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Copyright © 2008 The Biophysical Society. All rights reserved.
Biophysical Journal, Volume 95, Issue 5, 2160-2171, 1 September 2008

doi:10.1529/biophysj.107.124909

Biophysical Theory and Modeling

Modeling Study of the Effects of Membrane Surface Charge on Calcium Microdomains and Neurotransmitter Release

Luigi Catacuzzeno^,  , Bernard Fioretti and Fabio Franciolini

Dipartimento di Biologia Cellulare e Ambientale, Università di Perugia, Perugia, Italy

Address reprint requests to Luigi Catacuzzeno, Dipartimento di Biologia Cellulare e Ambientale, Università di Perugia, via Pascoli 1, I-06123 Perugia, Italy.

The entry of Ca²⁺ ions through voltage-gated Ca²⁺ channels is necessary for triggering vesicle fusion and neurotransmitter release 1,2. Electron microscopy studies of vertebrate synapses provide strong evidence for strict colocalization between synaptic vesicles and transmembrane particles, thought to be Ca²⁺ channels, with an estimated channel-vesicle distance of ∼10–30nm 3. Several studies point to a direct physical and functional interaction between voltage-gated Ca²⁺ channels and several vesicle-associated proteins implicated in neurotransmitter release 3,4,5. Because of the limited rate of diffusion of Ca²⁺ ions within the cytoplasm and small Ca²⁺ channel-vesicle distances, the Ca²⁺ concentration ([Ca]) at the sensor protein is thought to attain levels much higher than those present in the bulk. Due to the small spatial scale of these Ca²⁺ signals (Ca²⁺ microdomains), it has been impossible thus far to investigate their properties experimentally by direct Ca²⁺ imaging. Therefore, information regarding their shape and dynamics has remained elusive.

Theoretical modeling is thus the only viable approach for making inferences about the detailed properties of the Ca²⁺ microdomains and their linkage to neurotransmitter release. Computer simulations of Ca²⁺ microdomains, based essentially on Fick's law of diffusion, have demonstrated that at a few tens of nanometers from an open Ca²⁺ channel, where the sensor protein is supposedly located, the [Ca] reaches levels >100μM 6,7,8. In addition, the predicted [Ca] profiles show steep spatial gradients, such that within several hundred nanometers from the channel the enhanced [Ca] level has decreased to near bulk levels. Because of the steep Ca²⁺ microdomain gradients, these models predict that release of a docked vesicle would be mainly controlled by the colocalized Ca²⁺ channel, even in the presence of other Ca²⁺ channels located farther away from the vesicle 9,10.

Although some functional data, such as those derived from lower vertebrate synapses, essentially support these theoretical predictions 11,12,13, recent results on the mammalian calyx of Held and cortical synapses seem to suggest a different scenario 14,15. First, at the release Ca²⁺ sensor of the rat calyx of Held, a much smaller [Ca] (10–25μM) was found to be sufficient to achieve both the amount and the kinetics of the transmitter release under physiological conditions 1,16. Second, experiments aimed at evaluating the sensitivity of neurotransmitter release to Ca²⁺ channel blockers and fast Ca²⁺ buffers indicate that remote Ca²⁺ channels may also contribute to the fusion of a single vesicle, with some of them possibly being as far as 100nm from the release Ca²⁺ sensor 16,17. These recent observations cannot be reproduced by theoretical models that assume a strong colocalization between Ca²⁺ channels and release vesicles, as the structural evidence suggests 9. Alternatives include consideration of a less strict channel-vesicle colocalization at the Calyx of Held synapse in the rat 9. However, conclusions about Ca²⁺ channel-vesicle topography based on interpretation of available functional data strongly rely on the validity of the employed theoretical models of Ca²⁺ microdomains. These should take into account all the relevant physical processes or parameters necessary to give a correct prediction of the [Ca] profile near an open Ca²⁺ channel.

