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J Neurosci. Author manuscript; available in PMC 2010 March 30.

Published in final edited form as:

PMCID: PMC2784595

NIHMSID: NIHMS155978

The publisher's final edited version of this article is available free at J Neurosci

See other articles in PMC that cite the published article.

Recently there has been significant interest and progress in the study of spatio-temporal dynamics of Ca^{2+} that triggers exocytosis at a fast chemical synapse, which requires understanding the contribution of individual calcium channels to the release of a single vesicle. Experimental protocols provide insight into this question by probing the sensitivity of exocytosis to Ca^{2+} influx. While varying extracellular or intracellular Ca^{2+} concentration assesses the intrinsic biochemical Ca^{2+} cooperativity of neurotransmitter release, varying the number of open Ca^{2+} channels using pharmacological channel block or the tail current titration probes the cooperativity between individual Ca^{2+} channels in triggering exocytosis. Despite the wide use of these Ca^{2+} sensitivity measurements, their interpretation often relies on heuristic arguments. Here we provide a detailed analysis of the Ca^{2+} sensitivity measures probed by these experimental protocols, present simple expressions for special cases, and demonstrate the distinction between the Ca^{2+} *current cooperativity*, defined by the relationship between exocytosis rate and the whole-terminal Ca^{2+} current magnitude, and the underlying Ca^{2+} *channel cooperativity*, defined as the average number of channels involved in the release of a single vesicle. We find simple algebraic expressions that show that the two are different but linearly related. Further, we use 3D computational modeling of buffered Ca^{2+} diffusion to analyze these distinct Ca^{2+} cooperativity measures, and demonstrate the role of endogenous Ca^{2+} buffers on such measures. We show that buffers can either increase or decrease the Ca^{2+} current cooperativity of exocytosis, depending on their concentration and the single-channel Ca^{2+} current.

An important open question in the understanding of neurotransmitter vesicle exocytosis is the degree to which individual Ca^{2+} channels cooperate during exocytosis of a single vesicle at a given synaptic terminal. This question is referred to as the domain overlap problem (Schneggenburger and Neher, 2005), and has been addressed using an extension of the method commonly employed to dissect the Ca^{2+} sensitivity of exocytosis. When Ca^{2+} concentration at the release site [Ca^{2+}]_{Int} is varied directly using caged-Ca^{2+} compounds (Bollmann et al., 2000; Schneggenburger and Neher, 2000; Beutner et al., 2001), or more indirectly by changing the extracellular [Ca^{2+}] (Dodge and Rahamimoff, 1967), the resulting relationship between exocytosis rate *R* and [Ca^{2+}]_{Int} is an indication of intrinsic biochemical Ca^{2+} sensitivity of exocytosis. This relationship is usually fit to a power function, ** R** ~

Despite the wide use of this Ca^{2+} *current cooperativity* analysis, most studies rely on heuristic arguments to interpret the resulting data, and to-date only a few modeling studies have analyzed this experimental protocol, the Monte-Carlo studies of (Shahrezaei et al., 2006) and (Luo et al., 2008), and the studies of (Zucker and Fogelson, 1986), (Bertram et al., 1999) and (Meinrenken et al., 2003) that relied on deterministic solutions to the mass-action buffered Ca^{2+} diffusion equations. Here we extend the work of (Bertram et al., 1999), and show that a careful re-examination of the problem reveals an interesting distinction and non-trivial relationships between the Ca^{2+} current cooperativity **m _{ICa}** and the underlying channel cooperativity,

Results in Figs. 2, ,8-128-12 involve deterministic 3D simulations of buffered Ca^{2+} diffusion. We assume that the binding of Ca^{2+} to the endogenous buffers is described by simple mass action kinetics with one-to-one stoichiometry:

Probabilities of distinct configurations of the release site with two channels per vesicle. Denoting the open (unblocked) channel probability by p_{o}, the probability of both channels being blocked is P(0)=(1−p_{o})^{2}, whereas the probability of one **...**

$$B+C{a}^{2+}\underset{{k}_{\text{off}}}{\overset{{k}_{\text{on}}}{\rightleftharpoons}}\text{CaB}$$

(1)

where *k*_{on} and *k*_{off} are, respectively, the binding and the unbinding rates of the Ca^{2+} buffer, *B*. This leads to the following reaction-diffusion equations for the Ca^{2+} concentration, and the concentrations of the free (unbound) buffer:

$$\{\begin{array}{l}\frac{\partial C}{\partial t}={D}_{Ca}{\nabla}^{2}C+R(C,B)-{k}_{\text{uptake}}\left(C-{C}_{\text{bgr}}\right)+\frac{{i}_{Ca}(t)}{2F}\sum _{j=1}^{{N}_{\text{channels}}}\delta (r-{r}_{j})\\ \frac{\partial B}{\partial t}={D}_{B}{\nabla}^{2}B+R(C,B)\end{array}$$

(2)

Here, *C* = [Ca^{2}] and *B* are concentrations of Ca^{2+} and the buffer, respectively, *k*_{uptake}=4 s^{−1} is the rate of Ca^{2+} uptake by internal Ca^{2+} stores, and *R* is the reaction term describing the mass-action kinetics given by scheme (1):

$$R(C,B)=-{k}_{\text{on}}C\phantom{\rule{0.2em}{0ex}}B+{k}_{\text{off}}\phantom{\rule{0.2em}{0ex}}({B}_{\text{total}}-B).$$

(3)

B_{total} denotes the total concentration of the buffer; *D _{B}* and

We adopt the cooperative Ca^{2+} binding scheme of (Heidelberger et al., 1994), with parameter values from (Felmy et al., 2003):

$$X\underset{{k}^{\text{off}}}{\overset{5C{a}^{2+}{k}^{\text{on}n}}{{\displaystyle \rightleftharpoons}}}\text{CaX}\underset{2b{k}^{\text{off}}}{\overset{4C{a}^{2+}{k}^{\text{on}}}{{\displaystyle \rightleftharpoons}}}C{a}_{2}X\underset{3{b}^{2}{k}^{\text{off}}}{\overset{3C{a}^{2+}{k}^{\text{on}}}{{\displaystyle \rightleftharpoons}}}C{a}_{3}X\underset{4{b}^{3}{k}^{\text{off}}}{\overset{2C{a}^{2+}{k}^{\text{on}}}{{\displaystyle \rightleftharpoons}}}C{a}_{4}X\underset{5{b}^{4}{k}^{\text{off}}}{\overset{C{a}^{2+}{k}^{\text{on}}}{{\displaystyle \rightleftharpoons}}}C{a}_{5}X\stackrel{\gamma}{\to}\text{Fused}$$

(4)

These reactions are driven by the Ca^{2+} time course found by integrating Eqs. (2)-(3); the binding and unbinding rates are set to *k ^{on}* =0.116 μM

All spatial Ca^{2+} diffusion simulations (Figs. 1, ,77--11)11) were performed using the *CalC* (“Calcium Calculator”) software (Matveev, 2008). *CalC* uses the Alternating-Direction Implicit finite-difference method to solve the buffered diffusion equations (Eqs. (2)-(3)), with second order accuracy in space and time. To preserve the accuracy of the method in the presence of the non-linear buffering term, equations for [Ca^{2+}] and [B] are solved on separate time grids, shifted with respect to each other by half a time step. *CalC* uses an adaptive time-step method, with a non-uniform spatial grid that has greater density of points close to the Ca^{2+} channel array. Grid size is adjusted to limit the numerical error to about 5% (grid of 60 × 60 × 50 points). *CalC* integrates the ordinary differential equations derived from Eq. (4) using the 4^{th} order adaptive Runge-Kutta method. *CalC* is freely available from http://www.calciumcalculator.org, and runs on all commonly used computational platforms (UNIX, Mac OS X, and Windows/Intel). To ensure reproducibility of this work, the commented simulation script files generating the data reported here are available at the *CalC* web site.

