INTRODUCTION

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

TISSUE PERFUSION is normally characterized by a marked degree of spatial and temporal heterogeneity. Average local perfusion in the heart, for example, may vary by at least sixfold under resting conditions, with the consequence that some microregions are permanently vulnerable to hypoxia (21, 62). Superimposed time-dependent fluctuations in perfusion that can encompass a wide range of frequencies and amplitudes within the same vascular bed are also apparent in myocardial elements as large as 1 cm³ and even at the segmental and lobar levels in the pulmonary circulation (18, 57). Such temporal variability results principally from spontaneous rhythmic fluctuations in vascular tone and diameter, a phenomenon known as vasomotion, which can be observed at all levels of the circulation. From a functional point of view, the asynchronous opening and closing of resistance arteries in different parts of the same vascular network ensures that all tissue elements receive intermittent perfusion, and at the microcirculatory level, vasomotion is thought to enhance fluid filtration and lymphatic drainage (see Refs. 23 and 34 for review). Theoretical simulations suggest that large-amplitude oscillations in vessel diameter can increase time-averaged hydraulic conductance (55), so that vasomotion may also represent an adaptive homeodynamic response in pathological states such as hypoxia and hemorrhagic shock, in which it becomes particularly pronounced (34).

Although the complex waveforms of vasomotion are often highly irregular, there is accumulating evidence that they are not simply random in origin but are generated by nonlinearity in the mechanisms regulating smooth muscle tone. Indeed, fluctuations in pressure and flow in isolated rabbit ear resistance arteries exhibit patterns that are generic in nonlinear systems, including period-doubling, quasiperiodic, and intermittent routes for the transition from regular to irregular dynamics, so that vasomotion may be classified as a deterministic or "chaotic" process (11, 14, 24-28). Nonlinearity may confer specific biological advantages as chaotic fluid flow accelerates diffusion-limited chemical reactions and mass-transfer processes and could therefore enhance the delivery of oxygen and nutrients and the removal of metabolites (23). Chaos may also allow large changes in state to occur with minimal energy expenditure through exploitation of the high sensitivity of nonlinear systems to perturbation, thus enhancing the flexibility and efficiency of vascular control (23, 26).

To gain insights into the nonlinear interactions that underlie vasomotion at the cellular level, in the present study we have constructed a mathematical model of the ion transport systems that participate in the regulation of vascular tone. The model simulates fluctuations in free cytosolic Ca²⁺ concentration ([Ca²⁺]_i) within a single myocyte, with changes in [Ca²⁺]_i translated into changes in force generation via the latch state hypothesis of Hai and Murphy (29). The mathematical formulation was assisted by the construction of a geometric phase space portrait of the dynamics of experimental signals from isolated rabbit ear resistance arteries by the technique of time-delayed coordinate embedding (11, 23). The complexity of this representation, which is known as an "attractor," can be quantified as a nonintegral fractal correlation dimension (here denoted as D_C) following nonlinear analysis with the algorithm of Grassberger and Procaccia (22). D_C provides an estimate of the minimum number of participating dynamic control variables when rounded to the nearest upper integer, a phase space of at least three dimensions, i.e., D_C > 2 being required to classify the behavior of a system as chaotic (22). In general, the correlation dimension of fluctuations in pressure and flow in rabbit arteries takes a value between 2 and 4 (14, 24-27), thus characterizing vasomotion as a low-dimensional chaotic process that can in theory be modeled with a "minimal" approach involving just four independent variables. This conclusion is confirmed by pharmacological analysis, which has revealed the participation of "slow" intracellular and "fast" membranous Ca²⁺ oscillators with periods on the order of 1-5 min and 5-30 s, respectively (11, 25). In the present model, two variables were therefore assigned to each subsystem to permit independent oscillatory activity. The selection of these variables was guided by the changes in D_C observed on administration of specific pharmacological probes (see below).

