Compatibility between itinerant synaptic receptors and stable postsynaptic structure

Cellular diffusion processes in singularly perturbed domains

There are many processes in cell biology that can be modeled in terms of particles diffusing in a two-dimensional (2D) or three-dimensional (3D) bounded domain Ω ⊂ R d containing a set of small subdomains or interior compartments U j , j = 1 , … , N (singularly-perturbed diffusion problems). The domain Ω could represent the cell membrane, the cell cytoplasm, the cell nucleus or the extracellular volume, while an individual compartment could represent a synapse, a membrane protein cluster, a biological condensate, or a quorum sensing bacterial cell. In this review we use a combination of matched asymptotic analysis and Green's function methods to solve a general type of singular boundary value problems (BVP) in 2D and 3D, in which an inhomogeneous Robin condition is imposed on each interior boundary ∂ U j . This allows us to incorporate a variety of previous studies of singularly perturbed diffusion problems into a single mathematical modeling framework. We mainly focus on steady-state solutions and the approach to steady-state, but also highlight some of the current challenges in dealing with time-dependent solutions and randomly switching processes.

Asymptotic analysis of particle cluster formation in the presence of anchoring sites

The aggregation or clustering of proteins and other macromolecules plays an important role in the formation of large-scale molecular assemblies within cell membranes. Examples of such assemblies include lipid rafts, and postsynaptic domains (PSDs) at excitatory and inhibitory synapses in neurons. PSDs are rich in scaffolding proteins that can transiently trap transmembrane neurotransmitter receptors, thus localizing them at specific spatial positions. Hence, PSDs play a key role in determining the strength of synaptic connections and their regulation during learning and memory. Recently, a two-dimensional (2D) diffusion-mediated aggregation model of PSD formation has been developed in which the spatial locations of the clusters are determined by a set of fixed anchoring sites. The system is kept out of equilibrium by the recycling of particles between the cell membrane and interior. This results in a stationary distribution consisting of multiple clusters, whose average size can be determined using an effective mean-field description of the particle concentration around each anchored cluster. In this paper, we derive corrections to the mean-field approximation by applying the theory of diffusion in singularly perturbed domains. The latter is a powerful analytical method for solving two-dimensional (2D) and three-dimensional (3D) diffusion problems in domains where small holes or perforations have been removed from the interior. Applications range from modeling intracellular diffusion, where interior holes could represent subcellular structures such as organelles or biological condensates, to tracking the spread of chemical pollutants or heat from localized sources. In this paper, we take the bounded domain to be the cell membrane and the holes to represent anchored clusters. The analysis proceeds by partitioning the membrane into a set of inner regions around each cluster, and an outer region where mean-field interactions occur. Asymptotically matching the inner and outer stationary solutions generates an asymptotic expansion of the particle concentration, which includes higher-order corrections to mean-field theory that depend on the positions of the clusters and the boundary of the domain. Motivated by a recent study of light-activated protein oligomerization in cells, we also develop the analogous theory for cluster formation in a three-dimensional (3D) domain. The details of the asymptotic analysis differ from the 2D case due to the contrasting singularity structure of 2D and 3D Green's functions.

Quantifying postsynaptic receptor dynamics: insights into synaptic function

The molecular composition of presynaptic and postsynaptic neuronal terminals is dynamic, and yet long-term stabilizations in postsynaptic responses are necessary for synaptic development and long-term plasticity. The need to reconcile these concepts is further complicated by learning- and memory-related plastic changes in the molecular make-up of synapses. Advances in single-particle tracking mean that we can now quantify the number and diffusive properties of specific synaptic molecules, while statistical thermodynamics provides a framework to analyse these molecular fluctuations. In this Review, we discuss the use of these approaches to gain quantitative descriptions of the processes underlying the turnover, long-term stability and plasticity of postsynaptic receptors and show how these can help us to understand the balance between local molecular turnover and synaptic structural identity and integrity.

