Barriers to diffusion in dendrites and estimation of calcium spread following synaptic inputs

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

Reinforcing Interdisciplinary Collaborations to Unravel the Astrocyte "Calcium Code"

In this review article, we present the major insights from and challenges faced in the acquisition, analysis and modeling of astrocyte calcium activity, aiming at bridging the gap between those fields to crack the complex astrocyte “Calcium Code”. We then propose strategies to reinforce interdisciplinary collaborative projects to unravel astrocyte function in health and disease.

Influence of T-Bar on Calcium Concentration Impacting Release Probability

The relation of form and function, namely the impact of the synaptic anatomy on calcium dynamics in the presynaptic bouton, is a major challenge of present (computational) neuroscience at a cellular level. The Drosophila larval neuromuscular junction (NMJ) is a simple model system, which allows studying basic effects in a rather simple way. This synapse harbors several special structures. In particular, in opposite to standard vertebrate synapses, the presynaptic boutons are rather large, and they have several presynaptic zones. In these zones, different types of anatomical structures are present. Some of the zones bear a so-called T-bar, a particular anatomical structure. The geometric form of the T-bar resembles the shape of the letter “T” or a table with one leg. When an action potential arises, calcium influx is triggered. The probability of vesicle docking and neurotransmitter release is superlinearly proportional to the concentration of calcium close to the vesicular release site. It is tempting to assume that the T-bar causes some sort of calcium accumulation and hence triggers a higher release probability and thus enhances neurotransmitter exocytosis. In order to study this influence in a quantitative manner, we constructed a typical T-bar geometry and compared the calcium concentration close to the active zones (AZs). We compared the case of synapses with and without T-bars. Indeed, we found a substantial influence of the T-bar structure on the presynaptic calcium concentrations close to the AZs, indicating that this anatomical structure increases vesicle release probability. Therefore, our study reveals how the T-bar zone implies a strong relation between form and function. Our study answers the question of experimental studies (namely “Wichmann and Sigrist, Journal of neurogenetics 2010”) concerning the sense of the anatomical structure of the T-bar.

Calcium modeling of spine apparatus-containing human dendritic spines demonstrates an “all-or-nothing” communication switch between the spine head and dendrite

Dendritic spines are highly dynamic neuronal compartments that control the synaptic transmission between neurons. Spines form ultrastructural units, coupling synaptic contact sites to the dendritic shaft and often harbor a spine apparatus organelle, composed of smooth endoplasmic reticulum, which is responsible for calcium sequestration and release into the spine head and neck. The spine apparatus has recently been linked to synaptic plasticity in adult human cortical neurons. While the morphological heterogeneity of spines and their intracellular organization has been extensively demonstrated in animal models, the influence of spine apparatus organelles on critical signaling pathways, such as calcium-mediated dynamics, is less well known in human dendritic spines. In this study we used serial transmission electron microscopy to anatomically reconstruct nine human cortical spines in detail as a basis for modeling and simulation of the calcium dynamics between spine and dendrite. The anatomical study of reconstructed human dendritic spines revealed that the size of the postsynaptic density correlates with spine head volume and that the spine apparatus volume is proportional to the spine volume. Using a newly developed simulation pipeline, we have linked these findings to spine-to-dendrite calcium communication. While the absence of a spine apparatus, or the presence of a purely passive spine apparatus did not enable any of the reconstructed spines to relay a calcium signal to the dendritic shaft, the calcium-induced calcium release from this intracellular organelle allowed for finely tuned “all-or-nothing” spine-to-dendrite calcium coupling; controlled by spine morphology, neck plasticity, and ryanodine receptors. Our results suggest that spine apparatus organelles are strategically positioned in the neck of human dendritic spines and demonstrate their potential relevance to the maintenance and regulation of spine-to-dendrite calcium communication.

