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Edited by: Mark Potse, Inria Bordeaux – Sud-Ouest Research Center, France

Reviewed by: Marina Scardigli, European Laboratory for Non-Linear Spectroscopy (LENS), Italy; Aslak Tveito, Simula Research Laboratory, Norway

This article was submitted to Cardiac Electrophysiology, a section of the journal Frontiers in Physiology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

T-tubules are invaginations of the lateral membrane of striated muscle cells that provide a large surface for ion channels and signaling proteins, thereby supporting excitation–contraction coupling. T-tubules are often remodeled in heart failure. To better understand the electrical behavior of T-tubules of cardiac cells in health and disease, this study addresses two largely unanswered questions regarding their electrical properties: (1) the delay of T-tubular membrane depolarization and (2) the effects of T-tubular sodium current on T-tubular potentials. Here, we present an elementary computational model to determine the delay in depolarization of deep T-tubular membrane segments as the narrow T-tubular lumen provides resistance against the extracellular current. We compare healthy tubules to tubules with constrictions and diseased tubules from mouse and human, and conclude that constrictions greatly delay T-tubular depolarization, while diseased T-tubules depolarize faster than healthy ones due to tubule widening. Increasing the tubule length non-linearly delays the depolarization. We moreover model the effect of T-tubular sodium current on intraluminal T-tubular potentials. We observe that extracellular potentials become negative during the sodium current transient (up to −40 mV in constricted T-tubules), which feedbacks on sodium channel function (self-attenuation) in a manner resembling ephaptic effects that have been described for intercalated discs where opposing membranes are very close together. The intraluminal potential and sodium current self-attenuation however greatly depend on sodium current conductance. These results show that (1) the changes in passive electrical properties of remodeled T-tubules cannot explain the excitation–contraction coupling defects in diseased cells; and (2) the sodium current may modulate intraluminal potentials. Such extracellular potentials might also affect excitation–contraction coupling and macroscopic conduction.

Transverse (T-)tubules are deep invaginations of the lateral membrane of skeletal and cardiac muscle cells. In mammalian ventricular cardiomyocytes, T-tubules form a complex network throughout the cell, especially in species with high heart rates such as mice (

A first question that has hardly been addressed concerns the delay after which deep segments of T-tubules depolarize. Based on measurements of dextran diffusion out of T-tubules and corresponding modeling of this diffusion process,

A second question concerns the role played by T-tubular sodium current. Although the existence of a T-tubular pool of sodium channels is still under debate (

Taken together, this work contributes to the fundamental understanding of passive and active electrical behavior of cardiac T-tubules in health and disease.

Our model approximated one T-tubule as a cylinder of length

Delay of membrane depolarization in murine and human T-tubules.

Parameters for T-tubular model of healthy and post-myocardial (MI) T-tubules of human and mouse.

_{e} (Ω cm) |
_{m} (μF/cm^{2}) |
_{m} (mS/cm^{2}) |
||||

Healthy mouse | 98.5_{(}_{1}_{)} |
9_{(}_{2}_{)} |
100 | 2 | 0.143 | 101 |

Constrictions | 9.85_{(}_{1}_{)} |
0.45^{∗}/9^{∗∗} |
100 | 2 | 0.143 | 5^{∗}/101^{∗∗} |

Post-MI mouse | 108_{(}_{1}_{)} |
9_{(}_{2}_{)} |
100 | 1 | 0.143 | 101 |

Healthy human | 147_{(}_{3}_{)} |
5_{(}_{4}_{)} |
100 | 2 | 0.143 | 21 |

Post-MI human | 218_{(}_{3}_{)} |
5_{(}_{4}_{)} |
100 | 1 | 0.143 | 21 |

^{∗}, parameters of each of the constrictions from the five-constrictions model;

^{∗∗}, parameters of the overall constricted model. Numbers between brackets correspond to the following publications: (1),