Dependence of m_{ICa} and m_{CH} on the open channel probability for different values of total buffer concentration, in the idealized case of two equidistant channels. Total current is 0.05pA, and the distance from the two channels to the release site is 20 **...**

We first examine the intrinsic biochemical Ca^{2+} cooperativity of neurotransmitter release corresponding to the exocytosis scheme of (Felmy et al., 2003) (Eq. (4)) that we use in this study. There are two ways to measure the intrinsic cooperativity: either by directly varying intracellular [Ca^{2+}]_{int}, for instance using caged-Ca^{2+} compounds combined with Ca^{2+} imaging to measure [Ca^{2+}]_{int}, or by varying the extracellular Ca^{2+} concentration, [Ca^{2+}]_{ext}. Therefore, we distinguish between two cooperativity measures, **n**_{int} and **n**_{ext}. Generally, cooperativity is defined as the exponent of an assumed power-law relationship

$$R\sim {\left({[C{a}^{2+}]}_{\text{int}}\right)}^{{\text{n}}_{\text{int}}},R\sim {\left({[C{a}^{2+}]}_{\text{ext}}\right)}^{{\text{n}}_{\text{ext}}}$$

(5)

However, release follows a sigmoidal Hill function of Ca^{2+}, so we prefer to use the slope of the log-log dependence of release on [Ca^{2+}]:

$${\text{n}}_{\text{int}}=\frac{d\text{log}\phantom{\rule{0.2em}{0ex}}R}{d\text{log}\phantom{\rule{0.1em}{0ex}}{[C{a}^{2+}]}_{\text{int}}},\phantom{\rule{0.5em}{0ex}}{\text{n}}_{\text{ext}}=\frac{d\text{log}\phantom{\rule{0.2em}{0ex}}R}{d\text{log}\phantom{\rule{0.1em}{0ex}}{[C{a}^{2+}]}_{\text{ext}}}$$

(6)

Here **n=n**_{int} represents the true biochemical Ca^{2+}-cooperativity of exocytosis. Although a direct measurement of the intrinsic Ca^{2+} cooperativity has been successfully carried out at some synapses, notably at the calyx of Held (Bollmann et al., 2000; Schneggenburger and Neher, 2000), in many preparations the Ca^{2+} cooperativity of exocytosis is assessed indirectly, by varying the extracellular [Ca^{2+}] concentration. The resulting indirect measure, **n _{ext}**, is expected to provide an accurate estimate of the true biochemical Ca

While the small cooperativity at high [Ca^{2+}] influx is caused by the saturation of the release machinery, the small cooperativity at low values of [Ca^{2+}] influx is due to the effect of non-zero background [Ca^{2+}] ([Ca^{2+}]_{bgr}=0.1μM); small non-zero background release rate is present even in the absence of Ca^{2+} influx, so the peak Ca^{2+} has to be significantly higher than this background value in order to lead to a release rate increase.

When release rate is plotted as a function of the single-channel current, *i*_{Ca} (panels C and D), the influence of the buffer is revealed. For a given range of peak *i*_{Ca} the cooperativity can vary from near 1 to more than 4, depending on the buffer concentration. Also, increasing the buffer concentration can either decrease the cooperativity (over a range of small *i*_{Ca} values) or increase it (over a range of large *i*_{Ca} values). In the former case, an increase in buffer diminishes the already weak effects on release of the opening of a Ca^{2+} channel, relative to the background release, so the apparent cooperativity declines. In the latter case there is saturation of release sites which is partially relieved by an increase in the buffer concentration, resulting in an increase in the apparent cooperativity when the buffer concentration is increased. In either case, the buffer changes the relationship between i_{Ca} and [Ca]_{int} by absorbing free Ca^{2+} ions. Thus, experimental measurements of biochemical cooperativity obtained by varying [Ca]_{ext} (and thus i_{Ca}) should include a range of variation sufficient to contain the peak in the cooperativity curve (panel D), since otherwise the biochemical cooperativity will be underestimated.

We now turn to the main question of how one can measure the extent of Ca^{2+} channel domain overlap in the triggering of release. The number of channels contributing to exocytosis of a single vesicle can be assessed by measuring the sensitivity of exocytosis to a partial pharmacological block of Ca^{2+} channels, rather than the uniform variation in [Ca^{2+}]_{int} or [Ca^{2+}]_{ext} that is used to probe the biochemical cooperativity of exocytosis (Yoshikami et al., 1989; Mintz et al., 1995; Wu et al., 1999). We are primarily concerned with the case of non-selective block, whereby all Ca^{2+} channels involved in release are uniformly affected but will briefly restate and extend the results for selective channel block previously discussed in (Bertram et al., 1999).

An alternative approach is provided by the tail current protocol, whereby the number of open channels is varied while keeping the driving force constant by applying hyperpolarizing pulses following activating depolarizing steps of different duration (Quastel et al., 1992; Gentile and Stanley, 2005). Assuming power-law scaling between release rate and the macroscopic (rather than single-channel) Ca^{2+} current, *I*_{Ca}, the Ca^{2+} *current cooperativity* of exocytosis would be defined as the exponent **m** of this relationship, *R* ~ (*I _{Ca}*)

$$\text{m}=\frac{\text{log}(R/{R}_{0})}{\text{log}({I}_{Ca}/{I}_{0})}=\frac{\text{log}(R/{R}_{0})}{\text{log}({p}_{o})}$$

(7)

Here *p _{o}* is the fraction of channels that are unaffected by the channel blocker, or the fraction of channels that open in response to depolarization in the tail current protocol. Below we will refer to this exponent as the

$${\text{m}}_{\text{ICa}}=\frac{d\text{log}\phantom{\rule{0.2em}{0ex}}R}{d\text{log}\phantom{\rule{0.2em}{0ex}}{I}_{Ca}}=\frac{d\text{log}\phantom{\rule{0.2em}{0ex}}P(R)}{d\text{log}\phantom{\rule{0.2em}{0ex}}{p}_{\text{o}}}$$

(8)

where *P*(*R*) is the probability of release. Although the above two definitions of current cooperativity can differ significantly, we will see that both measures attain their lower bound of 1 in the limit of single-channel coupling of each vesicle, or in the limit of strong channel block (*p*_{o}→0), and attain the upper bound of **n** (biochemical cooperativity) in the limit of many overlapping domains, which is only possible when a small fraction of channels is blocked (*p*_{o}→1) (see Figure 3). The current cooperativity defined by Eq. (8) describes the sensitivity of release rate to the total Ca^{2+} current when it is varied by varying the channel open fraction *p*_{o}. It does *not* represent the sensitivity of release to the *single-channel* Ca^{2+} current, *i*_{Ca}.

