Modeling and analysis of calcium signaling events leading to long-term depression in cerebellar Purkinje cells

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.

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.

Redundancy principle and the role of extreme statistics in molecular and cellular biology

The paradigm of chemical activation rates in cellular biology has been shifted from the mean arrival time of a single particle to the mean of the first among many particles to arrive at a small activation site. The activation rate is set by extremely rare events, which have drastically different time scales from the mean times between activations, and depends on different structural parameters. This shift calls for reconsideration of physical processes used in deterministic and stochastic modeling of chemical reactions that are based on the traditional forward rate, especially for fast activation processes in living cells. Consequently, the biological activation time is not necessarily exponentially distributed. We review here the physical models, the mathematical analysis and the new paradigm of setting the scale to be the shortest time for activation that clarifies the role of population redundancy in selecting and accelerating transient cellular search processes. We provide examples in cellular transduction, gene activation, cell senescence activation or spermatozoa selection during fertilization, where the rate depends on numbers. We conclude that the statistics of the minimal time to activation set kinetic laws in biology, which can be very different from the ones associated to average times.

Spontaneous calcium signaling of cartilage cells: from spatiotemporal features to biophysical modeling

Intracellular calcium ([Ca²⁺]_i) oscillation is a fundamental signaling response of cartilage cells under mechanical loading or osmotic stress. Chondrocytes are usually considered as nonexcitable cells with no spontaneous [Ca²⁺]_i signaling. This study proved that chondrocytes can exhibit robust spontaneous [Ca²⁺]_i signaling without explicit external stimuli. The intensity of [Ca²⁺]_i peaks from individual chondrocytes maintain a consistent spatiotemporal pattern, acting as a unique "fingerprint" for each cell. Statistical analysis revealed lognormal distributions of the temporal parameters of [Ca²⁺]_i peaks, as well as strong linear correlations between their means and sds. Based on these statistical findings, we hypothesized that the spontaneous [Ca²⁺]_i peaks may result from an autocatalytic process and that [Ca²⁺]_i oscillation is controlled by a threshold-regulating mechanism. To test these 2 mechanisms, we established a multistage biophysical model by assuming the spontaneous [Ca²⁺]_i signaling of chondrocytes as a combination of deterministic and stochastic processes. The theoretical model successfully explained the lognormal distribution of the temporal parameters and the fingerprint feature of [Ca²⁺]_i peaks. In addition, by using antagonists for 10 pathways, we revealed that the initiation of spontaneous [Ca²⁺]_i peaks in chondrocytes requires the presence of extracellular Ca²⁺, and that the PLC-inositol 1,4,5-trisphosphate pathway, which controls the release of calcium from the endoplasmic reticulum, can affect the initiation of spontaneous [Ca²⁺]_i peaks in chondrocytes. The purinoceptors and transient receptor potential vanilloid 4 channels on the plasma membrane also play key roles in the spontaneous [Ca²⁺]_i signaling of chondrocytes. In contrast, blocking the T-type or L-type voltage-gated calcium channel promoted the spontaneous calcium signaling. This study represents a systematic effort to understand the features and initiation mechanisms of spontaneous [Ca²⁺]_i signaling in chondrocytes, which are critical for chondrocyte mechanobiology.-Zhou, Y., Lv, M., Li, T., Zhang, T., Duncan, R., Wang, L., Lu, X. L. Spontaneous calcium signaling of cartilage cells: from spatiotemporal features to biophysical modeling.

Disentangling astroglial physiology with a realistic cell model in silico

Electrically non-excitable astroglia take up neurotransmitters, buffer extracellular K+ and generate Ca2+ signals that release molecular regulators of neural circuitry. The underlying machinery remains enigmatic, mainly because the sponge-like astrocyte morphology has been difficult to access experimentally or explore theoretically. Here, we systematically incorporate multi-scale, tri-dimensional astroglial architecture into a realistic multi-compartmental cell model, which we constrain by empirical tests and integrate into the NEURON computational biophysical environment. This approach is implemented as a flexible astrocyte-model builder ASTRO. As a proof-of-concept, we explore an in silico astrocyte to evaluate basic cell physiology features inaccessible experimentally. Our simulations suggest that currents generated by glutamate transporters or K+ channels have negligible distant effects on membrane voltage and that individual astrocytes can successfully handle extracellular K+ hotspots. We show how intracellular Ca2+ buffers affect Ca2+ waves and why the classical Ca2+ sparks-and-puffs mechanism is theoretically compatible with common readouts of astroglial Ca2+ imaging.

Reproducibility in Computational Neuroscience Models and Simulations

OBJECTIVE

Like all scientific research, computational neuroscience research must be reproducible. Big data science, including simulation research, cannot depend exclusively on journal articles as the method to provide the sharing and transparency required for reproducibility.

METHODS

Ensuring model reproducibility requires the use of multiple standard software practices and tools, including version control, strong commenting and documentation, and code modularity.

RESULTS

Building on these standard practices, model-sharing sites and tools have been developed that fit into several categories: 1) standardized neural simulators; 2) shared computational resources; 3) declarative model descriptors, ontologies, and standardized annotations; and 4) model-sharing repositories and sharing standards.

