A two-dimensional finite element model of intercellular cAMP signaling through gap junction channels

Abstract

While cyclic adenosine monophosphate (cAMP) is typically considered an intracellular signal, it has been shown to spread between adjacent cells through connexin-based gap junction channels, promoting gap junctional intercellular communication (GJIC). Gap junction-mediated signaling is critical for the coordinated function of many tissues, and have been linked with cardiovascular disease, neurogenerative disease, and cancers. In particular, it plays a complex role in tumor suppression or promotion. This work introduces a two-dimensional finite element model that can describe intercellular cAMP signaling in the presence of gap junctions on membrane interfaces. The model was utilized to simulate cAMP transfer through one and two gap junction channels on the interface of a cluster of two pulmonary microvascular endothelial cells. The simulation results were found to generally agree with what has been observed in the literature in terms of GJIC. The research outcomes suggest that the proposed model can be employed to evaluate the permeability properties of a gap junction channel if its cAMP volumetric flow rate can be experimentally measured.

1. Introduction

Cyclic adenosine monophosphate (cAMP) is a common second messenger involved in the regulation of a wide range of cellular functions such as cell proliferation, cell metabolism, and cell movement, e.g., Conti and Beavo (2007); Mehats et al. (2002). While cAMP has been widely investigated as an intracellular signal, some studies have found that cAMP signals in one cell in a cluster may diffuse across interfaces to adjacent cells through connexin-based gap junction channels, thus promoting intercellular communication. These junctions are clusters of transmembrane channels composed of connexin (Cx) proteins that allow the passage of ions and small molecules, including second messengers, between adjacent cells, e.g., Bruzzone et al. (1996).

Following the first evidence of cAMP diffusion between cells through gap junction channels reported in Lawrence et al. (1978), numerous studies have been conducted to demonstrate the gap junctional intercellular communication (GJIC) of cAMP, e.g., Bedner et al. (2006); Fonseca et al. (2022); Gupta et al. (2016); Kim at al. (2021); Ponsioen et al. (2007). While there has been a large number of numerical studies on modeling intracellular signaling of second messengers, e.g., Williamson et al. (2009); Oliveira et al. (2010); Feinstein et al. (2012); Saucerman et al. (2014); Stone et al. (2019); Warren et al. (2021), research on numerical modeling of intercellular cAMP signaling via gap junction channels is lacking. In this work, we extended our previously developed two-dimensional (2-D) finite element models of cAMP intracellular cAMP signaling (Stone et al., 2019) to simulate cAMP communication between adjacent cells via gap junctions.

The finite element method (FEM) is a powerful numerical technique that can be used to solve complex engineering problems. It has also found applications in the field of biology. FEM has been extensively used in biomechanics to study the mechanical behavior of biological tissues, e.g., Freutel et al. (2014). It has been utilized to simulate fluid flow in biological systems, e.g., Oshima et al. (2001), the transport of drugs in biological tissues, e.g., Gudnason et al. (2018), or second messenger (such as cAMP and Ca2+) signaling, e.g., Stone et al. (2019); Warren et al. (2021); Naik and Pardasani (2017); Naik (2020). However, to the best knowledge of the authors, FEM has not been utilized to simulate intercellular cAMP signaling through gap junction channels.

For the finite element (FE) model developed in this work, a cellular cluster is modeled as a multi-domain having domain-domain interfaces where gap junction channels may be located. Each domain in the model is discretized with three-node triangular elements. However, a gap junction is represented by a two-node line element on the interface where the junction is situated. Each node belongs to one of the two domains sharing this interface, and the line element is characterized by the cAMP volumetric flow rate of the gap junction channel it models. This flow rate can be used to determine the permeability of the gap junction channel if its cross-section area normal to the cAMP flow is known.

The verification of the proposed model was partially performed using an idealized two-cell cluster where each cell is a circular domain, as analytical solutions for this geometry are available for verifying the numerical results. The verified model was then employed to simulate intracellular cAMP signaling through gap junction channels in a cluster of two interconnected pulmonary microvascular endothelial cells (PMVECs).

Experimental findings over the past half-century have suggested that intercellular communication through Cx-based gap junction channels can play different roles, either as tumor-promoting or tumor-suppressing factors, e.g., Mehta et al. (1999); Boucher et al. (2018); Wu and Wang (2019); Chen et al. (2021). The availability of numerical tools, such as the FE technique developed in this work, could potentially assist experimentalists in shedding more light on the roles of Cx-based gap junction channels in promoting or suppressing the growth of tumors in different types of cancer. These tools could also be useful in the future development of cancer treatment pharmaceuticals.

