Conditions for Kir-induced bistability of membrane potential in capillary endothelial cells

Abstract

A simplified model for electrophysiology of endothelial cells is used to examine the conditions that can lead to bistability of membrane resting potential. The model includes the effects of inward-rectifying potassium (Kir) ion channels, whose current-voltage relationship shows an interval of negative slope and whose maximum conductance is dependent on the extracellular potassium concentration. The background current resulting from other types of channels is assumed to be linearly related to membrane potential. A method is presented for identifying the boundaries in the parameter space for the background currents of the regions of bistability. It is shown that these regions are relatively narrow and depend on extracellular potassium concentration. The results are used to define conditions leading to transitions between depolarized and hyperpolarized membrane states. These behaviors can influence the properties of conducted responses, in which changes in membrane potential are propagated along blood vessel walls. Conducted responses are important in the local regulation of blood flow in the brain and other tissues.

Graphical Abstract

1. Introduction

The resting membrane potentials of endothelial and vascular smooth muscle cells strongly affect blood vessel function, including contraction and dilation [1,2]. Under some conditions, vascular cells can exhibit two levels of stable resting potential. A bimodal distribution of membrane potentials has been observed in smooth muscle cells [3,4]. Evidence also exists for bistability of membrane potential in endothelial cells [5]. In vascular conducted responses, changes in membrane potential are propagated along vessel walls via endothelial cells, which are connected by gap junctions [6]. The typical state of endothelial cells is depolarized, with a membrane potential around −40 mV. A shift to a hyperpolarized state in arteriolar endothelial cells can trigger relaxation of smooth muscle cells, resulting in vasodilation in skeletal muscle [7] and brain [8].

These conducted responses often show exponential decay of the change in membrane potential with distance travelled, as expected if the gap junctions and the endothelial cell membrane function as fixed electrical resistances. However, some observations indicate propagation of conducted responses without exponential decay [9]. Such behavior can be accounted for by bistability of endothelial cell resting potential, such that a transition between the two states propagates as a travelling wave [10,11].

Conducted responses play an important role in the local control of blood flow [12]. In the brain, for example, increases in neuronal activity lead to local increases of cerebral blood flow (CBF). This phenomenon is referred to as neurovascular coupling (NVC), and involves the dilation of upstream arterioles so that the metabolic oxygen demands of neural tissue are met [13]. The physiological importance of conducted responses has focused attention on the electrophysiology of endothelial cells, including the occurrence of bistable membrane potentials.

The resting membrane potential of cells is largely controlled by potassium channels in the membrane. Several types of potassium channels have been identified in endothelial cells [14]. Among these, inward-rectifying potassium (Kir) channels are particularly relevant to the phenomena considered here, for two reasons. Firstly, the conductance of Kir channels depends on the extracellular K⁺ concentration [15]. An increase in extracellular K⁺, resulting from neuronal activity, can increase the conductance of endothelial Kir channels, causing hyperpolarization and potentially initiating a conducted response [11]. Secondly, Kir channels exhibit a highly nonlinear voltage-current relationship, including a negative slope over a range of membrane potentials. When such channels are combined with channels with linear voltage-current relationships, the possibility exists for a cell to have two stable resting membrane potentials [3,10,11]. In the present study, we develop a simplified mathematical model for the electrophysiology of an endothelial cell and use the model to examine the cellular conditions that lead to this bistable behavior.

2. Methods

The model for an isolated endothelial cell is shown schematically in Figure 1. The outward potassium current in the inward-rectifying potassium channels, i_Kir, is given by

$$i_{{}_{Kir}}=G_{{}_{Kir}}(V_{{}_m}-E_{{}_K})$$ (1)

where G_Kir is the conductance of the channel per endothelial cell and V_m is the resting membrane potential (inside – outside). Here, E_K is the reversal (Nernst) potential for K⁺, given approximately in mV by

$$E_{{}_K}=-61.5{\log }_{{}_{10}}\left([{\mathrm{K}}^{{}^{{}_{+}}}{]}_{{}_i}∕[{\mathrm{K}}^{{}^{{}_{+}}}{]}_{{}_o}\right)$$ (2)

where [K⁺]_i and [K⁺]_o are the intracellular and extracellular potassium concentrations. Parameter values are given in Table 1. The voltage-dependent conductance is represented by a Boltzmann-type equation [16]

$$G_{{}_{Kir}}=\frac{G_{{}_{Kir,0}}}{1+\text{exp}[(V_{{}_m}-V_{{}_{0.5}})∕k]}$$ (3)

where V_0.5 = E_K + 25 mV is V_m at half maximal inactivation and k = 7 mV defines the steepness of the sigmoidal curve describing the relation between G_Kir and V_m [11]. The maximal Kir conductance G_Kir,0 shows an approximate square-root dependence on [K⁺]_o given by

$$G_{{}_{Kir,0}}={\overline{G}}_{{}_{Kir}}([{\mathrm{K}}^{{}^{{}_{+}}}{]}_{{}_o}{)}^{{}^{{}_{1∕2}}}$$ (4)

where ${\overline{G}}_{{}_{Kir}}=0.18\text{nS}∕{\text{mM}}^{0.5}$ [11]. All conductances are expressed per endothelial cell.