Distinction between Ca^{2+} *current cooperativity* (m_{ICa}) vs. *channel* cooperativity (m_{CH}) in the case of two equidistant channels per vesicle, for different values of release site saturation, r=(R2)/P(R1). Cooperativity measures are plotted **...**

The Ca^{2+} *current cooperativity* measures **m _{ICa}** and

To clearly demonstrate the distinction between **m _{CH}** and

In the case of two equidistant channels, the current cooperativity given by Eq. (8) can be calculated as

$${\text{m}}_{\text{ICa}}=\frac{d\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}R}{d\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}{I}_{Ca}}=\frac{d\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}P(R)}{d\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}{p}_{o}}=\frac{{p}_{o}}{P(R)}\frac{d\phantom{\rule{0.2em}{0ex}}P(R)}{d{p}_{o}},$$

(9)

where we take into account that the release rate *R* is proportional to release probability, *P*(*R*), given by:

$$\begin{array}{c}P(R)=P(R|1)P(1)+P(R|2)P(2)=2P(R|1)\left({p}_{0}(1-{p}_{0})+\frac{r}{2}{p}_{0}^{2}\right)\hfill \\ \phantom{\rule{2.3em}{0ex}}r\equiv \frac{\text{P}\left(\text{R}|\text{2}\right)}{\text{P}\left(\text{R}|\text{1}\right)}=\{\begin{array}{c}1,\text{full saturation}\\ \sim {2}^{\text{n}},\text{no saturation}\end{array}\hfill \end{array}$$

(10)

where *P*(*Rk*) is the conditional probability of release given that *k* channels are open, *P*(*k*) is the probability that *k* channels are open (see Figure 2), and the release ratio parameter *r* quantifies the increase in release produced by the opening of two channels, compared to the case of a single open channel. This is a crucial parameter in the problem, reaching its lowest value of 1 when the release is completely saturated by the opening of a single channel, and attaining its highest value when the release is far from saturation, in which case it approximately equals 2^{n}, where **n** is the actual biochemical cooperativity (number of Ca^{2+} binding sites per vesicle). In the notation of (Bertram et al., 1999), *r* = 1 / *f*_{1} and ρ = 1 − *p _{o}*. Plugging Eq. (10) into Eq. (9) yields:

$${\text{m}}_{\text{ICa}}=\frac{{p}_{0}}{{p}_{0}\left(1-{p}_{0}\right)+\frac{r}{2}\phantom{\rule{0.2em}{0ex}}{{p}_{0}}^{2}}\left(1-2{p}_{0}+r{p}_{0}\right)=\frac{1+\left(r-2\right){p}_{0}}{1+\left(r-2\right)\frac{{p}_{0}}{2}}$$

(11)

A somewhat different expression is obtained with the logarithmic definition, Eq. (7):

$${\text{m}}_{\text{ICa},\text{log}}=\frac{\text{log}(R/{R}_{0})}{\text{log}\phantom{\rule{0.4em}{0ex}}{p}_{o}}=1+\frac{\text{log}({p}_{o}+2(1-{p}_{o})/r)}{\text{log}\phantom{\rule{0.4em}{0ex}}{p}_{o}}$$

(12)

where the release rate *R* is again taken to be proportional to the probability of release, *P*(*R*), and *R _{0}* is the release rate at zero block fraction (

We next derive an expression for the channel cooperativity, which we define as the average number of channels that open to produce a single release event, weighted by the amount of Ca^{2+} that each open channel delivers to the vesicle site; this quantifies the number of channels contributing to exocytosis at a given site. In the case of equidistant channels, each channel provides on average the same amount of Ca^{2+} to the vesicle, so the channel cooperativity equals the average number of channels that open per release event. Denoting *P*(*kR*) the probability that *k* channels were open when a release event occurred, and applying Bayes' formula for conditional probabilities, in the case of two equidistant channels we obtain

$$\begin{array}{c}{\text{m}}_{\text{CH}}=1\cdot P(1|R)+2\cdot P(2|R)=\frac{P(R|1)P(1)}{P(R)}+2\cdot \frac{P(R|2)P(2)}{P(R)}=\hfill \\ =\frac{P(R|1)P(1)+2\cdot P(R|2)P(2)}{P(R|1)P(1)+P(R|2)P(2)}=\frac{P(1)+2rP(2)}{P(1)+rP(2)}=\frac{1-{p}_{o}+r{p}_{o}}{1-{p}_{o}+\frac{r}{2}{p}_{o}}=\frac{1+(r-1)\phantom{\rule{0.2em}{0ex}}{p}_{o}}{1+\left(r-2\right)\phantom{\rule{0.2em}{0ex}}\frac{{p}_{0}}{2}}\hfill \end{array}$$

(13)

This definition easily generalizes to the case of an arbitrary number of equidistant channels (see Appendix A).

Our expressions for *current cooperativity*, Eq. (11), and *channel cooperativity*, Eq. (13), are very similar, but distinct. Figure 3 shows the dependence of these cooperativity measures on channel open probability in the case of two equidistant channels, under several conditions. In Figure 3A,B the release site is not near Ca^{2+} saturation, so the opening of a second channel has a maximal effect on release, *r* = 2** ^{n}**. The different measures of cooperativity all increase with

As Figure 3 shows, **m _{CH}** approaches the number of available channels (here,

From this analysis, we see that **m _{ICa}** provides an accurate estimate for

Figure 3. Distinction between Ca^{2+} *current cooperativity* (m_{ICa}) vs. *channel* cooperativity (m_{CH}) in the case of two equidistant channels per vesicle, for different values of release site saturation, r=(R2)/P(R1). Cooperativity measures are plotted against the open channel probability (fraction), *p*_{o}. m_{ICa} equals m_{ICa,log} (solid gray curve) only in the case *r*=2, corresponding to panel D (solid gray and black curves overlap). Note that m_{ICa,log} was previously examined in Fig. 7 of (Bertram et al., 1999), in the case *r*≥2, using a different notation, 1/*r*=(*f*_{(1)}+*f*_{(2)})/2, *ρ*=1−*p*_{o}, *n*=m_{ICa,log}.

Figure 4 shows the behavior of the three distinct cooperativity measures for three fixed values of *p*_{o} and a range of release ratio values, *r*. Notice that all measures of cooperativity decrease as the release site becomes saturated (*r*→1) and that **m _{ICa}** and

Dependence of Ca^{2+} channel and Ca^{2+} current cooperativity measures on the release ratio, *r*, for different levels of open channel fraction, *p*_{o}.