_{e}) is set at 100 Ω cm. Conductance of resting membrane (

_{m}) is set at 0.143 mS/cm

^{2}(

_{K1}and leak currents. Capacitance (

_{m}) was set at 2 μF/cm

^{2}in healthy cells to simulate microfolds, and at 1 μF/cm

^{2}to simulate the loss of microfolds in disease (

T-tubules are typically tortuous. Unless the tubules are extremely convolved, i.e., as long as the radius of the tubule remains much smaller than the curvature of the tortuosity, tortuosity can be modeled as an increased length. Moreover, glycosaminoglycans, or collagen as observed in T-tubules from diseased hearts (_{e}) were systematically varied. In this T-tubular model, sodium current in the different nodes was modeled according to _{Na} (normalized to membrane capacitance) is given by

where _{Na} is the maximal conductance of the sodium current [set unless specified otherwise to 23 mS/μF (_{m} is the membrane potential, _{Na} is the Nernst potential of sodium (set to +55 mV), _{Na} of 23 mS/μF was determined in whole-cell recordings in chick embryonic heart cells (_{Na} of 19.4 mS/μF determined in rat ventricular cardiomyocytes (_{Na} values are also systematically varied in selected simulations.

The sodium current gating variables were governed by differential equations of the form

with _{∞} and τ_{y} being functions of voltage given explicitly in _{∞} and τ_{y}.

A voltage pulse from −85 to −20 mV was applied at the mouth of the tubule (−80 to +20 mV for simulations without sodium current). This situation mimics a cell that is perfectly voltage clamped at the level of its bulk membrane. The value of −20 mV was chosen to elicit a large sodium current. Calcium channels were not included in the model as these channels open only when the majority of sodium channels have already inactivated.

The intracellular potential was considered spatially uniform, set to the value given by the voltage clamp protocol. This assumption is justified based on the following estimation of the potential generated by the current emanating from a long cylinder in a conductive medium. We consider a cylinder of radius _{m}⋅_{Na}, where _{m} is the membrane capacitance per unit area in μF/cm^{2} and _{Na} is the sodium current density in absolute value, in μA/μF). By applying the principle of charge conservation and Gauss’ theorem, and taking advantage of the radial symmetry of the problem (

For _{Na} = 500 μA/μF (quite large), _{m} = 2 μF/cm^{2}, ρ = 200 Ω⋅cm, and _{m} = 1 μF/cm^{2}, and ρ = 200 Ω⋅cm, one obtains a potential of 35 μV, which is indeed the range registered from neuronal preparations by electrode arrays (

The model was implemented numerically using a one-dimensional finite-difference scheme. Membrane potential (_{m}) and the gating variables of the sodium current were integrated using the forward Euler method using a constant time step of 0.25 ns. Simulations were implemented and run in MATLAB (version 2015a, The MathWorks, Natick, MA, United States). The MATLAB source code producing the figures is provided in the

First, we set out to answer the question how long it takes to charge the membrane as a capacitor and depolarize the T-tubules in the absence of sodium current. In other words: what is the delay between depolarization at the plasma membrane and deep in the T-tubules?

When we consider an electrophysiological experiment in which a cardiomyocyte is voltage-clamped, a voltage step at the pipette site will first induce a capacitive current into the cell membrane, which will cause depolarization (

When the tubule length was systematically varied in the murine and human T-tubular models, the delay until −40 mV was reached in the deepest T-tubular segment exhibited a non-linear dependence on tubule length (

Varying parameters in the murine and human T-tubule models. In the models of mouse healthy (red solid line), mouse post-myocardial infarction (MI) (red dashed line), human healthy (black solid line), and human post-MI (black dashed line) T-tubules, we varied tubule length

Classical cable theory (

where _{m} is the conductance of the membrane per unit area, and ρ_{e} is the resistivity of extracellular space. It follows that the length constant for the healthy mouse T-tubule (characteristics specified in

where Δ_{0} = 100 mV is the voltage applied at the tubule mouth and

For the healthy mouse T-tubule, Eq. 5 yields a voltage drop of 4.72 mV, while Eq. 6 yields only 0.117 mV. This corresponds to the negligible voltage drop shown in _{mouse post–MI} = 194 μm; λ_{human healthy} = 227 μm; λ_{human post–MI} = 276 μm), the voltage drops are negligible in all cases (mouse post-MI: Δ_{inf} = 4.52 mV, Δ_{sealed} = 0.107 mV; human healthy: Δ_{inf} = 2.18 mV, Δ_{sealed} = 0.0243 mV; human post-MI Δ_{inf} = 1.7938 mV, Δ_{sealed} = 0.0164 mV).

