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A Model for Shear Stress Sensing and Transmission in Vascular Endothelial Cells

A Model for Shear Stress Sensing and Transmission in Vascular Endothelial Cells
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  Biophysical Journal Volume 84 June 2003 4087–4101 4087 A Model for Shear Stress Sensing and Transmissionin Vascular Endothelial Cells Bori M. Mazzag, John S. Tamaresis, and Abdul I. Barakat Department of Mechanical and Aeronautical Engineering, University of California, Davis, California ABSTRACT Arterial endothelial cell (EC) responsiveness to flow is essential for normal vascular function and plays a role inthe development of atherosclerosis. EC flow responses may involve sensing of the mechanical stimulus at the cell surface withsubsequent transmission via cytoskeleton to intracellular transduction sites. We had previously modeled flow-induceddeformation of EC-surface flow sensors represented as viscoelastic materials with standard linear solid behavior (Kelvinbodies). In the present article, we extend the analysis to arbitrary networks of viscoelastic structures connected in series and/or parallel. Application of the model to a system of two Kelvin bodies in parallel reveals that flow induces an instantaneousdeformation followed by creeping to the asymptotic response. The force divides equally between the two bodies when theyhave identical viscoelastic properties. When one body is stiffer than the other, a larger fraction of the applied force is directed tothe stiffer body. We have also probed the impact of steady and oscillatory flow on simple sensor-cytoskeleton-nucleusnetworks. The results demonstrated that, consistent with the experimentally observed temporal chronology of EC flowresponses, the flow sensor attains its peak deformation faster than intracellular structures and the nucleus deforms more rapidlythan cytoskeletal elements. The results have also revealed that a 1-Hz oscillatory flow induces significantly smaller deformations than steady flow. These results may provide insight into the mechanisms behind the experimental observationsthat a number of EC responses induced by steady flow are not induced by oscillatory flow. INTRODUCTION By virtue of their location at the interface between thebloodstream and the vascular wall, arterial endothelial cells(ECs)arecontinuouslyexposedtoahighlydynamicshear(or frictional) stress environment. The ability of ECs to respondand adapt to changes in fluid mechanical shear stress isessential for fundamental processes including vasoregulationin response to acute changes in blood flow and arterial wallremodeling in response to chronic hemodynamic alterations(Pohl et al., 1986; Langille and O’Donnell, 1986). Further-more, abnormalities and/or inadequacies in endothelialresponsiveness to shear stress are involved in the develop-ment of atherosclerosis (Nerem, 1992; Davies, 1995).Recent research has established that shear stress intricatelyregulates EC structure and function. This occurs via a coordi-nated sequence of biological events that begins with veryrapid responses that include activation of flow-sensitive K 1 and Cl  channels (Olesen et al., 1988; Jacobs et al., 1995;Barakat et al., 1999; Nakao et al., 1999) and of GTP-bindingproteins (G-proteins) (Gudi et al., 1996, 1998), changes incell membrane fluidity (Haidekker et al., 2000; Butler et al.,2001) and intracellular pH (Ziegelstein et al., 1992), andmobilization of intracellular calcium (Dull and Davies, 1991;Ando et al., 1988; Shen et al., 1992; Geiger et al., 1992).These rapid responses are followed by stimulation of mitogen-activated protein kinase signaling (Traub and Berk,1998), activation of a host of gene and protein regulatoryresponses (Malek and Izumo, 1994; Resnick and Gimbrone,1995; Garcia-Cardena et al., 2001), and induction of exten-sive cytoskeletal remodeling that ultimately leads to cellular elongation in the direction of the applied shear stress (Deweyet al., 1981; Nerem et al., 1981; Eskin et al., 1984).Beyond being merely responsive to shear stress, ECs re-spond differently to different types of shear stress. For ins-tance, while steady shear stress induces intracellular calciumoscillations and morphological