INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL ANALYSIS RESULTS DISCUSSION APPENDIX 1 APPENDIX 2 REFERENCES

The elevation of the cytoplasmic calcium concentration is a central step in many intracellular signal transduction pathways (e.g., Thomas et al., 1996; Berridge, 1997). Recently, it has been observed in various systems that calcium signals can also mediate intercellular communication by eliciting or coordinating calcium signals in surrounding cells, for example in the liver (Robb-Gaspers and Thomas, 1995; Patel et al., 1999), and the astrocyte networks of the central nervous system (Cornell-Bell et al., 1990; Giaume and Venance, 1998). Two general pathways of intercellular calcium signaling have been identified: the diffusion of cytoplasmic messenger molecules through gap junctions (e.g., Charles et al., 1992; Giaume and McCarthy, 1996; Tjordmann et al., 1997; Toyofuku et al., 1998; Domenighetti et al., 1998), and the secretion of extracellular messengers (Hassinger et al., 1996; Schlosser et al., 1996).

Both the calcium-releasing messenger inositol 1,4,5-trisphosphate (IP₃) and calcium can participate in the gap-junctional mode of transmission (Sáez et al., 1989; Christ et al., 1992). In some systems, such as the airway epithelium, a stimulus applied to a single cell elevates IP₃ through activation of phospholipase C (PLC). IP₃ is thought to diffuse from the stimulated cell and trigger calcium release in surrounding cells. (Sanderson, 1995; Sneyd et al., 1995). In other systems, an external signal linked to PLC activation is applied globally, so that IP₃ concentration increases in practically all cells. Under these conditions, calcium has been hypothesized to act as an intercellular mediator, e.g., in pancreatic acini (Yule et al., 1996), chondrocytes (D'Andrea and Vittur, 1997), hepatocytes (Robb-Gaspers and Thomas, 1995), and the blowfly salivary gland (Zimmermann and Walz, 1999). This appears feasible, as calcium liberation from the endoplasmic reticulum (ER) can be activated by calcium in the presence of IP₃ receptor calcium release channels (IP₃R) that are sensitized by IP₃ (Bezprozvanny and Ehrlich, 1995), and potentially also ryanodine receptor channels (RyR; Meissner, 1994). We refer to this phenomenon for both sensitized IP₃R and RyR as calcium-induced calcium release (CICR). In the presence of CICR, calcium influx through gap junctions may trigger calcium release in a cell and in this way a regenerative intercellular calcium wave could spread. Given the occurrence of CICR and gap junctions in many systems, this may be a basic mechanism of intercellular calcium signaling. However, up to now little is known about the requirements on the various cellular calcium transport processes that would enable gap-junctional calcium fluxes to propagate a calcium signal.

The understanding of the interaction of these processes can be greatly facilitated by mathematical modeling. Recently, models based on a CICR/gap-junctional calcium diffusion mechanism were developed for the formation of intercellular spiral waves of calcium in hippocampal slices (Wilkins and Sneyd, 1998), and for the synchronization of calcium oscillations in hepatocyte couplets (Höfer, 1999). A common finding of these mainly numerical studies is the existence of a critical junctional calcium permeability, which must be exceeded for intercellular wave propagation or synchronization. The calcium kinetics in the two models assume two different mechanisms of the decline of calcium concentration, both of which have been implicated in experimental studies: slow inactivation of the IP₃R (Wilkins and Sneyd, 1998) and decrease of the total calcium content of the cell (Höfer, 1999). However, it appears that the calcium wavefronts primarily propagate through the interaction of CICR and calcium diffusion, so that the waves should have common properties irrespective of the particular dynamics in their wakes. In the present paper we thus study a model of calcium elevation through CICR coupled to cytoplasmic and gap-junctional calcium diffusion.