	EXPERIMENTAL BACKGROUND TO THE MODEL

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Intracellular Oscillator

View larger version (33K): [in this window] [in a new window]   Fig. 1.   Schematic of key ion fluxes that contribute to 2 oscillatory subsystems of model. Internal stores [i.e., sarcoplasmic reticulum (SR)] possessing a ryanodine-sensitive Ca2+-induced Ca2+ release (CICR) mechanism are essential for genesis of intracellular Ca2+ oscillations, whereas stores possessing inositol 1,4,5-trisphosphate (InsP3) receptors are considered to sequester Ca2+ avidly and serve as a large "sink/source" that is permanently full. Agonists stimulate Ca2+ influx through voltage- and receptor-operated Ca2+ channels (VOCC and ROCC, respectively) and also cause InsP3-induced Ca2+ release (IICR) from stores into cytosol. Cyclic changes in membrane potential (Vm) and Ca2+ influx via VOCC underpin activity of membrane oscillator, with negative feedback on Vm provided by transport systems that promote hyperpolarization. These include Ca2+-activated K+ channels, Na+-K+-ATPase, and Cl channels. Ca2+-ATPase pumps () mediate uptake by stores and contribute to Ca2+ extrusion from cell. Na+/Ca2+ exchange can promote Ca2+ extrusion or influx, depending on instantaneous value of Vm. Coupling between 2 oscillatory subsystems is effected through cytosolic Ca2+ concentration.

Ca²⁺ release from stores. The sarcoplasmic reticulum (SR) of vascular smooth muscle is anatomically and functionally heterogeneous and possesses Ca²⁺- and InsP₃-sensitive Ca²⁺ stores (32, 38, 66). Compounds that block sequestration of Ca²⁺ by the SR Ca²⁺-ATPase, such as cyclopiazonic acid and thapsigargin, deplete both types of stores, whereas the alkaloid ryanodine selectively impairs the functionality of a specific Ca²⁺-release channel (7, 38). In rabbit ear arteries, ryanodine appears to target a subcomponent of the SR that is closely coupled to the contractile machinery of the smooth muscle cell as it suppresses the activity of an intracellular oscillator without affecting average perfusion pressure (25, 27). By contrast, cyclopiazonic acid and thapsigargin induce a depressor response and initially transform chaotic behavior to highly specific patterns of nonlinear response known as mixed-mode oscillations before high concentrations suppress vasomotion completely (11, 27). These observations suggest that Ca²⁺ is released from distinct store subtypes in rabbit ear arteries so that, in the formulation of the present model, CICR and IICR were assumed to originate from separate stores, as in the Goldbeter-Dupont-Berridge model of Ca²⁺ spiking (20).

Voltage-operated Ca²⁺ influx. VOCCs provide the dominant pathway for Ca²⁺ influx in vascular smooth muscle, and the L-type Ca²⁺ antagonist verapamil can suppress contraction in rabbit ear arteries (11, 25). The open state probability of VOCCs is a steep sigmoidal function of cell membrane potential (51), and their contribution to vasomotion was modeled as a noninactivating whole cell conductance with a Bolzmann-like activation term, the slope of which as a function of voltage was taken from Nelson et al. (51). In addition to regulating VOCC conductance by causing membrane depolarization, constrictor agonists may also promote the opening of such channels directly by shifting their activation curve to the left, such that open state probability is higher at a given voltage (51).

Receptor-operated Ca²⁺ influx. A receptor-gated cationic channel that is insensitive to membrane depolarization and to inhibitors of VOCCs, is not modulated by [Ca²⁺]_i or other second messengers, and possesses a threefold higher selectivity for Ca²⁺ than for Na⁺ has been identified in membrane patches from rabbit ear artery myocytes (6). Agonists such as ATP and norepinephrine thus promote an inward flux of monovalent and divalent cations, such that ~10% of the current stimulated under conditions of voltage clamp is carried by Ca²⁺ (3, 5, 6). Because the nature of the ROCC remains poorly defined (51), in the present model Ca²⁺ movements via this pathway were simplified as a constant term, the magnitude of which was directly related to agonist concentration. Under certain conditions, constrictor agonists may stimulate Ca²⁺ influx into vascular smooth muscle without causing depolarization (9), and it was therefore assumed that receptor-operated Ca²⁺ influx does not modulate membrane potential.

