^{2+}microdomains within tubular-sarcoplasmic reticular junctions of amphibian skeletal muscle

A finite element analysis modelled diffusional generation of steady-state Ca^{2+} microdomains within skeletal muscle transverse (T)-tubular-sarcoplasmic reticular (SR) junctions, sites of ryanodine receptor (RyR)-mediated SR Ca^{2+} release. It used established quantifications of sarcomere and T-SR anatomy (radial diameter ^{2+} influx densities,^{2+} signals following muscle fibre depolarization. ^{2+}], [Ca^{2+}]_{edge,} matched reported corresponding ^{2+}] elevations given diffusional boundary ^{2+} diffusion coefficients ^{2+}]_{edge} upon ^{2+}] upon Ca^{2+} release through their entire test voltage range. The resulting model consistently predicted elevated steady-state T-SR junctional ~ µM-[Ca^{2+}] elevations radially declining from maxima at the T-SR junction centre along the entire axial T-SR distance. These [Ca^{2+}] heterogeneities persisted through 10^{4}- and fivefold, variations in ^{2+}] values, whilst preserving the flux conservation (^{2+}] T-SR microdomains that could regulate cytosolic and membrane signalling molecules including calmodulin and RyR, These findings directly fulfil recent experimental predictions invoking such Ca^{2+} microdomains in observed regulatory effects upon Na^{+} channel function, in a mechanism potentially occurring in similar restricted intracellular spaces in other cell types.

Intracellular endoplasmic or sarcoplasmic reticular (SR) membrane systems gating store Ca^{2+} release into the cytosol following surface membrane activation, often involving ryanodine receptor (RyR) activation, occur widely amongst cell types. These include both excitable (skeletal, cardiac and smooth muscle, and cerebellar Purkinje^{1–3}, hippocampal^{4} and other central nervous system neurones^{5,6}) and non-excitable, including thrombocyte, cell types^{7,8}. These intracellular membranes often form appositions with surface membrane with proximities (< 10–30 nm) permitting direct protein–protein/lipid interaction^{9,10} though not accommodating entire organelles. Their intervening electron-dense cytosol could also reflect local concentrations of proteins, lipids or ions. Electron microscopic sections can reveal parallel alignments extending over ~ 100–400 nm distances without fusion of the component membranes potentially offering restricted diffusion spaces permitting ion, including Ca^{2+}, accumulation and microdomain formation.

In skeletal and cardiac muscle, following surface membrane propagation, Na^{+} channel mediated action potentials are conducted into the cellular interior at regular intervals along the muscle length through electrically continuous transverse (T-) tubular membranes. At specific regions, these come geometrically close (~ 12 nm) to, whilst remaining electrically isolated from, terminal cisternal membranes of the SR Ca^{2+} store. The resulting T-SR triad and dyad junctions are strategic to excitation–contraction coupling^{11–13}. In cardiac muscle, tubular membrane Ca^{2+} influx through voltage-activated dihydropyridine receptors (DHPR2; Cav1.2) activates SR Ca^{2+} release by cardiac-type SR RyR2^{14}. In skeletal muscle, Ca_{v}1.1 DHPR1 conformational changes themselves allosterically activate directly coupled RyR1^{15}.

Released Ca^{2+} then diffuses into the cytosol activating nearby myofilaments and consequent muscle contraction, prior to its SR Ca^{2+}-ATPase mediated re-sequestration at longitudinal as opposed to T-SR membrane sites. However, recent evidence implicates Ca^{2+} in further regulatory functions involving T and SR membrane proteins. Skeletal and cardiac muscle Na^{+} channels (Na_{v}1.4 and Na_{v}1.5) possess potential Ca^{2+} and calmodulin (CaM) binding modulatory sites^{16}. Here Ca^{2+} can bind directly to the Na_{v} C-terminal domain at one or more of its EF-like hand motifs^{17}, or indirectly, through Ca^{2+}/CaM binding, to its isoleucine-glutamine (IQ) domain region following initial Ca^{2+} binding to EF-hand motifs on CaM^{18,19} or to a site between Na_{v} domains III and IV^{20}. Voltage-gated Na^{+} channels additionally contain sites phosphorylatable by Ca^{2+}/CaM-regulated CaM kinase II (CaMKII)^{21}, and by protein kinase C^{22}. In vitro single-cell patch clamp studies reported that elevating [Ca^{2+}]_{i} to ~ 2 µM by rapid Ca^{2+} photo-release or overspill from neighbouring Ca^{2+} channels, reduced Na^{+} current, _{Na}, in Na_{v}1.4-transfected HEK293 cells and skeletal muscle cell lines. These effects were abrogated by intracellular BAPTA^{23}, or mutations involving either the CaM Ca^{2+}-binding-EF hands or the Nav1.4 C-terminal IQ domain^{19,23,24}.

Intact native murine skeletal muscle fibres were also studied by a loose patch clamping technique that minimized perturbations in intracellular Ca^{2+} signalling^{25}. Activation or inhibition of SR Ca^{2+} release then respectively reduced or increased Na^{+} current, _{Na}. The latter were accomplished by direct pharmacological RyR1 activation by caffeine or 8‐(4‐chlorophenylthio)‐2′‐O‐methyladenosine‐3′,5′‐cyclic monophosphate (8-CPT), or RyR1 inhibition by dantrolene. Dantrolene pretreatment further abrogated the downregulatory effects of caffeine or 8-CPT on _{Na}^{26,27}. Murine cardiac muscle showed similar effects when SR Ca^{2+} release was enhanced either by 8-CPT challenge, or with the RyR2-P2328S genetic modification^{28,29}. In contrast, cyclopiazonic acid (CPA) increased skeletal muscle _{Na}, while preserving its time courses, steady-state half-maximum voltages and steepness factors. CPA pre-treatment also abrogated the effects of caffeine^{30}. Yet previous Ca^{2+} fluorescence studies performed in rat soleus and oesophageal striated muscle using fluo-3-AM and fura-PE3-AM had reported that both caffeine and CPA increased bulk cytosolic [Ca^{2+}]^{31–34}. However, in contrast to modifying RyR-mediated SR Ca^{2+} release, CPA inhibits SR Ca^{2+}-ATPase (SERCA)-mediated cytosolic Ca^{2+} re-uptake^{31,32}. The consequent store Ca^{2+} depletion would then be expected to reduce, rather than increase, RyR1-mediated Ca^{2+} influx into the T-SR junction^{30}.

These paradoxical findings prompted suggestions that RyR1-mediated Ca^{2+} release took place into a microdomain in the vicinity of both the SR RyR1 and the T-tubular membrane Na_{v}1.4 and that the local elevation in Ca^{2+} concentration, [Ca^{2+}]_{TSR}, would then modify Na_{v}1.4 function^{30}. Such a hypothesis would predict contrasting increases and decreases in local microdomain [Ca^{2+}] following caffeine, and dantrolene or CPA challenge. The narrow, ~ 12 nm T-SR junctions that could form spaces with restricted intracellular diffusion close to the RyR1 Ca^{2+}-release sites might be implicated in such microdomain formation. These could result in changes in local in vivo [Ca^{2+}]_{TSR}, distinct from those of [Ca^{2+}]_{i} in the remaining bulk cytosol. This could explain the contrasting actions of RyR1 agonists and RyR1 or SERCA antagonists on _{Na} through correspondingly contrasting effects on local Ca^{2+} or Ca^{2+}/CaM levels, to which the Na_{v}1.4 would be directly or indirectly exposed, despite their similar effects on bulk cytosolic [Ca^{2+}]_{i}^{33,34}.

