INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The discovery of the synaptic vesicle associated proteins that are involved in exocytosis, such as the SNAP receptors synaptotagmin and synaptobrevin and the soluble N-ethyl maleimide-sensitive fusion protein attachment protein (SNAPs) syntaxin and SNAP 25, have greatly changed the conceptual framework within which quantal transmission can now be considered (Südhof, 1995). Indeed, synaptotagmin may well be the calcium sensor that triggers transmitter release on the arrival of an action potential (Brose et al., 1992). Of particular interest has been the discovery of a tight coupling between the voltage-dependent calcium channel that gates the entry of calcium for triggering exocytosis (whether N-type or P/Q-type) and a complex consisting of syntaxin and synaptotagmin (O'Connor et al., 1993; Yoshida et al., 1992; el Far et al., 1995; Martin-Moutot et al., 1996). The existence of a secretory unit of this kind has important implications for quantal transmission (Bennett, 1996). It is natural to associate a secretory unit with the concept of the "calcium microdomain," the region of high calcium concentration of the order of 100 µM that is calculated to occur within a distance ~25-50 nm from an open calcium channel (Simon and Llinás, 1985; Zucker and Fogelson, 1986). It has been suggested that these are the distances to be expected between the calcium channel and the calcium sensor among the vesicle associated proteins, probably synaptotagmin, although there is no direct evidence for these conjectures at this time (Yoshikami et al., 1989; Stanley, 1993).

There is evidence that secretory units significantly increase the efficiency of transmission at some synapses. Only one calcium channel need open in the avian ciliary ganglion terminal for a quantum of transmitter to be released, indicating that the calcium within a single microdomain is sufficient to trigger secretion (Stanley, 1993). Uncoupling the connection of domains II and III of the alpha I subunit of the N-type channel with syntaxin/SNAP 25 in the secretory unit leads to fragmentation of the units in motor nerve terminals, with a consequent 25% drop in evoked quantal release (Rettig et al., 1997). In addition, the observation that relatively slow calcium chelators such as ethylene glucol-bis(-aminoethyl ether)-N,N,N',N'-tetraacetic acid (EGTA) do not affect transmission at motor nerve terminals, whereas a fast chelator such as 1,2-bis(2-aminophenoxy)ethane-N,N,N,N-tetraacetic acid acetoxymethyl ester (BAPTA) does (Robataille et al., 1993), suggests that it is the high and relatively fast calcium transient in the microdomains that is responsible for triggering exocytosis in motor nerve terminals. Furthermore, the results of voltage clamp studies of transmitter release in the stellate ganglion of the squid indicate that single nonoverlapping calcium microdomains exist in the terminals, presumably within secretory units (Augustine et al., 1991). It would seem, then, that the calcium in microdomains of secretory units in both preganglionic and motor nerve terminals is dominant for triggering secretion. The importance attached to calcium in microdomains has prompted the present study.

A Monte Carlo description is given of the spatial distribution of calcium ions after they move out of an open channel and bind to the fixed and mobile buffers in region of domains after the opening of the channel. Calculations are given for channels having the properties of N-type calcium channels and opening spontaneously or under an action potential. These results are compared with the solutions of the transport equations that describe the buffered diffusion of calcium in the presence of rapid stationary and mobile calcium buffers (Wagner and Keizer, 1994). They are also compared with two approximations to the solutions, namely the rapid buffering approximation (Wagner and Keizer, 1994; Smith et al., 1996; Smith, 1996) and the linearized steady-state approximation (Neher, 1986; Pape et al., 1995; Naraghi and Neher, 1997). In addition, an analysis is given of the calcium changes in the volume of a varicosity or bouton in the presence of an indicator consequent on the opening of a single calcium channel, either spontaneously or under an action potential, and these results compared with those obtained from calculations that use deterministic equations to describe calcium movements in terminals (Sinha et al., 1997). Finally, the most appropriate set of parameters arrived at for the description of calcium dynamics in a microdomain is used in a Monte Carlo analysis of the probability of quantal release from vesicles. These are arranged in different spatial arrays about a channel and open either spontaneously or under an action potential.

