Pascual Medina
email: pascual.medina@uv.es

María Gloria Segarra

Dolores Mauricio

José M Vila

Pascual Chuan

Salvador Lluch

ROLE OF Ca-ACTIVATED K CHANNELS AND Na,K-ATPase IN PROSTAGLANDIN E-AND E-INDUCED INHIBITION OF THE ADRENERGIC RESPONSE IN HUMAN VAS DEFERENS

Keywords: Electrical field stimulation, Human vas deferens, Norepinephrine, Potassium channels, Prostaglandins, Smooth muscle

HAL is

Prostaglandins are a family of biologically lipid acids synthesized by cyclooxygenase from a common precursor, arachidonic acid. Previous studies have indicated that prostaglandin E 1 (PGE 1 ) and PGE 2 , and other prostanoids, influence adrenergic neurotransmission in the vas deferens from several species [1][2][3][4]. In human vas deferens, the motor innervation is mainly noradrenergic [5,6] and PGE 1 and PGE 2 have been reported to inhibit neurogenic contractions by decreasing the release of norepinephrine from sympathetic nerve endings [7]. The mechanisms whereby PGE 1 and PGE 2 modify neurotransmitter release have received much attention but remain poorly understood. Prostaglandins of the E series inhibit transmitter release by an action on stimulus-secretion coupling and more specifically on the availability of Ca 2+ for the release mechanism [8].

K + channels modulate the adrenergic contractile responses in human vas deferens [START_REF] Medina | Modulation of adrenergic responses of human vas deferens by K + channel inhibitors[END_REF] and several agents have been shown to inhibit neurotransmitter release from nerve endings through a mechanism that involves the opening of prejunctional K + channels, membrane hyperpolarization and reduction in Ca 2+ influx via voltage-activated Ca 2+ channels [10,11]. In vas deferens, it has been demonstrated that K + channels are involved in the prejunctional inhibitory effects of atrial natriuretic factor [12],  2adrenoceptor agonists [13] and sildenafil, an inhibitor of phosphodiesterase 5 [14].

Activation of K + channels stimulates cellular K + efflux [15]. In vascular smooth muscle, it has been demonstrated that an increased extracellular K + concentration ([K + ] o ) evokes rapid hyperpolarization and relaxation [16] by stimulating an ouabain-sensitive Na + ,K + -ATPase [17]. In rat vas deferens, increased extracellular K + concentrations lower than those necessary to induce contraction of the smooth muscle inhibit electrically-induced A c c e p t e d M a n u s c r i p t 4 contractions [18]. A role for increased [K + ] o and Na + ,K + -ATPase in modulation of adrenergic neurotransmission in the human vas deferens has not yet been investigated.

Therefore, the present study was designed to examine the effects of increasing [K + ] o and Na + , K + -ATPase on adrenergic contractions of human vas deferens and the contribution of K + channels and Na + ,K + -ATPase to the inhibitory effects of PGE 1 and PGE 2 on adrenergic neurotransmission.

A c c e p t e d M a n u s c r i p t Committee of our institution and informed consent was obtained from each subject before the study. The specimens were placed in chilled isotonic NaCl, and were divided into ring preparations 3-4 mm long.

Isolated vas deferens preparation

Ring preparations were suspended between two L-shaped stainless steel pins. One pin was fixed to the organ bath wall while the other was connected to a strain gauge (Grass FT03, Grass Instruments Division Astromed, Inc., West Warwick, RI, U.S.A.). Changes in isometric force were recorded by use of Chart v3.4/s software and a MacLab/8e data acquisition system (ADInstruments, East Sussex, U.K.). Each preparation was set up in a 4-ml bath containing modified Krebs-Henseleit solution of the following millimolar composition: NaCl, 115; KCl, 4.6; KH 2 PO 4 1.2; MgCl 2 •6H 2 O, 1.2; CaCl 2 , 2.5; NaHCO 3 , 25; glucose, 11.1; and disodium EDTA, 0.01. The solution was equilibrated with 95% O 2 and 5% CO 2 . The preparations were allowed to equilibrate for 1-h and during this time tension was adjusted to a final tension of 19.6 mN.

Electrical field stimulation was provided by a Grass S88 stimulator (Grass Instruments Division Astromed, Inc., West Warwick, RI, U.S.A.) via two platinum electrodes positioned on each side and parallel to the axis of the ring. Single square wave pulses (0.25 ms pulse duration, 20 Hz, at a supramaximal voltage of 20 Vcm -1 ) were used. The

Experimental procedure.

To study the neuromodulatory action of prostanglandins, when electrically induced phasic contractions were stable (after 15-20 minutes), PGE 1 or PGE 2 (10 -8 -10 -6 M) were added cumulatively to the preparations and the effects of electrical field stimulation were recorded. To examine the role of K + channel activation in the effects of PGE 1 and PGE 2 on electrical field stimulation induced contractions, concentration-response curves to PGE 1 and PGE 2 were established in the presence of one of the following inhibitors: TEA (10 -3 M), a nonspecific K + channel blocker [20], glibenclamide (10 -5 M), a selective blocker of ATP-sensitive K + (K ATP ) channels [21], charybdotoxin (10 -7 M), an inhibitor of Ca 2+ -activated K + (K Ca ) channels of both large-conductance (K Ca 1.1) [22] and intermediate-conductance (K Ca 3.1) [23], iberiotoxin (10 -7 M), an inhibitor of K Ca 1.1 channels [24], or apamin (10 -6 M), an inhibitor of K Ca of small-conductance (K Ca 2) channels [25].

Concentration-response curves to norepinephrine were obtained in a cumulative manner in the absence (control) and in the presence of PGE 1 (10 -6 M) or PGE 2 (10 -6 M) from separate preparations. K + channels activation stimulates K + efflux [15]. As increases in the [K + ] o evoke rapid vascular smooth muscle relaxation and hyperpolarization [16], we studied the adrenergic responses in the presence of increased [K + ] o (10.8-25.8 To determine whether Na + ,K + -ATPase activity is involved in the adrenergic responses of the human vas deferens, the contractions to electrical field stimulation and norepinephrine were recorded before and 15 min after the addition of ouabain (10 -7 -10 -6 M), a Na + ,K + -ATPase inhibitor.

In another group of experiments, the influence of Na + ,K + -ATPase on the PGE 1 -, PGE 2or increasing K + concentrations -induced inhibition of contractions evoked by electrical field stimulation was examined by exposing vas deferens rings to ouabain (10 -6 M) for 15 min before the addition of PGE 1 (10 -8 -3x10 -6 M), PGE 2 (10 -8 -3x10 -6 M) or [K + ] o (10.8-25.8 mM) and the responses to electrical stimulation were recorded.

Drugs

The following drugs were used: norepinephrine hydrochloride, prazosin hydrochloride, tetrodotoxin, tetraethylammonium bromide (TEA), charybdotoxin, iberiotoxin, apamin, glibenclamide and ouabain (Sigma Chemical Co. St. Louis, MO, USA). All drugs were dissolved in Krebs solution, except for glibenclamide and ouabain which were dissolved initially in dimethyl sulphoxide and further diluted in Krebs solution to the proper final concentration. Stock solutions of the drugs were freshly prepared every day, and kept on ice throughout the experiment.

Data analysis

All values are expressed as means ± S.E.M. Contractions are reported as absolute values (mN) or as percentages of control responses. pD 2 values (negative logarithm of the molar concentration at which half-maximum contraction occurs) were determined from 1). Moreover, pretreatment with either ouabain (10 -6 M) or increasing [K + ] o to 25.8 mM were without effect on norepinephrine-induced contractions (Figure 5 and Table 1).

A c c e p t e d M a n u s c r i p t The present study demonstrates a role for K + channels and Na + ,K + -ATPase in the prejunctional inhibitory effects of PGE 1 and PGE 2 on adrenergic responses of human vas deferens. It has been demonstrated that PGE 1 and PGE 2 selectively depress neurogenic contractions in human vas deferens without modifying the contractions to norepinephrine [7]. The lack of effect of PGE 1 and PGE 2 on norepinephrine-induced contractions suggests that postjunctional inhibitory mechanisms are not involved in the neuromodulatory action of these prostaglandins. The inhibitory effect on neurogenic contractions has been attributed to decreased norepinephrine release from nerve endings in the human vas deferens [7].

The mechanisms whereby PGE 1 and PGE 2 induce inhibition of neurotransmitter release remain poorly understood. Several agents could inhibit neurotransmitter release from nerve endings in vas deferens through a mechanism that involves activation of K + channels. In human vas deferens, charybdotoxin-sensitive, iberiotoxin-insensitive, K + channels modulate the adrenergic contractile response by interfering with Ca 2+ entry through dihydropyridine Ca 2+ channels [START_REF] Medina | Modulation of adrenergic responses of human vas deferens by K + channel inhibitors[END_REF]. Furthermore, it has been shown that K + channels are involved in the prejunctional inhibitory effects of atrial natriuretic factor in the rabbit isolated vas deferens [12] and in the  2 -adrenoceptor-mediated inhibition in rat vas deferens [13]. In the human vas deferens, sildenafil, an inhibitor of phosphodiesterase 5, inhibits adrenergic contractions through activation of presynaptic K Ca 1.1 channels [14].

Glibenclamide has been shown to block K ATP channels [21] with an IC 50 in the micromolar range for smooth muscle preparations [26]. In rat vas deferens, the inhibitory action of calcitonin-gene-related peptide (CGRP) on neurotransmission was A c c e p t e d M a n u s c r i p t 12 at least in part mediated by the activation of K ATP channels in sympathetic nerve terminals as shown by the prevention of the neuromodulatory effects of CGRP in the presence of glibenclamide 10 -5 M [27]. Our results in human vas deferens demonstrate that glibenclamide at concentration as high as 10 -5 M did not modify the inhibition induced by PGEs on adrenergic responses. These results rule out the involvement of K ATP channels activation in the inhibitory effects of PGEs on neurogenic contractions of the human vas deferens.