Direct experimental explorations for such [Ca^{2+}]_{TSR} microdomains possibly using fluorescent Ca^{2+} indicator methods need to address the small dimensions and dispersed nature of the T-SR compartment^{35–37}. The present complementary approach applies diffusional modelling techniques^{38} to explore the physical parameters permitting accumulation or depletion of released SR Ca^{2+} within the T-SR junction. It demonstrated that established anatomical and physiological features related to skeletal muscle excitation–contraction coupling are physically compatible with generation of significant Ca^{2+} microdomains in both activated and resting muscle fibres. We then discuss their possible physiological effects both in myocyte T-SR junctions and in similar membrane appositions in other cell types.

We employed anatomical, optical and electron microscope quantifications of sarcomere and T-SR junction structure from amphibian twitch fibres as the muscle type for which the fullest data are available^{39–44}. This provided the required details of T-tubular-sarcoplasmic reticular (T-SR) junction anatomy for the modelling studies (Table _{T}/_{S}, yielded the sarcomeric surface membrane area:

Structural characteristics of amphibian skeletal muscle fibres and transverse tubular-sarcoplasmic reticular (T-SR) junctions.

Name of variable | Definition | Value (physiological units) | Value (SI units) | ||||
---|---|---|---|---|---|---|---|

Length of sarcomere | 3.65 μm | 3.65 × 10 | |||||

Diameter of fibre | 100 μm | 1.00 × 10 | |||||

Surface membrane capacitance |
| 1.0 μF/cm | 1.00 × 10 | ||||

T-Tubular membrane capacitance |
| 5.0 μF/cm | 5.00 × 10 | ||||

Ratio of T-tubular to surface membrane capacitance | ( | 5.0 | 5.0 | ||||

Sarcomere surface membrane area |
| 1147 μm | 1.147 × 10 | ||||

Sarcomere tubular membrane area |
| 5733 μm | 5.733 × 10 | ||||

Sarcomere volume |
| 2.87 × 10 | 2.87 × 10 | ||||

Proportion of T-tubular membrane area apposed to triad junctions | 0.3 | 0.3 | |||||

Width of T-SR Junction | 12 nm | 1.2 × 10 | |||||

Diameter of sarcoplasmic reticular (SR) terminal cisternae | 220 nm | 2.20 × 10 | |||||

SR membrane area of T-SR junction | π | 38,013.27 nm | 3.8013 × 10 | ||||

Area at edge of T-SR junction | π | 8293.804 nm | 8.2938 × 10 | ||||

Ratio of volume of T-SR spaces to that of whole cell |
| 7.20 × 10 | 7.20 × 10 | ||||

Tubular membrane area abutted by T-SR junction |
| 1720 μm | 1.720 × 10 | ||||

Total number of T-SR junctions in one sarcomere |
| 4.5248 × 10 | 4.5248 × 10 | ||||

Total number of T-SR junctions in unit volume of muscle |
| 1.5784 × 10 | 1.5784 × 10 |

Flux properties, previous experimental results | |||||||
---|---|---|---|---|---|---|---|

Test membrane potential, | − 45 | − 40 | − 30 | − 20 | −10 | 0 | |

Maximum rate of [Ca | 3.55 | 18.0 | 90.0 | 120.0 | 170.0 | 180.0 | |

Peak cytosolic calcium concentration, [Ca | 0.135 | 0.511 | 1.711 | 2.427 | 2.858 | 3.161 | |

Ca | 5.92 × 10 | 3.00 × 10 | 1.50 × 10 | 2.00 × 10 | 2.83 × 10 | 3.00 × 10 | |

Ca | 2.25 × 10 | 1.14 × 10 | 5.70 × 10 | 7.60 × 10 | 1.07 × 10 | 1.14 × 10 | |

Ca | 4 × 10 | 4 × 10 | 4 × 10 | 4 × 10 | 4 × 10 | 4 × 10 | |

Exit length, | 9.2 | 9.2 | 9.2 | 9.2 | 9.2 | 9.2 |

Morphometric electron micrographic estimates of the proportion _{T} accounted for by T-SR junctions^{44} then gave the total T-tubular membrane area contributing to T-SR junction structures _{TSR} = _{T}

T-SR junction structure represented using a formal geometric model divided into finite elements. Left: Geometrical representation of formalized T-SR junction represented as two disks, respectively of tubular (T) and sarcoplasmic reticular (SR) membranes, of diameter,

The finite element analysis of steady state Ca^{2+} diffusion through a single T-SR space used the circular ends of the geometry defined above to represent its respective T-tubular and SR membranes. The rim separating their edges connected the T-SR and remaining intracellular spaces. ^{2+} influx density _{influx} across the SR membrane face of each individual T-SR junction:

In contrast, the T-tubular face represented a zero-flux boundary surface. The Ca^{2+} then diffuses through and leaves the T-SR space at the rim with diffusion coefficient

Away from the boundaries where

At the

This edge region thus incorporates a first-order efflux Neumann BC describing diffusional Ca^{2+} efflux _{efflux} leaving the T-SR space across its edge area π

Its efflux density _{efflux} is proportional to the edge Ca^{2+} concentration [Ca^{2+}]^{TSR}_{edge}:

This gives efflux equation:

Its Fick’s Law constant ^{2+} diffusion coefficient ^{2+} from a well stirred cytosolic compartment by SERCA activity without saturation. The

Diffusional processes within the T-SR junction were represented by superimposing a finite element mesh upon the T-SR junction geometry using PDE Toolbox, dividing that 3D geometry into fine tetrahedral elements of specified maximum length ^{2+}] gradients and in duplicate runs matching different mesh sizes to validate those used in the reconstructions. The FEM and MATLAB solved the equations specified for the system and specified input parameters producing a [Ca^{2+}] dataset in the form of an array, whose spatial resolution was determined by the fineness of the mesh, set by the maximum element length

between influx and efflux boundaries, yielding the condition (see Eqns. 3 and 9):

This condition was used in checks for the steady state condition in detailed explorations of the effect of specific parameters that follow.

The modelling process first sought three-dimensional (3D) reconstructions of radial steady state [Ca^{2+}] distributions resulting from diffusional processes using the computational T-SR parameters listed in Table ^{2+} release influx densities into each individual T-SR junction, _{influx,} used in the MATLAB program PDE Toolbox, could be related to previously reported^{45} experimental initial rates of SR Ca^{2+} release d[Ca^{2+}]/d^{2+} flux into the sarcomere cytosolic volume

Antipyrylazo III absorbances in amphibian skeletal muscle fibres subject to voltage clamp steps from a − 90 mV resting to a 0 mV test membrane potential^{45} reported a value of d[Ca^{2+}]/d^{3} s). The Ca^{2+} then diffuses through the T-SR space down its resulting concentration gradients with the diffusion constant ^{7} nm^{2}/s previously reported for amphibian skeletal muscle^{46–48}. The Ca^{2+} finally leaves the T-SR junction space effluxing into surrounding cytosol at the edge of the T-SR junction across diffusional area π^{2+}]^{TSR}_{edge}. This proved close and proportional to reported experimental peak cytosolic Ca^{2+} concentration, [Ca^{2+}]_{max} at the explored 0 mV test voltage with the use of an exit length value ^{45}.

