Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress

Bulletin of Mathematical Biology, Vol. 59, No. 6, pp. 1029-1046, 1997 Elsevier Science Inc.

�9 1997 Society for Mathematical Biology 0092-8240/97 $17.00 + 0.00

S0092-8240(97)00052-9

M O D E L FOR THE A L I G N M E N T OF ACTIN FILAMENTS IN E N D O T H E L I A L CELLS SUBJECTED TO FLUID SHEAR STRESS

�9 A. SUCIU, G. CIVELEKOGLU, Y. TARDY and J.-J. MEISTER Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSE-Ecublens, 1015 Lausanne, Switzerland ( E.mail : tardy @ dprnail.epfl.ch )

Cultured vascular endothelial cells undergo significant morphological changes when subjected to sustained fluid shear stress. The cells elongate and align in the direction of applied flow. Accompanying this shape change is a reorganization at the intracellular level. The cytoskeletal actin filaments reorient in the direction of the cells' long axis. How this external stimulus is transmitted to the endothelial cytoskeleton still remains unclear. In this article, we present a theoretical model accounting for the cytoskeletal reorganization under the influence of fluid shear stress. We develop a system of integro-partial-differential equations describing the dynamics of actin filaments, the actin-binding proteins, and the drift of transmembrane proteins due to the fluid shear forces applied on the plasma membrane. Numerical simulations of the equations show that under certain conditions, initially randomly oriented cytoskeletal actin filaments reorient in structures parallel to the externally applied fluid shear forces. Thus, the model suggests a mechanism by which shear forces acting on the cell membrane can be transmitted to the entire cytoskeleton via molecular interactions alone. �9 1997 Society for Mathematical Biology

1. Introduction. Vascular endothelial cells (ECs) form a monolayer lining the inner wall of blood vessels. This specific cell structure serves as an interface between the flowing blood and the underlying components of the vessel wall. It plays an important role in the exchange of gases and nutrients between the blood and the wall tissue as well as in the regulation of the vascular resistance by its action on the underlying smooth-muscle cells that control and determine the diameter of the vessels. During the past few decades, it has become obvious that the endothelium is also involved in the genesis of cardiovascular diseases--in particular, atherosclerosis is the most frequent one (National Institutes of Health-U.S., 1981). Mechanical forces induced by the pulsatile blood flow have been proposed to be responsible for the localization of atherosclerotic plaques i n specific areas of the cardiovascular system (Fry, 1976; Ku et al., 1985). This has led numerous researchers to study the response of endothelial cells to

1029

1030 A. SUCIU et al.

mechanical forces. In particular, investigators have studied the effects of shear stress due to blood flow on the morphological and functional behavior of endothelial cells.

Endothelial cells elongate and align in the direction of flow when subjected to sustained flow-induced shear stress, both in uivo and in vitro (Dewey et aL, 1981; Langille and Adamson, 1981). This change in cell shape is accompanied by a reorganization of the entire cytoskeleton, and in particular, of the highly dynamic actin filaments (Franke et al., 1984; Levesque and Nerem, 1985; Wechezak et aL, 1985; Girard et al., 1993). Under static (i.e., no-flow) conditions, ECs display a cobblestone appearance with a nearly circular cell shape. Thick bundles of actin filaments, commonly referred to as stress fibers, are preferentially assembled in randomly oriented short bands lining the contour of the cells in their basal parts. The cytoplasmic space between these parallel bundles is filled with a distributed network composed of single actin filaments displaying random orientations (Satcher, 1993). Sustained exposure to fluid shear stress reori- ents the actin bundles parallel to the flow direction, spanning the whole cell from the basal to the apical surface. The actin filaments constituting the distributed network also reorient, and the network develops a compressed appearance, as if it were bent in the direction of the applied flow (Satcher, 1993). The degree of the reversible reorganization of the actin cytoskeleton is dependent on the time of exposure and the magnitude of the fluid shear stress. Exposure to larger fluid shear stress leads to faster and finally more pronounced reorientation of the actin cytoskeleton (Wechezak et aL, 1985; Ookawa et al., 1993). Despite the strong relationship between the mechanical stimulus applied on ECs and their internal reorganization, the mechanisms underlying the cytoskeletal remodeling from an unordered to a highly ordered state remain unknown (Ando and Kamiya, 1993; Davies and Tripathi, 1993; Girard et al., 1993). In this article, we propose a mechanism accounting for the dynamics of endothelial actin cytoskeleton reorganization under the influence of fluid shear stress.

A similar goal was addressed in Sherratt and Lewis (1993). Their work concerned a spontaneous alignment of actin filaments as a response to local anisotropies of external or internal stress fields. In this nondynamic model, the spontaneous response of the actin meshwork to stress fields depends on a single parameter representing the resistance of the cytoskeletal network of actin filaments and binding proteins. Thus, the role of actin-binding proteins and their interactions with microfilaments was not considered explicitly in this model. The role of dynamic molecular dynamics in cytoskeletal reorganization has been previously considered in theoretical frameworks. In Dufort and Lumsden (1993), actin polymerization and depolymerization, as well as interactions between regulatory proteins and microfilaments, were studied in a cellular automaton model, displaying both temporal and spatial dynamics of actin cytoskeleton. In another model, by

MODEL FOR ALIGNMENT OF ACTIN FILAMENTS 1031

Civelekoglu and Edelstein-Keshet (1994), the formation and switch between various cytoskeletal structures mediated by activation or inactivation of different types of actin-binding proteins were considered. These two latter models provide great insight into how molecular mechanisms alone can regulate cytoskeletal reorganization. However, they do not account for the effect of external mechanical stimuli such as fluid shear stress, which constitutes the main goal of the present work.