A non-obvious relationship reveals itself if the channel cooperativity is plotted against the current cooperativity, as in Figure 5. Even though **m _{CH}** and

Channel cooperativity m_{CH} and current cooperativity m_{ICa} are linearly related. As the release ratio is increased from 1 (saturating release case) to 2^{8} (super-cooperative release), the point (m_{ICa},m_{CH}) moves along a line with slope that depends on *p*_{o} **...**

$${\text{m}}_{\text{CH}}=(1-{p}_{0}){\text{m}}_{\text{ICa}}+2\phantom{\rule{0.1em}{0ex}}{p}_{0}$$

(14)

as can be verified by substituting in the formulas for **m _{CH}** and

$${\text{m}}_{\text{CH}}-{\text{m}}_{\text{ICa}}={p}_{0}(2-{\text{m}}_{\text{ICa}}).$$

(15)

In the next subsection we will show that **m _{ICa}** is bounded by the number of available channels (here

Figure 5 also confirms the conclusions above on the relationships between **m _{CH}** and

The results summarized by Figs. 3--55 for two equidistant channels may be extended to M channels, and the derivation of **m _{CH}** and

Dependence of m_{CH} and m_{ICa} on the open channel fraction in the case of M=5 equidistant channels, described by Eq. (30) of Appendix A.

In particular, Figs. 3--66 demonstrate that neither **m _{ICa}** nor

Current cooperativity **m _{ICa}** is also bounded by the biochemical cooperativity of exocytosis,

$${\text{m}}_{\text{ICa}}\le \text{min}(\text{M},\text{n})$$

(16)

The generalized definition of **m _{CH}** for

$${\text{m}}_{\text{CH}}=(1-{p}_{0}){\text{m}}_{\text{ICa}}+{p}_{0}\phantom{\rule{0.1em}{0ex}}\text{M}$$

(17)

Eq. (17) may be re-written as

$${\text{m}}_{\text{CH}}-{\text{m}}_{\text{ICa}}={p}_{0}(\text{M}-{\text{m}}_{\text{ICa}})\ge 0\phantom{\rule{0.1em}{0ex}},$$

(18)

demonstrating that **m _{ICa}** ≤

If the number of channels is greater than the number of Ca^{2+} binding sites **n**, the Monte Carlo simulation-based measure of channel cooperativity introduced by (Shahrezaei et al., 2006) (see also (Luo et al., 2008)) cannot reach the upper limit **M** that bounds our measure **m**_{CH}. They defined channel cooperativity as the average number of channels contributing Ca^{2+} ions to the Ca^{2+} binding sites in any given release event and calculated it by tracking the source of each of the ions that bind to the release sites. The latter measure, which we will denote **m**_{MC}, is bounded by the number of binding sites, since the total number of channels that contribute an ion to a release event cannot exceed the number of Ca^{2+} binding sites. It is also bounded by the number of channels, **M**, so we have **m**_{MC} ≤ min (**M**,**n**), the same bound obeyed by **m _{ICa}**, which may have encouraged the notion that the two are equivalent (Shahrezaei et al., 2006).

In the initial definition of **m**_{CH} introduced with Eq. (13), there were only two channels, which was less than the tacitly assumed number of binding sites, so **m _{CH}** and

Instead of fixing the value of the crucial parameter *r* = *P*(*R*2) / *P*(*R*1) by hand as in the previous section, we now use computer simulations of Ca^{2+} diffusion to model the current cooperativity protocol. That is, we compute *P*(*R*1) and *P*(*R*2) by solving the Ca^{2+} diffusion equations (Eqs. (2)-(3)) and using the Ca^{2+} binding scheme at the release site given by Eq. (4). Such direct simulation of Ca^{2+} diffusion will also enable us to demonstrate the effect of Ca^{2+} buffers on **m _{CH}** and

Figure 7 shows the dependence of **m _{CH}** and

In Figure 7, both **m _{ICa}** and

In contrast, Figure 8 illustrates that a further increase in *B*_{total} results in a *decrease* of release ratio, and hence, a *reduction* of channel and current cooperativities. This is because for *B*_{total}>1mM the residual background release rate provides a more significant contribution to evoked release, so the number of open channels becomes less important. This results in a reduced release ratio. The reduction in *r* for large *B*_{total} is reflected in decreasing values of **m _{CH}** and

Finally, Figure 9 summarizes our results on the effect of buffers on Ca^{2+} channel and current cooperativities of exocytosis, examining different values of the single-channel Ca^{2+} current, *i*_{Ca} (panels A-C) and channel-vesicle distance (panels D-F). Note that the values of **m _{CH}** and

In the previous sections we focused on the situation of two channels equidistant from a vesicle. We now generalize these results for the case of two non-equidistant channels.

We begin with the generalization of the current cooperativity expressions **m**_{ICa} given by Eqs. (9)-(11), using labels “10”, “01” and “11” to reflect the opening of the proximal channel, the distal channel, and both channels together, respectively: P(01)=P(10)=p_{o}(1−p_{o}), P(11)=p_{o}^{2}. The release probability expression becomes

$$\begin{array}{c}P(R)=P(R|10)P(10)+P(R|01)P(01)+P(R|11)P(11)\hfill \\ \phantom{\rule{2em}{0ex}}=\left[P(R|10)+P(R|01)\right]\left\{{p}_{o}(1-{p}_{o})+\frac{r}{2}{p}_{o}^{2}\right\},\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}r=\frac{2P(R|11)}{P(R|10)+P(R|01)}\hfill \end{array}$$

(19)

Note that the parameter *r* agrees with its definition in Eq. (10) in the limit of equidistant channels, and in the notation of (Bertram et al., 1999) is identical to 1/*f*_{1}. Inserting Eq. (19) into Eq. (9) yields:

$${\text{m}}_{\text{ICa}}=\frac{d\phantom{\rule{0.1em}{0ex}}\text{log}\phantom{\rule{0.1em}{0ex}}P(R)}{d\phantom{\rule{0.1em}{0ex}}\text{log}\phantom{\rule{0.2em}{0ex}}{p}_{o}}=\frac{{p}_{o}}{{p}_{o}(1-{p}_{o})+\frac{r}{2}{p}_{o}^{2}}\left\{1-2{p}_{o}+r{p}_{o}\right\}=\frac{1+(r-2)\phantom{\rule{0.1em}{0ex}}{p}_{o}}{1+\left(r-2\right)\phantom{\rule{0.1em}{0ex}}\frac{{p}_{o}}{2}}$$

(20)

This expression is equivalent to Eq. (11), given the above generalized definition of *r*. In particular, once again it is clear that **m _{ICa}** is bounded by the number of channels,

To generalize the channel cooperativity measure **m**_{CH} given by Eq. (13), we quantify the contribution of each channel to the Ca^{2+} domain in the vicinity of the vesicle by measuring the Ca^{2+} concentration at the release site resulting from the opening of one channel at a time. That is, we let Ca_{10} and Ca_{01} denote the Ca^{2+} concentrations at the release site when only the proximal or distal channel is open, respectively. Note that *Ca*_{01}/*Ca*_{10} < 1. We define the average number of channels contributing to release when both channels are open as 1 + *Ca*_{01}/*Ca*_{10}. Thus,

$${\text{m}}_{\text{CH}}=1\cdot P(10|R)+1\cdot P(01|R)+\left[1+\frac{C{a}_{01}}{C{a}_{10}}\right]P(11|R)$$

