A Brownian model of glutamate diffusion in excitatory
synapses of hippocampus
Francesco Ventriglia *, Vito Di Maio
Istituto di Cibernetica,CNR,Via Toiano6,I-80072Arco Felice(NA),Italy
Abstract
We simulated the diffusion of glutamate, following the release of a single vesicle from a pre-synaptic terminal, in the synaptic cleft by using a Brownian diffusion model based on Langevin equations. The synaptic concentration time course and the time course of quantal excitatory post-synaptic current have been analyzed. The results showed that they depend on the number of receptors located at post-synaptic membrane. Their time course are dependent both on the total number of the post-synaptic receptors and on the eccentricity of the pre-synaptic glutamate vesicle. © 2000 Elsevier Science Ireland Ltd. All rights reserved.
Keywords:Brownian diffusion model; Langevin equations; Glutamate; Synaptic transmission; Hippocampus
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1. Introduction
The understanding of the mechanisms underly-ing synaptic transmission is of fundamental im-portance for the modern Neurobiology. This importance becomes also more apparent in the study of hippocampus where synaptic transmis-sion, more evidently then in other brain regions, produces those phenomena as the long-term po-tentiation (LTP) and the long-term depression (LTD) that are considered the building blocks of memory formation and learning.
A simulation model based on the kinetic theory of neural system (Ventriglia, 1994, 1998) has been developed in order to investigate neural activity of the CA3 sub-field of hippocampus. Although
real-istic from a biological point of view, in this model synaptic activity has been considered only by a theoretical point of view. Because the understand-ing of activity of hippocampus sub fields cannot neglect a detailed treatment of synaptic transmis-sion, we started the study of synaptic diffusion in excitatory synapses of hippocampus (Ventriglia and Di Maio, 2000).
The main excitatory neurotransmitter of brain, and of hippocampus in particular, is the amino acid glutamate. In the hippocampus, synapses formed by the granular cells of the Dentate Girus on the pyramidal cells and synapses made by the Schaffer’s collaterals of pyramidal neurons, both on the inhibitory inter-neurons and on pyramidal neurons release glutamate. The detailed regulation of these synapses is responsible for LTP (Yeckel et al., 1999) and LTD (Carroll et al., 1999) gener-ation. Post-synaptic electrical activity is carried on
* Corresponding author.
E-mail address:[email protected] (F. Ventriglia).
by ionotropic (i.e. associated with fully functional ion channels) receptors. These receptors are divided into two main groups:N-methyl-D
-aspar-tate-sensitive (NMDARs) and a -amino-3-hydr-oxy-5-methyl-4-isoxazolepropionic-sensitive (AM-PARs). When bound by glutamate they carry a depolarizing inward current named excitatory post-synaptic current (EPSC). Glutamate molecules are packed in small vesicle placed at the pre-synaptic terminal. A single vesicle contains a quantum of molecules that is usually assumed constant in the same synaptic terminal. The EPSC produced by a single vesicle release is named quantal-EPSC. NMDARs and AMPARs play a different role in post-synaptic response. AMPARs when bound by glutamate give a fast response, but have a lower affinity for the neurotransmiter than NMDARs (Lester et al., 1990). On the other hand, NMDARs are blocked by Mg2+ ions that
are removed only after a relevant membrane de-polarization (Bliss and Collingridge, 1993). In ad-dition, NMDARs activation is Ca2+ dependent.
AMPARs and NMDARs are co-localized at the post-synaptic membrane to form the post-synaptic density (PSD) (Takumi et al., 1999). LTP and LTD consist mainly of long-term modifications of post-synaptic electrical activity. The basic mecha-nisms by which these LTP and LTD are regulated are, however, still matters of debate. Regulation of synaptic transmission can act both at pre- and post-synaptic level influencing LTP and LTD for-mation. Mainly, pre-synaptic regulation is thought to act modifying the probability of vesicle release following pre-synaptic electrical activity (for a review see Miller, 1998). At the post-synap-tic level, regulation can be carried through mecha-nisms involving post-synaptic receptors both of ionotropic and metabotropic type. The number of post-synaptic receptors, their spatial organization, the relative number of AMPARs and NMDARs, their affinity for glutamate and the different dy-namics of binding can be modulated (Gomperts et al., 1988; Clements et al., 1992; Ottersen and Landsen, 1997).
