosensory candidate for most cells subject to fluid flow
is fluid shear stress. There is an extensive literature on the
biochemical and ultrastructural effects of fluid shear on cells since
the study by Dewey et al. (11) first demonstrated that
vascular endothelial cells could be grown in culture and exposed to
fluid shear under carefully controlled conditions. A wide variety of
ultrastructural and intracellular biochemical responses to fluid shear
have been documented involving Ca2+, second messengers, and
NO release (10). Vascular endothelial cells are exposed to
fluid shear stresses that are typically 10-20 dyn/cm2
on the arterial side of the circulation and 1-2
dyn/cm2 on the venous side. If the surface of the brush
border cells did not have microvilli, then the fluid shear stress would
vary from about 1 to 5 dyn/cm2 for Poiseuille flow over the
normal range of flow rates, which lies in the same range as for
vascular endothelium. However, the presence of the microvilli produces
a unique flow structure in the vicinity of the brush border, wherein
the fluid shear stress at the base of the microvilli is several hundred
times smaller than the drag forces on the microvilli themselves. The
dynamics of this fluid flow and the forces generated by both the fluid shear stresses acting near the tips of the microvilli and the axial
flow past the main body of the microvillus will be examined. In
particular, we are interested in the force and torque (bending moment)
distribution on each microvillus, and we hypothesize that it is the
bending moment produced by these forces that acts as a mechanical
transducer for the sensing of the axial fluid flow along the tubule.
These hydrodynamic forces will then be used as input into an elastic
model for the bending deformation of the actin cytoskeleton of the
microvillus to predict the displacement of the microvilli tips. We
speculate that this bending moment is transmitted to the cytoskeleton
in the cell interior, where it is converted into a biochemical response.
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METHODS |
Ultrastructural Model
The idealized ultrastructural model for the proximal
tubule is sketched at the cellular and tubular cross-sectional scales in Fig. 1. Figure 1A is a
schematic of a proximal tubule cell from an S2 segment in the distal
part of the convoluted tubule or the beginning of the straight segment
(35). It is in this region that Maunsbach et al.
(36) examined the effect of flow on proximal tubule
ultrastructure. The cells in this segment are characterized by densely
spaced microvilli, numerous mitochondria, and extensive interdigitation
of the lateral membrane near the basal surface. The microvilli are even
more dense and longer in the S1 segment and significantly less dense in
the S3 segment, which comprises most of the straight tubule. Of
particular interest in this study is how this difference in
ultrastructure affects the flow, forces, and torques on the
microvilli. Figure 1B is our macroscopic model of the entire
tubule shown in transverse cross section. The luminal radius of the
tubule excluding the microvilli is RL, and the
height of the brush border is L. The total radius of the
tubule, R0, is thus RL + L. From a hydrodynamic viewpoint the fluid flow is subtle,
since the microvilli form a closely spaced array in which there is a
small, but nonnegligible axial bulk flow through the microvilli, which
is driven by the axial pressure gradient in the tubule lumen. In
addition, there are two thin interaction layers, one near the
microvilli tips and one at the base of the microvilli, where the fluid
flow must adjust to satisfy the no-slip boundary conditions at the
apical membrane of the proximal tubule cell in Fig. 1A or
the fluid shear matching condition at the microvilli tips. The
interaction layer near the base of the microvilli determines the fluid
shearing stress on the apical membrane of the proximal tubule cells and the interaction layer near the tips of the microvilli determines the
forces on the tips and, as we show later, provides the dominant contribution to the hydrodynamic torque on the microvillus. A central
question in determining the torque is the drag distribution. In
essence, one wishes to determine how the small forces acting over most
of the length of the microvilli due to the pressure-driven bulk flow
compared with the much larger drag forces that are felt in the highly
localized region near the microvilli tips.

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Fig. 1.
A: schematic diagram of an epithelial cell in
proximal tubule S2 segment. The cells in this segment are characterized
by densely spaced microvilli (a), numerous mitochondria
(b), and extensive interdigitation of the lateral membrane
near the basal surface (c). B: idealized
mathematical model of tubule cross-sectional geometry, with brush
border of thickness L.
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|
Figure 2A is an enlarged
sketch of the transverse section of the brush border showing the
cross-sectional geometry of the microvilli. Ultrastructural studies
have shown that the microvilli form a closely spaced hexagonal array
(37) where the open gap,
, between microvilli can vary
significantly with fluid flow rate (36) and as previously
noted with location on the S1, S2, and S3 segments. This hexagonal
array will be further subdivided into repetitive periodic units, each
containing the equivalent of a single microvillus, as shown in Fig.
2B. Ultrastructural studies (20, 44, 37) have
shown the existence of longitudinal actin microfilaments of 6-7 nm
diameter extending the length of the microvillus but no microtubules or
intermediate filaments. In our model we assume that these actin
filaments provide a structural rigidity that prevents the microvilli
from deforming significantly during flow from a hydrodynamic
standpoint. By this we mean that the microvilli can undergo small
deformations in which the axial microfilaments might serve as strain
transducers but that these small strains do not significantly affect
spacing of the microvilli and hence the fluid dynamic forces
and torques acting on them.

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Fig. 2.
A: idealized model of brush border in
transverse section showing hexagonal microvillus array. B:
repetitive periodic unit of the hexagonal microvillus array. Note that
this unit contains the equivalent of one microvillus.
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Mathematical Model
The hydrodynamic problem for the flow past the microvilli tips is
a classic unsolved problem in the fluid mechanics literature. Rigorous
hydrodynamic solutions have been obtained for the two-dimensional Stokes flow past periodic fiber arrays which are infinite in extent. One of the best known of these solutions, that of Sangani and Acrivos
(46), describes the flow transverse to an infinite
hexagonal array of circular fibers whose cross-sectional geometry is
shown in Fig. 2A. The more difficult problem, which we wish
to analyze in the present study, is the three-dimensional flow in the
transition region in the vicinity of the microvilli tips. In
particular, we wish to examine the flow transition that occurs between
the shear flow in the tubule lumen and the fiber-containing region where the flow asymptotically approaches the behavior described by the
Sangani and Acrivos solution. This region is of greatest concern to us
since it is the primary determinant of the forces and torques on the microvilli.
