Diffusion laws in dendritic spines

Control of Ca2+ signals by astrocyte nanoscale morphology at tripartite synapses

Much of the Ca2+ activity in astrocytes is spatially restricted to microdomains and occurs in fine processes that form a complex anatomical meshwork, the so-called spongiform domain. A growing body of literature indicates that those astrocytic Ca2+ signals can influence the activity of neuronal synapses and thus tune the flow of information through neuronal circuits. Because of technical difficulties in accessing the small spatial scale involved, the role of astrocyte morphology on Ca2+ microdomain activity remains poorly understood. Here, we use computational tools and idealized 3D geometries of fine processes based on recent super-resolution microscopy data to investigate the mechanistic link between astrocytic nanoscale morphology and local Ca2+ activity. Simulations demonstrate that the nano-morphology of astrocytic processes powerfully shapes the spatio-temporal properties of Ca2+ signals and promotes local Ca2+ activity. The model predicts that this effect is attenuated upon astrocytic swelling, hallmark of brain diseases, which we confirm experimentally in hypo-osmotic conditions. Upon repeated neurotransmitter release events, the model predicts that swelling hinders astrocytic signal propagation. Overall, this study highlights the influence of the complex morphology of astrocytes at the nanoscale and its remodeling in pathological conditions on neuron-astrocyte communication at so-called tripartite synapses, where astrocytic processes come into close contact with pre- and postsynaptic structures.

Protein drift-diffusion dynamics and phase separation in curved cell membranes and dendritic spines: Hybrid discrete-continuum methods

We develop methods for investigating protein drift-diffusion dynamics in heterogeneous cell membranes and the roles played by geometry, diffusion, chemical kinetics, and phase separation. Our hybrid stochastic numerical methods combine discrete particle descriptions with continuum-level models for tracking the individual protein drift-diffusion dynamics when coupled to continuum fields. We show how our approaches can be used to investigate phenomena motivated by protein kinetics within dendritic spines. The spine geometry is hypothesized to play an important biological role regulating synaptic strength, protein kinetics, and self-assembly of clusters. We perform simulation studies for model spine geometries varying the neck size to investigate how phase-separation and protein organization is influenced by different shapes. We also show how our methods can be used to study the roles of geometry in reaction-diffusion systems including Turing instabilities. Our methods provide general approaches for investigating protein kinetics and drift-diffusion dynamics within curved membrane structures.

Post-Synapses in the Brain: Role of Dendritic and Spine Structures

Brain synapses are neuronal structures of the greatest interest. For a long time, however, the knowledge about them was variable, and interest was mostly focused on their pre-synaptic portions, especially neurotransmitter release from axon terminals. In the present review interest is focused on post-synapses, the structures receiving and converting pre-synaptic messages. Upon further modulation, such messages are transferred to dendritic fibers. Dendrites are profoundly different from axons; they are shorter and of variable thickness. Their post-synapses are of two types. Those called flat/intended/aspines, integrated into dendritic fibers, are very frequent in inhibitory neurons. The spines, small and stemming protrusions, connected to dendritic fibers by their necks, are present in almost all excitatory neurons. Several structures and functions including the post-synaptic densities and associated proteins, the nanoscale mechanisms of compartmentalization, the cytoskeletons of actin and microtubules, are analogous in the two post-synaptic forms. However other properties, such as plasticity and its functions of learning and memory, are largely distinct. Several properties of spines, including emersion from dendritic fibers, growth, change in shape and decreases in size up to disappearance, are specific. Spinal heads correspond to largely independent signaling compartments. They are motile, their local signaling is fast, however transport through their thin necks is slow. When single spines are activated separately, their dendritic effects are often lacking; when multiple spines are activated concomitantly, their effects take place. Defects of post-synaptic responses, especially those of spines, take place in various brain diseases. Here alterations affecting symptoms and future therapy are shown to occur in neurodegenerative diseases and autism spectrum disorders.

Voltage compartmentalization in dendritic spines in vivo

[Figure: see text].

Nanoscale molecular architecture controls calcium diffusion and ER replenishment in dendritic spines

Dendritic spines are critical components of neuronal synapses as they receive and transform synaptic inputs into a succession of calcium-regulated biochemical events. The spine apparatus (SA), an extension of smooth endoplasmic reticulum, regulates slow and fast calcium dynamics in spines. Calcium release events deplete SA calcium ion reservoir rapidly, yet the next cycle of signaling requires its replenishment. How spines achieve this replenishment without triggering calcium release remains unclear. Using computational modeling, calcium and STED superresolution imaging, we show that the SA replenishment involves the store-operated calcium entry pathway during spontaneous calcium transients. We identified two main conditions for SA replenishment without depletion: a small amplitude and a slow timescale for calcium influx, and a close proximity between SA and plasma membranes. Thereby, spine’s nanoscale organization separates SA replenishment from depletion. We further conclude that spine’s receptor organization also determines the calcium dynamics during the induction of long-term synaptic changes.

The role of molecular diffusion within dendritic spines in synaptic function

Spines are tiny nanoscale protrusions from dendrites of neurons. In the cortex and hippocampus, most of the excitatory postsynaptic sites reside in spines. The bulbous spine head is connected to the dendritic shaft by a thin membranous neck. Because the neck is narrow, spine heads are thought to function as biochemically independent signaling compartments. Thus, dynamic changes in the composition, distribution, mobility, conformations, and signaling properties of molecules contained within spines can account for much of the molecular basis of postsynaptic function and regulation. A major factor in controlling these changes is the diffusional properties of proteins within this small compartment. Advances in measurement techniques using fluorescence microscopy now make it possible to measure molecular diffusion within single dendritic spines directly. Here, we review the regulatory mechanisms of diffusion in spines by local intra-spine architecture and discuss their implications for neuronal signaling and synaptic plasticity.

Geometric Control of Frequency Modulation of cAMP Oscillations due to Calcium in Dendritic Spines

The spatiotemporal regulation of cyclic adenosine monophosphate (cAMP) and its dynamic interactions with other second messengers such as calcium are critical features of signaling specificity required for neuronal development and connectivity. cAMP is known to contribute to long-term potentiation and memory formation by controlling the formation and regulation of dendritic spines. Despite the recent advances in biosensing techniques for monitoring spatiotemporal cAMP dynamics, the underlying molecular mechanisms that attribute to the subcellular modulation of cAMP remain unknown. In this work, we model the spatiotemporal dynamics of calcium-induced cAMP signaling pathway in dendritic spines. Using a three-dimensional reaction-diffusion model, we investigate the effect of different spatial characteristics of cAMP dynamics that may be responsible for subcellular regulation of cAMP concentrations. Our model predicts that the volume/surface ratio of the spine, regulated through the spine head size, spine neck size, and the presence of physical barriers (spine apparatus), is an important regulator of cAMP dynamics. Furthermore, localization of the enzymes responsible for the synthesis and degradation of cAMP in different compartments also modulates the oscillatory patterns of cAMP through exponential relationships. Our findings shed light on the significance of complex geometric and localization relationships for cAMP dynamics in dendritic spines.

Subunit exchange enhances information retention by CaMKII in dendritic spines

Molecular bistables are strong candidates for long-term information storage, for example, in synaptic plasticity. Calcium/calmodulin-dependent protein Kinase II (CaMKII) is a highly expressed synaptic protein which has been proposed to form a molecular bistable switch capable of maintaining its state for years despite protein turnover and stochastic noise. It has recently been shown that CaMKII holoenzymes exchange subunits among themselves. Here, we used computational methods to analyze the effect of subunit exchange on the CaMKII pathway in the presence of diffusion in two different micro-environments, the post synaptic density (PSD) and spine cytosol. We show that CaMKII exhibits multiple timescales of activity due to subunit exchange. Further, subunit exchange enhances information retention by CaMKII both by improving the stability of its switching in the PSD, and by slowing the decay of its activity in the spine cytosol. The existence of diverse timescales in the synapse has important theoretical implications for memory storage in networks.