A subsequent paper (Bennett et al., 2000) gives a Monte Carlo analysis of the probability of quantal release for the case where many secretory units are present at a nerve terminal and a number of their associated calcium channels open upon the arrival of a nerve impulse.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

In this section we outline the theory of calcium diffusion and buffering in a three-dimensional space, and then describe how this system can be simulated numerically using a Monte Carlo approach. The use of Monte Carlo methods in the context of transmitter release, diffusion, and binding was pioneered by Bartol et al. (1991) and subsequently applied in a number of related studies (Faber et al., 1992; Bennett et al., 1995b,c, 1996, 1997b, 1998). The following adapts the method to the case of calcium entering via a single channel, then diffusing and interacting with both fixed and mobile buffers, and binding to vesicle associated proteins. The region considered is a cubical box with 1-µm sides (Fig. 1). The plasmalemma is represented by the 1 µm × 1 µm base, lying in the xy plane, and calcium ions enter via a channel situated at the center of this plane. The other sides of the box represent the membrane walls of the terminal, and all six sides can contain calcium pumps that are incorporated into the Monte Carlo scheme as described below.

View larger version (13K): [in this window] [in a new window]   FIGURE 1   The Monte Carlo simulation uses a cubic box with 1-µm side length. The plasmalemma is represented by the 1 µm × 1 µm base; calcium ions enter via a channel situated in the center of the base. Calcium pumps are situated on all six walls of the box.

The Monte Carlo simulation method is an alternative to solving the reaction-diffusion equations, which are deterministic differential equations giving the temporal and spatial dependence of the average concentrations of the free calcium and of the various buffer complexes that may be present. A detailed comparison between the differential equation and the Monte Carlo approaches for the case of transmitter diffusion and binding has been given by Bartol et al. (1991) and many of the same arguments apply in the present case. In particular, the Monte Carlo approach is closer to the physical situation in that it gives some indication of the size of the stochastic fluctuations that are likely to occur. These can be very significant in the neighborhood of an open channel, where although the average calcium concentration can reach the order of 100 µM, this relatively high concentration is due to the presence of only a few calcium ions in a small volume. (In fact, for a concentration of 100 µM a cubic box with 100-nm sides would contain only ~60 ions.)

Calcium channels

Calcium channels in the plasmalemma can undergo either spontaneous or evoked opening. In the spontaneous case, a channel opens for a certain time T and during this time admits a constant current i_c; thus the input calcium current is a rectangular pulse. If the channel is assumed to have only two states, open and closed, then T is exponentially distributed (Bennett et al., 1995c, 1997a); that is, it has the density function

(1)

where  is a constant. Thus T has mean and standard deviation equal to 1/.

For the case of evoked release under an action potential, calcium channels open at different times for different durations, with therefore different driving forces on the calcium entry through the channels. A quantitative description of this has been given in the case of N-type calcium channels (Bennett et al., 1997a; see especially Fig. 4 in that paper). There, the way in which a single channel opens under a Hodgkin-Huxley action potential was investigated in detail with the opening and closing times, t_open and t_close, being modeled as nonhomogeneous Poisson processes with rate parameters that depend on the action potential, and hence are functions of time (compare Clay and DeFelice, 1983). The single-channel calcium current, i_c(t), was also expressed as a function of the potential. The total charge q that enters through a single channel during the course of an action potential can be found as

(2)

where g(t) is 1 if the channel is open and 0 otherwise; the second expression follows upon the assumption that a given channel opens at most once during the course of a single action potential. (Simulations showed that multiple openings were rare: Bennett et al., 1997a.)

Differential equations for calcium diffusion and buffering

The conventional way to characterize calcium diffusion and buffering is via differential equations that describe the spatial and temporal evolution of the various ionic and molecular species present. The buffering is assumed to be governed by the reaction

(3)

where Ca²⁺ represents free calcium ions, B_i represents unbound buffer molecules, and CaB_i represents calcium bound to buffer; i = f for fixed buffer and i = m for mobile buffer. The system is then governed by the equations (see, for example, Wagner and Keizer, 1994; Naraghi and Neher, 1997):

(4)

(5)

(6)

(7)

(8)

where (t) is the calcium source density, assumed to be at the origin [(r) is a Dirac delta function] and D are the various diffusion coefficients. These equations can be solved directly, using suitable numerical techniques (see, for example, Smith et al., 1996). Also, a number of approximations have been developed. One of these, the rapid buffering approximation (Wagner and Keizer, 1994; Smith et al., 1996; Smith, 1996) utilizes the fact that the buffering kinetics act on a time scale that is much faster than the time scale for diffusion (see Appendix A), leading to the conclusion that the reaction described by Eq. 3 rapidly attains equilibrium, so that

(9)

where [B_i]_T = [B_i] + [CaB_i] is the total buffer concentration and K_i = k_i/k_i⁺ is the dissociation constant for buffer i. Use of this approximation enables the set of Eqs. 4-8 to be replaced by a single differential equation describing the transport of calcium (Wagner and Keizer, 1994) and this has advantages for numerical solution (Smith et al., 1996) and for mathematical analysis (Smith, 1996).