There have been several prior studies that have examined related
interface problems. The one that is the closest to the present study is
the model proposed by Mokady et al. (38) for the shear flow over an endothelial glycocalyx. In this model the endothelial glycocalyx is treated as a layer of porous matrix whose fibers are
arranged in a periodic horizontal square array parallel to the
underlying solid boundary and transverse to the flow direction. A
numerical technique developed in Larson and Higdon (30) is used to solve the Stokes equations for viscous flow around each fiber
and the decay in the average horizontal velocity as one penetrates the
matrix layer. This problem differs from the present one in several
fundamental aspects. In our problem the microvilli (the fiber array in
Fig. 1B) are vertical rather than horizontal, the flow is
confined in a circular cylinder rather than unbounded, the fiber
fraction is much greater (this is typically 0.01 to 0.02 for the
endothelial glycocalyx), and the interface is cylindrical rather than
planar. Also, our fiber array is hexagonal rather than square, but this
adds no further difficulty since the solutions in Sangani and Acrivos
(46) for the infinite fiber array describe both fiber geometries.
The most important difference cited above is the fiber orientation. The
force on a horizontal fiber transverse to the flow direction is
uniform, whereas in our problem the most important feature is the
variation of the force along the microvillus length. This type of
problem has not been treated previously, to our knowledge. The presence
of a confining cylindrical boundary at R = R0 creates an axial pressure gradient that
drives a small, but important, flow across the microvilli. The
magnitude of this flow determines the relative importance of the drag
forces over the main body of the microvillus as opposed to the drag
force on the microvilli tips in the interface region described above.
The problem for the detailed flow around the horizontal fibers in
Mokady et al. (38) is two-dimensional, in contrast to the
present flow geometry, which is fully three-dimensional. We thus seek a
simplified solution approach in which we try to avoid the necessity for
obtaining detailed solutions for the three-dimensional velocity field
surrounding each microvillus. The key to this simplification is to
devise an approach that first describes just the decay in the average velocity with distance from the microvillus tip and then to use this
solution in a model that describes the local fluid flow around and the
local variation of the axial force along the microvillus.
The simplified solution approach that we have adopted combines
effective medium theory (Brinkman equation) and the rigorous hydrodynamic solutions in Sangani and Acrivos for Stokes flow past the
infinite hexagonal fiber array. We first consider the solution to the
axisymmetric flow problem sketched in Fig. 1B. This flow is
divided into two regions, a core flow and a flow through the fiber
(microvillus) array sketched in Fig. 2A. The flow in the
core region or lumen is described by the Navier-Stokes equation for
unidirectional axial flow in a circular cylinder. For this flow the
inertial terms vanish, and the simplified equation can be
written in axisymmetric (R, Z) coordinates as
|
(1)
|
where Uc is the fluid velocity in the core,
and µ is the fluid viscosity. In the margin region
RL < R < R0 with microvilli, we use a Brinkman equation
for flow through a porous medium (6)
|
(2)
|
Here Um is the local average axial velocity
in the microvillus array, and Kp is the
two-dimensional Darcy permeability for an infinite fiber array. The
expression for Kp will be derived shortly using
the solution for the drag coefficient for an individual fiber in
Sangani and Acrivos (46). Equation 2 differs
from Eq. 1 in that it contains a distributed body force or
Darcy term, µUm/Kp, due
to the drag on the microvilli. The second term in Eq. 2 is
the same as the viscous term in Eq. 1. This term is only important near boundaries or interfaces where the gradient in the axial
velocity is large. In the region removed from boundaries, the Darcy
term will dominate, except in the limit where the microvilli are
sparse. Both the Stokes and Brinkman equations are linear, and thus it
is a simple matter to include the radial component of the velocity to
account for fluid absorption. This absorption also leads to a decay in
the axial velocity and a decrease in the axial pressure gradient in the
tubule lumen. This radial flow is not of importance in the present
study, since it is directed along the axis of the microvillus and thus
does not produce a drag or bending moment on the microvillus.
Equations 1 and 2 can be cast in dimensionless
form by introducing the dimensionless variables
|
(3)
|
where the characteristic length, velocity, and pressure are
defined by
and i = c or m depending on whether one is
describing the core flow or brush-border flow, respectively. Here,
RL is the luminal radius, L is the
height of the microvilli, Q is the flow in the tubule, A is
the cross-sectional area of the tubule, and U0
is the average velocity.
The dimensionless form of Eq. 2 for axisymmetric flow
|
(4)
|
contains a single dimensionless parameter
given by
|
(5)
|
The denominator,
, in the
definition of
is a characteristic length describing the thickness
of the fiber interaction layers at the base and tip of the microvilli. One can show, as did Tsay and Weinbaum (49), that this
characteristic thickness is of the same order as the microvillus
spacing
. Thus one anticipates that the solution of Eq. 4
will entail thin regions whose thickness is of order
near the base
and tip of the microvilli where there will be steep velocity gradients
and the velocity must adjust to satisfy boundary and matching
conditions. Since
is of order 0.1 µm and
R0 is typically 20-30 µm depending on flow rate,
is of order several hundred. We will, therefore, be
interested in the solutions to Eq. 4 in the large
limit.
There are two boundary conditions, one at the center of the tubule,
r = 0, where we require that the velocity be symmetric
|
(6)
|
and one on the wall (apical membrane), r = 1, where we require that the no-slip boundary condition be satisfied
|
(7)
|
In addition, we require that velocity and shear stress be
continuous at the edge, r = rL,
of the brush border
|
(8)
|
where
m and
c are given by µ ×
Um/
R and µ ×
Uc/
R, respectively.
Solution for Velocity Field and Mass Flow
The general solution to Eq. 4 is given by
|
(9)
|
where I0(r) and
K0(r) are zero-order modified-Bessel
functions, dp/dz is the dimensionless pressure gradient, and
C1 and C2 are arbitrary
integration constants.
The general solution to Eq. 1, which satisfies boundary
condition Eq. 6, can be written in dimensionless form as
|
(10)
|
where C3 is an integration constant.