Probing the Interplay between Dendritic Spine Morphology and Membrane-Bound Diffusion

Dendritic spines are protrusions along neuronal dendrites that harbor the majority of excitatory postsynapses. Their distinct morphology, often featuring a bulbous head and small neck that connects to the dendritic shaft, has been shown to facilitate compartmentalization of electrical and cytoplasmic signaling stimuli elicited at the synapse. The extent to which spine morphology also forms a barrier for membrane-bound diffusion has remained unclear. Recent simulations suggested that especially the diameter of the spine neck plays a limiting role in this process. Here, we examine the connection between spine morphology and membrane-bound diffusion through a combination of photoconversion, live-cell superresolution experiments, and numerical simulations. Local photoconversion was used to obtain the timescale of diffusive equilibration in spines and followed by global sparse photoconversion to determine spine morphologies with nanoscopic resolution. These morphologies were subsequently used to assess the role of morphology on the diffusive equilibration. From the simulations, we could determine a robust relation between the equilibration timescale and a generalized shape factor calculated using both spine neck width and neck length, as well as spine head size. Experimentally, we found that diffusive equilibration was often slower, but rarely faster than predicted from the simulations, indicating that other biological confounders further reduce membrane-bound diffusion in these spines. This shape-dependent membrane-bound diffusion in mature spines may contribute to spine-specific compartmentalization of neurotransmitter receptors and signaling molecules and thereby support long-term plasticity of synaptic contacts.

Analysis of diffusion in curved surfaces and its application to tubular membranes

Diffusion of particles in curved surfaces is inherently complex compared with diffusion in a flat membrane, owing to the nonplanarity of the surface. The consequence of such nonplanar geometry on diffusion is poorly understood but is highly relevant in the case of cell membranes, which often adopt complex geometries. To address this question, we developed a new finite element approach to model diffusion on curved membrane surfaces based on solutions to Fick's law of diffusion and used this to study the effects of geometry on the entry of surface-bound particles into tubules by diffusion. We show that variations in tubule radius and length can distinctly alter diffusion gradients in tubules over biologically relevant timescales. In addition, we show that tubular structures tend to retain concentration gradients for a longer time compared with a comparable flat surface. These findings indicate that sorting of particles along the surfaces of tubules can arise simply as a geometric consequence of the curvature without any specific contribution from the membrane environment. Our studies provide a framework for modeling diffusion in curved surfaces and suggest that biological regulation can emerge purely from membrane geometry.

Residence times of receptors in dendritic spines analyzed by stochastic simulations in empirical domains

Analysis of high-density superresolution imaging of receptors reveals the organization of dendrites at nanoscale resolution. We present here an apparently novel method that uses local statistics extracted from short-range trajectories for the simulations of long-range trajectories in empirical live cell images. Based on these empirical simulations, we compute the residence time of a receptor in dendritic spines that accounts for receptors' local interactions and geometrical membrane organization. We report here that depending on the type of the spine, the residence time varies from 1 to 5 min. Moreover, we show that there exists transient organized structures, previously described as potential wells that can regulate the trafficking of receptors to dendritic spine: the simulation results suggest that receptor trafficking is regulated by transient structures.

Spines slow down dendritic chloride diffusion and affect short-term ionic plasticity of GABAergic inhibition

Cl⁻ plays a crucial role in neuronal function and synaptic inhibition. However, the impact of neuronal morphology on the diffusion and redistribution of intracellular Cl⁻ is not well understood. The role of spines in Cl⁻ diffusion along dendritic trees has not been addressed so far. Because measuring fast and spatially restricted Cl⁻ changes within dendrites is not yet technically possible, we used computational approaches to predict the effects of spines on Cl⁻ dynamics in morphologically complex dendrites. In all morphologies tested, including dendrites imaged by super-resolution STED microscopy in live brain tissue, spines slowed down longitudinal Cl⁻ diffusion along dendrites. This effect was robust and could be observed in both deterministic as well as stochastic simulations. Cl⁻ extrusion altered Cl⁻ diffusion to a much lesser extent than the presence of spines. The spine-dependent slowing of Cl⁻ diffusion affected the amount and spatial spread of changes in the GABA reversal potential thereby altering homosynaptic as well as heterosynaptic short-term ionic plasticity at GABAergic synapses in dendrites. Altogether, our results suggest a fundamental role of dendritic spines in shaping Cl⁻ diffusion, which could be of relevance in the context of pathological conditions where spine densities and neural excitability are perturbed.

