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AU-KBC Research Centre, MIT Campus of Anna University, Chromepet, Chennai, India 600044
Correspondence: Address reprint requests to S. V. Ramanan, Tel.: 91-44-223-4885; Fax: 91-44-2-223-1034; E-mail: ramanan{at}au-kbc.org.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: MEMBRANE CURRENTSACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: MEMBRANE CURRENTSACKNOWLEDGEMENTSREFERENCES |
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). This autoregulatory response is achieved by two different mechanisms (2), namely: the metabolic mechanism, in which metabolites, such as adenosine and pO2, act on the blood vessel; and the myogenic mechanism, in which the change in the transmural pressure itself acts on the vessel to maintain a constant blood flow. The myogenic response is independent of neural, metabolic, and hormonal influences and seems to be an inherent characteristic of smooth muscle, being especially pronounced in arterioles (3).
Over the last several decades, numerous investigators have demonstrated the importance of the myogenic response in the local regulation of blood flow and capillary pressure, and in the generation of basal vascular tone (4). The myogenic mechanism has been shown to play a significant role in autoregulation in arteries isolated from various tissues, including cerebral (5) and coronary arteries (6).
The mechanism underlying the myogenic response is thought to be as follows (7). It has been established that increased intravascular pressure causes a graded membrane potential depolarization of smooth muscle cells that line the arterial wall in various tissues (8,9). These tissues include rat middle cerebral arteries (9) and rabbit cerebral arteries (7,10). This depolarization, which probably results from the opening of stretch-activated TRC channels (11), causes voltage-dependent calcium channels to open. The resultant increase in cytosolic calcium, through a series of signaling processes (12), finally activates myosin light chain kinases resulting in the contraction of the cell and constriction of arterial diameter. The increased calcium itself also activates the release of calcium from the sarcoplasmic reticulum as calcium sparks. These sparks activate calcium-activated large potassium (BK) channels, which in turn hyperpolarize the cell. This mechanism acts as a feedback loop to regulate the steady-state membrane potential (7).
The experimental information above has formed the basis for many models of myogenic response. Most of the models reported so far take into account only the mechanical aspects of myogenic response such as phosphorylation, cross-bridge formation, force development, length-tension relationship, vessel resistance, and vessel diameter. For example, the force equilibrium model proposed in Borgstrom et al. (13,14) describes the responses of myogenic vascular resistance to changes in transmural pressure. The kinetic model of cross-bridge phosphorylation and the regulation of latch state in smooth muscle are described in Hai and Murphy (15). Lee et al. (16) described a biomechanical model that was based on the assumption that the arteriolar wall exhibits viscoelastic properties. A minimal model of arterial vasomotion, including the nonlinear interaction of intracellular and membrane calcium oscillators, is developed in Parthimos et al. (17). However, none of the above models encapsulate the cellular electrochemical properties that form the basis of the myogenic response.
The only electrochemical model of smooth muscle is a kinetic model that incorporates membrane channels and transporters (18) as well as mechanical components of cell response during the development of tension. The results of the model (18) were compared (19) against the experimental results reported in Knot et al. (7) and Knot and Nelson (10). In these experiments, the myogenic response was studied in intact cannulated cerebral arteries. The calcium levels, the membrane potential, and the arterial diameter of pressurized small cerebral arteries were simultaneously measured. These data led to the observation that cytosolic calcium depended only on the membrane potential. The results in Yang et al. (19) mimic the experimental data for steady-state membrane potential and arterial diameter, but not for intracellular calcium.
The steady-state myogenic response is reached in a path-independent manner, as reflected, e.g., in the fact that it is the same when either pressure steps or pressure ramps are used to elicit tone (20). We have therefore attempted in this work to model the smooth muscle only in the steady state. The model is a purely electrochemical representation of the changes in the vascular smooth muscle cell in response to applied pressure. We do not model the mechanical responses of the cell in response to the changes in calcium and potential. However, the model can be used to calculate changes in the membrane potential as well as the calcium concentrations in response to applied pressures, as well as in response to channel and transporter antagonists.