The matching conditions, Eqs. 7 and 8, determine
the unknown constants C1,
C2, and C3. These
expressions for the Ci and their asymptotic
expressions, in the limit
> 1, are given by
|
(11A)
|
|
(11B)
|
|
(11C)
|
The dimensional water flux in the tubule is given by
|
(12A)
|
Equation 12A is the integral of the velocity over the
entire cross section of the tubule. Since the brush- border flow
contributes very little to the total flux, the second integral
describing this flux can be neglected, and Eq. 12A reduces
to
|
(12B)
|
From Eq. 11, C3 is of
O(
), and the second term of Q in Eq. 12B is of
order 1/
smaller than the first. The first term in Eq. 12B is the same as for the Poiseuille flow in a tube of
dimensionless radius rL. The second term in
Eq. 12B describes a small bulk flow due to the small slip
velocity at the microvilli tips.
By introducing the dimensionless water flux
one can write the dimensionless pressure gradient to order 1/
as
|
(13)
|
Here q is the dimensionless flux, Q/Q0, and
rL is the dimensionless position of the edge of
the brush border.
Darcy Permeability Coefficient Kp
The expression for the two-dimensional Darcy permeability
coefficient can be derived by examining a periodic unit in the
hexagonal microvillus array shown in Fig. 2B. The
distributed or body force per unit volume, F, is the Darcy
term in Eq. 2.
The total pressure force acting on a periodic unit, ABCD, in
Fig. 2B, whose length and height are (2a +
) and (2a +
)
/2, respectively, is
|
(14)
|
whereas Darcy's law requires that
|
(15)
|
From Eqs. 14 and 15, the pressure force
acting on the periodic unit ABCD can be written as
|
(16)
|
Note, however, from Fig. 2B, that this is also the
local drag force F per unit length on a single microvillus,
since the periodic unit for the hexagonal array comprises a single
cylindrical fiber or microvillus. Thus from Eq. 16,
F is given by
|
(17)
|
and Kp can be written in terms of
F as
|
(18)
|
Sangani and Acrivos (46) have obtained a numerical
solution for the Stokes flow past the periodic fiber array in Fig.
2B. They showed that the dimensionless drag,
F/µUm, can be expressed by
|
(19)
|
where c, the solid fraction, is defined by
|
(20)
|
The expression in Eq. 19 is valid for
c < 0.4. For control flow conditions where
= 74.1 nm and a = 45 nm, c = 0.27. From Eqs. 18, 19, and 20
|
(21)
|
Equation 21 is rigorously valid only for
two-dimensional flow. It is, however, also a reasonable approximation
for the local drag along a microvillus where Um
is a function of R. A rigorous justification of this
approximation would require a much more complicated three-dimensional
analysis similar to that in Tsay and Weinbaum (49) where
the radial pressure gradient and velocity are considered.
Drag and Shear Force per Unit Tubule Length
The total drag force on the microvilli per unit tubule
length is given by
|
(22)
|
Equation 22 is the integrated drag on all
microvilli that lie in the annular region RL < R < R0 due to thin effective
hydrodynamic resistance, the Darcy term
µUm/Kp in Eq. 2. Using Eqs. 1 and 2, one can write
Eq. 22 in the equivalent form
where i = c or m in the last integral.
The latter integral can be readily evaluated
|
(23)
|
From Eq. 23 it is clear that the pressure
drop per unit tubule length (first term on right-hand side of
Eq. 23) is balanced by the shear force at wall,
Fs (second term on right-hand side of
Eq. 23), and the drag force, Fd, on
the microvilli per unit tubule length.
The shear force per unit tubule length is given by
|
(24)
|
From Eq. 23, the ratio
of the drag force to the
shear force per unit tubule length is expressed by
|
(25)
|
Note that in Eq. 25,
depends only on the brush
border and microvillus geometry.
Drag and Torque on a Single Microvillus
To obtain the drag force distribution fd
acting on each microvillus, we divide the total drag on all the
microvilli per unit length, Fd in Eq. 22, by n, number of microvilli per unit length of
tubule, and integrate between RL, the microvilli
tip and any position R
|
(26)
|
When R is equal to the radius of the tubule, Eq. 26 gives the total drag force acting on a single microvillus.
From Eq. 26 the local force per unit length of microvillus
is given by
|
(27)
|
The bending moment per unit length acting on the base of the
microvillus is given by
where
= R0
R is the lever arm of the force element. The integrated
torque distribution is defined by
|
(28)
|
T(R) is the torque acting on a single
microvillus between its tip and any position R. When
R is equal to the radius of the tubule, Eq. 28
gives the total torque acting on a single microvillus.
Elastic Model for Bending of Microvillus
Fortunately, there is sufficient information on the
structure of the F-actin cytoskeleton of the microvillus and the
elastic properties of an individual actin filament to construct from
first principles a model for its bending deformation due to its
hydrodynamic loading. These ultrastructural studies summarized by
Maunsbach (37) indicate that there are between 6 and 10 long axial microfilaments randomly distributed in each cross section.
Most of the axial microfilaments are clustered in the central region of
the microvillus cross section rather than near its periphery. In
our idealized ultrastructural model shown in Fig.
3A, we have assumed that on average seven such filaments are arranged in an hexagonal array with
six of the filaments equally spaced on a circle that is the half-radius
of the cross section. This is a close approximation to the electron
micrographs shown in Maunsbach (37). Each microfilament is
7 nm in diameter. Although the spacing of the transverse linker molecules, fimbrin and
-actinin, is not known to our knowledge, it
is reasonable to assume that their primary function is that of spacer
molecules that hold the long axial filaments in a nearly parallel
array. The bending moment on the microvillus due to the distributed
hydrodynamic drag is, therefore, borne entirely by its seven axial
elements. This structure is analogous to a reinforced concrete beam in
which the bending resistance of the concrete between its axial steel
reinforcing rods is neglected. Fig. 3B is a sketch showing
the geometry of the deformed microvillus with just its central
microfilament.

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Fig. 3.
A: idealized structural model for the
arrangement of axial F-actin filaments in microvillus cross section.
The 7 F-actin filaments form a hexagonal array in which the 6 off-axis
filaments are equally spaced on a circle that is the half radius of the
cross section; a is the radius of the microvillus membrane,
and rf is the radius of the actin filament.
B: geometry of the deformed microvillus showing axial
deflection of its central filament. The mechanical loading is a
combination of a concentrated force, Pm, acting at the tip,
accounting for approximately three-fourths of the total force and a
uniform loading, qm, accounting for approximately
one-fourth of the total force.
