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

Biophysical Modeling of Synaptic Plasticity

Dendritic spines are small, bulbous compartments that function as postsynaptic sites and undergo intense biochemical and biophysical activity. The role of the myriad signaling pathways that are implicated in synaptic plasticity is well studied. A recent abundance of quantitative experimental data has made the events associated with synaptic plasticity amenable to quantitative biophysical modeling. Spines are also fascinating biophysical computational units because spine geometry, signal transduction, and mechanics work in a complex feedback loop to tune synaptic plasticity. In this sense, ideas from modeling cell motility can inspire us to develop multiscale approaches for predictive modeling of synaptic plasticity. In this article, we review the key steps in postsynaptic plasticity with a specific focus on the impact of spine geometry on signaling, cytoskeleton rearrangement, and membrane mechanics. We summarize the main experimental observations and highlight how theory and computation can aid our understanding of these complex processes.

Biophysical Modeling of Actin-Mediated Structural Plasticity Reveals Mechanical Adaptation in Dendritic Spines

Synaptic plasticity is important for learning and memory formation; it describes the strengthening or weakening of connections between synapses. The postsynaptic part of excitatory synapses resides in dendritic spines, which are small protrusions on the dendrites. One of the key features of synaptic plasticity is its correlation with the size of these spines. A long-lasting synaptic strength increase [long-term potentiation (LTP)] is only possible through the reconfiguration of the actin spine cytoskeleton. Here, we develop an experimentally informed three-dimensional computational model in a moving boundary framework to investigate this reconfiguration. Our model describes the reactions between actin and actin-binding proteins leading to the cytoskeleton remodeling and their effect on the spine membrane shape to examine the spine enlargement upon LTP. Moreover, we find that the incorporation of perisynaptic elements enhances spine enlargement upon LTP, exhibiting the importance of accounting for these elements when studying structural LTP. Our model shows adaptation to repeated stimuli resulting from the interactions between spine proteins and mechanical forces.

Spatiotemporal modelling reveals geometric dependence of AMPAR dynamics on dendritic spine morphology

The modification of neural circuits depends on the strengthening and weakening of synaptic connections. Synaptic strength is often correlated to the density of the ionotropic, glutamatergic receptors, AMPARs, (α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors) at the postsynaptic density (PSD). While AMPAR density is known to change based on complex biological signalling cascades, the effect of geometric factors such as dendritic spine shape, size and curvature remain poorly understood. In this work, we developed a deterministic, spatiotemporal model to study the dynamics of AMPARs during long-term potentiation (LTP). This model includes a minimal set of biochemical events that represent the upstream signalling events, trafficking of AMPARs to and from the PSD, lateral diffusion in the plane of the spine membrane, and the presence of an extrasynaptic AMPAR pool. Using idealized and realistic spine geometries, we show that the dynamics and increase of bound AMPARs at the PSD depends on a combination of endo- and exocytosis, membrane diffusion, the availability of free AMPARs and intracellular signalling interactions. We also found non-monotonic relationships between spine volume and the change in AMPARs at the PSD, suggesting that spines restrict changes in AMPARs to optimize resources and prevent runaway potentiation. KEY POINTS: Synaptic plasticity involves dynamic biochemical and physical remodelling of small protrusions called dendritic spines along the dendrites of neurons. Proper synaptic functionality within these spines requires changes in receptor number at the synapse, which has implications for downstream neural functions, such as learning and memory formation. In addition to being signalling subcompartments, spines also have unique morphological features that can play a role in regulating receptor dynamics on the synaptic surface. We have developed a spatiotemporal model that couples biochemical signalling and receptor trafficking modalities in idealized and realistic spine geometries to investigate the role of biochemical and biophysical factors in synaptic plasticity. Using this model, we highlight the importance of spine size and shape in regulating bound AMPA receptor dynamics that govern synaptic plasticity, and predict how spine shape might act to reset synaptic plasticity as a built-in resource optimization and regulation tool.

