Mitochondrial calcium handling within the interstitial cells of Cajal

Abstract

The interstitial cells of Cajal (ICC) drive rhythmic pacemaking contractions in the gastrointestinal system. The ICC generate pacemaking signals by membrane depolarizations associated with the release of intracellular calcium (Ca²⁺) in the endoplasmic reticulum (ER) through inositol-trisphosphate (IP₃) receptors (IP₃R) and uptake by mitochondria (MT). This Ca²⁺ dynamic is hypothesized to generate pacemaking signals by calibrating ER Ca²⁺ store depletions and membrane depolarization with ER store-operated Ca²⁺ entry mechanisms. Using a biophysically based spatio-temporal model of integrated Ca²⁺ transport in the ICC, we determined the feasibility of ER depletion timescale correspondence with experimentally observed pacemaking frequencies while considering the impact of IP₃R Ca²⁺ release and MT uptake on bulk cytosolic Ca²⁺ levels because persistent elevations of free intracellular Ca²⁺ are toxic to the cell. MT densities and distributions are varied in the model geometry to observe MT influence on free cytosolic Ca²⁺ and the resulting frequencies of ER Ca²⁺ store depletions, as well as the sarco-endoplasmic reticulum Ca²⁺ ATP-ase (SERCA) and IP₃ agonist concentrations. Our simulations show that high MT densities observed in the ICC are more relevant to ER establishing Ca²⁺ depletion frequencies than protection of the cytosol from elevated free Ca²⁺, whereas the SERCA pump is more relevant to containing cytosolic Ca²⁺ elevations. Our results further suggest that the level of IP₃ agonist stimulating ER Ca²⁺ release, subsequent MT uptake, and eventual activation of ER store-operated Ca²⁺ entry may determine frequencies of rhythmic pacemaking exhibited by the ICC across species and tissue types.

spatially localized increases in intracellular Ca²⁺ concentrations are key triggers for numerous cellular functions ranging from cardiac cell signal transduction to neurotransmitter release with implications for disease when they fail to function; see Ref. 5, for instance. We here consider Ca²⁺ dynamics within pacemaker cells of the gastrointestinal (GI) tract, the interstitial cells of Cajal (ICC) (a glossary of all abbreviations can be found in Table 1). They were established as the GI pacemakers (44) long after their initial discovery in the 19th century (12), and the performance and structure of the ICC network are intimately related to proper GI motility and function. Disruption of the network can result in conditions such as constipation, gastroparesis, or achalasia (33, 66). The pacemaking signals themselves in the ICC are membrane depolarizations also known as slow waves (SW) that rhythmically occur at different intrinsic frequencies depending on species and tissue type. They can range in humans from three cycles per minute (cpm) in the stomach to 8–12 cpm in the intestine (43) (63), and these depolarizations coordinate contractions in surrounding smooth muscle tissue. The physiological location, spatial scales, and the relative paucity of experimental data of the ICC continue to challenge experimentalists and theoretical explorations. It is known, however, that unitary potentials, or spontaneous-transient depolarizations (SD), of the ICC membrane generate the SW in some way, yet the mechanisms responsible for producing SD themselves are not clearly understood. Membrane channels in the ICC such as the nonspecific cation conductance (NSCC) (63) and the Ca²⁺-activated chloride channels (the ANO1) (84) are likely involved in SD production, but their roles are unclear.

Table 1.

Glossary

Acronym	NCBI Gene Name	Full Expression
ATP		Adenosine triphosphate
	ANO1	Ca2+-activated chloride channel
ER		endoplasmic reticulum
GI		Gastrointestinal
ICC		interstitial cells of Cajal
IP3		Inositol-trisphosphate
IP3R	ITPR1	Inositol-trisphosphate receptor channel
	LetM1	MT H+/Ca2+ antiporter
MT		Mitochondria
MCU	MCU	MT uniporter
NCX	NCX	Sodium-calcium exchanger
NSCC	TRPC	Nonspecific cation conductance
PMCA	ATP2B2	Plasma membrane calcium ATP-ase
PMU		Pacemaking unit
SD		Spontaneous depolarizations
SERCA	ATP2A2	Sarco-endoplasmic reticulum ATP-ase
SW		Slow waves
CC		Closed PMU, cluster (15) MT configuration
C5		Closed PMU, 5 MT configuration
C1		Closed PMU, 1 MT configuration
OC		Open PMU, cluster (15) MT configuration
O5		Open PMU, 5 MT configuration
O1		Open PMU, 1 MT configuration

Daniel et al. (20) showed depletions of the endoplasmic reticulum (ER) Ca²⁺ reservoir reduce the frequency of intestinal smooth muscle contractions (19) and suggested that the pacing of ICC is related to recycling Ca²⁺ into the ER from sequestered stores in caveolae (20). It is known that Ca²⁺ transport via the intracellular inositol-trisphosphate (IP₃) receptors (IP₃R) on the ER into the cytosol (73, 78) and mitochondria (MT) uptake of cytosolic Ca²⁺ (38, 78) is involved in generation of SW activity. Disruption of MT sodium-calcium exchangers (NCX) reduces the frequency of the Ca²⁺ oscillations (45), albeit modestly and eventually on a time scale of minutes; notably, the MT NCX are also essential to the function of store-operated Ca²⁺ entry (SOCE) (52). We thus hypothesize that depletions of the ER Ca²⁺ reservoir and subsequent activation of SOCE are fundamental to pacemaking of the ICC. The eventual depletion of ER Ca²⁺ stores and activation of ER SOCE mechanisms, such as the stromal interaction molecules (STIM) on the ER membrane sensing intraluminal ER Ca²⁺ and their plasma membrane (PM) complements, either the Orai or transient receptor potential channels (TRPC) (11), are essential to intracellular ICC Ca²⁺ oscillations. These Ca²⁺ oscillations would then, in turn, determine the electrical SW activity observed by interaction with Ca²⁺-sensitive ion channels. Depletions of the ER Ca²⁺ reservoir by virtue of the ER-MT Ca²⁺ transport dynamic in a biophysically based model are of comparable timescales to those observed in ICC SW pacemaking and can be calibrated by the level of IP₃ stimulus, MT number, and ER pump density.

These calibrations are complicated somewhat by the effects of elevated [Ca²⁺]_i. Our previous study showed that steady and persistent stimulation of the IP₃R elevated [Ca²⁺]_i within a spatially confined arrangement of ER, MT, and PM called a pacemaking unit (PMU) gives sustained high and toxic [Ca²⁺]_i in the bulk cytosolic interior (51). Those results hinged on the performance of MT Ca²⁺ removal and PMU diffusive confinement of Ca²⁺. The ICC are known to express high densities of MT (14, 19), but it is also unclear whether the spatial arrangement of MT and ER into PMUs (63) is common. Furthermore, MT Ca²⁺ uptake is typically considered primarily via the high-capacity, low-sensitivity uniporter (MCU) shown to activate only at very high levels of [Ca²⁺]_i, (K_1/2 of 19 mM) (40). However, another apparently ubiquitous high-sensitivity and low-capacity MT Ca²⁺ transporter, the Ca²+/H⁺ antiporter known as the LetM1, may substantially affect the resulting [Ca²⁺]_i. The LetM1, characterized in HeLa cells (37), is suggested to be an independent pathway for MT Ca²⁺ uptake (74) or may be another component of the MCU (22). Nevertheless, the LetM1 potentially transforms the role of MT in cytosolic Ca²⁺ handling from highly localized Ca²⁺ uptake where MCU are situated near [Ca²⁺] “hotspots” to a more global [Ca²⁺]_i maintenance role.

We therefore present a biophysically based spatial model of the ICC ER-MT Ca²⁺ transport dynamic with a variety of MT populations and spatial arrangements and the first reported inclusion of a LetM1 in an MT Ca²⁺ transport model. This includes comparisons between geometries with a diffusively confining PMU region or without. Our objectives with this particular model are twofold: determination of the importance of MT Ca²⁺ handling, distribution, and spatial arrangements of a PMU to 1) calibration of time scales for ER Ca²⁺ depletions, and 2) protection of the cytosolic interior from persistent and dangerous levels of free Ca²⁺. Although we do not include membrane potentials or channel fluxes, characterizing the Ca²⁺ transport dynamic in the ICC will illuminate the subsequent impact of Ca²⁺ interactions on membrane ion transporters. Elucidating the subtle interplay between ER and MT Ca²⁺ transport calibrating ER Ca²⁺ reserve depletions while protecting the bulk interior from Ca²⁺ toxicity will provide insights into the key mechanism involved in GI pacemaking.

MATERIALS AND METHODS

For computational expedience, we restrict ourselves to two spatial dimensions with a focus on the pacemaking or PMU region itself and not the entire ICC. We also use homogenization techniques to represent the convoluted ER geometry impact on Ca²⁺ diffusion (29). Homogenization allows cytosolic and ER Ca²⁺ to coexist at each point in space with ER geometry attenuating luminal Ca²⁺ diffusion. Only the bulk cytosolic interior has Ca²⁺ homogenized; otherwise, cytosolic and ER Ca²⁺ exist in distinct domains. The PMU and surrounding homogenized regions are coupled by appropriate conditions on adjoining boundaries. The number and distribution of MT domains and the geometric confinement of our idealized PMU were varied to facilitate investigating MT impact on the Ca²⁺ dynamic as well as relevance of spatial distribution and PMU configuration.