Local accumulation times in a diffusion-trapping model of receptor dynamics at proximal axodendritic synapses

The lateral diffusion and trapping of neurotransmitter receptors within the postsynaptic membrane of a neuron play a key role in determining synaptic strength and plasticity. Trapping is mediated by the reversible binding of receptors to scaffolding proteins (slots) within a synapse. In this paper we introduce a method for analyzing the transient dynamics of proximal axodendritic synapses in a diffusion-trapping model of receptor trafficking. Given a population of spatially distributed synapses, each of which has a fixed number of slots, we calculate the rate of relaxation to the steady-state distribution of bound slots (synaptic weights) in terms of a set of local accumulation times. Assuming that the rates of exocytosis and endocytosis are sufficiently slow, we show that the steady-state synaptic weights are independent of each other (purely local). On the other hand, the local accumulation time of a given synapse depends on the number of slots and the spatial location of all the synapses, indicating a form of transient heterosynaptic plasticity. This suggests that local accumulation time measurements could provide useful information regarding the distribution of synaptic weights within a dendrite.

The biophysical basis underlying the maintenance of early phase long-term potentiation

The maintenance of synaptic changes resulting from long-term potentiation (LTP) is essential for brain function such as memory and learning. Different LTP phases have been associated with diverse molecular processes and pathways, and the molecular underpinnings of LTP on the short, as well as long time scales, are well established. However, the principles on the intermediate time scale of 1-6 hours that mediate the early phase of LTP (E-LTP) remain elusive. We hypothesize that the interplay between specific features of postsynaptic receptor trafficking is responsible for sustaining synaptic changes during this LTP phase. We test this hypothesis by formalizing a biophysical model that integrates several experimentally-motivated mechanisms. The model captures a wide range of experimental findings and predicts that synaptic changes are preserved for hours when the receptor dynamics are shaped by the interplay of structural changes of the spine in conjunction with increased trafficking from recycling endosomes and the cooperative binding of receptors. Furthermore, our model provides several predictions to verify our findings experimentally.

Reciprocal stabilization of glycine receptors and gephyrin scaffold proteins at inhibitory synapses

Postsynaptic scaffold proteins immobilize neurotransmitter receptors in the synaptic membrane opposite to presynaptic vesicle release sites, thus ensuring efficient synaptic transmission. At inhibitory synapses in the spinal cord, the main scaffold protein gephyrin assembles in dense molecule clusters that provide binding sites for glycine receptors (GlyRs). Gephyrin and GlyRs can also interact outside of synapses, where they form receptor-scaffold complexes. Although several models for the formation of postsynaptic scaffold domains in the presence of receptor-scaffold interactions have been advanced, a clear picture of the coupled dynamics of receptors and scaffold proteins at synapses is lacking. To characterize the GlyR and gephyrin dynamics at inhibitory synapses, we performed fluorescence time-lapse imaging after photoconversion to directly visualize the exchange kinetics of recombinant Dendra2-gephyrin in cultured spinal cord neurons. Immuno-immobilization of endogenous GlyRs with specific antibodies abolished their lateral diffusion in the plasma membrane, as judged by the lack of fluorescence recovery after photobleaching. Moreover, the cross-linking of GlyRs significantly reduced the exchange of Dendra2-gephyrin compared with control conditions, suggesting that the kinetics of the synaptic gephyrin pool is strongly dependent on GlyR-gephyrin interactions. We did not observe any change in the total synaptic gephyrin levels after GlyR cross-linking, however, indicating that the number of gephyrin molecules at synapses is not primarily dependent on the exchange of GlyR-gephyrin complexes. We further show that our experimental data can be quantitatively accounted for by a model of receptor-scaffold dynamics that includes a tightly interacting receptor-scaffold domain, as well as more loosely bound receptor and scaffold populations that exchange with extrasynaptic pools. The model can make predictions for single-molecule data such as typical dwell times of synaptic proteins. Taken together, our data demonstrate the reciprocal stabilization of GlyRs and gephyrin at inhibitory synapses and provide a quantitative understanding of their dynamic organization.