An overall rate constant describing the dependence of the summed Ca^{2+} fluxes upon [Ca^{2+}]^{TSR}_{edge} could be determined using the predicted number of T-SR junctions in unit muscle volume,

The total Ca^{2+} efflux into unit muscle volume is then:

The constant of proportionality describing this linear dependence on [Ca^{2+}]^{TSR}_{edge} is then.

This resulting rate constant is smaller than but comparable to experimental rate constants describing eventual SR resequestration of the released cytosolic Ca^{2+}. Thus, previous experimental studies suggested rate constants for such unsaturable SR Ca^{2+-}ATPase mediated Ca^{2+} uptake around 22.3 ± 8.14/s under similarly steady state conditions where Ca^{2+} binding to remaining, saturable, fast-exchanging cytosolic binding sites was constant^{49}.

Figure ^{2+}] through the radial geometry of the T-SR junction. The false colour maps reconstruct perspective steady-state [Ca^{2+}] at the (A) tubular and (B) SR membrane faces, and (C) en face [Ca^{2+}] at the tubular membrane face of the T-SR junction. Between elements where a solution has not been calculated, the result is interpolated automatically according to the coarseness of the mesh. These maps show a significant [Ca^{2+}] heterogeneity or microdomain across the radial axis resulting in a > fivefold difference between centre and edge T-SR junction [Ca^{2+}]. These resulting Ca^{2+} microdomains extended the entire T-SR distance mapping [Ca^{2+}] gradients onto the tubular membrane surface. The colour map (Fig. ^{2+}] at that point, displaying [Ca^{2+}] nonuniformities extending across the T-tubular membrane.

Sarcoplasmic reticulum Ca^{2+} release produces Ca^{2+} microdomains in the T-SR model. (A-C) Heat map with overlaid 12 nm finite element mesh demonstrating steady state radial [Ca^{2+}] gradients following application of Ca^{2+} influx density, _{influx}, expected from a depolarizing step from the resting to a 0 mV test voltage in voltage clamped amphibian muscle fibre as described in previous reports^{45}. Highest [Ca^{2+}] is at the T-SR centres with ~ fivefold concentration reductions between the centre and edge in both (^{2+} microdomains extend the entire axial T-SR distance. This is demonstrated in heat maps of: (^{2+}] (vertical, ^{2+}] taken along the ^{2+}] plotted against radial distance along the T-tubular membrane in both ^{2+}] heatmap through midline axial section through the T-SR junction. Vertical axis plots axial (^{2+}]. Note uniformity in concentration along the ^{2+}] as represented in the heat maps. (^{2+}] solutions at selected positions within the T-SR junction. (^{2+}] at the edge (black), centre (blue) of the T-SR junction and midway between these two (red lines) plotted against time following initiation of the Ca^{2+} influx, _{influx} from the SR face of the T-SR junction. Initial instabilities in plots observed in the edge traces reflect stiff-stable properties of the solutions. (^{2+}] with radial distance from the centre of the T-SR junction parallel to the radial axis at the T-tubular membrane, the SR membrane, or halfway between these. (^{2+}] plotted with time following the onset of Ca^{2+} influx, but with its termination after 0.5 ms.

Figure ^{2+} microdomain extending to the edge of the T-SR junction with [Ca^{2+}] radially falling ~ fivefold over ~ 100 nm from the T-SR junction centre. Figure ^{2+}] within a section taken along a midline axial plane through the T-SR junction. It illustrates diffusional effects following SR Ca^{2+} release along the bottom of the frame and the resulting nearly axially uniform [Ca^{2+}] and radial [Ca^{2+}] gradients running towards the T-SR junction edge.

The Ca^{2+} microdomain represented above was quantified from the array of [Ca^{2+}] values at each element with time following imposition of the Ca^{2+} influx density _{influx}; missing data between element solutions were interpolated from nearby nodal solutions. To quantify [Ca^{2+}] with time in the microdomain, each finite element in the mesh was assigned a number referenced using a command retrieving the solution from the node closest to a specified 3-dimensional Cartesian co-ordinate (_{influx,} a timescale ~ 1–2 orders of magnitude shorter than experimentally measured Ca^{2+} transients in vivo^{45}. The resulting steady state [Ca^{2+}] at axial distances within the T-SR junction could additionally be obtained by interpolation for plotting against radial distance from the centre of the T-SR junction (Fig. ^{2+}] timecourses with imposition and termination of the Ca^{2+} fluxes confirmed that [Ca^{2+}] recovered back to its initial starting value, as expected for a first-order diffusional system (Fig. ^{2+}] microdomain characteristics in close agreement, validating the computational parameters used in our finite element analysis (Supplementary Table

The T-SR junction model was then tested against varying _{influx} holding constant the remaining diffusional, _{influx} values used were calculated from previous experimental d[Ca^{2+}]/d^{2+}]_{edge} changes were then compared with corresponding experimental peak changes in intracellular Ca^{2+} concentrations [Ca^{2+}]_{max}, at the same test voltage^{45}.

Firstly, all the test voltages were associated with T-SR junction Ca^{2+} microdomains (Fig. ^{2+} influx to reach steady state [Ca^{2+}] values at the centre and edge of the T-SR junction, and halfway between these points. Figure ^{2+}] with radial distance close to the T-tubular (superscript ‘T’) and SR membranes (superscript ‘SR’), and halfway along their axial distance (superscript ‘TSR’). [Ca^{2+}] declined from its maximum values with radial distance from the T-SR junction centre (subscripts below: ‘centre’) towards the T-SR edge (subscript ‘edge’) with similar readouts close to the T-tubular (subscript: ‘T’) and SR membranes (subscript: ‘SR’), and half-way (subscript: ‘50%’) along their axial distance.

Ca^{2+} microdomains reconstructed using varied Ca^{2+} flux densities derived from experimental Ca^{2+} transients in response to test voltage steps. (^{2+} microdomains with [Ca^{2+}] magnitudes graded with experimentally derived Ca^{2+} influx densities _{influx} corresponding to varying experimental test voltage steps applied to amphibian muscle fibres^{45}. (^{2+}] changes with time following onset of the imposed _{influx} and (^{2+}] upon radial distances from the centre of the T-SR junction. The [Ca^{2+}] across the T-SR junction falls with reducing depolarizing steps at all points in both the steady state and with time. For comparison, the ^{2+} flux densities and resulting Ca^{2+} concentrations following the graded test voltage steps. Thus: (^{2+}] (filled symbols) achieved following test voltage steps ^{45} compared with [Ca^{2+}]_{edge} (open symbols) in response to Ca^{2+} flux densities, _{influx} determined from reported rates of increase in [Ca^{2+}], d[Ca^{2+}]/d^{2+}]_{edge} from the reconstructed T-SR junction against reported cytoplasmic [Ca^{2+}]^{45}. The points fit a linear regression model with a gradient close to 1 (1.02) and intercepts close to the origin. (^{2+} influx across the SR membrane _{influx} against computed [Ca^{2+}]_{edge} expressed in SI units. The points fit a linear regression model with a gradient and zero intercept matching that expected from the conservation condition. The matching computed and observed [Ca^{2+}] in (