On the other hand, we tested the effects of TEA, a nonspecific K + channel blocker [20] on the neuromodulatory effects of PGE 1 and PGE 2 . It is accepted that blockade with 10 -3 M TEA is sufficient to almost fully inhibit K Ca 1.1 channels [28]. In our experiments millimolar concentrations of TEA prevented the inhibitory effects of PGE 1 and PGE 2 on neurogenic contractions, suggesting a role of K Ca channels in this effect.

To further characterize the member of K Ca channel group involved in the neuromodulatory effects of PGEs we used more selective blockers of K Ca channels.

Apamin has proved to be extremely valuable as a highly specific blocker for which the only known receptors are the K Ca channels of small conductance (K Ca 2) [29]. In our experiments, apamin 10 -6 M, a concentration expected to block completely all subtypes of K Ca 2 channels, failed to alter the inhibition caused by PGE 1 and PGE 2 on neurogenic contractions in human vas deferens. These data indicate that K Ca 2 channels are not involved in the neuromodulatory effects of PGE 1 and PGE 2 .

Charybdotoxin, a blocker of both K Ca 1.1 and K Ca 3.1 channels [22,23], prevented the inhibitory effects of PGE 1 and PGE 2 on sympathetic contractions of the human vas deferens. These results demonstrate that the K + channels involved are sensitive to charybdotoxin. Therefore, we tested the effect of iberiotoxin, a selective blocker of K Ca 1.1 channels [24], on the effects of PGE 1 and PGE 2 on adrenergic contractions. The A previous study in rat vas deferens demonstrated that increasing [K + ] o , at concentrations lower than those necessary to induce contraction of the smooth muscle, completely inhibited the electrically-induced contractions [18], an effect attributed to a desensitization of the rat vas deferens to electrically-induced contractions. Studies of vascular smooth muscle showed that increased [K + ] o in the range 6-16 mM produce hyperpolarization and vasodilatation by stimulating an ouabain-sensitive Na + ,K + -ATPase [17]. In guinea-pig vas deferens relatively high concentrations of ouabain, an inhibitor of Na + ,K + -ATPase, is reported to cause a gradual increase in the norepinephrine release from the peripheral adrenergic neurons [30] and an increase in the sensitivity to norepinephrine [31]. Particularly for the vas deferens, species is thought to play an important role in Na + ,K + -ATPase activity and function [31].

However, the effects of inhibition of Na + ,K + -ATPase on adrenergic responses of human vas deferens have not been investigated. As the resting tension of human vas deferens was not changed by ouabain in our experiments, we suggest that Na + ,K + -ATPase is not normally active in the relaxed vas deferens smooth muscle. Furthermore, ouabain had no effect on norepinephrine-mediated contractions at concentration of 10 -6 M, but increased neurogenic contractions. These results indicate that the enhanced neurogenic contractions induced by ouabain involves the inhibition of Na + ,K + -ATPase activity of the adrenergic nerve endings.

Increasing [K + ] o over the range that inhibits electrical field stimulation-induced adrenergic contractions should also stimulate the Na + ,K + -ATPase by increasing the amount of extracellular K + that binds to the pump [32]. In human vas deferens, ouabain stimulates the Na + ,K + -ATPase in the range that inhibits adrenergic contractions.

Because Na + ,K + -ATPase activity is ultimately responsible for most of the resting membrane potential, the ability of increased extracellular K + to regulate adrenergic contractions could in part be due to its effect on Na + ,K + -ATPase activity. There are two effects of the pump on membrane potential: (i) an electrogenic effect, due to the 3Na + /2K + exchange (this is the smaller, but more rapid effect) and (ii) by maintaining the transmembrane K + gradient the membrane potential is created (this is the more powerful, but slower effect). Our results suggest that in the inhibitory effects of K + on the adrenergic contractions of human vas deferens only the first effect is relevant because in our experimental conditions it is not possible to increase the transmembrane K + gradient by increasing [K + ] o .

However, involvement of other mechanisms/pathways cannot be excluded. In rat vas deferens it has been demonstrated that PGE 2 is released from the epithelium in response to ATP stimulation and it is responsible for mediating the membrane hyperpolarization and inhibition of contraction of smooth muscle cells by activating cAMP-dependent K + channels [37]. However, our results show that PGE 2 inhibited neurogenic contractions but did not exert inhibitory effects on norepinephrine-induced contractions, indicating that PGE 2 acts directly on sympathetic nerve endings.

Recently, it has been demonstrated the proximity and functional interaction between the

An important function of the vas deferens is the transport of spermatozoa from the epididymis for inclusion in the semen. Since  1 -adrenoceptors activation by norepinephrine is required for normal contraction of the vas deferens and consequent sperm ejaculation [38], it seems likely that increased formation of PGE 2 in physiological and pathological conditions such as inflammation [39], may cause aperistalsis of the adrenergically innervated vas deferens by activating K Ca 1.1 channels and Na + ,K + -ATPase and impairment of male fertility. Although the present studies provide evidence that activation of presynaptic KCa1.1 channels inhibits adrenergic contractions of vas deferens, future studies are necessary to elucidate the potential therapeutic modification of vas deferens motility, via pharmacologic intervention of these channels.

The findings of the present study support the view that the presynaptic inhibitory effect of PGE 1 and PGE 2 is mediated via activation of large-conductance K Ca channels.

Increasing [K + ] o induces reversible inhibition of adrenergic vas deferens contractions.

These inhibitory effects on neurogenic contractions of human vas deferens are mediated by activation of Na + ,K + -ATPase. Prostaglandins are a family of biologically lipid acids synthesized by cyclooxygenase from a common precursor, arachidonic acid. Previous studies have indicated that prostaglandin E 1 (PGE 1 ) and PGE 2 , and other prostanoids, influence adrenergic neurotransmission in the vas deferens from several species [1][2][3][4]. In human vas deferens, the motor innervation is mainly noradrenergic [5,6] and PGE 1 and PGE 2 have been reported to inhibit neurogenic contractions by decreasing the release of norepinephrine from sympathetic nerve endings [7]. The mechanisms whereby PGE 1 and PGE 2 modify neurotransmitter release have received much attention but remain poorly understood. Prostaglandins of the E series inhibit transmitter release by an action on stimulus-secretion coupling and more specifically on the availability of Ca 2+ for the release mechanism [8]. K + channels modulate the adrenergic contractile responses in human vas deferens [START_REF] Medina | Modulation of adrenergic responses of human vas deferens by K + channel inhibitors[END_REF] and several agents have been shown to inhibit neurotransmitter release from nerve endings through a mechanism that involves the opening of prejunctional K + channels, membrane hyperpolarization and reduction in Ca 2+ influx via voltage-activated Ca 2+ channels [10,11]. In vas deferens, it has been demonstrated that K + channels are involved in the prejunctional inhibitory effects of atrial natriuretic factor [12],  2adrenoceptor agonists [13] and sildenafil, an inhibitor of phosphodiesterase 5 [14].

Activation of K + channels stimulates cellular K + efflux [15]. In vascular smooth muscle, it has been demonstrated that an increased extracellular K + concentration ([K + ] o ) evokes rapid hyperpolarization and relaxation [16] by stimulating an ouabain-sensitive Na + ,K + -ATPase [17]. In rat vas deferens, increased extracellular K + concentrations lower than those necessary to induce contraction of the smooth muscle inhibit electrically-induced

Isolated vas deferens preparation

Ring preparations were suspended between two L-shaped stainless steel pins. One pin was fixed to the organ bath wall while the other was connected to a strain gauge (Grass FT03, Grass Instruments Division Astromed, Inc., West Warwick, RI, U.S.A.). Changes in isometric force were recorded by use of Chart v3.4/s software and a MacLab/8e data acquisition system (ADInstruments, East Sussex, U.K.). Each preparation was set up in a 4-ml bath containing modified Krebs-Henseleit solution of the following millimolar composition: NaCl, 115; KCl, 4.6; KH 2 PO 4 1.2; MgCl 2 •6H 2 O, 1.2; CaCl 2 , 2.5; NaHCO 3 , 25; glucose, 11.1; and disodium EDTA, 0.01. The solution was equilibrated with 95% O 2 and 5% CO 2 . The preparations were allowed to equilibrate for 1-h and during this time tension was adjusted to a final tension of 19.6 mN.

Electrical field stimulation was provided by a Grass S88 stimulator (Grass Instruments Division Astromed, Inc., West Warwick, RI, U.S.A.) via two platinum electrodes positioned on each side and parallel to the axis of the ring. Single square wave pulses (0.25 ms pulse duration, 20 Hz, at a supramaximal voltage of 20 Vcm -1 ) were used. The

Experimental procedure.

To study the neuromodulatory action of prostanglandins, when electrically induced phasic contractions were stable (after 15-20 minutes), PGE 1 or PGE 2 (10 -8 -10 -6 M) were added cumulatively to the preparations and the effects of electrical field stimulation were recorded. To examine the role of K + channel activation in the effects of PGE 1 and PGE 2 on electrical field stimulation induced contractions, concentration-response curves to PGE 1 and PGE 2 were established in the presence of one of the following inhibitors: TEA (10 -3 M), a nonspecific K + channel blocker [20], glibenclamide (10 -5 M), a selective blocker of ATP-sensitive K + (K ATP ) channels [21], charybdotoxin (10 -7 M), an inhibitor of Ca 2+ -activated K + (K Ca ) channels of both large-conductance (K Ca 1.1) [22] and intermediate-conductance (K Ca 3.1) [23], iberiotoxin (10 -7 M), an inhibitor of K Ca 1.1 channels [24], or apamin (10 -6 M), an inhibitor of K Ca of small-conductance (K Ca 2) channels [25].

Concentration-response curves to norepinephrine were obtained in a cumulative manner in the absence (control) and in the presence of PGE 1 (10 -6 M) or PGE 2 (10 -6 M) from separate preparations. K + channels activation stimulates K + efflux [15]. As increases in the [K + ] o evoke rapid vascular smooth muscle relaxation and hyperpolarization [16], we studied the adrenergic responses in the presence of increased [K + ] o (10.8-25.8 mM) to check the possibility that increasing [K + ] o may inhibit adrenergic contractions. At the end of each series, the physiological concentration of K + was restored by washing out with Krebs- To determine whether Na + ,K + -ATPase activity is involved in the adrenergic responses of the human vas deferens, the contractions to electrical field stimulation and norepinephrine were recorded before and 15 min after the addition of ouabain (10 -7 -10 -6 M), a Na + ,K + -ATPase inhibitor.