Figure 1.

Simplified models for endothelial cell electrophysiology including Kir channels. A. Lumped model with a single linear background current with variable reversal potential. B. Model with two linear background currents with different reversal potentials.

Table 1.

Parameter values and ranges considered

Parameter	Symbol	Value(s)	Source
Intracellular potassium concentration	[K+]i	128 mM	[11]
Extracellular potassium concentration	[K+]o	3 to 12 mM
Potassium current reversal potential	EK	−100 to −63 mV	Eq. (2)
Background current reversal potential	Ebg	−60 to +10 mV
Maximal Kir conductance per cell	GKir,0	0.3 to 0.6 nS	Eq. (4), [11]
Background conductance per cell	Gbg	0 to 0.4 nS
Leakage conductance per cell	GL	0 to 0.3 nS
Non-voltage-dependent potassium conductance per cell	GKNV	0 to 0.15 nS
Membrane capacitance per cell	C	8 pF	[11]

The background (non-Kir) membrane current is assumed to depend linearly on membrane potential. Two models for the background current are considered. In the lumped background model (Figure 1A), a single background current is assumed, i_bg = G_bg (V_m − E_bg), with E_bg denoting the reversal potential and G_bg denoting the conductance. In the two-component background model (Figure 1B), the background current consists of a leakage current i_L = G_LV_m with conductance G_L and anon-voltage-dependent potassium current i_KNV = G_KNV(V_m − E_K) with conductance G_KNV [4]. The reversal potential of the leakage current is set to zero, to approximate the overall reversal potential for non-potassium currents [4].

The membrane potential is governed by the dynamical equation C_mdV_m/dt = −i_i, where C_m is membrane capacitance per cell, i_i = i_KIR + i_bg for the lumped background model, and i_i = i_KIR + i_L + i_KNV for two-component background model. When the endothelial cell is at a resting membrane potential, i_i = 0 and the two models give i_KIR = −i_bg and i_KIR = −i_L −i_KNV.

Figure 2 shows plots of i_Kir and −i_bg (or equivalently −i_L −i_KNV) as functions of V_m, for several parameter values. Intersections of these curves correspond to solutions of i_i = 0 . The properties of equation (2) imply that at least one and no more than three intersections exist in all cases. The stability of these equilibria can be deduced by considering the case in which V_m is slightly more negative than its value at the intersection. If one intersection is present (Figure 2A and C), the Kir current is less than the negative of the passive current, and so i_i is negative and the cell returns to its equilibrium state. Therefore, this equilibrium is stable. If three intersections are present (Figure 2B), the same argument shows that the equilibria corresponding to the intersections on the left and right are stable. For the middle equilibrium, however, i_i is positive for a negative perturbation of V_m, and so this equilibrium is unstable. In this case, the membrane potential is bistable.

Figure 2.

Voltage current characteristics of Kir channel, showing origin of bistability. Blue curve shows current (in pA per cell) as a function of membrane potential when [K⁺]_o = 3 mM. Red curve shows negative of background current, i_bg = G_bg(V_m − E_bg), where G_bg = 0.1 nS per cell. A. E_bg = −25 mV, one stable equilibrium depolarized state. B. E_bg = −40 mV, two stable states and one intermediate unstable state. C. E_bg = −60 mV, one stable hyperpolarized state. D. Marginal states. Red line: G_bg = 0.1 nS, E_bg = −35 mV. Orange line: G_bg = 0.12 nS, E_bg = −50 mV.

The main goal of this study is to identify regions in parameter space that lead to bistability. On the boundaries of such regions, the background conductance line must be tangent to the to i_Kir curve (Figure 2D). The boundary in parameter space of the bistable region can therefore be parameterized in terms of V_m by finding the tangent line to the to i_Kir curve at each V_m value. For the lumped background model, the conductance G_bg is given by the slope of the tangent line and the reversal potential E_bg is given by the V_m-axis intercept. The corresponding parameters for the two-component background model are G_KNV = G_bgE_bg / E_k and G_L = G_bg − G_KNV.

To gain further insight into the two-component background model, resting membrane potential values V_m that give rise to an equilibrium state were plotted for each point in the parameter space, for fixed [K⁺]_o values. This creates a surface over the G_KNV-G_L plane. When bistability exists, the surface appears folded, with unstable equilibrium solutions forming an intermediate surface between two layers of stable solutions. The resulting plots are used to examine the conditions that can lead to transitions between depolarized and hyperpolarized membrane potentials. Computations were performed using MATLAB (MathWorks, Natick MA).

3. Results

The boundaries defining regions of bistability in parameter space are shown in Figure 3 for values of [K⁺]_o from 3 to 12 mM, representing the approximate physiological range. Each boundary has a cusp corresponding to the inflection point of the i_Kir curve, where the slope of the tangent line is maximized. As [K⁺]_o is increased, the boundaries shift in the direction of higher background conductance values.

Figure 3.