Secondly, quantifications of Ca^{2+} microdomains arising from the range of _{influx} values and corresponding test voltages showed that these were similar in form, consistent with the linearity expected of diffusional processes. This emerged from comparing ratios between [Ca^{2+}] at the T-SR centre with those at the edge and halfway between (50%) at the T-tubular and SR membranes, and within the T-SR space. Thus [Ca^{2+}]^{T}_{50%}/[Ca^{2+}]^{T}_{centre}, [Ca^{2+}]^{TSR}_{50%}/[Ca^{2+}]^{TSR}_{centre}, and [Ca^{2+}]^{SR}_{50%}/[Ca^{2+}]^{SR}_{centre} values were all close to 79% and [Ca^{2+}]^{T}_{edge}/[Ca^{2+}]^{T}_{centre}, [Ca^{2+}]^{TSR}_{edge}[Ca^{2+}]^{TSR}_{centre} and [Ca^{2+}]^{SR}_{edge}/[ Ca^{2+}]^{SR}_{centre} were all close to 14%. Similarly, the radial distances from the T-SR centre over which [Ca^{2+}] fell by 50% of its centre-edge range were all 77.67, 77.90 and 77.76 nm whether close to the T-tubular membrane, the SR membrane and halfway between these respectively.

Thirdly, progressively larger _{influx} values corresponding to the increasing experimental d[Ca^{2+}]/dt with progressive depolarization correspondingly predicted increased steady state [Ca^{2+}] whether at the centre, 50% from or at the T-SR edge, or close to T-tubular or SR membranes or between these. They produced substantially higher, ~ 10’s of µM [Ca^{2+}] values at the T-SR centre than the ~ µM [Ca^{2+}] values at the edge.

Both the individual values of, and the relationships between, the modelled _{influx} at differing test voltages and [Ca^{2+}]_{edge} closely matched the corresponding reported experimental relationship between d[Ca^{2+}]/d^{2+}]_{max}. Figure _{influx} values gave [Ca^{2+}]_{edge} values similar to those of the corresponding reported experimental [Ca^{2+}]_{max}^{45}. Figure ^{2+}]_{edge}, and [Ca^{2+}]_{max} with close to unity gradient and zero intercept. Furthermore, combining the Ca^{2+} influx and efflux equality condition, _{influx} = _{efflux}, and the efflux equation (Eqs.

This yielded data points replicating the predicted linear dependence between _{influx} and [Ca^{2+}]_{edge}^{–17} m^{3}/s (Fig. _{influx} reductions similarly (A) converged monotonically to steady state [Ca^{2+}] measurable at selected radial distances from the T-SR junction centre yielding (B) uniform radial [Ca^{2+}] gradients along the T-SR distance visualized as (C) radial [Ca^{2+}] gradients extending the full axial T-SR junction distance (Supplementary Table ^{2+}]^{T}_{50%}/[Ca^{2+}]^{T}_{centre}, [Ca^{2+}]^{TSR}_{50%}/[Ca^{2+}]^{TSR}_{centre}, and [Ca^{2+}]^{SR}_{50%}/[Ca^{2+}]^{SR}_{centre} as well as the [Ca^{2+}]^{T}_{edge}/[Ca^{2+}]^{T}_{centre}, [Ca^{2+}]^{TSR}_{edge}^{/}[Ca^{2+}]^{TSR}_{centre} and [Ca^{2+}]^{SR}_{edge}/[Ca^{2+}]^{SR}_{centre} remained essentially constant (79% and 14%). Furthermore, radial distances from the T-SR centre over which [Ca^{2+}] fell by 50% of its centre-edge range, whether close to the T-tubular membrane, the SR membrane and half way between these respectively remained similar (77.67, 77.90 and 77.76 nm). [Ca^{2+}] levels through the T-SR junction fell proportionally with 10- and 100-fold _{influx} reductions: [Ca^{2+}]^{T}_{centre} fell from ~ 0.43 μM to 0.043 μM and 0.0043 μM respectively, as expected from the relationship established in Fig.

Skeletal muscle T-SR junction properties vary with physiological conditions, both within and between muscle fibres, individual muscles and muscle types. Furthermore, such surface-cytoplasmic membrane appositions also occur in and vary amongst other cell types. Nevertheless, the Ca^{2+} microdomains robustly persisted with wide variations involving previously experimentally reported Ca^{2+} diffusion constants, _{influx}.

Firstly, we extended the modelling beyond the reported skeletal muscle cytosolic Ca^{2+} diffusion coefficient ^{7} nm^{2}/s^{46–48}. Higher reported values reach 5.2 × 10^{8} nm^{2}/s in other cell types^{50} (cf:^{51–54}) and 1 × 10^{9} nm^{2}s at infinite dilution in vitro^{55,56} (Supplementary Table ^{44}. Solutions with ^{5} and 1 × 10^{9} nm^{2}/s at constant _{influx}, continued to converge with correspondingly varied relative timescales (Fig. ^{2+}] gradients extending the full axial distance between T and SR membranes (Fig. ^{2+}] with distance from the T-SR centre (Fig. ^{2+}] at the edge and halfway from the centre, to [Ca^{2+}] at the T-SR centre whether close to T-tubular or SR or between the two membranes, remained at 79% and 14% respectively. Distances for [Ca^{2+}] to fall to 50% between centre and edge values similarly remained unchanged. However, diffusion coefficient value markedly influenced [Ca^{2+}] at the centre of the T-SR junction.

Ca^{2+} microdomain characteristics persist with variation in Ca^{2+} diffusion coefficient. (A-C) Exploring effects of variations in reported in vitro and in vivo Ca^{2+} diffusion coefficient values (units: nm^{2}/s), ^{2+} influx density, _{influx,} T-SR diameter, ^{2+}] with time following onset of imposed _{influx}. (^{2+}] with radial distance from the centre of the T-SR junction. (^{2+}] across a midline axial slice; values of [Ca^{2+}] scaled linearly with maximum [Ca^{2+}] at centre of T-SR junction to illustrate features of microdomain characteristics and radial concentration gradients. (^{2+}] at the centre, the edge, and halfway between these of the T-SR junction, and diffusion coefficient, ^{2+}]^{TSR}_{centre}, [Ca^{2+}]^{TSR}_{50%} and [Ca^{2+}]^{TSR}_{edge}, against diffusion coefficient, ^{2+}]^{TSR}_{centre}, [Ca^{2+}]^{TSR}_{50%} and [Ca^{2+}]^{TSR}_{edge}, and ^{2+} influx density, _{influx,} T-SR diameter,

Nevertheless, even the highest ^{9} nm^{2}/s free diffusion value at infinite dilution predicted near µM-[Ca^{2+}] microdomains. Conversely, successive tenfold reductions in ^{5} nm^{2}/s resulted in limiting ~ mM hypothetical TSR [Ca^{2+}] levels approaching reported in situ free SR [Ca^{2+}] (~ 3.6 mM; assuming a 6.1 mM SR [calsequestrin] with a 1.1 mM Ca^{2+} binding constant^{57}). Plotting log_{10}[Ca^{2+}] values at the centre, edge of the T-SR junction, and halfway between these against log_{10}(^{2+}] profiles shown by the Ca^{2+} microdomains (Fig. ^{2+}]-^{–13} mol/(ms) agreeing with the steady state

