It is well-established that conduction in cardiac tissue is regulated by ionic currents and gap junction (GJ) coupling (Shaw and Rudy, 1997; Kucera et al., 2002). In ventricular tissue, the voltage-gated sodium (Na⁺) channel, Na_v1.5, is primarily responsible for generating the depolarizing Na⁺current (I_Na), and connexin43 (Cx43) is the primary GJ protein, facilitating the passive current flow between adjacent cells (Veeraraghavan et al., 2014), both of which mediate conduction. Altering either I_Naor GJ coupling can lead to changes in conduction and ultimately increase the risk of arrhythmias (Quan and Rudy, 1990; Shaw and Rudy, 1997; Rohr et al., 1998).

We (Veeraraghavan et al., 2015; Veeraraghavan and Gourdie, 2016; Mezache et al., 2020) and others (Kucera et al., 2002; Rhett et al., 2012; Agullo-Pascual et al., 2014; Leo-Macias et al., 2016) have shown that Na_v1.5channels preferentially localize at the intercalated disc (ID), the area of cell-cell junctions in cardiac tissue. Multiple in silico studies have hypothesized that I_Naat the ID can be altered via Na⁺nanodomain signaling at the intercellular cleft space (Kucera et al., 2002; Sperelakis, 2002; Mori et al., 2008; Lin and Keener, 2010; Wei et al., 2016; Tveito et al., 2017; Weinberg, 2017; Hichri et al., 2018; Jæger et al., 2019; Wei and Tolkacheva, 2020). This, in turn, can modulate cell-cell coupling through a mechanism known as ephaptic coupling (EpC). In this paper we consider two primary effects of EpC: electrical field effects and Na⁺depletion in the intercellular cleft (i.e., the narrow extracellular space between electrically coupled cells at the ID). To elaborate briefly, EpC is governed by the following: Na⁺influx during the action potential upstroke in an “upstream” or pre-junctional depolarizing cell during a propagating electrical wave decreases the electrical potential within the intercellular cleft. This reduction of the potential within the intercellular cleft then depolarizes the “downstream” or post-junctional apposing cell from the extracellular, rather than the intracellular, side of the cell membrane. Additionally, Na⁺influx reduces the Na⁺concentration within the intercellular cleft, which governs the flux of the Na⁺channels at the ID in both cells. The width or volume of the intercellular cleft space is one of the key properties governing the magnitude of these effects. When the intercellular cleft is narrow, both the elevated transmembrane potential (V_m) and locally depleted Na⁺concentration within the intercellular cleft reduce the Na⁺current driving force and, therefore, the Na⁺current. This reduction in I_Nahas been termed “self-attenuation” and has been shown to slow conduction velocity (CV) (Kucera et al., 2002; Sperelakis, 2002; George et al., 2016; Hichri et al., 2018).

Several key properties governing EpC and conduction overall are known to change during development: Cells in neonatal myocardium do not have fully formed IDs, and Na⁺channels and GJs are distributed diffusively throughout the sarcolemma (Fromaget et al., 1992; Vreeker et al., 2014). Consistent with reduced Na⁺channel expression, it has been shown that pediatric cardiomyocytes produce a reduced I_Na, compared to adult cardiomyocytes (Cai et al., 2011; Cordeiro et al., 2013). Cx43 is essentially undetectable until 23 weeks in utero and remain randomly distributed on the sarcolemma in neonatal cardiomyocytes (Peters et al., 1994; Hirschy et al., 2006; Vreeker et al., 2014; Swift et al., 2020). Vreeker et al. showed that Cx43 tends to relocate to the lateral membrane around 5 months postnatal and does not begin to preferentially localize at the ID until around 2.5–5 years old, with full preferential localization occurring at roughly 7 years of age (Vreeker et al., 2014). Na_v1.5channels, however, reside on the lateral membrane in neonatal cardiomyocytes and begin to begin to preferentially localize at the ID around 5 months postnatal, much earlier than Cx43 (Harrell et al., 2007; Vreeker et al., 2014). Additionally, studies have shown that adult cardiomyocytes are larger than neonatal cardiomyocytes (Cordeiro et al., 2013; Vreeker et al., 2014; Swift et al., 2020). This is especially important given that cell size broadly influences all electrical activity in the cell by altering surface area, cell volume, membrane capacitance, ion channel expression, etc. (Kato et al., 1996; Spach et al., 2000).

