Dendritic spine geometry can localize GTPase signaling in neurons

Stress-mediated dysregulation of the Rap1 small GTPase impairs hippocampal structure and function

The effects of repeated stress on cognitive impairment are thought to be mediated, at least in part, by reductions in the stability of dendritic spines in brain regions critical for proper learning and memory, including the hippocampus. Small GTPases are particularly potent regulators of dendritic spine formation, stability, and morphology in hippocampal neurons. Through the use of small GTPase protein profiling in mice, we identify increased levels of synaptic Rap1 in the hippocampal CA3 region in response to escalating, intermittent stress. We then demonstrate that increased Rap1 in the CA3 is sufficient in and of itself to produce stress-relevant dendritic spine and cognitive phenotypes. Further, using super-resolution imaging, we investigate how the pattern of Rap1 trafficking to synapses likely underlies its effects on the stability of select dendritic spine subtypes. These findings illuminate the involvement of aberrant Rap1 regulation in the hippocampus in contributing to the psychobiological effects of stress.

Numerical investigations of the bulk-surface wave pinning model

The bulk-surface wave pinning model is a reaction-diffusion system for studying cell polarisation. It is constituted by a surface reaction-diffusion equation, coupled to a bulk diffusion equation with a non-linear boundary condition. Cell polarisation arises as the surface component develops specific patterns. Since proteins diffuse much faster in the cell interior than on the membrane, in the literature, the bulk component is often assumed to be spatially homogeneous. Therefore, the model can be reduced to a single surface equation. However, in real applications a spatially non-uniform bulk component might be an important player to take into account. In this paper, we study, through numerical computations, the role of the bulk component and, more specifically, how different bulk diffusion rates might affect the polarisation response. We find that the bulk component is indeed a key factor in determining the surface polarisation response. Moreover, for certain geometries, it is the spatial heterogeneity of the bulk component that triggers the polarisation response, which might not be possible in a reduced model. Understanding how polarisation depends on bulk diffusivity might be crucial when studying models of migrating cells, which are naturally subject to domain deformation.

Sensing the shape of a cell with reaction diffusion and energy minimization

Some dividing cells sense their shape by becoming polarized along their long axis. Cell polarity is controlled in part by polarity proteins, like Rho GTPases, cycling between active membrane-bound forms and inactive cytosolic forms, modeled as a "wave-pinning" reaction-diffusion process. Does shape sensing emerge from wave pinning? We show that wave pinning senses the cell's long axis. Simulating wave pinning on a curved surface, we find that high-activity domains migrate to peaks and troughs of the surface. For smooth surfaces, a simple rule of minimizing the domain perimeter while keeping its area fixed predicts the final position of the domain and its shape. However, when we introduce roughness to our surfaces, shape sensing can be disrupted, and high-activity domains can become localized to locations other than the global peaks and valleys of the surface. On rough surfaces, the domains of the wave-pinning model are more robust in finding the peaks and troughs than the minimization rule, although both can become trapped in steady states away from the peaks and valleys. We can control the robustness of shape sensing by altering the Rho GTPase diffusivity and the domain size. We also find that the shape-sensing properties of cell polarity models can explain how domains localize to curved regions of deformed cells. Our results help to understand the factors that allow cells to sense their shape-and the limits that membrane roughness can place on this process.

Computation, wiring, and plasticity in synaptic clusters

Synaptic clusters on dendrites are extraordinarily compact computational building blocks. They contribute to key local computations through biophysical and biochemical signaling that utilizes convergence in space and time as an organizing principle. However, these computations can only arise in very special contexts. Dendritic cluster computations, their highly organized input connectivity, and the mechanisms for their formation are closely linked, yet these have not been analyzed as parts of a single process. Here, we examine these linkages. The sheer density of axonal and dendritic arborizations means that there are far more potential connections (close enough for a spine to reach an axon) than actual ones. We see how dendritic clusters draw upon electrical, chemical, and mechano-chemical signaling to implement the rules for formation of connections and subsequent computations. Crucially, the same mechanisms that underlie their functions also underlie their formation.