After the arriving of a pre-synaptic spike (or spontaneously), a vesicle of neurotransmitter fuses with the cell membrane and the molecules of the neurotransmitter diffuse in the synaptic cleft and
bind to the receptors at the post-synaptic mem-brane producing the EPSC. Large variability has been found either in the production of EPSCs (produced by more quanta) and of quantal EPSCs (Faber et al., 1992; Jonas et al., 1993; Liu et al., 1999). A large debate arose concerning the causes of quantal EPSC variability. On one side there is the possibility that glutamate can spillover the synaptic cleft activating extra-synaptic receptors (Kullmann and Asztely, 1998). Such spillover, should produce activation of NMDARs that have high affinity for glutamate and not of AMPARs since the latter have a low glutamate affinity and would bound very rarely the few molecules arriv-ing from the extra-synaptic space. This theory, however, can explain EPSCs variability and not quantal EPSCs variability. The finding that some synapses are silent for AMPAR, that the number of post-synaptic receptors can vary as a function of synaptic activity and that post-synaptic recep-tors can be reorganized at the PSD, support a post-synaptic hypothesis for quantal EPSC vari-ability (Gomperts et al., 1988; Forti et al., 1997; Ottersen and Landsen, 1997; Nusser et al., 1998; Takumi et al., 1999). Of particular interest for the LTP phenomenon is the issue of the saturation of the post-synaptic receptors in response to the release of neurotransmitters from a single vesicle. In fact, whereas the lack of saturation can address the origin of LTP towards pre-synaptic mecha-nisms, such as variability of the amount of neuro-transmitters released, the receptor saturation can
shift importance towards post-synaptic
mechanisms.
who used different models and/or different val-ues for simulation parameters (Faber et al., 1992; Agmon and Edelstein, 1997; Kruk et al., 1997). The use by these authors of a much higher number of transmitter molecules in a quantal release, with respect to the number that can be computed by the data reported by Shikorski and Stevens (1997) for CA1 synapses in the adult mouse, is of particular interest. If, as reported by these last authors, the external vesicle mean diameter is 35.2 nm, assuming a vesicular membrane thickness of 6 nm, the inner vesicle diameter will be 23.2 nm. By using a vesicular glutamate concentration of 130 mM (in the range between 60 and 210 mM, Clements et al., 1992), we obtained a number of molecules per vesicle of about 500 (Ventriglia and Di Maio, 2000) against a common usage of 1000 – 10 000 molecules. In addition, the temporal step we used for simulation is several order of mag-nitudes lower than that used by other authors. We also introduced in our model the passage of molecules through the fusion pore that is impor-tant for the diffusion time-course of glutamate.
Here we present a new series of computa-tional experiments in which we analyze new causes for quantal EPSC variability. We tested how the variability of post-synaptic responses can be influenced by the AMPARs density and by the eccentricity of the pre-synaptic vesicle.
2. Model
The Brownian model for the synaptic diffu-sion used in the present work was described in Ventriglia and Di Maio (2000). The consider-ation of an ensemble of Brownian particles
de-scribing the diffusion of the pool of
neurotransmitters contained in a single vesicle is the main idea of that article. Describing the dy-namics of the single neurotransmitter molecules by appropriate Langevin equations, we were able to avoid the space discretization necessary when Monte Carlo or diffusion equations meth-ods are used and we were allowed to deal with the geometry of the synaptic structures in a very simple and direct manner. By using time
dis-cretized Langevin equations, we simulated the release of a single vesicle of neurotransmitter, the diffusion of neurotransmitter molecules in the synaptic cleft, their binding on post-synaptic receptors, their re-uptake, and their spillover.
We assumed that, at time t=0, the neuro-transmitter molecules, as Brownian particles, were all inside the synaptic vesicle (a sphere) and they were distributed uniformly in space and according to a Maxwell distribution in the velocity. At a time ts\0, the simulated arrival
of a pre-synaptic spike produced the opening of a hole (the fusion pore) in the vesicle membrane fused with the pre-synaptic membrane. In our model, the fusion pore is assumed as a cylindri-cal space with a fixed diameter and fixed height. The opening of the fusion pore allowed the neu-rotransmitter molecule, according to their erratic Brownian motion, to diffuse in the synaptic cleft that we simulated as a flat cylinder (synaptic cylinder) with prefixed height and diameter (20 and 400 nm, respectively). A pre-synaptic active zone with a diameter 220 nm (with docked vesi-cles) and a juxtaposed PSD of the same
diame-ter containing AMPARs and NMDARs
completed the geometry of our model forming a cylinder (PSD cylinder) included in the synaptic cylinder. The motion of each Brownian particle, with mass m, position ri, and velocity 7i, can be
described by the following Langevin equation:
r;i=6i (1a)
m6;
i= −g6i+2og&0xFFFD;bi(t) (1b)
where the friction parameter g and the term of stochastic force (in Eq. (1b)) are due to the in-teraction of each neurotransmitter molecule with the molecules of water assumed to fill all the model space where neurotransmitter molecules move. The index i stands for the ith of the N molecules contained in a vesicle. As stochastic force we supposed a white Gaussian noise (ji(t)jj(t+D)=dijd(D)) with intensity 2og.