The model (Fig. 1) incorporates L-type calcium channels, calcium pumps, inward rectifiers, sodium-calcium exchangers, sodium-potassium pumps, and stretch currents. We have specifically considered processes with long time constants that would be important in setting steady-state myogenic tone. The model has an inbuilt feedback loop for the potential (7) through the dependence of calcium-activated maxi-K potassium channels on calcium sparks. The intrinsic channel and pump parameter values in the model are very close to those reported in the literature. The model is able to mimic the experimental data on the steady-state characteristics of vascular smooth muscle cells under a variety of experimental conditions, including pressure, channel and pump block, and variation in extracellular ionic concentrations.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: MEMBRANE CURRENTSACKNOWLEDGEMENTSREFERENCES |
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Calcium fluxes across the sarcoplasmic reticulum (SR)
Calcium pumps and release mechanisms such as ryanodine receptors and IP3 receptors are present in the sarcoplasmic reticulum (SR), and affect calcium dynamics. The activity of the pumps and channels in the SR membrane may vary with the intracellular calcium concentration ([Ca]i). However, the SR is a finite source, and in the steady state, uptake and release across the SR membrane must be equal. This implies that the calcium concentration inside the SR is not a free variable in the steady state, but is determined by [Ca]i.
It is known (21) that calcium sparks, which originate in the SR, activate calcium-activated BK channels and thereby influence the membrane potential. However, it has been shown experimentally that the frequency of the sparks is a function only of the membrane potential under the physiological range of transmural pressures (22,23). We therefore do not explicitly include the SR in the calculation of [Ca]i.
Ionic currents
The complete set of equations that describe the ionic currents through the various channels, transporters and pumps in the model are given in the Appendix. The parameters in these equations are of two kinds:
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Voltage-operated L-type calcium channel, Il
); these are based on whole-cell current data in smooth muscle from rat cerebral resistance arteries. The equations in Rubart et al. (24) do not take into account the observation that high [Ca]i blocks LTCCs in VSMCs with an IC50 of 380 nM and a Hill coefficient of 3 (25); however, with these parameter values, the model results for [Ca]i were far below the experimental results. We used the values in the L-R model for cardiac myocytes (26,27), namely an IC50 of 600 nM and a Hill coefficient of 2.
Stretch-operated channels, Is,c
,28,29) indicate that pressure-induced depolarization results from the opening of stretch-activated nonspecific cation channels. Single-channel and macroscopic currents have been measured during mechanical stimulation of vascular smooth muscle from coronary arteries (30). The stretch current is primarily carried by sodium ions, but with contributions from potassium and calcium ions (31). We represent the current across the open channel based on the GHK equation, as previously proposed (18). The pressure- (or stretch-) dependence of the current is modeled as a Boltzmann (18). There are no experimental data on the parameters of the Boltzmann that describe the pressure-current relation. These two intrinsic parameters alone were also estimated by fitting. As VSMCs exhibit a basal tone, we seeded the half-maximal stretch to correspond to the stress induced by a basal isobaric pressure of 60 mmHg (
1/2 = 240 mmHg). The relationship between stress and transmural pressure is described below. The parameter k defines the operational stress-sensing range of the stretch channel; we seeded this range to a stress of 140 mmHg (or 35 mmHg of pressure). ). Using this information with the GHK equation gave a seed conductance gs of 0.14 pA/mM.
Calcium extrusion pump, Icp or ICaP
). Ca-ATPase pumps are stimulated by calmodulin, which itself binds four calcium ions (33). The binding is known to be cooperative, with a Hill coefficient of 4 and a half-maximal concentration of 300 nM (34). We have assumed that the pump activity is linearly related to the concentration of the calcium-calmodulin complex. ). [Ca]i at low transmural pressures is 100 nM (10). This yields a seed value for the maximal flux I0,cp of 8.2 pA.
Sodium calcium exchanger, INCX
,37). We have therefore followed the formalism used in modeling the NCX in cardiac myocytes (38,39). The model also takes into account the observation that high extracellular sodium ([Na]o) blocks the reverse-mode activity of the NCX with a Hill coefficient of 2 and a Kd of 60 mM (40). 
) has a value of 2.5 x 10–4 pA/(mM)4 in cardiac myocytes (39). was seeded to 2 x 10–5 to reflect the observation that the Na-Ca exchanger plays only a supportive role when [Na]i is kept at physiological levels (37,41).
