| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
* Department of Mathematics, University of Auckland, New Zealand; Department of Pharmacology and Physiology, University of Rochester, Medical Center, Rochester, New York; and Department of Mathematics, University of Auckland, New Zealand
Correspondence: Address reprint requests to J. Sneyd, Fax: 64-9-3737-457; E-mail: sneyd{at}math.auckland.ac.nz.
 |
ABSTRACT |
|---|
|
INTRODUCTION |
|---|
; Wäsle and Edwardson, 2002). Ca2+ release induced by inositol trisphosphate (IP3) generally takes the shape of [Ca2+]i oscillations, the period and amplitude of which depend on the agonist concentration. In pancreatic acinar cells Ca2+ waves are generally initiated in the apical region and spread over the cell to form a global intracellular wave (Fogarty et al., 2000b; Nathanson et al., 1992; Straub et al., 2000). Moreover, this Ca2+ wave can propagate from one cell to another. Similar intercellular Ca2+ waves have been reported in a wide range of different cell types, such as airway epithelial cells (Sanderson, 1995), astrocytes (Giaume and Venance, 1998), and in the intact liver (Thomas et al., 1995; Robb-Gaspers and Thomas, 1995). In a pancreatic acinus, Ca2+ signals appear to travel in a wavelike manner between coupled cells and thereby serve as a means of intercellular communication (Petersen and Petersen, 1991; Stauffer et al., 1993; Yule et al., 1996).
Intercellular Ca2+ waves have been extensively studied in other cell types. Early models of intercellular Ca2+ waves in epithelial cells (Sneyd et al., 1995) proposed the intercellular diffusion of IP3 as the principal wave-generating mechanism, whereas recent experimental work has pointed out the importance of extracellular diffusing messengers, at least in some cell types (Clair et al., 2001; Yule et al., 1996). Two recent studies by G. Dupont and her colleagues (Clair et al., 2001; Dupont et al., 2000a, 2000b), and T. Höfer (Höfer, 1999; Höfer et al., 2001) have been published on multiplets of hepatocytes, the former maintaining the crucial importance of IP3 as a signaling messenger and the latter showing that junctional Ca2+ diffusion is the most significant for synchronization. These studies in hepatocytes are not yet satisfactorily resolved. However, one thing of which we are certain is that a single model cannot possibly hope to be appropriate for all cell types in which such intercellular waves are observed. Despite this, models tend to explain the coordination of intercellular waves by a combination of one or more of three general mechanisms: 1), the diffusion of Ca2+ through gap junctions; 2), the diffusion of IP3 through gap junctions; and 3), an extracellular diffusing messenger.
It is our goal to determine the principal mechanisms by which intercellular waves in pancreatic acinar cells are coordinated. Acinar cells are known to be connected by gap junctions (Ngezahayo and Kolb, 1993; Petersen and Petersen, 1991; Stauffer et al., 1993; Yule et al., 1996), and it has been claimed that the intercellular diffusion of Ca2+ and/or IP3 could be important mechanisms by which a group of acinar cells could coordinate their responses (Ngezahayo and Kolb, 1993; Petersen and Petersen, 1991; Stauffer et al., 1993; Yule et al., 1996). Furthermore it has been suggested that transmission of the signal from the most agonist-sensitive cells to the neighboring cells could explain why preparations of acini secrete digestive enzymes more efficiently than single cells (Yule et al., 1996). Intriguingly, increasing the frequency of the Ca2+ oscillations by modulation of gap-junctional permeability appears to increase the rate of enzyme secretion (Stauffer et al., 1993). However, despite numerous experimental studies, the precise role of either Ca2+ or IP3 diffusion remains unclear.
Yule et al. (1996) investigated intercellular Ca2+ signaling in connected clusters of acinar cells. Their observations can be briefly described as follows:
Based on these data, it was proposed that Ca2+ acts as a co-agonist with IP3 to enhance the Ca2+-releasing action of IP3 receptors and that diffusion of the two molecules through gap junctions underlie intercellular signaling in acinar cells.
Some of these results were studied by Sneyd et al. (2003), whereas others we have checked in the present work. However, we do not show detailed pictures of those results here. Essentially, given the fact that the model was constructed precisely to behave in this manner, the fact that it can qualitatively reproduce those results comes as no surprise, and tells us little about the cells that we did not already know. Here, instead, we study a much more complex and less well-understood problem—that of intercellular synchronization, which was the subject of point 1, above.