To quantify how constrictions change the delay of depolarization deep in a T-tubule, we incorporated constrictions into our model of an unbranched cylindrical T-tubule. We used the parameters for the healthy mouse T-tubule as these parameters led to the longest depolarization delay (10 μs to depolarize the innermost T-tubule segment to −40 mV). We introduced five 450-nm-long constrictions with a 10-fold diameter reduction to 19.7 nm, their centers spaced 1.8 μm apart (_{v} channels (−40 mV) in the deepest tubular segment was reached ∼75 μs later than at the surface (

Delay of membrane depolarization in murine T-tubules with constrictions. In the healthy murine T-tubule (

Constricting the tubule 10 times to a diameter of 19.7 nm over its full length (

Additionally, we observed a voltage drop of 0.84 mV in the tubule with five constrictions (

As a next step, we investigated the effect of putative voltage-gated sodium (Na_{v}) channels on tubular depolarization (see _{Na} model in the non-constricted T-tubule are presented in _{Na} (principle of charge conservation). The negative extracellular potential contributes to the depolarization of the membrane a few mV beyond −20 mV (_{Na} = 55 mV), which diminishes the driving force of the sodium current (_{m}–_{Na}) and thus the sodium current itself (∼15% reduction in an unconstructed T-tubule, ∼35% in a T-tubule with five constrictions, and ∼40% in an overall constricted T-tubule) (

Modeling sodium current in a healthy murine T-tubule. A voltage-gated sodium current [formulated according to ^{3}, ^{3}

Modeling sodium current in a murine T-tubule with five constrictions. ^{3}, ^{3}

Modeling sodium current in an overall constricted murine T-tubule. ^{3}, ^{3}

Constricting the T-tubule delays the onset of the sodium current from ∼0.03 (no constrictions) to ∼0.15 ms (overall constriction) and ∼2 ms (five constrictions) (

Results using the _{Na} model by _{Na} density and a less negative extracellular potential. However, the degree of sodium current self-attenuation was similar (

Varying sodium current parameters in the healthy murine T-tubular model with and without constrictions. The Luo–Rudy–Livshitz sodium current (_{Na}) model [blue (_{Na} model (orange). For the tubule without constrictions _{e} _{Na}. _{Na} was systematically varied using the following values: 2.3, 4.6, 6.9, 9.2, 11.5, 13.8, 16.1, 18.4, 20.7, 23, 34.5, and 46 mS/μF. Self-attenuation was quantified as the smallest peak _{Na} in the tubule (in absolute value) normalized by the largest peak _{Na}.

In _{Na}) and the sodium current model on the delay after which −40 mV is reached, the minimal extracellular potential and on _{Na} self-attenuation. Firstly, _{Na} hardly affects the delay to −40 mV in the deepest segment of the T-tubules without and with constrictions (_{Na} induces a more negative extracellular potential (

The choice of sodium current model hardly influences the time after which −40 mV is reached (_{Na} however affects extracellular potentials stronger in the Luo–Rudy–Livshitz model than in the

This work addressed hitherto incompletely explored passive and active electrical properties of T-tubules. We show that (1) the depolarization delay in deep T-tubular membrane is negligible in our models; (2) the voltage drop along a passive T-tubule is negligible; (3) the depolarization delay depends mainly on tubule length; and (4) sodium current interacts with intraluminal potentials, which modifies its kinetics and makes it smaller (self-attenuation) in deep T-tubular segments in a _{Na}-dependent fashion.