In another group of experiments, the influence of Na + ,K + -ATPase on the PGE 1 -, PGE 2or increasing K + concentrations -induced inhibition of contractions evoked by electrical field stimulation was examined by exposing vas deferens rings to ouabain (10 -6 M) for 15 min before the addition of PGE 1 (10 -8 -3x10 -6 M), PGE 2 (10 -8 -3x10 -6 M) or [K + ] o (10.8-25.8 mM) and the responses to electrical stimulation were recorded.

Drugs

The following drugs were used: norepinephrine hydrochloride, prazosin hydrochloride, tetrodotoxin, tetraethylammonium bromide (TEA), charybdotoxin, iberiotoxin, apamin, glibenclamide and ouabain (Sigma Chemical Co. St. Louis, MO, USA). All drugs were dissolved in Krebs solution, except for glibenclamide and ouabain which were dissolved initially in dimethyl sulphoxide and further diluted in Krebs solution to the proper final concentration. Stock solutions of the drugs were freshly prepared every day, and kept on ice throughout the experiment. Figure 1A shows representative tracings of the effects of PGE 1 (10 -8 -10 -6 M) on EFSevoked contractions of the vas deferens rings in the absence (control) and in the presence of K + channel blockers and Figure 1B and C summarizes the results from all the experiments. PGE 1 induced concentration-dependent inhibition of EFS-evoked contractions with a pD 2 value of 7.35±0.10 (Figure 1B andC). Apamin (10 -6 M) and glibenclamide (10 -5 M) did not modify the inhibitory effect of PGE 1 on neurogenic contractions whereas charybdotoxin (10 -7 M), TEA (10 -3 M) and iberiotoxin (10 -7 M) reduced significantly the inhibitory effects of PGE 1 . The effects of PGE 2 on neurogenic contractions and the inhibition of these effects by K + channel blockers were similar to those observed with PGE 1 (Figure 2). Figure 3 (A andB) shows that the increase in [K + ] o in the Krebs-Henseleit solution caused concentration-dependent inhibition of the EFS-induced contractions with a maximum inhibitory effect of 96%. The inhibition was reversible after washout of the excess of K + . Ouabain (10 -7 -10 -6 M) had no significant effect on the basal tone of the preparations but increased contractile responses to EFS (Figure 3C andD). Ouabain (10 -6 M) counteracted the inhibitory effects of PGE 1 , PGE 2 and increasing [K + ] o on EFS-induced contractions (Figure 4).

A c c e p t e d

M a n u s c r i p t 10 Cumulative addition of norepinephrine (10 -6 -3x10 -4 M) induced repetitive phasic, concentration-dependent contractions with a pD 2 value of 5.01±0.10. Pretreatment with PGE 1 (10 -6 M) or PGE 2 (10 -6 M) did not affect the contractions induced by norepinephrine (Table 1). Moreover, pretreatment with either ouabain (10 -6 M) or increasing [K + ] o to 25.8 mM were without effect on norepinephrine-induced contractions (Figure 5 and Table 1).

A c c e p t e d M a n u s c r i p t The present study demonstrates a role for K + channels and Na + ,K + -ATPase in the prejunctional inhibitory effects of PGE 1 and PGE 2 on adrenergic responses of human vas deferens. It has been demonstrated that PGE 1 and PGE 2 selectively depress neurogenic contractions in human vas deferens without modifying the contractions to norepinephrine [7]. The lack of effect of PGE 1 and PGE 2 on norepinephrine-induced contractions suggests that postjunctional inhibitory mechanisms are not involved in the neuromodulatory action of these prostaglandins. The inhibitory effect on neurogenic contractions has been attributed to decreased norepinephrine release from nerve endings in the human vas deferens [7].

The mechanisms whereby PGE 1 and PGE 2 induce inhibition of neurotransmitter release remain poorly understood. Several agents could inhibit neurotransmitter release from nerve endings in vas deferens through a mechanism that involves activation of K + channels. In human vas deferens, charybdotoxin-sensitive, iberiotoxin-insensitive, K + channels modulate the adrenergic contractile response by interfering with Ca 2+ entry through dihydropyridine Ca 2+ channels [START_REF] Medina | Modulation of adrenergic responses of human vas deferens by K + channel inhibitors[END_REF]. Furthermore, it has been shown that K + channels are involved in the prejunctional inhibitory effects of atrial natriuretic factor in the rabbit isolated vas deferens [12] and in the  2 -adrenoceptor-mediated inhibition in rat vas deferens [13]. In the human vas deferens, sildenafil, an inhibitor of phosphodiesterase 5, inhibits adrenergic contractions through activation of presynaptic K Ca 1.1 channels [14].

Glibenclamide has been shown to block K ATP channels [21] with an IC 50 in the micromolar range for smooth muscle preparations [26]. In rat vas deferens, the inhibitory action of calcitonin-gene-related peptide (CGRP) on neurotransmission was On the other hand, we tested the effects of TEA, a nonspecific K + channel blocker [20] on the neuromodulatory effects of PGE 1 and PGE 2 . It is accepted that blockade with 10 -3 M TEA is sufficient to almost fully inhibit K Ca 1.1 channels [28]. In our experiments millimolar concentrations of TEA prevented the inhibitory effects of PGE 1 and PGE 2 on neurogenic contractions, suggesting a role of K Ca channels in this effect.

To further characterize the member of K Ca channel group involved in the neuromodulatory effects of PGEs we used more selective blockers of K Ca channels.

Apamin has proved to be extremely valuable as a highly specific blocker for which the only known receptors are the K Ca channels of small conductance (K Ca 2) [29]. In our experiments, apamin 10 -6 M, a concentration expected to block completely all subtypes of K Ca 2 channels, failed to alter the inhibition caused by PGE 1 and PGE 2 on neurogenic contractions in human vas deferens. These data indicate that K Ca 2 channels are not involved in the neuromodulatory effects of PGE 1 and PGE 2 .

Charybdotoxin, a blocker of both K Ca 1.1 and K Ca 3.1 channels [22,23], prevented the inhibitory effects of PGE 1 and PGE 2 on sympathetic contractions of the human vas deferens. These results demonstrate that the K + channels involved are sensitive to charybdotoxin. Therefore, we tested the effect of iberiotoxin, a selective blocker of K Ca 1.1 channels [24], on the effects of PGE 1 and PGE 2 on adrenergic contractions. The A previous study in rat vas deferens demonstrated that increasing [K + ] o , at concentrations lower than those necessary to induce contraction of the smooth muscle, completely inhibited the electrically-induced contractions [18], an effect attributed to a desensitization of the rat vas deferens to electrically-induced contractions. Studies of vascular smooth muscle showed that increased [K + ] o in the range 6-16 mM produce hyperpolarization and vasodilatation by stimulating an ouabain-sensitive Na + ,K + -ATPase [17]. In guinea-pig vas deferens relatively high concentrations of ouabain, an inhibitor of Na + ,K + -ATPase, is reported to cause a gradual increase in the norepinephrine release from the peripheral adrenergic neurons [30] and an increase in the sensitivity to norepinephrine [31]. Particularly for the vas deferens, species is thought to play an important role in Na + ,K + -ATPase activity and function [31].

However, the effects of inhibition of Na + ,K + -ATPase on adrenergic responses of human vas deferens have not been investigated. As the resting tension of human vas deferens was not changed by ouabain in our experiments, we suggest that Na + ,K + -ATPase is not normally active in the relaxed vas deferens smooth muscle. Furthermore, ouabain had no effect on norepinephrine-mediated contractions at concentration of 10 -6 M, but increased neurogenic contractions. These results indicate that the enhanced neurogenic contractions induced by ouabain involves the inhibition of Na + ,K + -ATPase activity of the adrenergic nerve endings.

Increasing [K + ] o over the range that inhibits electrical field stimulation-induced adrenergic contractions should also stimulate the Na + ,K + -ATPase by increasing the amount of extracellular K + that binds to the pump [32]. In human vas deferens, ouabain Because Na + ,K + -ATPase activity is ultimately responsible for most of the resting membrane potential, the ability of increased extracellular K + to regulate adrenergic contractions could in part be due to its effect on Na + ,K + -ATPase activity. There are two effects of the pump on membrane potential: (i) an electrogenic effect, due to the 3Na + /2K + exchange (this is the smaller, but more rapid effect) and (ii) by maintaining the transmembrane K + gradient the membrane potential is created (this is the more powerful, but slower effect). Our results suggest that in the inhibitory effects of K + on the adrenergic contractions of human vas deferens only the first effect is relevant because in our experimental conditions it is not possible to increase the transmembrane K + gradient by increasing [K + ] o .

However, involvement of other mechanisms/pathways cannot be excluded. Recently, it has been demonstrated the proximity and functional interaction between the KCa1.1 channel and Na + ,K + -ATPase [33]. This interplay is suggested to depend on the proper microenviroment. Previous studies demonstrated that inhibition of Na + ,K + -ATPase and subsequent accumulation of intracellular Na + decreases K Ca 1.1 current [33,[START_REF] Liang | Intracellular Na + modulates large conductance Ca 2+ -activated K + currents in human umbilical vein endothelial cells[END_REF]. Another possibility is that inhibition of Na + ,K + -ATPase by ouabain leads to an accumulation of Na + close to the microdomains harboring K Ca 1.1 channels, which inhibits the channel function.

Furthermore, ouabain blocked the inhibitory effects of PGE 1 and PGE 2 on sympathetic contractions in human vas deferens, suggesting that activation of Na + ,K + -ATPase plays an important role in the presynaptic inhibition of the norepinephrine release induced by these prostaglandins. Consistent with our data, an acute stimulatory effect of PGE 1 and A c c e p t e d M a n u s c r i p t 15 PGE 2 on Na + ,K + -ATPase activity was observed in rabbit renal proximal tubule cells and cortical collecting tubule of the nephron [35,36].