A second approximation, valid for small calcium entry in the presence of a large concentration of mobile buffer, is the linearized steady-state approximation first given by Neher for the case where [B_m] and [CaB_m] are assumed constant (Neher, 1986) and later generalized to include the diffusion of B_m (Stern, 1992; Pape et al., 1995; Naraghi and Neher, 1997). This approximation gives the leading term in a perturbation expansion, the unperturbed state being the equilibrium state before calcium entry influx. It leads to the steady state calcium concentration being given by (Pape et al., 1995; Naraghi and Neher, 1997)

(10)

where c₀  [Ca²⁺]₀ is the background calcium level,  is the constant source density, assumed to be into a hemisphere,  is a space constant given by 1/² = 1/_c² + 1/_m² where _c =  is the characteristic length for binding of diffusing calcium, and _m =  is the corresponding quantity for the mobile buffer (D_c  D_Ca, D_m  D_Bm). If _{c m}, the last term in Eq. 10 can be neglected and  can be replaced by _c in the second term, which recovers the earlier expression of Neher (1986).

Monte Carlo method

The following sections give details of the Monte Carlo simulation scheme. This is implemented via a computer program written in Fortran and run on a Digital Alpha workstation. The pseudo-code used is given in Appendix B.

Diffusion of calcium

(11)

(12)

(13)

1

1

(14)

(15)

Fixed buffer

(16)

(17)

(18)

Mobile buffer

(19)

(20)

Calcium pumps

At the plasmalemma and at any other terminal boundary the diffusing molecules are mirror-reflected as described above (unless they bind to vesicles or pumps, as described below). In a number of calculations, a cubic box with 1-µm sides is used to represent the terminal, and all its sides are reflecting walls. In other calculations, where only effects near the plasmalemma are being considered, the volume considered is a cube with 1-µm sides, but only the side representing the plasmalemma is a reflecting wall; the other sides are transparent, and once a molecule passes through them it is lost and no longer included in the bookkeeping.

Calcium pumps occur both in the plasmalemma and in the other walls of the terminal. The pumping is represented as a "tunneling" process in which the cytoplasmic calcium ion first binds (irreversibly) to the pump, and then upon unbinding is ejected from the terminal:

(21)

This process must now be incorporated into the Monte Carlo scheme. Following Bartol et al.'s treatment of receptor and of esterase binding, it is assumed that the membrane walls are "tiled" with squares of side length 1/, where _P is the pump density and that each square contains one pump. To bind to an available pump a calcium ion must first hit the square containing that pump; this will be determined by the line joining its position at the beginning and at the end of the interval t. Assuming that it hits, it will then bind with a probability p₊, which is related to the corresponding binding rate k₁ by (cf. Bartol et al., 1991, Eq. 6a)

(22)

If the ion fails to bind, or if it hits a square where the pump molecule is already bound, then it undergoes specular reflection, as described above. The unbinding probability per timestep does not depend on the calcium diffusion and is simply given by

(23)

The rate coefficients k₁ and k₂ are not directly measurable. To get estimates it is necessary to relate them to the usual parameters describing lumped calcium extrusion through a plasma membrane. The standard expression for the flux due to pumping is V_P[Ca²⁺]_actual/(K_P + [Ca²⁺]_actual), where V_P and K_P are constants (Sala and Hernandez-Cruz, 1990; Kargacin and Fay, 1991; Nowycky and Pinter, 1993). If we assume that there is also a constant inward leak that just balances the pump at the resting calcium level c₀, then [Ca²⁺]_excess satisfies

(24)

where _P = V_PK_P/(K_P + c₀) and _P = K_P + c₀. This can be related to the Monte Carlo scheme by applying the Michaelis-Menten approximation to Eq. 21 (e.g., Murray, 1993), leading to the following expression for the rate of calcium extrusion:

(25)

where [Ca²⁺]_excess is the excess calcium concentration inside the terminal, defined by Eq. 15. Comparing Eq. 25 with Eq. 24 gives k₂ = _P/_P and k₁ = k₂/_P. Note that in the above way of assigning parameter values the pump density, _P, is, within reasonable limits, an arbitrary parameter. Once _P has been given a value, k₁ and k₂ are then chosen to give the desired pumping rate.