One potentially relevant factor that has not been incorporated into existing models of Ca²⁺ microdomains and neurotransmitter release is electrostatic force, which could markedly influence the distribution of Ca²⁺ ions within the cell. It is known that the inner leaflet of mammalian plasma membrane carries a significant negative charge due to the negatively charged phospholipid headgroups and amino acids of membrane proteins facing the cytoplasm 18,19. In addition recent studies suggest that phosphoinositides, carrying a substantial negative headgroup charge, concentrate at the presynaptic active zones 20,21, raising the possibility that the negative surface charge density at release sites may well be even higher than in other regions of the plasma membrane. It is also established that these negative charges generate a negative electrostatic potential that persists for several nanometers within the electrolytic solution bathing the membrane. This negative potential will attract cations and repel anions, and will have an impact on the distribution of ions in the immediate vicinity.

In this article, we explore a possible role of membrane surface charges in shaping the Ca²⁺ microdomain around an open Ca²⁺ channel. This has been achieved by using a computational model that calculates the ionic concentration profiles and the electrostatic potential near the membrane, in the presence of Ca²⁺ influx. We have also tested how the changes in the Ca²⁺ microdomain properties induced by membrane surface charges will reflect on the Ca²⁺-dependent neurotransmitter release. Our results indicate that surface charges make a tangible contribution toward shaping Ca²⁺ microdomains, thus changing the predicted neurotransmitter release.

Our model consists of a flat plasma membrane bathed by an electrolytic aqueous solution. The negative charges at the surface of the membrane, mainly due to phospholipid headgroups, are represented as a uniformly smeared negative surface charge density, σ_T, which in our calculations varies up to −0.1 C/m², a value within the range experimentally determined for the inner leaflet of plasma membranes 19. Theoretical Poisson-Boltzmann calculations indicate that the approximation of discrete membrane charges with a uniformly smeared membrane charge has realistic consequences in the assessed profile of the electrostatic potential, at least for physiologically relevant membrane phospholipid compositions 22. The plasma membrane also contains one or more Ca²⁺ channels through which Ca²⁺ ions can enter the electrolytic solution at a rate that depends on the unitary Ca²⁺ current, i_Ca. The solution in contact with the membrane contains K⁺, Ca²⁺, Mg²⁺, and Cl⁻ ions in addition to a mobile buffer, B²⁻, that binds Ca²⁺ ions in accordance with the following 1:1 reaction scheme

(Scheme 1)

(1)

The concentration profiles near the membrane are calculated by solving the flux conservative equation, applied to each ion present in the electrolytic solution. We considered the steady-state form of the flux conservative equation, which for ion j reads

(2)

(3)

The term F_j in Eq. (2) accounts for changes in particle concentration due to chemical reactions. In our model, the only chemical reaction considered is the binding of Ca²⁺ to the mobile Ca²⁺ buffer ((Scheme 1)). Accordingly, F_j=k₁ [Ca] [B] −k₋₁ [CaB] when j represents either Ca²⁺ or B²⁻ and F_j=k₋₁ [CaB] −k₁ [Ca] [B] when j represents CaB. For the other species present in solution, F_j=0.

In our model, most of the particles residing in the electrolytic solution possess a charge. Their concentration profiles will thus be also determined by the electrostatic potential, V, according to Eq. (3). The value of V is determined by considering all the charges present in the system, including the charged particle in the electrolytic solution and the surface charge on the membrane, by solving the following Poisson equation

(4)

(5)

It is well known that monovalent and divalent cations present in the solution can bind to negatively charged phospholipids, reducing the effective surface charge density, σ. The term “binding” is here used to describe interactions between ions and phospholipids that exceed those made through the mean electrostatic field. Following the work of McLaughlin et al. 18, we assume that K⁺ ions bind to the negatively charged phospholipids P⁻ according to the scheme

(Scheme 2)

(Scheme 3)

(6)

The system to be solved consists of coupled partial differential equations, namely, the flux conservative equation for each mobile species (Eq. (2)) and the Poisson equation of electrostatics (Eq. (4)). Individually, each of these equations was solved using a finite-difference approach, and implementing the successive over-relaxation method described in Press et al. 25. The solution of the overall system was found by applying the following iterative scheme, similar to that applied for the solution of the same set of equations in calculations of ion channel permeation (PNP theory 26).

