Effect of gap junction distribution on impulse propagation in a monolayer of myocytes: a model study

Abstract

Aims

To use microstructural computer models to study how four features of myocardial architecture affect propagation: brick wall tissue structures, jutting at cell ends, gap junction distribution and conductance along cell borders, and increased structural discontinuity.

Methods and results

Simulations of longitudinal and transverse plane wave propagation and point propagation were performed in several two-dimensional (2D) microstructural models of adult cardiac tissue. Conduction velocities and maximum upstroke velocities were measured for a range of gap junction conductances and distributions. In tissue models with normal to low connectivity, brick wall architecture and jutting decrease cell-to-cell delay, increase longitudinal conduction velocity, and decrease longitudinal maximum upstroke velocity. Transverse conduction velocity also increases if the overlap or jutting introduces additional lateral (side-to-side) connections between myocytes. Both end-to-end and side-to-side interplicate gap junctions increase longitudinal and transverse conduction velocity; however, side-to-side interplicate gap junctions have the greatest influence on transverse conduction velocity and longitudinal and transverse maximum upstroke velocity.

Conclusion

The complex structure of myocardium creates additional pathways of current flow that enhance both longitudinal and transverse propagation. These alternative pathways of current help to maintain conduction as connectivity between cells decreases.

Introduction

Over the past few decades, experimental and modelling studies have established that electrical propagation in cardiac tissue is strongly influenced not only by ionic properties but also by myocardial architecture. As a result, an increasing number of new experimental therapies for cardiac disease have focused on restoring normal cardiac structure and function by implanting tissue constructs with controlled architecture or by administering drugs that target intercellular coupling. ^{1 , 2} In order to gain a better understanding of the effect of myocardial architecture on impulse propagation in normal and diseased hearts, researchers have developed computer models of varying complexity that give a more realistic representation of cardiac structure.

One of the earliest developments in realistic modelling of cardiac structure was the shift from a continuous view that represents cardiac tissue as a fully connected medium to a discontinuous view that represents cardiac tissue as groups of cells that are only interconnected through low resistance pathways. ³ Comparisons of one-dimensional (1D) continuous and discontinuous fibres have shown that for high gap junction resistances, the discontinuous fibre behaves very different from the continuous fibre. ^{3 , 4} In particular, conduction velocity is not inversely proportional to the square of axial resistivity and the maximum rate of rise of the action potential increases as gap junction resistance increases. ³ In two dimensions, the transition from a continuous to discontinuous cardiac structure is more challenging to model. Leon and Roberge ⁵ developed a parallel cable network in which each row of cardiac cells is represented as a continuous fibre and multiple rows of cardiac cells are connected in parallel by a regular pattern of transverse resistors. This approach is computationally efficient and allows for a more discontinuous representation of transverse propagation; however, the assumption of continuous propagation in the longitudinal direction is not valid for high gap junction resistances. In recent years, Spach et al . have constructed a detailed two-dimensional model (2D) of cardiac tissue which incorporates features of myocardial architecture such as variable cell shapes, overlap between cells, and jutting at cell borders. Using neonatal and adult models of cardiac tissue, they have been able to reproduce conduction velocities and maximum rate-of-rise values that are very similar to those that have been observed experimentally. One of their most interesting findings from this model is that propagation speed and action potential waveshape in the transverse direction are significantly affected not only by gap junction distribution but also by cell size. ^{6 , 7}

Although a number of detailed computer models of myocardial architecture have been proposed, there has been no systematic comparison of these representations using a common framework. Consequently, questions remain about how various features of the architecture contribute to impulse propagation and, more importantly, how changes in the shape and interconnection of cells lead to conduction block and arrhythmia. This study investigates the role of four components of myocardial architecture, namely, brick wall tissue structure, jutting at cell ends, gap junction distribution and conductance along cell borders, and structural discontinuities (intercellular clefts), on longitudinal and transverse conduction velocity and action potential morphology. The results suggest that loss of transverse coupling that can arise in disease can slow propagation velocity along the length of the fibre, increasing the likelihood for conduction block.

Methods

Tissue structure

To explore the effect of specific components of cell architecture on electrophysiological properties, we constructed three subgroups of 2D models that represent cardiac tissue as groups of myocytes interconnected through discrete gap junction resistances. All of the tissue models are shown in Figure  1 A–H .