During the past decade it has become increasingly clear that abnormal synaptic plasticity is a major hallmark of neurological and cognitive disorders. Developing a better understanding of the synaptic plasticity process, which describes the ability of neurons to adapt their contacts in an activity-dependent manner, will lead to improved treatment of many neurological and cognitive disorders. It is known that calcium-dependent events such as synaptic transmission, intracellular calcium release, and calcium wave propagation, are required for many types of synaptic plasticity expression. However, the biological significance of these processes in neurons of the adult human cortex remains unknown. Due to technical limitations and ethical concerns, experimental data addressing this biologically and clinically relevant topic are not available. Therefore, we have implemented a computational model to study the intracellular calcium dynamics in realistic human dendritic spines based on detailed morphological reconstructions. With our model and simulations, we have established the morphological and biological requirements for the propagation of calcium from spines into the dendrites. Our results suggest a critical role for the calcium-storing spine apparatus organelle in regulating calcium homeostasis and propagation in human dendritic spines.

High rates of calcium-free diffusion in the cytosol of living cells

Calcium (Ca²⁺) is a universal second messenger that participates in the regulation of innumerous physiological processes. The way in which local elevations of the cytosolic Ca²⁺ concentration spread in space and time is key for the versatility of the signals. Ca²⁺ diffusion in the cytosol is hindered by its interaction with proteins that act as buffers. Depending on the concentrations and the kinetics of the interactions, there is a large range of values at which Ca²⁺ diffusion can proceed. Having reliable estimates of this range, particularly of its highest end, which corresponds to the ions free diffusion, is key to understand how the signals propagate. In this work, we present the first experimental results with which the Ca²⁺-free diffusion coefficient is directly quantified in the cytosol of living cells. By means of fluorescence correlation spectroscopy experiments performed in Xenopus laevis oocytes and in cells of Saccharomyces cerevisiae, we show that the ions can freely diffuse in the cytosol at a higher rate than previously thought.

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.

Conductance of porous media depends on external electric fields

In obstacle-filled media, such as extracellular or intracellular lumen of brain tissue, effective ion-diffusion permeability is a key determinant of electrogenic reactions. Although this diffusion permeability is thought to depend entirely on structural features of the medium, such as porosity and tortuosity, brain tissue shows prominent nonohmic properties, the origins of which remain poorly understood. Here, we explore Monte Carlo simulations of ion diffusion in a space filled with overlapping spheres to predict that diffusion permeability of such media decreases with stronger external electric fields. This dependence increases with lower medium porosity while decreasing with radial (two-dimensional or three-dimensional) compared with homogenous (one-dimensional) fields. We test our predictions empirically in an electrolyte chamber filled with microscopic glass spheres and find good correspondence with our predictions. A theoretical insight relates this phenomenon to a disproportionately increased dwell time of diffusing ions at potential barriers (or traps) representing geometric obstacles when the field strength increases. The dependence of medium ion-diffusion permeability on electric field could be important for understanding conductivity properties of porous materials, in particular for the accurate interpretation of electric activity recordings in brain tissue.

Mechanosensation in traumatic brain injury

Traumatic brain injury (TBI) is distinct from other neurological disorders because it is induced by a discrete event that applies extreme mechanical forces to the brain. This review describes how the brain senses, integrates, and responds to forces under both normal conditions and during injury. The response to forces is influenced by the unique mechanical properties of brain tissue, which differ by region, cell type, and sub-cellular structure. Elements such as the extracellular matrix, plasma membrane, transmembrane receptors, and cytoskeleton influence its properties. These same components also act as force-sensors, allowing neurons and glia to respond to their physical environment and maintain homeostasis. However, when applied forces become too large, as in TBI, these components may respond in an aberrant manner or structurally fail, resulting in unique pathological sequelae. This so-called "pathological mechanosensation" represents a spectrum of cellular responses, which vary depending on the overall biomechanical parameters of the injury and may be compounded by repetitive injuries. Such aberrant physical responses and/or damage to cells along with the resulting secondary injury cascades can ultimately lead to long-term cellular dysfunction and degeneration, often resulting in persistent deficits. Indeed, pathological mechanosensation not only directly initiates secondary injury cascades, but this post-physical damage environment provides the context in which these cascades unfold. Collectively, these points underscore the need to use experimental models that accurately replicate the biomechanics of TBI in humans. Understanding cellular responses in context with injury biomechanics may uncover therapeutic targets addressing various facets of trauma-specific sequelae.