A novel stochastic simulation approach enables exploration of mechanisms for regulating polarity site movement

Cells polarize their movement or growth toward external directional cues in many different contexts. For example, budding yeast cells grow toward potential mating partners in response to pheromone gradients. Directed growth is controlled by polarity factors that assemble into clusters at the cell membrane. The clusters assemble, disassemble, and move between different regions of the membrane before eventually forming a stable polarity site directed toward the pheromone source. Pathways that regulate clustering have been identified but the molecular mechanisms that regulate cluster mobility are not well understood. To gain insight into the contribution of chemical noise to cluster behavior we simulated clustering using the reaction-diffusion master equation (RDME) framework to account for molecular-level fluctuations. RDME simulations are a computationally efficient approximation, but their results can diverge from the underlying microscopic dynamics. We implemented novel concentration-dependent rate constants that improved the accuracy of RDME-based simulations, allowing us to efficiently investigate how cluster dynamics might be regulated. Molecular noise was effective in relocating clusters when the clusters contained low numbers of limiting polarity factors, and when Cdc42, the central polarity regulator, exhibited short dwell times at the polarity site. Cluster stabilization occurred when abundances or binding rates were altered to either lengthen dwell times or increase the number of polarity molecules in the cluster. We validated key results using full 3D particle-based simulations. Understanding the mechanisms cells use to regulate the dynamics of polarity clusters should provide insights into how cells dynamically track external directional cues.

Cells localize polarity molecules in a small region of the plasma membrane forming a polarity cluster that directs functions such as migration, reproduction, and growth. Guided by external signals, these clusters move across the membrane allowing cells to reorient growth or motion. The polarity molecules continuously and randomly shuttle between the cluster and the cell cytosol and, as a result, the number and distribution of molecules at the cluster constantly changes. Here we present an improved stochastic simulation algorithm to investigate how such molecular-scale fluctuations induce cluster movement across the cell membrane. Unexpectedly, cluster mobility does not correlate with variations in total molecule abundance within the cluster, but rather with changes in the spatial distribution of molecules that form the cluster. Cluster motion is faster when polarity molecules are scarce and when they shuttle rapidly between the cluster and the cytosol. Our results suggest that cells control cluster mobility by regulating the abundance of polarity molecules and biochemical reactions that affect the time molecules spend at the cluster. We provide insights into how cells harness random molecular behavior to perform functions important for survival, such as detecting the direction of external signals.

Three-dimensional stochastic simulation of chemoattractant-mediated excitability in cells

During the last decade, a consensus has emerged that the stochastic triggering of an excitable system drives pseudopod formation and subsequent migration of amoeboid cells. The presence of chemoattractant stimuli alters the threshold for triggering this activity and can bias the direction of migration. Though noise plays an important role in these behaviors, mathematical models have typically ignored its origin and merely introduced it as an external signal into a series of reaction-diffusion equations. Here we consider a more realistic description based on a reaction-diffusion master equation formalism to implement these networks. In this scheme, noise arises naturally from a stochastic description of the various reaction and diffusion terms. Working on a three-dimensional geometry in which separate compartments are divided into a tetrahedral mesh, we implement a modular description of the system, consisting of G-protein coupled receptor signaling (GPCR), a local excitation-global inhibition mechanism (LEGI), and signal transduction excitable network (STEN). Our models implement detailed biochemical descriptions whenever this information is available, such as in the GPCR and G-protein interactions. In contrast, where the biochemical entities are less certain, such as the LEGI mechanism, we consider various possible schemes and highlight the differences between them. Our simulations show that even when the LEGI mechanism displays perfect adaptation in terms of the mean level of proteins, the variance shows a dose-dependence. This differs between the various models considered, suggesting a possible means for determining experimentally among the various potential networks. Overall, our simulations recreate temporal and spatial patterns observed experimentally in both wild-type and perturbed cells, providing further evidence for the excitable system paradigm. Moreover, because of the overall importance and ubiquity of the modules we consider, including GPCR signaling and adaptation, our results will be of interest beyond the field of directed migration.