changes in ECs, purelyoscillatory flow (zero net flow rate) does not elicit either of these responses (Helmlinger et al., 1991, 1995). A number of shear stress-responsive genes also exhibit differentialresponsiveness to different types of shear stress (Chappellet al., 1998; Lum et al., 2000; Garcia-Cardena et al., 2001).The notion of differential responsiveness is especially sig-nificant in light of the observation that early atheroscleroticlesions localize preferentially in arterial regions exposed tolow and/or oscillatory shear stress while regions subjected tohigh and unidirectional shear stress remain largely spared(Nerem, 1992; Ku et al., 1985; Asakura and Karino, 1990).The mechanisms by which ECs respondto shear stress andby which they discriminate among different types of shear stress remain to be elucidated. A working model for EC flowresponsiveness has been proposed and is schematicallydepicted in our Fig. 1 (see also Davies and Tripathi, 1993;Davies,1995).Thismodel,whichisinspiredbythetensegrityhypothesis of cellular mechano-responsiveness (Ingber,1993;Ingberetal.,1994),postulatesthatthefluidmechanicalstimulus is sensed by structures at the EC surface that act asflow sensors. Flow sensors may be discrete transmembranemolecules,clustersof such molecules,subdomains of the cellmembrane, or even the entire membrane. Once sensed, the Submitted April 21, 2002, and accepted for publication January 21, 2003. Address reprint requests to Abdul I. Barakat, Ph.D., Dept. of Mechanical &Aeronautical Engineering, University of California, One Shields Ave.,Davis, CA 95616. Tel.: 530-754-9295; Fax: 530-752-4158;   2003 by the Biophysical Society0006-3495/03/06/4087/15 $2.00  flow signal is transmitted via cytoskeleton to variousintracellular sites including the nucleus, cell-cell adhesionproteins, and focal adhesion sites where it is transduced toa biochemical response. This proposed mechanotransductionscheme should be viewed as complementary to the more wellcharacterized receptor-mediated signaling pathways that exist in vascular ECs.We recently developed a model of the deformation of anEC-surface flow sensor in response to different types of shear stress (Barakat, 2001). The flow sensor was modeled asa viscoelastic material with standard linear solid behavior.The results demonstrated that the peak sensor deformationwas considerably larger for steady and nonreversingpulsatileflow than for purely oscillatory flow. It was hypothesizedthat this may constitute a mechanism by which ECs dif-ferentially respond to different types of flow. Becausethe mechanotransduction model shown in Fig. 1 postulatesthat flow sensors on the EC surface are directly coupled tocytoskeletal elements that are in turn coupled to variousintracellular structures, it is essential to extend the results of our previous model to include these couplings. The present article extends the one-body analysis to networks of viscoelastic bodies that represent coupled systems of cell-surface sensors and various intracellular structures. Thedeformations of the various network components in responseto different types of flow and the sensitivities of thesedeformations to various model parameters are presented. MATHEMATICAL DEVELOPMENTFormulation of governing equations Our goal is to develop a mathematical framework to describethe deformations of EC-surface flow sensors and coupledintracellular structures in response to either steady or purelyoscillatory flow. Similar to our previous analysis (Barakat,2001), each structure is modeled as a viscoelastic body withstandard linear solid behavior (Kelvin body) and character-ized by its own set of viscoelastic parameters. As describedelsewhere (Fung, 1981; Barakat, 2001), a Kelvin body con-sists of a linear spring with spring constant   k  1  (spring 1) inparallel with a Maxwell body, which consists of a linear spring with spring constant   k  2  (spring 2) in series witha dashpot with coefficient of viscosity  m . Kelvin bodies aregeneral linear viscoelastic models, and they have beenfrequently used to represent the mechanical behavior of various tissues. Of specific interest to the present formula-tion, recent experimental studies have expressed the vis-coelastic properties of cell nuclei (Guilak et al., 2000),cytoskeletal elements (Sato et al., 1996), and transmembraneproteins (Bausch et al., 1998), in terms of the parameters of a Kelvin body.We begin by deriving the formulation describing thedeformations of   n -Kelvin bodies connected either in series or in parallel and discuss solutions to the governing equationsunder both steady and oscillatory forcing functions. Wesubsequently describe simple networks that consist of com-binations of Kelvin bodies connected in series and paralleland that are used to model mechanical signal transmissionin ECs. Finally, we discuss the implications of the results tooverall EC responsiveness to different types of shear stress. Kelvin bodies in series Fig. 2  A  depicts a system of   n -Kelvin bodies connected inseries. For a forcing function  F  ( t  ) applied to this series of  n -bodies, the force experienced by each body in the serieswill be  F  ( t  ); thus,  F  1 ð t  Þ¼  F  2 ð t  Þ¼¼  F  n ð t  Þ¼  F  ð t  Þ :  (1)The deformation of the entire series is given simply as thesum of the individual deformations: u ð t  Þ¼ + ni ¼ 1 u i ð t  Þ :  (2)As shown elsewhere (Fung, 1981; Barakat, 2001), thedeformation  u i ( t  ) of the  i th Kelvin body given a forcingfunction  F  ( t  ) across this body is governed by the followingfirst-order linear differential equation:  F  1 m i k  2i _  F  F  ¼ k  1i u i 1 m i  1 1 k  1i k  2i   _ uu i ;  (3)where  _  F  F   and  _ uu i  are the time derivatives of   F   and  u i ,respectively. We consider that the force  F  ( t  ) due to fluid flowis applied suddenly at   t   ¼  0 as a step function and that thisforce is sustained for the entire time period considered.Because we wish to investigate the effects of both steady andoscillatory flow, two types of forcing functions are consid-ered. For steady flow, the forcing function takes the form:  F  ð t  Þ¼  F  0 ;  (4) FIGURE 1 Schematic diagram of the working model for endothelial shear stress sensing and transmission. Cell-surface flow sensors (which may bediscrete structures, cell membrane microdomains, or the entire cellmembrane) detect a flow stimulus and transmit it directly via cytoskeletonto various intracellular transduction sites including the nucleus, cell-celladhesion proteins, and focal adhesion sites on the abluminal cell surface. 4088 Mazzag et al.Biophysical Journal 84(6) 4087–4101  whereas for oscillatory flow, the equivalent expression is:  F  ð t  Þ¼  F  0  cos v t  ;  (5)where  v  is the angular frequency of oscillation. For bothtypes of forcing functions, the applied force at   t  ¼ 0 is  F  (0) ¼  F  0 . For a suddenly applied force  F  0 , the appropriate initialcondition for the  i th body in the series is (Barakat, 2001): u i ð 0 Þ¼  F  ð 0 Þ k  1i 1 k  2i :  (6)For a given forcing function (either Eq. 4 or 5), Eq. 3 canbe solved subject to the initial condition in Eq. 6 to yieldthe deformation  u i ( t  ) for each body in the series. Thedeformations of the individual bodies are independent of oneanother and can therefore be solved for separately. Analyticsolutions to the deformation of a single Kelvin body for bothsteady and oscillatory flow are given elsewhere (Barakat,2001). For the sake of consistency with the formulation of the Kelvin bodies connected in parallel, we have opted towrite down the problem in matrix notation as: d  ~ uudt   ¼  D ~ uu 1 ~ cc ;  (7)with the initial condition ~ uu ð 0 Þ¼ ~ uu 0 ;  (8)where we define ~ uu [ u 1  u n 266664377775 ;  D [  k  11 k  21 m 1 ð k  11 1 k  21 Þ  0     0   0   k  1i k  2i m i ð k  1i 1 k  2i Þ    0     0   k  1n k  2n m n ð k  1n 1 k  2n Þ 266664377775 ;~ cc [ 1 ð k  11 1 k  21 Þ k  21 m 1  F  1  _  F  F     1 ð k  1n 1 k  2n Þ k  2n m n  F  1  _  F  F   26666643777775 ;  and  ~ uu 0 [  F  ð 0 Þ k  11 1 k  21   F  ð 0 Þ k  1n 1 k  2n 266664377775 : Kelvin bodies in parallel We can now develop the formulation for   n -Kelvin bodiescoupled in parallel (Fig. 2  B ). There are two fundamentaldifferences between the parallel and series formulations.First, in the parallel case all  n -bodies are constrained todeform equally so that  u 1 ð t  Þ¼ u 2 ð t  Þ¼¼ u n ð t  Þ :  (9)Secondly, the total force acting on the  n -body system isthe sum of the forces acting on the individual