CONCLUSION

A number of complementary innovations have been proposed to enhance sharing, transparency, and reproducibility. The individual user can be encouraged to make use of version control, commenting, documentation, and modularity in development of models. The community can help by requiring model sharing as a condition of publication and funding.

SIGNIFICANCE

Model management will become increasingly important as multiscale models become larger, more detailed, and correspondingly more difficult to manage by any single investigator or single laboratory. Additional big data management complexity will come as the models become more useful in interpreting experiments, thus increasing the need to ensure clear alignment between modeling data, both parameters and results, and experiment.

Visualization of Ca2+ Filling Mechanisms upon Synaptic Inputs in the Endoplasmic Reticulum of Cerebellar Purkinje Cells

The endoplasmic reticulum (ER) plays crucial roles in intracellular Ca(2+) signaling, serving as both a source and sink of Ca(2+), and regulating a variety of physiological and pathophysiological events in neurons in the brain. However, spatiotemporal Ca(2+) dynamics within the ER in central neurons remain to be characterized. In this study, we visualized synaptic activity-dependent ER Ca(2+) dynamics in mouse cerebellar Purkinje cells (PCs) using an ER-targeted genetically encoded Ca(2+) indicator, G-CEPIA1er. We used brief parallel fiber stimulation to induce a local decrease in the ER luminal Ca(2+) concentration ([Ca(2+)]ER) in dendrites and spines. In this experimental system, the recovery of [Ca(2+)]ER takes several seconds, and recovery half-time depends on the extent of ER Ca(2+) depletion. By combining imaging analysis and numerical simulation, we show that the intraluminal diffusion of Ca(2+), rather than Ca(2+) reuptake, is the dominant mechanism for the replenishment of the local [Ca(2+)]ER depletion immediately following the stimulation. In spines, the ER filled almost simultaneously with parent dendrites, suggesting that the ER within the spine neck does not represent a significant barrier to Ca(2+) diffusion. Furthermore, we found that repetitive climbing fiber stimulation, which induces cytosolic Ca(2+) spikes in PCs, cumulatively increased [Ca(2+)]ER. These results indicate that the neuronal ER functions both as an intracellular tunnel to redistribute stored Ca(2+) within the neurons, and as a leaky integrator of Ca(2+) spike-inducing synaptic inputs.

Investigation of sex differences in the expression of RORA and its transcriptional targets in the brain as a potential contributor to the sex bias in autism

Background

Autism spectrum disorder (ASD) is a neurodevelopmental condition characterized by significant impairment in reciprocal social interactions and communication coupled with stereotyped, repetitive behaviors and restricted interests. Although genomic and functional studies are beginning to reveal some of the genetic complexity and underlying pathobiology of ASD, the consistently reported male bias of ASD remains an enigma. We have recently proposed that retinoic acid-related orphan receptor alpha (RORA), which is reduced in the brain and lymphoblastoid cell lines of multiple cohorts of individuals with ASD and oppositely regulated by male and female hormones, might contribute to the sex bias in autism by differentially regulating target genes, including CYP19A1 (aromatase), in a sex-dependent manner that can also lead to elevated testosterone levels, a proposed risk factor for autism.

Methods

In this study, we examine sex differences in RORA and aromatase protein levels in cortical tissues of unaffected and affected males and females by re-analyzing pre-existing confocal immunofluorescence data from our laboratory. We further investigated the expression of RORA and its correlation with several of its validated transcriptional targets in the orbital frontal cortex and cerebellum as a function of development using RNAseq data from the BrainSpan Atlas of the Developing Human Brain. In a pilot study, we also analyzed the expression of Rora and the same transcriptional targets in the cortex and cerebellum of adult wild-type male and female C57BL/6 mice.

Results

Our findings suggest that Rora/RORA and several of its transcriptional targets may exhibit sexually dimorphic expression in certain regions of the brain of both mice and humans. Interestingly, the correlation coefficients between Rora expression and that of its targets are much higher in the cortex of male mice relative to that of female mice. A strong positive correlation between the levels of RORA and aromatase proteins is also seen in the cortex of control human males and females as well as ASD males, but not ASD females.

Conclusions

Based on these studies, we suggest that disruption of Rora/RORA expression may have a greater impact on males, since sex differences in the correlation of RORA and target gene expression indicate that RORA-deficient males may experience greater dysregulation of genes relevant to ASD in certain brain regions during development.

Electronic supplementary material

The online version of this article (doi:10.1186/2040-2392-6-7) contains supplementary material, which is available to authorized users.