Metabolic Compartmentalization at the Leading Edge of Metastatic Cancer Cells

Despite advances in targeted therapeutics and understanding in molecular mechanisms, metastasis remains a substantial obstacle for cancer treatment. Acquired genetic mutations and transcriptional changes can promote the spread of primary tumor cells to distant tissues. Additionally, recent studies have uncovered that metabolic reprogramming of cancer cells is tightly associated with cancer metastasis. However, whether intracellular metabolism is spatially and temporally regulated for cancer cell migration and invasion is understudied. In this review, we highlight the emergence of a concept, termed “membraneless metabolic compartmentalization,” as one of the critical mechanisms that determines the metastatic capacity of cancer cells. In particular, we focus on the compartmentalization of purine nucleotide metabolism (e.g., ATP and GTP) at the leading edge of migrating cancer cells through the uniquely phase-separated microdomains where dynamic exchange of nucleotide metabolic enzymes takes place. We will discuss how future insights may usher in a novel class of therapeutics specifically targeting the metabolic compartmentalization that drives tumor metastasis.

Molecular Basis of Mitochondrial and Peroxisomal Division Machineries

Mitochondria and peroxisomes are ubiquitous subcellular organelles that are highly dynamic and possess a high degree of plasticity. These organelles proliferate through division of pre-existing organelles. Studies on yeast, mammalian cells, and unicellular algae have led to a surprising finding that mitochondria and peroxisomes share the components of their division machineries. At the heart of the mitochondrial and peroxisomal division machineries is a GTPase dynamin-like protein, Dnm1/Drp1, which forms a contractile ring around the neck of the dividing organelles. During division, Dnm1/Drp1 functions as a motor protein and constricts the membrane. This mechanochemical work is achieved by utilizing energy from GTP hydrolysis. Over the last two decades, studies have focused on the structure and assembly of Dnm1/Drp1 molecules around the neck. However, the regulation of GTP during the division of mitochondrion and peroxisome is not well understood. Here, we review the current understanding of Dnm1/Drp1-mediated divisions of mitochondria and peroxisomes, exploring the mechanisms of GTP regulation during the Dnm1/Drp1 function, and provide new perspectives on their potential contribution to mitochondrial and peroxisomal biogenesis.

Comprehensive Imaging of Sensory-Evoked Activity of Entire Neurons Within the Awake Developing Brain Using Ultrafast AOD-Based Random-Access Two-Photon Microscopy

Determining how neurons transform synaptic input and encode information in action potential (AP) firing output is required for understanding dendritic integration, neural transforms and encoding. Limitations in the speed of imaging 3D volumes of brain encompassing complex dendritic arbors in vivo using conventional galvanometer mirror-based laser-scanning microscopy has hampered fully capturing fluorescent sensors of activity throughout an individual neuron's entire complement of synaptic inputs and somatic APs. To address this problem, we have developed a two-photon microscope that achieves high-speed scanning by employing inertia-free acousto-optic deflectors (AODs) for laser beam positioning, enabling random-access sampling of hundreds to thousands of points-of-interest restricted to a predetermined neuronal structure, avoiding wasted scanning of surrounding extracellular tissue. This system is capable of comprehensive imaging of the activity of single neurons within the intact and awake vertebrate brain. Here, we demonstrate imaging of tectal neurons within the brains of albino Xenopus laevis tadpoles labeled using single-cell electroporation for expression of a red space-filling fluorophore to determine dendritic arbor morphology, and either the calcium sensor jGCaMP7s or the glutamate sensor iGluSnFR as indicators of neural activity. Using discrete, point-of-interest scanning we achieve sampling rates of 3 Hz for saturation sampling of entire arbors at 2 μm resolution, 6 Hz for sequentially sampling 3 volumes encompassing the dendritic arbor and soma, and 200–250 Hz for scanning individual planes through the dendritic arbor. This system allows investigations of sensory-evoked information input-output relationships of neurons within the intact and awake brain.

