INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

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.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Ultrastructural Model

View larger version (40K): [in this window] [in a new window]   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.

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.

View larger version (14K): [in this window] [in a new window]   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.

Mathematical Model

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 U_c is the fluid velocity in the core, and µ is the fluid viscosity. In the margin region R_L < R < R₀ with microvilli, we use a Brinkman equation for flow through a porous medium (6)

(2)

Here U_m is the local average axial velocity in the microvillus array, and K_p is the two-dimensional Darcy permeability for an infinite fiber array. The expression for K_p 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, µU_m/K_p, 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, R_L 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 = r_L, of the brush border

(8)

where _m and _c are given by µ × U_m/R and µ × U_c/R, respectively.

Solution for Velocity Field and Mass Flow

(9)

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 C_i 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 r_L. 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 r_L is the dimensionless position of the edge of the brush border.

Darcy Permeability Coefficient K_p

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 K_p 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/µU_m, 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 U_m 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

(22)

(23)

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

(26)

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

View larger version (12K): [in this window] [in a new window]   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.

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, r_f, is small compared with the radius of the microvillus a, the contribution of each filament to I is the area of the filament r_f² 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, q_m, over most of the length of the microvillus and a concentrated force, P_m, 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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Velocity Field

View larger version (10K): [in this window] [in a new window]   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.

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.

View larger version (14K): [in this window] [in a new window]   Fig. 5.   Dimensionless drag coefficient (F/µU) and tip velocity vs. open gap  between microvilli.

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

View larger version (9K): [in this window] [in a new window]   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

3

View larger version (13K): [in this window] [in a new window]   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.

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.

View larger version (13K): [in this window] [in a new window]   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.

Effect of Flow on Drag and Torque

Bending Deformation of the Microvillus

View larger version (14K): [in this window] [in a new window]   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.