(21)

This heuristic approximation is particularly reasonable in the absence of buffering, since binding probability is proportional to Ca^{2+} concentration, and it reduces to Eq. (13) in the limit of equidistant channels, Ca_{10}=Ca_{01}. From Eqs. (19) and (21), we obtain:

$$\begin{array}{c}{\text{m}}_{\text{CH}}=\frac{P(R|10)P(10)+P(R|01)P(01)}{P(R)}+\left[1+\frac{C{a}_{01}}{C{a}_{10}}\right]\frac{P(R|11)P(11)}{P(R)}=\hfill \\ =\frac{{p}_{o}(1-{p}_{o})+\left[1+\frac{C{a}_{01}}{C{a}_{10}}\right]\frac{r}{2}\phantom{\rule{0.1em}{0ex}}{p}_{o}^{2}}{{p}_{o}(1-{p}_{o})+\frac{r}{2}\phantom{\rule{0.1em}{0ex}}{p}_{o}^{2}}=1+\frac{C{a}_{01}/C{a}_{10}}{1+1/(r\phantom{\rule{0.2em}{0ex}}f)}\hfill \end{array}$$

(22)

where the new parameter *f*=p_{o}/[2(1-p_{o})] is the ratio of two-channel to single-channel opening probabilities. In the limit of moving one channel far from the vesicle while keeping the other channel fixed, **m _{CH}** as defined by Eq. (22) approaches 1, which agrees with the intuition that the closer channel dominates.

In contrast to the case of equidistant channels, the sign of the difference between **m _{ICa}** and

$${\text{m}}_{\text{CH}}-{\text{m}}_{\text{ICa}}=\frac{C{a}_{01}/C{a}_{10}+2/r-1}{1+\frac{1}{r\phantom{\rule{0.1em}{0ex}}f}}$$

(23)

In particular, **m _{ICa}** overestimates

Dependence of m_{ICa} and m_{CH} on the total buffer concentration, in the case of two channels placed at distances of 20nm and 30nm from the release site, with a Ca^{2+} current of 0.1pA through each of the channels. Note the high sensitivity to buffer concentration, **...**

These results are further illustrated in Figure 9G-I, where the distance to the remote channel is varied along the horizontal parameter axis. Note that the entire Figure 10 corresponds to the white vertical line in panels G-I of this Figure. The lack of similarity between the behavior of **m _{CH}** (Figure 9H) and

In the case of non-equidistant channels, as in the case of equidistant channels, **m _{ICa}** and

Finally, we note that the dependence of channel cooperativity on buffer concentration and distance allows us to infer the effect of diffusional barriers, which are likely to exert significant influence on the Ca^{2+} concentration microdomains at the active zone (Kits et al., 1999; Glavinovic and Rabie, 2001; Shahrezaei and Delaney, 2004). Although we have not explicitly calculated the effect of diffusion barriers, their influence on current and channel cooperativities can be understood in terms of the parameter dependence shown in Figs. 7--11.11. Namely, if the barriers uniformly shield a population of channels from the vesicle, this effect is analogous to an increase in the diffusional distance of each channel from the release site, which may either decrease or increase **m _{CH}** and

Above we have considered the case of non-selective channel block, whereby the open channel fraction *p*_{o} characterizes the open probability of all channels contributing to exocytosis. However, Ca^{2+} current cooperativity measurements are also used to probe the participation of different Ca^{2+} channel subtypes in exocytosis, by blocking one type of Ca^{2+} channel at a time (Dunlap et al., 1995; Mintz et al., 1995; Reid et al., 1998; Wu et al., 1999; Scheuber et al., 2004). This situation was analyzed in detail by (Bertram et al., 1999) (see p. 742 therein) in terms of cooperativity measure **m _{ICa,log}** (denoted “n” in that paper). Here we extend those results to the

$$\begin{array}{l}P(R)=P(R|10)P(10)+P(R|01)P(01)+P(R|11)P(11)=P(R|11)\left\{{p}_{1}{p}_{2}+{f}_{10}{p}_{1}(1-{p}_{2})+{f}_{01}{p}_{2}(1-{p}_{1})\right\}\\ \text{where}\phantom{\rule{1em}{0ex}}{f}_{10}=\frac{P(R|10)}{P(R|11)},\phantom{\rule{0.5em}{0ex}}{f}_{01}=\frac{P(R|01)}{P(R|11)}\end{array}$$

If only the distal channel is affected by the block, *p*_{1} is constant but *p*_{2} varies. Assuming for simplicity that both channel contribute equally to total Ca^{2+} current, *I*_{Ca} is proportional to *p*_{1}+*p*_{2}, and we obtain

$${\text{m}}_{\text{ICa}}=\frac{d\phantom{\rule{0.2em}{0ex}}\text{log}\phantom{\rule{0.1em}{0ex}}P(R)}{d\phantom{\rule{0.2em}{0ex}}\text{log}({p}_{1}+{p}_{2})}=\frac{{p}_{1}+{p}_{2}}{P(R)}\frac{dP(R)}{d{p}_{2}}=({p}_{1}+{p}_{2})\frac{{p}_{1}-{f}_{10}{p}_{1}+{f}_{01}(1-{p}_{1})}{{p}_{1}\phantom{\rule{0.1em}{0ex}}{p}_{2}+{f}_{10}\phantom{\rule{0.1em}{0ex}}{p}_{1}(1-{p}_{2})+{f}_{01}\phantom{\rule{0.1em}{0ex}}{p}_{2}(1-{p}_{1})}$$

As the distance from the release site to the blocked (distal) channel increases, then *f*_{10}→1, and *f*_{01}→0, and according to the above expression, **m _{ICa}**→0. Moreover,

Generalizing **m _{CH}** to the non-selective case transforms Eq. (22) to

$$\begin{array}{c}{\text{m}}_{\text{CH}}=\frac{P(R|10)P(10)+P(R|01)P(01)}{P(R)}+\left[1+\frac{C{a}_{01}}{C{a}_{10}}\right]\frac{P(R|11)P(11)}{P(R)}\hfill \\ \phantom{\rule{2em}{0ex}}=\frac{{f}_{10}\phantom{\rule{0.1em}{0ex}}{p}_{1}(1-{p}_{2})+{f}_{01}\phantom{\rule{0.1em}{0ex}}{p}_{2}(1-{p}_{1})+{p}_{1}\phantom{\rule{0.1em}{0ex}}{p}_{2}\left[1+C{a}_{01}/C{a}_{10}\right]}{{f}_{10}\phantom{\rule{0.1em}{0ex}}{p}_{1}(1-{p}_{2})+{f}_{01}\phantom{\rule{0.1em}{0ex}}{p}_{2}(1-{p}_{1})+{p}_{1}\phantom{\rule{0.1em}{0ex}}{p}_{2}}=1+\frac{C{a}_{01}/C{a}_{10}}{1+{f}_{10}(1-{p}_{2})/{p}_{2}+{f}_{01}(1-{p}_{1})/{p}_{1}}\hfill \end{array}$$