In this article, we modify and extend the model proposed in Civelekoglu and Edelstein-Keshet (1994) describing cytoskeletal remodeling via molecular mechanisms, to account for an external mechanical stimulus. Toward this goal, we propose a mechanism for the transduction of fluid shear stress on actin filaments composing the submembraneous actin cytoskeleton. Numerical simulations of the resulting system of integro-partial-differential equations show how the alignment phenomenon depends on the magnitude of fluid shear stress and on the relative amount of filaments that interact with the cell membrane. Finally, we suggest experiments to test the validity of the assumptions and the predictions of the model.

2. The Model.

2.1. The modeling approach. The dynamics of the actin cytoskeleton are governed by a great number of interacting molecules which regulate the formation and binding of filaments in various configurations, leading to an organization at the macromolecular level (Pollard and Cooper, 1986; Cooper, 1991). To describe cytoskeletal reorganization under the influence of fluid shear stress, we adapt the mathematical description of the dynamics of the actin filaments and the binding proteins given in Civelekoglu and Edelstein-Keshet (1994), and further describe the attachment of actin filaments to the cell membrane via transmembrane proteins, and the effect of an external force exerted on the membrane. In the model, we view the actin cytoskeleton as a pool of interacting molecules. As we are interested in the orientation of the actin filaments and not in their spatial distribution, we restrict our attention to the distribution of filaments in one angular dimension. The same approach was used in Edelstein-Keshet and Ermen- trout (1990) and Civelekoglu and Edelstein-Keshet (1994), providing an easier mathematical treatment of complex biological situations. During the process of cytoskeletal reorganization, no significant variation in the total amount of actin filaments is observed (Satcher, 1993). Therefore, we do not explicitly include the polymerization or depolymerization of actin filaments, and we assume a constant average filament length. That is, we consider a treadmilling case in which net polymerization equals net depolymerization (Wegner, 1976).

1032 A . S U C I U et al.

2.2. Basic assumptions and definitions. We distinguish among four different types of actin filaments (Fig. 1 and Table 1). Filaments of type L and M L are those which are not connected to others by actin-binding proteins, and thus are called "free." Filaments of type B and M8 are those connected to each other via parallel binding proteins and are therefore called "bound." Free filaments are analogous to the distributed actin network, whereas bound ones represent the actin bundles (stress fibers). Further- more, we distinguish between filaments that are not connected to the cellular membrane (types L and B) and those that are via membrane-binding proteins (types M L and M8). The actin filaments are found at tached to the cell membrane with their barbed (fast-polymerizing) end forming a nearly orthogonal connection with it (Cooper, 1991). This is reflected in the model by restricting the variables of type M L and M B to orientations between 0 ~ and 180 ~ A complete list of the variables and the parameters used in the model is given in Tables 1 and 2.

We describe the following dynamic properties of actin filaments in the equations:

1. Binding and de tachment of filaments to and from the cell membrane . 2. Reorientat ion of actin filaments induced by fluid shear stress. 3. Binding and unbinding of actin filaments via parallel binding proteins. 4. Random reorientat ion of free actin filaments.

�9 T ~ x , , � 9 L(0,t) , �9

B(0,t) , ~ "

I~ " - " ~ I / I~ M '0,t)

= 180 ~ CeU m e m b r a n e 0 = 0 ~

Figure 1. Four types of actin filaments representing the actin cytoskeleton and the bundling proteins. L = free filaments not attached to the membrane; B = bound filaments not attached to the membrane; M L = free filaments attached to the membrane; M B = bound filaments attached to the membrane; V = bundling proteins; �9 = transmembrane proteins to which the filaments of type M L and M B are connected. The angle of orientation of a filament, 0, is the angle it forms with the cell membrane.

MODEL FOR ALIGNMENT OF ACTIN FILAMENTS

Table 1. Variables of the model

1033

t 0

L(O, t)

B(O, t)

ML(O, t)

Ms(O, t)

K(A O)

Time Angle of orientation with respect to the direction of the

cell membrane and the flow Concentration of free actin filaments at orientation 0 and time t,

with 0 ~< 0 ~< 360 ~ Concentration of bound actin filaments at orientation 0 and time t,

with 0 ~< 0 ~< 360 ~ Concentration of membrane-attached free actin filaments at

orientation 0 and time t, with 0 ~< 0 ~< 180 ~ Concentration of membrane-attached bound actin filaments at

orientation 0 and time t, with 0 ~< 0 ~< 180 ~ Angle-dependent rate constant for binding of a filament in presence

of parallel actin-binding proteins

In the following subsections we describe the mathematical formulation of these dynamic processes and introduce the model parameters.

2.2.1. Binding and detachment of filaments to and from the cell membrane. The polar attachment of actin filaments to the plasma membrane can be accounted for by an angle-dependent attachment function x(O). In the model, we assume the following form of the attachment function to reflect this polarity: x(O) --- K.sin 0 (0 ~< 0~< 180~ Here, K is a constant representing the maximal attachment rate. As this attachment process is reversible, we assume a constant dissociation rate, qJ (Table 2).