Though the term noise usually carries negative connotations, it can also contribute positively to the characteristic dynamics of a system. In biological systems, where noise arises from the stochastic interactions between molecules, its study is usually confined to genetic regulatory systems in which copy numbers are small and fluctuations large. However, noise can have important roles when the number of signaling molecules is large. The extension of pseudopods and the subsequent motion of amoeboid cells arises from the noise-induced trigger of an excitable system. Chemoattractant signals bias this triggering thereby directing cell motion. To date, this paradigm has not been tested by mathematical models that account accurately for the noise that arises in the corresponding reactions. In this study, we employ a reaction-diffusion master equation approach to investigate the effects of noise. Using a modular approach and a three-dimensional cell model with specific subdomains attributed to the cell membrane and cortex, we explore the spatiotemporal dynamics of the system. Our simulations recreate many experimentally-observed cell behaviors thereby supporting the biased-excitable network hypothesis.

Describing the movement of molecules in reduced-dimension models

When addressing spatial biological questions using mathematical models, symmetries within the system are often exploited to simplify the problem by reducing its physical dimension. In a reduced-dimension model molecular movement is restricted to the reduced dimension, changing the nature of molecular movement. This change in molecular movement can lead to quantitatively and even qualitatively different results in the full and reduced systems. Within this manuscript we discuss the condition under which restricted molecular movement in reduced-dimension models accurately approximates molecular movement in the full system. For those systems which do not satisfy the condition, we present a general method for approximating unrestricted molecular movement in reduced-dimension models. We will derive a mathematically robust, finite difference method for solving the 2D diffusion equation within a 1D reduced-dimension model. The methods described here can be used to improve the accuracy of many reduced-dimension models while retaining benefits of system simplification.

How cells determine the number of polarity sites

The diversity of cell morphologies arises, in part, through regulation of cell polarity by Rho-family GTPases. A poorly understood but fundamental question concerns the regulatory mechanisms by which different cells generate different numbers of polarity sites. Mass-conserved activator-substrate (MCAS) models that describe polarity circuits develop multiple initial polarity sites, but then those sites engage in competition, leaving a single winner. Theoretical analyses predicted that competition would slow dramatically as GTPase concentrations at different polarity sites increase toward a 'saturation point', allowing polarity sites to coexist. Here, we test this prediction using budding yeast cells, and confirm that increasing the amount of key polarity proteins results in multiple polarity sites and simultaneous budding. Further, we elucidate a novel design principle whereby cells can switch from competition to equalization among polarity sites. These findings provide insight into how cells with diverse morphologies may determine the number of polarity sites.

The role of molecular diffusion within dendritic spines in synaptic function

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

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

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

Geometric principles of second messenger dynamics in dendritic spines

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

Brain proteome changes in female Brd1+/- mice unmask dendritic spine pathology and show enrichment for schizophrenia risk

Genetic and molecular studies have implicated the Bromodomain containing 1 (BRD1) gene in the pathogenesis of schizophrenia and bipolar disorder. Accordingly, mice heterozygous for a targeted deletion of Brd1 (Brd1^+/- mice) show behavioral phenotypes with broad translational relevance to psychiatric disorders. BRD1 encodes a scaffold protein that affects the expression of many genes through modulation of histone acetylation. BRD1 target genes have been identified in cell lines; however the impact of reduced Brd1 levels on the brain proteome is largely unknown. In this study, we applied label-based quantitative mass spectrometry to profile the frontal cortex, hippocampus and striatum proteome and synaptosomal proteome of female Brd1^+/- mice. We successfully quantified between 1537 and 2196 proteins and show widespread changes in protein abundancies and compartmentalization. By integrative analysis of human genetic data, we find that the differentially abundant proteins in frontal cortex and hippocampus are enriched for schizophrenia risk further linking the actions of BRD1 to psychiatric disorders. Affected proteins were further enriched for proteins involved in processes known to influence neuronal and dendritic spine morphology e.g. regulation of cytoskeleton dynamics and mitochondrial function. Directly prompted in these findings, we investigated dendritic spine morphology of pyramidal neurons in anterior cingulate cortex and found them significantly altered, including reduced size of small dendritic spines and decreased number of the mature mushroom type. Collectively, our study describes known as well as new mechanisms related to BRD1 dysfunction and its role in psychiatric disorders, and provides evidence for the molecular and cellular dysfunctions underlying altered neurosignalling and cognition in Brd1^+/- mice.