Secondly, T-SR distances also vary with physiological conditions (Supplementary Table ^{58}. They increase to 20.15 nm and 29.60 nm with fatiguing low-frequency intermittent stimulation^{59} and exposure to hypotonic solutions^{60} respectively. Computational solutions modelling these variations in _{influx}, ^{2+} microdomains with increased [Ca^{2+}] at the T-SR centre declining with radial distance. However, differences between [Ca^{2+}] values close to the SR, the T-tubular membranes, and within the intervening space occurred at the greater T-SR distances (Fig. ^{2+}] non-uniformities (Fig. ^{2+} microdomain radial profiles through the entire T-SR junctional distance observed in the computations varying _{influx} and

Altered Ca^{2+} microdomain characteristics with increased T-SR distances.. (^{2+} influx density, _{influx} diffusion coefficient, ^{2+}] with time following onset of imposed _{influx}. (^{2+}] with radial distance from the centre of the T-SR junction; note significant differences in the [Ca^{2+}] magnitudes close to the T-tubular, the sarcoplasmic reticular (SR) membranes, and the intervening T-SR region. (^{2+}] across a midline axial slice; values of [Ca^{2+}] are scaled linearly with maximum concentration at centre of T-SR junction [Ca^{2+}] to illustrate features of microdomain characteristics and the radial concentration gradients. Note marked pluming of the Ca^{2+} microdomain with increasing T-SR distance. (^{2+}] at the centre, the edge, and halfway between these upon T-SR distance, ^{2+}]^{TSR}_{centre}, [Ca^{2+}]^{TSR}_{50%} and [Ca^{2+}]^{TSR}_{edge}, against T-SR distance, _{influx} and T-SR diameter,

Quantification of these effects as ^{2+}]^{T}_{50%}/[Ca^{2+}]^{T}_{centre} and [Ca^{2+}]^{T}_{edge}/[Ca^{2+}]^{T}_{centre} remained at ~ 79% and ~ 14% respectively. Similarly at the midpoint between T-tubular and SR membranes, [Ca^{2+}]^{TSR}_{50%}/[Ca^{2+}]^{TSR}_{centre} and [Ca^{2+}]^{TSR}_{edge}/[Ca^{2+}]^{TSR}_{centre} were ~ 78% and ~ 13% respectively. However, close to SR membrane, [Ca^{2+}]^{SR}_{50%}/[Ca^{2+}]^{SR}_{centre} and [Ca^{2+}]^{SR}_{edge}/[Ca^{2+}]^{SR}_{centre} increased to ~ 80% and ~ 18% respectively. Furthermore, the axial nonuniformities in [Ca^{2+}] close to the SR and T-tubular membranes involved both the centre and edge of the T-SR junction. Thus, [Ca^{2+}]^{SR}_{centre}/[Ca^{2+}]^{T}_{centre} increased from 1.005 to 1.133 and [Ca^{2+}]^{SR}_{edge}/[Ca^{2+}]^{T}_{edge} increased from 1.031 to 1.565.

In addition, radial distances for [Ca^{2+}] to fall to 50% as a fraction of the total T-SR radius showed contrasting patterns at the T-tubular and SR membranes with increasing _{centre}/(T-SR radius) then fell from 0.763 to 0.748 whereas _{edge}/(T-SR radius) increased from 0.765 to 0.793. Finally, both steady state centre and edge [Ca^{2+}], whether at the T-tubular or SR membranes, or within the T-SR junction, decreased. Thus, [Ca^{2+}]^{T}_{centre} and [Ca^{2+}]^{T}_{edge} fell from 43.965 and 6.190 to 8.428 and 1.081 respectively. Nevertheless, even the lowest values of [Ca^{2+}]^{T}_{edge} resulted in µM-[ Ca^{2+}] changes. Finally, [Ca^{2+}]_{edge} inversely depended upon ^{–11} mol/m^{2}) approximated predictions from parameter values in the steady state conservation condition ^{–11} mol/m^{2}.

Thirdly, significant variations in effective areas of membrane appositions occur not only between skeletal muscle T-tubular and SR membrane but also occur in and between other cell types. We quantified these effects successively reducing T-SR junction diameters, _{influx}, ^{2+} microdomains in which [Ca^{2+}] declined with radial distance from the T-SR junction centre. These radial gradients accompanied significant axial [Ca^{2+}] variations at the T-tubular and SR membranes and the intervening space (Fig.

Calcium microdomain stability breaks down at decreased T-SR diameters. (^{2+} influx density, _{influx}, diffusion coefficient, ^{2+}] with time following onset of imposed _{influx}. Note initial instabilities close to the onset of imposition of _{influx} at the reduced T-SR diameters. (^{2+}] with radial distance from the centre of the T-SR junction; note significant differences in the regions close to the T-tubular and sarcoplasmic reticular (SR) membranes, and the intervening T-SR region with decreasing T-SR diameter, ^{2+}] across a midline axial slice; values of [Ca^{2+}] are scaled linearly with maximum concentration at centre of T-SR junction [Ca^{2+}] to illustrate microdomain characteristics and radial concentration gradients. Marked pluming of Ca^{2+} microdomain with decreasing T-SR diameter. (^{2+}] at the centre, the edge, and halfway between these upon T-SR diameter. Relationships emerging from plots of [Ca^{2+}]^{TSR}_{centre}, [Ca^{2+}]^{TSR}_{50%} and [Ca^{2+}]^{TSR}_{edge}, against TSR diameter, _{influx} and T-SR distance

Quantifications of these alterations in Ca^{2+} microdomain characteristics first demonstrated changes in [Ca^{2+}]^{T}_{50%}/[Ca^{2+}]^{T}_{centre} and [Ca^{2+}]^{T}_{edge}/[Ca^{2+}]^{T}_{centre} from 78 and 14% respectively at ^{2+}] differences between T-SR regions close to the T-tubular and SR membranes with [Ca^{2+}]^{SR}_{centre}/[Ca^{2+}]^{T}_{centre} and [Ca^{2+}]^{SR}_{edge}/[Ca^{2+}]^{T}_{edge} at 1.021 and 1.111 at ^{2+}] fell to half of its maximum value as a proportion of T-SR radius (_{centre}/(T-SR radius)) and (_{edge}/(T-SR radius)) increased from 0.761 to 0.768 and 0.950 to 1.000 respectively. Finally, [Ca^{2+}]^{T}_{centre} and [Ca^{2+}]^{T}_{edge} fell from 21.876 and 3.025 to 1.047 and 0.473 respectively. Thus, even a ^{2+}] microdomain differences. [Ca^{2+}]_{edge} positively correlated with ^{4} to 1.35 × 10^{4} mol/m^{4} in agreement with steady state flux conservation, giving ^{4} mol/m^{4} (Fig.