2. Governing equations

As with the intracellular cAMP signaling pathway, the intercellular pathway is governed by the following set of partial differential equations (Stryer, 1995; Rich et al., 2001; Williamson et al., 2009; Feinstein et al., 2012; Stone et al., 2019) generally describing cAMP diffusion, synthesis (AC activity) and degradation (PDE activity):

$$\frac{\partial C}{\partial t}=\{\begin{array}{cc}D{\nabla }^{2}C\hfill & \text{if}t<t_s\hfill \\ D{\nabla }^{2}C+E_{AC}(x,y)\hfill & \text{if}{t}_s\le t<t_d\hfill \\ D{\nabla }^{2}C+E_{AC}(x,y)-M(C)\hfill & \text{if}t\ge t_d\hfill \end{array}\phantom{\}}$$ (1)

where the cAMP concentration $C=C(t,x,y)$ is a function of time $t$ and location ($x,y$) in 2-D space, $D$ denotes the effective diffusion coefficient, $E_{AC}$ is the cAMP synthesis function, $M(C)$ is the Michaelis-Menten reaction rate for cAMP degradation, and $t_s$ and $t_d$ are the time points when AC and PDE activities are initiated, respectively. Inside a given cell, it is assumed that $t_s\le t_d$ as increases in PDE activity (in addition to that at baseline) are typically triggered by increased cAMP production by AC activity.

By adopting the steady-state assumption in Michaelis-Menten kinetics of PDE enzymes, it may be written

$$M(C)=\frac{V_{\max }C}{K_M+C}$$ (2)

where $V_{\max }$ denotes the maximum cAMP hydrolysis rate, and $K_M$ is the Michaelis-Menten constant for cAMP binding to PDE.

Possible boundary conditions (BCs) include prescribed cAMP concentration $C$ and/or concentration flux along part or the entire plasmalemmal or perinuclear region of a cell. The BC equation used to simulate AC activities in these regions is given by

$$D\frac{\partial C}{\partial n}=\beta$$ (3)

where $\mathit{n}$ is the normal vector to the boundary representing the region in question and $\beta$ is a concentration flux quantifying the amount of AC activity.

For more details of the above governing equations and a numerical implementation of these equations using finite element analysis (FEA) with three-node triangular elements for cAMP signaling within a single cell, the reader is referred to Stone et al. (2019).

3. Finite element implementation for GJIC of cAMP

As shown in Stone et al. (2019), the linear system of algebraic equations needed for solving for the unknown nodal cAMP concentrations in a single cell at time $t_i$ is given by

$$[K_r]\{c_r{\}}_i=\{F_r{\}}_i$$ (4)

where $[K_r]$ is a symmetric matrix, $\{c_r{\}}_i$ is the vector of unknown nodal concentrations at time $t_i$ and $\{F_r{\}}_i$ denotes a vector whose elements at time $t_i$ depend on the nodal concentrations found at the previous time, i.e. , $t_{i-1}$.

Without any loss of generality, consider the FEA of a model of two interconnected cells as shown in Figs. 1(a) or 1(b). If there is no GJIC (no gap junction on the interface, Fig. 1(a)), cAMP signaling within each cell has no effect on the other. Two distinct nodes should be employed at a given location on the interface (for example, nodes $\gamma$ and $\delta$ at the upper location or nodes $\mu$ and $\nu$ at the lower location in Fig. 1(a) where $\gamma$ and $\mu$ belong to cell 1 while $\delta$ and $\nu$ belong to cell 2) to obtain the cAMP concentrations at that location as these concentrations are independent from one another (no GJIC). The unknown nodal concentrations $\{c_r^{(1)}{\}}_i$ in cell 1 or $\{c_r^{(2)}{\}}_i$ in cell 2 at time $t_i$ can be independently found by solving the corresponding system of equations of the form (4). Note that the cAMP concentrations at nodes $\gamma$ and $\mu$ are contained in vector $\{c_r^{(1)}{\}}_i$ while those at nodes $\delta$ and $\nu$ are in vector $\{c_r^{(2)}{\}}_i$. By putting these two systems together one gets the following unified system:

$$\left[\begin{array}{cc}\hfill K_r^{(1)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill K_r^{(2)}\hfill \end{array}\right]{\left\{\begin{array}{c}\hfill c_r^{(1)}\hfill \\ \hfill c_r^{(2)}\hfill \end{array}\right\}}_i={\left\{\begin{array}{c}\hfill F_r^{(1)}\hfill \\ \hfill F_r^{(2)}\hfill \end{array}\right\}}_i$$ (5)

where superscripts (1) and (2) refer to cells 1 and 2, respectively. This system shows that there is no coupling between the cAMP concentrations in the two cells because of the zero-matrices along the anti-diagonal which indicates the absence of cAMP transfer across their interface (absence of GJIC).