Crosstalk between biochemical signalling network architecture and trafficking governs AMPAR dynamics in synaptic plasticity

Synaptic plasticity involves modification of both biochemical and structural components of neurons. Many studies have revealed that the change in the number density of the glutamatergic receptor AMPAR at the synapse is proportional to synaptic weight update; an increase in AMPAR corresponds to strengthening of synapses while a decrease in AMPAR density weakens synaptic connections. The dynamics of AMPAR are thought to be regulated by upstream signalling, primarily the calcium-CaMKII pathway, trafficking to and from the synapse, and influx from extrasynaptic sources. Previous work in the field of deterministic modelling of CaMKII dynamics has assumed bistable kinetics, while experiments and rule-based modelling have revealed that CaMKII dynamics can be either monostable or ultrasensitive. This raises the following question: how does the choice of model assumptions involving CaMKII dynamics influence AMPAR dynamics at the synapse? To answer this question, we have developed a set of models using compartmental ordinary differential equations to systematically investigate contributions of different signalling and trafficking variations, along with their coupled effects, on AMPAR dynamics at the synaptic site. We find that the properties of the model including network architecture describing different stability features of CaMKII and parameters that capture the endocytosis and exocytosis of AMPAR significantly affect the integration of fast upstream species by slower downstream species. Furthermore, we predict that the model outcome, as determined by bound AMPAR at the synaptic site, depends on (1) the choice of signalling model (bistable CaMKII or monostable CaMKII dynamics), (2) trafficking versus influx contributions and (3) frequency of stimulus. KEY POINTS: The density of AMPA receptors (AMPARs) at the postsynaptic density of the synapse provides a readout of synaptic plasticity, which involves crosstalk between complex biochemical signalling networks including CaMKII dynamics and trafficking pathways including exocytosis and endocytosis. Here we build a model that integrates CaMKII dynamics and AMPAR trafficking to explore this crosstalk. We compare different models of CaMKII that result in monostable or bistable kinetics and their impact on AMPAR dynamics. Our results show that AMPAR density depends on the coupling between aspects of biochemical signalling and trafficking. Specifically, assumptions regarding CaMKII dynamics and its stability features can alter AMPAR density at the synapse. Our model also predicts that the kinetics of trafficking versus influx of AMPAR from the extrasynaptic space can further impact AMPAR density. Thus, the contributions of both signalling and trafficking should be considered in computational models.

Dendritic spine morphology regulates calcium-dependent synaptic weight change

Dendritic spines act as biochemical computational units and must adapt their responses according to their activation history. Calcium influx acts as the first signaling step during postsynaptic activation and is a determinant of synaptic weight change. Dendritic spines also come in a variety of sizes and shapes. To probe the relationship between calcium dynamics and spine morphology, we used a stochastic reaction-diffusion model of calcium dynamics in idealized and realistic geometries. We show that despite the stochastic nature of the various calcium channels, receptors, and pumps, spine size and shape can modulate calcium dynamics and subsequently synaptic weight updates in a deterministic manner. Through a series of exhaustive simulations and analyses, we found that the calcium dynamics and synaptic weight change depend on the volume-to-surface area of the spine. The relationships between calcium dynamics and spine morphology identified in idealized geometries also hold in realistic geometries, suggesting that there are geometrically determined deterministic relationships that may modulate synaptic weight change.

Green fluorescent cAMP indicator of high speed and specificity suitable for neuronal live-cell imaging

Cyclic adenosine monophosphate (cAMP) is a canonical intracellular messenger playing diverse roles in cell functions. In neurons, cAMP promotes axonal growth during early development, and mediates sensory transduction and synaptic plasticity after maturation. The molecular cascades of cAMP are well documented, but its spatiotemporal profiles associated with neuronal functions remain hidden. Hence, we developed a genetically encoded cAMP indicator based on a bacterial cAMP-binding protein. This indicator "gCarvi" monitors [cAMP]_i at 0.2 to 20 µM with a subsecond time resolution and a high specificity over cyclic guanosine monophosphate (cGMP). gCarvi can be converted to a ratiometric probe for [cAMP]_i quantification and its expression can be specifically targeted to various subcellular compartments. Monomeric gCarvi also enables simultaneous multisignal monitoring in combination with other indicators. As a proof of concept, simultaneous cAMP/Ca²⁺ imaging in hippocampal neurons revealed a tight linkage of cAMP to Ca²⁺ signals. In cerebellar presynaptic boutons, forskolin induced nonuniform cAMP elevations among boutons, which positively correlated with subsequent increases in the size of the recycling pool of synaptic vesicles assayed using FM dye. Thus, the cAMP domain in presynaptic boutons is an important determinant of the synaptic strength.

