# Explicit Theoretical Analysis of How the Rate of Exocytosis Depends on Local Control by Ca2+ Channels

Authors: Francesco Montefusco; Morten Gram Pedersen
Journal: Computational and Mathematical Methods in Medicine
 (2018)

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## Abstract

Hormones and neurotransmitters are released from cells by calcium-regulated exocytosis, and local coupling between Ca2+ channels (CaVs) and secretory granules is a key factor determining the exocytosis rate. Here, we devise a methodology based on Markov chain models that allows us to obtain analytic results for the expected rate. First, we analyze the property of the secretory complex obtained by coupling a single granule with one CaV. Then, we extend our results to a more general case where the granule is coupled with n CaVs. We investigate how the exocytosis rate is affected by varying the location of granules and CaVs. Moreover, we assume that the single granule can form complexes with inactivating or non-inactivating CaVs. We find that increasing the number of CaVs coupled with the granule determines a much higher rise of the exocytosis rate that, in case of inactivating CaVs, is more pronounced when the granule is close to CaVs, while, surprisingly, in case of non-inactivating CaVs, the highest relative increase in rate is obtained when the granule is far from the CaVs. Finally, we exploit the devised model to investigate the relation between exocytosis and calcium influx. We find that the quantities are typically linearly related, as observed experimentally. For the case of inactivating CaVs, our simulations show a change of the linear relation due to near-complete inactivation of CaVs.


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## Body

## 1. Introduction

Molecules, e.g., neurotransmitters and proteins, are released from the cell by exocytosis [1]. In this paper, we focus on regulated exocytosis in the endocrine cells that release different kinds of hormones regulating various physiological processes [2]. When hormone secretion is defectively regulated, several diseases may develop. For example, in diabetes, the two main pancreatic hormones, insulin and glucagon, are not released appropriately for fine-tuning glucose homeostasis [3, 4]. Therefore, it is crucial to achieve a better understanding of the main mechanisms underlying hormone exocytosis that determines the control of different physiological processes.In most endocrine cells, the hormones are contained in secretory granules that, in response to a series of cellular mechanisms culminating with an increase in the intracellular Ca2+ levels, fuse with the cell membrane and release the hormone molecules. The main mechanisms regulating hormone exocytosis are shared with exocytosis of synaptic vesicles underlying neurotransmitter release in neurons [1, 5]. The granules contain v-SNARE proteins that can form the so-called SNARE complexes with t-SNAREs inserted in the cell membrane [1]. SNARE complexes interact with other proteins, notably, Ca2+-sensing proteins such as synaptotagmins, which trigger exocytosis upon Ca2+ binding. Therefore, the local Ca2+ concentration at the Ca2+ sensor of the exocytotic machinery is a key factor determining the probability rate of exocytosis of the secretory granule [6].Recently, we have devised a detailed model of Ca2+ dynamics and exocytosis for the glucagon-secreting pancreatic alpha-cells and showed how exocytosis is dependent on calcium dynamics, in particular, on calcium levels surrounding the Ca2+ channels (CaVs) [7], the so-called nanodomains [8]. Here, in order to characterize the local interactions between the single granule and the surrounding CaVs, we will exploit a strategy that is similar to the methodology devised in our recent paper to describe the large conductance BK potassium current that is controlled locally by CaVs [9]. We showed that the number and the type of CaVs coupled with the BK channel affect the electrical activity of neurons and other excitable cells, such as pancreatic beta-cells and pituitary cells. Therefore, we will implement mathematical modelling for characterizing the local interactions between granules and CaVs and, specifically, Markov chain models that could provide important insight into the exocytosis rate. In particular, by using the Markov chain theory [10], we will achieve analytic results for the expected rate and show how coupling different numbers and types of CaVs with the granule determines different responses.


## 2. Methods

### 2.1. CaV Channel Model

We model the Ca2+ channel by using the 3-state Markov chain of Figure 1(a), where C corresponds to the closed state, O to the open state, and B to the inactivated (blocked) state of the calcium channel [11]. Then, the CaV model takes values in the state space S=C,O,B and its transition rate or generator matrix MCaV is given by(1)MCaV=−αα0β−β−δδ0γ−γ,where α and β represent the voltage-dependent Ca2+ channel opening rate and closing rate, respectively, and have the following forms:(2)αV=α0e−α1V,βV=β0e−β1V.Figure 1
Markov chain models for Ca2+ channel (CaV), exocytosis of single granule and granule-CaV complex. (a) Markov chain model for CaV, where C is the closed state, O the open state, and B the inactivated or blocked state. (b) Markov chain model for exocytosis of a single granule adjacent to the plasma membrane, where G0 correspond to the state with no bound Ca2+ ions, G1 with one, G2 with two, and G3 with three. (c) Markov chain model for the approximated exocytosis model where the dynamics of states G2 and G3 are described by the auxiliary variable G23 using quasi-steady state approximation for the corresponding ODE model. (d) Markov chain model for the granule-CaV complex where the granule dynamics are described by the model shown in panel (c) and the CaV dynamics by the model shown in panel (a).
(a)
(b)
(c)
(d)The rate for channel inactivation,δ, is Ca2+-dependent and has the following form:(3)δ=δ0×CaCaV,where CaCaV is the Ca2+ concentration at the Ca2+ sensor for inactivation and is given using reaction-diffusion theory [8, 12, 13] by(4)CaCaV=iCamax8πrCaDCaFexp−rCaDCa/kB+Btotal,where iCamax=g¯CaV−VCa is the single-channel Ca2+ current with g¯Ca the single-channel conductance and VCa the reverse potential, and rCa represents the distance of the sensor for Ca2+-dependent inactivation from the channel pore. Finally, γ is the constant reverse reactivation rate. Table 1 reports the parameter values for the CaV model defined by above equations.Table 1
Model parameters.
Parameter
Value
Unit
CaV model parameters
α
0
0.6
ms−1
α
1
−
0.1
mV−1
β
0
0.2
ms−1
β
1
0.0375
mV−1
γ
0.002
ms−1
δ
0
0.0025
μM−1·ms−1
Parameters for calculating Ca2+ concentration at different distances
r
Ca
7
nm
r
G
10, 20, 30, 50
nm
D
Ca
250
μm2·s−1
F
9.6485
C mol−1
k
B
500
μM−1·s−1
B
total
30
μM
V
Ca
60
mV
g
¯
Ca
2.8
pS
Ca
c
0.1
μM
Ca
b
0.1
μM
Exocytosis model parameters
k
+
1.85
μM−1·s−1
k
−
50
s−1
u
1000
s−1The deterministic description of the 3-state Markov chain model for the CaV channel is given by the following ODE system:(5)dcdt=βo−αc,dodt=αc+γb−β+δo,b=1−c−o=1−h,where the italic lowercase letters represent the corresponding state variables of the ODE model (h represents the fraction of Ca2+ channels not inactivated).Finally, in order to investigate the relationship between exocytosis and Ca2+ loading, we compute the total charge entering via the Ca2+ channel at a given step voltage with time window, ts, as(6)QCa=∫0tsoτ⋅iCamaxdτ.


