Organization of the presynaptic site

Topography of the presynaptic active zone

In the most general version of our model, the presynaptic AZ consisted of N_C independent Ca²⁺ channels and N_V independent vesicular release sites of the readily releasable pool (RRP), all positioned arbitrarily. We, however, concentrated on the analysis of several characteristic scenarios of the AZ topography. Data from electron and STED immunofluorescence microscopy jointly showed that presynaptic Ca²⁺ channels are concentrated within the plasma membrane underneath the presynaptic density at mature IHC AZs (Wong et al., 2014, i.e., the original article of chapter 2 in the present dissertation). A typical presynaptic density area was approx-imated by a 80 nm×400 nm stripe. Our previous experimental and modeling work suggested a particular Ca²⁺ channel distribution within the presynaptic density. Importantly, the aver-age effective coupling distance R_c between the Ca²⁺ channels and Ca²⁺ sensors of exocytosis was estimated to be ∼ 20 nm (Pangrˇsiˇc et al., 2015, i.e., the original article of chapter 3 in the present dissertation). The number of the effective channels contributing to [Ca²⁺] at the vesicle fusion sensors was ∼2 per each vesicle on average (Wong et al., 2014). These findings are compatible with the so-called Ca²⁺-nanodomain coupling regime of Ca²⁺ influx to vesicle fusion (Eggermann et al., 2012; Tarr et all., 2013). It is important to note, however, that the numbers mentioned above represent an average synapse, with no information about the heterogeneity across different synapses. Thus, in this work, we also considered the opposite, so-called, Ca²⁺-microdomain coupling regime between the channels and vesicular release sites.

The defining property of this scenario is a large number of Ca²⁺-channels contributing Ca²⁺ at any vesicular release site (Eggermann et al., 2012).

To represent the Ca²⁺-nanodomain coupling regime, we assumed an idealized limiting case when each vesicle release site is associated with one particular Ca²⁺ channel, which contributes to [Ca²⁺] only at that “coupled” vesicle (Fig. 4.1C). To represent the Ca²⁺-microdomain cou-pling scenario, an idealized limiting case when [Ca²⁺] at each particular release site is due to population averaged contribution of Ca²⁺ channels at the whole presynaptic site was assumed (Fig. 4.1C). Indeed, when the effective number of contributing channels to [Ca²⁺] at a par-ticular location is large, the fluctuations of [Ca²⁺] due to the stochastic opening and closing of the channels are averaged out. In addition to the idealized Ca²⁺-nanodomain and Ca²⁺ -microdomain coupling regimes, we also studied two more scenarios (but only when analyzing experimental ISI distributions). For one of them, it was assumed that each vesicular release site is associated to a finite number of private Ca²⁺ channels (Fig. 4.1C). We called it as

interme-diate Ca²⁺-domain coupling scenario, because it represents an intermediate case between the limiting scenarios of Ca²⁺-nanodomain (N_C = 1) and Ca²⁺-microdomain (N_C =∞) coupling.

In another additional scenario, one or a few channels were coupled to several vesicles at the same time (Fig. 4.1C), thus we named it as shared Ca²⁺-domain coupling scenario.

Dynamics of presynaptic Ca²⁺ channel gating

The stochastic process of opening and closing of presynaptic voltage-gated Ca²⁺ channels was described by a continuous time Markov chain in our model. In the basic version of the model, we considered a two state scheme of the channel gating (Fig. 4.1D), which is characterized by one opening (k₊₁) and one closing (k−1) rates. In agreement to experimental findings (see, e.g., Neef et al., 2009; Zampini et al., 2013), V_m dependence of the channel open probability (p_o) was described by a Boltzmann function,

p_o(V_m) = p¯_o

1 +e^{−(Vm−V0.5)/kV}. (4.1)

Here, ¯p_o is the maximal open probability, V_0.5 is the half activation voltage, and k_V is the voltage sensitivity. In vitro estimates of V_0.5 and k_V averaged over many IHC synapses are respectively ∼ −30 mV and ∼ 7 mV (see, e.g., Johnson et al., 2005; Neef et al., 2009; Frank et al., 2010). The value of ¯p_o is more debatable. Whole cell recordings indicated ¯p_o ∼ 0.4, while single channel recordings pointed to ¯po ∼ 0.2 (Zampini, et al., 2013) or even ¯po < 0.1 (Zampini, et al., 2014). It follows from the whole cell and single channel recordings that a variation of V_m in the physiological range mainly affects the channel opening, not the closing rates (see Supplementary Material 1.1). Thus, we assumed k−1(V_m) ≡ const. This, together with Eq. (4.1), constrained the voltage dependence ofk₊₁:

k₊₁(V_m) = k−1·p¯_o

1−p¯_o+e^{−(Vm−V0.5)/kV}. (4.2) k−1, V_0.5, k_V, and ¯p_o were considered as free parameters in the model.

To better understand the influence of the details of the channel gating dynamics on the presynaptic vesicle release, we also explored kinetic schemes of channel gating with several intermediate closing states (Fig. 4.1D). Currently available single channel recordings are too cursory to properly constrain the kinetic model of Ca²⁺ channel gating at IHC synapses. How-ever, the dwell time statistics indicate the presence of channels in two different modes (see Zampini, et al., 2011, 2013, 2014). At least one of those modes is apparently described by three, not two gating states (two intermediate closed states followed by one open state). Inac-tivation of presynaptic Ca²⁺ influx in IHCs is weak (see, e.g., Johnson et al., 2005, Cui, et al., 2007, Neef et al., 2009) and was not taken into account in our work.

Dynamics of presynaptic [Ca²⁺]

The dynamics of the [Ca²⁺] at presynaptic AZs (denoted by [Ca²⁺] for later use) which arises

Among them is the geometry of the intracellular space, the kinetics of presynaptic Ca²⁺ channel gating, as well as the diffusion and binding reactions of Ca²⁺to intracellular Ca²⁺buffers. How-ever, we found that a few important approximations can be applied to describe the dynamics of [Ca²⁺] in the case of IHC ribbon synapses.

First, we noted that the build-up and collapse of [Ca²⁺] domains within the active zone tightly follows opening and closing of the presynaptic Ca²⁺ channels. Fig. 4.2A shows examples of [Ca²⁺] dynamics due to Ca²⁺ influx through a single channel, 20 nm and 100 nm from its mouth (seeMethods for the details of the model). In the presence of 0.5 mM BAPTA + 2 mM MgATP, a conservative estimate of the intrinsic Ca²⁺ buffering strength of IHC (Pangrˇsiˇc, et al., 2015, i.e., the original article of chapter 3 in the present dissertation), the correspondence between the normalized increment of [Ca²⁺] (blue line) and the channel state (grey line) is nearly ideal. Quantitatively, the equilibration times of the [Ca²⁺] increment are much shorter than the opening or closing times of the channel. If the distance from the channel is further increased, the dynamics of [Ca²⁺] increasingly deviates from that of the Ca²⁺ channel gating.

Indeed, the equilibration time of [Ca²⁺] increases with the distance from the channel. This is illustrated in Fig. 4.2B where times necessary to reach 95 % of the steady state level a plotted as functions of the distance from the channel for various Ca²⁺ buffering conditions. However, the contribution of the channel to [Ca²⁺] decreases exponentially fast with the distance in the presence of Ca²⁺ buffers (Naraghi & Neher 1997) and becomes irrelevant at large distances. For example, the steady state level of [Ca²⁺] during continued Ca²⁺ influx decreases from 25.1µM to 0.6µM when the distance is increased from 20 nm to 100 nm in the presence of 0.5 mM BAPTA + 2 mM MgATP.

To show how well the mentioned argument applies to [Ca²⁺] at realistic AZs, we tested the dynamics of the presynaptic [Ca²⁺] at the sensors of exocytosis for two opposing IHC AZ scenarios considered in (Wong et al., 2014). In the so-called scenario M1, the presynaptic Ca²⁺ channels were distributed randomly in the area of the presynaptic density. In the so-called scenario M3, the presynaptic Ca²⁺ channels were placed in the physical contact with the RRP vesicles (one channel per vesicle). Further details of the model are given inMethods. The upper sub-panels in Fig. 4.2C, D show the time-averaged [Ca²⁺] profiles at the level of the plasma membrane atVm =−20 mV in a steady state for particular realizations of scenarios M1 and M3.

We considered a random snapshot of the states of the presynaptic Ca²⁺ channel population in a steady state and estimated the time required for [Ca²⁺] at the sensors of exocytosis to reach 95 % of the steady state level after stimulus onset. Depolarizations to -40 mV, -30 mV, and -20 mV from an initial state with all Ca²⁺ channels closed were considered. This time, t_0.95, was then averaged over all 14 vesicular release sites within the AZ, 50 different realizations of the channel population states, and 10 different realizations of the AZ scenarios. Fig. 4.2E illustrates how t_0.95 (mean ±s.d.) depends on V_m. Note that V_m = −20 mV elicits the near maximal Ca²⁺ influx level (Wong et al., 2014). Consistent with our above considerations of the single channel system, [Ca²⁺] equilibration was fast (t only tens of µs even for scenario

Figure 4.2: Properties of [Ca²⁺] dynamics at IHC active zones. (A) Exemplary trajectories of [Ca²⁺] evolution across time due to opening and closing of a single channel in the presence of 0.5 mM BAPTA + 2 mM MgATP (blue) and in the absence of [Ca²⁺] buffers (green). The upper and the lower subpanels show [Ca²⁺] 20 nm and 100 nm away from the channel, respectively. (B) The time it takes to reach 95 % of the steady state level of [Ca²⁺] after opening of a Ca²⁺channel at different distances from the channel and in different Ca²⁺ buffering conditions. (C) Time-averaged steady state [Ca²⁺] (upper panel) and free [BAPTA] (lower panel) at the level of the plasma membrane at Vm = −20 mV for AZ scenario M1. (D) The same as (C) but for AZ scenario M3. (E) Time it takes to reach 95 % of the steady state level of [Ca²⁺] at the Ca²⁺ sensors of exocytosis for AZ scenarios M1 and M3 at different V_m. (F) Relative deviations between [Ca²⁺] at the Ca²⁺ sensors of exocytosis obtained by solving generic and linearized reaction diffusion equations. (G) Linear fits of the dependencies of time-averaged steady state [Ca²⁺] at the Ca²⁺ sensors of exocytosis on the single channel current

M1). As follows from Fig. 4.2A, B such short times allow to treat the build-up and collapse of increments of [Ca²⁺] as instantaneous upon the channel closing and opening.

The speed of [Ca²⁺] equilibration depends on the concentrations and kinetic properties of the intracellular mobile Ca²⁺ buffers. In the absence of mobile buffers, the equilibration is slowed (Fig. 4.2B, C, green line). However, even the presence of a small amount of a fast mobile Ca²⁺ buffer strongly speeds the equilibration. The light blue curve in Fig. 4.2B shows the dependence of t0.95 on the distance from a single channel in the presence of 0.1 mM BAPTA.

Another important factor shaping the equilibration times of [Ca²⁺] after the channel opening or closing are the immobile Ca²⁺ buffers. These buffers do not influence the steady state levels of [Ca²⁺], but slow down the equilibration (Naraghi & Neher 1997). Concentrations and [Ca²⁺] binding properties of immobile [Ca²⁺] buffers in IHCs are not known. However, even assuming the strongest immobile buffering described so far – 4 mM, k_on = 10⁸ M⁻¹s⁻¹, and K_D = 10⁻⁴M (Xu et al. 1997, Matthews & Dietrich 2015) – the equilibration times would still be rather small (Fig. 4.2B, brown line) for the Ca²⁺-nanodomain and intermediate Ca²⁺-domain coupling scenarios (Fig. 1C). Please note that [Ca²⁺] is constant in a steady state for AZs in the idealized Ca²⁺-microdomain coupling regime by definition. Thus, immobile Ca²⁺ buffers have no effect then.

The second important feature of [Ca²⁺] dynamics at the presynaptic AZs we found is the near-linear superposition of Ca²⁺ domains arising from Ca²⁺ influx through separate presynap-tic channels. To show this, we first calculated the steady state spatial profiles of [Ca²⁺] for the AZ scenarios M1 and M3 by solving the generic reaction diffusion equations (seeMethods).

Then, we calculated the steady state profiles of [Ca²⁺] by considering each open presynaptic Ca²⁺ channel in isolation and summing up the spatial [Ca²⁺] profiles corresponding to each open channel. Fig. 4.2F shows the average relative difference between [Ca²⁺] at the Ca²⁺ sen-sors of exocytosis calculated by using the superposition ([Ca²⁺]_L) and by solving the generic reaction-diffusion equations ([Ca²⁺]_N). The top points of the vertical bars shown in Fig. 4.2F mark the upper limits of the 0 – 95 percentile range of the discrepancy measure. The averaging was performed in the same way as when calculating t_0.95 described above. The discrepancy between the two estimates is small even for the near-maximal Ca²⁺ influx at V_m = −20 mV.

Mathematically, these findings are equivalent to the fact that the reaction-diffusion equations describing the dynamics of [Ca²⁺] can be linearized to a very good approximation. The re-sults shown in Fig. 4.2 were obtained by assuming RRP vesicles and synaptic ribbons being permeable to Ca²⁺ ions and buffer molecules. However, the same conclusion remained valid when these organelles were treated as diffusion barriers (as described in “Additional results”

of chapter 2 in the present dissertation). The average relative discrepancy, equivalent to that shown in Fig. 4.2, did not exceed 10% then (data not shown).

An important condition for the validity of the linear approximation is the fact that the concentrations of the mobile Ca²⁺-unbound buffer molecules are rather uniform at the AZ compared to [Ca²⁺] (Fig. 4.2C, D). This renders the concentrations of the Ca²⁺-unbound buffer

molecules quite insensitive to the particular combinations of open channels within the AZ. Thus, the reaction-diffusion dynamics can be linearized around the average concentration of the Ca²⁺ -unbound buffer within the AZ for a given overall Ca²⁺ influx. This would remain reasonable even if the buffer was considerably depleted due to incoming Ca²⁺. However, then the equations linearized for one particular overall Ca²⁺ influx could not be generalized to different Ca²⁺influx levels. On the other hand, if buffer depletion is not too strong, it is possible to linearize the dynamics around one chosen Ca²⁺ influx level such that the linearization works well within the whole range of the stimulus intensities of our interest. This turns out to be a very realistic situation in the case of the IHC AZs. Fig. 4.2G shows how average value of [Ca²⁺] at the Ca²⁺ sensor of exocytosis depends on the single channel current (filled circles) for scenarios M1 and M3. Here, i_Ca = 100 % corresponds to V_m = −20 mV at which the overall Ca²⁺ influx is maximal. For both scenarios, the dependence nearly perfectly follows the linear relation y = k · x shown by solid lines in Fig. 4.2G. This means that linearization of the reaction diffusion equations at one particular i_Ca value works well within the whole range of stimulus intensities of our interest.

Guided by the above results, we assumed in this work that: 1) the dynamics of [Ca²⁺] is instantaneous upon opening and closing of the presynaptic Ca²⁺ channels, 2) the overall [Ca²⁺] at the presynaptic site is equal to the sum of the contributions of each presynaptic Ca²⁺ channel when considered separately from each other. If these assumptions are taken into account, then [Ca²⁺] at the Ca²⁺ sensors of exocytosis are fully described by the conductance states of the presynaptic Ca²⁺ channels at a given moment and steady state levels of [Ca²⁺] due to each channel in isolation. We denoted the contributions to [Ca²⁺] at the vesicular release sites by each presynaptic channel as ∆[Ca²⁺]^s_i,j, where, in the most general case,i= 1, N_C andj = 1, N_V. In this work, we treated ∆[Ca²⁺]^s_i,j as free parameters which determine the dynamics of fusion of presynaptic vesicles. When calculating vesicle release rate dependence on V_m, ∆[Ca²⁺]^s_i,j were assumed to be independent of V_m. This is justified by the large difference between the resting membrane potential of IHCs (-45 mV, Cody & Russell, 1987) and the reversal potential of Ca²⁺ current (∼ +40 mV, see, e.g., Johnson et al. 2005; Frank et al., 2010) with respect to the range of V_m values where the release rate of a synapse varies considerably (∼ 10 mV see section 4.4.4).

Dynamics of presynaptic vesicle release and replenishment

Based on presynaptic membrane capacitance measurements combined with photolysis of caged Ca²⁺, it was suggested that fusion of the RRP vesicles at IHC synapses happens upon binding of 4 – 5 Ca²⁺ ions to the Ca²⁺-sensor of exocytosis (Beutner et al., 2001). In our work, the original Markov chain scheme of Ca²⁺ dependent vesicle fusion proposed by (Beutner et al., 2001) was used as the base model for vesicle fusion (Fig. 4.1E). This scheme is characterized by four parameters: the intrinsic Ca²⁺ binding rate (k_on), the Ca²⁺ unbinding rate (k_{of f}), the

(γ). In vitro estimates of these parameters, based on presynaptic exocytic responses of many IHCs, each containing roughly a dozen synapses, were respectively 0.028µM⁻¹ms⁻¹, 2.2 ms⁻¹, 0.4, and 1.7 ms⁻¹. Note that the apparent Ca²⁺-binding rate, which we denoted by k_rel, is the product of k_on and [Ca²⁺] at the sensor of exocytosis. In our model, we varied all these parameters, including the number of the Ca²⁺-binding sites.

The kinetics and Ca²⁺ dependence of the replenishment of RRP vesicles at IHC synapses, as well as other ribbon synapses, are currently poorly understood. In vitro patch clamp record-ings indicated strong positive dependence of the RRP recovery rate on intracellular [Ca²⁺] at auditory hair cell synapses (Moser & Beutner, 2000; Spassova et al., 2004). In fact, the RRP recovery was abolished by application of 2.5 – 5 mM Ca²⁺ buffer EGTA in the paired pulse stimulation paradigm. These findings suggested that the RRP recovery depends on distant [Ca²⁺], several hundred nanometers away from the presynaptic Ca²⁺ channels (Moser & Beut-ner, 2000; Spassova et al., 2004). Nevertheless, no quantitative relations between intracellular [Ca²⁺] and RRP recovery have been reported so far. Vesicle replenishment to a partially de-pleted RRP during ongoing stimulation likely deviates from that after cessation of a stimulus.

Given these uncertainties, vesicle replenishment was treated as a one step process characterized by one rate parameter k_rep in our model. At fixed V_m, [Ca²⁺] is constant in time at locations distant from the presynaptic channels because the fluctuations due to opening/closing of indi-vidual Ca²⁺ channels are averaged out. Thus, k_rep is a well defined function of V_m in a steady state. In the basic version of the model we simply assumed k_rep(V_m) ≡ k_rep = const. More generally, the k_rep vs V_m relation was described by a Boltzmann function,

k_rep(V_m) =

k˜_rep

1 +e^{−(Vm−V˜0.5)/kV}, (4.3) where ˜krep is the maximum replenishment rate, ˜V0.5the half activation voltage,kV is the voltage sensitivity (the same as for po(Vm)).

In the most basic version of the model considered, vesicle fusion was directly induced upon irreversible binding of one Ca²⁺ ion (Fig. 4.1E). Such a vesicular release site was characterized by only two states, empty (fused) or filled with a vesicle, and was called as a 2-state site. We used a termn_V-state vesicular release site to refer to those sites defined byn_V ≥3 states. Note that, for these sites, n_V =m+ 2, where mis the number of Ca²⁺-binding sites per Ca²⁺ sensor of exocytosis.