The model considers a linear cell array and accounts for the following calcium transport and binding processes: 1) CICR from the calcium stores of the ER; 2) removal from the cytoplasm; 3) buffering by calcium binding to proteins, lipids, and other molecules; 4) cytoplasmic diffusion; and 5) in the perijunctional space, calcium fluxes across the gap junctions. Many of the parameters of these processes can vary, and their variation may affect intercellular wave propagation. In the presence of IP₃R, the maximal rate of CICR is a function of the IP₃ concentration that depends on the amount of PLC-activating external agonist. The composition of calcium buffers in the cell can be regulated and also altered experimentally (e.g., Wang et al., 1997). Moreover, all parameters may vary with cell type and conditions, particularly pertinent to the study of intercellular signals being the regulation of gap-junctional permeability (Bruzzone et al., 1996).

The analysis of the model will focus on the conditions under which intercellular calcium waves can occur, and on how the occurrence and properties of the waves depend on the parameters of the calcium transport processes in the cell. We are aware that a detailed representation of these processes requires considerably more complex models. However, the present paper is aimed at elucidating characteristics of the CICR/calcium diffusion mechanism in terms of basic cellular parameters. The results may inform experimental studies and more detailed modeling approaches to specific systems.

	MODEL

TOP ABSTRACT INTRODUCTION MODEL ANALYSIS RESULTS DISCUSSION APPENDIX 1 APPENDIX 2 REFERENCES

Model equations and parameters

Consider the linear array of cells depicted in Fig. 1. The concentration of cytoplasmic calcium, in the ith cell, [Ca_cyt,i²⁺] = u_i(x, t), i = 0, 1, ..., n, is governed by the rates of calcium release from the ER and removal from the cytoplasm, f(u_i), and by cytoplasmic diffusion with an effective diffusion coefficient D,

(1)

where L denotes the length of a cell and x is mapped for each cell individually to the interval (0, L). The function h(x) refers to the spatial distribution of calcium release/uptake sites.

View larger version (5K): [in this window] [in a new window]   FIGURE 1   Linear array of cells of uniform length, L, coupled by gap junctions.

During an intercellular calcium signal, calcium remains elevated for some time, usually some tens of seconds, as the signal spreads to neighboring cells, usually within seconds. We study the advance of the front of the signal and, for simplicity, assume that a (quasi-) stationary state of high calcium concentration is attained in the wake of the front. This can be described by the rate expression

(2)

with v_m and K_a denoting the maximal rate and half-saturation constant of CICR, respectively, and k being a "lumped" rate constant of calcium removal from the cytoplasm. Hill coefficients between 1.2 and 3.5 have been used to fit data for IP₃R (Bezprozvanny and Ehrlich, 1995; Dufour et al., 1997); subsequently we take  = 2. To carry out mathematical analysis of the model, we also consider a piecewise linear (p.w.l.) approximation to Eq. 2, corresponding to the limit   ,

(3)

where H(·) is the Heaviside step function. Equations 2 and 3 have previously been used as simple rate expressions for CICR (e.g., Murray, 1993). The width of a cytoplasmic calcium wavefront is about one to several micrometers. On this scale, we assume a homogeneous distribution of calcium release and uptake sites, h(x) = 1. However, before a calcium wave is initiated, a distance d between the gap junctions and the ER may have to be bridged solely by diffusion. To reflect this in a simple manner, we take

(4)

where d is the distance between gap junctions and calcium stores, 0  d < L/2.

The intercellular calcium fluxes are assumed proportional to the concentration differences across the gap junctions,

(5)

where P is the effective gap-junctional calcium permeability.

Cytoplasmic calcium is bound to many molecules, leading to a substantial buffering of its concentration. Equations 1 and 5 include the effect of buffering via a rapid-equilibrium approximation, assuming that calcium binding is fast compared to the rates of CICR, calcium removal, and buffer molecule diffusion, and that the buffers are not saturated by calcium binding. These are reasonable assumptions for a large class of buffering molecules (Neher and Augustine, 1992; Wagner and Keizer, 1994). In Appendix 1 it is shown that the calcium dynamics are then governed by Eqs. 1 and 5 with an effective rate of calcium release/removal f(u), an effective diffusion coefficient D, and an effective junctional calcium permeability P, defined as

(6)

where B_j, K_j, and D_j denote the total concentration of the calcium binding sites of the buffer species j, their dissociation constant, and the diffusion coefficient of the buffer species j, respectively. D₀, f₀(u), and P₀ are the respective values the diffusion coefficient of calcium, its release/removal rate, and its junctional permeability would attain in the absence of buffers. Values of _j B_j/K_j range between 20 and 100 in various cell types (Neher and Augustine, 1992; Daub and Ganitkevitch, 2000).