In rat vas deferens it has been demonstrated that PGE 2 is released from the epithelium in response to ATP stimulation and it is responsible for mediating the membrane hyperpolarization and inhibition of contraction of smooth muscle cells by activating cAMP-dependent K + channels [37]. However, our results show that PGE 2 inhibited neurogenic contractions but did not exert inhibitory effects on norepinephrine-induced contractions, indicating that PGE 2 acts directly on sympathetic nerve endings.

An important function of the vas deferens is the transport of spermatozoa from the epididymis for inclusion in the semen. Since  1 -adrenoceptors activation by norepinephrine is required for normal contraction of the vas deferens and consequent sperm ejaculation [38], it seems likely that increased formation of PGE 2 in physiological and pathological conditions such as inflammation [39], may cause aperistalsis of the adrenergically innervated vas deferens by activating K Ca 1.1 channels and Na + ,K + -ATPase and impairment of male fertility. Although the present studies provide evidence that activation of presynaptic KCa1.1 channels inhibits adrenergic contractions of vas deferens, future studies are necessary to elucidate the potential therapeutic modification of vas deferens motility, via pharmacologic intervention of these channels.

The findings of the present study support the view that the presynaptic inhibitory effect of PGE 1 and PGE 2 is mediated via activation of large-conductance K Ca channels.

Increasing [K + ] o induces reversible inhibition of adrenergic vas deferens contractions.

These inhibitory effects on neurogenic contractions of human vas deferens are mediated by activation of Na + ,K + -ATPase.

5 2 .

 2 MATERIAL AND METHODS Segments (15-20 mm long) of the epididymal part of the vas deferens were taken from 45 healthy men without previous history of psychiatric or physical illness, medication use, or substance abuse (mean age, 39, range 30-44 years) who were sterilized by elective vasectomy. The study was approved by the Human Ethics

  5 s and the stimulation interval 180 s. The stimulation parameters used elicit contractile responses that are abolished by tetrodotoxin (10 -6 M) or prazosin (10 -6 M)[19].

  mM) to check the possibility that increasing [K + ] o may inhibit adrenergic contractions. At the end of each series, the physiological concentration of K + was restored by washing out with Krebs-A c c e p t e d M a n u s c r i p t 7 Henseleit solution and electrical stimulation was repeated to confirm the reversibility of the effects of high K + concentrations on neurogenic response.

  response curves by non-linear regression analysis. The responses obtained in each subject were averaged to yield a single value. Differences between untreated and treated groups were assessed by one-way analysis of variance (ANOVA), and then Bonferroni's test was performed. Statistical significance was accepted at P < 0.05.A c c e p t e d M a n u s c r i p t 9 3. RESULTS Electrical field stimulation (EFS) induced phasic contraction in segments of the vas deferens which was abolished by tetrodotoxin (10 -6 M) and prazosin (10 -6 M), thus indicating that the smooth muscle contraction was due to the release of norepinephrine from adrenergic nerves acting on  1 adrenoceptors (results not shown).

Figure

  Figure 1A shows representative tracings of the effects of PGE 1 (10 -8 -10 -6 M) on

Figure 3 (

 3 Figure 3 (A and B) shows that the increase in [K + ] o in the Krebs-Henseleit solution

  iberiotoxin in the present study provides evidence for K Ca 1.1 channels as a significant component of PGE 1 and PGE 2 inhibitory effects on neurogenic contractions of the human vas deferens.

  -induced inhibition of adrenergic contractions, indicating that increasing [K + ] o

  KCa1.1 channel and Na + ,K + -ATPase [33]. This interplay is suggested to depend on the proper microenviroment. Previous studies demonstrated that inhibition of Na + ,K + -ATPase and subsequent accumulation of intracellular Na + decreases K Ca 1.1 current [33,34]. Another possibility is that inhibition of Na + ,K + -ATPase by ouabain leads to an accumulation of Na + close to the microdomains harboring K Ca 1.1 channels, which inhibits the channel function. Furthermore, ouabain blocked the inhibitory effects of PGE 1 and PGE 2 on sympathetic contractions in human vas deferens, suggesting that activation of Na + ,K + -ATPase plays an important role in the presynaptic inhibition of the norepinephrine release induced by these prostaglandins. Consistent with our data, an acute stimulatory effect of PGE 1 A c c e p t e d M a n u s c r i p t 15 and PGE 2 on Na + ,K + -ATPase activity was observed in rabbit renal proximal tubule cells and cortical collecting tubule of the nephron [35,36].

22 FIGURE LEGENDS Figure 1 .

 22LEGENDS1 FIGURE LEGENDS

Figure 2 .

 2 Figure 2. Graphical representation of the inhibition of EFS-induced contractions by

Figure 3 .

 3 Figure 3. A) Representative tracing and B) bar graph showing the inhibitory effects of increasing extracellular K + concentration ([K + ] o ) from 5.8 to 25.8 mM on

Figure 4 .

 4 Figure 4. Inhibition of electrical field stimulation (EFS)-induced contractions by prostaglandin E 1 (PGE 1 ), prostaglandin E 2 (PGE 2 ) and increasing extracellular K + concentration ([K + ] o ) from 10.8 to 25.8 mM in the absence (control) and in the

Figure 5 . 2 ABSTRACT

 52 Figure 5. A) Recordings illustrating the contractile effects of norepinephrine in the absence (control, extracellular K + concentration [K + ] o = 5.8 mM) and in the presence of [K + ] o of 25.8 mM or ouabain (10 -6 M). Concentration-response curves for norepinephrine in the absence (control, [K + ] o = 5.8 mM) and in the presence of B) [K + ] o

  ]. A role for increased [K + ] o and Na + ,K + -ATPase in modulation of adrenergic neurotransmission in the human vas deferens has not yet been investigated.Therefore, the present study was designed to examine the effects of increasing [K + ] o and Na + , K + -ATPase on adrenergic contractions of human vas deferens and the contribution of K + channels and Na + ,K + -ATPase to the inhibitory effects of PGE 1 and PGE 2 on adrenergic neurotransmission.

  mm long) of the epididymal part of the vas deferens were taken from 45 healthy men without previous history of psychiatric or physical illness, medication use, or substance abuse (mean age, 39, range 30-44 years) who were sterilized by elective vasectomy. The study was approved by the Human Ethics Committee of our institution and informed consent was obtained from each subject before the study. The specimens were placed in chilled isotonic NaCl, and were divided into ring preparations 3-4 mm long.

  5 s and the stimulation interval 180 s. The stimulation parameters used elicit contractile responses that are abolished by tetrodotoxin (10 -6 M) or prazosin (10 -6 M)[19].

A c c e p t e d M a n u s c r i p t 7 Henseleit

 7 solution and electrical stimulation was repeated to confirm the reversibility of the effects of high K + concentrations on neurogenic response.

2. 4

 4 Data analysis All values are expressed as means ± S.E.M. Contractions are reported as absolute values (mN) or as percentages of control responses. pD 2 values (negative logarithm of the molar concentration at which half-maximum contraction occurs) were determined from A c c e p t e d M a n u s c r i p t 8 individual concentration-response curves by non-linear regression analysis. The responses obtained in each subject were averaged to yield a single value. Differences between untreated and treated groups were assessed by one-way analysis of variance (ANOVA), and then Bonferroni's test was performed. Statistical significance was accepted at P < 0.05.

  Electrical field stimulation (EFS) induced phasic contraction in segments of the vas deferens which was abolished by tetrodotoxin (10 -6 M) and prazosin (10 -6 M), thus indicating that the smooth muscle contraction was due to the release of norepinephrine from adrenergic nerves acting on  1 adrenoceptors (results not shown).

  part mediated by the activation of K ATP channels in sympathetic nerve terminals as shown by the prevention of the neuromodulatory effects of CGRP in the presence of glibenclamide 10 -5 M[27]. Our results in human vas deferens demonstrate that glibenclamide at concentration as high as 10 -5 M did not modify the inhibition induced by PGEs on adrenergic responses. These results rule out the involvement of K ATP channels activation in the inhibitory effects of PGEs on neurogenic contractions of the human vas deferens.

  iberiotoxin in the present study provides evidence for K Ca 1.1 channels as a significant component of PGE 1 and PGE 2 inhibitory effects on neurogenic contractions of the human vas deferens.

  -induced inhibition of adrenergic contractions, indicating that increasing [K + ] o stimulates the Na + ,K + -ATPase in the range that inhibits adrenergic contractions.

20 ( 22 FIGURE LEGENDS Figure 1 .

 2022LEGENDS1 FIGURE LEGENDS

Figure 2 .

 2 Figure 2. Graphical representation of the inhibition of EFS-induced contractions by

Figure 3 .

 3 Figure 3. A) Representative tracing and B) bar graph showing the inhibitory effects of increasing extracellular K + concentration ([K + ] o ) from 5.8 to 25.8 mM on electrical field

A

  c c e p t e d M a n u s c r i p t 23 response elicited by EFS before the addition of K + or ouabain. Values are mean ± S.E.M. Numbers in parentheses are the number of subjects in each group. *P<0.05 versus control.

Figure 4 .

 4 Figure 4. Inhibition of electrical field stimulation (EFS)-induced contractions by prostaglandin E 1 (PGE 1 ), prostaglandin E 2 (PGE 2 ) and increasing extracellular K + concentration ([K + ] o ) from 10.8 to 25.8 mM in the absence (control) and in the presence

Figure 5 .

 5 Figure 5. A) Recordings illustrating the contractile effects of norepinephrine in the absence (control, extracellular K + concentration [K + ] o = 5.8 mM) and in the presence of [K + ] o of 25.8 mM or ouabain (10 -6 M). Concentration-response curves for norepinephrine in the absence (control, [K + ] o = 5.8 mM) and in the presence of B) [K + ] o

ACKNOWLEDGEMENTS This work was supported by the Ministerio de Ciencia e Innovación and Conselleria de Sanidad de la Generalitat Valenciana. ACKNOWLEDGEMENTS This work was supported by the Ministerio de Ciencia e Innovación and Conselleria de Sanidad de la Generalitat Valenciana.