### 2.2. Exocytosis Model

We assume a single granule, adjacent to the plasma membrane and primed for exocytosis, that can be in one of four different states depending on the number of Ca2+ ions bound to the Ca2+ sensor on the granule, likely synaptotagmin [14]: in G0 with no bound Ca2+ ions, or in G1 with one, or in G2 with two, or in G3 with three bound ions. Once it is in G3, the granule can fuse with the membrane and release its hormone content, assuming the final state Y [6, 15]. Therefore, we use a five-state Markov chain model for describing exocytosis as shown in Figure 1(b), where the model takes values in the state space S=G0,G1,G2,G3,Y, and its transition rate or generator matrix MG is given by(7)MG=−3kCa3kCa000k−−2kCa−k−2kCa0002k−−kCa−2k−kCa0003k−−u−3k−u00000,where(8)kCa=k+×CaGrepresents the Ca2+ binding rate, with CaG the Ca2+ concentration at the granule sensor given by Equation (4) with r=rG being the distance from the CaV to the Ca2+ sensor on the granule. In the following, the distance from the CaV to the granule means the distance from the CaV to the Ca2+ sensor on the granule, which will be of the order of tens of nm. For comparison, secretory granules have diameters on the order 100–500 nm [16–19]. We assume a constant number of Ca2+ sensor molecules, which is therefore included in the binding parameter kCa. The parameter k− is the unbinding rate, and u is the fusion rate. Table 1 reports the parameter values.The deterministic description of the 5-state Markov chain model for exocytosis is given by the following ODE system:(9)dg0dt=−3kCag0+k−g1,(10)dg1dt=−2kCa+k−g1+3kCag0+2k−g2,(11)dg2dt=−kCa+2k−g2+2kCag1+3k−g3,(12)dg3dt=−u+3k−g3+kCag2,(13)y=1−g0−g1−g2−g3.For the above ODE model of Equations (9)–(13), we exploit quasi steady-state approximation for state g3, since its dynamics are fastest (the value of u is much higher than those of the other parameters). Then, by renaming the state variables as(14)g23=g2+g3,by setting Equation (12) equal to zero yielding(15)g3=Ag23,withA=kCakCa+3k−+u,and by summing Equations (11) and (12), we achieve a single ODE model for describing the dynamics of state variable g2 and g3 as follows:(16)dg23dt=−2k−1−A+uAg23+2kCag1.The corresponding Markov chain model takes values in the state spaceS=G0,G1,G23,Y (Figure 1(c)) and is described by the following generating matrix, MGap:(17)MGap=−3kCa3kCa00k−−2kCa−k−2kCa002k−1−A−2k−1−A−uAuA0000.Note that stateY of the Markov chain described by MGap is an absorbing state: the process can never leave Y after entering it, reflecting that fusion is an irreversible process. Then MGap can be rewritten as(18)MGap=D3×3d3×101×30,where(19)D3×3=−3kCa3kCa0k−−2kCa−k−2kCa02k−1−A−2k−1−A−uA,describes only the transitions between the transient states G0, G1, and G23 and d=0,0,uAT is a vector containing the transition intensities from the transient states to the absorbing state Y. The row vector 0∈R1×3 consists entirely of 0’s since no transitions from Y to the transient states can occur. The remaining element of the matrix MGap is 0 and gives the transition rate out of the absorbing state.Using phase-type distribution results for Markov chains [10], we obtain an explicit formula for calculating the expected event rate λY to reach the absorbing state Y, given the initial probability row vector π for the transient states (π=πG0,πG1,πG23), as(20)λY=1π−D−11,where 1∈R3×1.


### 2.3. Granule-CaV Complex Model with 1 : 1 and 1 :n Stoichiometries

#### 2.3.1. 1 : 1 Stoichiometry

By coupling the CaV and exocytosis models, we obtain the 12-state Markov chain model of Figure1(d). The model takes values in the state space(21)S=CG0,OG0,BG0,CG1,OG1,BG1,CG23,OG23,BG23,CY,OY,BY,and its transition matrix, DG:CaV, is as follows:(22)DG:CaV9×9=MCaV−3diagkCac,kCao,kCac3diagkCac,kCao,kCac03×3diagk−,k−,k−MCaV−diag2kCac+k−,2kCao+k−,2kCac+k−2diagkCac,kCao,kCac03×32k−diag1−Ac,1−Ao,1−AcMCaV−diag2k−1−Ac+uAc,2k−1−Ao+uAo,2k−1−Ac+uAc,where MCaV is defined by Equation (1), kCacAc by Equation (8) (Equation (15)) with CaG=Cac, i.e., the concentration at the granule when the associated CaV is closed (or inactivated, i.e., Cac=Cab), and kCaoAo by Equation (8) (Equation (15)) with CaG=Cao, i.e., the concentration at the granule when the associated CaV is open, computed by Equation (4). Then, the expected exocytosis rate for the single granule, λY1, can be estimated by using Equation (20), assuming initially the granule in state G0 and the CaV closed, i.e., the complex in the state CG0π=1,01×8, as(23)λY1=1π−DG:CaV−11,where 1∈R9×1.We also consider the particular case with non-inactivating CaV (i.e., the Ca2+ channel can be only in C or in O). In this case, MCaV∈R2×2 and is defined by Equation (1) with δ=γ=0, and then DG:CaV, given by Equation (22), belongs to R6×6.