Vesicles and exocytosis

The standard assumption will be that calcium ions bind to the vesicle-associated proteins according to the scheme (Heidelberger et al., 1994; Heinemann et al., 1994)

(26)

where S_i, i = 1, ..., 4 denotes the state with i sites occupied, S_ex denotes the state with four sites occupied after the conformational change, µ_i(t) (_i(t)) are the rates of attachment (detachment) of a calcium ion at the ith step, and (t) is the rate for the final step. These attachment rates µ_i are taken to be proportional to the excess calcium concentration at the position of the vesicle-associated protein, and the detachment rates _i are taken to be constants, independent of time:

(27)

and the conformational change rate  is taken to be a time-independent constant. A further simplification is to assume no cooperativity in binding or unbinding, in which case k_i^a = k^a and k_i^d = k^d, i = 1, ... , 4. The above single-affinity scheme for the binding of calcium to the vesicle-associated proteins has been used to describe calcium-dependent exocytosis at synapses formed by goldfish retinal bipolar cells (Heidelberger et al., 1994) and bovine chromaffin cells (Heinemann et al., 1994). A more complex scheme may be used in which the binding sites have different affinities, although an investigation of such kinetics indicates that the results obtained for the probability of secretion are not much different from those for the single-affinity case (see Fig. 7 in Bennett et al., 1997a).

To incorporate this into the Monte Carlo scheme, we again follow Bartol et al.'s treatment of receptor binding and assume that the plasmalemma is "tiled" with squares whose size is determined by the geometric placement and the density of the vesicle array being modeled. The vesicle binding site is represented as a disk of radius r_V placed at the center of selected tiles. To bind, a calcium ion must first hit the disk [which will be determined by whether its free path during the interval t would pass through the tile, this path being the straight line from point (x, y, z) to point (x + x, y + y, z + z)]; it will then bind with probability p₊, which is related to the corresponding forward binding rate k₊ by (compare Eq. 22):

(28)

where a_V = r_V² is the area of the vesicle disk. If it does not bind it is reflected in the usual way. The binding probabilities change according to how many calcium ions are already bound; for the binding of the ith calcium ion, where i = 1, ... , 4, p₊ is given by Eq. 28 with k₊ = (5  i)k^a.

The unbinding per timestep does not depend on the calcium concentration and is given by (cf. Eq. 18)

(29)

where k = ik^d, i = 1, ... , 4 is the relevant rate constant. The conformation change is treated using a similar formula.

Other schemes have also been proposed with different Ca²⁺ stoichiometries, requiring the binding of either three or two Ca²⁺ before the final conformational change leading to exocytosis (Heinemann et al., 1993; 1994). These will be used as a comparison with the four-binding case given by Eq. 26.

Parameter choice

The main parameter values used in the calculations are listed in Table 1. Some comments on these choices can be found in the Discussion section.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The Monte Carlo results are divided into two main sections. The first of these is concerned with establishing the calcium transients that occur immediately about a single open N-type calcium channel; in addition, the transients of free calcium and of indicator calcium in the volume of a varicosity or bouton are considered. The second section is concerned with using the information provided by the previous section to establish how a calcium transient at a secretory unit triggers vesicle exocytosis following an action potential, when these vesicles are arranged in different configurations with respect to the channel.

In all cases the terminal contains a mobile buffer with the characteristics of calmodulin and a fixed buffer with the characteristics of calbindin (see Table 1). In addition, a calcium indicator is sometimes present, which is either furaptra, representative of a low-affinity indicator, or fura-2, representative of a high-affinity indicator. Unless otherwise mentioned, all values of free calcium and of calcium bound to the various buffers are the means of at least five trials, where each trial is a Monte Carlo simulation using the same parameters and initial conditions, but different random numbers. If standard deviations are also given, then at least 10 trials were used. (The expression is used for the standard deviation; the alternative expression involving n  1 in place of n will give values differing by at most 5%, and this difference would be undetectable on the graphs.)