R J Allen

I D L Bogle

A J Ridley

A Model of Localised Rac1 Activation in Endothelial Cells Due to Fluid Flow

Keywords: Stokes Flow, Endothelial Cell, Rho GTPases, Src, Shear Stress

Endothelial cells respond to fluid flow by elongating in the direction of flow. Cytoskeletal changes and activation of signalling molecules have been extensively studied in this response, including: activation of receptors by mechano-transduction, actin filament alignment in the direction of flow, changes to cell-substratum adhesions, actin-driven lamellipodium extension, and localised activation of Rho GT-Pases. To study this process we model the force over a single cell and couple this to a model of the Rho GTPases, Rac and Rho, via a Kelvin-body model of mechanotransduction. It is demonstrated that a mechano-transducer that can respond to the normal component of the force is likely to be a necessary component of the signalling network in order to establish polarity. Furthermore, the rate-limiting step of Rac1 activation is predicted to be conversion of Rac-GDP to Rac-GTP, rather than activation of upstream components. Modelling illustrates that the aligned endothelial cell morphology could attenuate the signalling network.

Introduction

Endothelial cells (ECs) respond to fluid flow by elongating in the direction of flow [START_REF] Davies | Flow-mediated endothelial mechanotransduction[END_REF][START_REF] Ballerman | Shear stress and the endothelium[END_REF][START_REF] Wojciak-Stothard | Shear stress-induced endothelial cell polarization is mediated by Rho and Rac but not Cdc42 or PI 3-kinases[END_REF][START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF][START_REF] Mccue | Shear stress regulates forward and reverse planar cell polarity of vascular endothelium in vivo and in vitro[END_REF]. This response is believed to be crucial for the correct function of the endothelium. Atherosclerotic plaques usually localize in regions of turbulent blood flow (for example, at sites of arterial branching). In these regions the endothelium has increased permeability and ECs take on a rounded morphology. This is in contrast to their ellipsoidal shape in regions of smooth flow [START_REF] Weinbaum | Effect of cell turnover and leaky junctions on arterial macromolecular transport[END_REF][START_REF] Ross | Atherosclerosis -an inflammatory disease. Mechanism of Disease[END_REF]. Hence, there is a correlation (although not necessarily causation) between EC morphology and atherosclerotic plaques.

In order to respond to fluid flow, ECs must be able to transduce the mechanical force of the fluid into a biochemical signal. This mechano-transduction is likely to be mediated by activation of proteins via conformational changes due to increased tension or compression. A number of different proteins have been implicated in mechano-transduction, including cell-cell adhesion proteins such as PECAM-1 [START_REF] Chiu | Mechanotransduction in an extracted cell model: Fyn drives stretch-and flow-elicited pecam-1 phosphorylation[END_REF][START_REF] Tzima | A mechanosensory complex that mediates the endothelial cell response to fluid shear stress[END_REF], stretch-activated ion channels [START_REF] Martinac | Mechanosensitive ion channels: molecules of mechanotransduction[END_REF], and integrin-associated proteins in focal adhesions (FA). Integrins are transmembrane hetero-dimers that anchor the cell to the extracellular matrix (ECM). Hence if the cell is to remain adherent they must experience an increase in tensile force following the onset of fluid flow. This force is transmitted through the cytoskeleton to FAs. Integrins have been shown to respond to fluid flow [START_REF] Tzima | Activation of integrins in endothelial cells by fluid shear stress mediates Rhodependent cytoskeletal alignment[END_REF] and to be locally activated at the downstream edge [START_REF] Goldfinger | Localized 4 integrin phosphorylation directs shear stressinduced endothelial cell alignment[END_REF]. However, this may be due to increased integrin-ECM ligation. The role of FAs and integrins is further supported by early FA remodelling [START_REF] Davies | Quantitative studies of endothelial cell adhesion: directional remodelling of focal adhesion sites in response to flow forces[END_REF] and fast activation of the integrin-associated tyrosine kinase Src (Wang et al., 2005a). p130Cas, an adaptor protein, is a known mechano-transducer associated with FAs, and can be activated in response to mechanical stretching [START_REF] Sawada | Force sensing by mechanical extension of the src family kinase substrate p130cas[END_REF]. It is also activated through Src-induced tyrosine phosphorylation [START_REF] Okud | Shear stress stimulation of p130cas tyrosine phosphorylationrequires calcium-dependent c-src activation[END_REF].

Mechano-transduction and cellular mechanics have been extensively modelled previously [START_REF] Mazzag | A model for shear stress sensing and transmission in vascular endothelial cells[END_REF][START_REF] Stamenovic | A microstructural approach to cytoskeletal mechanics based on tensegrity[END_REF][START_REF] Schwarz | Focal adhesions as mechanosensors: The two-spring model[END_REF][START_REF] Herant | The mechanics of neutrophils: synthetic modeling of three experiments[END_REF]. For example, force transduction through ECs has been modelled using Kelvin bodies in parallel and in series [START_REF] Mazzag | A model for shear stress sensing and transmission in vascular endothelial cells[END_REF] and focal adhesions have been modelled using a two spring model [START_REF] Schwarz | Focal adhesions as mechanosensors: The two-spring model[END_REF]. Cell structure has been described by a tension-integrity mode, and a tensegrity model has been applied to a discrete system of struts (microtubules) and cables (actin filaments) [START_REF] Stamenovic | A microstructural approach to cytoskeletal mechanics based on tensegrity[END_REF]. This model demonstrated that stiffness of the model cell increased with pre-stress generated by actin-myosin contraction. Aspects of neutrophil mechanics have been successfully described using a two-phase model of reactive interpenetrating flows [START_REF] Herant | The mechanics of neutrophils: synthetic modeling of three experiments[END_REF].

The Rho GTPases Rho and Rac regulate both FA formation and endothelial shape changes in response to fluid flow. Rho GTPases regulate cytoskeletal dynamics and contribute to many cellular responses including cell migration and cytokinesis [START_REF] Ridley | Rho gtpases and cell migration[END_REF][START_REF] Narumiya | Rho gtpases in animal cell mitosis[END_REF][START_REF] Ridley | Rho gtpases and actin dynamics in membrane protrusions and vesicle trafficking[END_REF][START_REF] Jaffe | Rho GTPases: Biochemistry and biology[END_REF]. Rho GTPases switch between an inactive guanosine diphosphate (GDP) bound form and an active guanosine triphosphate (GTP) bound form which interacts with downstream effectors. Switching between these two forms usually occurs on the membrane and is regulated by guanine nucleotide exchange factors (GEFs), GTPase-activating proteins (GAPs) and guanine dissociation inhibitors (GDIs; see figure 1).

RhoA leads to acto-myosin contraction via Rho kinases (ROCKs) and myosin light chain (MLC) phosphorylation [START_REF] Riento | ROCKs: multifunctional kinases in cell biology[END_REF][START_REF] Kawano | Phosphorylation of myosin binding submit (mbs) of myosin phosphatase by Rho-kinase in vivo[END_REF]. In response to fluid flow, RhoA has been reported to contribute to cell shape change by transiently inducing cell contraction, subsequently allowing cells to elongate in the direction of flow [START_REF] Wojciak-Stothard | Shear stress-induced endothelial cell polarization is mediated by Rho and Rac but not Cdc42 or PI 3-kinases[END_REF]. Rac activation is localised to the downstream edge in ECs responding to fluid flow [START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF]. In response to integrin activation, Src has been shown to activate a Rac GEF, Vav2 [START_REF] Garrett | VEGF-induced Rac1 activation in endothelial cells is regulated by the guanine nucleotide exchange factor Vav2[END_REF], and p190RhoGAP (a GAP for Rho) [START_REF] Arthur | Integrin engagement suppresses rhoa activity via a c-src dependent mechanism[END_REF]. Thus, in this context, activation of Rac and inhibition of Rho is hypothesised to occur. In addition, it is believed that inhibitory crosstalk between Rho and Rac leads to separate zones of activation [START_REF] Jilkine | Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk[END_REF][START_REF] Narang | Spontaneous polarization in eukaryotic gradient sensing: A mathematical model based on mutual inhibition of frontness and backness pathways[END_REF].

In order to investigate how the interplay betweeen mechanical forces and EC morphology can influence downstream signalling, the work presented here models the mechanical force, transduction of the mechanical force into a biochemical signal and the downstream signaling to Rac1. We will describe three coupled computational models of these processes: fluid flow over an EC attached to a planar surface, mechano-transduction on the basal surface and signalling to Rac1. The flow model predicts the force on the surface of the cell. This force is coupled to a biochemical model via a mechano-transduction model.

The Flow Model

To establish a possible mechanism linking the force on an EC to localised Rac activation, the fluid flow, u, is modelled over a single cell attached to a planar surface (z = 0) by the Stokes equation:

-∇P + μ∇ 2 u + ρb = 0 ( 1
)
where P is the pressure and μ and ρ are the viscosity and density of the fluid respectively. On the characteristic length and time scales of this problem the Reynolds number is very small. Thus, assuming Newtonian characteristics, the fluid is well modelled by equation 1. Here the Stokes equation is solved by using a boundary integral representation, which has previously been used in this context [START_REF] Hazel | Vascular endothelial cells minimize the total force on their nuclei[END_REF]Wang and Dimitrakopoulos, 2006a,b).

In this representation the fluid flow at u j (x 0 ) is given by:

u j (x 0 ) = u ∞ j (x 0 ) - 1 8πμ S f i (x 0 )G ij (x, x 0 )dS (2)
where u ∞ (x 0 ) = (kz, 0, 0) is the flow far away from the cell, S is the cell surface, f i is the surface force and G ij is a Green's function that vanishes if x ∈ S [START_REF] Pozrikidis | Boundary integral and singularity methods for linearized viscous flow[END_REF][START_REF] Blake | A note on the image system for a stokeslet in a no-slip boundary[END_REF][START_REF] Higdon | Stokes flow in arbitrary two-dimensional domains: shear flow over ridges and cavities[END_REF]. Intuitively equation 2 represents a sum of point forces weighted so that the boundary conditions are satisfied. It is these weights that are the unknowns.