The first subgroup consists of tissue models with varying degrees of overlap between cells in neighbouring rows. In the uniform (UN) model ( Figure  1 A ), the myocytes are represented as rectangular boxes, have uniform length and width, and are stacked directly on top of each other. In the brick wall (BW) model ( Figure  1 B ), the myocytes are represented as rectangular boxes, have uniform widths and uniform lengths (except at the borders) and have overlap in a pattern similar to a brick wall. In the random brick wall (RBW) model ( Figure  1 C ), the myocytes are represented as rectangular boxes, have uniform widths but variable lengths and have overlap.

The second subgroup consists of tissue models with jutting at the cell ends. The uniform model with jutting (UNJ) ( Figure  1 D ) and the random brick wall model with jutting (RBWJ) ( Figure  1 E ) are similar to the (UN) and (RBW) models, respectively, except that the myocytes have a regular stairstep pattern of jutting at the cell ends.

The third subgroup consists of tissue models with randomly shaped cells and varying degrees of transverse coupling. In the random (RAND) model ( Figure  1 F ), each myocyte is represented as a random stairstep unit with irregular jutting at the cell borders. Both the length and width of the cells are varied, and the myocytes are stacked in a semi-brick wall configuration. This model is defined having normal structural discreteness. In the random model with 50% structural discreteness (RANDwSD50) ( Figure  1 G ), long, narrow structural discontinuities are added into the RAND model by removing all of the transverse connections between 50% of the cell–cell lateral borders. The length of the intercellular clefts ranges from the length of one half of a cell (∼77 μm) to the length of three cells (∼432 μm). In the random model with 100% structural discreteness (RANDwSD100) ( Figure  1 H), long, narrow structural discontinuities are added into the RAND model by removing all of the transverse connections between 100% of the cell–cell lateral borders.

The random mesh is generated using an iterative cell growth procedure. First, a rectangular domain is partitioned into an n by m grid which contains a total of n times m elements. The average length ( l ) and width ( w ) of the cell are specified by the user. A single element (seed) is selected for each cell at intervals of l along the horizontal axis and intervals of w along the vertical axis. During each iteration, all of the myocytes in the grid are allowed to ‘grow’ by one element into the surrounding area that has not already been assigned to a myocyte. Three criteria are imposed during the growing process: ¹ a growing cell must always be interconnected; ² the ends of the cell should jut outward; ³ and a cell should not exceed the maximum cell size specified by the user. The cell growth process continues until no cells can expand into the surrounding area.

The uniform and brick wall tissue models (UN, UNJ, BW, RBW, and RBWJ) were 0.1728 cm × 0.1728 cm, contained 46 656 nodes and ∼864 myocytes. The random models (RAND and RANDwSD) were 0.72 cm × 0.72 cm, contained 810 000 nodes, and ∼15 000 myocytes.

Myocyte representation

Each myocyte is divided into elements with length and width of 8 μm as shown in Figure  2 A . Similar to the method of Spach et al ., ⁶ the membrane is represented as two parallel surfaces separated by an intracellular space with a depth of 11.3 μm. The nominal surface area is 128 μm ² and was increased to 242 μm ² to account for the irregular surface of the cell membrane. The average cell volume was 39 053 μm ³ , the average cell length was 144 μm, and the average cell width was 24 μm. The myocyte shapes were generated randomly and are only an approximation of cell shapes that may occur in ventricular cardiac tissue. Other tissue parameters are given in Table  1 .

Gap junction distribution

The discrete gap junctions connecting myocytes together were distributed in patterns observed experimentally in neonatal and adult cardiac tissue. ^6–8 In the neonatal gap junction configuration shown in Figure  2 B , punctuate gap junctions were spaced every 8 μm, and the cells were 100% coupled together at the ends of cells (end-to-end) and along the lateral sides of cells (side to side). In the adult gap junction configuration shown in Figure  2 C , the cells were coupled end to end by plicate gap junctions in the longitudinal direction and by interplicate gap junctions in the transverse direction. This end-to-end region between myocytes is defined as the intercalated disc. The cells were randomly coupled along 15% of the side-to-side borders by combined plicate junctions. In this study, we also distinguish between end-to-end interplicates, which are located inside the intercalated disc, and side-to-side interplicates, which are located at the edge of the intercalated disc. The position of side-to-side interplicate is designated by arrow in Figure  3 B . The range of individual values of gap junction conductance used in this study ( g_j : 0.01–1 μS) was consistent with values measured experimentally and with values used in other cellular models of cardiac tissue. ⁶