Methodologies for the modeling and simulation of biochemical networks, illustrated for signal transduction pathways: a primer

Biochemical networks depict the chemical interactions that take place among elements of living cells. They aim to elucidate how cellular behavior and functional properties of the cell emerge from the relationships between its components, i.e. molecules. Biochemical networks are largely characterized by dynamic behavior, and exhibit high degrees of complexity. Hence, the interest in such networks is growing and they have been the target of several recent modeling efforts. Signal transduction pathways (STPs) constitute a class of biochemical networks that receive, process, and respond to stimuli from the environment, as well as stimuli that are internal to the organism. An STP consists of a chain of intracellular signaling processes that ultimately result in generating different cellular responses. This primer presents the methodologies used for the modeling and simulation of biochemical networks, illustrated for STPs. These methodologies range from qualitative to quantitative, and include structural as well as dynamic analysis techniques. We describe the different methodologies, outline their underlying assumptions, and provide an assessment of their advantages and disadvantages. Moreover, publicly and/or commercially available implementations of these methodologies are listed as appropriate. In particular, this primer aims to provide a clear introduction and comprehensive coverage of biochemical modeling and simulation methodologies for the non-expert, with specific focus on relevant literature of STPs.

Computational neurobiology is a useful tool in translational neurology: the example of ataxia

Hereditary ataxia, or motor incoordination, affects approximately 150,000 Americans and hundreds of thousands of individuals worldwide with onset from as early as mid-childhood. Affected individuals exhibit dysarthria, dysmetria, action tremor, and diadochokinesia. In this review, we consider an array of computational studies derived from experimental observations relevant to human neuropathology. A survey of related studies illustrates the impact of integrating clinical evidence with data from mouse models and computational simulations. Results from these studies may help explain findings in mice, and after extensive laboratory study, may ultimately be translated to ataxic individuals. This inquiry lays a foundation for using computation to understand neurobiochemical and electrophysiological pathophysiology of spinocerebellar ataxias and may contribute to development of therapeutics. The interdisciplinary analysis suggests that computational neurobiology can be an important tool for translational neurology.

Integration of modeling with experimental and clinical findings synthesizes and refines the central role of inositol 1,4,5-trisphosphate receptor 1 in spinocerebellar ataxia

A suite of models was developed to study the role of inositol 1,4,5-trisphosphate receptor 1 (IP3R1) in spinocerebellar ataxias (SCAs). Several SCAs are linked to reduced abundance of IP3R1 or to supranormal sensitivity of the receptor to activation by its ligand inositol 1,4,5-trisphosphate (IP3). Detailed multidimensional models have been created to simulate biochemical calcium signaling and membrane electrophysiology in cerebellar Purkinje neurons. In these models, IP3R1-mediated calcium release is allowed to interact with ion channel response on the cell membrane. Experimental findings in mice and clinical observations in humans provide data input for the models. The SCA modeling suite helps interpret experimental results and provides suggestions to guide experiments. The models predict IP3R1 supersensitivity in SCA1 and compensatory mechanisms in SCA1, SCA2, and SCA3. Simulations explain the impact of calcium buffer proteins. Results show that IP3R1-mediated calcium release activates voltage-gated calcium-activated potassium channels in the plasma membrane. The SCA modeling suite unifies observations from experiments in a number of SCAs. The cadre of simulations demonstrates the central role of IP3R1.

Quantitative properties and receptor reserve of the DAG and PKC branch of G(q)-coupled receptor signaling

G_q protein–coupled receptors (G_qPCRs) of the plasma membrane activate the phospholipase C (PLC) signaling cascade. PLC cleaves the membrane lipid phosphatidylinositol 4,5-bisphosphate (PIP₂) into the second messengers diacylgycerol (DAG) and inositol 1,4,5-trisphosphate (IP₃), leading to calcium release, protein kinase C (PKC) activation, and in some cases, PIP₂ depletion. We determine the kinetics of each of these downstream endpoints and also ask which is responsible for the inhibition of KCNQ2/3 (K_V7.2/7.3) potassium channels in single living tsA-201 cells. We measure DAG production and PKC activity by Förster resonance energy transfer–based sensors, and PIP₂ by KCNQ2/3 channels. Fully activating endogenous purinergic receptors by uridine 5′triphosphate (UTP) leads to calcium release, DAG production, and PKC activation, but no net PIP₂ depletion. Fully activating high-density transfected muscarinic receptors (M₁Rs) by oxotremorine-M (Oxo-M) leads to similar calcium, DAG, and PKC signals, but PIP₂ is depleted. KCNQ2/3 channels are inhibited by the Oxo-M treatment (85%) and not by UTP (<1%), indicating that depletion of PIP₂ is required to inhibit KCNQ2/3 in response to receptor activation. Overexpression of A kinase–anchoring protein (AKAP)79 or calmodulin (CaM) does not increase KCNQ2/3 inhibition by UTP. From these results and measurements of IP₃ and calcium presented in our companion paper (Dickson et al. 2013. J. Gen. Physiol.http://dx.doi.org/10.1085/jgp.201210886), we extend our kinetic model for signaling from M₁Rs to DAG/PKC and IP₃/calcium signaling. We conclude that calcium/CaM and PKC-mediated phosphorylation do not underlie dynamic KCNQ2/3 channel inhibition during G_qPCR activation in tsA-201 cells. Finally, our experimental data provide indirect evidence for cleavage of PI(4)P by PLC in living cells, and our modeling revisits/explains the concept of receptor reserve with measurements from all steps of G_qPCR signaling.