To solve for the unknowns the cell surface is discretised into n triangles (figure 2). The surface is defined by:

z = ⎧ ⎪ ⎨ ⎪ ⎩ H (R 2 -r 2 ) R 2 when r 2 < R 2 0 o t h e r w i s e (3)
where H is the maximum height of the EC, R = 20 μ is the typical radius of an approximately circular (in the xy plane) EC and r 2 = ax 2 + b 2 . Unless otherwise noted, a = b = 1, figure 2. It is supposed that the EC is not adapted to flow, and the height of the EC is taken to be H = 6 μm. Depending on conditions, and EC type, ECs vary in height from 1-12 μm [START_REF] Wee | Icam-1/lfa-1 interaction contributes to the induction of endothelial cell-cell separation: implication for enhanced leukocyte diapedesis[END_REF][START_REF] Barbee | Shear stress-induced reorganization of the surface topography of living endothelial cells imaged by atomic force microscopy[END_REF].

To facilitate the solution of a system of partial differential equations (to model reaction-diffusion), the discretization is chosen such that the projection of the surface nodes onto the plane z = 0 gives a hexagonal lattice, with each hexagon composed of four of the surface element triangles. In the following results the area of the hexagons is 1.66 μm 2 , corresponding to a side length of 0.8 μm. Equation 2 is solved by applying the no-slip boundary condition on S; u(x 0 ) = 0 if x 0 ∈ S. A linear problem for the unknown surface force can be formed by applying the discrete version of equation 2 at n points on S, x 0 . The surface force is assumed constant over surface elements. The linear problem for the f i (x 0 ) is solved by an iterative Krylov subspace method [START_REF] Saad | Krylov subspace methods for solving large unsymmetric linear systems[END_REF].

The numerical method was validated on a hemisphere surface (in contrast to the surface described by equation 3) and compared to a known approximate analytical solution for the non-dimensional total force on this surface due to shear flow (F = 4.30π, [START_REF] Price | Slow linear shear flow past a hemispherical bump in a plane wall[END_REF]) and was found to be accurate up to an error of 1.7% for 3000 surface elements.

Results

All parameters are given in table 1. Figure 3 gives the pattern of shear stress on the cell surface as a result of flow in a large artery. Under these conditions the maximum shear stress is 10 dynes cm -2 , which is similar to the wall shear stress calculated in in vitro flow chamber experiments [START_REF] Wojciak-Stothard | Shear stress-induced endothelial cell polarization is mediated by Rho and Rac but not Cdc42 or PI 3-kinases[END_REF][START_REF] Tzima | Activation of integrins in endothelial cells by fluid shear stress mediates Rhodependent cytoskeletal alignment[END_REF][START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF].

It is evident that the force component normal to the surface of the cell could give rise to the spatial heterogeneity via separate regions of tension and compression (figure 4). For a symmetric cell, the tangential surface component is symmetrical with respect to the upstream and downstream points of the cell (figure 3). Although the surface component could be crucial for signalling via ion channels, this signalling would (assuming uniform distributions of channels) initially lead to global changes in the cell.

Force Transduction & GTPase signalling

The force on the cell surface is coupled to the network (figure 7), which is modelled below, by treating integrins as a Kelvin body, figure 5. Previously, [START_REF] Ferko | Finiteelement stress analysis of a multicomponent model of sheared and focallyadhered endothelial cells[END_REF] showed that heterogeneity in intra-cellular could play an important role in the sub-cellular localization of mechano-transduction. However, here we make the simplifying assumptions that the cytosol/cytoskeletal medium is uniform and that force is transmitted to the basal surface locally. Both shear and normal forces are borne by cytoskeletal elements and, potentially, could activate mechano-transducers either at the basal surface or at ion channels in the membrane. However, the hypothesis we aim to test is whether a response to the force normal to the surface could be responsible for breaking the symmetry in the upstream and downstream signaling regions. Hence, to test whether the magnitude of the normal force could stretch a mechano-transducer by a biologically relevant amount we coupled the Kelvin body model, figure 5, to the force due to fluid flow.

The equation governing the Kelvin body [START_REF] Mazzag | A model for shear stress sensing and transmission in vascular endothelial cells[END_REF], given that the force is constant, is

ν 1 + h 1 h 2 dz e dt = h 1 z e -F ( 4
)
where F is the force on the body and z e is its extension. Only tensile forces are considered and, to simplify the model, a smooth concentration of integrins is assumed; that is, clustering is ignored. Interestingly, [START_REF] Ferko | Finiteelement stress analysis of a multicomponent model of sheared and focallyadhered endothelial cells[END_REF] illustrated that focal adhesions focus and amplify shear forces on the apical surfaces of ECs. Hence, the coarse grained approach applied here provides a conservative estimate for the force applied at the basal surface. The parameters for modelling integrins as a Kelvin body have been fitted to experimental data [START_REF] Bausch | Measurement of local viscoelasticity and forces in living cells by magnetic tweezers[END_REF]. The normal force on the cell surface is of the order of pN, leading to an extension of the mechano-transducer in the order of nanometres, figure 6.

This model is coupled to a model of Rho GTPase interaction, figure 7, by up-regulation of Src activation, figure 8. The coupling, between extension (z e ) and the biochemical network (figure 7) is via the parameter k + (z e ), which -in the model-is the rate of Src activation. Src has been shown to be activated (either directly or indirectly) by both shear stress and applied mechanical force (Wang et al., 2005b;[START_REF] Jalali | Shear stress activates p60src-Ras-MAPK signaling pathways in vascular endothelial cells[END_REF][START_REF] Fleming | Role of PECAM-1 in the shear-stress-induced activation of akt and the endothelial nitric oxide synthase (enos) in endothelial cells[END_REF]. Furthermore, there is evidence that stretch activation exhibits behaviour similar to figure 8. For example, in response to mechanical stretching PECAM-1 has switch like, saturating, activation [START_REF] Chiu | Mechanotransduction in an extracted cell model: Fyn drives stretch-and flow-elicited pecam-1 phosphorylation[END_REF]. However, for quantitative parameter estimation for the Src activation coupling (figure 8) it was necessary to use a course fitting to other experimental data, see appendix.

In the model, the resulting Src activation activates the Rho GTPase Rac1. Rho GTPases have previously been modelled in in a variety of contexts [START_REF] Jilkine | Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk[END_REF][START_REF] Goryachev | Computational model explains high activity and rapid cycling of Rho GTPases within protein complexes[END_REF][START_REF] Sakumura | A molecular model for axon guidance based on cross talk between rho gtpases[END_REF][START_REF] Marèe | Polarization and movement of keratocytes: A multiscale modelling approach[END_REF]. [START_REF] Civelekoglu-Scholey | Model of coupled transient changes of Rac, Rho, adhesions and stress fibers alignment in endothelial cells responding to shear stress[END_REF] modelled GTPase activation in this context, but with broader scope and less details of the activation of Rho GTPases themselves.

There is no feedback in this system so once activated there is no return to basal values. The reactions governing the network in figure 7 are:

Src k - k + (ze) Src* Src* + GEF R k 2 k 1 Src* + GEF* R RacGDP + GEF* R k 4 k 3 RacGDP-GEF* R k 5 Rac + GEF* R RacGTP + GAP* R k 7 k 6 RacGTP-GAP* R k 8 RacGDP + GAP* R RhoGTP + GAP R k 10 k 9 RhoGTP + GAP* R Src* + GAP ρ k 12 k 11 Src* + GAP* ρ RhoGTP + GAP* ρ k 14 k 13 RhoGTP-GAP* ρ k 15 RhoGDP + GAP* ρ RhoGDP + GEF* ρ k 17 k 16 RhoGDP-GEF* ρ k 18 Rho + GEF* ρ RacGTP + GEF* ρ k 20 k 19 RacGTP + GEF ρ GAP ρ k 22 k 21 GAP* ρ GEF R k 24 k 23 GEF* R GAP R k 26 k 25 GAP* R GEF ρ k 28 k 27 GEF* ρ (5)
where the subscript R refers to Rac and ρ to Rho. For example GAP ρ is a GAP for Rho. Active forms are denoted by an asterisk. The nucleotide free GTPases are assumed to quickly bind to GTP (due to the high intracellular GTP:GDP ratio).

Reactions (5) are modelled mathematically as second-order mass action reaction diffusion equations. For example, for a complex A j interacting with complexes A i 0 ≤ i ≤ n the time evolution of the concentration of A i at time t and position x, A i (x, t), is modelled as

∂A i ∂t = n j=0 α j A j + n k=0 n j=0 β jk A j A k + D i ∇ 2 A i
Applying this equation to reactions (5) gives a partial differential equation for each species in figure 7 with α j and β jk . Clearly most of the α j and β jk are zero since most species do not interact in this scheme (figure 7) -the non-zero values correspond to the reaction rates (k 1 to k 28 ).

As a generalisation only two values are taken for D i , corresponding to whether A i is solely membrane bound (GEFs, RacGTP, RhoGTP, Src), or membrane and cytosol bound. Here the GDIs are not modelled explicitly. [START_REF] Jilkine | Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk[END_REF] proposed a quasi-steady state between GDI bound and unbound GT-Pases. The effect of this is a modified diffusion coefficient to take into account the diffusion on the membrane and in the cytosol for the GDP bound GT-Pase. This is the approach followed here. It is also assumed that the GAPs and GEFs are specific for either Rho or Rac but not both [START_REF] Rossman | Gef means go: turning on rho gtpases with guanine nucleotide-exchange factors[END_REF][START_REF] Tcherkezian | Current knowledge of the large rhogap family of proteins[END_REF]; in reality this need not be the case. The effect of this additional level of coupling is a direction of future research.

The PDEs were integrated on a 2-D projection of the cell surface mesh. This projection was designed so that the projected mesh defined a hexagonal grid.

The PDEs were integrated on this grid by a simple Euler method with the spatial derivative approximated by central differences. On the time-scale of interest it was assumed that the cell was impermeable. Hence, no flux boundary conditions were implemented using reflective conditions at the cell edge.