Finally, the recent reports that had suggested an existence of T-SR junction Ca^{2+} microdomains included evidence suggesting background ryanodine receptor (RyR)-mediated Ca^{2+} fluxes modulating Na_{v}1.4 function even in resting skeletal muscle^{27,30}. Available Fura-2 studies suggest resting cytosolic [Ca^{2+}] between 0.06 and 0.14 μM^{61}. The highest reported, Fluo-3, studies reported up to 0.30 μM^{62,63} and the lowest Ca^{2+}-sensitive microelectrode measurements, 0.038 μM^{64} (Supplementary Table _{influx} from the influx and efflux equations for _{influx} and _{efflux} respectively, assuming the conservation condition _{influx} = _{efflux}. Approximating these [Ca^{2+}] values to ^{2+}] range (Fig. ^{2+} microdomains with radial concentration profiles (Fig. ^{2+}] at the edge and halfway between edge and centre, and the centre (79% and 14%), and _{centre}, _{50%}, and _{rim} values (77.67, 77.76 and 77.90 nm) all identical to corresponding values at higher _{influx}. Finally, the [Ca^{2+}]^{T}_{centre,}, [Ca^{2+}]^{TSR}_{centre} and [Ca^{2+}]^{SR}_{centre} approached 1 µM [Ca^{2+}] concentrations, giving 0.268, 0.270 and 0.274 at the lowest and 0.990, 0.995 and 1.010 µM at the highest resting [Ca^{2+}].

Ca^{2+} microdomains attain micromolar [Ca^{2+}] levels at edge [Ca^{2+}] corresponding to resting [Ca^{2+}]. Effects of variations in Ca^{2+} influx density, _{influx,} generating T-SR junction edge Ca^{2+} concentrations, [Ca^{2+}]_{edge} achieving previously reported resting [Ca^{2+}] levels studied at constant values of diffusion coefficient, ^{2+}] with time following onset of imposed _{influx}. (^{2+}] with radial distance from the centre of the T-SR junction. (^{2+}] across a midline axial slice; values of [Ca^{2+}] are scaled linearly with maximum concentration at centre of T-SR junction [Ca^{2+}] to illustrate persistence and features of microdomains including their radial concentration gradients.

Together, the present findings substantiate recent experimental reports predicting physiologically significant T-SR Ca^{2+} microdomains in skeletal muscle. These had implicated RyR-mediated SR Ca^{2+} release in modulating tubular Na_{v}1.4 function in both activated and resting skeletal muscle^{27,30}. The spatial [Ca^{2+}] data arrays obtained from the model of activated (Fig. ^{2+}] (Fig. ^{2+}] with increasing depolarization (Fig. ^{2+}] levels corresponding to dissociation constants, _{d} of typical modulatory proteins including calmodulin. Furthermore, the tested values of resting [Ca^{2+}] similarly predicted exposures of significant proportions of T-tubular membrane to lower, but nevertheless significant, 0.5 µM [Ca^{2+}] (Fig.

Proportions of T-SR junction T-tubular membrane area exposed to varied tested microdomain [Ca^{2+}]. Results shown for (^{2+}] values.

Recent experimental reports implicated hypothetical Ca^{2+} microdomains in paradoxical Ca^{2+}-mediated effects on skeletal muscle Na_{v}1.4 activation following pharmacological manoeuvres increasing or decreasing RyR-mediated SR Ca^{2+} release^{26,27,30}. They went on to suggest that T-SR junctional sites of close membrane proximity, key to excitation contraction coupling, might form diffusion-restricted, ultrastructurally dispersed intracellular subcompartments. Although accounting for only a small proportion of total cytosolic volume, these might sustain regulatory local [Ca^{2+}] heterogeneities in the vicinity of the tubular Na_{v}1.4^{35–37}. These could potentially cause local ~ μM-[Ca^{2+}] alterations previously reported to modify Na^{+} channel function^{23} arising from direct Ca^{2+}, or indirect, Ca^{2+}-calmodulin mediated, binding to Na_{v}1.4^{18–20,23,24}. Our present modelling studies accordingly explored and characterized conditions required for such Ca^{2+} microdomain formation within these T-SR junctional structures.

^{11–13,39–44}. [Ca^{2+}] gradients through the resulting formalized geometrical model of a typical T-SR junction in both resting and stimulated muscle fibres were then resolved by finite element method (FEM) solutions of basic Fick diffusion equations. Their boundary conditions first included a RyR-mediated Ca^{2+} release into the T-SR space by a uniform Ca^{2+} influx density _{influx} across its SR face. Subsequent Ca^{2+} diffusion with a diffusion coefficient established from previous experimental reports from amphibian myoplasm was then mapped through the radially symmetric T-SR junctional space. The second, efflux, boundary condition at the edge of the modelled junction was similarly described by a first order [Ca^{2+}]-dependent process into a well-stirred cytosolic space of infinitely large volume. The latter formalism further matched previous reported eventual first order steady state SERCA-mediated resequestration of the released cytosolic Ca^{2+}^{49}.

The boundary conditions simulated conditions both of full and of graded activation by previously reported experimental voltage clamp steps from resting to both 0 mV and varying test potentials and the resulting alterations in cytosolic [Ca^{2+}]^{45}. At the influx boundary, Ca^{2+} influx densities _{influx} for each voltage were determined directly from the corresponding reported maximum rates of SR Ca^{2+} release, d[Ca^{2+}]/d^{2+}] at the edge of the T-SR junction, [Ca^{2+}]_{edge}, was first matched to the corresponding experimental maximum cytosolic concentration [Ca^{2+}]_{max}, by optimising the single free parameter giving exit length _{influx} through different test voltages. In all events, further corrections for any discrepancies arising from a small [Ca^{2+}]_{edge} > [Ca^{2+}]_{max} would tend to enhance rather than reduce the computed [Ca^{2+}] magnitudes.

^{2+}^{30}. The computational solutions following step impositions of _{influx} converged to give steady state T-SR junctional Ca^{2+} microdomains. Their heatmap representations demonstrated radial [Ca^{2+}] gradients extending the entire axial T-SR junction distance. The microdomains were quantified radially at the centre, at distances halfway to, and at the edge of the T-SR junction, in axial regions close to the T-tubular and SR membranes and in the intervening T-SR junction space. Furthermore, varying _{influx} to reflect previously experimentally reported d[Ca^{2+}]/d^{2+}]_{edge} closely approximating those of the corresponding experimental [Ca^{2+}]_{max}. The accordingly linear, [Ca^{2+}]_{edge}–[Ca^{2+}]_{max} relationship had unity gradient and zero intercept. Furthermore, the [Ca^{2+}]_{edge} values themselves depended linearly on the corresponding Ca^{2+} influx _{influx} terms with a gradient fulfilling predictions from the geometrical terms in the equation for the corresponding Ca^{2+} efflux.

All these _{influx} conditions consistently generated T-SR junction Ca^{2+} microdomains containing physiologically significant, ~ μM-[Ca^{2+}], heterogeneities graded with imposed depolarization. These could locally elevate [Ca^{2+}] from ~ 0.3–0.4 µM at activation threshold, to 17–20 µM at full activation, relative to the remaining bulk extra-junctional cytosolic [Ca^{2+}]. These microdomains all extended through the entire axial T-SR distance. They showed similar normalised [Ca^{2+}] profiles with radial distance, in which [Ca^{2+}] declined > fivefold from its maximum in the centre to the edge of the T-SR junction. The spatial [Ca^{2+}] dependences persisted with proportionately reduced [Ca^{2+}] even with 10 and 100-fold _{influx} reductions below threshold levels for observed Ca^{2+} release^{45}. These modelling studies adopting established quantifications for baseline T-SR and sarcomere membrane structure, and diffusion coefficient values, ^{2+}]/d^{2+}]_{max} in activated amphibian skeletal muscle.