Delayed rectifier channel, Idr
,43). ), the peak conductance gdr was seeded to 7 nS.
Inward rectifier channel, Iir
), except that the inverse slope of the Boltzmann was increased from 6.9 to 8.1 mV. Outward currents through the IR channel in VSMCs are less well characterized, as they are quite small (44). Fig. 2 shows the open probability Pir in the model (dashed lines) as a function of the membrane potential (Vm, positive to EK); also plotted are the pure Boltzmann probability pir below (solid lines) as well as the probability in the L-R model (dotted lines) for cardiac myocytes (26,27). The open probability Pir differs from a Boltzmann through the incorporation of an extra term 
. The effect of the term is to introduce a small dip in the outward current at 
15 mV positive of Ek; this changes the stability of the model slightly at very low pressures and is discussed in Results, below. 
(44).
Calcium-activated BK channel, Ibk
) and Wang and Mathers (45). BK channels are activated by cytoplasmic calcium, and are generally colocalized with calcium spark release channels (46). During the spark, BK channels are exposed to a mean calcium concentration of 
10 µM, 50–100 times the averaged [Ca]i (22). The sparking frequency P'bk increased as a Boltzmann function of the potential Vm in the myogenic response range (23). The spark duration, 
, is 20–50 ms and does not depend on potential (22,46). We have changed the dependence of the sparking frequency on Vm into a dependence on [Ca]i, based on the correspondence between calcium and membrane potential (10). ,46).
Na-K pump, INaK
,27,39). However, the Na and K half-saturation constants have been set to values reported for Na-K pumps in VSMCs (47). 
was seeded to 20 pA (47). VSMCs have a capacitance of 25 pF; the normalized value of 0.8 pA/pF may be compared to the corresponding value of 2.25 pA/pF in cardiac myocytes (39).
Background currents, Ib,c
).
Transmural pressure and stress on the membrane
In experiments on intact arteries, changes in intracellular calcium, voltage, and arterial diameter were measured against a range of transmural pressures from 10 to 100 mmHg (10). As we model only cellular and not tissue response in this work, we need to convert the applied transmural pressure, a tissue parameter, into a cellular parameter, namely membrane stress, which is the stimulus for the myogenic response (3,12,20,48). However, in these experiments, the myogenic response was such that the arterial diameter remained relatively unchanged over the range of applied pressures. According to Laplace's law, /P = r/w, where the radius r is 60 µm, and the wall thickness w is 15 µm. At constant diameter, this law implies that the stress varies linearly with pressure. We note that the stress as calculated by Laplace's law is the net wall stress in the vessel. However, we need to relate the wall stress to the membrane stress sensed by the stretch-sensitive channels. The total wall stress may be considered to be counterbalanced by two parallel components:
e: Tension due to stretching (strain) of the passive extracellular matrix. A measure of the strain is provided by arterial diameter. Experimentally, this strain is only weakly altered by stress, since arterial diameter changes only slightly with transmural pressure when the vessel has myogenic tone. We therefore assume that e does not change with pressure P. c: Tension due to intracellular components. This includes active force generation developed by myofibrils as well as passive forces from the cytoskeleton and the viscous cytoplasm. The change in net VSMC cell length is small with variations in transmural pressure, as VSMCs are circumferentially oriented; changes in cell length due to contraction of the force-generating elements would be counterbalanced by stretching of the passive cytoskeletal structures. As these intracellular components are connected in series, they would exert equal forces (= c) in the steady state, where c = –e. It is likely that stretch-operated channels are coupled either to the cytoskeleton or to integrins (49), and that the stress that regulates their activity is c. As noted before, we have followed (18) in modeling the dependence of the open probability on c through a Boltzmann relationship.
The stress e felt by the stress-sensitive channels differs from the wall stress = (r/w)P only by the constant factor e, which can be absorbed into the constant half-maximal tension parameter 1/2. We have used a conversion factor of 4 (= r/w) in converting from applied pressure to stress. A different conversion factor would not affect the results, though the stretch channel parameters 1/2 and k would have to be appropriately rescaled.