Our study is based on the previous model of Ca2+ dynamics in pancreatic acinar cells due to Sneyd et al. (2003), which we describe in detail in Appendix 1. From this model we construct two qualitatively different models of a triplet of coupled acinar cells.
Model 1: point oscillators
We begin by modeling a triplet of cells as a system of three coupled oscillators. Each oscillator is described by a purely temporal model with no spatial gradients of Ca2+ or IP3 (i.e., a system of ordinary differential equations). The cells are coupled by transmembrane diffusion of Ca2+ or IP3, or both. We perform a bifurcation analysis of this coupled system of ordinary differential equations and describe the various possible kinds of oscillations that occur in such a triplet of coupled oscillators. Our results here are generic; there is nothing special about pancreatic cell oscillators to distinguish them from any other triplet of coupled oscillators. Nevertheless, the exact behavior must be established before the results can be compared to the spatially distributed model.
Model 2: spatially distributed cells
Next, we construct a spatially distributed version of the model, in which each cell is modeled as a two-dimensional continuum, coupled to its neighbors along each common border. This model thus uses partial differential equations and is solved numerically using a finite element method. The various oscillatory modes that appear are complicated by the nature of the oscillation within each individual cell. Nevertheless, useful comparisons can be made to the point-oscillator model. In effect, the spatially distributed model provides an important check on whether the point-oscillator model can say anything useful about the actual physiological system.
Using these two models,
). However, we found that the stability of each of these modes was different in the two models. Most interestingly, we found that in the spatially distributed model the most stable phase-locked mode was one where two cells oscillate in synchrony, with the third cell phase-locked but with a different phase. The phase-locked oscillation in which each cell is exactly 1/3 out of phase was not a stable solution for the spatially distributed model, although it was stable in the point-oscillator model. Our model thus predicts that regular intercellular waves, moving at constant speed around a ring of cells, would not, in general, be observed. For the readers' convenience, we summarize the terminology we use here as
= 0 between the cells). |
MODEL 1: POINT-OSCILLATOR CELLS |
|---|
Description of the model
We approximate a cluster of three pancreatic acinar cells with a bi-directional ring of three identical, symmetrically and diffusively coupled oscillators. A schematic diagram of Model 1 is presented in Fig. 1 A. The model for the dynamics of each cell in the cluster is based on a recent model of intracellular Ca2+ dynamics in pancreatic acinar cells, published by Sneyd et al. (2003). Although, in the original model, the cell was assumed to have distinct apical and basal regions, with parameter values that were determined by fitting to experimental data, we do not incorporate this complexity at this stage of the modeling. Instead, we first study the behavior of three coupled point-oscillators in which each oscillator is modeled using the apical parameters of the full model. We will use different parameters for the apical and basal regions only in the spatially distributed model (Model 2).
|
where the superscript i takes the values 1,2,3, and refers to each of the three cells in the triplet. The cells are coupled through diffusion of Ca2+ with coupling strength, 
(s–1). A detailed description and justification of each of the model components can be found in Sneyd et al. (2003). The system of ordinary differential Eqs. 12–20 corresponding to this model is given in Appendix 1. If each 
is the same (i.e., 
), then the model describes the response of three identical cells, with identical levels of agonist stimulation and thus identical [IP3]. By varying the levels of stimulation in each individual cell of the cluster, i.e., by choosing different values of in each cell, we can study synchronization in a group of coupled cells, each with a different intrinsic period.
Numerical method
The model equations (Eqs. 12–20) were solved numerically and the bifurcation analysis was performed using the software package XPPAUT (Ermentrout, 2002), which contains a front-end for AUTO (Doedel et al., 1997).
Results
Steady states
The model of three identical diffusively coupled cells has a symmetric steady state, characterized by identical variable values for all cells. In all the results we discuss here, we use the parameters given in Table 1. The parameters pst and are treated as bifurcation parameters. To model a heterogeneous cluster of cells, we assume that pst, which denotes the level of agonist stimulation, is different in different cells. In this case, each model cell has a different resting [IP3]. Because of the complexity of studying multidimensional bifurcation surfaces, the more complicated case, where each cell has different parameter values, is not considered here.
|
of the Jacobian may be factorized into two parts, such that
 | (1) |
The factor F() corresponds to the characteristic polynomial of the single cell, and is therefore of ninth degree, because of Eqs. 12–20. The other factor, D(), is of ninth degree as well. The possibility of such a factorization was also demonstrated in systems describing glycolytic oscillations in yeast cells (Wolf and Heinrich, 1997). The solutions of Eq. 1 determine the stability of the steady state.