Typical ranges for the parameters are given in Table 1; the values may vary with the particular system. In the case of IP₃R, v_m increases with the IP₃ concentration. D specifically depends on the concentration of diffusive calcium buffers, for which D_j > 0. This reflects that calcium may diffuse bound to buffers and then be released. P₀ depends on the number, distribution, type, and state of the gap junctions, and on the permeating molecule. The quoted values of P₀, obtained for molecules other than calcium and in specific cell types, are taken to indicate the accessible range. _j B_j/K_j of 20-100 yields P between 0.01 and 0.15 µm/s (Eq. 6).

Limitations of the model

The rate functions Eqs. 2 and 3 are substantially simplified compared to other models of cellular calcium dynamics. Of relatively little consequence is the neglect of a small leak flux of calcium into the cytoplasm, causing the resting calcium level in the model to be zero, rather than at a concentration of 50-100 nM. Inclusion of a leak would not affect the results, so that we have not done so. It is also assumed that the calcium concentration in the ER is negligibly reduced by calcium release, rendering the release rate a function of only cytoplasmic calcium. Moreover, the inactivation of the IP₃R by very high calcium concentrations is not included. As a consequence of the latter two assumptions, the model can describe the leading front, but not the falling phase, of a calcium signal. Such a description is reasonable if the time scales of intercellular spread and slower decline of the signal separate. The model is found to overestimate the peak calcium concentrations for values of v_m and k that give realistic intracellular calcium wave speeds, reaching 1-3 µM, rather than the measured values of 500 nM  1 µM, probably owing to the neglect of inactivating processes. The various calcium removal processes from the cytoplasm (primarily uptake into ER and mitochondria, near the plasma membrane also efflux from the cell) have been lumped into a single linear removal rate, ku. In real systems, saturation effects at higher calcium concentration will play a role; however, at least for small concentration changes, such as through gap-junctional calcium influx, saturating rate laws can be linearized. An assumption implicit in the spatially one-dimensional formulation of the calcium diffusion fluxes in Eqs. 1 and 5 is an overall homogeneous distribution of gap junction channels across the plasma membranes at cell contacts.

	ANALYSIS

TOP ABSTRACT INTRODUCTION MODEL ANALYSIS RESULTS DISCUSSION APPENDIX 1 APPENDIX 2 REFERENCES

The analysis is simplified by obtaining suitable parameter groupings from the model parameters. We introduce the scaled time  = kt, space  = x/L, and calcium concentration  = u/K_a. For continuity, we will use the symbol u(, ) instead of for the scaled concentration; it will be set apart from the unscaled concentration by the independent variables. The model takes the form

(7)

(8)

(9a)

(9b)

with the smooth and p.w.l. kinetics, respectively; h(; l) is given by Eq. 4 with l = d/L. The three dimensionless parameter groupings are

(10)

With Eq. 6 we obtain

(11)

where v_m,0 and k₀ are the values of the rate constants in the absence of buffering. The behavior of the scaled model is independent of the concentrations of non-diffusive calcium buffers (for which D_j = 0).

Equations 7 and 8 can be approximated by a simpler set of equations for the average cytoplasmic calcium concentrations, U_i() = ₀¹ u_i(, )d, if the diffusion rate over the length of the cell is much greater than the rates of calcium release and of the junctional fluxes. In the corresponding limit /  0 and p/  0, Eqs. 7 and 8 can be shown to be equivalent to

(12)

From Table 1 and assuming a cell length of 10 µm, the following ranges for , , and p are obtained: 5 <  < 50, 0.1 <  < 0.4, and 0.001 < p < 0.015. Therefore, the compartmental approximation Eq. 12 is not reasonable for our model. It is useful as a point of reference in the analysis.