Endoplasmic Reticulum in Metaplasticity: From Information Processing to Synaptic Proteostasis

The ER (endoplasmic reticulum) is a Ca²⁺ reservoir and the unique protein-synthesizing machinery which is distributed throughout the neuron and composed of multiple different structural domains. One such domain is called EMC (endoplasmic reticulum membrane protein complex), pleiotropic nature in cellular functions. The ER/EMC position inside the neurons unmasks its contribution to synaptic plasticity via regulating various cellular processes from protein synthesis to Ca²⁺ signaling. Since presynaptic Ca²⁺ channels and postsynaptic ionotropic receptors are organized into the nanodomains, thus ER can be a crucial player in establishing TMNCs (transsynaptic molecular nanocolumns) to shape efficient neural communications. This review hypothesized that ER is not only involved in stress-mediated neurodegeneration but also axon regrowth, remyelination, neurotransmitter switching, information processing, and regulation of pre- and post-synaptic functions. Thus ER might not only be a protein-synthesizing and quality control machinery but also orchestrates plasticity of plasticity (metaplasticity) within the neuron to execute higher-order brain functions and neural repair.

Computational investigation of the dynamic control of cAMP signaling by PDE4 isoform types

Cyclic adenosine monophosphate (cAMP) is a generic signaling molecule that, through precise control of its signaling dynamics, exerts distinct cellular effects. Consequently, aberrant cAMP signaling can have detrimental effects. Phosphodiesterase 4 (PDE4) enzymes profoundly control cAMP signaling and comprise different isoform types of which the enzymatic activity is modulated by differential feedback mechanisms. Because these feedback dynamics are non-linear and occur coincidentally, their effects are difficult to examine experimentally, but can be well simulated computationally. Through understanding the role of PDE4 isoform types in regulating cAMP signaling, PDE4-targeted therapeutic strategies can be better specified. Here, we established a computational model to study how feedback mechanisms on different PDE4 isoform types lead to dynamic, isoform-specific control of cAMP signaling. Ordinary differential equations describing cAMP dynamics were implemented in the VirtualCell (VCell) environment. Simulations indicated that long PDE4 isoforms exert the most profound control on oscillatory cAMP signaling, as opposed to the PDE4-mediated control of single cAMP input pulses. Moreover, elevating cAMP levels or decreasing PDE4 levels revealed different effects on downstream signaling. Together these results underline that cAMP signaling is distinctly regulated by different PDE4 isoform types and that this isoform-specificity should be considered in both computational and experimental follow-up studies to better define PDE4 enzymes as therapeutic targets in diseases in which cAMP signaling is aberrant.

Regenerative Calcium Currents in Renal Primary Cilia

Polycystic kidney disease (PKD) is a leading cause of end-stage renal disease. PKD arises from mutations in proteins, one a Ca2+-conducting channel, expressed in the primary cilia of renal epithelial cells. A common hypothesis is that Ca2+ entering through ciliary ion channels may reduce cystogenesis. The cilia have at least two Ca2+-conducting channels: polycystin-2 (PC2) and TRPV4 (transient receptor potential (TRP) cation channel, subfamily V, member 4), but how substantially they can increase intraciliary Ca2+ is unknown. By recording channel activities in isolated cilia, conditions are identified under which the channels can increase free Ca2+ within the cilium by at least 500-fold through regenerative (positive-feedback) signaling. Ca2+ that has entered through a channel can activate the channel internally, which increases the Ca2+ influx, and so on. Regenerative signaling is favored when the concentration of the Ca2+ buffer is reduced or when a slower buffer is used. Under such conditions, the Ca2+ that enters the cilium through a single PC2 channel is sufficient to almost fully activate that same channel. Regenerative signaling is not detectable with reduced external Ca2+. Reduced buffering also allows regenerative signaling through TRPV4 channels, but not through TRPM4 (TRP subfamily M, member 4) channels, which are activated by Ca2+ but do not conduct it. On a larger scale, Ca2+ that enters through TRPV4 channels can cause secondary activation of PC2 channels. I discuss the likelihood of regenerative ciliary Ca2+ signaling in vivo, a possible mechanism for its activation, and how it might relate to cystogenesis.

Can biophysical models of dendritic spines be used to explore synaptic changes associated with addiction?

