Calcium oscillations in a triplet of pancreatic acinar cells

Abstract

We use a mathematical model of calcium dynamics in pancreatic acinar cells to investigate calcium oscillations in a ring of three coupled cells. A connected group of cells is modeled in two different ways: 1), as coupled point oscillators, each oscillator being described by a spatially homogeneous model; and 2), as spatially distributed cells coupled along their common boundaries by gap-junctional diffusion of inositol trisphosphate and/or calcium. We show that, although the point-oscillator model gives a reasonably accurate general picture, the behavior of the spatially distributed cells cannot always be predicted from the simpler analysis; spatially distributed diffusion and cell geometry both play important roles in determining behavior. In particular, oscillations in which two cells are in synchrony, with the third phase-locked but not synchronous, appears to be more dominant in the spatially distributed model than in the point-oscillator model. In both types of model, intercellular coupling leads to a variety of synchronous, phase-locked, or asynchronous behaviors. For some parameter values there are multiple, simultaneous stable types of oscillation. We predict 1), that intercellular calcium diffusion is necessary and sufficient to coordinate the responses in neighboring cells; 2), that the function of intercellular inositol trisphosphate diffusion is to smooth out any concentration differences between the cells, thus making it easier for the diffusion of calcium to synchronize the oscillations; 3), that groups of coupled cells will tend to respond in a clumped manner, with groups of synchronized cells, rather than with regular phase-locked periodic intercellular waves; and 4), that enzyme secretion is maximized by the presence of a pacemaker cell in each cluster which drives the other cells at a frequency greater than their intrinsic frequency.

FIGURE 1

(A) Schematic diagram of Model 1, the point-oscillator model. (B) Schematic diagram of the single-cell model.

FIGURE 2

(A) Bifurcation diagram of the single-cell model, showing the maximum of the periodic orbits as a function of p_st. (B) Bifurcation diagram of Model 1 for ɛ = 0.6, showing the maximum of the periodic orbits as a function of p_st. (C) Magnified view of branches b₄ and b₅ for ɛ = 0.03. (D) Two-parameter bifurcation diagram in (p_st, ɛ)-space. (HB, Hopf bifurcation; BP, branch point; L, saddle node of periodics; TR, Torus bifurcation point.) The broken lines denote instability. In these computations, as in all other computations on Model 1, we use the apical parameter values.

FIGURE 3

(A) Typical asymmetric phase-locked oscillations in Model 1 for ɛ = 0.6 and p_st = 20.6. (B) Typical symmetric phase-locked oscillations in Model 1 for ɛ = 0.03 and p_st = 20.

FIGURE 4

Two-parameter bifurcation diagram in (p_st,ɛ)-space. (A) Loci of the Hopf bifurcation points of Model 1 in the case of a heterogeneous cluster of three cells. (B) Locus of the first pair of Hopf bifurcation points of Model 1 for triplets of cells with varying degrees of heterogeneity. (C) Locus of the second pair of Hopf bifurcation points of Model 1 for triplets of cells with varying degrees of heterogeneity. (D) Locus of the third pair of Hopf bifurcation points of Model 1 for triplets of cells with varying degrees of heterogeneity.

FIGURE 5

Typical oscillations in Model 1 in the case of weak coupling. A and B correspond to mild heterogeneity, and C and D to greater heterogeneity. (A) Uncoupled cells; (B) typical asymmetric phase-locked oscillations in Model 1 with weak coupling. Here, the common frequency is determined by the fastest cell in the cluster. These oscillations arise via a Torus bifurcation when the branch of periodic orbits originating at HB₅ gains stability. (C) Uncoupled cells; (D) typical phase-locked (1:2:2) oscillations in Model 1 with weak coupling. These oscillations correspond to a branch of periodic orbits arising via a period doubling bifurcation on the branch originating at HB₅.

FIGURE 6

Comparison between the typical oscillations in Model 1 at near-threshold (C and D) level of stimulation and at higher (A and B) stimulation level. (A and C) Uncoupled cells. (B and D) Strongly coupled cells, showing synchronized oscillations. Here the common frequency and amplitude are an average of those of the individual cells. These oscillations correspond to the branch of periodic orbits corresponding to the pair of Hopf bifurcation points HB₁ and HB₆ in Fig. 4 B. A and B use and C and D use and Note that in C and D, c is plotted on a log scale.

FIGURE 7

(A) Experimental image of a cluster of three pancreatic acinar cells; (B) the mesh upon which we solve the equations of Model 2 using a finite element method. (C) Experimental image of the cluster showing the approximate positions of the apical and basal regions that were used in the model.

FIGURE 8

Two-parameter diagram in (p_f,c_f)-space showing approximate regions where synchronous, asynchronous, and phase-locked behavior occurs. Using the parameters p₁ = 10, p₂ = 20, and p₃ = 30, we solved the model equations until approximate steady-state behavior was obtained. Since effects of hysteresis were not investigated, and due to limited resolution in p_f and c_f, the boundaries are only approximate.

FIGURE 9

Intercellular oscillations in the triplet where each cell is assumed to be homogeneous, i.e., the parameters do not vary within each cell and are assumed to be those of the apical region (Table 1). The concentration of IP₃ is constant (p_st = 40) over the three cells, and the cells are identical. Ca²⁺ gap-junctional permeability is set to c_f = 1.2. (A) Traces taken from the middle of the apical region; (B) traces taken from the middle of the basal region. (Note that the parameter values are the same for each region but the responses are different because of geometrical factors.)

FIGURE 10

Intercellular oscillations in the triplet where each cell is assumed to be heterogeneous, i.e., the apical and basal regions have different parameter values. Traces are taken approximately from the regions denoted by the arrows in Fig. 7 C. Initially p_int = 17 over the three cells. In each of the cells the parameter p_st is chosen so that the three cells tend to different steady states of [IP₃] (). 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.