Approaches and tools for modeling signaling pathways and calcium dynamics in neurons

Signaling pathways are cascades of intracellular biochemical reactions that are activated by transmembrane receptors, and ultimately lead to transcription in the nucleus. In neurons, both calcium permeable synaptic and ionic channels as well as G protein coupled receptors initiate activation of signaling pathway molecules that interact with electrical activity at multiple spatial and time scales. At small temporal and spatial scales, calcium modifies the properties of ionic channels, whereas at larger temporal and spatial scales, various kinases and phosphatases modify the properties of ionic channels, producing phenomena such as synaptic plasticity and homeostatic plasticity. The elongated structure of neuronal dendrites and the organization of multi-protein complexes by anchoring proteins imply that the spatial dimension must be explicit. Therefore, modeling signaling pathways in neurons utilizes algorithms for both diffusion and reactions. The small size of spines coupled with small concentrations of some molecules implies that some reactions occur stochastically. The need for stochastic simulation of many reaction and diffusion events coupled with the multiple temporal and spatial scales makes modeling of signaling pathways a difficult problem. Several different software programs have achieved different aspects of these capabilities. This review explains some of the mathematical formulas used for modeling reactions and diffusion. In addition, it briefly presents the simulators used for modeling reaction-diffusion systems in neurons, together with scientific problems addressed.

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.

Genome-wide identification of transcriptional targets of RORA reveals direct regulation of multiple genes associated with autism spectrum disorder

BACKGROUND

We have recently identified the nuclear hormone receptor RORA (retinoic acid-related orphan receptor-alpha) as a novel candidate gene for autism spectrum disorder (ASD). Our independent cohort studies have consistently demonstrated the reduction of RORA transcript and/or protein levels in blood-derived lymphoblasts as well as in the postmortem prefrontal cortex and cerebellum of individuals with ASD. Moreover, we have also shown that RORA has the potential to be under negative and positive regulation by androgen and estrogen, respectively, suggesting the possibility that RORA may contribute to the male bias of ASD. However, little is known about transcriptional targets of this nuclear receptor, particularly in humans.

METHODS

Here we identify transcriptional targets of RORA in human neuronal cells on a genome-wide level using chromatin immunoprecipitation (ChIP) with an anti-RORA antibody followed by whole-genome promoter array (chip) analysis. Selected potential targets of RORA were then validated by an independent ChIP followed by quantitative PCR analysis. To further demonstrate that reduced RORA expression results in reduced transcription of RORA targets, we determined the expression levels of the selected transcriptional targets in RORA-deficient human neuronal cells, as well as in postmortem brain tissues from individuals with ASD who exhibit reduced RORA expression.

RESULTS

The ChIP-on-chip analysis reveals that RORA1, a major isoform of RORA protein in human brain, can be recruited to as many as 2,764 genomic locations corresponding to promoter regions of 2,544 genes across the human genome. Gene ontology analysis of this dataset of genes that are potentially directly regulated by RORA1 reveals statistically significant enrichment in biological functions negatively impacted in individuals with ASD, including neuronal differentiation, adhesion and survival, synaptogenesis, synaptic transmission and plasticity, and axonogenesis, as well as higher level functions such as development of the cortex and cerebellum, cognition, memory, and spatial learning. Independent ChIP-quantitative PCR analyses confirm binding of RORA1 to promoter regions of selected ASD-associated genes, including A2BP1, CYP19A1, ITPR1, NLGN1, and NTRK2, whose expression levels (in addition to HSD17B10) are also decreased in RORA1-repressed human neuronal cells and in prefrontal cortex tissues from individuals with ASD.

CONCLUSIONS

Findings from this study indicate that RORA transcriptionally regulates A2BP1, CYP19A1, HSD17B10, ITPR1, NLGN1, and NTRK2, and strongly suggest that reduction of this sex hormone-sensitive nuclear receptor in the brain causes dysregulated expression of these ASD-relevant genes as well as their associated pathways and functions which, in turn, may contribute to the underlying pathobiology of ASD.

A common framework of signal processing in the induction of cerebellar LTD and cortical STDP

Cerebellar long-term depression (LTD) and cortical spike-timing-dependent synaptic plasticity (STDP) are two well-known and well-characterized types of synaptic plasticity. Induction of both types of synaptic plasticity depends on the spike timing, pairing frequency, and pairing numbers of two different sources of spiking. This implies that the induction of synaptic plasticity may share common frameworks in terms of signal processing regardless of the different signaling pathways involved in the two types of synaptic plasticity. Here we propose that both types share common frameworks of signal processing for spike-timing, pairing-frequency, and pairing-numbers detection. We developed system models of both types of synaptic plasticity and analyzed signal processing in the induction of synaptic plasticity. We found that both systems have upstream subsystems for spike-timing detection and downstream subsystems for pairing-frequency and pairing-numbers detection. The upstream systems used multiplication of signals from the feedback filters and nonlinear functions for spike-timing detection. The downstream subsystems used temporal filters with longer time constants for pairing-frequency detection and nonlinear switch-like functions for pairing-numbers detection, indicating that the downstream subsystems serve as a leaky integrate-and-fire system. Thus, our findings suggest that a common conceptual framework for the induction of synaptic plasticity exists despite the differences in molecular species and pathways.