The IP₃R channels are stochastically modeled with the Hybrid-Gillespie method (60) coupled to a reaction-diffusion system solved with a customized finite-element method routine. Our focus is on local PMU Ca²⁺ dynamics as modulated by MT number and distribution, resulting impact on surrounding bulk cytosolic [Ca²⁺], and time scales for ER reserve depletion. Dynamically varying membrane voltage or membrane channel fluxes such as T-type Ca²⁺ channels are excluded due to our focus on the intracellular Ca²⁺ dynamic operating before any subsequent channel influx activity.

Geometry

Two of the six total geometries deployed are presented in Fig. 1. PMU region and surrounding ICC subregion extents are based on Fig. 4F from Ref. 63 with some variations, such as a solo, five, or fifteen MT regions. Two different PMU configurations were also applied. One includes a narrow distance of 15 nm between the MT and PM boundary forming a “closed” PMU. Another configuration with a larger distance of 125 nm is built (an “open” PMU, see Fig. 1). We use three MT densities and distributions (14, 5, and a solo MT) and two different PMU configurations (the closed and open) giving a total of six geometries. MT region distributions and sizes are somewhat arbitrary yet intended to represent intersections of our two-dimensional ICC with MT in varying orientations and extents of radii from 60 to 220 nm.

Fig. 1.

Two-dimensional (2D) interstitial cells of Cajal (ICC)-pacemaking unit (PMU) geometry. ICC geometry dimensions based on microscopy image of PMU within a myenteric ICC cell (62). ICC PMU and a surrounding region length set at 2 μm and height set at 750 nm. 3 regions, Ω₁, Ω₂, and Ω₃, represent the PMU, endoplasmic reticulum (ER), and bulk cytosol, respectively, with the PMU subdivided into PMU proper, Ω_1a, and ER-mitochondria (MT) gap, Ω_1b. Bulk cytosol region Ω₃ with homogenized ER and cytosolic calcium (see text). Two geometric configurations were utilized: 1) closed PMU (top) with PMU MT and plasma membrane (PM) distance at 15 nm and 2) open PMU (bottom) with this distance at 125 nm. Interior MT regions with 4 (bottom) or 14 (top) or a solo (not shown) MT. A total of 6 geometries were used: 3 open PMU, and 3 closed PMU geometries with 1, 5, and 15 MT. Note, MT areas exclude any dynamic variables and are effectively Ca²⁺ sinks. Interior circular MT area radii range from 60 nm to 220 nm with arrangements and sizes somewhat arbitrary and based on microscopy images (62). A Neumann insulation condition is applied to boundary δΩ₁; δΩ₂ represents the PM with a PM Ca²⁺ pump, leakage, and a sodium-calcium exchanger (NCX). δΩ₃ is the homogenization interface with solution and gradient matching, and δΩ₈ is the MT transporter boundary with MT NCX, the Ca²⁺/H⁺ antiporter (known as LetM1) and the low-sensitivity uniporter (MCU). PMU extents estimated from microscopy images and set to 750 nm length and 250 nm height. ER-MT gap Ω_1b is continuous with region Ω₁, and width is set as per observation (59). Note close apposition of inositol-trisphosphate (IP₃) receptors (IP₃R) and discrete MCU on boundaries δΩ₆ and δΩ₇, respectively, within this gap. Boundaries are labeled δΩ_3a and δΩ_3b for PMU cytosolic and ER region homogenization interfaces, respectively. Sarco-endoplasmic reticulum Ca²⁺ ATP-ase (SERCA) and MT transport on boundaries δΩ₅ and δΩ₄, respectively.

Fig. 4.

IP₃ (0.1 μM) and SERCA (75 μM) simulations transporter variation comparison. Plot configuration is similar to previous; see Fig. 3 for legend and label description. Results with SERCA density increased to 75 μM with reduced [IP₃] at 0.1 μM presented. Top: only 1 significant variation of average bulk [Ca²⁺]_i occurs with MT inactivation: O5 configuration with no PMU LetM1 activity. 2 other variants in the O1 configuration are nearly significant, with values of P = 0.065 both with no gap MCU or combined with LetM1 inactivation throughout the PMU. Otherwise, only sealing of the ER-MT gap gives substantial reductions as in the closed PMU with 15 MT (CC), but gap sealing in the open PMU has a negligible effect (P > 0.8) except in the 5 MT (O5) case. High similarities emerge for the base variation between the different configurations: P values are over 0.8 for C1 and OC (as noted), C1 and C5, and finally C1 and O1. Others are moderately similar, as with C5 and O1 (P = 0.59) and O5 and O1 (P = 0.48). Standard deviations of the 6 simulation averages in each variant are overall modest; the greatest is about 400 nM without PMU MCU in the O1; otherwise all are less than 200 nM with the smallest at 14 nM (CC, GapSealed). The CC configuration displays little variation of averages with σ₂ < 100 nM in every variant. Bottom: disabling MT Ca²⁺ uptake throughout the PMU significantly increases depletion time in 2 closed PMU versions (C5 and C1). Otherwise, only the C1 shows significant impacts, with gap transporters turned off and without MCU in the PMU. Nearly significant increases in depletion times occur with the PMU LetM1 inactive in the CC and O5 configurations (both P = 0.065), and without ER-MT gap MCU uptake in the C1 (P = 0.065). The 5 MT are sensitive to gap sealing, where both PMU types give nearly significant increases with P just over 0.06. Only 1 substantially similar base suite of times emerges between the C5 and O5; another moderately similar pair are the 15 MT (CC and OC) with a P of around 0.4.

Geometry regions are labeled Ω₁, Ω₂, and Ω₃ for the PMU, ER, and homogenized bulk cytosol/ER, respectively. We do not dynamically represent the MT Ca²⁺; MT storage capacity of Ca²⁺ is instead treated as a sink, and we thus assume a fixed MT [Ca²⁺]. The exterior boundary δΩ₁ has a no-flux insulation condition; other boundaries δΩ₂–δΩ₈ have calcium-dependent transporter fluxes. Homogenized and nonhomogenized region interfaces are along their shared boundaries: Ω₁ and Ω₂ for cytosolic Ca²⁺, and Ω₁ and Ω₃ for ER Ca²⁺; necessary solution and gradient matching is performed as previously detailed (29). Each boundary flux term is described in Appendix A.

Model Equations

Cytosolic and lumenal ER [Ca²⁺] are represented as c and ce, respectively in units of μM (see Eq. 1 and 5); b and be are the corresponding calcium buffer concentrations in their respective regions. IP₃ is only a parameter and not a diffusive variable. Only a single calcium buffer inside the PMU is here representative of mobile cytosolic buffering and a single mobile buffer in the ER lumen as well.

Reaction diffusion equations for each region, Ω_k, are shown below (see Eq. 1, 5, and 8). Region Ω₃ is the only homogenized domain. See Appendix A for details of the boundary flux and reaction terms depicting calcium transporters and buffering. Briefly, we deploy a model of the MCU based on observations (40), include the LetM1 as characterized (37), and utilize a recent IP₃R model (64). The PM NCX and the MT NCX are represented by the Higgins et al. (35) model. In our model, the MT NCX removes MT matrix Ca²⁺; because we represent MT Ca²⁺ as a fixed value, this is a constant flux parameter trickling into the cytosol.

Model summary.

See Table 2 for a model geometry region and corresponding biological domain summary; each boundary flux term is also given in Table 3.

Table 2.

Region summary

Region	Biological Domain
Ω1	PMU cytosolic
Ω1a	PMU proper
Ω1b	ER-MT gap
Ω2	ER
Ω3	Homogenized cytosol and ER

Table 3.

Boundary term summary

Boundary	Condition
δΩ1	No-flux, insulation condition
δΩ2	PMCA: JδΩ2PMCA, PM Leak: JδΩ2PmLeak PM NCX: JδΩ2PMNCX
δΩ3a	ER-homogenized region continuity (constraints in Eq. 7 and 10)
δΩ3b	PMU cytosolic-homogenized region continuity (constraints in Eq. 4 and 10)
δΩ4	MT NCX: JδΩ4NCX, MT LetM1: JδΩ4LetM1
δΩ5	ER SERCA: JδΩ5SERCA
δΩ6	ER IP3R channel: JδΩ6IP3R
δΩ7	Discrete MT uniporter (MCU): JδΩ7MCU
δΩ8	MT flux density: JδΩ8MT

Region Ω₁ (PMU cytosol).