The kinetics of CICR and calcium removal exhibit bistability, provided  > 2 for Eq. 9a and  > 1 for Eq. 9b (Fig. 2). The upstroke of a calcium signal is represented by the transition from the rest state u = 0 to the excited state u_a; a moving transition front corresponds to the front of a calcium wave.

View larger version (12K): [in this window] [in a new window]   FIGURE 2   Bistable kinetics of ER calcium release and removal g(u), with sigmoid CICR function (Eq. 9a), solid line, and step-function CICR (Eq. 9b), dashed line;  = 10. The states u = 0 and u = ua correspond to the rest state of low cytoplasmic calcium and the elevated calcium level following the triggering of CICR, respectively. Quantitatively comparable results for both types of rate functions are obtained below if the threshold for CICR in the p.w.l. kinetics is adjusted to be equal to the unstable steady state of the smooth rate function by assuming a smaller Ka for Eq. 9b, and correspondingly a larger value of .

Suppose that initially the calcium concentration in all cells is at the rest state, u_i(x, 0) = 0, and that a local stimulus is applied in cell 0, at x = 0,

(13)

The calcium level is thought to remain elevated sufficiently long to assume s constant. Local initiation has been observed in liver lobules at sites of particularly high hormone sensitivity (Tordjmann et al., 1998), and in the blowfly salivary gland after calcium injection (Zimmermann and Walz, 1999). If the stimulus triggers a regenerative intercellular calcium wave, eventually all cells of the array become activated. However, it may be possible that signal propagation fails at some distance from the point of initiation, because the gap-junctional calcium influx into a cell becomes too small to excite CICR. In such a case the spatial range of the signal remains limited. The asymptotic behavior,   , in the limit of a semi-infinite cell array, 0  i < , yields

(14a)

(14b)

for regenerative intercellular calcium waves and spatially limited calcium signals, respectively. Here the u_i() denote stationary concentrations. If a stationary profile satisfying the boundary conditions 13 and 14b exists, we have observed that the u_i(, ) approach this solution after application of a local stimulus s. Therefore, regenerative intercellular calcium waves are not found in this case. The solutions to Eq. 12 with bistable kinetics have an analogous property; it can be used to obtain a condition for regenerative wave propagation (Keener, 1987).

The calculations can be done explicitly with the p.w.l. kinetics Eq. 9b. To guarantee initiation of a calcium signal within a cell, we assume s > 1, and spreading intracellular calcium waves to exist, yielding, from the condition ₀^u_a g (u) du > 0 (cf. Murray, 1993),  > 2. Stationary solutions to Eqs. 7 and 8 with 9b may satisfy Eqs. 13 and 14b, if calcium in cells up to cell m, m  0, is above the CICR threshold, while in the remaining cells it is below:

(15)

Letting u_i/ = 0 in Eq. 7 yields the following ansatz for the calcium profiles,

(16)

where _i =  if 0  i  m and zero otherwise,  = e^(12l)/, and continuity of u_i and u_i/ at  = l and  = 1  l is ensured. Joining the solutions for neighboring cells through Eqs. 8, one obtains a linear system of difference equations for _i and _i, of the form (_i+1, _i+1)^T = A(_i, _i)^T; the matrix A results from evaluating Eqs. 8 with Eq. 16 for i < m and i > m + 1. It is solved by

(17)

(18)

with

(19)

₁ = (  )/(1/  ), and ₂ = (  1)/(/  1). It is straightforward to show that  is real and 0    1. In Eq. 18 we have applied the boundary condition Eq. 14b, excluding ⁱ-terms.

Using Eqs. 17 and 18 with Eq. 16 to evaluate the gap-junctional flux conditions Eq. 8 between cells m and m + 1, and the left boundary condition Eq. 13, a linear system of equations for b₁, b₂, and B₁, as a function of m, is obtained. This generally has a unique solution. In this way, the coefficients _i and _i in Eq. 16 are found in terms of the model parameters, with the spatial range of the signal m to be determined. Consistency of the solution with relation 15 yields m such that

(20)

are fulfilled.