As the remote channel is separated from the release site, Ca_{01}→0, *f*_{01}→0, and *f*_{10}→1, so the channel cooperativity approaches unity. Thus, in contrast to **m _{ICa}**,

The notion of cooperativity between release rate and Ca^{2+} concentration was initially introduced to characterize the intrinsic chemical properties of the putative release mechanism (Dodge and Rahamimoff, 1967). More recently the cooperativity concept has been extended to quantify the sensitivity of the release rate to the whole-terminal Ca^{2+} current, I_{Ca}, whereby the number of open channels is varied while the single-channel Ca^{2+} current is kept constant. The exponent of the resulting non-linear relationship between release rate and I_{Ca} has been termed “current cooperativity” and denoted here **m _{ICa}**. A similar technique measures the sensitivity of exocytosis rate to the fraction of available channels by titrating the channel opening probability using tail currents (Gentile and Stanley, 2005). These techniques allow one to address the question of how many channels contribute to the exocytosis of a single vesicle at a given synaptic terminal (reviewed in (Gentile and Stanley, 2005)). However, the validity of

Here we have rigorously defined the various measures of cooperativity, both conceptually and mathematically. Our goal was not to establish the number of channels contributing to a single release event, which is likely to vary between different types of synapses, but to clarify the concepts and formulas so that such determinations can be made rigorously and consistently from experimental data. In particular, we showed that the Ca^{2+} current cooperativity, **m _{ICa}**, and the underlying Ca

Although we have shown **m _{ICa}** and

The definitions of **m _{ICa}** and

For the case of two non-equidistant channels, we found the expression for **m _{ICa}** (Eq. (20)) to be identical to that for the equidistant channel case (Eq. (11)), and constructed a heuristic generalization of

Neither **m _{CH}** nor

Finally, we addressed the conflicting results on the effects of exogenous buffers on channel cooperativity gleaned from prior modeling work. The study of (Bertram et al., 1999) suggested that the effect of buffer addition will depend strongly on its Ca^{2+}-binding properties. Namely, a saturable buffer would increase the cooperativity regardless of channel arrangement, due to the inter-channel interactions in the saturation of the buffer. On the other hand, a high concentration of non-saturable buffer was shown to decrease the simulated Ca^{2+} current cooperativity, since its only effect would be to intercept Ca^{2+} ions from each of the channels, restricting the extent of domain overlap. In contrast, the Monte-Carlo study of (Shahrezaei et al., 2006) suggested that buffers can only reduce channel cooperativity, regardless of the buffering properties. However, the latter study did not include a full sensitivity analysis with respect to various buffering parameters, due to the computational expense of Monte Carlo simulations, and did not distinguish between channel and current cooperativity. Here we addressed this open question by simulating the buffered Ca^{2+} diffusion for a range of total buffer concentration values and other simulation parameters, and the results in Figs. 7--99 show that **m _{CH}** and

This work was supported by the National Science Foundation grant DMS-0817703 (to V.M.), DMS-0613179 (to R.B.), and the Intramural Research Program of the National Institutes of Health, NIDDK (A.S.).

Here we generalize Eq. (14) to the case of **M** equidistant channels. Denoting *P*(*Rk*) the probability of release given *k* open channels, the release probability is given by

$$P(R)=\sum _{k=1}^{\text{M}}P(R|k)P(k)=\sum _{k=1}^{M}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{\text{p}}_{\text{o}}^{\text{k}}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{o})}^{\text{M}-k},\phantom{\rule{0.7em}{0ex}}\text{where}\phantom{\rule{0.4em}{0ex}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)=\frac{\text{M}!}{k!(\text{M}-k)!}$$

(24)

Now, the current cooperativity is given by:

$$\begin{array}{c}{\text{m}}_{\text{ICa}}=\frac{{p}_{o}}{P(R)}\frac{dP(R)}{d{p}_{o}}=\frac{1}{P(R)}{p}_{o}\frac{d}{d{p}_{o}}\sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{\text{p}}_{\text{o}}^{\text{k}}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{o})}^{\text{M}-k}\hfill \\ \phantom{\rule{12.2em}{0ex}}=\frac{1}{P(R)}\sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{\text{p}}_{\text{o}}^{\text{k}}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{o})}^{\text{M}-k}\left[k-\frac{{p}_{o}(\text{M}-k)}{1-{p}_{o}}\right]\hfill \\ \phantom{\rule{12.2em}{0ex}}=\frac{1}{P(R)}\sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\hfill \\ k\hfill \end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{\text{p}}_{\text{o}}^{\text{k}}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{o})}^{\text{M}-k}\frac{k-{p}_{o}\text{M}}{1-{p}_{o}}\hfill \end{array}$$

(25)

Multiplying by (1-*p*_{o}), we obtain:

$$\begin{array}{c}(1-{p}_{0}){\text{m}}_{\text{ICa}}=\frac{1}{P(R)}\sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{p}_{0}^{k}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{0})}^{\text{M}-k}\phantom{\rule{0.2em}{0ex}}\left[k-{p}_{0}\text{M}\right]\hfill \\ \phantom{\rule{5.3em}{0ex}}=\frac{1}{P(R)}\left\{\sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{p}_{0}^{k}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{0})}^{\text{M}-k}k\right\}-\frac{{p}_{0}\text{M}}{P(R)}P(R)={\text{M}}_{\text{CH}}-{p}_{0}\text{M}\hfill \end{array}$$

where the channel cooperativity **m _{CH}** is given as the generalization of Eq. (13):

$${\text{m}}_{\text{CH}}=\frac{1}{P(R)}\sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right)P(R|k)\phantom{\rule{0.1em}{0ex}}{p}_{0}^{k}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{0})}^{\text{M}-k}k$$

(26)

Here it is assumed that all *k* open equidistant channels contribute equally to release. Thus

$${\text{m}}_{\text{CH}}=(1-{p}_{0}){\text{m}}_{\text{ICa}}+{p}_{0}\text{M}$$

(27)

which generalizes Eq. (14)

To obtain closed-form expressions for **m _{CH}** and

$$P(R)=\sum _{k=1}^{\text{M}}P(R|k)P(k)\propto \sum _{k=1}^{\text{M}}\left(\begin{array}{c}\text{M}\\ k\end{array}\right){k}^{\text{n}}\phantom{\rule{0.1em}{0ex}}{\text{p}}_{\text{o}}^{\text{k}}\phantom{\rule{0.1em}{0ex}}{(1-{p}_{o})}^{\text{M}-k}={\left[{\left[{p}_{o}\frac{d}{d{p}_{o}}\right]}^{\text{n}}{({p}_{o}+{q}_{o})}^{\text{M}}\right]}_{{q}_{o}=1-{p}_{o}}$$