Numerical methods
The equations in the Appendix were solved to satisfy the requirement that the fluxes for all three cations be zero in the steady state. This was achieved by varying Vm, [Na]i, [Ca]i, and [K]i for given external parameters, e.g., applied pressure or external ionic concentrations, subject to charge balance considerations. The variation was done such that the quantity 
was minimized, where
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The symbol Ic refers to the total current carried by the cation c. The minimization was done using the simplex algorithm as implemented in the GNU scientific library. Standard extracellular ionic concentrations are given in Table 3, and these are used unless otherwise specified in the text. Seed values used for the minimization are listed in Table 4 for the state variables, Vm, [Na]i, [Ca]i, and [K]i. Two seed values were tried for the membrane potential, due to the presence of dual bistable states in the model (see Results). Programs were written in C and compiled against the BLAS and GSL libraries.
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In the second time-dependent simulation, we have compared the kinetics from experiment and model in response to changes in extracellular concentrations of [Na]o (see Alterations in [Na]o, below). The transient response is calculated with the same adiabatic assumption that was used above when (say) the Na-K pump is blocked. That is, kinetics due to rapid channel gating is ignored, and all channels are assumed to have steady-state behavior appropriate to the current resting potential. Thus, only changes in intracellular ionic activity and membrane potential are followed with time. In all these kinetic simulations, we assume a 2-pL cell volume (50). We emphasize that any similarity in kinetics between experimental data and model is only indicative of compatibility.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: MEMBRANE CURRENTSACKNOWLEDGEMENTSREFERENCES |
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The results of the model are broadly in agreement with the experimental data.
Estimation of parameters and justification
As discussed in Materials and Methods, values for intrinsic parameters, i.e., those parameters which describe the dependence of membrane currents on ionic concentrations, membrane potential, and [Ca]i, are taken directly from the literature. However, for the stretch channel, the slope and half-maximum values of the Boltzmann dependence of current upon membrane tension are not experimentally characterized. Other variable (extrinsic) parameters are those that relate to the numbers of channels, pumps, and transporters, e.g., peak conductance and maximum pump current. There are thus a total of 10 variable parameters in the model (see Table 2). The seed values for these parameters were set as described in the individual subsections above in Materials and Methods for the channels and pumps.
Fig. 3, A and B, show the experimental data for [Ca]i and membrane potential Vm (as solid squares) (10). The experimental data covers six different applied transmural pressures from 10 to 100 mmHg (Fig. 6 in (10)). The model parameters were estimated by minimizing the difference between the predictions of the model (solid lines) and the experimental data on [Ca]i and Vm at these six transmural pressures.
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), was therefore fixed at 20 pA. The final estimates for the variable parameters in the model are shown in Table 2. Almost all the parameter values are within a factor of 2 with respect to the seed values. One major difference between the seed and final values is in the LTCC conductance, gl. There are two explanations for this difference:
The estimate of the maximal flux across the NCX also differs from the seed value. However, this is perhaps not significant given that the seed value was chosen somewhat arbitrarily in the absence of relevant experimental data.
There are two parameters describing the Boltzmann response of the stretch channel to membrane tension—the slope k and the half-maximum tension 1/2. The value for the slope k corresponds to a transmural pressure of 27 mmHg, which is approximately half the isobaric pressure (60 mmHg). However, the transmural pressure corresponding to the tension where the channel is half-maximally activated is 137 mmHg, which is more than twice the isobaric pressure. The current through the stretch channel would thus almost linearly increase with membrane tension (or pressure) in physiological conditions. In fact, this accords with experimental data where the open probability Pm of the channel is a linear function of the applied stress (30).
Sodium as a function of applied pressure
Fig. 3 C shows the variation predicted by the model in [Na]i with pressure (in solid lines); [Na]i changed from 5.93 mM at low pressures (10 mmHg) to 16.92 mM at high pressures (100 mmHg). The sodium concentration of 8.56 mM in the model at a transmural pressure of 60 mM Hg is comparable to experimentally observed basal values of 10–11 mM (51–53). The rise of [Na]i with increasing transmural pressure is an experimentally testable prediction of the model. It may be noted that ventricular myocyte models (26,27) exhibit similar changes in [Na]i if it is clamped at a depolarized state (data not shown).