Electrical Compartmentalization in Neurons

The dendritic tree of neurons plays an important role in information processing in the brain. While it is thought that dendrites require independent subunits to perform most of their computations, it is still not understood how they compartmentalize into functional subunits. Here, we show how these subunits can be deduced from the properties of dendrites. We devised a formalism that links the dendritic arborization to an impedance-based tree graph and show how the topology of this graph reveals independent subunits. This analysis reveals that cooperativity between synapses decreases slowly with increasing electrical separation and thus that few independent subunits coexist. We nevertheless find that balanced inputs or shunting inhibition can modify this topology and increase the number and size of the subunits in a context-dependent manner. We also find that this dynamic recompartmentalization can enable branch-specific learning of stimulus features. Analysis of dendritic patch-clamp recording experiments confirmed our theoretical predictions.

The functional organization of excitation and inhibition in the dendrites of mouse direction-selective ganglion cells

Recent studies indicate that the precise timing and location of excitation and inhibition (E/I) within active dendritic trees can significantly impact neuronal function. How synaptic inputs are functionally organized at the subcellular level in intact circuits remains unclear. To address this issue, we took advantage of the retinal direction-selective ganglion cell circuit, where directionally tuned inhibition is known to shape non-directional excitatory signals. We combined two-photon calcium imaging with genetic, pharmacological, and single-cell ablation methods to examine the extent to which inhibition ‘vetoes’ excitation at the level of individual dendrites of direction-selective ganglion cells. We demonstrate that inhibition shapes direction selectivity independently within small dendritic segments (<10µm) with remarkable accuracy. The data suggest that the parallel processing schemes proposed for direction encoding could be more fine-grained than previously envisioned.

Increased Confinement and Polydispersity of STIM1 and Orai1 after Ca2+ Store Depletion

STIM1 (a Ca²⁺ sensor in the endoplasmic reticulum (ER) membrane) and Orai1 (a pore-forming subunit of the Ca²⁺-release-activated calcium channel in the plasma membrane) diffuse in the ER membrane and plasma membrane, respectively. Upon depletion of Ca²⁺ stores in the ER, STIM1 translocates to the ER-plasma membrane junction and binds Orai1 to trigger store-operated Ca²⁺ entry. However, the motion of STIM1 and Orai1 during this process and its roles to Ca²⁺ entry is poorly understood. Here, we report real-time tracking of single STIM1 and Orai1 particles in the ER membrane and plasma membrane in living cells before and after Ca²⁺ store depletion. We found that the motion of single STIM1 and Orai1 particles exhibits anomalous diffusion both before and after store depletion, and their mobility-measured by the radius of gyration of the trajectories, mean-square displacement, and generalized diffusion coefficient-decreases drastically after store depletion. We also found that the measured displacement distribution is non-Gaussian, and the non-Gaussian parameter drastically increases after store depletion. Detailed analyses and simulations revealed that single STIM1 and Orai1 particles are confined in the compartmentalized membrane both before and after store depletion, and the changes in the motion after store depletion are explained by increased confinement and polydispersity of STIM1-Orai1 complexes formed at the ER-plasma membrane junctions. Further simulations showed that this increase in the confinement and polydispersity after store depletion localizes a rapid increase of Ca²⁺ influx, which can facilitate the rapid activation of local Ca²⁺ signaling pathways and the efficient replenishing of Ca²⁺ store in the ER in store-operated Ca²⁺ entry.

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.