The friction term g is dependent on the abso-lute temperature according to the following equation
g=kBT
In these equations kB is the Boltzmann constant,
T is the absolute temperature in Kelvin, Dis the diffusion coefficient of glutamate ando=kBT.
The free Brownian motion of neurotransmitter molecules was limited only by the interaction with the structures limiting the model space. The pre-synaptic surface contained receptors that could absorb neurotransmitter molecules (re-uptake). The collisions of a molecule with the pre-synaptic surface could then produce, with a prefixed prob-ability PR, the re-uptake of the molecule. The
molecules crossing the lateral boundaries of the synaptic cleft were lost by spillover. We assumed that the surface distribution of the receptors on the PSD was uniform for both types, and, more-over, as Agmon and Edelstein (1997), we postu-lated that each receptor was located within a small square and the totality of them formed a square matrix representing the PSD. When a neu-rotransmitter molecule hit the post-synaptic sur-face it could bind, with a probability PB, to the
receptor enclosed in the square containing the contact point. We assumed that each receptor, both AMPAR and NMDAR, had two binding sites for glutamate (Clements et al., 1998) and that the probability to bind to the second site was one half that to bind to the first one. After a molecule bound a receptor, it remained attached for a timetbinding. When a second molecule bound
a receptor, it could go to the open state with a probability (Po) of 0.71 (Jonas et al., 1993) and in
this case an ionic current started to flow inside the post-synaptic cell. Dynamics of channel conduc-tance and unbinding of molecules from the recep-tor were not considered in the present experiment since tbinding is much higher than our simulation
time.
The following time discretized Langevin equa-tions for diffusion were implemented in a parallel FORTRAN program by using MPI (message passing interface) routines
m &0xFFFD;bt
(2b) where vt is a random vector with three
compo-nents, each having a Gaussian distribution with
mean value m=0 and S.D. s=1; D is the time step. The program ran on 12 processors one of which was the root processors and the other were the slaves. The parallel program attributed at each slave processor a prefixed number of molecules and stochastic paths were computed for all the molecules. A parallel random number generator was used to compute, at each time step, the random vector vt, and the re-uptake and binding
probabilities. The space position ri(t)=(xi(t),
yi(t),zi(t)) of each molecule was saved as a
func-tion of time at every 50 000 iterafunc-tions. The post-synaptic binding co-ordinates and time of binding were saved at the time of occurrence.
3. Simulation and results
Two different causes of quantal EPSC variabil-ity, one pre-synaptic and one post-synaptic, were investigated by using the program described above. The glutamate time-course in the synaptic cleft, the collective binding properties of the AM-PAR population in the PSD and the related com-puted quantal EPSC were analyzed under two different assumptions. The common values of the model parameters are presented in Table 1. Due to the small computational time step — 40 fs, each experiment ran for 4×109 iterations
(corre-sponding to 160 ms) to obtain for the computed EPSC a time course comparable with the risingpe-riod of those found in electrophysiological experi-ments. The computed quantal EPSC for
AM-Table 1
Common parameters of the model used in all the experiments
T 298 K Temperature
D
Glutamate diffusion coefficient 7.6×10−6cm2
s−1
2.4658025×10−25 m
Molecular mass of glutamate
kg Simulation time step D 40×10−15s
dpore
Diameter of fusion pore 4 nm 12 nm Height of fusion pore hpore
Open probability of AMPA Popen 0.71
Fig. 1. Comparison of concentration profile and quantal EPSC produced in experiment with variable position of the fusion pore (A) and variable number of AMPARs (B).
PARs was obtained by summing the unitary cur-rent (Iu=0.67 pA, Jonas et al., 1993) of each
opened receptor in the time course.