Figure 1:

Two different schematics shown in subfigures (a) and (b) were considered to model interconnected cells. Two pairs of nodes, ($\gamma ,\delta$) and ($\mu ,\nu$), are placed at the two locations of interest on the interface. Nodes $\gamma$ and $\mu$ belong to Cell 1 while $\delta$ and $\nu$ belong to Cell 2: (a) A model of intracellular cAMP signaling (no gap junction on the interface), i.e. no connection between nodes $\gamma$ and $\delta$, and nodes $\mu$ and $\nu$; (b) A model of intercellular cAMP signaling where there are two gap junction channels at the two locations of interest. The cAMP transfers across the two junctions are evaluated by using the two interface elements $\gamma -\delta$ and $\mu -\nu$.

For the coefficient matrix of system (5), the submatrix corresponding to the cAMP concentrations at nodes $\gamma$ and $\delta$ ($C_{\gamma }$ and $C_{\delta }$) can be found as

$$\left[\begin{array}{cc}\hfill k_{\gamma \gamma }\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill k_{\delta \delta }\hfill \end{array}\right]$$ (6)

In this work, to model GJIC of cAMP we propose to use a line/interface element similar to the linear heat-conduction element to connect the pairs of overlapping nodes on the interface (for example, the pairs ($\gamma ;\delta$) and ($\mu ;\nu$) as shown in Fig. 1(b)) making the concentrations at the pairs of nodes related. For the interface element $\gamma -\delta$ in Fig. 1(b)), this relation is described by the following cAMP transfer matrix:

$${\overline{k}}_{\gamma \delta }=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -1\hfill \\ \hfill -1\hfill & \hfill 1\hfill \end{array}\right]$$ (7)

where ${\overline{k}}_{\gamma \delta }$ is the cAMP volumetric flow rate for the interface element $\gamma -\delta$. In the absence of GJIC (Fig. 1(a)), ${\overline{k}}_{\gamma \delta }$ = 0. If there is a total transfer of cAMP across the interface at the pair of nodes ($\gamma ;\delta$), ${\overline{k}}_{\gamma \delta }\to \infty$ which results in $C_{\gamma }=C_{\delta }$ as expected.

In this case, the submatrix (6) becomes

$$\left[\begin{array}{cc}\hfill k_{\gamma \gamma }+{\overline{k}}_{\gamma \delta }\hfill & \hfill -{\overline{k}}_{\gamma \delta }\hfill \\ \hfill -{\overline{k}}_{\gamma \delta }\hfill & \hfill k_{\delta \delta }+{\overline{k}}_{\gamma \delta }\hfill \end{array}\right]$$ (8)

This new submatrix, which is the result of adding the cAMP transfer matrix (7) to submatrix (6), shows a coupling of $C_{\gamma }$ and $C_{\delta }$ (the entries on the anti-diagonal of submatrix (8) are non-zero).

4. Verification of the proposed FEA Model

To verify this model, we used 2-D geometries of two two-cell clusters: an idealized cluster of two interconnected hollow circular cells and a biologically-accurate cluster of two interconnected cultured PMVECs as shown in Fig. 2. The PMVEC cluster image in Fig. 2(b) was determined from confocal microscopy experiments and acquired using a Nikon A1R spectral microscope as described previously in Annamdevula et al. (2018). To be able to use analytical solutions available for the hollow circular geometry to validate our FEA model, a slight contact between the cells was made as this almost preserves the hollow circular geometry of both cells . The contact results in a straight-line interface discretized by the pairs of nodes (1; 150), (2; 151) and (3; 152) as depicted by the red dots in Fig. 2(a). The outer and inner radii of the left hollow circular cell are 15 $\mu \mathrm{m}$ and 8 $\mu \mathrm{m}$, while those for the right cell are 10 $\mu \mathrm{m}$ and 5 $\mu \mathrm{m}$, respectively.

Figure 2:

Two geometries for clusters of interconnected cells were considered for this study – an idealized geometry of two interconnected circular cells shown in subfigure (a) and a biologically-accurate geometry of two interconnected cultured PMVECs shown in subfigures (b) and (c): (a) FEA mesh for 2-D model of two interconnected circular cells. The yellow nodes 121, 134, 144, 3 in the left cell, and 152, 157, 167, 179 in the right cell are cellular locations where numerical results are compared with the analytical solution. The interface between the two cells is discretized using three pairs of red nodes (1;150), (2;151), and (3;152); (b) Confocal microscopy image of two interconnected cultured PMVECs; (c) FEA mesh for 2-D model of two interconnected cultured PMVECs. Gap junction channels are assumed to be located at the pairs of red nodes (2;421) and (10;429) on the interface between the left and right cells. Localized cytosolic AC activity is assumed to occur at the location of element # 65.