Lifetime of a structure evolving by cluster aggregation and particle loss, and application to postsynaptic scaffold domains

The dynamics of several mesoscopic biological structures depend on the interplay of growth through the incorporation of components of different sizes laterally diffusing along the cell membrane, and loss by component turnover. In particular, a model of such an out-of-equilibrium dynamics has recently been proposed for postsynaptic scaffold domains, which are key structures of neuronal synapses. It is of interest to estimate the lifetime of these mesoscopic structures, especially in the context of synapses where this time is related to memory retention. The lifetime of a structure can be very long as compared to the turnover time of its components and it can be difficult to estimate it by direct numerical simulations. Here, in the context of the model proposed for postsynaptic scaffold domains, we approximate the aggregation-turnover dynamics by a shot-noise process. This enables us to analytically compute the quasistationary distribution describing the sizes of the surviving structures as well as their characteristic lifetime. We show that our analytical estimate agrees with numerical simulations of a full spatial model, in a regime of parameters where a direct assessment is computationally feasible. We then use our approach to estimate the lifetime of mesoscopic structures in parameter regimes where computer simulations would be prohibitively long. For gephyrin, the scaffolding protein specific to inhibitory synapses, we estimate a lifetime longer than several months for a scaffold domain when the single gephyrin protein turnover time is about half an hour, as experimentally measured. While our focus is on postsynaptic domains, our formalism and techniques should be applicable to other biological structures that are also formed by a balance of condensation and turnover.

Linking glutamate receptor movements and synapse function

Regulation of neurotransmitter receptor content at synapses is achieved through a dynamic equilibrium between biogenesis and degradation pathways, receptor stabilization at synaptic sites, and receptor trafficking in and out synapses. In the past 20 years, the movements of receptors to and from synapses have emerged as a series of highly regulated processes that mediate postsynaptic plasticity. Our understanding of the properties and roles of receptor movements has benefited from technological advances in receptor labeling and tracking capacities, as well as from new methods to interfere with their movements. Focusing on two key glutamatergic receptors, we review here our latest understanding of the characteristics of receptor movements and their role in tuning the efficacy of synaptic transmission in health and brain disease.

Inhibitory Receptor Diffusion Dynamics

The dynamic modulation of receptor diffusion-trapping at inhibitory synapses is crucial to synaptic transmission, stability, and plasticity. In this review article, we will outline the progression of understanding of receptor diffusion dynamics at the plasma membrane. We will discuss how regulation of reversible trapping of receptor-scaffold interactions in combination with theoretical modeling approaches can be used to quantify these chemical interactions at the postsynapse of living cells.

A model of operant learning based on chaotically varying synaptic strength

Operant learning is learning based on reinforcement of behaviours. We propose a new hypothesis for operant learning at the single neuron level based on spontaneous fluctuations of synaptic strength caused by receptor dynamics. These fluctuations allow the neural system to explore a space of outputs. If the receptor dynamics are altered by a reinforcement signal the neural system settles to better states, i.e., to match the environmental dynamics that determine reward. Simulations show that this mechanism can support operant learning in a feed-forward neural circuit, a recurrent neural circuit, and a spiking neural circuit controlling an agent learning in a dynamic reward and punishment situation. We discuss how the new principle relates to existing learning rules and observed phenomena of short and long-term potentiation.

Alpha subunit-dependent glycine receptor clustering and regulation of synaptic receptor numbers

Accumulation of glycine receptors at synapses requires the interaction between the beta subunit of the receptor and the scaffold protein gephyrin. Here, we questioned whether different alpha subunits could modulate the receptors' diffusion and propensity to cluster at spinal cord synapses. Using quantitative photoactivated localisation microscopy we found that alpha-1 and alpha-3 containing glycine receptors display the same α₃:β₂ stoichiometry and gephyrin binding. Despite these similarities, alpha-3 containing receptors are less mobile and cluster at higher density compared to alpha-1, with 1500 versus 1100 complexes µm^-2, respectively. Furthermore, we identified a subunit-specific regulation of glycine receptor copy numbers at synapses: when challenged with interleukin 1β, the synaptic occupancy of alpha-1 but not alpha-3 receptors was reduced. This mechanism may play a role in the cell-type dependent regulation of glycinergic currents in response to interleukin 1β and highlights the capacity of the alpha subunits to affect receptor-gephyrin binding at synapses.