Dynamics of a single cell
The Ca2+ dynamics of an isolated pancreatic acinar cell have been well studied experimentally. It has been observed that physiological doses of cholecystokinin induce baseline spikes of Ca2+, whereas acetylcholine produces faster, sinusoidal, oscillations (Cancela et al., 2002; Cancela, 2001; Lawrie et al., 1993; Yule et al., 1991). Like many other cell types exhibiting IP3-induced Ca2+ oscillations (Berridge and Dupont, 1994) there exists a critical agonist dose above which a pancreatic acinar cell responds with Ca2+ oscillations. In addition, the period of the oscillations decreases with increasing agonist stimulation.
In the case of a single cell, Model 1 reproduces these experimental observations. The dynamics of a single cell as pst varies is summarized in Fig. 2 A. The steady state is stable for small and large values of pst, and there is an intermediate region of values between the two Hopf bifurcation points (HB1 and HB2), where the steady state loses stability. The bifurcation at HB2 is supercritical, i.e., it gives rise to a stable branch of periodic solutions and, as pst decreases, this branch loses stability in a neighborhood of HB1. The Hopf bifurcation HB1 is supercritical as well but the branch that originates from HB1 has no physiological significance. The behavior of our system in a neighborhood of HB1 is very complex and beyond the scope of this study. Therefore it is omitted in Fig. 2 A and the bifurcation diagram is not complete. However, the branch that originates from HB2 corresponds to stable oscillations in [Ca2+] with physiological period and amplitude. Moreover the period of the oscillations increases as pst, and hence the agonist dose, decreases.
|
; Baesens et al., 1991; Takamatsu et al., 2001; Yoshimoto et al., 1993). Ashwin et al. (1990), in particular, studied three identical oscillators with weak symmetric coupling, whereas Baesens et al. (1991) performed a detailed study of three non-identical (i.e., with three different intrinsic frequencies) weakly coupled oscillators. In contrast, Takamatsu et al. (2001) and Yoshimoto et al. (1993) investigated specifically designed experimental systems of three coupled oscillators. Although most of the various modes of collective behavior found theoretically, as well as observed experimentally, are predicted by symmetric bifurcation theory (Golubitsky and Stewart, 2002), this theory does not apply when the differences between the oscillators are large.
Thus, as pointed out in Strogatz and Stewart (1993), the possible types of collective behavior in a system of three diffusively and all-to-all coupled oscillators (in the case of three cells all-to-all and nearest-neighbor coupling schemes coincide) could be classified according to the strength of coupling and the relative differences between the individual oscillators in the following way:
Homogeneous cluster of three cells
The homogeneous cluster of three cells can be regarded as a D3(S3) symmetric bi-directional ring of three identical nonlinear oscillators with identical, diffusive two-way nearest-neighbor coupling (Fig. 1 A). According to the theory of symmetry-breaking bifurcations (Golubitsky and Stewart, 2002), there are several oscillatory branches, each corresponding to a different isotropy subgroup of D3 x S1. In other words, there are a number of possible oscillatory modes, predicted by the equivariant Hopf bifurcation theory, in which a bi-directional ring of three cells could synchronize.
To study a homogeneous cluster of three cells we assume that 
is the same for each cell. Our point-oscillator model then exhibits most of the predicted oscillatory behaviors. Fig. 2 B presents the bifurcation diagram of three identical coupled cells for a fixed value of = 0.6 (s–1) by using pst as the bifurcation parameter. This scheme is valid for all three variables c(i), i = 1,2,3 since the type of the dynamical behavior, i.e., whether it is a steady-state or oscillatory state, is the same for all variables for any given values of pst and , due to the D3-symmetry properties of the system.
Fig. 2 B clearly demonstrates that the dynamics of three coupled cells is much more complex than that of a single cell. The steady state in Fig. 2 B loses stability inside a region of pst values, bounded by the pair of Hopf bifurcation points, 
and 
Note that, because the coupled oscillators are identical, 
and 
have the same values as in the case of a single cell (as shown in Fig. 2 A, where they are labeled HB1 and HB2). In the three-cell model a second pair of Hopf bifurcation points, 
and 
result from the factor D(). These new Hopf bifurcations occur on the unstable branch of the symmetric steady state.