Histological and microscopy studies of healthy myocardium indicate that the interplicate regions of myocytes contain a larger percentage of gap junctional area (∼80%), and consequently have a larger conductance than the plicate regions. ^9–11 In most of the simulations in this study, we assume that all gap junctions have the same conductance values; however, we also run a series of simulations in the RAND and RBW tissue structures to determine how the removal of different types of gap junctions ( g_j = 0) affects propagation. The RANDwSD model ( Figure  1 G and H ) is essentially a special case in which we remove a certain percentage of side-to-side interplicate gap junctions in order to simulate different levels of structural discontinuity that occur in cardiac disease. All results from these simulations were compared with a nominal case that had an adult gap distribution with all conductances set equal to 0.154 μS.

Membrane properties

The membrane ionic currents ( I_ion ) were represented using LR1 dynamics, ¹² with modified calcium ( d,f ) kinetics. The sodium current, and consequently the action potential upstroke, was unchanged in the modified version. The maximum upstroke velocity in a membrane patch for both the original and the modified models was 420 V/s.

Computational details

Simulations were performed in each tissue structure (UN, UNJ, BW, RBW, RBWJ, and RAND) for both neonatal and adult gap junction distributions for three different gap junction conductances: 0.01, 0.10, and 1 μS. Longitudinal and transverse plane waves were generated by stimulating the cells along the left border or the top border of the sheet with a set of intracellular current pulses. In addition, a point stimulus was generated in the RAND and RANDwSD models by applying intracellular current pulses to 110 nodes in a square region in the centre of the sheet. All stimuli were 2 ms in duration and ∼1.5–2 times threshold intensity.

Longitudinal upstroke velocities were calculated using membrane nodes along a line parallel to the longitudinal axis of the 2D sheet. Transverse upstroke velocities were calculated using membrane nodes along a line parallel to the transverse axis of the 2D sheet. Measurements in the 0.1728 cm × 0.1728 cm model were taken from points within the centre of the tissue to minimize boundary effect and the effect of stimulus artefact. Conduction velocities in the random models were calculated from two membrane nodes that were 0.36 cm apart and at least 0.18 cm from the borders of the sheet.

This study uses a monodomain approximation which assumes that the conductivity of the extracellular medium is infinite. A semi-implicit scheme with a conjugate gradient solver was used to solve the system of equations. ¹³ The time step was kept constant at 2 μs, and output data were recorded every 10 μs. All simulations used the CardioWave software platform. ¹⁴

Results

Effect of brick wall tissue structure

Simulations were performed using the first subgroup of tissue models (UN, BW, and RBW) to explore how the overlap of cells affects propagation. Table  2 summarizes the longitudinal and transverse conduction velocities and maximum upstroke velocities for UN, BW, and RBW tissue structures with gap junction conductances ( g_j = 0.01, 0.1, 1 μS) in both neonatal and adult gap junction distributions.

Neonatal gap distribution

When compared with the neonatal UN tissue model, the neonatal BW models show increased longitudinal conduction velocity and no significant change in transverse conduction velocity. When gap junction conductance is large ( g_j = 1 μS), the UN, BW, and RBW tissue models show little difference in either longitudinal or transverse conduction velocity.

The action potentials in the neonatal UN tissue models have larger maximum upstroke velocities for longitudinal propagation than for transverse propagation. In contrast, the action potentials in the neonatal BW and RBW tissue structures have maximum upstroke velocities that are higher in the transverse direction than in the longitudinal direction. As gap junction conductance increases, the difference between longitudinal and transverse maximum upstroke velocities decreases for all tissue structures.

Comparing the results from the first subgroup with the more realistic RAND model in the third subgroup shows that at low gap junction conductances ( g_j ≤ 0.1 μS), the action potentials in the RAND model have a lower longitudinal conduction velocity than the brick wall models (BW, RBW), but much higher than the UN model and have transverse conduction velocities that are slightly higher than in both the UN and the BW models. The longitudinal maximum upstroke velocities are slightly higher than those in the BW case but lower than the UN case. The transverse maximum upstroke velocity is approximately the same in all of the models. At high gap junction conductances ( g_j = 1 μS), however, the RAND case is comparable with the BW cases.