We subdivide Ω₁ into two subregions: Ω_1a and Ω_1b, the PMU proper and a 10-nm-wide strip between the ER and MT (the ER-MT gap), respectively (Fig. 1). The ER and MT form tightly proximal regions with physical contact and tethering calibrating this 10-nm distance (16). The PMU proper region, Ω_1a, has cytosolic calcium, c, with buffer, b, and buffer activity is included with the source term Φ_c. No homogenization is applied in this region; hence no ER calcium or ER buffering appears:

$$\begin{array}{c}\frac{\partial c}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_c{\nabla }^{2}c\hspace{0.17em}+\hspace{0.17em}{\Phi }_c(c,b)\\ \frac{\partial b}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_b{\nabla }^{2}b\hspace{0.17em}-\hspace{0.17em}{\Phi }_c(c,b)\end{array}$$ (1)

Region Ω_1b equations are identical to Eq. 1 except buffers are omitted. Calmodulin bound with calcium is 6.5 nm in length (79), and protruding ion channel structures (18) along with ER-MT tethering proteins (59) preclude buffer activity in the 10-nm-wide strip. The gap equation is thus:

$$\frac{\partial c}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_c{\nabla }^{2}c$$ (2)

REGION Ω₁ FLUXES.

Both c and ce fluxes here are given by:

$$D_c\frac{\partial c}{\partial n}\hspace{0.17em}=-\hspace{0.17em}{J}_{PMCA}^{\delta \Omega 2}\hspace{0.17em}-\hspace{0.17em}{J}_{PMNCX}^{\delta \Omega 2}\hspace{0.17em}+\hspace{0.17em}{J}_{MTNCX}^{\delta \Omega 4}\hspace{0.17em}-\hspace{0.17em}{J}_{LetM1}^{\delta \Omega 4}\hspace{0.17em}-\hspace{0.17em}{J}_{SERCA}^{\delta \Omega 5}\hspace{0.17em}+\hspace{0.17em}{J}_{IP3R}^{\delta \Omega 6}\hspace{0.17em}-\hspace{0.17em}{J}_{MCU}^{\delta \Omega 7}\hspace{0.17em}+\hspace{0.17em}{J}_{PMLeak}^{\delta \Omega 2}$$ (3)

$$D_{ce}\frac{\partial ce}{\partial n}\hspace{0.17em}=\hspace{0.17em}{J}_{SERCA}^{\delta \Omega 5}\hspace{0.17em}-\hspace{0.17em}{J}_{IP3R}^{\delta \Omega 6}\hspace{0.17em}$$

whereby n denotes the appropriate boundary unit normal. Fluxes to and from the ER region, Ω₂, are the sarco-endoplasmic reticulum Ca²⁺ ATP-ase (SERCA) and IP₃R channels applied on boundaries δΩ₅ and δΩ₆ and denoted by J_SERCA^δΩ5 and J_SERCA^δΩ₆. Fluxes to and from the MT are the individual MCU on 10-nm-wide subdomains, whereas the MT NCX and the LetM1 are distributed over most of the MT boundary. On boundaries δΩ₇ and δΩ₄, these are denoted by J_MCU^δΩ7, J_MTNCX^δΩ4, and J_LetM1^δΩ4. The PM ATP-ase calcium pumps (PMCA) and the NCX (PM NCX) are on boundary δΩ₂ and denoted by J_PMCA^δΩ2 and J_PMNCX^δΩ2, respectively. To ensure a physiological steady state, we include a calcium influx across the PM balancing pump and exchanger activity; this leak is denoted by J_PMLeak^δΩ2.

Homogenization matching conditions are applied at interfaces between the homogenized region Ω₁ and nonhomogenized region Ω₃. These include the top of the ER-MT gap, the space between the ER and PM (left PMU), and between the MT and PM (right PMU); we denote these boundaries as δΩ_3a. Diffusive variable solutions and gradients are matched at these boundaries to ensure continuity (29). The constraints

$$\begin{array}{c}{c}_{\Omega 1}=c_{\Omega 3}\hfill \\ b_{\Omega 1}=b_{\Omega 3}\hfill \\ D_c\frac{\partial c_{\Omega 1}}{\partial n}=D_{Hc}\frac{\partial c_{\Omega 3}}{\partial n}\\ D_b\frac{\partial b_{\Omega 1}}{\partial n}=D_{Hb}\frac{\partial b_{\Omega 3}}{\partial n}\end{array}$$ (4)

are thus enforced where D_c, D_Hc, D_b, and D_Hb represent nonhomogenized and homogenized cytosolic calcium and buffer diffusion coefficients, respectively. For c and b, however, these coefficients are identical (D_c = D_Hc, etc.); they effectively cancel in Eq. 4.

Region Ω₂ (ER lumen).

Only ER calcium, ce, and buffers, be, are included here; buffering is depicted with term Φ_ce. No cytosolic calcium (c) or buffers (b) are present:

$$\begin{array}{c}\frac{\partial ce}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_{ce}{\nabla }^{2}ce+{\Phi }_{ce}(ce,be)\\ \frac{\partial be}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_{be}{\nabla }^{2}be-{\Phi }_{ce}(ce,be)\end{array}$$ (5)

REGION Ω₂ FLUXES.

PMU cytosolic region, Ω₁, fluxes are given by

$$D_{ce}\frac{\partial ce}{\partial n}\hspace{0.17em}=\hspace{0.17em}{J}_{SERCA}^{\delta \Omega 5}\hspace{0.17em}-\hspace{0.17em}{J}_{IP3R}^{\delta \Omega 6}$$ (6)

on boundaries δΩ₅ and δΩ₆, for SERCA and IP₃R activity with the terms J_SERCA^δΩ5 and J_IP3R^δΩ6, respectively. The boundary δΩ₆ is set at 10 nm wide as is the boundary δΩ₇ for discrete MCUs.

The interface adjoining Ω₂ with the homogenized Ω₃, denoted as δΩ_3b, requires matching conditions analogous to those given in Eq. 4:

$$\begin{array}{c}c{e}_{\Omega 2}\hspace{0.17em}=\hspace{0.17em}c{e}_{\Omega 3}\\ D_{ce}\frac{\partial c{e}_{\Omega 2}}{\partial n}\hspace{0.17em}=\hspace{0.17em}{D}_{Hce}\frac{\partial c{e}_{\Omega 3}}{\partial n}\\ b{e}_{\Omega 2}\hspace{0.17em}=\hspace{0.17em}b{e}_{\Omega 3}\\ D_{be}\frac{\partial b{e}_{\Omega 2}}{\partial n}\hspace{0.17em}=\hspace{0.17em}{D}_{Hbe}\frac{\partial b{e}_{\Omega 3}}{\partial n}\end{array}$$ (7)

Unlike for the cytosolic variables, diffusion coefficients for ce and be in the regions Ω₂ and Ω₃ differ here; see Table 4.

Table 4.

Model parameters

Item	Symbol	Value	Reference
Cytosolic calcium	c	100 nM
ER lumenal calcium	ce	500 μM	(82)
Cytosolic buffer (total)	bTOT	75 μM	(35)
ER lumenal buffer (total)	beTOT	6 mM	(35)
MT calcium (constant)	CMT	310 nM	(26)
Intracellular sodium (constant)	Ncyt	10.1 μM	(25)
MT sodium (constant)	NMT	Ncyt/3	(56)
IP3	P	0.1 or 0.6 μM
Diffusion, cytosolic calcium	Dc	0.223 μm2/ms	(1)
Diffusion, ER Lumenal calcium	Dce	0.223 μm2/ms	(1)
Homogenized diffusion, ER Lumenal calcium	DHce	0.11 μm2/ms	Calculated
Diffusion, cytosolic buffer	Db	0.08 μm2/ms	(50)
Diffusion, ER buffer	Dbe	0.027 μm2/ms	(50, 77)
homogenized Diffusion, ER buffer	DHbe	0.013 μm2/ms	Calculated
SERCA density	ρS	15–75 μM	(8)
LetM1 density	PL	50 pumps/μm2	Estimated

Region Ω₃ (ER and cytosol, homogenized).

Homogenization is applied to this region only. The technique allows efficient simulation of distinct calcium compartments (i.e., cytosol and ER lumen) in the same space. Both c and ce diffuse and react with buffers b and be, respectively; we thus have the following:

$$\begin{array}{l}\frac{\partial c}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_c{\nabla }^{2}c\hspace{0.17em}+\hspace{0.17em}{\Phi }_c(c,b)\hspace{0.17em}-\hspace{0.17em}{\Phi }_{SERC}(c,ce)\\ \frac{\partial ce}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_{Hce}{\nabla }^{2}ce\hspace{0.17em}+\hspace{0.17em}{\Phi }_{ce}(ce,be)\hspace{0.17em}+\hspace{0.17em}{\Phi }_{SERC}(c,ce)\\ \frac{\partial b}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_b{\nabla }^{2}b\hspace{0.17em}-\hspace{0.17em}{\Phi }_c(c,b)\\ \frac{\partial be}{\partial t}\hspace{0.17em}=\hspace{0.17em}{D}_{Hbe}{\nabla }^{2}be\hspace{0.17em}-\hspace{0.17em}{\Phi }_{ce}(ce,be)\end{array}$$ (8)

Diffusion coefficients are constant for all variables; D for ce and be are effective diffusion rates as per homogenization. The homogenized D for cytosolic species, c and b, is negligibly different, so cytosolic values are used, and source terms Φ_c and Φ_ce depict calcium buffer reactions.

The SERCA pump flux term, J_SERCA^δΩ5, and the volumetric SERCA, Φ_SERC, are related by an ER volume and surface scaling factor, γ_Vol/Surf, (from ER in rat basophilic leukemia cells, Ref. 50). The SERCA model flux and source term versions are detailed in Appendix A.

REGION Ω₃ FLUXES.