Branch b1 (Fig. 2 B), originating from the rightmost Hopf bifurcation point, 
loses its stability for a range of pst values bounded by a pair of symmetry breaking bifurcation points, BP1 and BP2. On b1 all three cells oscillate with identical amplitudes and in synchrony (phase shift = 0 between the cells). The symmetry breaking bifurcation points, BP1 and BP2, give rise to two more branches, b2 and b3, which have stable and unstable parts and connect the points BP1 and BP2. Both branches b2 and b3 originate via subcritical bifurcations and become stable through saddle-node-of-periodics bifurcations for a region of pst values between L21 and L22, and L31 and L32, respectively. Because of the D1-symmetry we have pst(L21) = pst(L31) and pst(L22) = pst(L32). Those branches correspond to asymmetric phase-locked oscillations, when two of the cells oscillate in phase and the third cell oscillates out of phase with the other two cells (with phase shift 0 < < T/3) and slightly different amplitudes determined by b2 and b3 (for an example, see Fig. 3 A). The range of pst values where this type of collective behavior exists becomes smaller as the coupling strength, , increases, and is replaced by stable synchronous oscillations. The bifurcation diagram of the whole system for values of > 2.133 (s–1) looks exactly like the bifurcation diagram of a single cell ( = 0). We discuss this further below.
|
When the coupling strength, , is small, branch b4 becomes stable via a Torus bifurcation, for a range of pst values bounded by TR1 and TR2 (Fig. 2 C), and corresponds to symmetric phase-locked oscillations with phase shift = T/3 between each pair of cells and identical amplitudes (for an example, see Fig. 3 B). As increases, branch b4 loses stability, whereas branch b5 is found to be unstable for all values of pst and .
The main bifurcations 
and Lkj (j = 1, 2 and k = 2, 3) explained above are shown in Fig. 2 D, where, in addition to pst, the parameter is varied. These two-parameter continuations divide the (pst, ) plane into different regions corresponding to different dynamical behaviors. In the regions left of the line HBsyn1 and right of the line the system will always approach a stable steady state. Between the lines 
the steady state is unstable and the system will tend toward one of the limit cycles representing synchronous, phase-locked, or asynchronous oscillations. In Fig. 2 D the lines Lkj (j = 1,2 and k = 2,3) are represented by a single line because pst(L21) = pst(L31) and pst(L22) = pst(L32) for all values of . Above the line, Lkj-only stable synchronous oscillations are possible, whereas below 
and Lkj (j = 1,2 and k = 2,3), stable asymmetric phase-locked, and stable or unstable (depending on the value of ) symmetric phase-locked oscillations, may exist.
Fig. 2, B–D, reveals that there are parameter regions where different stable oscillatory modes may coexist, depending on the initial conditions. The simultaneous occurrence of stable portions of branches b1, b2, and b3 in a neighborhood of the bifurcation points BP1 and BP2, or stable parts of branches b2, b3, and b4 at low values of , accounts for bi-rhythmicity in the present model. Moreover, for very small regions of pst and -values, the coexistence of stable portions of branches b1, b2, b3, and b4 indicates that there exists even a tri-rhythmicity of asynchronous oscillations and two kinds of phase-locked behavior—symmetric and asymmetric.
Heterogeneous cluster of three cells
Real cells are not identical. For instance, there is experimental evidence that different pancreatic acinar cells respond differently to the same level of agonist stimulation (Ngezahayo and Kolb, 1993). When differences between the cells are small, i.e., they oscillate with similar frequencies, the results from the analysis of the homogeneous cluster of three cells will be a good approximation for the collective behavior of a heterogeneous cluster. However, this is not the case when those differences are larger.
To investigate the situation when the differences among the individual cells are significant, we study Model 1 using different values for 
for each individual cell. Since each cell will thus have a different intrinsic frequency, to respond in synchrony the frequencies of the different cells in the cluster must adjust to those of their neighbors. Our analysis shows that this can be accomplished by gap-junctional diffusion of Ca2+. For that purpose, we study Model 1 again, using pst and as the main bifurcation parameters.
Of course, there are many other ways in which the cluster of cells could be made heterogeneous. Varying any of the parameters from cell to cell would accomplish the task. However, the ultimate effect of such changes is effectively expressed in the intrinsic frequency of the cells. No matter how the intrinsic frequency is changed, the qualitative effect will be the same. Thus for our purposes it is sufficient merely to vary to obtain these different intrinsic frequencies.
