Brownian diffusion of AMPA receptors is sufficient to explain fast onset of LTP
© Tolle and Le Novère; licensee BioMed Central Ltd. 2010
Received: 18 May 2009
Accepted: 16 March 2010
Published: 16 March 2010
Long-Term Potentiation (LTP) of synapses is thought to be due in part to a change in AMPA Receptor trafficking leading to an increase in the number of AMPA Receptors at the synapse. LTP onset occurs within seconds after the induction signal. A particle-based stochastic simulation software is used to investigate the effect of Brownian diffusion of glutamate receptors on receptor incorporation into the synaptic specialisation and the time-course of LTP expression. The model of the dendritic spine includes receptors diffusing within the membrane, scaffold molecules within the synaptic specialisation capable of binding receptors and a molecular picket-fence surrounding the synaptic membrane area, all features found within the biological system.
During simulations, receptors accumulate rapidly at the post-synaptic density (PSD) from the extra-synaptic membrane under a number of biologically observed conditions. The time of half-saturation, t1/2, defined as the time-point at which half the available scaffold proteins are occupied with receptors, is found to be 710 ms. Different scaffold distributions are shown to have little effect on this time-course. Decreasing the probability of escape of receptors from the PSD domain, thus localising receptors closer to the scaffold proteins, substantially decreases t1/2. A decrease of escape probability from 1 to 0 brings about a non-linear decrease in t1/2 from 710 ms to 390 ms. Release-location of receptors within the spine is found to affect the initial rate of receptor incorporation. We simulate three possible sources of receptors: (i) receptors distributed within the spine extra-synaptic membrane; (ii) receptors from exocytotic vesicles released to the synaptic spine; and (iii) receptors entering the spine from the dendritic shaft through the spine neck. Receptors released from exocytotic vesicles initially accumulate faster than receptors released from the other two sources. A model of glutamate release and glutamate-receptor interaction shows that newly inserted receptors make a substantial contribution to a glutamate evoked response within the observed time-frame.
Fast accumulation of AMPA Receptors is consistent with experimentally observed fast onset of LTP expression.
Fast excitatory synaptic transmission in the vertebrate brain is mediated by the α-amino-3-hydroxy-5-methyl-isoxazolepropionic-sensitive subtype of ionotropic glutamate receptors (AMPARs). These receptors are found enriched at the Post-Synaptic Density (PSD), a protein-rich, electron dense, layer located opposite the pre-synaptic active zone . Far from being static entities, AMPARs undergo movement and trafficking by lateral diffusion within the membrane, as well as to and from intra-cellular stores by endo-/exocytosis [2–4]. The movement of AMPARs has implications for the maintenance of synaptic strength during resting state, for synapse formation during synaptogenesis, and for synaptic remodelling during synaptic plasticity .
Synaptic plasticity is the capacity of the synapse to alter the efficacy of its transmission. One of the best studied forms of synaptic plasticity is Long-Term Potentiation (LTP), an activity-driven long lasting increase in synaptic strength, considered to be one of the molecular bases of learning and memory [6, 7]. LTP expression is thought to be due to the modulation of the conductance of AMPARs present at the synaptic specialisation [8, 9], a change in AMPAR trafficking leading to an increase in the number of AMPARs at the synapse [10–12], or both. The increase in signal amplitude brought about by LTP is detectable within approximately 10 seconds following the LTP induction event  and, if caused by an increase in receptor number, has been estimated to involve only a small number of additional AMPARs . The small window of time within which an increase in signal amplitude becomes detectable places constraints on the mechanism of LTP expression. The source of AMPAR molecules for incorporation into the PSD is one such constraint. Additional receptors are thought to come from intracellular stores which are exocytosed to the neuronal membrane . However, the exact locus of exocytosis has been difficult to pinpoint, with previous experiments suggesting either a site peripheral to the PSD  or at the nerve cell body . Recent experiments point to the locus being on the dendritic shaft, close to the spine, but not the spine itself , while other suggest that the receptors incorporated in the synapse come from the extra-synaptic membrane (ESM) of the spine . No exocytosis directly to the synapse or indeed to the dendritic spine membrane has been shown.