Individual currents
Fig. 4, A–C, shows the contributions of the calcium, potassium, and sodium fluxes in the model over a range of transmural pressures. It may be noted that the fluxes through the Na-Ca exchanger (NCX) in the forward mode are relatively small in comparison to other fluxes for both sodium and calcium. The relative contribution of the inward rectifier to K+ flux as compared to the other three potassium channels is higher at small applied pressures (
40 mmHg) (54); this is explored in detail below (please see Bistable States). Overall, [Ca]i and Vm vary with transmural pressure in a manner such that ionic fluxes have an almost linear dependence on pressure.
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), 65 mM in an isometrically stretched vascular preparation after 1 h (51), or an increase of 6.3 mM in 10 min in cultured VSMCs after ouabain-induced block (53).
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), the relatively small increase in [Na]i at low transmural pressures (to 26 mM at 10 mmHg) after Na-K pump block is due to NCX reverse-mode activity. However, the model predicts that the membrane potential would be hyperpolarized. This may seem counterintuitive, since the Na-K pump is electrogenic, and blocking the pump should apparently depolarize the membrane. Nevertheless, the prediction of a hyperpolarization at low pressures upon Na-K pump blockade is a testable prediction of the model.
It should be noted that the predicted hyperpolarization at low pressures would be observed only in the steady state. Fig. 3, A and B, also show the instantaneous response after block of the Na-K pump in gray lines. It may be seen that the membrane depolarizes almost uniformly by 6–7 mV over the entire range of transmural pressures, while [Ca]i also increases uniformly by 30–50 nM. These results are compatible with the 17 nM immediate increase in [Ca]i observed in isolated detrusor cells (55) upon blocking the Na-K pump with strophanthidin (100 µM).
In isometrically stretched mesenteric vessels, 1 mM ouabain reduced sodium efflux by approximately half (51). The model predicts a similar reduction in sodium efflux, upon Na-K pump block, at 70 mmHg transmural pressure (Fig. 4 C).
Bistable states
Two stable states exist in the model over a range of transmural pressures. Fig. 5 shows these two states, which we label as the upper and lower states, as solid and dashed lines, respectively; [K]o was set to 4.8 mM. In the pressure range of 0–35 mmHg where both states coexist, the lower state is hyperpolarized by 15–17 mV with respect to the upper state and [Ca]i differs by 50–70 nM. The bistable behavior of the model at low pressures appears to be qualitatively consistent with observations in unstretched strips from the spiral modiolar artery (56). In this preparation, the resting potentials had a bimodal distribution, with two stable levels at –40 and –75 mV. These were, arespectively, labeled low-RP and high-RP states.
Fig. 6 shows the unfolding of the two stable states as external [K]o is varied from 1.5 mM to 15 mM. The range of pressures over which the two states coexist reduces with increasing [K]o, with an overlap of almost zero when [K]o is 15 mM. These dual stable states arise in the model due to the bell-shape of outward currents through the IR channel (Fig. 2). Fig. 6 C shows steady-state [Ca]i (in dotted lines) when Iir is set to zero in the model; it can be seen that the lower state disappears. Two pieces of experimental evidence support this interpretation:
); and ), blocking the IR channel with barium causes a leftward shift in the activation curve of tone with pressure. The relaxed diameter of renal arterial vessels (18 µm) is substantially smaller than that of cerebral resistance arteries (100 µm); nevertheless, the results of the model upon IR channel block agree qualitatively with the experimental data.
The model also offers a possible explanation for the experimentally observed differences in graded versus nongraded response upon stepwise changes in pressure (10,57) at [K]o = 6 mM (Fig. 6 C). In the experimental protocol followed in Knot and Nelson (10), the arteries were equilibrated at 60 mmHg, where only the upper state is stable. It is possible that these arteries remained in the upper state as pressure was changed, resulting in a graded depolarization at lower pressures. In the protocol in Osol et al. (57), the arteries were equilibrated at 10 mmHg, where only the lower state is stable. As pressure is increased beyond 45 mmHg, the lower state is destabilized in the model and the system would shift abruptly to the upper state. This behavior seems to mimic the experimental observations (57), where calcium increases suddenly as the transmural pressure is increased from 50 to 60 mmHg.