bodies so that   F  ð t  Þ¼  + ni ¼ 1  F  i ð t  Þ :  (10)How the total applied force  F  ( t  ) divides among theindividual bodies depends on the particular parameter valuesof each of the Kelvin bodies and is generally not knowna priori. We denote the force splitting coefficient (i.e., thefraction of the total force) for the  i th body by  a i ( t  ), so that theforce  F  i ( t  ) exerted on this body is given by  F  i ð t  Þ¼ a i ð t  Þ  F  ð t  Þ ;  (11)and the individual force splitting coefficients add up to unity.This can equivalently be expressed as1   + n  1i ¼ 1 a i ð t  Þ   ¼ a n  ¼  F  n ð t  Þ  F  ð t  Þ  :  (12)For   i ¼ 1 ; 2 ;  . . .  ; n  1 ;  the governing constitutive relationbetween the applied force and the resulting deformationresembles that given in Eq. 3 and has the form FIGURE 2 Schematic diagram of   n -Kelvin bodies coupled (  A ) in seriesand (  B ) in parallel. Each body consists of a linear spring with spring constant  k  1  in parallel with a Maxwell body, which consists of a linear spring withspring constant   k  2  in series with a dashpot with coefficient of viscosity  m . Shear Stress Transmission in Endothelium 4089Biophysical Journal 84(6) 4087–4101  a i  F  1 m i k  2i d dt  ð a i  F  Þ¼ k  1i u 1 m i  1 1 k  1i k  2i   _ uu :  (13)The initial condition equivalent to that in Eq. 6 is u ð 0 Þ¼ a i ð 0 Þ  F  ð 0 Þ k  1i 1 k  2i :  (14)For the  n th body, the equivalent expressions are1   + n  1i ¼ 1 a i    F  1 m n k  2n d dt   1   + n  1i ¼ 1 a i    F    ¼ k  1n u 1 m n  1 1 k  1n k  2n   _ uu ;  (15)and u ð 0 Þ¼ 1   + n  1i ¼ 1 a i ð 0 Þ    F  ð 0 Þ k  1n 1 k  2n :  (16)Eqs. 13 and 15 can be rearranged as follows. For  i ¼ 1 ; 2 ;  . . .  ; n  1 ; m i  1 1 k  1i k  2i   _ uu  m i k  2i d dt  ð a i  F  Þ¼ k  1i u 1 a i  F  ; (17)and for  i ¼ n ;  m n  1 1 k  1n k  2n   _ uu 1 m n k  2n + n  1i ¼ 1 d dt  ð a i  F  Þ¼ k  1n u   + n  1i ¼ 1 a i  F  1  F  1 m n k  2n _  F  F  : (18)Note that   u (0), the initial deformation, is the same for all of the bodies; therefore, we get   n -equations in the  n -unknowns u (0) and  a i (0) for   i  ¼  1 ;  . . .  ; n  1 :  Once these unknownsare obtained, the force splitting coefficient of the  n th body, a n (0), is determined directly from the constraint given in Eq.12. Eqs. 14 and 16 can now be combined and rearrangedto yield u ð 0 Þ¼  F  ð 0 Þ + ni ¼ 1 ð k  1i 1 k  2i Þ :  (19)Thus, Eq. 14 yields a i ð 0 Þ¼  F  ð 0 Þð k  1i 1 k  2i Þ + ni ¼ 1 ð k  1i 1 k  2i Þ :  (20)Now, the constitutive relations given by Eqs. 17 and 18 incombination with the initial conditions given by Eqs. 19 and20 can be cast in matrix form as d  ~ uudt   ¼  A  1  D ~ uu 1  A  1 ~ cc ;  (21)and ~ uu ð 0 Þ¼ ~ uu 0 ;  (22)where ~ uu [ u ð t  Þ a 1 ð t  Þ  F  ð t  Þ a n  1 ð t  Þ  F  ð t  Þ 2666666437777775 ;  A [ m 1  1 1 k  11 k  21     m 1 k  21 0      0 m 2  1 1 k  12 k  22    0   m 2 k  22 0     0     0   m i k  2i 0     m ð n  1 Þ  1 1 k  1 ð n  1 Þ k  2 ð n  1 Þ    0      0   m ð n  1 Þ k  2 ð n  1 Þ m n  1 1 k  1n k  2n    m n k  2n      m n k  2n 26666666666643777777777775 ;  D [  k  11  1 0      0  k  12  0 1 0     0   0 1 0     k  1 ð n  1 Þ  0       1  k  1n   1       1 26666666666643777777777775 ;  and ~ cc [ 0  0  F  1  m n k  2n _  F  F  2666666666437777777775 : Eqs. 21 and 22 are the governing differential equation andinitial condition, respectively, that must be solved for theunknown vector  ~ uu ð t  Þ , an  n 3 1 vector whose entries are thedeformation  u ( t  ) (which is the same for all  n -bodies) and a i ( t  )  F  ( t  ). Note that we are solving for   a i ( t  )  F  ( t  ) and not explicitly for the force splitting coefficients  a i ( t  ). There aretwo practical reasons for this. When solving for   a i ( t  ) directly,particular choices of the flow, for example the oscillatoryflow of Eq. 5, result in the problem becoming singular periodically. Also, if   A  and  D  are defined differently to solvefor   a i ( t  ) explicitly, they will be functions of time, and thisconsiderably slows down the computations. To avoid theseproblems, we compute  u ( t  ) and  a i ( t  )  F  ( t  ). Because  F  ( t  ) isa known function, it is always possible to find  a i ( t  ) if needed. Model EC networks Now that we have obtained the formulation