#### 2.3.2. 1 :n Stoichiometry

In the following, we assume the case where the granule is coupled with more than one CaV. In particular, by consideringk Ca2+ channels, we have a Markov chain model with nS=∑i=0kk+1−i=k2/2+3k/2+1 possible states describing the k CaVs. In particular, the CaVs model takes values in the state space S=Ck−i−jOiBj with j∈0,…,k and i∈0,…,k−j, and its generating matrix, MkCaV, is given by(24)MkCaVnS×nS=M0k+1×k+101×kδdiag1,…,k0⋯⋯⋯⋯⋯00k×1γIk⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯00k+1−j×1jγIk+1−jMjk+1−j×k+1−j01×k−jδdiag1,…,k−j0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱δ⋮⋯⋯⋯⋯⋯0kγMk1×1,where(25)M0k+1×k+1=−kαkα0⋯⋯⋯⋯⋯0β⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯0i−1β−k−i−1α−i−1β+δk−i−1α0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱α0⋯⋯⋯⋯⋯0kβ−kβ+δ,(26)Mjk+1−j×k+1−j=−k−jα−jγk−jα0⋯⋯⋯⋯⋯0β⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯0i−1β−k−i−1−jα−i−1β+δ−jγk−i−1−jα0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱α0⋯⋯⋯⋯⋯0k−jβ−k−jβ+δ−jγ,and Mk1×1=−kγ.Then, by coupling the CaVs and exocytosis models, we obtain a4nS-state Markov chain model. The model takes values in the state space S=Ck−i−jOiBjGl,…,Ck−i−jOiBjG23,Ck−i−jOiBjY, with j∈0,…,k, i∈0,…,k−j and l∈0,1, and its transition matrix, DG:kCaV, can be written as(27)DG:kCaV3nS×3nS=MkCaV−KCa1KCa10nS×nSk−InSMkCaV−KCa2−k−InSKCa20nS×nS2k−InS−DAMkCaV−2k−InS−DA−uDA,where(28)KCa1nS×nS=3kCac003kCaodiag1,…,k⋱3kCac003kCaodiag1,…,k−j⋱3kCac,(29)KCa2nS×nS=2kCac002kCaodiag1,…,k⋱2kCac002kCaodiag1,…,k−j⋱2kCac,(30)DAnS×nS=Ac00Aodiag1,…,k⋱Ac00Aodiag1,…,k−j⋱Ac.Then, the expected exocytosis rate for the single granule coupled withk CaVs, λYk, can be estimated by using Equation (20), assuming initially the granule in state G0 and the k CaVs closed, i.e., the complex is initially in state CkG0 π=1,01×3nS−1, which yields(31)λYk=1π−DG:kCaV−11,where 1∈R3nS×1.For the particular case with non-inactivating CaVs channels,MkCaV=M0 by Equations (24) and (25) with δ=γ=0, and then, DG:kCaV, given by Equation (27), belongs to R3k+1×3k+1.In order to compare the rate for a granule coupled with different numberk of CaVs, we define the relative rate, ρλk, as(32)ρλk=λYkλYn,with k=1,…,n. Moreover, in order to compare the rate at different distances from the granule to CaVs, we define the relative distance rate, ρλd, as(33)ρλd=λYdλYdmin,where λYd is the rate computed at a given distance rG and λYdmin, the rate computed at rG=10 nm.


## 2.1. CaV Channel Model

We model the Ca2+ channel by using the 3-state Markov chain of Figure 1(a), where C corresponds to the closed state, O to the open state, and B to the inactivated (blocked) state of the calcium channel [11]. Then, the CaV model takes values in the state space S=C,O,B and its transition rate or generator matrix MCaV is given by(1)MCaV=−αα0β−β−δδ0γ−γ,where α and β represent the voltage-dependent Ca2+ channel opening rate and closing rate, respectively, and have the following forms:(2)αV=α0e−α1V,βV=β0e−β1V.Figure 1
Markov chain models for Ca2+ channel (CaV), exocytosis of single granule and granule-CaV complex. (a) Markov chain model for CaV, where C is the closed state, O the open state, and B the inactivated or blocked state. (b) Markov chain model for exocytosis of a single granule adjacent to the plasma membrane, where G0 correspond to the state with no bound Ca2+ ions, G1 with one, G2 with two, and G3 with three. (c) Markov chain model for the approximated exocytosis model where the dynamics of states G2 and G3 are described by the auxiliary variable G23 using quasi-steady state approximation for the corresponding ODE model. (d) Markov chain model for the granule-CaV complex where the granule dynamics are described by the model shown in panel (c) and the CaV dynamics by the model shown in panel (a).
(a)
(b)
(c)
(d)The rate for channel inactivation,δ, is Ca2+-dependent and has the following form:(3)δ=δ0×CaCaV,where CaCaV is the Ca2+ concentration at the Ca2+ sensor for inactivation and is given using reaction-diffusion theory [8, 12, 13] by(4)CaCaV=iCamax8πrCaDCaFexp−rCaDCa/kB+Btotal,where iCamax=g¯CaV−VCa is the single-channel Ca2+ current with g¯Ca the single-channel conductance and VCa the reverse potential, and rCa represents the distance of the sensor for Ca2+-dependent inactivation from the channel pore. Finally, γ is the constant reverse reactivation rate. Table 1 reports the parameter values for the CaV model defined by above equations.Table 1
Model parameters.
Parameter
Value
Unit
CaV model parameters
α
0
0.6
ms−1
α
1
−
0.1
mV−1
β
0
0.2
ms−1
β
1
0.0375
mV−1
γ
0.002
ms−1
δ
0
0.0025
μM−1·ms−1
Parameters for calculating Ca2+ concentration at different distances
r
Ca
7
nm
r
G
10, 20, 30, 50
nm
D
Ca
250
μm2·s−1
F
9.6485
C mol−1
k
B
500
μM−1·s−1
B
total
30
μM
V
Ca
60
mV
g
¯
Ca
2.8
pS
Ca
c
0.1
μM
Ca
b
0.1
μM
Exocytosis model parameters
k
+
1.85
μM−1·s−1
k
−
50
s−1
u
1000
s−1The deterministic description of the 3-state Markov chain model for the CaV channel is given by the following ODE system:(5)dcdt=βo−αc,dodt=αc+γb−β+δo,b=1−c−o=1−h,where the italic lowercase letters represent the corresponding state variables of the ODE model (h represents the fraction of Ca2+ channels not inactivated).Finally, in order to investigate the relationship between exocytosis and Ca2+ loading, we compute the total charge entering via the Ca2+ channel at a given step voltage with time window, ts, as(6)QCa=∫0tsoτ⋅iCamaxdτ.