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|
According to the elementary theory for the bending of beams
(4), the deflection y of the microvillus (see
Fig. 3B), satisfies the fourth order equation
|
(29)
|
Here x = R0
R, E is the Young's modulus of the individual
actin filaments, I is the moment of inertia of the cross
section, and D(x) is the distributed axial load
due to the hydrodynamic drag obtained from Eq. 27.
I depends on the orientation of the bending axis. However,
one can show that for the geometry in Fig. 3A the moment of
inertia varies insignificantly with any axis passing through the origin
because of the hexagonal symmetry. Since the radius of an actin
filament, rf, is small compared with the radius
of the microvillus a, the contribution of each filament to
I is the area of the filament
rf2 times the distance from the
axis of rotation. Thus I for the cross-sectional geometry in
Fig. 3A is
|
(30)
|
The drag force distribution or the beam loading is given
by Eq. 27. However, as will be shown in the
RESULTS, this loading can be well approximated by the
superposition of two simple loads, a uniform load, qm, over
most of the length of the microvillus and a concentrated force,
Pm, acting at its tip. This allows us to write a greatly
simplified expression for the loading, which enables us to integrate
the beam equation analytically. This simplified loading distribution is
given by
|
(31)
|
Four boundary conditions are needed to define the
boundary value problem for the deflection of the microvillus. At
x = 0, the base of the microvillus, it is reasonable to
assume that the longitudinal actin filaments are anchored to more rigid
supporting structures in the intracellular cytoskeleton. These could be
either more complex actin networks near the apical membrane,
microtubules, or intermediate filaments. The nature of this anchoring
is not yet known. If this anchoring is relatively inflexible and
treated as a rigid support, then the displacement and slope of the
microvillus relative to the vertical will vanish
|
(32)
|
At the free end of the beam, x = L, there is no bending moment, and the vertical shear force
does not vanish. We require that
|
(33)
|
The solution to the boundary value problem defined by Eqs.
29, 31, 32, and 33 is given by
|
(34)
|
At the free end of the beam, x = L, the
maximum deflection is achieved. From Eq. 34, it is given by
|
(35)
|
 |
RESULTS |
Velocity Field
In Fig. 4 we have plotted the
velocity profiles in the tubule lumen, the transition layer in the
vicinity of the microvilli tips and in the brush border. Three curves
are required because the velocity scale varies so greatly between
regions. This behavior is characteristic of solutions in the large
limit. The solutions in Fig. 4 as well as Fig. 6 are for a control
value of Q of 30 nl/min. The edge of the brush border is located at a
dimensionless r of 0.847. The centerline velocity in the
lumen is 1.66 mm/s, and the profile in the core is a parabola typical
of Poiseuille flow except near the interface with the brush border,
where the velocity drops to very small values over a distance of a few
tenths of a micron as observed in Fig. 4B. At the edge of
the brush border, r = 0.847, the velocity is only 4.11 µm/s or roughly 1/400 of the centerline velocity. The velocity then
falls off rapidly as one enters the lateral spaces between the
microvilli and, as shown in Fig. 4C, asymptotically
approaches a nearly constant value of only 10 nm/s in the central
region of the brush border. As seen in Fig. 4B the thickness
of the transition region at the microvilli tip where the velocity
decays to the nearly vanishing bulk flow velocity is ~0.011 in
dimensionless r units. This corresponds to a distance of
0.180 µm in physical units or about 7% of the microvilli length.
This is about 2.4 times the open gap between microvilli, which for the
control flow is 74 nm. As observed in Fig. 4C, a transition
layer of comparable thickness exists at the base of the microvilli
where the velocity decays from the miniscule bulk flow velocity in the
brush border to zero at the apical membrane, so as to satisfy no-slip
conditions. Because of the miniscule magnitude of the bulk flow in the
brush border, the velocity in Fig. 4C is plotted in
nanometers per second, where the velocity at the edge of the boundary
layer at the apical membrane is about 10 nm/s. There is a slight
decrease in the bulk flow velocity across the brush border due to the
curvature of the radial coordinate system. This small bulk flow is
driven by the axial pressure gradient in the lumen of the tubule.

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Fig. 4.
Velocity profiles in lumen (A), in tip region
(B), and in central region of brush border (C).
Tip is located at r = 0.847. Note that velocity at
microvilli tip is ~1/400 the center line velocity in core, and bulk
flow in brush border is ~1/400 of the tip velocity.
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|
In Fig. 5 we have plotted the tip
velocity vs. the open gap
between microvilli and also the
dimensionless drag coefficient given by Eq. 19. It is clear
that there is a nearly linear relationship between the tip velocity and
. This linearity can easily be derived from asymptotic analysis, see
the APPENDIX. This effective slip velocity at the
microvilli tips gives rise to the second term in the expression for Q
in Eq. 12B.
The miniscule bulk flow in the brush border contributes negligibly to
the total flow in the tubule. However, it plays a very important role
in determining the total force on each microvillus and the torque that
it experiences. Similarly, it might seem at first glance that the
details of the velocity profile in the thin transition region near the
tips of the microvillus are not significant. Since the forces in Stokes
flow are proportional to the velocity, it is the integral of this
velocity profile that determines the contribution of the tip region to
the total force and torque on the microvillus. The central question is
how this integrated drag on the tips of the microvilli compares with
the much smaller forces due to the bulk flow but which act over most of
the length of the microvillus.
Drag vs. Shear Force
With a bulk flow velocity in the brush border that is five orders
of magnitude smaller than the centerline velocity in the tubule, one
wonders how the shear force acting on the apical membrane at the base
of the microvilli compares with the drag force on the microvilli. As
noted earlier, nearly all previous studies on the effect of fluid
forces on vascular endothelial cells have examined the effect of fluid
shear on the cell's cytoskeleton and intracellular biochemical
responses. Since both these forces scale linearly with the velocity,
the ratio of the drag force to the fluid shearing force would be
independent of the tubule flow rate, if the microvilli geometry did not
change with flow rate. The ultrastructural study by Maunsbach et al.