A critical situation occurs if the calcium concentration in cell m + 1 just reaches the CICR threshold at the location of the calcium stores: u_m+1(l) = 1. This condition separates the case of the (m + 1)st cell being not excited from the case of it becoming excited. For u_m+1(l) one obtains

(21)

where l_1,2 = 1 ± l/ and





If only a limited number of cells becomes excited, then u_m+1(l) < 1 for some finite value of m. Conversely, if lim_m u_m+1(l) > 1, then the solution satisfies Eq. 14a instead of Eq. 14b, since u_i  , as i  . In this case, we expect the stimulus to evoke nondecaying intercellular calcium waves. The critical condition separating the two cases is

(22)

Taking the limit m   in Eq. 21, we observe that the condition for propagation, Eq. 22, depends on the cellular parameters , p, , and l, but not on the size of the initiating stimulus s.

In the special case l = 0 (no gap between gap junctions and calcium stores), Eq. 22 yields

(23)

In the limit   , Eq. 23 can be solved for the critical junctional permeability p_c required for wave propagation. The limiting critical permeability is obtained as

(24)

in agreement with direct analysis of the compartmental Eq. 12 (cf. Keener, 1987). Recall that  > 2. The parameter estimates suggest the limit of small  to be appropriate for calcium waves. By expanding the square root in Eq. 19 for large T, we obtain from Eq. 23,

(25)

The critical junctional permeability is a monotonically increasing function of , initially given by Eq. 25 and approaching p_c, (cf. Fig. 3 a). Equation 25 was also obtained in the analysis of a pair of "semi-infinite" cells (Wilkins and Sneyd, 1998).

View larger version (48K): [in this window] [in a new window]   FIGURE 3   Critical gap-junctional permeability Pc required for intercellular calcium wave propagation. (a) Pc for p.w.l. kinetics (Eq. 9b) according to Eq. 23 (solid lines) and Eq. 25 (dashed lines); (b) Pc for smooth kinetics (Eq. 9a), calculated numerically at the points indicated. Results for three different values of  as given at the curves; values in (a) are chosen to obtain the same CICR thresholds as in (b) (cf. Fig. 2). (c) Space-time plot of the calcium concentration for intercellular wave (Eq. 9a with  = 15, D = 10 µm2/s, P = 0.03 µm/s, + in b); (d) propagation failure (P = 0.005 µm/s, × in b). The leftmost cell was stimulated and the first 5 of an array of 10 cells are shown. The nondimensional  and p have been converted to D and P, assuming L = 10 µm, k = 1/s; calcium concentration scale in dimensionless units.

It is also of interest how fast CICR becomes excited by gap-junctional calcium influx. Assume that in a cell calcium is at its resting concentration, while in a neighboring cell it has attained an elevated level, with concentration û immediately at the gap-junctional contact of the two cells. To be specific, let l = 0 and the cell to become excited at its left end,  = 0. The initial calcium rise before the triggering of CICR, u(, ) < 1, can be approximated by Eq. 7 subject to the initial and boundary conditions

(26)

with g(u) = u. Following, e.g., Crank (1975), the solution of this problem is obtained, in dimensional quantities, as

(27)

The steady state (x) is given by

(28)

indicating that the right boundary condition in Eqs. 26 is a reasonable approximation if L(k/D)^1/2 1. We found good agreement between Eq. 27 and numerical solutions on a finite domain with L(k/D)^1/2 = 4.