## 2.2. Exocytosis Model

We assume a single granule, adjacent to the plasma membrane and primed for exocytosis, that can be in one of four different states depending on the number of Ca2+ ions bound to the Ca2+ sensor on the granule, likely synaptotagmin [14]: in G0 with no bound Ca2+ ions, or in G1 with one, or in G2 with two, or in G3 with three bound ions. Once it is in G3, the granule can fuse with the membrane and release its hormone content, assuming the final state Y [6, 15]. Therefore, we use a five-state Markov chain model for describing exocytosis as shown in Figure 1(b), where the model takes values in the state space S=G0,G1,G2,G3,Y, and its transition rate or generator matrix MG is given by(7)MG=−3kCa3kCa000k−−2kCa−k−2kCa0002k−−kCa−2k−kCa0003k−−u−3k−u00000,where(8)kCa=k+×CaGrepresents the Ca2+ binding rate, with CaG the Ca2+ concentration at the granule sensor given by Equation (4) with r=rG being the distance from the CaV to the Ca2+ sensor on the granule. In the following, the distance from the CaV to the granule means the distance from the CaV to the Ca2+ sensor on the granule, which will be of the order of tens of nm. For comparison, secretory granules have diameters on the order 100–500 nm [16–19]. We assume a constant number of Ca2+ sensor molecules, which is therefore included in the binding parameter kCa. The parameter k− is the unbinding rate, and u is the fusion rate. Table 1 reports the parameter values.The deterministic description of the 5-state Markov chain model for exocytosis is given by the following ODE system:(9)dg0dt=−3kCag0+k−g1,(10)dg1dt=−2kCa+k−g1+3kCag0+2k−g2,(11)dg2dt=−kCa+2k−g2+2kCag1+3k−g3,(12)dg3dt=−u+3k−g3+kCag2,(13)y=1−g0−g1−g2−g3.For the above ODE model of Equations (9)–(13), we exploit quasi steady-state approximation for state g3, since its dynamics are fastest (the value of u is much higher than those of the other parameters). Then, by renaming the state variables as(14)g23=g2+g3,by setting Equation (12) equal to zero yielding(15)g3=Ag23,withA=kCakCa+3k−+u,and by summing Equations (11) and (12), we achieve a single ODE model for describing the dynamics of state variable g2 and g3 as follows:(16)dg23dt=−2k−1−A+uAg23+2kCag1.The corresponding Markov chain model takes values in the state spaceS=G0,G1,G23,Y (Figure 1(c)) and is described by the following generating matrix, MGap:(17)MGap=−3kCa3kCa00k−−2kCa−k−2kCa002k−1−A−2k−1−A−uAuA0000.Note that stateY of the Markov chain described by MGap is an absorbing state: the process can never leave Y after entering it, reflecting that fusion is an irreversible process. Then MGap can be rewritten as(18)MGap=D3×3d3×101×30,where(19)D3×3=−3kCa3kCa0k−−2kCa−k−2kCa02k−1−A−2k−1−A−uA,describes only the transitions between the transient states G0, G1, and G23 and d=0,0,uAT is a vector containing the transition intensities from the transient states to the absorbing state Y. The row vector 0∈R1×3 consists entirely of 0’s since no transitions from Y to the transient states can occur. The remaining element of the matrix MGap is 0 and gives the transition rate out of the absorbing state.Using phase-type distribution results for Markov chains [10], we obtain an explicit formula for calculating the expected event rate λY to reach the absorbing state Y, given the initial probability row vector π for the transient states (π=πG0,πG1,πG23), as(20)λY=1π−D−11,where 1∈R3×1.


## 2.3. Granule-CaV Complex Model with 1 : 1 and 1 :n Stoichiometries

### 2.3.1. 1 : 1 Stoichiometry

By coupling the CaV and exocytosis models, we obtain the 12-state Markov chain model of Figure1(d). The model takes values in the state space(21)S=CG0,OG0,BG0,CG1,OG1,BG1,CG23,OG23,BG23,CY,OY,BY,and its transition matrix, DG:CaV, is as follows:(22)DG:CaV9×9=MCaV−3diagkCac,kCao,kCac3diagkCac,kCao,kCac03×3diagk−,k−,k−MCaV−diag2kCac+k−,2kCao+k−,2kCac+k−2diagkCac,kCao,kCac03×32k−diag1−Ac,1−Ao,1−AcMCaV−diag2k−1−Ac+uAc,2k−1−Ao+uAo,2k−1−Ac+uAc,where MCaV is defined by Equation (1), kCacAc by Equation (8) (Equation (15)) with CaG=Cac, i.e., the concentration at the granule when the associated CaV is closed (or inactivated, i.e., Cac=Cab), and kCaoAo by Equation (8) (Equation (15)) with CaG=Cao, i.e., the concentration at the granule when the associated CaV is open, computed by Equation (4). Then, the expected exocytosis rate for the single granule, λY1, can be estimated by using Equation (20), assuming initially the granule in state G0 and the CaV closed, i.e., the complex in the state CG0π=1,01×8, as(23)λY1=1π−DG:CaV−11,where 1∈R9×1.We also consider the particular case with non-inactivating CaV (i.e., the Ca2+ channel can be only in C or in O). In this case, MCaV∈R2×2 and is defined by Equation (1) with δ=γ=0, and then DG:CaV, given by Equation (22), belongs to R6×6.


### 2.3.2. 1 :n Stoichiometry

In the following, we assume the case where the granule is coupled with more than one CaV. In particular, by consideringk Ca2+ channels, we have a Markov chain model with nS=∑i=0kk+1−i=k2/2+3k/2+1 possible states describing the k CaVs. In particular, the CaVs model takes values in the state space S=Ck−i−jOiBj with j∈0,…,k and i∈0,…,k−j, and its generating matrix, MkCaV, is given by(24)MkCaVnS×nS=M0k+1×k+101×kδdiag1,…,k0⋯⋯⋯⋯⋯00k×1γIk⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯00k+1−j×1jγIk+1−jMjk+1−j×k+1−j01×k−jδdiag1,…,k−j0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱δ⋮⋯⋯⋯⋯⋯0kγMk1×1,where(25)M0k+1×k+1=−kαkα0⋯⋯⋯⋯⋯0β⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯0i−1β−k−i−1α−i−1β+δk−i−1α0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱α0⋯⋯⋯⋯⋯0kβ−kβ+δ,(26)Mjk+1−j×k+1−j=−k−jα−jγk−jα0⋯⋯⋯⋯⋯0β⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯0i−1β−k−i−1−jα−i−1β+δ−jγk−i−1−jα0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱α0⋯⋯⋯⋯⋯0k−jβ−k−jβ+δ−jγ,and Mk1×1=−kγ.Then, by coupling the CaVs and exocytosis models, we obtain a4nS-state Markov chain model. The model takes values in the state space S=Ck−i−jOiBjGl,…,Ck−i−jOiBjG23,Ck−i−jOiBjY, with j∈0,…,k, i∈0,…,k−j and l∈0,1, and its transition matrix, DG:kCaV, can be written as(27)DG:kCaV3nS×3nS=MkCaV−KCa1KCa10nS×nSk−InSMkCaV−KCa2−k−InSKCa20nS×nS2k−InS−DAMkCaV−2k−InS−DA−uDA,where(28)KCa1nS×nS=3kCac003kCaodiag1,…,k⋱3kCac003kCaodiag1,…,k−j⋱3kCac,(29)KCa2nS×nS=2kCac002kCaodiag1,…,k⋱2kCac002kCaodiag1,…,k−j⋱2kCac,(30)DAnS×nS=Ac00Aodiag1,…,k⋱Ac00Aodiag1,…,k−j⋱Ac.Then, the expected exocytosis rate for the single granule coupled withk CaVs, λYk, can be estimated by using Equation (20), assuming initially the granule in state G0 and the k CaVs closed, i.e., the complex is initially in state CkG0 π=1,01×3nS−1, which yields(31)λYk=1π−DG:kCaV−11,where 1∈R3nS×1.For the particular case with non-inactivating CaVs channels,MkCaV=M0 by Equations (24) and (25) with δ=γ=0, and then, DG:kCaV, given by Equation (27), belongs to R3k+1×3k+1.In order to compare the rate for a granule coupled with different numberk of CaVs, we define the relative rate, ρλk, as(32)ρλk=λYkλYn,with k=1,…,n. Moreover, in order to compare the rate at different distances from the granule to CaVs, we define the relative distance rate, ρλd, as(33)ρλd=λYdλYdmin,where λYd is the rate computed at a given distance rG and λYdmin, the rate computed at rG=10 nm.