(36) revealed that the open separation between the
microvilli increased from 62.1 nl to 90.4 nm as the tubule flow was
increased from 5 to 45 nl/min. One is thus interested not only in the
ratio of the drag to fluid shear force at control conditions, where
this separation distance is 74.1 nm, but also how this ratio changes
more generally with microvilli spacing. The results of Eq. 25 are plotted in Fig. 6. One
observes that the ratio of the drag to fluid shear force increases from
~360 to 580 as the open spacing between microvilli decreases from
90.4 to 62.1 nm and that there is a rapid fall off in this ratio as the
distance between microvilli increases. The density of the microvilli in
the S3 segment is approximately one-quarter that in the S2 segment and
thus could be twice the control value cited above, or 150 nm. Even for
this large spacing, the ratio of the drag to shear force is nearly 200.

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Fig. 6.
Plot of Eq. 25 for the ratio of the drag
force, Fd, on the brush border microvilli and
the shear stress, Fs, on the apical membrane per
unit length of tubule. Note depends only on tubule and brush border
geometry and is independent of both viscosity and flow rate; varies
between 360 and 580 for the values of measured in Maunsbach et al.
(36).
|
|
Drag and Torque Distribution on a Single Microvillus
In Fig. 7 we have plotted Eq. 26 for the integrated drag distribution starting at the
microvillus tip, r = 0.847, to the base of the
microvillus. One observes a sharp break in this curve at approximately
r = 0.858. At this location the average velocity differs by less than 1% from the bulk flow velocity in the interior of
the brush border. The integrated drag at this location is 73.8% of the
total drag on the microvillus, although this drag acts on only the 7%
(0.18 µm) of the microvillus length near its tip. In summary, the
drag due to the bulk flow in the interior of the brush border
contributes ~1/4 of the total drag, and the flow near the tip
contributes 3/4 of the total drag. This highly asymmetric distribution, as we see next, provides for a large amplification in the
torque experienced by the microvillus. The total drag force on the
microvillus for a control flow of 30 nl/min is 7.38 × 10
3 pN.

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Fig. 7.
Integrated drag force distribution along the microvillus
starting from the microvillus tip. Note that tip boundary layer is
about 7% (0.18 µm) of microvillus length, and 73.8% of the total
drag force is concentrated in the tip region.
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|
In Fig. 8 we have plotted Eq. 28 for the integrated torque distribution starting at the
microvillus tip proceeding to the base of the microvillus. One again
observes a sharp break in this curve at approximately r = 0.858. The integrated torque at this location is 86.2% of the total
torque on the microvillus, although this torque, like the drag, acts
only on the 7% of the microvillus length near its tip. In summary, the
torque due to the bulk flow in the interior of the brush border
contributes only about 1/7 of the total torque, and the flow near the
tip contributes 6/7.

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Fig. 8.
Integrated torque distribution along the microvillus
starting from the microvillus tip. Note that 86.2% of the total torque
occurs in the tip boundary layer, outer 0.18 µm of microvillus.
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|
Effect of Flow on Drag and Torque
In Table 1 we have summarized the
effects of changing the flow rate using the ultrastructural data
provided by Maunsbach et al. (36). One observes that the
drag force on the microvilli does not vary linearly with flow
rate if the changes in lumen diameter and microvilli separation with
flow rate are accounted for. In fact, one notes that there is only a
slightly more than twofold increase in drag when the flow is increased
from 5 to 45 nl/min. If we assume that the length of the microvilli do
not change, then, since three-quarters of the drag acts near the tips of the microvilli, the torque should increase roughly in proportion to
the fluid drag on the microvilli tips. This nonlinear behavior, which
is observed in experiments with individual perfused tubules in situ, is
associated with an increase in hydrostatic pressure in the lumen of the
tubule. This increase in tubule lumen pressure will cause a distension
of the perfused nephron, since neighboring nephrons will not be at
elevated transepithelial hydrostatic pressure. In vivo, this distension
probably does not occur, since neighboring nephrons would all be at the
same transepithelial pressure.
Bending Deformation of the Microvillus
The bending deformation of the microvillus and the deflection of
its tip are given by Eqs. 34 and 35,
respectively. The two key parameters in these expressions are the
moment of inertia, I, given by Eq. 30 and the
Young's modulus, E, of the individual actin filaments in
the microvillus cross section shown in Fig. 3A. Realistic
values for both of these parameters are available. In calculating
I, we have used the measured values,
rf = 3.5 nm and a = 45 nm
(37). The Young's modulus for an F-actin microfilament has been estimated from its bending modulus. This has been determined from force measurements obtained by the micromanipulation of single actin filaments by Kishino and Yanagida (23). The
calculated value for E is 1.44 × 109
dyn/cm2. In Fig. 9 we have
plotted the deformed shapes of microvilli of four different lengths
from 1.5 to 3.0 µm for a control flow of 30 nl/min when the open gap
between microvilli is 74.1 nm. Equation 35
shows that the maximum deflection of the microvilli, which is
primarily determined by the concentrated drag forces near its tip, is
proportional to the third power of the microvilli length. This accounts
for the large increase in the maximum deflection with microvilli
length. For a microvillus of 2.5 µm length in an S2 segment, the tip
deflection is 3.78 nm, whereas for a 1.5-µm microvillus, the tip
deflection is about one-fifth this value. Thus, at a control flow rate
of 30 nl/min, the maximum tip deflection varies from about 1 to 5% of
the microvilli diameter 90, nm.

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Fig. 9.
Microvillus deflection for control flow, 30 nl/min, for
microvilli of four different lengths from 1.5 to 3.0 µm.
2RL = 27.7 µm and = 74.1 nm for all
microvilli.
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|
As noted earlier, while the lengths of the microvilli decrease as one
proceeds from the S1 to the S2 and S3 segments, the spacing of the
microvilli increase and
for the S3 segment is about twice that for
the S2 segment (52). One can show from Eq. 23
that the hydrodynamic loading of an S3 microvillus at its tip,
Pm, is roughly four times that for the S2 microvillus. This suggests that the length of the microvilli may decrease as their spacing increases so that the deflection of the microvilli tips does
not vary greatly along the length of the tubule.
 |
DISCUSSION |
Glomerulotubular balance was first established in rat kidney via
micropuncture (48), where spontaneous variation in
glomerular filtration (over a range of flows from 15 to 60 nl/min) was
accompanied by a constant fractional proximal reabsorption. During
periods of low filtration, this aspect of proximal tubule transport
ensures preservation of adequate delivery of sodium to the distal
nephron (preserving acid and potassium excretion). The mechanisms for balanced tubular reabsorption appear to include both peritubular capillary effects and luminal factors (13, 14, 19, 53). With respect to peritubular factors, any increase in filtration fraction must result in an increased peritubular protein concentration, and increased peritubular capillary protein concentration enhances proximal reabsorption (7, 17). The mechanism is thought to involve changes in interstitial hydrostatic pressure (25,
26), with effects on both the tight junctions (31)
and on the cells themselves. Nevertheless, a mathematical model of
proximal nephron, which incorporated the observed effect of peritubular
Starling forces to modulate reabsorption, demonstrated that the
observed constancy of fractional reabsorption could only be achieved
when alterations in luminal flow, independent of peritubular Starling forces, could also influence proximal reabsorption (51).