In the first series of computational experiments, the position of the pre-synaptic vesicle varied with respect to the axis of the synaptic cylinder (X0=
0), to study the effects of vesicle eccentricity. For this computational series, we made a first experi-ment with the vesicle centered at X0=0 (used as
reference) and three experiments with theX0point
shifted along theX-axis. The three positions were respectively atX0=30, X0=60 and X0=90 nm.
We fixed 157 AMPARs and 157 NMDARs on the PSD. The results are presented in Fig. 1A and Fig. 2A and B. In Fig. 1A, the time-courses of the total neurotransmitter concentration, computed for the PSD cylindrical volume and of the EPSCs due to AMPARs activation are shown for all the
experiments. This figure shows that the shifting of the vesicle towards the boundary of the active zone modifies the amplitude of concentration time course of the neurotransmitter and of quantal EPSCs. In Fig. 2, the upper part of each subfigure (A, B and C) represents a frame containing the total AMPARs bound after 160 ms and the com-puted quantal EPSC time course. The lower part represents frames containing the total receptors bound by two molecules at different time inter-vals. By looking at Fig. 2A (X0=0) it has to be
noted that after a time shorter than the simulated time all the neurotransmitters are bound, spilled over or re-uptaken while receptors are not yet saturated. An annulus of no-bound receptors is present in the PSD. Fig. 2B, related to the case of X0=90 nm, shows that a part of the annulus is
PSD larger than the one needed for saturation is present. With respect to this observation, a rea-sonable hypothesis could be that the annulus is the result of an evolutionary process tending to reduce the quantal EPSC variability merely due to the eccentricity of the vesicles.
In the second series of experiments, the number of AMPARs was varied whereas the number of NMDARs and the diameter of the PSD remained fixed (respectively 157 NMDARs and 220 nm for the PSD). The number of AMPARs varied along four experiments being 20, 70, 98 and 157 respec-tively. These values reflected data from literature (Forti et al., 1997; Nusser et al., 1998; Takumi et al., 1999). Because the PSD area was kept con-stant, AMPAR density decreased by decreasing the number of AMPARs while the size of each component the post-synaptic matrix containing a receptor increased. Results of this series of experi-ments are shown in Fig. 1B and Fig. 2A and C. In Fig. 2C the extreme case, only 20 AMPARs, is presented. The reduction of quantal EPSC due to a reduced number of AMPARs has been found as expected. The concentration profile time course is also influenced by the total number of receptors as can be noted by comparing Fig. 1A and B where it is apparent the reduction of the concen-tration while EPSC and AMPARs increase. This effect is notable because an important factor in reducing the synaptic concentration of free gluta-mate consists in its binding to the receptors.
Finally, a single computational experiment has been carried on to explore a case in which the number of molecules contained in the synaptic vesicle was higher than usual. For this experi-ment, we used the upper value of the range of vesicular concentration for glutamate (i.e. 210 mM). This concentration value gives a number of 900 molecules for the inner diameter we assumed for the vesicle. In this simulation we used 157 AMPARs and 157 NMDARs while the vesicle was docked at X0=0. The results of this
simula-tion (data not presented) showed again no satura-tion of the receptors placed on the PSD. Although reduced, the annulus of free receptors was yet present.
4. Discussion
The results of the present computational experi-ments confirmed that the synaptic geometry plays an important role in the process of neurotransmit-ter diffusion and in the production of quantal EPSC. This effect is mainly visible when small vesicles are present as in the case of CA1 and CA3 excitatory synapses of hippocampus. In these cases, the number of molecules inside the vesicle is a limiting factor to the saturation of post-synaptic receptors. If the number of diffusing molecules is so close to the number of receptors placed on the PSD, then the probability of saturation is very low. This can be an important factor in determin-ing a stochastic variability of quantal EPSC in the same synapses during different trials. To get satu-ration, a number of molecules much higher than that found by using data from Shikorski and Stevens (1997) must be used and this could be the reason because other authors described post-synaptic receptors saturation. The problem of concentration of glutamate in the vesicles need to be fixed more precisely since it would be an important factor for a better understanding of LPT and LTD phenomena in the hippocampus. Although significant, vesicle eccentricity seems to play a minor role in amplitude and time course regulation of quantal EPSC. Because an annulus of free receptors remains at the PSD after diffu-sion, it plays a conservative role in maintaining a low variability when the released vesicle is far from the center of the active zone.
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