In the heterogeneous case, the steady state generally has three pairs of Hopf bifurcations (Fig. 4 A), arising when pairs of eigenvalues cross the imaginary axis. As in the single cell, the steady state is stable for small and large values of pst, and in the intermediate region between the Hopf bifurcation points, HB1 and HB6 (which arise when the first pair of eigenvalues crosses the imaginary axis), it loses stability. There are two more pairs of Hopf bifurcation points, HB2 and HB4, and, HB3 and HB5, originating respectively from the second and the third pair eigenvalues. The system (Eqs. 12–20) exhibits oscillatory behavior between HB1 and HB6. Fig. 4 summarizes the loci of the Hopf bifurcation points existing in the system for three different degrees of heterogeneity (
and 
) in the pst–-plane.
|
If the cells are uncoupled, i.e., there is no gap-junctional diffusion of Ca2+, each cell oscillates with its intrinsic frequency (Fig. 5, A and C, and Fig. 6, A and C). Similar asynchronous behavior is observed for non-zero but very small ( 
0) coupling. When the coupling is weak (for our system, < 0.8 (s–1)) there are several possible stable oscillations. The branch originating at HB4 appears to be unstable for all values of pst and . In the case of weak coupling the branch arising from HB5, initially unstable, becomes stable via a Torus bifurcation and gives phase-locked (1:1:1) behavior with the common period determined by the fastest cell in the cluster, as shown in Fig. 5, A and B. Such increases in the frequency of a coupled triplet of pancreatic acinar cells compared to the single cell have been reported in Petersen and Petersen (1991). Furthermore, period-doubling bifurcations on the same branch (originating at HB5) give rise to phase-locked behavior different from (1:1:1). Fig. 5 D illustrates an example of such a behavior, with ratios (1:2:2), i.e., where the slowest oscillator drops out of every other cycle. When the coupling strength is small the branch originating from HB6 is stable only for very large values of pst and loses stability via a Torus bifurcation approximately at pst(HB5).
|
|
, who showed that increasing gap-junctional permeability with 
synchronizes the amplitude and frequency of the cholecystokinin evoked [Ca2+]i oscillations. |
MODEL 2: SPATIALLY DISTRIBUTED CELLS |
|---|
 TOPABSTRACTINTRODUCTIONMODEL 1: POINT-OSCILLATOR CELLS MODEL 2: SPATIALLY DISTRIBUTED... DISCUSSION AND CONCLUSIONSAPPENDIX 1: SUMMARY OF...APPENDIX 2: FACTORIZATION OF...ACKNOWLEDGEMENTSREFERENCES |
|---|
; Fogarty et al., 2000a; Kasai et al.,1993). Our initial numerical simulations of Model 2 will not be so complex, however, but will instead just assume that each cell has the same parameters throughout (parameters corresponding to the apical region). We call such cells homogeneous. This initial simplified approach will facilitate comparison with the results of Model 1, for which we used the apical parameters. However, we will then discuss simulations of the full model, in which each cell has distinct apical and basal regions. These model cells we call heterogeneous.
Description of the model
As before, we use the model of Sneyd et al. (2003), who assumed that the apical and basal regions of the cells have different densities of IP3 receptors (IPRs) and ryanodine receptors (RyRs), and are separated by a mitochondrial buffer band (Tinel et al., 1999; Straub et al., 2000) (Fig. 7 C). The parameter kf controls the density of IPR, and v1 controls the density of RyR. The parameter values are determined by fitting to experimental data (Sneyd et al., 2003). Both Ca2+ and IP3 are assumed to diffuse, with constant diffusion coefficients of Dc and Dp, respectively. We do not include explicit buffers, and hence all the diffusion coefficients, as well as all the Ca2+ fluxes, are defined as effective diffusion coefficients and fluxes.
|
with a time constant of 1/ß s. As before, we model agonist stimulation by an increase in 
[IP3], denoted by p, then obeys the reaction-diffusion equation
 | (2) |
is assumed to be spatially homogeneous within each cell, i.e., each part of the cell tends toward the same [IP3] after agonist stimulation. However, it is important to note that, although within each cell pst is not spatially varying, nevertheless the steady-state distribution of IP3 will not be spatially homogeneous within each cell. Because each cell has a different value of pst, intercellular diffusion of IP3 will cause steady-state spatial gradients of IP3, particularly close to the intercellular boundaries. The same is true of the resting distribution of Ca2+. Due to the asymmetric geometry of the acinus, and the fact that each cell has different apical and basal regions, spatial Ca2+ gradients will be introduced at rest. Thus the steady state must be found numerically, by starting with reasonable, random, initial conditions, and integrating until the steady state is reached. All the simulations of Model 2 use this spatially heterogeneous steady state as the initial condition.