We found that this iterative scheme was stable provided that the Poisson equation solved at each iteration was modified as 28

(7)

For each condition tested, we first solved the ion concentration profiles and electrostatic potential in the absence of Ca²⁺ influx. Due to the symmetry of the system in all directions parallel to the membrane, this solution was found by applying the iterative scheme described above to one-dimensional versions of Eqs. (2), which describe the spatial profiles of the variables along the z-direction, normal to the membrane. To find this solution, the following boundary conditions were used: for the flux conservative equation, f at z=0 (at the membrane) was settled to zero (reflective boundary condition), whereas at the other extreme (very far from the membrane), the ion concentrations were fixed to their given bulk values. In the Poisson equation, at z=0, dV/dz=−σ/ɛɛ₀, and at z=L, V=0. The concentration and potential profiles found under these equilibrium conditions were then used as starting values for the subsequent two- or three-dimensional computations made in the presence of Ca²⁺ influx. In this case, the following boundary conditions were used: for the flux conservative equation, f_j=0 at z=0 everywhere and for all ion species. At the Ca²⁺ channel location, f_Ca=−i_Ca/(z_CaFA), where A is the area of the membrane facing the computational box containing the channel. At the other boundaries, ion concentrations were fixed to their equilibrium values, as determined from the previous one-dimensional computation. For the Poisson equation, dV/dz=−σ/ɛɛ₀ at the membrane, whereas at the other boundaries, V was fixed to the equilibrium value, found in the one-dimensional computation.

Depending on the number of Ca²⁺ channels present in the membrane, we used either two- or three-dimensional versions of Eqs. (2). In the presence of only one open Ca²⁺ channel, the system has cylindrical symmetry, allowing its solution on a two-dimensional grid, including a z direction normal to the membrane and an r direction parallel to the membrane, and expressing the distance from the Ca²⁺ channel. The minimum grid-element size had a width (Δr) of 0.01nm and a height (Δz) of 0.01nm, with its origin centered at the channel site. The dimensions of each adjacent grid element were increased by increasing Δr by 20% in the r direction, and Δz by 20% in the z direction. The computational box had a total dimension of 1.36μm in both the z and r directions, and was composed of a total of 56×56 grid elements. We verified that reducing the size of the computational grids and assuming constantly spaced elements did not appreciably change the solution.

In the presence of more than one Ca²⁺ channel, the system loses its cylindrical symmetry. Therefore, in this case, we used a three-dimensional version of Eqs. (2), solved along the x and y directions parallel to the membrane, and the z direction normal to the membrane. The dimensions of the grid elements were settled in the following way. Δz had the same values used in the two-dimensional computations, whereas the values of Δx and Δy had values of 256, 128, 64, 32, 16, 8, 4, 2, 1, 2, 4, 8, 4, 2, 1, 2, 4, 8, 4, 2, 1, 2, 4, 8, 16, 32, 64, 128, and 256nm. The computational box had a total dimension of 1.031μm in both the x and y directions, and 1.36μm in the z direction, and was composed of a total of 14,297 grid elements. The six Ca²⁺ channels were placed in correspondence with the grid elements having Δx=Δy=1nm in contact with the membrane.

To check the accuracy of the code, we tested our output against several limiting cases for which analytical solutions exist. We first tested the Poisson equation and the flux conservative equation of our two-dimensional code separately. The Poisson equation was tested by considering a system containing no ions in solution and a flat charged membrane at z=0, by imposing V=0 at the computational grids farther from the membrane. For this system, the analytical solution for the electrostatic potential profile is

(8)

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Figure 1

Comparison between existing analytical solutions (solid lines) and the output of our code (open symbols) for different simple systems. (A) A charged planar surface (σ_T=−0.05 C/m²) with no ions in solution. (B) The flux conservative equation tested in the absence of buffer and with 10mM of Ca²⁺ buffer. (C and D) A 1:1 electrolytic solution in contact with a charged membrane (σ_T=−0.05 C/m²).

The two-dimensional flux conservative equation was tested by computing the [Ca] profile in the presence of an open Ca²⁺ channel, in the absence of Ca²⁺ buffers, and not considering electrostatics. The analytical solution of this system is

(9)

(10)

Figure 1B shows that in this case also our code gives results essentially identical to those obtained with Eq. (10).