We hypothesize that the developmental-associated increase in both Na⁺and GJ current will increase CV from the neonatal stage to the adult and that this regulation will be influenced by the relative strength of EpC at different developmental stages. In the paper, we perform a wide parameter investigation, varying age-associated parameters including gap junctional conductance (f_gap), cell size (S), Na⁺channel density (ρ_Na), and Na⁺channel ID localization (ID_Na) and measure CV in simulated cardiac tissue. To our knowledge, no studies have investigated the interdependence of these parameters on conduction within a health myocardium during development from neonatal to adult tissue. While one study investigated the changes on impulse conduction in the canine myocardium, it was limited to a comparison of 8 week old postnatal to adult purkinje fibers (Rosen et al., 1981). Thus, in our study, we investigate conduction through ventricular tissue in a range of developmental stages and conditions. We find that CV has a biphasic dependence on cell size, in a manner that depends on GJ coupling. In addition, we find that ephaptic effects play a larger role in conduction for larger cells with low GJ coupling. Interestingly, simulations predict that variability in cellular properties in intermediate developmental stage between neonatal cardiomyocytes and adult cardiomyocytes can lead to a wide range of possible CV values, but this range narrows to a normal range with adult tissue-associated parameters.

Full details of the computational model are provided in Supplementary Material. Briefly, we simulate a 50-cell cable of guinea pig ventricular myocytes (Livshitz and Rudy, 2007) that incorporates a Markov chain model for the wild-type (WT) Na⁺channel (Clancy et al., 2002), shown in Figure 1. We note that a Markov chain formulation for I_Na was not necessary to reproduce our results, as prior studies have shown EpC effects are reproduced by Hodgkin-Huxley Na⁺channel models (Kucera et al., 2002; Weinberg, 2017; Moise et al., 2021), but rather is used to facilitate an appropriate comparison with our prior work simulating Na_v1.5gain-of-function mutations (Nowak et al., 2021), which were also modeled with a Markov chain. As in our previous work (Greer-Short et al., 2017; Weinberg, 2017; Nowak et al., 2020, 2021) and work performed by others (Kucera et al., 2002), we account for non-uniform Na⁺subcellular localization by spatially discretizing each cell into two ID membrane patches at the ends of the cell and axial membrane patches along the length of the cell. The number of axial patches varied with the size of the cell, as described below, with each axial patch fixed in length (L_p = 10 μm).

Cells are coupled via GJs and EpC: gap junctional coupling is represented via gap junctional conductances coupling ID nodes of adjacent cells (g_gap, represented by resistor R_gap in the electrical circuit). EpC is represented by a T-shaped junction of two intercellular cleft resistances (R_cl) and a radial bulk (R_radial), which are proportional and inversely proportional to intercellular cleft width w, respectively. Intracellular nodes are coupled with a myoplasmic resistance (R_myo) (Figure 1A). The nominal value for g_gap is defined as ggap0=1266nS, and changes in GJ coupling are accounted for by adjusting the GJ scaling factor f_gap (between 0 and 1), such that ggap=fgapggap0.

A wide range of macroscopic GJ conductance values have been measured experimentally, with values ranging from the low 10s of nanosiemens up to approximately 2,000 nS (Weingart, 1986; Wittenberg et al., 1986; White et al., 1990; Moreno et al., 1994; Kwak and Jongsma, 1996; Verheule et al., 1997; Kucera et al., 2002; Valiunas et al., 2002; Desplantez et al., 2007; McCain et al., 2012; Nielsen et al., 2012). While studies have shown increases in Cx43 expression with age, to our knowledge, no studies have quantified GJ conductances throughout development. Therefore, we performed simulations with GJ conductance spanning from the low to high end of physiological measurements and hypothesized that such a range is qualitatively similar to different developmental stages and consistent with Cx43 expression changes. Importantly, the values chosen were consistent with experimental measurements of conduction for neonatal and adult myocardium (Rosen et al., 1981; George et al., 2019; King et al., 2021). We note that age-associated changes in GJ localization are represented as changes in GJ coupling (i.e., changes in f_gap), as all GJs are located at cell ends in the one-dimensional tissue model.

We account for dynamic [Na⁺] in three spaces: (i) the ID ([Na+]idisc), mediated by INadisc and intracellular diffusion, (ii) the intercellular cleft ([Na+]ecleft), with volume proportional to w and mediated by ID Na⁺current INadisc and passive diffusion with the bulk extracellular space, and (iii) the axial intracellular space ([Na+]iax), mediated by axial Na⁺current INaax and intracellular diffusion. The cable was paced at one end with a specified basic cycle length (BCL). Unless otherwise stated, for all simulations, we utilize a BCL = 500 ms or a pacing rate of 2 Hz, which is normal pacing for the guinea pig model that is utilized for this study.