Effective treatments that prevent or reduce drug relapse vulnerability should be developed to relieve the high burden of drug addiction on society. This will only be possible by enhancing the understanding of the molecular mechanisms underlying the neurobiology of addiction. Recent experimental data have shown that dendritic spines, small protrusions from the dendrites that receive excitatory input, of spiny neurons in the nucleus accumbens exhibit morphological changes during drug exposure and withdrawal. Moreover, these changes relate to the characteristic drug-seeking behavior of addiction. However, due to the complexity of the dendritic spines, we do not yet fully understand the processes underlying their structural changes in response to different inputs. We propose that biophysical models can enhance the current understanding of these processes by incorporating different, and sometimes, discrepant experimental data to identify the shared underlying mechanisms and generate experimentally testable hypotheses. This review aims to give an up-to-date report on biophysical models of dendritic spines, focusing on those models that describe their shape changes, which are well-known to relate to learning and memory. Moreover, it examines how these models can enhance our understanding of the effect of the drugs and the synaptic changes during withdrawal, as well as during neurodegenerative disease progression such as Alzheimer's disease.

Systems modeling predicts that mitochondria ER contact sites regulate the postsynaptic energy landscape

Spatiotemporal compartmentation of calcium dynamics is critical for neuronal function, particularly in postsynaptic spines. This exquisite level of Ca²⁺ compartmentalization is achieved through the storage and release of Ca²⁺ from various intracellular organelles particularly the endoplasmic reticulum (ER) and the mitochondria. Mitochondria and ER are established storage organelles controlling Ca²⁺ dynamics in neurons. Mitochondria also generate a majority of energy used within postsynaptic spines to support the downstream events associated with neuronal stimulus. Recently, high resolution microscopy has unveiled direct contact sites between the ER and the mitochondria (MERCs), which directly channel Ca²⁺ release from the ER into the mitochondrial membrane. In this study, we develop a computational 3D reaction-diffusion model to investigate the role of MERCs in regulating Ca²⁺ and ATP dynamics. This spatiotemporal model accounts for Ca²⁺ oscillations initiated by glutamate stimulus of metabotropic and ionotropic glutamate receptors and Ca²⁺ changes in four different compartments: cytosol, ER, mitochondria, and the MERC microdomain. Our simulations predict that the organization of these organelles and inter-organellar contact sites play a key role in modulating Ca²⁺ and ATP dynamics.

We further show that the crosstalk between geometry (mitochondria and MERC) and metabolic parameters (cytosolic ATP hydrolysis, ATP generation) influences the neuronal energy state. Our findings shed light on the importance of organelle interactions in predicting Ca²⁺ dynamics in synaptic signaling. Overall, our model predicts that a combination of MERC linkage and mitochondria size is necessary for optimal ATP production in the cytosol.

Vertebrate cells differentially interpret ciliary and extraciliary cAMP

Hedgehog pathway components and select G protein-coupled receptors (GPCRs) localize to the primary cilium, an organelle specialized for signal transduction. We investigated whether cells distinguish between ciliary and extraciliary GPCR signaling. To test whether ciliary and extraciliary cyclic AMP (cAMP) convey different information, we engineered optogenetic and chemogenetic tools to control the subcellular site of cAMP generation. Generating equal amounts of ciliary and cytoplasmic cAMP in zebrafish and mammalian cells revealed that ciliary cAMP, but not cytoplasmic cAMP, inhibited Hedgehog signaling. Modeling suggested that the distinct geometries of the cilium and cell body differentially activate local effectors. The search for effectors identified a ciliary pool of protein kinase A (PKA). Blocking the function of ciliary PKA, but not extraciliary PKA, activated Hedgehog signal transduction and reversed the effects of ciliary cAMP. Therefore, cells distinguish ciliary and extraciliary cAMP using functionally and spatially distinct pools of PKA, and different subcellular pools of cAMP convey different information.

Computational modeling approaches to cAMP/PKA signaling in cardiomyocytes

The cAMP/PKA pathway is a fundamental regulator of excitation-contraction coupling in cardiomyocytes. Activation of cAMP has a variety of downstream effects on cardiac function including enhanced contraction, accelerated relaxation, adaptive stress response, mitochondrial regulation, and gene transcription. Experimental advances have shed light on the compartmentation of cAMP and PKA, which allow for control over the varied targets of these second messengers and is disrupted in heart failure conditions. Computational modeling is an important tool for understanding the spatial and temporal complexities of this system. In this review article, we outline the advances in computational modeling that have allowed for deeper understanding of cAMP/PKA dynamics in the cardiomyocyte in health and disease, and explore new modeling frameworks that may bring us closer to a more complete understanding of this system. We outline various compartmental and spatial signaling models that have been used to understand how β-adrenergic signaling pathways function in a variety of simulation conditions. We also discuss newer subcellular models of cardiovascular function that may be used as templates for the next phase of computational study of cAMP and PKA in the heart, and outline open challenges which are important to consider in future models.