The new nanophysiology: regulation of ionic flow in neuronal subcompartments

Cable theory and the Goldman-Hodgkin-Huxley-Katz models for the propagation of ions and voltage within a neuron have provided a theoretical foundation for electrophysiology and been responsible for many cornerstone advances in neuroscience. However, these theories break down when they are applied to small neuronal compartments, such as dendritic spines, synaptic terminals or small neuronal processes, because they assume spatial and ionic homogeneity. Here we discuss a broader theory that uses the Poisson-Nernst-Planck (PNP) approximation and electrodiffusion to more accurately model the constraints that neuronal nanostructures place on electrical current flow. This extension of traditional cable theory could advance our understanding of the physiology of neuronal nanocompartments.

Barriers in the brain: resolving dendritic spine morphology and compartmentalization

Dendritic spines are micron-sized protrusions that harbor the majority of excitatory synapses in the central nervous system. The head of the spine is connected to the dendritic shaft by a 50-400 nm thin membrane tube, called the spine neck, which has been hypothesized to confine biochemical and electric signals within the spine compartment. Such compartmentalization could minimize interspinal crosstalk and thereby support spine-specific synapse plasticity. However, to what extent compartmentalization is governed by spine morphology, and in particular the diameter of the spine neck, has remained unresolved. Here, we review recent advances in tool development - both experimental and theoretical - that facilitate studying the role of the spine neck in compartmentalization. Special emphasis is given to recent advances in microscopy methods and quantitative modeling applications as we discuss compartmentalization of biochemical signals, membrane receptors and electrical signals in spines. Multidisciplinary approaches should help to answer how dendritic spine architecture affects the cellular and molecular processes required for synapse maintenance and modulation.

Input transformation by dendritic spines of pyramidal neurons

In the mammalian brain, most inputs received by a neuron are formed on the dendritic tree. In the neocortex, the dendrites of pyramidal neurons are covered by thousands of tiny protrusions known as dendritic spines, which are the major recipient sites for excitatory synaptic information in the brain. Their peculiar morphology, with a small head connected to the dendritic shaft by a slender neck, has inspired decades of theoretical and more recently experimental work in an attempt to understand how excitatory synaptic inputs are processed, stored and integrated in pyramidal neurons. Advances in electrophysiological, optical and genetic tools are now enabling us to unravel the biophysical and molecular mechanisms controlling spine function in health and disease. Here I highlight relevant findings, challenges and hypotheses on spine function, with an emphasis on the electrical properties of spines and on how these affect the storage and integration of excitatory synaptic inputs in pyramidal neurons. In an attempt to make sense of the published data, I propose that the raison d'etre for dendritic spines lies in their ability to undergo activity-dependent structural and molecular changes that can modify synaptic strength, and hence alter the gain of the linearly integrated sub-threshold depolarizations in pyramidal neuron dendrites before the generation of a dendritic spike.

Shape-induced asymmetric diffusion in dendritic spines allows efficient synaptic AMPA receptor trapping

Dendritic spines are the primary postsynaptic sites of excitatory neurotransmission in the brain. They exhibit a remarkable morphological variety, ranging from thin protrusions, to stubby shapes, to bulbous mushroom shapes. The remodeling of spines is thought to regulate the strength of the synaptic connection, which depends vitally on the number and the spatial distribution of AMPA-type glutamate receptors (AMPARs). We present numerical and analytical analyses demonstrating that this shape strongly affects AMPAR diffusion. We report a pronounced suppression of the receptor exit rate out of spines with decreasing neck radius. Thus, mushroomlike spines become highly effective at retaining receptors in the spine head. Moreover, we show that the postsynaptic density further enhances receptor trapping, particularly in mushroomlike spines local exocytosis in the spine head, in contrast to release at the base, provides rapid and specific regulatory control of AMPAR concentration at synapses.

Super-resolution 2-photon microscopy reveals that the morphology of each dendritic spine correlates with diffusive but not synaptic properties

The structure of dendritic spines suggests a specialized function in compartmentalizing synaptic signals near active synapses. Indeed, theoretical and experimental analyses indicate that the diffusive resistance of the spine neck is sufficient to effectively compartmentalize some signaling molecules in a spine for the duration of their activated lifetime. Here we describe the application of 2-photon microscopy combined with stimulated emission depletion (STED-2P) to the biophysical study of the relationship between synaptic signals and spine morphology, demonstrating the utility of combining STED-2P with modern optical and electrophysiological techniques. Morphological determinants of fluorescence recovery time were identified and evaluated within the context of a simple compartmental model describing diffusive transfer between spine and dendrite. Correlations between the neck geometry and the amplitude of synaptic potentials and calcium transients evoked by 2-photon glutamate uncaging were also investigated.