The impact of variation in luminal flow rate on tubular reabsorption
has been termed "perfusion-absorption balance" (53). This effect has been best demonstrated in rat microperfusion studies (2, 18, 40, 43). With respect to specific solutes,
flow-dependent reabsorption has been found to influence the transport
of glucose (24), bicarbonate (1, 9, 33), and
chloride (16, 54). One of the best illustrations of this
phenomenon is the micropuncture data of Chan et al. (9),
in which a threefold increase in luminal perfusion rate (with trivial
changes in luminal HCO3
concentration) produced a
doubling of the rate of bicarbonate reabsorption. The underlying
mechanism for flow-dependent changes in reabsorption has not been
established. One consideration has been an unstirred layer effect
within the brush border (42). The morphological
observations of Maunsbach et al. (36) demonstrated that
lower tubule flow rates are associated with diminished tubule distention and a compaction of the brush border microvilli.
Nevertheless, model calculations indicate it is unlikely that there is
any appreciable convective stirring within this pile (3),
nor should the diffusion barrier between the bulk luminal fluid and the
cell membrane pose any significant hindrance to
Na+/H+ exchange (27). Two
relatively recent studies raise the possibility that increases in axial
flow velocity recruit new transporters into the luminal membrane.
Preisig (41) examined recovery of cellular pH from an
acute acid load in vivo. With increases in luminal flow rate, the pH
recovery mediated by Na+/ H+ exchange was
enhanced. Maddox et al. (34) subjected rats to acute
changes in vascular volume to obtain hydropenic, euvolemic, and
volume-expanded groups, with respective grouping according to
decreased, normal, and increased GFR. When brush-border membrane vesicles were prepared from each of these groups and
Na+/H+ kinetic parameters were assessed, it was
found that the Vmax determinations stratified in
parallel with GFR.
Ultimately, perfusion-absorption balance must derive from an afferent
sensor of fluid flow rate in series with a cascade of effector steps
which insert or activate new membrane transporters. The calculations of
this report indicate that the microvilli that constitute the proximal
tubule brush border are physically suitable to function as such a
sensor. The second component of this system may well be the actin
cytoskeleton, which is abundant within and beneath the proximal tubule
brush border (35). Indeed, the cytoskeleton has been
implicated in volume regulation in a variety of cells (39), and specifically in proximal tubule, cytoskeletal
disruption impairs the cell solute loss (volume regulatory decrease)
following hypotonic cell swelling (32). Direct
interactions of the cytoskeleton with a variety of membrane
transporters, including ion channels, cotransporters, and the
Na+-K+-ATPase, have been identified
(8). Thus specific interaction between the proximal tubule
cytoskeleton and the apical cell membrane Na+/H+ exchanger remains a critical feature to
demonstrate, to maintain plausibility for this scheme of signal
transduction. Such a connection is provided by the proposal of
Lamprecht et al. (29) in which the
Na+/H+ exchanger is linked via
Na+/H+ exchanger regulatory factor to
ezrin, a kinase anchoring protein, which is attached to the actin
cytoskeleton. Indeed, gross disruption of the actin cytoskeleton has
recently been shown to impair function of the proximal tubule
Na+/H+ exchanger, when expressed in other cells
(28). It is not known, however, whether more subtle
changes, such as perturbation of cytoskeletal stresses, can also affect
this transporter. It must also be acknowledged that more
extensive feedback loops may be important, from flow changes to
membrane transporter density via gene activation. Such responses have
been identified in renal tubular cells in culture, in which changes in
shear stress have been implicated in increased number of microvilli and
in increased abundance of integral membrane proteins (22).
Obviously, this response cannot be invoked to account for acute
(micropuncture) observation of perfusion-absorption balance.
The analysis in the present study indicates that from both a
hydrodynamic and structural standpoint, the microvilli are ideally suited to serve as mechanotransducers. Previously, it has been widely
believed that the primary role of the brush border was to greatly
amplify the luminal membrane area required for water and solute
exchange. This does not explain why the microvilli have such striking
uniformity in length, why the spacing between microvilli is so regular,
why the microvilli decrease in length as their density decreases, what
determines the density and distribution of the axial actin
microfilaments in the microvilli, and how these microfilaments are
anchored to the intracellular cytoskeleton near the apical membrane.
These questions lead to a new view of the brush-border epithelial cells
not only as a transport organ, but as a cell with specialized
ultrastructural components that could have mechanosensory functions.
We first consider the hydrodynamic function of the brush border. Except
for the thin interaction region near the microvilli tips, the axial
velocity within the brush border is five orders of magnitude smaller
than the average axial velocity within the lumen of the tubule. In
contrast, one can estimate that if one-half the filtrate is reabsorbed
along the entire length (1 cm) of the proximal tubule, then the
radially directed average velocity along the axes of the microvilli is
0.24 µm/s. This is a tiny fraction (0.15 × 10
3)
of the axial velocity in the lumen. The radially directed flow due to
reabsorption is, therefore, about 24 times the axially directed bulk
flow within the brush border, but this radial flow produces no drag or
torque on the microvilli since it is directed along the microvillus
axis. The miniscule axial bulk flow contributes negligibly to the total
flow in the tubule, yet it determines the nature of the hydrodynamic
forces on both the microvilli and the apical membrane.