We finally tested our iterative scheme, including the solution of both the flux conservative and Poisson equations, by considering a system that includes a charged surface bathed by a 1:1 electrolyte in the absence of Ca²⁺ influx. The analytical solution of this system, derived by Gouy and Chapman (reviewed by McLaughlin 29), is

(11)

(12)

Finally, we checked the three-dimensional version of our iteration scheme by comparing its solution, obtained with only one open Ca²⁺ channel, with that obtained with the two-dimensional iteration scheme (data not shown).

A large variety of Ca²⁺-triggered release models have been assessed experimentally and employed in modeling studies. Here we use a release model derived from Ca²⁺ uncaging experiments at the rat calyx of Held synapse 15, including five independent Ca²⁺ binding sites

(Scheme 4)

(13)

In our model vesicle fusion is triggered by one Ca²⁺ channel, described by the following two-state kinetic scheme 31,32

Table 1 gives descriptions and numerical values for the parameters used in this study. The parameters K₁, K₂, and K₃, describing the equilibrium constants for ion binding to phospholipid membranes, were taken from studies assessing ion binding affinity to phosphatidylserine-containing membranes 18,24. As in Shahrezaei and Delaney 33, we considered an endogenous buffer with a rather high affinity, fast kinetics, and slow diffusion 34. We used 0.5mM of this buffer, equivalent to a buffer capacity of 250. Total surface charge density (σ_T) was considered a variable parameter whose numerical values ranged between 0 and 0.1 C/m², in accordance with the surface charge densities found for the inner leaflet of mammalian plasma membrane 19. Finally, we considered a Ca²⁺ current of 0.2 pA and a Ca²⁺ diffusion coefficient of 220μm²/s. The sensitivity of the output of the model to variations of the main parameters is presented in Supplementary Material, Data S1 .

Fig. 2 shows results obtained from simulations taken in the absence of Ca²⁺ influx and with a membrane surface charge density of −0.05 C/m², a value comparable to those found for the inner leaflet of mammalian plasma membranes 19,29. The negative charge localized at the surface of the membrane generates a negative electrostatic potential that decays to near zero within a few nanometers from the membrane (Figure 2A). As shown in Figure 2B, this negative electrostatic potential causes an increase in the [Ca] near the membrane, according to the Boltzmann relationship valid under equilibrium conditions:

(14)

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Figure 2

Diffuse double layer near a charged membrane in the absence of Ca²⁺ influx. (A) Electrostatic potential (V) as a function of the distance from the membrane (z), obtained from a simulation carried out in the absence of Ca²⁺ influx and with a membrane surface charge density of −0.05 C/m². (B) [Ca] as a function of distance from the membrane, obtained from the same simulation used in A. (C) Relationship between the [Ca] at 0.1, 1, and 10nm from the membrane and [Ca]_bulk, obtained from a series of simulations carried out in the absence of Ca²⁺ influx and with a surface charge density of −0.05 C/m². (D) Dependence of the effective (unbound) charge density (σ) and electrostatic potential (V at z=0) on the [Ca]_bulk, obtained from the same simulations used in C.

Fig. 3 compares simulations of Ca²⁺ microdomains obtained in the presence of an open Ca²⁺ channel, when the membrane has no surface charge, and with a membrane carrying a surface charge of −0.05 C/m² and −0.1 C/m². Figure 3A shows the [Ca] profiles along the z axis (normal to the membrane) at a radial distance of r=0 (in front of the channel), and Figure 3BD, shows the [Ca] profiles along the r axis (parallel to the membrane) at distances of 0.1, 1, and 10nm from the membrane. It is evident that inclusion of a negative surface charge significantly changes the [Ca] profiles. Very close to the membrane (z=0.1nm), the [Ca] ismuch higher in the presence of negative surface charges (Figure 3B) due to the strong attraction exerted by the negative electrostatic potential at these locations (Figure 3Binset). Farther from the membrane, at z=1nm, the [Ca] in the presence of surface charge is still higher for relatively high radial distances from the channel (Figure 3C), in accordance with the existence of a still significantly negative electrostatic potential (Figure 3C, inset). However, at smaller distances from the Ca²⁺ channel, the [Ca] estimated in the presence of surface charge is smaller than in the absence of surface charge. This result suggests that besides an obvious attractive tendency on Ca²⁺ ions, surface charges exert an additional action that tends to reduce Ca²⁺ accumulation near the point of Ca²⁺ influx. This effect is even more evident by looking at the [Ca] profiles obtained at 10nm from the membrane, where the electrostatic potential is decayed close to zero (Figure 3D, inset). In this case, the [Ca] in the presence of surface charge is lower than the [Ca] obtained without surface charge for radial distances from the channel <∼40nm. For higher radial distances, surface charges have only minor effects on the [Ca] (Figure 3D).