2.2.2. Reorientation of actin filaments induced by fluid shear stress. Actin filaments of the endothelial cytoskeleton exhibit numerous contacts with the apical cell membrane (Satcher, 1993). They are attached to integral membrane proteins embedded in the phospholipid bilayer, directly or indirectly through one or more intermediary proteins (Geiger, 1989; Kauf- mann et aL, 1992; Luna and Hitts, 1992; Pavalko and Otey, 1994). These transmembrane proteins, or chains of proteins, physically link the extracel- lular milieu to the cytoskeleton. Their function as mechanoreceptors, responsible for the transmission of forces across the cell membrane, has been reported in the literature (e.g., Wang et al., 1993).

Integral membrane proteins are highly mobile molecules and are known to undergo Brownian motion within the plasma membrane (Ishihara and Jacobson, 1993; Kusumi et al., 1993). Therefore, we propose that under the effect of fluid shear forces, these transmembrane proteins are dragged along the direction of flow. We further assume that the translocation of these proteins in turn induces a rotation of the actin filaments that are attached to them at the inner surface of the membrane (Fig. 2). The form of the resulting angular velocity associated with this drift, as well as its


%

e4

;>

~ m ~ ~ ~ . =

~ . .= m ~ ~ ~,~

~.~,~ = ~ ~ ~ ~ . : , ~: ~ ~ ~

= ~ = ,- =~ " ~ ' ~ , . ~ . ~

~ ~:.~ ~

T T

" ~ : I . : : I .

0


/ ~ ~0) ** v sin(e)

0 = 180 ~ 'U - ~ 0 = 0 ~ \

Blood flow ~"~ y l

Figure 2. The mechanism leading to the rotation of filaments: The fluid shear force r applied to the luminal surface of the endothelial cells drags the transmembrane protein along the liquid-like membrane. This induces a rotation of the filament attached to it on the cytoplasmic side. The angular drift velocity to(O) of this rotational motion is proportional to r and to sin O, where 0 is the angle between the filament and the cell membrane.

dependence on both fluid shear stress and the average filament length is given in Appendix A. Assuming that the flux associated with this angular drift of filaments is a product of density My(O, t) (respectively M~(O, t)) and velocity tOML(O) (respectively tOMB(O)), it can be expressed as:

JML( O, t) = toML ( 0 ) "ML( O, t) = VML" sin 0"My( O, t)

JMB( O, t) = tOMB( 0 )" MB( O, t) = VMB" sin 0"MB( O, t)

(1)

(2)

where VML (respectively VMB) represents the maximal amplitude of the angular velocity induced by the fluid shear stress, ~- (Appendix A). This leads to a convectional drift for the filaments of type M c and M B in the angular space as follows:

,9 0 --~ML( O,t) = VO{JMc( O)} = -~{VML sin O.ML( O,t)}

d O V, -~MB( O,t) = VO{JMB( O) } = --~{ M. sin O'M.( O,t)}

(3)

(4)

We assume that VMB << VML , since the rotational motion of filaments of t y p e M s is considerably more restricted than those of type M E , owing to the attachment of filaments of type M B with other filaments.

2.2.3. Binding and unbinding of actin filaments via parallel binding proteins. Actin filaments in ECs are often found to be organized in parallel with each other. This suggests that in the endothelial cytoskeleton, actin-binding proteins promoting parallel attachment of microfilaments are present in higher concentrations or have higher binding affinities than those which promote orthogonal attachments (Weeds, 1982; Wong et al., 1983; Stossel


et al., 1985). In Civelekoglu and Edelstein-Keshet (1994), it was shown that both orthogonal and parallel patterns can coexist. A higher concentration or binding affinity of parallel binding proteins always resulted in reorganization of filaments in bundles, demonstrating that at a low concentration, orthogonal binding proteins have little or no influence on this reorganization phenomenon. For this reason, we consider only the interactions between parallel binding proteins and actin filaments in this model.

The binding and alignment of filaments via parallel binding proteins, leading to exchange between the bound and the free filaments, were previously considered in Civelekoglu and Edelstein-Keshet (1994). Here, we use the same convolution integral to represent this process, such as:

0 2zr - -L (O , t )=-p f lL (O , t ) [ K(O-O')B(O',t)dO'=-pflL(K*~) (5) 3t Jo

where p and /3 are the concentration and binding affinity of parallel binding proteins, and K(O) is the angle-dependent kernel representing the probability of binding of two filaments via parallel binding proteins as given in Civelekoglu and Edelstein-Keshet (1994). The dissociation rate of the actin-binding proteins connecting the filaments is represented by a constant, ~b, and leads to an exchange between the bound and free filaments. The reader is referred to Civelekoglu and Edelstein-Keshet (1994) for a more detailed description of such terms.

2.2.4. Random reorientation of free actin filaments. As in Civelekoglu and Edelstein-Keshet (1994), we assume that the free filaments undergo rotational diffusion with diffusion coefficients tz/~ and tzML, respectively. This random reorientation is described for both types of free filaments (L and M c) by the Laplacian operator in one dimension, 0, as follows:

032 032 tzL-~L(O,t) and tzML-~ML(O,t)

Only the free filaments are allowed to undergo rotational diffusion, and not the bound filaments, since the frictional forces in the cytoplasm limit this effect for large macromolecules. Further, we allow different diffusion coefficients for free filaments (L) and free filaments attached to the membrane (ML) , /z L and /zMs respectively. Note that these filaments do not necessarily undergo random reorientation in the conventional sense; rather, this angular diffusion accounts for the randomizing influence of various biochemical processes such as cytoskeletal turnover or breaking and annealing of filaments.