These observations, in conjunction with the discovery that AMPARs diffuse by Brownian motion in the ESM , led to the suggestion that the ESM pool of AMPARs alone could act as the source for receptors during LTP . Although the density of extra-synaptic receptors is small compared to synaptic receptors , the large area of ESM compared to synaptic membrane area gives rise to a large source of extra-synaptic receptors. In effect, the synapse acts as a diffusion-trap for the receptors within the ESM upon an LTP induction signal. Activity within a synapse, as well as an increase in intracellular calcium, as occurs during the early stages of LTP induction, have been shown to reduce the movement of AMPARs in the plasma membrane [2, 20].
A number of previous models were designed to investigate the diffusion of AMPARs in the synaptic membrane [21–23]. Earnshaw and Bressloff used a two compartment ODE model of the spine to investigate the effect of various trafficking parameters, such as the rate of exocytosis and endocytosis and exchange of receptors from the PSD to the ESM, on number of receptors in the PSD over the timescale of minutes . In a subsequent model, the authors gain insight into the diffusion of receptors along the dendrite, with spines acting as diffusion traps . The model of Holcman and Triller uses a Markovian model to determine the steady state behaviour of the synapse, and to illustrate how synaptic strength can be maintained despite the dynamics of the receptors . The authors further examine how modulation of the dendritic spine size affects the number of receptors over time scales of many seconds.
These models have either used ODE models or abstract representation of the synaptic specialization and operate on timescales of seconds to minutes. None of the models deal on the timescale of milliseconds or takes account of the microstructure of the spine and the relative positioning of the interacting components. Yet geometry and spatial parameters are important when dealing with the diffusion in the PSD [20, 24]. Particle-based monte-carlo simulations have frequently been used in the past to study movement and aggregation of membrane receptors [25, 26].
We use an in-house developed particle-based stochastic simulation software (see accompanying paper) to investigate the effect of Brownian diffusion of AMPARs on receptor incorporation into the synapse and the time-course of LTP expression. A model of the dendritic spine is detailed, including AMPARs in the ESM, scaffold molecules capable of binding AMPARs in the PSD and a molecular picket-fence surrounding the PSD. We use the software and model to show that the diffusion-trap model for LTP expression is compatible with the experimentally observed time-course of LTP. Diffusion and incorporation of AMPAR from the ESM is sufficient to explain the fast onset of LTP. We analyse the response of the system to alterations in some of the numerical parameters which influence the binding of receptors to scaffold molecules, such as the diffusion coefficient of AMPARs and the AMPAR/scaffold binding radius. As would be expected from a diffusion-reaction system, an increase in either the diffusion coefficient of AMPARs or the binding radius both lead to more rapid accumulation of AMPARs at the synapse. Increasing the number of scaffold elements relative to the number of AMPAR molecules additionally increases the rate of AMPAR capture. In contrast, changes in the distribution of AMPAR binding scaffold elements in the PSD were found to have little effect on the time-course of AMPAR capture. Furthermore, we evaluate the effect of confinement of AMPAR to a PSD micro-domain on receptor incorporation and find that confinement of the AMPARs to the PSD area increases the rate of AMPAR capture by the scaffold element, by trapping AMPARs in the vicinity of scaffold elements. Release location of AMPAR is also found to have an effect on the time-course of receptor capture.
All simulations are performed using Meredys, an in-house developed, particle-based stochastic simulation software. Models are described using an implementation of NeuroML . Meredys uses Monte Carlo algorithms to simulate molecular diffusion and reaction in a bounded simulation volume. A detailed description of the software is found in an accompanying paper.
Receptor incorporation into the PSD
0.5 μ m3
3.05 μ m2
calculated from above
Area of PSD
0.27 μ m2
calculated from above
Radius of PSD
calculated from above
Area of ESM
2.78 μ m2
calculated from above
Radius of AMPAR head particle
r AMPAR_ head
Radius of AMPAR tail particle
Radius of Scaffold particle
Diffusion Coefficient of AMPAR
0.45 μ m2/s
Receptor density in ESM
20 μ m-2
Effect of Biophysical Parameters
Reaction Rates 1.
