Modeling traveling calcium waves in cellular structures

Abstract

We report a parametric simulation study of traveling calcium waves in two classes of cellular structures: dendrite-like processes and an idealized cell body. It is motivated by the hypothesis that calcium waves may participate in spatiotemporal sensory processing; accordingly, its objective is to elucidate the dependence of traveling wave characteristics (e.g., propagation speed and amplitude) on various anatomical and physiological parameters. The models include representations of inositol trisphosphate and ryanodine receptors (which mediate transient calcium entry into the cytoplasm from the endoplasmic reticulum), as well as other entities involved in calcium transport or reactions. These support traveling cytoplasmic calcium waves, which are fully regenerative for significant ranges of model parameters. We also observe Hopf bifurcations between stable and unstable regimes, the latter being characterized by periodic calcium spikes. Traveling waves are possible in unstable processes during phases with sufficiently high calcium levels in the endoplasmic reticulum. Damped and abortive waves are observed for some parameter values. When both receptor types are present and functional, we find wave speeds on the order of 100 to several hundred micrometers per second and cytosolic calcium transients with amplitudes of tens of micromolar; when ryanodine receptors are absent, these values are on the order of tens of micrometers per second and 1–6 micromolar. Even with significantly downgraded channel conductance, ryanodine receptors can significantly impact wave speeds and amplitudes. Receptor areal densities and the diffusion coefficient for cytoplasmic calcium are the parameters to which wave characteristics are most sensitive.

Supplementary Information

The online version contains supplementary material available at 10.1007/s10827-025-00898-2.

Introduction

Calcium waves are a phenomenon that has been observed in cell biology for various cell types, and may be characterized as transient increases in intracellular calcium levels that vary periodically in time and/or space. In the nervous system, calcium waves have been found in both glia and neurons. Such waves may be transmitted between cells by several mechanisms, such that they can extend over local networks of cells. Calcium waves in both glia and neurons are mediated primarily by receptors with trans-membrane ion channels, whose states are dependent on the concentrations of calcium and/or other ligands.

In this work, we are motivated by the hypothesis that traveling calcium waves (whether occurring in networks of glia or neurons) may play a role in modulating spatiotemporal patterns of activity in parts of the nervous system that process sensory signals. Inspired by this hypothesis, we conducted a numerical study of traveling calcium waves in model systems that incorporate the relevant receptors and other transport and reaction mechanisms found in cells in which calcium waves have been observed. Although intercellular transmission of calcium waves is clearly significant in the context of this hypothesis, for tractability we limited this study to single cells, specifically the morphological structures (i.e., dendrites or cellular processes, and cell bodies) that interpose between the input and output structures of cells, and did not consider mechanisms of wave initiation or intercellular transmission. The primary objective of the study is to evaluate the dependence of wave propagation on the parameters associated with the contributing elements – for example, the densities of membrane proteins that mediate calcium entry or removal, or certain morphological parameters – with the goal of understanding these dependencies and their implications.

In our modeling effort, we include two different types of receptors that have been implicated in this phenomenon: inositol 1,4,5-trisphosphate receptors (InP3Rs) and ryanodine receptors (RyRs), which are expressed in both glia (in particular, astrocytes), and in neurons. These receptors reside in the membrane of the endoplasmic reticulum (ER) (Solovyova & Verkhratsky, 2003) (as well as the nuclear and mitochondrial membranes), and release calcium into the cytoplasm from the ER. InP3Rs have an established role in calcium waves in astrocytes (Holtzclaw et al., 2002; Sheppard et al., 1997; Simpson et al., 1998; Verkhratsky et al., 2012; Wang et al., 2006), although any role for RyRs is more controversial and complex: authors have reported contrasting results on the degree of functional calcium transport by RyRs in astrocytes (Beck et al., 2004; Fiacco & McCarthy, 2006; Hua et al., 2004; Langley & Pierce, 1994; Matyash et al., 2002; Simpson et al., 1998; Verkhratsky et al., 2002; Wang et al., 2006). Conversely, fully functional RyRs as well as InP3Rs have been implicated in calcium transients in neurons (Hertle & Yeckel, 2007; McPhersonx et al., 1991; Ross, 2015; Sharp et al., 1993; Seymour-Laurent and Barish, 1995), although the distribution of the two receptor types varies considerably with cell type (see, e.g., Berridge, 1998).

Additionally, non-uniform spatial distributions of receptors within cells have been noted (Fitzpatrick et al., 2009; Horne, 1999; Manita & Ross, 2009; Ross, 2015; Seymour-Laurent & Barish, 1995; Simpson et al., 1998; Ur-Rahman et al., 2009). Amplifying properties have been attributed to areas with concentrations of receptors (Manita & Ross, 2009; Yagodin et al., 1994), and such areas are implicated in localized calcium “puffs” or “sparks” in various cell types (Denizot et al., 2019; Horne, 1999; Parker & Yao, 2007; Ross, 2015; Solovey & Dawson, 2010; Spiro, 1999; Swillens et al., 1999).

Although this study is not limited to glia per se, glial networks would be an attractive candidate substrate as an adjunct to spatiotemporal processing, given the bidirectional interaction that is now known to take place between neurons and glia (Araque et al., 2001a; Auld & Robitaille, 2003; Fields & Stevens-Graham, 2008; Laming et al., 2000; Perea & Araque, 2005), and the fact that a glial (as opposed to neural) model would be uncomplicated by electrical excitability and the propagation of significant membrane potentials throughout the cells. Since the phenomenon of calcium waves were first described in astrocytes (Cornell-Bell et al., 1990), further study led to the view that chemical signaling occurs in glia and that they play a dynamic role in the function of the nervous system (Allen and Barres, 2009; Allen & Lyons, 2018; Douglas et al., 2013; Fields & Stevens-Graham, 2008; Laming et al., 2000; Mattson et al., 2000; Paniccia et al., 2022; Perea & Araque, 2005; Smith, 1992; Tasker et al., 2012). Roles in both neuromodulation (Araque et al., 2001b; Pacholko et al., 2020; Santello & Volterra, 2009; Smith, 1994; Tritsch & Bergles, 2007) and vasomodulation (e.g., Attwell et al., 2010) have been proposed based on findings that glia can respond to and release chemical transmitters, and the fact that there is close superposition of processes of various glial cell types with neural processes and synapses (Araque et al., 1999; Aten et al., 2022; Halassa et al., 2007; Perea & Araque, 2005; Reichenbach et al., 2010; Santello & Volterra, 2009; Ventura & Harris, 1999), and microvasculature (e.g., Attwell et al., 2010). With respect to their active interactions with neurons, the descriptive term perhaps most often used to characterize glial function is ‘regulation’ (Ben Haim and Rowitch, 2016; Fields and Burnstock, 2006; Fields et al., 2008; Laming et al., 2000; Tasker et al., 2012) – although there seems to be a growing recognition that specific computational roles need to be considered for glia in order to realistically integrate them into systems neuroscience (Kastanenka et al., 2019; Perea & Araque, 2005).

Materials and methods

Mathematical models of both InP3Rs and RyRs are included in our models (although in some suites of simulations, RyRs are omitted, or their calcium transport rate is downgraded, in order to represent respectively a lack, or reduction, of receptor functionality). These receptors act in parallel with other mechanisms for calcium transport across membranes, detailed below. When the receptors are inactive, the net effect of these other mechanisms is to maintain low calcium levels in the cytoplasm (< 100 nM) with much higher levels in the ER and extracellular spaces (hundreds of μM) (review: Burdakov et al., 2005), whereas when calcium channels are activated, the result is significant dynamic increases in cytoplasmic calcium.

Morphological model configurations

Model morphologies comprise membrane-enclosed volumes that represent cells or parts of cells. We chose to implement idealized geometries for these representations, rather than empirically-based morphologies that would require custom grids to solve the governing equations. The configurations include:

• A pair of homogeneous compartments, representing cytoplasm and enclosed ER, embedded in an extracellular space. This configuration has no associated characteristic spatial dimensions. It is used for examining dynamical regimes, including stability and equilibrium states, or instability, as associated with sets of model parameters;

• A model for elongated cellular processes, consisting of a uniform, elongate cytoplasmic cylinder embedded in extracellular space, with an enclosed cylindrical ER. This configuration is used for examining the calcium dynamics (particularly the propagation of traveling calcium waves) in such processes;

• Variations on the process model allowing examination of tapered processes and junctions of process ramifications, as well as non-uniform receptor distribution;

• A model for a cell body, with attachments for cellular processes. The cell body morphology is idealized, so that it can be covered by a grid defined on cylindrical coordinates. It is discoidal in shape, with variable thickness maximal in the center, containing an enclosed, elaborated ER, an excluded central region representing the cell nucleus, and 'roots' for junction with cellular processes. A folded or stacked ER morphology is implicitly assumed, with individual folds that are narrow in the radial dimension and extensive in the circumferential and thickness dimensions. ER elaborations are not explicitly modeled; rather, it is simply assumed that there is ER adjacent to each cytoplasmic volume element in the cell body, and ER volume is neglected. The process roots are the same thickness as the cell body at the perimeter, but are broader along its circumference (i.e., are elliptical in cross-section) and taper to a cylindrical cross-section distally. They are assigned a cylindrical ER, although one that is assumed to be “rougher”, i.e., with higher ER surface-area-to-volume ratio, than the smooth cylindrical ER in the process model. This configuration is designed to give an idea of the spatiotemporal dynamics of traveling waves in a (simplified) cell body.

Morphologies are illustrated in Appendix 1.

Functional model elements

The calcium transport mechanisms incorporated into some or all of the model configurations include:

• InP3Rs;

• RyRs;

• Sarco-endoplasmic reticulum ATPase (SERCA) pumps, which reside in the ER membrane and transport calcium from the cytoplasm into the ER.

• Plasma membrane calcium ATPase (PMCA) pumps, related to SERCA pumps, but which reside in the plasma membrane (PM), and transport calcium from the cytoplasm to the extracellular space;

• Sodium-calcium exchange (NCX) pumps, which reside in the PM and transport calcium from the cytoplasm to the extracellular space (and sodium in the opposite direction);

• Non-specific PM calcium leakage;

• Non-specific ER membrane calcium leakage;

• Diffusion of calcium ions, both within the cytoplasm and within the ER.

Model elements that may play a role in triggering or modulating calcium transport include:

• InP3 influx into the terminal regions of cellular processes, which is supposed to be the end result of a reaction chain initiated by release of neurotransmitters or gliotransmitters in the vicinity of these terminations;

• Generation and degradation of InP3. InP3 molecules are generated from precursor molecules attached to the internal side of the plasma membrane, in a reaction catalyzed by calcium and/or G_α protein molecules that are produced as a result of external inputs, and they are degraded by reaction in the cytoplasm;

• Diffusion of InP3 within the cytoplasm (which is responsible for its transport from the plasma membrane to the ER membrane where it acts upon InP3Rs);

• Activation and deactivation of protein kinase C (PKC);

• Inhibitory modulation of the production of InP3 by activated PKC;

• Buffering of free calcium in the cytoplasm by reagents such as calbindin (CalB);

A schematic representation of these elements in situ is shown in Fig. 1. We outline the basis of models used for these elements in the following sections, including choices of selected baseline parameter values. Governing equations are given in Appendix 2.