Figures 2(a) and 2(c) show FEA meshes for the hollow circular cluster and the PMVEC cluster, respectively. These meshes were constructed using Distmesh (Persson and Strang, 2004). The hollow circular cluster mesh consists of a total of 273 nodes and 419 three-node triangular elements (including the three pairs of nodes on the interface as mentioned above, 229 elements in the left cell and 190 elements in the right cell). The PMVEC cluster mesh has 432 nodes and 683 elements in total (14 pairs of nodes on the interface, 362 elements in the left cell and 321 elements in the right cell). The above numbers of nodes and elements were determined by incrementally increasing the mesh densities within the program Distmesh until the meshes in Figs. 2(a) and 2(c) pass the mesh convergence tests. The data used in running the proposed FEA model in this work were based on those reported in Feinstein et al. (2012) and Stone et al. (2019) for PMVECs. As in Stone et al. (2019), no cAMP transport between the cytoplasm and nucleus is assumed in this work. Unless otherwise specified, no-flux BCs along the perinuclear region are employed to enforce this assumption.

The verification of the model was performed for the implementation of each component of cAMP signaling (synthesis, diffusion and degradation). As the analytical solutions employed were developed for single domain cases, the verifications in this work were independently done in each cell of a cluster without GJIC (no gap junction on the interface). For the sake of brevity, only a verification of the implementation of cAMP diffusion is presented.

In this section, we described a verification of the FE implementation of the diffusion component in the governing equation (1), as well as the independent operation of the individual cells in the case of no GJIC. The verification was performed using the hollow circular cluster mesh shown in Fig. 2(a) as the analytical solutions for diffusion in a hollow circular domain are available (Carslaw and Jaeger, 1959). Here, PDE activity is assumed to be absent, and different diffusion coefficients, initial conditions and AC activities are independently applied to the left and right cells. Both AC activities start at the same time. The effective diffusion coefficient in the right cell ($D$ = 30 $\mu$m²/s) is three times larger than that in the left cell. The initial condition in the left cell is $C_0$ = 0.05 $\mu$M and that value in the right cell is 0.1 $\mu$M.

Figure 3(a) depicts a very good agreement between the numerical and analytical solutions for the cAMP responses at nodes 121, 134, 144 and 3 along the cytosolic thickness of the left cell where AC activity was uniformly generated at the plasma membrane (see Fig. 2(a) for the locations of the four nodes). This AC synthesis is modeled by applying a positive flux $\beta$ = 4 $\mu$M·$\mu$m/s in the plasmalemmal region and zero flux in the perinuclear region. More details on how to determine $\beta$ from a given magnitude of cAMP synthesis $E_{AC}$ can be found in Stone et al. (2019).

Figure 3:

The FEA model was verified using the cluster of two interconnected circular cells (Fig. 2(a)) without GJIC. Data represent the FE vs analytical solutions for the time courses of cAMP concentration at different cellular locations in response to different diffusion coefficients and BCs concurrently applied on both cells of the cluster: (a) Verification in the left cell where $D$ = 10 $\mu$m²/s and $\beta$ = 4 $\mu$M·$\mu$m/s at the plasma membrane; (b) Verification in the right cell where $D$ = 30 $\mu$m²/s, $C_p$ = 3 $\mu$M at the plasma membrane and $C_n$ = 1 $\mu$M at the perinuclear region.

Figure 3(b) shows the numerical vs analytical results for the cAMP concentrations at two cytosolic nodes 157 and 167 (see Fig. 2(c) for the locations of these nodes). The results appear to excellently agree with each other. Note that AC activity in the right cell was created by suddenly imposing constant concentrations $C_p$ = 3 $\mu$M and $C_n$ = 1 $\mu$M at the plasmalemmal and perinuclear regions, respectively. Due to this type of AC activity, the cAMP concentrations reach a steady state as shown in Fig. 3(b). The speed at which the concentrations achieve their steady-state value is proportional to the magnitude of the diffusion coefficient. The steady state concentration at node 157 is higher than that at node 167 as node 157 is closer to the plasma membrane where the imposed concentration is larger.