Detailed characteristics of such membrane appositions vary amongst muscle or cell types, and with physiological and physical conditions. Nevertheless, _{influx}^{2+}

First, alterations in ^{2+} buffering capacities, κ = (Δ[bound Ca]/Δ[free Ca])), of its contained immobile and mobile buffers, and the diffusion coefficient _{mobile} of the mobile buffer. Assuming the rapid buffer approximation, the resulting steady state _{Ca2+} by the expression^{65–67}:

^{2+} binding sites including negatively charged membrane bilayer phospholipid groups^{68} and Ca^{2+}-binding domains in Ca^{2+} dependent ion channel, cytoskeletal, transport motor and membrane-associated Ca^{2+} binding kinase proteins^{69}. This would generate local, steady state equilibria between Ca^{2+} binding and Ca^{2+} diffusion: depleted Ca^{2+}-free immobile buffer cannot be replaced by diffusion from remote sites. Immobile buffer would then leave steady-state Ca^{2+} microdomains unaffected^{67}. Whilst its action could alter the [Ca^{2+}] kinetics, our modelled step impositions of Ca^{2+} influxes increased T-SR free [Ca^{2+}] to steady state values over ~ 0.2 ms exponential timecourses. These were 1–2 orders of magnitude shorter than those of experimentally observed Ca^{2+} transients^{45,70}. In contrast, _{mobile}. Our computations explored a wide range of conditions extending from limiting maximal ^{2+} dilution without buffer^{55} then progressively reducing ^{2+}] values approaching and exceeding SR [Ca^{2+}]. They included physiologically realistic values corresponding to known in vivo Ca^{2+} buffering capacities of ~50 and ~100 in the respective absence and presence of 1 mM MgATP^{51,71}, and empirical in vivo skeletal muscle values^{46–48}.

These explorations demonstrated persistent [Ca^{2+}] heterogeneities despite large increases or decreases in diffusion coefficient ^{7} nm^{2}/s)^{46–48}, including values reported either in vitro^{55,56} or in other cell types^{50–54}. On the one hand, even the highest reported, in vitro, ^{9} nm^{2}/s, value corresponding to infinite dilution^{55} persistently yielded [Ca^{2+}] microdomains approaching μM-[Ca^{2+}] at the T-SR junction centre. On the other, the electron-densities within T-SR junctions could reflect local protein, lipid or ion concentrations reducing ^{44}. Here, progressive tenfold decreases down to ^{5} nm^{2}/s predicted ~ mM hypothetical [Ca^{2+}] actually approaching reported in situ free SR [Ca^{2+}] (~ 3.6 mM; assuming a 6.1 mM SR [calsequestrin] with a 1.1 mM Ca^{2+} binding constant^{57}). These correspondingly slowed the formation time courses of such microdomains. Nevertheless, in the steady state, through this entire explored ^{2+} microdomains persisted with unchanged radial [Ca^{2+}] profiles extending the full distance between T and SR membranes confirming the linearity condition in this model system. [Ca^{2+}]_{edge} varied inversely with

Secondly, whilst averaging ^{58} to 20 nm with fatiguing stimulation^{59,60}. Varying ^{2+}] nonuniformities themselves varying along the axial distance between SR and T-tubular membranes, to extents increasing with increasing ^{2+}]_{edge} varied with ^{2+} microdomain heatmaps again giving tapering plume-like forms at the smallest diameters. These were quantified as increased radial non-uniformities and marked axial [Ca^{2+}] differences between regions close to the SR and T-tubular membranes and the intervening space along the axial T-SR distance. Falls in [Ca^{2+}] with radial distance from the T-SR junction centre and [Ca^{2+}] at the SR relative to the T-tubular membranes became less marked with decreasing ^{2+}]_{edge} increased with ^{2+}] heterogeneities between their periphery and centre persisted even with more than 100% increases in T-SR distance or 75% reductions in T-SR diameter from control values derived from established morphometric data.

The previous experimental reports had also invoked background, RyR-mediated influxes of Ca^{2+} in Ca^{2+} microdomain generation in resting in addition to activated muscle^{27,30}. Accordingly, ^{2+}_{influx} values calculated from previously reported cytosolic [Ca^{2+}] values in resting muscle. These previous Fura-red fluorescence studies had suggested experimental resting [Ca^{2+}] ranging between 0.060 to 0.140 μmol/dm^{3} (ref. ^{61}). We further explored further reduced resting [Ca^{2+}] limits of ~ 0.038 μmol/dm^{3} from Ca^{2+}-sensitive microelectrode studies^{64} and possible higher 0.300 μmol/dm^{3} limits^{62,63}. The resulting T-SR modelling continued to predict Ca^{2+} microdomains with their characteristic spatial characteristics. Furthermore, microdomain [Ca^{2+}] levels at the T-SR centre, whether close to the T-tubular or SR membranes or in the intervening space, approached or attained ~ μM-[Ca^{2+}].

The present findings taken together could be used to reconstruct the proportions of T-tubular membrane area and therefore of resident Na_{v}1.4 exposed to successively greater levels of T-SR junction microdomain [Ca^{2+}] in both activated and resting muscle. Successively greater proportions of activated T-tubular membrane became exposed to successively higher, 0.1 to 10 µM [Ca^{2+}] with increasing depolarization. In addition, significant proportions of even resting T-tubular membrane remained exposed to significant, ~ 0.5 µM [Ca^{2+}]. These findings therefore provide a physical basis for the previous suggestions implicating Ca^{2+} microdomain formation in observed modifications in Na_{v}1.4 function^{27,30}. Ca^{2+}-CaM binding takes place with ~ µM [Ca^{2+}] dissociation constants^{72,73}. Feedback µM-[Ca^{2+}] levels arising from SR Ca^{2+} release could therefore potentially modify both skeletal^{26,27,30} and cardiac muscle^{28,29} Na_{v}1.4 or Na_{v}1.5 through direct or indirect, Ca^{2+}-calmodulin (Ca^{2+}-CaM) mediated, actions on their C-terminal domains^{16,18–20,23,24}. Such concentrations further match the photorelease-induced 1–2 µM cytosolic [Ca^{2+}] elevations previously reported to modify Na_{v}1.4 function^{23}.

