Geometric principles of second messenger dynamics in dendritic spines

Reviewing mathematical models of sperm signaling networks

Dave Garbers' work significantly contributed to our understanding of sperm's regulated motility, capacitation, and the acrosome reaction. These key sperm functions involve complex multistep signaling pathways engaging numerous finely orchestrated elements. Despite significant progress, many parameters and interactions among these elements remain elusive. Mathematical modeling emerges as a potent tool to study sperm physiology, providing a framework to integrate experimental results and capture functional dynamics considering biochemical, biophysical, and cellular elements. Depending on research objectives, different modeling strategies, broadly categorized into continuous and discrete approaches, reveal valuable insights into cell function. These models allow the exploration of hypotheses regarding molecules, conditions, and pathways, whenever they become challenging to evaluate experimentally. This review presents an overview of current theoretical and experimental efforts to understand sperm motility regulation, capacitation, and the acrosome reaction. We discuss the strengths and weaknesses of different modeling strategies and highlight key findings and unresolved questions. Notable discoveries include the importance of specific ion channels, the role of intracellular molecular heterogeneity in capacitation and the acrosome reaction, and the impact of pH changes on acrosomal exocytosis. Ultimately, this review underscores the crucial importance of mathematical frameworks in advancing our understanding of sperm physiology and guiding future experimental investigations.

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.

Cristae formation is a mechanical buckling event controlled by the inner mitochondrial membrane lipidome

Cristae are high-curvature structures in the inner mitochondrial membrane (IMM) that are crucial for ATP production. While cristae-shaping proteins have been defined, analogous lipid-based mechanisms have yet to be elucidated. Here, we combine experimental lipidome dissection with multi-scale modeling to investigate how lipid interactions dictate IMM morphology and ATP generation. When modulating phospholipid (PL) saturation in engineered yeast strains, we observed a surprisingly abrupt breakpoint in IMM topology driven by a continuous loss of ATP synthase organization at cristae ridges. We found that cardiolipin (CL) specifically buffers the inner mitochondrial membrane against curvature loss, an effect that is independent of ATP synthase dimerization. To explain this interaction, we developed a continuum model for cristae tubule formation that integrates both lipid and protein-mediated curvatures. This model highlighted a snapthrough instability, which drives IMM collapse upon small changes in membrane properties. We also showed that cardiolipin is essential in low-oxygen conditions that promote PL saturation. These results demonstrate that the mechanical function of cardiolipin is dependent on the surrounding lipid and protein components of the IMM.

Cristae formation is a mechanical buckling event controlled by the inner membrane lipidome

Cristae are high curvature structures in the inner mitochondrial membrane (IMM) that are crucial for ATP production. While cristae-shaping proteins have been defined, analogous mechanisms for lipids have yet to be elucidated. Here we combine experimental lipidome dissection with multi-scale modeling to investigate how lipid interactions dictate IMM morphology and ATP generation. When modulating phospholipid (PL) saturation in engineered yeast strains, we observed a surprisingly abrupt breakpoint in IMM topology driven by a continuous loss of ATP synthase organization at cristae ridges. We found that cardiolipin (CL) specifically buffers the IMM against curvature loss, an effect that is independent of ATP synthase dimerization. To explain this interaction, we developed a continuum model for cristae tubule formation that integrates both lipid and protein-mediated curvatures. The model highlighted a snapthrough instability, which drives IMM collapse upon small changes in membrane properties. We also showed that CL is essential in low oxygen conditions that promote PL saturation. These results demonstrate that the mechanical function of CL is dependent on the surrounding lipid and protein components of the IMM.

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.

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

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

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

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

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.

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.

Morphological principles of neuronal mitochondria

In the highly dynamic metabolic landscape of a neuron, mitochondrial membrane architectures can provide critical insight into the unique energy balance of the cell. Current theoretical calculations of functional outputs like adenosine triphosphate and heat often represent mitochondria as idealized geometries, and therefore, can miscalculate the metabolic fluxes. To analyze mitochondrial morphology in neurons of mouse cerebellum neuropil, 3D tracings of complete synaptic and axonal mitochondria were constructed using a database of serial transmission electron microscopy (TEM) tomography images and converted to watertight meshes with minimal distortion of the original microscopy volumes with a granularity of 1.64 nanometer isotropic voxels. The resulting in-silico representations were subsequently quantified by differential geometry methods in terms of the mean and Gaussian curvatures, surface areas, volumes, and membrane motifs, all of which can alter the metabolic output of the organelle. Finally, we identify structural motifs present across this population of mitochondria, which may contribute to future modeling studies of mitochondrial physiology and metabolism in neurons.