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Figure 3

Effects of surface charge on the submembrane [Ca] profile in the presence of Ca²⁺ influx. [Ca] profiles along the z directions at r=0 (A), and along the r direction at 0.1, 1, and 10nm from the membrane (B–D, respectively) for a diffusion model without surface charge (solid lines) and in the presence of surface charge densities of −0.05 and −0.1 C/m² (dashed and dotted lines, respectively). For all models, i_Ca=−0.2 pA. The insets show the spatial profiles of the electrostatic potential for the same simulations.

A possible explanation for the observed reduction in the [Ca] at small radial distances from the channel could be that a preferential surface diffusion of positively charged ions along the plane of the membrane is promoted by the negative surface charge 35,36. Surface charges would indeed favor Ca²⁺ ions spreading out very efficiently along the lateral direction while allowing a much slower redistribution intothe third dimension normal to the membrane. To verify this hypothesis, we looked at the effects of surface charges on the electrodiffusive fluxes in both the lateral and normal directions. As shown in Fig. 4, the presence of surface charges markedly increases the lateral flux (estimated at the computational boxes in contact with the membrane) while producing a significant decrease of the electrodiffusive flux along the direction normal to the membrane (estimated for the computational boxes right in front of the channel). These results suggest that in the presence of surface charge, Ca²⁺ ions preferentially travel along the lateral direction and very close to the membrane, thus reducing their accumulation at points more distant from the membrane.

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Figure 4

Changes of the Ca fluxes in the presence of surface charge. The electrodiffusive fluxes of Ca²⁺ ions assessed (A) in the radial direction, at z=0, and (B) along the direction normal to the membrane at the inner face of the channel (r=0). The data are obtained from the same simulations presented in Fig. 3.

To better understand the effects of surface charges on the spatial profile of the Ca²⁺ microdomain, in Fig. 5 we plotted the [Ca] profiles of Fig. 3, normalized to the [Ca] present at 0.1nm from the point of Ca²⁺ influx. These plots suggest that a relevant effect of negative surface charges is to decrease the steepness of the Ca²⁺ microdomain decay in the r direction, with the result that the [Ca] remains elevated for longer distances from the point of Ca²⁺ influx (Figure 5BD). By contrast, surface charges cause an increase in the steepness in the z direction, normal to the membrane (Figure 5A). The changes in the shape of Ca²⁺ microdomains due to the presence of a surface charge can also be appreciated by the isoconcentration profiles shown in Figure 5EF. It is evident that the Ca²⁺ microdomain, symmetrically distributed along the r and z directions in the absence of surface charges, flattens out in the presence of negative surface charges.

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Figure 5

Effects of surface charges on the spatial profiles of the Ca²⁺ microdomain. (A–D) Profiles of the [Ca] normalized to its value at 0.1nm from the point of Ca²⁺ influx, taken from the same simulation shown in Fig. 3. (E and F) Isoconcentration profiles around an open Ca²⁺ channel obtained for [Ca]s of 10μM and 100μM, respectively, in the presence of surface charge densities of 0, −0.05, and −0.1 C/m².