It was assumed that the transmission of the force on the cell due to fluid flow is fast relative to Src activation. Hence, the mechano-transducer extension (figure 6) is taken to occur instantaneously. This extension (predicted by the flow model coupled with the mechano-transduction model) leads to Src activation and is the input signal to the reaction network, figure 7.

Results

Applying these equations on the cell (which is now a 2-D projection) gives localised Src and Rac activation in the downstream edge, figure 9. Using the derived parameters (see appendix) the time-course of Src activation is comparable to previously reported [START_REF] Fleming | Role of PECAM-1 in the shear-stress-induced activation of akt and the endothelial nitric oxide synthase (enos) in endothelial cells[END_REF][START_REF] Jalali | Shear stress activates p60src-Ras-MAPK signaling pathways in vascular endothelial cells[END_REF]. The whole cell response for Rac, figure 9(d), is slightly less than reported [START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF]. However, the model predicts maximal Rac activation within 25 minutes, which is comparable to the observed response [START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF]. Interestingly, the rates of GEF R activation due to active Src reach new steady states relatively quickly, and it is actually the rates of GAP and GEF catalysed GDP-GTP exchange which are the rate limiting ones. The pattern of Rac activation, figure 9(b), is generated by localised Src activation; however it is maintained due to mutual Rho and Rac inhibition (though with removal of this cross-talk a similar, albeit less distinct, pattern as shown in figure 9(b) occurs -not shown) and slow diffusion of the active membrane bound RacGTP. It is also crucial that the RacGDP-GEF intermediary is membrane bound. Figure 10 shows the same equations solved whilst taking this intermediary as diffusing both on the membrane and in the cytosol. With this higher diffusion rate, even of this relatively short-lived intermediary, the Rac activation pattern is destroyed, figure 10 Previous experimental work has shown activation of Rac and inhibition of Rho returning to basal levels after an extended period of time [START_REF] Tzima | Activation of integrins in endothelial cells by fluid shear stress mediates Rhodependent cytoskeletal alignment[END_REF][START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF][START_REF] Wojciak-Stothard | Shear stress-induced endothelial cell polarization is mediated by Rho and Rac but not Cdc42 or PI 3-kinases[END_REF]. In this current model there is no feedback, so the new steady states are maintained. However it is not necessarily the case that there must be direct biochemical feedback. It is possible that the change in morphology itself is enough to damp the system back to basal levels, i.e. as the cell polarises the signal from the flow is attenuated. After long term exposure to shear-stress ECs flatten to approximately 50% of their original height, which reduces the total and maximum shear on their surface [START_REF] Barbee | Shear stress-induced reorganization of the surface topography of living endothelial cells imaged by atomic force microscopy[END_REF]). This could switch off the signalling network. However a cell elongated (and flattened) in one direction would not align with the flow in a different direction if this was the only effect. The EC orientation may play an important role in modulating the signal. Cells orientated perpendicular (a = 1/3 and b = 3, equation 3) to the flow have a higher response (as measured by Rac-GTP) than parallel (a = 3 and b = 1/3, equation 3) and symmetric cells (Figure 11). This effect may be amplified by clustering of integrins: if they are localised in FA then the location of FAs with respect to the up and downstream regions of the cell could be important.

Discussion

Exactly how ECs interpret the force due to fluid flow and how the components of the process interact remains an open question. Here, coupling models of fluid flow, mechano-transduction and Rho GTPase activation we have tested a plausible mechanism of interpreting the flow by a mechano-transducer and the downstream signal this would induce in the Rho GTPases RhoA and Rac1. The strength of this work is the coupling of the multi-scale models from fluid flow through to localized biochemical signaling.

The hypothesis presented here is that there is a mechano-transducer that is sensitive to the force normal to surface. We suggest that this mechanotransducer is activated in a region localized near the applied force, and tested whether this could lead to spatial localization of Rac1 activity (i.e. given the pattern of normal force and diffusion of some key components of the signalling network, can local activation of Rac1 be initiated). If our hypothesis had failed using this model then we would also expect it to fail on inclusion of a more complete consideration of cytoskeletal biomechanics and shear force.

The mechano-transduction model is a simplified first approximation and only one component of the force was considered. This model could be developed to allow force in any direction. However, to do so would require a more sophisticated treatment of cytoskeletal mechanics-which was beyond the scope of this work. This would be a particularly interesting future direction; the heterogeneity, and complexity, of cellular structure may lead to unexpected conclusions regarding the transmission of force through ECs. For example, more realistic models of large focal adhesions in discrete locations, rather than the coarse grained continuous adhesions assumed here, could potentially lead to distinct conclusions.

This work implies that the heterogeneity of the force normal to the cell surface could give rise to the polarized signalling response. It has previously been suggested that the normal force could play an important role in EC signalling (Wang and Dimitrakopoulos, 2006a,b). Response to the normal force may be a necessary component of the process of EC polarisation and alignment, although it is probably not sufficient. It is probable that mechano-transduction via other pathways could supplement the model here. For example, it is possible that the ion channels could either respond to the normal component of the force in some manner, perhaps due to cytoskeleton coupling (Barakat et al., 2006), or are pre-oriented asymmetrically. Precisely how ECs transduce and respond to a mechanical force is unclear. However, modelling has also shown if cells respond to the normal force then a change in the morphology of the cell itself may attenuate signalling. In particular, the orientation of cells with respect to the cell can, potentially, lead to very different signalling. This suggests a plausible mechanism for cells sensing the direction of the flow and whether, given their current morphology, they should respond to the flow.

Here, a plausible mechanism for signal transducted is presented. Src is taken as a a read-out of FA signalling, but it should be noted that it is likely that there are many intermediaries linking FA signalling and Rho GTPase activity. Hence, further work is required to establish, either computationally or experimentally, the complete nature of FA signalling in response to fluid flow.

Components of integrin complexes are known to respond to force. Although the exact nature of the mechano-transduction is unclear.

Src, a tyrosine kinase associated with integrins, was taken as a biochemical measure of EC response to force and hypothesised to link directly to Rac GEF and Rho GAP activation. This hypothesis was tested with a simple model of Rho GTPase cross-talk and cycling (between GTP-and GDP-bound forms). Modelling showed that this hypothesis could lead to GTPase activation comparable to the reported levels for whole cell averages and qualitatively similar to localized regions of Rac activation.

It is evident that the Rho GTPases make excellent candidates for triggering local cytoskeleton events. Acting as switches they can be highly regulated by GEFs and GAPs and, furthermore, their association with the plasma membrane (once activated) means that the activation is kept in locally distinct areas due to slow diffusion. Thus to generate localised patterns of Rho GT-Pase activation, it seems that either a bistability is established or that it is the GEF that is activated rather than a GAP inactivated. In the latter case the faster diffusing GAP will result in activation being a more global phenomenon, similar to figure 10. The mechanism presented here is similar to a more general suggestion for local activation of GTPases [START_REF] Bement | Rho gtpase activity zones and transient contractile arrays[END_REF].

It is also demonstrated that measures of activation across the whole cell can not distinguish between different hypotheses; more local measures of protein activity in response to fluid flow are required to make such a determination.

In principle this hypothesis could be valid in an alternative form with a different read out from integrin complex signalling and a different link between GEFs and GAPs and the mechano-transducer. However, it does seem clear that the normal component of the force on the EC must play a role in initiating the heterogeneity in signalling, however the model here could be augmented by different pathways resulting from either integrin, ion channel or cell junction mechano-transduction.

is a good approximation near the arterial wall. The viscosity of the fluid was taken to be 1cP.

GTPase Model Parameters

Parameters were found from (in order of preference) experimental data, from previous modelling work or were estimated.

For the force activation of Src, reaction 5, the parameters were estimated by comparison with an experimental time course [START_REF] Jalali | Shear stress activates p60src-Ras-MAPK signaling pathways in vascular endothelial cells[END_REF] for Src activation due to shear stress. The form of k + (z e ) was taken as k + (z e ) = k + + k m /(1 + e 10-ze ). To estimate k + , k m and k -the level of Src was assumed to be in a steady state S (in the absence of diffusion):

S = k + + km 1+e 10-ze k + + km 1+e 10-ze + k -
By integrating this function over a square (as an approximation to the cell) and assuming a linear relationship (in the downstream half of the cell) between z e and the distance from the centre of the cell an expression for the fold increase in Src can be reached. Comparing this with the experimental observations [START_REF] Jalali | Shear stress activates p60src-Ras-MAPK signaling pathways in vascular endothelial cells[END_REF][START_REF] Fleming | Role of PECAM-1 in the shear-stress-induced activation of akt and the endothelial nitric oxide synthase (enos) in endothelial cells[END_REF] of a characteristic 2-5 fold increase in activated Src leads to a relationship between k + , k m and k -.

A second relationship can be drawn by comparing the order of activation of Src in the model with experimental data. Solving the ODE for reaction 5, gives a characteristic time for Src to be maximally activated; the solution has a -D.exp(-(k + + k m + k -)t) term, so as t → ∞ the activated Src level approaches its maximum. This maximum is observed within 4-10 minutes, hence (k + + k m + k -)t must be large for t in this range. To finalise the parameter values a third relationship is assumed, following previous authors [START_REF] Fuß | Deactivation of Src family kinases: Hypothesis testing using a monte carlo sensitivity analysis of systems-level properties[END_REF]: initially the steady state level of activated Src is ≈1% of the total, so 100k + = k -. These relationship are sufficient to estimate k + = 0.00006s -1 , k -= 0.006s -1 and k m = 0.0147s -1 .

The parameters for the Kelvin body, figure 5, were taken from previously reported modelling and experimental work [START_REF] Mazzag | A model for shear stress sensing and transmission in vascular endothelial cells[END_REF][START_REF] Bausch | Measurement of local viscoelasticity and forces in living cells by magnetic tweezers[END_REF]. Note that these parameters were derived by attaching a microbead to the cell surface, with an estimated area of approximately 6 μm 2 attached to the cell [START_REF] Bausch | Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead mircorheometry[END_REF]. Hence, to correctly apply these parameters to the Kelvin body the f i (note that this is a force per area, equation 2) have to be scaled by 6 μm 2 .