2.3. Model equations. The following system of integro-partial-differential equations depicts the temporal evolution of the molecular interactions of four types of filaments together with the influence of an extemally applied force, as described above:

0 - - L ( O, t) = - p i lL( K*fl ) - p[3L( K ' L ) - p i lL( K * M L) - p i lL( K * M B) Ot

+ r - x ( O ) L + OM L + l . % - - O 2 L

(6.1) 002

0 - - B ( O, t) = p i l L ( K ' L ) + p f l B ( K * L ) + p f lML(K*L) + p f lMB(K*L) 3t

- ~oB - x ( O ) B + O M B (6.2)

O 3 --~ ML( O , t ) = - flpML( K* L ) + q~MB - OM L + x ( O ) L - --~ ( O)ML( O ) ML )

02ML +/~ML 002 (6.3)

0 0 - ~ M u ( O,t) = oflML( K * L ) - r B - OM B + x ( O)B - ~-~( OJMB( O)MB )

(6.4)

with

Mtota l = foa~r {L(O, t ) + B ( O , t ) + ML(O,t ) + MB(O, t ) }dO= constant. (6.5)

The system [Eqs. (6.1-6.4)] together with the conservation equation [Eq. (6.5)] forms a mass-balance equation including the continuum representa- tion of a random walk, exchange between different types of filaments via interactions, and a convective flux of the filaments in angular space. Because we begin to observe the system starting at a given initial distribution, Equations (6.1-6.4) form an initial value problem with the following Dirichlet and Neumann boundary conditions:

L( 0 = 0, t) = L( 0 = 360 ~ t)

B ( O = O,t) = B ( 0 = 360~ t)

0 t) 0 = 0 , 1 8 0 ~ 09 t) 0 = 0 , 1 8 0 ~ -~ML(O, = --~MB(O, =0

(7)

(8)

(9)


Note that we have periodic boundary conditions for the filaments of types L and B [Eqs. (7) and (8)], and no-flux conditions for the filaments of types M L and M~, which are limited to orientations O, with 0 ~< 0 ~< 180 ~ owing to their polarity in binding to the membrane [Eq. (9)].

2.4. Numerical simulations. The system of equations (6.1-6.4) was nondimensionalized, and numerical solutions were found using finite dif- ference schemes with a fixed time step At, a forward time centered space scheme for the Laplacian and an upwind differencing scheme for the convection terms (Press et al., 1987). The Courant condition was used as the stability criterion for the employed numerical methods (Press et al., 1987). The parameters used in the simulations were calculated based on values found in biological literature in raw form or taken from previous models (Pollard and Cooper, 1986; Dufort and Lunsden, 1993). Table 2 lists the standard values of the parameters. The time step At corresponds to the time scale of the rate constants of biomolecular interactions (in the order of 0.01-0.1 s). Typically, after several thousand iterations, the simulations yielded stable stationary solutions.

3. Results. Here, we summarize the results of the numerical simulations in which the initial densities of all four filament types were uniform with small random deviations. The shape of the distributions of the stationary state of the system depends only on the set of parameters chosen, and not on the specific shape of the initial distributions. For the same set of parameters, initially nonuniform filament distributions evolve toward the same stationary solution as those started at uniform initial distributions.

3.1. Filaments reorientation. The time evolution of the angular densities of the four types of actin filaments leading to the reorientation of all four types in orientations close to 180 ~ is presented in Fig. 3, which shows the initial homogeneous filament distributions, the state of the system at an intermediate time point, and the final configuration which is the stationary solution of the system. Note that in the final configuration, the membrane- attached filaments (M E and MB) have accumulated in orientations close to 180 ~ The filaments of type B and L, which are not attached to the membrane, have also accumulated in orientations close to 180 ~ . This anisotropy is more pronounced for bound filaments and less developed for free filaments, owing to rotational diffusion which competes with reorientation, as a result of the external force.

These results correspond to the experimentally observed reorientation of the cytoskeletal actin filaments in endothelial cells after sustained exposure to flow (Wechezak et al., 1985; Ookawa et al., 1993). The angular distributions of the four types of actin filaments in the final state of the system are

e,/

E ~s r : O

t L 0.1


0,1

0

0

. . . . L !'+ ,,,', , ~ , , ' , , / ' , ,,+ ,,',; y__.;+,; +,

. . . . . interm. " ~ . ."

--station. -- ' , ,~.~X I I i I i I I~

9 0 1110 2 7 0 3110 O [don)

"-"-": i~ittiae~l" A B

i ' ~o ;~o zro 3~o O [de0]

.•p 0.8

i

0 . 4

0 . 2

:Y

..,.~

| I i i 9O 180

0 [aeo]

M L

- - + i n i U a l

. . . . . interm. stat ion.

I a ZTO SIlO

0 . $ S

O.Z8

O.Zl

0.14

0.07

v ,/,~ v t::~:l , / " I] ;"~ ..

i i i i 9 0 1 8 0

e [d~l

M B

- - - i n i t i a l . . . . . in term. - - s t a t i o n .