An interesting consequence of the reduced steepness of [Ca] profiles along the direction parallel to the membrane is an increased overlap of Ca²⁺ microdomains originating from distinct Ca²⁺ channels. This is better appreciated in the three-dimensional simulations shown in Fig. 6, where we consider six Ca²⁺ channels (1–6) located 21nm from each other (Figure 6A), and compare the [Ca] profiles near channel 1 when all six channels are open and when channels 2–6 are turned off (Figure 6B). This allows us to assess the contribution of neighbor channels to the Ca²⁺ microdomain established by channel 1 (Figure 6C). As shown in Figure 6BC, the contribution of channels 2–6 to the total [Ca] close to channel 1 is significantly increased when a negative surface charge is introduced, indicating an increased Ca²⁺ microdomain overlap.

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Figure 6

Effects of surface charge on Ca²⁺ microdomain overlap. (A) Schematic representation of the Ca²⁺ channel topography considered in our three-dimensional simulations. Circles indicate the positions of Ca²⁺ channels, and the arrow indicates the direction where the [Ca] profiles shown in B and C were estimated. (B) [Ca] as a function of the distance from Ca²⁺ channel 1 (cf. A) for three-dimensional simulations considering a surface charge density of 0, −0.05 and −0.1 C/m². The solid curves represent the [Ca] profiles obtained when all six Ca²⁺ channels are open, whereas dotted curves represent the same profiles when only Ca²⁺ channel 1 is open. (C) Fractional contribution of Ca²⁺ channels 2–6 to [Ca] as a function of the distance from Ca²⁺ channel 1 (cf. A).

Experimental estimates of submembrane [Ca] near a point of Ca²⁺ influx is usually performed by the so-called reverse approach 14,15,37. Specifically, the activity of Ca²⁺-dependent proteins placed in close proximity to the site of Ca²⁺ entry, such as Ca²⁺-activated K⁺ channels or the Ca²⁺ sensor of the release machinery, is first evaluated in the absence of Ca²⁺ influx, by applying different [Ca]_bulks under equilibrium conditions. In the presence of Ca²⁺ influx, the [Ca] reached at the sensor protein is then assumed to be equal to the [Ca]_bulk that under equilibrium conditions (no Ca²⁺ influx) would give the same degree of sensor protein activation, a quantity we here call apparent [Ca] at the sensor protein, [Ca]_SP*. Obviously, because of the Ca²⁺ accumulation near a negatively charged membrane, for a sensor protein located very close to the membrane (where the electrostatic potential is still significantly negative (cf. Fig. 2)) the [Ca]_bulk needed to attain a given protein activity (i.e., the [Ca]_SP*) will be lower than the actual [Ca] in contact with the sensor protein. It follows that the appropriate output of our model to be compared with experimental data deriving from the reverse approach is the Ca²⁺ microdomain profile given in terms of the [Ca]_SP*. The converted Ca²⁺ microdomain profiles, assessed from [Ca] versus [Ca]_bulk relationships similar to those shown in Figure 2C, are shown in Fig. 7. These profiles indicate that the [Ca]_SP* in the presence of asurface charge is significantly smaller than the [Ca]_SP* in the absence of a surface charge (obviously, in the absence of surface charge, [Ca]=[Ca]_SP*) for small distances from the channel, in both the r and z directions. At distances farther from the Ca²⁺ channel, no substantial differences are observed in the [Ca]_SP* profiles of the two models. These results indicate that membrane surface charges tend to reduce the estimated [Ca] at the sensor protein, provided that a strong Ca²⁺ channel-vesicle colocalization exists.

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Figure 7

[Ca]_SP* profiles at different surface charge densities. [Ca]_SP* profiles along the z and r directions obtained from three different simulations considering a surface charge density of 0, −0.05, and −0.1 C/m².