Reaction Rate (in Ms-1)
Binding radius (in nm)
Reaction Rates 2.
Reaction Rate (in Ms-1)
Diffusion Coefficient (in μ m2/s)
Effect of Scaffold Distribution and Density
Effect of Confinement
Release location of AMPARs
The Model of Glutamate Signalling
These results show that the model can simulate glutamate signalling effectively, comparing well with published results for both previous models and laboratory experiments.
AMPA Receptor Capture during Glutamate Release
The above results show that diffusion and incorporation of AMPAR can rapidly increase the number of receptors within the PSD. However, the early incorporation of receptors may not immediately translate into an increase in excitatory post-synaptic current (EPSC) strength. It has been pointed out that the majority of receptors activated during an EPSC are done so by an initial 'spike' of glutamate concentration close to the glutamate release site . In addition, spacing between receptors has a marked effect on the height of the signal - as the spacing between receptors increases, the height of the response drops . It is expected that the accumulation of receptors occurs first at the periphery of the PSD, as the scaffold elements present there are first encountered by a diffusing AMPAR upon reaching the synapse. As a consequence, the effect of this incorporation on the EPSC needs to be further investigated.
The signalling model simulation simulates 10 ms of the glutamate signal. At the end of the signalling simulation, the state of the synaptic receptors is noted and merged with the state of the remaining receptors, taken from the output at the end of the preceding incorporation simulation. The whole procedure is then repeated.
223.4 +/- 63.29 nm
224.5 +/- 60.77 nm
221.5 +/- 63.35 nm
220.9 +/- 63.35 nm
220.1 +/- 62.9 nm
219.2 +/- 64.18 nm
217.9 +/- 64.63 nm
214.7 +/- 67.08 nm
Discussion and Conclusions
We present a biophysical realistic model to investigate the effect of AMPAR movement in the post-synaptic membrane during the initial phase of LTP expression. The effect of AMPAR diffusion parameters, and PSD scaffold composition and geometry, on the incorporation of receptors into the PSD is analysed. Further, the effect of receptor incorporation into the synapse on the post-synaptic signal are examined. The model system incorporates AMPARs diffusing in the membrane, scaffold proteins, capable of binding AMPARs, distributed within the PSD, and glutamate release from postsynaptic stores and interacting with membrane receptors. Knowledge of the distribution of receptors within the synaptic membrane [35, 36] was used in the construction of the models. The diffusive behaviour of AMPARs, as observed in particle-tracking experiments , was also incorporated in the models. None of the models of AMPAR diffusion to date have probed the effect of the different distributions of scaffold elements on the incorporation of AMPARs at the synapse. Yet, theoretical models have shown that the placement of traps can affect the rate of diffusion-limited processes substantially .
The model and accompanying simulation results support the hypothesis that AMPARs can come from the pool of extrasynaptic receptors to cause LTP expression within the allotted time and by random diffusion alone. For the range of measured diffusion coefficient and a range of binding radii, AM-PARs can accumulate within the PSD within the time frame of LTP expression . The response of the model to changes in the ratio of scaffold elements to AMPARs, different initial distributions of both scaffold elements within the PSD and AMPARs within the ESM, and a change in the confinement of AMPARs to the PSD area is analysed. The time of half-saturation, t1/2, was used as a measure of the speed of binding. It is dependent on the diffusion coefficient of the receptors, the binding radius of the receptor-scaffold interaction, the number of interacting components, as well as the average initial distance of the receptors from the scaffold elements. This distance, in turn, is dependent on the receptor and the scaffold initial distributions.
AMPAR movement in the PSD is thought to be affected mainly by two factors: (i) interaction with scaffold molecules, and (ii) entrance/exit rates of receptors to/from the PSD. The exact nature of the protein responsible for anchoring AMPARs to the PSD during LTP induction remains elusive. The search is made more complicated by the difficulty in differentiating between molecules responsible for targeting AMPARs to the PSD as compared to molecules responsible for maintaining AMPARs at the PSD . Either may also be different for different AMPAR subtypes , or may not even bind to AMPARs at all, but to their auxiliary proteins instead [43, 44]. As a consequence, it is difficult to estimate the affinity of AMPARs for scaffold elements, or the density and distribution of scaffold proteins in the PSD. Several plausible models are considered in this study.