There is no IP₃R activity here and no boundary efflux from the ER. Instead, only the PMCA, an associated leakage across the PM, and the PM NCX calcium extrusion are located on boundary δΩ₂. A combined MT transport term is applied on boundary δΩ₈:

$$D_c\frac{\partial c}{\partial n}\hspace{0.17em}=-\hspace{0.17em}{J}_{PMCA}^{\delta \Omega 2}\hspace{0.17em}-\hspace{0.17em}{J}_{PMNCX}^{\delta \Omega 2}\hspace{0.17em}+\hspace{0.17em}{J}_{PMLeak}^{\delta \Omega 2}\hspace{0.17em}+\hspace{0.17em}{J}_{MT}^{\delta \Omega 8}$$ (9)

The term J_MT^δΩ8 merges all MT transport on the boundary δΩ₈ including MCU flux density distributed over the boundary of all MT domains. Contrast this with the MCU depiction in J_MCU^δΩ7 of Eq. 3, where the flux is discretely applied to a single 10-nm subdomain. Note, this term is usually a net influx of MT Ca²⁺ into the cytosol due to the steady action of the MT NCX pumping Ca²⁺ out of the MT matrix; unless the MCU or LetM1 remove Ca²⁺, this steady trickle into the cytosol results. Assumptions about the ER structure are required for homogenization. A periodic structure represents the complex ER geometry, and, based on the estimated ratio of cytosolic to ER volumes, effective diffusion coefficients are computed (29). Estimates of ICC volume include 1,431 μm³ (81), and, combined with estimated MT and ER volumes of 13% and 10% of total, respectively (15), the cytosolic volume to ER volume ratio is 7.7. Only ER lumenal diffusive species have effective diffusion coefficients applied; all other diffusion coefficients are virtually identical across regions. The continuity constraints

$$\begin{array}{c}c{e}_{\Omega 1}\hspace{0.17em}=\hspace{0.17em}c{e}_{\Omega 3}\hfill \\ b{e}_{\Omega 1}\hspace{0.17em}=\hspace{0.17em}b{e}_{\Omega 3}\hfill \\ D_{ce}\frac{\partial c{e}_{\Omega 1}}{\partial n}\hspace{0.17em}=\hspace{0.17em}{D}_{Hce}\frac{\partial c{e}_{\Omega 3}}{\partial n}\\ D_{be}\frac{\partial b{e}_{\Omega 1}}{\partial n}\hspace{0.17em}=\hspace{0.17em}{D}_{Hbe}\frac{\partial b{e}_{\Omega 3}}{\partial n}\end{array}$$ (10)

are thus enforced on boundaries δΩ_3a and δΩ_3b.

RESULTS

Simulations were performed with variant numbers of MT on two different spatial configurations, either with a small 15-nm diffusive barrier or a larger opening of 125 nm (see Fig. 1). The IP₃ and SERCA protein levels were varied then compared over the suites of MT number and spatial configurations. For simplicity, we refer to a “base” suite of MT Ca²⁺ transporter activity, defined in Table 4, and note whenever they are changed.

0.6. μM IP₃, 15 μM SERCA

Concentrations of IP₃ in the ICC are not well known; prior models used constant levels of 1.0 μM (25) or an initial dynamic value of 0.258 μM (81). A relatively moderate and constant [IP₃] of 0.6 μM was applied; note, IP₃ in our model is a fixed parameter. ICC SERCA protein densities are not known either; however, SERCA concentrations in cardiac cells range from 15 to 75 μM (8). These upper and lower values were used for testing SERCA influence in our idealized ICC.

At a high [IP₃] and low SERCA density, the two PMU configurations demonstrate noticeable differences, particularly for the solo MT, where, with an open PMU, the bulk cytosolic average [Ca²⁺] is dramatically higher overall (Fig. 2). Both PMU types with 15 MT give roughly equivalent average bulk [Ca²⁺]_i. At the lower SERCA density, more MT are simply more effective at Ca²⁺ containment. Notice as well, with an open PMU, the combined PMU and bulk cytosolic [Ca²⁺]_i are reduced somewhat compared with the closed PMU, reflecting impact of confinement of Ca²⁺ within the PMU.

Fig. 2.

IP₃ (0.6 μM) and SERCA (15 μM) simulations. Bulk cytosol and ER Ca²⁺. Due to IP₃R model stochasticity, 6 simulations were performed, and averages were computed. Comparisons of the closed (A) with open (B) PMU geometries over all MT densities are shown. Legend indicates MT density plotted: 1MT is the solo MT (black), 5MT is the 5 MT (gray), and CMT is the cluster of 15 MT (light gray). Note, [Ca²⁺]_i plots either include both PMU and bulk interior (total [Ca²⁺]_i, top) or the bulk cytosol alone (bulk [Ca²⁺]_i, middle), and top panes are presented quasi-isometrically for clarity. A constant [IP₃] of 0.6 μM shows steady IP₃R activity with varying Ca²⁺ penetration into the interior. Both PMU types exhibit reduced [Ca²⁺]_i in the bulk interior. Means of the average bulk cytosolic [Ca²⁺]_i for the 15 and 5 MT are 0.71 and 0.9 μM with a closed PMU (A, middle) and 0.88 and 1.24 μM for the open PMU (B, middle), respectively. Peak [Ca²⁺] near the IP₃R channel mouth typically reach around 45 μM in both the PMU and ER-MT gap (not shown). Bottom: [Ca²⁺] (C) are with a gap-sealed open PMU geometry; inset: arrow where the ER-MT gap interface with the bulk cytosol was blocked. ER Ca²⁺ depletions compared without the gap-sealed (B, bottom) suggest that PMU Ca²⁺ egress via ER-MT gap is responsible for similar times to deplete between the 5 MT and CMT configurations.

ER Ca²⁺ store losses over the 1-s time scale vary with MT number and show interesting effects of geometry. Depletions with a single MT are similar with open or closed PMUs, whereas more MT with a closed PMU generally removes more ER Ca²⁺. Alternatively, an open PMU with 5 and 15 MT exhibit similar ER Ca²⁺ losses, whereas sealing the ER-MT gap of the open PMU (at location indicated) alters this. Preventing direct Ca²⁺ flow from the gap to the bulk interior illustrates the importance of MT proximity to the IP₃R despite exposure of Ca²⁺ into the bulk interior through the flank of the open PMU.

The sealing of the ER-MT gap also assists greatly in containment of [Ca²⁺]_i elevations as a result of the greater distance Ca²⁺ must traverse between IP₃R and the interior MT through the large open PMU aperture. Figure 3, top, shows the means of bulk interior [Ca²⁺]_i averages statistically compared (Wilcoxon rank-sum, α = 0.05, n₁ = n₂ = 6) across MT number and PMU configuration. Further comparisons were performed with simulations, where we selectively disabled the LetM1, the MCU, or both in the PMU and/or the ER-MT gap. At this relatively high [IP₃] and low SERCA density, only ER-MT gap sealing significantly reduces bulk cytosolic Ca²⁺. A single MT in either PMU configuration poorly contains elevations and displays wide variations in averages, whereas increasing MT number reduces the spread of means. More MT also decreases sensitivity to disabling of the MT Ca²⁺ uptake in the PMU; effective protection from bulk [Ca²⁺]_i elevations is with simply more MT. Action of MT Ca²⁺ uptake in the PMU (either ER-MT gap or throughout) is not as crucial as overall MT number and corresponding Ca²⁺ removal; with either no LetM1 or MCU activity in the PMU, no significant differences emerge. Bottling up Ca²⁺ within the closed PMU does assist the smaller MT numbers in containing bulk [Ca²⁺]_i as evidenced by 5 MT performing comparably to the full cluster of 15 surrounding an open PMU, in addition to exhibiting lower overall [Ca²⁺]_i.

Fig. 3.