Ca²⁺ extrusion ATPase. In rabbit ear arteries, inhibition of the membrane Ca²⁺ extrusion pump with the vanadate ion evokes a large pressor response and a small decrease in D_C (14). In different simulations, linear or parabolic activation curves were used to approximate the dependence of the pump kinetics on [Ca²⁺]_i over the physiologically relevant range (15), and its voltage dependence was formulated as a linear function of membrane potential, the slope of which was such as to give a 40% greater rate of Ca²⁺ efflux at 0 mV than at 100 mV, in accord with observations that the rate of extracellular Na⁺ concentration ([Na⁺]_o)-independent Ca²⁺ efflux from vascular myocytes increases linearly with membrane potential (16). The pump exchanges Ca²⁺ for H⁺, but the stoichiometry of the exchange may vary with local pH (15, 16), and under different experimental conditions it has been reported to be electrogenic (16) or electroneutral (53). The possible influence of Ca²⁺ extrusion via this mechanism on membrane potential was not incorporated in the simulations presented.

Na⁺/Ca²⁺ exchange. Reductions in [Na⁺]_o inhibit Na⁺/Ca²⁺ exchange and progressively reduce D_C for vasomotion in rabbit ear arteries to <2 (14). The exchanger can act as an efflux (forward mode) or an influx (reverse mode) pathway for Ca²⁺ movements, with the instantaneous direction of the Ca²⁺ flux depending on the difference between membrane potential and the reversal potential of the exchanger, which may vary between 30 and 45 mV according to [Ca²⁺]_i (4, 54). Inasmuch as the stoichiometry of the exchange is 3 Na⁺:1 Ca²⁺, below its reversal potential this transport system will promote [Ca²⁺]_i extrusion associated with depolarization due to the net movement of charge; above the reversal potential these effects will change direction. Consequently, in mathematical simulations the signs for the contribution of this mechanism to the rate of change of [Ca²⁺]_i and membrane potential will always be opposed. Modulation of exchange activity by local changes in Na⁺ concentration was ignored in the formulation of the present model.

Membrane Oscillator

To generate oscillatory electrical membrane activity that contributes to rhythmic mechanical responses, mechanisms that promote hyperpolarization must operate to oppose voltage-dependent Ca²⁺ influx. Experimental evidence suggests that a spectrum of ion channels, pumps, and exchangers may act in concert to provide such negative feedback (14). Reductions in the activity of Ca²⁺-activated K⁺ (K_Ca) channels, Cl channels, the Na⁺-K⁺-ATPase, or Na⁺/Ca²⁺ exchange thus attenuate the membrane component of rabbit ear artery vasomotion, with each class of intervention separately causing D_C to fall to <2 (14). Although it is therefore clear that multiple channels, pumps, and exchangers contribute to the regulation of tone in these vessels, it is likely that the entrainment of weakly contributing variables results in the suppression of a number of degrees of freedom of the system with a corresponding reduction in its overall dynamic complexity (44). To construct a minimal four-dimensional model that conforms to experimental findings, the open state probability of the K_Ca channel was chosen as the fourth independent mathematical control variable. Membrane potential can fluctuate between 0 and 50 mV during vasomotion (48, 64), so that potassium ions, with reversal potential less than 80 mV (52), are the only species able to drive the membrane to sufficiently low potentials. Other membrane transport systems that influence D_C were included in the model as modulatory fluxes.

