Theoretical Study of the Function of the IP 3 Receptor / BK Channel Complex in a Single Neuron

Large conductance calcium-activated potassium (BK) channels carry out many functions in the central nervous system. These channels open in response to increased cytosolic calcium ([Ca]cyt) concentration. The influx of calcium ions to the cytosol can occur through voltage-gated calcium channels (VGCCs) on the plasma membrane and/ or through IP3 receptors (IP3-Rs) and ryanodine receptors (RyRs) on the endoplasmic reticulum membrane. The BK channel/IP3-R/RyR interaction has been widely reported in smooth muscle but scarcely investigated in relation to neurons. The aim of this study was to theoretically explore the function of the BK/IP3-R complex by means of a computational model of a neuron that replicates the interaction between the release of Ca2+ from the endoplasmic reticulum (through IP3-Rs) and the opening of the BK channels. The mathematical models are based on the Hodgkin-Huxley formalism and the Goldbeter model. These models were implemented on Visual Basic® and differential equations were solved numerically. Distinct conditions were contemplated for BK conductance and the efflux of endoplasmic Ca2+ to the cytosol. An abrupt rise in [Ca]cyt (≥ 5 μM) and short duration (spark) was found to activate BK channels and either pause or stop the action potential train.

The BK channels have several distinctive characteristics. (1) They are homotetramers with two regulatory domains containing two high-affinity Ca 2+ binding sites. (2) They are voltage and calcium-dependent, requiring both membrane depolarization and calcium for their activation. Ca 2+ binding and voltage sensor activation act almost independently to enhance channel opening [6] . BK channels can open in the absence of calcium but are more sensitive to calcium at depolarizing voltage steps [2] . Hence, their sensitivity to calcium is strongly dependent on the membrane potential. The dissociation constant (Kd) for calcium is in the micromolar range at -60 mV and in the nanomolar range at +20 to +40 mV [7] . (3) BK channels are fast activating (on the order of 1 ms or less) compared to IK and SK channels, which have a slow activation time (lasting hundreds of milliseconds or over a second, respectively) [8] .
BK channels have various functions in the central nervous system. At the soma of many neuronal cells, they control the speed of action potential repolarization and mediate the rapidity of afterhyperpolarization. Therefore, they can influence spike frequency adaptation [9] [10] . They are often physically associated with voltage-gated calcium channels (VGCCs), thus forming microdomains with them [11] [12] . Due to their presence in nerve terminals and their co-assembly with VGCCs at active zones, BK channels are particularly suitable for regulating the release of neurotransmitters, increase the duration of PA, prevent backpropagation in dendrites, and produce a decrease in firing frequency [9] [13] [14] .
The concentration of free and bound [Ca 2+ ] cyt is reported to be approximately 100 nM and 10 μM, respectively [15] . While a localized increase in [Ca 2+ ] cyt has been evidenced in some studies, others show spatio-temporal calcium signaling restricted to nano and microdomains in neurons [16] and smooth muscle [17] [18] .

Endoplasmic reticulum (ER)
Because of containing a high concentration of Ca 2+ binding proteins, the ER is the main Ca 2+ storage organelle in cells. Indeed, the total amount of Ca 2+ may be >1 mM, while the concentration of free [Ca 2+ ] ER (Ca 2+ in the ER) has been quantified at 100-700 μM [19] .
To maintain equilibrium, mechanisms of influx and efflux of Ca 2+ are activated on the ER membrane. There are two types of processes related to the efflux of Ca 2+ from the ER, being the Ca 2+ -induced Ca 2+ release (CICR) and IP 3 -induced Ca 2+ release (IICR) processes. The [Ca 2+ ] cyt interacts with ryanodine receptors and IP 3 receptors (RyRs, IP 3 -Rs) in the former and IP 3 with its receptors (IP 3 -Rs) in the latter, in both cases to release Ca 2+ from the ER [20] . On the other hand, Ca 2+ is recaptured into the ER by the activity of smooth endoplas-mic reticulum Ca 2+ -ATP ase (SERCA) pumps [21] . Thus, a low concentration of [Ca 2+ ] cyt (50 -100 nM) is maintained by the coordinated action of the inflow of Ca 2+ to the ER through pumps on the ER membrane, and the efflux of Ca 2+ from the cytosol to the extracellular space through pumps (PMCA) on the plasma membrane [22] .