Geometric principles of second messenger dynamics in dendritic spines

Dendritic spines are small, bulbous protrusions along dendrites in neurons and play a critical role in synaptic transmission. Dendritic spines come in a variety of shapes that depend on their developmental state. Additionally, roughly 14-19% of mature spines have a specialized endoplasmic reticulum called the spine apparatus. How does the shape of a postsynaptic spine and its internal organization affect the spatio-temporal dynamics of short timescale signaling? Answers to this question are central to our understanding the initiation of synaptic transmission, learning, and memory formation. In this work, we investigated the effect of spine and spine apparatus size and shape on the spatio-temporal dynamics of second messengers using mathematical modeling using reaction-diffusion equations in idealized geometries (ellipsoids, spheres, and mushroom-shaped). Our analyses and simulations showed that in the short timescale, spine size and shape coupled with the spine apparatus geometries govern the spatiotemporal dynamics of second messengers. We show that the curvature of the geometries gives rise to pseudo-harmonic functions, which predict the locations of maximum and minimum concentrations along the spine head. Furthermore, we showed that the lifetime of the concentration gradient can be fine-tuned by localization of fluxes on the spine head and varying the relative curvatures and distances between the spine apparatus and the spine head. Thus, we have identified several key geometric determinants of how the spine head and spine apparatus may regulate the short timescale chemical dynamics of small molecules that control synaptic plasticity.

Fast calcium transients in dendritic spines driven by extreme statistics

Fast calcium transients (<10 ms) remain difficult to analyse in cellular microdomains, yet they can modulate key cellular events such as trafficking, local ATP production by endoplasmic reticulum-mitochondria complex (ER-mitochondria complex), or spontaneous activity in astrocytes. In dendritic spines receiving synaptic inputs, we show here that in the presence of a spine apparatus (SA), which is an extension of the smooth ER, a calcium-induced calcium release (CICR) is triggered at the base of the spine by the fastest calcium ions arriving at a Ryanodyne receptor (RyR). The mechanism relies on the asymmetric distributions of RyRs and sarco/ER calcium-ATPase (SERCA) pumps that we predict using a computational model and further confirm experimentally in culture and slice hippocampal neurons. The present mechanism for which the statistics of the fastest particles arriving at a small target, followed by an amplification, is likely to be generic in molecular transduction across cellular microcompartments, such as thin neuronal processes, astrocytes, endfeets, or protrusions.

The First 100 nm Inside the Pre-synaptic Terminal Where Calcium Diffusion Triggers Vesicular Release

Calcium diffusion in the thin 100 nm layer located between the plasma membrane and docked vesicles in the pre-synaptic terminal of neuronal cells mediates vesicular fusion and synaptic transmission. Accounting for the narrow-cusp geometry located underneath the vesicle is a key ingredient that defines the probability and the time scale of calcium diffusion to bind calcium sensors for the initiation of vesicular release. We review here the time scale, the calcium binding dynamics and the consequences for asynchronous versus synchronous release. To conclude, three-dimensional modeling approaches and the associated coarse-grained simulations can now account efficiently for the precise co-organization of vesicles and Voltage-Gated-Calcium-Channel (VGCC). This co-organization is a key determinant of short-term plasticity and it shapes asynchronous release. Moreover, changing the location of VGCC from few nanometers underneath the vesicle modifies significantly the release probability. Finally, by modifying the calcium buffer concentration, a single synapse can switch from facilitation to depression.

Fluorescence correlation spectroscopy experiments to quantify free diffusion coefficients in reaction-diffusion systems: The case of Ca^{2+} and its dyes

Many cell signaling pathways involve the diffusion of messengers that bind and unbind to and from intracellular components. Quantifying their net transport rate under different conditions then requires having separate estimates of their free diffusion coefficient and binding or unbinding rates. In this paper, we show how performing sets of fluorescence correlation spectroscopy (FCS) experiments under different conditions, it is possible to quantify free diffusion coefficients and on and off rates of reaction-diffusion systems. We develop the theory and present a practical implementation for the case of the universal second messenger, calcium (Ca^{2+}) and single-wavelength dyes that increase their fluorescence upon Ca^{2+} binding. We validate the approach with experiments performed in aqueous solutions containing Ca^{2+} and Fluo4 dextran (both in its high and low affinity versions). Performing FCS experiments with tetramethylrhodamine-dextran in Xenopus laevis oocytes, we infer the corresponding free diffusion coefficients in the cytosol of these cells. Our approach can be extended to other physiologically relevant reaction-diffusion systems to quantify biophysical parameters that determine the dynamics of various variables of interest.