Fig. 1.

Schematic diagram of the model elements involved in mass transport and reaction in situ. Membranes are indicated by double lines, with plasma membrane and associated elements designated in blue, and ER in red

Receptors

Light has been thrown on the complex behavior of InP3 receptors (Bezprozvanny et al., 1991) by recent work on single-channel recordings (Wagner & Yule, 2012). Opening of receptor calcium channels is mediated by the ligand inositol 1,4,5-trisphosphate (InP3), but is also modulated by intracellular calcium, as well as adenosine triphosphate (ATP). We necessarily model the calcium effect, since we are interested in intracellular calcium transients, but we assume that ATP concentration remains consistent and moderately elevated so that its modulatory effect can be neglected.

The central feature of calcium modulation is a modal or bell-shaped dependence of open probability of InP3R channels on cytoplasmic calcium levels: channel opening is inhibited at low and at high calcium concentrations, but facilitated at intermediate levels (roughly speaking, in the range from around 100 nM to a few μM) (Bezprozvanny et al., 1991; Ionescu et al., 2007). This implements a positive feedback mechanism, but one that can be shut down if cytoplasmic calcium reaches sufficient levels. As receptors undergo calcium-dependent activation or deactivation, they appear to switch between two modes: one with high open probability, and one with low (Ionescu et al., 2007). We capture the dependence of InP3R state on both InP3 and calcium with a simplified and modified version of a kinetic model proposed by Siekmann et al. (2012) and Cao et al. (2013), as implemented for Type 2 InP3Rs (the isoform found in vertebrate astrocytes: Holtzclaw et al., 2002; Petravicz et al., 2008). The Siekmann-Cao-Sneyd (SCS) model features two clusters of states, one labeled 'Park' corresponding to the low-open-probability mode, and a second called 'Drive' corresponding to the high-open-probability mode. Transitions between the modes is implemented by InP3- and calcium-dependent rates governing transitions between a pair of states, one in each cluster.

Another feature of the InP3R is a long refractory period that affects re-opening once inhibition by high calcium levels has occurred. We model this by modifying the basic SCS model, eliminating a ‘closed’ state with low residency from the ‘Drive’ cluster and adding a 'closed' state to the 'Park' cluster, with one of the two associated transition rates being calcium-dependent.

The open probability of the SCS model in the ‘Park’ mode, although low, is still large enough to support significant leakage of calcium into the cytoplasm. In our version of the model, we reduce the open probability in ‘Park’ by 80%, by adjusting a transition rate from a closed state to the open state from the value given by Cao et al. (2013).

At physiological levels of [Ca²⁺] in the ER lumen, the gradient-driven unitary calcium current through an open InP3R channel is estimated to be in the range of 0.1pA – 0.15pA (Mak & Foskett, 2015; Smith & Parker, 2009; Vais et al., 2010). For the divalent calcium ion, this corresponds to a molar flow rate of 5.2E-13 μmol.s⁻¹ – 7.8E-13 μmol.s⁻¹. In our model, we take the unitary calcium transport rate for an open InP3 receptor channel to be proportional to trans-membrane concentration difference, with a value of 6.5E-13 μmol.s⁻¹ when ER calcium is undepleted.

For the areal density of InP3Rs in the ER membrane, we select a nominal figure of 10 μm⁻² for the baseline value. This is roughly consistent with figures from the lowest-density ER membrane regions of Purkinje cells (Satoh et al., 1990), with density estimates from patch-clamp of Xenopus oocyte nuclear membrane (Mak & Foskett, 1994), and with estimates obtained from volume densities for non-Purkinje cerebellar neurons (Parys et al., 1995) in conjunction with morphological data for astrocytes in feline cortex (Williams et al., 1980). This density results in relatively small numbers of InP3Rs per element in the ER volume elements used for numerical analysis in our model processes. The receptors are therefore distributed as discrete units in most experiments with these processes. For each element, the product of this density parameter and the area of ER membrane is rounded to an integer to determine the number of receptors. This procedure results in a baseline density of 10.56 μm⁻². An exception to discretization is made when the morphological parameters r_d (the process radius) or vol_R (the volume ratio of ER to cytoplasm) are varied. In these cases, the baseline areal density of InP3Rs is maintained without discretization.

The ligand InP3 is produced by a reaction at the inner surface of the plasma membrane, in which the bound precursor phosphatidylinositol 4,5-bisphosphate (PI-4,5-P₂) is cleaved to form diacylglycerol (DAG) and free inositol 1,4,5-trisphosphate (InP3) (Berridge & Irvine, 1984; McLaughlin et al., 2002). This occurs at a rate that depends directly on intracellular calcium concentration – thus implementing a second positive feedback mechanism. In addition, its production is inhibited by activated (calcium-bound) protein kinase C (PKC), and thus inhibition driven by increased calcium levels in the cell (thereby providing a negative feedback). InP3 diffuses to the ER membrane where it affects InP3R activation, and it is degraded by bulk reaction in the cytoplasm. We model its effect on the InP3R by a modulation of the transition rates between ‘Park’ and ‘Drive’ modes, in a manner similar to Siekmann et al. (2012) and Cao et al. (2013).

Ryanodine receptors are represented using a version of a kinetic model proposed by Keizer and Levine (1996) and used by Breit and Quiesser in a study of calcium waves in neurons (2018), but modified to improve numerical stability at high levels of cytoplasmic calcium. Like InP3Rs, RyRs display calcium-enhanced channel opening (reviews: Coronado et al., 1994; Meissner, 2004), which is implemented in the model with two calcium-dependent transition rates. RyRs are also inhibited by high intracellular calcium levels, but in this case requiring concentrations on the order of 100 μM (Gomez & Yamaguchi, 2014; Xu & Meissner, 1998). In a set of preliminary simulations, we did not find cytoplasmic calcium concentrations in the vicinity of RyRs reached these levels, and thus RyR inhibition by cytoplasmic calcium was not implemented in our study.

The unitary calcium current associated with an open channel in a fully-functional RyR is appreciably larger than that of the InP3R channel, with varying estimates depending on membrane potential as well as the concentration gradient (Fill & Copello, 2002). Based on estimates for single-channel gradient-driven current from experiments in cardiac and smooth muscle (Mejía-Alvarez et al., 1999), we selected a value of 3.0E-12 μmol.s⁻¹ for the calcium transport rate associated with an open RyR channel with an undepleted ER. As with the InP3R, we assume the calcium flow through an open channel is proportional to the trans-membrane concentration difference.

RyRs are subject to a variety of modulatory substances (review: Simpson et al., 1995; Zalk et al., 2007), which might plausibly contribute to their differing behavior in neurons versus glia. In our simulations, in addition to experiments including both InP3Rs and RyRs, we did simulations in which RyRs were not present in the ER membrane, and others in which their calcium conductance was downgraded. This latter measure was intended to mimic possible receptor characteristics in astrocytes. Because we found little information on how their kinetics might be modified, we simply reduced the transport rate associated with the open channel states while retaining the same kinetic framework. In addition, because a study involving InP3 knockouts in mouse astrocytes (Srinivasan et al., 2015) showed that this procedure nearly abolished calcium responses in neural cell bodies (but not in the distal processes), we elected not to include RyRs in our cell body model.

Given the high permeability associated with RyR channels, we select a relatively low areal density of 0.4 μm⁻² as a nominal baseline value. As with InP3Rs, RyRs are modeled as discrete units in most experiments. In our process models, the product of their desired density and an ER membrane area element is typically less than unity. Accordingly, an RyR is assigned to the elements with an integer frequency obtained by rounding the inverse of this product. (For example, there is an RyR assigned to one of every five axial ER elements in our baseline cellular process configuration. This corresponds to an actual baseline density of 0.4223 μm⁻²).

When the process radius is varied, RyR distribution is likewise discrete, with radius values set by choosing integer RyR frequencies and computing the corresponding radii that yield exactly the baseline RyR density in the ER membrane. However, when the ER/cytoplasm volume ratio is varied, the baseline frequency is maintained, but the RyR calcium transport rate parameter for each volume element is rescaled to the value it would assume if the baseline density were exactly maintained. In this way, the sole effect of varying ER volume can be examined.

Calcium pumps

Plasma membrane calcium pumps include low-affinity, high-clearance-rate NCX pumps and high-affinity, low-clearance-rate PMCA pumps. NCX pumps are assumed to operate in the forward regime, and are therein modeled with a first-order Hill-type equation. The form and parameters are similar to those used by Breit and Queisser (2018) and Graupner (2003). Reflecting their relatively low calcium affinity, NCX have a relatively large half-maximum concentration constant, here set to 1.8 μM. The unitary maximum calcium clearance rate is set to 2.5E-15 μmol.s⁻¹, and the areal density to 14 μm⁻² as a baseline value. Because this number is relatively small, the number of NCX pumps per element of PM is discretized in the same manner as described for InP3Rs, and this discretization is likewise abandoned when the morphological parameters are varied.

PMCA pumps (Brini et al., 2012) are modeled with a second-order Hill equation, also after Breit and Queisser (2018) and Graupner (2003; see also Graupner et al., 2005). PMCAs are regulated by a number of substances including in particular calmodulin (Carafoli, 1992), which can increase calcium affinity in isoforms 1 and 4 by up to several orders of magnitude (Enyedi et al., 1987) (by way of reference, multiple isoforms, including 1,2, and 4 are present in vertebrate astrocytes (Fresu et al., 1999)). We treat PMCA as being in an up-regulated state in our model, but the very low half-maximum concentration constant (60 nM) used by Breit and Quiesser, which is reported in a study of high-affinity rat PMCA 2 isoforms (Elwess et al., 1997) is smaller than most other values reported in the literature for PMCAs (e.g., Keizer & Levine, 1996; Korkotian et al., 2004). Accordingly, we set this parameter to an intermediate value of 0.24 μM. The unitary maximum calcium clearance rate is set to 1.7E-17 μmol.s⁻¹ (Graupner, 2003), and baseline areal density for the pumps is taken to be 500 μm⁻² (Breit and Queisser, 2018).

The SERCA pump is modeled with a first-order Hill-type equation, but modified after Sneyd et al. (2003) to account for dependence on ER luminal calcium levels. The unitary calcium clearance rate of SERCA is quite low, possibly as small as 3.5 calcium ions per second (Gómez-Viquez et al., 2003; Lytton et al., 1992). We use 7.0E-18 μmol.s⁻¹ for this rate. A relatively high calcium affinity is imputed to SERCA pumps (discussed further in Sect. 3.3 in below), with the half-maximum concentration constant set to 0.1 μM. Areal density is set to balance the InP3R and non-specific ER membrane leakage so as to approximate the desired value for ER luminal [Ca²⁺] under quiescent conditions, which we select to be 400 μM. This requires SERCA density of 4,000 μm⁻².

Calcium leakage

The rate of non-specific plasma membrane calcium leakage is assumed to be constant, reflecting an assumption that extracellular calcium concentration remains much higher than cytoplasmic calcium and varies little with intracellular state. The baseline value of PM leakage flux is set to 5E-16 μmol.s⁻¹.μm⁻². The non-specific ER membrane calcium leakage rate is taken as proportional to the trans-membrane [Ca²⁺] difference (which can vary significantly with state), and is also set to 5.0E-16 μmol.s⁻¹.μm⁻² at the nominal quiescent ER calcium concentration of 400 μM.

