An aggregation-removal model for the formation and size determination of post-synaptic scaffold domains

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.

Nonequilibrium cluster-cluster aggregation in the presence of anchoring sites

Nonequilibrium cluster-cluster aggregation of particles diffusing in or at the cell membrane has been hypothesized to lead to domains of finite size in different biological contexts, such as lipid rafts, cell adhesion complexes, or postsynaptic domains in neurons. In this scenario, the desorption of particles balances a continuous flux to the membrane, imposing a cutoff on possible aggregate sizes and giving rise to a stationary size distribution. Here, we investigate the case of nonequilibrium cluster-cluster aggregation in two dimensions where diffusing particles and/or clusters remain fixed in space at specific anchoring sites, which should be particularly relevant for synapses but may also be present in other biological or physical systems. Using an effective mean-field description of the concentration field around anchored clusters, we derive an expression for their average size as a function of parameters such as the anchoring site density. We furthermore propose and solve appropriate rate equations that allow us to predict the size distributions of both diffusing and fixed clusters. We confirm our results with particle-based simulations and discuss potential implications for biological and physical systems.

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.

Stable but not rigid: Chronic in vivo STED nanoscopy reveals extensive remodeling of spines, indicating multiple drivers of plasticity

Excitatory synapses on dendritic spines of pyramidal neurons are considered a central memory locus. To foster both continuous adaption and the storage of long-term information, spines need to be plastic and stable at the same time. Here, we advanced in vivo STED nanoscopy to superresolve distinct features of spines (head size and neck length/width) in mouse neocortex for up to 1 month. While LTP-dependent changes predict highly correlated modifications of spine geometry, we find both, uncorrelated and correlated dynamics, indicating multiple independent drivers of spine remodeling. The magnitude of this remodeling suggests substantial fluctuations in synaptic strength. Despite this high degree of volatility, all spine features exhibit persistent components that are maintained over long periods of time. Furthermore, chronic nanoscopy uncovers structural alterations in the cortex of a mouse model of neurodegeneration. Thus, at the nanoscale, stable dendritic spines exhibit a delicate balance of stability and volatility.

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.

Self-Organized Critical Coexistence Phase in Repulsive Active Particles

We revisit motility-induced phase separation in two models of active particles interacting by pairwise repulsion and uncover new qualitative features: the resulting dense phase contains gas bubbles distributed algebraically up to a typically extremely large cutoff scale. At large enough system size and/or global density, all the gas may be contained inside the bubbles, at which point the system is microphase separated with a finite cutoff bubble scale. We further observe that the ordering is clearly anomalous, with different dynamics for the coarsening of the dense phase and of the gas bubbles. This self-organized critical phenomenology is reproduced by a "reduced bubble model" that implements the basic idea of reverse Ostwald ripening put forward in Tjhung et al. [Phys. Rev. X 8, 031080 (2018)PRXHAE2160-330810.1103/PhysRevX.8.031080].

Computational Modeling of Inhibitory Transsynaptic Signaling in Hippocampal and Cortical Neurons Expressing Intrabodies Against Gephyrin

GABAergic transmission regulates neuronal excitability, dendritic integration of synaptic signals and oscillatory activity, thought to be involved in high cognitive functions. By anchoring synaptic receptors just opposite to release sites, the scaffold protein gephyrin plays a key role in these tasks. In addition, by regulating GABA_A receptor trafficking, gephyrin contributes to maintain, at the network level, an appropriate balance between Excitation (E) and Inhibition (I), crucial for information processing. An E/I imbalance leads to neuropsychiatric disorders such as epilepsy, schizophrenia and autism. In this article, we exploit a previously published computational method to fit spontaneous synaptic events, using a simplified model of the subcellular pathways involving gephyrin at inhibitory synapses. The model was used to analyze experimental data recorded under different conditions, with the main goal to gain insights on the possible consequences of gephyrin block on IPSCs. The same approach can be useful, in general, to analyze experiments designed to block a single protein. The results suggested possible ways to correlate the changes observed in the amplitude and time course of individual events recorded after different experimental protocols with the changes that may occur in the main subcellular pathways involved in gephyrin-dependent transsynaptic signaling.