FCS experiments to quantify Ca2+ diffusion and its interaction with buffers

Ca²⁺ signals are ubiquitous. One of the key factors for their versatility is the variety of spatio-temporal distributions that the cytosolic Ca²⁺ can display. In most cell types Ca²⁺ signals not only depend on Ca²⁺ entry from the extracellular medium but also on Ca²⁺ release from internal stores, a process which is in turn regulated by cytosolic Ca²⁺ itself. The rate at which Ca²⁺ is transported, the fraction that is trapped by intracellular buffers, and with what kinetics are thus key features that affect the time and spatial range of action of Ca²⁺ signals. The quantification of Ca²⁺ diffusion in intact cells is quite challenging because the transport rates that can be inferred using optical techniques are intricately related to the interaction of Ca²⁺ with the dye that is used for its observation and with the cellular buffers. In this paper, we introduce an approach that uses Fluorescence Correlation Spectroscopy (FCS) experiments performed at different conditions that in principle allows the quantification of Ca²⁺ diffusion and of its reaction rates with unobservable (non-fluorescent) Ca²⁺ buffers. To this end, we develop the necessary theory to interpret the experimental results and then apply it to FCS experiments performed in a set of solutions containing Ca²⁺, a single wavelength Ca²⁺ dye, and a non-fluorescent Ca²⁺ buffer. We show that a judicious choice of the experimental conditions and an adequate interpretation of the fitting parameters can be combined to extract information on the free diffusion coefficient of Ca²⁺ and of some of the properties of the unobservable buffer. We think that this approach can be applied to other situations, particularly to experiments performed in intact cells.

Hybrid Markov-mass action law model for cell activation by rare binding events: Application to calcium induced vesicular release at neuronal synapses

Binding of molecules, ions or proteins to small target sites is a generic step of cell activation. This process relies on rare stochastic events where a particle located in a large bulk has to find small and often hidden targets. We present here a hybrid discrete-continuum model that takes into account a stochastic regime governed by rare events and a continuous regime in the bulk. The rare discrete binding events are modeled by a Markov chain for the encounter of small targets by few Brownian particles, for which the arrival time is Poissonian. The large ensemble of particles is described by mass action laws. We use this novel model to predict the time distribution of vesicular release at neuronal synapses. Vesicular release is triggered by the binding of few calcium ions that can originate either from the synaptic bulk or from the entry through calcium channels. We report here that the distribution of release time is bimodal although it is triggered by a single fast action potential. While the first peak follows a stimulation, the second corresponds to the random arrival over much longer time of ions located in the synaptic terminal to small binding vesicular targets. To conclude, the present multiscale stochastic modeling approach allows studying cellular events based on integrating discrete molecular events over several time scales.

Induction and Consolidation of Calcium-Based Homo- and Heterosynaptic Potentiation and Depression

The adaptive mechanisms of homo- and heterosynaptic plasticity play an important role in learning and memory. In order to maintain plasticity-induced changes for longer time scales (up to several days), they have to be consolidated by transferring them from a short-lasting early-phase to a long-lasting late-phase state. The underlying processes of this synaptic consolidation are already well-known for homosynaptic plasticity, however, it is not clear whether the same processes also enable the induction and consolidation of heterosynaptic plasticity. In this study, by extending a generic calcium-based plasticity model with the processes of synaptic consolidation, we show in simulations that indeed heterosynaptic plasticity can be induced and, furthermore, consolidated by the same underlying processes as for homosynaptic plasticity. Furthermore, we show that by local diffusion processes the heterosynaptic effect can be restricted to a few synapses neighboring the homosynaptically changed ones. Taken together, this generic model reproduces many experimental results of synaptic tagging and consolidation, provides several predictions for heterosynaptic induction and consolidation, and yields insights into the complex interactions between homo- and heterosynaptic plasticity over a broad variety of time (minutes to days) and spatial scales (several micrometers).