The approach to the steady state is monotonic at any space point. The characteristic time for the transition to the steady state is in the range of 0.5 s for the parameters of Table 1. It is possible to obtain the parameter dependence of the transition time, t_s. According to the definition of transition time by (Lloréns et al., 1999), we take t_s(x) = ₀ (1  u(x, t)/(x))dt, yielding t_s(0) = /[2k(P + )] at the gap junctions.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL ANALYSIS RESULTS DISCUSSION APPENDIX 1 APPENDIX 2 REFERENCES

Condition for propagation of intercellular waves

Intuitively, intercellular calcium wave propagation requires sufficiently strong gap-junctional coupling of the cells. In the model, an explicit condition for intercellular regenerative wave propagation is established for p.w.l. kinetics (Eq. 9b). We consider first the case of gap junctions and calcium stores being very close (d = 0 in Eq. 4). Equation 23 then yields a critical gap-junctional calcium permeability for wave propagation, P_c. P_c is a function of the other parameters; in Fig. 3 a its dependence on the effective calcium diffusivity is depicted. For better comparison with experimental data, in this and all following figures the nondimensional values p and  have been converted to P and D, assuming L = 10 µm and k = 1/s. For realistic values of the cytoplasmic calcium diffusivity, the propagation condition is approximated by

(29)

(Eq. 25 with dimensional parameters). Corresponding numerical results for the rate equation 9a give the same kind of critical curve (Fig. 3 b, Appendix 2). For P > P_c, a local stimulus triggers a regenerative intercellular calcium wave (Fig. 3 c). It consists of a succession of intracellular waves punctuated by gap-junctional delays, as observed experimentally. If P < P_c, no regenerative intercellular waves exist (Fig. 3 d). The values for P_c fall in the range of effective calcium permeabilities indicated by experimental data (Table 1).

Interestingly, the critical permeability for intercellular waves decreases with decreasing effective calcium diffusivity D, P_c ~  for small D. This dependence indicates that CICR is triggered locally in the perijunctional space. For small values of D, inflowing calcium remains more localized (cf. Eq. 28), and smaller gap-junctional calcium influx is required to reach the CICR threshold near the gap junctions than for large D.

In the example for the failure of intercellular wave propagation given in Fig. 3 d, the signal does not propagate beyond the stimulated cell. However, a certain number of neighboring cells might still be excited, if the propagation condition 23 is not satisfied. In the analysis of the p.w.l. model, this is given by the number m (m  0), determined by the inequalities 20. To calculate m, values of the calcium concentration s in the stimulated cell are chosen comparable to the calcium amplitude reached upon triggering of CICR, u_a =  in the p.w.l. model. Then parameter regions of seizable extent exist for only two cases of finite m: no cells and 1 cell, the immediate neighbor of the stimulated cell, becoming excited. In a third large region, the propagation condition is satisfied and all cells become excited via a regenerative calcium wave (Fig. 4). All other regions with m > 1 cells excited exist in between the upper boundary of the one-cell region and the boundary to intercellular waves (cf. Fig. 4 b for m = 2), but their size is negligible.

View larger version (23K): [in this window] [in a new window]   FIGURE 4   Range of spatially limited signals. (a) In the D-P plane for  = 100 and the applied stimulus s = 200 (p.w.l. kinetics); (b) magnification of a region of (a). Conversion of p and  to dimensional form as in Fig. 3.

This behavior depends on the value of s; for considerably larger variation in s (100-fold and more), regions with m > 1 become noticeable. Also, for values of  just above 2 the critical permeabilities are much larger, and with this the regions for finite cell numbers becoming excited are enlarged. However, for realistic ranges for s and P, intercellular wave propagation in the model is practically an all-or-none phenomenon. Regenerative calcium waves are triggered if the propagation condition is satisfied; otherwise the signal remains restricted to the stimulated cell or its neighbors. This is also found in numerical simulations with smooth kinetics.

Speed of propagation

A central quantity to be compared with experimental data is the speed of calcium wave propagation. For intercellular signals the speed is determined by the time taken by the wavefront to traverse the cell and the time spent to cross the gap junctions. Referring to these as intracellular delay, _cyt, and gap-junctional delay, _gj, respectively, the overall speed of propagation becomes

(30)

An estimate of the intracellular wave speed is obtained by calculating the speed at which a travelling front would propagate according to Eq. 7 if no cell boundaries were present (e.g., Murray, 1993). This yields

(31)

where const. = /(  2) for the p.w.l. kinetics Eq. 9b.