IP₃ (0.6 μM) and SERCA (15 μM) simulations. Transporter variation comparisons. All MT and PMU type configurations are compared at the base parameters (see Table 4). Legend entries indicate transporter activity variants. “MT Off” indicates that the LetM1 and the MCU are both inactive in either the ER-MT gap (region Ω_1b) or throughout the PMU (region Ω_1a). The LetM1 or the MCU alone were also inactivated in the regions indicated. “LetM1 Off PMU” is then where the LetM1 alone is inoperative within the PMU, and “MCU Off Gap” is where the MCU is alone inoperative in the ER-MT gap. Note, MT transporters are fully operational in the bulk cytosolic interior (region Ω₃) in all variants. “GapSealed” is with all transporters operational but the ER-MT gap closed (see Fig. 2C). Geometry and MT configurations given in x-axis: initial C or O indicates closed or open PMU, and second digit MT number as in a Cluster of 15. C5 thus denotes 5 MT and a closed PMU. Six simulations for each of the 7 variants were performed (n = 6). The semi-log (y-axis) plots thus show 43 simulations over 6 geometries in a total of 258 simulations. Top: means of individual simulation bulk [Ca²⁺]_i displayed over time t = 100–1,000 ms as indicated. Maximum and minimum bulk [Ca²⁺]_i (upper and lower bar extents, respectively) over all n = 6 simulations are included. First 100 ms excluded here during initial IP₃R model transient phase; thus plots are typically above baseline [Ca²⁺]_i of 100 nM. Statistical comparisons between base suite with all transport active and each transporter variation within PMU/MT number are computed (Wilcoxon rank-sum test, n₁ = n₂ = 6; P < 0.05); statistically significant differences are marked in gray. Gap sealing provides the only significant results. Further comparisons of base variants performed across the PMU/MT number; P values showing the greatest similarity of interest between the open and closed PMU types are shown with double arrow and P value. Here, closed PMU with 5 MT is similar to the open PMU with 15 MT due to closed PMU diffusive confinement. Standard deviations for solo MT higher in general (maximum σ of 2.8 μM in the C1, LetM1PE Uni PE variant, minimum 58 nM for CC, GapSealed). Bottom: marker plot showing times to empty the ER Ca²⁺ reserve by 50% from initial value of 500 μM. Simulations extend to 1,000 ms; times to deplete reserves extrapolated with exponential decays fit to the average ER [Ca²⁺]. Markers indicate individual simulation estimated depletion time; grey markers are statistically significant deviations from the base suite (Wilcoxon rank-sum, P < 0.05). Gap-sealed geometries without substantial differences are here except for the O5 configuration. Depletion times are sensitive to LetM1 and MCU inactivation throughout the PMU (LetM1 PE Uni PE). Double arrow and P value note highest similarity between base parameters of interest; clusters of 15 MT in the open or closed PMU configuration (OC and CC, respectively) display comparable depletion times of about 1–2 s.

Performing suites of six simulations up to 10 s or more over all MT transporter variants is computationally expensive, yet the time scales of interest are longer than 1 s. We thus compared exponential fits of our 1-s results with pilot simulations out to 5 s or more until ER Ca²⁺ depletions to 50% of initial [Ca²⁺]_ER occurred. Such fits of the shorter time scales result in averages comparable to the longer time scale simulations (not shown); hence, extrapolations of times to deplete with such exponential fits are presented (Fig. 3, bottom). Note, the 50% figure is based on the STIM1 half-maximal activation constant of 210 μM ER [Ca²⁺] (10).

With more MT active, we see correspondingly shorter ER depletion times reflecting greater MT Ca²⁺ removal. If, however, there is no MT Ca²⁺ uptake in the PMU, substantial increases in depletion times occur (P < 0.05) unless there is only a single MT. Disabling just the PMU MCU also increases average depletion times, but only one instance is significant and when assisted by a closed PMU. Alternatively, if only the LetM1 antiporter is turned off in the PMU, no significant increases occur at all, suggesting that both the LetM1 and the MCU are required for effective MT Ca²⁺ removal. Notably, location of the MCU either within the ER-MT gap or the PMU proper exposes them to different Ca²⁺ levels (up to 45 or 35 μM, respectively), yet MCU situated in the gap appears to have the most influence on depletion times only in the solo MT case.

Impact of open or closed PMU is not as apparent here as with bulk [Ca²⁺]_i. Comparable depletion times are only due to MT number; high P values (P > 0.9) occur only with identical MT densities. Evidently, the number of MT transporters and proximity to ER IP₃R channels determine times to 50% Ca²⁺ reserve reduction, given that depletion time extrapolations are a sufficient estimate. Sealing of the ER-MT gap is again influential here but not as dramatic as with containment of cytosolic Ca²⁺, where we see modest overall increases in depletion times.

This [IP₃] and SERCA density result in rather high average [Ca²⁺]_i, particularly with a single MT. Increasing MT to 15 with correspondingly more Ca²⁺ removal still gives rather high averages, whereas limiting Ca²⁺ escape into the interior with ER-MT gap sealing reduces this further. Our estimate of the LetM1 density at 50 pumps/μm² may be too low. We thus increased this density up to as much as 200/μm² but did not observe bulk [Ca²⁺]_i averages under 200 nM, at this SERCA concentration. The plasma membrane NCX is also noted with a rather high density in the ICC (45), and our estimate based on cardiac cell NCX activity may also be too low (see Appendix, section Other fluxes J_SERCA, J_PMNCX, J_PMCA, J_PMLeak, J_MT). We thus increased it up to a factor of 4, yet mean bulk [Ca²⁺]_i remained above 300 nM. Meanwhile, estimated times to 50% ER depletion are also somewhat rapid. The time scales of depletions at around 1.5–10.0 s, or 6–40 cpm, situate when SOCE may activate at roughly the same cpm for some recorded intestinal SW such as canine at around 10–12 or murine at 20–25 cpm (2, 41), yet this is still too rapid for other tissue types (e.g., human gastric).

0.1. μM IP₃, 75 μM SERCA

Type I IP₃R open probabilities (P_O) at 100 nM are 80% less than at [IP₃] of 1 μM (71); we fit key transition rates in the IP₃R model and set [IP₃] at this reduced level. We also increased SERCA density by a factor of 5 to 75 μM, the upper estimated range for cardiac cells (8). Bulk [Ca²⁺]_i averages are considerably lower, and deviations within simulation suites are noticeably reduced (Fig. 4, top). Differences between MT number and PMU type configurations are also diminished, yet diffusive confinement of the closed PMU still assists fewer MT in performing comparably with more MT around an open PMU; e.g., the 5 MT, closed PMU compares well with the 15 MT, open PMU at this lower [IP₃]; this occurs at either the 75 or even a reduced 15 μM SERCA (not shown, P = 0.94). One MT was comparable to five times more MT at the higher [IP₃] and is similar to 15 times more MT (P = 1.0) at this lower IP₃ stimulus.

Disabling MT Ca²⁺ uptake in the PMU has varied effects. With the LetM1 turned off, averages are significantly reduced (5 MT, open PMU), and disabling the MCU has no significant effect (all P > 0.13). Blocking the ER-MT gap is again most salient at this 75 μM SERCA, where one configuration lowers average bulk [Ca²⁺]_i to under 200 nM (15 MT, closed PMU). Note, only at this low [IP₃] and high SERCA density do bulk [Ca²⁺]_i averages fall below 200 nM in any of our simulations, without varying LetM1 or PM NCX densities. If we again increase LetM1 density to 200 pumps/μm², bulk [Ca²⁺]_i falls below 200 nM, unlike with a stronger IP₃ stimulus and lower SERCA. By contrast, increasing PM NCX density by a factor of 4 still exhibits bulk [Ca²⁺]_i over 300 nM at this higher SERCA level.

Estimated times to drain ER Ca²⁺ reserves increase at this [IP₃] and SERCA density. One MT now requires up to 80 s without MCU uptake in the ER-MT gap, and times double to about 3 s with 15 MT and a closed PMU (Fig. 4, bottom). Without any PMU MT Ca²⁺ uptake, three significant increases occur (P < 0.05) but none with 15 MT. Gap sealing variously increases depletion times, but 1 MT with a closed PMU shows exceptional reductions. In this case, the IP₃R open more frequently (roughly 50–100 more events) providing more PMU [Ca²⁺] of on average around 1 μM for the lone MT to remove. Slowest mean depletion times at this [IP₃] and SERCA density are around 3 s or 20 cpm, analogous to murine intestinal SW of about 25 cpm (41). However, this ER Ca²⁺ depletion rate is again rather quick compared with slower ICC SW observed, e.g., the human stomach at 3 cpm (53).

Elevated Bulk [Ca²⁺]_i Exposure

Our simulated bulk [Ca²⁺]_i are quite high as with 1 MT averages over 10 μM at the higher IP₃ level and lower SERCA level; we observe the same high Ca²⁺ concentrations resulting with a low [IP₃] at 0.1 μM and SERCA set at the lower range (not shown). The different MT densities are compared across all combinations of SERCA and IP₃ levels and shown in Fig. 5A, where all mean bulk [Ca²⁺]_i exceed the 100 nM baseline. PMU type is more relevant here, whereas MT number has some effect, but differences are most dependent on SERCA and IP₃ levels, e.g., with high SERCA and low IP₃, the solo MT is comparable to the 15. Average bulk [Ca²⁺]_i are nevertheless well over the 100 nM baseline.

Fig. 5.

Bulk cytosolic Ca²⁺ overview. Box plots of bulk [Ca²⁺]_i excluding PMU concentrations are shown. Whiskers for each box plot include extrema. All MT transporters are fully operational throughout; SERCA and IP₃ concentrations vary: 15S and 75S indicate 15 and 75 μM for SERCA, and 0.1P and 0.6P indicate 100 and 600 nM IP₃, respectively. Means of the spatially averaged bulk [Ca²⁺]_i for each of n = 6 run over all PMU type and MT density shown in A. Pairs of the open and closed PMU types (gray and black, respectively) aligned with the parameter variants are noted in x-axis label with 3 panels showing MT density (15, 5, and 1 MT). Note plot of the means for each average bulk [Ca²⁺]_i excludes maximal and minimal levels for each simulation as plotted previously; only means shown here. Further note the semilogarithmic (y-axis) scale. B: distribution of contiguous time spans for each individual simulation where the bulk [Ca²⁺]_i are greater than cytotoxic level of 300 nM computed and collected into box plots for all configurations. For clarity, time spans less than 10 ms are discarded. Legend entries and columns are identical to A, yet note linear (y-axis) scale. Note significant effects of PMU type with the 5 MT configuration: P = 0.06 and 0.049, 15 μM SERCA with 0.6 μM IP₃ and 75 μM SERCA with 0.1 μM IP₃, respectively.