Models of calcium dynamics in cerebellar granule cells

Intracellular calcium dynamics is critical for many functions of cerebellar granule cells (GrCs) including membrane excitability, synaptic plasticity, apoptosis, and regulation of gene transcription. Recent measurements of calcium responses in GrCs to depolarization and synaptic stimulation reveal spatial compartmentalization and heterogeneity within dendrites of these cells. However, the main determinants of local calcium dynamics in GrCs are still poorly understood. One reason is that there have been few published studies of calcium dynamics in intact GrCs in their native environment. In the absence of complete information, biophysically realistic models are useful for testing whether specific Ca(2+) handling mechanisms may account for existing experimental observations. Simulation results can be used to identify critical measurements that would discriminate between different models. In this review, we briefly describe experimental studies and phenomenological models of Ca(2+) signaling in GrC, and then discuss a particular biophysical model, with a special emphasis on an approach for obtaining information regarding the distribution of Ca(2+) handling systems under conditions of incomplete experimental data. Use of this approach suggests that Ca(2+) channels and fixed endogenous Ca(2+) buffers are highly heterogeneously distributed in GrCs. Research avenues for investigating calcium dynamics in GrCs by a combination of experimental and modeling studies are proposed.

Computational analysis of calcium signaling and membrane electrophysiology in cerebellar Purkinje neurons associated with ataxia

Background

Mutations in the smooth endoplasmic reticulum (sER) calcium channel Inositol Trisphosphate Receptor type 1 (IP3R1) in humans with the motor function coordination disorders Spinocerebellar Ataxia Types 15 and 16 (SCA15/16) and in a corresponding mouse model, the IP3R1^{delta18/delta18} mice, lead to reduced IP3R1 levels. We posit that increasing IP3R1 sensitivity to IP3 in ataxias with reduced IP3R1 could restore normal calcium response. On the other hand, in mouse models of the human polyglutamine (polyQ) ataxias, SCA2, and SCA3, the primary finding appears to be hyperactive IP3R1-mediated calcium release. It has been suggested that the polyQ SCA1 mice may also show hyperactive IP3R1. Yet, SCA1 mice show downregulated gene expression of IP3R1, Homer, metabotropic glutamate receptor (mGluR), smooth endoplasmic reticulum Ca-ATP-ase (SERCA), calbindin, parvalbumin, and other calcium signaling proteins.

Results

We create a computational model of pathological alterations in calcium signaling in cerebellar Purkinje neurons to investigate several forms of spinocerebellar ataxia associated with changes in the abundance, sensitivity, or activity of the calcium channel IP3R1. We find that increasing IP3R1 sensitivity to IP3 in computational models of SCA15/16 can restore normal calcium response if IP3R1 abundance is not too low. The studied range in IP3R1 levels reflects variability found in human and mouse ataxic models. Further, the required fold increases in sensitivity are within experimental ranges from experiments that use IP3R1 phosphorylation status to adjust its sensitivity to IP3. Results from our simulations of polyglutamine SCAs suggest that downregulation of some calcium signaling proteins may be partially compensatory. However, the downregulation of calcium buffer proteins observed in the SCA1 mice may contribute to pathology. Finally, our model suggests that the calcium-activated voltage-gated potassium channels may provide an important link between calcium metabolism and membrane potential in Purkinje cell function.

Conclusion

Thus, we have established an initial platform for computational evaluation and prediction of ataxia pathophysiology. Specifically, the model has been used to investigate SCA15/16, SCA1, SCA2, and SCA3. Results suggest that experimental studies treating mouse models of any of these ataxias with appropriately chosen peptides resembling the C-terminal of IP3R1 could adjust receptor sensitivity, and thereby modulate calcium release and normalize IP3 response. In addition, the model supports the hypothesis of IP3R1 supersensitivity in SCA1.

Combined computational and experimental approaches to understanding the Ca(2+) regulatory network in neurons

Ca(2+) is a ubiquitous signaling ion that regulates a variety of neuronal functions by binding to and altering the state of effector proteins. Spatial relationships and temporal dynamics of Ca(2+) elevations determine many cellular responses of neurons to chemical and electrical stimulation. There is a wealth of information regarding the properties and distribution of Ca(2+) channels, pumps, exchangers, and buffers that participate in Ca(2+) regulation. At the same time, new imaging techniques permit characterization of evoked Ca(2+) signals with increasing spatial and temporal resolution. However, understanding the mechanistic link between functional properties of Ca(2+) handling proteins and the stimulus-evoked Ca(2+) signals they orchestrate requires consideration of the way Ca(2+) handling mechanisms operate together as a system in native cells. A wide array of biophysical modeling approaches is available for studying this problem and can be used in a variety of ways. Models can be useful to explain the behavior of complex systems, to evaluate the role of individual Ca(2+) handling mechanisms, to extract valuable parameters, and to generate predictions that can be validated experimentally. In this review, we discuss recent advances in understanding the underlying mechanisms of Ca(2+) signaling in neurons via mathematical modeling. We emphasize the value of developing realistic models based on experimentally validated descriptions of Ca(2+) transport and buffering that can be tested and refined through new experiments to develop increasingly accurate biophysical descriptions of Ca(2+) signaling in neurons.