BK channel -IP 3 receptor interaction
The BK channel/IP 3 -R microdomain has received less attention, and its role is controversial. IP 3 -Rs are localized in the ER membrane and the BK channels in the plasma membrane. The BK channels and IP 3 receptors are very close to one another [23] . The ER membrane is believed to be initially generated as part of the nuclear envelope, which then expands and morphs into a complex reticulum that can extend to distant cellular compartments such as the axons, dendrites, and dendritic spines of neurons, but with a similar morphology and there is closeness between the ER membrane and the cytoplasmatic membrane [24] [25] [26] . The cisternae of the ER are classified in accordance with their proximity to the plasma membrane. Type I is the farthest from the plasma membrane, while type II and III are nearer, frequently following its profile [26] .
Pan et al., reported the interaction between BK channels and IP 3 -Rs in human embryonic kidney cells (HEK293) [27] . Neuropeptide galanin activates galanin receptors (GalR2s), and IP 3 -Rs are activated through the protein kinase G pathway. The increase in [Ca 2+ ] cyt is due to Ca 2+ efflux from the ER through IP 3 -Rs. The authors demonstrated that the rise in the level of [Ca 2+ ] cyt comes from the ER but did not quantify this change.
In arterial smooth muscle cells, relaxation and contraction are regulated by calcium released from the sarcoplasmic reticulum. The flow of calcium from the ER to the cytosol (induced by IP 3 and ryanodine) activates the BK channels, thus facilitating a negative feedback mechanism in opposition to vasoconstric-tion [17] . There is evidence of the proximity of BK channels and calcium release sites. This is further supported by co-immunoprecipitation experiments [28] . As a consequence, such channels would be exposed to a high calcium concentration (>10 μM, in the order of 1-100 μM).
In neurons, Irie and Trussell [23] described a nanodomain between RyRs on the ER membrane and plasma membrane VGCCs (voltage-gated Ca 2+ channels), and another one between RyRs and BK channels in the soma of cartwheel inhibitory interneurons of the dorsal cochlear nucleus. Through the VGCC-RyR interaction, the latter receptors trigger the release of Ca 2+ . The internal increase in calcium acts on plasma membrane BK channels to control action potential activity and shape the burst. The interaction of the nanodomains and the Ca 2+ transients must be very rapid (in a millisecond timescale), and thus arise only tens of nanometers from the plasma membrane [23] . IP 3 is highly mobile in the cytosol. It is synthesized in the plasma membrane and diffuses into the cell where it encounters its specific receptors (IP 3 -Rs) on the ER [29] . In neocortical pyramidal neurons, IP 3 produces calcium waves via activation of metabotropic glutamate receptors. When measured with non-buffering low-affinity Ca 2+ indicators, such waves have a peak amplitude of over 5 μM [30] and propagate with a velocity of ~100 μm/s [31] . According to Ross [31] , the release of Ca 2+ from ER has been less studied because it is not associated with specific changes in the membrane potential. As can be seen in this work, the impact of the release of Ca 2+ from ER on the membrane potential was researched indirectly through the BK channels.
The importance of the interaction between BK channels and voltage-gated calcium channels has been demonstrated in the release of neurotransmitters, where they play a regulatory role that prevents excito-toxicity [32] ; in the smooth muscle of blood vessels, where it regulates blood pressure and plays an important role in preventing hypertension [33] . In these cases, the prevention mechanism is a negative feedback system. The experimental study is facilitated because both channels are found on the same membrane, and voltage clamping and transfections can be performed to combine different types of VGCC channels with the BK channel, etc. On the other hand, the experimental study of the BK/IP 3 -R complex is more difficult, the channels are in different membranes and consequently, voltage clamping cannot be performed. This justifies a theoretical study of the BK/IP 3 -R. It is unknown whether, at a somatic level in neurons, the BK/IP 3 -R interaction is present as a protective mechanism. Based on the reported studies, the coexistence of BK and IP 3 -R channels at the somatic level is known [23] and of the proximity of the cell membrane and the endoplasmic reticulum and contact sites between these structures [24] , necessary conditions for the presence of BK/ IP 3 -R. The proposed hypothesis is that: in neurons, at the somatic level, there is a BK/IP 3 -R interaction. If this is so, what would its role be in neuronal activity? What would be the mechanisms involved? It is not known whether an abrupt and short-term outflow of Ca 2+ could activate BK channels in the neuron's soma; it is also not known whether there is a commitment between the number of BK channels present in the membrane and the response of the neuron to [Ca 2+ ] cyt .  [41] . Thirdly, some models (e.g., the one created by Blackwell and Kotaleski, in [39] ) also incorporate second messengers within the biochemical reactions that are triggered by metabotropic glutamate receptor (mGluR) activation and lead to IP 3 production. Each group of models has different timescales.

Modeling of Ca
The mathematical models are related to the molecular interaction of IP 3 with the IP 3 -R and the Ca 2+activated channel, which are activated sequentially for the release of Ca 2+ from the ER. These models have employed different mathematical techniques [40] . The current contribution focuses on the interaction of [Ca 2+ ] ER with BK channels. Detailed molecular kinetics of the mechanisms of release of Ca 2+ from the ER is not essential for the purpose of this work. The phenomenological model of Goldbeter of such Ca 2+ release was herein found to be sufficiently accurate and appropriate in its timescale and was combined with the phenomenological model of Hodgkin and Huxley. The latter formulation describes the electrical activity of neurons [41] . Based on the aforementioned models, the present study theoretically explored the function of a BK/IP 3 -R microdomain.