Adult gap distribution

When the gap junction distribution is switched from neonatal to adult, the longitudinal conduction velocity of the UN tissue model remains the same. The longitudinal conduction velocities in the BW and RBW tissue models decrease but are still larger than that observed in the UN tissue model. As expected, the reduced transverse coupling between cells slows transverse conduction velocities for all tissue structures by as much as 70%. The BW and RBW tissue models have slightly higher transverse conduction velocities compared with the UN model.

When gap junction conductance is low, tissue structures with adult gap distributions have larger transverse maximum upstrokes than tissue structures with neonatal gap distributions. The BW and RBW models also have larger longitudinal maximum upstroke velocities. The largest differences between longitudinal and transverse maximum upstroke velocities occur in brick wall tissue models with mid-range gap junction conductances ( g_j = 0.1 μS).

A comparison between the UN, BW, and RBW models and the RAND model shows that the RAND models have longitudinal conduction velocities that are slightly higher than both the UN and the RAND tissue structures and longitudinal maximum upstroke velocities that are lower than the UN and brick wall models. As the gap junction conductance increases, the difference between the RAND and RBW longitudinal maximum upstrokes becomes almost negligible. The RAND tissue models have transverse conduction velocities that are 20–30% larger than observed in the UN, BW, and RBW models. At low gap junction conductances, the RAND models also have lower transverse maximum upstrokes than the UN, BW, and RBW models. Similarly to the brick wall models, the largest differences between the RAND longitudinal and transverse maximum upstroke velocities occur for mid-range gap junction conductances ( g_j =0.01 μS).

Effect of jutting at cell borders

Because the more realistic RAND model includes jutting at the end that is not seen in the more idealized models (UN, BW, and RBW), simulations were performed in the second subgroup of tissue models (UNJ, RBWJ) in order to investigate the role of the stairstep interface. Table  2 also summarizes the longitudinal and transverse conduction velocities and maximum upstrokes for the UNJ, RBWJ tissue structures with gap junction conductances ( g_j = 0.01, 0.1, 1 μS) in both neonatal and adult gap junction distributions.

The results show that tissue structures with jutting at cell ends (UNJ, RBWJ) have larger longitudinal conduction velocities than tissue structures without jutting (UN, RBW). The most significant increases occur in UNJ tissue models and in RBWJ tissue structures with reduced transverse coupling. In RBWJ models with adult gap distributions, jutting also slightly decreases longitudinal maximum upstroke.

To better analyse the effect of the coupling in and near the jutting region of the cell, the gap conductance was increased to a nominally normal value (i.e. g_j = 0.154 μS) to obtain more realistic propagation. Table  3 shows the longitudinal and transverse conduction velocities and maximum upstrokes for the nominal case for the UN, RBWJ, and RAND tissue structures. The simulations showed that conduction velocity and waveshape were both affected when junctions were removed in the intercalated disc region.

Longitudinal conduction velocity in the RBWJ and RAND models decreases when plicate, end-to-end interplicate, or side-to-side interplicate gap junctions are removed from the tissue. When the plicate junction is removed from the UN tissue structure, longitudinal propagation blocks completely. Longitudinal propagation in brick wall structures, however, does not block when the plicate junction is removed, since the interplicate junctions maintain coupling. Longitudinal maximum upstroke velocity increases in the RBWJ and RAND models when combined plicates, plicates, and side-to-side interplicates are removed. The most significant increases in longitudinal maximum upstroke velocities occur in the RBWJ and RAND models when both end-to-end and side-to-side interplicates are removed from the tissue. Longitudinal maximum upstroke jumps from 246 to 274 V/s in the RBWJ case and from 254 to 284 V/s in the RAND case.