Spatial modeling of cell signaling networks

The shape of a cell, the sizes of subcellular compartments, and the spatial distribution of molecules within the cytoplasm can all control how molecules interact to produce a cellular behavior. This chapter describes how these spatial features can be included in mechanistic mathematical models of cell signaling. The Virtual Cell computational modeling and simulation software is used to illustrate the considerations required to build a spatial model. An explanation of how to appropriately choose between physical formulations that implicitly or explicitly account for cell geometry and between deterministic versus stochastic formulations for molecular dynamics is provided, along with a discussion of their respective strengths and weaknesses. As a first step toward constructing a spatial model, the geometry needs to be specified and associated with the molecules, reactions, and membrane flux processes of the network. Initial conditions, diffusion coefficients, velocities, and boundary conditions complete the specifications required to define the mathematics of the model. The numerical methods used to solve reaction-diffusion problems both deterministically and stochastically are then described and some guidance is provided in how to set up and run simulations. A study of cAMP signaling in neurons ends the chapter, providing an example of the insights that can be gained in interpreting experimental results through the application of spatial modeling.

Virtual NEURON: a strategy for merged biochemical and electrophysiological modeling

Because of its highly branched dendrite, the Purkinje neuron requires significant computational resources if coupled electrical and biochemical activity are to be simulated. To address this challenge, we developed a scheme for reducing the geometric complexity; while preserving the essential features of activity in both the soma and a remote dendritic spine. We merged our previously published biochemical model of calcium dynamics and lipid signaling in the Purkinje neuron, developed in the Virtual Cell modeling and simulation environment, with an electrophysiological model based on a Purkinje neuron model available in NEURON. A novel reduction method was applied to the Purkinje neuron geometry to obtain a model with fewer compartments that is tractable in Virtual Cell. Most of the dendritic tree was subject to reduction, but we retained the neuron's explicit electrical and geometric features along a specified path from spine to soma. Further, unlike previous simplification methods, the dendrites that branch off along the preserved explicit path are retained as reduced branches. We conserved axial resistivity and adjusted passive properties and active channel conductances for the reduction in surface area, and cytosolic calcium for the reduction in volume. Rallpacks are used to validate the reduction algorithm and show that it can be generalized to other complex neuronal geometries. For the Purkinje cell, we found that current injections at the soma were able to produce similar trains of action potentials and membrane potential propagation in the full and reduced models in NEURON; the reduced model produces identical spiking patterns in NEURON and Virtual Cell. Importantly, our reduced model can simulate communication between the soma and a distal spine; an alpha function applied at the spine to represent synaptic stimulation gave similar results in the full and reduced models for potential changes associated with both the spine and the soma. Finally, we combined phosphoinositol signaling and electrophysiology in the reduced model in Virtual Cell. Thus, a strategy has been developed to combine electrophysiology and biochemistry as a step toward merging neuronal and systems biology modeling.

Postsynaptic signal transduction models for long-term potentiation and depression

More than a hundred biochemical species, activated by neurotransmitters binding to transmembrane receptors, are important in long-term potentiation (LTP) and long-term depression (LTD). To investigate which species and interactions are critical for synaptic plasticity, many computational postsynaptic signal transduction models have been developed. The models range from simple models with a single reversible reaction to detailed models with several hundred kinetic reactions. In this study, more than a hundred models are reviewed, and their features are compared and contrasted so that similarities and differences are more readily apparent. The models are classified according to the type of synaptic plasticity that is modeled (LTP or LTD) and whether they include diffusion or electrophysiological phenomena. Other characteristics that discriminate the models include the phase of synaptic plasticity modeled (induction, expression, or maintenance) and the simulation method used (deterministic or stochastic). We find that models are becoming increasingly sophisticated, by including stochastic properties, integrating with electrophysiological properties of entire neurons, or incorporating diffusion of signaling molecules. Simpler models continue to be developed because they are computationally efficient and allow theoretical analysis. The more complex models permit investigation of mechanisms underlying specific properties and experimental verification of model predictions. Nonetheless, it is difficult to fully comprehend the evolution of these models because (1) several models are not described in detail in the publications, (2) only a few models are provided in existing model databases, and (3) comparison to previous models is lacking. We conclude that the value of these models for understanding molecular mechanisms of synaptic plasticity is increasing and will be enhanced further with more complete descriptions and sharing of the published models.

Use of virtual cell in studies of cellular dynamics

The Virtual Cell (VCell) is a unique computational environment for modeling and simulation of cell biology. It has been specifically designed to be a tool for a wide range of scientists, from experimental cell biologists to theoretical biophysicists. The models created with VCell can range from the simple, to evaluate hypotheses or to interpret experimental data, to complex multilayered models used to probe the predicted behavior of spatially resolved, highly nonlinear systems. In this chapter, we discuss modeling capabilities of VCell and demonstrate representative examples of the models published by the VCell users.