The model system uses the binding radius, the maximum distance two molecules can approach each other before reacting, as a measure of the affinity of AMPARs for the scaffold binding molecules, as detailed by Andrews and Bray . The binding radius is derived from Smoluchowski's theory for reaction rates , and in the algorithm is calculated from the reactants diffusion coefficients, the reactions experimental reaction rate, and the Brownian Dynamics algorithms step length. For diffusion limited reactions this is equal to the sum of the molecular radii of the interacting components . As the exact nature of the protein-protein interaction trapping AMPARs at the synapse is unknown and experimental reaction rates are missing, a range of possible binding radii is tested. All the radii fit into a biologically meaningful range. The results indicate that for all the binding radii tested incorporation still proceeds rapidly within the seconds range (figure 1a). Whether, and how, an LTP induction stimulus can rapidly regulate the anchor sites remains to be determined. A likely model is that anchor molecules are already present at the synapse, and "activated" by the rise in Ca2+ brought about by Ca2+ influx through the NMDAR. Such a model would be consistent with the observed decrease in receptor mobility following Ca2+uncaging .
Rates of reactions in the above system also depend on the diffusion coefficient of the reacting entities. The effect of the AMPAR diffusion coefficient on the time course of receptor incorporation are seen in figure 1c. Factors influencing the diffusion coefficient of a protein in a membrane include the radius of the proteins membrane spanning region and the viscosity of the membrane among other factors. A number of diffusion coefficients have been measured for AMPARs within the neuronal plasma membrane using single-molecule fluorescent microscopy , possibly reflecting the heterogeneity of the lipid environment in the neuronal membrane , as well as the association of AMPARs with other membrane spanning proteins .
A number of possible distributions for AMPARs at the PSD, ranging from uniform  to annular  or patchy , have been determined. The exact ultrastructure of the PSD has not been determined, but presumably the observed distribution of AMPARs reveals the underlying distribution of AMPAR binding scaffold proteins in the PSD. As the placement of traps in different spatial arrangements can have a substantial effect on the rate of diffusion-limited processes such as the diffusion to capture , all of the above distributions were tested. Distribution of scaffold molecules within the PSD has little effect on the time course of receptor capture (Figure 5). Although the annular distribution displayed a slightly slower rate after an initial period, this is most likely due to the larger number of receptors closer to the edge of the PSD domain in the uniform and patch distribution compared to the annular distribution. Regardless of distribution, scaffold elements do saturate rapidly.
In the model, corralled diffusion within the PSD area is examined. The restriction to diffusion is uni-directional only, with AMPARs allowed to enter the PSD area but restricted in exit from the PSD. This restriction localises the AMPARs to the PSD and hence in the vicinity of the scaffold molecules. The effect of the PSD corral on the incorporation of receptors is noticeable for the duration of the measurements, with a more secure corral leading to an increase in the initial rate of receptor incorporation, as well as a lower t1/2. Whether a similar mechanism is utilised in vivo remains to be seen. Receptors have been shown to undergo confined diffusion [24, 28] once they enter the synapse. Even the synapse itself appears to contain sub-domains . The exact reason for this is as yet unknown, though models suggest that synaptic strength can vary strongly depending on the correlation of post-synaptic receptor placement and presynaptic glutamate release [14, 47]. It should be noted that the experimental data for AMPAR diffusion does not allow for the differentiation between confined diffusion and obstacle-impeded diffusion . Although the above model assumes diffusion within a corral, both processes probably influence synaptic AMPAR diffusion in vivo.
The source of the AMPARs required for LTP expression may be receptors present in intracellular stores [10, 49]. However, the locus of receptor exocytosis has not yet been determined. Various methods used have placed the location of exocytosis into the spine but peripheral to the PSD , in the dendrite close to the spine but not the spine itself , or at the nerve-cell body . All of these scenarios require the AMPARs to translocate to the PSD. The latter two depend on AMPARs entering the spine through the spine neck. If the spine neck can act as a diffusion barrier  then this may require the utilization of motor proteins accounting for the observation that myosin Va is required for AMPAR insertion into the synapse . In either case, the release location of AMPARs affects the time-course of receptor incorporation. Exocytosis closer to the PSD greatly increases the initial rate of receptor capture to the PSD. The rates for the three release distributions tested converge as the remaining receptors in each simulation series reach diffusional equilibrium.