I a

2 7 0

Figure 3. Three steps in the t ime evolution of the angular densities of four types of actin filaments. L = free filaments; B = bound filaments; M L = membrane- at tached free filaments; and M B = membrane-a t tached bound filaments. Plotted are angular filament densities versus the angle of orientation, 0, of filaments. Dashed lines = initial homogeneous filament distribution. Dot ted lines = state of the system at an in termediate time. Solid lines = final stationary solution after N = 1200 iterations. The parameter values used for these numerical simulations are the standard values listed in Table 2. The initial density for all four types of filaments is a homogeneous distribution with 20% random devia- tion. The mean levels are L(0 , t = 0) = B(0 , t = 0) = ML(0 , t = 0) = MB(0, t = 0) = 0.25 /xM. The intermediary and stationary state correspond to N = 600 and 1200 iterations, respectively, with t ime step At = 0.075. The angular grid size is z lO= 10 o.

SIlO

analogous to the angular distribution of actin filaments in cells exposed to flow; i.e., filaments are found preferentially parallel to the direction of flow, corresponding to 0 ~ and 180 ~ , in the model. The distribution of free filaments qualitatively reproduces the experimentally observed compression of the distributed actin network (Satcher, 1993). In addition, the more pronounced parallel reorientation of bound filaments close to 180 ~ represents the higher degree of parallel alignment of actin bundles (stress fibers) in the direction of flow in flow-exposed cells.


The fact that filaments which are not attached to the membrane (and thus not directly exposed to the fluid shear forces) also reorient parallel to flow demonstrates that the effect of fluid shear forces on the membrane can be transmitted into the entire cytoskeleton via molecular interactions between filaments and the binding proteins alone.

The simulations also show that the peaks at 180 ~ in the final distributions disappear when the external force is removed (i.e., VML = VMB = 0), leading to homogeneous distributions in most cases. This indicates that this reorientation process is reversible, which was previously observed experimentally (Ando and Kamiya, 1993).

3.2. Varied amplitude of shear stress. We also investigated the response of the cytoskeleton as a function of the magnitude of the applied shear stress. For this purpose, numerical solutions of the equations were found for three different values of the maximal angular velocities, namely, Vlo w, Vmedium , and Vhig h (Fig. 4). Recall that VML and VMB represent the maximal flux velocities proportional to fluid shear stress z.

The accumulation of all types of filaments in orientations close to 180 ~ is already observed in the case of the medium-level velocity (Vmeaium); however, it is clearly more pronounced with a 10-fold increase in velocity (Vmgh), indicating the dependence of cytoskeletal reorganization on the magnitude of the fluid shear stress. The accumulation close to 180 ~ is especially pronounced for the membrane-attached filaments and for the bound filaments, indicating the high degree of parallel alignment of microfilament bundles in the cytoskeleton. However, even in the case of high velocity, the angular distribution of the free filaments, representing the distributed actin network, maintains a certain degree of angular isotropy due to diffusion. For a 10-fold decrease in velocity (Vlow), no reorientation of filaments was observed, and the final stationary state of the system only slightly deviated from the initial distribution. This suggests the existence of a shear-stress threshold for cytoskeletal reorganization. Note, however, the accumulation of filaments of type L and B to angles between 0 ~ and 180 ~ and a slight accumulation of filaments of type M L and M s around 90 ~ as a result of the polarity of attachment of filaments to the membrane and the form of the attachment function X(0).

These results indicate the dependence of cytoskeletal reorganization on the magnitude of the fluid shear stress applied to the luminal surface of the endothelial cells, in agreement with experimental observations (Wechezak et al., 1985; Ookawa et aL, 1993). In addition, the existence of a shear-stress threshold for the occurrence of cytoskeleton remodeling as a response to applied flow corresponds with the experimental observation that endothelial cells align only above a certain shear stress level (Dewey et al., 1981; Zhao et al., 1995). It seems that below this level, the intrinsic molecular dynamics of the cytoskeleton predominates the effect of external forces.


�9 l ~ 0 . U

r

t~ ~=

0 P

IJ.

0.22

- - : i n i t i a l ' - / ' ~ - " - ..... viow / \ L . . . . . V medium / \ . / . , , \

...... ,.

" , o �9 o j . f ."----7" . . . . . . . . .

j i"

#t t"

I , I , I

O [degl

S

I, ~=

r162 1

- - - initial ..... V l o w

. . . . . V medium V high

#.,r

180 270 $110

O [ d ~ l

, ~ 1.7S

i 1.S

4-I

i 1.25

4:: o.Ts

E 0.15

0

I ,1 M' 1 l- . . . . . ,,:. I I - - - i n i t i a l -I t . . - " .-" "'/.-.J . . . . . V low 1 F"': - - - . . - z - - - / - - t . . . . . v m~ium 4

. . . . . . "" ~ V high

0 90 180 270 360

O [deg]

~ 0.42

; 0.2!

X

i~.~7~ ~ ~i B - - - - ; 7 - - , . , .:'"'" i / "4 ..... v Io* �9 . '" / I . . . . . V medium , , . - -"" ~ V high

90 180 270 SSO

6 ld~ l

Figure 4. Dependence of the reorientation of the four types of actin filaments on the magnitude of the angular velocities of membrane-attached filaments (VMr and VMB). Shown are the angular distributions of filaments at the stationary state of the system obtained for N = 1200 iterations, for three different values of VML and VMB: dash-dotted lines for medium velocity (Vmecli,m) , solid lines for high velocity (Vhigh) , and dotted lines for low velocity (I/tow)" VML = Vmedium = 0.17 rad/s; Vhig h = 1.7 rad/s; Vzo w = 0.017 rad/s; and VMB is 10% of VML in all three cases. All other parameters are as in Fig. 3. The initial filament distributions are the same for all three cases.