PKC and PLA2: probing the complexities of the calcium network

Lipid signaling and phosphorylation cascades are fundamental to calcium signaling networks. In this review, we will discuss the recent laboratory findings for the phospholipase A(2) (PLA(2))/protein kinase C (PKC) pathway within cellular calcium networks. The complexity and connectivity of these ubiquitous cellular signals make interpretation of experimental results extremely challenging. We present here computational methods which have been developed to conquer such complex data, and how they can be used to make models capable of accurately predicting cellular responses within multiple calcium signaling pathways. We propose that information obtained from network analysis and computational techniques provides a rich source of knowledge which can be directly translated to the laboratory benchtop.

Spine neck geometry determines spino-dendritic cross-talk in the presence of mobile endogenous calcium binding proteins

Dendritic spines are thought to compartmentalize second messengers like Ca2+. The notion of isolated spine signaling, however, was challenged by the recent finding that under certain conditions mobile endogenous Ca(2+)-binding proteins may break the spine limit and lead to activation of Ca(2+)-dependent dendritic signaling cascades. Since the size of spines is variable, the spine neck may be an important regulator of this spino-dendritic crosstalk. We tested this hypothesis by using an experimentally defined, kinetic computer model in which spines of Purkinje neurons were coupled to their parent dendrite by necks of variable geometry. We show that Ca2+ signaling and calmodulin activation in spines with long necks is essentially isolated from the dendrite, while stubby spines show a strong coupling with their dendrite, mediated particularly by calbindin D28k. We conclude that the spine neck geometry, in close interplay with mobile Ca(2+)-binding proteins, regulates the spino-dendritic crosstalk.

Computational study of non-homogeneous distribution of Ca(2+) handling systems in cerebellar granule cells

The spatiotemporal distribution of cytosolic free calcium concentration ([Ca(2+)](i)) in cerebellar granule cells (GrCs) is thought to be critical in defining the occurrence and direction of long-term changes in synaptic strength at cerebellar mossy fiber-GrC synapses. Despite this, the mechanisms responsible for shaping Ca(2+) transients in GrCs are not well understood. To investigate the interplay between Ca(2+) entry, extrusion, buffering and dendritic morphology in shaping Ca(2+) elevations in GrCs, we developed a model of Ca(2+) regulation in these cells and examined the requirements for reproducing fluorescence responses to depolarization and synaptic stimulation previously described in the literature. Two conclusions can be drawn from our simulation results. First, a significant progressive decrease in the amplitudes of depolarization-evoked fluorescence transients from the dendritic endings (digits) toward the soma of GrCs, can be reproduced in the model only if the density of Ca(2+) channels is considerably higher or the concentration of endogenous buffers is much lower in the digits than in the parent dendrites. In contrast, heterogeneities in the distribution of Ca(2+) pumps or in cytosolic fractional volume cannot account for the formation of [Ca(2+)](i) gradients in GrCs. Second, much lower amplitudes of fluorescence transients induced by depolarization and synaptic stimulation than expected from typical measurements of Ca(2+) and NMDA receptor-mediated currents can be reconciled with a pronounced slowing of the decay of fluorescence responses in the digits of GrCs after introducing a high-affinity Ca(2+) indicator if a high-capacity immobile Ca(2+) buffer (presumably plasma membrane-associated) is suggested to be present in the soma and apical part of digits. Mitochondria also are likely to modulate synaptically evoked Ca(2+) responses in GrCs. The alternative hypotheses are thoroughly discussed and research avenues for their testing in future experiments are proposed.

Analysis of phosphatidylinositol-4,5-bisphosphate signaling in cerebellar Purkinje spines

A 3D model was developed and used to explore dynamics of phosphatidylinositol-4,5-bisphosphate (PIP2) signaling in cerebellar Purkinje neurons. Long-term depression in Purkinje neurons depends on coincidence detection of climbing fiber stimulus evoking extracellular calcium flux into the cell and parallel fiber stimulus evoking inositol-1,4,5-trisphosphate (IP3)-meditated calcium release from the endoplasmic reticulum. Experimental evidence shows that large concentrations of IP3 are required for calcium release. This study uses computational analysis to explore how the Purkinje cell provides sufficient PIP2 to produce large amounts of IP3. Results indicate that baseline PIP2 concentration levels in the plasma membrane are inadequate, even if the model allows for PIP2 replenishment by lateral diffusion from neighboring dendrite membrane. Lateral diffusion analysis indicates apparent anomalous diffusion of PIP2 in the spiny dendrite membrane, due to restricted diffusion through spine necks. Stimulated PIP2 synthesis and elevated spine PIP2 mediated by a local sequestering protein were explored as candidate mechanisms to supply sufficient PIP2. Stimulated synthesis can indeed lead to high IP3 amplitude of long duration; local sequestration produces high IP3 amplitude, but of short duration. Simulation results indicate that local sequestration could explain the experimentally observed finely tuned timing between parallel fiber and climbing fiber activation.