Interactions between calmodulin and neurogranin govern the dynamics of CaMKII as a leaky integrator

Calmodulin-dependent kinase II (CaMKII) has long been known to play an important role in learning and memory as well as long term potentiation (LTP). More recently it has been suggested that it might be involved in the time averaging of synaptic signals, which can then lead to the high precision of information stored at a single synapse. However, the role of the scaffolding molecule, neurogranin (Ng), in governing the dynamics of CaMKII is not yet fully understood. In this work, we adopt a rule-based modeling approach through the Monte Carlo method to study the effect of Ca2+ signals on the dynamics of CaMKII phosphorylation in the postsynaptic density (PSD). Calcium surges are observed in synaptic spines during an EPSP and back-propagating action potential due to the opening of NMDA receptors and voltage dependent calcium channels. Using agent-based models, we computationally investigate the dynamics of phosphorylation of CaMKII monomers and dodecameric holoenzymes. The scaffolding molecule, Ng, when present in significant concentration, limits the availability of free calmodulin (CaM), the protein which activates CaMKII in the presence of calcium. We show that Ng plays an important modulatory role in CaMKII phosphorylation following a surge of high calcium concentration. We find a non-intuitive dependence of this effect on CaM concentration that results from the different affinities of CaM for CaMKII depending on the number of calcium ions bound to the former. It has been shown previously that in the absence of phosphatase, CaMKII monomers integrate over Ca2+ signals of certain frequencies through autophosphorylation (Pepke et al, Plos Comp. Bio., 2010). We also study the effect of multiple calcium spikes on CaMKII holoenzyme autophosphorylation, and show that in the presence of phosphatase, CaMKII behaves as a leaky integrator of calcium signals, a result that has been recently observed in vivo. Our models predict that the parameters of this leaky integrator are finely tuned through the interactions of Ng, CaM, CaMKII, and PP1, providing a mechanism to precisely control the sensitivity of synapses to calcium signals. Author Summary not valid for PLOS ONE submissions.

3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries

Recent advances in electron microscopy have enabled the imaging of single cells in 3D at nanometer length scale resolutions. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. Enabling such simulations requires watertight meshing of electron micrograph images into 3D volume meshes, which can then form the basis of computer simulations of such processes using numerical techniques such as the finite element method. In this paper, we describe the use of our recently rewritten mesh processing software, GAMer 2, to bridge the gap between poorly conditioned meshes generated from segmented micrographs and boundary marked tetrahedral meshes which are compatible with simulation. We demonstrate the application of a workflow using GAMer 2 to a series of electron micrographs of neuronal dendrite morphology explored at three different length scales and show that the resulting meshes are suitable for finite element simulations. This work is an important step towards making physical simulations of biological processes in realistic geometries routine. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery at the interface of geometry and cellular processes. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open.

Computational Modeling Reveals Frequency Modulation of Calcium-cAMP/PKA Pathway in Dendritic Spines

Dendritic spines are the primary excitatory postsynaptic sites that act as subcompartments of signaling. Ca²⁺ is often the first and most rapid signal in spines. Downstream of calcium, the cyclic adenosine monophosphate (cAMP)/protein kinase A (PKA) pathway plays a critical role in the regulation of spine formation, morphological modifications, and ultimately, learning and memory. Although the dynamics of calcium are reasonably well-studied, calcium-induced cAMP/PKA dynamics, particularly with respect to frequency modulation, are not fully explored. In this study, we present a well-mixed model for the dynamics of calcium-induced cAMP/PKA dynamics in dendritic spines. The model is constrained using experimental observations in the literature. Further, we measured the calcium oscillation frequency in dendritic spines of cultured hippocampal CA1 neurons and used these dynamics as model inputs. Our model predicts that the various steps in this pathway act as frequency modulators for calcium, and the high frequency of calcium input is filtered by adenylyl cyclase 1 and phosphodiesterases in this pathway such that cAMP/PKA only responds to lower frequencies. This prediction has important implications for noise filtering and long-timescale signal transduction in dendritic spines. A companion manuscript presents a three-dimensional spatial model for the same pathway.