Intracellular reaction and diffusion

The movement of calcium in the cytoplasm is impeded relative to aqueous solution by two significant effects: crowding by intracellular entities (Biess et al., 2011); and buffering reactions that tie up free calcium (Allbritton et al., 1992; Biess et al., 2011; Gilabert, 2012; Wang et al., 1997). These effects are often lumped together in the literature, but are separately accounted for here. Diffusion is modeled by Fick’s Law, with diffusion coefficients reduced from values for aqueous solution. Baseline values are recorded in Table 2 and Table 3 below. In the cell body model, diffusion in the radial direction is reduced still further due to the assumed disposition of ER elaborations, per Table 3.

Table 2.

Cellular process-specific model parameters

Parameter	Value	Units	Description	Fixed / Baseline
rd	0.5	μm	Radius, cellular process	B
volR	0.1	–	Volume ratio, ER/cytoplasm	B
DCac	72	μm2.s−1	Effective Ca diffusion coefficient, cytoplasm	B
DCae	144	μm2.s−1	Effective Ca diffusion coefficient, ER	B
DInP3	150	μm2.s−1	InP3 diffusion coefficient	B

Table 3.

Cell body-specific model parameters. The morphology of the cell is described in the text, and shown in further detail in Appendix 1. Diffusion in the cell body is assumed to be non-isotropic, with diffusion in the radial direction most significantly limited due to the extensive disposition of the ER in the circumferential and thickness dimensions. The baseline diffusion reduction factors for the other directions are assumed to match those values for the cellular processes; distinct values are used for radial diffusion. A subscript ‘b’ appended to various parameter names indicate these are associated with the cell and not attached processes

Parameter	Value	Units	Description	Fixed / Baseline
rb	6	μm	Radius, cell body	F
rn	2.4	μm	Radius, cell nucleus	F
tbo	3	μm	Cell thickness, outer edge	F
tbo	6	μm	Cell thickness, center	F
np	6	–	Number of attached cellular processes	B
lr	2	μm	Length of process roots	F
rrmin	1.5	μm	Minor axis of root at attachment	F
rrmax	2.1	μm	Major axis of root at attachment	F
volRb	0.15	–	Volume ratio, cell ER/cytoplasm	B
kEb	4	–	Fold factor, cell body ER	B
kEr	kEb/2	–	Fold factor, process root ER	F
DCacr	48	–	Radial Ca diffusion coefficient, body	B
DCaer	96	–	Radial Ca diffusion coefficient, ER	B
DInP3r	100	–	Radial InP3 diffusion coefficient, body	B
fCoVar	1	–	Scaling variable used to co-vary DCacr, DCaer and DInP3r linearly	B

Although multiple substances are involved with calcium buffering in vivo, we model the effect by a reversible binding reaction of calcium with a single reagent. While it has been shown that mobility and kinetics of unbound buffers can have some effect on calcium wave propagation speed (Kupferman et al., 1997), we assume that the effect of primary significance is tying up of free calcium present in the solvent, and simply treat the interaction as a bulk reaction and the reagent as locally immobile. The total available reagent concentration is set to 40 μM, the value estimated by Müller et al. for concentration of the buffer Calbindin in hippocampal neurons (Müller et al., 2005). The dissociation rate for the buffered complex Calbindin.Ca is set to 19 s⁻¹ per (Müller et al., 2005), and forward reaction (buffering) rate, which is assumed proportional to the free calcium concentration and unbound reagent concentrations, is set to a baseline value of 1.9 s⁻¹.μM⁻¹, which at equilibrium results in 80% of cytoplasmic calcium buffered, at low concentration levels.

Calcium buffering reactions reduce effective mass transport of calcium in the cytoplasm significantly. To examine the combined effects of crowding and buffering, we performed simulations of one-dimensional diffusion in an (effectively) semi-infinite medium with a fixed-concentration boundary condition, and examined the concentration profile in the medium as a function of time. The combination of diffusion with our baseline coefficient of 72μm².s⁻¹ (roughly 30% of its value in aqueous solution) and the calcium buffering model produces free calcium profiles similar to reaction-free diffusion with coefficients on the order of 20μm².s⁻¹ – consistent with results reported in the literature (e.g., Allbritton et al., 1992).

InP3 is modeled as generated at the plasma membrane (i.e., with production proportional to membrane area) by a reaction with a calcium-dependent rate, and degraded by a first-order reaction in the bulk cytoplasm. InP3 generation is also modeled as inversely dependent on concentration of activated PKC. InP3 diffusion in the cytoplasm also is modeled by Fick’s Law, with baseline diffusion coefficient set to 150 μm².s⁻¹, a smaller value than the ~ 280μm².s⁻¹ reported for Xenopus oocyte cytoplasm (Allbritton et al., 1992), but larger than the effective value of ~ 100μm².s⁻¹ derived in a subsequent analysis (Ornelas-Guevara et al., 2023).

PKC is found in two subtypes, PKC delta (not sensitive to Ca2 +) and PKC alpha (sensitive to Ca2 +). Data from protein and RNA expression studies support the claim that both PKC alpha and delta are expressed in astrocytes and neurons (Human protein atlas; Linnarsson lab, 2015; Sjöstedt et al., 2020; Zeisel et al., 2015). We include a model for PKC subtype alpha, with dynamics modeled as a reversible bulk reaction with calcium-dependent forward kinetics (i.e., activation), first-order deactivation, and a finite total pool available.

Instantiation of governing equations for morphological models

Governing equations are discretized for numerical solution, with the process and cell body models divided by grids that define discrete volume elements. Trans-membrane transport by the mechanisms described above is treated as comprising sources or sinks which are localized to the cytoplasmic volume elements adjacent to either the ER or the PM, and to ER elements (which are all adjacent to cytoplasm). Likewise, reactions are regarded as sources or sinks of reactants / products associated with each cytoplasmic volume element for bulk reactions, or elements adjacent to PM for surface reaction. Diffusion equations are applied to both calcium and InP3 in the cytoplasm, and to calcium only in the ER.

For cellular processes, when RyRs are not present, a two-dimensional diffusion model with axial and radial components is used in the cytoplasm. Circumferential diffusion is neglected due to assumed symmetry, with uniform distribution of channels and pumps assumed around the circumference of the PM or ER. However, because RyRs are assumed to be sparsely distributed, a three-dimensional diffusion model is used when they are present, with the axial and circumferential locations of individual receptors explicitly specified. The cytoplasmic grid defines annular volume elements in the two-dimensional case, or segmented annular elements in the three-dimensional case (see Fig. 10 in Appendix 1). The baseline axial length is 0.5 μm, and three radial divisions are present, with variable dimensions according to process and ER radii. In the three-dimensional model, there are six circumferential divisions. A one-dimensional model with axial diffusion only is used in in the ER, as in prior studies (e.g., Breit and Queisser, 2018). The ER is thus divided into cylindrical axial elements only, each also 0.5 μm long and aligned with a cytoplasmic annulus. Table 2 below lists the model parameters specific to the cellular process model.

Fig. 10.

Model morphologies. Panel a illustrates a segment of a cellular process model with the grid used for numerical analysis superimposed in dashed lines. The circumferential grid is present only when RyRs are included in a simulation. Panel b is a three-dimensional representation of the cell body model with process roots. Panel c shows the circumferential and radial grids in a top / plan view, and Panel d, the grids in the circumferential and thickness directions in a central cross-section. Panel e is a top / plan view of the cell body illustrating (schematically) the assumed disposition of segments of ‘folds’ of ER with respect to the coordinate directions. Diffusion within the ER is supposed to be relatively more limited in the radial direction than in the circumferential or thickness dimensions

In uniform processes, RyRs are instantiated with a uniform axial period and (due to their low areal density) are always separated by at least two elements axially. In an initial set of simulations, a sample of eight processes was defined in which circumferential placement of RyRs among the six ER-adjacent annular segments was selected randomly, with all parameters set at baseline values. The placement configuration that gave wave speed closest to the sample mean was selected for use in subsequent simulations.

Diffusion in the idealized cell body model is set up as a three-dimensional problem, with radial circumferential, and axial components. The cytoplasm is divided into 48-element grids in the circumferential dimension, six in the radial dimension, and ten in the axial direction. For individual volume elements, circumferential arc length at the central radii ranges from 0.35 μm to 0.74 μm, the radial extent is 0.6 μm, and the axial extent ranges from 0.3 μm to 0.6 μm. These remain fixed, as the cell body is set to a fixed, representative size. Each cytoplasmic volume element in the cell body is assumed to be adjacent to an ER element due to the folded structure.

The process roots are set up similarly to the processes themselves, with two-dimensional (axial and radial) diffusion in the cytoplasm and one-dimensional (axial) only in the ER. Volume elements are 0.5 μm long in the axial dimension, and there are three in the radial dimension in the cytoplasm. Constants for computation of diffusion in the cell body and roots are set up to account, respectively, for the tapers in the cell thickness and the root widths. Diffusion across the junction between the roots and cell body are treated with a boundary-value approach: a boundary value is computed by equating the fluxes across the boundary on each side as the area-weighted average of the fluxes associated with each volume element there. This approximation results in a single boundary value that is imposed on all volume elements that contact the junction. This procedure is applied to both the cytoplasmic and ER compartments. Table 3 below lists the model parameters specific to the cell body model.

Illustration of the cell body is also given in Appendix 1.

Simulation procedures and conventions

Simulations were performed with MATLAB (Mathworks, Natick, MA, USA). The Laplacian operator governing diffusion was implemented with a second-order scheme derived from the MOLE library (Castillo & Grone, 2003; Corbino et al., 2024). Temporal integration was performed by simple quadrature.

Simulations of calcium transients were run with particular choices of parameters, including morphological parameters for the process model. A ‘baseline’ set of parameters was defined, some of which were fixed throughout the study, while others were varied. Suites of simulations generally involved variations of an individual parameter while others remained at the baseline values. Morphology-independent parameters are listed in Table 1, whereas parameters particular to the process and cell body models are listed in Table 2 and Table 3, respectively. Baseline values are given in the tables.

Table 1.