Confinement of IP₃R-released Ca²⁺ in a submembrane PMU appears to mitigate this. The closed PMU assists fewer MT in performing comparably with more MT around an open PMU; e.g., the 5 MT, closed PMU compares well with the 15 MT, open PMU at both high (Fig. 3, top) and low [IP₃] combined with a low 15 μM SERCA (not shown, P = 0.94). One MT with a closed PMU may also be similar to five (Fig. 3, top) or even 15 times more MT (Fig. 4, top), yet a solo MT is overall inadequate though, confirming our prior study (51). The open PMU alternatively permits more exposure of Ca²⁺ to the internal MT and their uptake mechanisms. We performed a simple comparison of the PMU configurations with no IP₃R activity and an initially elevated PMU [Ca²⁺]_i (Fig. 6). The interior MT cannot assist the PMU MT until Ca²⁺ exits the PMU, and, notably, position of an open IP₃R influences the Ca²⁺ levels at each PMU egress. For instance, the open PMU exhibits higher [Ca²⁺]_i levels at the large right flanking opening if a gap channel is active, reducing overall concentrations at the other egress points (see annotations, Fig. 6). Sealing the ER-MT gap in our simulations with active IP₃R holds average bulk [Ca²⁺]_i below 200 nM, yet this is likely a nonphysiological configuration. ER and MT tethering exquisitely calibrates the gap to 10 nm, and varying it disrupts inter-organelle crosstalk (16). The gap-sealing effects we see may also be simply artifacts of our two-dimensional geometry as well; a sealed three-dimensional ER-MT gap would require fusion of their membranes over a rather large cytosolic space.

Fig. 6.

PMU geometry impact. Surface plots showing [Ca²⁺]_i throughout the cytosol for tests with no IP₃R activity; instead, Ca²⁺ influx was elevated at the PMU PM boundary such that maximum concentrations are up to 400 nM in the closed PMU geometry (A) and for comparison the same flux parameters in the open PMU (B). All transporters are active throughout the simulation at the base parameter values (see Table 4) but with no IP₃R activity. The closed PMU naturally confines the Ca²⁺ more effectively, whereas the open permits more exposure of Ca²⁺ to MT uptake situated within the bulk interior, resulting in a slight decrease of overall Ca²⁺ levels by about 25 nM. Annotations indicate [Ca²⁺]_i at 3 egress points from the PMU (right: furthest from the ER at PM; left, proximal to ER at PM; gap, the ER-MT gap) for a test of PMU type effect on Ca²⁺ diffusion into the bulk interior. Only 2 channels were activated with no stochastic model: a simple Heaviside function opened and closed a channel for 10 ms either in the PMU (1st number) or the ER-MT gap (2nd number). Proximity clearly dictates peak [Ca²⁺]_i. Higher concentrations occur for channel activity closest to the egress point, and the closed PMU generally shows higher levels with 1 exception: the right flank where a gap channel influx into the PMU gives slightly higher levels at the large opening in the open PMU.

The ICC are observed with high basal levels of [Ca²⁺]_i (72), yet persistent and high [Ca²⁺]_i damages the cell and is manifest by PM blebbing at around 300–400 nM [Ca²⁺]_i (70) after periods of minutes (65); eventual exhaustion of MT ATP production leads to apoptosis (27). Cells do exploit oscillating [Ca²⁺]_i for signaling (6) while also protecting from persistent and toxic [Ca²⁺]_i. We thus consider time spans where bulk [Ca²⁺]_i is over 300 nM, within our simulated time scales. When IP₃R cease activity, Ca²⁺ is driven below the 300 nM level with varying effectiveness, where averages (n = 6) are in many instances over 300 nM throughout, but individual simulations do recover below this threshold (not shown). Time spans of bulk [Ca²⁺]_i over the 300 nM level are computed and compared in Fig. 5B. PMU type emerges as influential here only with 5 MT, whereas SERCA effects are more salient now; the solo MT at low SERCA is over the toxic level for effectively the duration. Five MT further prove less effective than the 15 when challenged by a strong IP₃ stimulus and a low SERCA. Preventing sustained Ca²⁺ elevations beyond 200 ms requires only a high SERCA density: regardless of PMU type, MT density or IP₃ level, at 75 μM SERCA all bulk [Ca²⁺]_i elevations over 300 nM are held (on average) to under 200 ms.

DISCUSSION

Characterizing the fundamental GI pacemaking mechanism requires understanding the underlying oscillating ICC Ca²⁺ dynamic. To that end, a biophysical spatio-temporal model of integrated ER and MT Ca²⁺ transport in the ICC including a stochastic IP₃R model under fixed [IP₃] and dynamic [Ca²⁺]_i stimuli was constructed. Various MT numbers, spatial distributions, and diffusive confinement effects on 1) ER Ca²⁺ reservoir depletion times and 2) bulk [Ca²⁺]_i elevation containment were compared. Simulations at two IP₃ and SERCA levels were performed on six different geometries of two PMU types (closed and open) and three quantities of MT (1, 5, and 15). Effects of geometry, MT number, and transporter activity were then statistically compared for significance.

Time Scales of ER Depletion

Computational constraints and stochastic IP₃R effects limited our simulated time scale to 1 s; we thus estimated when the ER Ca²⁺ reservoir depletes by 50%, or 250 μM, with exponential decay extrapolation. This level corresponds with a K_1/2 of about 210 μM for activation of SOCE mechanisms such as STIM (10). Estimations for all base configurations are presented as depletion cpm in Fig. 7. A wide range of frequencies emerge that increase along with MT number.

Fig. 7.

Overall ER depletion comparison. Box plots show estimated cycles per minute of ER Ca²⁺ depletions, based on exponential curve-fits of ER [Ca²⁺] extrapolated out to time of 50% reductions. Each pair of columns presents the 2 different PMU configurations (open or closed) aligned with the 4 variations of parameters for SERCA density (S, either 15 or 75 μM) or IP₃ concentration (P, either 0.6 or 0.1 μM) in 8 individual plots for each MT number (15, 5, or solo MT). All MT transporters were fully operational. Whiskers for each box plot include extrema.

MT number, SERCA, and IP₃R levels calibrate depletion time scales, whereas PMU type appears less important; average times are comparable between the open and closed versions. For 5 MT with low SERCA and high IP₃, PMU type appears influential, yet it is not statistically significant (P = 0.13). Lower IP₃R Ca²⁺ release and removal by MT produce lower cpm, especially with strong SERCA Ca²⁺ uptake. Observed SW frequencies vary considerably across tissue and species type ranging from human gastric at around 3 cpm (53), canine intestinal at 10–12 cpm (2), to 25 cpm for murine (41). Our simulations predict depletion cycles over these ranges, but the slowest cpm are with the 1 MT that cannot alone maintain low bulk [Ca²⁺]_i. Furthermore, the ICC exhibit a high MT density (61), so a solo MT configuration is physiologically unlikely.

With more MT, Fig. 7 suggests that [IP₃] below 100 nM would set depletion times around the lower experimentally observed cpm, consistent with inhibited phospholipase C lowering IP₃ production and Ca²⁺ oscillation frequencies (45). ICC [IP₃] are not well known, but Xenopus laevis levels change from tens of nanomolar to up over a micromolar (46); the ICC may conceivably display similar ranges. At low fixed [IP₃], the ICC would have slow ER depletion rates comparable to gastric or colonic SW frequencies, whereas a stronger IP₃ stimulus would empty the ER faster at rates comparable to the intestinal cpm. By setting different fixed IP₃ production levels, the ICC could produce a variety of pacemaking frequencies.

Containment of Cytosolic Ca²⁺

ER depletion frequencies must be weighed against any persistent and toxic levels of Ca²⁺. Simulated bulk [Ca²⁺]_i are quite high as with 1 MT averages over 10 μM. The different MT densities across SERCA and IP₃ levels exhibit mean bulk [Ca²⁺]_i exceeding the 100 nM baseline (Fig. 5A). PMU type and IP₃ stimulus are more relevant here, whereas MT number has some effect, but differences depend on SERCA and IP₃ levels, e.g., with high SERCA and low IP₃, the solo MT is comparable to the 15. Average bulk [Ca²⁺]_i are nevertheless well over the 100 nM mark.

The PMU and diffusive confinement assists in containment although MT position is important as well. The 15 MT distributions placed diffusive barriers at the PMU exits, whereas the 5 MT with its more distant interior MT permitted more Ca²⁺ into the bulk cytosol. Despite this greater Ca²⁺ exposure though, in comparisons with elevated [Ca²⁺]_i in an open PMU and no IP₃R activity, the 5 MT number with less Ca²⁺ uptake capacity simply cannot lower [Ca²⁺]_i as effectively as the cluster of 15 MT (not shown). Hence, although the closed PMU indeed assists in some containment, it appears that MT number and distribution around the ER IP₃R release sites and not a PMU proper are more relevant to overall Ca²⁺ containment.