Fig. 7 plots the potential Vm as a function of [Ca]i for all values of [K]o shown in Fig. 6. Values from the upper state are plotted as open circles, and values for the lower state as solid circles. The points from both upper and lower curves are similar where there is an overlap, in the range of potentials from –60 mV to –30 mV. This implies that changes in [K]o, or pressure, do not change the dependence of [Ca]i on Vm, as has been noted (10).
In a common experimental protocol, pressurized cannulated arteries are exposed to increasing [K]o in a stepwise fashion (10,54,58). Under these conditions, the artery dilates at some [K]o ranging from 7 to 10 mM, and this response is usually not graded with [K]o. The abrupt dilation is accompanied by a hyperpolarization and a decrease in [Ca]i. This experimental scenario is akin to taking a cross-section at a fixed pressure across the panels in Fig. 6. Fig. 8 shows this cross section for both upper and lower stable states at a transmural pressure of 60 mmHg (in solid lines), along with experimental data from Knot and Nelson (10) (as solid squares). While the model qualitatively mimics the behavior and shape of the experimental data, it can be seen from Fig. 8 that the predicted hyperpolarization and change in [Ca]i are less than that observed.
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), a step change of [K]o from 5 mM to 1.5 mM induced vasoconstriction at low pressures and vasorelaxation at high pressures. In Fig. 6, the panels A and B for steady-state [Ca]i, when [K]o is 1.5 mM and 4.8 mM, respectively, may be compared. The lower state disappears at low [K]o, implying that reduction in [K]o at low pressures would cause the system to move from the lower to the upper state, inducing depolarization and an increase in [Ca]i (by 50 nM). At high pressures (>40 mmHg), in the model, the system remains on the upper state as [K]o is lowered to 1.5 mM; the upper state itself hyperpolarizes by 5 mV, and there is a small decrease in [Ca]i. The model is thus qualitatively consistent with the experimental observations reported in Chilton and Loutzenhiser (54).
Nongraded dilation upon increased [K]o is also seen experimentally when the Na-K pump is blocked by ouabain (0.1 mM) (58). In the model, as shown in Fig. 8 in dashed black lines, blocking the Na-K pump results in a graded response in [Ca]i upon [K]o increase. However, the instantaneous behavior of the model (gray lines) shows nongraded behavior. Moreover, increasing the ouabain concentration (to 0.5 mM) abruptly reversed the [K]o induced dilation in cerebral arteries (58), implying that Na-K pump block was only partial at the lower ouabain dose (of 0.1 mM). With partial block of the Na-K pump, the model exhibits dual stable states, with an abrupt reduction in [Ca]i as [K]o is increased (data not shown).
The model predicts an increase in [Na]i when the artery is abruptly relaxed by increasing [K]o. This is an experimentally testable consequence of the model. At a transmural pressure of 60 mmHg and with [K]o at 12 mM, for instance, steady-state [Na]i values in upper and lower states are 7.3 and 9.4 mM, respectively. The model also predicts a lowering of [Na]i as [K]o is further increased; for example, changing [K]o from 16 to 61 mM reduces [Na]i from 7.6 to 3.0 mM.
As noted in Materials and Methods, we have assumed that membrane stress is directly proportional to transmural pressure. This was predicated on the observation (Fig. 6 b of (10)) that arterial diameter is fairly constant at 120 µm for transmural pressures from 10 to 100 mmHg. However, as alterations in [K]o do change arterial diameter, all of the results shown here for a single smooth muscle cell may not be directly applicable to the intact vessel. A complete mechano-electro-chemical model would be needed to address this issue; it is considered further in Discussion, below.
Effect of channel blockers
Table 5 compares experimental data and model results for [Ca]i and Vm from pressurized arteries when various channels are blocked pharmacologically. The experimental data derive from observations in cerebral arteries (7,10,42,59). The majority of the experimental results relate only to Vm; the [Ca]i data, where available, is shown in parentheses in Table 5. The results from the model considered here hold for cannulated arteries only under the assumption of constant arterial diameter, while many channel blockers, especially those for LTCCs, change arterial diameter drastically. Thus any similarity between the model results and the experimental data can be seen only as indicative.