Ionic current through K_Ca channels. Multiple K_Ca channel isoforms may coexist in the same artery type (52). In a detailed study, Toro et al. (65) reconstructed the K_Ca channels present in pig coronary artery in lipid bilayers and distinguished different subtypes on the basis of their voltage and Ca²⁺ sensitivity. Frequency histograms of prevalence demonstrated two main populations of "maxi" K_Ca channels with conductances of 245 and 295 pS, each population consisting of two isoforms. The 245-pS subtypes had half-maximal activation potentials of 80 and 6 mV, whereas the 295-pS subtypes had half-maximal activation potentials of 28 and 66 mV (65). Subsidiary channels were also found at ~50 and ~400 pS and also exhibited differences in sensitivity to voltage and [Ca²⁺]_i. The voltage slope coefficients of these channels were generally similar, with the channel exhibiting greatest sensitivity to Ca²⁺ half-maximally activated at ~1.2 µM Ca²⁺ (65). In the present minimal model, representative values of these parameters were selected to define the activation characteristics of a single "composite" K_Ca subtype. The Hill coefficient for the Ca²⁺ dependence of the activation sigmoidal was taken as 2 to conform with experimental findings (65). The kinetics of channel inactivation were described by a first-order rate equation with an exponential decay constant. Membrane potential and [Ca²⁺]_i may also influence the rate of K_Ca channel closure (40, 52), but this additional complexity was not incorporated in the model. Although voltage-dependent K⁺ channels are functionally active and modulate perfusion pressure in rabbit ear arteries, they do not contribute to the dynamical complexity of vasomotion, perhaps reflecting their insensitivity to [Ca²⁺]_i, and were therefore not included in the model (14). ATP-sensitive K⁺ channels influence neither tone nor vasomotion in rabbit ear arteries under well-oxygenated conditions (25).

Ionic current through Na⁺-K⁺-ATPase. Under normal physiological circumstances the Na⁺-K⁺-ATPase is thought to contribute to vascular smooth muscle cell potential by hyperpolarizing the membrane by up to 10 mV (1). Inasmuch as changes in the local concentration of K⁺ and Na⁺ were not included as an integral dynamical feature of the model, the contribution of this pump was simplified as a constant hyperpolarizing flux.

Ionic current through Cl channels. A depolarizing Ca²⁺-activated outward Cl flux has been reported in rabbit ear arteries (3, 39), and reductions in extracellular Cl concentration and niflumic acid, an inhibitor of Ca²⁺-activated Cl channels, suppress the activity of the membrane oscillator involved in vasomotion in rabbit ear arteries (14). At membrane potentials more negative than the Nernst reversal potential (approximately 25 mV), efflux of Cl promotes depolarization of vascular smooth muscle, whereas at potentials more positive than this value, efflux of chloride ions results in hyperpolarization (39). Modulation of channel activity by Ca²⁺ concentration was not specifically incorporated in the present minimal model, however, and Cl fluxes were represented simply as a constant term multiplied by the electrochemical gradient for Cl.

	METHODS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Experimental

Mathematical Formulation

Intracellular oscillator.

Ca²⁺ fluxes into the cytosol

Ca²⁺ uptake by stores

Membrane oscillator.

Relationship between ion fluxes and membrane potential

Open state probability of K_Ca channels

(1d)

All symbols are defined in Table 1, which summarizes the fixed physiological values of the parameters (e.g., slope functions and reversal potentials) used for the simulations.

Ca²⁺ Buffering and Volume Scaling

(1c')

Calcium ions that enter the cytosol are rapidly bound by intracellular proteins (60). Indeed, experiments in gonadotrophs and chromaffin cells suggest that ~1 in 100 calcium ions remains free in the cytosol (43). Changes in [Ca²⁺]_i may thus be related to the two electrogenic membrane Ca²⁺ currents included in the model, I_Ca and I_Na/Ca, by expressions of the form, d[Ca²⁺]_i/dt = fI, where f is the fraction of calcium ions that are not buffered,  = 1/ZV_cytF is a scaling factor that converts current to rate of change in [Ca²⁺]_i, V_cyt is the volume of the cytosol, F is Faraday's constant, and Z = 2 for I_Ca and 1 for I_Na/Ca (since the stoichiometry of this exchanger is 3 Na⁺:1 Ca²⁺). From Eq. 1, a and c', it follows that for each of these currents g = G/f = ZV_cytFG/f and that  = V_cytF/fC_m. The formulation of Eq. 1c thus introduces a scaling factor, , that involves the parameters of V_cyt, C_m, and the fraction of calcium ions that remains free (note that the divalent nature of Ca²⁺, i.e., Z = 2 is accounted for by the coefficient 2G_Ca in Eq. 1c; for other mechanisms included in the model, Z = 1). This "lumped" approach provides a convenient formulation for simulations, inasmuch as the contributing parameters are known only approximately. Indeed, to obtain realistic oscillations that matched experimental data, it was necessary to adjust  over the broad range of 0.2-8 (Table 2). In practice,  exerts an important influence on the relative periods of the two contributing oscillators, which can exhibit significant variation between different experimental preparations.