Transverse conduction velocity in the UN, RBWJ, and RAND models decreases anywhere from 20 to 40% when the combined plicate is removed and ∼40–50% when the side-to-side interplicates are removed. In the UN model, transverse conduction velocity decreases 5% when the plicate gap junction is removed. Transverse maximum upstroke velocity increases most significantly when side-to-side interplicate junctions are removed from the tissue. Transverse maximum upstroke velocity jumps from 277 to 326 V/s in the RBWJ case and from 267 to 298 V/s in the RAND case. In the UN model, transverse maximum upstroke velocity increases when the plicate gap junction is removed because there is no longitudinal current flow to adjacent cells. The largest differences between the longitudinal and transverse maximum upstroke velocities also occur when the side-to-side interplicate is removed; the RBWJ model has a 27% difference between the longitudinal and transverse maximum upstroke velocities (257 and 326 V/s, respectively).

Effect of lateral coupling on ratio of longitudinal to transverse conduction velocities

The results in Table  2 suggest that the longitudinal conduction velocity is affected by the degree of transverse coupling depending on the degree of overlap. Simulations were performed in the third subgroup ( Figure  1 F–H ) to investigate how lateral decoupling of cells, often seen in disease, affects propagation. Table  4 shows the longitudinal and transverse conduction velocities and maximum upstrokes for the nominal case ( g_j = 0.154 μS) with normal structural discontinuity and for the RANDwSD50 and RANDwSD100 tissue structures with increased structural discontinuity. Results are given for longitudinal and transverse plane wave propagation and point propagation.

When the amount of structural discontinuity in the tissue increases to 100%, longitudinal conduction velocity decreases from 50.0 to 42 cm/s and longitudinal maximum upstroke velocity increases from 254 to 278 V/s. Transverse conduction velocity decreases from 16.8 to 7.7 cm/s, whereas transverse maximum upstroke velocity increases from 267 to 308 V/s.

Conduction velocity measured along the x (longitudinal) axis of the elliptic wavefront was 10–14% slower than longitudinal conduction velocity measured using plane wavefronts. In both cases, the ratio of longitudinal to transverse conduction velocities increases as structural discontinuity increases. The difference between the point and plane wave CV ratios, however, increases from 0.3 in the normal case to 0.8 in the RANDwSD100 case. Figure  3 shows that as the degree of structural discontinuity increases, the path of the wavefront becomes more tortuous, the shape of both longitudinal and transverse action potentials becomes more irregular, and the cell-to-cell activation delays become more pronounced.

To better illustrate the effects of lateral coupling on longitudinal propagation, a simulation was performed on the RBW model in which a single row of the brick wall cells was isolated from the rest of the tissue. When transverse current flow was eliminated, the longitudinal conduction velocity and maximum upstroke of the isolated fibre were identical to the UN case (Table  2 ). Figure  4 shows that fibres that are coupled together in a brick wall configuration have shorter cell-to-cell delays than fibres that are laterally isolated from neighbouring fibres. The gap junction delay in the RBW tissue structure was 0.21 ms, whereas the gap junction delay in the isolated fibre was 0.27 ms.

Discussion

This study illustrates how the complex myocardial architecture creates pathways of current flow that can significantly affect both longitudinal and transverse propagation. Table  2 shows that under conditions of low coupling, the longitudinal propagation in the BW model is much faster than in the UN model. The BW model used in this paper is structurally different from the parallel cable models used by other researchers in previous studies because it allows for current flow between rows of cells during longitudinal propagation. The parallel cable model corresponds more with the UN model in this study because they both incorporate interconnected cables (or rows of cells) with identical structures and do not allow transverse current flow during propagation of longitudinal plane waves. ^{5 , 15} Bursac et al . ¹⁶ showed that in a monolayer of neonatal rat cells, decreasing lateral coupling by selectively engineering gaps between cardiac fibres reduced longitudinal conduction velocity when the conduction velocity ratio was greater than 3.5. Similarly to other studies, we have found that longitudinal propagation is slower for an elliptic wavefront than for a plane wavefront. One explanation that has been proposed for this observation is that the increased current load created by the steep curvature of the wavefront slows longitudinal conduction velocity. ¹⁷ In general, the results suggest that conditions that reduce transverse coupling will slow longitudinal propagation, independent of changes in gap junction conductance.