Activity Dependent and Independent Determinants of Synaptic Size Diversity

The extraordinary diversity of excitatory synapse sizes is commonly attributed to activity-dependent processes that drive synaptic growth and diminution. Recent studies also point to activity-independent size fluctuations, possibly driven by innate synaptic molecule dynamics, as important generators of size diversity. To examine the contributions of activity-dependent and independent processes to excitatory synapse size diversity, we studied glutamatergic synapse size dynamics and diversification in cultured rat cortical neurons (both sexes), silenced from plating. We found that in networks with no history of activity whatsoever, synaptic size diversity was no less extensive than that observed in spontaneously active networks. Synapses in silenced networks were larger, size distributions were broader, yet these were rightward-skewed and similar in shape when scaled by mean synaptic size. Silencing reduced the magnitude of size fluctuations and weakened constraints on size distributions, yet these were sufficient to explain synaptic size diversity in silenced networks. Model-based exploration followed by experimental testing indicated that silencing-associated changes in innate molecular dynamics and fluctuation characteristics might negatively impact synaptic persistence, resulting in reduced synaptic numbers. This, in turn, would increase synaptic molecule availability, promote synaptic enlargement, and ultimately alter fluctuation characteristics. These findings suggest that activity-independent size fluctuations are sufficient to fully diversify glutamatergic synaptic sizes, with activity-dependent processes primarily setting the scale rather than the shape of size distributions. Moreover, they point to reciprocal relationships between synaptic size fluctuations, size distributions, and synaptic numbers mediated by the innate dynamics of synaptic molecules as they move in, out, and between synapses.SIGNIFICANCE STATEMENT Sizes of glutamatergic synapses vary tremendously, even when formed on the same neuron. This diversity is commonly thought to reflect the outcome of activity-dependent forms of synaptic plasticity, yet activity-independent processes might also play some part. Here we show that in neurons with no history of activity whatsoever, synaptic sizes are no less diverse. We show that this diversity is the product of activity-independent size fluctuations, which are sufficient to generate a full repertoire of synaptic sizes at correct proportions. By combining modeling and experimentation we expose reciprocal relationships between size fluctuations, synaptic sizes and synaptic counts, and show how these phenomena might be connected through the dynamics of synaptic molecules as they move in, out, and between synapses.

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.

How mRNA Localization and Protein Synthesis Sites Influence Dendritic Protein Distribution and Dynamics

Proteins drive the function of neuronal synapses. The synapses are distributed throughout the dendritic arbor, often hundreds of micrometers away from the soma. It is still unclear how somatic and dendritic sources of proteins shape protein distribution and respectively contribute to local protein changes during synaptic plasticity. Here, we present a unique computational framework describing for a given protein species the dendritic distribution of the mRNA and the corresponding protein in a dendrite. Using CaMKIIα as a test case, our model reveals the key role active transport plays in the maintenance of dendritic mRNA and protein levels and predicts the short and long timescales of protein dynamics. Our model reveals the fundamental role of mRNA localization and dendritic mRNA translation in synaptic maintenance and plasticity in distal compartments. We developed a web application for neuroscientists to explore the dynamics of the mRNA or protein of interest.

Design and implementation of multi-signal and time-varying neural reconstructions

Several efficient procedures exist to digitally trace neuronal structure from light microscopy, and mature community resources have emerged to store, share, and analyze these datasets. In contrast, the quantification of intracellular distributions and morphological dynamics is not yet standardized. Current widespread descriptions of neuron morphology are static and inadequate for subcellular characterizations. We introduce a new file format to represent multichannel information as well as an open-source Vaa3D plugin to acquire this type of data. Next we define a novel data structure to capture morphological dynamics, and demonstrate its application to different time-lapse experiments. Importantly, we designed both innovations as judicious extensions of the classic SWC format, thus ensuring full back-compatibility with popular visualization and modeling tools. We then deploy the combined multichannel/time-varying reconstruction system on developing neurons in live Drosophila larvae by digitally tracing fluorescently labeled cytoskeletal components along with overall dendritic morphology as they changed over time. This same design is also suitable for quantifying dendritic calcium dynamics and tracking arbor-wide movement of any subcellular substrate of interest.

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.