It was found that the GAP and GEF activation rates were not the rate limiting steps in network, figure 7. However, the rate-limiting step of conversion from the GDP-to GTP-bound form of Rac is dependent on GEF activity. Hence, provided the GEF activation rate was sufficiently fast then the rate-limiting Rac activation could be initiated early enough so that it occurred on a similar time scale to observed [START_REF] Tzima | Activation of Rac1 by shear stress in endothelial cells mediates both cytoskeletal reorganization and effects on gene expression[END_REF]. Hence, it was necessary to have the Rac GEF activating quickly, on the order of a minute. Similarly, the Rho GAP was also required to become activated quickly. These criterion determined the rates for the Rho GAP, and Rac GEF. Similar rates were taken for the corresponding Rac GAP and Rho GEF. However, they were altered slightly so that there were similar basal levels of the active form (otherwise the basal level of Src gives a bias to the Rho GAP and the Rac GEF). Rates were finally fixed by demanding a low ratio initially of active to inactive forms, and by (following the onset of flow) demanding the new level of active GAP or GEF was significantly less than the total (to avoid saturation). Initial total concentration of Rac and Rho (either active, inactive or bound in intermediary complexes) was taken to be 10 μM of each, which is a typical cellular amount [START_REF] Jilkine | Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk[END_REF][START_REF] Goryachev | Computational model explains high activity and rapid cycling of Rho GTPases within protein complexes[END_REF]. Total concentration of RacGAP and RhoGAP, RhoGEF and RacGEF was taken to be 1 μM, 1 μM 2 μM and 2 μM; with corresponding (initial) active concentrations 0.46 μM, 0.75 μM, 0.41 μM and 0.24 μM.

The parameters governing GTPase cycling between GTP and GDP bound states were found to be the most significant in governing the response; fortunately experimental data was available for this data [START_REF] Goryachev | Computational model explains high activity and rapid cycling of Rho GTPases within protein complexes[END_REF][START_REF] Zhang | Characterization of the interactions between the small GTPase Cdc42 and its GTPase-activating roteins and utative effectors[END_REF][START_REF] Zhang | The role of Mg 2+ cofactor in the guanine nucleotide exchange and GTP-hydrolysis reactions of rho family gtp-binding proteins[END_REF].

Fig. 1. Rho GTPase regulation by GAPs, GEFs and GDIs. In the inactive state (GDP bound) the GTPase interaction with the plasma membrane is inhibited by interaction with a GDI, which in turn prevents activation. The GEF removes GDP from the GTPase. Subsequent binding of GTP occurs quickly (because of the high ratio of GTP to GDP). The rate of hydrolysis of RhoGTP is increased by binding of a GAP [START_REF] Dermardirossian | GDIs: central regulatory molecules in rho GTPase activation[END_REF][START_REF] Bos | GEFs and GAPs: Critical elements in the control of small G protiens[END_REF]. Fig. 6. The force normal to the cell surface is applied to the Kelvin body, figure 5, to give a predicted pattern of integrin deformation. Note that compressive forces were assumed not to affect mechano-transducter signalling and, for simplicity, integrin clustering was coarse-grained into a smooth distribution. c) Fold Src increase over the whole cell. This is fixed by parameter estimation (Appendix) to give a similar response as found experimentally [START_REF] Fleming | Role of PECAM-1 in the shear-stress-induced activation of akt and the endothelial nitric oxide synthase (enos) in endothelial cells[END_REF][START_REF] Jalali | Shear stress activates p60src-Ras-MAPK signaling pathways in vascular endothelial cells[END_REF]. (d) Average RacGTP concentration over the whole cell.

  (a), whereas the whole cell measure of Rac activation, figure 10(b), is broadly similar to figure 9(d). Whole cell profiles of activation therefore characterise the system poorly, and thus to test and generate hypotheses of EC responses to fluid flow it is necessary to have experimental data that quantifies local activation of proteins.

Figure 11 (

 11 Figure 11(d) shows Rho inhibition across the whole cell: the concentration distribution of Rho-GTP is more homogenous (not shown) than the Rac-GTP distribution because Src activates GAP ρ which can diffuse relatively quickly through the cell. Some experiments report rapid transient activation of Rho[START_REF] Wojciak-Stothard | Shear stress-induced endothelial cell polarization is mediated by Rho and Rac but not Cdc42 or PI 3-kinases[END_REF], but this could be through a different pathway (either via integrins or a different mechano-transduction mechanism).

Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .

 2345 Fig. 2. An individual cell's (attached to a plane at z = 0) surface discretised into a triangle mesh, such that the projection on the x -y plane of the mesh nodes gives the vertices of hexagons. The fluid flow is in the positive x direction: u = (kz, 0, 0), z ≥ 0.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Rho, Rac and Src network. Force on the cell leads (indirectly) to Src tyrosine phosphorylation. Phosphorylated Src can activate a Rho GAP as well as a Rac GEF, leading to Rac activation and Rho inhibition. Rho and Rac mutually inhibit each other, RhoGTP by activating a Rac GAP and RacGTP inhibits Rho by inhibiting a Rho GEF.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. The effect of allowing the RacGDP-GEF intermediary (equations 5) to diffuse in the cytosol and on the membrane as oppose to just on the membrane as in figure 9(b). A localised pattern of activation is not generated, though the average [RacGTP] is similar as in 9(d). (a) Concentration distribution of RacGTP, (b) Average RacGTP concentration over the whole cell.

Table 1 Model

 1

	Parameter	Description	Value	Units	Source
	Dm	Membrane diffusion	0.1	μm 2 s -1	Marèe 2006
	Dmc	Membrane and cytosol diffusion	10	μm 2 s -1	Marèe 2006
	k	Shear rate	280	s -1	See Appendix
	μ	Fluid viscosity	1 × 10 -6	g μm -1 s -1	See Appendix
	ν	Dashpot viscosity	6.33 × 10 -5	Pa m s	Mazzag 2003
	h 1	Spring constant	1.25 × 10 -3	Pa m	Mazzag 2003
	h 2	Spring constant	1.61 × 10 -3	Pa m	Mazzag 2003
	k + (ze)	Extension dependent Src activation	fig 8	s -1	See Appendix
	k -	Src inactivation rate	0.006	s -1	See Appendix
	k 1	GEF activation	0.83	μM -1 s -1	See Appendix
	k 2	GEF inactivation	0	μM -1 s -1	See Appendix
	k 3	Rac GEF binding rate	0.0034	μM -1 s -1	B. Zhang, 2000
	k 4	Rac GEF unbinding rate	1.1 × 10 -6	s -1	B. Zhang, 2000
	k 5	Rac GDP disassociation	0.094	s -1	B. Zhang, 2000
	k 6 ,	Rac GAP binding	0.017	μM -1 s -1	Goraychev, 2006
	k 7	Rac GAP unbinding	0.05	s -1	Goraychev, 2006
	k 8	RacGTP GAP hydrolysis	8.3	s -1	Goraychev, 2006
	k 9	Rac GAP activation	0.001	μM -1 s -1	See Appendix
	k 10	Rac GAP inactivation	0	μM -1 s -1	See Appendix
	k 11	Rho GAP activation	0.8	μM -1 s -1	See Appendix
	k 12	Rho GAP inactivation	0	μM -1 s -1	See Appendix
	k 13	Rho GAP binding	0.017	μM -1 s -1	= k 6
	k 14	Rho GAP unbinding	0.05	μs -1	= k 7
	k 15	RhoGTP GAP hydrolysis	8.3	μs -1	= k 8
	k 16	Rho GEF binding rate	0.0034	μM -1 s -1	= k 3
	k 17	Rho GEF unbinding rate	1.1 × 10 -6	s -1	= k 4
	k 18	Rho GDP dissassociation rate	0.094	s -1	= k 5
	k 19	Rho GEF activation	0	μM -1 s -1	See Appendix
	k 20	Rho GEF inactivation	0.0001	μM -1 s -1	See Appendix
	k 21	basal Rho GAP activation	0.002	s -1	See Appendix
	k 22	basal Rho GAP inactivation	0.009	s -1	See Appendix
	k 23	basal Rac GEF activation	0.00045	s -1	See Appendix
	k 24	basal Rac GEF inactivation	0.09	s -1	See Appendix
	k 25	basal Rac GAP activation	0.002	s -1	See Appendix
	k 26	basal Rac GAP inactivation	0.004	s -1	See Appendix
	k 27	basal Rho GEF activation	0.00045	s -1	See Appendix
	k 28	basal Rho GEF inactivation	0.015	s -1	See Appendix

Parameters. See references for accuracy in parameter values. Parameters estimated in the appendix are determined up to an order of magnitude.

1. We develop a multi-scale model of the endothelial response to fluid flow. 2. We couple models of fluid flow, mechano-transduction and Rho GTPase signaling. 3. We hypothesize that the normal force on the cell surface establishes signaling polarity. 4. We show that signaling can be attenuated by cell morphologies aligned with the direction of fluid flow. 5. Conversion of Rac-GDP to Rac-GTP is likely to be the rate limiting step in Rac activation.

Appendix

See the table below for the parameters used in this work.

Flow Parameters

The shear rate is estimated by modelling the flow for a steady Newtonian fluid in a large cylinder as a Poiseuille flow:

where u(r) is the flow velocity at r, x is a unit vector along the cylinder, a is the radius of the cylinder and r is the distance from the centre of the cylinder and Q is the volumetric flow rate . The mean volumetric flow rate through the common carotid artery (CCA) has been measured as approximately 6 ml s -1 [START_REF] Marshall | Carotid flow rates and flow division at the bifurcation in healthy volunteers[END_REF], and the inner radius as 3mm [START_REF] Polak | Compensatory increase in common carotid artery diameter[END_REF]. Equation 6 is used to estimate u at 6μm; approximately the height of an endothelial cell. The shear rate k is found by comparing this solution with a simpler model for the flow, u(x) = kzx, simple shear flow, which we assume