(28)

Current cooperativity is obtained using Eq. (9), while channel cooperativity is obtained as

$${\text{m}}_{\text{CH}}=\frac{1}{P(R)}\sum _{k=1}^{\text{M}}k\phantom{\rule{0.2em}{0ex}}P(R|k)P(k)=\frac{1}{P(R)}{\left[{\left[{p}_{o}\frac{d}{d{p}_{o}}\right]}^{\text{n}+1}{({p}_{o}+{q}_{o})}^{\text{M}}\right]}_{{q}_{o}=1-{p}_{o}}$$

(29)

Note in particular that P(R) is a polynomial of order **M** in *p*_{o}, and therefore its logarithmic derivative, **m _{ICa}**=dlog P(R)/dlog

The resulting functional form of **m _{CH}** and

$$\begin{array}{c}\text{n}=1:(\text{linear realtionship between release and}\phantom{\rule{0.1em}{0ex}}[{\text{Ca}}^{2+}]:{\text{m}}_{\text{ICa}}=1,\phantom{\rule{0.3em}{0ex}}{\text{m}}_{\text{CH}}=1+{p}_{o}(\text{M}-1)\hfill \\ \text{n}=2:{\text{m}}_{\text{ICa}}=\frac{1+2{p}_{o}(\text{M}-1)}{1+{p}_{o}(\text{M}-1)},{\text{m}}_{\text{CH}}=\frac{1+3{p}_{o}(\text{M}-1)+{p}_{o}^{2}(\text{M}-1)(\text{M}-2)}{1+{p}_{o}(\text{M}-1)}\hfill \\ \text{n}=3:\{\begin{array}{l}{\text{m}}_{\text{ICa}}=\frac{1+6{p}_{o}(\text{M}-1)+3{p}_{o}^{2}(\text{M}-1)(\text{M}-2)}{1+3{p}_{o}(\text{M}-1)+{p}_{o}^{2}(\text{M}-1)(\text{M}-2)}\\ {\text{m}}_{\text{CH}}=\frac{1+7{p}_{o}(\text{M}-1)+6{p}_{o}^{2}(\text{M}-1)(\text{M}-2)+{p}_{o}^{3}(\text{M}-1)(\text{M}-2)(\text{M}-3)}{1+3{p}_{o}(\text{M}-1)+{p}_{o}^{2}(\text{M}-1)(\text{M}-2)}\hfill \end{array}\hfill \\ \text{n}=4:\{\begin{array}{l}{\text{m}}_{\text{ICa}}=\frac{1+{p}_{o}(\text{M}-1)\left(14+{p}_{o}(\text{M}-2)\left(18+4{p}_{o}(\text{M}-3)\right)\right)}{1+{p}_{o}(\text{M}-1)\left(7+{p}_{o}(\text{M}-2)\left(6+{p}_{o}(\text{M}-3)\right)\right)}\\ {\text{m}}_{\text{CH}}=\frac{1+{p}_{o}(\text{M}-1)\left(15+{p}_{o}(\text{M}-2)\left(25+{p}_{o}(\text{M}-3)\left(10+{p}_{o}(\text{M}-4)\right)\right)\right)}{1+{p}_{o}(\text{M}-1)\left(7+{p}_{o}(\text{M}-2)\left(6+{p}_{o}(\text{M}-3)\right)\right)}\hfill \end{array}\hfill \\ \text{n}=5:\{\begin{array}{l}{\text{m}}_{\text{ICa}}=\frac{1+{p}_{o}(\text{M}-1)\left(30+{p}_{o}(\text{M}-2)\left(75+{p}_{o}(\text{M}-3)\left(40+5{p}_{o}(\text{M}-4)\right)\right)\right)}{1+{p}_{o}(\text{M}-1)\left(15+{p}_{o}(\text{M}-2)\left(25+{p}_{o}(\text{M}-3)\left(10+{p}_{o}(\text{M}-4)\right)\right)\right)}\\ {\text{m}}_{\text{CH}}=\frac{1+{p}_{o}(\text{M}-1)\left(31+{p}_{o}(\text{M}-2)\left(90+{p}_{o}(\text{M}-3)\left(65+{p}_{o}(\text{M}-4)\left(15+{p}_{o}(\text{M}-5)\right)\right)\right)\right)}{1+{p}_{o}(\text{M}-1)\left(15+{p}_{o}(\text{M}-2)\left(25+{p}_{o}(\text{M}-3)\left(10+{p}_{o}(\text{M}-4)\right)\right)\right)}\hfill \end{array}\hfill \end{array}$$

(30)

As a check of these expressions, setting **M**=2 yields Eqs. (11) and (13), with *r*=2** ^{n}**.

We find the following limiting behavior for *p _{o}*→1, for any values of

$${\text{m}}_{\text{ICa}}=\text{M}\left[1-{\left(1-\frac{1}{\text{M}}\right)}^{\text{n}}\right]+O({q}_{0});\phantom{\rule{0.5em}{0ex}}{\text{m}}_{\text{CH}}=\text{M}\left[1-{\left(1-\frac{1}{\text{M}}\right)}^{\text{n}}{q}_{0}\right]+O({q}_{0}^{2})$$

(31)

Where *q*_{o}=1−*p*_{o}. Note in particular that **m _{CH}** approaches the channel number

$${\text{m}}_{\text{ICa}}\le \text{M}\left[1-{\left(1-\frac{1}{\text{M}}\right)}^{\text{n}}\right]=\text{M}\left[1-\left(1-\frac{\text{n}}{\text{M}}+\frac{\text{n}(\text{n}-1)}{2{\text{M}}^{2}}-\mathrm{...}\right)\right]=\text{n}\left[1-\frac{\text{n}-1}{2\text{M}}+\mathrm{...}\right]$$

(32)

Therefore, m_{ICa} ≤ min(n,M).

If the release is partially saturated by the opening of several channels, the above expressions should be understood as upper bounds on the corresponding values of m_{ICa} and m_{CH}. The case of complete saturation by a single channel corresponds to n=0, and both cooperativity measures can be derived from the release probability, *P*(*R*)=1−(1−*p*_{o})^{M}:

$$\text{n}=0:\phantom{\rule{0.3em}{0ex}}{\text{m}}_{\text{ICa}}=\text{M}\frac{{p}_{0}{(1-{p}_{0})}^{\text{M}-1}}{1-{(1-{p}_{0})}^{\text{M}}},\phantom{\rule{0.3em}{0ex}}{\text{m}}_{\text{CH}}=\text{M}\frac{{p}_{0}}{1-{(1-{p}_{0})}^{\text{M}}}$$