Overall, there is reasonable agreement between the trends in the model and the experimental data. In many cases, there is also quantitative agreement between data and model. However, there are two major exceptions:
). The depolarization in the model with DR channel block was only 6 mV, with [Ca]i increasing from 223 to 252 nM. This is a major difference between the model and the experimental data. However, blocking DR and BK channels simultaneously in the model depolarized the cell by 23 mV, comparable to the experimental data.
Alterations in [Na]o
A reduction in extracellular sodium affects the model in three ways: 1), the voltage-dependence of the Na-K pump is reduced (60); 2), the reverse-mode activity of the NCX is increased (37), with a Hill coefficient of 2 and a Kd of 60 mM (40); and 3), the stretch-activated sodium influx is lowered due to the shift in sodium reversal potential. At very low [Na]o (below 10 mM), the sodium flux across the stretch channel is so small that the cell becomes unresponsive to pressure.
The solid line in Fig. 9 A shows [Ca]i as a function of time (with the adiabatic assumption) when [Na]o is suddenly lowered to 5 mM at t = 0. In this simulation, [K]o was set to 4.7 mM. There is a small increase in [Ca]i (25 nM) that persists in the steady state. These results are in accord with experimental data in aortic VSMCs (61). Fig. 9 A also shows the response (dashed line) when the Na-K pump is first blocked, and the cell is subsequently challenged by abruptly lowering [Na]o to 5 mM at t = 0. As noted earlier, blocking the Na-K pump itself increases baseline [Ca]i. With lowered [Na]o, [Ca]i is transiently increased by 200 mM, and is elevated by 26 nM after 10 min. This compares with the experimental data, after block of calcium release from the SR (Fig. 8 in (61)), where [Ca]i transiently increased by 120 nM, and remains elevated after 10 min by 26 nM. It should be noted that experimental responses are very different when the SR release is not blocked, and there seems to be some indication of a secondary phenomenon that increases [Ca]i again after 7–8 min.
The forward-mode activity of the NCX was evaluated in Batlle et al. 61) by testing the effect of ionomycin on aortic VSMCs. We simulated this effect by increasing background calcium channel activity by a factor of 7.5, when basal [Ca]i increased by 100 nM, comparable to the data (Fig. 10 in (61)). Lowering [Na]o in the presence of ionomycin transiently increased model [Ca]i by 140 nM; steady-state [Ca]i was 123 nM above basal level. In the experimental data (Fig. 11 in (61)), increases in [Ca]i with nominally zero [Na]o were much higher, 350 nM transiently, and 150 nM in the steady state. As noted above, the difference between model and experiment may arise partly from transient SR fluxes and a secondary calcium-influx component, both of which are not taken into account in the model.
Fig. 9 B shows the results of simulations obtained under a different protocol (51,52), where the Na-K pump was blocked at t = 0 ([K]o = 6 mM). Blocking the Na-K pump results in a gradual increase in [Na]i (dashed line in Fig. 9 B). Also plotted in Fig. 9 B (solid line) is the peak of the transient increase in calcium upon brief application of 5 mM [Na]o. The relationship between [Ca]i and [Na]i in this protocol is almost linear, which may correlate to the observed linearity between [Na]i and force (52).
Fig. 9 C shows the change in [Ca]i when the cell is challenged briefly by various [Na]o after a steady state has been reached with Na-K pump block. The shape of the curve is similar to the observed-force [Na]o with this protocol (panels B and C of Fig. 9 may be compared with panels b and c of Fig. 1 in (51)).
The results from the simulations at low [Na]o do not agree with several experimental observations:
). In the model, at zero [Na]o, the cell hyperpolarizes to values close to the potassium reversal potential. ,64) indicate a large constriction when extracellular sodium is lowered, which is not seen in the model. ), inhibiting NCX activity with KB-R7943 hyperpolarized the membrane from –35 mV to –43 mV with an accompanying large vasodilation, at 80 mmHg. In the model, under these conditions, the hyperpolarization was 1 mV, and intracellular calcium levels remained unchanged. ) upon decreasing [Na]o from 140 to 30 mM. The changes in the model were much smaller (20 nM); this difference may be attributable to calcium release from the SR, as has been noted above. |
DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: MEMBRANE CURRENTSACKNOWLEDGEMENTSREFERENCES |
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The magnitude of the macroscopic conductances of the channels and the maximum flux through the various pumps and transporters are comparable to estimates in the literature (see Table 2). The resulting currents are of the order of pA, and are consistent with the high input resistances (
10 G
) characteristic of vascular smooth muscle cells. The various intrinsic parameters (such as potential activation slopes, half-activation voltages, or calcium dependence) were either fixed directly from previous estimates in VSMCs or were taken from parameters in ventricular cell models (26,27,39).