In skeletal muscle, elevated T-SR junctional microdomain [Ca^{2+}] could inhibit tubular Na_{v}1.4 function following normal sustained activity^{59,60} and contribute to particular clinical skeletal myopathies^{74}. A myotonic hyperexcitability disorder disrupting Ca^{2+}-mediated inhibition of Na_{v}1.4 function has been associated with Na_{v}1.4 C-terminal EF hand-like domain mutations^{75,76}. A condition associated with increased myotube diameters and resting [Ca^{2+}]_{i}, and decreased RyR1-mediated Ca^{2+} release reflecting possible abnormalities in triad junction formation and maintenance is associated with another, junctophilin (JP2), mutation^{77}. In murine cardiac muscle, Na_{v}1.5 inhibition followed increased SR Ca^{2+} release following pharmacological challenge^{28,29} and in RyR2-P2328S variants modelling the pro-arrhythmogenic catecholaminergic polymorphic ventricular tachycardia^{78–80}. In these examples, the underlying in vivo source of microdomain Ca^{2+} would likely remain the RyR-mediated Ca^{2+} release modelled here, as opposed to T-tubular Ca_{v}1.1 or Ca_{v}1.2 channel mediated entry of extracellular Ca^{2+}. Thus, early Ca^{2+} skeletal muscle voltage clamp currents, _{Caf} (~ 25 µA/cm^{2})^{81} and later cell attached patch clamped cardiomyocyte _{CaL} (~ 10 pA/pF^{82}; would contribute _{influx} terms (~ 8.64 × 10^{–7} and ~ 6.91 × 10^{–8} mol/m^{2}/s assuming similar _{T}/_{S} and _{influx}, at comparable test voltages. The larger skeletal muscle late _{Ca} (80 µA/cm2, giving _{influx} ~ 2.76 × 10^{–6} mol/m^{2}/s) evolves over time courses (hundreds of ms) more prolonged than excitation contraction coupling and shows rapid off kinetics on action potential repolarization^{83,84}.

Microdomain µM-[ Ca^{2+}] could also regulate other signalling biomolecules. They are involved in a bell-shaped in vitro open probability relationship for single channel RyR activation and inhibition^{14}. Here, cardiac and neuronal, RyR2 and RyR3 are more Ca^{2+}-sensitive than RyR1 but all are activated over the ~ 1 µM [Ca^{2+}] predicted in the present analysis^{85,86}. However, under their respective in vivo physiological [ATP] and [Mg^{2+}] conditions, cardiac^{87} but not skeletal muscle^{88} RyR activation involves Ca^{2+}-induced Ca^{2+} release. Skeletal muscle RyR activation instead involves direct allosteric coupling with either T-tubular Ca_{v}1.1-L-type Ca^{2+} channel voltage sensors^{15} or possibly other adjacent SR RyRs themselves coupled to such Ca_{v}1.1^{89,90}. ~ µM Ca^{2+}-CaM may also exert other cytosolic effects as on glyceraldehyde 3-phosphate dehydrogenase^{91} or itself provide local signaling domains^{36}.

Closely apposed membranes potentially mediating localized Ca^{2+} signalling involving Ca^{2+}-dependent proteins that would similarly permit divergent signalling at different sites also occur in widespread other cell types^{9,10}. At smooth muscle SR-plasma membrane appositions^{92}, local Ca^{2+} could modulate repolarizing Ca^{2+}-activated K^{+} channel activity^{93}. They also occur in neurons^{6}; cerebellar Purkinje and hippocampal neurones similarly signal using RyR- Ca^{2+} release channels^{1–4}. Amongst non-excitable cells, multiple 20–30 nm diameter membrane invaginations in thrombocyte open canalicular systems (OCS)^{7,8} form vacuolar structures apposed to membranes of the Ca^{2+}-storing deep tubular system (DTS) previously compared with muscle T-SR junctions^{94}. These gate inositol trisphosphate receptor rather than RyR mediated Ca^{2+} fluxes into the junction upon agonist stimulation. The theoretical analysis here thus combined available experimental anatomical and physiological data and diffusion theory to predict significant [Ca^{2+}] heterogeneities in the skeletal muscle T-SR-junction. Its findings might next prompt investigations of structure and function in these further examples. These might be complemented by direct experimental fluorescent Ca^{2+} indicator [Ca^{2+}]_{TSR} measurements^{35–37} were these to be able to address the small dimensions and dispersed nature of the microcompartments concerned.

Diffusional mechanics and its consequent temporal and spatial [Ca^{2+}] patterns were computationally examined in a model T-SR junction. This involved replicating the geometry within and through which the diffusional processes occurred, pre-processing through meshing and definition of physical conditions including loads, initial and boundary conditions, generation of system solutions and post-processing of the results (Supplementary Fig.

The underlying T-SR junction geometry was reconstructed virtually for use in a finite element analysis solving partial differential equations (PDEs) for the resulting diffusional processes with their accompanying boundary conditions (BCs) (See Supplementary File for software archive). The finite element method (FEM) described the complex geometry as a collection of subdomains (elements) by superimposing upon this geometry a mesh of tetrahedral elements joined at their vertices (nodes) and edges. The subdivision accurately represents this complex geometry, permits inclusion of dissimilar material properties, and provides a straightforward representation of the total solution whilst capturing local, microdomain, effects.

Specified BCs provided values of the field and related variables, in the present case, the normal derivatives of the field variable in the form of a Neumann-type BC. The FEM equations are formulated such that at the nodal connections, the value of the field variable at any connection is the same for each element connected to the node. Solutions at the edges of each adjacent element are therefore equal, ensuring continuity of field variables between elements, avoiding physically unacceptable gaps or voids in the solution^{38}. The original PDE problem is accordingly represented within each element with simpler equations approximating the solution to the original equations. Stationary linear problems whose coefficients are independent of the solution or its gradient yield a linear system of equations. In this case our PDE is time-dependent and hence the system of simpler equations is a set of ordinary differential equations (ODEs) then passed onto MATLAB solvers for numerical integration for solution. The FEM approximates the solution by minimizing the associated error function, automatically finding the linear combination of basis functions closest to the solution

Meshing and application of the FEM used Partial Differential Equation Toolbox (version 3.5 installed on 8th October 2020 by MathWorks) within MATLAB. This provides functions for solving structural mechanics, heat transfer and general PDEs using the FEM (

This entirely theoretical study did not involve animal procedures.

We thank the Medical Research Council (MR/M001288/1), Wellcome Trust (105727/Z/14/Z), and British Heart Foundation (PG/14/79/31102, PG/19/59/34582 and Cambridge Centre for Research Excellence) for their generous support. We also thank Joseph Bardsley for guidance in use of graphics software.

H.R.M. introduced use of the MATLAB system. C.L.-H.H. parametrized the T-SR model. O.J.B. developed, iterated, and ran the programmes including generating the graphical output. H.R.M. and C.L.-H.H. supervised project execution. C.L.-H.H. conceived the project and wrote the paper in conjunction with O.J.B and H.R.M.

The data that support the findings of this study are available from the corresponding author upon reasonable request and are furthermore summarized in the Supplementary file.

The authors declare no competing interests.

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^{2+}stores, is required for the induction of LTD in cultured cerebellar Purkinje cells

^{2+}release channels in rat brain

^{2+}recycling potentiates thrombin-evoked Ca

^{2+}signals in human platelets

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^{2+}-dependent shape change independently of platelet aggregation

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^{2+}and Ba

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^{2+}-independent luminescence of aequorin

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^{2+}buffers

^{2+}diffusion in microdomains and its implications for calculation of [Ca

^{2+}] at the mouth of a calcium channel

^{2+}regulation defective?

^{2+}signalling in skeletal muscle

^{2+}release channels (ryanodine receptors)

^{2+}/ calmodulin-dependent manner

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