Regions in parameter space giving rise to bistability, lying within the cusped curves, for [K⁺]_o = 3, 6, 9 and 12 mM. Conductances are in nS per cell. A. Lumped-background model. B. Two-component background model. Red dot corresponds to G_KNV = 0.05 nS and G_L = 0.1 nS.

Figure 4 shows the variation of equilibrium V_m values in the G_KNV-G_L plane for the two-component background model, for values of [K⁺]_o from 3 to 12 mM. In these plots, the region where the surface folds over itself represents bistability, with the upper and lower surfaces being stable and the middle surface being unstable. At low levels of [K⁺]_o, the membrane is relatively depolarized for most parameter values. Increasing [K⁺]_o leads to hyperpolarization over most of the parameter space.

Figure 4.

Dependence of equilibrium membrane potential V_m on conductances G_KNV and G_L in the two-component background model, for [K⁺]_o = 3, 6, 9 and 12 mM. Conductances are in nS per cell. Red dots correspond to G_KNV = 0.05 nS and G_L = 0.1 nS. For the case [K⁺]_o = 9 mM, the green curve shows the effect of increasing G_L from 0.1 to 0.2 nS, resulting in a transition from the hyperpolarized state (red dot) to a depolarized state (green dot).

4. Discussion

As a consequence of the nonlinear conductance properties of Kir channels, endothelial cells can have two stable membrane resting potentials under some conditions. This may contribute to the occurrence of non-decaying conducted responses in microvessels. Because conducted responses plays a critical role in flow regulation, the question arises: under what conditions can this bistability occur? In addressing this question, the two-component background model is used, as it more closely represents the actual situation with multiple conductances, each with its own reversal potential. In the two-component background model, the regions of bistability are relatively small and narrow in the parameter space of conductances, for any given extracellular potassium concentration (Figure 3B). In general, bistability is more prevalent when Kir channels dominate membrane conductance. This can be seen by reference to Figure 2: if the line representing −i_bg has a shallower slope, it can intersect the i_Kir curve three times for a wider range of E_bg. When the connections of endothelial cells to their neighbors via gap junctions are considered, the ranges of parameter values that allow non-decaying conducted responses are even narrower [10,11]. These results suggest that non-decaying conducted responses are unlikely to occur consistently, and that decaying responses are the more general case in the microcirculation.

The present model can be used to gain insight into the conditions that lead to a transition between depolarized and hyperpolarized states of the endothelial cell membrane. First we consider the effects of changes in [K⁺]_o, as can result from neuronal activity. The values G_KNV = 0.05 nS and G_L = 0.1 nS per cell, indicated by red dots in Figures 3 and 4, correspond to a depolarized state when [K⁺]_o = 3 mM, to a bistable state when [K⁺]_o = 6 mM, and to a hyperpolarized state when [K⁺]_o = 9 mM. The actual state in the bistable case depends on the history of [K⁺]_o. If the concentration is increased from 3 to 6 mM, the membrane remains in the depolarized state, and further increases in [K⁺]_o would be needed to stimulate hyperpolarization. If [K⁺]_o decreased from a 9 to 6 mM, the membrane remains hyperpolarized. This shows that hysteresis is associated with bistability [11].

The results (Figure 3B) show that the occurrence of bistability, for a given level of [K⁺]_o, depends sensitively on the leakage conductance GL, which represents the effects of non-potassium currents. Transient receptor potential cation channel subfamily V member 4 (TRPV4) channels are present in endothelial cell membranes and contribute to this current, being permeable to sodium and calcium ions. A signaling mechanism that can be triggered by neurotransmitters reaching the cell membrane results in opening of TRPV4 channels, corresponding to an increase in G_L [17]. This represents a potential mechanism by which hyperpolarization of the endothelial cell membrane can be reversed after being initially induced by an increase in [K⁺]_o. This represents one of several mechanisms by which TRPV4 channels may contribute to regulation of blood flow [18]. An example of the effect of increasing G_L is illustrated in Figure 4, for [K⁺]_o = 9 mM. If G_L is gradually increased from 0.1 to 0.2 nS, starting from a hyperpolarized state, the predicted membrane potential jumps abruptly from hyperpolarized (V_m = −45 mV) to depolarized (V_m = −19 mV) when G_L = 0.18 nS.

The model presented here is simplified in many respects, and is intended to provide insight into the conditions that lead to bistability in endothelial cell membrane potential and the physiological implications of this phenomenon. The model does not include effects of the coupling of endothelial cells by gap junctions, which, as already noted, further restricts the range of parameters leading to bistability. Many types of channels can affect membrane potential and may have nonlinear properties that are not represented in this model. The reversal potential for the background current is likely non-zero and dependent on the states of the various channel types.

Highlights.

• Inward-rectifying potassium (Kir) channels can cause two stable membrane potentials

• The conditions allowing such bistability are explored using mathematical models

• Bistability can occur for a narrow region within the possible parameter space

• Bistability can cause non-decaying conducted responses along blood vessel walls

• The results imply limitations on the occurrence of non-decaying conducted responses