However, regarding protecting the cytosol from toxic Ca²⁺, aspects of geometry, MT number, and IP₃R stimulus appear secondary to the overriding importance of the SERCA. A highly expressed SERCA density maintains effective protection of the cell from [Ca²⁺]_i elevations above 300 nM over durations of 200 ms (Fig. 5B). Although MT assist in controlling [Ca²⁺]_i, high SERCA densities are more effective. Given observations of high basal ICC [Ca²⁺]_i (72) likely due to persistent IP₃R release, we thus predict that ICC express high levels of SERCA level comparable to upper ranges observed in cardiac cells (8).

Implications for ICC Calcium Oscillations and Pacemaking

This study of ER Ca²⁺ depletion times and toxic [Ca²⁺]_i containment is in the context of ICC SW pacemaking. Remarks that ER Ca²⁺ store depletions may set time scales (20) for observed intracellular ICC Ca²⁺ oscillations (55, 69) partly inspired this work. Our simulations show depletion frequencies within the range of experimental observations for some SW activity (2, 41, 53). Observations of a closed PMU may be an artifact of high ICC MT density (68), possibly similar to cardiac cell MT “firewalls” for Ca²⁺ spark confinement (4, 47). However, because the closed PMU does not appear as relevant to containment as SERCA, we suggest a pacing unit localized at the ER and PM cellular junction (3) instead of a closed PMU with MT per se. This ER-PM junction is the site of ER Ca²⁺ sensors (STIM) and membrane Ca²⁺ channels (Orai or TRPC) interfacing and activating SOCE (21, 58, 83). The NSCC so prominently featured in the review of Sanders et al. (63) may be some variety of TRPC (39, 75) potentially situated in caveolae (19) typically proximal to the ER-PM junction. ER Ca²⁺ depletions may drive STIM protein and NSCC assembly at the ER-PM junction triggering Ca²⁺ influx.

Furthermore, our results suggest that MT proximity and not a closed PMU influences time scales of ER depletions; an ER-PM junction may or may not be proximal to MT. The picture is complicated, however, by observations that MT Ca²⁺ transport modulates SOCE (28), whereas a subtle ER and MT Ca²⁺ interaction is emerging (54). Establishing precisely how an ER-PM junctional pacing site depends on, or is merely influenced by, MT proximity awaits future investigation.

The ICC exhibit high MT density (14, 19), and the simulations with 15 MT are the most appropriate and effective at controlling [Ca²⁺]_i especially combined with a high SERCA level. In light of this, resulting ER depletions with 15 MT of 20 to 40 cpm at low (0.1 μM) and high (0.6 μM) [IP₃], respectively, suggest how the ICC modulate intracellular Ca²⁺ oscillations and resulting SW frequencies. Lower [IP₃] reduces IP₃R Ca²⁺ release and ER depletion cpm. Slow observed SW are gastric at around 3 cpm, considerably slower than with 15 MT in our model. We suggest then, at [IP₃] below 100 nM, slower depletion frequencies are obtainable; this is conceivably accessible to extrinsic modulation via acetylcholine receptors. By activating phospholipase C and ramping up IP₃ production and IP₃R Ca²⁺ release, pacing frequency will rise, as already suggested (45). However, ICC [IP₃] are not known; the Ca²⁺ oscillations may be controlled by oscillations of IP₃ itself (45, 72), but this awaits experimental confirmation.

LetM1 Antiporter Impact

This was the first Ca²⁺-dynamic model inclusion of the MT LetM1 H⁺/Ca²⁺ antiporter. The LetM1 is active at near baseline [Ca²⁺]_i with a K_1/2 of 400 nM (37) as opposed to the MCU with a 19 mM K_1/2 (40); this pair of transporters enables MT to handle a wide range of [Ca²⁺]_i loads. Varying LetM1 activity in the PMU region gives mixed results, however. Only one statistically significant effect of LetM1 inactivation emerges: with 5 MT and an open PMU (0.1 μM IP₃, 75 μM SERCA see Fig. 4, top) where overall [Ca²⁺]_i averages drop as a result of a few hundred fewer IP₃R opening events; notably, removal of outliers negates significance. By contrast, the LetM1 disabled throughout the PMU, and bulk interior significantly elevates bulk [Ca²⁺]_i by 70–80 nM (0.1 μM IP₃, 75 μM SERCA, P = 0.04, not shown). Alternatively, significant decreases of bulk [Ca²⁺]_i to 150 nM require LetM1 densities up to 200 pumps/μm² (same IP₃/SERCA, not shown). Regarding ER depletion times, effects of inactivating the PMU LetM1 alone are negligible. Some significant increases arise when removing outliers (e.g., 15 μM SERCA, 0.6 μM IP₃, Fig. 3, bottom), but ER depletion time scales are not sensitive to just the PMU LetM1. Disabling or increasing density of the LetM1 everywhere is statistically negligible for ER Ca²⁺ depletion times. However, in tandem with the MCU, noticeable effects emerge only at lower SERCA densities. When the MCU and LetM1 are both disabled, depletion times significantly rise at a higher IP₃ stimulus (see Fig. 3, bottom) and are nearly statistically significant at the lower [IP₃] but only with maximal number of MT and a closed PMU (P = 0.13). High densities of SERCA outcompeting the MCU and LetM1 for free cytosolic Ca²⁺, however, eclipse their combined effects on ER depletions. Overall, the LetM1 demonstrates some interesting effects such as modest IP₃R modulation, but it is primarily involved in baseline [Ca²⁺]_i maintenance working in combination with the MCU as indicated by its K_1/2 with potentially substantial transport capacity given a high enough physiological density.

Limitations

These results should be interpreted in light of mechanisms excluded yet known to be expressed in the ICC. These include ER Ca²⁺ depletion responses like SOCE, ion influxes, and membrane depolarizations triggering channels such as the ANO1 (30). We also excluded PM NCX reversal modes suggested to be involved in SW (45) or voltage-gated channels like the Ca²⁺ T-type implicated in entrainment of spatially distinct pacing sites (42, 72). These results are thus limited to mechanisms generating observed intracellular Ca²⁺ oscillations before membrane channel flux (55, 69). Furthermore, despite computational expediency, two-dimensional geometries cannot fully represent three-dimensional cellular systems. This is most salient here with sealing of the ER-MT gap that is physiologically unlikely. Also, a fixed [IP₃] excludes known dynamic interactions with Ca²⁺ as in negative feedback effects (57) with implications for whether [IP₃] themselves are oscillating as suggested (45, 72). Nevertheless, [IP₃] and several other key aspects such as SERCA or NCX density in the ICC await experimental determination.

Conclusion

Proximity to the ER release sites determines effectiveness of MT Ca²⁺ removal and subsequent time scales of ER depletions. We also propose that ICC Ca²⁺ oscillations are generated with steady [IP₃]-stimulating ER IP₃R release and MT uptake, consistent with observations of IP₃R and MT relevance to SW pacing (38, 67, 78). If this is the case, then for slower SW frequencies such as gastric activity (53), [IP₃] should be well under 100 nM. Eventual ER reserve depletions and SOCE activations would depolarize ER-PM junctional membranes or localized SW pacing clocks. Such depolarizations may be the observed SD events fundamental to SW pacemaking (23, 36) that may include ANO1 or T-type Ca²⁺ channel activity possibly in response to SOCE. Ca²⁺ channels proximal to SOCE sites are likely recruited to achieve observed levels of ICC Ca²⁺ oscillations because SOCE alone may not affect global [Ca²⁺]_i (58). Localized membrane depolarization would then heighten ANO1 sensitivity to local elevated [Ca²⁺]_i (80) and facilitate entrainment of spatially distinct pacing sites (42). Synchronizing spatially distinct intracellular Ca²⁺ clocks is likely via membrane voltage because SW propagate through ICC networks orders of magnitude faster than Ca²⁺ diffusion (72, 76). However feasible this mechanism may be, we still require further experimental confirmation of ICC parameters such as the [IP₃].

GRANTS

This work was supported in part by grants from the Health Research Council, New Zealand and the NIH (DK64775).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

Author contributions: S.A.M. conception and design of research; S.A.M. performed experiments; S.A.M. analyzed data; S.A.M. and L.K.C. interpreted results of experiments; S.A.M. prepared figures; S.A.M. drafted manuscript; S.A.M. and L.K.C. edited and revised manuscript; L.K.C. approved final version of manuscript.

Appendix A

Models of individual boundary flux transporters and calcium-buffer reactions are provided here. The solution method is also briefly described.

Boundary Fluxes

The boundary flux terms represent calcium transporters known or likely expressed in ICC (Fig. 1) and are based on published models.

IP₃R: J_IP3R.

Resident in ICC, type I IP₃R are related to SW generation (36, 78). A recent Type I IP₃R model (64) is deployed. The model includes a two-phase representation: “drive” and “park,” reflecting the active or quiescent state of the ion channel. There are six states overall: four closed (C1, C2, C3, and C4) and two open states (O4 and O6); see Ref. 64 for more detail. Ligand concentrations (Ca²⁺, IP₃, and ATP) influence two state transitions (q24 and q42) between closed states (C2 and C4) in the drive and park phases. Channel data utilized for the model derivation are with a few data points for [IP₃] and [ATP], but an extensive curve-fitting effort (13) estimated [IP₃] dependencies of q24 and q42 down to 0.6 μM; we also, however, required lower [IP₃] dependencies at 0.1 μM (see Table 5). We thus extended the data in Ref. 64 based on steady-state measurements (71); Type I IP₃R P_O drops by 80% for decreases in [IP₃] from 1.0 to 0.1 μM. Scaling down P_O from that given in Ref. 64 at 1.0 μM IP₃ and fitting Ca²⁺-dependent transitions q24 and q42, we then model IP₃R activity at a fixed 0.1 μM [IP₃]. Constant transition rates are given in Table 5.