Here we generalize Eqs. (19)-(22) to the case of **M** non-equidistant channels. Denoting *P*(*Rk*) the probability of release given a particular configuration “*k*” of open channels among the given **M** channel, release probability is given by

$$P(R)=\sum _{k=1}^{L}P(R|k){p}_{o}^{N(k)}{(1-{p}_{o})}^{\text{M}-N(k)}$$

(33)

where the sum extends over *L*=2^{M} possible configurations, and *N*(k) denotes the number of channels that are open in a particular configuration. Contrary to the equidistant channel case, the values *P*(*Rk*) cannot be determined using any simplifying assumptions, but have to be computed numerically, using a 3D model of Ca^{2+} diffusion and binding. We then obtain

$$\begin{array}{c}{\text{m}}_{\text{ICa}}=\frac{d\phantom{\rule{0.1em}{0ex}}\text{log}\phantom{\rule{0.1em}{0ex}}P(R)}{d\phantom{\rule{0.1em}{0ex}}\text{log}\phantom{\rule{0.1em}{0ex}}{p}_{0}}=\frac{{p}_{0}}{P(R)}\sum _{k=1}^{L}P(R|k){p}_{o}^{N(k)}{(1-{p}_{o})}^{\text{M}-N(k)}\left[\frac{N(k)}{{p}_{0}}-\frac{M-N(k)}{1-{p}_{0}}\right]\hfill \\ \phantom{\rule{8em}{0ex}}=\left[\frac{1}{P(R)}\sum _{k=1}^{L}N(k)P(R|k){p}_{o}^{N(k)}{(1-{p}_{o})}^{\text{M}-N(k)-1}\right]-\frac{{p}_{0}M}{1-{p}_{0}}\hfill \end{array}$$

(34)

Denoting *C _{i}*(

$${\text{m}}_{\text{CH}}=\frac{1}{P(R)}\sum _{k=1}^{L}P(R|k)\left(\frac{1}{{C}_{\text{max}}(k)}\sum _{i=1}^{N(k)}{C}_{i}(k)\right)\phantom{\rule{0.1em}{0ex}}{p}_{o}^{N(k)}{(1-{p}_{o})}^{\text{M}-N(k)}$$

(35)

Note that this generalizes Eq. (21) for **M**=2, in which case the sum extends over configurations *k*={“00”, “01, “10”, “11”}, among which *k*=“00” does not contribute since *P*(*R*00)=0, while for *k*=“01” and *k*=“10”, only one channel is open, so *C*_{max}(*k*) is identical to the sum *C*_{i}(*k*), yielding a factor of unity in the brackets; finally, in the state *k*=“11”, the factor in the brackets equals (C_{01}+C_{10})/C_{10}, in agreement with Eq. (21).

For the sake of simplicity, the derivation of the Ca^{2+} channel and the Ca^{2+} current cooperativities of release for the case of two channels (Eqs. (11)-(13)) ignores the contribution of background release rate due to the resting [Ca^{2+}]. In other words, we assume P(R0)=0. However, for small values of intracellular Ca^{2+} at the release site, these expressions are modified as follows:

$$\begin{array}{c}P(R)=P(R|0)P(0)+P(R|1)P(1)+P(R|2)P(2)=P(R|1)\left(\epsilon {(1-{p}_{o})}^{2}+2{p}_{o}(1-{p}_{o})+r{p}_{o}^{2}\right)\hfill \\ \phantom{\rule{2em}{0ex}}\epsilon \equiv \frac{\text{P}\left(\text{R}|\text{0}\right)}{\text{P}\left(\text{R}|\text{1}\right)},r\equiv \frac{\text{P}\left(\text{R}|2\right)}{\text{P}\left(\text{R}|\text{1}\right)}=\{\begin{array}{c}1,\text{full saturation}\\ \sim {2}^{\text{n}},\text{no saturation}\end{array}\hfill \end{array}$$

where *P*(*R*0) is the background resting release rate (release probability under condition that both channels are closed). Plugging this into Eq. (9), we obtain:

$${\text{m}}_{\text{ICa}}=2{p}_{o}\frac{-\epsilon \left(1-{p}_{o}\right)+1-2{p}_{o}+r{p}_{o}}{\epsilon {\left(1-{p}_{o}\right)}^{2}+2{p}_{o}\left(1-{p}_{o}\right)+r{p}_{o}^{2}}=2\frac{-\epsilon \left(1-{p}_{o}\right)+1+(r-2){p}_{o}}{\epsilon {\left(1-{p}_{o}\right)}^{2}/{p}_{o}+2+(r-2){p}_{o}}$$

(36)

Similarly, Eq. (13) becomes

$${\text{m}}_{\text{CH}}=\frac{P(R|1)P(1)+2\cdot P(R|2)P(2)}{P(R)}=\frac{2{p}_{o}(1-{p}_{o})+2r{p}_{o}^{2}}{\epsilon {(1-{p}_{o})}^{2}+2{p}_{o}(1-{p}_{o})+r{p}_{o}^{2}}=2\frac{1+(r-1){p}_{o}}{\epsilon {\left(1-{p}_{o}\right)}^{2}/{p}_{o}+2+(r-2){p}_{o}}$$

For the case of non-equidistant channels, Eq. (19) is modified as

$$\begin{array}{c}P(R)=P(R|00)P(00)+P(R|10)P(10)+P(R|01)P(01)+P(R|11)P(11)\hfill \\ \phantom{\rule{2.2em}{0ex}}=\frac{P(R|10)+P(R|01)}{2}\left\{\epsilon {(1-{p}_{o})}^{2}+2{p}_{o}(1-{p}_{o})+r{p}_{o}^{2}\right\}\hfill \\ \text{where}\phantom{\rule{0.5em}{0ex}}r=\frac{2P(R|11)}{P(R|10)+P(R|01)},\phantom{\rule{0.3em}{0ex}}\epsilon \equiv \frac{2P(R|00)}{P(R|10)+P(R|01)}\hfill \end{array}$$

With this redefinition of parameters *r* and ε, Eq. (36) for **m _{ICa}** remains valid, while the expression for current cooperativity, Eq. (22), is transformed to:

$$\begin{array}{l}{\text{m}}_{\text{CH}}=\frac{P(R|10)P(10)+P(R|01)P(01)}{P(R)}+\left[1+\frac{C{a}_{01}}{C{a}_{10}}\right]\frac{P(R|11)P(11)}{P(R)}=\\ =\frac{2{p}_{o}(1-{p}_{o})+r\left[1+\frac{C{a}_{01}}{C{a}_{10}}\right]\phantom{\rule{0.1em}{0ex}}{p}_{o}^{2}\phantom{\rule{0.1em}{0ex}}}{\epsilon {(1-{p}_{o})}^{2}+2{p}_{o}(1-{p}_{o})+r{p}_{o}^{2}}=\frac{C{a}_{01}/C{a}_{10}r{p}_{o}+2+(r-2){p}_{o}}{\epsilon {\left(1-{p}_{0}\right)}^{2}/{p}_{o}+2+(r-2){p}_{o}}=\frac{C{a}_{01}/C{a}_{10}r\phantom{\rule{0.2em}{0ex}}f+1+r\phantom{\rule{0.2em}{0ex}}f}{\epsilon /(4f)+1+r\phantom{\rule{0.2em}{0ex}}f}\end{array}$$

where $f=\frac{{p}_{o}}{2(1-{p}_{o})}$ (cf. Eq. (22))

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