Calcium microdomains

Spontaneous opening

View larger version (28K): [in this window] [in a new window]   FIGURE 2   The mean and standard deviation of the spatial calcium concentration profile near a single spontaneously opening channel at the end of the open time of the channel. The channel opens deterministically and admits a constant current. The calculations are performed using the Monte Carlo scheme, which gives the position of each calcium ion at each time step; these positions are converted to densities using hemispherical bins of 20-nm thickness. (A) Results for the average open time of an N-type channel of 0.2 ms (see Fig. 5 A in Bennett et al., 1995a) with a current of 2 pA; the time course of this current is shown as an inset in Aa. (B) Results for a relatively long open time of 0.8 ms and current of 2 pA; the time course is shown as an inset in Ba. In each case, panels a-c give the concentrations of free calcium ([Ca2+]), of calcium bound to the mobile buffer ([CaBm]), and the calcium bound to the fixed buffer ([CaBf]), respectively. (C) and (D) show the corresponding results using log ordinates. The parameters used in the calculations are given in Table 1. In each graph, the heavy line shows the mean concentration (in µM) and the thin lines show one standard deviation away from the mean; these are the result of 10 Monte Carlo simulations, each run starting from the same initial conditions with the same calcium influx (but using different random numbers).

Evoked opening

15

View larger version (29K): [in this window] [in a new window]   FIGURE 3   Frequency distribution of the total calcium charge that passes through a single N-type channel when it opens under an action potential. The total charge that enters during a single action potential is found using Eq. 2, where the opening and closing times, topen and tclose, and the single-channel current, ic(t), are calculated as in Bennett et al. (1997a). The histogram shows the frequency of occasions during five thousand action potentials at which a particular charge (pA · ms) is passed by the channel. (1 pA · ms = 1015 Coulombs and corresponds to the charge on 3120 calcium ions.) The histogram is well-fitted by an exponential distribution with mean 0.52 and standard deviation 0.50 pA · ms.

View larger version (28K): [in this window] [in a new window]   FIGURE 4   As for Fig. 2, except that the single channel now opens under an action potential. (A) Results for the average open time of an N-type channel of 0.84 ms (see Fig. 4 B in Bennett et al., 1997a) with a current of 0.62 pA. (B) Results for a relatively long open time of 1.0 ms and current of 1.5 pA according to the mean charge in Fig. 3.

Rapid buffering and linearized diffusion approximation

View larger version (18K): [in this window] [in a new window]   FIGURE 5   Comparison between the Monte Carlo (MC) calculation of the calcium profile near a single channel opening spontaneously (continuous heavy lines) and the results using the deterministic differential Eqs. 4-8 (dashed lines). Also shown are the results obtained using the rapid buffering approximation (RBA) given by Eq. 9 (dotted lines), and in A the thin continuous line is calculated using the linearized steady-state approximation (LSS) given by Eq. 10. Calculations are for the long open time of an N-type channel of 0.8 ms with a current of 2 pA. A-C give the concentrations of free calcium ([Ca2+]), of calcium bound to the mobile buffer ([CaBm]), and of calcium bound to the fixed buffer ([CaBf]), respectively. The parameters used in the calculations are given in Table 1. Log ordinates.

Ca²⁺ transients of a varicosity or bouton

The question arises as to whether it is possible to detect the changes in calcium concentration in the entire volume of a varicosity or bouton when a calcium channel opens either spontaneously or under an action potential. Monte Carlo calculations have been made of calcium transients for the average open times of both the spontaneous and evoked opening of a calcium channel. The channel is placed in the center of one of the six walls of a cubic volume with 1-µm side (Fig. 1), these being the approximate dimensions of small varicosities and boutons. In this case it is important to allow for the action of calcium pumps, as the only way that calcium ions can be removed from this volume, after being released from the endogenous buffers, is through such a mechanism. To this end we have modeled such pumps so that they may be included in the Monte Carlo simulations and distributed them over all six walls of the cubic volume (see Methods). Fig. 6 A shows the results for the spontaneous opening (0.2 ms and 2 pA), and Fig. 6 B those for the evoked opening (0.84 ms and 0.62 pA), for the case where no indicator is present. In each case, curve a gives the mean and standard deviation of the free calcium, b gives the calcium bound to the fixed buffer, and c gives the calcium bound to the endogenous mobile buffer. As only the number of free calcium ions and those attached to buffer within the 1 µm³ volume are determined, spatial inhomogeneities in the concentration of these throughout the volume during the first few milliseconds after channel closure are not considered. Upon channel closure, there is a rapid decrease in free calcium (over microseconds), as it is bound principally to the fixed buffer (Fig. 6). Thereafter, the decline in free calcium is primarily due to the calcium ions released from the buffers finding the pumps on the walls of the 1 µm³ volume. This free calcium falls to within 1/e of its initial value in ~15 ms (Fig. 6), a time that is mostly governed by the probability that once an ion hits the wall of the volume it will be extruded by a pump.