Buffering also occurs within the SR, and there is evidence from COS-7 cells that only ~1 in 400 calcium ions is free in this compartment, compared with 1 in 100 in the cytosol, because of the presence of specialized Ca²⁺-buffering proteins (36, 41). For the purposes of modeling, it therefore becomes necessary to define "effective volumes" available for Ca²⁺ movements (12) that are equivalent to 100 and 400 times the physical volumes of these compartments (V^e_cyt and V^e_SR), respectively, whereas the ratio of their actual volumes (V_SR/V_cyt) is <1:4, because the SR occupies <20% of the cytosol. It follows that V^e_SR ~ V^e_cyt, which permits the assumption that Ca²⁺ concentration changes in the cytosol and stores resulting from intracellular movements between these two compartments are directly related according to

(2a)

as is implicit in Eq. 1, a and b, if transmembrane fluxes are ignored. A more general relationship of the form

(2b)

could take into account structural differences between cells.

Parameter Estimation

5

12

12

1

In the model the contribution of Ca²⁺ influx through VOCCs to d[Ca²⁺]_i/dt is G_Ca(z  z_Ca1)/{1 + e^{[(zz_Ca2)/R_Ca]}}, which reduces to 0.02G_Ca µM/s for the fixed values z_Ca2 and R_Ca used in the simulations at an "average" membrane potential of 40 mV (Fig. 2). In practice, G_Ca was varied over the range 1-16 (Table 2), giving values of d[Ca²⁺]_i/dt = 0.02-0.32 µM/s, which encompass the value expected from the above order-of-magnitude calculation. To reflect the dominance of Ca²⁺ influx via VOCCs, this flux was generally made up to four times larger (depending on the instantaneous value of membrane potential) than the combined Ca²⁺ flux entering the cytosol via InsP₃-induced release from stores and receptor-operated channels.

View larger version (35K): [in this window] [in a new window]   Fig. 2.   A: phase plane plot for intracellular oscillator. Trajectories of free cytosolic Ca2+ concentration ([Ca2+]i) vs. Ca2+ concentration in SR ([Ca2+]SR; x vs. y) generate a limit cycle oscillation that is traversed in direction shown by arrows. Three principal regimens can be identified. Dashed and dotted curves, x and y nullclines, respectively; oscillations occur only when their intersection () is within "active ranges" of both (highlighted). B: oscillations in [Ca2+]i in time domain. In this simulation, membrane oscillator was "switched off" to reflect dynamics of [Ca2+]i oscillator in isolation. Consequently, signal generated is periodic in form. C and D: phase space plot for membrane oscillator and its associated temporal waveform, respectively. Dashed line, z nullcline; dotted line, w nullcline. , Equilibrium point; active range of each nullcline is highlighted. Limit cycle is traversed in direction shown by arrows. In this simulation, intracellular oscillator was switched off. KCa, Ca2+-activated K+ channel; Vm, membrane potential.

The magnitude of the Ca²⁺ flux via VOCCs employed in the above calculation can be used to estimate the rate of change of membrane potential during the upstroke of the vasomotion cycle. If it is assumed that the cell has a C_m of 1 µF/cm² (43) and a surface area of ~10⁵ cm², then total C_m  10⁵ µF. For a flux of n = 10⁶ ions/s, the current carried by Ca²⁺ is I_Ca = 2ne = 3.2 × 10¹³ C/s [where the electronic charge (e) = 1.6 × 10¹⁹ C]. It follows that dz/dt = I_Ca/C_m  32 mV/s. Given that opposing ion transport systems that promote membrane hyperpolarization are simultaneously active, this value is consistent with experimental measurements that range from 4 to 10 mV/s: ~4 mV/s (Fig. 1 in Ref. 64), ~5 mV/s (Fig. 2 in Ref. 17), and ~10 mV/s (Fig. 2 in Ref. 70).