A crude estimate for the time to reach the CICR threshold by gap-junctional calcium influx can be obtained for the p.w.l. kinetics from Eq. 27 by solving u(0, t) = 1 for t, at some value of û. Expanding u(0, t) for small times and taking û = , one finds explicitly,

(32)

This expression gives an indication of the parameter dependence of _gj. In particular, Eq. 32 predicts an increase of _gj with calcium diffusivity, whereas Eq. 31 yields a decrease of _cyt. Correspondingly, we find numerically that the overall wave speed exhibits a maximum at intermediate values of D, for both types of kinetics in Eq. 9 (Fig. 5, a and b). The numerical results for the delays _gj and _cyt are shown in Fig. 5 c.

View larger version (19K): [in this window] [in a new window]   FIGURE 5   Speed of intercellular calcium waves and intracellular and gap-junctional delays. Wave speed (a) for piecewise linear kinetics ( = 225); (b) for smooth kinetics ( = 15), for the values of P indicated at the curves (in µm/s); (c) intercellular (gj, open symbols) and intracellular (cyt, filled symbols) delays for (b). Conversion of p and d to dimensional form as in Fig. 3.

In the rat liver, intercellular wave speeds and gap-junctional delays ranging between ~5-50 µm/s and 2-12 s, respectively, have been reported (Robb-Gaspers and Thomas, 1995); similar values are found in hepatocyte couplets (Combettes et al., 1994). In the blowfly salivary gland, intercellular speeds range between 10 and 20 µm/s (Zimmermann and Walz, 1999). Values for the wave speed in the model fall in the lower range of these experimental data (the cell length of 10 µm assumed in the calculations is actually rather low; somewhat larger speeds than in Fig. 5 would result for larger cells, as _gj is practically independent of cell length). The numerical values for the gap-junctional delay are also in agreement with the experimental data.

Experimental measurements of _cyt and _gj can be used in the model to estimate P and D. Computing the functions _cyt(, p) and _gj(, p) numerically, at fixed , we found that the level curves _cyt = const. and _gj = const. in the (, p) plane have unique pairwise intersections. Hence a pair of values for the intracellular and intercellular delays corresponds to unique values for  and p, conditional on a choice of  (Table 2). The resulting estimates of D agree with experimental data (Allbritton et al., 1992), and also P is within the accessible range, particularly for the larger intercellular delays of a few seconds.

View this table: [in this window] [in a new window]   TABLE 2   Effective calcium diffusion coefficient and gap-junctional permeabilities estimated from the model by assuming cyt = 0.5 s and various gj;  = 15, l = 0; conversion of p and  to dimensional form as in Fig. 3

Effects of parameter changes

The model allows the consequences of changes in cellular parameters to be evaluated. We focus on the effects of varying calcium buffer concentration and the cytoplasmic IP₃ level on the capacity for regenerative wave propagation and on the gap-junctional delay.

The results of changing the concentration of a calcium buffer depend on whether the buffer is diffusive (D_j > 0) or stationary (D_j = 0). Of the nondimensional parameter groupings Eq. 10,  depends on diffusive buffers while  and p are independent of buffering. Therefore, the propagation condition Eq. 23 is independent of the concentration of stationary calcium buffers. By contrast, adding diffusive calcium buffer to a cell increases the critical calcium permeability and may thus shift the system from the regime of regenerative intercellular waves to failure of propagation (cf. Fig. 3). For the gap-junctional delay, we have approximately _gj ~ (D₀ +  D_jB_j/K_j)(1 +  B_j/K_j) from Eq. 32. Generally, addition of both diffusive and stationary calcium buffers causes an increase of _gj, which is greater for larger buffer diffusivity. If the calcium signals are of finite duration, it is conceivable that also changing the stationary buffer concentration may have effects other than altering the gap-junctional delay and wave speed.