Universal (morphology-independent) model parameters. (Rate parameters for receptor kinetics, other than r_Ig, are fixed and are given in Appendix 2.) ‘Baseline’ values are for parameters that may be varied during the study; ‘Fixed’ parameters are constant throughout. References for sources: B&Q = Breit and Queisser (2018); G = Graupner (2003); K&O = Kang and Othmer (2009); M&a = Müller et al. (2005)

Parameter	Value	Units	Description	Fixed / Baseline	Source
Ce0	400	μM	Nominal ER [Ca2+]	–	see text
kbCf	1.9	s−1.μM−1	Calcium buffering rate constant	B	see text
kbCb	19	s−1	Calcium buffering dissociation rate	B	M&a
rlx	5.0E-16	μmol.s−1.μm−2	PM Ca leakage rate constant	B	see text
rle	5.0E-16/Ce0	μmol.s−1.μm−2.μM−1	ER membrane Ca leakage rate constant	B	see text
CalB0	40	μM	Buffering reagent concentration	F	M&a
RrI	10	μm−2	InP3R areal density	B	see text
IrI	6.5E-13/Ce0	μmol.s−1.μM−1	InP3R calcium flow rate constant	F	see text
rmIf	1.4E-15	μmol.s−1.μm−2.μM−1	InP3 production rate constant	B	K&O
kmIb	2.5	s−1	InP3 degradation rate	B	K&O
kiPI	0.0943	–	Constant, InP3 inhibition by PKC	B	K&O
PKC0	1.0	μM	Available PKC concentration	B	K&O
kiPf	0.6	s−1.μM−1	Binding (activation) rate constant, PKC	B	K&O
kiPb	0.5	s−1	Dissociation (deactivation) rate, PKC	B	K&O
RrR	0.4	μm−2	RyR areal density	B	see text
IrR	3.0E-12/Ce0	μmol.s−1.μM−1	RyR unitary Ca flow rate constant	F	see text
RpN	14	μm−2	NCX calcium pump areal density	B	see text
IpN	2.5E-15	μmol.s−1	NCX unitary calcium flow constant	F	G
kpNC	1.8	μM	NCX half-max. concentration constant	F	G
RpP	500	μm−2	PMCA calcium pump areal density	B	B&Q
IpP	1.7E-17	μmol.s−1	PMCA unitary calcium flow rate	F	G
kpPC	0.24	μM	PMCA half-max. concentration const	F	see text
RpS	4000	μm−2	SERCA calcium pump areal density	B	see text
IpS	7.0E-18*Ce0	μmol.s−1.μM	SERCA unitary calcium flow constant	F	see text
kpSC	0.10	μM	SERCA half-max. concentration const	F	See text
rIg	70	s−1	Gating delay rate constant, InP3R	B	see text

Prior to each simulation of a process or cell body, the single compartment-pair model of cytoplasm and ER was run with the assigned parameter set, to determine stability and quiescent values for the state variables, which were used to establish initial conditions for all volume elements. When parameters were spatially non-uniform (as was the case in tapered processes or the cell body), average parameter values were used in these single-compartment pair simulations. (This means that local equilibria might differ slightly from the imposed initial conditions.)

Cellular processes

Isolated cellular process segments were simulated in order to characterize calcium waves in the structures as parameters were varied. Uniform segments of 100 μm length were used, with waves initiated at one of the two ends. Amplitudes are measured at the radial compartments adjacent to the PM (presumably the location of any effectors that would be activated by elevated calcium), and averaged over the circumference when RyRs are present. With no RyRs, wave speed is derived from traversal time over an axial distance of approximately 2.5 μm about the center of the process; when RyRs are present, most of the distance from the center to the (non-activated) end is used. This effectively averages over various relative placements of successive RyRs about the circumference. With RyRs, amplitude is defined in terms of maximal excursions occurring at axial locations of the RyRs. Waves were initiated from the quiescent state, except for a limited (4 μm-long) region at the initiating end in which initial conditions of elevated calcium and InP3 were imposed to trigger a wave. These simulations were run with sealed-end boundary conditions. However, similar results could be obtained by allowing InP3 (or InP3 and calcium) to flow into the distal end, and applying a pulse of these reagents. This method was used with tapered processes.

In tapered process simulations, process radius varies from 0.15 μm to 1.5 μm. Morphology-independent parameters were maintained at baseline values in these simulations, with RyRs either absent or present. When present, they are placed at a variable axial frequency determined by the baseline density and the local ER membrane area of the volume elements, and at random circumferential locations. The tapered region is interposed between a pair of uniform segments of 15 μm length, and respective radii consistent with the distal and proximal ends of the tapered region. The taper ratio is regarded as a parameter, and therefore length of the tapered region depended on its value, ranging between 18 μm and 40 μm. Because waves change in amplitude and shape while propagating, with widely varying time required to reach maximum calcium levels after calcium begins rising locally, speed measurements were based on the times at which the local waveform reached half-maximum values.

In branched processes, for simplicity’s sake, the radial compartments in the distal branches are treated as adjacent to the corresponding radial compartments in the proximal segment (i.e., there was no ‘mixing’ in the radial direction associated with the branch). Each joint consists of a distal pair with half the cross-sectional area of the proximal segment, and the distal radius was regarded as a parameter. Waves were initiated in either one or both distal segments. The traversal time for a wave (the difference in time at which peak calcium is achieved at set locations in an active distal segment and the proximal segment) is calculated for each case, and compared to the sum of the traversal times for the distal and proximal segments as computed from their individual propagation velocities. This allows the calculation of the delay or advancement of a wave due to the presence of a branch. These simulations were run with all morphology-independent parameters at baseline values, without RyRs, and for various process radii.

We implemented processes with non-uniform receptor densities by dividing the basic process model into a variable number of axial segments, in each of which the density of InP3Rs could be independently assigned. RyRs were omitted from these versions of the model, and morphology and other parameters were maintained at the baseline values.

Cell body

The cell body model was simulated using segments of processes attached to each of the cell roots. Each segment is 10 μm long, with same radius as the outer root (1.5 μm) to which it is coupled. The segments have only InP3Rs and any non-morphological parameters remain at the baseline values regardless of the values assumed by analogous parameters in the cell body and roots – insuring that the processes can support propagation of a consistent traveling wave. This configuration was used to excite a calcium wave in the cell body, by distal excitation of one or more attached processes (via a pulse of InP3 and calcium at the distal end), and was likewise used to observed the initiation of calcium waves in unstimulated process segments. Cell bodies are assigned two to six roots with attached processes; results are reported for the six-process model. These simulations were run with variations in several non-morphological parameters for the cell body.

During individual simulations, the delay times were measured between the arrival of an incoming calcium wave at the volume element adjacent to the process root (Fig. 7A, Measurement 1) and a similarly-positioned element adjacent to the root of each unexcited process (Fig. 7A, Measurements 2–4). The arrival of a wave at each of the elements was defined as the time of the first peak calcium concentration above a minimum level of 0.05 μM. The distance between these elements around the circumference of the cell, divided by the delay time, is used as a measure of wave speed. In addition, the amplitudes of calcium waves were noted at various elements within the body.

Fig. 7.

Wave propagation speed and amplitude in the process model as a function of various remaining parameters. The abscissas represent the scaling of each parameter relative to its baseline value. Curves obtained by varying different parameters are coded by color, as indicated in the legends. The designation ‘CeCtl’ for the parameter r_lx indicates that Ce_i was controlled during those experiments. Panel a gives wave speed as a function of parameter values for processes with RyRs; Panel b, amplitude for processes with RyRs, Panel c, speed for processes without RyRs; and Panel d, amplitude for processes without RyRs. The dependencies for variations of Ce_i (here labeled C_e0) are included in each of the graphs in the same format, for reference

Results

Stability

We find that our simulated systems may or may not have stable equilibria, depending on parameter values: consistent with results from other researchers (Chang et al., 2013; Handy et al., 2017; Lavrentovich & Hemkin, 2008; Sneyd et al., 2017; Zhou et al., 2019) we observe that they undergo a Hopf bifurcation at which stability of a fixed point (occurring at the quiescent, low-cytoplasmic-calcium state) is lost and limit cycle behavior ensues. These limit cycles take the form of periodic calcium spikes in the cytoplasm, with concomitant depletion of ER calcium. (Such cycles are often referred to as calcium waves, although they are synchronous throughout a uniform cellular process.) As our interest is in traveling waves, we did not examine bifurcations or the nature of oscillatory solutions in detail. However, it is worth noting that destabilization occurs when one parameter in particular – the non-specific PM calcium leakage flux – is increased from its baseline value, a finding that is consistent with results from other studies (e.g., Lavrentovich & Hemkin, 2008; Liu & Li, 2016).

Features of calcium dynamics associated with RyRs

Because the unitary currents of fully functional RyRs are relatively high, they tend to generate local micro-domains of elevated calcium when active RyRs are present in the process model. These are evident in simulations of process dynamics, to be seen in the examples below. Because they are not inhibited by high cytoplasmic calcium levels, we find calcium-induced calcium release by RyRs is reduced primarily by depletion of ER calcium.

Traveling waves in cellular processes

The equations implementing the cellular process model admit solutions in the form of a solitary traveling wave of elevated cytoplasmic calcium (or a pair of such waves moving in opposite directions, if initiated at a central location in the process). Traveling waves may be fully regenerative, by which we mean that they maintain a constant amplitude in a uniform process after initial transients have subsided, or they may be damped, indicating that their amplitude decreases as they propagate. Damped waves may die out completely at some point in the model process, and we refer to these as abortive waves. Other temporal solutions are possible as well, depending on initial conditions – we have observed single calcium spikes followed by settling to a stable equilibrium, for example. However, we focus primarily on traveling wave solutions in systems with stable quiescent equilibria.

Figure 2 illustrates the spatial form of traveling calcium waves in the model process with the baseline parametric configuration, with both receptors (Panel a), and with only InP3Rs (Panel c) present. These are both fully regenerative waves that propagate the length of the process. Movie M1 in the Supplementary Material depicts the propagation of these waves. Micro-domains of elevated calcium around RyRs are evident in Panel a, and these give the traveling wave a saltatory character as it travels.

Fig. 2.

Snapshots of traveling calcium waves in the model cellular process with the baseline parameters. The abscissas indicates axial position, with the waves initiated at left and traveling to the right. Panels a and c show circumferentially averaged cytoplasmic calcium concentrations. The blue and green curves depict calcium levels at each of three radial grid locations; green is adjacent to the ER, cyan is central, and blue is adjacent to the PM. The red curve shows InP3 concentration adjacent to the ER. Panels b and d show corresponding ER calcium concentrations. For Panels a and b, both InP3Rs and RyRs are present, with one RyR per five axial volume elements; for Panels c and d, only InP3Rs are present

Interestingly, traveling waves can occur in unstable as well as stable processes, if they are initiated at a phase in the limit cycle for which ER calcium concentration is sufficiently high. (An example is shown in Movie M2 in the Supplementary Material, which depicts a traveling calcium wave in a cellular process, with the far end subsumed by a calcium spike before the wave reaches it.)

When only RyRs are present in a model process, traveling calcium waves are also possible. With all non-InP3R parameters at baseline, fully-regenerative, saltatory waves are observed to propagate with a speed of over 300 μm.s⁻¹ and a maximum amplitude of 63 μM – but in this configuration, the equilibrium ER calcium concentration is greater than 1.25 mM, as a consequence of the absence of InP3Rs in the membrane. This observation illustrates that the ‘leakiness’ of InP3Rs (as modeled) in the inactive state plays a significant role in setting equilibrium ER calcium levels.

Recovery from transient calcium increases depends on several factors: depletion of ER calcium (particularly when RyRs are present), buffering of calcium after it enters and diffuses into the cytoplasm, and for InP3Rs, inhibition by increased cytosolic calcium levels. Calcium pumps, which have a modest but measurable effect on wave characteristics, play a lesser role in short-term reduction of cytoplasmic calcium, but are of course central to longer-term restoration of quiescent equilibrium. In the long term, recharge of ER calcium is also assisted by store-dependent as well as store-independent influx from the extracellular space (which is not modeled here, since on the time scales we consider, it is expected to have little effect on the characteristics of propagating waves. In addition, accurate modeling of store-operated calcium entry remains difficult due to the difficulty of experimental isolation of the process (Saftenku, 2022)).