Cooperative stochastic binding and unbinding explain synaptic size dynamics and statistics

Synapses are dynamic molecular assemblies whose sizes fluctuate significantly over time-scales of hours and days. In the current study, we examined the possibility that the spontaneous microscopic dynamics exhibited by synaptic molecules can explain the macroscopic size fluctuations of individual synapses and the statistical properties of synaptic populations. We present a mesoscopic model, which ties the two levels. Its basic premise is that synaptic size fluctuations reflect cooperative assimilation and removal of molecules at a patch of postsynaptic membrane. The introduction of cooperativity to both assimilation and removal in a stochastic biophysical model of these processes, gives rise to features qualitatively similar to those measured experimentally: nanoclusters of synaptic scaffolds, fluctuations in synaptic sizes, skewed, stable size distributions and their scaling in response to perturbations. Our model thus points to the potentially fundamental role of cooperativity in dictating synaptic remodeling dynamics and offers a conceptual understanding of these dynamics in terms of central microscopic features and processes.

Neurons communicate through specialized sites of cell–cell contact known as synapses. This vast set of connections is believed to be crucial for sensory processing, motor function, learning and memory. Experimental data from recent years suggest that synapses are not static structures, but rather dynamic assemblies of molecules that move in, out and between nearby synapses, with these dynamics driving changes in synaptic properties over time. Thus, in addition to changes directed by activity or other physiological signals, synapses also exhibit spontaneous changes that have particular dynamical and statistical signatures. Given the immense complexity of synapses at the molecular scale, how can one hope to understand the principles that govern these spontaneous changes and statistical signatures? Here we offer a mesoscopic modelling approach—situated between detailed microscopic and abstract macroscopic approaches—to advance this understanding. Our model, based on simplified biophysical assumptions, shows that spontaneous cooperative binding and unbinding of proteins at synaptic sites can give rise to dynamic and statistical signatures similar to those measured in experiments. Importantly, we find cooperativity to be indispensable in this regard. Our model thus offers a conceptual understanding of synaptic dynamics and statistical features in terms of a fundamental biological principle, namely cooperativity.

Stochastic lattice model of synaptic membrane protein domains

Neurotransmitter receptor molecules, concentrated in synaptic membrane domains along with scaffolds and other kinds of proteins, are crucial for signal transmission across chemical synapses. In common with other membrane protein domains, synaptic domains are characterized by low protein copy numbers and protein crowding, with rapid stochastic turnover of individual molecules. We study here in detail a stochastic lattice model of the receptor-scaffold reaction-diffusion dynamics at synaptic domains that was found previously to capture, at the mean-field level, the self-assembly, stability, and characteristic size of synaptic domains observed in experiments. We show that our stochastic lattice model yields quantitative agreement with mean-field models of nonlinear diffusion in crowded membranes. Through a combination of analytic and numerical solutions of the master equation governing the reaction dynamics at synaptic domains, together with kinetic Monte Carlo simulations, we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations, we show that the stochastic reaction-diffusion dynamics of synaptic receptors and scaffolds provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the observed single-molecule trajectories, and spatial heterogeneity in the effective rates at which receptors and scaffolds are recycled at the cell membrane. Our work sheds light on the physical mechanisms and principles linking the collective properties of membrane protein domains to the stochastic dynamics that rule their molecular components.

Self-assembly and plasticity of synaptic domains through a reaction-diffusion mechanism

Signal transmission across chemical synapses relies crucially on neurotransmitter receptor molecules, concentrated in postsynaptic membrane domains along with scaffold and other postsynaptic molecules. The strength of the transmitted signal depends on the number of receptor molecules in postsynaptic domains, and activity-induced variation in the receptor number is one of the mechanisms of postsynaptic plasticity. Recent experiments have demonstrated that the reaction and diffusion properties of receptors and scaffolds at the membrane, alone, yield spontaneous formation of receptor-scaffold domains of the stable characteristic size observed in neurons. On the basis of these experiments we develop a model describing synaptic receptor domains in terms of the underlying reaction-diffusion processes. Our model predicts that the spontaneous formation of receptor-scaffold domains of the stable characteristic size observed in experiments depends on a few key reactions between receptors and scaffolds. Furthermore, our model suggests novel mechanisms for the alignment of pre- and postsynaptic domains and for short-term postsynaptic plasticity in receptor number. We predict that synaptic receptor domains localize in membrane regions with an increased receptor diffusion coefficient or a decreased scaffold diffusion coefficient. Similarly, we find that activity-dependent increases or decreases in receptor or scaffold diffusion yield a transient increase in the number of receptor molecules concentrated in postsynaptic domains. Thus, the proposed reaction-diffusion model puts forth a coherent set of biophysical mechanisms for the formation, stability, and plasticity of molecular domains on the postsynaptic membrane.