## 2.3.1. 1 : 1 Stoichiometry

By coupling the CaV and exocytosis models, we obtain the 12-state Markov chain model of Figure1(d). The model takes values in the state space(21)S=CG0,OG0,BG0,CG1,OG1,BG1,CG23,OG23,BG23,CY,OY,BY,and its transition matrix, DG:CaV, is as follows:(22)DG:CaV9×9=MCaV−3diagkCac,kCao,kCac3diagkCac,kCao,kCac03×3diagk−,k−,k−MCaV−diag2kCac+k−,2kCao+k−,2kCac+k−2diagkCac,kCao,kCac03×32k−diag1−Ac,1−Ao,1−AcMCaV−diag2k−1−Ac+uAc,2k−1−Ao+uAo,2k−1−Ac+uAc,where MCaV is defined by Equation (1), kCacAc by Equation (8) (Equation (15)) with CaG=Cac, i.e., the concentration at the granule when the associated CaV is closed (or inactivated, i.e., Cac=Cab), and kCaoAo by Equation (8) (Equation (15)) with CaG=Cao, i.e., the concentration at the granule when the associated CaV is open, computed by Equation (4). Then, the expected exocytosis rate for the single granule, λY1, can be estimated by using Equation (20), assuming initially the granule in state G0 and the CaV closed, i.e., the complex in the state CG0π=1,01×8, as(23)λY1=1π−DG:CaV−11,where 1∈R9×1.We also consider the particular case with non-inactivating CaV (i.e., the Ca2+ channel can be only in C or in O). In this case, MCaV∈R2×2 and is defined by Equation (1) with δ=γ=0, and then DG:CaV, given by Equation (22), belongs to R6×6.


## 2.3.2. 1 :n Stoichiometry

In the following, we assume the case where the granule is coupled with more than one CaV. In particular, by consideringk Ca2+ channels, we have a Markov chain model with nS=∑i=0kk+1−i=k2/2+3k/2+1 possible states describing the k CaVs. In particular, the CaVs model takes values in the state space S=Ck−i−jOiBj with j∈0,…,k and i∈0,…,k−j, and its generating matrix, MkCaV, is given by(24)MkCaVnS×nS=M0k+1×k+101×kδdiag1,…,k0⋯⋯⋯⋯⋯00k×1γIk⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯00k+1−j×1jγIk+1−jMjk+1−j×k+1−j01×k−jδdiag1,…,k−j0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱δ⋮⋯⋯⋯⋯⋯0kγMk1×1,where(25)M0k+1×k+1=−kαkα0⋯⋯⋯⋯⋯0β⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯0i−1β−k−i−1α−i−1β+δk−i−1α0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱α0⋯⋯⋯⋯⋯0kβ−kβ+δ,(26)Mjk+1−j×k+1−j=−k−jα−jγk−jα0⋯⋯⋯⋯⋯0β⋱⋱0⋯⋯⋯⋯⋮0⋱⋱⋱0⋯⋯⋯⋮⋮⋯0i−1β−k−i−1−jα−i−1β+δ−jγk−i−1−jα0⋯⋮⋮⋯⋯⋯0⋱⋱⋱0⋮⋯⋯⋯⋯0⋱⋱α0⋯⋯⋯⋯⋯0k−jβ−k−jβ+δ−jγ,and Mk1×1=−kγ.Then, by coupling the CaVs and exocytosis models, we obtain a4nS-state Markov chain model. The model takes values in the state space S=Ck−i−jOiBjGl,…,Ck−i−jOiBjG23,Ck−i−jOiBjY, with j∈0,…,k, i∈0,…,k−j and l∈0,1, and its transition matrix, DG:kCaV, can be written as(27)DG:kCaV3nS×3nS=MkCaV−KCa1KCa10nS×nSk−InSMkCaV−KCa2−k−InSKCa20nS×nS2k−InS−DAMkCaV−2k−InS−DA−uDA,where(28)KCa1nS×nS=3kCac003kCaodiag1,…,k⋱3kCac003kCaodiag1,…,k−j⋱3kCac,(29)KCa2nS×nS=2kCac002kCaodiag1,…,k⋱2kCac002kCaodiag1,…,k−j⋱2kCac,(30)DAnS×nS=Ac00Aodiag1,…,k⋱Ac00Aodiag1,…,k−j⋱Ac.Then, the expected exocytosis rate for the single granule coupled withk CaVs, λYk, can be estimated by using Equation (20), assuming initially the granule in state G0 and the k CaVs closed, i.e., the complex is initially in state CkG0 π=1,01×3nS−1, which yields(31)λYk=1π−DG:kCaV−11,where 1∈R3nS×1.For the particular case with non-inactivating CaVs channels,MkCaV=M0 by Equations (24) and (25) with δ=γ=0, and then, DG:kCaV, given by Equation (27), belongs to R3k+1×3k+1.In order to compare the rate for a granule coupled with different numberk of CaVs, we define the relative rate, ρλk, as(32)ρλk=λYkλYn,with k=1,…,n. Moreover, in order to compare the rate at different distances from the granule to CaVs, we define the relative distance rate, ρλd, as(33)ρλd=λYdλYdmin,where λYd is the rate computed at a given distance rG and λYdmin, the rate computed at rG=10 nm.