ER calcium depletion associated with calcium waves is depicted in Fig. 2, Panels b and d. It is particularly profound when RyRs are present (Panel b). The reduction in ER concentration necessary to support a given peak concentration of free cytoplasmic calcium is dependent on the volume ratio parameter (i.e., it differs by a factor of ten in the baseline case) – but in addition, several times this much calcium must flow into the cytoplasm to account for buffering effects.

The role of InP3Rs in setting the quiescent state dominates both the non-specific calcium leakage and the RyR permeability in the ER membrane, in spite of the fact that we have re-parameterized the SCS model to reduce open probability in the ‘Park’ mode. This is the reason that a high SERCA density is required to maintain the quiescent ER [Ca²⁺] near 400 μM. (We note, however, that the baseline density of 4000 μm⁻² is not physiologically implausible, as SERCA densities up to 30,000 μm⁻² have been reported for smooth muscle (Lytton et al., 1989).) It is also the reason that we imputed a relatively high calcium affinity to SERCA pumps, with a half-maximum concentration constant k_pSC = 0.1 μM, versus the value of 0.18 μM reported by Sneyd et al. (2003) and used by Breit and Queisser (2018) in their modeling effort. This reduces the effect of InP3R leakiness, by pumping more calcium back into the ER.

The dependence of quiescent ER calcium levels on model parameters such as InP3R density presents a challenge to the interpretation of simulation results, because initial ER calcium by itself affects wave characteristics (e.g., amplitude and propagation speed), independent of channel permeabilities. To examine (or to eliminate) this effect, we ran various sets of simulations in which a virtual source/sink of calcium of fixed magnitude was applied to each volume element in the ER, in order to independently control quiescent/initial ER calcium concentration Ce_i without introducing additional conductance/permeability. Figure 3 illustrates results of experiments in which Ce_i was varied from trial to trial by this method, with all other parameters maintained at baseline values. Both wave amplitude and speed are nearly linearly related to Ce_i for the fully regenerative cases, especially when RyRs are present (Pearson correlation coefficient > 0.995 in both cases).

Fig. 3.

Dependence of calcium wave amplitude and propagation speed on initial/quiescent ER calcium concentration Ce_i, with all other parameters fixed at the baseline values. Panel a is for the case when both RyRs and InP3s are present in the ER membrane, and Panel b for InP3Rs only. Data points depicted as dots indicate fully regenerative waves, and open circles indicate the wave was damped, with amplitude and velocity measured as it crossed the center of the process. A violet diamond on the x-axis indicates that an abortive wave that did not reach the center of the process was observed at that value for Ce_i, and a red diamond indicates an unstable process with limit cycle behavior (i.e., periodic calcium spikes). The black triangles on the abscissas are placed at the baseline value Ce₀

Several parameters in addition to InP3R density, namely SERCA density and PM leakage, significantly influence quiescent ER calcium, and therefore the effects of changes in pump rate or leakage rate per se are entangled with the effect of changes in Ce_i. In order to disentangle these effects, in a number of simulations (to be indicated) we used the virtual source/sink method to force Ce_i to within 0.5% of the nominal 400 μM value, for the parameter variations under consideration. Among these simulations was one in which the SERCA parameter k_pSC was increased to 0.18 μM with other parameters at baseline, in order to examine the effects on wave characteristics. We found that wave speed was increased by about 8% and amplitude by about 3% by this change. (When k_pSC was set to appreciably higher values, the virtual source/sink method could not compensate for the increased flow of calcium from the ER without causing instability.)

Wave speed and amplitude depend on model parameters

At the baseline parameter values, including a process radius of 0.5 μm, the observed wave speed is 172 μm/s and amplitude 11.9 μM when RyRs are present, and 79.2 μm/s and amplitude 3.78 μM when they are not. Below we present results from experiments with the cellular process model as various parameters are varied about baseline values.

Figure 4 depicts wave speed and amplitude as a function of morphological parameters.

Fig. 4.

Wave propagation speed and amplitude as a function of morphological parameters process radius r_d (at left) and cytoplasm/ER volume ratio vol_R (at right). Other parameters remain at baseline values. Panels a and c are for processes that include both RyRs and InP3Rs, whereas Panels b and d are for processes that include InP3Rs only. The black triangles on the abscissas are placed at the baseline value of each parameter

The variation of wave amplitude with radius shows a roughly 1/r_d dependence over much of its range: this is to be expected as it reflects the ratio of ER membrane area to cytoplasmic volume. The same is true for propagation speed when no RyRs are present in the membrane (Panel b in Fig. 4). The speed dependence is more complex with RyRs, as shown in Panel a: when their areal density is fixed, RyRs occur more frequently in the axial direction as the radius increases, such that diffusion of calcium admitted by RyRs between the receptors begins to play a significant role in speeding up wave propagation. Conversely, when these receptors are widely separated, channel opening in RyRs is triggered primarily by InP3R-mediated transients.

The experimental variations in volume ratio, as described in the Modeling section, are designed to show the effects of increases or decreases in ER size without discrete changes in the numbers of receptors. These results show that a relatively larger ER supports greater wave amplitudes and speeds, consistent with more available calcium. When RyRs are present, wave amplitude is particularly sensitive to the volume ratio, because RyRs significantly deplete ER calcium and more is available from a larger ER.

Figure 5 depicts dependence of wave speed and amplitude on receptor densities.

Fig. 5.

Wave propagation speed and amplitude as a function of receptor areal densities. Other parameters remain at baseline values. Panels a and b are for variations in InP3 density; solid lines indicate simulations in which Ce_i is set at the nominal value by the virtual source/sink method; dashed lines, runs for which this control is not included and Ce_i varies with receptor density. RyRs were present for the Panel a results, and absent for Panel b. The red diamonds indicate instability was observed at those values of the density, and open circles indicate the wave was damped, with amplitude and velocity measured as it crossed the center of the process. Panel c shows dependence on RyR density, both with (solid lines) and without (dashed lines) InP3Rs. The violet diamond indicates a damped wave that did not cross the center of the process was observed for that value of the density, for the simulation without InP3Rs. Other parameters remain at baseline values

With RyRs at the baseline density, wave speed and amplitude are only weakly dependent on InP3R density R_rI, when Ce_i is controlled. (If not, they both decrease with increasing InP3R density as a result of the concomitant reduction in Ce_i.) The weak dependence is because the RyRs dominate the wave characteristics. However, when RyRs are not present, then the speed and amplitude of purely InP3R-mediated waves show a significant, direct density-dependence, when Ce_i is controlled. (If not, this dependence is virtually eliminated.)

A strong direct dependence of wave speed and amplitude is seen when RyR density R_rR is varied. Propagation speeds approaching 1 mm/s can be supported with high RyR densities. However, it is the presence of InP3Rs that allows propagating waves to be sustained at low RyR densities. If InP3Rs are not present (and Ce_i is controlled), abortive waves are seen at RyR densities below the baseline value.

Figure 6 depicts the effect of downgrading the RyR calcium transport rate on wave speed and amplitude.

Fig. 6.

Wave propagation speed and amplitude as a function of the scaling of the transport rate of calcium through open RyR channels. All other model parameters are at baseline values. The vertical arrow indicates the scaling for which the unitary calcium current through an RyR channel would match that through an InP3R channel

This illustrates that even when RyR functionality is significantly downgraded, the presence of these receptors can still have a significant effect on wave characteristics. When the unitary RyR calcium current is scaled to the same value as that for an InP3R, the presence of RyRs results in a 25% increase in wave speed and a 36% increase in amplitude – even though in the baseline configuration they are outnumbered twenty-five to one by InP3Rs. This outsized effect reflects the fact that the receptor is not inhibited by elevated cytoplasmic calcium.

Results from experiments in which the remaining parameters were varied are summarized in Fig. 7. Each parameter is normalized with respect to its baseline value; thus the slopes of each graph represent the relative sensitivity of the wave speed or amplitude to the varied parameter. For reference, the dependence on Ce_i is also plotted in the same format, to give an idea of scale. In the vicinity of the baseline configurations, sensitivity to Ce_i exceeds that for any of the remaining parameters.

Those parameters include:

• r_lx and r_le, the ER and plasma membrane leakage flux parameters, respectively. Ce_i is controlled in the first case but not in the second (since ER membrane leakage is dominated by the InP3 channels). Sensitivity to both is low and of opposite sign. Not depicted is the fact that the process destabilizes for values of r_lx slightly greater than four times the reference value.

• D_Cac, D_Cae, and D_InP3, the diffusion coefficients for cytoplasmic calcium, ER calcium, and cytoplasmic InP3, respectively. There is a pronounced positive sensitivity of wave speed to D_Cac both with and without RyRs present, although when they are absent this dependence decreases rapidly as D_Cac increases and other factors (such as inhibition of InP3Rs) limit the speed. Amplitudes are likewise relatively sensitive, but more so at lower values. Sensitivity to calcium diffusivity in the ER is quite low in all cases, and negative (possibly reflecting the increased spread of local calcium depletion in the ER). Sensitivity to the diffusivity of InP3 is weak when RyRs are present (since they do not depend on InP3), but when they are absent it is fairly pronounced for both speed and amplitude, particularly at lower values.

• R_{p_PM}, a plasma membrane pump scaling factor by which both plasma membrane pump rates are covaried, and R_pS, the SERCA pump rate. Sensitivities to both are low to moderate in all cases, and negative since all pumps remove calcium from the cytoplasm.

• r_mIf, the calcium-dependent production rate of InP3. Sensitivity of both speed and amplitude to this parameter is relatively low when RyRs are present (again since they dominate the wave characteristics), but relatively pronounced when they are absent.

• r_Ig, the InP3R configurational change rate that governs the gating variable dynamics. Sensitivity to this parameter is weak in all cases, except when it is small and RyRs are not present.

Not included in Fig. 7 are results for parameters related to PKC-mediated inhibition of InP3 production. A cursory examination of the model taken from Kang and Othmer (2009) shows that even if all available PKC were activated, it would only result in a reduction of about 10% in the rate of InP3 production – and thus would be expected to have minimal effect on calcium transients. Indeed, sensitivities of wave speed or amplitude to the parameters k_iPf (the activation rate for PKC), PKC₀ (the total PKC concentration), or k_iPi (scaling of the inverse modulation of InP3 production), are found to be small. An increase in any one of these by a factor of ten results in a decrease in speed or magnitude by less than 3% from baseline results, in processes with InP3s only. It is only when two or more are substantially increased that a significant effect is observed. Thus we conclude that if this model is to explain reported effects of PKC on calcium dynamics (e.g., Uchino et al., 2004), its parameterization must be significantly altered.

Damped waves are observed when certain parameters – including InP3R density, cytosolic calcium and InP3 diffusion coefficients, and quiescent/initial ER calcium concentration – assume sufficiently small values (several such cases are noted in the figures). When any of those parameters is reduced to very small values, as might be expected, there can be a complete failure of wave propagation. Damped waves were observed primarily in processes without RyRs. In addition to varying the relevant parameters with the process diameter fixed at baseline, we performed simulations to identify transition from regenerative to damped waves in processes with larger diameters. These indicated that such transitions occur at progressively greater parameter values as the process diameter increases. This suggests it is possible for regenerative waves in small-diameter peripheral processes to die out when they reach larger, more central segments, even with the same parameters in both regions.