Signaling Microdomains in the Spotlight: Visualizing Compartmentalized Signaling Using Genetically Encoded Fluorescent Biosensors

How cells muster a network of interlinking signaling pathways to faithfully convert diverse external cues to specific functional outcomes remains a central question in biology. Through their ability to convert dynamic biochemical activities to rapid and precise optical readouts, genetically encoded fluorescent biosensors have become instrumental in unraveling the molecular logic controlling the specificity of intracellular signaling. In this review, we discuss how the use of genetically encoded fluorescent biosensors to visualize dynamic signaling events within their native cellular context is elucidating the different strategies employed by cells to organize signaling activities into discrete compartments, or signaling microdomains, to ensure functional specificity.

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.

Three-Dimensional Model of Sub-Plasmalemmal Ca2+ Microdomains Evoked by the Interplay Between ORAI1 and InsP3 Receptors

Ca²⁺ signaling plays an essential role in T cell activation, which is a key step to start an adaptive immune response. During the transition from a quiescent to a fully activated state, Ca²⁺ microdomains characterized by reduced spatial and temporal extents are observed in the junctions between the plasma membrane (PM) and the endoplasmic reticulum (ER). Such Ca²⁺ responses can also occur in response to T cell adhesion to other cells or extracellular matrix proteins in otherwise unstimulated T cells. These non-TCR/CD3-dependent Ca²⁺ microdomains rely on d-myo-inositol 1,4,5-trisphosphate (IP₃) signaling and subsequent store operated Ca²⁺ entry (SOCE) via the ORAI/STIM system. The detailed molecular mechanism of adhesion-dependent Ca²⁺ microdomain formation remains to be fully elucidated. We used mathematical modeling to investigate the spatiotemporal characteristics of T cell Ca²⁺ microdomains and their molecular regulators. We developed a reaction-diffusion model using COMSOL Multiphysics to describe the evolution of cytosolic and ER Ca²⁺ concentrations in a three-dimensional ER-PM junction. Equations are based on a previously proposed realistic description of the junction, which is extended to take into account IP₃ receptors (IP₃R) that are located next to the junction. The first model only considered the ORAI channels and the SERCA pumps. Taking into account the existence of preformed clusters of ORAI1 and STIM2, ORAI1 slightly opens in conditions of a full ER. These simulated Ca²⁺ microdomains are too small as compared to those observed in unstimulated T cells. When considering the opening of the IP₃Rs located near the junction, the local depletion of ER Ca²⁺ allows for larger Ca²⁺ fluxes through the ORAI1 channels and hence larger local Ca²⁺ concentrations. Computational results moreover show that Ca²⁺ diffusion in the ER has a major impact on the Ca²⁺ changes in the junction, by affecting the local Ca²⁺ gradients in the sub-PM ER. Besides pointing out the likely involvement of the spontaneous openings of IP₃Rs in the activation of SOCE in conditions of T cell adhesion prior to full activation, the model provides a tool to investigate how Ca²⁺ microdomains extent and interact in response to T cell receptor activation.

The role of molecular diffusion within dendritic spines in synaptic function

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

Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry

Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.

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.

Serial Prefrontal Pathways Are Positioned to Balance Cognition and Emotion in Primates

The delicate balance among primate prefrontal networks is necessary for homeostasis and behavioral flexibility. Dorsolateral prefrontal cortex (dlPFC) is associated with cognition, while the most ventromedial subgenual cingulate area 25 (A25) is associated with emotion and emotional expression. Yet A25 is weakly connected with dlPFC, and it is unknown how the two regions communicate. In rhesus monkeys of both sexes, we investigated how these functionally distinct areas may interact through pregenual anterior cingulate area 32 (A32), which is strongly connected with both. We found that dlPFC innervated the deep layers of A32, while A32 innervated all layers of A25, mostly targeting spines of excitatory neurons. Approximately 20% of A32 terminations formed synapses on inhibitory neurons in A25, notably the powerful parvalbumin inhibitory neurons in the deep layers, and the disinhibitory calretinin neurons in the superficial layers. By innervating distinct inhibitory microenvironments in laminar compartments, A32 is positioned to tune activity in columns of A25. The circuitry of the sequential pathway indicates that when dlPFC is engaged, A32 can dampen A25 output through the parvalbumin inhibitory microsystem in the deep layers of A25. A32 thus may flexibly recruit or reduce activity in A25 to maintain emotional equilibrium, a process that is disrupted in depression. Moreover, pyramidal neurons in A25 had a heightened density of NMDARs, which are the targets of novel rapid-acting antidepressants. Pharmacologic antagonism of NMDARs in patients with depression may reduce excitability in A25, mimicking the effects of the neurotypical serial pathway identified here.SIGNIFICANCE STATEMENT The anterior cingulate is a critical hub in prefrontal networks through connections with functionally distinct areas. Dorsolateral and polar prefrontal areas that are associated with complex cognition are connected with the anterior cingulate in a pattern that allows them to indirectly control downstream activity from the anterior cingulate to the subgenual cingulate, which is associated with heightened activity and negative affect in depression. This set of pathways provides a circuit mechanism for emotional regulation, with the anterior cingulate playing a balancing role for integration of cognitive and emotional processes. Disruption of these pathways may perturb network function and the ability to regulate cognitive and affective processes based on context.