Order-dependent coincidence detection in cerebellar Purkinje neurons at the inositol trisphosphate receptor

Associative long-term depression (LTD) at cerebellar parallel fiber-Purkinje cell synapses is sensitive to the temporal order in which the parallel fiber is coactivated with the climbing fiber input, but how order sensitivity is achieved is unknown. Here we show that the cerebellar inositol-1,4,5-trisphosphate (IP3) receptor, whose activation is required for LTD induction, is sensitive in situ to the order of presentation of its coagonists, IP3 and cytoplasmic calcium. By focally photolyzing a novel caged IP3 compound in dendritic spines, we find that pairing IP3 with climbing fiber-mediated calcium entry leads to a large calcium release transient if the climbing fiber is activated up to 100 ms before or up to 500 ms after IP3 uncaging. This asymmetric timing window for coactivation follows the kinetics of calcium removal and IP3 unbinding from the receptor and is not limited by IP3 metabolism. IP3 receptor binding thus acts as an eligibility trace that can drive temporal order-dependent calcium release and LTD induction in Purkinje cells and event order-dependent sensory plasticity in the whole animal.

Ca2+ requirements for cerebellar long-term synaptic depression: role for a postsynaptic leaky integrator

Photolysis of a caged Ca(2+) compound was used to characterize the dependence of cerebellar long-term synaptic depression (LTD) on postsynaptic Ca(2+) concentration ([Ca(2+)](i)). Elevating [Ca(2+)](i) was sufficient to induce LTD without requiring any of the other signals produced by synaptic activity. A sigmoidal relationship between [Ca(2+)](i) and LTD indicated a highly cooperative triggering of LTD by Ca(2+). The duration of the rise in [Ca(2+)](i) influenced the apparent Ca(2+) affinity of LTD, and this time-dependent behavior could be described by a leaky integrator process with a time constant of 0.6 s. A computational model, based on a positive-feedback cycle that includes protein kinase C and MAP kinase, was capable of simulating these properties of Ca(2+)-triggered LTD. Disrupting this cycle experimentally also produced the predicted changes in the Ca(2+) dependence of LTD. We conclude that LTD arises from a mechanism that integrates postsynaptic Ca(2+) signals and that this integration may be produced by the positive-feedback cycle.

DANGER, a Novel Regulatory Protein of Inositol 1,4,5-Trisphosphate-Receptor Activity

We report the cloning and characterization of DANGER, a novel protein which physiologically binds to inositol 1,4,5-trisphosphate receptors (IP(3)R). DANGER is a membrane-associated protein predicted to contain a partial MAB-21 domain. It is expressed in a wide variety of neuronal cell lineages where it localizes to membranes in the cell periphery together with IP(3)R. DANGER interacts with IP(3)R in vitro and co-immunoprecipitates with IP(3)R from cellular preparations. DANGER robustly enhances Ca(2+)-mediated inhibition of IP(3) RCa(2+) release without affecting IP(3) binding in microsomal assays and inhibits gating in single-channel recordings of IP(3)R. DANGER appears to allosterically modulate the sensitivity of IP(3) RtoCa(2+) inhibition, which likely alters IP(3)R-mediated Ca(2+) dynamics in cells where DANGER and IP(3)R are co-expressed.

Homer proteins control neuronal differentiation through IP(3) receptor signaling

Neurons expand, sustain or prune their dendritic trees during ontogenesis [Cline, H.T. (2001). Dendritic arbor development and synaptogenesis. Curr. Opin. Neurobiol. 11, 118-126; Wong, W.T. and Wong, R.O.L. (2000) Rapid dendritic movements during synapse formation and rearrangement. Curr. Opin. Neurobiol. 10, 118-124] which critically depends on neuronal activity [Wong, W.T., Faulkner-Jones, B.E., Sanes, J.R. and Wong, R.O.L. (2000) Rapid dendritic remodeling in the developing retina: dependence on neurotransmission and reciprocal regulation by Rac and Rho. J. Neurosci. 20, 5024-5036; Li, Z., Van Aelst, L. and Cline, H.T. (2000) Rho GTPases regulate distinct aspects of dendritic arbor growth in Xenopus central neurons in vivo. Nat. Neurosci. 3, 217-225; Wong, W.T. and Wong, R.O.L. (2001) Changing specificity of neurotransmitter regulation of rapid dendritic remodeling during synaptogenesis. Nat. Neurosci. 4, 351-352.] and sub-cellular Ca(2+) signals [Lohmann, C., Myhr, K.L. and Wong, R.O. (2002) Transmitter-evoked local calcium release stabilizes developing dendrites, Nature 418, 177-181.]. The role of synaptic clustering proteins connecting both processes is unclear. Here, we show that expression levels of Vesl-1/Homer 1 isoforms critically control properties of Ca(2+) release from intracellular stores and dendritic morphology of CNS neurons. Vesl-1L/Homer 1c, an isoform with a functional WH1 and coiled-coil domain, but not isoforms missing these features were capable of potentiating intracellular calcium signaling activity indicating that such regulatory interactions function as a general paradigm in cellular differentiation and are subject to changes in expression levels of Vesl/Homer isoforms.