3.3. Varied amounts of filaments. We also investigated cases in which the relative amount of membrane-attached and total amount of filaments, m, was varied to analyze the sensitivity of the model to the binding affinity of actin filaments to transmembrane proteins. This relative amount m is defined as:

~Tr 1

MMB +MML J0 ML(O,t) + MB(O,t)dO m = = (10)

Mtotal Mtotal

with Mtota 1 given in Equation (6.5).


In Fig. 5, we show the final s ta t ionary solut ions o f the m o d e l equa t ions for t h r e e d i f fe ren t values o f the ra t io m - - n a m e l y , m = 0.33, 0.1, and 0.01. In all t h r ee cases, b o t h types o f m e m b r a n e - a t t a c h e d f i laments a ccu m u la t ed close to 180 ~ owing to the i r d i rec t exposu re to ex te rna l forces. H o w e v e r , only fo r the h igh and m e d i u m level o f m was the re a p r o n o u n c e d o r i en ta - t ion o f f i laments o f types L and B n e a r 180 ~ F o r a low va lue o f m, f i laments o f type L and B did no t devia te m u c h f r o m the i r initial h o m o g e - n e ous dis tr ibut ion: T h o s e o f type L exhib i ted a small t e n d e n c y to reor ien t , whe reas those o f type B showed an i r regu la r pa t te rn .

This resul t d e m o n s t r a t e s tha t r eo rgan i za t i on o f the act in cy toske le ton u n d e r the inf luence o f ex te rna l forces is sensit ive to the re la t ive a m o u n t o f

~ 0 . 7

.~ 0.e

~ 0.s E ~ 0.4

lu

0.2

0.1

0 0

..... initial . . . . m - 0.33 i ~

- - - - m - 0 . 1 0 i i I

" ~ m - 0 .01 t \

/ i ~ " ~ . ~ ._- . . . . . . ~ ~ - ,

~ . o ~ / /

, f -

g I * I

L

180

O [degl

\ , %

%

2 StlO

~ 0 . 9

E "~ 0.11

,,In

0.3

0 0

- , , !

..... initial i t I I - - - m - 0 . 3 3 ! i B - - - ' m - 0 1 0 J l ~ I

1 ' , I ~ m - 0 . 0 1 I I 1,1

t , " t

;I i'

90 1~0 170 3 6 4 3

0 [(leg]

~ 0.6

i 0.41;

E l i=

C

. ~ 0.15

I

J /I i !s

i t I l

t : t :

t ! / /

. . . . c . - " . - "

90 ISO e [aegl

M L

..... initial - - ' m - 0 . 3 3 . . . . . m - 0 . 1 0 ~ m - 0 . 0 1

| 270 380

0.4

0.1

i o.a

o

E

o 0

. . . . . . . . . . . . . . . . . . 1-" "~ !

I I I ! ;

t ; t ;

I �9

i I 7 s .~

90 180 e [deg]

M B

. . . . . initial - - ' m - 0 . 3 3 . . . . . m - 0 . 1 0 ~ m - 0 . 0 1

i , 270

Figure 5. Dependence of the reorientation of four types of actin filaments on the parameter m, representing the relative amount of membrane-attached filaments [Eq. (6.5)]. The stationary solution of the system after N = 2400 iterations for three different values of m is shown, m = 0.33 in dashed lines; m = 0.1 in dash-dotted lines; and m = 0.01 in solid lines. VML and VMB are the high-velocity magnitudes used in Fig. 4. All other parameters are as in Fig. 3. The upper parts of the peaks for the filament densities of type M L, M B, and B are not shown, for better visibility of details of the graphics. The actual plots for these filaments are identical with the plots for Vhig h in Fig. 4.

SIlO


membrane-attached filaments. For the parameters used in the simulations, we found that a relatively low amount of membrane-attached filaments, < 10% of the total amount, was sufficient to transmit the effect of the external force into the entire cytoskeleton.

4. Discussion. The model presented in this article describes the reorganization of the actin cytoskeleton in endothelial cells exposed to flow-induced shear stress. To our knowledge, this is the first dynamic model to account for both the molecular interactions of the actin cytoskeleton and the effect of an external force. The description of the filament alignment phenomenon in ECs with a 10-parameter model represents a great simplifica- tion of the many biochemical and physical processes involved. However, the model reproduces qualitatively the main features reported in the literature.

Gathering realistic values for the parameters used in our model was a difficult task, for two reasons. First, values found in the literature were measured under different conditions and for different cell types; and second, since the model is a two-dimensional (2D) analogue of the real situation, 3D values adapted from the literature can be used only to get a relative order of magnitude of the parameters of the model. Therefore, extrapolating quantitative results from the model, such as the real time scale for example, is difficult and beyond the scope of this article.

The key element of the model is the physical mechanism responsible for the transmission of the external forces to the cytoskeleton. We assumed that this is due to a drag of transmembrane proteins within the liquid-like membrane caused by fluid shear forces, leading to a rotation of actin filaments attached to them and eventually setting the directional remodeling of the cytoskeleton. This assumption is based on existing knowledge about the mobility of integral proteins within the plasma membrane and their interactions with the actin cytoskeleton (Kusumi et al., 1993). In this simple mechanism, the net translocation of these proteins results from a balance between the fluid shear forces propelling the membrane proteins and rectifying their random thermal motion preventing them from diffusing backward, and the load force opposing the motion, owing to the association of integral membrane proteins with the cytoskeleton.