The contribution of newly incorporated receptors to the glutamate evoked signal is measured. It has been proposed that only a few extra open AMPARs may be necessary to increase the amplitude of the signal for LTP . The same model suggests that 80% of the current is carried by channels in a 240 nm diameter region around the release site. The model presented shows that receptors captured to the synapse following a diffusion/trap model are first incorporated at peripheral binding sites within the PSD, assuming uniformly distributed anchor molecules. It is conceivable that the sequestering of receptors by binding sites at the periphery of the PSD could lead to insufficient proximity of newly acquired receptors to the glutamate release site for the receptors to participate in the signal. However the model shows that distant receptors still contribute to the glutamate signal. In addition, the model suggests that newly acquired receptors contribute to the signal very early on in the incorporation process. This observation is consistent with the idea of extrasynaptic receptors acting as the source of new receptors during LTP expression.
Many questions remain to be answered, and as more data becomes available, the details of the model will change and be refined. The kinetics of the interaction of receptors with scaffold proteins should be further investigated. Anomalous diffusion has been observed for receptors diffiusing in the synapse, and attributed to confinement [2, 5]. However the causes of anomalous diffusion can be many and, as previously mentioned, the available data does not point conclusively to diffusion within a confined domain . Transient interactions can lead to similar behaviour. Research suggests that the PSD itself may be divided into specific sub-domains which impact on the glutamate evoked signal . This division of the synapse into subcompartments requires more examination. Especially the organisation of receptors in a sub-domain on the EPSC, how receptor concentrations can be controlled at the level of the sub-domain, and the effect of sub-domain correlation with the glutamate release site on the EPSC need to be addressed. Mobility of receptors within sub-domain and exchange between sub-domains, as well as exit and entrance from synapse are clearly factors affecting the incorporation of receptors into the synapse and the resulting increase in glutamate evoked current.
Model of the dendritic spine
The model used to describe the receptor movements in the dendritic spine includes the compartmentalisation of the dendritic spine plasma membrane into distinct membrane domains, diffusion of receptors within the plasma membrane, and the presence of scaffold molecules in the synaptic area capable of binding the receptors. The effect of changing various parameters on AMPAR accumulation at the PSD are investigated in this study. What follows is the description of an incorporation reference model used as the prototype for the subsequent construction of specific models. The various parameter values used in the reference model are given in Table 1.
The plasma membrane of the synaptic spine is modelled as a square with a surface area, A spine . The two membrane compartments that comprise the plasma membrane of the dendritic spine are the ESM and the synaptic plasma membrane corresponding to the PSD.
Molecules within the membrane
AMPAR entities are embedded in the membrane where they are allowed to diffuse freely. The density of AMPARs in the ESM is taken from values reported in the literature . A cytoplasmic tail part allows AMPARs to interact with the scaffold entities located below the plasma membrane. Scaffold molecules are represented as separate, static entities. Scaffold entities are placed just below the PSD membrane domain to allow interaction with the tail region of receptor entities. The molecular nature of the anchoring site for AMPARs at the PSD is still not determined, and may well depend on the state of the individual synapse, as well as on the subtype of AMPAR [31, 32]. Since the identity of the AMPAR binding scaffold is not known there are no experimentally observed values for the density of the scaffold elements within the PSD. We investigate the effect of changing scaffold density. For the incorporation reference model, however, we assume that the number of anchors is equal to the number of free AMPARs in the ESM.
Distribution of Molecules in the Membrane
Each scaffold entity initial placement in the PSD was determined as follows:
Polar coordinates for the position of the entity in the simulation volume where created by drawing two random numbers, R and ϕ from U(0,1) and U(0,2π), respectively and transformed into Cartesian coordinates by x = cos(ϕ) * * radius psd and y = sin(ϕ) * * radiuspsd.