As shown in Figure  4 , the action potentials propagating in the decoupled fibre approach that seen in the uniform (UN) structure. Surprisingly, even when side-to-side connections are significantly reduced as in the case of the RANDwSD100 model, the tissue is still can maintain a transverse conduction velocity of 6–7 cm/s. This is because the transverse wavefront is able to travel longitudinally and ‘zig-zag’ through the few side-to-side interconnections left in the cell. This ‘zig-zag’ conduction has been documented experimentally in studies of infarcted papillary muscles in the heart. ¹⁸

The simulations also showed that another structural feature that affects longitudinal propagation velocity is the jutting at the end of the cells. Jutting helps to ¹ increase the number of interplicate gap junctions in the intercalated disc region and ² increase overlap between rows of cells. Both of these factors help to increase longitudinal conduction velocity in the adult UNJ model ( g_j = 0.01 μS) from 36.0 to 49.1 cm/s and to reduce longitudinal maximum upstroke from 322 to 273 V/s. The effect on transverse conduction is significantly less, increasing slightly or not at all in the UNJ and RBWJ structures. This may in part be due to the limited model of cell jutting we use in the UNJ and RBWJ models. The RAND tissue structure incorporates larger regions of jutting not only at the ends of cells but also along the lateral borders of the cell. The jutting along the lateral borders increases the number of side-to-side interplicate gap junctions and most likely accounts for the larger transverse conduction velocities in the RAND tissue structure. When jutting is added to the UN model, the longitudinal conduction velocity becomes as large or larger than that in the RBWJ tissue model. A portion of this increase is caused by small amounts of overlap at the ends of cells. When the overlap is removed, the longitudinal conduction velocity in the adult UNJ model ( g_j = 0.1 μS) decreases from 49.1 to 44.3 cm/s, thus indicating that even small amounts of overlap between cells can enhance longitudinal propagation.

In the various models studied, both types of interplicate gap junctions are important for determining longitudinal conduction velocity and maximum upstroke velocities of the action potentials. This is consistent with our earlier findings that transverse current flow reduces cell-to-cell delay and thus facilitates longitudinal propagation. The transverse conduction properties are determined primarily by the number and distribution of combined plicate gap junctions and side-to-side interplicate junctions. The end-to-end interplicate gap junctions, however, seem to contribute very little to transverse propagation. This indicates that the distribution and orientation of gap junction plaques within intercalated disc region may play an important role in determining conduction velocity and maximum upstroke velocity.

One notable discrepancy between the adult RAND model and experimental observations in thin sections of adult cardiac tissue is the small difference (∼5%) between longitudinal and transverse maximum upstroke. In comparison, experimental studies of healthy adult tissue have shown much larger directional differences of ∼20–30%, and other microstructural modelling studies by Spach et al . ^{3 , 6} have shown directional differences of ∼12%. The simulation results presented here indicate that in a monodomain, the directional differences in maximum upstroke increase as the conduction velocity ratio increases, and the largest directional differences in maximum upstroke occur in brick wall models with reduced side-to-side connections. It is important to note that the RAND tissue structure is only a hypothetical arrangement of a monolayer of cardiac cells, and it may be possible to increase directional differences in maximum upstroke velocities by increasing the brick wall overlap between cells, adjusting gap junction conductances within the intercalated disc, or reducing the percentage of side-to-side connections. Other factors have also been shown to affect the maximum upstroke. Henriquez and Plonsey ¹⁹ showed that directional differences in maximum upstroke could also arise due to transverse loading in the tissue depth in the presence of an adjacent bath. Because cultures of adult cells with adult coupling are not yet feasible, it is unknown how large the differences might be in a true monolayer of cells compared with a tissue slice.

Although the models considered in this study do not fully consider the effects of the interstitial space on electrical propagation in cardiac tissue and are perhaps more relevant to 2D cardiac cell cultures, the simulations reveal that features of the myocardial architecture such as overlap and jutting help to enhance conduction when connectivity between cells is reduced. Although these features help to maintain impulse conduction from cell to cell, they can also enable slow, tortuous conduction that can increase susceptibility to cardiac arrhythmias. Future studies with these microstructural models will focus on the role that myocardial architecture in healthy and diseased tissue plays in determining the dynamics of more complicated patterns of electrical activation such as premature excitation and spiral waves.

Conflict of interest : none declared

Funding

National Institutes of Health (R01HL076767 to C.H.); National Science Foundation Graduate Fellowship to M.L.H.

Acknowledgements

Special thanks to Sandy Henriquez for her help with visualization using AVS Express.

References