Benzodiazepine-dependent stabilization of GABA(A) receptors at synapses

GABA(A) receptors constitutively enter and exit synapses by lateral diffusion in the plane of the neuronal membrane. They are trapped at synapses through their interactions with gephyrin, the main scaffolding protein at inhibitory post-synaptic densities. Previous work has shown that the synaptic accumulation and diffusion dynamics of GABA(A)Rs are controlled via excitatory synaptic activity. However, it remains unknown whether GABA(A)R activity can itself impact the surface trafficking of the receptors. Here we report the effects of GABA(A)R agonists, antagonists and allosteric modulators on the receptor's surface dynamics. Using immunocytochemistry and single particle tracking experiments on mouse hippocampal neurons, we show that the agonist muscimol decreases GABA(A)R and gephyrin levels at synapses and accelerates the receptor's lateral diffusion within 30–120 min of treatment. In contrast, the GABA(A)R antagonist gabazine increased GABA(A)R amounts and slowed down GABA(A)R diffusion at synapses. The response to GABA(A)R activation or inhibition appears to be an adaptative regulation of GABAergic synapses. Surprisingly, the positive allosteric modulator diazepam abolished the regulation induced by muscimol, and this effect was observed on α1, α2, α5 and γ2 GABA(A)R subunits. Altogether these results indicate that diazepam stabilizes synaptic GABA(A)Rs and thus prevents the agonist-induced regulation of GABA(A)R levels at synapses. This occurred independently of neuronal activity and intracellular calcium and involved GABA(A)R–gephyrin interactions, suggesting that the changes in GABA(A)R diffusion depend on conformational changes of the receptor. Our study provides a new molecular mechanism involved in the adaptative response to changes in GABA(A)R activity and benzodiazepine treatments.

Mapping the energy and diffusion landscapes of membrane proteins at the cell surface using high-density single-molecule imaging and Bayesian inference: application to the multiscale dynamics of glycine receptors in the neuronal membrane

Protein mobility is conventionally analyzed in terms of an effective diffusion. Yet, this description often fails to properly distinguish and evaluate the physical parameters (such as the membrane friction) and the biochemical interactions governing the motion. Here, we present a method combining high-density single-molecule imaging and statistical inference to separately map the diffusion and energy landscapes of membrane proteins across the cell surface at ~100 nm resolution (with acquisition of a few minutes). Upon applying these analytical tools to glycine neurotransmitter receptors at inhibitory synapses, we find that gephyrin scaffolds act as shallow energy traps (~3 kBT) for glycine neurotransmitter receptors, with a depth modulated by the biochemical properties of the receptor-gephyrin interaction loop. In turn, the inferred maps can be used to simulate the dynamics of proteins in the membrane, from the level of individual receptors to that of the population, and thereby, to model the stochastic fluctuations of physiological parameters (such as the number of receptors at synapses). Overall, our approach provides a powerful and comprehensive framework with which to analyze biochemical interactions in living cells and to decipher the multiscale dynamics of biomolecules in complex cellular environments.

Gephyrin-independent GABA(A)R mobility and clustering during plasticity

The activity-dependent modulation of GABA-A receptor (GABA(A)R) clustering at synapses controls inhibitory synaptic transmission. Several lines of evidence suggest that gephyrin, an inhibitory synaptic scaffold protein, is a critical factor in the regulation of GABA(A)R clustering during inhibitory synaptic plasticity induced by neuronal excitation. In this study, we tested this hypothesis by studying relative gephyrin dynamics and GABA(A)R declustering during excitatory activity. Surprisingly, we found that gephyrin dispersal is not essential for GABA(A)R declustering during excitatory activity. In cultured hippocampal neurons, quantitative immunocytochemistry showed that the dispersal of synaptic GABA(A)Rs accompanied with neuronal excitation evoked by 4-aminopyridine (4AP) or N-methyl-D-aspartic acid (NMDA) precedes that of gephyrin. Single-particle tracking of quantum dot labeled-GABA(A)Rs revealed that excitation-induced enhancement of GABA(A)R lateral mobility also occurred before the shrinkage of gephyrin clusters. Physical inhibition of GABA(A)R lateral diffusion on the cell surface and inhibition of a Ca(2+) dependent phosphatase, calcineurin, completely eliminated the 4AP-induced decrease in gephyrin cluster size, but not the NMDA-induced decrease in cluster size, suggesting the existence of two different mechanisms of gephyrin declustering during activity-dependent plasticity, a GABA(A)R-dependent regulatory mechanism and a GABA(A)R-independent one. Our results also indicate that GABA(A)R mobility and clustering after sustained excitatory activity is independent of gephyrin.