Traveling waves in processes with branches, tapers, and non-uniform receptor distribution

For the tapered process, average speed in the tapered segment was computed for waves traveling in both directions (‘centripetal’ indicating from small diameter to large, and ‘centrifugal’, vice-versa). Three different taper values, 3.33%, 5%, and 7.5%, were evaluated at the baseline parameter set. Results were compared to the average speed computed from fixed-diameter experiments, computed by numerical integration of the curves shown in Fig. 4. We found that mean centripetal speeds were smaller than centrifugal when RyRs were present along with InP3Rs, but the opposite was true when only InP3Rs were present. Results are presented in Table 4.

Table 4.

Mean propagation speed results from tapered dendrites. “Predicted Speed” is the value estimated from fixed-radius process model results

Receptor	Taper	Centripetal speed (μm/s)	Centrifugal speed (μm/s)	Predicted speed (μm/s)
Both	3.33%	172.3	192.5	188.6
Both	5%	168.4	196.7
Both	7.50%	163.9	212.0
InP3R only	3.33%	50.44	47.88	52.38
InP3R only	5%	55.05	49.71
InP3R only	7.50%	62.42	52.44

The presence of a branch in a cellular process was observed to affect the transmission time of a centripetal traveling calcium wave that traverses the branch. According to the criterion described in Sect. 2.4.1, we found that if just one of the two distal segments meeting at the branch is activated, a net delay from + 5 ms for distal process radius = 0.15 μm, to + 24 ms for radius = 0.5 μm, is induced. For larger-diameter processes, this delay increases rapidly; for example at r_d = 0.833 μm, it is + 144 ms. If both distal segments are simultaneously activated, rather than a delay, there is a net advance for smaller radii (e.g., –15 ms for radius = 0.5 μm); but as radius increases, the effect transitions to a delay, (e.g. + 10 ms at r_d = 0.833 μm).

Additional simulations with tapered processes confirm, as suggested in the prior section, that it is possible for a regenerative wave in the distal portion to become damped and die out in the thicker proximal part of the process.

We expected a priori that the effect of heterogeneous receptor distributions on wave propagation in a cellular process can be largely inferred from the receptor density-dependence seen in uniform processes (as illustrated in Fig. 5). To verify this, we performed a limited number of simulations of processes with non-uniform receptor densities. We found that variations in wave speed and amplitude in such processes were generally predictable based on the results from the uniform-density model (although complex calcium profiles could evolve at boundaries between regions with significantly different densities). Damped waves propagating in low-density regions were found capable of initiating fully-regenerative waves on transition to high-density regions, so long as forward wave propagation prevailed when the wave reached the boundary. We did also find, however, that regions entirely devoid of receptors could put a stop to wave propagation even when as short as a few micrometers in axial extent. Movie M4 in the Supplementary Material illustrates some characteristic responses in a process with variable InP3R density.

Traveling waves in a cell body model

The cell body model is intended to convey some idea of calcium wave propagation through a structure with morphology roughly comparable to a real cell such as an astrocyte, with an ER of high surface-area-to-volume ratio (and only InP3Rs), and with attached cellular processes. For baseline parameters, we found such waves can be triggered by an incoming wave in a single attached process, and they propagate through the body to the roots of each of the attached unstimulated processes, in which they may trigger outgoing waves. Figure 8 shows a snapshot of a wave (with calcium levels indicated by a colormap) during propagation in a cell body with six attached processes, in a top / plan view of the structure, along with several views of waveforms as functions of time. These waveforms, which are at elements bounding the process roots, show complex dynamics with a multimodal shape. Movie M3 in the Supplementary Material shows the time history of wave propagation throughout this structure with the baseline parameters.

Fig. 8.

Illustrations of calcium wave propagation in the cell body model. Panel a shows a snapshot of a propagating wave in the plan view, color-mapped with lighter colors indicating greater concentrations. Panel b depicts calcium waves at the outermost root elements as a function of time relative to wave initiation in process #1. The vertical line at the peak of the wave reaching root #1 is the time reference for computing the delays (and from those the speeds) of waves reaching the other roots, also depicted. Panel c depicts waveforms as function of time, with InP3R areal density as a parameter

Wave propagation depends on model parameters

The parameter dependence and features of calcium waves in the cell body model were found to be qualitatively consistent with the analogous waves described in greater detail for the process model. The wave speed (computed according to the procedure described in Sect. 2.4.2) at the baseline parameters is approximately 45 μm/s. Figure 9 depicts dependence of wave speed on various non-morphometric parameters, in the same format as in Fig. 7. Wave amplitudes in the internal regions of the cell body were found to be on the order of 4 μM – consistent with waves in processes of radius on the order of 0.5 μm, but greater than in the process segments that attach to and convey waves to the cell body. This reflects the assumed high surface area-to-volume ratio of the ER, and its proximity to all cytoplasmic volume elements.

Fig. 9.

Speed of calcium wave propagation in the cell body model (as defined in the text) as a function of selected model parameters, as indicated in the legend. Speeds are calculated based on propagation times from the root of the stimulated process to a the most distal unstimulated process; b the intermediate unstimulated process; and c the adjacent unstimulated process

As in the process model, unstable dynamics, and instances of damped and abortive waves could be observed for certain ranges of analogous critical parameters.

Discussion

This work examines the dependence of traveling calcium wave characteristics – with primary focus on propagation speed and amplitude – as a function of various parameters that characterize the morphology of elongate cellular processes, and parameters that characterize the receptors, channels, and pumps that contribute to those wave properties, in both processes and a cell body model. We also note that these systems can undergo a Hopf bifurcation and destabilize, with oscillations consisting of periodic calcium spikes. (Interestingly, traveling waves are possible in unstable processes during phases of the oscillations when ER calcium levels are high.)

In our cellular processes model, we find that traveling wave solutions are supported by the presence of InP3Rs, RyRs, or both simultaneously. These waves propagate axially along the process. Wave speeds of tens to on the order of 100 μm/s, and amplitudes in the range of one to six μM, are realized with InP3 receptors alone and with physiological levels of store calcium in the model ER. With the addition of fully functional RyR receptors (or with such receptors alone), wave speeds of several hundred μm/s and amplitudes of tens of μM are possible.

In considering how these figures compare with empirical results, we note that in general, calcium waves in astrocytes are regarded as mediated primarily by InP3Rs, and in neurons primarily by RyRs. Reported wave speeds in astrocytes tend to be in the range of less than 10 μm/s to as high as 40 μm/s (Dani et al., 1992; Fiacco & McCarthy, 2006; Newman, 2001; Scemes, 2000; Scemes & Giaume, 2006; Venance et al., 1997; Yagodin et al., 1994). The values in this range are smaller than those we observed in model processes with InP3s only – but almost all reported results are for intercellular waves and thus affected by delays in intercellular transmission. For neurons, calcium wave speeds on the order of 100 μm/s to 200 μm/s are reported (Charles et al., 1996; Ross, 2015), consistent with our results when RyRs are present. Our results are likewise roughly consistent with results from other modeling studies (e.g., Höfer et al., 2002, and Kang & Othmer, 2009, for astrocytes; Breit and Queisser, 2018, for neurons).

Further results of interest include the finding that initial/quiescent calcium stores strongly affect not just the amplitude but the speed of traveling waves in the cellular process model. The amplitude dependence is perhaps intuitively understandable, given that calcium flux is proportional to the store-to-cytoplasm concentration difference, but the speed dependence, although noted by others (e.g., Sneyd et al., 1993), is less so. With only InP3Rs present, a doubling of ER calcium concentration results in roughly a doubling of both amplitude and wave speed. When RyRs are present, the dependence is even more marked, with an increase by a factor of nearly three with doubling of ER calcium – perhaps less surprising since the wave is terminated primarily by ER calcium depletion. The dependence on process radius (which also determines the radius of the smooth ER in the cylindrical model) is also notable, with a decreasing dependence of both speed and amplitude when only InP3Rs present. This is presumably due to the increased radial diffusion distance from ER to plasma membranes for both Ca²⁺ and InP3. When RyRs are also present, amplitude likewise shows an inverse dependence, but the speed dependence is nonmonotone due to a decreased axial separation between individual RyRs as diameter (and thus surface area) of the ER increases. This affects calcium-induced calcium release simply by decreasing the diffusion distance. Because of the dependence on available calcium, relative ER volume has an influence as well. In tapered processes, speed decreases as a wave propagates from smaller- to larger-diameter regions (with the opposite effect for propagation in the opposite direction). The ranges of wave speeds mentioned in the previous paragraph also apply to the average speeds observed in tapered processes with radii varying from 150 nm to 1.5 μm.

Non-morphological parameters that strongly influence wave characteristics include InP3R density in the ER membrane (primarily when functional RyRs are not present) and RyR receptor density, as might be expected. A moderately strong influence is exerted by the calcium diffusion coefficient, and (when only InP3Rs are present), the InP3 diffusion coefficient, in the cytoplasm. Other parameters exert a lesser influence on wave characteristics per se, although they are instrumental in setting the quiescent calcium and InP3 levels in the cytoplasm.

Destabilization of the model process can occur with increasing plasma membrane leakage of calcium, or (at the baseline level of plasma membrane leakage) high InP3R densities, and high levels of initial/quiescent ER calcium. Although instability was not observed with varying RyR densities, these were quite low in all cases, and we expect that instability would likely be possible if they were sufficiently high. Waves in stable processes were fully regenerative for significant ranges of the model parameters. However, damped waves with various rates of decay, and abortive waves that failed to propagate from the initiation site, were observed with sufficiently low values of receptor densities, initial / quiescent ER calcium, and calcium diffusion coefficient in the cytoplasm, consistent with results from a prior modeling study (Guisoni et al., 2015).

Of particular interest with respect to calcium waves in astrocytes are the results of simulations in which the calcium transport rate of RyRs was downgraded. RyRs are believed to be present in these cells but their degree of functionality is questionable. Simulations with significantly reduced transport rates show that sparsely-distributed RyRs can still enhance wave speeds and amplitudes relative to processes with InP3Rs alone, due to the fact that the RyRs are not inhibited by physiological levels of cytoplasmic calcium.

In several of our suites of simulations, we artificially controlled initial/quiescent ER calcium Ce_i in order to separate the effects of calcium transport by various entities during wave propagation, from their effects in determining Ce_i. This was a particular issue with the InP3R kinetic model that we used (Cao et al., 2013; Siekmann et al., 2012), which was formulated to explain InP3R behavior in particular in vitro experiments. We speculate that inactive InP3Rs in vivo are likely to display less pronounced calcium leakage, if only based on energetic considerations. If this is the case, physiological levels of ER calcium could be maintained with a lower density of SERCA pumps than used in our simulations. These could have lower calcium affinities (more in line with reported values) than those we used to avoid stability issues induced by calcium leakage with artificial Ce_i control – and this would likely result in modest increases in wave speeds relative to the values we have reported.