Impact of morphology on diffusive dynamics on curved surfaces

Diffusive processes on nonplanar substrates are deeply relevant for cellular function and transport and increasingly used to probe and characterize the behavior of proteins in membranes. We present analytical and numerical analyses of in-plane diffusion of discrete particles on curved geometries reflecting various generic motifs in biology and explore, in particular, the effect that the shape of the substrate has on the characteristic time scales of diffusive processes. To this end, we consider both collective measures (the relaxation of concentration profiles towards equilibrium) and single-particle measures (escape rates and first passage times of individual diffusing molecules): the first relevant for the correct interpretation of FRAP experiments in curved environments; the second, for single-particle tracking probes. Each of these measures is sensitively affected by the morphology of the substrate, and we find that the exit rate out of a domain is not uniquely set by the size of its boundary, illustrating the general principle we reveal: By varying the shape of a substrate, Nature can control the diffusive time scales in a microenvironment without changing the bare substrate properties.

The role of dendritic spine morphology in the compartmentalization and delivery of surface receptors

Since AMPA receptors are major molecular players in both short- and long-term plasticity, it is important to identify the time-scales of and factors affecting the lateral diffusion of AMPARs on the dendrite surface. Using a mathematical model, we study how the dendritic spine morphology affects two processes: (1) compartmentalization of the surface receptors in a single spine to retain local chemistry and (2) the delivery of receptors to the post-synaptic density (PSD) of spines via lateral diffusion following insertion onto the dendrite shaft. Computing the mean first passage time (MFPT) of surface receptors on a sample of real spine morphologies revealed that a constricted neck and bulbous head serve to compartmentalize receptors, consistent with previous works. The residence time of a Brownian diffusing receptor on the membrane of a single spine was computed to be ∼ 5 s. We found that the location of the PSD corresponds to the location at which the maximum MFPT occurs, the position that maximizes the residence time of a diffusing receptor. Meanwhile, the same geometric features of the spine that compartmentalize receptors inhibit the recruitment of AMPARs via lateral diffusion from dendrite insertion sites. Spines with narrow necks will trap a smaller fraction of diffusing receptors in the their PSD when considering competition for receptors between the spines, suggesting that ideal geometrical features involve a tradeoff depending on the intent of compartmentalizing the current receptor pool or recruiting new AMPARs in the PSD. The ultimate distribution of receptors among the spine PSDs by lateral diffusion from the dendrite shaft is an interplay between the insertion location and the shape and locations of both the spines and their PSDs. The time-scale for delivery of receptors to the PSD of spines via lateral diffusion was computed to be ∼ 60 s.

Live-cell superresolution imaging by pulsed STED two-photon excitation microscopy

Two-photon laser scanning microscopy (2PLSM) allows fluorescence imaging in thick biological samples where absorption and scattering typically degrade resolution and signal collection of one-photon imaging approaches. The spatial resolution of conventional 2PLSM is limited by diffraction, and the near-infrared wavelengths used for excitation in 2PLSM preclude the accurate imaging of many small subcellular compartments of neurons. Stimulated emission depletion (STED) microscopy is a superresolution imaging modality that overcomes the resolution limit imposed by diffraction and allows fluorescence imaging of nanoscale features. Here, we describe the design and operation of a superresolution two-photon microscope using pulsed excitation and STED lasers. We examine the depth dependence of STED imaging in acute tissue slices and find enhancement of 2P resolution ranging from approximately fivefold at 20 μm to approximately twofold at 90-μm deep. The depth dependence of resolution is found to be consistent with the depth dependence of depletion efficiency, suggesting resolution is limited by STED laser propagation through turbid tissue. Finally, we achieve live imaging of dendritic spines with 60-nm resolution and demonstrate that our technique allows accurate quantification of neuronal morphology up to 30-μm deep in living brain tissue.