## 3. Results and Discussion

We analyze the behavior of the devised exocytosis model where the single granule is coupled withk Ca2+ channels by using phase-type distribution results for Markov chains [10] (see Methods). First, we assume that a granule is coupled with one CaV and, then, we extend the results to a more general case with k CaVs. Moreover, we consider for both the cases (1 or k CaVs) that the granule forms complexes with inactivating or non-inactivating CaVs. This scenario reflects, e.g., what it is observed in pancreatic beta-cells where the two main high voltage-activated Ca2+ channels, the L- and P/Q-type Ca2+ channels, are examples of inactivating and non-inactivating CaVs, respectively [20].


### 3.1. Granule Coupled with One Inactivating (or Non-Inactivating) CaV

Figure2(a) shows the expected exocytosis rate, λY1, computed by Equation (23), for a granule at different distances from an inactivating CaV channel. Independently of the distance to the CaV, the exocytosis rate has a bell-shaped relation to voltage, as seen experimentally [20–22]. The same holds true in the case of non-inactivating CaV (Figure 2(b)). As the distance between the granule and the Ca2+ channel increases, the expected rate decreases substantially and nonlinearly (for instance, in Figure 2(a), compare the red and blue lines for rG=20 nm and rG=10 nm, respectively). This is clearer from Figure 2(c), showing the relative distance rate ρλd defined by Equation (33) for different values of rG. Note that increasing the distance by a factor of two corresponds to a more than five-fold reduction of the exocytosis rate (the relative ratio is less than 0.2, see the red plot in Figure 2(c)). This steep dependence of the distance to the channel is because the calcium levels drop rapidly, moving away from the channel [8, 23].Figure 2
Expected exocytosis rate for single granule coupled with one (inactivating or non-inactivating) CaV. (a, b) Expected exocytosis rateλY1 for the granule at different distances rG from one inactivating (a) or non-inactivating (b) CaV: rG=10 nm (blue curves), rG=20 nm (red), rG=30 nm (green), and rG=50 nm (magenta). Note the different scales on the y-axes. The insert in (b) is a zoom on the lower, left part of the figure for comparison with (a). (c, d) Relative rate ρλd computed at different distances (rG=20 nm (red), rG=30 nm (green), and rG=50 nm (magenta)) of the granule from the inactivating (c) or not-inactivating (d) CaV and compared to the case with rG=10 nm.
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(d)We perform a similar analysis for the case where a granule is coupled with a non-inactivating CaV (Figure2(b)). We note an increase about of two orders of magnitudes for the exocytosis rate compared to the case with a granule coupled with an inactivating CaV (Figures 2(a) and 2(b)): the exocytosis proceeds more rapidly since the triggering Ca2+ signal is increased due to non-inactivation of Ca2+ currents. Also in this case, the degree of decrease for the rate is much higher than the relative increase for the distance (Figure 2(d)). However, the benefit in terms of ρλd by reducing the distance is slightly less than that obtained with inactivating CaV (compare Figures 2(c) and 2(d)): for the case with inactivating CaV, it seems that moving away from the channel, ρλd decreases more due to the inactivation of CaV that determines a further drop of calcium levels.


### 3.2. Granule Coupled withk Inactivating (or Non-Inactivating) CaVs

Figures3(a)–3(d) show the expected exocytosis rate λYk computed by Equation (31), for a granule coupled with different numbers of inactivating CaVs and at fixed distances between the granule and the CaVs. It is clear that increasing the number of CaVs coupled with the granule determines a rise of the exocytosis rate. Moreover, as the number of CaVs coupled with the granule increases, the rise in the rate is more pronounced when the distance of the granule from the CaVs is small. This is evident by considering the relative rate ρλk defined by Equation (32) (Figure 3(e)). For instance, consider the cyan curves computed for k=4 with different types of lines denoting the different distances of the granule from the CaVs. In this case, the number of CaVs decreases by a factor of 2 (from 8 to 4) while the exocytosis rate drops more than threefold for rG=20 nm (dashed cyan line, ρλk<0.3, for V>−10 mV) and more than fivefold for rG=10 nm (solid cyan line, ρλk<0.2, for V>−10 mV).Figure 3
Expected exocytosis rate for single granule coupled withk (inactivating or non-inactivating) CaVs. (a–d) and (f–i) Expected exocytosis rate λYk for the granule at fixed distance rG from k (k=1 (blue curves), k=2 (magenta), k=4 (cyan), and k=8 (black)) inactivating/not-inactivating CaVs: rG=10 nm (a/f), 20 nm (b/g), 30 nm (c/h), and 50 nm (d/i). The insert in (f) is a zoom on the lower, left part of the figure for comparison with (a). (e) and (j) Relative rate ρλk obtained from the granule coupled with k inactivating/not-inactivating CaVs (e/j), for k=1 (blue), k=2 (magenta), and k=4 (cyan) and compared to the case with n=8 CaVs. The different type of line corresponds to ρλk computed at fixed distance: rG=10 nm (solid line), rG=20 nm (dashed), rG=30 nm (dash-dotted), and rG=50 nm (dotted).
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(j)As done for the case with one CaV, we performed the same analysis withk non-inactivating CaVs coupled with the granule (Figures 3(f)–3(i)). Also in this case, it is clear that increasing the number of CaVs determines a rise of the exocytosis rate for the granule. Surprisingly and in contrast with the case with inactivating CaVs, as the number of non-inactivating CaVs increases, the relative rise in exocytosis rate is much higher at larger distances from the CaVs, as shown in Figure 3(j) reporting the relative rate ρλk. In case the number of CaVs is reduced from 8 to 4, the exocytosis rate decreases by 2–2.5-fold when the granule is near the CaVs (see the solid cyan curve for rG=10 nm, 0.4<ρλk<0.5 with −20  mV <V<40 mV), while it goes down fivefold when the granule is far from CaVs (see the dotted cyan curve for rG=50 nm, 0.2<ρλk<0.3 with −20 mV <V<40 mV). It seems that when the granule is surrounded by more non-inactivating CaVs, it is not necessary that the granule is very close to the CaVs for triggering exocytosis.