Our cell body model was intended to allow examination of wave propagation in a structure with a representative morphology, and folded ER of high surface-area-to-volume ratio, with only InP3Rs present in the ER membrane. Calcium waves initiated in a cell body by a wave incoming from an attached process are found to propagate largely in the circumferential direction (around the excluded nuclear area), but with some radial component. With the baseline parameters, velocities around the outer circumference are on the order of tens of μm/s, and wave amplitudes are on the order of 4 μM, similar to those seen in a process with radius 0.5 μm.

Although we have not considered natural mechanisms of initiation or intercellular transmission of traveling calcium waves, our results nevertheless suggest that such waves could be a useful adjunct for spatiotemporal neural processing for temporal sequences with time scales on the order of hundreds of milliseconds to seconds, if the underlying brain structures have spatial extents on the order of 100 μm to millimeters.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Appendix 1 Model morphologies

The morphologies of the models used are illustrated in Fig. 10.

Implementation of the folded ER morphology in the cell body is implicit, with ER membrane assumed to be adjacent to each cytoplasmic volume element in the structure. (This effectively ignores the volume occupied by the ER itself.) Fig. 10 Panel e gives a schematic representation of the ER arrangement, to convey the general idea. The process roots are assumed to have higher ER surface-area-to-volume ratio than the smooth cylindrical process model, although the specific morphology is likewise not explicitly given.

Appendix 2 Governing Equations

The governing equations for the models are as follows. The rate of change of concentration d[Z]/dt of a particular chemical species Z is given individually for each transport mechanism or reaction associated with a compartment, with the understanding that all such components are summed to yield the total rate of change of [Z]. Parameters other than receptor kinetic constants are described in Table 1; kinetics parameters for InP3 and RyR receptors are given respectively in Tables 5 and 6. Derived parameters are described in the text.

Table 5.

Kinetic parameters of InP3R model. The column “Original Value” refers to parameter values given in Siekmann et al. (2012)

Parameter	Value	Units	Original Value?
rI12	1240	s−1	Y
rI21	88	s−1	Y
rI26	10,500	s−1	Y
rI62	4010	s−1	Y
rI45	2.2	s−1	N
rI54	3330	s−1	Y
rI34	0.05	s−1	N
rI43	1000	s−1	N
rI42	50	s−1	N
rI24	400	s−1	N
rIG	70	s−1	N
km	0.08	μM	N
kp	3.5	μM	N
kbh	0.5	μM	N

Table 6.

Kinetic parameters of RyR model. The column “Original Value” refers to parameter values given in Breit and Quiesser (2018)

Parameter	Value	Units	Original Value?
rR31	28.2	s−1	Y
rR43	385.9	s−1	Y
rR23	0.1	s−1	Y
rR32	1.75	s−1	Y
kR13	150	s−1.μM−4	N
kR34	1.75	s−1.μM−3	N
rsat34	4166	s−1	N
rsat13	12,610	s−1	N
Ccsat	3.028	μM	N

The kinetics of the inositol trisphosphate receptor model involve four ‘closed’ and two ‘open’ states. The kinetic equations, modified from Siekmann et al. (2012), are

$$\frac{\mathit{d}{\mathit{C}}_{\mathit{I}1}}{\mathit{d}\mathit{t}}={\mathit{r}}_{\mathit{I}21}\bullet {\mathit{C}}_{\mathit{I}2}-{\mathit{r}}_{\mathit{I}12}\bullet {\mathit{C}}_{\mathit{I}1}$$

$$\frac{\mathit{d}{\mathit{C}}_{\mathit{I}2}}{\mathit{d}\mathit{t}}={\mathit{r}}_{\mathit{I}12}\bullet {\mathit{C}}_{\mathit{I}1}+{\mathit{\varphi }}_{\mathit{I}42}\bullet {\mathit{C}}_{\mathit{I}4}+{\mathit{r}}_{\mathit{I}62}\bullet {\mathit{O}}_{\mathit{I}6}-{(\mathit{r}}_{\mathit{I}21}+{\mathit{\varphi }}_{\mathit{I}24}+{\mathit{r}}_{\mathit{I}26})\bullet {\mathit{C}}_{\mathit{I}2}$$

$$\frac{\mathit{d}{\mathit{C}}_{\mathit{I}3}}{\mathit{d}\mathit{t}}={\mathit{\varphi }}_{\mathit{I}43}\bullet {\mathit{C}}_{\mathit{I}4}-{\mathit{r}}_{\mathit{I}34}\bullet {\mathit{C}}_{\mathit{I}3}$$

$$\frac{\mathit{d}{\mathit{C}}_{\mathit{I}4}}{\mathit{d}\mathit{t}}={{{\mathit{\varphi }}_{\mathit{I}24}\bullet {\mathit{C}}_{\mathit{I}2}+\mathit{r}}_{\mathit{I}34}\bullet {\mathit{C}}_{\mathit{I}3}+{\mathit{r}}_{\mathit{I}54}\bullet {\mathit{O}}_{\mathit{I}5}-(\mathit{\varphi }}_{\mathit{I}42}+{\mathit{\varphi }}_{\mathit{I}43}+{\mathit{r}}_{\mathit{I}45})\bullet {\mathit{C}}_{\mathit{I}4}$$

$$\frac{\mathit{d}{\mathit{O}}_{\mathit{I}5}}{\mathit{d}\mathit{t}}={\mathit{r}}_{\mathit{I}45}\bullet {\mathit{C}}_{\mathit{I}4}-{\mathit{r}}_{\mathit{I}54}\bullet {\mathit{O}}_{\mathit{I}5}$$

$$\frac{\mathit{d}{\mathit{O}}_{\mathit{I}6}}{\mathit{d}\mathit{t}}={\mathit{r}}_{\mathit{I}26}\bullet {\mathit{C}}_{\mathit{I}2}-{\mathit{r}}_{\mathit{I}62}\bullet {\mathit{O}}_{\mathit{I}6},$$

where the symbol C is used to designate closed state probabilities, and O open states. After Siekmann et al. (2012), the rate functions designated with the symbol f are based on concentration-dependent gating variables m_I, h_I, and m_Ip. The dependence of the first two also includes single-pole dynamics. The forms selected for our simplified model are

$$\left({\mathit{r}}_{\mathit{I}\mathit{G}},+,\frac{\mathit{d}}{\mathit{d}\mathit{t}}\right){\mathit{m}}_{\mathit{I}}=\frac{{\mathit{r}}_{\mathit{I}\mathit{G}}\bullet {\left[{\mathit{C}\mathit{a}}^{2+}\right]}^6}{{\left[{\mathit{C}\mathit{a}}^{2+}\right]}^6+{{\mathit{k}}_{\mathit{m}}}^6}$$

$$\left({\mathit{r}}_{\mathit{I}\mathit{G}},+,\frac{\mathit{d}}{\mathit{d}\mathit{t}}\right){\mathit{h}}_{\mathit{I}}=\frac{{\mathit{r}}_{\mathit{I}\mathit{G}}\bullet {{\mathit{k}}_{\mathit{h}}}^6}{{\left[{\mathit{C}\mathit{a}}^{2+}\right]}^6+{{\mathit{k}}_{\mathit{h}}}^6}$$

$${\mathit{m}}_{\mathit{I}\mathit{p}}=\frac{{\left[\mathit{I},\mathit{n},\mathit{P},3\right]}^3}{{\left[\mathit{I},\mathit{n},\mathit{P},3\right]}^3+{{\mathit{k}}_{\mathit{b}\mathit{p}}}^3},$$

where the concentrations [Ca²⁺] and [InP3] are those prevailing on the cytoplasmic side of the ER membrane, and r_IG is a rate constant. Based on these variables, the f functions are

$${\mathit{\varphi }}_{\mathit{I}24}={\mathit{r}}_{\mathit{I}24}\bullet \left(1,-,{\mathit{m}}_{\mathit{I}},\bullet ,{\mathit{h}}_{\mathit{I}},\bullet ,{\mathit{m}}_{\mathit{I}\mathit{p}}\right)$$

$${\mathit{\varphi }}_{\mathit{I}42}={\mathit{r}}_{\mathit{I}42}\bullet {\mathit{m}}_{\mathit{I}}\bullet {\mathit{h}}_{\mathit{I}}\bullet {\mathit{m}}_{\mathit{I}\mathit{p}}$$

$${\mathit{\varphi }}_{\mathit{I}43}={\mathit{r}}_{\mathit{I}43}\bullet \left(1,-,{\mathit{h}}_{\mathit{I}}\right).$$

The rate of increase of cytoplasmic [Ca²⁺] via open InP3R channels in an ER-adjacent volume element is modeled as simply proportional to the total open probability:

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}={\mathit{k}}_{\mathit{r}\mathit{I}}\bullet ({\mathit{O}}_{\mathit{I}5}+{\mathit{O}}_{\mathit{I}6})\bullet (\left[{\mathit{C}\mathit{a}}^{2+},{]}_{\mathit{E}},-,\left[{\mathit{C}\mathit{a}}^{2+}\right]\right),$$

where [Ca²⁺]_E is the local ER calcium concentration, and the constant

$${\mathit{k}}_{\mathit{r}\mathit{I}}=\left(\frac{{\mathit{A}}_{\mathit{E}\mathit{M}}}{\mathit{V}}\right)\bullet {\mathit{R}}_{\mathit{r}\mathit{I}}{\mathit{I}}_{\mathit{r}\mathit{I}},$$

where (A_EM/V) is the ER membrane area-to-volume ratio of the element, and the computation may or may not involve rounding as indicated in the text.

The rate of increase of [InP3] in a PM-adjacent volume element is

$$\frac{\mathit{d}[\mathit{I}\mathit{n}\mathit{P}3{]}^{+}}{\mathit{d}\mathit{t}}=\left(\frac{{\mathit{A}}_{\mathit{P}\mathit{M}}}{\mathit{V}}\right)\bullet \frac{{\mathit{r}}_{\mathit{m}\mathit{I}\mathit{f}}\bullet \left[{\mathit{C}\mathit{a}}^{2+}\right]}{(1+{\mathit{k}}_{\mathit{i}\mathit{P}\mathit{I}}\bullet [\mathit{C}\mathit{a}.\mathit{P}\mathit{K}\mathit{C}])},$$

where Ca.PKC indicates activated (bound) PKC, and (A_PM/V) is the plasma membrane area-to-volume ratio of the element. The degradation rate of [InP3] in any cytoplasmic volume is

$$\frac{\mathit{d}[\mathit{I}\mathit{n}\mathit{P}3{]}^{-}}{\mathit{d}\mathit{t}}=-{\mathit{k}}_{\mathit{m}\mathit{I}\mathit{b}}\bullet \left[\mathit{I},\mathit{n},\mathit{P},3\right].$$

Activation and deactivation of PKC in a PM-adjacent volume element are governed by

$$\frac{\mathit{d}[\mathit{C}\mathit{a}.\mathit{P}\mathit{K}\mathit{C}]}{\mathit{d}\mathit{t}}={\mathit{k}}_{\mathit{i}\mathit{P}\mathit{f}}\bullet \left[{\mathit{C}\mathit{a}}^{2+}\right]\bullet ({\mathit{P}\mathit{K}\mathit{C}}_0-\left[\mathit{C},\mathit{a},.,\mathit{P},\mathit{K},\mathit{C}\right])-{\mathit{k}}_{\mathit{i}\mathit{P}\mathit{b}}\bullet \left[\mathit{C},\mathit{a},.,\mathit{P},\mathit{K},\mathit{C}\right],$$

where PKC₀ is the total available PKC concentration.