The PSD disk was divided into 5 concentric circles each of thickness . Each segment had a probability associated with it of a receptor being placed within it determined from the experimental data of Kharazia & Weinberg . The scaffold entities are placed uniformly (see above) within each segment.
The PSD was composed of 5 disks of radius 96 nm, corresponding to the confinement radius measured in active synapses , arranged as a pentagon, with the centres of the disks 194.4 nm from the centre of the PSD. Receptors were placed as for the Uniform distribution above. Receptors that did not fall into any of the 5 disks were replaced.
AMPAR entity initial placement in the ESM area was determined as follows:
Each Entity Cartesian coordinates were determined by drawing X and Y from U(- , ). The distance of (x,y) from the origin was calculated and if found to be less than radius PSD , the point was discarded and a new pair of random numbers created.
Polar coordinates for the position of the each entity in the simulation volume where created by drawing one random number, ϕ from U(0,2π). Coordinates where transformed into Cartesian coordinates by x = cos(ϕ) * d full and y = sin(ϕ) * d full Where d full is 872.5 nm from the PSD centre, corresponding the point of contact with the spine neck.
Three point sources were randomly determined by drawing one random number, ϕ from U(0,2π). Coordinates were transformed into Cartesian coordinates by x = cos(ϕ) * d half and y = sin(ϕ) * d half Where d half is 583.95 nm from the PSD centre, corresponding to a point half way between the edge of the PSD and the point of contact with the spine neck. The first two points determined the initial position of 18 AMPAR and the last point determined the position of 19 AMPAR.
The Model of Glutamate Signalling
Kinetic constants. Kinetic constants for AMPAR
Temperature adjusted value (Q10= 3.0)
4.59 * 106 M-1s-1
23.85 * 106 M-1s-1
4.26 * 103 s-1
22.14 * 103 s-1
28.4 * 106 M-1s-1
147.57 * 106 M-1s-1
3.26 * 103 s-1
16.94 * 103 s-1
1.27 * 106 M-1s-1
6.6 * 106 M-1s-1
4.24 * 103 s-1
22.03 * 103 s-1
2.89 * 103 s-1
15.02 * 103 s-1
Receptor movement in the synaptic spine was simulated using the Meredys simulation software (Available at: http://www.ebi.ac.uk/compneursrv/meredys.html). All simulations were run on a Centos 4.2 Linux LSF Cluster. The individual hosts used were a mixture of 32 bit and 64 bit machines. The cluster contains approximately 470 CPU cores across 130 machines. Each run simulated the movement of receptors across the dendritic spine membrane. The parameters for the 'prototypical' reference model are outlined in Table 1. Each change in a parameter from the reference model as indicated in the text was tested with a simulation series. A simulation series consisted of a total of at least 30 individual simulations. The random number generator of the simulation software was seeded with a different values for each simulation. Results obtained were averaged over the number of simulations in a series. Each simulation was run for at least 5 * 106 iterations, and each iteration had a step length of 1 μ s, amounting to a total simulated time of at least 5 s. Output was captured in text files analysed with Perl scripts. The NeuroML input files of the reference model for Meredys used for the simulation can be found in the supplementary material.
Calculation of reaction rates
The Smoldyn algorithm used in Meredys requires reaction rates to be supplied to in order to determine an appropriate binding radius, σ. These rates are calculated from the desired binding radius, the step-length and the diffusion coefficients of the interacting entities by in-house developed software utilising the Smoldyn algorithm. Tables 2 & 3 show the input rates and the resulting binding radius.
Determination of MSD plot
Where (x i (0), y i (0) is a particles initial position, and (x i (t), y i (t) is a particles position at time t. N is the total number of particles and i is the particle index.
Construction of the trace
The trace of a molecule within the membrane was constructed from simulation output file using in-house built software for converting Meredys position output information into a trace file. The program takes the position of a particle for successive iteration steps and connects them with straight lines.
We thank D. Choquet for insightful discussion, Sarah Birch for careful reading of the manuscript and Anton Enright for aid with the creation of the images. Both DPT and NL were supported by the European Molecular Biology Laboratory.
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