The results of our computational model show that the delay in T-tubular depolarization and T-tubular sodium current depend on the exact geometry of the T-tubule, notably on spatial variations in tubule diameter and the presence of constrictions. The exact location of the constrictions are also expected to modulate these factors.

We observed that the activation threshold of L-type voltage-gated calcium channels in deep T-tubular segments is attained within a very short time after the cell surface is excited when the “exit resistance” of capacitive current is taken into account (∼0.01 ms for a mouse T-tubule without constrictions and up to ∼0.10 ms with overall constriction). Thus, the delay of depolarization of mouse and human T-tubules is sufficiently short to ensure excitation–contraction coupling. The T-tubular delay represents no major latency for the sodium current considering that the conduction time along a 100-μm-long cell with a macroscopic conduction velocity of 100 cm/s would be 100 μs (

Human T-tubules depolarize quicker than murine T-tubules because they are relatively wide and short. On a cross-section of a cardiomyocyte, human T-tubules look like spokes on a wheel, and do not form intricate networks like in murine cells (_{v}1.2 trafficking (

Without dilatations and constrictions, our model gives a negligible voltage drop of 0.117 mV (or 0.117%) at steady state from mouth to deep T-tubular node. This is comparable to previously reported results: for a rat T-tubule of 6.84 μm (

Moreover, tubules exhibit branching, which we did not consider. Several confocal and super-resolution imaging studies have demonstrated the complex T-tubular topology (

Our model is considerably simpler than a number of previous models of T-tubules. The model of

In addition, in our model, we did not incorporate irregular tubular shapes (e.g., non-circular or variable cross-section shapes, highly convolved tubules) or ion concentration changes. The first would require a finite-element modeling approach and the second would require the implementation of ion fluxes using the Nernst–Planck equation. Both are numerically much more elaborate and demanding. Rather, our aim was to develop a model answering the following two questions: (1) how long does it take for a depolarization to propagate into the T-tubule and (2) how does the presence of the sodium current influence this depolarization. The simplicity of our model can be regarded as an advantage as it permits new insights with modest computational effort.

When inserting the sodium current in our computational model, we found that the sodium current self-attenuates deep in the T-tubules, and the extracellular potential becomes negative (

The self-attenuation of the sodium current will not affect the calcium current as this effect dies out before the calcium channels open. Moreover, we assumed stable calcium concentrations during our simulations, as we focused on the characterization of the sodium current in our study and ran short simulations of 1 ms, given that the sodium current lasts about 1 ms (

Interestingly, sodium current showed faster activation and inactivation kinetics in the deeper segments of the tubule due to the negative extracellular potentials, and the resulting more positive transmembrane potentials. Given that length and constrictions influence the sodium current, changing the T-tubular geometrical pattern, for instance during heart failure (

Given the self-attenuation of the sodium current, it may be interesting to investigate whether the late sodium current in deep T-tubular segments is also quenched, which has been predicted to occur at the intercalated disc (

Importantly, the Luo–Rudy–Livshitz model [developed initially for the guinea pig action potential (

It would of course be interesting to see how the results are influenced when the intracellular potential is part of the dynamics. Such an investigation would require a detailed model of the extracellular space, the intracellular space and the membrane, such as the finite element model of

Taken together, our study gives important insights in the passive and active electrical behaviors of cardiac T-tubules. We show that biophysical properties of the sodium current as well as T-tubular depolarization greatly depend on T-tubular geometry. When investigating T-tubular structure and function in health and disease, considering these behaviors may be worthwhile to understand the functional consequences of structural remodeling, especially in context of the T-tubular sodium current.

All datasets generated for this study and the respective source codes are included in article/

SV contributed to the conceptualization, the preparation and the writing of the original draft. SV and JK contributed to the methodology and visualization. SV, HA, and JK contributed to the reviewing and editing of the manuscript. HA and JK contributed to the supervision. JK contributed to the software and formal analysis.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors express their gratitude to Dr. Jean-Sébastien Rougier for thorough feedback on the manuscript. This manuscript has been released as a pre-print at

The Supplementary Material for this article can be found online at: