An image-based model of calcium waves in differentiated neuroblastoma cells

Impact of Interdependent Ca2+ and IP3 Dynamics On ATP Regulation in A Fibroblast Model

The vital participation of Ca2+ in human organ functions such as muscular contractions, heartbeat, brain functionality, skeletal activity, etc, motivated the scientists to thoroughly research the mechanisms of calcium (Ca2+) signalling in distinct human cells. Ca2+, inositol triphosphate (IP3), and adenosine triphosphate (ATP) play important roles in cell signaling and physiological processes. ATP and its derivatives are hypothesized to be important in the pathogenic process that leads to fibrotic illnesses like fibrosis. Fluctuations in Ca2+ and IP3 in a fibroblast cell influence ATP production. To date, no evidence of coupled Ca2+ and IP3 mechanics regulating ATP generation in a fibroblast cell during fibrotic disease has been found. The current work suggests an integrated mechanism for Ca2+ and IP3 dynamics in a fibroblast cell that regulates ATP generation. Simulation has been carried out using the finite element approach. The mechanics of interdependent systems findings vary dramatically from the results of basic independent system mechanics and give fresh information about the two systems' activities. The numerical results provide new insights into the impacts of disturbances in source influx, the serca pump, and buffers on interdependent Ca2+ and IP3 dynamics and ATP synthesis in a fibroblast cell. According to the findings of this study, fibrotic disorders cannot be attributed solely to disruptions in the processes of calcium signaling mechanics but also to disruptions in IP3 regulation mechanisms affecting the regulation of calcium in the fibroblast cell and ATP release.

How Merkel cells transduce mechanical stimuli: A biophysical model of Merkel cells

Merkel cells combine with Aβ afferents, producing slowly adapting type 1(SA1) responses to mechanical stimuli. However, how Merkel cells transduce mechanical stimuli into neural signals to Aβ afferents is still unclear. Here we develop a biophysical model of Merkel cells for mechanical transduction by incorporating main ingredients such as Ca2+ and K+ voltage-gated channels, Piezo2 channels, internal Ca2+ stores, neurotransmitters release, and cell deformation. We first validate our model with several experiments. Then we reveal that Ca2+ and K+ channels on the plasma membrane shape the depolarization of membrane potentials, further regulating the Ca2+ transients in the cells. We also show that Ca2+ channels on the plasma membrane mainly inspire the Ca2+ transients, while internal Ca2+ stores mainly maintain the Ca2+ transients. Moreover, we show that though Piezo2 channels are rapidly adapting mechanical-sensitive channels, they are sufficient to inspire sustained Ca2+ transients in Merkel cells, which further induce the release of neurotransmitters for tens of seconds. Thus our work provides a model that captures the membrane potentials and Ca2+ transients features of Merkel cells and partly explains how Merkel cells transduce the mechanical stimuli by Piezo2 channels.

Heterogeneous off-target impact of ion-channel deletion on intrinsic properties of hippocampal model neurons that self-regulate calcium

How do neurons that implement cell-autonomous self-regulation of calcium react to knockout of individual ion-channel conductances? To address this question, we used a heterogeneous population of 78 conductance-based models of hippocampal pyramidal neurons that maintained cell-autonomous calcium homeostasis while receiving theta-frequency inputs. At calcium steady-state, we individually deleted each of the 11 active ion-channel conductances from each model. We measured the acute impact of deleting each conductance (one at a time) by comparing intrinsic electrophysiological properties before and immediately after channel deletion. The acute impact of deleting individual conductances on physiological properties (including calcium homeostasis) was heterogeneous, depending on the property, the specific model, and the deleted channel. The underlying many-to-many mapping between ion channels and properties pointed to ion-channel degeneracy. Next, we allowed the other conductances (barring the deleted conductance) to evolve towards achieving calcium homeostasis during theta-frequency activity. When calcium homeostasis was perturbed by ion-channel deletion, post-knockout plasticity in other conductances ensured resilience of calcium homeostasis to ion-channel deletion. These results demonstrate degeneracy in calcium homeostasis, as calcium homeostasis in knockout models was implemented in the absence of a channel that was earlier involved in the homeostatic process. Importantly, in reacquiring homeostasis, ion-channel conductances and physiological properties underwent heterogenous plasticity (dependent on the model, the property, and the deleted channel), even introducing changes in properties that were not directly connected to the deleted channel. Together, post-knockout plasticity geared towards maintaining homeostasis introduced heterogenous off-target effects on several channels and properties, suggesting that extreme caution be exercised in interpreting experimental outcomes involving channel knockouts.

Beyond analytic solution: Analysis of FRAP experiments by spatial simulation of the forward problem

Fluorescence redistribution after photobleaching is a commonly used method to understand the dynamic behavior of molecules within cells. Analytic solutions have been developed for specific, well-defined models of dynamic behavior in idealized geometries, but these solutions are inaccurate in complex geometries or when complex binding and diffusion behaviors exist. We demonstrate the use of numerical reaction-diffusion simulations using the Virtual Cell software platform to model fluorescence redistribution after photobleaching experiments. Multiple simulations employing parameter scans and varying bleaching locations and sizes can help to bracket diffusion coefficients and kinetic rate constants in complex image-based geometries. This approach is applied to problems in membrane surface diffusion as well as diffusion and binding in cytosolic volumes in complex cell geometries. In addition, we model diffusion and binding within phase-separated biomolecular condensates (liquid droplets). These are modeled as spherical low-affinity binding domains that also define a high viscosity medium for exchange of the free fluorescently labeled ligand with the external cytosol.

Mechanistic insights of neuronal calcium and IP3 signaling system regulating ATP release during ischemia in progression of Alzheimer's disease

The mechanisms of calcium ([Ca²⁺]) signaling in various human cells have been widely analyzed by scientists due to its crucial role in human organs like the heartbeat, muscle contractions, bone activity, brain functionality, etc. No study is reported for interdependent [Ca²⁺] and IP₃ mechanics regulating the release of ATP in neuron cells during Ischemia in Alzheimer's disease advancement. In the present investigation, a finite element method (FEM) is framed to explore the interdependence of spatiotemporal [Ca²⁺] and IP₃ signaling mechanics and its role in ATP release during Ischemia as well as in the advancement of Alzheimer's disorder in neuron cells. The results provide us insights of the mutual spatiotemporal impacts of [Ca²⁺] and IP₃ mechanics as well as their contributions to ATP release during Ischemia in neuron cells. The results obtained for the mechanics of interdependent systems differ significantly from the results of simple independent system mechanics and provide new information about the processes of the two systems. From this study, it is concluded that neuronal disorders cannot only be simply attributed to the disturbance caused directly in the processes of calcium signaling mechanics, but also to the disturbances caused in IP₃ regulation mechanisms impacting the calcium regulation in the neuron cell and ATP release.

Ten steps to investigate a cellular system with mathematical modeling

Cellular and intracellular processes are inherently complex due to the large number of components and interactions, which are often nonlinear and occur at different spatiotemporal scales. Because of this complexity, mathematical modeling is increasingly used to simulate such systems and perform experiments in silico, many orders of magnitude faster than real experiments and often at a higher spatiotemporal resolution. In this article, we will focus on the generic modeling process and illustrate it with an example model of membrane lipid turnover.

Ca2+ entry through NaV channels generates submillisecond axonal Ca2+ signaling

Calcium ions (Ca²⁺) are essential for many cellular signaling mechanisms and enter the cytosol mostly through voltage-gated calcium channels. Here, using high-speed Ca²⁺ imaging up to 20 kHz in the rat layer five pyramidal neuron axon we found that activity-dependent intracellular calcium concentration ([Ca²⁺]_i) in the axonal initial segment was only partially dependent on voltage-gated calcium channels. Instead, [Ca²⁺]_i changes were sensitive to the specific voltage-gated sodium (Na_V) channel blocker tetrodotoxin. Consistent with the conjecture that Ca²⁺ enters through the Na_V channel pore, the optically resolved I_Ca in the axon initial segment overlapped with the activation kinetics of Na_V channels and heterologous expression of Na_V1.2 in HEK-293 cells revealed a tetrodotoxin-sensitive [Ca²⁺]_i rise. Finally, computational simulations predicted that axonal [Ca²⁺]_i transients reflect a 0.4% Ca²⁺ conductivity of Na_V channels. The findings indicate that Ca²⁺ permeation through Na_V channels provides a submillisecond rapid entry route in Na_V-enriched domains of mammalian axons.

Nerve cells communicate using tiny electrical impulses called action potentials. Special proteins termed ion channels produce these electric signals by allowing specific charged particles, or ions, to pass in or out of cells across its membrane. When a nerve cell ‘fires’ an action potential, specific ion channels briefly open to let in a surge of positively charged ions which electrify the cell. Action potentials begin in the same place in each nerve cell, at an area called the axon initial segment. The large number of sodium channels at this site kick-start the influx of positively charged sodium ions ensuring that every action potential starts from the same place.

Previous research has shown that, when action potentials begin, the concentration of calcium ions at the axon initial segment also increases, but it was not clear which ion channels were responsible for this entry of calcium. Channels that are selective for calcium ions are the prime candidates for this process. However, research in squid nerve cells gave rise to an unexpected idea by suggesting that sodium channels may not exclusively let in sodium but also allow some calcium ions to pass through. Hanemaaijer, Popovic et al. therefore wanted to test the routes that calcium ions take and see whether the sodium channels in mammalian nerve cells are also permeable to calcium.

Experiments using fluorescent dyes to track the concentration of calcium in rat and human nerve cells showed that calcium ions accumulated at the axon initial segment when action potentials fired. Most of this increase in calcium could be stopped by treating the neurons with a toxin that prevents sodium channels from opening. Electrical manipulations of the cells revealed that, in this context, the calcium ions were effectively behaving like sodium ions. Human kidney cells were then engineered to produce the sodium channel protein. This confirmed that calcium and sodium ions were indeed both passing through the same channel.

These results shed new light on the relationship between calcium ions and sodium channels within the mammalian nervous system and that this interplay occurs at the axon initial segment of the cell. Genetic mutations that ‘nudge’ sodium channels towards favoring calcium entry are also found in patients with autism spectrum disorders, and so this new finding may contribute to our understanding of these conditions.

Intracellular calcium stores mediate metaplasticity at hippocampal dendritic spines

Key points

Calcium (Ca²⁺) entry mediated by NMDA receptors is considered central to the induction of activity‐dependent synaptic plasticity in hippocampal area CA1; this description does not, however, take into account the potential contribution of endoplasmic reticulum (ER) Ca²⁺ stores.

The ER has a heterogeneous distribution in CA1 dendritic spines, and may introduce localized functional differences in Ca²⁺ signalling between synapses, as suggested by experiments on metabotropic receptor‐dependent long‐term depression.

A physiologically detailed computational model of Ca²⁺ dynamics at a CA3–CA1 excitatory synapse characterizes the contribution of spine ER via metabotropic signalling during plasticity induction protocols.

ER Ca²⁺ release via IP₃ receptors modulates NMDA receptor‐dependent plasticity in a graded manner, to selectively promote synaptic depression with relatively diminished effect on LTP induction; this may temper further strengthening at the stronger synapses which are preferentially associated with ER‐containing spines.

Acquisition of spine ER may thus represent a local, biophysically plausible ‘metaplastic switch’ at potentiated CA1 synapses, contributing to the plasticity–stability balance in neural circuits.

Abstract

Long‐term plasticity mediated by NMDA receptors supports input‐specific, Hebbian forms of learning at excitatory CA3–CA1 connections in the hippocampus. There exists an additional layer of stabilizing mechanisms that act globally as well as locally over multiple time scales to ensure that plasticity occurs in a constrained manner. Here, we investigated the role of calcium (Ca²⁺) stores associated with the endoplasmic reticulum (ER) in the local regulation of plasticity at individual CA1 synapses. Our study was spurred by (1) the curious observation that ER is sparsely distributed in dendritic spines, but over‐represented in larger spines that are likely to have undergone activity‐dependent strengthening, and (2) evidence suggesting that ER motility at synapses can be rapid, and accompany activity‐regulated spine remodelling. We constructed a physiologically realistic computational model of an ER‐bearing CA1 spine, and examined how IP₃‐sensitive Ca²⁺ stores affect spine Ca²⁺ dynamics during activity patterns mimicking the induction of long‐term potentiation and long‐term depression (LTD). Our results suggest that the presence of ER modulates NMDA receptor‐dependent plasticity in a graded manner that selectively enhances LTD induction. We propose that ER may locally tune Ca²⁺‐based plasticity, providing a braking mechanism to mitigate runaway strengthening at potentiated synapses. Our study provides a biophysically accurate description of postsynaptic Ca²⁺ regulation, and suggests that ER in the spine may promote the re‐use of hippocampal synapses with saturated strengths.

Calcium (Ca²⁺) entry mediated by NMDA receptors is considered central to the induction of activity‐dependent synaptic plasticity in hippocampal area CA1; this description does not, however, take into account the potential contribution of endoplasmic reticulum (ER) Ca²⁺ stores.

The ER has a heterogeneous distribution in CA1 dendritic spines, and may introduce localized functional differences in Ca²⁺ signalling between synapses, as suggested by experiments on metabotropic receptor‐dependent long‐term depression.

A physiologically detailed computational model of Ca²⁺ dynamics at a CA3–CA1 excitatory synapse characterizes the contribution of spine ER via metabotropic signalling during plasticity induction protocols.

ER Ca²⁺ release via IP₃ receptors modulates NMDA receptor‐dependent plasticity in a graded manner, to selectively promote synaptic depression with relatively diminished effect on LTP induction; this may temper further strengthening at the stronger synapses which are preferentially associated with ER‐containing spines.

Acquisition of spine ER may thus represent a local, biophysically plausible ‘metaplastic switch’ at potentiated CA1 synapses, contributing to the plasticity–stability balance in neural circuits.

Three dimensional coupled reaction–diffusion modeling of calcium and inositol 1,4,5-trisphosphate dynamics in cardiomyocytes

Nanoparticles have shown great promise in improving cancer treatment efficacy by changing the intracellular calcium level through activation of intracellular mechanisms. One of the mechanisms of the killing of the cancerous cell by a nanoparticle is through elevation of the intracellular calcium level. Evidence accumulated over the past decade indicates a pivotal role for the IP₃ receptor mediated Ca²⁺ release in the regulation of the cytosolic and the nuclear Ca²⁺ signals. There have been various studies done suggesting the role of IP₃ receptors (IP₃R) and IP₃ production and degradation in cardiomyocytes. In the present work, we have proposed a three-dimensional unsteady-state mathematical model to describe the mechanism of cardiomyocytes which focuses on evaluation of various parameters that affect these coupled dynamics and elevate the cytosolic calcium concentration which can be helpful to search for novel therapies to cure these malignancies by targeting the complex calcium signaling process in cardiomyocytes. Our study suggests that there are other factors involved in this signaling which can increase the calcium level, which can help in finding treatment for cancer. The cytosolic calcium level may be controlled by IP₃ signaling, leak, source influx of calcium (σ) and maximum production of IP₃ (V_P). We believe that the proposed model suggests new insight into finding treatment for cancer in cardiomyocytes through elevation of the cytosolic Ca²⁺ concentration by various parameters like leak, σ, V_P and especially by other complex cell signaling dynamics, namely IP₃ dynamics.

We propose a three-dimensional unsteady-state mathematical model to describe the mechanism of cardiomyocytes.

Active dendrites regulate the spatiotemporal spread of signaling microdomains

Microdomains that emerge from spatially constricted spread of biochemical signaling components play a central role in several neuronal computations. Although dendrites, endowed with several voltage-gated ion channels, form a prominent structural substrate for microdomain physiology, it is not known if these channels regulate the spatiotemporal spread of signaling microdomains. Here, we employed a multiscale, morphologically realistic, conductance-based model of the hippocampal pyramidal neuron that accounted for experimental details of electrical and calcium-dependent biochemical signaling. We activated synaptic N-Methyl-d-Aspartate receptors through theta-burst stimulation (TBS) or pairing (TBP) and assessed microdomain propagation along a signaling pathway that included calmodulin, calcium/calmodulin-dependent protein kinase II (CaMKII) and protein phosphatase 1. We found that the spatiotemporal spread of the TBS-evoked microdomain in phosphorylated CaMKII (pCaMKII) was amplified in comparison to that of the corresponding calcium microdomain. Next, we assessed the role of two dendritically expressed inactivating channels, one restorative (A-type potassium) and another regenerative (T-type calcium), by systematically varying their conductances. Whereas A-type potassium channels suppressed the spread of pCaMKII microdomains by altering the voltage response to TBS, T-type calcium channels enhanced this spread by modulating TBS-induced calcium influx without changing the voltage. Finally, we explored cross-dependencies of these channels with other model components, and demonstrated the heavy mutual interdependence of several biophysical and biochemical properties in regulating microdomains and their spread. Our conclusions unveil a pivotal role for dendritic voltage-gated ion channels in actively amplifying or suppressing biochemical signals and their spatiotemporal spread, with critical implications for clustered synaptic plasticity, robust information transfer and efficient neural coding.

The spatiotemporal spread of biochemical signals in neurons and other cells regulate signaling specificity, tuning of signal propagation, along with specificity and clustering of adaptive plasticity. Theoretical and experimental studies have demonstrated a critical role for cellular morphology and the topology of signaling networks in regulating this spread. In this study, we add a significantly complex dimension to this narrative by demonstrating that voltage-gated ion channels on the plasma membrane could actively amplify or suppress the strength and spread of downstream signaling components. Given the expression of different ion channels with wide-ranging heterogeneity in gating kinetics, localization and density, our results point to an increase in complexity of and degeneracy in signaling spread, and unveil a powerful mechanism for regulating biochemical-signaling pathways across different cell types.

Spine-to-Dendrite Calcium Modeling Discloses Relevance for Precise Positioning of Ryanodine Receptor-Containing Spine Endoplasmic Reticulum

The endoplasmic reticulum (ER) forms a complex endomembrane network that reaches into the cellular compartments of a neuron, including dendritic spines. Recent work discloses that the spine ER is a dynamic structure that enters and leaves spines. While evidence exists that ER Ca²⁺ release is involved in synaptic plasticity, the role of spine ER morphology remains unknown. Combining a new 3D spine generator with 3D Ca²⁺ modeling, we addressed the relevance of ER positioning on spine-to-dendrite Ca²⁺ signaling. Our simulations, which account for Ca²⁺ exchange on the plasma membrane and ER, show that spine ER needs to be present in distinct morphological conformations in order to overcome a barrier between the spine and dendritic shaft. We demonstrate that RyR-carrying spine ER promotes spine-to-dendrite Ca²⁺ signals in a position-dependent manner. Our simulations indicate that RyR-carrying ER can initiate time-delayed Ca²⁺ reverberation, depending on the precise position of the spine ER. Upon spine growth, structural reorganization of the ER restores spine-to-dendrite Ca²⁺ communication, while maintaining aspects of Ca²⁺ homeostasis in the spine head. Our work emphasizes the relevance of precise positioning of RyR-containing spine ER in regulating the strength and timing of spine Ca²⁺ signaling, which could play an important role in tuning spine-to-dendrite Ca²⁺ communication and homeostasis.

Disentangling astroglial physiology with a realistic cell model in silico

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

A biophysically detailed computational model of urinary bladder small DRG neuron soma

Bladder small DRG neurons, which are putative nociceptors pivotal to urinary bladder function, express more than a dozen different ionic membrane mechanisms: ion channels, pumps and exchangers. Small-conductance Ca2+-activated K+ (SKCa) channels which were earlier thought to be gated solely by intracellular Ca2+ concentration ([Ca]i) have recently been shown to exhibit inward rectification with respect to membrane potential. The effect of SKCa inward rectification on the excitability of these neurons is unknown. Furthermore, studies on the role of KCa channels in repetitive firing and their contributions to different types of afterhyperpolarization (AHP) in these neurons are lacking. In order to study these phenomena, we first constructed and validated a biophysically detailed single compartment model of bladder small DRG neuron soma constrained by physiological data. The model includes twenty-two major known membrane mechanisms along with intracellular Ca2+ dynamics comprising Ca2+ diffusion, cytoplasmic buffering, and endoplasmic reticulum (ER) and mitochondrial mechanisms. Using modelling studies, we show that inward rectification of SKCa is an important parameter regulating neuronal repetitive firing and that its absence reduces action potential (AP) firing frequency. We also show that SKCa is more potent in reducing AP spiking than the large-conductance KCa channel (BKCa) in these neurons. Moreover, BKCa was found to contribute to the fast AHP (fAHP) and SKCa to the medium-duration (mAHP) and slow AHP (sAHP). We also report that the slow inactivating A-type K+ channel (slow KA) current in these neurons is composed of 2 components: an initial fast inactivating (time constant ∼ 25-100 ms) and a slow inactivating (time constant ∼ 200-800 ms) current. We discuss the implications of our findings, and how our detailed model can help further our understanding of the role of C-fibre afferents in the physiology of urinary bladder as well as in certain disorders.

TRPM8 inhibits endothelial cell migration via a non-channel function by trapping the small GTPase Rap1

Endothelial cell adhesion and migration are critical steps of the angiogenic process, whose dysfunction is associated with tumor growth and metastasis. The TRPM8 channel has recently been proposed to play a protective role in prostate cancer by impairing cell motility. However, the mechanisms by which it could influence vascular behavior are unknown. Here, we reveal a novel non-channel function for TRPM8 that unexpectedly acts as a Rap1 GTPase inhibitor, thereby inhibiting endothelial cell motility, independently of pore function. TRPM8 retains Rap1 intracellularly through direct protein-protein interaction, thus preventing its cytoplasm-plasma membrane trafficking. In turn, this mechanism impairs the activation of a major inside-out signaling pathway that triggers the conformational activation of integrin and, consequently, cell adhesion, migration, in vitro endothelial tube formation, and spheroid sprouting. Our results bring to light a novel, pore-independent molecular mechanism by which endogenous TRPM8 expression inhibits Rap1 GTPase and thus plays a critical role in the behavior of vascular endothelial cells by inhibiting migration.

Degeneracy in the regulation of short-term plasticity and synaptic filtering by presynaptic mechanisms

KEY POINTS

We develop a new biophysically rooted, physiologically constrained conductance-based synaptic model to mechanistically account for short-term facilitation and depression, respectively through residual calcium and transmitter depletion kinetics. We address the specific question of how presynaptic components (including voltage-gated ion channels, pumps, buffers and release-handling mechanisms) and interactions among them define synaptic filtering and short-term plasticity profiles. Employing global sensitivity analyses (GSAs), we show that near-identical synaptic filters and short-term plasticity profiles could emerge from disparate presynaptic parametric combinations with weak pairwise correlations. Using virtual knockout models, a technique to address the question of channel-specific contributions within the GSA framework, we unveil the differential and variable impact of each ion channel on synaptic physiology. Our conclusions strengthen the argument that parametric and interactional complexity in biological systems should not be viewed from the limited curse-of-dimensionality standpoint, but from the evolutionarily advantageous perspective of providing functional robustness through degeneracy.

ABSTRACT

Information processing in neurons is known to emerge as a gestalt of pre- and post-synaptic filtering. However, the impact of presynaptic mechanisms on synaptic filters has not been quantitatively assessed. Here, we developed a biophysically rooted, conductance-based model synapse that was endowed with six different voltage-gated ion channels, calcium pumps, calcium buffer and neurotransmitter-replenishment mechanisms in the presynaptic terminal. We tuned our model to match the short-term plasticity profile and band-pass structure of Schaffer collateral synapses, and performed sensitivity analyses to demonstrate that presynaptic voltage-gated ion channels regulated synaptic filters through changes in excitability and associated calcium influx. These sensitivity analyses also revealed that calcium- and release-control mechanisms were effective regulators of synaptic filters, but accomplished this without changes in terminal excitability or calcium influx. Next, to perform global sensitivity analysis, we generated 7000 randomized models spanning 15 presynaptic parameters, and computed eight different physiological measurements in each of these models. We validated these models by applying experimentally obtained bounds on their measurements, and found 104 (∼1.5%) models to match the validation criteria for all eight measurements. Analysing these valid models, we demonstrate that analogous synaptic filters emerge from disparate combinations of presynaptic parameters exhibiting weak pairwise correlations. Finally, using virtual knockout models, we establish the variable and differential impact of different presynaptic channels on synaptic filters, underlining the critical importance of interactions among different presynaptic components in defining synaptic physiology. Our results have significant implications for protein-localization strategies required for physiological robustness and for degeneracy in long-term synaptic plasticity profiles.

The Effect of Gap Junctional Coupling on the Spatiotemporal Patterns of Ca2+ Signals and the Harmonization of Ca2+-Related Cellular Responses

Calcium ions (Ca²⁺) are important mediators of a great variety of cellular activities e.g. in response to an agonist activation of a receptor. The magnitude of a cellular response is often encoded by frequency modulation of Ca²⁺ oscillations and correlated with the stimulation intensity. The stimulation intensity highly depends on the sensitivity of a cell to a certain agonist. In some cases, it is essential that neighboring cells produce a similar and synchronized response to an agonist despite their different sensitivity. In order to decipher the presumed function of Ca²⁺ waves spreading among connecting cells, a mathematical model was developed. This model allows to numerically modifying the connectivity probability between neighboring cells, the permeability of gap junctions and the individual sensitivity of cells to an agonist. Here, we show numerically that strong gap junctional coupling between neighbors ensures an equilibrated response to agonist stimulation via formation of Ca²⁺ phase waves, i.e. a less sensitive neighbor will produce the same or similar Ca²⁺ signal as its highly sensitive neighbor. The most sensitive cells within an ensemble are the wave initiator cells. The Ca²⁺ wave in the cytoplasm is driven by a sensitization wave front in the endoplasmic reticulum. The wave velocity is proportional to the cellular sensitivity and to the strength of the coupling. The waves can form different patterns including circular rings and spirals. The observed pattern depends on the strength of noise, gap junctional permeability and the connectivity probability between neighboring cells. Our simulations reveal that one highly sensitive region gradually takes the lead within the entire noisy system by generating directed circular phase waves originating from this region.

The calcium ion (Ca²⁺), a universal signaling molecule, is widely recognized to play a fundamental role in the regulation of various biological processes. Agonist–evoked Ca²⁺ signals often manifest as rhythmic changes in the cytosolic free Ca²⁺ concentration (c_cyt) called Ca²⁺ oscillations. Stimuli intensity was found to be proportional to the oscillation frequency and the evoked down-steam cellular response. Stochastic receptor expression in individual cells in a cell population inevitably leads to individually different oscillation frequencies and individually different Ca²⁺-related cellular responses. However, in many organs, the neighboring cells have to overcome their individually different sensitivity and produce a synchronized response. Gap junctions are integral membrane structures that enable the direct cytoplasmic exchange of Ca²⁺ ions and InsP₃ molecules between neighboring cells. By simulations, we were able to demonstrate how the strength of intercellular gap junctional coupling in relation to stimulus intensity can modify the spatiotemporal patterns of Ca²⁺ signals and harmonize the Ca²⁺-related cellular responses via synchronization of oscillation frequency. We demonstrate that the most sensitive cells are the wave initiator cells and that a highly sensitive region plays an important role in the determination of the Ca²⁺ phase wave direction. This sensitive region will then also progressively determine the global behavior of the entire system.

Routes of Ca2+ Shuttling during Ca2+ Oscillations: FOCUS ON THE ROLE OF MITOCHONDRIAL Ca2+ HANDLING AND CYTOSOLIC Ca2+ BUFFERS

In some cell types, Ca²⁺ oscillations are strictly dependent on Ca²⁺ influx across the plasma membrane, whereas in others, oscillations also persist in the absence of Ca²⁺ influx. We observed that, in primary mesothelial cells, the plasmalemmal Ca²⁺ influx played a pivotal role. However, when the Ca²⁺ transport across the plasma membrane by the “lanthanum insulation method” was blocked prior to the induction of the serum-induced Ca²⁺ oscillations, mitochondrial Ca²⁺ transport was found to be able to substitute for the plasmalemmal Ca²⁺ exchange function, thus rendering the oscillations independent of extracellular Ca²⁺. However, in a physiological situation, the Ca²⁺-buffering capacity of mitochondria was found not to be essential for Ca²⁺ oscillations. Moreover, brief spontaneous Ca²⁺ changes were observed in the mitochondrial Ca²⁺ concentration without apparent changes in the cytosolic Ca²⁺ concentration, indicating the presence of a mitochondrial autonomous Ca²⁺ signaling mechanism. In the presence of calretinin, a Ca²⁺-buffering protein, the amplitude of cytosolic spikes during oscillations was decreased, and the amount of Ca²⁺ ions taken up by mitochondria was reduced. Thus, the increased calretinin expression observed in mesothelioma cells and in certain colon cancer might be correlated to the increased resistance of these tumor cells to proapoptotic/pronecrotic signals. We identified and characterized (experimentally and by modeling) three Ca²⁺ shuttling pathways in primary mesothelial cells during Ca²⁺ oscillations: Ca²⁺ shuttled between (i) the endoplasmic reticulum (ER) and mitochondria, (ii) the ER and the extracellular space, and (iii) the ER and cytoplasmic Ca²⁺ buffers.

Variability in State-Dependent Plasticity of Intrinsic Properties during Cell-Autonomous Self-Regulation of Calcium Homeostasis in Hippocampal Model Neurons

How do neurons reconcile the maintenance of calcium homeostasis with perpetual switches in patterns of afferent activity? Here, we assessed state-dependent evolution of calcium homeostasis in a population of hippocampal pyramidal neuron models, through an adaptation of a recent study on stomatogastric ganglion neurons. Calcium homeostasis was set to emerge through cell-autonomous updates to 12 ionic conductances, responding to different types of synaptically driven afferent activity. We first assessed the impact of theta-frequency inputs on the evolution of ionic conductances toward maintenance of calcium homeostasis. Although calcium homeostasis emerged efficaciously across all models in the population, disparate changes in ionic conductances that mediated this emergence resulted in variable plasticity to several intrinsic properties, also manifesting as significant differences in firing responses across models. Assessing the sensitivity of this form of plasticity, we noted that intrinsic neuronal properties and the firing response were sensitive to the target calcium concentration and to the strength and frequency of afferent activity. Next, we studied the evolution of calcium homeostasis when afferent activity was switched, in different temporal sequences, between two behaviorally distinct types of activity: theta-frequency inputs and sharp-wave ripples riding on largely silent periods. We found that the conductance values, intrinsic properties, and firing response of neurons exhibited differential robustness to an intervening switch in the type of afferent activity. These results unveil critical dissociations between different forms of homeostasis, and call for a systematic evaluation of the impact of state-dependent switches in afferent activity on neuronal intrinsic properties during neural coding and homeostasis.

Ca2+ dysregulation in the endoplasmic reticulum related to Alzheimer's disease: A review on experimental progress and computational modeling

Alzheimer's disease (AD) is a devastating, incurable neurodegenerative disease affecting millions of people worldwide. Dysregulation of intracellular Ca(2+) signaling has been observed as an early event prior to the presence of clinical symptoms of AD and is believed to be a crucial factor contributing to its pathogenesis. The progressive and sustaining increase in the resting level of cytosolic Ca(2+) will affect downstream activities and neural functions. This review focuses on the issues relating to the increasing Ca(2+) release from the endoplasmic reticulum (ER) observed in AD neurons. Numerous research papers have suggested that the dysregulation of ER Ca(2+) homeostasis is associated with mutations in the presenilin genes and amyloid-β oligomers. These disturbances could happen at many different points in the signaling process, directly affecting ER Ca(2+) channels or interfering with related pathways, which makes it harder to reveal the underlying mechanisms. This review paper also shows that computational modeling is a powerful tool in Ca(2+) signaling studies and discusses the progress in modeling related to Ca(2+) dysregulation in AD research.

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

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

Quantitative interactions between the A-type K+ current and inositol trisphosphate receptors regulate intraneuronal Ca2+ waves and synaptic plasticity

The A-type potassium current has been implicated in the regulation of several physiological processes. Here, we explore a role for the A-type potassium current in regulating the release of calcium through inositol trisphosphate receptors (InsP3R) that reside on the endoplasmic reticulum (ER) of hippocampal pyramidal neurons. To do this, we constructed morphologically realistic, conductance-based models equipped with kinetic schemes that govern several calcium signalling modules and pathways, and constrained the distributions and properties of constitutive components by experimental measurements from these neurons. Employing these models, we establish a bell-shaped dependence of calcium release through InsP3Rs on the density of A-type potassium channels, during the propagation of an intraneuronal calcium wave initiated through established protocols. Exploring the sensitivities of calcium wave initiation and propagation to several underlying parameters, we found that ER calcium release critically depends on dendritic diameter and that wave initiation occurred at branch points as a consequence of a high surface area to volume ratio of oblique dendrites. Furthermore, analogous to the role of A-type potassium channels in regulating spike latency, we found that an increase in the density of A-type potassium channels led to increases in the latency and the temporal spread of a propagating calcium wave. Next, we incorporated kinetic models for the metabotropic glutamate receptor (mGluR) signalling components and a calcium-controlled plasticity rule into our model and demonstrate that the presence of mGluRs induced a leftward shift in a Bienenstock-Cooper-Munro-like synaptic plasticity profile. Finally, we show that the A-type potassium current could regulate the relative contribution of ER calcium to synaptic plasticity induced either through 900 pulses of various stimulus frequencies or through theta burst stimulation. Our results establish a novel form of interaction between active dendrites and the ER membrane, uncovering a powerful mechanism that could regulate biophysical/biochemical signal integration and steer the spatiotemporal spread of signalling microdomains through changes in dendritic excitability.

Models of calcium dynamics in cerebellar granule cells

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

Multilevel complexity of calcium signaling: Modeling angiogenesis

Intracellular calcium signaling is a universal, evolutionary conserved and versatile regulator of cell biochemistry. The complexity of calcium signaling and related cell machinery can be investigated by the use of experimental strategies, as well as by computational approaches. Vascular endothelium is a fascinating model to study the specific properties and roles of calcium signals at multiple biological levels. During the past 20 years, live cell imaging, patch clamp and other techniques have allowed us to detect and interfere with calcium signaling in endothelial cells (ECs), providing a huge amount of information on the regulation of vascularization (angiogenesis) in normal and tumoral tissues. These data range from the spatiotemporal dynamics of calcium within different cell microcompartments to those in entire multicellular and organized EC networks. Beside experimental strategies, in silico endothelial models, specifically designed for simulating calcium signaling, are contributing to our knowledge of vascular physiology and pathology. They help to investigate and predict the quantitative features of proangiogenic events moving through subcellular, cellular and supracellular levels. This review focuses on some recent developments of computational approaches for proangiogenic endothelial calcium signaling. In particular, we discuss the creation of hybrid simulation environments, which combine and integrate discrete Cellular Potts Models. They are able to capture the phenomenological mechanisms of cell morphological reorganization, migration, and intercellular adhesion, with single-cell spatiotemporal models, based on reaction-diffusion equations that describe the agonist-induced intracellular calcium events.

A three-dimensional chemo-mechanical continuum model for smooth muscle contraction

Based on two fields, namely the placement and the calcium concentration, a chemo-mechanically coupled three-dimensional model, describing the contractile behaviour of smooth muscles, is presented by means of a strain energy function. The strain energy function (Schmitz and Böl, 2011) is additively decomposed into a passive part, relating to elastin and collagen, and an active calcium-driven part related to the chemical contraction of the smooth muscle cells. For the description of the calcium phase the four state cross-bridge model of Hai and Murphy (Hai and Murphy, 1988) has been implemented into the finite element method. Beside three-dimensional illustrative boundary-value problems demonstrating the features of the presented modelling concept, simulations on an idealised artery document the applicability of the model to more realistic geometries.

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

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

Spatial modeling of cell signaling networks

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

Developing models in virtual cell

This Teaching Resource provides lecture notes, slides, and a student assignment for a two-part lecture on mathematical modeling using the Virtual Cell environment. The lectures discuss the steps involved in developing and running simulations using Virtual Cell, with particular focus on spatial partial differential equation models. We discuss how to construct both ordinary differential equation models, in which the cytoplasm is considered a well-mixed cellular compartment, and partial differential equation models, which calculate how chemical species change as a function of both time and location. The Virtual Cell environment is especially well suited for models that explore spatial specificity of cellular reactions. Partial differential equation models in Virtual Cell can give rise to simulations using predefined cellular geometries, which enable direct comparison with imaging data. These models address questions regarding the regulatory capability arising from spatial organization of the cell. Examples are provided of studies that have successfully exploited the Virtual Cell software to address the spatial contribution to signaling.

Fast calcium waves

Calcium waves are propagated in five main speed ranges which cover a billion-fold range of speeds. We define the fast speed range as 3-30μm/s after correction to a standard temperature of 20°C. Only waves which are not fertilization waves are considered here. 181 such cases are listed here. These are through organisms in all major taxa from cyanobacteria through mammals including human beings except for those through other bacteria, higher plants and fungi. Nearly two-thirds of these speeds lie between 12 and 24μm/s. We argue that their common mechanism in eukaryotes is a reaction-diffusion one involving calcium-induced calcium release, in which calcium waves are propagated along the endoplasmic reticulum. We propose that the gliding movements of some cyanobacteria are driven by fast calcium waves which are propagated along their plasma membranes. Fast calcium waves may drive materials to one end of developing embryos by cellular peristalsis, help coordinate complex cell movements during development and underlie brain injury waves. Moreover, we continue to argue that such waves greatly increase the likelihood that chronic injuries will initiate tumors and cancers before genetic damage occurs. Finally we propose numerous further studies.

Stochastic hybrid modeling of intracellular calcium dynamics

Deterministic models of biochemical processes at the subcellular level might become inadequate when a cascade of chemical reactions is induced by a few molecules. Inherent randomness of such phenomena calls for the use of stochastic simulations. However, being computationally intensive, such simulations become infeasible for large and complex reaction networks. To improve their computational efficiency in handling these networks, we present a hybrid approach, in which slow reactions and fluxes are handled through exact stochastic simulation and their fast counterparts are treated partially deterministically through chemical Langevin equation. The classification of reactions as fast or slow is accompanied by the assumption that in the time-scale of fast reactions, slow reactions do not occur and hence do not affect the probability of the state. Our new approach also handles reactions with complex rate expressions such as Michaelis-Menten kinetics. Fluxes which cannot be modeled explicitly through reactions, such as flux of Ca(2+) from endoplasmic reticulum to the cytosol through inositol 1,4,5-trisphosphate receptor channels, are handled deterministically. The proposed hybrid algorithm is used to model the regulation of the dynamics of cytosolic calcium ions in mouse macrophage RAW 264.7 cells. At relatively large number of molecules, the response characteristics obtained with the stochastic and deterministic simulations coincide, which validates our approach in the limit of large numbers. At low doses, the response characteristics of some key chemical species, such as levels of cytosolic calcium, predicted with stochastic simulations, differ quantitatively from their deterministic counterparts. These observations are ubiquitous throughout dose response, sensitivity, and gene-knockdown response analyses. While the relative differences between the peak-heights of the cytosolic [Ca(2+)] time-courses obtained from stochastic (mean of 16 realizations) and deterministic simulations are merely 1%-4% for most perturbations, it is specially sensitive to levels of G(βγ) (relative difference as large as 90% at very low G(βγ)).

A multiscale hybrid approach for vasculogenesis and related potential blocking therapies

Solid tumors must recruit and form new blood vessels for maintenance, growth and detachments of metastases. Discovering drugs that block malignant angiogenesis is thus an important approach in cancer treatment and has given rise to multiple in vitro and in silico models. The present hybrid individual cell-based model incorporates some underlying biochemical events relating more closely the classical Cellular Potts Model (CPM) parameters to subcellular mechanisms and to the activation of specific signaling pathways. The model spans the three fundamental biological levels: at the extracellular level a continuous model describes secretion, diffusion, uptake and decay of the autocrine VEGF; at the cellular level, an extended lattice CPM, based on a system energy reduction, reproduces cell dynamics such as migration, adhesion and chemotaxis; at the subcellular level, a set of reaction-diffusion equations describes a simplified VEGF-induced calcium-dependent intracellular pathway. The results agree with the known interplay between calcium signals and VEGF dynamics and with their role in malignant vasculogenesis. Moreover, the analysis of the link between the microscopic subcellular dynamics and the macroscopic cell behaviors confirms the efficiency of some pharmacological interventions that are currently in use and, more interestingly, proposes some new therapeutic approaches, that are counter-intuitive but potentially effective.

Fluorescence perturbation techniques to study mobility and molecular dynamics of proteins in live cells: FRAP, photoactivation, photoconversion, and FLIP

The technique of fluorescence recovery after photobleaching (FRAP) was introduced in the mid-1970s to study the diffusion of biomolecules in living cells. For several years, it was used mainly by a small number of biophysicists who had developed their own photobleaching systems. Since the mid-1990s, FRAP has gained increasing popularity because of the conjunction of two factors: First, photobleaching techniques are easily implemented on confocal laser-scanning microscopes (CLSMs), and so FRAP has become available to anyone who has access to such equipment. Second, the advent of green fluorescent protein (GFP) has allowed easy fluorescent tagging of proteins and their observation in living cells. Thanks both to the versatility of modern CLSMs, which allow control of laser intensity at any point of the image, and to the development of new fluorescent probes, additional photoperturbation techniques have emerged during the last few years. After the photoperturbation event, one observes and then analyzes how the fluorescence distribution relaxes toward the steady state. Because the photochemical perturbation of suitable fluorophores is essentially irreversible, changes of fluorescence intensity in the perturbed and unperturbed regions are due to the exchange of tagged molecules between those regions. This article first discusses the materials required for performing FRAP experiments on a CLSM and the software for data analysis. It then describes general considerations on how to perform FRAP experiments as well as the necessary controls. Finally, different possible ways to analyze the data are presented.

Design and analysis of molecular relay channels: an information theoretic approach

In this paper, we consider a molecular relay channel in which signal molecules are transmitted by a sender of communication, propagated, amplified, removed in the channel, and sensed by the receiver of communication that decodes the signal molecules. To understand and characterize the communication capacity of the molecular relay channel, we develop an information communication model that consists of the transmitter, channel, and receiver. Mutual information is used to quantify the amount of information that is transfered from the transmitter through the channel to the receiver. The method employed and results presented in this paper may help elucidate design principles of biological systems as well as help in the design and engineering of synthetic biological systems from the perspective of information theory.

Use of virtual cell in studies of cellular dynamics

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

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

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

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

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

Initiation and propagation of a neuronal intracellular calcium wave

The ability to image calcium movement within individual neurons inspires questions of functionality including whether calcium entry into the nucleus is related to genetic regulation for phenomena such as long term potentiation. Calcium waves have been initiated in hippocampal pyramidal cells with glutmatergic signals both in the presence and absence of back propagating action potentials (BPAPs). The dendritic sites of initiation of these calcium waves within about 100 microm of the soma are thought to be localized near oblique junctions. Stimulation of synapses on oblique dendrites leads to production of inositol 1,4,5-trisphosphate (IP(3)) which diffuses to the apical dendrite igniting awaiting IP(3) receptors (IP(3)Rs) and initiating and propagating catalytic calcium release from the endoplasmic reticulum. We construct a reduced mathematical system which accounts for calcium wave initiation and propagation due to elevated IP(3). Inhomogeneity in IP(3) distribution is responsible for calcium wave initiation versus subthreshold or spatially uniform suprathreshold activation. However, the likelihood that a calcium wave is initiated does not necessarily increase with more calcium entering from BPAPs. For low transient synaptic stimuli, timing between IP(3) generation and BPAPs is critical for calcium wave initiation. We also show that inhomogeneity in IP(3)R density can account for calcium wave directionality. Simulating somatic muscarinic receptor production of IP(3), we can account for the critical difference between calcium wave entry into the soma and failure to do so.

Modelling reaction kinetics inside cells

In the past decade, advances in molecular biology such as the development of non-invasive single molecule imaging techniques have given us a window into the intricate biochemical activities that occur inside cells. In this chapter we review four distinct theoretical and simulation frameworks: (i) non-spatial and deterministic, (ii) spatial and deterministic, (iii) non-spatial and stochastic and (iv) spatial and stochastic. Each framework can be suited to modelling and interpreting intracellular reaction kinetics. By estimating the fundamental length scales, one can roughly determine which models are best suited for the particular reaction pathway under study. We discuss differences in prediction between the four modelling methodologies. In particular we show that taking into account noise and space does not simply add quantitative predictive accuracy but may also lead to qualitatively different physiological predictions, unaccounted for by classical deterministic models.

Systems biology of macrophages

Cells and tissues function in context. Under a given growth or survival medium they perform tasks, replicate and die. Given a stimulus they respond by invoking myriad biomolecular networks that result in a specified cellular outcome. At any given instant it can be argued that the cell is in a "state" defined by its components--their concentrations and locations, the interactions between components--that are modulated in space and time, and the complex circuitry--that involves a large number of interacting networks and a snapshot of the dynamical processes--such as gene expression, cell cycle, transport of components, etc. At present, we can measure, using high and low throughput methods, several cellular components in a context-dependent manner and obtain a partial picture of cellular networks and dynamical processes. Are these measurements sufficient to answer important biological questions and help reconstruct a systems-level understanding of a mammalian cell? This chapter will address systems biology strategies developed to address this question and demonstrate the power of integration of diverse cellular data for answering interesting biological questions in macrophages. We will use this systems biology approach to address the following questions: (1) How good are macrophage cell lines in addressing phenotypic biology of primary macrophages? (2) How do signals associated with inflammatory molecules regulate gene transcription in macrophages? (3) How can we combine proteomic and other cellular measurements to characterize the repertoire of upstream signaling networks invoked by macrophages? (4) How do designed knockdowns of proteins influence cellular phenotypes?

A bidomain threshold model of propagating calcium waves

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca(2+)) in living cells. Modeling Ca(2+) release as a threshold process allows the explicit construction of traveling wave solutions to probe the dependence of Ca(2+) wave speed on physiologically important parameters such as the threshold for Ca(2+) release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca(2+) resequestration from the cytosol to the ER, and the total [Ca(2+)] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca(2+) waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called 'tango waves' and the dependence of Ca(2+) wave speed on the total [Ca(2+)].

A kinetic model for calcium dynamics in RAW 264.7 cells: 1. Mechanisms, parameters, and subpopulational variability

Calcium (Ca(2+)) is an important second messenger and has been the subject of numerous experimental measurements and mechanistic studies in intracellular signaling. Calcium profile can also serve as a useful cellular phenotype. Kinetic models of calcium dynamics provide quantitative insights into the calcium signaling networks. We report here the development of a complex kinetic model for calcium dynamics in RAW 264.7 cells stimulated by the C5a ligand. The model is developed using the vast number of measurements of in vivo calcium dynamics carried out in the Alliance for Cellular Signaling (AfCS) Laboratories. Ligand binding, phospholipase C-beta (PLC-beta) activation, inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) dynamics, and calcium exchange with mitochondria and extracellular matrix have all been incorporated into the model. The experimental data include data from both native and knockdown cell lines. Subpopulational variability in measurements is addressed by allowing nonkinetic parameters to vary across datasets. The model predicts temporal response of Ca(2+) concentration for various doses of C5a under different initial conditions. The optimized parameters for IP(3)R dynamics are in agreement with the legacy data. Further, the half-maximal effect concentration of C5a and the predicted dose response are comparable to those seen in AfCS measurements. Sensitivity analysis shows that the model is robust to parametric perturbations.

Where does all the PIP2 come from?

Despite its very low concentration in the plasma membrane, PIP(2) is the precursor for the important second messenger InsP(3) and, independently, is a key modulator of membrane signalling molecules such as ion channels. However, it has been difficult to determine the spatial and temporal characteristics of PIP(2) and InsP(3) during a cell signalling event. Our laboratory used bradykinin stimulation of N1E-115 neuroblastoma cells to infer the InsP(3) dynamics from calcium imaging studies, biochemical analysis and InsP(3) uncaging. We have used computational modelling with Virtual Cell to help analyse and interpret experimental data on the details of the calcium release process as well as to build a comprehensive image-based model of agonist-induced calcium release in a neuronal cell. These data provided a constraint for the further investigation of how low levels of cellular PIP(2) could provide sufficient InsP(3) for calcium release. Using biochemical assays, quantitative imaging of GFP-based probe translocation and computational analysis, it was shown that PIP(2) synthesis is stimulated concomitant with its hydrolysis. This mechanism should be important not just for consideration of PIP(2) as a precursor of InsP(3), but for any pathway that can be directly or indirectly modulated by PIP(2).

Characterization of local calcium signals in tubular networks of endoplasmic reticulum

To explain the large time and space scales of elementary calcium events in Xenopus oocytes it is assumed that the Ca2+ source is located on tubules of the endoplasmic reticulum, which provide local barriers for diffusion. The event duration, width and signal mass dependence on the total quantity of released Ca2+ is determined at different orientations of the scan line and different ionic currents. Excellent agreement with published data is obtained with on- and off-rate constants of the fluorescent indicator of 15 microM(-1) s(-1) and 2.55 s(-1), respectively. It is found that one signal mass unit, calculated with the classical method that assumes spherical symmetry of the cytosolic space surrounding the release site, corresponds to 0.189 fC of released Ca2+ in the presence of a tubular network. It is estimated that release Ca2+ currents and amounts are randomly distributed, with averages of 0.165 pA and 3.66 fC per event and average release duration of 22.2 ms. The total quantity of liberated Ca2+ and the release current amplitude in the presence of endoplasmic reticulum tubules is predicted to be about one order of magnitude lower than estimated within the isotropic diffusion formalism. This could have implications in muscle cell Ca2+ imaging as well.

A three-dimensional model of intercellular calcium signaling in epithelial cells

We have developed a fully three-dimensional (3D) model of calcium signaling in epithelial cells based on a set of reaction diffusion equations that are solved on a large-scale finite-element code in three dimensions. We have explicitly included the cellular compartments including the cell nucleus, cytoplasm, and gap junctions. The model allows for buffering of free Ca2+, calcium-induced calcium release, and the explicit inclusion of mobile buffers. To make quantitative comparisons to experimental results, we used fluorescence microscopy images of cells to generate an accurate mesh describing cell morphology. We found that Ca2+ wave propagation through the tissue is a function of both initial conditions used to start the wave and various geometrical parameters that affect propagation such as gap junction density and distribution, and the presence of nuclei. The exogenous dyes used in experimental imaging also affect wave propagation.