The two-dimensional version of the model consists of two reaction-diffusion equations, Eqs. 28 and 29, describing the reaction and diffusion of cytosolic Ca2+ and IP3 respectively, coupled to a system of seven ordinary differential equations, Eqs. 13–19, for the dynamics of the Ca2+ concentration in the ER, the IPR, and the RyR. A summary of the model equations is given in Appendix 1, and the parameter values are given in Table 1.
Incorporation of gap junctions
Gap junctions in pancreatic acinar cells are permeable to both Ca2+ and IP3 and allow the diffusion of diverse small-sized molecules between neighboring cells (Ngezahayo and Kolb, 1993; Petersen and Petersen, 1991; Stauffer et al., 1993; Yule et al., (1996). Therefore we incorporate gap-junctional diffusion of both molecules in our model. We assume that at each internal cell boundary the flux is dependent on the concentration difference across the membrane as well as on the permeability of the gap junctions to Ca2+ and IP3, respectively. Thus, at each intercellular boundary
 | (3) |
 | (4) |
Numerical method
The model equations (Eqs. 28 and 29) were solved using a standard Galerkin Finite Element method and the rest of the model equations were solved by a backward Euler method in two spatial dimensions on a finite element mesh (Fig. 7 B) based on a real image of a triplet of pancreatic acinar cells (Fig. 7 A). No-flux boundary conditions were applied on the external borders of each cell and the cells were connected by the internal boundary conditions, Eqs. 3 and 4. Explicit gap junctions were not included in the numerical simulations; it was assumed that IP3 and Ca2+ can diffuse between cells at any grid point on the internal borders. The equations were solved on a mesh with 1301 grid points (nodes) corresponding to 1239 elements within each cell. The location of the apical, the mitochondria buffer, and the basal regions were approximately determined by comparing to the experimental images for each of the cells (Fig. 7 C).
Results
Dynamics of a single cell
The dynamics of a single spatially distributed cell is discussed in detail in Sneyd et al. (2003). For completeness we will outline here some of the main results from this analysis. Due to the lower receptor densities, the basal responses are smaller than in the apical region, which agrees well with experimental data (Kasai et al., 1993; Straub et al., 2000). Furthermore, model simulations show that Ca2+ rises in the apical region first, followed by a wave-spread across the basal region as has been experimentally observed (Fogarty et al., 2000b; Kasai et al., 1993; Lawrie et al., 1993; Leite et al., 2002; Nathanson et al., 1992; Straub et al., 2000; Thorn et al., 1993; Thorn, 1996). At low agonist stimulation mitochondrial uptake can eliminate an intracellular Ca2+ wave, but only does so for a narrow range of parameter values. This is consistent with the relatively narrow range of IP3 concentrations in which Ca2+ signals confined to the apical region in pancreatic acinar cells have been observed (Straub et al., 2000). The model predicts that there are two distinct mechanisms of wave propagation. At low [IP3] the Ca2+ wave is transmitted from the apical to the basal region by an active wave, dependent on diffusion of Ca2+ between release sites, whereas at high [IP3] the intracellular phase waves result from differences in the intrinsic frequencies of the oscillations in the apical and basal regions. In this second case, the wave is called a kinematic wave, and does not depend on Ca2+ diffusion.
Dynamics of three coupled cells
When the cells are identical each cell eventually ends up with the same [IP3] and thus the diffusion of IP3 through gap junctions has no influence on the long-term collective behavior of the cluster. However, by assuming that each cell has a different value of we are able to generate intercellular [IP3] gradients. This is consistent with the suggestion made in Yule et al. (1996) that differences in the intrinsic frequencies of oscillations among the cells in a triplet are most likely due to differences in the ability of different cells to produce IP3. Such differences have also been identified as gradients in agonist sensitivity, which implies differences in [IP3] between connected cells.