To our knowledge, this particular process has not yet been observed experimentally. Therefore, we suggest the following experiments to test the validity of the force transduction mechanism proposed in our model:

1. Measure the drift of transmembrane proteins under the influence of flow-induced shear forces.

2. Verify that the inhibition of the interactions between the actin cytoskeleton and the transmembrane proteins can block the proposed transduction mechanism.


Profound effects of other external mechanical stimuli on the remodeling of endothelial cells, such as pulsatile flow or cyclic strain, have been previously reported in the literature (Helmlinger et al., 1991; Zhao et al., 1995). In the framework of this model, the effect of such mechanical stimuli on the endothelial cytoskeleton can also be accounted for. To this end, it will be necessary to modify the equations to respect the nonautonomous nature of a pulsatile flow or to include a spatial variable to account for the cyclic strain.

In Civelekoglu and Edelstein-Keshet (1994), it was shown that the molecular interactions alone are sufficient to promote the formation of cytoskeletal patterns such as bundles or orthogonal networks of filaments. However, in that report, the directions of the patterns that formed, e.g., the main axis of the bundles, was found to be dependent on the initial conditions and could not be predicted by the model. In this article, we show that a slight perturbation induced by fluid shear stress on a small fraction of filaments successfully triggers this reorganization phenomenon and eventually determines the main axis of the forming cytoskeletal patterns, such as the one given by the direction of flow.

In summary, our model suggests that a mechanistic drift of actin filaments anchored in the membrane combined with the cytoskeletal dynamics may be sufficient to explain the remodeling of endothelial cells as a response to fluid shear stress.

APPENDIX

Rotation of Actin Filaments Induced by Fluid Shear Stress. We estimated the angular velocity oJ(0) of the shear stress-induced rotation of an actin filament by methods of classical mechanics and polymer dynamics (Doi and Edwards, 1986). The combination of the fluid shear force and the frictional forces in the cytoplasm results in a complex behavior of oJ(0) as a function of 0.

In the case of high frictional forces both perpendicular and parallel to the rotating filament, reflecting its entanglement in the cytoskeleton and its lateral confinement with respect to the membrane, this functional behavior can be approximated as being proportional to sin 0. We assumed this particular case in the framework of the model with the following angular velocity for the rotation of filaments:

r = VuL-sin 0c~ -sin 0r .sin 0 (A1) L " . ~ s

Here, z denotes the value of the fluid shear acting on the cell membrane, 0 is the angle between the filament and the cell membrane, r/s is the viscosity of the fluid phase of the cytoplasm (Luby-Phelps et al., 1986), and L is the length of an actin filament considered as a rigid rod. [The average length of actin filaments in endothelial cells is on the order of i /~m (Satcher, 1993), which is significantly shorter than their persistence length (Kiis et al., 1993; Janmey et al., 1994)]. Here, we consider that the force, F~, acting on the end of a filament is proportional to z, and to the surface A (exposed to shear stress) of the transmembrane protein attached to the filament. The friction coefficient ~friction, accounting for the

MODEL FOR ALIGNMENT OF ACrlN FILAMENTS 1045

frictional forces in the cytoplasm due to the entanglement of an actin filament with its surrounding, depends highly on the microfilament concentration and on the length L of the filaments with an exponent a in the order of 6 (Doi and Edwards, 1986).

This work was supported by the CHUV-EPFL-UNIL Biomedical Engineer- ing Collaboration Research Program.

REFERENCES

Ando, J. and A. Kamiya. 1993. Blood flow and vascular endothelial cell function. Front. Med. Biol. Eng. 5, 245-264.

Burridge, K., G. Nuckolls, C. Otey, F. Pavalko, K. Simon and C. Turner. 1990. Actin-membrane interaction in focal adhesions. Cell Diff. Dev. 32, 337-342.

Civelekoglu, G. and L. Edelstein-Keshet. 1994. Modeling the dynamics of F-actin in the cell. Bull. Math. Biol. 56, 587-616.

Cooper, J. A. 1991. The role of actin polymerization in cell motility. Ann. Rev. Physiol. 53, 585-605.

Davies, P. F. and S. C. Tripathi. 1993. Mechanical stress mechanisms and the cell (an endothelial paradigm). Circ. Res. 72, 239-245.

Dewey, C. F., S. R. Bussolari, M. A. Gimbrone and P. F. Davies. 1981. The dynamic response of vascular endothelial cells to fluid shear stress. J. Biomech. Eng. 103, 177-185.

Doi, M. and S. F. Edwards. 1986. The Theory of Polymer Dynamics. Oxford: Clarendon Press. Dufort, P. A. and C. J. Lumsden. 1993. Cellular automaton model of the actin cytoskeleton.

Cell Motil. Cytoskel. 25, 87-104. Edelstein-Keshet, L. and G. B. Ermentrout. 1990. Models for contact-mediated pattern

formation: cells that form parallel arrays. J. Math. Biol. 29, 33-58. Franke, R. P., M. Grille, H. Schnittler and D. Drenckhahn. 1984. Induction of human

vascular endothelial stress fibers by fluid shear stress. Nature 307, 648-649. Fry, D. L. 1976. Hemodynamic forces in atherogenesis. In Cerebrovascular Diseases,

P. Scheinberg (Ed). New York: Raven Press, pp. 77-95. Geiger, B. 1989. Cytoskeleton-associated cell contacts. Curr. Opin. Cell Biol. 1, 103-109. Girard, P. R., G. Helmlinger and R. M. Nerem. 1993. Shear stress effects on the morphology

and cytomatrix of cultured vascular endothelial cells. In Physical Forces and the Mam- malian Cell, J. A. Frangos (Ed). San Diego, CA: Academic Press, pp. 193-222.