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.

Microdomain [Ca(2+)] Fluctuations Alter Temporal Dynamics in Models of Ca(2+)-Dependent Signaling Cascades and Synaptic Vesicle Release

Ca(2+)-dependent signaling is often localized in spatially restricted microdomains and may involve only 1 to 100 Ca(2+) ions. Fluctuations in the microdomain Ca(2+) concentration (Ca(2+)) can arise from a wide range of elementary processes, including diffusion, Ca(2+) influx, and association/dissociation with Ca(2+) binding proteins or buffers. However, it is unclear to what extent these fluctuations alter Ca(2+)-dependent signaling. We construct Markov models of a general Ca(2+)-dependent signaling cascade and Ca(2+)-triggered synaptic vesicle release. We compare the hitting (release) time distribution and statistics for models that account for [Ca(2+)] fluctuations with the corresponding models that neglect these fluctuations. In general, when Ca(2+) fluctuations are much faster than the characteristic time for the signaling event, the hitting time distributions and statistics for the models with and without Ca(2+) fluctuation are similar. However, when the timescale of Ca(2+) fluctuations is on the same order as the signaling cascade or slower, the hitting time mean and variability are typically increased, in particular when the average number of microdomain Ca(2+) ions is small, a consequence of a long-tailed hitting time distribution. In a model of Ca(2+)-triggered synaptic vesicle release, we demonstrate the conditions for which [Ca(2+)] fluctuations do and do not alter the distribution, mean, and variability of release timing. We find that both the release time mean and variability can be increased, demonstrating that Ca(2+) fluctuations are an important aspect of microdomain Ca(2+) signaling and further suggesting that Ca(2+) fluctuations in the presynaptic terminal may contribute to variability in synaptic vesicle release and thus variability in neuronal spiking.

Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximation

The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudinal diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.

Molecular mechanisms underlying neuronal synaptic plasticity: systems biology meets computational neuroscience in the wilds of synaptic plasticity

Interactions among signaling pathways that are activated by transmembrane receptors produce complex networks and emergent dynamical behaviors that are implicated in synaptic plasticity. Temporal dynamics and spatial aspects are critical determinants of cell responses such as synaptic plasticity, although the mapping between spatiotemporal activity pattern and direction of synaptic plasticity is not completely understood. Computational modeling of neuronal signaling pathways has significantly contributed to understanding signaling pathways underlying synaptic plasticity. Spatial models of signaling pathways in hippocampal neurons have revealed mechanisms underlying the spatial distribution of extracellular signal-related kinase (ERK) activation in hippocampal neurons. Other spatial models have demonstrated that the major role of anchoring proteins in striatal and hippocampal synaptic plasticity is to place molecules near their activators. Simulations of yet other models have revealed that the spatial distribution of synaptic plasticity may differ for potentiation versus depression. In general, the most significant advances have been made by interactive modeling and experiments; thus, an interdisciplinary approach should be applied to investigate critical issues in neuronal signaling pathways. These issues include identifying which transmembrane receptors are key for activating ERK in neurons, and the crucial targets of kinases that produce long-lasting synaptic plasticity. Although the number of computer programs for computationally efficient simulation of large reaction-diffusion networks is increasing, parameter estimation and sensitivity analysis in these spatial models remain more difficult than in single compartment models. Advances in live cell imaging coupled with further software development will continue to accelerate the development of spatial models of synaptic plasticity.