### 3.3. Relationship between Ca2+ Influx and Exocytosis

To investigate the relationship between exocytosis and Ca2+ loading, we consider a set of scenarios where the granule is coupled with different number of non-inactivating or inactivating CaVs, placed very close (10 nm) or far (100 nm) from the granule. Figure 4(a) shows the calcium current at V=0 mV, for different numbers of non-inactivating CaVs, while Figure 4(b) shows the corresponding cases with inactivating CaVs. In the latter, it is evident how the calcium influx drops after few tens of ms due to the inactivation of the CaVs. Figures 4(c) and 4(d) show the probability of exocytosis pY (pY=PSt=Y) vs. the integral of the Ca2+ current, QCa, defined by Equation (6), for the granule placed close to the CaV cluster, for different numbers of CaVs (rG=10 nm). For the case of non-inactivating CaVs (Figure 4(c)), pY raises linearly with QCa, with slope that increases with the number of CaVs and then saturates due to the depletion of the granule pool as pY approaches 1 (see also [24]). For inactivating CaVs, we note a change of the slope of the linearity between pY and QCa that is not only due to depletion (when y≥0.5) but also to near-complete inactivation of CaVs, in particular after 50 ms (Figure 4(d)). Figures 4(e) and 4(f) show pY vs. QCa when the granule is placed far from CaVs (rG=100 nm). Due to the distance to CaVs, the calcium concentration at the granule increases only modestly; hence, a greater calcium influx QCa is needed to allow the granule to move through the Markov chain from N0 to Y and undergoes exocytosis. This causes an evident initial delay for the granule to be released, resulting in an initial convex relation between pY and QCa. After this initial phase, for the case of non-inactivating CaVs (Figure 4(e)), pY raises linearly with QCa with slope depending on the number of CaVs. For higher QCa, the slope of pY slightly decreases in the case with k=8 CaVs reflecting slight depletion of the granule pool (pY≈0.5 at QCa=500 fC). For inactivating CaVs (Figure 4(f)), as for the case with rG=10 nm, we note a change of the linearity between pY and QCa that is due to CaV inactivation.Figure 4
Relationship between Ca2+ influx and exocytosis rate. (a, b) Ca2+ current for the different number of non-inactivating (a) and inactivating (b) CaVs. (c, d) Probability of exocytosis, pY, vs. integral of Ca2+ currents, QCa, computed at V=0 mV by increasing the integration time, ts, from 1 to 500 ms, for the granule at distance rG=10 nm from k not-inactivating (c) or inactivating (d) CaVs. (e, f) Legends as in (c, d) with rG=100 nm. In each panel, the different colors represent different number of CaVs: blue curve for k=1; magenta for k=2; cyan for k=4; and black for k=8. The inserts in (c–f) show a zoom-in of pY vs. QCa on lower QCa values for the granule coupled with different number k of CaVs: k=4 and 8 in (c) and k=1, 2, 4, and 8 in (d–f).
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## 3.1. Granule Coupled with One Inactivating (or Non-Inactivating) CaV

Figure2(a) shows the expected exocytosis rate, λY1, computed by Equation (23), for a granule at different distances from an inactivating CaV channel. Independently of the distance to the CaV, the exocytosis rate has a bell-shaped relation to voltage, as seen experimentally [20–22]. The same holds true in the case of non-inactivating CaV (Figure 2(b)). As the distance between the granule and the Ca2+ channel increases, the expected rate decreases substantially and nonlinearly (for instance, in Figure 2(a), compare the red and blue lines for rG=20 nm and rG=10 nm, respectively). This is clearer from Figure 2(c), showing the relative distance rate ρλd defined by Equation (33) for different values of rG. Note that increasing the distance by a factor of two corresponds to a more than five-fold reduction of the exocytosis rate (the relative ratio is less than 0.2, see the red plot in Figure 2(c)). This steep dependence of the distance to the channel is because the calcium levels drop rapidly, moving away from the channel [8, 23].Figure 2
Expected exocytosis rate for single granule coupled with one (inactivating or non-inactivating) CaV. (a, b) Expected exocytosis rateλY1 for the granule at different distances rG from one inactivating (a) or non-inactivating (b) CaV: rG=10 nm (blue curves), rG=20 nm (red), rG=30 nm (green), and rG=50 nm (magenta). Note the different scales on the y-axes. The insert in (b) is a zoom on the lower, left part of the figure for comparison with (a). (c, d) Relative rate ρλd computed at different distances (rG=20 nm (red), rG=30 nm (green), and rG=50 nm (magenta)) of the granule from the inactivating (c) or not-inactivating (d) CaV and compared to the case with rG=10 nm.
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(d)We perform a similar analysis for the case where a granule is coupled with a non-inactivating CaV (Figure2(b)). We note an increase about of two orders of magnitudes for the exocytosis rate compared to the case with a granule coupled with an inactivating CaV (Figures 2(a) and 2(b)): the exocytosis proceeds more rapidly since the triggering Ca2+ signal is increased due to non-inactivation of Ca2+ currents. Also in this case, the degree of decrease for the rate is much higher than the relative increase for the distance (Figure 2(d)). However, the benefit in terms of ρλd by reducing the distance is slightly less than that obtained with inactivating CaV (compare Figures 2(c) and 2(d)): for the case with inactivating CaV, it seems that moving away from the channel, ρλd decreases more due to the inactivation of CaV that determines a further drop of calcium levels.