Helmlinger, G., R. V. Geiger, S. Schreck and R. M. Nerem. 1991. Effects of pulsatile flow on cultured vascular endothelial cell morphology. J. Biomech. Eng. 113, 123-131.

Ishihara, A. and K. Jacobson. 1993. A closer look at how membrane proteins move. Biophys. J. 65, 1754-1755.

Janmey, P. A., S. Hvidt, J. K~is, A. M. D. Lerche, E. Sackmann, M. Schliwa and T. P. Stossel. 1994. The mechanical properties of actin gel. J. Biol. Chem. 269, 32503-32513.

K~is, J., H. Strey, M. B~irmann and E. Sackmann. 1993. Direct measurement of the wave-vector-dependent bending stiffness of freely flickering actin filaments. Europhys. Lett. 21, 865-870.

Kaufmann, S., J. K~is, W. H. Goldmann, E. Sackmann and G. Isenberg. 1992. Talin anchors and nucleates actin filaments at lipid membranes. FEBS Lett. 314, 203-205.

Ku, D. N., D. P. Giddens, C. K. Zarins and S. Glagov. 1985. Pulsatile flow and atherosclerosis in the human carotid bifurcation. Arteriosclerosis 5, 293-302.

Kusumi, A., Y. Sako and M. Yamamoto. 1993. Confined lateral diffusion of membrane receptors as studied by single particle tracking (nanovid microscopy): effects of calcium- induced differentiation in cultured epithelial cells. Biophys. J. 65, 2021-2040.

Langille, B. L. and S. L. Adamson. 1981. Relationship between blood flow direction and endothelial cell orientation at arterial branch sites in rabbits and mice. Circ. Res. 48, 481-488.

Levesque, M. J. and R. M. Nerem. 1985. The elongation and orientation of cultured endothelial cells in response to shear stress. J. Biomech. Eng. 107, 341-347.


Luby-Phelps, K., D. L. Taylor and F. Lanni. 1986. Probing the structure of cTtoplasm. J. Cell Biol. 102, 2015-2022.

Luna, E. J. and A. L. Hitt. 1992. Cytoskeleton-plasma membrane interactions. Science 258, 955-964.

Meyer, R. K. and U. Aebi. 1990. Bundling of actin filaments by alpha-actin depends on its molecular length. J. Cell Biol. 110, 2013-2024.

National Institutes of Health-United States. 1981. Arteriosclerosis 1981. Bethesda, MD: National Heart, Lung and Blood Institute, National Institutes of Health.

Ookawa, K., M. Sato and N. Ohshima. 1993. Time course changes in cytoskeletal structures of cultured endothelial cells. Front. Med. Biol. Eng. 5, 121-125.

Otey, C. A., F. M. Pavalko and K. Burridge. 1990. An interaction between alpha-actinin and the fl-l-integrin subunit in vitro. J. Cell Biol. 111, 721-729.

Pavalko, F. M. and C. A. Otey. 1994. Role of adhesion molecule cytoplasmic domains in mediating interactions with the cytoskeleton. Proceedings of the Society for Experimental Biology & Medicine 205, 282-293.

Pollard, T. D. and J. A. Cooper. 1986. Actin and actin-binding proteins: a critical evaluation of mechanisms and functions. Ann. Rev. Biochem. 55, 987-1035.

Press, W. H., B. P. Flannery, S. A. Teukolsky and W. T. Vetterling. 1987. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge University Press.

Satcher, R. L. 1993. A mechanical model of vascular endothelium. Ph.D. thesis, MIT, Cambridge, MA.

Sherratt, J. A. and J. Lewis. 1993. Stress-induced alignment of actin filaments and the mechanics of cytogel. Bull. Math. Biol. 55, 637-654.

Stossel, T. P., C. Chaponnier, R. M. Ezzell, et al. 1985. Nonmuscle actin-binding proteins. Annu. Rev. Cell Biol. 1, 353-402.

Wang, N., J. P. Butler and D. E. Ingber. 1993. Mechanotransduction across the cell surface and through the cytoskeleton. Science 260, 1124-1127.

Wechezak, A. R., R. F. Viggers and L. R. Sauvage. 1985. Fibronectin and F-actin redistribu- tion in cultured endothelial ceils exposed to shear stress. Lab. Invest. 53, 639-647.

Weeds, A. 1982. Actin-bonding proteins: regulators of cell architecture and motility. Nature 296, 811-816.

Wegner, A. 1976. Head to tail polymerization of actin. J. Mol. Biol. 108, 139-150. Wong, A. J., T. D. Pollard and I. M. Herman. 1983. Actin filament stress fibers in vascular

endothelial cells in vivo. Science 167, 867-869. Zhao, S., A. Suciu, T. Ziegler, J. E. Moore, E. Bfirki, J.-J. Meister and H. R. Brunner. 1995.

Synergistic effects of fluid shear stress and cyclic circumferential stretch on vascular endothelial cell morphology and cytoskeleton. Arterioscler. Thromb. Vasc. Biol. 15, 1781-1786.

R e c e i v e d 25 N o v e m b e r 1996

R e v i s e d ve r s ion a c c e p t e d 30 Apr i l 1997

Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress

Documents

Transcript of Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress