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Glutamate–AMPAR interaction in a model of synaptic transmission
brain research 1536 (2013) 168–176
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Review
Glutamate–AMPAR interaction in a model of synaptic transmission Francesco Ventriglian, Vito Di Maio Istituto di Cibernetica “E. Caianiello” del CNR, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy
ar t ic l e in f o
abs tra ct
Article history:
Over the last several years we have investigated the excitatory synaptic response by means
Accepted 26 April 2013
of a mathematical model based on a detailed description of the synapse geometry, the
Available online 3 May 2013
Brownian motion of Glutamate molecules and their binding to postsynaptic receptors.
Keywords:
Recently, the basic model has been modified for the numbers, the size and the 3D structure
Glutamate synaptic response
of receptors according to new data from the literature. Some results of simulations
Synaptic parameters
performed with the updated model are shown here. They were aimed to study the
Binding probability
synaptic response in relation to the binding probability, to the probable height of the
Computer simulation
receptors in the synaptic cleft, and to the space-time distribution of Glutamate/Receptor collisions. A first series of simulations permitted to determine a possible range of values for the binding probability of Glutamate to receptors. Other simulations, investigating the changes induced on the synaptic response by the variations of the height of AMPA receptors in synaptic cleft, allowed to identify the height producing the higher amplitude peak of the mEPSCs. Finally, two new statistical descriptors for analyzing the synaptic response were presented. The first is based on the study of the space distribution of the number of Glutamate/Receptor collisions. Simulations investigating the effects of an increasing eccentricity of the releasing vesicle allowed assessing this method. The second one considers the inter-collision times between Glutamate molecules and binding sites. The results of some of the last simulations demonstrated its capacity to highlight the subtleties and the randomness underlying the activation of the receptors. This article is part of a Special Issue entitled Neural Coding 2012. & 2013 Elsevier B.V. All rights reserved.
Contents 1. 2.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2.1. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2.2. Brownian model for Glutamate diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2.3. Binding probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.4. EPSC computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.5. Technical details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
n
Corresponding author. Fax: +39 081 8675 326 E-mail addresses: [email protected], [email protected] (F. Ventriglia), [email protected] (V. Di Maio).
0006-8993/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.brainres.2013.04.051
brain research 1536 (2013) 168–176
169
3.
Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.1. Possible range of values for the binding probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.2. Effects on the mEPSC by changes of the Receptor Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.3. Effects of vesicle eccentricity on glutamate/AMPAR collision number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.4. Waiting times among collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
1.
Introduction
The main drive of the cerebral activity, the excitatory synapse, is always under investigation because of its role in information transmission between brain neurons. Despite a huge amount of experimental data and computational results, the comprehension of the excitatory synaptic function still remains blurred for many aspects. Mathematical models and computer simulations analyzed the contributions of single synaptic components in shaping the postsynaptic response and in determining the stochastic variability observed experimentally (Agmon and Edelstein, 1997; Rabie et al., 2006; Kleppe and Robinson, 1999; Trommershäuser et al., 2001; Uteshev and Pennefather, 1996; Wahl et al., 1996). Along this research line, we investigated a hippocampal excitatory synapse by means of a mathematical model based on the description of Brownian motion of Glutamate molecules within the synaptic cleft and of their interaction with the components of the synapse (Ventriglia and Di Maio, 2000). The results of our simulations, performed on a parallel computer by using an ultra-fast time scale (simulation step of 40 fs), demonstrated that intrinsic random variations in basic pre-synaptic elements of the synapse can reproduce the observed stochastic variability of the miniature excitatory postsynaptic current (mEPSC) (Ventriglia and Di Maio, 2000, 2003; Ventriglia, 2004). Based on a more precise geometrical description of the synaptic structure, our more recent simulations considered also the effects of structural elements, previously not described, such as filaments extending across the synaptic cleft outside the volume delimited by the Active Zone (AZ) and the postsynaptic Density (PSD) (Zuber et al., 2005; Ichimura and Hashimoto, 1988; Ventriglia, 2011). The results showed that their presence induced a small, but significant, increase in the response of the synapse. New published data demonstrated that the size of postsynaptic receptors is much larger than previously supposed while their number is much smaller (in particular the last statement is true for the NMDA receptors—NMDARs) (Armstrong and Gouaux, 2000; Araç et al., 2007; Malinow and Malenka, 2002; Santos et al., 2009; Tichelaar et al., 2004). By utilizing this new information we could deal better with a problem arising from the hypothesis supposed by the larger part of authors about the computation of the binding of Glutamate molecules to receptors. It is, in fact, based on the assumption of stationarity for Glutamate concentration in the synaptic cleft for times of the order of milliseconds. In real conditions, on the contrary, concentration of Glutamate is far from being stationary and its clearance from synaptic cleft is much faster (Forti et al., 1997; Ventriglia, 2004, 2011). For these reasons a computation of the Glutamate binding on the base of receptor collisions and
binding probability was proposed in recent papers (Ventriglia, 2004, 2011). In a last paper, by a geometrical reasoning and comparisons among the amplitude peaks of mEPSCs computed by our computer simulation and those present in the literature, measured by electrophysiology setups, we attempted to find a more valid value for the binding probability (Ventriglia and Di Maio, 2012). These computations did not eliminate the necessity of new computer simulations whose results are presented in this paper. The first series of simulations were carried out to investigate the possibility to identify a reasonable range of values for the binding probability. Also, according the new data about the 3D dimensions of AMPA receptors (AMPARs), some of the new simulations were carried on with the aim to obtain a better estimate of the height of the portion of receptors protruding in the synaptic cleft. In such a case we observed the changes produced on the synaptic response by the variation of this synaptic parameter. Other simulations were utilized to demonstrate the validity of two new statistical descriptors that we proposed for the analysis of the synaptic function. The first is based on the study of the space distribution of the number of collisions of Glutamate molecules on receptor binding sites. The second one describes the inter-collision time between the subsequent visits made by single Glutamate molecules on the binding sites of the different AMPA receptors, until the binding.
2.
The model
2.1.
Geometry
Our model of synapse has the form of two concentric cylinders with a common height of 20 nm. The larger one, with a diameter of 480 nm (if not otherwise specified), encloses all the synaptic cleft. Included in it, a virtual cylinder with a diameter of 220 nm (if not otherwise specified), having bases on AZ and on PSD, simulates the active synaptic space. On the top of AZ, a sphere having an inner diameter of 26.6 nm (Schikorski and Stevens, 1997) and containing Nm ¼ 775 Glutamate molecules simulates the releasing vesicle, while AMPA and NMDA receptors are modeled as small cylinders protruding in the synaptic cleft from the PSD zone, with a diameter of 14 nm. They are placed on the vertexes of a grid spanning the entire PSD (see Fig. 1). Two small circles, located randomly along a virtual ring on the exposed face of the receptor, simulate the binding sites for Glutamate; their dimensions are equal to those of the cross-section of a Glutamate molecule. Filaments within the cleft fill the annular space, external at the AZ/PSD synaptic volume, until the
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Fig. 1 – Grids of receptors and filaments. The results of the simulation related to Fig. 4 were used to obtain the path values of glutamate molecules. They have been used to show the receptor and filament positions in the synaptic cleft, as holes in the fabric woven by the flying Glutamate molecules. Receptors are more evident in the left panel (large dots in the central zone), while filaments are more visible on the right one (smaller dots in the peripheral zone). The PSD zone has a diameter of 320 nm— the largest used in this article, and accommodates 168 AMPARs and 20 NMDARs. The spacing of the two grids – Receptors and Filaments – are equal, 20 nm.
boundary of the synaptic cleft. They are simulated by thin cylinders (diameter of 7 nm) connecting pre- and postsynaptic membranes, arranged according to an ordered structure (see Fig. 1). The grid spacing of filaments and receptors is large 22 nm (or 20 nm—in some cases), according to literature data (Zuber et al., 2005; Ichimura and Hashimoto, 1988). The releasing vesicle is either centered on AZ or at a distance of 45 or 90 nm with respect to the center (eccentricity). We assume that the arrival of an Action Potential (AP) opens an expanding fusion pore within the two lipid bilayers of the cell membrane and the vesicle membrane, connecting the inner volume of the docked vesicle and the synaptic cleft. The fusion pore has been simulated as a cylinder with a length of 12 nm.
2.2.
Brownian model for Glutamate diffusion
The starting time of our computer simulation, t¼ 0, is coincident with the instant at which the area of the expansion pore is equal to the cross-section of a Glutamate molecule. We assumed that at this time the Glutamate molecules contained in the vesicle have an uniform space distribution and their velocity is distributed according to a Maxwell distribution. The opening of the fusion pore allows them to begin their travel toward the synaptic cleft due to Brownian motion. This motion has been described by Langevin equations (Ventriglia and Di Maio, 2000, 2003; Ventriglia, 2004), which, in the discretized time form, appear as ri ðt þ ΔÞ ¼ ri ðtÞ þ vi ðtÞΔ vi ðt þ ΔÞ ¼ vi ðtÞ−γ
vi ðtÞ Δþ m
ð1Þ pffiffiffiffiffiffiffiffiffiffi 2ϵγΔ Ωi m
ð2Þ
where i is the ith molecule (i ¼ 1…N; N being the total number of Glutamate molecules), Δ is the time step and Ωi is a random vector with three components, each having a Gaussian distribution with mean value μ ¼ 0 and standard deviation s ¼ 1. The other parameters are:
m, the molecular mass; γ, a friction term, which depends on the absolute temperature: γ ¼ kB T=D, where kB is the Boltzmann constant, T is the absolute temperature in Kelvin degrees, D is the diffusion coefficient of Glutamate; ϵ ¼ kB T. The value of the diffusion coefficient D for Glutamate was computed for a temperature of 37 1C (Table 1). These equations were implemented in a parallel FORTRAN program by using MPI (message passing interface) paralleling routines. The free movement of Glutamate molecules was restricted only by the collisions with the structures delimiting the model space such as the inner surface of the vesicle and of the fusion pore, the surfaces of the other synaptic structures and of the pre- and post-synaptic membranes. To obtain the greatest space sensibility, no spatial grid has been used. Therefore, a continuous range of space values could be explored by the discretized model. An extremely short time step Δ was chosen, equal to 40 10−15 s (i.e., 40 fs). It allowed an accurate description of the position of Glutamate molecules and of their collisions on synaptic structures, albeit in a nonquantum description. The molecules were considered dimensionless (points) during their free movement or in case of collision with the synaptic surfaces. Conversely, in case of collision with a hole of almost equal size, as the upper and lower openings of the fusion pore or the binding sites of receptors, a spherical or ovoid shape has been used. Since intra-synaptic transporters are very rare, we supposed that Glutamate molecules were absorbed in the synaptic cleft only with a very low probability (PR). Instead, the escaping of molecules (spillover) was due mainly to the exit through the lateral borders of the synaptic cleft. We considered that, due to the high concentration of extrasynaptic transporters near the synaptic boundary, molecules reaching that place had a very low probability to return back and consequently an absorbing boundary was imposed. Hence, the molecules absorbed within the synaptic cleft or crossing the lateral surface were not more dealt with by the program.
brain research 1536 (2013) 168–176
2.3.
Binding probability
Glutamate molecules hitting the receptors contained in the PSD can either bind or be reflected. To compute the binding probability, we used a method based on geometrical considerations we have proposed in previous papers (Ventriglia, 2011; Ventriglia and Di Maio, 2012). We used this method since the traditional method is based on the assumption of stationarity for glutamate concentration in the synaptic cleft of the order of milliseconds (Clements et al., 1992; Jonas et al., 1993), while in real conditions this is not true. In fact, when the neurotransmitters are released by a vesicle their flow is far from the stationarity (this can be deduced also looking at the Fig. 6 or Fig. 7 in this paper) and, moreover, the diffusion process of Glutamate within the synaptic cleft is much faster of what hypothesized (the neurotransmitters remain in the cleft for hundreds, and not thousands, of microseconds) (Forti et al., 1997). In summary, by assuming an elongated shape for a Glutamate molecule and a circular shape of binging sites (see Fig. 2), and by considering that a Glutamate molecule can bind a receptor only when correctly oriented (Armstrong and Gouaux, 2000), we computed a preliminary value of the binding probability (B′P ) as the ratio between the volume of a particular spherical cone including all the useful (for the binding) orientations of the Glutamate molecule and that of the unit sphere including all its possible orientations (see Ventriglia, 2011; Ventriglia and Di Maio, 2012 for a more detailed explanation). Since the binding could be also ruled by some possible, unknown frictional (or quantum) elements, the value of probability (B′P ) so computed was reduced by an arbitrary value of the 20% (BP ¼ 0:80 B′P ). Hence, by varying the angle of the cone, we could observe the effect of different binding probabilities on the synaptic response. In order not to introduce too many unknown, arbitrary parameters in the model, the same probability value was utilized to compute the binding with both AMPA and NMDA receptors.
2.4.
171
EPSC computation
Because under normal conditions the ionic channels of NMDA receptors are blocked by magnesium ions, only currents which passed through AMPA receptor have been used to compute the mEPSC. The schema used for receptor activation has been derived from the usual Markov chain with three states, namely: Basal (B)–closed, Active (A)–open, Desensitized (D)–closed, each with three sub-states 0, 1, 2, denoting the unbound, the singly-bound and the doublebound states, respectively (Clements et al., 1992; Jonas and Spruston, 1994). However, the importance of the transition to (and from) the desensitized state D in shaping EPSC time course is very controversial and Jonas and Spruston (1994) retain that the desensitization is not a major determinant of EPSC decay. Since we are interested mainly in the study of the rising phases and of the peak amplitudes of the EPSC, the transitions to D states were neglected in our computer simulations. In the simplified kinetic model of receptor activity utilized here the transitions occur in a linear cascade B0 ⇌B1 ⇌B2 ⇌A2 . It is well known that to open its ionic channel a receptor needs to be bound to at least two molecules of Glutamate simultaneously, in normal conditions. The transitions between the receptor state B2 and A2 cause random variations in the synaptic response, since the opening and the closing of a double-bound receptor follow probabilistic rules. When a receptor goes into the open state, it remains open for a random period of time τO and then it passes into the closed state and vice versa. A random variable τC is related to the period of closed state. The probability of receptor opening PO is defined as the ratio between the total opening time and the sum of the total opening and total closing time. The random variables τO and τC are distributed according to the following probability density function (PDF) of exponential form PðτÞ: PðτÞ ¼ αe−ατ
ð3Þ
with a mean value τ¼
1 : α
ð4Þ
The values of α for the two random variables are chosen in such a way that PO ¼
Fig. 2 – Spherical cone enclosing all the possible directions of glutamate molecules able to produce a binding on a receptor site. The angle of the cone represented here has a value of 1201, the largest one ever used by us – in previous articles (Ventriglia, 2004, 2011) – corresponding to a BP ¼ 0.25.
τO : τO þ τC
ð5Þ
The opening probability utilized was PO ≈0:83, different by the value PO ¼ 0:71, computed by Jonas et al. (1993). Also, the transitions from the closed state B2 to the closed state B1 obey to stochastic laws causing random variations of the synaptic response. A negative exponential distribution, with a mean value τ I , is used for these transitions. A computer program, which worked off line with respect to the simulation diffusion parallel program, used the AMPAR double binding times (i.e., the times at which the AMPARs went into the state B2) to compute the random transitions to/ from the state A2 and to the state B1 to produce single EPSC responses. The current, flowing through a generic kth AMPA receptor, was null during the closing periods, and reached a peak IMr during the opening periods. The peak current was chosen as a random variable distributed according to a
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Gaussian distribution with mean value I Mr ¼ −1:70 pA and SD¼ 0.3 pA. These values and the value of the opening probability PO were chosen because they produced the best fit with a standard mEPSC (Forti et al., 1997), that we used as reference mEPSC time course (Ventriglia and Di Maio, 2000). The time course of the computed synaptic current (Icomp ðtÞ) was obtained by adding all the single AMPA receptor currents for a time of 6 ms ( n 0 if receptor is closed ð6Þ Icomp ðtÞ ¼ ∑ Ik ðtÞ; Ik ðtÞ ¼ if receptor is opened I Mr k¼1 where n is the number of receptors and Ik ðtÞ is the current of the kth receptor at time t. For each computational experiments, 1000 current time courses –raw mEPSCs – were produced and their values have been averaged in computed mEPSCs. The amplitude peaks were used for comparisons with those of the mEPSCs recorded by experimental setups (Auger and Marty, 2000; Forti et al., 1997).
2.5.
Technical details
The program was implemented on a parallel computer, based on a cluster of four dual-processor workstations. Due to the heavy computational task, the diffusion-binding simulations were carried out for the shortest useful time, i.e., 13 109 iterations, corresponding to 520 μs (in this case each simulation lasted 4 days). At this time all the free Glutamate molecules were exited from the synaptic cleft, while the diameter of the fusion pore had reached a value of 4.61 nm. The space position (ri ðtÞð ¼ ðxi ðtÞ; yi ðtÞ; zi ðtÞÞÞ) at each discrete time for each Glutamate molecule together with the AMPA and NMDA receptor binding times were computed. The space positions of Glutamate molecules were saved on disk every 50,000 iterations (2 ns) and the receptor binding times were saved on a matrix to be used for the computation of the mEPSC. Moreover, the time and the place of a collision between Glutamate molecules and the receptor binding sites were saved to compute the space distribution of the collision number for each molecule and for each receptor.
3.
Simulation and results
3.1.
Possible range of values for the binding probability
In a previous paper we noted the possibility to define a lower limit for the probability of binding of Glutamate to AMPARs. In fact, we carried out two series of three simulations which could be related to AMPA receptor trafficking. Two values were used for the binding probability (BP ≃0:014 and BP ≃0:005—corresponding to spherical cone angles of 301 and 181) and three different values for the number of AMPARs. Results showed that, while the series linked to the greater binding probability gave rise to computed mEPSCs with amplitude peaks growing steadily as the number of AMPARs, the series with the smaller BP exhibited computed mEPSCs having amplitude peaks with an erratic behavior (see Fig 8 in Ventriglia and Di Maio, 2012). Hence, in the attempt to obtain a better approximate value for the binding probability, a new series of computer
Fig. 3 – High dispersion shown by the amplitude peak of the computed mEPSC in simulations with a BP corresponding to a spherical cone angle of 181, 154 AMPARs, 18 NMDARs and different seeds for the random number generator (only green lines, the nongreen lines are related to simulations with lower number of receptors).
simulations was performed to investigate more fully the case with BP ≃0:005 (spherical cone angle of 181), whose results are presented here. A vesicle centered at X0 ¼ 0 on AZ, with inner radius of 13.3 nm and an overall width of 38.6 nm, releasing 775 Glutamate molecules (vesicular concentration of about 130 mM) was considered. On the PSD grid were placed 154 AMPARS and 18 NMDARs in positions which did not change along the series. Receptors and Filaments had a grid spacing of 20 nm, the radius of the AZ/PSD space was equal to 160.0 nm, while the entire synaptic cleft had a radius of 240.0 nm. In this simulation series, all the geometric/biological parameters of the synaptic system remained unmodified, only the initial seed of the random number generator changed from a simulation to the other. These conditions were such that at time 0 only the position and the velocity of Glutamate molecules within the vesicle differed. The simulation results showed a large variance of the computed EPSC amplitude peak (see Fig. 3). At last, a single simulation with a very low binding probability was carried out: BP ≃0:0015—spherical cone angle of 101. The same geometrical parameters of the previous series of simulations (for the angle of 181) were used, with exception of the number of AMPARs and NMDARs which were increased to 168 and 20, respectively. A clear cut result was obtained: only two AMPA receptors were double bound and the peak amplitude of the computed mEPSC was only 1 pA (see Fig. 4), that is a value far below the lowest recorded (see Figure 4 in Forti et al., 1997). We can conclude that a reasonable value for BP seems to be in the range 0.014–0.005 (and that a realistic value should be more close to the higher value).
3.2. Effects on the mEPSC by changes of the Receptor Height In the last few years, some new data were obtained by experimental research which define better the 3D structure (and dimensions) of AMPA receptors (Mayer, 2011; Nakagawa, 2010; Sobolevsky et al., 2009; Tichelaar et al., 2004). Experimental investigations by electron microscopy at 20 Å
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Fig. 4 – Computed mEPSC for a very low binding probability BP ¼ 0.0015221209, (spherical cone angle of 101). The geometric parameters were the same of the simulations with a BP related to an angle of 181. Only the PSD zone was larger (diameter of 320 nm), as well as the number of receptors, both for AMPARs (168) and for NMDARs (20).
173
Fig. 6 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 10 nm. The Z-axis shows the number of hits on both receptor sites, while the receptor position is reported on X,Y axes.
Fig. 7 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 8 nm. Fig. 5 – Computed mEPSC from simulations with three different portions of receptors protruding within the synaptic cleft: 6 nm, 8 nm and 10 nm.
resolution on a fully assembled iGluR showed that a receptor has an elongated shape with dimensions of 170 Å 140 Å 110 Å and a twofold symmetry centered on its longitudinal axis (Tichelaar et al., 2004). If we exclude the portion of receptor embedded within the postsynaptic membrane which has a thickness of almost 6 nm, the length of the portion protruding in the cleft from the PSD is unknown. Then, we investigated the effects produced on the synaptic response by the change of the height of this portion. Three different heights were simulated: 6 nm (the same that we used in all our previous simulations), 8 nm and 10 nm. For these simulations the other parameters were: Vesicle releasing 775 Glutamate molecules at a position X0 ¼ 0; Radius of AZ/PSD¼110.0 nm and Radius of entire synaptic cleft¼ 240.0 nm; BP ≃0:014; 55 AMPARs and 13 NMDARs allocated on the PSD grid; Grid spacing for Receptors and Filaments ¼22 nm. The results, shown by computed mEPSCs, are reported in Fig. 5. We can observe that the peak of maximum amplitude has been obtained with a protrusion of 8 nm. In Figs. 6 and 7 the difference between the case of protrusion of 10 nm and
8 nm is demonstrated also by the space distributions of Glutamate hits on receptor binding sites.
3.3. Effects of vesicle eccentricity on glutamate/AMPAR collision number Experimental data showed that the docking of neurotransmitter vesicles on AZ occurs in different positions (see Schikorski and Stevens, 1997, for example), and our previous simulations demonstrated the decreasing of the response (mEPSC) with the increasing of the eccentricity of the releasing vesicle. Herein, we analyzed more deeply this phenomenon – which has important implications for the synaptic function – by computing, also, the space distribution of number of collisions of Glutamate molecules on receptor binding sites. In these computational experiments the receptor protrusion height in the synaptic cleft was of 8 nm, and the releasing vesicle was positioned at X0 ¼ 0 nm (center of the synapse), or at X0 ¼ 45 nm or at X0 ¼ 90 nm from the center. The other parameters of the simulations were: Released Molecules¼ 775; BP ≃0:014 (angle 301); Number of AMPARs, 55; Number of NMDARs, 13. In Fig. 8 the computed mEPSCs for the three positions are shown. While, in Fig. 7 and in Figs. 9 and 10 the space distribution of number of collisions on receptor binding
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sites are presented, for the same vesicle positions. These last three figures demonstrated the causes of the modification due to the eccentricity of the response peak amplitudes, reported in Fig. 8.
3.4.
Fig. 8 – Computed mEPSC from simulations with a receptor protrusion of 8 nm and three different docking positions for the releasing vesicle: X0 ¼ 0 nm (center of the synapses), X0 ¼ 45 nm and X0 ¼ 90 nm from the center.
Waiting times among collisions
From one of the immediately above simulations (that in which X0 ¼ 90 nm) we extracted some information to analyze the dynamics of the binding process of a few Glutamate molecules also by means of the time elapsed between subsequent collisions on the binding sites, until the final binding, of (also different) receptors (Waiting Times). Three sequences of waiting times between successive impacts on the AMPARs binding sites are shown in Figs. 11–13. In Fig. 11 (related to molecule number 715) sites of four different receptors were visited (111 hits). Fig. 12 shows a simple situation in which only two receptors were visited (molecule No. 709 and 94 hits), while a more complex series of events occurred with the third molecule (No. 81), Fig. 13. In this case, the molecule visited five receptors, with one (receptor No. 74) encountered twice in different epochs (181 hits). The three molecules showed different behaviors also for the duration of the periods from the first impact to the final one: 240.355.305, 45.884.630, 225.321.371 time steps (9:61422984 μs, 1:8353852 μs, 9:01285490 μs), respectively. These figures demonstrated clearly the randomness of the events (bindings) inducing eventually the receptor activation and the consequent random variability of the synaptic response.
4.
Discussion
The stochastic variability of the excitatory postsynaptic current is clearly influenced by many parameters related to the synaptic structure and to the dynamic interaction between neurotransmitter molecules and receptors. By considering Fig. 9 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 8 nm and eccentricity of X0 ¼ 45 nm.
Fig. 10 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 8 nm and eccentricity of X0 ¼ 90 nm.
Fig. 11 – Sequence of waiting times. The horizontal axis shows the values of the decimal logarithm of the time differences between two successive impacts. Molecule No. 715. Four different receptors were visited. First impact, at time ¼ 190:349398720 μs (after 4.758.734.968 time steps), on receptor 13, then passing through receptors 26 and 27, and finally binding on receptor 56 at time ¼ 199:963628560 μs, after 240.355.305 time steps from the impact on first site.
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some updates to parameters of our model, related to the dimensions of receptors (much larger than supposed), to their number (smaller – mainly for NMDARs), to the ratio AMPARs/ NMDARs, we performed some simulations to study with greater details the dynamics of the binding of Glutamate molecules to postsynaptic receptors (Ventriglia and Di Maio, 2012). Some
Fig. 12 – Molecule No. 709. Only two receptors were visited. First impact, at time ¼ 117:3394581 μs, on receptor 66, then passing to receptor 58 and final binding at time ¼ 119:1748433 μs.
Fig. 13 – Molecule No. 81. In this more complex case six receptors were visited. First impact, at time ¼ 138:2470155 μs (after 3.456.175.387 time steps), on receptor 55, then passing through receptors 74, 95 (only three hits), 85, 74 again, and finally going to receptor 63 with final binding at time ¼ 147:2598704 μs, after 225.321.371 time steps from the impact on first site.
175
results of those simulations showed the possibility to define a lower limit for the binding probability of Glutamate to receptors and induced us to deepen the investigation. Among all the simulations presented in this article, some were dedicated to the study the effects of a specific binding probability BP ¼0.005—related to a solid angle of 181. The results showed a large variance of the computed mEPSC amplitude peak at the changing of only the initial seed of the random number generator and demonstrated the irregularity of the synaptic response related to this binding probability. Moreover, from the results of a simulation with a very low binding probability, corresponding to a spherical cone angle of 101, we computed a mEPSC whose amplitude peak (about 1 pA) was much lower than the inferior limit of experimentally measured mEPSCs (5–64 pA; see Forti et al., 1997, for example). This demonstrates the unlikelihood of that probability. Thus, a possible range of values for the binding probability, BP, seems to be 0.005–0.014 and, however, a better approximate value must be searched for the inferior limit, at least. Results of simulations related to the portion of the receptor protruding from the PSD showed that the maximal efficiency is obtained for a protrusion of 8 nm, in correspondence of which the highest value of the mEPSC peak amplitude is obtained. This value was about 10% higher than that related to 6 nm, previously used by us (Ventriglia, 2004, 2011; Ventriglia and Di Maio, 2012). If factors related to the chemical structure of the AMPA receptor or to its physical links with the synaptic membrane or with the entire synapse, do not forbid this length, metabolic and functional considerations on the high energetic costs of the activity of the neural tissue concur to sustain the validity of this protrusion height (Attwell and Laughlin, 2001; Niven and Laughlin, 2008). In such a case, next simulations should consider this value for the height of the receptors. Another series of simulations was devoted to investigate more thoroughly the effects of the eccentricity of the releasing vesicle on the synaptic response. The study was based on the analysis of the behavior of the space distribution of the collision number of Glutamate molecules on receptor binding sites. The decreasing of the mEPSC amplitude peak with the increasing of the eccentricity was clearly related to the changes showed by this distribution. In fact, by Figs. 7, 9 and 10 we see that the peak of the collision distribution was always located at the position of the releasing vesicle, and that the increase of the eccentricity induced a shift of this distribution – and of its peak – toward the periphery of the post-synaptic density. In this way, the collision distribution assumed a form increasingly distorted, in which points (receptors) having a lower collision number occupied larger and larger portions of the synaptic space containing the
Table 1 – Simulation parameters. Temperature Glutamate diffusion coefficient at 310.16 K (37 1C) Molecular mass of glutamate Simulation time step Length of fusion pore Areal pore velocity Re-uptake probability
T D m Δ hpore Vareal Pr
310.16 K 10.0 10−6 cm2 s−1 2.4658025 10−25 kg 40 10−15 s 12 nm 31.4 nm2 ms−1 3 10−6
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receptors. These results showed a reduction of the volume of the collision distribution. Since the AMPA receptor activation depends on the magnitude of the number of collisions suffered by the receptor, through a binding probability, it is evident that the eccentricity of the releasing vesicle, by causing a lower global number of collisions, induces the production of mEPSCs with lower peak amplitudes. From some simulations of the last series above, the sequences of collisions of all Glutamate molecule on receptor binding sites, until the eventual binding, were extracted. They were used to assess the utility of the analysis of the intercollision waiting times to obtain more precise information on the binding process. We have reported here, in Figs. 11–13, three examples that showed in a visible way the great variability of the distribution of the waiting times and, at the same time, the different series of events occurred to the various neurotransmitters. In such a way, the complexity of the receptor activation and some of the fundamental elements of the randomness of the synaptic response became apparent.
references
Agmon, N., Edelstein, A.L., 1997. Collective binding properties of receptor arrays. Biophys. J. 72, 1582–1594. Araç, D., Boucard, A.A., Ozkan, E., Strop, P., Newell, E., Südhof, T.C., Brunger, A.T., 2007. Structures of neuroligin-1 and the neuroligin-1/neurexin-1 beta complex reveal specific protein– protein and protein–Ca2þ interactions. Neuron 56, 992–1003. Armstrong, N., Gouaux, E., 2000. Mechanisms for activation and antagonism of an AMPA-sensitive glutamate receptor: crystal structures of the GluR2 Ligand Binding Core. Neuron 28, 165–181. Attwell, D., Laughlin, S.B., 2001. An energy budget for signaling in the grey matter of the brain. J. Cereb. Blood Flow Metab. 21, 1133–1145. Auger, C., Marty, A., 2000. Quantal currents at single-site central synapses. J. Physiol. 526, 3–11. Clements, J.D., Lester, R.A., Tong, J., Jahr, C.E., Westbrook, G.L., 1992. The time course of glutamate in the synaptic cleft. Science 258, 11498–11501. Forti, L., Bossi, M., Bergamaschi, A., Villa, A., Malgaroli, A., 1997. Loose-patch recordings of single quanta at individual hippocampal synapses. Nature 388, 874–878. Ichimura, T., Hashimoto, P.H., 1988. Structural components in the synaptic cleft captured by freeze-substitution and deep etching of directly frozen cerebellar cortex. J. Neurocytol. 17, 3–12. Jonas, P., Major, G., Sakmann, B., 1993. Quantal components of unitary EPSCs at the mossy fibre synapse on CA3 pyramidal cells of rat hippocampus. J. Physiol. 472 C, 615–663. Jonas, P., Spruston, N., 1994. Mechanisms shaping glutamatemediated excitatory postsynaptic currents in the CNS. Curr. Opin. Neurobiol. 4, 366–372. Kleppe, I.C., Robinson, H.P., 1999. Determining the activation time course of synaptic AMPA receptors from openings of colocalized NMDA receptors. Biophys. J. 77, 1418–1427.
Malinow, R., Malenka, R.C., 2002. AMPA receptor trafficking and synaptic plasticity. Annu. Rev. Neurosci. 25, 103–126. Mayer, M.L., 2011. Structure and mechanism of glutamate receptor ion channel assembly, activation and modulation. Curr. Opin. Neurobiol. 21, 283–290. Nakagawa, T., 2010. The biochemistry, ultrastructure, and subunit assembly mechanism of AMPA receptors. Mol. Neurobiol. 42, 161–184. Niven, J.E., Laughlin, S.B., 2008. Energy limitation as a selective pressure on the evolution of sensory systems. J. Exp. Biol. 211 (Pt 11), 1792–1804. Rabie, H.R., Rong, J., Glavinović, M.I., 2006. Monte Carlo simulation of release of vesicular content in neuroendocrine cells. Biol. Cybern. 94, 483–499. Santos, S.D., Carvalho, A.L., Caldeira, M.V., Duarte, C.B., 2009. Regulation of AMPA receptors and synaptic plasticity. Neuroscience 158, 105–125. Schikorski, T., Stevens, C.F., 1997. Quantitative ultrastructural analysis of hippocampus excitatory synapses. J. Neurosci. 17, 5858–5867. Sobolevsky, A.I., Rosconi, M.P., Gouaux, E., 2009. X-ray structure, symmetry and mechanism of an AMPA-subtype glutamate receptor. Nature 462, 745–756. Tichelaar, W., Safferling, M., Keinnen, K., Stark, H., Madden, D.R., 2004. The three-dimensional structure of an ionotropic glutamate receptor reveals a dimer-of-dimers assembly. J. Mol. Biol. 344, 435–442. Trommershäuser, J., Titz, S., Keller, B.U., Zippelius, A., 2001. Variability of excitatory currents due to single-channel noise, receptor number and morphological heterogeneity. J. Theor. Biol. 208, 329–343. Uteshev, V.V., Pennefather, P.S., 1996. A mathematical description of miniature postsynaptic current generation at central nervous system synapses. Biophys. J. 71, 1256–1266. Ventriglia, F., Di Maio, V., 2000. A Brownian simulation model of glutamate synaptic diffusion in the femtosecond time scale. Biol. Cybern. 83, 93–109. Ventriglia, F., Di Maio, V., 2003. Stochastic fluctuations of the quantal EPSC amplitude in computer simulated excitatory synapses of hippocampus. Biosystems 71, 195–204. Ventriglia, F., 2004. Saturation in excitatory synapses of hippocampus investigated by computer simulations. Biol. Cybern. 90, 349–359. Ventriglia, F., 2011. Effect of filaments within the synaptic cleft on the response of excitatory synapses simulated by computer experiments. Biosystems 104, 14–22. Ventriglia, F., Di Maio, V., 2012. Effects of AMPARs trafficking and glutamate-Receptors binding probability on stochastic variability of EPSC. Biosystems (Available online 14 February 2013). Wahl, L.M., Pouzat, C., Stratford, K.J., 1996. Monte Carlo simulation of fast excitatory synaptic transmission at a hippocampal synapse. J. Neurophysiol. 75, 597–608. Zuber, B., Nikonenko, I., Klauser, P., Muller, D., Dubochet, J., 2005. The mammalian central nervous synaptic cleft contains a high density of periodically organized complexes. Proc. Natl. Acad. Sci. USA 102, 19192–19197.
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Review
Glutamate–AMPAR interaction in a model of synaptic transmission Francesco Ventriglian, Vito Di Maio Istituto di Cibernetica “E. Caianiello” del CNR, Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy
ar t ic l e in f o
abs tra ct
Article history:
Over the last several years we have investigated the excitatory synaptic response by means
Accepted 26 April 2013
of a mathematical model based on a detailed description of the synapse geometry, the
Available online 3 May 2013
Brownian motion of Glutamate molecules and their binding to postsynaptic receptors.
Keywords:
Recently, the basic model has been modified for the numbers, the size and the 3D structure
Glutamate synaptic response
of receptors according to new data from the literature. Some results of simulations
Synaptic parameters
performed with the updated model are shown here. They were aimed to study the
Binding probability
synaptic response in relation to the binding probability, to the probable height of the
Computer simulation
receptors in the synaptic cleft, and to the space-time distribution of Glutamate/Receptor collisions. A first series of simulations permitted to determine a possible range of values for the binding probability of Glutamate to receptors. Other simulations, investigating the changes induced on the synaptic response by the variations of the height of AMPA receptors in synaptic cleft, allowed to identify the height producing the higher amplitude peak of the mEPSCs. Finally, two new statistical descriptors for analyzing the synaptic response were presented. The first is based on the study of the space distribution of the number of Glutamate/Receptor collisions. Simulations investigating the effects of an increasing eccentricity of the releasing vesicle allowed assessing this method. The second one considers the inter-collision times between Glutamate molecules and binding sites. The results of some of the last simulations demonstrated its capacity to highlight the subtleties and the randomness underlying the activation of the receptors. This article is part of a Special Issue entitled Neural Coding 2012. & 2013 Elsevier B.V. All rights reserved.
Contents 1. 2.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2.1. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 2.2. Brownian model for Glutamate diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 2.3. Binding probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.4. EPSC computation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 2.5. Technical details. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
n
Corresponding author. Fax: +39 081 8675 326 E-mail addresses: [email protected], [email protected] (F. Ventriglia), [email protected] (V. Di Maio).
0006-8993/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.brainres.2013.04.051
brain research 1536 (2013) 168–176
169
3.
Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.1. Possible range of values for the binding probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.2. Effects on the mEPSC by changes of the Receptor Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.3. Effects of vesicle eccentricity on glutamate/AMPAR collision number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.4. Waiting times among collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
1.
Introduction
The main drive of the cerebral activity, the excitatory synapse, is always under investigation because of its role in information transmission between brain neurons. Despite a huge amount of experimental data and computational results, the comprehension of the excitatory synaptic function still remains blurred for many aspects. Mathematical models and computer simulations analyzed the contributions of single synaptic components in shaping the postsynaptic response and in determining the stochastic variability observed experimentally (Agmon and Edelstein, 1997; Rabie et al., 2006; Kleppe and Robinson, 1999; Trommershäuser et al., 2001; Uteshev and Pennefather, 1996; Wahl et al., 1996). Along this research line, we investigated a hippocampal excitatory synapse by means of a mathematical model based on the description of Brownian motion of Glutamate molecules within the synaptic cleft and of their interaction with the components of the synapse (Ventriglia and Di Maio, 2000). The results of our simulations, performed on a parallel computer by using an ultra-fast time scale (simulation step of 40 fs), demonstrated that intrinsic random variations in basic pre-synaptic elements of the synapse can reproduce the observed stochastic variability of the miniature excitatory postsynaptic current (mEPSC) (Ventriglia and Di Maio, 2000, 2003; Ventriglia, 2004). Based on a more precise geometrical description of the synaptic structure, our more recent simulations considered also the effects of structural elements, previously not described, such as filaments extending across the synaptic cleft outside the volume delimited by the Active Zone (AZ) and the postsynaptic Density (PSD) (Zuber et al., 2005; Ichimura and Hashimoto, 1988; Ventriglia, 2011). The results showed that their presence induced a small, but significant, increase in the response of the synapse. New published data demonstrated that the size of postsynaptic receptors is much larger than previously supposed while their number is much smaller (in particular the last statement is true for the NMDA receptors—NMDARs) (Armstrong and Gouaux, 2000; Araç et al., 2007; Malinow and Malenka, 2002; Santos et al., 2009; Tichelaar et al., 2004). By utilizing this new information we could deal better with a problem arising from the hypothesis supposed by the larger part of authors about the computation of the binding of Glutamate molecules to receptors. It is, in fact, based on the assumption of stationarity for Glutamate concentration in the synaptic cleft for times of the order of milliseconds. In real conditions, on the contrary, concentration of Glutamate is far from being stationary and its clearance from synaptic cleft is much faster (Forti et al., 1997; Ventriglia, 2004, 2011). For these reasons a computation of the Glutamate binding on the base of receptor collisions and
binding probability was proposed in recent papers (Ventriglia, 2004, 2011). In a last paper, by a geometrical reasoning and comparisons among the amplitude peaks of mEPSCs computed by our computer simulation and those present in the literature, measured by electrophysiology setups, we attempted to find a more valid value for the binding probability (Ventriglia and Di Maio, 2012). These computations did not eliminate the necessity of new computer simulations whose results are presented in this paper. The first series of simulations were carried out to investigate the possibility to identify a reasonable range of values for the binding probability. Also, according the new data about the 3D dimensions of AMPA receptors (AMPARs), some of the new simulations were carried on with the aim to obtain a better estimate of the height of the portion of receptors protruding in the synaptic cleft. In such a case we observed the changes produced on the synaptic response by the variation of this synaptic parameter. Other simulations were utilized to demonstrate the validity of two new statistical descriptors that we proposed for the analysis of the synaptic function. The first is based on the study of the space distribution of the number of collisions of Glutamate molecules on receptor binding sites. The second one describes the inter-collision time between the subsequent visits made by single Glutamate molecules on the binding sites of the different AMPA receptors, until the binding.
2.
The model
2.1.
Geometry
Our model of synapse has the form of two concentric cylinders with a common height of 20 nm. The larger one, with a diameter of 480 nm (if not otherwise specified), encloses all the synaptic cleft. Included in it, a virtual cylinder with a diameter of 220 nm (if not otherwise specified), having bases on AZ and on PSD, simulates the active synaptic space. On the top of AZ, a sphere having an inner diameter of 26.6 nm (Schikorski and Stevens, 1997) and containing Nm ¼ 775 Glutamate molecules simulates the releasing vesicle, while AMPA and NMDA receptors are modeled as small cylinders protruding in the synaptic cleft from the PSD zone, with a diameter of 14 nm. They are placed on the vertexes of a grid spanning the entire PSD (see Fig. 1). Two small circles, located randomly along a virtual ring on the exposed face of the receptor, simulate the binding sites for Glutamate; their dimensions are equal to those of the cross-section of a Glutamate molecule. Filaments within the cleft fill the annular space, external at the AZ/PSD synaptic volume, until the
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Fig. 1 – Grids of receptors and filaments. The results of the simulation related to Fig. 4 were used to obtain the path values of glutamate molecules. They have been used to show the receptor and filament positions in the synaptic cleft, as holes in the fabric woven by the flying Glutamate molecules. Receptors are more evident in the left panel (large dots in the central zone), while filaments are more visible on the right one (smaller dots in the peripheral zone). The PSD zone has a diameter of 320 nm— the largest used in this article, and accommodates 168 AMPARs and 20 NMDARs. The spacing of the two grids – Receptors and Filaments – are equal, 20 nm.
boundary of the synaptic cleft. They are simulated by thin cylinders (diameter of 7 nm) connecting pre- and postsynaptic membranes, arranged according to an ordered structure (see Fig. 1). The grid spacing of filaments and receptors is large 22 nm (or 20 nm—in some cases), according to literature data (Zuber et al., 2005; Ichimura and Hashimoto, 1988). The releasing vesicle is either centered on AZ or at a distance of 45 or 90 nm with respect to the center (eccentricity). We assume that the arrival of an Action Potential (AP) opens an expanding fusion pore within the two lipid bilayers of the cell membrane and the vesicle membrane, connecting the inner volume of the docked vesicle and the synaptic cleft. The fusion pore has been simulated as a cylinder with a length of 12 nm.
2.2.
Brownian model for Glutamate diffusion
The starting time of our computer simulation, t¼ 0, is coincident with the instant at which the area of the expansion pore is equal to the cross-section of a Glutamate molecule. We assumed that at this time the Glutamate molecules contained in the vesicle have an uniform space distribution and their velocity is distributed according to a Maxwell distribution. The opening of the fusion pore allows them to begin their travel toward the synaptic cleft due to Brownian motion. This motion has been described by Langevin equations (Ventriglia and Di Maio, 2000, 2003; Ventriglia, 2004), which, in the discretized time form, appear as ri ðt þ ΔÞ ¼ ri ðtÞ þ vi ðtÞΔ vi ðt þ ΔÞ ¼ vi ðtÞ−γ
vi ðtÞ Δþ m
ð1Þ pffiffiffiffiffiffiffiffiffiffi 2ϵγΔ Ωi m
ð2Þ
where i is the ith molecule (i ¼ 1…N; N being the total number of Glutamate molecules), Δ is the time step and Ωi is a random vector with three components, each having a Gaussian distribution with mean value μ ¼ 0 and standard deviation s ¼ 1. The other parameters are:
m, the molecular mass; γ, a friction term, which depends on the absolute temperature: γ ¼ kB T=D, where kB is the Boltzmann constant, T is the absolute temperature in Kelvin degrees, D is the diffusion coefficient of Glutamate; ϵ ¼ kB T. The value of the diffusion coefficient D for Glutamate was computed for a temperature of 37 1C (Table 1). These equations were implemented in a parallel FORTRAN program by using MPI (message passing interface) paralleling routines. The free movement of Glutamate molecules was restricted only by the collisions with the structures delimiting the model space such as the inner surface of the vesicle and of the fusion pore, the surfaces of the other synaptic structures and of the pre- and post-synaptic membranes. To obtain the greatest space sensibility, no spatial grid has been used. Therefore, a continuous range of space values could be explored by the discretized model. An extremely short time step Δ was chosen, equal to 40 10−15 s (i.e., 40 fs). It allowed an accurate description of the position of Glutamate molecules and of their collisions on synaptic structures, albeit in a nonquantum description. The molecules were considered dimensionless (points) during their free movement or in case of collision with the synaptic surfaces. Conversely, in case of collision with a hole of almost equal size, as the upper and lower openings of the fusion pore or the binding sites of receptors, a spherical or ovoid shape has been used. Since intra-synaptic transporters are very rare, we supposed that Glutamate molecules were absorbed in the synaptic cleft only with a very low probability (PR). Instead, the escaping of molecules (spillover) was due mainly to the exit through the lateral borders of the synaptic cleft. We considered that, due to the high concentration of extrasynaptic transporters near the synaptic boundary, molecules reaching that place had a very low probability to return back and consequently an absorbing boundary was imposed. Hence, the molecules absorbed within the synaptic cleft or crossing the lateral surface were not more dealt with by the program.
brain research 1536 (2013) 168–176
2.3.
Binding probability
Glutamate molecules hitting the receptors contained in the PSD can either bind or be reflected. To compute the binding probability, we used a method based on geometrical considerations we have proposed in previous papers (Ventriglia, 2011; Ventriglia and Di Maio, 2012). We used this method since the traditional method is based on the assumption of stationarity for glutamate concentration in the synaptic cleft of the order of milliseconds (Clements et al., 1992; Jonas et al., 1993), while in real conditions this is not true. In fact, when the neurotransmitters are released by a vesicle their flow is far from the stationarity (this can be deduced also looking at the Fig. 6 or Fig. 7 in this paper) and, moreover, the diffusion process of Glutamate within the synaptic cleft is much faster of what hypothesized (the neurotransmitters remain in the cleft for hundreds, and not thousands, of microseconds) (Forti et al., 1997). In summary, by assuming an elongated shape for a Glutamate molecule and a circular shape of binging sites (see Fig. 2), and by considering that a Glutamate molecule can bind a receptor only when correctly oriented (Armstrong and Gouaux, 2000), we computed a preliminary value of the binding probability (B′P ) as the ratio between the volume of a particular spherical cone including all the useful (for the binding) orientations of the Glutamate molecule and that of the unit sphere including all its possible orientations (see Ventriglia, 2011; Ventriglia and Di Maio, 2012 for a more detailed explanation). Since the binding could be also ruled by some possible, unknown frictional (or quantum) elements, the value of probability (B′P ) so computed was reduced by an arbitrary value of the 20% (BP ¼ 0:80 B′P ). Hence, by varying the angle of the cone, we could observe the effect of different binding probabilities on the synaptic response. In order not to introduce too many unknown, arbitrary parameters in the model, the same probability value was utilized to compute the binding with both AMPA and NMDA receptors.
2.4.
171
EPSC computation
Because under normal conditions the ionic channels of NMDA receptors are blocked by magnesium ions, only currents which passed through AMPA receptor have been used to compute the mEPSC. The schema used for receptor activation has been derived from the usual Markov chain with three states, namely: Basal (B)–closed, Active (A)–open, Desensitized (D)–closed, each with three sub-states 0, 1, 2, denoting the unbound, the singly-bound and the doublebound states, respectively (Clements et al., 1992; Jonas and Spruston, 1994). However, the importance of the transition to (and from) the desensitized state D in shaping EPSC time course is very controversial and Jonas and Spruston (1994) retain that the desensitization is not a major determinant of EPSC decay. Since we are interested mainly in the study of the rising phases and of the peak amplitudes of the EPSC, the transitions to D states were neglected in our computer simulations. In the simplified kinetic model of receptor activity utilized here the transitions occur in a linear cascade B0 ⇌B1 ⇌B2 ⇌A2 . It is well known that to open its ionic channel a receptor needs to be bound to at least two molecules of Glutamate simultaneously, in normal conditions. The transitions between the receptor state B2 and A2 cause random variations in the synaptic response, since the opening and the closing of a double-bound receptor follow probabilistic rules. When a receptor goes into the open state, it remains open for a random period of time τO and then it passes into the closed state and vice versa. A random variable τC is related to the period of closed state. The probability of receptor opening PO is defined as the ratio between the total opening time and the sum of the total opening and total closing time. The random variables τO and τC are distributed according to the following probability density function (PDF) of exponential form PðτÞ: PðτÞ ¼ αe−ατ
ð3Þ
with a mean value τ¼
1 : α
ð4Þ
The values of α for the two random variables are chosen in such a way that PO ¼
Fig. 2 – Spherical cone enclosing all the possible directions of glutamate molecules able to produce a binding on a receptor site. The angle of the cone represented here has a value of 1201, the largest one ever used by us – in previous articles (Ventriglia, 2004, 2011) – corresponding to a BP ¼ 0.25.
τO : τO þ τC
ð5Þ
The opening probability utilized was PO ≈0:83, different by the value PO ¼ 0:71, computed by Jonas et al. (1993). Also, the transitions from the closed state B2 to the closed state B1 obey to stochastic laws causing random variations of the synaptic response. A negative exponential distribution, with a mean value τ I , is used for these transitions. A computer program, which worked off line with respect to the simulation diffusion parallel program, used the AMPAR double binding times (i.e., the times at which the AMPARs went into the state B2) to compute the random transitions to/ from the state A2 and to the state B1 to produce single EPSC responses. The current, flowing through a generic kth AMPA receptor, was null during the closing periods, and reached a peak IMr during the opening periods. The peak current was chosen as a random variable distributed according to a
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Gaussian distribution with mean value I Mr ¼ −1:70 pA and SD¼ 0.3 pA. These values and the value of the opening probability PO were chosen because they produced the best fit with a standard mEPSC (Forti et al., 1997), that we used as reference mEPSC time course (Ventriglia and Di Maio, 2000). The time course of the computed synaptic current (Icomp ðtÞ) was obtained by adding all the single AMPA receptor currents for a time of 6 ms ( n 0 if receptor is closed ð6Þ Icomp ðtÞ ¼ ∑ Ik ðtÞ; Ik ðtÞ ¼ if receptor is opened I Mr k¼1 where n is the number of receptors and Ik ðtÞ is the current of the kth receptor at time t. For each computational experiments, 1000 current time courses –raw mEPSCs – were produced and their values have been averaged in computed mEPSCs. The amplitude peaks were used for comparisons with those of the mEPSCs recorded by experimental setups (Auger and Marty, 2000; Forti et al., 1997).
2.5.
Technical details
The program was implemented on a parallel computer, based on a cluster of four dual-processor workstations. Due to the heavy computational task, the diffusion-binding simulations were carried out for the shortest useful time, i.e., 13 109 iterations, corresponding to 520 μs (in this case each simulation lasted 4 days). At this time all the free Glutamate molecules were exited from the synaptic cleft, while the diameter of the fusion pore had reached a value of 4.61 nm. The space position (ri ðtÞð ¼ ðxi ðtÞ; yi ðtÞ; zi ðtÞÞÞ) at each discrete time for each Glutamate molecule together with the AMPA and NMDA receptor binding times were computed. The space positions of Glutamate molecules were saved on disk every 50,000 iterations (2 ns) and the receptor binding times were saved on a matrix to be used for the computation of the mEPSC. Moreover, the time and the place of a collision between Glutamate molecules and the receptor binding sites were saved to compute the space distribution of the collision number for each molecule and for each receptor.
3.
Simulation and results
3.1.
Possible range of values for the binding probability
In a previous paper we noted the possibility to define a lower limit for the probability of binding of Glutamate to AMPARs. In fact, we carried out two series of three simulations which could be related to AMPA receptor trafficking. Two values were used for the binding probability (BP ≃0:014 and BP ≃0:005—corresponding to spherical cone angles of 301 and 181) and three different values for the number of AMPARs. Results showed that, while the series linked to the greater binding probability gave rise to computed mEPSCs with amplitude peaks growing steadily as the number of AMPARs, the series with the smaller BP exhibited computed mEPSCs having amplitude peaks with an erratic behavior (see Fig 8 in Ventriglia and Di Maio, 2012). Hence, in the attempt to obtain a better approximate value for the binding probability, a new series of computer
Fig. 3 – High dispersion shown by the amplitude peak of the computed mEPSC in simulations with a BP corresponding to a spherical cone angle of 181, 154 AMPARs, 18 NMDARs and different seeds for the random number generator (only green lines, the nongreen lines are related to simulations with lower number of receptors).
simulations was performed to investigate more fully the case with BP ≃0:005 (spherical cone angle of 181), whose results are presented here. A vesicle centered at X0 ¼ 0 on AZ, with inner radius of 13.3 nm and an overall width of 38.6 nm, releasing 775 Glutamate molecules (vesicular concentration of about 130 mM) was considered. On the PSD grid were placed 154 AMPARS and 18 NMDARs in positions which did not change along the series. Receptors and Filaments had a grid spacing of 20 nm, the radius of the AZ/PSD space was equal to 160.0 nm, while the entire synaptic cleft had a radius of 240.0 nm. In this simulation series, all the geometric/biological parameters of the synaptic system remained unmodified, only the initial seed of the random number generator changed from a simulation to the other. These conditions were such that at time 0 only the position and the velocity of Glutamate molecules within the vesicle differed. The simulation results showed a large variance of the computed EPSC amplitude peak (see Fig. 3). At last, a single simulation with a very low binding probability was carried out: BP ≃0:0015—spherical cone angle of 101. The same geometrical parameters of the previous series of simulations (for the angle of 181) were used, with exception of the number of AMPARs and NMDARs which were increased to 168 and 20, respectively. A clear cut result was obtained: only two AMPA receptors were double bound and the peak amplitude of the computed mEPSC was only 1 pA (see Fig. 4), that is a value far below the lowest recorded (see Figure 4 in Forti et al., 1997). We can conclude that a reasonable value for BP seems to be in the range 0.014–0.005 (and that a realistic value should be more close to the higher value).
3.2. Effects on the mEPSC by changes of the Receptor Height In the last few years, some new data were obtained by experimental research which define better the 3D structure (and dimensions) of AMPA receptors (Mayer, 2011; Nakagawa, 2010; Sobolevsky et al., 2009; Tichelaar et al., 2004). Experimental investigations by electron microscopy at 20 Å
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Fig. 4 – Computed mEPSC for a very low binding probability BP ¼ 0.0015221209, (spherical cone angle of 101). The geometric parameters were the same of the simulations with a BP related to an angle of 181. Only the PSD zone was larger (diameter of 320 nm), as well as the number of receptors, both for AMPARs (168) and for NMDARs (20).
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Fig. 6 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 10 nm. The Z-axis shows the number of hits on both receptor sites, while the receptor position is reported on X,Y axes.
Fig. 7 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 8 nm. Fig. 5 – Computed mEPSC from simulations with three different portions of receptors protruding within the synaptic cleft: 6 nm, 8 nm and 10 nm.
resolution on a fully assembled iGluR showed that a receptor has an elongated shape with dimensions of 170 Å 140 Å 110 Å and a twofold symmetry centered on its longitudinal axis (Tichelaar et al., 2004). If we exclude the portion of receptor embedded within the postsynaptic membrane which has a thickness of almost 6 nm, the length of the portion protruding in the cleft from the PSD is unknown. Then, we investigated the effects produced on the synaptic response by the change of the height of this portion. Three different heights were simulated: 6 nm (the same that we used in all our previous simulations), 8 nm and 10 nm. For these simulations the other parameters were: Vesicle releasing 775 Glutamate molecules at a position X0 ¼ 0; Radius of AZ/PSD¼110.0 nm and Radius of entire synaptic cleft¼ 240.0 nm; BP ≃0:014; 55 AMPARs and 13 NMDARs allocated on the PSD grid; Grid spacing for Receptors and Filaments ¼22 nm. The results, shown by computed mEPSCs, are reported in Fig. 5. We can observe that the peak of maximum amplitude has been obtained with a protrusion of 8 nm. In Figs. 6 and 7 the difference between the case of protrusion of 10 nm and
8 nm is demonstrated also by the space distributions of Glutamate hits on receptor binding sites.
3.3. Effects of vesicle eccentricity on glutamate/AMPAR collision number Experimental data showed that the docking of neurotransmitter vesicles on AZ occurs in different positions (see Schikorski and Stevens, 1997, for example), and our previous simulations demonstrated the decreasing of the response (mEPSC) with the increasing of the eccentricity of the releasing vesicle. Herein, we analyzed more deeply this phenomenon – which has important implications for the synaptic function – by computing, also, the space distribution of number of collisions of Glutamate molecules on receptor binding sites. In these computational experiments the receptor protrusion height in the synaptic cleft was of 8 nm, and the releasing vesicle was positioned at X0 ¼ 0 nm (center of the synapse), or at X0 ¼ 45 nm or at X0 ¼ 90 nm from the center. The other parameters of the simulations were: Released Molecules¼ 775; BP ≃0:014 (angle 301); Number of AMPARs, 55; Number of NMDARs, 13. In Fig. 8 the computed mEPSCs for the three positions are shown. While, in Fig. 7 and in Figs. 9 and 10 the space distribution of number of collisions on receptor binding
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sites are presented, for the same vesicle positions. These last three figures demonstrated the causes of the modification due to the eccentricity of the response peak amplitudes, reported in Fig. 8.
3.4.
Fig. 8 – Computed mEPSC from simulations with a receptor protrusion of 8 nm and three different docking positions for the releasing vesicle: X0 ¼ 0 nm (center of the synapses), X0 ¼ 45 nm and X0 ¼ 90 nm from the center.
Waiting times among collisions
From one of the immediately above simulations (that in which X0 ¼ 90 nm) we extracted some information to analyze the dynamics of the binding process of a few Glutamate molecules also by means of the time elapsed between subsequent collisions on the binding sites, until the final binding, of (also different) receptors (Waiting Times). Three sequences of waiting times between successive impacts on the AMPARs binding sites are shown in Figs. 11–13. In Fig. 11 (related to molecule number 715) sites of four different receptors were visited (111 hits). Fig. 12 shows a simple situation in which only two receptors were visited (molecule No. 709 and 94 hits), while a more complex series of events occurred with the third molecule (No. 81), Fig. 13. In this case, the molecule visited five receptors, with one (receptor No. 74) encountered twice in different epochs (181 hits). The three molecules showed different behaviors also for the duration of the periods from the first impact to the final one: 240.355.305, 45.884.630, 225.321.371 time steps (9:61422984 μs, 1:8353852 μs, 9:01285490 μs), respectively. These figures demonstrated clearly the randomness of the events (bindings) inducing eventually the receptor activation and the consequent random variability of the synaptic response.
4.
Discussion
The stochastic variability of the excitatory postsynaptic current is clearly influenced by many parameters related to the synaptic structure and to the dynamic interaction between neurotransmitter molecules and receptors. By considering Fig. 9 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 8 nm and eccentricity of X0 ¼ 45 nm.
Fig. 10 – Space distributions of glutamate hits on receptor sites for a receptor protrusion of 8 nm and eccentricity of X0 ¼ 90 nm.
Fig. 11 – Sequence of waiting times. The horizontal axis shows the values of the decimal logarithm of the time differences between two successive impacts. Molecule No. 715. Four different receptors were visited. First impact, at time ¼ 190:349398720 μs (after 4.758.734.968 time steps), on receptor 13, then passing through receptors 26 and 27, and finally binding on receptor 56 at time ¼ 199:963628560 μs, after 240.355.305 time steps from the impact on first site.
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some updates to parameters of our model, related to the dimensions of receptors (much larger than supposed), to their number (smaller – mainly for NMDARs), to the ratio AMPARs/ NMDARs, we performed some simulations to study with greater details the dynamics of the binding of Glutamate molecules to postsynaptic receptors (Ventriglia and Di Maio, 2012). Some
Fig. 12 – Molecule No. 709. Only two receptors were visited. First impact, at time ¼ 117:3394581 μs, on receptor 66, then passing to receptor 58 and final binding at time ¼ 119:1748433 μs.
Fig. 13 – Molecule No. 81. In this more complex case six receptors were visited. First impact, at time ¼ 138:2470155 μs (after 3.456.175.387 time steps), on receptor 55, then passing through receptors 74, 95 (only three hits), 85, 74 again, and finally going to receptor 63 with final binding at time ¼ 147:2598704 μs, after 225.321.371 time steps from the impact on first site.
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results of those simulations showed the possibility to define a lower limit for the binding probability of Glutamate to receptors and induced us to deepen the investigation. Among all the simulations presented in this article, some were dedicated to the study the effects of a specific binding probability BP ¼0.005—related to a solid angle of 181. The results showed a large variance of the computed mEPSC amplitude peak at the changing of only the initial seed of the random number generator and demonstrated the irregularity of the synaptic response related to this binding probability. Moreover, from the results of a simulation with a very low binding probability, corresponding to a spherical cone angle of 101, we computed a mEPSC whose amplitude peak (about 1 pA) was much lower than the inferior limit of experimentally measured mEPSCs (5–64 pA; see Forti et al., 1997, for example). This demonstrates the unlikelihood of that probability. Thus, a possible range of values for the binding probability, BP, seems to be 0.005–0.014 and, however, a better approximate value must be searched for the inferior limit, at least. Results of simulations related to the portion of the receptor protruding from the PSD showed that the maximal efficiency is obtained for a protrusion of 8 nm, in correspondence of which the highest value of the mEPSC peak amplitude is obtained. This value was about 10% higher than that related to 6 nm, previously used by us (Ventriglia, 2004, 2011; Ventriglia and Di Maio, 2012). If factors related to the chemical structure of the AMPA receptor or to its physical links with the synaptic membrane or with the entire synapse, do not forbid this length, metabolic and functional considerations on the high energetic costs of the activity of the neural tissue concur to sustain the validity of this protrusion height (Attwell and Laughlin, 2001; Niven and Laughlin, 2008). In such a case, next simulations should consider this value for the height of the receptors. Another series of simulations was devoted to investigate more thoroughly the effects of the eccentricity of the releasing vesicle on the synaptic response. The study was based on the analysis of the behavior of the space distribution of the collision number of Glutamate molecules on receptor binding sites. The decreasing of the mEPSC amplitude peak with the increasing of the eccentricity was clearly related to the changes showed by this distribution. In fact, by Figs. 7, 9 and 10 we see that the peak of the collision distribution was always located at the position of the releasing vesicle, and that the increase of the eccentricity induced a shift of this distribution – and of its peak – toward the periphery of the post-synaptic density. In this way, the collision distribution assumed a form increasingly distorted, in which points (receptors) having a lower collision number occupied larger and larger portions of the synaptic space containing the
Table 1 – Simulation parameters. Temperature Glutamate diffusion coefficient at 310.16 K (37 1C) Molecular mass of glutamate Simulation time step Length of fusion pore Areal pore velocity Re-uptake probability
T D m Δ hpore Vareal Pr
310.16 K 10.0 10−6 cm2 s−1 2.4658025 10−25 kg 40 10−15 s 12 nm 31.4 nm2 ms−1 3 10−6
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receptors. These results showed a reduction of the volume of the collision distribution. Since the AMPA receptor activation depends on the magnitude of the number of collisions suffered by the receptor, through a binding probability, it is evident that the eccentricity of the releasing vesicle, by causing a lower global number of collisions, induces the production of mEPSCs with lower peak amplitudes. From some simulations of the last series above, the sequences of collisions of all Glutamate molecule on receptor binding sites, until the eventual binding, were extracted. They were used to assess the utility of the analysis of the intercollision waiting times to obtain more precise information on the binding process. We have reported here, in Figs. 11–13, three examples that showed in a visible way the great variability of the distribution of the waiting times and, at the same time, the different series of events occurred to the various neurotransmitters. In such a way, the complexity of the receptor activation and some of the fundamental elements of the randomness of the synaptic response became apparent.
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