Results from verification testing in Figs. 3(a) and 3(b) demonstrate a successful numerical implementation of the cAMP diffusion equations, as well as independent operation of the individual cells in the absence of GJIC in the proposed FEA model. It should be noted that the above conditions (absence of PDE activity in combination with either AC activity uniformly generated at the plasma membrane or AC activity created by suddenly imposing different constant concentrations at the plasma membrane and perinuclear region) were chosen due to the availability of analytical solutions that can be used to verify the FEA model developed. The biological parameter values employed in the verifications were selected within the range of those reported in Feinstein et al. (2012) for PMVECs.

5. Numerical simulations

After verifying that the FEA model generated accurate results as shown in the previous section, we next proceeded to simulate a series of intracellular and intercellular cAMP signaling scenarios, each with increasing levels of sophistication, that allowed visualization of the cAMP gradients that may form during cell signaling events, and in which cAMP medicated cell-cell communication via gap junctions may occur.

The numerical simulations presented in the following sections were done using the PMVEC cluster mesh shown in Fig. 2(c). For all of these simulations, the same initial condition $C_0$ = 0.05 $\mu$M was assumed for both cells. The numerical results for the simulations in Sections 5.1 and 5.2 are displayed in the form of color-filled maps of cAMP distribution within the cellular cluster at some typical time points. These maps can effectively be biologically interpreted as they (a) show how cAMP signals transmit across the gap junction channels at the interface; (b) show the spatial distribution of cAMP levels, represented by different colors, within the cluster; (c) reveal the specific regions of the cluster that are involved in a particular physiological response to a stimulus such as AC activity.

5.1. A single gap junction on the interface

This section describes the use of the proposed FEA model in simulating intercellular cAMP signaling within the cluster of two interconnected cultured PMVECs aforementioned due to a localized AC activity in its left (donor) cell and the presence of a gap junction channel on the interface of the two cells. This represents an idealized scenario where cAMP signals generated in one cell may influence cAMP signal levels and spatial distribution in an adjacent cell, as has been reported in Ponsioen et al. (2007).

The single gap junction was assumed to be at the pair of nodes (2; 421) on the interface (see Fig. 2(c)) and its cAMP volumetric flow rate was taken to be $\overline{k}$ = 100; $\mu$m³/s. The cAMP transfer across this gap junction occurred after a localized AC activity ($E_{AC}$ = 15 $\mu$M/s) at the location represented by element # 65 in the left cell (see Fig. 2(c)) had started at time $t$ = 30 s. PDE activities were presumed to be activated immediately in both cells following the local AC activity. The parameters used to describe diffusion and PDE activity were $D$ = 15 $\mu$m²/s, $V_{\max }$ = 0.295 $\mu$M/s, $K_m$ = 2 $\mu$M, and $D$ = 10 $\mu$m²/s, $V_{\max }$ = 0.02 $\mu$M/s, $K_m$ = 3 $\mu$M for the left and right cells, respectively.

The cAMP concentration distribution in the cluster over time is depicted in Figs. 4(a) to 4(d). At time $t$ = 5 s (Fig. 4(a)), the concentration in both cells was still at the initial level of 0.05 $\mu$M/s. At time $t$ = 35 s (Fig. 4(b)), the concentration significantly increased around the location of the localized AC activity that had taken place five seconds earlier. GJIC effects were evidenced from Figs. 4(c) ($t$ = 75 s) and 4(d) ($t$ = 300 s) as some amount of cAMP can be clearly seen to be transferred from the left cell to the right cell through the gap junction. Figure 4(d) shows the steady-state cAMP distribution in the cluster.

Figure 4:

Simulation results for GJIC with a single gap junction in the cluster of two interconnected cultured PMVECs were plotted as color-filled maps of cAMP distribution at different time points. The gap junction ($\overline{k}$ = 100 $\mu$m³/s) is located at the the pair of nodes (2; 421) in the cluster (Fig. 2(c)) which is subjected to a localized AC activity in left/donor cell: (a) Result at $t$ = 5 s; (b) Result at $t$ = 35 s; (c) Result at $t$ = 75 s; (d) Result at $t$ = 300 s (Left/donor cell: Localized cytosolic AC activity with $E_{AC}$ = 15 $\mu$M/s at $t$ = 30 s on element #65, $D$ = 15 $\mu$m²/s, $V_{\max }$ = 0.295 $\mu$M/s, $K_m$ = 2 $\mu$M; Right/acceptor cell: $D$ = 10 $\mu$m²/s, $V_{\max }$ = 0.02 $\mu$M/s, $K_m$ = 3 $\mu$M).