We are interested in whether the results from the analysis of the point-oscillator model, where we have used apical parameter values, could be related to the results from the analysis of the spatially distributed model. Therefore we begin by neglecting the spatial heterogeneity inside each cell and use only apical parameter values over the three cells. In this way the cells differ in size and shape, and are spatially distributed, but have the same parameters throughout. Later we include the structural heterogeneity of each cell by assuming different parameters in the apical and basal regions.
Homogeneous cells; only apical parameter values
First we study the simpler case where either Ca2+ or IP3 gap-junctional flux is assumed to be zero, i.e., the cells in the cluster are assumed to communicate only through one of the two possible messengers. Intercellular diffusion of IP3 alone fails to coordinate Ca2+ oscillations even for unrealistically large values of IP3 permeability (computations not shown). In contrast, setting pf = 0 (µm s–1), i.e., preventing IP3 diffusion between cells, does not prevent the long-term development of synchronized oscillations (computations not shown). We conclude that Ca2+ is necessary and sufficient for synchronizing Ca2+ oscillations. However, it has been experimentally verified (Ngezahayo and Kolb, 1993; Petersen and Petersen, 1991; Stauffer et al., 1993; Yule et al., 1996) that IP3 also diffuses through gap junctions. So, if the junctional flux of IP3 is not the reason for the long-term spatial organization of Ca2+ signals in coupled cells, then does it contribute in this process and, if so, to what extent?
We suggest that IP3 gap-junctional diffusion contributes to synchronization of Ca2+ oscillations in connected cells by decreasing the differences in the individual oscillators' periods. Hence smaller amounts of Ca2+ diffusing through gap junctions (represented by smaller values of cf) will be enough to coordinate the oscillations in [Ca2+]. (By coordinated behavior we mean either synchronous or phase-locked.) This is illustrated in Fig. 8, where the respective regions of phase-locking and synchronization in the two parameter space (pf,cf) are outlined. It is clearly seen that increasing IP3 permeability leads to a decrease in the value of cf which is sufficient to coordinate the three cells in the cluster. It is worthwhile to note that qualitatively the same result follows from similar simulations with the point-oscillator model in which IP3 as well as Ca2+ is allowed to diffuse (computations not shown).
|
|
). This happens not only in the case of triplets of cells but also in bigger clusters, where a group of cells responds in synchrony followed by another group (unpublished data). However, this model prediction needs to be confirmed by more extensive analysis of data. Comparison of Fig. 10, A and C, demonstrates again the role of IP3 gap-junctional diffusion in smoothing the differences between the cells in the cluster. Comparing Fig. 10, A and C, we see that for one and the same strength of Ca2+ coupling (cf = 0.9 µm s–1) the type of collective behavior depends on the amount of IP3 diffusing through gap junctions. When the coupling strength for IP3 is smaller (pf = 2.0 µm s–1) the differences between the cells are larger and hence they fail to phase-lock (1:1:1) (Fig. 10 C). Increasing the coupling strength for IP3 (pf = 3.0 µm s–1) leads to asymmetric phase locking, as shown in Fig. 10 A. These theoretical results agree very well with the experimental observations (Ngezahayo and Kolb, 1993) that gap-junctional diffusion modulates the collective oscillatory behavior in coupled pancreatic acinar cells.
|
|
DISCUSSION AND CONCLUSIONS |
|---|
) and studied the dynamics of three cells coupled in a ring. Such a geometric arrangement corresponds to what is commonly found in experiment. Our goal was to study whether intercellular diffusion of Ca2+ could synchronize intercellular waves, and what role intercellular diffusion of IP3 plays in such synchronization. To do this, we constructed two qualitatively different models. Model 1 modeled each cell as a point oscillator, with no spatial dependence, an approach that is commonly used in the study of coupled oscillators. Model 2 assumed each cell to be spatially distributed, coupled by Ca2+ and/or IP3 diffusion through gap junctions along each common border. Since this is a more realistic scenario, our goal was to determine whether analysis of the simpler Model 1 could give any insight into the behavior of the more complex Model 2.