The opening probability of the IP₃R increases with IP₃ concentration (Bezprozvanny and Ehrlich, 1995). Therefore the maximal rate of CICR, v_m, is an increasing function of the IP₃ concentration, and the effects of changes in IP₃ can be inferred qualitatively by changing v_m. The critical junctional calcium permeability for regenerative intercellular wave propagation decreases with increasing v_m (Fig. 6; Eq. 29). Alternatively, for a certain P, there will be a critical v_m and a corresponding IP₃ threshold. In the regime of wave propagation, the gap-junctional delay decreases with increasing IP₃ concentration (Fig. 7 a). For periodic calcium waves in the rat liver, Robb-Gaspers and Thomas (1995) indeed observed the gap-junctional delay to become smaller with increasing vasopression concentration. The numerically calculated dependence _gj(v_m) qualitatively follows the prediction of Eq. 32, as does the dependence of _gj on the junctional calcium permeability (Fig. 7 b).

View larger version (19K): [in this window] [in a new window]   FIGURE 6   Critical gap-junctional calcium permeability Pc as a function of the maximal rate of CICR, vm, for two different values of the calcium diffusivity as indicated at the curves (D in µm2/s). Numerical result with kinetics Eq. 9a.

View larger version (23K): [in this window] [in a new window]   FIGURE 7   Dependence of the gap-junctional delay gj on (a) maximal CICR rate vm, and (b) on gap-junctional calcium permeability P. Numerical result with kinetics Eq. 9a; P = 0.05 µm/s in (a), vm = 3 µM/s in (b); D = 10 and 20 µm2/s as indicated.

Asymmetric calcium signaling through gap junctions

Intercellular calcium signaling has been observed between different cell types, with potentially different values of cellular parameters (Hirata et al., 1998). Parameters can also vary within a cell population. Heterogeneity in calcium buffering, gap-junctional permeability, and CICR rate may affect the intercellular propagation of calcium signals. In particular, the model suggests mechanisms by which a calcium signal can spread from one type of cells to another, but not vice versa.

Consider two cell types, A and B, which differ with respect to the composition of cytoplasmic calcium buffers, and thus in their effective calcium diffusion coefficients. According to the propagation condition, signal propagation may be possible for a low calcium diffusivity (e.g., cell type A), but impossible for a larger diffusivity (cell type B). If a signal is evoked in cell type A, a calcium wave can spread among these cells but cannot cross the boundary to type B (transition from low to sufficiently high cytoplasmic calcium diffusivity is not possible). However, if a calcium signal is evoked in a cell B bordering a cell A, it will cross to this cell and spread among the A cells (transition from high to low calcium diffusivity is possible). If, in addition, the higher diffusivity among the cells B is compensated by a larger junctional permeability, than between A and B cells, then the situation depicted in Fig. 8 may arise. Here, calcium waves can propagate among A cells, B cells, from cells B to A, but not from cells A to B. Also, other constellations can result in asymmetric gap-junctional calcium signaling. For example, a smaller CICR rate constant v_m (smaller ) and a larger junctional permeability in B cells than in A cells can yield qualitatively the same result as in Fig. 8. These hypothetical mechanisms do not require asymmetry of gap-junctional permeabilities.

View larger version (28K): [in this window] [in a new window]   FIGURE 8   Asymmetric gap-junctional calcium signaling. (a) Array of four cells of two types with different junctional permeabilities (A: 0.02 µm/s, B: 0.03 µm/s) and effective cytoplasmic diffusion coefficients for calcium (A: 5 µm2/s, B: 15 µm2/s); junctional permeability between cells A2 and B1 0.02 µm/s;  = 10 in A and B. (b) Critical curve for wave propagation taken from Fig. 3; propagation is possible between cells of the same type (, ) and from B to A (), but not from A to B (). Accordingly, a calcium signal elicited in cell A1 propagates to A2 but not to B1 (c), whereas a signal elicited in cell B2 spreads to cells B1, A2, and A1 (d). Calcium concentration scale in dimensionless units.

Finite distance between gap junctions and calcium stores

Up to now, we have focused on the case l = d/L = 0. The impact of a finite distance between gap junctions and the calcium release/uptake sites is illustrated in Fig. 9 for the p.w.l. form of g(u); analogous results are obtained numerically with smooth g(u). The junctional permeability required for intercellular waves