We next assessed the effect of the cAMP volumetric flow rate on the amount of cAMP passage across the gap junction channel. To this end, two additional values of $\overline{k}$ were assumed for this gap junction, namely, $\overline{k}$ = 10 $\mu$m³/s and 1000 $\mu$m³/s. This simulation is important as prior studies have reported a range of cAMP channel permeabilities, e.g., Bedner et al. (2006); Kanaporis et al. (2008), but the effects of channel permeability on dynamic intercellular cAMP signaling and overall cAMP spatial distributions are not fully known. Figures 5(a) and 5(b) depict the steady-state cAMP distributions within the cluster for $\overline{k}$ = 10 $\mu$m³/s and $\overline{k}$ = 1000 $\mu$m³/s, respectively. These results are plausible as they show that a lower value of cAMP flow rate $\overline{k}$ leads to a lesser amount of cAMP being transferred across the gap junction. It can also be observed that the steady-state cAMP distribution for $\overline{k}$ = 1000 $\mu$m³/s (Fig. 5(b)) is very similar to that for $\overline{k}$ = 100 $\mu$m³/s (Fig. 4(d)). This suggests that a value of 100 $\mu$m³/s for $\overline{k}$ is sufficient to simulate the maximum transfer of cAMP across the gap junction channel in this simulation.

Figure 5:

Simulation results for GJIC with a single gap junction in the cluster of two interconnected cultured PMVECs were plotted as color-filled maps of cAMP distribution for different values of $\overline{k}$. The gap junction is located at the the pair of nodes (2; 421) in the cluster (Fig. 2(c)) which is subjected to a localized AC activity in left/donor cell: (a) Result for $\overline{k}$ = 10 $\mu$m³/s; (b) Result for $\overline{k}$ = 1, 000 $\mu$m³/s (Left/donor cell: Localized cytosolic AC activity with $E_{AC}$ = 15 $\mu$M/s at $t$ = 30 s on element #65, $D$ = 15 $\mu$m²/s, $V_{\max }$ = 0.295 $\mu$M/s, $K_m$ = 2 $\mu$M; Right/acceptor cell: $D$ = 10 $\mu$m²/s, $V_{\max }$ = 0.02 $\mu$M/s, $K_m$ = 3 $\mu$M).

The results of this simulation, as shown in Figs. 5(a) and 5(b), are in line with what was observed in Kanaporis et al. (2008): Cx43 permeability results in a rapid passage of a sufficient quantity of cAMP to trigger intracellular responses from the acceptor cell before PDE activity occurs.

5.2. Two gap junctions on the interface

The sophistication of the intercellular cAMP signaling model was further increased to allow simulation of multiple gap junction channels, as may occur in many typical scenarios where PMVECs and many cell types, are cultured in a confluent monolayer or three dimensional cultures. In this section, we presented how the proposed FEA model can be utilized to simulate intercellular cAMP signaling within a cluster of two interconnected cultured PMVECs having two gap junction channels at the interface. Specifically, the simulation focuses on cAMP signaling across these two gap junctions as a result of a uniform AC activity that occurred in the plasma membrane of the right (donor) cell and different PDE activities in the left (acceptor) cell.

In this simulation, we assumed that the two gap junction channels (see Fig. 2(c)) were located at the pairs of nodes (2; 421) and (10; 429), where $\overline{k}$ = 10 $\mu$m³/s and 100 $\mu$m³/s, respectively. It’s worth noting that the second (lower) gap junction has a cAMP volumetric flow rate that is ten times higher than that of the first (upper) one.

Figures 6(a) to 6(d) show the cAMP distribution in the PMVEC cluster at four different time points. Five seconds after the start of a uniform AC activity ($\beta$ = 1.4 $\mu$M·$\mu$m/s) in the plasmalemmal region of the right cell, some accumulation of cAMP in this region can be seen in Fig. 6(a). Cyclic AMP quickly started to diffuse through the two gap junction channels but this passage was only clearly visible in Fig. 6(c) at time $t$ = 50 s. A larger concentration of cAMP around the two gap junctions in the acceptor cell was found at the equilibrium state as depicted in Fig. 6(d) at time $t$ = 300 s. As the lower gap junction channel has a higher cAMP volumetric flow rate than the upper one, a higher cAMP concentration can be seen around this gap junction in the left cell.