Synaptic bistability due to nucleation and evaporation of receptor clusters

We introduce a bistability mechanism for long-term synaptic plasticity based on switching between two metastable states that contain significantly different numbers of synaptic receptors. One state is characterized by a two-dimensional gas of mobile interacting receptors and is stabilized against clustering by a high nucleation barrier. The other state contains a receptor gas in equilibrium with a large cluster of immobile receptors, which is stabilized by the turnover rate of receptors into and out of the synapse. Transitions between the two states can be initiated by either an increase (potentiation) or a decrease (depotentiation) of the net receptor flux into the synapse. This changes the saturation level of the receptor gas and triggers nucleation or evaporation of receptor clusters.

Ankyrin G restricts ion channel diffusion at the axonal initial segment before the establishment of the diffusion barrier

Ion channel immobilization by ankyrin G is regulated by casein kinase 2 in immature hippocampal neurons.

In mammalian neurons, the precise accumulation of sodium channels at the axonal initial segment (AIS) ensures action potential initiation. This accumulation precedes the immobilization of membrane proteins and lipids by a diffusion barrier at the AIS. Using single-particle tracking, we measured the mobility of a chimeric ion channel bearing the ankyrin-binding motif of the Nav1.2 sodium channel. We found that ankyrin G (ankG) limits membrane diffusion of ion channels when coexpressed in neuroblastoma cells. Site-directed mutants with decreased affinity for ankG exhibit increased diffusion speeds. In immature hippocampal neurons, we demonstrated that ion channel immobilization by ankG is regulated by protein kinase CK2 and occurs as soon as ankG accumulates at the AIS of elongating axons. Once the diffusion barrier is formed, ankG is still required to stabilize ion channels. In conclusion, our findings indicate that specific binding to ankG constitutes the initial step for Nav channel immobilization at the AIS membrane and precedes the establishment of the diffusion barrier.

Synaptic stability and plasticity in a floating world

A fundamental feature of membranes is the lateral diffusion of lipids and proteins. Control of lateral diffusion provides a mechanism for regulating the structure and function of synapses. Single-particle tracking (SPT) has emerged as a powerful way to directly visualize these movements. SPT can reveal complex diffusive behaviors, which can be regulated by neuronal activity over time and space. Such is the case for neurotransmitter receptors, which are transiently stabilized at synapses by scaffolding molecules. This regulation provides new insight into mechanisms by which the dynamic equilibrium of receptor-scaffold assembly can be regulated. We will briefly review here recent data on this mechanism, which ultimately tunes the number of receptors at synapses and therefore synaptic strength.

Cellular transport and membrane dynamics of the glycine receptor

Regulation of synaptic transmission is essential to tune individual-to-network neuronal activity. One way to modulate synaptic strength is to regulate neurotransmitter receptor numbers at postsynaptic sites. This can be achieved either through plasma membrane insertion of receptors derived from intracellular vesicle pools, a process depending on active cytoskeleton transport, or through surface membrane removal via endocytosis. In parallel, lateral diffusion events along the plasma membrane allow the exchange of receptor molecules between synaptic and extrasynaptic compartments, contributing to synaptic strength regulation. In recent years, results obtained from several groups studying glycine receptor (GlyR) trafficking and dynamics shed light on the regulation of synaptic GlyR density. Here, we review (i) proteins and mechanisms involved in GlyR cytoskeletal transport, (ii) the diffusion dynamics of GlyR and of its scaffolding protein gephyrin that control receptor numbers, and its relationship with synaptic plasticity, and (iii) adaptative changes in GlyR diffusion in response to global activity modifications, as a homeostatic mechanism.