The nearly vanishing axial bulk flow velocity in the brush border in
Fig. 5C was previously predicted in Basmadjian et al. (3), using a semi-empirical Carman-Kozeny equation to
estimate the hydraulic resistance. This semi-empirical relation was
widely used before the more accurate solution of Sangani and Acrivos (46) (Eq. 19) was derived. In this
earlier study Basmadjian and coworkers (3) derived an
expression for the ratio of the flow in the brush border to the flow in
the core. For control flow in the present study, this ratio is
~5 × 10
6, a value that is similar to the
prediction in Basmadjian et al. (3) for a dense
microvillus array. These investigators were primarily interested in
unstirred layer effects due to the axial flow rather than the forces
and torques due to the flow in the thin transition layer near the
microvilli tips. This thin boundary layer, which is the primary focus
of the present study, was neglected since it contributed
insignificantly to the transport processes in the brush border.
Figure 6 shows that, despite the very small axial bulk flow through the
brush border, the drag forces on the microvilli are at least 200 times
greater than the shear forces acting on the underlying apical membrane.
Proximal tubule epithelial cells would never be able to detect fluid
shear forces in the same manner as vascular endothelial cells, although
the flow rate in the tubule is quite similar to the flow rate in a
blood vessel of comparable diameter. One of the remarkable features of
the circulation is that, although the blood vessels undergo more than
20 generations of branching, the fluid shear stress is nearly constant
throughout the entire arterial side of the vasculature
(21). Furthermore, relatively modest changes in fluid
shear stress are readily detected by endothelial cells
(10). These wall shear variations lead to short-term
biochemical responses and, in large vessels, endothelial-dependent adaptive remodeling after longer exposure to changes in wall shear stress (55). Proximal tubule epithelial cells thus need a
different machinery to sense fluid flow rates, and hydrodynamic drag
would appear to be a more logical candidate since it is several hundred times greater than fluid shear.
An unusual feature of the brush border microvilli is their remarkable
local uniformity of length. Such uniformity would not be a prerequisite
for a transport function associated with unstirred layer effects. It
would, however, be essential if the tips of the microvilli all needed
to sense the flow at the edge of the brush border. The detailed
velocity profile near the tips of the microvilli in Fig. 4B
provides an intriguing and rational argument for this structure. One
observes that the axial velocity decays over a length that is less than
0.2 µm or roughly twice the open spacing
between the tips. A
microvillus that is only slightly shorter than its neighbors would
experience virtually no axial flow near its tip, and the hydrodynamic
torque on this microvillus would be only about one-seventh that of its
neighbors. There would be a small torque due to the bulk flow in the
interior of the brush border, but as noted in Fig. 8, this torque is
about one-seventh that due to the drag at the microvilli tips. The
deflection of this microvillus would be much smaller than its neighbors
(see Eq. 35).
The arguments just advanced for uniformity in microvilli length
also apply for the unusual hexagonal regularity in microvilli spacing
sketched in Fig. 2A. If this spacing were not regular, then
each microvilli tip would experience a very different drag force.
Figure 5 is a plot of Eq. 19 where the solid fraction,
c, has been related to the open spacing
between the
microvilli using Eq. 20. It is evident from Fig. 5 that the
dimensionless drag coefficient, F/µU, on the
microvillus tip is very sensitive to the spacing
and that the
stimulus from each microvillus would vary greatly if their distribution
were random. Such regularity in ultrastructure would only make sense if
the microvilli were relatively stiff and their deformation due to
hydrodynamic forces was small compared with their spacing.
It is clear from Fig. 5 that the drag force coefficient falls off
rapidly as the open gap
increases. However, Fig. 5 also shows that
the velocity at the microvillus tip,
Um(RL), increases nearly
linearly with
as the microvilli density decreases. The net result
is that the drag force per unit length at the microvilli tips is
relatively insensitive to
, and one can show there is a weak minimum
(not shown) at
= 130 nm. The depth of the penetration of the
tip boundary layer, however, also increases with
, so that the
integrated drag force on the tip increases as
decreases. This
suggests that several competing factors determine the optimum spacing
and length of the microvilli. The amplification of absorptive area
along the apical surface decreases from 36 to 15 from the S1 to the S3
segment in mammalian tubules (37), and this correlates with volume reabsorption. However, a given amplification can be achieved by either long microvilli that are less dense or short microvilli that are more dense. It is here that mechanical forces could
play an important role. In our hypothesized mechanosensory system, one would anticipate that either the torque on the microvillus or its tip deflection are the critical measures of flow rate. Thus, if
one wanted to keep torque constant, then one could decrease microvillus
length while increasing the separation of the microvilli so as to
increase the value of
Um(RL) and the drag at
the microvilli tip. Similarly, a different flow-microvillus length
relationship would emerge if one were to maximize deflection of the
microvillus tip in Eq. 35. This optimal design
requires further study.
From a mechanosensory viewpoint the stimulus could be either the
bending deformation of the microvillus or the cytoskeletal structures
that anchor the axial actin microfilaments to the intracellular cytoskeleton. We shall examine each of these possibilities. The model
for microvillus bending provides insight into the first of these
possibilities. The predicted deformations of the microvilli in Fig. 9
are based on realistic models of the actin microfilament structure in
Fig. 3A, taken from Maunsbach (37), and a
reliable estimate of the Young's modulus of individual F-actin
filaments in Kishino and Yanagida (23). The predicted
maximum deflections of the microvilli at normal flow rates, 30 nl/min,
are 1 to 5% of the microvilli diameter depending on microvilli length.
The microvilli, therefore, act as remarkably stiff bristles whose deformation under flow is less than 0.2% of their length. This supports our initial assumption that the fluid flow through the microvilli and the resulting forces and torques can be calculated neglecting this deformation. Although this bending deformation is
small, we shall show next that it is sufficient to produce an
intracellular biochemical response.
There is a large literature on the biochemical response of cells grown
on elastic substrates that are then subjected to stretch. These studies
typically show that biochemical responses are elicited only for strains
(change in length over initial length) that are 1% or greater. This
could be the opening of stretch-sensitive ion channels in the membrane
or the release of Ca2+ from intracellular stores. When the
microvillus bends, there is a relative motion of the axial actin
filaments, and they slide past one another much like the pages of a
paperback book when you bend its cover. The relative motion of the
front and back cover is of the same order as the edge deflection and
thus the relevant strain is the edge deflection divided by the total
thickness of its pages. In the microvillus, this relative motion would
be transmitted via cross-linking molecules such as ezrin, which link the actin cytoskeleton to proteins in the plasma membrane, or fimbrin
and
-actinin, which link the deforming actin fibers. The order of
magnitude of the strain on these cross-linking molecules would be the
tip deformation divided by the microvillus diameter, which was 1 to
5%. These strains fall in the range where intercellular biochemical
responses are elicited on cells stretched in vitro. In the present case
a plausible candidate for the membrane protein is NHE3, the
Na+/H+ exchanger located in the microvillus
membrane (29).