In addition to GJ coupling, we perform simulations in which we adjust several key age-associated properties: (1) We vary Na⁺channel localization at the ID (ID_Na) between 0.1 and 1, to account for Na⁺channel redistribution, where ID_Na = 1 represents 100% or all Na⁺ channels localized at the ID. (2) We define and vary a cell size scaling factor (S) between 0.2 and 1 to account for cell size growth, where 100% cell size represents an adult cell. The cell geometry is assumed to be cylindrical, with radius r and length L, defined as r = Sr₀ and L = n_pL_p, where the nominal adult radius r₀ = 11 μm, the number of axial patches np=Snp0, and the maximum axial membrane patches np0=10. Note, we only consider values of Ssuch that n_p is a whole number, and that nominal adult cell length L0=np0Lp=100 μm. Additionally, note that since S scales both cell length and radius, cell membrane surface area is scaled by S², e.g., S = 0.4 corresponds with surface area scaled by a factor of 0.16. (3) We also vary the Na⁺channel density (ρ_Na) between 0.2 and 1 to account for age-associated changes in Na⁺channel expression, where 100% ρ_Narepresents full expression of Na⁺channels on the cell membrane. The total cellular Na⁺conductance is proportional to both ρ_Naand total cell surface area, such that we can define a normalized total Na⁺channel conductance (GNa=ρNaS2, also between 0 and 1), where the total Na⁺channel conductance (in physical units) is equal to G_Na, scaled by the nominal total Na⁺channel conductance (GNa0=21.78 mS/cm2). Thus, for ρ_Naand Sof 100%, total Na⁺ channel conductance is GNa0. (4) Finally, we vary the intercellular cleft width (w) from 10 to 40 nm, consistent with intercellular cleft width ranges measured at the ID in our previous work (Veeraraghavan and Gourdie, 2016; Greer-Short et al., 2017; Nowak et al., 2020, 2021).

In this study, we investigate the regulation of key age-dependent properties, specifically cell size, Na⁺channel density and localization, and GJ coupling, on cardiac conduction. We summarize our key findings: CV is consistently increased by increased Na⁺current density, across all conditions. However, simulations predict that CV biphasically depends on cell size, depending on the strength of GJ coupling. That is, CV increases with increasing cell size for high GJ coupling, yet decreases with increasing cell size for low GJ coupling. As a consequence, CV and total cell Na⁺channel conductance are well-correlated in cardiac tissue with high GJ coupling, but not correlated with CV for low GJ coupling. We predict that the role of EpC governing conduction changes during development, such that neonatal tissue is less sensitive to changes in EpC due to smaller Na⁺channel ID localization. We postulate that even though GJ coupling is low during this early stage, the small cell size and therefore small membrane capacitance is such that the lower Na⁺current density is still sufficient to maintain robust conduction in the myocardium. However, conduction is very slow during this stage due to the relatively high ratio of cell-cell junctions for a given length of tissue. These findings are consistent with Swift et al. (2020), who found slower atrioventricular conduction in neonatal and early postnatal rats, compared with adult rat myocardium.

As development progresses, model predictions are consistent with faster conduction. Further, simulations predict that a wide range of conduction velocities are possible during intermediate developmental stages due to variability in cellular/tissue properties and the relative timing of developmental changes. However, despite similar variability in parameter values, this variability in CV prediction narrows in adult tissue and is consistent with experimental measures (George et al., 2019; King et al., 2021). Interestingly, we see a larger influence of EpC effects in larger cell sizes with reduced GJ coupling, as would be the case in intermediate developmental stages and in adult myocardium for pathological conditions such heart failure (Smith et al., 1991; Peters et al., 1997; Yao et al., 2003; Akar et al., 2004; Poelzing and Rosenbaum, 2004), indicating a possible mechanism for maintained conduction during such transitional or diseased states. Additionally, this enhanced variability in conduction for intermediate developmental stages may be desirable, as this variability suggests an ability to adapt and modulate cardiac activity in response to developmental perturbations, which inherently vary significantly for different individuals.

Previous studies have investigated the properties of Na⁺channels and GJs and their roles in the developing heart. Work by Harrell et al. (2007) found that Na_v1.5mRNA was significantly more up-regulated in adult mouse hearts than in neonatal hearts. Similarly, Cordeiro et al. (2013) find that both peak and late I_Nais significantly smaller in neonatal canine cardiomyocytes compared to that of adults, and Cai et al. (2011) found the same in human atrial cardiomyocytes. Vreeker et al. (2014) show that the GJ protein Cx43 transitions from diffusely distributed on the cardiomyocyte membrane to highly associated with mechanical junctions at the ID during development in human samples, as similarly shown in rat myocardium (Angst et al., 1997). In contrast, Peters et al. (1994) found that GJs and adherens junctions have a highly correlated distribution over postnatal development of the ventricle in human samples. Incorporating developmental changes in GJ distribution, Spach et al. (2000) previously simulated a neonatal cardiomyocyte with diffuse GJs and smaller size and found that conduction was slower in the neonatal tissue, compared with adult myocardium with GJs primarily localized at the ID and larger size.