MATERIALS AND METHODS
In the development of a new model, consideration was given to the spatial structure of the soma and some key concepts related to the kinetics of Ca 2+ . The increase in Ca 2+ is located in the microdomain [16] formed mainly by the following factors [28] : the BK channels on the soma membrane [1] [44] and the Ca 2+and IP 3 -R-sensitive channels on the ER membrane [42] .
The two membranes are very close to one another [26] .
There is evidence from electron microscopy, with 3D reconstructions, of the proximity of the plasmatic membrane with the ER membrane and of numerous contact sites between these structures, mainly in the neuron soma [24] .

The Hodgkin and Huxley formalism and the BK model
The electrical activity of the neuron was reproduced with the Hodgkin and Huxley formalism (H-H model), consisting of an equation that represents the membrane potential (Equation 1) and others that define channel gating variables (Equations 2-5) [45] [46] . ". ".

Goldbeter Model
The mathematical model of Goldbeter (1990) was found to be adequate [32] because the time involved in the BK channel is on a millisecond scale [ [38] , and the other to Ca 2+ (the CICR) [48] .
Dupont and Goldbeter developed another model with a single compartment. In a single group, considering the existence of the same two types of channels (one sensitive to Ca 2+ and the other to IP 3 ), no oscillations occur unless the contribution of the IICR is insignificant compared to that of the CICR [49] . Oscillations in [Ca 2+ ] cyt are similar in the one-and two-compartment models [49] . The latter was employed in the present study since it does not require such an extreme decrease in the contribution of the IICR as needed in the one-compartment model. -sensitive mechanism (Y). The system is governed by the two kinetic equations [32] : Where V 0 is related to the flow of Ca 2+ into the cell, k to its flow out of the cell, V 1 to its flow into the cytosol from the IP 3 -sensitive mechanism, β to the saturation function of IP 3 -Rs (cooperative nature), and K f to the leaky transport of Y into Z. Finally, V 2 expresses the rate of the SERCA pump and V 3 the rate of transport of Ca 2+ released from the IP 3 -sensitive mechanism into the cytosol.    [49] . Since the ATPase pump (SERCA) binds to two calcium ions per molecule of ATP, its activity is expressed by using Michaelis-Menten kinetics and a value of 2 for the Hill coefficient (Equation 16) [50] . The system of equations was solved simultaneously by the numerical method with a fourth-order Runge-Kutta algorithm (dt = 0.01) written on Visual Basic [53] [ 54] .