for   n -Kelvinbodies coupled in either series or parallel, we can consider combinations of these two types of couplings to construct  4090 Mazzag et al.Biophysical Journal 84(6) 4087–4101  simple networks that model the EC mechanotransmissionscheme depicted in Fig. 1. Our analysis will permit determi-nation of the deformation of each component of the networkas a function of time as well as the division of the forceimparted by either steady or oscillatory flow among thenetwork components.The model in Fig. 1 suggests that EC cytoskeleton playsa central role in transmitting the flow signal from the cellsurface to various intracellular sites. Cellular cytoskeletonhas three primary components: actin filaments, microtubules,and intermediate filaments. Actin filaments provide impor-tant structural support and often associate into contractilebundles called stress fibers. Qualitatively, actin filaments aremore rigid than the other cytoskeletal elements and thusrupture at relatively low strain (Janmey et al., 1991); how-ever, actin is rapidly recycled and new filaments reformed.Microtubules, on the other hand, exhibit considerably greater flexibility and are therefore capable of withstanding highstrains (Janmey et al., 1991). Intermediate filaments are not very rigid at low strain but harden considerably at highstrain—ideal behavior for their primary role of providingmechanical support for the nucleus (Janmey et al., 1991).Fig. 3 illustrates two simple model networks that will beused to demonstrate the results of our analysis. The first (Fig.3  A ) consists of a flow sensor on the EC surface (Kelvin body1)that iscoupledtotheECnucleus(body4)viatwo identicalactin stress fibers (bodies 2 and 3). Indeed, stress fiber coupl-ing to structures in the cell membrane such as cell-surfaceintegrins has been demonstrated experimentally (Critchley,2000; Zamir and Geiger, 2001), and their coupling to thenuclear membrane has been speculated to occur (Daviesand Tripathi, 1993). It should be noted that although the flowsensor is depicted in Fig. 3  A  as a discrete structure on thecell surface, it can equally correspond to clusters of suchstructures, microdomains in the cell membrane, or even theentire membrane. Fig. 3  A  also illustrates the networkbreakdown for the simple four-body model cell. The twostress fibers are connected in parallel to one another, and theyare in series with the flow sensor on one side and the nucleuson the other.The second simulated network (Fig. 3  B ) is slightly morecomplex, and it is inspired by experimental evidence that dif-ferent components of cytoskeleton are coupled to one ano-ther, often via various linker proteins. This network consistsof a flow sensor (Kelvin body 1) connected in series to twoactin stress fibers (bodies 2 and 3) that are connected inparallel. Each of the stress fibers is subsequently coupled toa microtubule (bodies 4 and 5), and the two microtubulesareconnectedtothecellnucleus(body6).ThecorrespondingKelvinbodynetworkrepresentationisalsoshowninFig.3  B . Model parameter values For each type of viscoelastic body modeled, the values of thetwo spring constants  k  1  and  k  2  as well as the dashpot coeffi-cient of viscosity  m  must be specified. The baseline valuesused in the simulations are shown in Table 1. The values for actin filaments are based on micropipette aspiration studieson ECs (Sato et al., 1996). The values for the nucleus arebased on recent micropipette aspiration measurements that have demonstrated that the nucleus is three-to-four timesstiffer and approximately twice as viscous as the cytoplasm FIGURE 3 Model networks for endothelial shear stress transmission. (  A )Schematic and Kelvin body representation of a four-body networkconsisting of a flow sensor (body 1) connected to two actin stress fibers(bodies 2 and 3) that are in turn connected to the cell nucleus (body 4). (  B )Schematic and Kelvin body representation of a six-body network consistingof a flow sensor (body 1) connectedto two actin stress fibers (bodies 2 and 3)that are connected to the nucleus (body 6) via two microtubules (bodies4 and 5). Shear Stress Transmission in Endothelium 4091Biophysical Journal 84(6) 4087–4101
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