Table 5.

IP₃R parameters

Item	Symbol	Value	Reference
Maximum flux rate	VIP3Rmax	0.064 pA	Calculated; (9)
2D flux scaling	σCh	0.01 μm	Calculated
Transition rates	q12	1240/s	(64)
	q21	88
	q23	3
	q32	69
	q26	10,500
	q62	4,010
	q45	11
	q54	3,330

The IP₃R are stochastically modeled; in an open state, ion flux is scaled linearly by the ER lumenal and cytosolic [Ca²⁺] difference:

$$J_{IP3R}\hspace{0.17em}=\hspace{0.17em}{\sigma }_{Ch}{V}_{IP3R\max }(c-ce)S_{IPR}(c,p)$$ (A1)

S_IPR switches between 0 and 1 for either a closed or open channel state. Appropriate boundary flux scaling for the 2-D model is via σ_Ch that is based on surface-to-length ratios of physical and modeled geometries; σ_Ch is described in Two-dimensional flux scaling.

Uniporter: J_MCU.

MT [Ca²⁺] are fixed at 310 nM (26), and we assume a constant Ψ_MT; we thus omit MCU models including their variation (24) (see Table 6). Instead, we use the Hille function representation described by Kirichok et al. (40) with a K_1/2 of 19 μM and coefficient of 0.6 combined with a strong maximal current of 0.5–1.0 pA, comparable to the IP₃R. However, this formulation results in MCU activity at resting levels of [Ca²⁺]_i; an allosteric inactivation term is thus added (K_1/2 of 1.0 μM, n = 3.5):

$$J_{Uni}\hspace{0.17em}=\hspace{0.17em}{\sigma }_{Ch}{V}_{MaxUni}\left(\frac{1}{1+{\left(K_{1/2}^{act}/C{a}^{2+}\right)}^{n_{act}}}\right)\left(\frac{1}{1+{\left(K_{1/2}^{inact}/C{a}^{2+}\right)}^{n_{inact}}}\right)$$ (A2)

Table 6.

Uniporter parameters

Item	Symbol	Value	Reference
Maximal uptake rate	JMaxUni	0.75 pA	(40)
Half-maximal activation	K1/2	19 μM	(40)
Activation Hille exponent	n1/2	0.6	(40)
Half-maximal inactivation	K′1/2	1.0 μM	Calculated
Inactivation Hille exponent	n′1/2	3.5	Calculated
2D flux scaling	σCh	0.01 μm	Calculated

σ_Ch scales the MCU flux for our 2-D model; see Two-dimensional flux scaling for details.

LetM1 H⁺/Ca²⁺ antiporter J_LetM1.

LetM1 activity is represented by a Hill function (37); it operates near 100 nM [Ca²⁺]_i, and we set a K_1/2 of 400 nM with a Hille coefficient of 1:

$$J_{LetM1}\hspace{0.17em}=\hspace{0.17em}{\sigma }_{Letm1}{V}_{LetM1\max }\left(\frac{1}{1+\left(K_{1/2}/C{a}^{2+}\right)}\right)\hspace{0.17em}-\hspace{0.17em}{J}_{Letm1Base}$$ (A3)

Maximal pump rate is 1,700 ions/s (37) or roughly 0.27 fA, substantially less than the MCU at 0.5–1.0 pA. LetM1 density is unknown; based on a calculated 40 MCU per μm² (40), we estimate 50 LetM1 per μm², our baseline density. J_Letm1Base counters the Hille functional activity such that at baseline [Ca²⁺]_i net flux is zero. In the PMU geometry, the MCU are discrete transport sites, but they are combined with the other MT transporters in the bulk cytosolic interior (see below).

Other fluxes J_SERCA, J_PMNCX, J_PMCA, J_PMLeak, J_MT.

The SERCA, NCX, and PMCA pump models are all taken from published cardiac models (34) with some variation (see Tables 7, 8, and 9). Ratios of phosphate-dependent (ATP, ADP, and P_i) parameters in the SERCA model determine resting ER [Ca²⁺]; we set them to obtain a typical base of 500 μM ER [Ca²⁺] (50).

Table 7.

SERCA parameters

Item	Symbol	Value	Reference
Pump density	PSERCA	15–75 μM	(8)
ER surface-to- volume ratio	γVol/Surf	32	(50)
SERCA reaction rates	ks1	2.957e-3/ms	Calculated
	ks2	7.393e-10/ms	Calculated
	ks3	1.932e-6 μM−2 ·ms−1	Calculated
	ks4	0.111/ms	Calculated
	ks5	8.416e-7/ms	Calculated
	ks6	0.0317 μM2 ·ms−1	Calculated
2D flux scaling	σSERCA	7.2 μm	Calculated

Table 8.

NCX parameters

Item	Symbol	Value	Reference
Allosteric half-maximal constant	KmCaact	0.4 μM	(49)
Allosteric Hille coefficient	nncxHill	4	(49)

Table 9.

PM leak parameters

Item	Symbol	Value	Reference
Leakage rate	kPM	1.15e-7 μm μM·ms−1·mV−1	(35)
PM voltage	VPM	−70.1 mV	(25)

PM NCX are known to be expressed in the ICC (45), but densities are unknown. Using one estimate of cardiac cell NCX density of 0.26 pA/pF (31), we calculate a maximal Ca²⁺ removal of about 75 μM/s in our ICC geometry (assuming a 1-μm height). Furthermore, we assume ICC density as lower than the cardiac such that PM NCX remove about 50 μM of Ca²⁺/s and deploy this figure here.

In the bulk cytosolic region Ω₃, MT fluxes are combined into one term, J_MT, that includes the LetM1, MTNCX, and the MCU activity. This simplifies the spatial discretization of our geometries; distributing the MCU over the MT boundaries, instead of discrete MCU sites as in the PMU, substantially reduces number of unknowns in the computational problem. The MTNCX formulation is the same as utilized for the PM NCX (34) but with MT [Ca²⁺] as well as the sodium concentrations fixed resulting in a steady trickle of Ca²⁺ into the cytosol from the MT attenuated by any increases in [Ca²⁺]_i.

Two-dimensional flux scaling.

Two-dimensional geometries reduce computational expense but require flux term modifications for proper dimensions. The 3-D molar flux per unit area (mol/l² T) was scaled into a 2-D molar flux per unit length (mol/l T) by estimating surface-to-length ratios. IP₃R, SERCA, MCU, and LetM1 fluxes include such scaling (σ) for dimensional conversion; these are ratios of estimated physiological surface areas and corresponding model geometry boundary lengths. For example, σ_SERCA is calculated from estimated ICC PMU volume (1 μm³) (25), ICC ER volume fraction (10% of total volume) (15), and observed ER surface-to-volume ratio (32/μm) (50). The estimated PMU ER surface area (3.2 μm²) and the length of our 2-D ER geometry (443 nm, boundary δΩ₅ in Fig. 1) gives a σ_SERCA of 7.2 μm. Regarding σ_Ch, we estimate that IP₃R and MCU have 100-nm² area for channel pore mouths (18), and channel subdomains are set to 10 nm in our geometry (boundaries δΩ₆ and δΩ₇). The resulting surface-to-length ratio σ_Ch is then 0.010 μm. MT flux scalings derive from surface to circumference ratios from assumed spherical geometries for MT regions.

Reaction Terms

Reaction terms are taken directly from published cardiac models, where cytosolic and ER luminal Ca²⁺ buffers are combined into representative terms and are typical mass-action kinetics; see Ref. 35 for details (see Table 10).

Table 10.

Reaction parameters

Item	Symbol	Value	Reference
Cytosolic buffer forward rate	k+b	0.4096 μM/ms	(35)
Cytosolic buffer reverse rate	k−b	0.2456 μM/ms	(35)
ER buffer forward rate	k+be	2e-5 μM/ms	(35)
ER buffer reverse rate	k−be	1.26e-2 μM/ms	(35)

Solution Method

Model equations were numerically solved with an unstructured finite element method, implemented in MATLAB (48) and heavily vectorized for memory efficiency and performance. Geometry meshes were built with the spatial discretizer CUBIT (17). Fully implicit, adaptive time integration (2nd order Adams-Bashforth/Trapezoidal predictor-corrector) (32) and the hybrid Gillespie method (60) were combined to solve the coupled reaction-diffusion equations and the stochastic IP₃R model. A customized GMRES implementation in MATLAB solved the resulting systems of linear equations.

Statistical Testing

A Wilcoxon rank-sum testing scheme routine in MATLAB (48) provided comparison of [Ca²⁺]_i averages and the extrapolated times to 50% ER Ca²⁺ depletions. Significance threshold, α, was set at a P value of 0.05; all simulations were with n = 6 runs.