## 3.2. Granule Coupled withk Inactivating (or Non-Inactivating) CaVs

Figures3(a)–3(d) show the expected exocytosis rate λYk computed by Equation (31), for a granule coupled with different numbers of inactivating CaVs and at fixed distances between the granule and the CaVs. It is clear that increasing the number of CaVs coupled with the granule determines a rise of the exocytosis rate. Moreover, as the number of CaVs coupled with the granule increases, the rise in the rate is more pronounced when the distance of the granule from the CaVs is small. This is evident by considering the relative rate ρλk defined by Equation (32) (Figure 3(e)). For instance, consider the cyan curves computed for k=4 with different types of lines denoting the different distances of the granule from the CaVs. In this case, the number of CaVs decreases by a factor of 2 (from 8 to 4) while the exocytosis rate drops more than threefold for rG=20 nm (dashed cyan line, ρλk<0.3, for V>−10 mV) and more than fivefold for rG=10 nm (solid cyan line, ρλk<0.2, for V>−10 mV).Figure 3
Expected exocytosis rate for single granule coupled withk (inactivating or non-inactivating) CaVs. (a–d) and (f–i) Expected exocytosis rate λYk for the granule at fixed distance rG from k (k=1 (blue curves), k=2 (magenta), k=4 (cyan), and k=8 (black)) inactivating/not-inactivating CaVs: rG=10 nm (a/f), 20 nm (b/g), 30 nm (c/h), and 50 nm (d/i). The insert in (f) is a zoom on the lower, left part of the figure for comparison with (a). (e) and (j) Relative rate ρλk obtained from the granule coupled with k inactivating/not-inactivating CaVs (e/j), for k=1 (blue), k=2 (magenta), and k=4 (cyan) and compared to the case with n=8 CaVs. The different type of line corresponds to ρλk computed at fixed distance: rG=10 nm (solid line), rG=20 nm (dashed), rG=30 nm (dash-dotted), and rG=50 nm (dotted).
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(j)As done for the case with one CaV, we performed the same analysis withk non-inactivating CaVs coupled with the granule (Figures 3(f)–3(i)). Also in this case, it is clear that increasing the number of CaVs determines a rise of the exocytosis rate for the granule. Surprisingly and in contrast with the case with inactivating CaVs, as the number of non-inactivating CaVs increases, the relative rise in exocytosis rate is much higher at larger distances from the CaVs, as shown in Figure 3(j) reporting the relative rate ρλk. In case the number of CaVs is reduced from 8 to 4, the exocytosis rate decreases by 2–2.5-fold when the granule is near the CaVs (see the solid cyan curve for rG=10 nm, 0.4<ρλk<0.5 with −20  mV <V<40 mV), while it goes down fivefold when the granule is far from CaVs (see the dotted cyan curve for rG=50 nm, 0.2<ρλk<0.3 with −20 mV <V<40 mV). It seems that when the granule is surrounded by more non-inactivating CaVs, it is not necessary that the granule is very close to the CaVs for triggering exocytosis.


## 3.3. Relationship between Ca2+ Influx and Exocytosis

To investigate the relationship between exocytosis and Ca2+ loading, we consider a set of scenarios where the granule is coupled with different number of non-inactivating or inactivating CaVs, placed very close (10 nm) or far (100 nm) from the granule. Figure 4(a) shows the calcium current at V=0 mV, for different numbers of non-inactivating CaVs, while Figure 4(b) shows the corresponding cases with inactivating CaVs. In the latter, it is evident how the calcium influx drops after few tens of ms due to the inactivation of the CaVs. Figures 4(c) and 4(d) show the probability of exocytosis pY (pY=PSt=Y) vs. the integral of the Ca2+ current, QCa, defined by Equation (6), for the granule placed close to the CaV cluster, for different numbers of CaVs (rG=10 nm). For the case of non-inactivating CaVs (Figure 4(c)), pY raises linearly with QCa, with slope that increases with the number of CaVs and then saturates due to the depletion of the granule pool as pY approaches 1 (see also [24]). For inactivating CaVs, we note a change of the slope of the linearity between pY and QCa that is not only due to depletion (when y≥0.5) but also to near-complete inactivation of CaVs, in particular after 50 ms (Figure 4(d)). Figures 4(e) and 4(f) show pY vs. QCa when the granule is placed far from CaVs (rG=100 nm). Due to the distance to CaVs, the calcium concentration at the granule increases only modestly; hence, a greater calcium influx QCa is needed to allow the granule to move through the Markov chain from N0 to Y and undergoes exocytosis. This causes an evident initial delay for the granule to be released, resulting in an initial convex relation between pY and QCa. After this initial phase, for the case of non-inactivating CaVs (Figure 4(e)), pY raises linearly with QCa with slope depending on the number of CaVs. For higher QCa, the slope of pY slightly decreases in the case with k=8 CaVs reflecting slight depletion of the granule pool (pY≈0.5 at QCa=500 fC). For inactivating CaVs (Figure 4(f)), as for the case with rG=10 nm, we note a change of the linearity between pY and QCa that is due to CaV inactivation.Figure 4
Relationship between Ca2+ influx and exocytosis rate. (a, b) Ca2+ current for the different number of non-inactivating (a) and inactivating (b) CaVs. (c, d) Probability of exocytosis, pY, vs. integral of Ca2+ currents, QCa, computed at V=0 mV by increasing the integration time, ts, from 1 to 500 ms, for the granule at distance rG=10 nm from k not-inactivating (c) or inactivating (d) CaVs. (e, f) Legends as in (c, d) with rG=100 nm. In each panel, the different colors represent different number of CaVs: blue curve for k=1; magenta for k=2; cyan for k=4; and black for k=8. The inserts in (c–f) show a zoom-in of pY vs. QCa on lower QCa values for the granule coupled with different number k of CaVs: k=4 and 8 in (c) and k=1, 2, 4, and 8 in (d–f).
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## 4. Conclusions

In this paper, we devise a strategy that allows us to characterize the local interactions between granules and CaVs. The methodology is similar to our approach for modelling the local effect of CaVs on whole-cell BK currents [9]. We develop Markov chain models describing the dynamics of a single granule coupled with one or more inactivating (or non-inactivating) Ca2+ channels and use phase-type distribution results [10] for estimating the expected exocytosis rate.We investigate how the release probability of a granule can be affected by varying the number of CaVs and the distance of the (Ca2+ sensor of the) granule from CaVs. In particular, from our analysis, we find that the distance between the granule and CaVs is a major factor in determining the exocytosis rate, as we recently demonstrated and quantified explicitly [23]. Further and in agreement with experiments [23], the simulations presented here show that the increase of the number of CaVs coupled with the granule determines a much higher rise of the exocytosis rate, which in the case of inactivating CaVs is more pronounced when the granule is close to CaVs (≈10 nm), whereas for non-inactivating CaVs the highest relative increase in rate is obtained when the CaVs are far from CaVs (≈50  nm).We also study the relationship between Ca2+ influx and exocytosis. The results of the devised exocytosis model confirm that the granule secretion is generally linearly related to the integral of Ca2+current, as experimentally observed [25–29] and theoretically justified [24]. Surprisingly, for the case of inactivating CaVs, our analysis shows a change of the linear relation between pY and QCa due to near-complete inactivation of CaVs. This fact is due to the rather complex exocytosis model where the efficacy of Ca2+ influx in triggering exocytosis depends on the number of active CaVs, as clearly seen in the case of non-inactivating CaVs (Figures 4(c) and 4(e)), because of multiple steps of Ca2+ bindings before exocytosis. During inactivation, the effective number of CaVs declines, which has a similar effect as reducing the number of CaVs, and hence the slope of the relation between exocytosis and QCa decreases. This finding reinforces the notion that a concave relation between exocytosis and Ca2+ influx does not necessarily reflect pool depletion [24] and provides a new example of such a scenario.



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*Source: 5721097-2018-11-14.xml*