The results shown above suggest that the presence of a surface charge at the inner leaflet of the membrane may have a significant effect on the vesicle release probability. To verify this notion, we used a Ca²⁺-dependent release model, derived from experiments carried out on the calyx of Held 15, and looked at the effects of varying the membrane surface charge on the release probability. In these simulations, we assumed that the fusion of the vesicle is gated by only one Ca²⁺ channel placed at a radial distance of either 10 or 40nm from the vesicle Ca²⁺ sensor. In the model, vesicle fusion is evoked by an action-potential-like voltage waveform that controls the gating of a two-state voltage-gated Ca²⁺ channel 31. Since the Ca²⁺ affinity of the release model we used (as for any other existing release model) was derived experimentally in terms of the [Ca]_bulk under equilibrium conditions, it represents an apparent Ca²⁺ affinity, which, in the presence of surface charges and with the Ca²⁺ sensor located within the electrical double layer, may differ significantly from the true value. To overcome this problem, in our simulations we changed the Ca²⁺ binding rate constant of the model (k_on) to obtain an action-potential-induced release probability under control conditions of ∼0.2, a value similar to that estimated at the rat calyx of Held synapse 9. As shown in Fig. 8, we found that the effect of changing the surface charge density on the predicted neurotransmitter release varies depending on the position of the Ca²⁺ sensor relative to the membrane. When the Ca²⁺ sensor is placed very close to the membrane (i.e., within the electrical double layer), the evoked release probability increases with charge density. This is due to the increase of the [Ca] at these locations, promoted by the electrostatic attraction of Ca²⁺ ionsby the negative surface charge (cf. Fig. 3). By contrast, an inverse relationship between the surface charge density and the evoked neurotransmitter release is obtained when the Ca²⁺ sensor is located farther from the membrane (Figure 8D). As shown in Figure 3 and Figure 4, this is due to a reduction of the [Ca] at these locations caused by an increased lateral diffusion of Ca²⁺ ions promoted by the surface charges.

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Figure 8

Effects of membrane surface charges on modeled synaptic release. (A and B) Temporal profile of the membrane potential, the [Ca], and the release probability obtained for a Ca²⁺ sensor located at r=40nm and z=10nm, in the presence of a surface charge density of either 0 (A) or −0.05 C/m² (B). (C) Schematic representation of the model used to assess the effects of membrane surface charges on neurotransmitter release. (D) Release probability assessed from simulations similar to those shown in A and B for different locations of the Ca²⁺ sensor and surface charge densities (indicated). For each location considered, the Ca²⁺ binding rate constant to the Ca²⁺ sensor was chosen to give a release probability of ∼0.2 for a surface charge density of −0.05 C/m².

In this study, we looked at the effects of negative membrane surface charges on the properties of the Ca²⁺ microdomains that build up around open Ca²⁺ channels. We found that membrane surface charges, present at densities comparable to those found in the inner leaflet of plasma membranes, substantially modify the height and shape of the Ca²⁺ microdomain. Specifically, besides an expected attractive tendency on Ca²⁺ ions that results in an increased [Ca] very close to the membrane, surface charges exert an additional action that tends to reduce Ca²⁺ build-up around open Ca²⁺ channels. This causes a reduction in the [Ca] close to the point of Ca²⁺ influx, an effect that is particularly evident at distances from the membrane >1nm, where the electrostatic potential is low.

A possible explanation for this somehow unexpected result is that the negative surface charge facilitates lateral diffusion of the charged ions 35,36 (cf. Fig. 4)). Ca²⁺ ions, attracted by the negative surface charges, raise the electrostatic potential in the vicinity of the channel, thus generating an electrical gradient that in turn increases the diffusion of Ca²⁺ ions in the lateral direction. In this way, surface charges would enable a thin film of concentrated Ca²⁺ to spread out very efficiently along the two dimensions of the surface membrane, rather than moving away from the membrane. Under this interpretation, electrostatics drastically change the properties of Ca²⁺ diffusion away from the Ca²⁺ channel, and clearly distinguish two distinct steps: a rapid lateral spread along the negative surface potential, and a slower redistribution of the spread Ca²⁺ into the third dimension, normal to the membrane.

Our model also shows that negative surface charges tend to decrease the spatial steepness of the Ca²⁺ microdomain, resulting in an increased overlap of microdomains originating from different Ca²⁺ channels. This result suggests a greater contribution of distant Ca²⁺ channels to the release of a vesicle located very close to a point of Ca²⁺ influx. This could have particular relevance at synapses where vesicular release is controlled by multiple Ca²⁺ channels, such as hippocampal and cerebellar synapses 17,38,39.