View larger version (20K): [in this window] [in a new window]   FIGURE 6   The mean and standard deviation of the calcium concentration in a cubic volume of 1 µm3 about a single channel in a varicosity or bouton. The concentration is averaged over the whole cubic volume. (A) Results for the average spontaneous open time of an N-type channel of 0.2 ms with a current of 2 pA. (B) Results for the average evoked open time of an N-type channel of 0.84 ms with a current of 0.62 pA. In each case, curves a-c give the concentrations of free calcium ([Ca2+]), of calcium bound to the mobile buffer ([CaBm]), and the calcium bound to the fixed buffer ([CaBf]), respectively. In each graph the heavy line shows the mean concentration (in µM) and the thin lines show one standard deviation away from the mean. Log ordinates.

Effect of buffers and indicators on Ca²⁺

The above calculations have all been made using the standard characteristics of the endogenous buffers given in Table 1. In this section consideration is given to the effects of varying the parameters of the mobile and fixed buffers as well as those of the indicator concentration when furaptra is used as the indicator. The case for which Monte Carlo simulations have been carried out is that of a spontaneously opening channel (0.8 ms and 2 pA), which gives the calcium transients in a varicosity or bouton of volume 1 µm³ shown in Fig. 6. The temporal characteristics of the signal for calcium bound to furaptra, together with the peak free calcium transient, have been calculated in each case.

Fig. 7 shows the results of varying the characteristics of the mobile buffer and indicator on the time at half-peak of [CaB_i] (Fig. 7, A and B). A 100-fold increase in [B_m]_T produces only a threefold increase in the half-time of the [CaB_i] signal (Fig. 7 A); similar increases in k_m⁺ or K_m of the mobile buffer produce only about a twofold decrease in the half-time of the [CaB_i] transient (Fig. 7 A). There is only a very small standard deviation about these mean values, so that they have not been included in the graph. Less than a threefold change occurs in the half-time of the [CaB_i] as the result of a 100-fold increase in the values of [B_i]_T and its k_i⁺ or K_i values (Fig. 7 B); again there is very little variability associated with these values.

View larger version (25K): [in this window] [in a new window]   FIGURE 7   The effect of mobile buffer and indicator parameters on the half-time of the transient for calcium bound to the indicator furaptra ([CaBi]), upon the opening of a spontaneous calcium channel (0.8 ms and 2 pA) in a bouton or varicosity of volume 1 µm × 1 µm × 1 µm. Results from Monte Carlo simulations are given for a range of parameter values for mobile buffer (Bm) and for indicator (Bi). (A) The parameter values of the mobile buffer ([Bm]T; Km; km+) were changed while all other parameters were maintained at their standard values (Table 1). (B) The parameter values of the indicator ([Bi]T; Ki; ki+) were changed while all other parameters were maintained at their standard values (Table 1). Calcium pumps were present in the walls of the volume under consideration, as outlined in the Methods section.

Fig. 8 shows the results in the case of varying the characteristics of the fixed buffer and indicator concentration on the time at half-peak of [CaB_i] (Fig. 8, A and B). A 10-fold change in the values of [B_f]_T and its k_f⁺ can produce very significant changes (of the order of 10-fold), in the half-time of the [CaB_i] transient, at the standard values for the B_m buffer and of furaptra given in Table 1; there is only a very small variation about these mean values (Fig. 8 A). There is only a small change in the half-time of the [CaB_i] with a 100-fold change in the values of [B_i], with again very little variation associated with these values, [B_f] being kept at its standard value given in Table 1 (Fig. 8 B). These calculations show that the characteristics of the [CaB_i] transient are relatively little affected by changes in the parameters of the mobile buffer or the indicator compared with changes in the fixed buffe