Particle-based simulations of polarity establishment reveal stochastic promotion of Turing pattern formation

Polarity establishment, the spontaneous generation of asymmetric molecular distributions, is a crucial component of many cellular functions. Saccharomyces cerevisiae (yeast) undergoes directed growth during budding and mating, and is an ideal model organism for studying polarization. In yeast and many other cell types, the Rho GTPase Cdc42 is the key molecular player in polarity establishment. During yeast polarization, multiple patches of Cdc42 initially form, then resolve into a single front. Because polarization relies on strong positive feedback, it is likely that the amplification of molecular-level fluctuations underlies the generation of multiple nascent patches. In the absence of spatial cues, these fluctuations may be key to driving polarization. Here we used particle-based simulations to investigate the role of stochastic effects in a Turing-type model of yeast polarity establishment. In the model, reactions take place either between two molecules on the membrane, or between a cytosolic and a membrane-bound molecule. Thus, we developed a computational platform that explicitly simulates molecules at and near the cell membrane, and implicitly handles molecules away from the membrane. To evaluate stochastic effects, we compared particle simulations to deterministic reaction-diffusion equation simulations. Defining macroscopic rate constants that are consistent with the microscopic parameters for this system is challenging, because diffusion occurs in two dimensions and particles exchange between the membrane and cytoplasm. We address this problem by empirically estimating macroscopic rate constants from appropriately designed particle-based simulations. Ultimately, we find that stochastic fluctuations speed polarity establishment and permit polarization in parameter regions predicted to be Turing stable. These effects can operate at Cdc42 abundances expected of yeast cells, and promote polarization on timescales consistent with experimental results. To our knowledge, our work represents the first particle-based simulations of a model for yeast polarization that is based on a Turing mechanism.

Many cells need to generate and maintain biochemical signals in specific subcellular regions. This phenomenon is broadly called polarity establishment, and is important in fundamental processes such as cell migration and differentiation. A key polarity factor found in diverse organisms, including yeast and humans, is the protein Cdc42. In yeast, Cdc42-dependent polarization occurs through a self-reinforcing biochemical signaling loop. Directional cues can guide polarity establishment, but interestingly, yeast can polarize in the absence of such a cue. The mechanism thought to underlie this symmetry breaking involves the amplification of inhomogeneities in molecular distributions that arise from molecular-level fluctuations. We investigated the effects of random fluctuations on polarization by performing particle-based simulations of the Cdc42 signaling network. We found that fluctuations can facilitate polarization, allowing faster polarization, and polarization over a broader range of concentrations. Our observations may help understand how polarity works in other systems.

Biophysics of Biochemical Signaling in Dendritic Spines: Implications in Synaptic Plasticity

Dendritic spines are mushroom-shaped postsynaptic compartments that host biochemical signal cascades important for synaptic plasticity and, ultimately, learning and memory. Signaling events in spines involve a signaling network composed of hundreds of signaling proteins interacting with each other extensively. Synaptic plasticity is typically induced by Ca²⁺ elevation in spines, which activates a variety of signaling pathways. This leads to changes in the actin cytoskeleton and membrane dynamics, which in turn causes structural and functional changes of the spine. Recent studies have demonstrated that the activities of these proteins have a variety of spatiotemporal patterns, which orchestrate signaling activity in different subcellular compartments at different timescales. The diffusion and the decay kinetics of signaling molecules play important roles in determining the degree of their spatial spreading, and thereby the degree of the spine specificity of the signaling pathway.