The second possibility is that the sensory stimulus is the torque
exerted on the anchoring elements in the intracellular cytoskeleton. The ultrastructural arrangement of the microvilli is ideally designed to maximize this function. Nearly 90% of the torque is produced by
drag forces acting at the microvilli tips. The long lever arm accentuates the moments on the intracellular cytoskeleton, and the
stiffness of the microvilli allows one to transmit this torque without
a large deformation that would produce tip-tip interaction. Our basic
assumption in solving Eq. 29 is that the microvillus acts
like a cantilever beam that satisfies a rigid end condition (32), much like the roots of a tree fasten its trunk to
the ground.
The total drag force acting on the microvillus for a control flow of 30 nl/min is 7.4 × 10
3 pN. This is about two orders of
magnitude smaller than the drag forces applied by laser tweezers in
moving proteins laterally in the plane of the membrane. Typically,
forces less than 0.1 pN are too small to hold the proteins in the laser
trap when the tail of the protein interacts with the underlying
microfilaments in the membrane skeleton (45). Forces of
the order of 0.5 pN will cause a large deformation of the F-actin
microfilaments that form the fence of the microdomains that restrict
the diffusion of the membrane proteins. A significant rebound of the
protein is observed if it escapes from the optical trap at a
microdomain boundary. Thus the hydrodynamic forces predicted by our
model are small compared with the dragging forces that create large deformations of the membrane cytoskeleton. This is consistent with our
hypothesis that the longitudinal actin filaments do indeed provide
sufficient stiffness to prevent the microvilli from undergoing large
strains and that the underlying cytoskeleton beneath the apical
membrane, which provides the anchoring support, can withstand the
bending moments at the base of the microvilli.
One of the most attractive features of this new model is that it is
able to take full advantage of the available information that has been
gathered over the past few decades on the ultrastructure of the
microvilli, their actin cytoskeleton, and the physical parameters that
determine both the hydrodynamic and elastic behavior. The limitation
and uncertainty in the hydrodynamic model are minor since the spacing
and regularity of the microvilli has been well documented. Similarly,
the cytoskeletal model for the bending of the microvilli sketched in
Fig. 3, which is idealized in the spacing of the axial actin filaments,
is a quite realistic representation of the ultrastructure
(37). The greatest uncertainty in the prediction of the model
lies in the available measurements determining the Young's modulus
E of the individual actin filament. The estimation of
E has an order of magnitude uncertainty, since the
experimental techniques have been based on measurements of the flexural
rigidity EI of a single actin filament subject to either
micromanipulation or Brownian deformation (12, 15, 23) and
not direct measurement of E, which is a derived quantity
once the moment of inertia I of the actin filament has been
estimated. The measured values of EI have varied from
15 × 103 pN · nm2
(12) to 73 × 103
pN · nm2 (15) with the measurement of
17 × 103 pN · nm2
(23) used in our calculation being close to the former.
Since I varies as the fourth power of the effective radius
of the filament, small variations in assumed cross-sectional geometry
lead to substantial changes in I. In this study we have used
the calculation in Satcher and Dewey (47), which assumes
an effective radius of 3.5 nm, since this corresponds closely to the
measurements reported in Refs. 35 and 44. This gives a value of
E of 0.14 GPa. If the effective radius had been reduced 2.5 nm, as assumed in Dupuis et al. (12), then
E would be a factor of four larger and the microvilli tip
deflections a factor of four smaller.
The theoretical predictions of our model suggest a relatively simple
critical experiment that can be performed to test the basic hypothesis
that the microvilli serve a mechanosensory function, namely, to examine
the effect of viscosity on reabsorption. If the flow rate is kept
constant, but the viscosity of the perfusing fluid is increased, then
the drag on the microvilli will increase and produce a larger drag on
the microvilli tips. This should increase the torque on the microvilli
and, if our hypothesis is correct, produce the same effect as
increasing the flow rate. Additional biological information will be
necessary, however, to formulate a complete signaling mechanism.
Although the axial actin structure of the microvillus has been known
since the early 1970s, there is no equivalent information on how this
microfilament skeleton is anchored into the intracellular cytoskeleton.
Finally, much more needs to be learned about the relationship between
the sensing mechanism and the interactions that lead to the entry of
Na+ across the apical membrane. In particular, the specific
interactions between the proximal tubule cytoskeleton and the apical
cell membrane Na+/H+ exchanger need to be elucidated.
 |
APPENDIX |
Slip Velocity at Microvillus Tip
In the interior of brush border the bulk flow velocity
is uniform, and there is no shear force at all. The dimensionless
Brinkman Eq. 4 reduces to the Darcy equation, and the
dimensionless bulk velocity, um, is given by
|
(A1)
|
From Eq. 10 the dimensionless tip velocity is given
by
|
(A2)
|
In the large
limit C3 is given by
Eq. 11C, and the dimensional slip velocity from Eq. 3 can be written as
|
(A3)
|
Using the definition of
and rL, one
can write Eq. A3 as
|
(A4)
|
Tsay and Weinbaum (49) have shown that the Darcy
permeability coefficient Kp is closely
approximated by
|
(A5)
|
Substituting Eq. A5 into Eq. A4, the tip
velocity is given by
|
(A6)
|
where K is a constant. Thus the tip velocity varies
nearly linearly with
.
 |
ACKNOWLEDGEMENTS |
This research was performed in partial fulfillment of the
requirements for the PhD degree from the City University of New York by
P. Guo.
 |
FOOTNOTES |
A. M. Weinstein acknowledges the support of National Institute of
Diabetes and Digestive and Kidney Diseases Grant R01-DK-29857.
Address for reprint requests and other correspondence: S. Weinbaum, Dept. of Mechanical Engineering, The City College of New York, Convent Ave. at 138 St., New York, NY 10031 (E-mail:
weinbaum{at}me-mail.engr.ccny.cuny.edu).
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 18 January 2000; accepted in final form 23 May 2000.
 |
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