Chronic alcohol alters dendritic spine development in neurons in primary culture

Dendritic spines are specialised membrane protrusions of neuronal dendrites that receive the majority of excitatory synaptic inputs. Abnormal changes in their density, size and morphology have been associated with various neurological and psychiatric disorders, including those deriving from drug addiction. Dendritic spine formation, morphology and synaptic functions are governed by the actin cytoskeleton. Previous in vivo studies have shown that ethanol alters the number and morphology of spines, although the mechanisms underlying these alterations remain unknown. It has also been described how chronic ethanol exposure affects the levels, assembly and cellular organisation of the actin cytoskeleton in hippocampal neurons in primary culture. Therefore, we hypothesised that the ethanol-induced alterations in the number and shape of dendritic spines are due to alterations in the mechanisms regulating actin cytoskeleton integrity. The results presented herein show that chronic exposure to moderate levels of alcohol (30 mM) during the first 2 weeks of culture reduces dendritic spine density and alters the proportion of the different morphologies of these structures in hippocampal neurons, which affects the formation of mature spines. Apparently, these effects are associated with an increase in the G-actin/F-actin ratio due to a reduction of the F-actin fraction, leading to changes in the levels of the different factors regulating the organisation of this cytoskeletal component. The data presented herein indicate that these effects occur between weeks 1 and 2 of culture, an important period in dendritic spines development. These changes may be related to the dysfunction in the memory and learning processes present in children prenatally exposed to ethanol.

Control of flux by narrow passages and hidden targets in cellular biology

Critical biological processes, such as synaptic plasticity and transmission, activation of genes by transcription factors, or double-strained DNA break repair, are controlled by diffusion in structures that have both large and small spatial scales. These may be small binding sites inside or on the surface of the cell, or narrow passages between subcellular compartments. The great disparity in spatial scales is the key to controlling cell function by structure. We report here recent progress on resolving analytical and numerical difficulties in extracting properties from experimental data, from biophysical models, and from Brownian dynamics simulations of diffusion in multi-scale structures. This progress is achieved by developing an analytical approximation methodology for solving the model equations. The reported results are applied to analysis and simulations of subcellular processes and to the quantification of their biological functions.

Heterogeneity of AMPA receptor trafficking and molecular interactions revealed by superresolution analysis of live cell imaging

Simultaneous tracking of many thousands of individual particles in live cells is possible now with the advent of high-density superresolution imaging methods. We present an approach to extract local biophysical properties of cell-particle interaction from such newly acquired large collection of data. Because classical methods do not keep the spatial localization of individual trajectories, it is not possible to access localized biophysical parameters. In contrast, by combining the high-density superresolution imaging data with the present analysis, we determine the local properties of protein dynamics. We specifically focus on AMPA receptor (AMPAR) trafficking and estimate the strength of their molecular interaction at the subdiffraction level in hippocampal dendrites. These interactions correspond to attracting potential wells of large size, showing that the high density of AMPARs is generated by physical interactions with an ensemble of cooperative membrane surface binding sites, rather than molecular crowding or aggregation, which is the case for the membrane viral glycoprotein VSVG. We further show that AMPARs can either be pushed in or out of dendritic spines. Finally, we characterize the recurrent step of influenza trajectories. To conclude, the present analysis allows the identification of the molecular organization responsible for the heterogeneities of random trajectories in cells.

From abnormal hippocampal synaptic plasticity in down syndrome mouse models to cognitive disability in down syndrome

Down syndrome (DS) is caused by the overexpression of genes on triplicated regions of human chromosome 21 (Hsa21). While the resulting physiological and behavioral phenotypes vary in their penetrance and severity, all individuals with DS have variable but significant levels of cognitive disability. At the core of cognitive processes is the phenomenon of synaptic plasticity, a functional change in the strength at points of communication between neurons. A wide variety of evidence from studies on DS individuals and mouse models of DS indicates that synaptic plasticity is adversely affected in human trisomy 21 and mouse segmental trisomy 16, respectively, an outcome that almost certainly extensively contributes to the cognitive impairments associated with DS. In this review, we will highlight some of the neurophysiological changes that we believe reduce the ability of trisomic neurons to undergo neuroplasticity-related adaptations. We will focus primarily on hippocampal networks which appear to be particularly impacted in DS and where consequently the majority of cellular and neuronal network research has been performed using DS animal models, in particular the Ts65Dn mouse. Finally, we will postulate on how altered plasticity may contribute to the DS cognitive disability.