While these studies are crucial in understanding cardiac development, to our knowledge, our study is the first to combine all these key developmental changes occurring in conjunction with each other, specifically cellular size, gap junctional conductance, and Na⁺channel expression and distribution. Thus, we find that ephaptic effects are more pronounced in larger cells with low GJ coupling. We also find that CV biphasically depends on cell size in a manner dependent on GJ coupling: CV is relatively fast for both low GJ coupling and small cell size and high GJ coupling and large cell size. Interestingly, we find that CV is correlated total Na⁺conductance for high GJ coupling. Finally, by incorporating previous data on developmental changes (Peters et al., 1994; Spach et al., 2000; Harrell et al., 2007; Cai et al., 2011; Cordeiro et al., 2013; Vreeker et al., 2014), we predict how the variability in the ranges for conduction change in conjunction with developmental stages.

Recently, we investigated the age-dependent manifestation of a long QT type 3 (LQT3)-associated gain-of-function mutation in Na_v1.5 (Nowak et al., 2021). LQT3, while relatively rare, has a high mortality, reaching 49% (Vignati, 2007), and critically patients tend to remain asymptomatic until well after puberty (Beaufort-Krol et al., 2005; Wilde et al., 2016; Kutyifa et al., 2018). Interestingly, in pediatric patients with LQT syndromes, Na⁺channel blockers are commonly prescribed as a form of chronic management (Hanisch, 2001), and our results are consistent with the safety of this approach due to the weak dependence of conduction on overall total cell Na⁺channel conductance. In our study, we predicted that not only does the manifestation of arrhythmias depend on developmental stage, but that the sensitivity to changes in intercellular cleft width depended on these age-associated properties as well. Interestingly, Brugada syndrome (BrS), an inherited cardiac arrhythmia disorder caused a loss-of-function mutation in Na_v1.5, also often manifests later in life; the average first event occurs around 42 years old (Milman et al., 2017). Our study is consistent with this clinical manifestation, as simulations predict that in these early developmental stages with low GJ coupling, conduction is less sensitive to overall Na⁺conductance (Figure 4), due to the overall small cell sizes in neonatal tissue.

Finally, we acknowledge limitations of our study. To study developmental changes in Na⁺channel distribution, we incorporate this critical subcellular detail in the representation of cardiac tissue; however, our model is still a simplification of the complex cardiac tissue structure. Specifically, our model assumes a simplified cylindrical cell and uniform intercellular cleft, while the geometry of individual cells and the ID structure is known to be complex heterogeneous (Veeraraghavan et al., 2015). Future work will focus on investigating how ID structure changes in development and impacts conduction (Moise et al., 2021). Additionally, cardiac tissue is a three-dimensional structure, and our one-dimensional cable representation cannot fully represent all aspects of developmental changes, such as GJ localization along the lateral membrane. In a two- or three-dimensional tissue, GJ redistribution from the lateral membrane to the ID would also be expected to impact conduction heterogeneously dependent on the direction of wavefront propagation relative to the underlying tissue geometry, such that early developmental stages are associated with isotropic conduction while anisotropic conduction is associated with adult myocardium, consistent with previous work from Spach et al. (2000). Further, while we observe minimal differences in CV between normal and slow pacing rates, Entz et al. (2016) previously showed that CV slowing at faster pacing rates differed between longitudinal and transverse conduction, and further was regulated by extracellular ionic composition, suggesting that properties governing EpC may similarly modulate heart rate dependence. Additionally, expression of ion channels vary across individual patients, and these differences may impact conduction during development. Further, recent work from the Posnack lab has shown that several expression of several key ion channel and calcium handling proteins vary with developmental stage in rats (Swift et al., 2020). We are particularly interested in similar changes in human myocardium, and future work will incorporate such details as available and established in the literature.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

MN performed computational studies and wrote the original manuscript draft. RV contributed to the manuscript draft. SP and SW designed simulations, provided funding, and contributed to the manuscript draft. All authors contributed to the article and approved the submitted version.

This study was supported by funding from the National Institutes of Health, grant numbers R01HL138003 (SW, SP), R01HL102298 (SP), and R01HL148736 (RV).

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.

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