The kinetics of the ryanodine receptor model involve four ‘closed’ and two ‘open’ states. The kinetic equations, adapted from Breit and Queisser (2018), are

$$\frac{\mathit{d}{\mathit{C}}_{\mathit{R}1}}{\mathit{d}\mathit{t}}={\mathit{r}}_{\mathit{R}31}\bullet {\mathit{O}}_{\mathit{R}3}-{\mathit{\varphi }}_{\mathit{R}13}\bullet {\mathit{C}}_{\mathit{R}1}$$

$$\frac{\mathit{d}{\mathit{C}}_{\mathit{R}2}}{\mathit{d}\mathit{t}}={\mathit{r}}_{\mathit{R}32}\bullet {\mathit{O}}_{\mathit{R}3}-{\mathit{r}}_{\mathit{R}23}\bullet {\mathit{C}}_{\mathit{R}2}$$

$$\frac{\mathit{d}{\mathit{O}}_{\mathit{R}3}}{\mathit{d}\mathit{t}}={\mathit{\varphi }}_{\mathit{I}13}\bullet {\mathit{C}}_{\mathit{R}1}+{\mathit{r}}_{\mathit{R}23}\bullet {\mathit{C}}_{\mathit{R}2}+{\mathit{r}}_{\mathit{R}43}\bullet {\mathit{O}}_{\mathit{R}4}-\left({\mathit{r}}_{\mathit{R}31},+,{\mathit{r}}_{\mathit{R}32},+,{\mathit{\varphi }}_{\mathit{R}34}\right)\bullet {\mathit{O}}_{\mathit{R}3}$$

$$\frac{\mathit{d}{\mathit{O}}_{\mathit{R}4}}{\mathit{d}\mathit{t}}={\mathit{\varphi }}_{\mathit{R}34}\bullet {\mathit{O}}_{\mathit{R}3}-{\mathit{r}}_{\mathit{R}43}\bullet {\mathit{O}}_{\mathit{R}4}.$$

In this model, the rate functions f_R13 and f_R34 have respectively fourth- and third-order dependence on [Ca²⁺]. We retain this dependence at lower [Ca²⁺] levels, but subject these functions to saturation at higher [Ca²⁺] (i.e., levels at which the functions dominate the constant rates in the equations):

$${\mathit{\varphi }}_{\mathit{R}13}=\frac{{\mathit{k}}_{\mathit{R}13}\bullet {\left[{\mathit{C}\mathit{a}}^{2+}\right]}^4\bullet {\mathit{r}}_{\mathit{s}\mathit{a}\mathit{t}13}}{{{\mathit{k}}_{\mathit{R}13}\bullet \left[{\mathit{C}\mathit{a}}^{2+}\right]}^4+{\mathit{r}}_{\mathit{s}\mathit{a}\mathit{t}13}}$$

$${\mathit{\varphi }}_{\mathit{R}34}=\frac{{\mathit{k}}_{\mathit{R}34}\bullet {\left[{\mathit{C}\mathit{a}}^{2+}\right]}^3\bullet {\mathit{r}}_{\mathit{s}\mathit{a}\mathit{t}34}}{{{\mathit{k}}_{\mathit{R}34}\bullet \left[{\mathit{C}\mathit{a}}^{2+}\right]}^3+{\mathit{r}}_{\mathit{s}\mathit{a}\mathit{t}34}}.$$

The rate of change of cytoplasmic [Ca²⁺] through open RyR channels in an ER-adjacent volume element, where the ER membrane contains an RyR, is

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}={\mathit{k}}_{\mathit{R}\mathit{r}}\bullet ({\mathit{O}}_{\mathit{R}3}+{\mathit{O}}_{\mathit{R}4})\bullet (\left[{\mathit{C}\mathit{a}}^{2+},{]}_{\mathit{E}},-,\left[{\mathit{C}\mathit{a}}^{2+}\right]\right).$$

Because RyRs do not occur at a frequency of more than one per cytoplasmic compartment in any configuration of the process model, the constant k_Rr is defined as

$${\mathit{k}}_{\mathit{R}\mathit{r}}=\frac{{\mathit{I}}_{\mathit{R}\mathit{r}}}{\mathit{V}},$$

and the axial period of occurrence of RyRs is

$${\mathit{M}}_{\mathit{r}\mathit{R}}=\mathbf{r}\mathbf{o}\mathbf{u}\mathbf{n}\mathbf{d}\left(1,/,(,{\mathit{A}}_{\mathit{E}\mathit{M}},\bullet ,{\mathit{R}}_{\mathit{r}\mathit{R}},)\right),$$

where ‘round’ indicates rounding to the nearest integer.

The plasma membrane pumps clear calcium from PM-adjacent volume elements. NCX pump clearance is modeled with

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}={-\mathit{k}}_{\mathit{p}\mathit{N}}\bullet \frac{[{\mathit{C}\mathit{a}}^{2+}]}{{\mathit{k}}_{\mathit{p}\mathit{N}\mathit{C}}+[{\mathit{C}\mathit{a}}^{2+}]},$$

where

$${\mathit{k}}_{\mathit{p}\mathit{N}}=\left(\frac{{\mathit{A}}_{\mathit{P}\mathit{M}}}{\mathit{V}}\right)\bullet {\mathit{R}}_{\mathit{p}\mathit{N}}{\mathit{I}}_{\mathit{p}\mathit{N}},$$

and the computation involves rounding the number of pump units per compartment, as indicated in the text.

PMCA calcium clearance is modeled by

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}={-\mathit{k}}_{\mathit{p}\mathit{P}}\bullet \frac{{[{\mathit{C}\mathit{a}}^{2+}]}^2}{{{\mathit{k}}_{\mathit{p}\mathit{P}\mathit{C}}}^2+{[{\mathit{C}\mathit{a}}^{2+}]}^2},$$

where

$${\mathit{k}}_{\mathit{p}\mathit{P}}=\left(\frac{{\mathit{A}}_{\mathit{P}\mathit{M}}}{\mathit{V}}\right)\bullet {\mathit{R}}_{\mathit{p}\mathit{P}}{\mathit{I}}_{\mathit{p}\mathit{P}}.$$

SERCA pumps clear calcium from ER-adjacent volume elements. SERCA clearance is modeled by

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}={-\mathit{k}}_{\mathit{p}\mathit{S}}\bullet \frac{[{\mathit{C}\mathit{a}}^{2+}]}{{(\mathit{k}}_{\mathit{p}\mathit{S}\mathit{C}}+[{\mathit{C}\mathit{a}}^{2+}])\bullet [{\mathit{C}\mathit{a}}^{2+}{]}_{\mathit{E}}},$$

where

$${\mathit{k}}_{\mathit{p}\mathit{S}}=\left(\frac{{\mathit{A}}_{\mathit{E}\mathit{M}}}{\mathit{V}}\right)\bullet {\mathit{R}}_{\mathit{p}\mathit{S}}{\mathit{I}}_{\mathit{p}\mathit{S}}.$$

Calcium leakage from the ER into an ER-adjacent cytoplasmic compartment is modeled by

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}=\left(\frac{{\mathit{A}}_{\mathit{E}\mathit{M}}}{\mathit{V}}\right)\bullet {\mathit{r}}_{\mathit{l}\mathit{e}}\bullet (\left[{\mathit{C}\mathit{a}}^{2+},{]}_{\mathit{E}},-,\left[{\mathit{C}\mathit{a}}^{2+}\right]\right),$$

whereas leakage from the extracellular space into a PM-adjacent compartment by

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}=\left(\frac{{\mathit{A}}_{\mathit{P}\mathit{M}}}{\mathit{V}}\right)\bullet {\mathit{r}}_{\mathit{l}\mathit{x}}.$$

Buffering of calcium in the cytoplasm is modeled by

$$\frac{\mathit{d}[{\mathit{C}\mathit{a}}^{2+}]}{\mathit{d}\mathit{t}}={{\mathit{k}}_{\mathit{b}\mathit{C}\mathit{b}}\bullet \left[\mathit{C},\mathit{a},.,\mathit{C},\mathit{a},\mathit{l},\mathit{B}\right]-\mathit{k}}_{\mathit{b}\mathit{C}\mathit{f}}\bullet \left[{\mathit{C}\mathit{a}}^{2+}\right]\bullet \left({\mathit{C}\mathit{a}\mathit{l}\mathit{B}}_0,-,\left[\mathit{C},\mathit{a},.,\mathit{C},\mathit{a},\mathit{l},\mathit{B}\right]\right),$$

where CalB₀ is the total available concentration of the buffering agent (i.e., calbindin), [Ca.CalB] is the concentration of bound calcium, and which equation also implicitly describes the dynamics of free and bound calbindin.

The calcium flux supplied to a cytoplasmic volume element from an adjacent ER volume element must be subtracted from the store calcium in that ER volume element. This is accomplished by setting

$$\frac{\mathit{d}{[{\mathit{C}\mathit{a}}^{2+}]}_{\mathit{E}\mathit{R}}}{\mathit{d}\mathit{t}}=-\frac{1}{{\mathit{v}\mathit{o}\mathit{l}}_{\mathit{R}}}\bullet \frac{\mathit{d}{\left[{\mathit{C}\mathit{a}}^{2+}\right]}_{\mathit{c}\mathit{y}\mathit{t}\mathit{o}}}{\mathit{d}\mathit{t}},$$

where the subscripts ER and cyto refer respectively to the ER and cytoplasmic volume elements. When multiple cytoplasmic elements border an ER volume element around its circumference, their rates of change of [Ca²⁺] are summed on the right-hand side of this equation in order to compute d[Ca²⁺]_ER/dt.

Overall, the spatiotemporal dynamics of calcium and InP3 are modeled with the diffusion equation with a source term,

$$\frac{\partial [\mathit{Z}]}{\partial \mathit{t}}={{\mathit{D}}_{\mathit{Z}}\bullet \mathrm{\nabla }}^2\left[\mathit{Z}\right]+{\mathit{f}}_{\mathit{Z}},$$

where [Z] is the concentration of the diffusant (Ca²⁺ or InP3), D_Z is its diffusion coefficient in either cytosol or the ER, ∇² is the Laplacian operator, and f_Z is a source term that sums the molar rate of change of the diffusant due to all local transmembrane fluxes and reactions described above, in the compartment of interest. The Laplacian is computed in the cylindrical coordinates, with appropriate boundary conditions for dimensions other than circumferential.

Author contributions

PAS defined the models, wrote the basic scripts to implement the models, conducted all experiments on the cellular process model, and co-wrote the manuscript; BMBB wrote the framework code to control simulations, conducted all experiments on the cell body model, and co-wrote the manuscript.

Funding

This work was supported in part by the United States Air Force Office of Scientific Research (grants FA9550–16–1–0153 and FA9550–23–1–0714), and in part by the Swedish Research Council (grant VR 2018–03452).

Data availability

The raw data and software supporting the conclusions of this article are available via GitHub at https://github.com/pashoe4/Shoemaker-Bekkouche-code-data.

Declarations

Competing Interests

The authors declare that they have no competing interests with regard to this work.