A semiempirical approach was adopted to estimate the operating values of the remaining coefficients in Eq. 1c, inasmuch as, in practice, such variables as individual channel conductances and numbers of channel per cell are, in general, unknown. The model thus reproduces the experimental changes in membrane potential observed when the different ion transport systems are inhibited pharmacologically (see RESULTS). The coefficients chosen to reflect the magnitudes of net Ca²⁺ extrusion across the membrane by the Ca²⁺-ATPase and Na⁺/Ca²⁺ exchange and uptake by the SR also match relative estimates of these fluxes in vascular smooth muscle (see Inhibition of the membrane Ca²⁺-ATPase).

Integration of Ca²⁺ Dynamics by Latch Bridge Formation

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

(3a)

(3b)

(3c)

(3d)

(3e)

1

1

1

1

1

Numerical Methods

Analytic solution of Eq. 1, a-d, is impossible, inasmuch as the equations constitute a nonlinear fourth-order system. To gain insights into underlying mechanisms, each contributing oscillator was therefore first examined individually by reducing the four-dimensional system to two two-dimensional projections of the attractor. Phase space plots of the variable pairs x vs. y and z vs. w were thus obtained to evaluate the essential dynamical features of each subsystem and investigate systematically the consequences of changes in the coupling between them.

	RESULTS

TOP ABSTRACT INTRODUCTION EXPERIMENTAL BACKGROUND TO THE... METHODS RESULTS DISCUSSION REFERENCES

Stability Analysis

Equation 1, a and b. The equations describing the intracellular subsystem generate a limit cycle oscillation that exhibits three distinct regimens that are clearly identifiable in the phase plane plot of Fig. 2A and in the time domain representation of Fig. 2B. Along the slow section the dynamics are dominated by sequestration of Ca²⁺ entering the cell within the SR, with the result that [Ca²⁺]_i remains almost constant. This phase is followed by a "very fast" section, in which the SR empties abruptly into the cytosol. In the ensuing fast section, Ca²⁺ is extruded from the cell more rapidly than it enters from the extracellular space, so that [Ca²⁺]_i declines and [Ca²⁺]_SR remains low. When [Ca²⁺]_i becomes sufficiently low, uptake again dominates over extrusion, thus permitting sequestration of Ca²⁺ within the SR and allowing a new cycle to ensue.

(4a)

(4b)

(5)

(6a)

(6b)

(7a)

(7b)

Equation 1, c and d. A phase plane representation of the dynamics of the membrane oscillator given by Eq. 1, c and d, is shown in Fig. 2C. As in the case of the intracellular oscillator, the conditions necessary for oscillatory behavior can be obtained by considering the z and w nullclines, i.e., the loci of points where the rates of change of membrane potential and K_Ca channel open state probability are zero. The equilibrium point (z*,w*) is defined by (dz/dt)| = (dw/dt)| = 0.

(8a)

(8b)

(9)

Latch bridge dynamics. Figure 3 illustrates the integrating effect of the latch bridge mechanism on oscillations in [Ca²⁺]_i in which the parameters in Eq. 1 were selected to generate fast membrane oscillations with an amplitude ~20% of the slow intracellular oscillator. The top traces show how a high value of the latch bridge decay constant K₇ (0.2) selectively dampens fluctuations attributable to the membrane subsystem with little effect on the position of the extrema of the slow oscillations; i.e., the slow component of the force and signals remain almost in phase. Low values of K₇ (e.g., 0.01) cause further suppression of membrane oscillations, a phase shift to the right in the slow component and also a small rise in average tone. In physiological terms, high values of K₇ correspond to rapid decay of the latch bridge, so that the system is capable of responding to rapid changes in [Ca²⁺]_i. For low values of K₇ the latch decays slowly, allowing the system to function as an integrator of [Ca²⁺]_i.

View larger version (37K): [in this window] [in a new window]   Fig. 3.   Representative simulations showing force derived from Ca2+ oscillations by integrating system of Eq. 2 for different values of K7 (i.e., latch bridge decay constant).