Figure 6:

Simulation results for GJIC with two gap junctions in the cluster of two interconnected cultured PMVECs were plotted as color-filled maps of cAMP distribution at different time points. The gap junctions are located at the the pair of nodes (2; 421) with $\overline{k}$ = 10 $\mu$m³/s and (10; 429) with $\overline{k}$ = 100 $\mu$m³/s in the cluster (Fig. 2(c)) which is subjected to a uniform AC activity ($\beta$ = 1.4 $\mu$M·$\mu$m/s) in the plasmalemmal region of the right/donor cell: (a) Result at $t$ = 5 s; (b) Result at $t$ = 20 s; (c) Result at $t$ = 50 s; (d) Result at $t$ = 300 s (Left/acceptor cell: $D$ = 8 $\mu$m²/s, $V_{\max }$ = 0.05 $\mu$M/s, $K_m$ = 3 $\mu$M; Right/donor cell: $D$ = 6 $\mu$m²/s, $V_{\max }$ = 0.295 $\mu$M/s, $K_m$ = 2 $\mu$M).

After running the multi-gap junction model, we utilized this model to assess the effects of different PDE activities in the acceptor cell on overall intercellular cAMP signaling. This simulation corresponds to a range of possible biological scenarios in which PDE activities may be higher in one cell than another, for example due to differing activation or expression of different PDE subtypes, or due to co-culture conditions, or as may be expected in more complex organoid or tissue preparations. To implement this additional variation in PDE activity, we varied the amount of the acceptor cell $V_{\max }$ while keeping $K_m$ unchanged at 3 $\mu$M. Figures 7(a) and 7(b) show the results for $V_{\max }$ = 0.02 $\mu$M/s and 0.2 $\mu$M/s, respectively. These results indicate that, under smaller values of $V_{\max }$, i.e. lower PDE activity, the acceptor cell retains more amount of cAMP transferred from the donor cell via gap junctions. These numerical results also agree with what was observed in Ponsioen et al. (2007) with regards to a major factor that PDE activity plays in determining cAMP penetration into acceptor cells.

Figure 7:

Simulation results for GJIC with two gap junctions in the cluster of two interconnected cultured PMVECs were plotted as color-filled maps of cAMP distribution for different values of $V_{\max }$ (different PDE activities) in the left/acceptor. The gap junctions are located at the the pair of nodes (2; 421) with $\overline{k}$ = 10 $\mu$m³/s and (10; 429) with $\overline{k}$ = 100 $\mu$m³/s in the cluster (Fig. 2(c)) which is subjected to a uniform AC activity ($\beta$ = 1.4 $\mu$M·$\mu$m/s) in the plasmalemmal region of the right/donor cell: (a) Result for $V_{\max }$ = 0.02 $\mu$M/s; (b) Result for $V_{\max }$ = 0.2 $\mu$M/s (Left/acceptor cell: $D$ = 8 $\mu$m²/s, $K_m$ = 3 $\mu$M; Right/donor cell: $D$ = 6 $\mu$m²/s, $V_{\max }$ = 0.295 $\mu$M/s, $K_m$ = 2 $\mu$M).

5.3. cAMP amplitude transferred across gap junction channels

As a final step in demonstrating the utility of the proposed FEA model, in this section, we described how to employ the model in conjunction with experiments to predict the permeability of a Cx-based gap junction channel.

Experimental results reported in Ponsioen et al. (2007) showed that approximately 40% of the amplitude of response of cAMP in Rat-1 cells was transferred to adjacent cells through Cx43-based gap junction channels. By using the proposed FEA model, the percentage of cAMP steady-state amplitude $C^{\infty }$ transferred to an acceptor cell through the gap junction can be determined for a given cell type with a known cAMP volumetric flow rate $\overline{k}$.

Reconsider the GJIC simulation for the PMVEC cluster with a single gap junction channel located at the pair of nodes 2 (in the left/donor cell) and 421 (in the right/acceptor cell) as described in Section 5.1 where localized cytosolic AC activity occurred in the donor cell (see Figs. 5(a) and 5(b)). To gain an effective biological interpretation of cAMP signaling in this simulation, timecourse plots were used to visualize its dynamics over time on both the donor and acceptor sides of the gap junction. Another objective was to determine a suitable time when the steady-state amplitudes of cAMP at nodes 2 and 421, respectively denoted as $C_2^{\infty }$ and $C_{421}^{\infty }$, could be measured.

Figures 8(a) to 8(d) depict the time courses for $C_2$ and $C_{421}$ for four different values of $\overline{k}$, namely $\overline{k}$ = 0, 2.6, 10 and 300 $\mu$m³/s, respectively. According to these figures, $C_2^{\infty }$ and $C_{421}^{\infty }$ can be observed at time $t$ = 800 s. In this case, the ratios $C_{421}^{\infty }$/$C_2^{\infty }$ are found to be 0.031, 0.397, 0.723 and 0.988 for $\overline{k}$ = 0, 2.6, 10 and 300 $\mu$m³/s, respectively. In other words, approximately 3%, 40%, 72% and 99% of the cAMP amplitude were accordingly transferred to the right/acceptor cell through the gap junction channel.