Crawling and turning in a minimal reaction-diffusion cell motility model: Coupling cell shape and biochemistry

We study a minimal model of a crawling eukaryotic cell with a chemical polarity controlled by a reaction-diffusion mechanism describing Rho GTPase dynamics. The size, shape, and speed of the cell emerge from the combination of the chemical polarity, which controls the locations where actin polymerization occurs, and the physical properties of the cell, including its membrane tension. We find in our model both highly persistent trajectories, in which the cell crawls in a straight line, and turning trajectories, where the cell transitions from crawling in a line to crawling in a circle. We discuss the controlling variables for this turning instability and argue that turning arises from a coupling between the reaction-diffusion mechanism and the shape of the cell. This emphasizes the surprising features that can arise from simple links between cell mechanics and biochemistry. Our results suggest that similar instabilities may be present in a broad class of biochemical descriptions of cell polarity.

Adhesion-Dependent Wave Generation in Crawling Cells

Dynamic actin networks are excitable. In migrating cells, feedback loops can amplify stochastic fluctuations in actin dynamics, often resulting in traveling waves of protrusion. The precise contributions of various molecular and mechanical interactions to wave generation have been difficult to disentangle, in part due to complex cellular morphodynamics. Here we used a relatively simple cell type-the fish epithelial keratocyte-to define a set of mechanochemical feedback loops underlying actin network excitability and wave generation. Although keratocytes are normally characterized by the persistent protrusion of a broad leading edge, increasing cell-substrate adhesion strength results in waving protrusion of a short leading edge. We show that protrusion waves are due to fluctuations in actin polymerization rates and that overexpression of VASP, an actin anti-capping protein that promotes actin polymerization, switches highly adherent keratocytes from waving to persistent protrusion. Moreover, VASP localizes both to adhesion complexes and to the leading edge. Based on these results, we developed a mathematical model for protrusion waves in which local depletion of VASP from the leading edge by adhesions-along with lateral propagation of protrusion due to the branched architecture of the actin network and negative mechanical feedback from the cell membrane-results in regular protrusion waves. Consistent with our model simulations, we show that VASP localization at the leading edge oscillates, with VASP leading-edge enrichment greatest just prior to protrusion initiation. We propose that the mechanochemical feedbacks underlying wave generation in keratocytes may constitute a general module for establishing excitable actin dynamics in other cellular contexts.

Understanding the role of synaptopodin and the spine apparatus in Hebbian synaptic plasticity - New perspectives and the need for computational modeling

Synaptopodin (SP) is a proline-rich actin-associated protein essential for the formation of a spine apparatus (SA) in dendritic spines. The SA consists of stacks of smooth endoplasmic reticulum (sER) contiguous with the meshwork of somatodendritic ER. Spines of SP-deficient mice contain sER but no SA, demonstrating that SP is necessary for the assembly of ER cisterns into the more complex SA organelle. Although the SA was described decades ago, its function was difficult to investigate and remained elusive, in part because reliable markers for the SA were missing. After SP was identified as an essential component and a reliable marker of the SA, a role of SP/SA in hippocampal synaptic plasticity could be firmly established using loss-of-function approaches. Further studies revealed that SP/SA participate in the regulation of Ca²⁺-dependent spine-specific Hebbian plasticity and in activity-dependent changes in the spine actin cytoskeleton. In this review we are summarizing recent progress made on SP/SA in Hebbian plasticity and discuss open questions such as causality, spatiotemporal dynamics and complementarity of SP/SA-dependent mechanisms. We are proposing that computational modeling of spine Ca²⁺-signaling and actin remodeling pathways could address some of these issues and could indicate future research directions. Moreover, reaction-diffusion simulations could help to identify key feedforward and feedback regulatory motifs regulating the switch between an LTP and an LTD signaling module in SP/SA-containing spines, thus helping to find a unified view of SP/SA action in Hebbian plasticity.