RESULTS AND DISCUSSION
The module for inputting data into the simulator ( Figure 1) provides an interactive resource for experimental recordings, allowing for access to all the parameters of the Goldbeter model [49] . There are three types of variables: those that increase or decrease [Ca 2+ ] cyt , as well as the ones related to the degree of cooperativity of the activation process (n Hill , m Hill , and p Hill ).
Firstly, the influx of Ca 2+ to the cytosol depends on the rate of transport through the IP 3 -sensitive mechanism (r= V 1 *β) and the Ca 2+ -sensitive mechanism (V 3 ), the maximum rate of Ca 2+ pumping release from the ER store (V M3 ), the threshold constants for release and activation (K R and K A , respectively) (Figures 1 and 5 [53] .  Khodakhah and Ogden [54] reported that IP 3 triggers a release of Ca 2+ from the ER with an initial well-defined delay, which decreases as the concentration of IP 3 rises from the ER takes place through sparklets, sparks, blink, scintilla, puffs, and other forms [16] . Since such elementary events are produced in microdomains, the multiple forms of Ca 2+ release confer intracellular Ca 2+ signaling with a broad architecture in space, time, and intensity, which in turn underlies signaling efficiency, stability, specificity, and diversity [55] . The calcium buffers are instrumental in achieving temporal, spatial, and functional compartmentalization under these conditions, creating steep gradients in a close proximity of channels, until reaching an internal calcium concentration on the order of tens of micromoles [56] . The local [Ca 2+ ] cyt transient in this simulation corresponds to a Ca 2+ spark, characterized by Cheng and Lederer as having an approximate amplitude of 5 μM and a duration of 35 milliseconds in a space of 30 nm [55] . Under the current conditions, the duration of the spark is three times longer and a space of 90-100 nm would be expected. The activation of BK channels by spark coming from the ER has been described in smooth muscle [17] but not in neuronal cell bodies. The microdomains of Ca 2+ consist of very small spaces (nm) between structures (e.g., voltage-gated Ca 2+ channels and BK channels or IP 3 -Rs), thus involving very local increases in Ca 2+ [57] .
Microdomains between BK channels and IP 3 -Rs have been found in different cells, including neurons, with distances of 100 nm or less [23] [59] .    17) [32] . The dependence of the BK channel on the voltage is of an allosteric type. It has been proposed that the increase in [Ca 2+ ] cyt could occur almost simultaneously with a rise in voltage, as long as the calcium source is close, as in the case of the BK/VGCC complex [60] . In the present study, a BK/IP 3 -R complex is assumed. Each stimulus (voltage and Ca 2+ ) interacts with different parts of the channel [7] . The overall conductance of BK channels on the plasma membrane is determined by the conductance of each channel as well as the number and open state probability of these channels (Equation 19) [61] .
An increment in V M3 to 700 μM/s generates a rise in [Ca 2+ ] cyt to 7 μM. Under these conditions, the neuron stops firing immediately after the first action potential ( Figure 6). Although the model of Goldbeter contemplates the action of IP 3 on IP 3 -R and offers a good approximation of the experimental data, it does not provide a dose-response relationship. It has been documented in the literature that the higher the level of IP 3 , the greater the release of Ca 2+ from the ER [62] .
Taking the limitations of the Goldbeter model into account, this simulation reasonably resembles the IP 3sensitive Ca 2+ release from the [Ca 2+ ] ER pool.
According to the results, Ca 2+ released from the ER to the cytosol can effectively gate BK channels, and the main effect is to stop the spike train ( Figure 6). These findings are consistent with reported experimental data. The Ca 2+ released from ER stores, specifically through IP 3 Rs but not RyRs, produces pauses in the firing of spiny projection neurons due to the activation of BK and SK channels [63] . The author proposes two Ca 2+ signaling pathways: (1) action potential → VGCC → RyR → BK & SK → sAHP (slow afterhyperpolarization); and (2) mGluR/mAChR (metabotropic glutamate receptors / muscarinic acetylcholine receptors) → IP 3 → IP 3 R → BK & SK → firing pause. In the current contribution, a theoretical evaluation was made of the level of [Ca 2+ ] cyt , mainly considering the interaction of IP 3 -Rs with BK channels. The find- Under physiological conditions, the PMCA pump and the SERCA pump are responsible for avoiding an excess concentration of [Ca 2+ ] cyt , which could be toxic to the cell [22] [49] . Such an excess concentration of Ca 2+ would inhibit IP 3 -R [22] . Calcium oscillations have been demonstrated to move like waves with an amplitude in the order of nM [21] [36] and a range dependent on their diffusion and binding with chelating molecules [18] . These oscillations have been proposed as signals for various cellular processes, including synaptic plasticity, regulation of neurotransmission, cell differentiation, apoptosis, embryonic development, and secretion [21] . The abrupt influx of Ca 2+ into the cell by voltage-gated calcium channels in the plasma membrane activates BK channels and triggers a change in action potential frequency [18] . The results of the present simulation show how an abrupt, focused increase in [Ca 2+ ] cyt from the endoplasmic reticulum store, very similar to those described as "spark", favors the activation of BK channels and produces a pause in the action potential train. This would imply that, under certain conditions, a mechanism capable of stopping neuronal signaling is activated.

CONCLUSIONS
A theoretical study of the IP 3 -R/BK channel interaction was carried out by means of a simulator.
Consequently, it was possible to modify the parameters to regulate the concentration of Ca 2+ in the cytosol in order to monitor the effects. The variables considered are related to the flow of Ca 2+ from cytosol: out of the cytosol to the ER (the SERCA pump) and to extra-cellular space (the PMCA pump), as well as into the cytosol from the ER (through the IP 3 -Rs) and from extracellular space (the VGCCs). Additionally, the conductance of the BK channel and the parameters of the A rise in [Ca 2+ ] cyt activates the BK channels, leading to an immediate pause or stop of the spike train. This function will allow the neuron to generate a firing pattern in burts and under certain conditions be a possible mechanism for resetting or preventing sustained abnormal PA activities (7). The colocalization of the BK and IP 3 -R channels in a microdomain is a necessary condition for the manifestation of the following pathway in a neuron: IP 3 → IP 3 -R → Ca 2+ → BK channels → pause (stop). This sequence is the regulatory mechanism (8) For neurons with a low g BK , a high [Ca 2+ ] cyt concentration, and a greater difference between this parameter and free [Ca 2+ ] ER is necessary to activate BK channels. (9) In case of having an elevated level of g BK conductance, neurons activate the BK channels more easily. The aforementioned theoretical results help to explain the experimental data of Clements et al. [63] and are in accordance with the proposal made by these authors about the existence of Ca 2+ signaling pathways