Another finding of this study is that surface charges reduce the [Ca] estimated at the release Ca²⁺ sensor by a reverse approach, provided that a strong channel-vesicle colocalization exists. This revisitation would, for instance, make the estimated [Ca] at the Ca²⁺ sensor of the calyx of Held synapse (10–25μM, which is significantly lower than the previous theoretical predictions of 75–300μM at few tens of nanometer from a Ca²⁺ channel 7,8) in principle not in contrast with a strong channel-vesicle colocalization.

We have also found that changing the surface charge density will have opposite effects on the action-potential-evoked neurotransmitter release, depending on the distance of the Ca²⁺ sensor of the release machinery from the membrane. Specifically, if the Ca²⁺ sensor lies within the electrical double layer, the release probability increases with the negative surface charge density. This is caused by the electrostatic attraction of Ca²⁺ ions by the negative surface charges, which results in an increase of the [Ca] at these locations. By contrast, if the Ca²⁺ sensor is located at farther distances from the membrane, an increase of the negative surface charge density causes a decrease of the release probability. As we have seen, this is due to the preferential lateral diffusion of Ca²⁺ ions promoted by the surface charge, which results in a decrease of the [Ca] at locations relatively far from the membrane. Since experimental strategies to change the surface charge density are known 40, this theoretical prediction could be used to determine the distance of the Ca²⁺ sensor from the membrane in real synapses.

Experimental data on the properties of Ca²⁺ microdomains and neurotransmitter release are typically obtained from patched cells, using exogenous Ca²⁺ buffers such as EGTA and BAPTA, which are considered to give information on the degree of colocalization of the Ca sensor and the point of Ca influx 1. These two Ca²⁺ buffers have the same affinity for Ca ions, but BAPTA has a binding constant ∼100 times larger than that of EGTA, thus allowing the capture of Ca²⁺ ions in a very short time after their entry into the cytosol, and in this way reducing the [Ca] at points relatively close (<10nm) to the Ca source. This kinetic difference between BAPTA and EGTA is due to the different protonation states of their Ca²⁺ binding moieties at physiological pH, with EGTA almost totally protonated, whereas BAPTA shows a much lower pKa. In this respect, negative surface charges are expected to electrostatically attract protons close to the membrane, thus increasing the degree of protonation of BAPTA molecules close to the point of Ca²⁺ influx, and reducing its Ca²⁺ binding rate constant to values closer to those of EGTA. This effect could in theory explain a numberof experimental data showing that the Ca²⁺ buffering capability of BAPTA, when compared to that observed for EGTA, is not as high as expected 17,41,42,43.

The predictions of our model suggest that changes in surface charge density could be instrumental in producing a form of activity-dependent change in release probability. Specifically, it is well known that aminophospholipid translocases, the enzymes that regulate the negatively charged phospholipid composition of the two sides of the plasma membrane, have an activity strongly regulated by the [Ca]_i44, such that a significant decrease of negatively charged phosphatidylserine present in the inner leaflet of plasma membrane can be observed after an increase in the [Ca]_i45. It is possible that changes in surface charge density are thus used to modulate neurotransmitter release in response to a strong Ca²⁺ influx promoted by high-frequency neuronal stimulation.

We are aware that this model, based on a continuum approach that greatly simplifies the numerical calculus, has limitations. First, our model considers a homogeneous smeared surface charge density and a mean field approach for Ca ion diffusion. Although theoretical Poisson-Boltzmann calculations have shown that this approximation is realistic for physiologically relevant phospholipid compositions 22, at the smallest distance from the membrane considered in this article (0.1nm), the mean field assumption could give results significantly different from the real case of discrete membrane negative charges. A more realistic view should include only a few Ca ions and discrete plasma membrane charges interacting with and influencing the diffusive movement. Second, a more realistic synaptic geometry should include membrane-associated synaptic vesicles, already shown to sensibly modify Ca²⁺ diffusion near open Ca²⁺ channels 33,46, as well as their cytoplasmic surface charge. It should in addition consider the narrow escape problem 47 resulting from the confined space between the plasma membrane and the vesicle. All these limitations call for a more appropriate approach, such as that provided by Monte Carlo simulations.