Our results can be summarized as follows:
Our simulations show increases in the frequency of the Ca2+ oscillations among connected cells compared to the frequency of a single cell, as has been observed experimentally in small clusters of pancreatic acinar cells (Ngezahayo and Kolb, 1993; Petersen and Petersen, 1991; Stauffer et al., 1993). At moderate levels of Ca2+ gap-junctional diffusion the common frequency is determined by the fastest cell in the cluster, as occurs also in coupled hepatocytes (Höfer, 1999). In agreement with the experimental data from pancreatic acinar cells, increasing gap-junctional conductance not only tunes the phase difference of the Ca2+ oscillations (Ngezahayo and Kolb, 1993) but also equalizes the amplitude and frequency of these oscillations (Stauffer et al., 1993). Therefore our results are consistent with the presence of a pacemaker cell in coupled multiplets of pancreatic acinar cells, where gap-junctional communication increases their sensitivity range. In this way, as proposed in Yule et al. (1996), the augmentation of a threshold Ca2+ response generated in one cell, all over the whole cluster, may result in increased secretion. Such increased enzyme secretion has also been reported in Stauffer et al. (1993).
All of the qualitative results from Model 1 are generic to three coupled oscillators. The various kinds of coupled oscillations predicted by the model are not unique to our particular formulation of the intracellular Ca2+ dynamics. Thus any model with the same basic structure (i.e., any model in which Ca2+ oscillations occur at constant [IP3]) would be expected to give the same qualitative results, although, of course, the precise details of the parameter values and the intracellular oscillations would vary. In particular, our conclusions about the crucial importance of gap-junctional Ca2+ diffusion for coupling cells would remain unchanged by the use of a different model with the same overall structure. Furthermore, our conclusion that a point-oscillator model is only an indifferent guide to the behavior of a spatially distributed model is independent of the precise model details.
However, there is one aspect of our model that could possibly play an important role in determining multicellular behavior. A fundamental dynamical feature of our model is that the concentration of IP3 does not oscillate. Although we have simulated a gradient of [IP3] between individual cells, this is not a periodically changing gradient. It is not yet known whether oscillations in [IP3] are necessary for Ca2+ oscillations in pancreatic acinar cells; experimental evidence so far is inconclusive. If an oscillating [IP3] is a crucial feature of this cell type, then this will have an important effect on the predictions from a multicellular model. For instance, if IP3 oscillations can themselves be synchronized between cells, then it is very likely that gap-junctional diffusion of IP3 will be sufficient (although perhaps not necessary) for intercellular synchronization. Thus, definitive confirmation of our model predictions awaits first a definitive determination of the role of IP3 oscillations in the single cell responses.
One other caveat must also be mentioned. In studying the synchronization of multicellular behavior, we have concentrated on the long-time responses, i.e., the kinds of coupled oscillations that develop after many oscillation periods. In the short-term, cells can appear to be synchronized, even though over a longer timescale such synchronization gradually breaks down. This point is very important when comparing model predictions to experimental data. If the data is not collected over a long-enough time period, it is not possible to conclude anything about the long-term synchronization of the cells—in which case, comparison with the model predictions above is problematic.
The values we use for (the coupling strength for Ca2+ in Model 1) are in the same range as those estimated by Höfer (1999) in a similar study of hepatocytes. In particular, the value of < 0.8 (s–1) predicted by our model giving weak coupling between heterogeneous cells, agrees well with the value given in Höfer (1999). Furthermore, the values of pf (gap-junctional permeability to IP3) in Model 2, which affect the coordination of Ca2+ signal in our analysis, are in very good agreement with the values used in studies of intercellular Ca2+ waves in hepatocytes (Dupont et al., 2000b), astrocytes (Höfer et al., 2002), mixed glial cells (Sneyd et al., 1998), and airway epithelial cells (Sneyd et al., 1995). However, the value of cf (gap-junctional permeability to Ca2+) predicted by our Model 2, although consistent with the value in Sneyd et al. (1998), is significantly higher than the values estimated in Höfer et al. (2001, 2002). The reason for such a discrepancy may be in the different geometry that we have used as well as in the polarized nature of the pancreatic acinar cells. Unfortunately there are no direct experimental measurements of the Ca2+ gap-junctional conductance in pancreatic acinar cells. Therefore the question about the precise values of cf remains open and requires further experimental and modeling work.
|
APPENDIX 1: SUMMARY OF THE MODEL EQUATIONS |
|---|
and 
C and D use 
and 
Note that in C and D, c is plotted on a log scale.
). As the intercellular diffusion of IP3 increases, phase-locking of the apical regions becomes more pronounced. However, due to their different geometry and size, which results in a lower effective coupling strength, the basal regions show a much lesser degree of coordination.