Cell shape regulates subcellular organelle location to control early Ca2+ signal dynamics in vascular smooth muscle cells

The shape of the cell is connected to its function; however, we do not fully understand underlying mechanisms by which global shape regulates a cell's functional capabilities. Using theory, experiments and simulation, we investigated how physiologically relevant cell shape changes affect subcellular organization, and consequently intracellular signaling, to control information flow needed for phenotypic function. Vascular smooth muscle cells going from a proliferative and motile circular shape to a contractile fusiform shape show changes in the location of the sarcoplasmic reticulum, inter-organelle distances, and differential distribution of receptors in the plasma membrane. These factors together lead to the modulation of signals transduced by the M3 muscarinic receptor/Gq/PLCβ pathway at the plasma membrane, amplifying Ca2+ dynamics in the cytoplasm, and the nucleus resulting in phenotypic changes, as determined by increased activity of myosin light chain kinase in the cytoplasm and enhanced nuclear localization of the transcription factor NFAT. Taken together, our observations show a systems level phenomenon whereby global cell shape affects subcellular organization to modulate signaling that enables phenotypic changes.

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.

Structural LTP: from synaptogenesis to regulated synapse enlargement and clustering

Nature teaches us that form precedes function, yet structure and function are intertwined. Such is the case with synapse structure, function, and plasticity underlying learning, especially in the hippocampus, a crucial brain region for memory formation. As the hippocampus matures, enduring changes in synapse structure produced by long-term potentiation (LTP) shift from synaptogenesis to synapse enlargement that is homeostatically balanced by stalled spine outgrowth and local spine clustering. Production of LTP leads to silent spine outgrowth at P15, and silent synapse enlargement in adult hippocampus at 2hours, but not at 5 or 30min following induction. Here we consider structural LTP in the context of developmental stage and variation in the availability of local resources of endosomes, smooth endoplasmic reticulum and polyribosomes. The emerging evidence supports a need for more nuanced analysis of synaptic plasticity in the context of subcellular resource availability and developmental stage.

Structural synaptic plasticity across sleep and wake

Sleep-dependent synaptic plasticity is crucial for optimal cognition. However, establishing the direction of synaptic plasticity during sleep has been particularly challenging since data in support of both synaptic potentiation and depotentiation have been reported. This review focuses on structural synaptic plasticity across sleep and wake and summarizes recent developments in the use of 3-dimensional electron microscopy as applied to this field.

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.

An Open-Source Mesh Generation Platform for Biophysical Modeling Using Realistic Cellular Geometries

Advances in imaging methods such as electron microscopy, tomography, and other modalities are enabling high-resolution reconstructions of cellular and organelle geometries. Such advances pave the way for using these geometries for biophysical and mathematical modeling once these data can be represented as a geometric mesh, which, when carefully conditioned, enables the discretization and solution of partial differential equations. In this work, we outline the steps for a naïve user to approach the Geometry-preserving Adaptive MeshER software version 2, a mesh generation code written in C++ designed to convert structural data sets to realistic geometric meshes while preserving the underlying shapes. We present two example cases: 1) mesh generation at the subcellular scale as informed by electron tomography and 2) meshing a protein with a structure from x-ray crystallography. We further demonstrate that the meshes generated by the Geometry-preserving Adaptive MeshER software are suitable for use with numerical methods. Together, this collection of libraries and tools simplifies the process of constructing realistic geometric meshes from structural biology data.

Applications and Challenges of Machine Learning to Enable Realistic Cellular Simulations

In this perspective, we examine three key aspects of an end-to-end pipeline for realistic cellular simulations: reconstruction and segmentation of cellular structures; generation of cellular structures; and mesh generation, simulation, and data analysis. We highlight some of the relevant prior work in these distinct but overlapping areas, with a particular emphasis on current use of machine learning technologies, as well as on future opportunities.

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.