Figure 8:

Simulation results for GJIC with a single gap junction in the cluster of two interconnected cultured PMVECs (Fig. 2(c)) were shown as timecourse plots of cAMP amplitudes at nodes 2 and 421 of the gap junction channel in response to different values of its cAMP volumetric flow rates $\overline{k}$. The cluster is subjected to a localized AC activity in left/donor cell: (a) Result for $\overline{k}$ = 0; (b) Result for $\overline{k}$ = 2.6 $\mu$m³/s; (c) Result for $\overline{k}$ = 10 $\mu$m³/s; (d) Result for $\overline{k}$ = 300 $\mu$m³/s. This simulation is similar to the one whose results are shown in Fig. 5. $C_2^{\infty }$ and $C_{421}^{\infty }$ are the steady-state amplitutes (measured at $t$ = 800 s) at nodes 2 and 421, respectively. $C_{421}^{\infty }$/$C_2^{\infty }$ represents the percentage of the cAMP amplitude transferred to the right/acceptor cell through the gap junction channel. Left/donor cell: Localized cytosolic AC activity with $E_{AC}$ = 15 $\mu$M/s starting at time $t$ = 30 s and at the location of element #65, $D$ = 15 $\mu$m²/s, $V_{\max }$ = 0.295 $\mu$M/s, $K_m$ = 2 $\mu$M; Right/acceptor cell: $D$ = 10 $\mu$m²/s, $V_{\max }$ = 0.02 $\mu$M/s, $K_m$ = 3 $\mu$M.

This simulation suggests that once the percentage of the cAMP amplitude transferred to adjacent cells can be measured for a given Cx-based channel, the proposed FEA model can be employed to predict the cAMP volumetric flow rate, thus the permeability of the connexin channel.

6. Concluding remarks

In this study, an FEA model for combined intracellular and intercellular cAMP signaling through gap junction channels was developed. The model was validated for cases involving cellular clusters without GJIC, where some analytical solutions for cAMP synthesis, degradation, or diffusion are available. The model allows for the effects of cAMP diffusion through gap junction channels in biologically-accurate cellular geometries to be evaluated, and allows visualization of spatial cAMP distributions at any timepoint in the model as well as cAMP signal dynamics for any point within the cellular cluster. Hence, this model provides a unified framework that can be used to more fully simulate cAMP signaling in near-confluent or confluent monolayers, as is typically performed in a wide range of experimental studies, especially those using endothelial cells. The ability to visualize both spatial cAMP distributions as well as to interrogate cAMP signal dynamics for any discrete point within the cellular cluster is of high utility, as many studies have indicated that cAMP signal localization may play a key role in determining downstream cellular physiological response. Hence, we anticipate that this modeling approach will be of high utility for simulating and gaining insight into the complex mechanisms underlying intercellular communication in pulmonary endothelial cells, as well as a wide range of other cell lines with physiological and pathological responses involving the cAMP signaling pathway.

The model developed was employed to simulate GJICs via gap junctions at the interface of a cluster of two interconnected PMVECs. The simulation results are plausible and expected. In particular, they broadly agree with some findings reported in Ponsioen et al. (2007): (a) Cx-based channels allow a quick passage of cAMP to set off GJIC response from the acceptor before PDE activity takes place; (b) Acceptor cells with lower PDE activity pursue cAMP levels in donor cells more effectively. The model was also utilized to simulate different percentages of the cAMP amplitude transferred to the acceptor cell of a cluster of two PMVECs through a gap junction channel having different volumetric flow rates. Determining cAMP permeability of connexin channels has been a subject of interest in GJIC research. However, to the authors’ best knowledge, only relative cAMP permeabilities can be experimentally determined and the procedures have been time consuming (Bedner et al., 2006). As the percentage of the cAMP amplitude transferred across a Cx-based gap junction channel in a cellular cluster can be experimentally quantified, e.g., Ponsioen et al. (2007), the proposed FEA model can use that percentage to estimate the cAMP volumetric flow rate which is a measure of cAMP permeability for the gap junction channel in question. Moreover, based on evidence suggesting a relationship between Connexin (Cx)-based gap junction channels and both the growth and suppression of tumors, the proposed model has the potential to complement experimental studies to further identify the roles of these channels in suppressing or promoting the growth of tumors in different types of cancer.