The only significant change in an intrinsic parameter in the present model is the introduction of small changes in the description of outward currents across the IR channel. As these outward currents are rather small in VSMCs, they are not well-characterized experimentally; yet they play a significant role both in setting the resting membrane potential as well as in responses to changes in [K]o. In the model, the open probability of IR channels at potentials >15 mV positive to EK is reduced compared to a Boltzmann distribution. This change shifted the appearance of the lower state in the model to higher [K]o, and made the model results more compatible with the experimental data at a [K]o of 6 mM (7,10). It may be that the outward IR currents are best characterized by optimizing for fits between experimental data and model results at various [K]o values, although this would have to be done in the context of a complete mechano-electro-chemical model of the arterial wall.
The model does not include contributions from anion channels, such as the Ca-dependent chloride channels that have been documented in arteries elsewhere (65). These may play a role in setting myogenic tone in some tissues (23,66). We have also not included an explicit contribution for the K-ATP channel. However, this channel, which is voltage-independent (67), is believed to be characterized in a manner very similar to the K background current.
As noted earlier, the work of Yang et al. (18,19) is the only previous model of electrochemical processes in VSMCs. The model in Yang et al. (18,19) differs from the model considered here in some key aspects:
,19) concentrated primarily on fast kinetics in VSMCs, where slow processes such as the inactivation of the LTCC do not play a major role. However, slow processes are significant in the steady state. ,19) did not consider the calcium-dependence of maxi-K channels through calcium sparks. Maxi-K activation by [Ca]i plays an important role in setting steady-state [Ca]i. ,19). While this is perhaps not significant in fast kinetic models, extrinsic parameters cannot be set independently in steady-state models, where all transmembrane fluxes are constrained to be zero over a range of transmural pressures. In the present model, we have adjusted the values of extrinsic parameters to satisfy this constraint. It is likely that the factors noted above played a role in improving the agreement between the results from the current model and steady-state experimental data. The contributions of the various model components are shown In Fig. 3, and have been briefly discussed in Individual Currents, above. Some additional points are:
An interesting prediction of the model is the existence of dual stable states under a variety of external conditions. The existence of these states followed from the bell-shape of outward currents through the IR channel. When these states coexist, they are marked by different [Ca]i and Vm. This implies that the response of the cell would be nongraded when the system makes a transit from one to the other of these states in response to changes in external conditions, such as changes in pressure, external [K], or IR block. This qualitative prediction of a nongraded behavior agrees with experimental data (10,54,58).
The results of the model indicate that, in many experimental conditions, VSMC responses are accompanied by changes in intracellular [Na]. For example, intracellular [Na] in the model varies greater than threefold with a transmural pressure increase from 10 to 100 mmHg. Other experimental manipulations, such as blocking the Na-K pump and variation in external [K] and [Na], also affect [Na]i, as noted in Results, above. [Na]i also differs across coexisting stable states, and changes in [Na]i should be co-observable with nongraded cell responses in the steady state. It would appear that monitoring of [Na]i, simultaneously with [Ca]i and Vm, would be a useful tool in refining cellular models.
In this article, we have presented a basic model for steady-state electrochemical behavior in VSMCs. The results of the model broadly agree with experimental data under a variety of external conditions. The model can serve as a basis for more elaborate models, e.g., to predict time-dependent transients by including channel kinetics and calcium buffering. Again, the inclusion of downstream effects of changes in [Ca]i, such as muscle tension, as well as tension from passive extracellular components, would allow for a complete model that can be used to predict changes in arterial diameter with pressure.
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APPENDIX: MEMBRANE CURRENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIX: MEMBRANE CURRENTSACKNOWLEDGEMENTSREFERENCES |
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