Simulations of Pharmacological Interventions

Inhibition of CICR. Figure 4, A and D, illustrates the experimental effects of ryanodine on patterns of vasomotion that exhibit different relative contributions from the two participating oscillators. In Fig. 4A the responses are dominated by the intracellular subsystem, inasmuch as all activity was abolished by ryanodine. In Fig. 4D both subsystems are active and ryanodine selectively attenuated the contribution of the intracellular oscillator but not membrane activity. In neither artery was there a major change in mean perfusion pressure.

View larger version (33K): [in this window] [in a new window]   Fig. 4.   A-C: experimental trace and 2 simulations illustrating suppression of intracellular oscillator by ryanodine. D-F: experimental trace and 2 simulations showing effects of ryanodine when initial dynamics involved intracellular and membrane subsystems. Simulations in B and E were obtained by linearly decreasing coefficient C in Eq. 1, a and b, from 5,000 to 1,000 and from 6,250 to 1,250, respectively, over time period shown; in C and F, leak coefficient (L) was additionally increased from 0.025 to 0.055, again in a linear fashion. Traces in D-F illustrate a reverse period-doubling cascade (chaos  P2  P1). Changes in coefficients were initiated at points indicated by arrows.

(10)

View larger version (18K): [in this window] [in a new window]   Fig. 5.   Phase space interpretation of effects of ryanodine. On reducing C or reducing C and increasing L simultaneously (A and B, respectively), intersection of x and y nullclines () moves vertically upward until oscillatory behavior ceases relatively abruptly. L alters slopes of nullclines at high [Ca2+]i, but there is almost no change in shape and size of limit cycle.

Blockade of VOCCs. In some experimental preparations the Ca²⁺ antagonist verapamil abolishes all oscillatory activity, whereas in others the amplitude of the slow component of vasomotion is enhanced and membrane activity is attenuated (Fig. 6). These distinct classes of response are associated with a fall in mean perfusion pressure and could be successfully simulated by reducing the coefficient G_Ca, which determines the magnitude of Ca²⁺ influx via VOCCs in Eq. 1, a and c (Fig. 6, B and D). The variable outcomes in such simulations are explained by differences in the balance between Ca²⁺ influx via VOCCs and other pathways, i.e., via ROCCs, Ca²⁺ release from the InsP₃-sensitive store, or reverse-mode Na⁺/Ca²⁺ exchange (respectively represented by the coefficients A₀, A₁, and G_Na/Ca). When the net Ca²⁺ flux is small, blockade of VOCCs may prevent adequate replenishment of stores, so that vasomotion ceases. Conversely, if the net Ca²⁺ flux into the cytosol via VOCC-independent pathways is large, intracellular oscillations may persist even in the presence of verapamil. Indeed, the critical difference between the simulations shown in Fig. 6, B and D, is that the contribution of the Na⁺/Ca²⁺ exchanger is nearly fourfold greater and the contribution of A₀ + A₁ ~50% greater in the case where oscillatory behavior is maintained. Phase space analysis provides the dynamical explanation for this apparently paradoxical experimental behavior (Fig. 7). As a consequence of the hyperbolic shape of the active ranges of the x and y nullclines, verapamil moves the equilibrium point of Eq. 1, a, and b, to higher [Ca²⁺]_SR and lower [Ca²⁺]_i with two quite distinct effects: in Fig. 7A the equilibrium point is forced beyond the active oscillatory area, thus being transformed from an unstable focus to a stable focus or node that attracts trajectories, with the result that oscillations cease. Conversely, the equilibrium point may remain within the active region, as an unstable focus, even with G_Ca = 0 (Fig. 7B). Movement of the equilibrium point upward and to the left is accompanied by an increase in the amplitude of oscillatory behavior as mean [Ca²⁺]_i and force development decrease, consistent with the experimental trace shown in Fig. 6D. The increase in the size of the limit cycle is also consistent with the slight increment in the period of the slow component of vasomotion observed in Fig. 6, C and D.

View larger version (33K): [in this window] [in a new window]   Fig. 6.   A and C: experimental traces illustrating effects of verapamil, which may abolish vasomotion or induce large-amplitude oscillations. B and D: model-generated force oscillations simulating this paradoxical action of verapamil. These were obtained by linearly decreasing coefficient GCa in Eq